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1,116,691,500,715 | arxiv | \section{Introduction} \label{ch:introduccion}
In February, 2018, the USA government estimated that cybercrime costs raised up to between 57 and 109 billions of dollars in 2016~\cite{CEA2020}.
Cybercrime has been growing for the last decades as it becomes more profitable. The most common goal for cybercriminals is monetary gain and they commonly organize to form online criminal enterprises and businesses~\cite{Collier20}.
The virtual battlefield where such criminal activities operate allows miscreants to perpetrate crimes in countries with different extradition laws than those of the country where they reside.
This strategy frequently makes cybercrime hard to prosecute, in addition to other technical characteristics that difficult attribution \cite{ODNI2018, InfosecInstitute}.
In recent times, the underground economy has developed a myriad of approaches that allow cybercriminals to acquire high financial profits.
With the cybercrime growth and specialization, many cybercriminals offer their products in an ``as-a-service'' model, where the attacker can purchase the service through the internet with little technical knowledge.
These services reduce the entry level for new criminals and motivate newcomers into the underground ~\cite{Huang2018,Pastrana18}.
In 2017, Panda security analyzed around 15m binaries~\cite{Panda2018}. The most noticeable thing was that, upon reviewing the data collected, they realized that 99.1\% of the samples were only seen once, probably due to binary packing and encryption. Indeed, a common digital commodity offered in underground markets is malware~\cite{van2018}. Concretely,
one of the most popular variants offered is ransomware~\cite{Pwc2020}, where the attacker denies access to the data of its victims until a ransom is paid (hence the name of the threat). When these attacks affect companies or public organizations, they might provoke business interruptions, thus increasing the economic and social damage~\cite{Ghafur19}. Ransomware operators often partner up with other criminal groups, either in a customer-service relationship (offering the software for a fixed fee or via a subscription-based access to constant updates) or in a profit sharing scheme (one party is responsible for developing and maintaining the ransomware while the other distributes it, both sharing an arranged percentage of the revenues). Previous works show that criminals can run ransomware campaigns with little technical knowledge, making use of the available services, with an estimated return of investment of between 504\% and 12,682\%~ \cite{Huang2018}.
Due to the profitability and specialization of cybercrime, modern ransomware campaigns have improved their sophistication. First, techniques from well-established cryptography schemes, so-called hybrid cryptosystems, have been recently adopted in ransomware operations, combining symmetric and asymmetric cryptography. Second, modern ransomware perpetrators have incorporated another monetizing technique that further pushes the victims to pay a ransom: data leakage extortion. Apart from encrypting the files, ransomware operators now steal data from the infected systems and threaten the victims to leak it online if no ransom is paid. This extortion scheme was initiated by a threat actor named \textit{TWISTED SPIDER} in the last quarter of 2019 \cite{CrowdStrike2020}, and was quickly followed by other ransomware groups~\cite{Panda2020, Catalin2020, BankInfoSecurity20202}. In order to face the ransomware threat, and to be able to recover the hijacked files, it is important to understand the criminal ecosystem and also how the malware evolves and operates
In this work, we study a novel ransomware campaign, dubbed Avaddon, which was launched on June 2020 in underground forums as a Ransomware-as-a-service (RaaS). Since then, Avaddon has been linked to various cyberattacks in 2020, and incorporates a recent trend on Ransomware which is is to publicly \textit{`blame and shame'} victims that do not pay the ransom~\cite{Intel471}. At the time of this writing, more than 574GB of data from 23 companies have been leaked and exposed online\footnote{For ethical and legal reasons, we have not downloaded nor checked the veracity of the exposed data since otherwise this would cause additional harm to users, and such analysis is not of public interest for the community~\cite{Thomas17}}. In addition, Avaddon operators have recently started to blackmail victims by running DDoS attacks. Existing reports described different technical features of Avaddon \cite{SubexSecure2020, idransomware2020, HornetSecurity2020, TrendMicro2020_1}. However, as far as the authors know, no public decryption procedure is available to recover files from an infection. We aim at filling this gap by providing an in-deph analysis of Avaddon and proposing a decryption routine that can decrypt files in real time, thus minimizing the impact of such attack. In particular, we analyze one of the first variants observed in early June, although the proposed decryption method is still functional for the latest samples of Avaddon at the time of this writing. The main contributions of this work are the following:
\begin{itemize}
\item We analyze the Avaddon business ecosystem and provide an step-by-step analysis of its technical capabilities, using advanced static and dynamic analysis. This analysis can be generalized to grasp an overview of how modern ransomware operates, since their \textit{modus operandi} is similar. As a result of our study, we provide a set of indicators of compromise (IoCs) that may serve security analysts to develop further tools and countermeasures to detect Avaddon, such as signatures or heuristics.
\item We showcase a typographical error in the list of services that Avaddon checks to avoid re-infecting victims, which is also present in modern variants of another Ransomware family, i.e., MedusaLocker. Additionally, we highlight that some similarities on the code of both families hint that they are operated or developed by the same group.
\item We describe a method to recover the symmetric keys used for the encryption, thus allowing victims to recover the files from infected systems. Accordingly, we present and make publicly available a tool which can help victims to recover from these attacks in real time. While this tool was designed using the analysis of the first versions of Avaddon, we have confirmed that it still works with the most up-to-date versions of the ransomware, released in mid-January 2021.
\end{itemize}
The rest of this work is structured as follows. Section~\ref{ch:background} describes the Avaddon criminal ecosystem and how it operates and evolves in the underground economy. Next, Section \ref{sec:analysis} discusses the results obtained after reverse engineering the sample using static and dynamic analysis and provides details of the internals of the ransomware. Section \ref{ch:key_recovery} describes the method to recover the session key used to encrypt the system and describes the remediation tool to decrypt the infected files. We provide experimental results in Section~\ref{sec:results} by infecting a sandboxed environment and decrypting the file system with the proposed approach. Finally, we conclude and discuss the implications and limitations of this paper in Section~\ref{ch:conclusions}.
\section{Background and related work} \label{ch:background}
In this Section, we first provide background information about how Ransomware works and existing defensive mechanisms. Then, we present the criminal ecosystem behind Avaddon, including its evolution in the underground economy, and how this has been reflected in the wild, i.e., leading to real-world cyberattacks.
\subsection{The Ransomware threat}
Ransomware is a type of malware that interrupts the business of the victim or denies access to its data until a ransom is paid, by means of data encryption. This type of malware has direct financial implications and has promoted the growth of cybercrime, where it is employed as a profitable business model \cite{Brewer2016}.
Before the popularization of cryptocurrencies, such as Bitcoin, online payment methods were risky for malware authors. SMS text messages, pre-paid cards or premium rate telephone numbers could be traced back easier than Bitcoin \cite{Zetter2015}. With the use of Bitcoin or other cryptocurrencies to ask for ransoms, it became much harder to trace the payments sent to criminals. Still, the characteristics of some cryptocurrencies allow for tracking transactions (although not connecting them to the attacker). For instance, Huang \textit{et al.} were able to track over \$16 million in likely ransom payments made by 19,750 potential victims during a two-year period \cite{Huang20182}. Thus, criminals have adopted privacy-preserving cryptocurrencies such as Monero that hinder tracking~\cite{Pastrana2019}. These cryptocurrencies, in combination with the cybercrime specialization, have promoted the ransomware threats as a profitable business for cybercriminals~\cite{Richardson2017}.
Ransomware detection approaches often leverage classical malware detection methods adopted for ransomware-specific behaviors. In this way, ransomware activities can be split in 8 stages~\cite{Hull2019}: \textit{fingerprint}, \textit{propagate}, \textit{communicate}, \textit{map}, \textit{encrypt}, \textit{lock}, \textit{delete} and \textit{threaten}. Kharaz \textit{et al.} focused the detection on common tasks performed by ransomware, such as changing the desktop wallpaper~\cite{Kharaz2016}. Some efforts have also been made to capture cryptography keys at runtime in order to facilitate decryption of infected systems~\cite{Kolodenker2017}. Following recent trends in malware detection, some Machine Learning-based approaches have also been proposed specifically targeting ransomware detection~\cite{Sgandurra2016, Vinayakumar2017, Lee2019}.
\subsection{The ecosystem of Avaddon}
Avaddon\footnote{The name of the ransomware, Avaddon, may be derived from the Hebrew term Abaddon, the name of an angel of the abyss in the Bible, mainly associated with the meaning of ``destruction''~\cite{idransomware2020}.} is a ransomware that was offered as an affiliate program on June, 2020 in a Russian underground forum, only accessible by invitation or after the payment of a registration fee. Concretely, the operators were looking for partners for their campaign. Additionally, Avaddon was promoted on other underground forums afterwards.\footnote{Due to ethical reasons, and to avoid promoting the site, we do not provide the name of the forums.} Actors that become affiliates are equipped with both the ransomware binary and an administration panel from where they can control their infections. Access to the program is free and constrained only for reputed (and Russian-speaking) actors. In exchange for this, partners have to share part of the obtained revenues from the ransomware to the owners and operators. This share depends on the amount of infections, starting from 35\% and decreasing up to 15\% for larger volumes. Therefore, affiliates, who are only responsible for distributing and installing the malware on infected systems, gain 65\% of the revenues generated by the ransomware, without the need of operating the payment system~\cite{SubexSecure2020}. Such distribution often relies on botnets hired in a Pay-Per-Install scheme~\cite{Caballero2011}. Additionally, partners can purchase installs on RDP servers, which is another popular product traded in underground economy~\cite{KasperskyRDP16}. Thus, the supply chain needed to enter in this business does not require technical knowledge and it opens the barrier to any criminal entrepreneur~\cite{Bhalerao19}.
As a restriction in the affiliates program of Avaddon, it is forbidden to target victims in the Commonwealth of Independent States (CIS). We describe the mechanism used to achieve this restriction in Section \ref{subsec:language_checks}.
A few days after their publication on underground forums (on 2020-06-04 at around 14:00:00 UTC) Avaddon was observed in the wild. Concretely, it was distributed via mail in a malspam campaign~\cite{HornetSecurity2020},
that consisted in low-quality phishing emails that attached a malicious file. These emails hinted that a compromising photo of the victim had been leaked, inciting the victim to open the file out of fear. The attached file was a zip-compressed JavaScript file. This file tried to masquerade as a JPG photo, having the extension ``.jpg'' just before the ``.js'' extension (e.g., ``IMG123456.jpg.js''). Upon execution, the malicious file would download and execute the ransomware.
Allegedly, the first wave of this campaign targeted mostly Canada, concretely various education services~\cite{HornetSecurity2020}.
However, their targets varied later.
Indeed, as mentioned before, Avaddon was launched as a RaaS, which means that the targets are not chosen by the ransomware developers (apart from the ban on CIS victims) but by the affiliates.
Upon infecting a system, a ransom note is left to the victim with instructions on how to pay the ransom. The note would lead the victim to a Tor hidden service, where payment must be done in exchange for the decryptor. At the time of this writing, the payment service is still operative, confirming that the campaign is ongoing.
Shortly after Avaddon was first seen in the wild, Trend Micro conducted and released a technical analysis~\cite{TrendMicro2020_1}. The report offers an overview of the ransomware capabilities and \textit{modus operandi}. However, no decryption option is mentioned (it only indicates how to remove the ransomware). Regarding the decryption process upon paying the ransom, some stories from affected users state that it is unreliable and recovery is not ensured~\cite{DecryptStory12020}.
Two months after the initial release, on August, 2020, Avaddon was updated to incorporate a new trending technique to their features~\cite{TrendMicro2020_2}: extortion to victims. Following the model from other ransomware campaigns, Avaddon operators decided to threat victims to exfiltrate their data, by making it publicly available if they do not pay the ransom~\cite{BankInfoSecurity2020, BleepingComputer2020}. By the end of January, 2021, Avaddon has allegedly infected and leaked full dump data from 20 companies (totalling 574.46 GB of data) and is extorting (i.e., threatening of leaking data) 3 other companies which have been recently infected.
Finally, in January 2021 (concurrent to the writing of this paper), Avaddon included a new technique used for extortion: attacking their victims with Denial-of-Service Attacks~\cite{BleepingComputerDDoS}. Therefore, the threat to victims is now three-fold, i) data is first encrypted in the infected systems, so it becomes unavailable, ii) data is leaked publicly if the ransom is not paid, and iii) a DDoS attacks is performed to disrupt their businesses until the ransom is paid.
At the time of writing, we are not aware of any public decrypting tool for Avaddon. Additionally, various reports and recent complaints from Avaddon victims about their decryption support~\cite{AmigoA2020, PintSizeNore2020} show that the campaign is still operative.
In this paper, we fill this gap and release an open-source tool that automatically detects and decrypts files, which could be integrated in existing Antivirus solutions.
\section{Ransomware analysis}\label{sec:analysis}
In this section, we provide an analysis of the Avaddon ransomware, concretely one released as part of their initial advertising campaign on June 2020. We follow standard techniques for malware analysis, concretely static and dynamic analysis. To perform the aforementioned analysis, we utilize popular tools for binary analysis (i.e., Binary Ninja~\footnote{https://binary.ninja/}, x64dbg~\footnote{https://x64dbg.com/} and Pestudio~\footnote{https://www.winitor.com/}) and a virtual machine to run the ransomware safely. We build the virtual environment on top of VirtualBox~\footnote{https://www.virtualbox.org/} and install Windows 7 x64 in the virtual system.
The analyzed binary (MD5:\hash{c9ec0d9ff44f445ce5614cc87398b38d}) is a Portable Executable (PE) file.
The PE format describes the structure of executable programs in Windows Operating Systems (OS)~\cite{MSPEFormat}. PE files are mainly divided in two important pieces: headers and sections. While headers contain information about the program itself and data to be read by the OS in order to correctly load and execute the program, sections contain the actual code and data of the program. Additionally, we see that its size is 1.1 MB, so it is not a large program. Finally, we see that the compilation time field of the binary is set to June, 3, 2020, at 11:47:22 (UTC). Although this field is prone to be modified by malware authors in order to confuse analysts, the timestamp is similar to the time of the first appearances of Avaddon samples \cite{SubexSecure2020}, which confirms that we are indeed analyzing one of the first versions of Avaddon.
We describe the packing protections of the analyzed binary in Section \ref{subsec:packing_protections}. Next, we show the imported functions and the extracted strings in Sections \ref{subsec:imports} and \ref{subsec:strings}, respectively. In Section \ref{subsec:anti_analysis}, we show the anti-analysis techniques employed by the binary. Then, we show how the ransomware authors implemented a protection to not infect Commonwealth of Independent States (CIS) victims in Section \ref{subsec:language_checks} and analyze the privilege escalation techniques, step by step, in Section \ref{subsec:privilege_escalation}. The details of the persistence mechanism are showcased in Section \ref{subsec:persistence}. Following, interactions with other processes and services are presented in Section \ref{subsec:services_manipulation}. Finally, we expose the cryptography mechanisms used in the last two sections, concretely key management (Section \ref{subsec:key_generation}) and file encryption (Section \ref{subsec:file_encryption}).
\subsection{Packing protections} \label{subsec:packing_protections}
Looking at some properties of the PE file, we conclude that the sample is not packed. First, we find that the PE file contains 4 sections which have almost no differences in size between disk and memory. This in an indicator of the PE file not being packed, since the presence of a virtual section (i.e., a section that requests space in memory but does not occupy bytes in disk) is a common indicator of packing protections. Then, we find over 200 imported functions and several meaningful strings, which present some useful information about the capabilities of the ransomware. Often, packing protections attempt to hide imports and strings in order to difficult analysis. Finally, the entropy levels of the PE file are not high enough to hint the presence of a packer. The highest entropy level is reached in the ``.text'' section, whose entropy value is 6.62. However, this does not confirm the existence of packed or encrypted code, which often have values ranging from a minimum of 6.677 to a maximum of 6.926 \cite{Lyda2007}.
\subsection{Imported functions} \label{subsec:imports}
The Windows OS offers an Application Programming Interface (API) which abstracts many functionalities from developers, e.g to interact with files, processes, etc. This also provides an abstraction layer regarding the underlying hardware. In order to call those functions, programs need to know their location in memory. This need may be fulfilled in different ways, but the most common method consists on importing the required functions prior to execution. This is done by the OS loader before transferring control to the program. To do so, the PE file contains an Import Address Table (IAT) in the headers, which includes a list of functions to be imported by the OS loader. When the file is executed, the OS loads the file in memory and fills the IAT with the addresses of each requested function. Then, the program is able to call those functions because it now knows where each of these functions is allocated in memory. Therefore, the IAT provides useful information about the capabilities and intentions of the program.
The functions imported by the Avaddon sample analyzed show common capabilities of ransomware, such as encryption (e.g., \textit{CryptGenKey} or \textit{CryptEncrypt}), persistence (e.g., \textit{RegCreateKeyW}, \textit{StartServiceW}), anti-analysis (e.g., \textit{IsDebuggerPresent}) or activity control (e.g., \textit{DeleteService} or \textit{TerminateProcess}).
\subsection{Strings} \label{subsec:strings}
Looking for strings through a PE file allows analysts to identify capabilities of the binary, as well as looking at the IAT. Indeed, some imports will appear when searching for strings if they are imported by name (external functions may be imported by name or ordinal \cite{MSPEFormat}). Therefore, we proceed to extract all readable strings that have more than 4 characters in the whole file. Then, we filter the extracted strings and exclude those that are not meaningful (bytes that are part of code may non-intentionally form readable strings that are not meaningful). In this case, as aforementioned, we find enough meaningful strings to think that the PE file is not packed. Many of the strings found were paths to folders or files (e.g., ``\hash{C:\textbackslash Temp}''). While we initially can not know the actual purpose of those files, we hypothesize that some of them may be used to drop additional payloads or to move the PE file upon infection to a different location (we confirm this hypothesis in Section \ref{subsec:persistence}).
Among the strings that are present in the PE file, we observe two of them that refer to cryptography providers (i.e., ``\hash{Microsoft} \hash{Enhanced} \hash{Cryptographic} \hash{Provider} \hash{v1.0}'' and ``\hash{Microsoft} \hash{Enhanced} \hash{RSA} \hash{and} \hash{AES} \hash{Cryptographic} \hash{Provider}''). These strings are normally used to acquire cryptography contexts using the Windows API, which are later needed to perform some cryptography operations. Additionally, some strings indicate that the ransomware was developed in C++. C++ is an object-oriented language. Although this characteristic does not provide any information about the capabilities of the sample, the particularities of C++ programs must be taken into account in the analysis process. We will highlight some C++ properties that allow us to extract conclusions in the reverse engineering process, but discussing the differences between C++ and other languages at assembly level is out of the scope of this work. For more information, we refer to Sabanal \textit{et al.} \cite{sabanal2007}.
Interestingly, we find many strings that are Base64 encoded. However, upon decoding them, no legible string is recovered. Therefore, we suspect that these strings are obfuscated by other means (i.e., encoding or encryption) in order to hide their content. This is a common mechanism in malware samples. In such case, those strings may be of importance to understand additional capabilities of the malware that may not be retrieved without further analysis.
We thus confirm that these strings are indeed encrypted and are only decrypted at runtime on demand, i.e., when they are required by the program. First, global variables are created to hold the encrypted strings, making them accessible from every function in the binary. In Algorithm \ref{alg:initialize_global_var}, we show one of the functions (0x4012a0 in this case) that creates a global variable pointing to an encrypted string. There, the encrypted string and its size are pushed onto the stack (lines 1-2). Then, a global variable is created at 0x4f8a28 with the content of the encrypted string (lines 3-4). Finally, a destruction function is registered (lines 5-6). This function will be called when the process exits. The global variable is then referenced wherever this particular string is needed in the program, and each encrypted string has its own initialization function. These functions are the constructors of the global variables. We know that these are global variables in the source code because:
\begin{enumerate}
\item There is one global variable per encrypted string and one constructor function for each global variable.
\item Each global variable has a predefined address. These addresses are hardcoded in each constructor function.
\item After initializing the global variable, a destructor function is registered to be called upon terminating the program.
\end{enumerate}
\begin{algorithm}
\SetAlgoLined
\caption{One of the functions responsible for initializing a global variable with the value of an encrypted string.}
\label{alg:initialize_global_var}
0x4012a0: push 0x30\; \tcp{Size of the encrypted string}
0x4012a2: push 0x49e180\; \tcp{Encrypted string}
0x4012a7: mov ecx, 0x4f8a28\; \tcp{Global variable}
0x4012ac: call 0x40a390\; \tcp{Creates a global variable at ecx (0x4f8a28 in this case) with the string stored at the value previously pushed (0x49e180 in this case)}
0x4012b1: push 0x4874a0\; \tcp{Destructor}
0x4012b6: call \_atexit\; \tcp{Register the destructor function to be called when the process ends}
0x4012bb: pop ecx\;
0x4012bc: retn\;
\end{algorithm}
Once initialized, the strings are decrypted and used as needed by referencing the global variables. In Algorithm \ref{alg:use_decrypted_string}, we show an example of this procedure. In particular, a string containing some command line arguments is decrypted and used to create a process. In this case, the goal is to delete security backups. First, the decryption function is called passing the global variable as an argument (lines 1-3). This function returns a new string with the decrypted value, which is immediately used to create the aforementioned process (lines 4-5). The sequence of instructions described can be summarized in the following pseudo code:
\begin{itemize}
\item[]\textit{CommandLine = DecryptString(GlobalVariable);}
\item[]\textit{CreateProcess(CommandLine);}
\end{itemize}
\begin{algorithm}
\SetAlgoLined
\caption{Decryption of a global variable into a temporary register.}
\label{alg:use_decrypted_string}
0x40d110: mov edx, 0x4f8a28\; \tcp{Global variable that contains an encrypted string}
0x40d115: lea ecx, [esp+0x8]\; \tcp{Local variable that will hold the decrypted string}
0x40d119: call decrypt\_string\; \tcp{Decrypts the string at edx (the global variable) and stores the result in ecx (the local variable)}
0x40d11e: push eax\; \tcp{eax now contains the decrypted string (it is equal to [esp+0x8], the local variable) which, in this case, is a command line}
0x40d11f: call create\_process\; \tcp{Creates a process with the command line received as argument}
\end{algorithm}
The decryption function is located at address 0x40c780. First, the received string is decoded from Base64. Then, as shown in Algorithm \ref{alg:decrypt_string}, each character is decrypted by substracting 2 units from its value (line 3) and XOR-ing the result with 67 (line 5). These instructions are executed once for every character in the string.
\begin{algorithm}
\SetAlgoLined
\caption{Characters decryption.}
\label{alg:decrypt_string}
0x40c820: mov al, byte [esi]\; \tcp{Move the current character to al (the lower 8 bits of eax)}
0x40c822: mov edx, dword [ebp-0x1c]\;
0x40c825: sub al, 0x2\; \tcp{Substract two units from the character}
0x40c827: mov edi, dword [ebp-0x18]\;
0x40c82a: xor al, 0x43\; \tcp{XOR the result with 0x43}
0x40c82c: mov byte [ebp-0x30], al\;
0x40c82f: cmp edx, edi\;
0x40c831: jae 0x40c84d\;
\end{algorithm}
Since we know the address in which global variables are placed, we have automatically re-labeled them in order to improve the readability of the code for the analyst. To allow for reproducibility and to assist other analyses on this and similar malware samples, we publish a script that automates these tasks using the \textit{Binary Ninja} tool in our public repository~\footnote{\url{https://github.com/JavierYuste/AvaddonDecryptor}}.
\subsection{Anti-analysis techniques} \label{subsec:anti_analysis}
Successfully infecting a system critically depends on not being detected. Thus malware authors often implement different techniques to evade antivirus systems or sandboxes. Additionally, mechanisms are frequently put in place in order to delay analysts and, therefore, increment the time needed for building detection tools for the sample (e.g., signatures). In the case of Avaddon, the binary is not packed, which is a common obfuscation technique. However, we observe other anti-analysis techniques, described next.
\textbf{String obfuscation}. As mentioned in prior sections, various of the strings are encrypted, which may hide important functionality. This technique is commonly used to: i) evade detection, and ii) delay analysts. See Section~\ref{subsec:strings} for a detailed description of this obfuscation technique and the process used to decrypt the strings.
\textbf{Anti-debugging}. We found a call to \textit{IsDebuggerPresent} at offset 0x42e03d. Debuggers are programs designed to analyze other programs at runtime (i.e., processes), and are used by security analysts to dynamically inspect malware. Hence, malware authors often embed code in their programs that checks for debuggers and, if detected, terminates the process or changes their behavior. In particular, \textit{IsDebuggerPresent} is a function provided by the Windows API.
If a debugger was attached to the program, this function would return true and the binary would exit. To circumvent this protection, we consider two options:
\begin{enumerate}
\item Hook the call to \textit{IsDebuggerPresent} so it always returns false. By doing this, we bypass any check done by the malware, changing the code on the fly, and the debugger would not be detected by the sample.
\item Change a binary value in the the Process Environment Block (PEB), a data structure that holds information about the process. That structure is built by the OS when executing the program and is unique per process. Among other information, it contains a bit that indicates if a debugger has been attached. When a call to \textit{IsDebuggerPresent} is made, it returns the value of that bit. Therefore, changing the value in the PEB would successfully hide the debugger from that call and from any manual checks (the PEB may also be walked through manually by parsing its structure).
\end{enumerate}
In order to avoid further anti-debugging mechanisms that may parse the PEB (i.e., not using \textit{IsDebuggerPresent}), we decided to implement the second option.
\subsection{Language checks} \label{subsec:language_checks}
To avoid infecting systems in some countries, it is frequently observed that malware binaries implement techniques to check the country where the infected machine is located, so as to ensure that citizens from some regions are not affected. It is common to see that CIS victims are dodged in many malware samples, as it is the case for this one. The most popular approach is to check for the keyboard layouts and the OS language. In this sample, we found both checks for different layouts and languages (addresses 0x42e0ec and 0x42e0b6, respectively). In particular, we discovered checks for language locales (i.e., Russian and Ukrainian) and keyboard layouts (i.e., Russian, Sakha, Tatar and Ukrainian). If any of these keyboard layouts or OS locales is found, the binary exits without harming the landed system. That is, this sample of Avaddon ransomware is designed to avoid infecting Russian and Ukrainian systems. This, together with the fact that the malware was first advertised in a Russian underground forum, provides strong (though not conclusive) evidence that the origin of the malware is Russia.
\subsection{Privilege escalation} \label{subsec:privilege_escalation}
Malware authors often spend great resources in order to infect systems, e.g. to gain initial access and evade detection by AV software. However, having invested so much effort in those tasks, their immediate post-infection activities might fail due to the need for administrator privileges if the user becomes suspicious after being requested to concede those privileges. Therefore, reducing the number of clicks needed from the victim is critical.
Indeed, malware actions usually require administrator privileges in the infected system to accomplish some critical tasks (e.g., acquire persistence, infect system files or processes, etc.). In this particular case, escalating privileges is critical because the ransomware needs to i) acquire persistence through registry keys (Section \ref{subsec:persistence}), ii) stop processes and services (Section \ref{subsec:services_manipulation}), and iii) delete backups (Section \ref{subsec:file_encryption}).
The process implemented to elevate privileges in Avaddon is a well known User Account Control (UAC) bypass. Indeed, there are public open-source implementations~\cite{hfiref0x2017} and it is not uncommon to find this technique in different malware families~\cite{Morphisec2020, SecurityInBits2019}. Next, we briefly summarize this process and how it is implemented in Avaddon. First, three registry keys are added or modified (at offset 0x40ed20). Concretely, these keys are:
\begin{enumerate}
\item \hash{HKEY\_LOCAL\_MACHINE\textbackslash SOFTWARE\textbackslash Microsoft\textbackslash Windows\textbackslash CurrentVersion\textbackslash Policies\textbackslash System} \hash{EnableLUA = 0} (disables the ``administrator in Admin Approval Mode'' user type \cite{MSEnableLua})
\item \hash{HKEY\_LOCAL\_MACHINE\textbackslash SOFTWARE\textbackslash Microsoft\textbackslash Windows\textbackslash CurrentVersion\textbackslash Policies\textbackslash System} \hash{ConsentPromptBehaviorAdmin = 0} (this option allows the Consent Admin to perform an operation that requires elevation without consent or credentials \cite{MSConsentPrompt})
\item \hash{HKEY\_LOCAL\_MACHINE\textbackslash SOFTWARE\textbackslash Microsoft\textbackslash Windows\textbackslash CurrentVersion\textbackslash Policies\textbackslash System} \hash{EnableLinkedConnections = 1} (makes user mapped drives available to the administrator versions of those users \cite{MSLinkedConn})
\end{enumerate}
The first two registry key values allow the sample to elevate privileges without alerting the user, and the third enables the access to volumes of the current user when administrator privileges are acquired.
Then, the sample checks its privileges (offset 0x41a5c0). If it has administrator privileges, it continues its execution without running the rest of the UAC bypass. Otherwise, administrator privileges are obtained by running the following procedure (implemented at 0x40ef90):
\begin{enumerate}
\item First, a class ID (CLSID) is decrypted. This CLSID was stored in the binary as an encrypted string, as we described in Section \ref{subsec:strings}. The decrypted value is ``\hash{\{3E5FC7F9-9A51-4367-9063-A120244FBEC7\}}'', which corresponds to CMSTPLUA. For the rest of this section, we refer to it as CLSID\_CMSTPLUA.
\item Next, an IID is decrypted in the same way, obtaining the value ``\hash{\{6EDD6D74-C007-4E75-B76A-E5740995E24C\}}''. For the rest of this section, we refer to it as IID\_ICMLuaUtil.
\item Then, a third string is decrypted, which contains the value ``\hash{Elevation:Administrator!new:}''.
\item Once the three strings have been decrypted, a new string is built by concatenating ``\hash{Elevation:Administrator!new:}'' and CLSID\_CMSTPLUA.
\item Next, the execution calls the function \textit{CoGetObject} in order to obtain a pointer to CMLuaUtil. The parameters of the call are as follows:\\
\quad \hash{CoGetObject(}``\hash{Elevation:Administrator!new:\{3E5FC7F9-9A51-4367-9063-A120244FBEC7\}}''\hash{,} \hash{0x24,} \hash{\&IID\_ICMLuaUtil,} \hash{\&CMLuaUtil)}\\
At this point, user interaction might be needed to grant administrator privileges for the program in some systems. In some cases, this might be accompanied by social engineering techniques, e.g. instructions accompanying the phishing email where the malware is attached. In this case, we have not observed any particular behavior.
\item If the call is successful, CMLuaUtil now points to a structure that contains the address of a function named \textit{ShellExec} (CMLuaUtil$\xrightarrow{}$lpVtbl$\xrightarrow{}$ShellExec).
\item After obtaining the absolute path of the malware PE file (via a call to \textit{GetModuleFileNameW}) the binary executes itself with administrator privileges by calling \textit{ShellExec} with the following parameters:\\
\quad \hash{ShellExec(CMLuaUtil,} ``\hash{C:\textbackslash [...]\textbackslash sample.exe}''\hash{,} \hash{[...])}
\end{enumerate}
\subsection{Persistence and infection tracking} \label{subsec:persistence}
In order to survive across reboots, malware samples must be run automatically on infected systems after the initial foothold has been obtained~\cite{Lockheed2020}. Otherwise, they would need to infect the system again if further runs are required. In order to achieve persistence in the system, there exist many approaches. Usually, malware authors acquire persistence by adding registry keys, creating services or registering scheduled tasks. By doing so, the malware sample is automatically run by the OS (e.g., at scheduled times or at every reboot).
Additionally, malware samples often implement mechanisms to prevent re-infection of already-infected systems, thus to minimize the risks of detection or to prevent disruption of previous runs.
By looking at the imported functions (see Section \ref{subsec:imports}) we hypothesize that persistence may be acquired via registry keys or services. Then, using dynamic analysis we confirm that persistence is obtained by adding two registry keys. Upon inspecting the code of the binary, we locate the function responsible for acquiring persistence at address 0x40cf50. The only purpose of this function is to add the following registry keys:
\begin{itemize}
\item \hash{HKU\textbackslash S-1-5-21-2724635997-1903860598-4104301868-1000\textbackslash Software\textbackslash Microsoft\textbackslash Windows\textbackslash CurrentVersion\textbackslash Run\textbackslash update:} \hash{"C:\textbackslash Users\textbackslash \%User Profile\%\textbackslash AppData\textbackslash Roaming\textbackslash \%sample\%.exe"}
\item \hash{HKLM\textbackslash SOFTWARE\textbackslash Wow6432Node\textbackslash Microsoft\textbackslash Windows\textbackslash CurrentVersion\textbackslash Run\textbackslash update:} \hash{"C:\textbackslash Users\textbackslash \%User Profile\%\textbackslash AppData\textbackslash Roaming\textbackslash \%sample\%.exe"}
\end{itemize}
With those registry keys in the system, the PE file is executed at each system reboot (notice that a copy of the sample is dropped at runtime in \hash{"C:\textbackslash Users\textbackslash \%User Profile\%\textbackslash AppData\textbackslash Roaming\textbackslash \%sample\%.exe"}, where ``\hash{\%sample\%}'' is the name of the PE file).
To avoid re-infecting a system more than once, a mutex is created with the value \hash{\{2A0E9C7B-6BE8-4306-9F73-1057003F605B\}}. If this mutex is already present in the system, the binary exits and does not encrypt files. In addition, the ransomware takes measures to avoid encrypting already encrypted files, as we describe in Section \ref{subsec:file_encryption}. Thus, having mechanisms to prevent re-infection of a machine might be to avoid reinfecting victims that have already payed a ransom. Nevertheless, the fact that the presence of such mutex is checked allows to prevent Avaddon infections. By creating such mutex in a healthy system, Avaddon ransomware samples will not execute, acting as an Avaddon vaccine. However, not every sample of Avaddon uses the same mutex, as it may change among versions.
\subsection{Process and service manipulation} \label{subsec:services_manipulation}
In order to avoid being detected or neutralized, some malware samples try to stop anti-malware solutions. In order to do so, administrator privileges must be acquired. However, it is often easier to acquire administrator privileges without being detected than to encrypt the whole file system without rising awareness. In Section \ref{subsec:imports}, we highlighted that the PE file imported some functions that may indicate an attempt to control some anti-malware solutions by interacting with services and processes. Additionally, before attempting to encrypt files, it is important to stop processes that may be locking some files. For instance, ransomware authors may look to stop database processes that may be locking database files.
In this case, we find two functions (located at offsets 0x41a8f0 and 0x40c990 of the sample) that try to stop a list of services and processes, respectively, if found in the system. As expected, among those lists, we have found anti-malware solutions (e.g., ``DefWatch'') and databases (e.g., ``sqlservr'').
We notice that the name of one of the services is misspelled. Concretely, ``vmware-usbarbitator64'' is missing an `r' (and should instead be ``vmware-usbarbitrator64''). This typographical error was found in other ransomware family, MedusaLockker. This indicates that developers reuse code from other families~\cite{TAU2020, Zsigovits2020}. We are unaware on whether this is due to the same actor developing both families, or due to code reuse from one to another (though we have found no evidence of the source code of MedusaLocker being leaked).
Indeed, we notice that the Tactics, Techniques and Procedures (TTPs) of Avaddon are very similar to those of MedusaLocker if we compare our analyses with the report on MedusaLocker from Carbon Black’s Threat Analysis Unit \cite{TAU2020}. This is an interesting fact regarding the attribution of this campaign which might require further investigation if future families share this peculiarity.
\subsection{Key generation} \label{subsec:key_generation}
One of the most critical parts of a ransomware campaign is the encryption process. The keys used, how they are imported or generated, how they are exported, the encryption algorithm chosen, etc., are important decisions for malware developers. An error in this process may allow analysts to develop measures to recover encrypted files, completely neutralizing the campaign revenues. In this case, two keys are used in the encryption process in a so-called hybrid scheme. One key (the session key) is randomly generated in each execution and used to encrypt the files in the system. This key is used in a symmetric encryption scheme, AES256. Therefore, the same key must be used to decrypt the affected files. The second key is a public one, part of an asymmetric scheme, RSA1. This key is imported (it is present in the PE file) and used only to encrypt the previously generated key. Therefore, the session key can only be decrypted by the malware authors, since the private key of the asymmetric scheme is only known by them.
The whole process that we described in the previous paragraph is split in three functions in the PE sample. These functions, responsible for key management, are located at offsets 0x413600, 0x413a60 and 0x413f50 respectively.
\paragraph{Public key import} The function at 0x413600 is responsible for importing the public key. The import is made by calling the Windows API function \textit{CryptImportKey} with the following parameters:\\
\hash{CryptImportKey(hProv: CSP,} \hash{pbData:} \hash{Key} \hash{to} \hash{be} \hash{imported,} \hash{dwDataLen:} \hash{Length} \hash{of} \hash{the} \hash{key,} \hash{hPubKey:} \hash{0,} \hash{dwFlags:} \hash{0,} \hash{phKey:} \hash{Handle} \hash{to} \hash{the} \hash{imported} \hash{key} \hash{after} \hash{the} \hash{call)}\\
The key (which is Base64 encoded) is part of a RSA1 public/private pair. As per the documentation \cite{MSImportKey}, the parameter \textit{hPubKey} must be equal to 0 when the key to be imported is a public key (a \textit{PUBLICKEYBLOB} object). This detail indicates that the imported key is actually the public one of the pair.
\paragraph{Generated key} After importing the public key, a random key is generated. This randomly generated key (the session key) is used to encrypt the files of the system later, using an AES256 scheme. The function responsible of generating the session key is the one located at 0x413f50. To generate it, a function from the Windows API is called, \textit{CryptGenKey}, with the following parameters:\\
\hash{CryptGenKey(hProv:} \hash{CSP,} \hash{Algid:} \hash{CALG\_AES\_256,} \hash{dwFlags:} \hash{CRYPT\_EXPORTABLE,} \hash{phKey:} \hash{Handle} \hash{to} \hash{the} \hash{generated} \hash{key} \hash{after} \hash{the} \hash{call)}\\
The parameter \textit{Algid} indicates that the generated key is to be used in AES256. Additionally, notice that the flags passed to the function indicate that the key must be exportable. Once the key has been generated, it is exported and encrypted using the previously imported RSA1 key. The result is then included in the ransom note, in order to allow the ransomware operators to recover the encryption key and provide a decryption tool to those victims that decide to pay a ransom.
\paragraph{Keys destruction} Finally, the function located at 0x413f50 is the one responsible for securely destroying the keys. This function will destroy the public RSA1 key and the generated AES256 key. The purpose of this function is to ensure that they do not remain in memory after being used. However, this function is only called when the process exits, which only occurs when the infected system is shutdown (the ransomware process remains active to also encrypt new files). Therefore, the session key is never destroyed if the system is not powered off. This is a mistake from the malware perspective since, as long as the computer remains active, the key is kept in memory and thus can be retrieved using basic forensics techniques. In Section \ref{ch:key_recovery}, we will take advantage of this detail to describe and present a tool to recover the symmetric key generated and decrypt all the affected files.
\subsection{File encryption} \label{subsec:file_encryption}
In Section \ref{subsec:key_generation}, we presented the mechanism used to generate the key used to encrypt files. Additionally, we showed that the algorithm used to encrypt files is AES256, a symmetric encryption scheme. In this Section, we will describe the process followed to encrypt files in the infected system.
The first step performed by the ransomware is to delete backups so the original files cannot be restored by locally saved security copies. To achieve that goal, the function at 0x41a800 executes the following processes:
\begin{itemize}
\item \hash{wmic.exe} \hash{SHADOWCOPY} \hash{/nointeractive}
\item \hash{wbadmin} \hash{DELETE} \hash{SYSTEMSTATEBACKUP}
\item \hash{wbadmin} \hash{DELETE} \hash{SYSTEMSTATEBACKUP} \hash{-deleteOldest}
\item \hash{bcdedit.exe} \hash{/set} \hash{\{default\}} \hash{recoveryenabled} \hash{No}
\item \hash{bcdedit.exe} \hash{/set} \hash{\{default\}} \hash{bootstatuspolicy} \hash{ignoreallfailures}
\item \hash{vssadmin.exe} \hash{Delete} \hash{Shadows} \hash{/All} \hash{/Quiet}
\end{itemize}
In order to successfully execute those processes, administrator privileges are needed, which were obtained using the procedure that we described in Section \ref{subsec:privilege_escalation}. Finally, the contents of the recycle bin are deleted by calling the Windows API function \textit{SHEmptyRecycleBinW}.
Next, files are encrypted following a depth-first search approach. Microsoft SQL and Exchange folders are prioritized, being the first ones to be encrypted. Then, the root path is encrypted (i.e., \hash{C:\textbackslash \textbackslash *}). Finally, shared folders and mapped volumes are enumerated and encrypted (e.g., \hash{D:\textbackslash \textbackslash *}, \hash{Y:\textbackslash \textbackslash *}, or \hash{\textbackslash \textbackslash VBoxSvr\textbackslash \textbackslash shared\_folder\textbackslash \textbackslash *}). Therefore, the order in which folders are encrypted, following a depth-first approach, is the following:\\
\begin{enumerate}
\item \hash{C:\textbackslash \textbackslash Program} \hash{Files\textbackslash \textbackslash Microsoft\textbackslash \textbackslash Exchange} \hash{Server\textbackslash \textbackslash *}
\item \hash{C:\textbackslash \textbackslash Program} \hash{Files} \hash{(x86)\textbackslash \textbackslash Microsoft\textbackslash \textbackslash Exchange} \hash{Server\textbackslash \textbackslash *}
\item \hash{C:\textbackslash \textbackslash Program Files\textbackslash \textbackslash Microsoft} \hash{SQL} \hash{Server\textbackslash \textbackslash *}
\item \hash{C:\textbackslash \textbackslash Program} \hash{Files} \hash{(x86)\textbackslash \textbackslash Microsoft SQL Server\textbackslash \textbackslash *}
\item \hash{C:\textbackslash \textbackslash *}
\item Shared folders and mapped volumes
\end{enumerate}
For each file encountered, the process performs three checks before the actual encryption.
\begin{enumerate}
\item \textbf{Strings from a whitelist}. The path is checked to not contain specific strings (see Appendix \ref{appedix:list_whitelisted_strings} for the list of skipped strings). If the absolute path of the file contains one of those strings, the file is left untouched. This check is excluded for the first four folders searched, those that belong to Microsoft SQL and Exchange servers. Therefore, this check is applied only to searches initiated at the root folder (i.e., C:\textbackslash \textbackslash *) or shared folders and mapped volumes.
\item \textbf{File extensions}. The extension of the file is checked. The extensions that are excluded (not encrypted) are the following: \hash{bin}, \hash{ini}, \hash{sys}, \hash{dll}, \hash{lnk}, \hash{dat}, \hash{exe}, \hash{drv}, \hash{rdp}, \hash{prf}, \hash{swp}, \hash{mdf}, \hash{mds} and \hash{sql}.
\item \textbf{Prevent re-encryption}. The third test checks if the file has already been encrypted by Avaddon. To do so, a signature at the end of the file (that is left after encrypting a file by the ransomware, as we will describe later in this section) is read. In particular, the last 24 bytes of the file are read. If the file has been previously encrypted, it should contain the hexadecimal values 0x200 and 0x1030307 at offsets 8 and 16 in those 24 bytes.
\end{enumerate}
If none of these checks is positive then the file is encrypted.
The encryption process is done by the function located at virtual address 0x413bb0. This function receives a copy of the AES256 key (see Section \ref{subsec:key_generation}) and the name of the file to be encrypted. We present a high-level pseudo code (some function signatures have been simplified to avoid using pointers) extracted from the analyzed function in Algorithm \ref{alg:file_encryption}. First, the size needed for the buffer to hold the bytes after encryption is calculated (line 1). Then, the file contents are read in chunks of 0x100000 bytes (line 5) and encrypted in blocks of 0x2000 bytes (lines 8-9). However, although there exists a loop to read and encrypt the whole file, only the first 0x100000 bytes are encrypted. This is due to the last call to \textit{SetFilePointerEx}, which sets the file pointer to the end of the file (line 18). When there are only 0x2000 or less bytes left to be encrypted (line 13), the last chunk of bytes is encrypted (lines 14-15) and written to the file (line 16). Notice that the parameter \textit{Final} (line 15) in the call to the encryption routine is always set to \textit{False}. This parameter should be \textit{True} if the block to encrypt is the last block of the file. We will need to take this detail into account in Section \ref{ch:key_recovery}. Finally, 512 unused bytes and the signature are written at the end of the file to mark it as encrypted (lines 20-22)\;
\begin{algorithm}
\SetAlgoLined
\caption{Function responsible for encrypting files.}
\label{alg:file_encryption}
\KwIn{\textit{File}, file to be encrypted\linebreak
\textit{Key}, a duplicate of the AES256 key}
\medskip
\textit{buffer\_size} $\leftarrow$ CryptEncrypt(hKey: Key, Final: False, pbData: 0, pdwDataLen: 0x2000)\;
\textit{file\_size} $\leftarrow$ GetFileSizeEx(hFile: File)\;
\textit{file\_pointer} $\leftarrow$ 0\;
\Do{\textit{number\_of\_bytes\_read} $\geq$ 0x100000 \&\& \textit{file\_pointer} $<$ \textit{file\_size}}{
\textit{bytes\_read}, \textit{number\_of\_bytes\_read} $\leftarrow$ ReadFile(hFile: \textit{File}, offset: \textit{file\_pointer}, nNumberOfBytesToRead: 0x100000)\;
\textit{i} $\leftarrow$ 0\;
\Do{\textit{i} $\leq$ \textit{number\_of\_bytes\_read} - 0x2000}{
\textit{bytes\_to\_encrypt} $\leftarrow$ \textit{bytes\_read[i:i+0x2000]}\; \tcp{The file is encrypted in blocks of 0x2000 bytes}
\textit{encrypted\_bytes} $\leftarrow$ CryptEncrypt(hKey: \textit{Key}, Final: False, pbData: \textit{bytes\_to\_encrypt})\;
WriteFile(hFile: \textit{File}, lpBuffer: \textit{encrypted\_bytes})\;
\textit{i} = \textit{i} + 0x2000\;
}
\If{\textit{number\_of\_bytes\_read} - \textit{i} $<$ 0x2000}{
\textit{bytes\_to\_encrypt} $\leftarrow$ \textit{bytes\_read[i:]}\;
\textit{encrypted\_bytes} $\leftarrow$ CryptEncrypt(hKey: \textit{Key}, Final: False, pbData: \textit{bytes\_to\_encrypt})\;
WriteFile(hFile: \textit{File}, lpBuffer: \textit{encrypted\_bytes})\;
}
\textit{file\_pointer} $\leftarrow$ SetFilePointerEx(hFile: \textit{File}, liDistanceToMove: \textit{0}, dwMoveMethod: \textit{FILE\_END}) \; \tcp{This call sets the file pointer to the end of the file. This is done to stop processing more bytes from the file}
}
WriteFile(hFile: \textit{File}, lpBuffer: \textit{VictimID})\; \tcp{The Victim ID is written to the end of the file}
\textit{signature} $\leftarrow$ GetSignature()\;
WriteFile(hFile: \textit{File}, lpBuffer: \textit{signature})\; \tcp{The signature is also written at the end}
\end{algorithm}
Therefore, the process is summarized as:
\begin{enumerate}
\item Calculate the size of the buffer needed to hold an encrypted block of 0x2000 (8192) bytes.
\item Obtain the size of the file.
\item Encrypt the first 0x100000 bytes of the file in blocks of 0x2000 (8192) bytes.
\item Write the victim ID (512 bytes) and the signature (24 bytes) at the end of the file.
\end{enumerate}
Here, we show an example of a signature written at the end of an encrypted file and highlight its different fields:\\
\colorbox{orange}{4E 4D 00 00 00 00 00 00} 00 02 00 00 01 00 00 00 \colorbox{cyan}{07 03 03 01} 01 01 E2 02\\
First, in orange, the original length of the file is written (0x4e4d or 20045 bytes in this case). Then, a hard-coded magic number is written at offset 16 (cian). This value is checked prior to encrypting a file, as we discussed earlier in this section.
\section{Decryption of infected systems} \label{ch:key_recovery}
In Section \ref{subsec:key_generation}, we described the functions responsible for importing, generating and destroying the cryptographic keys needed by the ransomware. As we pointed out, the key used for encrypting the system was randomly generated. Additionally, it was encrypted using a public key before being exported. Therefore, we are not able to know the key that is generated beforehand or to decrypt it after it has been exported, since we do not have the associated private key needed. However, we also hinted that the function responsible for destroying the cryptographic material was in fact never called if the system was not powered off. This is due to the ransomware process remaining in the background in order to encrypt new files or drives as they are created or connected. Since the keys are not destroyed and the ransomware process does not exit, we are able to recover the generated key. The only requirement is the memory of the ransomware process (i.e., a full dump). If such dump of the process (or the whole system) has been obtained, we may recover the key. This is of paramount importance, since users, upon seeing a ransom note, might be tempted to power off or reboot their systems in order to reestablish their machines, and would lose the opportunity of obtaining the key and thus decrypting the files.
In order to recover the key, we leverage the knowledge acquired during the advanced analysis process (see Section \ref{sec:analysis}) to identify the structure that points to the desired key. When a key is generated by using the Windows cryptography API (i.e., cryptsp.dll and rsaenh.dll) the key is an object of type HCRYPTKEY, which has the following structure \cite{CodeGuru2020}:
\noindent struct HCRYPTKEY\\
\{ \\
\indent void* CPGenKey; \\
\indent void* CPDeriveKey; \\
\indent void* CPDestroyKey; \\
\indent void* CPSetKeyParam; \\
\indent void* CPGetKeyParam; \\
\indent void* CPExportKey; \\
\indent void* CPImportKey; \\
\indent void* CPEncrypt; \\
\indent void* CPDecrypt; \\
\indent void* CPDuplicateKey; \\
\indent HCRYPTPROV hCryptProv; \\
\indent magic\_s *magic; \\
\}; \\
The first 10 fields of the structure point to functions of the Windows API. The eleventh field, \textit{hCryptProv}, points to the provider of the key and the functions (this provider must be first acquired before the key is generated via \textit{CryptAcquireContext} or a similar function). Finally, the last field points to another structure. This pointer is XOR-ed with a constant value, 0xE35A172C. Therefore, after XOR-ing the pointer with that magic constant, it points to the following structure:
\noindent struct magic\_s \\
\{ \\
\indent key\_data\_s *key\_data; \\
\}; \\
which contains a pointer to the following structure:
\noindent struct key\_data\_s \\
\{ \\
\indent void *unknown; \\
\indent uint32\_t alg; \\
\indent uint32\_t flags; \\
\indent uint32\_t key\_size; \\
\indent void* key\_bytes; \\
\}; \\
The \textit{key\_data\_s} structure contains three fields whose values are known:
\begin{itemize}
\item \textit{alg} contains the algorithm ID of the algorithm for which the key has been generated. In this case, the value of this field is 0x00006610, which corresponds to AES256 \cite{MSAlgID}.
\item \textit{flags} contains the value of the \textit{flags} parameter passed in the call to \textit{CryptGenKey} at 0x48f024. Therefore, its value is 0x00000001.
\item \textit{key\_size}, as it name hints, contains the size of the key. In this case, the key is 32 bytes long (0x00000020).
\end{itemize}
Finally, the fifth field contains a pointer to the actual key. Since we know the value of 24 of the last 28 bytes that form the structure (skipping the first field) we can search for this 28-byte pattern in the memory of the process. We thus are able to obtain a pointer to the generated key that was used to encrypt the files and finally the key itself.
We recall that the only requisite is that the system has not been powered off since it was infected, in order to maintain the key in memory.
Now that we have the symmetric key generated by the ransomware, we are able to decrypt the infected files. However, to do so we need to implement the reverse operation than the one performed by the ransomware (see Algorithm \ref{alg:file_encryption}). To decrypt any given file, we first parse the signature at the end of the file. There, we obtain the original size of the encrypted file. Then, we truncate the file to eliminate both the signature and the block of 512 bytes appended at the end of the file by the ransomware (536 bytes in total, since the signature is 24 bytes in length). Once we have the truncated file, we proceed to decrypt the first 0x100000 bytes in blocks of 8192 (0x2000) bytes. Notice that, as we showed in Algorithm \ref{alg:file_encryption}, the \textit{Final} parameter in the \textit{CryptEncrypt} calls was never set to \textit{True}. According to the documentation, this parameter should be \textit{True} when the last block is encrypted. Although we do not know if this nonstandard behavior is intentional or not, we are forced to do the same in the decryption routine. Therefore, we always set the Final parameter to be False in the calls to \textit{CryptDecrypt}. Then, we copy the rest of the file as is. Finally, if the file was smaller than 0x100000 bytes, we truncate it once again, now to the original size recovered earlier from the signature appended at the end, to remove the padding bytes.
Obtaining a memory dump of a process can be done by standard forensic tools. Therefore, we open source the tool to recover the symmetric key from memory and decrypt the infected files:\\ \url{https://github.com/JavierYuste/AvaddonDecryptor}.
\section{Experimentation} \label{sec:results}
We test our proposal in a virtual environment running a Windows 7 x64 OS. In particular, we build this virtual machine on top of a virtualization solution named VirtualBox in a 1.60 GHz Intel Core i5-8250U CPU with 16 GB RAM computer. From the available hardware, we assign 2 cores and 4 GB of RAM to the aforementioned guest system.
Then, we execute Avaddon on the virtual machine and let it encrypt the whole system. When Avaddon has not utilized more than 0.5\% of the CPU time in the last 60 seconds, we understand that it has finished encrypting files and confirm the infection due to the presence of ransom notes and encrypted files through the whole file system.
After infecting the virtual machine, we proceed to decrypt all the affected files. First, we pause the ransomware process with Process Explorer, a tool from the SysInternals suite~\footnote{\url{https://docs.microsoft.com/en-us/sysinternals/}}. Note that we can freely drop executable files in the system before stopping Avaddon, since the \textit{exe} extension is excluded. Once the process is suspended, we can safely operate in the infected system. Next, we dump the memory of the ransomware process with ProcDump, which is also part of the SysInternals suite. Finally, we execute the proposed decryption tool, which we open source. This tool i) confirms the infection by extracting the signature appended at the end of encrypted files, ii) obtains the AES256 symmetric key from the dumped memory of the ransomware process iii) and decrypts the whole file system.
We show the results in Table \ref{tab:experimentation_results}. From 209,186 files that were present in the whole system, we found that 9,135 (4.3\%) were encrypted, making a total of 607 MB. Our proposed tool successfully decrypted all the affected files in 10 minutes and 35 seconds. Additionally, we have tested our tool with the most recent version of Avaddon, which was observed from a wild URL on mid-January 2021, when this paper was written. We confirm that the decryptor still works, since we were able to decrypt all the infected files.
We must note some considerations. First, it is important to not turn off the computer after infection, since the proposed approach needs the encryption key to be present in memory. Otherwise, this would be destroyed and could only be recovered by means of the official channel proposed by the criminals, i.e. paying the ransom. Second, the proposed tool needs the original version of at least one encrypted file to find the correct symmetric key. This, however, can be easily achieved, e.g. by obtaining known files present by default in the Windows OS version installed in the affected system.
\begin{table}[]
\centering
\begin{tabular}{|c|c|}
\hline
Files in the system & 209,186\\
\hline
Files encrypted by Avaddon & 9,135\\
\hline
Total size of files in the system & 46.85 GB\\
\hline
Total size of encrypted files & 607 MB\\
\hline
Time spent decrypting files & 558.54 s\\
\hline
Total time & 635.63 s\\
\hline
\end{tabular}
\caption{Results of the experimentation in a virtual environment.}
\label{tab:experimentation_results}
\end{table}
\section{Conclusions} \label{ch:conclusions}
Current approaches of cybercrime specialization, including new malware techniques, increase the threat of modern ransomware campaigns. In this work, we have analyzed a new ransomware, Avaddon, operated as a RaaS in a shared profit scheme, first seen on June 2020. Avaddon incorporates two techniques aimed at increasing their financial revenues which are growing in popularity: i) threatening victims that do not want to pay the ransom fee to leak personal data from infected systems, and ii) conducting DDoS attacks against them. Data leakage have affected at least 23 organizations whose information is allegedly exposed online.
While having proper attribution is difficult, our analysis suggests that the threat actor behind Avaddon is from a CIS country. Indeed the initial announcement of the ransomware was made in a Russian underground forum, and it implements a policy to prevent infection of CIS-based victims. Moreover, a typographic error found in one of the processes fingerprinted by Avaddon suggest that this family is related with a previous ransomware, i.e. MedusaLocker, where the same error is also present. Indeed, the \textit{modus operandi} of Avaddon, that we detailed in this work, is similar to that of MedusaLocker \cite{TAU2020} and the list of services to stop is almost identical in both cases.
By examining a sample obtained from the first campaign of Avaddon and describing its behaviors, we took a grasp on the general ``Cyber Kill Chain'' of ransomware threats (land, escalate privileges, deactivate defenses, acquire persistence, delete backups and encrypt files) and a detailed analysis of this ransomware in particular. Using an hybrid scheme, Avaddon attempts to hide the session key from defenders. However, due to the way in which cryptography keys are managed in this ransomware, we have developed a tool to recover the session key from the memory of the infected systems and decrypt all the affected files. The decryption tool also works with newer variants of the ransomware. The only requirement for this method to work is that the victim's computer is not powered off after the infection.
Due to novelty of the ransomware, the business model in terms of an affiliate program, and the ability to extortion and blackmail victims (by means of exfiltration and DDoS attacks), it is likely to expect new variants of Avaddon and similar ransomware samples improving their mechanisms and expanding in the future. Thus, we believe that the analysis and tools provided in this paper can contribute to guide future analyses of such variants and to improve existing mitigation mechanisms.
\section*{Acknowledgements}
This work was supported by the Comunidad de Madrid (P2018/TCS-4566, co-financed by European Structural Funds ESF and FEDER).
\section*{References}
|
1,116,691,500,716 | arxiv | \section{Introduction}
\label{sec:introduction}
Many engineering problems in medicine can be cast as inverse problems in which
a vector of latent biophysical characteristics $\gv{\omega}$ (e.g., biomarkers) are estimated
from a vector of $M$ noisy measurements or signals $\v{y}\in \mathbb{R}^M$, each measurement being performed with
experimentally-controlled acquisition parameters $\v{p}_i$ ($i=1,\dots,M$) defining the experimental protocol $\mathcal{P} \coloneqq \left\{\v{p}_i\right\}_{i=1}^{M}$.
We consider the case in which a generative model of the signal $S\left(\gv{\omega};\v{p}\right)$ is available but hard to invert and costly to evaluate. This occurs for instance when $S$ is very complex or non-differentiable, which includes most cases of $S$ being a numerical simulation rather than a closed-form model. Without loss of generality, the signal $S$ is assumed to arise from $K$ independent contributions weighted by weights $\nu_1,\dots,\nu_K$
\begin{equation}
S\left(\gv{\omega};\mathcal{P}\right) = \sum\limits_{k=1}^K \nu_k S_k\left(\gv{\omega}_k;\mathcal{P}\right),
\label{eq:linearity_S}
\end{equation}
where the $S_k$ are available and with $\gv{\omega}=\left[\gv{\omega}_1^T \dots \gv{\omega}_K^T \right]^T$. When no linear separation is possible or known then $K=1$.
Because deep neural networks (DNNs) have shown a tremendous ability to learn complex input-output mappings \cite{hornik1990universal,fakoor2013using, guo2016deep,lucas2018using,zhao2019object}, it is tempting to resort to a DNN to learn $\gv{\omega}$ directly from the data $\v{y}$ or from synthetic samples $S\left(\gv{\omega};\mathcal{P}\right)$. This is arguably the least interpretable approach and is referred to as the fully-learned, black-box approach. Little insight into the prediction process is available, which may deter medical professionals from adopting it. In addition, changes to the input such as the size $M$ (e.g., missing or corrupt measurements) or a modified experimental protocol $\mathcal{P}$ (e.g., hardware update) likely require a whole new network to be trained.
At the other end of the spectrum, the traditional, physics-based approach in that case is to perform dictionary fingerprinting. As described in Section~\ref{sec:fingerprinting}, this roughly consists in pre-simulating many ($\Ntot{}$) possible signals $S$ and finding the combination of $K\ll \Ntot{}$ responses that best explains the measured data $\v{y}$. This method is explainable and interpretable but it is essentially a brute-force discrete search. It is thus computationally intensive at inference time (which may be problematic for real-time applications) and does not scale well with problem size.
We propose an intermediate method, referred to as a hybrid approach, which aims to capture the best of both worlds. The measured signal $\v{y}$ is projected onto a basis of $\Ntot{}$ fingerprints, pre-simulated using the biophysical model $S$. The naturally sparse representation (in theory, $K\ll \Ntot{}$ weights are non zero) is then fed to a neural network with an architecture driven by the physics of the process for final prediction of the latent biophysical parameters $\gv{\omega}$.
We describe the theory for general applications and provide an illustrative example of the estimation of brain tissue properties from diffusion-weighted magnetic resonance imaging (DW-MRI) data, where the inverse problem is learned on simulated data.
We compare the three methods (dictionary fingerprinting, hybrid and fully-learned) in terms of efficiency and accuracy on the estimated brain properties. The explainability of the hybrid and fully-learned approaches are assessed visually by projecting intermediate activations in a 2D space.
\section{Related work}
Despite their success at solving many (underdetermined) inverse problems~\cite{lucas2018using,bai2020deep} and despite efforts to explain model predictions in healthcare~\cite{elshawi2020interpretability,stiglic2020interpretability}, DNNs still offer little accountability and are mathematically prone to major instabilities~\cite{gottschling2020troublesome}. There has thus been a growing trend toward incorporating engineering and physical knowledge into deep learning frameworks~\cite{lucas2018using}. One way to do so is to unfold well-known, often iterative algorithms in a DNN architecture~\cite{ye2017tissue,ye2019deep}, or to produce parameter updates based on signal updates from the forward model $S$~\cite{ma2020deep}. However those approaches require many evaluations of the forward model $S$, which is not always possible or affordable.
The latent variables $\gv{\omega}$ can be difficult to access. This is the case in our example problem where microstructural properties of the brain cannot be measured \textit{in vivo}. In such cases, DNNs can be trained on synthetic data $S\left(\gv{\omega};\mathcal{P}\right)$ and then applied to experimental measurements $\v{y}$, which is the approach taken with our illustrative example. Another option is via a self-supervised framework wherein the inverse mapping $S^{-1}$ is learned such that $S\left(S^{-1}\left(\v{y}\right)\right) \approx \v{y}$~\cite{senouf2019self}. However, the latter again requires many evaluations of $S$.
Our physics-based fingerprinting approach is based on the framework by~\cite{ma2013magnetic} for magnetic resonance fingerprinting (MRF) extended to the multi-contribution case~\cite{rensonnet2018assessing,rensonnet2019towards}. As dictionary sizes increased and inference via fingerprinting became slower, DNN methods have been proposed to accelerate the process~\cite{oksuz2019magnetic,golbabaee2019geometry}. However, these do not consider the case of multiple linear signal contributions as in Eq.~\eqref{eq:linearity_S}.
In the field of DW-MRI (our illustrative example), training of DNNs using high-quality data $\v{y}$, i.e. data acquired with a rich protocol $\mathcal{P}$, has been suggested in several works~\cite{golkov2016q,ye2017tissue,schwab2018joint,fang2019deep,ye2020improved}. The main limitation is that such enriched data may not always be available. It is worth noting that \cite{ye2020improved} also proposed Lasso bootstrap to quantify prediction uncertainty, as a way to strengthen interpretability.
\section{Methods}
\label{sec:methods}
\subsection{Running example: estimation of white matter microstructure from DW-MRI}
\label{sec:example}
In DW-MRI, $M$ volumes of the brain are acquired by applying magnetic gradients with different characteristics $\v{p}$ (intensity, duration, profile). In each voxel of the brain white matter, a measurement vector $\v{y}\in \mathbb{R}^M$ is obtained by compiling the $M$ DW-MRI values at this given voxel location.
As illustrated in Figure~\ref{fig:microstructure}, the white matter is mainly composed of long, thin fibers known as \emph{axons} which tend to run parallel to other axons, forming bundles called \emph{fascicles} or \emph{populations}. In a voxel at current clinical resolution, 2 to 3 populations of axons may intersect \cite{jeurissen2013investigating}. In this paper a value of $K=2$ populations is assumed for simplicity. We use a simple model of a population of parallel axons (Figure~\ref{fig:microstructure}) parameterized by a main orientation $\v{u}$, an axon radius index $r$ (of the order of $\SI{1}{\micro\meter}$) and an axon fiber density $f$ (in $[0, 0.9]$), which are properties involved in a number of neurological and psychiatric disorders \cite{chalmers2005contributors, mito2018fibre, andica2020neurocognitive}. Assuming the signal contribution of each population $k$ to be independent~\cite{rensonnet2018assessing}, the generative signal model $S$ in a voxel is
\begin{equation}
S\left(\gv{\omega};\mathcal{P}\right) = \phi_{\textrm{SNR}}\left(\sum\limits_{k=1}^K \nu_k S_{\textrm{MC}}\underbrace{(\v{u}_k, r_k, f_k}_{\gv{\omega}_k};\mathcal{P})\right),
\label{eq:signal_model_dwmri}
\end{equation}
where the signal of each population is modeled by an accurate but computationally-intensive Monte Carlo simulation $S_{\textrm{MC}}$ \cite{hall2009convergence,rensonnet2015hybrid}. The weights $\nu_k$ can be interpreted as the fraction of voxel volume occupied by each population. The model is stochastic with $\phi$ modeling corruption by Rician noise \cite{gudbjartsson1995rician} at a given signal-to-noise ratio (SNR). In most clinical settings, SNR is high enough for Rician noise to be well approximated by Gaussian noise. A least-squares data fidelity term $\left\|\v{y} - S\left(\gv{\omega};\mathcal{P}\right) \right\|_2^2 $ is thus popular in brain microstructure estimation as it corresponds to a maximum likelihood solution.
The acquisition protocol $\mathcal{P}$ is the MGH-USC Adult Diffusion protocol of the Human Connectome Project (HCP), which comprises $M=552$ measurements \cite{setsompop2013pushing}.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth, clip=True, trim=0cm 7.0cm 7.25cm 0cm]{fig_microstructure}
\caption{\textbf{Running example : estimation of white matter microstructure.} A voxel of white matter, shown here on a T1 anatomical scan of a healthy young adult from the Human Connectome Project \cite{van2012human}, contains crossing populations of roughly parallel axons. In our example model, each population is described by an orientation $\v{u}$, a volume fraction $\nu$, an axon radius index $r$ and an axon density index $f$ and contributes independently to the DW-MRI signal. The vector of biophysical parameters is $\gv{\omega}=\left[\v{u}_1^T,\nu_1, r_1, f_1, \v{u}_2^T, \nu_2, r_2, f_2\right]^T$. }
\label{fig:microstructure}
\end{figure}
\subsection{Physics-based dictionary fingerprinting}
\label{sec:fingerprinting}
Fingerprinting is a general approach essentially consisting in a sparse look-up in a vast precomputed dictionary containing all possible physical scenarios. A least squares data fidelity term is assumed here~\cite{bai2020deep} although the framework could be extended to other objective functions.
For each contribution $k$ in Eq.~\eqref{eq:linearity_S}, a sub-dictionary $\v{C}^k \in\mathbb{R}^{M\times N_k}$ is presimulated once and for all by calling the known $S_k$, to form the total dictionary $\mathcal{D}\coloneqq \left[\v{C}^1 \dots \v{C}^K\right] \in \mathbb{R}^{M\times \Ntot{}}$, where $\Ntot{}\coloneqq \sum_{k=1}^K N_k$. Each $\v{C}^k$ thus contains $N_k$ atoms or \emph{fingerprints} $\v{A}^k_{j_k}\coloneqq S_k\left(\gv{\omega}_{kj_k};\mathcal{P}\right) \in \mathbb{R}^M $, with $j_k=1,\dots,N_k$,
corresponding to a sampling of $N_k$ points in the space of biophysical parameters given a known experimental protocol $\mathcal{P}$. The dictionary look-up in this multi-contribution setting can be mathematically stated as
\begin{equation}
\begin{array}{lll}
\hat{\v{w}}= & \argmin\limits_{\v{w}\geq 0}
& \left\|\v{y}-\begin{bmatrix}\v{C}^1 \dots \v{C}^K\end{bmatrix}\cdot
\begin{bmatrix}\v{w}_1\\ \vdots \\ \v{w}_K\end{bmatrix}
\right\|_2^2\\
& & \\
& \text{subject to} & \left| \mathbf{w}_k \right|_{0}=1, \quad k=1,\dots,K, \\
\end{array}
\label{eq:sparse_optimization}
\end{equation}
where the sparsity constraints $\left|\cdot\right|_0 $ on the sub-vectors $\v{w}_k$ guarantee that only one fingerprint $\v{A}^k_{j_k}$ per sub-dictionary $\v{C}^k$ contributes to the reconstructed signal. No additional tunable regularization of the solution is needed as all the constraints (e.g., lower and upper bounds on latent parameters $\gv{\omega}$) are included in the dictionary $\mathcal{D}$. Equation~\eqref{eq:sparse_optimization} is solved exactly by exhaustive search, i.e. by selecting the optimal solution out of $\prod_{k=1}^K N_k$ independent non-negative linear least squares (NNLS) sub-problems of $K$ variables each
\begin{equation}
\begin{split}
(\hat{j}_1,\dots, \hat{j}_K)=& \\
\argmin\limits_{1\leq j_k \leq N_k}\quad & \min\limits_{\v{w}\geq 0} \left\| \v{y}-\begin{bmatrix}\v{A}^1_{j_1} \dots \v{A}^K_{j_K}\end{bmatrix}\cdot
\begin{bmatrix}w_1\\ \vdots \\ w_K \end{bmatrix} \right\|_2^2.
\end{split}
\label{eq:combinatorial_optimization}
\end{equation}
Each sub-problem is convex and is solved exactly by an efficient in-house implementation\footnote{\texttt{solve\_exhaustive\_posweights} function of the Microstructure Fingerprinting library available at \url{https://github.com/rensonnetg/microstructure_fingerprinting}. Written in Python 3 and optimized with Numba (\url{http://numba.pydata.org/}).} of the active-set algorithm~\citep[][chap. 23, p. 161]{lawson1995solving}, which empirically runs in $\mathcal{O}\left(K\right)$ time. The optimal biophysical parameters $\hat{\v{\omega}}_k$ are finally obtained as those of the optimal fingerprint $\hat{j}_k$ in each $\v{C}^k$ and the weights $\hat{\nu}_k$ are estimated from the optimal $\hat{w}_k$.
The main limitation of this approach is its computational runtime complexity $\mathcal{O}\left(N_1\dots N_K K\right)$ or $\mathcal{O}\left(N^K K\right)$ if $N_k=N $ $\forall k$. Additionally, the size $N_k$ of each sub-dictionary $\v{C}^k$ also increases rapidly as the number of biophysical parameters, i.e. as the number of entries in $\gv{\omega}_k$ increases.
In our running example, $K=2$ sub-dictionaries of size $N=782$ each are pre-computed using Monte Carlo simulations, corresponding to biologically-informed values of axon radius $r$ and density $f$. The orientations $\v{u}_1, \v{u}_2$ are pre-estimated using an external routine \cite{tournier2007robust} and directly included in the dictionary.
\subsection{Fully-learned neural network}
\label{sec:full_nn}
As depicted in Figure~\ref{fig:architectures}, we focus on a multi-layer perceptron (MLP) architecture with rectified linear unit (ReLU) non-linearities (not shown in Figure~\ref{fig:architectures}), which has the benefit of a fast forward pass for inference. In our brain microstructure example, training is performed on simulated data obtained by Eq.~\eqref{eq:signal_model_dwmri} with tissue parameters $\gv{\omega}$ drawn uniformly from biophysically-realistic ranges. A mean squared error (MSE) loss is used to match Eq.~\eqref{eq:sparse_optimization} and \eqref{eq:combinatorial_optimization}. Dropout \cite{srivastava2014dropout} is included in every layer during training and stochastic gradient descent with adaptive gradient (Adagrad) \cite{duchi2011adaptive} is used to estimate parameters.
\begin{figure*}[h!]
\centering
\includegraphics[width=0.60\textwidth, clip=true, trim=0cm 0cm 11.5cm 0cm]{fig_architectures}
\caption{\textbf{Interpretable hybrid vs fully-learned approach.} (a) The vector of measurements $\v{y}$ is decomposed by NNLS into a sparse representation in the space of physics-based fingerprints (i.e., many $w_{kj_k}$ are zero). The weights are given to a split multi-layer perceptron (MLP) followed by final fully-connected (fc) layers to predict the tissue parameters (axon radius $r$, axon fiber density $f$, relative volume of axon population $\nu$). (b) In the fully-learned approach, the input is directly fed to a DNN which makes the final prediction. Rectified linear units (ReLU) follow every fc layer in both networks.}
\label{fig:architectures}
\end{figure*}
\subsection{Hybrid method}
\label{sec:hybrid}
The proposed method is a combination of the fingerprinting and the end-to-end DNN approaches presented above.
\paragraph{First stage: NNLS.} The same sub-dictionaries $\v{C}^k$ as in Section~\ref{sec:fingerprinting} are computed. Instead of solving Eq.~\eqref{eq:sparse_optimization} with 1-sparsity contraints on the sub-vectors $\v{w}_k$, a single NNLS problem is solved
\begin{equation*}
\begin{array}{lll}
\hat{\v{w}}= & \argmin\limits_{\v{w}\geq 0}
& \left\|\v{y}-\mathcal{D}\cdot \v{w} \right\|_2^2,\\
\end{array}
\end{equation*}
where the optimization completely ignores the structure of the dictionary $\mathcal{D}=\begin{bmatrix}\v{C}^1 \dots \v{C}^K\end{bmatrix} \in \mathbb{R}^{M\times \Ntot{}} $. Unlike in Eq.~\eqref{eq:combinatorial_optimization}, only one reasonably large NNLS problem with $\sum_{k=1}^KN_k$ variables is solved rather than many small problems with $K$ variables. The runtime complexity of this algorithm is $\mathcal{O}\left(\Ntot{}\right)$ in practice and typically yields sparse solutions~\citep{slawski2011sparse}. In fact, if the model $S$ used to generate the dictionary were perfect for the measurements $\v{y}$ we would have $\left|\v{w}\right|_0=K \ll \Ntot{}$. There is however no guarantee that the true latent fingerprints in $\mathcal{D}$ will be among those attributed non-zero weights by the NNLS optimization. However, we expect those selected fingerprints with non-zero weights to have underlying properties $\hat{\gv{\omega}}$ close to and informative of the true latent biophysical properties. The second stage of the method can be seen as finding the right combination of these pre-selected features for an accurate final prediction.
In our running example, as in the fingerprinting approach, the orientations $\v{u}_1, \v{u}_2$ are pre-estimated using an external routine \cite{tournier2007robust} and directly included in the dictionary.
\paragraph{Second stage: DNN.}
The output $\hat{\v{w}}$ of the first-stage NNLS estimation is given to the neural network depicted in Figure~\ref{fig:architectures}. Its architecture exploits the multi-contribution nature of the problem: each sub-vector $\hat{\v{w}}_{k}$ of $\hat{\v{w}}$ is first processed by a ``split'' independent multi-layer perceptron (MLP) containing $N_k$ input units (blue in Figure~\ref{fig:architectures}).
Splitting the input has the advantage of reducing the number of model parameters while accelerating the learning of compartment-specific features by preventing coadaptation of the model weights~\cite{hinton2012improving}. A joint MLP (green in Figure~\ref{fig:architectures}) performs the final prediction of biophysical parameters. The output of all fully-connected layers is passed through a ReLU activation. Being a feed-forward network, inference is very fast once trained. The overall computational complexity of the hybrid method is therefore dominated by the first NNLS stage.
\section{Experimental results}
\subsection{Efficiency}
Table~\ref{tab:performance} specifies the values obtained during the fine-tuning of the meta parameters of the DNNs used in the fully-learned and hybrid approaches. As predicted by theory, the hybrid method is an order of magnitude faster than the physics-based dictionary fingerprinting for inference. However its NNLS first stage makes it slower than the end-to-end DNN solution. Training times were similar but the fully-learned approach had the advantage of bypassing the precomputation of the dictionary.
\begin{table*}[t]
\caption{Meta parameters and efficiency of the three methods.}
\label{tab:performance}
\vskip 0.15in
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{lccc}
\toprule
& Fingerprinting & fully-learned & Hybrid \\
\midrule
minibatch size & $\times$ & $\num{5000}$ & $\num{5000}$ \\
dropout rate & $\times$ & $\num{0.05}$ & $\num{0.1}$ \\
learning rate & $\times$ & $\num{5e-4}$ & $\num{1.5e-3}$\\
training samples & $\times$ & $\num{4e5}$ & $\num{4e5}$ \\
hidden units & $\times$ & $\num{3600}$ & $\num{1700}$ \\
parameters & $\times$ & $\num{3e6}$ & $\num{4.6e5}$ \\
precomputation time & $\approx \SI{2}{\day}$ & $\times$ & $\approx \SI{2}{\day}$ \\
inference time/voxel & $\SI{1.25}{\second}$ & $\SI{1.02e-4}{\second}$ & $\SI{1.45e-1}{\second} $ \\
inference complexity & $\BigO{N_1\dots N_K K}$ & $\BigO{1}$ & $\BigO{N_1+\dots +N_K}$ \\
\bottomrule
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vskip -0.1in
\end{table*}
\subsection{Accuracy}
$\num{15000}$ samples $\left(\gv{\omega},S\left(\gv{\omega};\mathcal{P}\right)\right)$ (never seen by our DNNs during training) were simulated using Eq.\eqref{eq:signal_model_dwmri} of our DW-MRI example, with biophysical parameters in realistic ranges and SNR levels 25, 50 and 100. In order to test the robustness of the approaches to parameter uncertainty, two scenarios were tested for the fingerprinting and hybrid methods. First, the population orientations $\v{u}_1,\v{u}_2$ were estimated using \citep{tournier2007robust} and therefore subject to errors. Second, the reference groundtruth orientations were directly included in the dictionary $\mathcal{D}$. This did not apply to the fully-learned model as it learned all parameters from the measurements $\v{y}$ directly.
Figure~\ref{fig:accuracy} shows that the hybrid approach (green lines) was more robust to uncertainty on $\v{u}$ than the physics-based fingerprinting method (blue lines), as the mean absolute errors (MAEs) only slightly increased when $\v{u}$ was misestimated (continuous lines). The fully-learned model exhibited the best overall performance. The poorer performance on the estimation of the radius index $r$ (middle row) is a well-known pitfall of DW-MRI as the signal only has limited sensitivity to $r$~\cite{clayden2015microstructural,sepehrband2016towards}. As the reference latent value of $\nu$ increased (x axis of Figure~\ref{fig:accuracy}), estimation of all tissue properties generally improved for all models.
\begin{figure}
\centering
\includegraphics[scale=0.38, clip=True, trim=0.60cm 3.0cm 0.6cm 3.1cm]{fig_accuracy}
\caption{\textbf{The proposed hybrid method exhibits high accuracy and robustness.} Its accuracy on the estimated biophysical parameters measured by mean absolute error (MAE) from the reference groundtruth value was close to that of the end-to-end DNN approach for $\nu$ and $f$ and offered similar robustness to parameter uncertainty (here misestimated population orientations $\v{u}$). The estimation of $r$ (middle row) is notoriously difficult from DW-MRI data and should be interpreted with caution.}
\label{fig:accuracy}
\end{figure}
\subsection{Explainability}
A total of $\num{11730}$ test samples were simulated using Eq.~\eqref{eq:signal_model_dwmri} in realistic ranges for $\nu,r,f$ and crossing angle between $\v{u}_1$ and $\v{u}_2$ at SNR 50. To ease visualization of the results, $r_1=r_2$ and $f_1=f_2$ was enforced. The activations, defined as the vector of output values of a fully-connected layer \emph{after} the ReLU activation, were inspected at different locations of the DNNs used in the hybrid and the fully-learned methods. The projection of these multi-dimensional vectors into a 2-dimensional space was performed using t-SNE embedding~\cite{van2008visualizing} for each of the $\num{11730}$ test samples. The idea of this low-dimensional projection was to conserve the inter-sample distances and topology of the high-dimensional space.
As shown in Figure~\ref{fig:activations}, the DNN in the second stage of the hybrid method seemed to learn the $r$ and $f$ properties after just the first layer, while it took the fully-learned model three layers to display a similar sample topology. At the end of the split MLP the $\nu$ parameter (marker shapes in Figure~\ref{fig:activations}) did not seem to have been learned but the different values of $\nu$ were then well separated at the end of the merged final MLP (see architecture in Figure~\ref{fig:architectures}).
\begin{figure*}[h!]
\centering
\includegraphics[width=0.85\textwidth, clip=true, trim=0cm 2cm 0cm 0cm]{fig_activations}
\caption{\textbf{The neural network of the hybrid approach is faster to learn useful signal representations.} Projection in 2-dimensional plane of network activations using t-SNE embedding~\cite{van2008visualizing}. Top row: network used in the hybrid approach. Activations after (a) the first layer, (b) the last layer of the split MLP; (c) the last layer of the final MLP before final prediction. The input layer has already learned the axon radius $r$ and density $f$ and the final layers learn the relative volumes $\nu$ after the parallel networks merge (Figure~\ref{fig:architectures}). Bottom row: network in the fully-learned approach. Activations after (d) the first layer, (e) an intermediate layer, (f) the final layer. The visualization suggests that the first layer hasn't learned any tissue parameter yet.}
\label{fig:activations}
\end{figure*}
\section{Discussion and conclusion}
An approach was proposed combining the efficiency of deep neural networks with the interpretability of traditional optimization for general inverse problems in which evaluation of the forward generative model $S$ is possible but expensive. The potential of the method was exemplified on a problem of white matter microstructure estimation from DW-MRI. The DNN used in the hybrid method receives as input the measurements $\v{y}$ expressed as a linear combination of a few representative fingerprints taken from a physically-realistic basis. Exploiting the multi-contribution nature of the physical process, this input is separated by signal contribution and fed to independent MLP branches (see Figure~\ref{fig:architectures}). The overall effect is an expedited learning process, as suggested in Figure~\ref{fig:activations} where latent parameters $\v{\omega}_k$ are learned directly, ``for free'', for each axon population $k$. Consequently, we could consider further reducing the number and/or size of layers in the split MLP part of the network used in our illustrative example.
Figure~\ref{fig:activations} also confirms that the weights $\nu$ of the independent signal contributions are only learned in the final, joint MLP of the DNN of the hybrid method. This is because a relative signal weight can only be defined with respect to the \emph{other} weights. The split branches of the network treat individual signal contributions independently, unaware of the other contributions, and thus cannot learn the relative weights $\nu$. This suggests that they more specifically focus on the properties $\v{\omega}_k$ of each contribution.
A limitation of the proposed hybrid approach is the need to simulate the dictionary $\mathcal{D}$ which can be costly as the number of latent biophysical parameters increases. With high-dimensional inputs such as medical images, it may also be necessary to consider better adapted convolutional network architectures in the second stage of the hybrid method. General decompositions such as PCA analysis could then be considered~\cite{harkonen2020ganspace}. Our exemplary experiments were performed on simulated data and further investigation is required to demonstrate that the method generalizes to experimental data.
Further work should also test whether the hybrid approach enables transfer learning from a protocol $\mathcal{P}$ to a new protocol $\mathcal{P}'$. In our DW-MRI example, this could happen with a scanner update or protocol shortened to accomodate pediatric imaging, for instance. While a new dictionary $\mathcal{D}'$ would be required for the fingerprinting and the hybrid method, the DNN trained in the hybrid approach should in theory still perform well without retraining. This is because fingerprints in $\mathcal{D}'$ would be linked to the same latent biophysical parameters $\gv{\omega}_{kj_k}$ (the number of rows $M'$ of $\mathcal{D}'$ would change, but not the number of columns $\Ntot{}$). Preliminary results (not shown in this paper) were encouraging. The DNN of the fully-learned approach would need to be retrained completely however.
Future work will also more closely inspect the inference process in the network via techniques such as LIME~\cite{ribeiro2016should}, guided back-propagation~\cite{selvaraju2017grad}, shapley values and derivatives~\cite{shapley201617,sundararajan2020many} or layer-wise relevance propagation~\cite{bohle2019layer}. This would complement our t-SNE inspection and further reinforce the confidence in the prediction of our approach.
We hope that these preliminary findings may contribute to incorporating more domain knowledge in deep learning models and ultimately encourage the more widespread adoption of machine learning solutions in the medical field.
\medskip
\FloatBarrier
\section*{Acknowledgements}
Computational resources have been provided by the supercomputing facilities of the Universit\'{e} catholique de Louvain (CISM/UCL) and the Consortium des \'{E}quipements de Calcul Intensif en F\'{e}d\'{e}ration Wallonie Bruxelles (C\'{E}CI) funded by the Fond de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under convention 2.5020.11 and by the Walloon Region.
|
1,116,691,500,717 | arxiv | \section{Introduction}
In this paper we will study the geometry of $G$-manifolds with finitely many non-principal orbits. Here, both the group $G$ and the manifold are smooth, compact and connected, and the action of $G$ on the manifold is smooth and effective.
The orbits of such a $G$-action are either principal, exceptional (that is, non-principal but with the same dimension as a principal orbit), or singular. The codimension of the principal orbits is the cohomogeneity of the $G$-space.
The motivation for studying the situation where the non-principal orbits are finite in number arises from the study of cohomogeneity-one manifolds where existence of invariant metrics with positive Ricci curvature is controlled by the fundamental group.
\begin{thm}[\cite{Be},\cite{GZ2}] A compact $G$-manifold of cohomogeneity $0$ or $1$ admits an invariant metric of positive Ricci curvature if and only if its fundamental group is finite.
\end{thm}
It is already observed in \cite{GZ2} that this can not carry over to cohomogeneity $\geq 4$. The situation in between, i.e. for cohomogneity $2$ and $3$ is essentially open. There are some partial results however, under stronger conditions. Thus in \cite{BW0} metrics of positive Ricci curvature are constructed on asystatic compact $G$-manifolds with finite fundamental group all of whose singular orbits are fixed points.
Cohomogeneity-one manifolds have been studied intensively in recent times. The reason that these objects form such a good family to study is that they have a simple topoplogical description, but form a large and rich class containing many interesting and important examples. A compact cohomogeneity one manifold is either a fibre bundle over a circle (in which case all orbits are principal), or has precisely two non-principal orbits. Of particular note is the role that cohomogeneity one manifolds continue to play in the search for new examples of manifolds with good curvature characteristics. If one considers invariant metrics, then symmetry reduces the problem of describing and analysing such metrics to one which has a reasonable chance of being tractable. For example, new families of manifolds with non-negative sectional curvature, including many exotic spheres in dimension seven, have been been discovered as a result of this approach \cite{GZ1}. The cohomogeneity one condition in the context of positive sectional curvature has attracted particular attention due to the work of Grove, Ziller, Wilking, Verdiani and others. (See for example \cite{GWZ}, \cite{GVWZ}, \cite{V1}, \cite{V2}.) In a recent development, Grove, Verdiani and Ziller \cite{GVZ} and independently Dearricott \cite{D} have announced the existence of a new cohomogeneity-one manifold with positive sectional curvature. Together with the recent announcement of a positive sectional curvature metric on the Gromoll-Meyer sphere by Petersen and Wilhelm \cite{PW}, these are the first new examples of manifolds admitting positive sectional curvature metrics for a number of years.
As a compact cohomogeneity one manifold has either zero or two non-principal orbits, the study of $G$-manifolds with finitely many non-principal orbits can be viewed as a natural generalisation of the cohomogeneity one situation.
\medskip
Topologically, $G$-manifolds with finitely many non-principal orbits have a rather simple structure. Let $M$ be such a $G$-manifold, and suppose the principal isotropy is $K$, so that principal orbits are all equivariantly diffeomorphic to $G/K$. Suppose the non-principal isotropy groups are $H_1,...H_p$, so the non-principal orbits are $G/H_1,...,G/H_p$. It is crucial that the non-principal isotropy groups $H_i$ act with only one orbit type on spheres. In cohomogeneity one their action is transitive. In higher cohomogeneity, $K$ is normal in $H_i$ and $L_i:=H_i/K$ acts freely on a sphere. By Theorem 6.2 in \cite{Br} the group $L_i$ must finite or one of the groups $\U(1)$, $\SU(2)$, $N_{\SU(2)}\U(1)$. For details on this and the subsequent topological facts about $G$-manifolds with finitely many non-principal orbits we refer to \cite{BW}. From now on we will assume that the cohomogeneity is at least $2$.
Let $N_1,...N_p$ be disjoint equivariant tubular neighbourhoods of the non-principal orbits. Then $M^0:=M-\cup_{i=1}^p N_i$ consists of principal orbits only and we have a $G/K$-fibre bundle $M^0\to B$ with structure group $W:=N_GK/K$. Each tubular neighbourhood $N$ admits a simple description. Let $T:=\partial N$. It is clear that $T$ is a sphere bundle over $G/H$. Let $L$ denote one of the groups $\U(1)$, $\SU(2)$, $N_{\SU(2)}\U(1)$, or a finite group $\Gamma\subset O(n+1)$ which acts freely on $S^n$ and $\alpha\colon L \rightarrow H/K$ be an isomorphism. This naturally defines an action of $L$ on $D^{n+1} \times G/K$ (where $z\in L$ sends $(x,gK)\mapsto (zx,g\alpha(z^{-1}))K$). We will use the symbol $\times_{\alpha}$ to indicate quotients under this action. Thus we have $T\cong S^n\times_{\alpha} G/K$ and $N\cong D^{n+1}\times_{\alpha} G/K$.
To each non-principal orbit corresponds a boundary component of $B$, $\partial_i B=S^{n_i}/L_i.$ Since $\partial_i B$ is a quotient of a sphere by a free action, it follows that in the case of a singular orbit $\partial_i B$ is a quaternionic projective space if $L_i=\SU(2)$, a complex projective space if $L_i=\U(1)$, and $\CP^{odd}/\Z_2$ if $L_i=N_{\SU(2)}\U(1)$. If the orbit is exceptional, we must have $L_i$ finite, and in odd cohomogeneity $\partial_i B=\RP^{n_i}$, $n_i$ even, and $L_i=\Z_2$. Notice that if there is a singular orbit, the cohomogeneity must be odd.
In contrast to the case of cohomogeneity one, any number of non-principal orbits can occur. Thus, for instance, there are actions of $\U(1)$ on $S^{2k+1}$ and of $\SU(2)$ on $S^{2k}$ with only one non-principal orbit. We will show that many of the examples of $G$-manifolds with finitely many non-principal orbits constructed in \cite{BW} admit an invariant metric with positive Ricci curvature. We recall a few examples:
\begin{example}\label{list} (`Doubles') Let $L$ be finite or one of $\U(1)$, $\SU(2)$, $N_{\SU(2)}\U(1)$. Let $G$ and $K$ be compact Lie groups, $K\subset G$ and $\alpha\colon L\to N_GK$ be injective. Let
$$M:= D^{n+1}\times_{\alpha} G/K\cup D^{n+1}\times_{\alpha} G/K$$
where we glue the common boundary $T=S^n\times_{\alpha} G/K$ via the identity. This
is a $G$ manifold with two identical singular orbits. The orbit space is the suspension of $S^n/L$.
\end{example}
Recall that for $p_1,p_2$ coprime, the Aloff-Wallach space $W_{p_1,p_2}$ is the quotient $\SU(3)/\set{\diag{z^{p_1},z^{p_2},z^{-p_1-p_2}}}{z\in\U(1)}$.
\begin{example}\label{dim11} Given any two Aloff-Wallach spaces $W_{p_1,p_2}$ and $W_{q_1,q_2}$, there is an 11-dimensional
$SU(3)$-manifold $M^{11}_{p_1p_2q_1q_2}$ of cohomogeneity three, orbit space $S^3$, and two singular orbits equal to the given Aloff-Wallach spaces.
Within this family there is an infinite sequence of pairwise non-homotopy equivalent manifolds for which each pair of singular orbits is non-homotopy equivalent. There is also an infinite sequence of pairwise non-homotopy equivalent `doubles', that is, manifolds with two identical singular orbits.
\end{example}
\begin{example}\label{dim13} Given Aloff-Wallach spaces $W_{p_1,p_2}$ and $W_{q_1,q_2}$, there is a 13-dimensional $\SU(3)$-manifold
$M^{13}_{p_1p_2q_1q_2}$ of cohomogeneity 5, orbit space $\Sigma \CP^2$, and two singular orbits equal to the given Aloff-Wallach manifolds if and only if
$p_1^2+p_1p_2+p_2^2=q_1^2+q_1q_2+q_2^2$. Within this family there is an infinite
sequence of pairwise non-homotopic manifolds for which each pair of singular orbits is non-homotopic. There is also
an infinite sequence of pairwise non-homotopic `doubles', that is, manifolds for which each pair of singular orbits is identical.
\end{example}
We now consider the geometry of invariant metrics on $G$-manifolds with finitely many non-principal orbits.
In \cite{LY} it was shown that the existence of a compact non-abelian Lie group action on a compact manifold means that the manifold admits a metric of positive scalar curvature. Moreover, the same construction actually yields an {\it invariant} metric of positive scalar curvature (a fact not pointed out in \cite{LY}, but observed, for example, in \cite{H}).
Thus if $M$ is a $G$-manifold of the type under consideration in this paper, provided $G$ is not a torus, then $M$ must admit an invariant metric of positive scalar curvature. \par
Of particular importance here is the fact that for cohomogeneity one, the space of orbits is one dimensional and so makes no contribution to the curvature. In higher cohomogeneities, this is no longer the case.
Indeed the space of orbits might have particularly bad Ricci curvature characteristics. Thus there seems little hope of being able to prove a positive Ricci curvature existence theorem of comparable generality to \cite{GZ2} in our situation.
It seems reasonable to expect that the Ricci curvature of the space of orbits will play an explicit role in any existence theorem.
In fact, we are able to prove the following:
\begin{thm}\label{thm-riccurvmain} Let $M$ be a compact $G$-manifold with finitely many singular orbits, for which the principal orbit $G/K$ has finite fundamental group.
Let $M^0$ be the manifold with boundary resulting from the removal of small invariant tubular neighbourhoods around the non-principal orbits, so $M^0$ is the total space of a $G/K$-bundle with base $B$. The boundary components of $B$ are all quotients of spheres by free actions of subgroups of the orthogonal group, and thus have a standard metric induced by the round metric of radius one.
If $B$ can be equipped with a Ricci positive metric such that
\begin{enumerate}
\item\label{cond-fbs} for each $i$, the metric on boundary component $\partial_i B$ is the standard metric scaled by a factor $\lambda_i^2$;
\item\label{cond-pricurv} the principal curvatures (with outward normal) at boundary component $\partial_i B$ are strictly greater than $-1/\lambda_i$;
\end{enumerate}
then $M$ admits a $G$-invariant metric with positive Ricci curvature.
\end{thm}
From this it is easy to deduce
\begin{cor}\label{cor-4.2}
All compact G-manifolds with two singular orbits, orbit space a suspension of either a projective space or $\CP^{odd}/\Z_2$ and principal orbit $G/K$ with $\pi_1(G/K)$ finite, admit invariant Ricci positive metrics.
\end{cor}
In particular we have:
\begin{cor}\label{cor-4.3}
The families $M^{11}_{p_1p_2q_1q_2}$ of example \ref{dim11}, and $M^{13}_{p_1p_2q_1q_2}$ of example \ref{dim13} all admit invariant metrics of positive Ricci curvature.
\end{cor}
We also have examples of manifolds with a single non-principal orbit and positive Ricci curvature:
\begin{thm}\label{one} For every $n\ge 2$ there is an $SU(n)$-manifold of dimension $n^2+2$ and cohomogeneity three with a single singular orbit and an invariant metric of positive Ricci curvature.
\end{thm}
We now turn our attention to the sectional curvature, and specifically non-negative sectional curvature.
It is not difficult to show that in certain special circumstances, we can obtain manifolds with invariant metrics of non-negative sectional curvature.
\begin{thm}\label{K} All $G$-manifolds with two identical singular orbits and orbit space a suspension of either a projective space or $\CP^{odd}/\Z_2$ admit invariant metrics with non-negative sectional curvature.
\end{thm}
We immediately obtain:
\begin{cor} There are infintely many homotopy types of manifolds in both the families $M^{11}_{p_1p_2q_1q_2}$ and $M^{13}_{p_1p_2q_1q_2}$ which admit invariant metrics of non-negative sectional curvature.
\end{cor}
This paper is laid out as follows. In section 2 we investigate the geometry of tubular neighbourhoods of non-principal orbits.
In section 3 we give proofs for the main results, with some of the more technical results postponed until section 4. We conclude with a collection of open problems in section \ref{sec-open}.
The authors would like to thank Dmitri Alekseevsky for encouraging us to study manifolds with finitely many singular orbits, and for his subsequent help. We would also like to thank Thomas P\"uttmann for reading a preliminary draft of this paper and for his valuable comments.
\section{Tubular neighbourhoods of non-principal orbits}
With the same notation as in the Introduction, let us first focus on the `regular' part $M^0$ of our $G$-manifold $M$. We will assume that
$B=M^0/G$ comes equipped with a Ricci positive metric satisfying the properties given in Theorem \ref{thm-riccurvmain}.
Now fix a bi-invariant
metric $g_0$ on $G$, and let $\nu>0$. The metric $\nu g_0$ induces a normal homogeneous metric (which we will also denote
$\nu g_0$) on $G/K$. By assumption, $\pi_1 G/K$ is finite, and it is well-known (see \cite{Be}) that the metric $\nu g_0$ on $G/K$
has positive Ricci curvature.
On $M^0$ we will introduce a submersion metric. (See \S9 of \cite{B} for more details about the construction of such metrics.)
For this we require three ingredients: a base metric, a fibre metric, and a horizontal distribution of subspaces.
In the current situation we have base and fibre metrics. Any choice of horizontal distribution then gives a
submersion metric, and it follows from (\cite{B}; 9.70) that this submersion metric will have positive Ricci curvature provided
the constant $\nu$ is chosen sufficiently small. From now on, we will assume that $M^0$ is equipped with such
a Ricci positive submersion metric. We are free, of course, to a smaller value of $\nu$ later on if required.
Let us now turn our attention to the tubular neighbourhoods of non-principal orbits. As discussed in the
Introduction, these all take the form
$$N:=D^{n+1} \times_{\alpha} G/K,$$ where $\alpha:L \rightarrow H/K$ is an isomorphism, $L=\U(1),N_{\SU(2)}\U(1),\SU(2)$ or
a finite subgroup of $O(n+1)$, and $H$ is the singular isotropy.
Our approach to constructing a metric on this neighbourhood is to define a metric $g_1$ on $D^{n+1}$, a metric $g_2$ on $G/K,$
and then consider the product metric $g_1+g_2$ on $\pro$. By making a suitable choice of $g_1$ and $g_2$, we can arrange for this product
metric to induce a well-defined metric $g_Q$ on the quotient $N$. Moreover, by a possibly more refined choice of starting metrics,
we can show that this induced metric can always have positive Ricci curvature. Of course, such
neighbourhoods must then be glued smoothly, and within positive Ricci curvature, into $M^0$ with its submersion metric. In particular,
this means that the $G/K$-fibres on the boundary of the tubular neighbourhood must have normal homogeneous metrics $\nu g_0.$
We first deal with the case of exceptional orbits.
\begin{thm}\label{ex} Consider a tubular neighbourhood $\quo$ of an isolated exceptional orbit $G/H$, where $\alpha:\Gamma \rightarrow H/K$
is an isomorphism from a finite subgroup $\Gamma \subset O(n+1).$
Fix a bi-invariant background metric $g_0$ on $G$, and let $\nu>0$. Given constants $\lambda>0$ and $0<\Lambda<1$,
there is a $G$-invariant Ricci positive metric $g_Q$ on $\quo$ such that the complement of the exceptional orbit has a submersion metric
with fibres isometric to $(G/K,\nu g_0)$ and base isometric to $((0,R] \times S^n/\Gamma, dr^2 + h^2(r)\sigma^2)$,
where $\sigma^2$ is the metric on $S^n/\Gamma$ induced by $ds^2_n$, and where $h(R)=\lambda$ and $h'(R)=\Lambda.$
\end{thm}
\pf Choose a function $h(r)$ such that $h(r)=\sin r$ for $r$ small, $h''(r)<0$ for all $r$, and $h(R)=\lambda$, $h'(R)=\Lambda$ for some $R>0.$
It is clear that we can make such a choice. Moreover, the resulting metric $dr^2+h^2(r)ds^2_n$ on $D^{n+1}$ will have positive Ricci curvature, and hence
so will the product metric $dr^2+h^2(r)ds^2_n+\nu g_0$ on $D^{n+1} \times G/K$. As $\Gamma$ acts isometrically on this product,
we obtain a well-defined metric $g_Q$ on the quotient $\quo$. As the quotient map is a finite covering, this induced metric is locally
isometric to $dr^2+h^2(r)ds^2_n+\nu g_0$, and hence has positive Ricci curvature. Moreover, the metric on each
$G/K$ fibre in $(D^{n+1}-\{0\}) \times_{\alpha} G/K$ is clearly isometric to $\nu g_0.$
\proofend
\smallskip
We next consider the case where $L=\U(1)$ or $\SU(2).$
\begin{thm}\label{tube} Consider a tubular neighbourhood $\quo$ of an isolated singular orbit $G/H$. Fix a bi-invariant background
metric $g_0$ on $G$ so that $\alpha:L \rightarrow H/K$ is an isometry, where $L=\U(1)$ or $L=\SU(2).$
Given constants $\lambda>0$, $0<\Lambda<1$ and $0<\e<\e_0$ (where $\e_0=\e_0(G,H,K,g_0)$ is
the constant from Observation 5.5), for any $\nu<\lambda\e/(1+\e),$
there is a $G$-invariant Ricci positive metric $g_Q$ on $\quo$ such that for some small $\iota>0$, the $\iota$-neighbourhood
of the boundary has a submersion metric with fibre $G/K$, base $[R-\iota,R] \times \P$, all fibres isometric to the normal
homogeneous metric induced by $\nu g_0$, and base metric $dr^2+h^2(r)ds^2_n$ with $h(R)=\lambda$ and $h'(R)=\Lambda.$
\end{thm}
The proof of this result is somewhat technical, and depends crucially on several explicit curvature calculations. For this reason, we postpone
the proof and the relevant computational lemmata until section 4.
It remains to consider the case when $L=N_{\SU(2)}\U(1).$
\begin{cor}\label{norm} The statement of Theorem \ref{tube} continues to hold in the case $L=N_{\SU(2)}\U(1).$
\end{cor}
\pf The isomorphism $\alpha:N_{\SU(2)}\U(1) \rightarrow H/K$ restricts to an isomorphism $\alpha_0:U(1) \rightarrow (H/K)_0$ where
$(H/K)_0$ is the identity component of $H/K$. By Theorem \ref{tube} we obtain a Ricci positive metric on $D^{n+1} \times_{\alpha_0} G/K$
with the desired properties. We now simply observe that $\quo$ is a $\Z_2$-quotient of $D^{n+1} \times_{\alpha_0} G/K$, with $\Z_2$
acting isometrically. Thus we obtain a Ricci positive metric on $\quo$, and it is clear that this metric has all the claimed properties.
\proofend
\section{Proofs of the main results}\label{pf}
\noindent {\bf Proof of Theorem \ref{thm-riccurvmain}.} We make use of an observation due to Perelman \cite{P}, both to control the form of the metric on the space of orbits $B$ near the boundary components, and then to allow smooth Ricci positive gluing with the tubular neighbourhoods of isolated singular orbits.
According to Perelman, given two Ricci positive manifolds with isometric boundary components, if the principal curvatures at one boundary component are (strictly) greater than the negatives of the corresponding principal curvatures for the other boundary component, the non-smooth metric which results from gluing the two boundary components together can be smoothed in an arbitrarily small neighbourhood of the join to produce a metric with global Ricci positivity. \par
With this in mind, we add small collars of the form $\P_i\times [0,\e_i]$ for some small $\e_i$ to each boundary component $\P_i$.
On this collar we assume a metric of the form $ds^2+\theta_i(s)g_{\P}$ with $g_{\P}$ the standard metric on $\P$.
In order for the boundary $\P_i$ to be isometric to the collar at $s=0$ we clearly need $\theta_i(0)=\lambda_i$.
If we choose $\theta_i$ such that $\theta_i''<0$ and $|\theta_i'|<1$, it is easy to check that the Ricci curvature of the collar will be positive.
Let $p_i$ be the infimum of the principal curvatures (with outward pointing normal) at the boundary component $\P_i$.
Recall that by the assumption (2) in the statement of the Theorem, $p_i>-1/\lambda_i$.
It is easy to see that the principal curvatures at the $s=0$ boundary component of the collar (again with outward normal) are all equal to $-\theta_i'(0)/\theta_i(0)$.
Therefore if $\theta_i'(0)<\lambda_i p_i$ for all $i$, then by Perelman we can join the collar to the space of orbits and smooth (in an arbitrarily small region) within Ricci positivity.
(Note that such a value of $\theta_i'(0)$ always exists: by assumption, $p_i>-1/\lambda_i$, so $p_i\ge -(1/\lambda)+c_i$ for some $c_i>0$.
Therefore the upper bound on $\theta_i'(0)$ is $-1+\lambda_ic_i>-1$, and so we have a well-defined non-empty interval $(-1,-1+\lambda_i c_i)$ from which to choose $\theta_i'(0)$.) Notice that by making a careful choice for $\theta$ and $\e_i$, we can also ensure that at the new boundary we create, the metric still satisfies conditions (1) and (2) in the statement of the Theorem.
Specifically, we need $\theta_i'(\e_i)>-1$ in order to satisfy these requirements.
The conclusion from this analysis is that without loss of generality we are free to make the following
\par\noindent{\it Assumption.} The metric in a small
neighbourhood of the boundary $\P_i$ is isometric with $(\P_i \times [0,\e_i];ds^2+\theta_i(s)g_{\P})$ for some $\e_i>0,$
and with all principal curvatures at the boundary equal to $\theta_i'(\e_i)/\theta_i(\e_i) \in (-1/\theta_i(\epsilon),0).$
\par
Now let us turn our attention to $M$ itself.
Recall that $M^0$ is the manifold with boundary resulting from the removal of small invariant tubular neighbourhoods $N_i$ around the non-principal orbits,
so $M^0$ is the total space of a $G/K$-bundle over a manifold with boundary $B$.
Consider the tubular neighbourhood $\quo$, and suppose we wish to glue this to boundary component $i$ of $M^0$. Let $\P$ be the corresponding
boundary component of $B.$
Equip $\quo$ with the metric $g_Q$ as in Theorem \ref{tube} or Corollary \ref{norm} in the singular cases, or as in Theorem \ref{ex} in the exceptional case, and where we have chosen $\lambda=\theta_i(\e_i)$ and $\Lambda=|\theta_i'(\e_i)|$, with $\theta_i$ as in the above assumption.
The scaling function $h$ in $g_Q$ will now glue with $\theta$ when the $r$ and $s$ parameters are suitably concatenated to create a $C^1$ scaling function.
As a pre-requisite for the smooth gluing of $\quo$ to $M^0$, we need this scaling function to be smooth.
We can easily achieve this by making a minor adjustment to $h(r)$ close to the boundary of $\quo$, and in particular for $r$ in the interval $(R-\iota,R]$ (with $\iota$ as in Theorem \ref{tube}) in the singular case.
Specifically, we can adjust $h$ to make the required alteration in the second derivative, whilst keeping the variation in both $h$ and $h'$ arbitrarily small.
Such an adjustment does not destroy positive Ricci curvature: this is easy to see, for instance by using Proposition \ref{prp-5.4}. \par
Now consider the boundary of $\quo$ and the corresponding boundary of $M^0$ as $G/K$-bundles over $\P$.
A further pre-requisite for smooth metric gluing is that the horizontal distributions of these bundles much match under the identification.
We are free to choose a horizontal distribution for $M^0$ (viewed as as $G/K$-bundle over $B$), so we choose this in a way so as to agree near the appropriate boundary component with that coming from $g_Q$.
Note that the horizontal distribution determined by $g_Q$ is induced from the standard horizontal distribution for the Hopf fibration
in the singular case and from $TS^n$ in the exceptional case. Thus it is independent of all choices involved in the construction of $g_Q$, and
in particular is independent of the functions $f$, $h$ and the constant $\nu$.
This last point is important, as it means we can select a horizontal distribution for $M^0$ (taking all boundary components into consideration) at the outset, and thus find a constant $\nu_0>0$ such that the submersion metric on $M^0$ has positive Ricci curvature for all $\nu<\nu_0$. \par
It remains consider the $G/K$-fibres on both sides of the join.
The metric on the fibres near the boundary of $\quo$ is, by the construction of $g_Q$, the normal homogeneous metric induced by $\nu g_0$ on $G$, and this agrees with the fibre metrics for $M^0$.
According to Theorem \ref{tube}, in the singular case we need $\nu<\lambda_i\e/(1+\e)$ for the Ricci positivity of tubular neighbourhood $N_i$, and for Ricci positivity of $M^0$ we need $\nu<\nu_0$ as above.
Therefore, our construction of a smooth global Ricci positive metric can be completed by choosing a value for $\nu$ which is less than $\nu_0$
in the exceptional case, and less than both $\nu_0$ and the minimum of the $\lambda_i\e/(1+\e)$ in the singular case.
\proofend
\pfo {Theorem \ref{one}.}
We form a tubular neighbourhood of the singular orbit by setting $N=D^4\times_{\alpha} SU(n)$ where $\alpha:U(1)\rightarrow SU(n)$ is any injective homomorphism.
Since $SU(n)$ is simply-connected, the boundary of this neighbourhood $\partial N=S^2\times SU(n)$.
We can therefore equivariantly glue a product $D^3\times SU(n)$ to this neighbourhood to produce a closed manifold.
The existence of an invariant Ricci positive metric now follows easily from Theorem \ref{thm-riccurvmain}.
The manifold $B$ in the statement of Theorem \ref{thm-riccurvmain} is simply $D^3$ in our situation, and thus the metric conditions on $B$ required by Theorem \ref{thm-riccurvmain} can easily be satisfied.
\proofend
\pfo {Theorem \ref{K}.}
Construct a metric $g$ on the tubular neighbourhood $N=\quo$ of a singular orbit as the quotient of a product metric $g_1+g_2$ on $\pro$ in the following way.
Suppose $D^{n+1}$ has radius $\pi/2$ and set $g_1=dr^2+\sin^2 r\ ds^2_n$.
(So $g_1$ is round with constant sectional curvature $1$.) Let $g_2$ be a normal homogeneous metric on $G/K$.
The product metric $g_1+g_2$ clearly has non-negative sectional curvature as the curvatures of both $g_1$ and $g_2$ are non-negative.
As Riemannian submersions are non-decreasing for the sectional curvature, it follows that $g$ also has non-negative sectional curvature.
The $G/K$ fibre metrics at the boundary of $N$ are not normal homogeneous, but this does not matter in the case of doubles as we wish to glue $N$ to an identical object.
Viewing a neighbourhood of $\partial N$ as an $S^n$-bundle over $G/H\times (\pi/2-\epsilon,\pi/2]$, the only issue we need to consider when gluing is the smoothness of the $S^n$ metrics across the join.
But this is clear as $\sin r$ for $r\in [0,\pi/2]$ concatenates smoothly with its `reverse' $\sin ((\pi/2)-s)$ for $s\in [0,\pi/2]$ when $r=\pi/2$ is identified with $s=0$.
\proofend
\section{Curvature computations}\label{sec-proof} In section 3 we gave the proof of Theorem \ref{thm-riccurvmain}, our main existence result for positive Ricci curvature. This proof depends crucially on Theorems \ref{ex}, \ref{tube} and Corollary \ref{norm}. However, as a result of its technical nature, the proof of
Theorem \ref{tube} was postponed. The aim of the current section is to establish this Theorem and to perform the pre-requisite curvature computations.
In order to do this, we must study metrics on
$$D^{n+1}\times_{\alpha} G/K,$$
where in this case $\alpha$ is a group isomorphism $L \rightarrow H/K$ with $L=\U(1)$ or $\SU(2)$. In the sequel, it will be convenient
to identify $\U(1)$ and $\SU(2)$ with the spheres $S^1$ respectively $S^3$. We will generally deal with both cases at once by writing $S^q.$
Recall that the group $S^q$ acts on the product $\pro$ as follows:
$$z(p,gK)=(zp,gK\alpha(z^{-1})),$$
where the $S^q$-action on $D^{n+1}$ is the standard Hopf action on the first factor of $(S^n\times [0,R])/(S^n\times\{0\})=D^{n+1}$, and
the expression $\quo$ denotes the quotient of $D^{n+1}\times G/K$ by this action. \par
Let $\{v_k\}$ denote a local orthonormal frame field for $S^q(1)$.
As $S^q$ acts via $\alpha$ on $G/K$ and directly on $S^n$, we obtain induced action fields which we will denote $\{v^{\alpha}\}$ and $\{v^{\ast}\}$ respectively.
Notice that $\{v^{\alpha}\}$ is a local frame field for $H/K$-orbits in $G/K$, and that $\{v^{\ast}\}$ is a local frame field for the fibres of the Hopf fibration. \par
We will construct a product metric $g_1+g_2$ on $\pro$ in such a way that it induces a well-defined Ricci positive $G$-invariant metric on $\quo$, with all the properties we will need to glue smoothly into $M^0$.
First of all, we concentrate on constructing $g_1$ for the $D^{n+1}$-factor.
It is technically easier if we remove the centre point of $D^{n+1}$ and view the space as $(0,R]\times S^n$ for some $R>0$.
We will construct metrics on $(0,R]\times S^n$, but the boundary conditions we impose will ensure that our metric extends smoothly to $D^{n+1}$.
\par
Let $g_{\P}$ denote the standard Fubini-Study metric on a complex or quaternionic projective space $\P$.
In the Lemma below, we make use of the O'Neill formulas for the Ricci curvature of a Riemannian submersion.
See \cite{B} \S9 for details of the formulas and definitions of the terms involved.
See \cite{B} \S9.59, or \cite{V} for a discussion about constructing submersion metrics.
\begin{lm}\label{lm-5.1}
Consider the `extended' Hopf fibration $S^q\hookrightarrow S^n\times (0,R]\rightarrow\P\times (0,R]$.
Equip the base with the metric $dr^2+h^2(r)g_{\P}$ and the fibres with the metric $f^2(r)ds^2_q$.
Introduce into the total space the horizontal distribution which is the obvious extension of the stadnard horizontal distribution for the Hopf fibration.
Let $g_1=g_1(f,h)$ denote the resulting submersion metric.
Then $g_1$ has the following Ricci curvatures (denoted $\r_1$):
\begin{align*}
&\r_1(\partial_r)=-\dim\P\frac{h''}{h}-q\frac{f''}{f}\\
&\r_1(X_i)=\frac{1}{h^2}\r_{\P}(\check{Y}_i)-\frac{h''}{h}-q\frac{f'h'}{fh}-2\frac{f^2}{h^4}\la A_{Y_i},A_{Y_i}\ra\\
&\r_1(v^{\ast}_k)=\frac{1}{{f^2}}\r_{S^q(1)}(v_k)-\frac{f''}{f}-(q-1)\left(\frac{f'}{f}\right)^2-\dim\P\frac{f'h'}{fh} +\frac{f^2}{h^4}\la Av^{\ast}_k,Av^{\ast}_k\ra\\
&\r_1(X_i,v^{\ast}_k)=\frac{f}{h^3}\la (\check{\delta} A)Y_i,v^{\ast}_k\ra
\end{align*}
where $\{X_i\}$ are an orthonormal spanning set of vector fields for the horizontal distribution on $S^n$, $Y_i=h^{-1}X_i$ and $\check{Y}_i$ is the projection of $Y_i$ to $\P$.
The terms $\la A_{Y_i},A_{Y_i}\ra$, $\la Av^{\ast}_k,Av^{\ast}_k\ra$ and $\la (\check{\delta} A)Y_i,v^{\ast}_k\ra$ are the standard terms for the Hopf fibration on $S^n(1)$.
All other mixed Ricci curvature terms vanish.
\end{lm}
\pf These formulas follow from \cite{W2}, Proposition 4.2.
\proofend
We can make the above formulas more explicit by substituting the appropriate values for $\r_{\P}$, $\r_{S^q(1)}$ and the $A$-tensor terms.
It is well known \cite{B}, p.258, that with their standard Fubini-Study metrics, both $\CP^m$ and $\HP^m$ are Einstein manifolds, with Einstein constants $2m+2$ and $4m+8$ respectively.
For the $A$-tensor terms we have:
\begin{lm}\label{lm-5.2}
For the standard Hopf fibration over $\CP^m$ or $\HP^m$ we have $\la Av^{\ast}_k,Av^{\ast}_k\ra$ equal to $\dim \CP^m$ respectively $\dim \HP^m$, $\la AY_i,AY_i\ra=q$ and $\la (\check{\delta} A)Y_i,v^{\ast}_k\ra=0$.
\end{lm}
\pf These expressions can be evaluated by applying the O'Neill formulas for the Ricci curvature \cite{B}, \S9.70, to the standard Hopf fibration, using the known constant Ricci curvature values for the base, total space and fibre.
The computations are all elementary.
We mention only $\la (\check{\delta} A)Y_i,v_k\ra$ in the case where the base is $\CP^m$.
For this, consider $\r(Y_i+v^{\ast}_k)=2(2m)$ as $\|Y_i+v^{\ast}_k\|=\sqrt{2}$.
We also have
\begin{align*}
\r(Y_i+v^{\ast}_k)&=\r(Y_i)+\r(v^{\ast}_k)+2\r(Y_i,v^{\ast}_k)\\
&=2(2m)+2\r(Y_i,v^{\ast}_k).
\end{align*}
Hence $\r(Y_i,v^{\ast}_k)=0$.
The O'Neill formulas show that $\r(Y_i,v^{\ast}_k)=\la (\check{\delta} A)Y_i,v^{\ast}_k\ra$ for the standard Hopf fibration as the fibres are totally geodesic, forcing all $T$-tensor terms to vanish.
Analogous arguments apply for the Hopf fibration over $\HP^m$.
\proofend
\begin{cor}\label{cor-5.3}
The Ricci curvatures of $g_1$ are given by
\begin{align*}
&\r_1(\partial_r)=-\dim\P \frac{h''}{h}-q\frac{f''}{f};\\ &\r_1(X_i)=\dim\P\left(\frac{1-(h')^2}{h^2}\right)+\frac{2^q}{h^2}-\frac{h''}{h}-\left(\frac{h'}{h}\right)^2 -q\left(\frac{f'h'}{fh}-2\frac{f^2}{h^4}\right);\\ &\r_1(v^{\ast}_k)=(q-1)\frac{1-(f')^2}{f^2}-\frac{f''}{f}+\dim\P\left(\frac{f^2}{h^4}-\frac{f'h'}{fh}\right),
\end{align*}
with all mixed curvature terms vanishing.
\end{cor}
Notice that the metric $g_1$ extends to give a well-defined metric on $D^{n+1}$, provided $f$ and $h$ satisfy suitable boundary conditions near $r=0$.
Specifially, we require $f(0)=h(0)=0$, $f'(0)=h'(0)=1$, and $f$ and $h$ should be odd at $r=0$.
These conditions will certainly be satisfied if $f(r)=h(r)=\sin r$ for $r\in [0,\delta]$ for some small $\delta$, and we will assume this to be the case.
The values of $R$ (the radius of $D^{n+1}$) and $\delta$ will be determined later.
\begin{prp}\label{prp-5.4}
The metric $g_1=g_1(f,h)$ on $D^{n+1}$ has all Ricci curvatures strictly positive if the functions $f$ and $h$ satisfy:
\begin{align}
\label{cond-prp-5.4-1} &f(r)=h(r)=\sin r\text{ for }r\text{ small};\\
\label{cond-prp-5.4-2} &f''\le 0, h''\le 0, f''+h'' <0, f'\ge 0,\text{ and }h'\ge 0;\\
\label{cond-prp-5.4-3} &f\le h,\text{ and }\frac{f'}{f}\le\frac{h'}{h};\\
\label{cond-prp-5.4-4} &(f/h)^3\ge f'h'.
\end{align}
\end{prp}
\pf As all mixed Ricci curvature terms vanish, it suffices to show that the expressions for $\r_1(\partial_r)$, $\r_1(X_i)$ and $\r_1(v^{\ast}_k)$ are all strictly positive.
The positivity of $\r_1(\partial_r)$ is clear because of \eqref{cond-prp-5.4-2}.
To see the positivity of $\r_1(X_i)$, consider the case $q=1$.
The first term in the expression for $\r_1(X_i)$ in Corollary \ref{cor-5.3} is strictly positive for $r>0$ as a consequence of \eqref{cond-prp-5.4-1} and \eqref{cond-prp-5.4-2}.
By \eqref{cond-prp-5.4-2}, the term $-h''/h\ge 0$ for all $r$.
Therefore these two terms taken together have a strictly positive sum for all $r$.
It therefore suffices to show the non-negativity of the sum of the remaining terms:
$$2h^{-2}+(h'/h)^2-f'h'f^{-1}h^{-1}-2f^2h^{-4}.$$
By (iii) this expression is greater than or equal to
$$2h^{-2}+(h'/h)^2-(h'/h)^2-2h^{-2}$$
as required.
The case $q=3$ is analogous.
The positivity of $\r_1(v^{\ast}_k)$ follows immediately if the final term in the expression in Corollary \ref{cor-5.3} is non-negative.
But this is guaranteed by \eqref{cond-prp-5.4-4}.
\proofend
We now turn our attention to the space $G/K$.
Let $g_0$ be a bi-invariant metric on $G$ which makes $\alpha:S^q\rightarrow H/K$ an isometry, assuming the round metric of radius $1$ on $S^q$.
Note that this is possible as any bi-invariant metric on $S^1$ or $S^3$ must be round.
Fix a metric $g_{\nu}=\nu g_0$ on $G$, for some constant $\nu$.
As $g_{\nu}$ is bi-invariant it must have non-negative sectional curvature, so in particular it has non-negative Ricci curvature.
Consider the corresponding normal homogeneous metric on $G/K$.
By the O'Neill formulas, this too has non-negative sectional and therefore non-negative Ricci curvatures.
In fact, by \cite{Be} our normal homogeneous metric must have strictly positive Ricci curvature, as $\pi_1(G/K)<\infty$.
Now scale this normal homogeneous metric in the direction of the $H$-orbits by a factor $\mu$.
(Recall that $G/K$ is the total space of a fibration $H/K\hookrightarrow G/K\rightarrow G/H$.) Call the resulting metric $g_2=g_2(\mu,\nu)$.
The following is clear from the openness of the $\r>0$ condition:
\begin{obs}\label{obs-5.5}
There exists $\e_0=\e_0(G,H,K,g_0)$ such that for all $\e<\e_0$, $g_2$ has strictly positive Ricci curvature when $\mu=1+\e$.
\end{obs}
Note that $\e_0$ is independent of $\nu$. Fixing a value of $\e<\e_0$, we immediately deduce:
\begin{cor}\label{cor-5.6}
The product metric $g_1+g_2(1+\epsilon,\nu)$ on $\pro$ has positive Ricci curvature, and is both $S^q$-invariant and left $G$-invariant.
\end{cor}
The $S^q$-invariance of $g_1+g_2$ gives
\begin{cor}\label{cor-5.7}
The metric $g_1+g_2$ induces a well-defined metric $g_Q$ on $\quo$.
\end{cor}
For the purposes of gluing tubular neighbourhoods of isolated singular orbits into the manifold $M^0$, we want to ensure that the $G/K$-fibres near the boundary of $(\quo,g_Q)$ have normal homogeneous metrics.
\begin{prp}\label{prp-5.8}
Regarding $S^n$ as the total space of the Hopf fibration $S^q\hookrightarrow S^n\rightarrow\P$, consider the round metric as a submersion metric over $\P$.
Now rescale the fibres of this submersion so they are all isometric to $(S^q,\lambda ds^2_q)$.
Let $g_0$ be the bi-invariant metric on $G$ which makes $\alpha\colon S^q\rightarrow H/K$ an isometry, assuming the round metric of radius $1$ on $S^q$.
Fix a metric $g_{\nu}=\nu g_0$ on $G$, for some constant $\nu$, and consider the corresponding normal homogeneous metric on $G/K$.
Scale this normal homogeneous metric in the direction of the $H$-orbits by a factor $\mu$.
The resulting product metric on $S^n\times G/K$ induces a $G$-invariant metric on the quotient $S^n\times_{\alpha} G/K$.
The $G$-orbits in this quotient are all isometric to $G/K$ with the normal homogeneous metric induced from $g_{\nu}$ precisely when
$$\lambda=\frac{\mu\nu}{\mu-1}.$$
\end{prp}
\pf The proof is just a Cheeger-type argument, analogous to that in \cite{C}.
\proofend
\begin{cor}\label{cor-5.9}
The $G/K$-fibres at the boundary of $(\quo;g_Q)$ have normal homogeneous metrics induced by $\nu g_0$ if the function $f$ used to define $g_Q$ takes the value $(1+\e)\nu/\e$ there.
\end{cor}
We now investigate the Ricci curvature of $\quo$.
Our principal strategy for showing Ricci positivity is as follows.
If $Z_1$ and $Z_2$ are horizontal vectors in the total space of a submersion, then the Ricci curvature of the projections $\r(\check{Z}_1,\check{Z_2})$ is related to $\r(Z_1,Z_2)$ by the following O'Neill formula \cite{B}, \S9.36c:
$$\r(Z_1,Z_2)=\r(\check{Z}_1,\check{Z_2})-2\la A_{Z_1},A_{Z_2}\ra -\la TZ_1,TZ_2\ra +\frac{1}{2}\left(\la\nabla_{Z_1}N,Z_2\ra +\la\nabla_{Z_2}N,Z_1\ra\right).$$
In particular, this means that for $Z=Z_1=Z_2$ we have
$$\r(\check{Z})=\r(Z)+2\la A_Z,A_Z\ra +\la TZ,TZ\ra -\la\nabla_Z N,Z\ra.$$
Clearly $\la A_Z,A_Z\ra\ge 0$ and $\la TZ,TZ\ra\ge0$.
Therefore, assuming $\r(Z)>0$, if $\la\nabla_Z N,Z\ra\le 0$ we must have $\r(\check{Z})>0$ also.
It is therefore crucial to understand the $\la\nabla_Z N,Z\ra$ term.
In order to do this, we must first identify the vector field $N$.
Recall that by definition (\cite{B} \S9.34), $N=\sum_k T_{U_k}U_k$, where $\{U_k\}$ is an orthonormal frame field for the vertical distribution.
Although we will not need to know such a vertical frame field explicitly for the computation of $N$, we will need such formulas later on.
With this in mind, set
$$U_k=\frac{1}{{\sqrt{\den}}}(v_k^{\ast},-v^{\alpha}).$$
\begin{lm}\label{lm-5.10}
For the metric $g_1+g_2$ on $\pro$ we have
$$N=\frac{-qff'}{\den}\partial_r.$$
\end{lm}\pf We can view the metric $g_1+g_2$ on $\pro$ as a submersion metric which has been created from a submersion with isometric, totally geodesic fibres by rescaling in fibre directions by $\den$.
All $T$-tensor terms for the totally geodesic submersion vanish.
The effect of the rescaling on all the quantities appearing the O'Neill formulas was computed in \cite{W1}.
The expression for $N$ following a metric rescale by a function $\theta$ defined on the base is given by
$$N=-\frac{\dim(\text{fibre})}{2\theta}\nabla\theta.$$
Setting $\theta=\den$ in this formula gives the desired expression.
\proofend
For convenience we will write $N=\phi\partial_r$ from now on, with
$$\phi=\phi(r)=\frac{-qff'}{\den}.$$ \par
In the following, metric quantities without subscript will refer to the product metric $g_1+g_2$ on $\pro$.
Recall that $g_Q$ is the induced metric on the quotient space $\quo$.
It is easy to see that the horizontal distribution in $\pro$ is
$$(\mathcal{H}\oplus 0)\oplus\left\{\left(\frac{\mu\nu}{\lambda} v^{\ast},v^{\alpha}\right) \,|\, v\in TS^q\right\}\oplus (0\oplus \mathfrak{m}),$$
where $\mathcal{H}$ denotes the horizontal distribution for the Hopf submersion metric on $S^n$, and where $\mathfrak{m}$ is the distribution of orthogonal complements to $H/K$ orbits in $G/K$.
Recall that the vector fields $\{X_i\}$ are an orthonormal basis for $\mathcal{H}\oplus 0$.
Let $\{w_j\}$ be an orthonomal frame field for $\mathfrak{m}$.
Setting $\Delta_k=\vk$, we have that $\{\Delta_k\}$ is an orthogonal frame field for $\{((\mu\nu/\lambda) v^{\ast},v^{\alpha})\}$.
However, note that $\{\Delta_k\}$ is not an orthonormal set as
$$\|\Delta_k\|^2=\frac{(1+\e)\nu(f^2+(1+\e)\nu)}{f^2}.$$
As before, the projections of any of these vectors to $\quo$ will be indicated by a \ $\check{}$\ .
Collectively, these projections form a local basis. \par
Although the vectors $\{X_i\}$ are orthonormal, it will sometimes be useful in subsequent calculations to write $X_i=hY_i$, so the projections of the $Y_i$ on the base are unit vector fields with respect to the Fubini-Study metric.
The significance of this is that $\{Y_i\}$ are independent of the $r$ parameter.
As a result,
$$[Y_i,\partial_r]=[Y_i,\Delta_k]=[Y_i,w_j]=[Y_i,N]=[Y_i,U_k]=0.$$
\begin{lm}\label{lm-5.11} The following formulas hold:
\begin{align*}
[X_i,N]&=\phi(h'/h)X_i \ ; \\
[\partial_r,N]&=\phi' \partial_r \ ; \\
[v^{\ast}_k,N]&=[v^{\alpha},N]=[w_j,N]=0 .
\end{align*}
\end{lm}
\pf For the first expression, we begin by writing the Lie bracket as $[hY_i,\phi\partial_r]$ and then using the fact that $[Y_i,\partial_r]=0$.
The second expression is an elementary calculation, and the vanishing of the final three terms is immediate since $v^{\ast}$, $v^{\alpha}$ and $w_j$ are all independent of $r$ and are tangent to different factors to $(0,R]$ in the product $(0,R]\times S^n\times G/K$.
\proofend
\begin{lm}\label{lm-5.12}
The vector field $N$ has the following covariant derivatives:
$$\nabla_{X_i} N=\phi\frac{h'}{h} X_i, \hskip 2cm\nabla_{w_j}N=0, \hskip 2cm\nabla_{\partial_r}N=\phi'\partial_r,$$
$$\nabla_{v_k^{\ast}}N=\phi f'f^{-1}v_k^{\ast}, \quad\nabla_{\Delta_k}N=\phi f'f^{-1}\frac{(1+\e)\nu}{(1+\e)\nu+f^2}\Delta_k.$$
\end{lm}
\pf We proceed using the Koszul formula:
$$2\la\nabla_A B,C\ra=A\la B,C\ra+B\la C,A\ra-C\la A,B\ra +[[A,B],C]-[[B,C],A]+[[C,A],B].$$
We compute each of the sixteen possible terms $\la\nabla_{\bullet} N,\star\ra$.
For the most part these are zero.
We briefly mention those which are not. \par
For $\nabla_{X_i} N$, the term $2\la\nabla_{X_i} N,X_j\ra =\la [X_i,N],X_j\ra -\la [N,X_j],X_i\ra$.
By Lemma \ref{lm-5.11} we see that each of the terms on the right-hand side is equal to $\phi(h'/h)X_i$, and hence $\la\nabla_{X_i} N,X_j\ra = \phi\frac{h'}{h}\delta_{ij}$. \par
For $\nabla_{\partial_r}N$, the term $2\la\nabla_{\partial_r}N, \partial_r\ra =\la [\partial_r,N],\partial_r\ra -\la [N, \partial_r], \partial_r\ra$.
By Lemma \ref{lm-5.11} both of these terms are equal to $\phi'\partial_r$, and hence $\la\nabla_{\partial_r}N, \partial_r\ra =\phi'$. \par
For $\nabla_{v_k^{\ast}}N$ the only non-zero expression is $2\la\nabla_{v_k^{\ast}}N, v_l^{\ast}\ra=N\la v_k^{\ast},v_l^{\ast}\ra$.
Now $\|v_k^{\ast}\|^2=f^2$, so $\la v_k^{\ast},v_l^{\ast}\ra=f^2\delta_{kl}$, giving $\la\nabla_{v_k^{\ast}}N, v_l^{\ast}\ra=\phi ff' \delta_{kl}$. \par
For $\nabla_{\Delta_k} N$, the only non-zero term is $\la\nabla_{\Delta_k} N,\Delta_k\ra$.
We have
\begin{align*}
2\la\nabla_{\Delta_k} N,\Delta_k\ra &= 2\la\nabla_{(1+\e)\nu f^{-2}v_k^{\ast}+v_k^{\alpha}}N,(1+\e)\nu f^{-2}v_l^{\ast}+v_l^{\alpha}\ra\\
&=2\left[\fra\right]^2\la\nabla_{v_k^{\ast}}N, v_l^{\ast}\ra \text{ as all other terms clearly vanish},\\ &=2\left[\fra\right]^2 \phi ff' \delta_{kl} \text{ as shown above}.
\end{align*}
Therefore $\la\nabla_{\Delta_k} N,\Delta_k\ra=(1+\e)^2\nu^2\phi f'/f^3$. \par
From the above results, the conclusion of the Lemma is easy to establish.
Note that for $\nabla_{\Delta_k}N$ we need to take care since $\Delta_k$ is not a unit vector.
Specifically, we have
\begin{align*}
\nabla_{\Delta_k}N&=(1+\e)^2\nu^2\phi\frac{f'}{f^3}\frac{1}{\|\Delta_k\|}\frac{\Delta_k}{\|\Delta_k\|}\\
&=\phi\frac{f'}{f}\frac{(1+\e)\nu}{(1+\e)\nu+f^2}\Delta_k.
\end{align*}
\proofend
\begin{cor}\label{cor-5.13}
For the metric $g_Q$ on $\quo$, we have $\r_Q(\check{w}_j)>0$ for each $j$.
\end{cor}
\pf By Observation \ref{obs-5.5} we have $\r(w_j)>0$ for all $j$.
Now $\r(w_j)$ and $\r_Q(\check{w}_j)$ are related by
$$\r_Q(\check{w}_j)=\r(w_j)+2\la A_{w_j},A_{w_j}\ra+\la Tw_j,Tw_j\ra -\la\nabla_{w_j}N,w_j\ra.$$
Thus the first three terms on the right-hand side are strictly positive, non-negative and non-negative respectively.
By Lemma \ref{lm-5.12} we see that the final term vanishes, which establishes the result.
\proofend
\begin{cor}\label{cor-5.14}
For every $i$ we have $\r_Q(\check{X}_i)>0$.
\end{cor}
\pf We argue as in the proof of Corollary \ref{cor-5.13} above.
The only difference this time is that the term $\la\nabla_{X_i} N,X_i\ra$ is not zero.
From Lemma \ref{lm-5.12} we have $\la\nabla_{X_i} N,X_i\ra = \phi\frac{h'}{h}$.
But recall that
$$\phi=-\frac{qff'}{\den}\partial_r,$$
so in particular we have $\phi(r)\le 0$ for all $r$.
This means that the $\la\nabla_{X_i} N,X_i\ra$ term is non-positive, and therefore makes a non-negative contribution to $\r_Q(\check{X}_i)$.
\proofend
\begin{cor}\label{cor-5.15}
For every $k$ we have $\r_Q(\check{\Delta}_k)>0$.
\end{cor}
\pf The proof is essentially the same as for Corollary \ref{cor-5.14}.
This time we have
$$\la\nabla_{\Delta_k} N,\Delta_k\ra=(1+\e)^2\nu^2\phi\frac{f'}{f^3}$$
from Lemma \ref{lm-5.12}, and this makes a non-negative contribution to $\r_Q(\check{\Delta}_k)>0$.
\proofend
\begin{lm}\label{lm-5.16}
We have $\la T\partial_r,T\partial_r\ra= q(f'f)^2[\den]^{-2}$.
All other terms of the form $\la T\bullet,T\star\ra$ vanish.
\end{lm}
\pf We can view the metric $g_1+g_2$ on $\pro$ as a submersion metric which has been created from a submersion with isometric, totally geodesic fibres by rescaling in fibre directions by $\den$.
All $T$-tensor terms for the totally geodesic submersion vanish.
The effect of this rescaling on all the quantities appearing the O'Neill formulas was computed in \cite{W1}.
For the term $\la TA,TB\ra$, the value following a metric rescale by a function $\theta$ defined on the base is given by
$$\frac{\dim(\text{fibre})}{4\theta^2}A(\theta)B(\theta).$$
Setting $\theta=\den$ and computing derivatives gives the result.
\proofend
\begin{lm}\label{lm-5.17}
For each $k$ we have
$$A_{\partial_r} U_k=\frac{ff'}{[\den]^{\frac{3}{2}}}\Delta_k.$$
\end{lm}
\pf As the $A$-tensor is linear in both entries,
$$A_{\partial_r} U_k=\frac{1}{\sqrt{\den}}A_{\partial_r} (v_k^{\ast},-v^{\alpha}).$$
By definition of the $A$-tensor, this quantity is
$$\frac{1}{\sqrt{\den}}\mathcal{H}\nabla_{\partial_r}(v_k^{\ast},-v^{\alpha}),$$
where $\mathcal{H}$ denotes the horizontal component. \par
We next compute the components of this vector in the various directions.
Using the formula for $\nabla_{v_k^{\ast}}N$ established in Lemma \ref{lm-5.12} we see that
\begin{align*}
\la\nabla_{\partial_r}(v_k^{\ast},-v^{\alpha}),\Delta_l\ra &=f'f^{-1}\la v_k^{\ast}, \Delta_l\ra\\ &=(1+\e)\nu f'f^{-1}\delta_{kl}.
\end{align*}
Similarly, for the other directions it is easy to see that the terms $\la\nabla_{\partial_r}(v_k^{\ast},-v^{\alpha}),X_i\ra$, $\la\nabla_{\partial_r}(v_k^{\ast},-v^{\alpha}),w_j\ra$ and $\la\nabla_{\partial_r}(v_k^{\ast},-v^{\alpha}),\partial_r\ra$ all vanish. \par
Bearing in mind the fact that $\Delta_k$ is not a unit vector, we deduce that
\begin{align*}
\nabla_{\partial_r}(v_k^{\ast},-v^{\alpha})&=(1+\e)\nu\frac{f'}{f}\frac{1}{\|\Delta_k\|}\frac{\Delta_k}{\|\Delta_k\|}\\
&=\frac{ff'}{[\den]^\frac{3}{2}}\Delta_k
\end{align*}
as required.
\proofend
\begin{cor}\label{cor-5.18}
$$\la A_{\partial_r},A_{\partial_r}\ra= q\frac{(1+\e)\nu (f')^2}{[\den]^2}.$$
\end{cor}
\pf Recall from \cite{B}, \S9.33 that
$$\la A_{\partial_r},A_{\partial_r}\ra=\sum_k\la A_{\partial_r}U_k,A_{\partial_r}U_k\ra.$$
Using the result of Lemma \ref{lm-5.17}, we immediately obtain the desired expression.
\proofend
We are now in a position to investigate $\r_Q (\check{\partial}_r)$.
\begin{prp}\label{prp-5.19}
$\r_Q (\check{\partial}_r)>0$.
\end{prp}
\pf We know that
$$\r_Q (\check{\partial}_r)=\r(\partial_r)+2\la A_{\partial_r},A_{\partial_r}\ra +\la T\partial_r,T\partial_r\ra -\la\nabla_{\partial_r}N,\partial_r\ra.$$
Using the formula for $\r(\partial_r)$ from Corollary \ref{cor-5.3}, the formulas of Corollary \ref{cor-5.18} and Lemma \ref{lm-5.16} for the next two terms on the right-hand side, and Lemma \ref{lm-5.12} to evaluate the final term we obtain the expression
\begin{align*}
\r_Q(\check{\partial}_r)=-&\dim\P\frac{h''}{h}-q\frac{f''}{f}+q\frac{(1+\e)\nu (f')^2}{[\den]^2}+ q\frac{(ff')^2}{[\den]^2}\cr\cr &+q\frac{(f')^2}{\den}+q\frac{ff''}{\den}-2q\frac{(ff')^2}{[\den]^2},
\end{align*}
where the final three terms are just $\phi'$ written out explicitly. \par
Collecting similar terms gives
$$-\dim\P\frac{h''}{h}-q\frac{f''}{f}\left[1-\frac{f^2}{\den}\right]+ q\frac{(ff')^2}{[\den]^2}\left[1+\frac{2(1+\e)\nu}{f^2}+\frac{\den}{f^2}-2\right].$$
Simplifying this gives
$$r_Q(\partial_r)=-\dim\P\frac{h''}{h}+q\frac{(1+\e)\nu}{\den}\left(\frac{3(f')^2}{\den}-\frac{f''}{f}\right).$$
As we are assuming that both $f$ and $h$ are concave down functions, at least one of which is strictly concave down for all $r$, we deduce that this expression is strictly positive as claimed.
\proofend
So far we have established that $\r_Q(\check{\partial}_r)$, $\r_Q(\check{X}_i)$, $\r_Q(\check{\Delta}_k)$ and $\r_Q(\check{w}_j)$ are all strictly positive.
However this is not sufficient to deduce that {\it all} Ricci curvatures of the metric $g_Q$ are strictly positive.
\begin{lm}\label{lm-5.20}
For any $a,b,c\in\R$, $\r_Q(a\check{X}_i+b\check{\Delta}_k+c\check{w}_j)>0$.
\end{lm}
\pf Using the same line of reasoning as employed in Corollaries \ref{cor-5.13}, \ref{cor-5.14} and \ref{cor-5.15}, it suffices to show that the expression $\la\nabla_{aX_i+b\Delta_k +c\check{w}_j} N,aX_i+b\Delta_k+c\check{w}_j\ra$, is non-positive.
But this follows easily from (the proof of) Lemma \ref{lm-5.12}.
\proofend
It remains to study Ricci curvatures of the form $\r_Q(\check{\partial}_r+\check{Z})$ for $Z\in\Span\{X_i\}\oplus\Span\{\Delta_k\}\oplus\Span\{w_j\}$.
By elementary linear algebra, Ricci curvatures of this form will be positive if and only if
$$\r_Q(\check{\partial}_r)\r_Q(\check{Z})>(\r_Q(\check{\partial}_r,\check{Z}))^2$$
for all $Z$.
\begin{prp}\label{prp-5.21}
For all $Z\in\Span\{X_i\}\oplus\Span\{\Delta_k\}\oplus\Span\{w_j\}$, we have
$$\r_Q(\check{\partial}_r)\r_Q(\check{Z})>(\r_Q(\check{\partial}_r,\check{Z}))^2.$$
\end{prp}
\pf We prove this proposition in several steps.
The first step is to establish the inequality
\[{pf-prp-5.21-d1}\r_Q(\check{\partial}_r)\ge 2\la A_{\partial_r},A_{\partial_r}\ra.\]
From the curvature formulas established in the proof of Proposition \ref{prp-5.19}, we see that our inequality is equivalent to
$$-\dim\P\frac{h''}{h}+q\frac{(1+\e)\nu}{\den}\left(\frac{3(f')^2}{\den}-\frac{f''}{f}\right)\ge 2q\frac{(1+\e)\nu(f')^2}{[\den]^2}.$$
As $h''/h\ge 0$ it suffices to show that
$$\frac{3(f')^2}{\den}-\frac{f''}{f}\ge\frac{2(f')^2}{\den}.$$
As $f''/f\ge 0$ it then suffices to show that
$$\frac{3(f')^2}{\den}\ge\frac{2(f')^2}{\den},$$
which is clearly true.
Thus \eqref{pf-prp-5.21-d1} is established. \par
In fact we can go further than this.
By assumption, at least one of $f$ or $h$ is strictly concave down for each $r$.
Thus the inequality \eqref{pf-prp-5.21-d1} can actually be replaced by
\[{pf-prp-5.21-d2}\r_Q(\check{\partial}_r) > 2\la A_{\partial_r},A_{\partial_r}\ra.\] \par
We next claim that for any $Z\in\Span\{X_i\}\oplus\Span\{\Delta_k\}\oplus\Span\{w_j\}$, we have
$$\r_Q(\check{Z})> 2\la A_Z,A_Z\ra.$$
To see this, note that by the O'Neill formulas,
$$\r_Q(\check{Z})-\r(Z)-\la TZ,TZ\ra +\la\nabla_Z N,Z\ra =2\la A_Z,A_Z\ra.$$
Now we know from Corollary \ref{cor-5.6} that $\r(Z)>0$, $\la TZ,TZ\ra=0$ by Lemma \ref{lm-5.16}, and $\la\nabla_Z N,Z\ra\le 0$ by Lemma \ref{lm-5.12}.
Thus the inequality follows. \par
It is an elementary consequence of the Cauchy-Schwarz inequality that
$$\la A_Z,A_{\partial_r}\ra^2\le\la A_Z,A_Z\ra\la A_{\partial_r},A_{\partial_r}\ra,$$
and in particular we have
$$4\la A_Z,A_{\partial_r}\ra^2\le [2\la A_Z,A_Z\ra] [2\la A_{\partial_r},A_{\partial_r}\ra].$$
Combining this with \eqref{pf-prp-5.21-d2} and the corresponding inequality for $\r_Q(\check{Z})$ gives
$$\r_Q(\check{Z}) \r_Q(\check{\partial}_r)>4\la A_Z,A_{\partial_r}\ra^2.$$ \par
The proof of the Proposition will now follow from our final claim: $\r_Q(\check{Z},\check{\partial}_r)=2\la A_Z,A_{\partial_r}\ra$.
To see this we use the O'Neill formula
$$\r_Q(\check{Z},\check{\partial}_r)=\r(Z,\partial_r)+ 2\la A_Z,A_{\partial_r}\ra+\la TZ,T\partial_r\ra -\frac{1}{2}\left[\la\nabla_Z N,\partial_r\ra +\la\nabla_{\partial_r} N,Z\ra\right].$$
By Lemma \ref{lm-5.16} we have $\la TZ,T\partial_r\ra=0$, and the vanishing of the final term follows from Lemma \ref{lm-5.12}.
Thus the claim, and hence the Proposition is established.
\proofend
We immediately deduce:
\begin{cor}\label{cor-5.22}
For all $Z\in\Span\{X_i\}\oplus\Span\{\Delta_k\}\oplus\Span\{w_j\}$, we have $\r_Q(\check{\partial}_r+\check{Z})>0$.
\end{cor}
We are now in a position to prove Theorem \ref{tube}. \par
\pfo {Theorem \ref{tube}.}
We show that the functions $f$ and $h$ can be chosen so that the metric $g_Q$ satisfies all the requiements of the Theorem.
By combining the results of Corollaries \ref{cor-5.13}, \ref{cor-5.14} and \ref{cor-5.15}, Proposition \ref{prp-5.19}, Lemma \ref{lm-5.20} and Corollary \ref{cor-5.22}, we see that all Ricci curvatures of the metric $g_Q$ are strictly positive, provided $f$ and $h$ satisfy the conditions of Proposition \ref{prp-5.4}.
Of course $g_Q$ is $G$-invariant by construction.
By Corollary \ref{cor-5.9}, we will obtain fibres in an $\iota$-neighbourhood of the boundary all isometric to the normal homogeneous metric induced by $\nu g_0$ (for any choice of $\nu$) if $f(r)=(1+\e)\nu/\e$ for all $r\in [R-\iota,R]$.
Choose a function $h(r)$ such that $h(r)=\sin r$ for $r$ small, $h''(r)<0$ for all $r$, and $h(R)=\lambda$, $h'(R)=\Lambda$ for some $R>0$.
It is clear that we can make such a choice.
Next, note that if we set $f(r)=h(r)$, the conditions laid out in Proposition \ref{prp-5.4} are all satisfied.
However, setting $f(r)=h(r)$ will not allow us to achieve the required $f'(r)=0$ for $r\in [R-\iota,R]$.
For any choice of $\delta$ such that $0<\delta<R$, let $f_0(r)=h(r)$ for $r\in [0,\delta]$ and $f_0(r)=h(\delta)$ for $r\in (\delta,R]$.
The function $f_0$ is clearly not smooth, however it is clear that we can smooth it in an arbitrarily small neighbourhood of $r=\delta$ to a function $f$, so that $f''\le 0$.
Provided the smoothing neighbourhood is sufficiently small, the functions $f$ and $h$ then satisfy the requirements of Proposition \ref{prp-5.4}.
Note that we can arrange for $f(R)$ to be any value less than $h(R)=\lambda$.
To complete the proof, it remains to show that $f$ can take the value $(1+\e)\nu/\e$ close to the boundary.
But $f$ can take any value less than $\lambda$ at the boundary, so provided $(1+\e)\nu/\e<\lambda$ this boundary condition can be achieved.
Rearranging, this gives $\nu<\lambda\e/(1+\e)$ as claimed.
\proofend
\section{Open problems}\label{sec-open}
We conclude the paper with a selection of geometric open problems.
\begin{ils}
\item\label{op-1} Do any manifolds with a single singular orbit and cohomogeneity greater than three admit an invariant metric with positive Ricci curvature?
Recall that by Theorem \ref{one} we can construct manifolds of cohomogeneity three with a single singular orbit and positive Ricci curvature.
The problem with extending this family into higher cohomogeneities is that it necessitates extending the Fubini-Study metric on $\CP^{(k-1)/2}$ over the disc bundle corresponding to the imaginary sub-bundle of the canonical quaternionic line bundle over $\HP^{(k-3)/4}$ in such a way that the extension satisfies the requirements of Theorem \ref{thm-riccurvmain}. It is not clear to the authors whether such an extension is possible.
\item If the answer to question \ref{op-1} is yes, then do any of these manifolds admit invariant metrics with non-negative sectional curvature?
\item The Ricci positive examples displayed in section 2 have at most two singular orbits.
Is it possible to find invariant Ricci positive metrics on manifold having more than two singular orbits? \smallskip \noindent The obvious candidates are those for which $B$ (the space of orbits obtained when tubular neighbourhoods of the singular orbits have been removed from the original manifold) is a $3$-sphere less some discs.
(Thus the boundary is a disjoint union of $2$-spheres, that is, $\CP^1$s.) It is easily checked that conditions \eqref{cond-fbs} and \eqref{cond-pricurv} of Theorem \ref{thm-riccurvmain} mean that while two discs can comfortably be removed, taking out three discs results in these conditions {\it just} failing to hold.
The same is true when $B$ is a $5$-sphere less some discs (so the boundary components are all equal to $S^4=\HP^1$.) It is not clear whether the failure of these obvious candidates is due to their special nature, or whether they represent a general phenomenon.
Indeed it might be possible that {\it no} manifold with more than two singular orbits can support an invariant metric with positive Ricci curvature.
\item Are there any simply-connected examples which do {\it not} admit an invariant metric with positive Ricci curvature? \smallskip \noindent As noted in the Introduction, there is very little chance of all simply-connected $G$-manifolds with finitely many singular orbits admitting invariant metrics with positive Ricci curvature, since the topology of the space of orbits can be highly non-trivial and must surely influence the possible curvatures which the manifold can display.
\item\label{op-5} Are there any examples of non-double manifolds among the families $M^{11}_{p_1p_2q_1q_2}$ or $M^{13}_{p_1p_2q_1q_2}$, or indeed any non-double examples of any kind, which admit invariant metrics of non-negative sectional curvature? \smallskip \noindent If we simply want to join tubular neighbourhoods of two different singular orbits to create our manifold, then the main problem is that the horizontal distributions arising from the metric construction process never seem to match.
A possible strategy here is to look for horizontal distributions which could be deformed so as to join smoothly, whilst preserving non-negative sectional curvature.
The authors have no idea when or how such a deformation might be possible.
\item Do any of the manifolds with two different singular orbits admit metrics of almost non-negative sectional curvature? \smallskip \noindent On the face of it, this question is more likely to have a positive answer than question \ref{op-5}.
The motivation for this question arises from \cite{ST}, where it is shown that every compact cohomogeneity one manifold admits such a metric.
One of the features of cohomogeneity one manifolds which is important here is that the space of orbits, being one-dimensional, makes no contribution to the curvature.
On the other hand, in our situation this is not the case, so such metrics will almost certainly be much less common (assuming they exist at all).
\end{ils}
|
1,116,691,500,718 | arxiv | \section{Introduction}{\label{sec_intro}}
The problem of the unitary implementation of the quantum dynamics of free fields under the evolution defined by sufficiently general, physically acceptable, spacetime slicings has received a lot of attention in the past. In \cite{Helfer} Helfer showed that, in general, the quantum dynamics cannot be unitarily implemented in curved spacetimes and hints that, even in the Minkowski spacetime, this may also happen when considering the evolution between arbitrary Cauchy surfaces. Torre and Varadarajan \cite{TorreVar,TorreVar1,Torre} showed that this is indeed the case by studying the free scalar field on toroidal spatial slices (Cho and Varadarajan \cite{Var1} discussed Einstein-Rosen waves with a similar philosophy). The most important conclusion of these works is the realization of the fact that, beyond 1+1 dimensions, there are obstructions to the unitary implementation of the dynamics for certain types of seemingly good choices of Cauchy surfaces on a Minkowski spacetime. The authors of the quoted papers point out that, in fact, this is a \textit{generic} feature of the evolution of free quantum fields in more than one spatial dimension. An exception to this behavior is found for 1+1 dimensional models defined on the circle (the only closed one-dimensional manifold). In this case it is possible to show that the evolution can be unitarily implemented for arbitrary families of smooth spacelike slicings.
There are some issues that are not covered in the works that we have just quoted, for instance, the characterization of the families of embeddings capable of supporting unitary evolution in the $1+n$ dimensional case ($n>1$) or the consideration of spatial manifolds with boundary. The study of the latter problem is the main subject of the present paper. We hope that a straightforward extension of the methods that we use here will let us gain useful insights on the former.
Even in the simple one-dimensional setting, the introduction of boundaries ---and the consequent necessity of specifying boundary conditions to completely determine the classical dynamics-- introduces interesting changes regarding the unitary implementability of the quantum dynamics. This is so because, as we show in detail, the unitarity requirement forces us to work within particular classes of embeddings that cannot be arbitrarily chosen, in analogy with the generic situation in higher dimensions. The \textit{complete} characterization of the classes of spacelike embeddings capable of supporting unitary quantum evolution of the free scalar field in a bounded one dimensional region is the main result of the paper.
From a technical point of view an important difference between our work and that of Torre and Varadarajan \cite{TorreVar,TorreVar1} lies in the fact that, if the spatial manifold is a circle, it suffices to consider evolution from a flat inertial hypersurface to an arbitrary (spacelike) one. This is so because the dynamics is always unitarily implementable in that case and, hence, between any two spacelike hypersurfaces. This is no longer true for the systems that we discuss in the paper, so we will need to consider the evolution between arbitrary hypersurfaces from the start.
Despite the fact that the final result, regarding the characterization of the suitable embeddings, is the same for all the boundary conditions, we show that there are interesting differences between Dirichlet and Neumann boundary conditions on one hand and Robin boundary conditions on the other. In fact, some of the difficulties of dealing with the higher dimensional cases are already present in the Robin case (which is quite interesting in its own right \cite{BMS}), hence, their satisfactory resolution suggests that a complete characterization of the embeddings capable of supporting unitary evolution for $n$-dimensional tori is possible.
We want to mention at this point that Agullo and Ashtekar \cite{AA} have recently proposed a novel approach to the study of unitarity in quantum field theories. The main ingredient of their approach is to use different quantizations for the different time slices. We do not follow this path here but our results are not incompatible with theirs. Rather, we suggest a possible way to \textit{select} spacetime slicings by demanding unitarity of the time evolution in the standard sense. It is important to point out that our results show that inertial slicings are always allowed but it is always possible to find more general spacetime slicings (containing any given Cauchy surface) that support unitary evolution.
The lay out of the paper is the following. After this introduction we study in section \ref{sec_Bogoliubov} the general form of the Bogoliubov coefficients for the evolution ---between two arbitrary Cauchy surfaces--- for a class of models incorporating the boundary conditions that we consider: Dirichlet, Neumann and Robin. We also give necessary and sufficient conditions to guarantee that the dynamics in each of these cases is unitary. Sections \ref{Dirichlet} and \ref{Robin} concentrate on the Dirichlet and Robin boundary conditions, respectively, and also discuss the Neumann boundary conditions as a particular subcase of the latter. We end the paper in section \ref{sec_conclusions} with our conclusions and some comments. A number of technical points on the embeddings that we use in the paper are discussed in appendix \ref{appendix_A}. The second appendix \ref{appendix_B} gives some information regarding the behaviour of some important quantities (i.e.\ eigenvalues of the Laplace operators, normalization coefficients and the like). Finally appendix \ref{appendix_C} provides the proof of the main mathematical result employed in the characterization of the families of embeddings that can support unitary evolution.
\section{Bogoliubov coefficients and the unitary implementability of the quantum evolution}{\label{sec_Bogoliubov}}
Let us consider the classical evolution of a free, massless\footnote{The massive case is simpler to deal with. Its treatment is very similar to the one of the massless Robin case so we will not discuss it here. We work in the massless case to allow for zero modes.}, real scalar field $\varphi$ defined on the manifold (with boundary) $\mathbb{R}\times [0,\pi]$ and subject to boundary conditions of Dirichlet, Neumann or Robin type. Here we are considering a spacetime (naturally embedded in $\mathbb{R}^2$) endowed with a Minkowskian metric $\eta$ of signature $(-,+)$ . When convenient we will use global coordinates $(t,x)$, denote the $t$ and $x$ derivatives as $\dot{\varphi}$ and $\varphi'$, respectively, and write $\eta=-\mathrm{d}t^2+\mathrm{d}x^2$. The field equations are, simply,
\begin{equation}
\Box\varphi=0
\label{waveequation}
\end{equation}
where the fields are subject to boundary conditions at $x=0$ and $x=\pi$.
As it is well known, the space of solutions to \eqref{waveequation} can be endowed with a symplectic structure given by
\begin{equation}\label{simplectic}
\Omega(\varphi_1,\varphi_2)=\int_\Sigma\sqrt{\gamma_\Sigma}(\varphi_2{\L}_{n_\Sigma}\varphi_1-\varphi_1{\L}_{n_\Sigma}\varphi_2)\,,
\end{equation}
where $\varphi_1$ and $\varphi_2$ are two solutions to the field equations, $\Sigma$ is \textit{any} Cauchy hypersurface in $\mathbb{R}\times [0,\pi]$, ${\L}_{n_\Sigma}$ denotes the Lie derivative along the future directed unit normal to $\Sigma$ and $\gamma_\Sigma$ is the metric induced by $\eta$ on $\Sigma$. The metric volume form is $\sqrt{\gamma_\Sigma} \mathrm{d}\sigma$ where $\mathrm{d}\sigma$ is a fixed volume form on $\Sigma$ (that we will omit in many formulas). A straightforward argument that we sketch in the following shows that $\Omega$ is independent of the choice of $\Sigma$ for all the types of boundary conditions that we will employ in the paper. Indeed, in terms of the symplectic current
\begin{equation}\label{symplectic_current}
J(\varphi_1,\varphi_2):=\varphi_1\ast \mathrm{d}\varphi_2-\varphi_2\ast \mathrm{d}\varphi_1
\end{equation}
(with the Hodge dual defined as $\ast \mathrm{d}t=\mathrm{d}x$ and $\ast \mathrm{d}x=\mathrm{d}t$) we can write
\begin{equation}\label{symplectic_form}
\Omega(\varphi_1,\varphi_2)=\int_\Sigma J(\varphi_1,\varphi_2)\,.
\end{equation}
As $\mathrm{d}J=0$, if we apply Stokes theorem and integrate $\mathrm{d}J$ on a region $R$ bounded by two non-intersecting spacelike hypersurfaces $\Sigma_1$, $\Sigma_2$ and $\partial:=R\cap(\{x=0\}\cup\{x=\pi\})$ we get
\begin{equation*}
\int_{\Sigma_2}J- \int_{\Sigma_1}J+ \int_{\partial}J=0\,.
\end{equation*}
To conclude it suffices to check that $\imath_\partial^*J=(\varphi_1\varphi_2^\prime-\varphi_2\varphi_1^\prime)\mathrm{d}t=0$ for Dirichlet, Neumann and Robin boundary conditions and, hence, the integrals of $J$ on $\Sigma_1$ and $\Sigma_2$ are equal.
The solutions to the field equations \eqref{waveequation} can be expanded in terms of eigenfunctions of the different Laplacians
\begin{equation}\label{expansion_solutions}
\varphi(t,x)=a_0(1-i t)Q_0(x)+a_0^*(1+i t)Q_0(x)+\sum_{k=1}^\infty (a_ke^{-i\omega_kt}+a_k^*e^{i\omega_kt})Q_k(x)\,,
\end{equation}
where the modes $Q_k(x)$ satisfy
\[Q_k^{\prime\prime}(x)=-\omega_k^2Q_k(x)\]
\textit{and the boundary conditions}. Here $Q_0$ denotes the zero mode that only appears in the Neumann case. In terms of the Fourier coefficients $a_k$ and $a^*_k$ the symplectic form reads
\begin{equation}\label{symplectic_form_modes}
\Omega(\varphi_1,\varphi_2)=-i\sum_{k=1}^\infty 2\omega_kc_k^2(a_{1k}a^*_{2k}-a_{2k}a^*_{1k})-2c_0^2i(a_{10}a^*_{20}-a_{20}a^*_{10})\,.
\end{equation}
In order to have the standard expression for the symplectic form we choose the normalization constants $c_k$ as
\begin{equation*}
c_0^2=\int_0^\pi Q_0^2(x)\mathrm{d}x=\frac{1}{2}\,,\quad c_k^2=\int_0^\pi Q_k^2(x)\mathrm{d}x=\frac{1}{2\omega_k}\,,
\end{equation*}
so that we have the orthogonality conditions
\begin{equation}\label{orthogonality}
\int_0^\pi Q_{k_1}(x)Q_{k_2}(x)\mathrm{d}x=\frac{1}{2\omega_{k_1}}\delta_{k_1k_2}\,, \quad \int_0^\pi Q_0(x)Q_k(x)\mathrm{d}x=0\,,\quad k_1,k_2,k\in\mathbb{N}\,,
\end{equation}
and \eqref{symplectic_form_modes} is normalized.
In order to quantize and compute the Bogoliubov coefficients we need to know the scalar product in the space of complexified solutions of the field equations $\mathcal{S}_{\mathbb{C}}$ induced by the symplectic structure \eqref{simplectic}
\begin{equation}\label{Complexified_solution_space}
\mathcal{S}_{\mathbb{C}}:=\{\varphi:\varphi=a_0^+\varphi_0^++\sum_{k=1}^\infty a_k^+\varphi_k^++a_0^-\varphi_0^-+\sum_{k=1}^\infty a_k^-\varphi_k^-\}
\end{equation}
where the positive ($s=+$) and negative ($s=-$) solutions are chosen as
\begin{equation*}
\varphi_k^s(t,x):=e^{-i s \omega_k t}Q_k(x)\,,\quad \varphi_0^s:=(1-i s t)Q_0(x)\,.
\end{equation*}
The sesquilinear form
\begin{equation}\label{scalarproduct_1}
\langle \varphi_1,\varphi_2\rangle:=-i\int_\Sigma\sqrt{\gamma_\Sigma}(\varphi_2{\L}_{n_\Sigma}\varphi_1^*-\varphi_1^*{\L}_{n_\Sigma}\varphi_2)
\end{equation}
defines a (positive definite) scalar product on the positive frequency subspace of $\mathcal{S}_{\mathbb{C}}$. In an arbitrary Cauchy hypersurface given by $\mathbf{X}(x)=(T(x),X(x))$ in inertial coordinates (see appendix \ref{appendix_A}), the expression for \eqref{scalarproduct_1} is
\begin{align}\label{scalarproduct_2}
\langle \varphi_{k_1}^{s_1},\varphi_{k_2}^{s_2}\rangle & =\int_0^\pi |\gamma_{\mathbf{X}}|^{1/2} e^{-i(s_2\omega_{k_2}-s_1\omega_{k_1})T}(s_2 \omega_{k_2}+s_1 \omega_{k_1})n_{\mathbf{X}}^0Q_{k_1}(X)Q_{k_2}(X) \\
- i \int_0^\pi & |\gamma_{\mathbf{X}}|^{1/2} e^{-i(s_2\omega_{k_2}-s_1\omega_{k_1})T}(s_2 \omega_{k_2}+s_1 \omega_{k_1})n_{\mathbf{X}}^1\big(Q_{k_1}^\prime(X)Q_{k_2}(X) -Q_{k_1}(X)Q_{k_2}^\prime(X)\big)\,,\nonumber
\end{align}
where we denote the unit normal to the hypersurface $\mathbf{X}(\Sigma)$ as $n_{\mathbf{X}}(x)=(n_{\mathbf{X}}^0(x),n_{\mathbf{X}}^1(x))$, and similar formulas apply when zero modes are present (i.e.\ for Neumann boundary conditions).
In terms of the modes $Q_k$, and taking into account that in $1+1$ dimensions the unit future pointing normal $n$ is given by
\[
n=(n^0,n^1)=\frac{1}{\sqrt{\gamma_{\mathbf{X}}}}(X',T')\,,
\]
the Bogoliubov coefficients for the field evolution between two spatial hypersurfaces $\mathbf{X}_1(\Sigma)$ and $\mathbf{X}_2(\Sigma)$ can be written in the form
\begin{align}\label{Bogo_coeffs}
\beta_{k_1 k_2}^{\bm{X}_1\bm{X}_2} &= \int_\Sigma \big(\omega_{k_1}X_1'Q_{k_1}(X_1)Q_{k_2}(X_2)-iT_1'Q'_{k_1}(X_1)Q_{k_2}(X_2) \big) e^{i\omega_{k_2}T_2+i\omega_{k_1}T_1} \\
& -\int_\Sigma \big(\omega_{k_2}X'_2Q_{k_1}(X_1)Q_{k_2}(X_2)-iT_2'Q_{k_1}(X_1)Q'_{k_2}(X_2) \big) e^{i\omega_{k_2}T_2+i\omega_{k_1}T_1}\,.\nonumber
\end{align}
Following \cite{Shale}, the necessary and sufficient condition that guarantees the unitary implementability of the quantum evolution given by the Cauchy slices associated with $\bm{X}_1$ to $\bm{X}_2$ is
\[\sum_{k_1,k_2}|\beta_{k_1 k_2}^{\bm{X}_1\bm{X}_2}|^2<\infty\,.\]
In the following sections we study when this condition is satisfied and characterize the families of embeddings that support unitary evolution for the different types of boundary conditions used in the paper.
\section{Dirichlet boundary conditions}\label{Dirichlet}
The normalized eigenfunctions of the Laplace operator with Dirichlet boundary conditions in the interval $[0,\pi]$ are
\begin{equation}\label{eigen_Dirichlet}
Q_k(x)=\frac{1}{\sqrt{\pi k}}\sin kx\,,\quad \omega_k=k\,,\quad k\in \mathbb{N}\,.
\end{equation}
There are no zero modes in this case. By introducing these in \eqref{Bogo_coeffs} we immediately obtain
\begin{align*}
\beta_{k_1 k_2}^{\bm{X}_1\bm{X}_2}
&= \frac{1}{4\pi \sqrt{k_1k_2}}\int_0^\pi (k_2X'_2-k_1X_1'-k_1T_1'+k_2T'_2) e^{i (k_1(T_1+X_1)+k_2(T_2+X_2))}\, \mathrm{d}\sigma \\
&+ \frac{1}{4\pi \sqrt{k_1k_2}}\int_0^\pi (k_2X'_2-k_1X_1'+k_1T_1'-k_2T'_2) e^{i (k_1(T_1-X_1)+k_2(T_2-X_2))}\, \mathrm{d}\sigma\\
&+\frac{1}{4\pi \sqrt{k_1k_2}}\int_0^\pi (k_1X'_1-k_2X_2'+k_1T_1'+k_2T'_2) e^{i (k_1(T_1+X_1)+k_2(T_2-X_2))}\, \mathrm{d}\sigma \\
&+\frac{1}{4\pi \sqrt{k_1k_2}}\int_0^\pi (k_1X'_1-k_2X_2'-k_1T_1'-k_2T'_2) e^{i (k_1(T_1-X_1)+k_2(T_2+X_2))}\, \mathrm{d}\sigma\,.
\end{align*}
In order to compute the first two integrals in the preceding expression we closely follow the procedure introduced by Torre and Varadarajan (see appendix in \cite{TorreVar}). For instance, for the first integral $I_1$ we use the change of variables $u=\lambda(T_1+X_1)+(1-\lambda)(T_2+X_2)$ and integrate by parts [$\lambda\in(0,1)$ is defined as $\lambda=\lambda(k_1,k_2):=k_1/(k_1+k_2)\,$]. This gives
\begin{align*}
&I_1=\frac{i}{4\pi\sqrt{k_1k_2}(k_1+k_2)}\left(e^{i(k_1T_1(0)+k_2T_2(0))}\frac{k_2(X_2^\prime(0)+T_2^\prime(0))-k_1(X_1^\prime(0)+T_1^\prime(0))}{\lambda (T_1^\prime(0)+X_1^\prime(0))+(1-\lambda)(T_2^\prime(0)+X_2^\prime(0))}\right.\\
&\left.-(-1)^{k_1+k_2}e^{i(k_1T_1(\pi)+k_2T_2(\pi))}\frac{k_2(X_2^\prime(\pi)+T_2^\prime(\pi))-k_1(X_1^\prime(\pi)+
T_1^\prime(\pi))}{\lambda(T_1^\prime(\pi)+X_1^\prime(\pi))+(1-\lambda)(T_2^\prime(\pi)+X_2^\prime(\pi))}+O\left(\frac{1}{k_1+k_2}\right)\right)\,.
\end{align*}
A completely analogous procedure can be used to compute the second integral. Finally the last two integrals can be trivially obtained in closed form. Putting all this together we find that the Bogoliubov coefficients in this case are
\begin{align}
\hspace*{-4mm}\beta_{k_1 k_2}^{\bm{X}_1\bm{X}_2}
= &\frac{i\sqrt{k_1k_2}}{\pi (k_1+k_2)^2} \left(
\frac{(-1)^{k_1+k_2}e^{i(k_1T_1(\pi)+k_2T_2(\pi))}(X_1'(\pi)T_2'(\pi)-T_1'(\pi)X_2'(\pi))}{(\lambda T'_1(\pi)+(1-\lambda)T'_2(\pi))^2-(\lambda X'_1(\pi)+(1-\lambda)X'_2(\pi))^2}
\right.\label{Bogo_Dirichlet}\\
&\left.
-\frac{e^{i(k_1T_1(0)+k_2T_2(0))}(X_1'(0)T_2'(0)-T_1'(0)X_2'(0))}{(\lambda T'_1(0)+(1-\lambda)T'_2(0))^2-(\lambda X'_1(0)+(1-\lambda)X'_2(0))^2}+O\left(\frac{1}{k_1k_2}\right)
\right)\,.\nonumber
\end{align}
The Bogoliubov coefficients for Dirichlet boundary conditions \eqref{Bogo_Dirichlet} have the form
\begin{equation}\label{Bogo-coeffs_AB}
\hspace*{-10mm}\beta_{k_1 k_2}^{\bm{X}_1\bm{X}_2}=\frac{\sqrt{k_1 k_2}}{\pi(k_1+k_2)^2}\left(\gamma^{\bm{X}_1\bm{X}_2}_{k_1k_2}+O\left(\frac{1}{k_1k_2}\right)\right)\,,
\end{equation}
where
\begin{equation}\label{gamma_Dirich}
\gamma^{\bm{X}_1\bm{X}_2}_{k_1k_2}:=i e^{i(k_1\alpha_{1\sigma}+k_2\alpha_{2\sigma})}f_\sigma(\lambda) V_\sigma\Big|_{\sigma=0}^{\sigma=\pi}
\end{equation}
\begin{align}
V_\sigma:=X_1'(\sigma)T_2'(\sigma) -T_1'(\sigma)X_2'(\sigma)\,,\label{Vsigma}
\end{align}
$\alpha_{i0}$ and $\alpha_{i\pi}$ ($i=1,\,2$) are constants defined in terms of the end-points of the embeddings
\begin{align}
\alpha_{10}:=&T_1(0)\,, \,\,\,\,\,\quad\quad \alpha_{20}:=T_2(0) \label{betas}\\
\alpha_{1\pi}:=&T_1(\pi)+\pi\,, \quad \alpha_{2\pi}:=T_2(\pi)+\pi \label{alfas}
\end{align}
and
\[
\lambda=\lambda(k_1,k_2):=\frac{k_1}{k_1+k_2}\in (0,1)
\]
and the functions $f_0,\,f_\pi:[0,1]\rightarrow \mathbb{R}$ are given by
\begin{align}
f_\sigma(\lambda) &=\frac{1}{(\lambda T_1'(\sigma)+(1-\lambda)T_2'(\sigma))^2-(\lambda X_1'(\sigma)+(1-\lambda)X_2'(\sigma))^2}\,.\label{f0}
\end{align}
As we show in appendix \ref{appendix_C} the necessary and sufficient condition to guarantee the convergence of $\sum_{k_1,k_2}|\beta_{k_1 k_2}^{\bm{X}_1\bm{X}_2}|^2$ is the vanishing of the coefficients $V_0$ and $V_\pi$ defined in \eqref{Vsigma}. This implies that
\begin{equation}\label{conditions_Dirichlet}
\frac{T_1'(\pi)}{X_1'(\pi)}= \frac{T_2'(\pi)}{X_2'(\pi)}\,,\quad \frac{T_1'(0)}{X_1'(0)}= \frac{T_2'(0)}{X_2'(0)}\,.
\end{equation}
The interpretation of these conditions is straightforward. Unitary evolution requires that the slope of the embedded surfaces both at $0$ and $\pi$ must be separately preserved under the evolution of the system. Notice, by the way, that for any embedding $X'(0)\neq0$ and $X'(\pi)\neq0$ (see appendix \ref{appendix_A}).
It is interesting to notice that (for non-zero modes) in the Neumann case the Bogoliubov coefficients differ only in a global sign. In this case
\[
Q_k(x)=\frac{1}{\sqrt{\pi k}}\cos kx\,,\quad \omega_k=k\,,\quad k\in\mathbb{N}\,,
\]
and hence $\beta_{k_1 k_2}^{\bm{X}_1\bm{X}_2}$ for $k_1\,,k_2\in \mathbb{N}$ are simply \eqref{Bogo_Dirichlet} multiplied by $-1$. It is straightforward to see that $\beta_{0 k_2}^{\bm{X}_1\bm{X}_2}=\beta_{k_10}^{\bm{X}_1\bm{X}_2}=0$ and the concrete value of $\beta_{00}^{\bm{X}_1\bm{X}_2}$ is irrelevant. From this we conclude that the Neumann case essentially reduces to the Dirichlet case as far as the unitarity of the evolution is concerned.
\section{Robin boundary conditions}\label{Robin}
Let us consider now the problem with Robin boundary conditions of the form
\begin{align}
&Q_k^\prime(0)-\kappa_0 Q_k(0)=0 \label{Robin0}\\
&Q_k^\prime(\pi)+\kappa_\pi Q_k(\pi)=0\label{RobinPi}
\end{align}
with $k\in\mathbb{N}$ and $\kappa_0,\kappa_\pi\geq 0$. There are zero modes if and only if $\kappa_0=\kappa_\pi=0$ (Neumann boundary conditions) so we will concentrate in the following in modes with $\omega>0$. Mixed Robin-Dirichlet boundary conditions can be treated in a straightforward way so we will not discuss them here.
The eigenfunctions of the Laplace operator subject to these boundary conditions satisfy the equation $Q_k^{\prime\prime}=-\omega_k^2Q_k$ with $\omega_k$ given by the positive solutions to the equation
\begin{equation}
\label{ec_robin}
(\omega_k^2-\kappa_0\kappa_\pi)\sin\pi\omega_k-(\kappa_0+\kappa_\pi)\omega_k\cos\pi\omega_k=0\,.
\end{equation}
These eigenfunctions can be written as a linear combination of $\sin\omega_k x$ and $\cos\omega_k x$. However, in order to keep the analogy with the Dirichlet case it is convenient to write them in the form
\begin{equation}
Q_k(x)=C_k\sin(\omega_k x+\phi_k)\,,\label{eigenfunctions_Robin}
\end{equation}
where we have absorbed the constants multiplying $\sin\omega_k x$ and $\cos\omega_k x$ in a phase $\phi_k\in (0,\pi/2]$ and an amplitude $C_k>0$ satisfying
\begin{align*}
&\sin \phi_k=\frac{\omega_k}{\sqrt{\omega_k^2+\kappa_0^2}}\,,\quad \cos \phi_k=\frac{\kappa_0}{\sqrt{\omega_k^2+\kappa_0^2}}\,,\quad \phi_k=\arctan\frac{\omega_k}{\kappa_0}\,,\\
& \frac{1}{C^2_k}=\pi\omega_k\left( 1+\frac{\kappa_0+\kappa_\pi}{\pi}\frac{\omega_k^2+\kappa_0\kappa_\pi}{(\omega_k^2+\kappa_0^2)(\omega_k^2+\kappa_{\pi}^2)}\right)\,.
\end{align*}
The Robin boundary conditions \eqref{Robin0}-\eqref{RobinPi} imply that
\begin{align*}
&\omega_k\cos\phi_k=\kappa_0\sin\phi_k\,,\\
&\omega_k\cos(\pi\omega_k+\phi_k)=-\kappa_\pi\sin(\pi\omega_k+\phi_k)\,,
\end{align*}
and, hence, we can derive the following identities (valid for $\sigma=0,\pi$):
\begin{align}
\hspace*{-0.5cm}\kappa_\sigma\sin(\omega_{k_1}\sigma+\omega_{k_2}\sigma+\phi_{k_1}+\phi_{k_2})=&(-1)^{\frac{\sigma}{\pi}}(\omega_{k_1}+\omega_{k_2})\cos(\omega_{k_1}\sigma+\phi_{k_1})\cos(\omega_{k_2}\sigma+\phi_{k_2})\,,\label{identidad1}\\
\hspace*{-0.5cm}\kappa_\sigma\sin(\omega_{k_1}\sigma-\omega_{k_2}\sigma+\phi_{k_1}-\phi_{k_2})=&(-1)^{\frac{\sigma}{\pi}}(\omega_{k_1}-\omega_{k_2})\cos(\omega_{k_1}\sigma+\phi_{k_1})\cos(\omega_{k_2}\sigma+\phi_{k_2})\,.\label{identidad2}
\end{align}
Using the form of the eigenfunctions \eqref{eigenfunctions_Robin}, the general expression for the Bogoliubov coefficients \eqref{Bogo_coeffs} and following the same steps as in the Dirichlet case we get
\begin{align*}
\beta_{k_1 k_2}^{\bm{X}_1\bm{X}_2}
&= \frac{C_{k_1}C_{k_2}}{4}\int_0^\pi (\omega_{k_2}X'_2-\omega_{k_1}X_1'-\omega_{k_1}T_1'+\omega_{k_2}T'_2) e^{i (\omega_{k_1}(T_1+X_1)+\omega_{k_2}(T_2+X_2))+\phi_{k_1}+\phi_{k_2})}\, \mathrm{d}\sigma \\
&+ \frac{C_{k_1}C_{k_2}}{4}\int_0^\pi (\omega_{k_2}X'_2-\omega_{k_1}X_1'+\omega_{k_1}T_1'-\omega_{k_2}T'_2) e^{i (\omega_{k_1}(T_1-X_1)+\omega_{k_2}(T_2-X_2))-\phi_{k_1}-\phi_{k_2})}\, \mathrm{d}\sigma\\
&+\frac{C_{k_1}C_{k_2}}{4}\int_0^\pi (\omega_{k_1}X'_1-\omega_{k_2}X_2'+\omega_{k_1}T_1'+\omega_{k_2}T'_2) e^{i (\omega_{k_1}(T_1+X_1)+\omega_{k_2}(T_2-X_2))+\phi_{k_1}-\phi_{k_2})}\, \mathrm{d}\sigma \\
&+ \frac{C_{k_1}C_{k_2}}{4}\int_0^\pi (\omega_{k_1}X'_1-\omega_{k_2}X_2'-\omega_{k_1}T_1'-\omega_{k_2}T'_2) e^{i (\omega_{k_1}(T_1-X_1)+\omega_{k_2}(T_2+X_2))-\phi_{k_1}+\phi_{k_2})}\, \mathrm{d}\sigma\,.
\end{align*}
As we can see the structure of these integrals is very similar to those appearing in the computation of the Bogoliubov coefficients for the Dirichlet case; they can be obtained along the same lines. They are
\begin{align}
\beta_{k_1 k_2}^{\bm{X}_1\bm{X}_2}
&=\frac{\sqrt{\omega_{k_1}\omega_{k_2}}}{\pi(\omega_{k_1}+\omega_{k_2})^2}\left(\gamma^{\bm{X}_1\bm{X}_2}_{k_1k_2}+O\left(\frac{1}{\omega_{k_1}\omega_{k_2}}\right)\right)\label{Bogo_Robin}
\end{align}
where
\begin{align}
\gamma^{\bm{X}_1\bm{X}_2}_{k_1 k_2}:=\pi\sqrt{\omega_{k_1}\omega_{k_2}}C_{k_1}C_{k_2}e^{i\psi_\sigma(k_1,k_2)}f_\sigma(\tau)\Big( &
iV_\sigma\cos(\omega_{k_1}\sigma+\phi_{k_1}+\omega_{k_2}\sigma+\phi_{k_2})
\label{gamma_Robin}\\&\hspace*{-2.3cm}+
\kappa_\sigma g_\sigma(\tau)\frac{\omega_{k_1}+\omega_{k_2}}{\omega_{k_1}\omega_{k_2}}\sin(\omega_{k_1}\sigma+\phi_{k_1}) \sin(\omega_{k_2}\sigma+\phi_{k_2})\Big)\Big|_{\sigma=0}^{\sigma=\pi}\,.
\nonumber
\end{align}
In analogy with the Dirichlet case we have introduced the notation
\[\tau=\tau(\omega_{k_1},\omega_{k_2}):=\frac{\omega_{k_1}}{\omega_{k_1}+\omega_{k_2}}\in (0,1)\,,\]
The expressions for $f_\sigma$ and $V_\sigma$ are given in \eqref{f0} and \eqref{Vsigma}. We have also introduced
\begin{align*}
&g_\sigma(\tau):=\tau (N_1^2(\sigma)-S(\sigma))-(1-\tau)(N_2^2(\sigma)-S(\sigma))\\
&\psi_\sigma(k_1,k_2):=\omega_{k_1}T_1(\sigma)+\omega_{k_2}T_2(\sigma)\,,
\end{align*}
and
\begin{align*}
&N^2_i(\sigma):=T_i^{\prime 2}(\sigma)-X^{\prime2}(\sigma)\,, i=1,2\\
&S(\sigma):=T_1'(\sigma)T_2'(\sigma)-X_1'(\sigma)X_2'(\sigma)\,.
\end{align*}
Notice that, modulo a global sign, \eqref{gamma_Robin} reduces to \eqref{gamma_Dirich} in the Neumann case ($\kappa_0=\kappa_\pi=0$). In order to see this, the equations appearing in appendix \ref{appendix_B} are useful.
The discussion of the convergence of the series
\[
\sum_{k_1,k_2}|\beta_{k_1 k_2}^{\bm{X}_1\bm{X}_2}|^2
\]
is lengthy but straightforward. We just point out the most important features of the analysis.
The modulus squared of the Bogoliubov coefficients $|\beta_{k_1 k_2}^{\bm{X}_1\bm{X}_2}|^2$ consists of three types of terms: those involving $V_\sigma^2$, those with $g_\sigma^2$ and terms with phases $e^{\pm i(\psi_\pi-\psi_0)}$. The terms proportional to $V_\sigma^2$, as in the Dirichlet case, diverge; those with $g_\sigma^2$ converge; and an argument that relies on the result proved in appendix \ref{appendix_C} shows that the terms involving the phases also converge. As a consequence of this, the necessary and sufficient condition guaranteing the unitarity of the dynamics is, as in the Dirichlet case, the vanishing of $V_\sigma$.
\section{Conclusions and comments}{\label{sec_conclusions}}
We have analyzed the dynamical evolution of a quantum free scalar field defined on a flat spacetime of the form $\mathbb{R}\times [0,\pi]$ satisfying Dirichlet, Neumann or Robin boundary conditions. Along the way we have characterized the equivalence classes of spacelike embeddings that support unitary evolution. These classes are labeled by the values of $T'(0)/X'(0)$ and $T'(\pi)/X'(\pi)$, in the sense that for any two embeddings $\bm{X}_1$, $\bm{X}_2$ satisfying $T_1'(0)/X_1'(0)=T_2'(0)/X_2'(0)$ and $T_1'(\pi)/X_1'(\pi)=T_2'(\pi)/X_2'(\pi)$, the field dynamics between $\bm{X}_1(\Sigma)$ and $\bm{X}_2(\Sigma)$ can be unitarily implemented. Notice that the simultaneity hypersurfaces defined by an inertial (free) observer always define a slicing of spacetime for which the dynamics can be unitarily implemented. The Schr\"odinger picture of the functional evolution is not available in general (as in the $1+n$ dimensional case) but it is well defined within the equivalence class of embeddings just mentioned.
Several comments are in order now: The main reason to remain within the usual Fock quantization ---as we do here--- is the ease to physically interpret the states as vectors in a Hilbert space with simple properties. It is in this framework that the phenomenon of the impossibility to unitarily implement quantum evolution shows up. As emphasized by a number of authors, there are other possible approaches to the quantization of the field models that we discuss in this paper that can be used to circumvent some of the problems associated with the lack of unitary evolution in a satisfactory way. For instance, in the algebraic approach to quantum field theory (see for example \cite{Wald} and the discussion in \cite{TorreVar1,Torre}), the $C^*$ algebra $\mathcal{A}$ of basic observables of a free theory is taken to be the Weyl algebra. $\mathcal{A}$ is generated by elements $W(\varphi)$, labeled
by points $\varphi$ of the covariant phase space (i.e.\ the space of solutions to the field equations), satisfying
\[
W(\varphi)^*= W(-\varphi)\,,\quad W(\varphi_1)W(\varphi_2)= e^{-i\Omega(\varphi_1,\varphi_2)}W(\varphi_1 + \varphi_2). \]
The states are positive, normalized, linear functions $\omega:\mathcal{A} \rightarrow \mathbb{C}$. In this approach \cite{TorreVar1}, if we consider two Cauchy hypersurfaces represented by the embeddings $\bm{X}_1$ and $\bm{X}_2$ and we assign the state $\omega$ to the Cauchy hypersurface $\bm{X}_1(\Sigma)$, the expectation value of the observable represented by element $W(\varphi)$ of the Weyl algebra at the Cauchy hypersurface $\bm{X}_2(\Sigma)$ is always well defined and is given by
\[
\langle W(\varphi)\rangle_{(\omega,\bm{X}_1, \bm{X}_2)} = \omega(W(\mathcal{T}^{-1}_{(\bm{X}_1,\bm{X}_2)}\varphi))\,.
\]
where $\mathcal{T}_{(\bm{X}_1,\bm{X}_2)}$ is a bijection on the solution space $\Gamma$ defined in terms of the map $\mathcal{I}_{\bm{X}}$ that associates a solution to the field equations to particular initial data given on the Cauchy surface $\bm{X}(\Sigma)$. Notice that on the Cauchy hypersurface $\bm{X}_1(\Sigma)$ the expectation value is given by
\[\langle W(\varphi)\rangle_{(\omega,\bm{X}_1,\bm{X}_1)} = \omega(W(\varphi)).\]
No unitarity problem arises in this context in the Heisenberg picture, as long as one is only interested in the evolution of observables in the Weyl algebra. Of course, in order to study the physics associated with observables that do not belong to this algebra it is necessary to use other methods.
A possible solution to this issue (partially anticipated by Torre and Varadarajan in \cite{TorreVar1}) has been suggested by Agullo and Ashtekar in \cite{AA}. Given a foliation $\bm{X}_t$, $t\in \mathbb{R}$, one can construct a $1$-parameter family of representations $\bm{R}_t$ on Hilbert spaces $(\mathcal{H}_t,\langle\cdot,\cdot\rangle_t)$ of the Weyl algebra $\mathcal{A}$ in such a way that \textit{there exists} a family of unitary operators $U(t_2,t_1): \mathcal{H}_{t_1}\rightarrow \mathcal{H}_{t_2}$ such that
\[
\langle \Phi_1 , \bm{R}_{t_1}(W(\mathcal{T}^{-1}_{(\bm{X}_{t_1},\bm{X}_{t_2})}\varphi)) \Psi_1\rangle_{t_1}=\langle U(t_2,t_1)\Phi_1 , \bm{R}_{t_2}(W(\varphi)) U(t_2,t_1)\Psi_1\rangle_{t_2}\,.
\]
In this approach evolution is always unitary by construction.
As we have shown in the paper, the presence of boundaries prevents us from reaching the conclusions of \cite{TorreVar1}. The main reason is that the argument used in that paper to show that the dynamics can be unitarily implemented relies on the possibility of performing successive integrations by parts involving smooth functions to show that the Bogoliubov coefficients decay sufficiently fast. In the examples that we have considered here the surface terms appearing after integration by parts spoil the simple argument of \cite{TorreVar1} and require a careful discussion.
The analysis that we have presented here is more complicated than the one of \cite{TorreVar1} because we are forced to consider the evolution between \textit{arbitrary} Cauchy hypersurfaces (and not just from an inertial one to an arbitrary one) as the dynamics between generic Cauchy hypersurfaces cannot be unitarily implemented.
An interesting application of our results is the polymer quantization of this class of models generalizing the results already obtained in \cite{Var2,LV1,LV2,LV3} for the circle. This may help understand the quantization of diff-invariant field theories in the presence of boundaries and may find applications in the study of black holes in loop quantum gravity \cite{BV} (where they are modelled with the help of spacetime boundaries called \textit{isolated horizons} \cite{AK}).
\section*{Acknowledgments}
The authors wish to thank Ivan Agullo and Madhavan Varadarajan for their valuable comments. This work has been supported by the Spanish MINECO research grant FIS2014-57387-C3-3-P. Juan Margalef-Bentabol is supported by a ``la Caixa'' fellowship and a Residencia de Estudiantes (MINECO) fellowship.
\begin{appendices}
\section{Some facts about the spatial embeddings}\label{appendix_A}
The spacelike embeddings $\mathbf{X}:=(T,X)$ that we use in the paper can be defined in terms of a pair of $C^\infty([0,\pi])$ functions\footnote{This guarantees, among other things, that the integrals that are left after the several integration by parts that we perform in the paper are well defined.} $T,X:[0,\pi]\rightarrow \mathbb{R}$ satisfying the conditions $X(0)=0$, $X(\pi)=\pi$. The fact that they are required to be spacelike means that
\begin{equation}\label{spacelike}
-T'(s)^2+X'(s)^2>0
\end{equation}
for all $s\in [0,\pi]$. As a consequence of this (and the required orientability) we have $X'(s)>0$ for all $s\in[0,\pi]$ and, hence, \eqref{spacelike} is equivalent to either of the following:
\begin{align}
\left|\frac{T'(s)}{X'(s)}\right|<1\,, &\quad\forall s\in[0,\pi]\,,\label{spacelike1}\\
-X'(s)<T'(s)<X'(s) \,, &\quad\forall s\in[0,\pi]\,.\label{spacelike2}
\end{align}
Integrating \eqref{spacelike2} on the interval $[0,\pi]$ and using $X(0)=0$, $X(\pi)=\pi$ it is straightforward to show that $|T(\pi)-T(0)|<\pi$ which implies that $\gamma_i:=T_i(\pi)-T_i(0)+\pi$, ($i=1,2$ and $\mathbf{X}_i$ are two spacelike embeddings) satisfy $0<|\gamma_i|<2\pi$, conditions that play a relevant role in the analysis presented in appendix \ref{appendix_C}.
If we have two spacelike embeddings $\mathbf{X}_i$, $i=1,2$ and take $\lambda\in(0,1)$, the conditions \eqref{spacelike2} imply
\begin{align*}
& -\lambda X_1'(\pi)<\lambda T_1'(\pi)< \lambda X_1'(\pi)\\
& -(1-\lambda) X_2'(\pi)<(1-\lambda)T_2'(\pi)<(1-\lambda)X_2'(\pi)\,.
\end{align*}
Adding them up we see that $|\lambda T_1'(\pi)+(1-\lambda)T_2'(\pi)|<|\lambda X_1'(\pi)+(1-\lambda)X_2'(\pi)|$. Taking into account that $T_i'(\pi)^2-X_i'(\pi)^2<0$, for $i=1,2$ we conclude that for all $\lambda\in[0,1]$ we have
\[
(\lambda T_1'(\pi)+(1-\lambda)T_2'(\pi))^2-(\lambda X_1'(\pi)+(1-\lambda)X_2'(\pi))^2<0\,.
\]
In the same way we can prove that
\[
(\lambda T_1'(0)+(1-\lambda)T_2'(0))^2-(\lambda X_1'(0)+(1-\lambda)X_2'(0))^2<0\,.
\]
With the help of these last two conditions it is straightforward to show that the functions $f_0$ and $f_\pi$ defined by (\ref{f0}) can be extended to the closed interval $[0,1]$ in such a way that they are infinitely differentiable, i.e.\ they are bounded in $[0,1]$ together with all their derivatives. The same conclusion applies to $h:=f_0f_\pi$. In the following the extensions of these functions will be denoted with the same root letter.
\section{Some details about the Robin case}\label{appendix_B}
In this appendix we give some complementary information about the eigenvalues and eigenfunctions \eqref{eigenfunctions_Robin} for Robin boundary conditions.
Asymptotically $\omega_k$ behaves as
\begin{equation}\label{asympt_omega}
\omega_{k+1}=k+\frac{\kappa_0+\kappa_\pi}{\pi k}-\left(\frac{(\kappa_0+\kappa_\pi)^2}{\pi^2}+\frac{\kappa_0^3+\kappa_\pi^3}{3\pi}\right)\frac{1}{k^3}+O\left(\frac{1}{k^5}\right)\,,\quad k\rightarrow\infty\,.
\end{equation}
when $k\rightarrow\infty$. It is also useful to know the asymptotic behavior in the same limit of $\phi_k$
\begin{equation}
\phi_{k+1}=\frac{\pi}{2}-\frac{\kappa_0}{k}+\frac{3\kappa_0^2+\kappa_0^3+3\kappa_0\kappa_\pi}{3k^3}+O\left(\frac{1}{k^5}\right)\,,\label{asympt_phi}
\end{equation}
and
\begin{equation}
C_{k+1}\sqrt{\pi\omega_{k+1}}=1-\frac{\kappa_0+\kappa_\pi}{2\pi k^2}+O\left(\frac{1}{k^4}\right)\,.\label{asympt_Rk}
\end{equation}
The following identities are also useful
\begin{align*}
& \cos(\pi \omega_{k_1}+\phi_{k_1}+\pi \omega_{k_2}+\phi_{k_2})=(-1)^{k_1+k_2}\frac{\kappa_\pi^2-\omega_{k_1}\omega_{k_2}}{\sqrt{(\omega_{k_1}^2+\kappa_\pi^2)(\omega_{k_2}^2+\kappa_\pi^2)}}\,, \\
& \cos(\phi_{k_1}+\phi_{k_2})=\frac{\kappa_0^2-\omega_{k_1}\omega_{k_2}}{\sqrt{(\omega_{k_1}^2+\kappa_0^2)(\omega_{k_2}^2+\kappa_0^2)}}\,,\\
&\sin(\pi\omega_k+\phi_k)=(-1)^k\frac{\omega_k}{\sqrt{\omega_k^2+\kappa_\pi^2}}\,,\\
&\cos(\pi\omega_k+\phi_k)=(-1)^k\frac{\kappa_\pi}{\sqrt{\omega_k^2+\kappa_\pi^2}}\,,\\
&\sin(\pi\omega_k)=(-1)^{k+1}\frac{(\kappa_0+\kappa_\pi)\omega_k}{\sqrt{(\omega_k^2+\kappa_0^2)(\omega_k^2+\kappa_\pi^2)}}\,,\\
&\cos(\pi\omega_k)=(-1)^{k+1}\frac{\omega_k^2-\kappa_0\kappa_\pi}{\sqrt{(\omega_k^2+\kappa_0^2)(\omega_k^2+\kappa_\pi^2)}}\,.
\end{align*}
\section{Series convergence of the Bogoliubov coefficients for Dirichlet and Neumann boundary conditions}\label{appendix_C}
Let us prove that the vanishing of $V_0$ and $V_\pi$, defined in (\ref{Vsigma}), are the necessary and sufficient conditions for the convergence of
\[\sum_{k_1,k_2}|\beta_{k_1 k_2}^{\bm{X}_1\bm{X}_2}|^2<\infty\,,\quad k_1,\,k_2\in\mathbb{N}\]
if the Bogoliubov coefficients have the form \eqref{Bogo-coeffs_AB}:
\[
\beta_{k_1 k_2}^{\bm{X}_1\bm{X}_2}=\frac{\sqrt{k_1 k_2}}{\pi(k_1+k_2)^2}\left(V_\pi e^{i(k_1\alpha_{1\pi}+k_2\alpha_{2\pi})}f_\pi(\lambda)-V_0e^{i(k_1\alpha_{10}+k_2\alpha_{20})}f_0(\lambda)+O\left(\frac{1}{k_1+k_2}\right)\right)\,.
\]
The only non-trivial part of the proof is showing that the conditions are necessary. To this end we need to consider the following double series given by the sum of the squares of the absolute values of the ``leading part'' of the Bogoliubov coefficients
\[
\sum_{k_1,k_2\in\mathbb{N}}\frac{k_1k_2}{(k_1+k_2)^4}\Big(V_\pi^2f_\pi^2(\lambda)+V_0^2f_0^2(\lambda)-2V_0V_\pi f_0(\lambda)f_\pi(\lambda)\mathrm{Re}\left(e^{ik_1(\alpha_{1\pi}-\alpha_{10})+ik_2(\alpha_{2\pi}-\alpha_{20})}\right)\Big)\,.
\]
As this is a series of positive terms it will converge if and only if the following ordinary (``simple'') series does
\begin{equation}\label{seriesimple}
\hspace*{-5mm}\sum_{j=2}^\infty \sum_{k=1}^{j-1}\frac{k(j-k)}{j^4}\left(V_\pi^2f_\pi^2\left(\frac{k}{j}\right)+V_0^2f_0^2\left(\frac{k}{j}\right)-2V_0V_\pi f_0\left(\frac{k}{j}\right)f_\pi\left(\frac{k}{j}\right)\mathrm{Re}\left(e^{ik(\gamma_1-\gamma_2)+i\gamma_2j}\right)\right).
\end{equation}
As we have shown in appendix \ref{appendix_A} the condition that the embeddings are spacelike implies that $0<|\gamma_1|,\, |\gamma_2|<2\pi$.
The gist of the argument is showing that the series of positive terms
\begin{equation}\label{series1}
\sum_{j=2}^\infty \sum_{k=1}^{j-1}\frac{k(j-k)}{j^4}f_0^2\left(\frac{k}{j}\right)\,,\quad\sum_{j=2}^\infty \sum_{k=1}^{j-1}\frac{k(j-k)}{j^4}f_\pi^2\left(\frac{k}{j}\right)\,,
\end{equation}
diverge, whereas
\begin{equation}\label{series2}
\sum_{j=2}^\infty \sum_{k=1}^{j-1}\frac{k(j-k)}{j^4} f_0\left(\frac{k}{j}\right)f_\pi\left(\frac{k}{j}\right)e^{ik(\gamma_1-\gamma_2)+i\gamma_2j}
\end{equation}
converges. Under these circumstances, the only way to guarantee the convergence of \eqref{seriesimple} is to have $V_0=V_\pi=0$ as, otherwise, \eqref{series2} cannot compensate the other two, positive and divergent terms involving \eqref{series1}.
As we have shown in appendix \ref{appendix_A} we have $f_0, f_\pi\in C^\infty[0,1]$ and, hence, both functions are bounded above and below in the closed interval $[0,1]$. This immediately implies the existence of constants $M_0>0$ and $M_\pi>0$ such that
\[\sum_{j=2}^\infty \sum_{k=1}^{j-1}\frac{k(j-k)}{j^4}f_0^2\left(\frac{k}{j}\right)>M_0\sum_{j=2}^\infty \sum_{k=1}^{j-1}\frac{k(j-k)}{j^4} \,,\quad\sum_{j=2}^\infty \sum_{k=1}^{j-1}\frac{k(j-k)}{j^4}f_\pi^2\left(\frac{k}{j}\right)>M_\pi\sum_{j=2}^\infty \sum_{k=1}^{j-1}\frac{k(j-k)}{j^4}\,.\]
Now, as
\[
\sum_{j=2}^\infty\sum_{k=1}^{j-1}\frac{k(j-k)}{j^4}=\frac{1}{6}\sum_{j=2}^\infty\frac{j^3-j}{j^4}
\]
diverges we conclude that the series \eqref{series1} diverge too.
Let us consider now the series \eqref{series2} and write it as
\[
\sum_{j=2}^\infty \frac{e^{ij\gamma_2}}{j^4}\sum_{k=1}^{j-1}k(j-k)e^{ik(\gamma_1-\gamma_2)}h\left(\frac{k}{j}\right)\,,
\]
with $h=f_0f_{\pi}$. The preceding series has the form
\[
\sum_{j=2}^\infty e^{i\gamma_2 j} A_j\,,
\]
with
\[
A_j=\frac{1}{j^4}\sum_{k=1}^{j-1}k(j-k)e^{i(\gamma_1-\gamma_2)k}h\left(\frac{k}{j}\right)\,.
\]
We use now Cauchy's convergence criterion. To this end we consider the sums
\[
\left|\sum_{j=n}^{n+p} e^{i\gamma_2 j}A_j\right|
\]
with $n\,,p\in\mathbb{N}$. Let us define
\[
S_k:=\sum_{\ell=0}^k e^{i\gamma_2\ell}\,,
\]
it is then straightforward to show that $|S_k|\leq 1/|\sin\frac{\gamma_2}{2}|$ (remember that $0<|\gamma_2|< 2\pi$). We then have
\begin{align}
& \left|\sum_{j=n}^{n+p} e^{i\gamma_2 j}A_j\right|=\left|\sum_{j=n}^{n+p} (S_j-S_{j-1})A_j\right|=\left|\sum_{j=n}^{n+p}S_jA_j-\sum_{j=n-1}^{n+p-1}S_jA_{j+1}\right|\label{bound1} \\
=&\left|S_{n+p}A_{n+p}-S_{n-1}A_n+\sum_{j=n}^{n+p-1}S_j(A_j-A_{j+1}) \right|\leq \frac{1}{|\sin\frac{\gamma_2}{2}|}\left(|A_{n+p}|+|A_n|+\sum_{j=n}^{n+p-1}|A_j-A_{j+1}|\right)\,.\nonumber
\end{align}
In the following we show how to find bounds for the terms appearing in the last term of \eqref{bound1}. We start with the following one for $|A_j|$
\begin{equation}\label{bound2}
|A_j|\leq\frac{1}{j^4}\sum_{k=1}^{j-1}k(j-k)\left|h\left(\frac{k}{j}\right)\right|\leq\frac{M}{j^4}\sum_{k=1}^{j-1}k(j-k)=\frac{M}{6}\frac{j^2-1}{j^3}\leq\frac{M}{6j}\,,
\end{equation}
where $M$ is an upper bound for $|h|$ in $[0,1]$. From \eqref{bound2} we immediately conclude that, as $n\rightarrow\infty$, we have
\begin{equation}\label{bound3}
|A_{n+p}|=O\left(\frac{1}{n}\right)\,, |A_n|=O\left(\frac{1}{n}\right)\,.
\end{equation}
Let us study now $\sum_{j=n}^{n+p-1}|A_{j+1}-A_j|$. In order to do this we consider
\begin{align*}
\left|A_{j+1}-A_j\right|&=\left|\frac{1}{(j+1)^4}\sum_{k=1}^jk(j+1-k)e^{i\alpha k}h\left(\frac{k}{j+1}\right)-\frac{1}{j^4}\sum_{k=1}^{j-1}k(j-k)e^{i\alpha k}h\left(\frac{k}{j}\right)\right|\\
&=\left|\frac{j}{(j+1)^4}e^{i\alpha j}h\left(\frac{j}{j+1}\right)+\sum_{k=1}^{j-1} \left( \frac{k(j+1-k)}{(j+1)^4}h\left(\frac{k}{j+1}\right)-\frac{k(j-k)}{j^4}h\left(\frac{k}{j}\right) \right)e^{i\alpha k} \right|\\
&\leq\frac{M}{j^3}+\sum_{k=1}^{j-1}\left|\frac{k(j+1-k)}{(j+1)^4}h\left(\frac{k}{j+1}\right)-\frac{k(j-k)}{j^4}h\left(\frac{k}{j}\right)\right|\,.
\end{align*}
(here and in the following $\alpha:=\gamma_1-\gamma_2$). Now, by using the mean value theorem (remember that $h\in C^\infty([0,1])$), we have
\[
h\left(\frac{k}{j}\right)=h\left(\frac{k}{j}-\frac{k}{j+1}+\frac{k}{j+1}\right)=h\left(\frac{k}{j+1}+\frac{k}{j(j+1)}\right)=h\left(\frac{k}{j+1}\right)+h'(\sigma_{jk})\frac{k}{j(j+1)}\,,
\]
with $\sigma_{jk}\in(0,1)$, so that
\begin{align}
\left|A_{j+1}-A_j\right|\leq&\frac{M}{j^3}+\sum_{k=1}^{j-1}\left(\left|h\left(\frac{k}{j+1}\right)\left(\frac{k(j+1-k)}{(j+1)^4}-\frac{k(j-k)}{j^4}\right)\right|+|h'(\sigma_{jk})|\frac{k^2(j-k)}{j^5(j+1)}\right)\nonumber\\
\leq &\frac{M}{j^3}+M\sum_{k=1}^{j-1}\left|\frac{k(j+1-k)}{(j+1)^4}-\frac{k(j-k)}{j^4}\right|+\tilde{M}\sum_{k=1}^{j-1}\frac{k^2(j-k)}{j^5(j+1)}\leq\frac{\tilde{N}}{j^2}\,.\label{inequality}
\end{align}
where $\tilde{M}$ is an upper bound of $|h'|$ in $[0,1]$ and $\tilde{N}$ is a positive constant. We have now
\[
\sum_{j=n}^{n+p-1}|A_j-A_{j+1}|\leq\sum_{j=n}^{\infty}|A_j-A_{j+1}|=O\left(\frac{1}{n}\right)\,.
\]
Finally, plugging this and \eqref{bound3} in \eqref{bound1} we see that as $n\rightarrow\infty$
\[
\left|\sum_{j=n}^{n+p} e^{i\gamma_2 j}A_j\right|=O\left(\frac{1}{n}\right)\,,
\]
and, hence,
\[
\sum_{j=n}^\infty e^{i\gamma_2 j}A_j
\]
converges.
\end{appendices}
|
1,116,691,500,719 | arxiv | \section{Introduction}\label{sec:intro}
We study certain \emph{separation probabilities} for products of permutations. The archetypal question can be stated as follows: \emph{in the symmetric group $\mathfrak{S}_n$, what is the probability that the elements $1,2,\ldots,k$ are in distinct cycles of the product of two $n$-cycles chosen uniformly randomly?} The answer is surprisingly elegant: the probability is $\frac{1}{k!}$ if $n-k$ is odd and $\frac{1}{k!} + \frac{2}{(k-2)!(n-k+1)(n+k)}$ if $n-k$ is even. This result was originally conjectured by B\'ona \cite{MBRF} for $k=2$ and $n$ odd. Subsequently, Du and Stanley proved it for all $k$ and proposed additional conjectures \cite{DS}. The goal of this paper is to prove these conjectures, and establish generalizations of the above result. Our approach is different from the one used in \cite{DS}.\\
Let us define a larger class of problems.
Given a tuple $A=(A_1,\ldots,A_k)$ of $k$ disjoint non-empty subsets of $\{1,\ldots,n\}$, we say that a permutation $\pi$ is \emph{$A$-separated} if no cycle of $\pi$ contains elements of more than one of the subsets $A_i$.
Now, given two integer partitions $\la,\mu$ of $n$, one can wonder about the probability $P_{\la,\mu}(A)$ that the product of two uniformly random permutations of cycle type $\la$ and $\mu$ is $A$-separated. The example presented above corresponds to $A=(\{1\},\ldots,\{k\})$ and $\la=\mu=(n)$. Clearly, the separation probabilities $P_{\la,\mu}(A)$ only depend on $A$ through the size of the subsets $\#A_1,\ldots,\#A_k$, and we shall denote $\psig_{\la,\mu}^{\al}:=P_{\la,\mu}(A)$, where $\alpha=(\#A_1,\ldots,\#A_k)$ is a composition (of size $m\leq n$). Note also that $\psig_{\la,\mu}^{\al}=\psig_{\la,\mu}^{\al'}$ whenever the composition $\al'$ is a permutation of the composition $\al$. Below, we focus on the case $\mu=(n)$ and we further denote $\psig_{\la}^{\al}:=\psig_{\la,(n)}^{\al}$.\\
In this paper, we first express the separation probabilities $\psig_{\la}^{\al}$ as some coefficients in an explicit generating function. Using this expression we then prove the following symmetry property: if $\alpha=(\al_1,\ldots,\al_k)$ and $\beta=(\be_1,\ldots,\be_k)$ are compositions of the same size $m\leq n$ and of the same length $k$, then
\begin{equation} \label{eq:probsep}
\frac{\psig_{\lambda}^\al}{\prod_{i=1}^k \alpha_i!}=\frac{\psig_{\lambda}^{\beta}}{\prod_{i=1}^k \beta_i!}.
\end{equation}
Moreover, for certain partitions $\la$ (including the cases $\la=(n)$ and $\la=2^N$) we obtain explicit expressions for the probabilities $\psig_{\la}^{\al}$ for certain partitions $\la$. For instance, the separation probability $\psig_{(n)}^\al$ for the product of two $n$-cycles is found to be
\begin{equation} \label{eq:probsigma}
\psig_{(n)}^\al = \frac{(n-m)!\prod_{i=1}^k \alpha_i!}{(n+k)(n-1)!}\left(\frac{(-1)^{n-m}\binom{n-1}{k-2}}{\binom{n+m}{m-k}}+ \sum_{r=0}^{m-k}\frac{(-1)^r\binom{m-k}{r} \binom{n+r+1}{m}}{\binom{n+k+r}{r}}\right).
\end{equation}
This includes the case $\al=1^k$ proved by Du and Stanley \cite{DS}.\\
Our general expression for the separation probabilities $\psig_{\la}^{\al}$ is derived using a formula obtained in \cite{MV} about \emph{colored} factorizations of the $n$-cycle into two permutations. This formula displays a symmetry which turns out to be of crucial importance for our method. Our approach can in fact be made mostly bijective as explained in Section~\ref{sec:maps}. Indeed, the formula obtained in \cite{MV} builds on a bijection established in \cite{SV}. An alternative bijective proof was given in \cite{BM1} and in Section~\ref{sec:maps} we explain how to concatenate this bijective proof with the constructions of the present paper.
\medskip
\noindent {\bf Outline.}
In Section~\ref{sec:strategy} we present our strategy for computing the separation probabilities. This involves counting certain colored factorizations of the $n$-cycle. We then gather our main results in Section~\ref{sec:results}. In particular we prove the symmetry property \eqref{eq:probsep} and obtain formulas for the separation probabilities $\psig_{\la}^{\al}$ for certain partitions $\la$ including $\la=(n)$ or when $\la=2^N$.
In Section~\ref{sec:fixedpoints}, we give formulas relating the separation probabilities $\psig_{\la}^{\al}$ and $\psig_{\la'}^{\al}$ when $\la'$ is a partition obtained from another partition $\la$ by adding some parts of size 1. In Section~\ref{sec:maps}, we indicate how our proofs could be made bijective. We gather a few additional remarks in Section~\ref{sec:conclusion}. \\
\noindent {\bf Notation.}
We denote $[n]:=\{1,2,\ldots,n\}$. We denote by $\#S$ the cardinality of a set $S$.
A \emph{composition} of an integer $n$ is a tuple $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_k)$ of positive integer summing to $n$. We then say that $\al$ has \emph{size} $n$ and \emph{length} $\ell(\al)=k$. An \emph{integer partition} is a composition such that the \emph{parts} $\al_i$ are in weakly decreasing order. We use the notation $\lambda \models n$ (resp. $\lambda \vdash n$) to indicate that $\la$ is a composition (resp. integer partition) of $n$. We sometime write integer partitions in multiset notation: writing $\la=1^{n_1},2^{n_2},\ldots,j^{n_j}$ means that $\la$ has $n_i$ parts equal to $i$.
We denote by $\mathfrak{S}_n$ the symmetric group on $[n]$. Given a partition $\lambda$ of $n$, we denote by $\mathcal{C}_{\lambda}$ the set of permutations in $\mathfrak{S}_n$ with cycle type $\lambda$. It is well known that $\#\mathcal{C}_{\lambda}=n!/z_{\lambda}$ where $z_{\lambda}=\prod_i i^{n_i(\lambda)} n_i(\lambda)!$ and $n_i(\lambda)$ is the number of parts equal to $i$ in $\lambda$.
We shall consider symmetric functions in an infinite number of variables $\xx=\{x_1,x_2,\ldots\}$. For any sequence of nonnegative integers, $\al=(\al_1,\al_2,\ldots,\al_k)$ we denote $\xx^{\al}:=x_1^{\al_1}x_2^{\al_2}\ldots x_k^{\al_k}$. We denote by $[\xx^{\al}]f(\xx)$ the coefficient of this monomial in a series $f(\xx)$.
For an integer partition $\la=(\la_1,\ldots,\la_k)$ we denote by $p_{\lambda}(\xx)$ and $m_{\lambda}(\xx)$ respectively the \emph{power symmetric function} and \emph{monomial symmetric function} indexed by $\lambda$ (see e.g. \cite{EC2}). That is, $p_{\lambda}(\xx)=\prod_{i=1}^{\ell(\lambda)} p_{\lambda_i}(\xx)$ where $p_k(\xx)=\sum_{i\geq 1} x_i^k$, and $m_{\lambda}(\xx)=\sum_{\alpha} \xx^\al$ where the sum is over all the distinct sequences $\al$ whose positive parts are $\{\la_1,\la_2,\ldots,\la_k\}$ (in any order). Recall that the power symmetric functions form a basis of the ring of symmetric functions. For a symmetric function $f(\xx)$ we denote by $[p_{\lambda}(\xx)]f(\xx)$ the coefficient of $p_{\lambda}(\xx)$ of the decomposition of $f(\xx)$ in this basis.
\section{Strategy} \label{sec:strategy}
In this section, we first translate the problem of determining the separation probabilities $\psig_\la^\al$ into the problem of enumerating certain sets $\mS_\la^\al$. Then, we introduce a symmetric function $\GG^\al_n(\xx,t)$ whose coefficients in one basis are the cardinalities $\#\mS_\la^\al$, while the coefficients in another basis count certain ``colored'' separated factorizations of the permutation $(1,\ldots,n)$. Lastly, we give exact counting formulas for these colored separated factorizations. Our main results will follow as corollaries in Section~\ref{sec:results}.\\
For a composition $\al=(\al_1,\ldots,\al_k)$ of size $m\leq n$, we denote by $\mA_n^\al$ the set of tuples $A=(A_1,\ldots,A_k)$ of pairwise disjoint subsets of $[n]$ with $\#A_i=\al_i$ for all $i$ in $[k]$. Observe that $\#\mA_n^\al=\binom{n}{\alpha_1,\alpha_2,\ldots,\alpha_k,n-m}$.\\
Now, recall from the introduction that $\psig_{\la}^{\al}$
is the probability for the product of a uniformly random permutation of cycle type $\la$ with a \emph{uniformly random} $n$-cycle to be $A$-separated for a \emph{fixed} tuple $A$ in $\mA_n^\al$. Alternatively, it can be defined as the probability for the product of a uniformly random permutation of cycle type $\la$ with a \emph{fixed} $n$-cycle to be $A$-separated for a \emph{uniformly random} tuple $A$ in $\mA_n^\al$ (since the only property that matters is that the elements in $A$ are randomly distributed in the $n$-cycle).
\begin{definition}
For an integer partition $\la$ of $n$, and a composition $\al$ of $m\leq n$, we denote by $\mS_\la^\al$ the set of pairs $(\pi,A)$, where $\pi$ is a permutation in $\mathcal{C}_\la$ and $A$ is a tuple in $\mA_n^\al$ such that the product $\pi\circ (1,2,...,n)$ is $A$-separated.
\end{definition}\label{def:setS}
From the above discussion we obtain for any composition $\al=(\al_1,\ldots,\al_k)$ of size $m$,
\begin{equation}\label{eq:probtoenumsig}
\psig_{\lambda}^\al = \frac{\#\mS_\la^\al}{\binom{n}{\alpha_1,\alpha_2,\ldots,\alpha_k,n-m}\#\mathcal{C}_{\lambda}}.
\end{equation}\\
Enumerating the sets $\mS_\la^\al$ directly seems rather challenging. However, we will show below how to enumerate a related class of ``colored'' separated permutations denoted by $\mT^\al_{\ga}(r)$. We define a \emph{cycle coloring} of a permutation $\pi\in\mathfrak{S}_n$ in $[q]$ to be a mapping $c$ from $[n]$ to $[q]$ such that if $i,j\in[n]$ belong to the same cycle of $\pi$ then $c(i)=c(j)$. We think of $[q]$ as the set of \emph{colors}, and $c^{-1}(i)$ as set of \emph{elements colored $i$}.
\begin{definition}\label{def:setT}
Let $\ga=(\ga_1,\ldots,\ga_\ell)$ be a composition of size $n$ and length $\ell$, and let $\alpha=(\al_1,\ldots,\al_k)$ be a composition of size $m\leq n$ and length $k$. For a nonnegative integer $r$ we define $\mT^\al_{\ga}(r)$ as the set of quadruples $(\pi,A,c_1,c_2)$, where $\pi$ is a permutation of $[n]$, $A=(A_1,\ldots,A_k)$ is in $\mA^\al_n$, and
\begin{compactitem}
\item[(i)] $c_1$ is a cycle coloring of $\pi$ in $[\ell]$ such that there are $\ga_i$ element colored $i$ for all $i$ in $[\ell]$,
\item[(ii)] $c_2$ is a cycle coloring of the product $\pi\circ(1,2,\ldots,n)$ in $[k+r]$ such that every color in $[k+r]$ is used and for all $i$ in $[k]$ the elements in the subset $A_i$ are colored $i$.
\end{compactitem}
\end{definition}
Note that condition (ii) in Definition~\ref{def:setT} and the definition of cycle coloring implies that the product $\pi\circ(1,2,\ldots,n)$ is $A$-separated.\\
In order to relate the cardinalities of the sets $\mS^\al_{\lambda}$ and $\mT^\al_{\ga}(r)$, it is convenient to use symmetric functions (in the variables $\xx=\{x_1,x_2,x_3,\ldots\}$).
Namely, given a composition $\al$ of $m\leq n$, we define
\[
\GG^\al_n(\xx,t) := \sum_{\lambda \vdash n} p_{\lambda}(\xx) \sum_{ (\pi,A) \in \mS_\la^\al} t^{\uc(\pi,A)},
\]
where the outer sum runs over all the integer partitions of $n$, and $\uc(\pi,A)$ is the number of cycles of the product $\pi\circ(1,2,\ldots,n)$ containing none of the elements in $A$. Recall that the power symmetric functions $p_{\lambda}(\xx)$ form a basis of the ring of symmetric functions, so that the contribution of a partition $\la$ to $\GG^\al_n(\xx,t)$ can be recovered by extracting the coefficient of $p_{\lambda}(\xx)$ in this basis:
\begin{equation} \label{genseriestosig}
\nsig^\al_{\lambda} = [p_{\lambda}(\xx)] \,\, \GG^\al_n(\xx,1).
\end{equation}
As we prove now, the sets $\mT^\al_{\ga}(r)$ are related to the coefficients of $\GG^\al_n(\xx,t)$ in the basis of monomial symmetric functions.
\begin{proposition}\label{prop:generating-function}
If $\al$ is a composition of length $k$, then
\begin{equation} \label{eq:colsepmap}
\GG_n^\al(\xx,t+k)=\sum_{\ga \vdash n} m_{\ga}(\xx) \sum_{r\geq 0}\binom{t}{r} \,\#\mT^\al_{\gamma}(r),
\end{equation}
where the outer sum is over all integer partitions of $n$, and $\displaystyle \binom{t}{r}:=\frac{t(t-1)\cdots (t-r+1)}{r!}$.
\end{proposition}
\begin{proof} Since both sides of \eqref{eq:colsepmap} are polynomial in $t$ and symmetric function in $\xx$ it suffices to show that for any nonnegative integer $t$ and any partition $\ga$ the coefficient of $\xx^\ga$ is the same on both sides of \eqref{eq:colsepmap}.
We first determine the coefficient $\displaystyle [\xx^{\ga}]\GG_n^\al(\xx,t+k)$ when $t$ is a nonnegative integer.
Let $\la$ be a partition, and $\pi$ be a permutation of cycle type $\la$. Then the symmetric function $p_{\la}(\xx)$ can be interpreted as the generating function of the cycle colorings of $\pi$, that is, for any sequence $\ga=(\ga_1,\ldots,\ga_{\ell})$ of nonnegative integers, the coefficient $[\xx^{\ga}]p_{\la}(\xx)$ is the number of cycle colorings of $\pi$ such that $\ga_i$ elements are colored $i$, for all $i> 0$. Moreover, if $\pi$ is $A$-separated for a certain tuple $A=(A_1,\ldots,A_k)$ in $\mA^\al_n$, then $(t+k)^{\uc(S,\pi)}$ represents the number of cycle colorings of the permutation $\pi\circ(1,2,\ldots,n)$ in $[k+t]$ (not necessarily using every color) such that for all $i\in[k]$ the elements in the subset $A_i$ are colored $i$. Therefore, for a partition $\ga$ and a nonnegative integer $t$, the coefficient $[\xx^{\ga}]\GG_n^\al(\xx,t+k)$ counts the number of quadruples $(\pi,A,c_1,c_2)$, where $\pi,A,c_1,c_2$ are as in the definition of $\mT^\al_{\ga}(t)$ except that $c_2$ might actually use only a subset of the colors $[k+t]$. Note however that all the colors in $[k]$ will necessarily be used by $c_2$, and that we can partition the quadruples according to the subset of colors used by $c_2$. This gives
$$[\xx^{\ga}]\GG_n^\al(\xx,t+k)=\sum_{ r\geq 0}\binom{t}{r} \,\#\mT^\al_{\gamma}(r).$$
Now extracting the coefficient of $\xx^{\ga}$ in the right-hand side of~\eqref{eq:colsepmap} gives the same result. This completes the proof.
\end{proof}
In order to obtain an explicit expression for the series $\GG^\al_n(\xx,t)$ it remains to enumerate the sets $\mT^\al_\ga(r)$ which is done below.
\begin{proposition}\label{prop:cardT}
Let $r$ be a nonnegative integer, let $\al$ be a composition of size $m$ and length $k$, and let $\ga$ be a partition of size $n\geq m$ and length $\ell$. Then the set $\mT^\al_\ga(r)$ specified by Definition~\ref{def:setT} has cardinality
\begin{equation}\label{eq:cardT}
\#\mT^\al_\ga(r)=\frac{n(n-\ell)!(n-k-r)!}{(n-k-\ell-r+1)!}\, \binom{n+k-1}{n-m-r},
\end{equation}
if $n-k-\ell-r+1\geq 0$, and 0 otherwise.
\end{proposition}
The rest of this section is devoted to the proof of Proposition~\eqref{prop:cardT}.
In order to count the quadruples $(\pi,A,c_1,c_2)$ satisfying Definition~\ref{def:setT}, we shall start by choosing $\pi,c_1,c_2$ before choosing the tuple $A$. For compositions $\ga=(\ga_1,\ldots,\ga_\ell)$, $\de=(\de_1,\ldots,\de_{\ell'})$ of $n$ we denote by $\mB_{\ga,\de}$ the set of triples $(\pi,c_1,c_2)$, where $\pi$ is a permutation of $[n]$, $c_1$ is a cycle coloring of $\pi$ such that $\ga_i$ elements are colored $i$ for all $i\in[\ell]$, and $c_2$ is a cycle coloring of the permutation $\pi\circ (1,2,\ldots,n)$ such that $\de_i$ elements are colored $i$ for all $i\in[\ell']$. The problem of counting such sets was first considered by Jackson \cite{J} who actually enumerated the union $\mathcal{B}^n_{i,j}:=\displaystyle \bigcup_{\ga,\de\models n,~\ell(\ga)=i,~\ell(\de)=j}\mB_{\ga,\de}$ using representation theory. It was later proved in \cite{MV} that
\begin{equation}\label{eq:colored-factorizations}
\#\mB_{\ga,\de}=\frac{n(n-\ell)!(n-\ell')!}{(n-\ell-\ell'+1)!},
\end{equation}
if $n-\ell-\ell'+1\geq 0$, and 0 otherwise.
The proof of~\eqref{eq:colored-factorizations} in \cite{MV} uses a refinement of a bijection designed in \cite{SV} in order to prove Jackson's formula for $\#\mathcal{B}^n_{i,j}$.
Another bijective proof of~\eqref{eq:colored-factorizations} is given in \cite{BM1}, and we shall discuss it further in Section~\ref{sec:maps} (a proof of~\eqref{eq:colored-factorizations} using representation theory can be found in \cite{EV}).\\
One of the striking features of the counting formula~\eqref{eq:colored-factorizations} is that it depends on the compositions $\ga$, $\de$ only through their lengths $\ell$, $\ell'$. This ``symmetry'' will prove particularly handy for enumerating $\mT^\al_\ga(r)$. Let $r$, $\al$, $\ga$ be as in Proposition~\ref{prop:cardT}, and let $\de=(\de_1,\ldots,\de_{k+r})$ be a composition of $n$ of length $k+r$.
We denote by $\mT^\al_{\ga,\de}$ the set of quadruples $(\pi,A,c_1,c_2)$ in $\mT^\al_\ga(r)$ such that the cycle coloring $c_2$ has $\de_i$ elements colored $i$ for all $i$ in $[k+r]$ (equivalently, $(\pi,c_1,c_2)\in\mB_{\ga,\de}$). We also denote $\dd^\al_\de:=\prod_{i=1}^k{\de_i \choose \al_i}$. It is easily seen that for any triple $(\pi,c_1,c_2)\in\mB_{\ga,\de}$, the number $\dd^\al_\de$ counts the tuples $A\in\mA_n^\al$ such that $(\pi,A,c_1,c_2)\in\mT^\al_{\ga,\de}$. Therefore,
$$ \#\mT^\al_\ga(r)=\sum_{\de\models n,~\ell(\de)=k+r}\#\mT^\al_{\ga,\de}=\sum_{\de\models n,~\ell(\de)=k+r}\dd^\al_\de\, \#\mB_{\ga,\de},$$
where the sum is over all the compositions of $n$ of length $k+r$. Using~\eqref{eq:colored-factorizations} then gives
$$ \#\mT^\al_\ga(r)=\frac{n(n-\ell)!(n-k-r)!}{(n-k-\ell-r+1)!}\sum_{\de\models n,~\ell(\de)=k+r}\dd^\al_\de$$
if $n-k-\ell-r+1\geq 0$, and 0 otherwise. In order to complete the proof of Proposition~\ref{prop:cardT}, it only remains to prove the following lemma.
\begin{lemma} \label{lem:nb-markings}
If $\al$ has size $m$ and length $k$, then
$$\sum_{\de\models n,~\ell(\de)=k+r}\dd^\al_\de=\binom{n+k-1}{n-m-r}.$$
\end{lemma}
\begin{proof}
We give a bijective proof illustrated in Figure~\ref{fig:bijmarking}. One can represent a composition $\de=(\de_1,\ldots,\de_{k+r})$ as a sequence of rows of boxes (the $i$th row has $\de_i$ boxes). With this representation, $\dd^\al_\de:=\prod_{i=1}^k{\de_i \choose \al_i}$ is the number of ways of choosing $\al_i$ boxes in the $i$th row of $\delta$ for $i=1,\ldots,k$. Hence $\sum_{\de\models n,~\ell(\de)=k+r}\dd^\al_\de$ counts \emph{$\al$-marked compositions} of size $n$ and length $k+r$, that is, sequences of $k+r$ non-empty rows of boxes with some marked boxes in the first $k$ rows, with a total of $n$ boxes, and $\al_i$ marks in the $i$th row for $i=1,\ldots,k$; see Figure~\ref{fig:bijmarking}.
Now $\al$-marked compositions of size $n$ and length $k+r$ are clearly in bijection (by adding a marked box to each of the rows $1,\ldots,k$, and marking the last box of each of the rows $k+1,\ldots,k+r$) with $\al'$-marked compositions of size $n+k$ and length $k+r$ \emph{such that the last box of each row is marked}, where $\al'=(\al_1+1,\al_2+1,\ldots,\al_k+1,1,1,\ldots,1)$ is a composition of length $k+r$. Lastly, these objects are clearly in bijection (by concatenating all the rows) with sequences of $n+k$ boxes with $m+k+r$ marks, one of which is on the last box. There are $\binom{n+k-1}{n-m-r}$ such sequences, which concludes the proof of Lemma~\ref{lem:nb-markings} and Proposition~\ref{prop:cardT}.
\end{proof}
\fig{width=\linewidth}{bijmarking}{A $(2,1,2)$-marked composition of size $n=12$ and length $5$ and its bijective transformation into a sequence $n+k=15$ boxes with $m+k+r=5+3+2=10$ marks, one of which is on the last box.}
\section{Main results} \label{sec:results}
In this section, we exploit Propositions~\ref{prop:generating-function} and~\ref{prop:cardT} in order to derive our main results. All the results in this section will be consequences of the following theorem.
\begin{theorem}\label{thm:GF}
For any composition $\al$ of $m\leq n$ of length $k$, the generating function $\GG_n^\al(\xx,t+k)$ in the variables $t$ and $\xx=\{x_1,x_2,\ldots\}$ has the following explicit expression in the bases $m_{\la}(\xx)$ and $\binom{t}{r}$:
\begin{equation}\label{eq:GF-explicit}
\GG_n^\al(\xx,t+k) = \sum_{r=0}^{n-m}\binom{t}{r}\binom{n+k-1}{n-m-r}\sum_{\la \vdash n,~ \ell(\la)\leq n-k-r+1} \frac{n(n-\ell(\la))!(n-k-r)!}{(n-k-r-\ell(\la)+1)!} ~ m_{\la}(\xx).
\end{equation}
Moreover, for any partition $\la$ of $n$, one has $\displaystyle \nsig_\la^\al=[p_\la(\xx)]\GG_n^\al(\xx,1)$ and $\displaystyle \psig_{\la}^\al=\frac{\nsig_\la^\al}{{n \choose \al_1,\al_2,\ldots,\al_k,n-m}\#\mathcal{C}_\la}$.
\end{theorem}
Theorem~\ref{thm:GF} is the direct consequence of Propositions~\ref{prop:generating-function} and~\ref{prop:cardT}. One of the striking features of~\eqref{eq:GF-explicit} is that the expression of $\GG_n^\al(\xx,t+k)$ depends on $\al$ only through its size and length.
This ``symmetry property'' then obviously also holds for $\nsig_{\la}^\al=[p_{\la}(\xx)]\GG_n^\al(\xx,1)$, and translates into the formula \eqref{eq:probsep} for separation probabilities as stated below.
\begin{corollary}\label{cor:sepsym}
Let $\la$ be a partition of $n$, and let $\al=(\al_1,\ldots,\al_k)$ and $\be=(\be_1,\ldots,\be_k)$ be compositions of the same size $m$ and length $k$. Then,
\begin{equation} \label{eq:sepsym}
\nsig_{\la}^\al=\nsig_{\la}^{\beta},
\end{equation}
or equivalently, in terms of separation probabilities, $\displaystyle ~\frac{\psig_{\la}^\al}{\prod_{i=1}^k \alpha_i!}=\frac{\psig_{\la}^{\beta}}{\prod_{i=1}^k \beta_i!}$.
\end{corollary}
We now derive explicit formulas for the separation probabilities for the product of a uniformly random permutation $\pi$, with particular constraints on its cycle type, with a uniformly random $n$-cycle. We focus on two constraints: the case where $\pi$ is required to have $p$ cycles, and the case where $\pi$ is a fixed-point-free involution (for $n$ even).
\subsection{Case when $\pi$ has exactly $p$ cycles}
Let $\mC(n,p)$ denote the set of permutations of $[n]$ having $p$ cycles. Recall that the numbers $c(n,p)=\#\mC(n,p)=[x^p]x(x+1)(x+2)\cdots(x+n-1)$ are called the \emph{signless Stirling numbers of the first kind}. We denote by $\psig^{\al}(n,p)$ the probability that the product of a uniformly random permutation in $\mC(n,p)$ with a uniformly random $n$-cycle is $A$-separated for a given set $A$ in $\mA_n^\al$. By a reasoning similar to the one used in the proof of \eqref{eq:probtoenumsig}, one gets
\begin{equation}\label{eq:probtoenumsigP}
\psig^{\al}(n,p)=\frac{1}{{n \choose \al_1,\al_2,\ldots,\al_k,n-m}c(n,p)}\sum_{\la \vdash n, \ell(\la)=p} \nsig_{\la}^\al.
\end{equation}
We now compute the probabilities $\psig^{\al}(n,p)$ explicitly.
\begin{theorem} \label{thm:casePcycles}
Let $\alpha$ be a composition of $m$ with $k$ parts. Then,
\begin{equation} \label{eq:probsigmaP}
\psig^{\al}(n,p) = \frac{(n-m)! \prod_{i=1}^k \alpha_i!}{c(n,p)}\sum_{r=0}^{n-m} \binom{1-k}{r} \binom{n+k-1}{n-m-r}\frac{c(n-k-r+1,p)}{(n-k-r+1)!},
\end{equation}
where $c(n,p)$ are signless Stirling numbers of the first kind.
\end{theorem}
For instance, Theorem \ref{thm:casePcycles} in the case $m=n$ gives the probability that the cycles of the product of a uniformly random permutation in $\mC(n,p)$ with a uniformly random $n$-cycle refine a given set partition of $[n]$ having blocks of sizes $\al_1,\al_2,\ldots,\al_k$. This probability is found to be
$$
\psig^{\al}(n,p)= \frac{\prod_{i=1}^k \alpha_i!}{c(n,p)}\,\frac{c(n-k+1,p)}{(n-k+1)!}.
$$
We now prove Theorem~\ref{thm:casePcycles}. Via~\eqref{eq:probtoenumsigP}, this amounts to enumerating $\mS^{\al}(n,p):=\bigcup_{\la \vdash n, \ell(\la)=p}\mS_{\la}^\al$, and using Theorem~\ref{thm:GF} one gets
\begin{eqnarray}\label{eq:formsigmaPeq1}
\nsig^\al(n,p)&=&\sum_{\la \vdash n,\ell(\la)=p} [p_{\la}(\xx)] \, \GG_n^\al(\xx,1)\nonumber \\
&=& \sum_{r=0}^{n-m}\binom{1-k}{r}\binom{n+k-1}{n-m-r}\sum_{\ell=1}^{n-k-r+1} \frac{n(n-\ell)!(n-k-r)!}{(n-k-r-\ell+1)!}\, A(n,p,\ell),
\end{eqnarray}
where $\displaystyle A(n,p,\ell):=\sum_{\mu \vdash n,~ \ell(\mu)=p} [p_{\mu}(\xx)] \sum_{\la \vdash n,~ \ell(\la)=\ell} m_{\la}(\xx)$.
The next lemma gives a formula for $A(n,p,\ell)$.
\begin{lemma}\label{lem:coeff-k-cycles}
For any positive integers $p,\ell\leq n$
\begin{equation}\label{eq:coeff-k-cycles}
\sum_{\mu \vdash n,~ \ell(\mu)=p} [p_{\mu}(\xx)] \sum_{\la \vdash n,~ \ell(\la)=\ell} m_{\la}(\xx) ~=~ {n-1 \choose \ell-1}\frac{(-1)^{\ell-p} c(\ell,p)}{\ell!},
\end{equation}
where $c(a,b)$ are the signless Stirling numbers of the first kind.
\end{lemma}
\begin{proof}
For this proof we use the principal specialization of symmetric functions, that is, their evaluation at $\xx=1^a:=\{1,1,\ldots,1,0,0\ldots\}$ ($a$ ones). Since $p_{\ga}(1^a)=a^{\ell(\ga)}$ for any positive integer $a$, one gets
$$\sum_{\la \vdash n,~ \ell(\la)=\ell} m_{\la}(1^a)=\sum_{p=1}^n a^p \sum_{\mu \vdash n,~ \ell(\mu)=p} [p_{\mu}(\xx)] \sum_{\la \vdash n,~ \ell(\la)=\ell} m_{\la}(\xx).
$$
The right-hand side of the previous equation is a polynomial in $a$, and by extracting the coefficient of $a^p$ one gets
\begin{equation}\label{eq:extract-coeff}
\sum_{\mu \vdash n,~ \ell(\mu)=p} [p_{\mu}(\xx)] \sum_{\la \vdash n,~ \ell(\la)=\ell} m_{\la}(\xx) =[a^p]\sum_{\la \vdash n,~ \ell(\la)=\ell} m_{\la}(1^a).\nonumber
\end{equation}
Now, for any partition $\la$, $m_{\la}(1^a)$ counts the $a$-tuples of nonnegative integers such that the positive ones are the same as the parts of $\la$ (in some order). Hence $\displaystyle \sum_{\la \vdash n,~ \ell(\la)=\ell} m_{\la}(1^a)$ counts the $a$-tuples of nonnegative integers with $\ell$ positive ones summing to $n$. This gives,
\begin{equation}\label{eq:extract-coeff2}
\sum_{\la \vdash n,~ \ell(\la)=\ell} m_{\la}(1^a)={n-1 \choose \ell-1}{a \choose \ell}.\nonumber
\end{equation}
Extracting the coefficient of $a^p$ gives~\eqref{eq:coeff-k-cycles} since $\displaystyle [a^p]{a \choose \ell}=\frac{(-1)^{\ell-p}\,c(\ell,p)}{\ell!}$.
\end{proof}
Using Lemma~\ref{lem:coeff-k-cycles} in~\eqref{eq:formsigmaPeq1} gives
\begin{equation} \label{eq:formsigmaPeq2}
\nsig^\al(n,p) = n! \sum_{r\geq 0}^{n-m} \binom{1-k}{r} \binom{n+k-1}{n-m-r} \sum_{\ell=1}^{n-k-r+1} \binom{n-k-r}{\ell-1}\frac{(-1)^{\ell-p}c(\ell,p)}{\ell!},
\end{equation}
which we simplify using the following lemma.
\begin{lemma}\label{lem:simplifyP}
For any nonnegative integer $a$, $\displaystyle \sum_{q=0}^{a} \binom{a}{q}\frac{(-1)^{q+1-p}\,c(q+1,p)}{(q+1)!} = \frac{c(a+1,p)}{(a+1)!}$.
\end{lemma}
\begin{proof}
The left-hand side equals $[x^p] \sum_{q=0}^a \binom{a}{q}\binom{x}{q+1}$. Using the Chu-Vandermonde identity this equals $[x^p] \binom{x+a}{a+1}$ which is precisely the right-hand side.
\end{proof}
Using Lemma~\ref{lem:simplifyP} in~\eqref{eq:formsigmaPeq2} gives
\begin{equation} \label{eq:formsigmaPeq3}
\nsig^\al(n,p)=n! \sum_{r=0}^{n-m} \binom{1-k}{r} \binom{n+k-1}{n-m-r}\frac{c(n-k-r+1,p)}{(n-k-r+1)!} ,
\end{equation}
which is equivalent to~\eqref{eq:probsigmaP} via~\eqref{eq:probtoenumsig}. This completes the proof of Theorem~\ref{thm:casePcycles}. \hfill $\square$\\
In the case $p=1$, the expression \eqref{eq:probsigmaP} for the probability $\psig^{\al}(1)=\psig^{\al}_{(n)}$ can be written as a sum of $m-k$ terms instead. We state this below.
\begin{corollary}\label{cor:alternative-sum}
Let $\alpha$ be a composition of $m$ with $k$ parts. Then the separation probabilities $\psig_{(n)}^\al$ (separation for the product of two uniformly random $n$-cycles) are
\begin{equation} \nonumber
\psig_{(n)}^\al= \frac{(n-m)!\prod_{i=1}^k \alpha_i!}{(n+k)(n-1)!}\left(\frac{(-1)^{n-m}\binom{n-1}{k-2}}{\binom{n+m}{m-k}}+ \sum_{r=0}^{m-k}\frac{(-1)^r\binom{m-k}{r} \binom{n+r+1}{m}}{\binom{n+k+r}{r}}\right).
\end{equation}
\end{corollary}
The equation in Corollary~\ref{cor:alternative-sum}, already stated in the introduction, is particularly simple when $m-k$ is small. For $\al=1^k$ (i.e. $m=k$) one gets the result stated at the beginning of this paper:
\begin{equation}
\label{probkones}
\psig^{1^k}_{(n)}=\begin{cases}
\frac{1}{k!} &\text{ if } n-k \text{ odd,}\\
\frac{1}{k!} + \frac{2}{(k-2)!(n-k+1)(n+k)} &\text{ if } n-k \text{ even.}
\end{cases}
\end{equation}\\
In order to prove Corollary~\ref{cor:alternative-sum} we start with the expression obtained by setting $p=1$ in \eqref{eq:probsigmaP}:
\begin{eqnarray}\label{eq:formsigmaPeq4}
\psig^\al_{(n)}&=&\frac{(n-m)!\prod_{i=1}^k \alpha_i!}{(n-1)!}\sum_{r=0}^{n-m} \binom{1-k}{r} \frac{1}{n-k-r+1} \binom{n+k-1}{n-m-r}\nonumber \\
&=&\frac{(n-m)!\prod_{i=1}^k \alpha_i!}{(n-1)!}[x^{n-m}](1+x)^{1-k}\sum_{r=0}^{n+k-1}\frac{x^r}{r+m-k+1}{n+k-1 \choose r}.
\end{eqnarray}
We now use the following polynomial identity.
\begin{lemma}\label{lem:change-summation}
For nonnegative integers $a,b$, one has the following identity between polynomials in~$x$:
\begin{equation}\label{eq:change-summation}
\sum_{i=0}^a\frac{x^i}{i+b+1}{a \choose i}=\frac{1}{(a+1)}
\left( \frac{1}{{a+b+1 \choose b}(-x)^{b+1}}- \sum_{i=0}^b\frac{{b \choose i}(x+1)^{a+i+1}}{{a+i+1 \choose i}(-x)^{i+1}}\right).
\end{equation}
\end{lemma}
\begin{proof}
It is easy to see that the left-hand side of~\eqref{eq:change-summation} is equal to $\frac{1}{x^{b+1}}\int_{0}^x (1+t)^at^bdt$. Now this integral can be computed via integration by parts. By a simple induction on $b$, this gives the right-hand side of~\eqref{eq:change-summation}.
\end{proof}
Now using~\eqref{eq:change-summation} in \eqref{eq:formsigmaPeq4}, with $a=n+k-1$ and $b=m-k$, gives
\begin{eqnarray}
\psig^\al_{(n)} &=& \frac{(n-m)!\prod_{i=1}^k \alpha_i!}{(n+k)(n-1)!} [x^{n-m}] \left( \frac{(1+x)^{1-k}}{\binom{n+m}{m-k}(-x)^{m-k+1} }-\sum_{r=0}^{m-k} \frac{\binom{m-k}{r} (1+x)^{n+r+1}}{\binom{n+k+r}{r} (-x)^{r+1}}\right)\nonumber\\
&= &\frac{(n-m)!\prod_{i=1}^k \alpha_i!}{(n+k)(n-1)!}\left(\frac{(-1)^{n-m}\binom{n-1}{k-2}}{\binom{n+m}{m-k}}+ \sum_{r=0}^{m-k}\frac{(-1)^r\binom{m-k}{r} \binom{n+r+1}{m}}{\binom{n+k+r}{r}}\right).\nonumber
\end{eqnarray}
This completes the proof of Corollary~\ref{cor:alternative-sum}. \hfill $\square$
\subsection{Case when $\pi$ is a fixed-point-free involution}\label{subsec:involution}
Given a composition $\alpha$ of $m\leq 2N$ with $k$ parts, we define
\[
H_{N}^{\alpha}(t):= \sum_{(\pi,A) \in \mS_{2^N}^{\alpha}} t^{\uc(\pi,A)},
\]
where $\uc(\pi,A)$ is the number of cycles of the product $\pi \circ (1,2,\ldots,2N)$ containing none of the elements of $A$ and where $\pi$ is a fixed-point-free involution of $[2N]$. Note that $H_{N}^{\alpha}(t) = [p_{2^N}(\xx)]\, G^{\alpha}_{2N}(\xx,t)$. We now give an explicit expression for this series.
\begin{theorem} \label{thm:GF-Maps}
For any composition $\alpha$ of $m\leq 2N$ of length $k$, the generating series $H_{N}^{\alpha}(t+k)$ is given by
\begin{equation}\label{eq:GF-Mapsexplicit}
H_{N}^{\alpha}(t+k) = N\sum_{r=0}^{\min(2N-m,N-k+1)} \binom{t}{r}\binom{2N+k-1}{2N-m-r} 2^{k+r-N} \frac{(2N-k-r)!}{(N-k-r+1)!}.
\end{equation}
Consequently the separation probabilities for the product of a fixed-point-free involution with a $2N$-cycle are given by
\begin{equation}\label{eq:separation-involution}
\psig_{2^N}^\al=\frac{\prod_{i=1}^k\al_i!}{(2N-1)!(2N-1)!!}\sum_{r=0}^{\min(2N-m,N-k+1)} \binom{1-k}{r}\binom{2N+k-1}{2N-m-r} 2^{k+r-N-1} \frac{(2N-k-r)!}{(N-k-r+1)!}.
\end{equation}
\end{theorem}
\begin{rmk}
It is possible to prove Theorem~\ref{thm:GF-Maps} directly using ideas similar to the ones used to prove Theorem~\ref{thm:GF} in Section~\ref{sec:strategy}. This will be explained in more detail in Section~\ref{sec:maps}. In the proof given below, we instead obtain Theorem~\ref{thm:GF-Maps} as a consequence of Theorem~\ref{thm:GF}.
\end{rmk}
The rest of this section is devoted to the proof of Theorem~\ref{thm:GF-Maps}. Since $H_{N}^{\alpha}(t) = [p_{2^N}(\xx)]\, G^{\alpha}_{2N}(\xx,t)$, Theorem~\ref{thm:GF} gives
\begin{eqnarray} \label{eq:fpfinv}
\lefteqn{H_{N}^{\alpha}(t+k)= }\\
& &
\sum_{r=0}^{2N-m} \binom{t}{r} \binom{2N+k-1}{2N-m-r} \sum_{s=0}^{N-k-r+1} \frac{2N(N-s)!(2N-k-r)!}{(N-k-r-s+1)!} [p_{2^N}(\xx)] \sum_{\la \vdash 2N,~ \ell(\la)=N+s}m_{\la}(\xx). \nonumber
\end{eqnarray}
We then use the following result.
\begin{lemma} \label{lemMaps}
For any nonnegative integer $s\leq N$,
\[
[p_{2^N}(\xx)] \sum_{\la \vdash 2N,~\ell(\la)=N+s} m_{\la}(\xx)~=~ \frac{(-1)^s}{2^s s!(N-s)!}.
\]
\end{lemma}
\begin{proof}
For partitions $\la,\mu$ of $n$, we denote $S_{\la,\mu}=[p_\la(\xx)]m_\mu(\xx)$ and $R_{\la,\mu}=[m_\la(\xx)]p_\mu(\xx)$. The matrices $S=(S_{\la,\mu})_{\la,\mu \vdash n}$ and $R=(R_{\la,\mu})_{\la,\mu \vdash n}$ are the transition matrices between the bases $\{p_{\la}\}_{\la, \vdash n}$ and $\{m_{\la}\}_{\la \vdash n}$ of symmetric functions of degree $n$, hence $S=R^{-1}$. Moreover the matrix $R$ is easily seen to be lower triangular in the \emph{dominance order} of partitions, that is, $R_{\la,\mu}=0$ unless $\la_1+\la_2+\cdots +\la_i \leq \mu_1+\mu_2+\cdots+ \mu_i$ for all $i\geq 1$ (\cite[Prop. 7.5.3]{EC2}). Thus the matrix $S=R^{-1}$ is also lower triangular in the dominance order.
Since the only partition of $2N$ of length $N+s$ that is not larger than the partition $2^N$ in the dominance order is $1^{2s}2^{N-s}$, one gets
\begin{equation}\label{eq:extract1}
[p_{2^N}(\xx)] \sum_{\la \vdash 2N,~\ell(\la)=N+s} m_{\la}(\xx) = [p_{2^N}(\xx)] \,m_{1^{2s}2^{N-s}}(\xx).
\end{equation}
To compute this coefficient we use the standard scalar product $\langle\cdot,\cdot\rangle$ on symmetric functions (see e.g. \cite[Sec. 7]{EC2}) defined by $\langle p_{\la},p_{\mu}\rangle = z_{\la}$ if $\la=\mu$ and 0 otherwise, where $z_{\la}$ was defined at the end of Section~\ref{sec:intro}.
From this definition one immediately gets
\begin{equation}\label{eq:extract2}
[p_{2^N}] \,m_{1^{2s}2^{N-s}} = \frac{1}{z_{2^N}}\langle p_{2^N}, m_{1^{2s}2^{N-s}}\rangle= \frac{1}{N!2^N}\langle p_{2^N}, m_{1^{2s}2^{N-s}}\rangle.
\end{equation}
Let $\{h_{\la}\}$ denote the basis of the complete symmetric functions. It is well known that $\langle h_{\la},m_{\mu}\rangle=1$ if $\la=\mu$ and 0 otherwise, therefore $\langle p_{2^N}, m_{1^{2s}2^{N-s}}\rangle=[h_{1^{2s}2^{N-s}}]p_{2^N}$. Lastly, since $p_{2^N}=(p_2)^N$ and $p_2= 2h_2-h_1^2$ one gets
\begin{equation}\label{eq:extract3}
\langle p_{2^N}, m_{1^{2s}2^{N-s}}\rangle=[h_{1^{2s}2^{N-s}}]p_{2^N}= [h_1^{2s}h_2^{N-s}]\, (2h_2-h_1^2)^N=2^{N-s}(-1)^s {N\choose s}.
\end{equation}
Putting together \eqref{eq:extract1}, \eqref{eq:extract2} and \eqref{eq:extract3} completes the proof.
\end{proof}
By Lemma~\ref{lemMaps}, Equation \eqref{eq:fpfinv} becomes
\begin{align*}
H_{N}^{\alpha}(t+k)&= \sum_{r=0}^{2N-m} \binom{t}{r} \binom{2N+k-1}{2N-m-r} \sum_{s=0}^{N-k-r+1} \frac{2N(N-s)!(2N-k-r)!}{(N-k-r-s+1)!} \frac{(-1)^s}{2^s s!(N-s)!}\\
&=2N\sum_{r=0}^{2N-m} \binom{t}{r} \binom{2N+k-1}{2N-m-r}\frac{(2N-k-r)!}{(N-k-r+1)!} \sum_{s=0}^{N-k-r+1} \binom{N-k-r+1}{s}\frac{(-1)^s}{2^s}\\
&=2N\sum_{r=0}^{\min(2N-m,N-k+1)} \binom{t}{r}\binom{2N+k-1}{2N-m-r} \frac{(2N-k-r)!}{(N-k-r+1)!} \frac{1}{2^{N-k-r+1}},
\end{align*}
where the last equality uses the binomial theorem. This completes the proof of Equation \eqref{eq:GF-Mapsexplicit}. Equation \eqref{eq:separation-involution} then immediately follows from the case $t=1-k$ of \eqref{eq:GF-Mapsexplicit} via \eqref{eq:probtoenumsig}.
This completes the proof of Theorem~\ref{thm:GF-Maps}. \hfill$\square$
\section{Adding fixed points to the permutation $\pi$}\label{sec:fixedpoints}
In this section we obtain a relation between the separation probabilities $\psig_{\la}^\al$ and $\psig_{\la'}^\al$, when the partition $\la'$ is obtained from $\la$ by adding some parts of size 1. Our main result is given below.
\begin{theorem}\label{thm:addfixpt}
Let $\la$ be a partition of $n$ with parts of size at least $2$ and let $\la'$ be the partition obtained from $\la$ by adding $r$ parts of size 1. Then for any composition $\al=(\al_1,\ldots,\al_k)$ of $m\leq n+r$ of length $k$,
\begin{equation} \label{eq:addfixpoints}
\#\mS^{\alpha}_{\la'} = \sum_{p=0}^{m-k} \left(\frac{n+p}{n} \binom{n+m+r-p}{n+m} + \frac{m-p}{n} \binom{n+m+r-p-1}{n+m} \right)\binom{m-k}{p} \#\mS^{(m-k-p+1,1^{k-1})}_{\la}.
\end{equation}
Equivalently, in terms of separation probabilities,
\begin{equation}
\label{eq:addfixpointsProb}
\sigma^{\alpha}_{\la'} = \frac{n!}{\binom{n+r}{\alpha_1,\ldots,\al_k,n+r-m}\binom{n+r}{r}}\sum_{p=0}^{m-k} \frac{\left(\frac{n+p}{n} \binom{n+m+r-p}{n+m} + \frac{m-p}{n} \binom{n+m+r-p-1}{n+m} \right) \binom{m-k}{p}}{(n-m+p)!(m-k-p+1)!}\, \sigma^{(m-k-p+1,1^{k-1})}_{\la}.
\end{equation}
\end{theorem}
For instance, when $\al=1^k$ Theorem~\ref{thm:addfixpt} gives
\begin{equation} \nonumber
\sigma^{1^k}_{\la'} = \frac{\binom{n+r-k}{r}}{\binom{n+r}{r}^2}\left( \binom{n+r+k}{n+k} + \frac{k}{n}\binom{n+r+k-1}{n+k} \right)\sigma^{1^k}_{\la}.\\
\end{equation}
The rest of the section is devoted to proving Theorem~\ref{thm:addfixpt}. Observe first that \eqref{eq:addfixpointsProb} is a simple restatement of \eqref{eq:addfixpoints} via~\eqref{eq:probtoenumsig} (using the fact that $\#\mC_{\la'}={n+r\choose n}\#\mC_{\la}$).
Thus it only remains to prove \eqref{eq:addfixpoints}, which amounts to enumerating $\mS_{\la'}^\al$. For this purpose, we will first define a mapping $\Psi$ from $\mS_{\la'}^\al$ to $\hmS_\la^\al$, where $\hmS_\la^\al$ is a set closely related to $\mS_{\la}^\al$. We shall then count the number of preimages of each element in $\hmS_\la^\al$ under the mapping $\Psi$. Roughly speaking, if $(\pi',A)$ is in $\mS_{\la'}^\al$ and the tuple $A=(A_1,\ldots,A_k)$ is thought as ``marking'' some elements in the cycles of the permutation $\om=\pi'\circ (1,2,\ldots,n+r)$, then the mapping $\Psi$ simply consists in removing all the fixed points of $\pi'$ from the cycle structure of $\om$ and transferring their ``marks'' to the element preceding them in the cycle structure of $\om$.
We introduce some notation. A \emph{multisubset} of $[n]$ is a function $M$ which associates to each integer $i\in[n]$ its \emph{multiplicity} $M(i)$ which is a nonnegative integer. The integer $i$ is said to be \emph{in} the multisubset $M$ if $M(i)>0$.
The \emph{size} of $M$ is the sum of multiplicities $\sum_{i=1}^nM(i)$. For a composition $\al=(\al_1,\ldots,\al_k)$, we denote by $\hat{\mA_n^\al}$ the set of tuples $(M_1,\ldots,M_k)$ of \emph{disjoint} multisubsets of $[n]$ (i.e., no element $i\in[n]$ is in more than one multisubset) such that the multisubset $M_j$ has size $\al_j$ for all $j\in[k]$. For $M=(M_1,\ldots,M_k)$ in $\hmA_n^\al$ we say that a permutation $\pi$ of $[n]$ is $M$-separated if no cycle of $\pi$ contains elements of more than one of the multisubsets $M_j$.
Lastly, for a partition $\la$ of $n$ we denote by $\hmS_\la^\al$ the set of pairs $(\pi,M)$ where $\pi$ is a permutation in $\mC_\la$, and $M$ is a tuple in $\hmA_n^\al$ such that the product $\pi\circ (1,2,\ldots,n)$ is $M$-separated.
We now set $\la,\la',\al,k,m,n,r$ to be as in Theorem~\ref{thm:addfixpt}, and define a mapping $\Psi$ from $\mS_{\la'}^\al$ to $\hmS_\la^\al$.
Let $\pi'$ be a permutation of $[n+r]$ of cycle type $\la'$, and let $e_1<e_2<\cdots<e_n\in[n+r]$ be the elements not fixed by $\pi'$. We denote $\varphi(\pi')$ the permutation $\pi$ defined by setting $\pi(i)=\pi(j)$ if $\pi'(e_i)=e_j$. Observe that $\pi$ has cycle type $\la$.
\begin{rmk}\label{rk:adding-fixed}
If $e_1<e_2<\cdots<e_n\in[n+r]$ are the elements not fixed by $\pi'$ and $\pi=\varphi(\pi')$, then the cycle structure of the permutation $\pi'\circ (1,2,\ldots,n+r)$ is obtained from the cycle structure of $\pi\circ (1,2,\ldots,n)$ by replacing each element $i\in[n-1]$ by the sequence of elements $F_i=e_i,e_i+1,e_i+2,\ldots,e_{i+1}-1$, and replacing the element $n$ by the sequence of elements $F_n=e_n,e_n+1,e_n+2,\ldots,n+r,1,2,\ldots,e_{1}-1$. In particular, the permutations $\pi\circ (1,2,\ldots,n)$ and $\pi'\circ (1,2,\ldots,n+r)$ have the same number of cycles.
\end{rmk}
Now given a pair $(\pi',A)$ in $\mS_{\la'}^\al$, where $A=(A_1,\ldots,A_k)$, we consider the pair $\Psi(\pi',A)=(\pi,M)$, where $\pi=\varphi(\pi')$ and $M=(M_1,\ldots,M_k)$ is a tuple of multisubsets of $[n]$ defined as follows: for all $j\in [k]$ and all $i\in[n]$ the multiplicity $M_j(i)$ is the number of elements in the sequence $F_i$ belonging to the subset $A_j$ (where the sequence $F_i$ is defined as in Remark~\ref{rk:adding-fixed}). It is easy to see that $\Psi$ is a mapping from $\mS_{\la'}^\al$ to $\hmS_{\la}^\al$.
We are now going to evaluate $\#\mS_{\la'}^\al$ by counting the number of preimages of each element in $\hmS_{\la}^\al$ under the mapping $\Psi$.
As we will see now, the number of preimages of a pair $(\pi,M)$ in $\hmS_{\la}^\al$ only depends on $M$.
\begin{lemma}\label{lem:transfer}
Let $(\pi,M)\in\hmS_{\la}^\al$, where $M=(M_1,\ldots,M_k)$. Let $s$ be the number of distinct elements appearing in the multisets $M_1,\ldots,M_k$, and let $x=\sum_{j=1}^kM_j(n)$ be the multiplicity of the integer~$n$. Then the number of preimages of the pair $(\pi,M)$ under the mapping $\Psi$ is
\begin{equation}\label{eq:transfer}
\#\Psi^{-1}(\pi,M)=
\left\{\begin{array}{ll}
\displaystyle {n+r+s \choose n+m} & \textrm{ if $x=0$,}\\[10pt]
\displaystyle x\,{n+r+s \choose n+m}+ {n+r+s-1 \choose n+m}&\textrm{ otherwise}.
\end{array}\right.
\end{equation}
\end{lemma}
\begin{proof}
We adopt the notation of Remark~\ref{rk:adding-fixed}, and for all $i\in[n]$ we denote $M_*(i)=\sum_{j=1}^kM_j(i)$ the multiplicity of the integer $i$. In order to construct a preimage $(\pi',A)$ of $(\pi,M)$, where $A=(A_1,\ldots,A_k)$, one has to
\begin{compactitem}
\item[(i)] choose for all $i\in[n]$ the length $f_i>0$ of the sequence $F_i$ (with $\sum_{i=1}^nf_i=n+r$),
\item[(ii)] choose the position $b\in[f_n]$ corresponding to the integer $n+r$ in the sequence $F_n$,
\item[(iii)] if $M_j(i)>0$ for some $i\in[n]$ and $j\in[k]$, then choose which $M_j(i)$ elements in the sequence $F_i$ are in the subset $A_j$.
\end{compactitem}
Indeed, the choices (i), (ii) determine the permutation $\pi'\in\mC_{\la'}$ (since they determine the fixed-points of $\pi'$, which is enough to recover $\pi'$ from $\pi$), while by Remark~\ref{rk:adding-fixed} the choice (iii) determines the tuple of subsets $A=(A_1,\ldots,A_k)$.
We will now count the number ways of making the choices (i), (ii), (iii) by encoding such choices as rows of (marked and unmarked) boxes as illustrated in Figure~\ref{fig:adding-fixed-bij}. We treat separately the cases $x=0$ and $x\neq 0$. Suppose first $x=0$. To each $i\in[n]$ we associate a row of boxes $R_i$ encoding the choices (i), (ii), (iii) as follows:
\begin{compactitem}
\item[(1)] if $i\neq n$ and $M_*(i)=0$, then the row $R_i$ is made of $f_i$ boxes, the first of which is marked,
\item[(2)] if $i\neq n$ and $M_*(i)>0$, then the row $R_i$ is made of $f_i+1$ boxes, with the first box being marked and $M_*(i)$ other boxes being marked (the marks represent the choice (iii)),
\item[(3)] the row $R_n$ is made of $f_n+1$ boxes, with the first box being marked and an additional box being marked and called \emph{special marked box} (this box represents the choice (ii)).
\end{compactitem}
There is no loss of information in concatenating the rows $R_1,R_2,\ldots,R_n$ given that $M$ is known (indeed the row $R_i$ starts at the $(i+N_i)$th marked box, where $N_i=\sum_{h<i}M_*(h)$\,). This concatenation results in a row of $n+r+s+1$ boxes with $n+m+1$ marks such that the first box is marked and the last mark is ``special''; see Figure~\ref{fig:adding-fixed-bij}. Moreover there are ${n+r+s \choose n+m}$ such rows of boxes and any of them can be obtained for some choices of (i), (ii), (iii). This proves the case $x=0$ of Lemma~\ref{lem:transfer}.\\
\fig{width=\linewidth}{adding-fixed-bij}{Example of choices (1),(2),(3) encoded by a sequence of boxes, some of which being marked (indicated in gray), with one mark being special (indicated with a cross). Here $n=6$, $k=2$, $r=11$, $x=0$ and the multisubsets $M_1,M_2$ are defined by $M_1(1)=1$, $M_2(3)=1$, $M_1(4)=3$, and $M_j(i)=0$ for the other values of $i,j$.}
We now suppose $x>0$. We reason similarly as above but there are now two possibilities for the row $R_n$, depending on whether or not the integer $n+r$ belongs to one of the subsets $A_1,\ldots,A_k$. In order to encode a preimage such that $n+r$ belong to one of the subsets $A_1,\ldots,A_k$ the condition $(3)$ above must be changed to
\begin{compactitem}
\item[(3')] the row $R_n$ is made of $f_n+1$ boxes, with the first box being marked and $x$ other boxes being marked, one of which being called \emph{special marked box}.
\end{compactitem}
In this case, concatenating the rows $R_1,R_2,\ldots,R_n$ gives a row of $n+r+s$ boxes with $n+m$ marks, with the first box being marked and one of the $x$ last marked boxes being special. There are $x {n+r+s-1 \choose n+m-1}$ such rows and each of them comes from a unique choice of (i), (ii) and (iii).
Lastly, in order to encode a preimage such that $n+r$ does not belong to one of the subsets $A_1,\ldots,A_k$ the condition $(3)$ above must be changed to
\begin{compactitem}
\item[(3'')] the row $R_n$ is made of $f_n+1$ boxes, with the first box being marked and $x+1$ other boxes being marked, one of which being called \emph{special marked box}.
\end{compactitem}
In this case, concatenating the rows $R_1,R_2,\ldots,R_n$ gives a row of $n+r+s$ boxes with $n+m+1$ marks, with the first box being marked and one of the $x+1$ last marked boxes being special. There are $(x+1) {n+r+s-1 \choose n+m}$ such rows and each of them comes from a unique choice of (i), (ii) and (iii).
Thus, in the case $x>0$ one has
$$\#\Psi^{-1}(\pi,M)=x {n+r+s-1 \choose n+m-1}+(x+1) {n+r+s-1 \choose n+m}=x\,{n+r+s \choose n+m}+ {n+r+s-1 \choose n+m}.$$
This completes the proof of Lemma \ref{lem:transfer}.
\end{proof}
We now complete the proof of Theorem~\ref{thm:addfixpt}. For any composition $\ga=(\ga_1,\ldots,\ga_k)$, we denote by $\hmS_\la^{\al,\ga}$ the set of pairs $(\pi,M)$ in $\hmS_\la^\al$, where the tuple $M=(M_1,\ldots,M_k)$ is such that for all $j\in[k]$ the multisubset $M_j$ (which is of size $\al_j$) contains exactly $\ga_j$ distinct elements.
Summing~\eqref{eq:transfer} gives
\begin{equation}\label{eq:sum-transfer}
\sum_{(\pi,M)\in\hmS_\la^{\al,\ga}}\!\!\!\#\Psi^{-1}(\pi,M)=\left((\EE(X)+\PP(X=0)){n+r+|\ga| \choose n+m}+\PP(X>0){n+r+|\ga|-1 \choose n+m}\right)\,\#\hmS_\la^{\al,\ga},
\end{equation}
where $X$ is the random variable defined as $X=\sum_{j=1}^kM_j(n)$ for a pair $(\pi,M)$ chosen uniformly randomly in $\hmS_\la^{\al,\ga}$, $\EE(X)$ is the expectation of this random variable, and $\PP(X>0)=1-\PP(X=0)$ is the probability that $X$ is positive.
\begin{lemma}\label{lem:cyclic}
With the above notation, $\displaystyle \EE(X)=\frac{m}{n}$, and $\displaystyle \PP(X>0)=\frac{|\ga|}{n}$.
\end{lemma}
\begin{proof}
The proof is simply based on a cyclic symmetry. For $i\in[n]$ we consider the random variable $X_i=\sum_{j=1}^kM_j(i)$ for a pair $(\pi,M)$ chosen uniformly randomly in $\hmS_\la^{\al,\ga}$. It is easy to see that all the variables $X_1,\ldots,X_n=X$ are identically distributed since the set $\hmS_\la^{\al,\ga}$ is unchanged by cyclically shifting the value of the integers $1,2,\ldots,n$ in pairs $(\pi,M)\in\hmS_\la^{\al,\ga}$. Therefore,
$$n\,\EE(X)=\sum_{i=1}^n\EE(X_i)=\EE\left(\sum_{i=1}^nX_i\right)=\EE(m)=m,$$
and
$$n\,\PP(X>0)=\sum_{i=1}^n\PP(X_i>0)=\EE\left(\sum_{i=1}^n1_{X_i>0}\right)=\EE\left(|\ga|\right)=|\ga|.$$
\end{proof}
We now enumerate the set $\hmS_\la^{\al,\ga}$.
Observe that any pair $(\pi,M)$ in $\hmS_\la^{\al,\ga}$ can be obtained (in a unique way) from a pair $(\pi,A)$ in $\mS_\la^{\ga}$ by transforming $A=(A_1,\ldots,A_k)$ into $M=(M_1,\ldots,M_k)$ as follows: for each $j\in[k]$ one has to assign a positive multiplicity $M_j(i)$ for all $i\in A_j$ so as to get a multisubset $M_j$ of size $\al_j$. There are $\binom{\al_j-1}{\ga_j-1}$ ways of performing the latter task, hence
$$\#\hmS_\la^{\al,\ga}=\prod_{i=1}^k \binom{\al_i-1}{\ga_i-1}\,\#\mS_\la^\ga.$$
Using this result and Lemma~\ref{lem:cyclic} in \eqref{eq:sum-transfer} gives
\begin{equation}\label{eq:sum-transfer2}\nonumber
\sum_{(\pi,M)\in\hmS_\la^{\al,\ga}}\#\Psi^{-1}(\pi,M)=\left(\frac{m+n-|\ga|}{n}{n+r+|\ga| \choose n+m}+\frac{|\ga|}{n}{n+r+|\ga|-1 \choose n+m}\right)\prod_{i=1}^k \binom{\al_i-1}{\ga_i-1}\,\#\mS_\la^\ga.
\end{equation}
Observe that the above expression is 0 unless $\ga$ is less or equal to $\al$ componentwise. Finally, one gets
\begin{equation}\label{eq:fixedpointseq2}
\#\mS^{\alpha}_{\la'} = \sum_{\gamma\leq\al,~ \ell(\gamma)=k} \left(\frac{m+n-|\ga|}{n}{n+r+|\ga| \choose n+m}+\frac{|\ga|}{n}{n+r+|\ga|-1 \choose n+m}\right)\prod_{i=1}^k \binom{\al_i-1}{\ga_i-1}\,\#\mS_\la^\ga,
\end{equation}
where the sum is over compositions $\ga$ with $k$ parts, which are less or equal to $\al$ componentwise. Lastly, by Corollary~\ref{cor:sepsym}, the cardinality $\#\mS^{\gamma}_{\la'}$ only depends on the composition $\al$ through the length and size of $\al$. Therefore, one can use \eqref{eq:fixedpointseq2} with $\al=(m-k+1,1^{k-1})$, in which case the compositions $\ga$ appearing in the sum are of the form $\ga=(m-k-p+1,1^{k-1})$ for some $p\leq m-k$. This gives \eqref{eq:addfixpoints} and completes the proof of Theorem~\ref{thm:addfixpt}.\hfill$\square$
\section{Bijective proofs and interpretation in terms of maps}\label{sec:maps}
In this section we explain how certain results of this paper can be interpreted in terms of \emph{maps}, and can be proved bijectively. In particular, we shall interpret the sets $\mT_{\ga,\de}^\al$ of ``separated colored factorizations'' (defined in Section~\ref{sec:strategy}) in terms of maps. We can then extend a bijection from \cite{OB:Harer-Zagier-non-orientable} in order to prove bijectively the symmetry property stated in Corollary~\ref{cor:sepsym}.
\subsection{Interpretations of (separated) colored factorizations in terms of maps}
We first recall some definitions about maps. Our \emph{graphs} are undirected, and they can have multiple edges and loops. Our \emph{surfaces} are two-dimensional, compact, boundaryless, orientable, and considered up to homeomorphism; such a surface is characterized by its genus. A connected graph is \emph{cellularly embedded} in a surface if its edges are not crossing and its \emph{faces} (connected components of the complement of the graph) are simply connected. A \emph{map} is a cellular embedding of a connected graph in an orientable surface considered up to homeomorphism. A map is represented in Figure \ref{fig:one-face-maps}.
By cutting an edge in its midpoint one gets two \emph{half-edges}. A map is \emph{rooted} if one of its half-edges is distinguished as the \emph{root}. In what follows we shall consider rooted bipartite maps, and consider the unique proper coloring of the vertices in black and white such that the root half-edge is incident to a black vertex.
\fig{width=.6\linewidth}{one-face-maps}{(a) A rooted bipartite one-face map. (b) A rooted bipartite tree-rooted map (the spanning tree is indicated by thick lines). The root half-edge is indicated by an arrow.}
By a classical encoding (see e.g. \cite{LZ}), for any partitions $\la,\mu$ of $n$, the solutions $(\pi_1,\pi_2)\in\mC_\la\times\mC_\mu$ of the equation $\pi_1\circ\pi_2=(1,2,\ldots,n)$ are in bijection with the rooted one-face bipartite maps such that black and white vertices have degrees given by the permutations $\la$ and $\mu$ respectively. That is, the number of black (resp. white) vertices of degree $i$ is equal to the number of parts of the partition $\lambda$ (resp. $\mu$) equal to $i$. Let $\ga=(\ga_1,\ldots,\ga_\ell)$, $\de=(\de_1,\ldots,\de_{\ell'})$ be compositions of $n$ and let $\al=(\al_1,\ldots,\al_k)$ be a composition of $m\leq n$. A rooted bipartite map is $(\ga,\de)$-colored if its black vertices are colored in $[\ell]$ (that is, every vertex is assigned a ``color'' in $[\ell]$) in such a way that $\ga_i$ edges are incident to black vertices of color $i$, and its white vertices are colored in $[\ell']$ in such a way that $\de_i$ edges are incident to white vertices of color $i$. Through the above mentioned encoding, the set $\mB_{\ga,\de}$ of colored factorizations of the $n$-cycles defined in Section~\ref{sec:strategy} corresponds to the set of $(\ga,\de)$-colored rooted bipartite one-face maps. Similarly, the sets $\mT_{\ga,\de}^\al$ of ``separated colored factorizations'' corresponds to the set of $(\ga,\de)$-colored rooted bipartite one-face maps with some marked edges, such that for all $i\in[k]$ exactly $\al_i$ marked edges are incident to white vertices colored $i$.
The results in this paper can then be interpreted in terms of maps.
For instance, one can interpret \eqref{eq:GF-explicit} in the case $m=k=0$ (no marked edges) as follows:
$$\sum_{\la\vdash n}\sum_{M\in\mB_\la} \!p_\la(\xx)\,t^{\#\textrm{white vertices}}=\GG_n^\emptyset(\xx,t) =\sum_{r=1}^{n}\sum_{\la \vdash n,~ \ell(\la)\leq n-r+1} \! \!m_{\la}(\xx)\binom{t}{r} \frac{n(n-\ell(\la))!(n-r)!}{(n-r-\ell(\la)+1)!} \binom{n-1}{n-r},$$
where $\mB_\la$ is the set of rooted bipartite one-face maps such that black vertices have degrees given by the partition $\la$.
The results in Subsection \ref{subsec:involution} can also be interpreted in terms of \emph{general} (i.e., non-necessarily bipartite) maps. Indeed, the set $\mM_N=\mB_{2^N}$ can be interpreted as the set of general rooted one-face maps with $N$ edges (because a bipartite map in which every black vertex has degree two can be interpreted as a general map upon contracting the black vertices).
Therefore one can interpret \eqref{eq:GF-Mapsexplicit} in the case $m=k=0$ (no marked edges) as follows:
\begin{equation}\label{eq:HZ}
\sum_{M\in \mM_N}t^{\#\textrm{vertices}}=H_{N}^{\emptyset}(t) = N\sum_{r=1}^{N+1} \binom{t}{r} 2^{r-N} \frac{(2N-r)!}{(N-r+1)!}\binom{2N-1}{2N-r}.
\end{equation}
This equation is exactly the celebrated Harer-Zagier formula \cite{HZ}.
\subsection{Bijection for separated colored factorizations, and symmetry}
In this section, we explain how some of our proofs could be made bijective. In particular we will use bijective results obtained in \cite{OB:Harer-Zagier-non-orientable} in order to prove the symmetry result stated in Corollary~\ref{cor:sepsym}.\\
We first recall the bijection obtained in \cite{OB:Harer-Zagier-non-orientable} about the sets $\mB_{\ga,\de}$.
We define a \emph{tree-rooted map} to be a rooted map with a marked spanning tree; see Figure~\ref{fig:one-face-maps}(b).
We say that a bipartite tree-rooted map is $(\ell,\ell')$\emph{-labelled} if it has $\ell$ black vertices labelled with distinct labels in $[\ell]$, and $\ell'$ white vertices labelled with distinct labels in $[\ell']$. It was shown in \cite{OB:Harer-Zagier-non-orientable} that for any compositions $\ga=(\ga_1,\ldots,\ga_\ell)$, $\de=(\de_1,\ldots,\de_{\ell'})$ of $n$, the set $\mB_{\ga,\de}$
is in bijection with the set of $(\ell,\ell')$-labelled bipartite tree-rooted maps such that the black (resp. white) vertex labelled $i$ has degree $\ga_i$ (resp. $\de_i$).
From this bijection, it is not too hard to derive the enumerative formula \eqref{eq:colored-factorizations} (see Remark~\ref{rmk:symmetry}).
We now adapt the bijection established in \cite{OB:Harer-Zagier-non-orientable} to the sets $\mT_{\ga,\de}^\al$ of ``separated colored factorizations''.
For a composition $\al=(\al_1,\ldots,\al_k)$, a $(\ell,\ell')$-labelled bipartite maps is said to be \emph{$\al$-marked} if $\al_i$ edges incident to the white vertex labelled $i$ are marked for all $i$ in $[k]$.
\begin{theorem}\label{thm:bij}
The bijection in \cite{OB:Harer-Zagier-non-orientable} extends into a bijection between the set $\mT_{\ga,\de}^\al$ and the set of $\al$-marked $(\ell,\ell')$-labelled bipartite tree-rooted maps with $n$ edges such that the black (resp. white) vertex labelled $i$ has degree $\ga_i$ (resp. $\de_i$).
\end{theorem}
We will now show that the bijection given by Theorem \ref{thm:bij} easily implies
\begin{equation}\label{eq:sym-sep-colored}
\#\mT_\ga^\al(r)=\#\mT_\ga^\be(r),
\end{equation}
whenever the compositions $\al$ and $\be$ have the same length and size. Observe that, in turn, \eqref{eq:sym-sep-colored} readily implies Corollary~\ref{cor:sepsym}.
By Theorem~\ref{thm:bij}, the set $\mT_\ga^\al(r)$ specified by Definition~\ref{def:setT} is in bijection with the set $\mTR_\ga^\al(r)$ of $\al$-marked $(\ell,k+r)$-labelled bipartite tree-rooted maps with $n$ edges such that the black vertex labelled $i$ has degree $\ga_i$. We will now describe a bijection between the sets $\mTR_\ga^\al(r)$ and $\mTR_\ga^\be(r)$ when $\al$ and $\be$ have the same length and size. For this purpose it is convenient to interpret maps as graphs endowed with a rotation system. A \emph{rotation system} of a graph $G$ is an assignment for each vertex $v$ of $G$ of a cyclic ordering of the half-edges incident to $v$. Any map $M$ defines a rotation system $\rho(M)$ of the underlying graph: the cyclic orderings are given by the clockwise order of the half-edges around the vertices. This correspondence is in fact bijective (see e.g. \cite{MT}): for any connected graph $G$ the mapping $\rho$ gives a bijection between maps having underlying graph $G$ and the rotation systems of $G$. Using the ``rotation system'' interpretation, any map can be represented in the plane (with edges allowed to cross each other) by choosing the clockwise order of the half-edges around each vertex to represent the rotation system; this is the convention used in Figures \ref{fig:marked-tree-rooted} and \ref{fig:easy-tree-rooted}.
\fig{width=.8\linewidth}{marked-tree-rooted}{Left: a $(3,1,1)$-marked $(4,5)$-labelled bipartite tree-rooted map. Right: the $(2,1,2)$-marked $(4,5)$-labelled bipartite tree-rooted map obtained by applying the mapping $\varphi_{1,3}$. In this figure, maps are represented using the ``rotation system interpretation'', so that the edge-crossings are irrelevant. The spanning trees are drawn in thick lines, the marked edges are indicated by stars, and the root half-edge is indicated by an arrow.}
We now prove \eqref{eq:sym-sep-colored} it is sufficient to establish a bijection between the sets $\mTR_\ga^\al(r)$ and $\mTR_\ga^\be(r)$ in the case $\al=(\al_1,\ldots,\al_k)$, $\be=(\be_1,\ldots,\be_k)$ with $\be_i=\al_i-1$, $\be_j=\al_j+1$ and $\al_s=\be_s$ for $s\neq i,j$. Let $M$ be an $\al$-marked $(\ell,\ell')$-labelled bipartite tree-rooted map. We consider the path joining the white vertices $i$ and $j$ in the spanning tree of $M$. Let $e_i$ and $e_j$ be the edges of this path incident to the white vertices $i$ and $j$ respectively; see Figure~\ref{fig:marked-tree-rooted}. We consider the first marked edge $e_i'$ following $e_i$ in clockwise order around the vertex $i$ (note that $e_i\neq e_i'$ since $\al_i=\be_i+1>1$). We then define $\varphi_{i,j}(M)$ as the map obtained by ungluing from the vertex $i$ the half-edge of $e_i'$ as well as all the half-edges appearing strictly between $e_i$ and $e_i'$, and gluing them (in the same clockwise order) in the corner following $e_j$ clockwise around the vertex $j$. Figure~\ref{fig:marked-tree-rooted} illustrates the mapping $\varphi_{1,3}$. It is easy to see that $\varphi_{i,j}(M)$ is a tree-rooted map, and that $\varphi_{i,j}$ and $\varphi_{j,i}$ are reverse mappings. Therefore $\varphi_{i,j}(M)$ is a bijection between $\mTR_\ga^\al(r)$ and $\mTR_\ga^\be(r)$. This proves \eqref{eq:sym-sep-colored}.
\begin{rmk} \label{rmk:symmetry}
By an argument similar to the one used above to prove \eqref{eq:sym-sep-colored}, one can prove that if $\ga,\ga',\de,\de'$ are compositions of $n$ such that $\ell(\ga)=\ell(\ga')$ and $\ell(\de)=\ell(\de')$ then $\mB_{\ga,\de}=\mB_{\ga',\de'}$ (this is actually done in a more general setting in \cite{BM1}). From this property one can compute the cardinality of $\mB_{\ga,\de}$ by choosing the most convenient compositions $\ga$, $\de$ of length $\ell$ and $\ell'$. We take $\ga=(n-\ell+1,1,1,\ldots,1)$ and $\de=(n-\ell'+1,1,1,\ldots,1)$, so that $\#\mB_{\ga,\de}$ is the number of $(\ell,\ell')$-labelled bipartite tree-rooted maps with the black and white vertices labelled 1 of degrees $n-\ell+1$ and $n-\ell'+1$ respectively, and all the other vertices of degree 1. In order to construct such an object (see Figure~\ref{fig:easy-tree-rooted}), one must choose the unrooted plane tree (1 choice), the labelling of the vertices ($(\ell-1)!(\ell'-1)!$ choices), the $n-\ell-\ell'+1$ edges not in the tree (${n-\ell \choose n-\ell-\ell'+1}{n-\ell' \choose n-\ell-\ell'+1}(n-\ell'-\ell'+1)!$ choices), and lastly the root ($n$ choices). This gives \eqref{eq:colored-factorizations}.
\end{rmk}
\fig{width=.3\linewidth}{easy-tree-rooted}{A tree-rooted map in $\mB_{\ga,\de}$, where $\ga=(8,1,1,1,1)$, $\de=(9,1,1,1)$. Here the map is represented using the ``rotation system interpretation'', so that the edge-crossings are irrelevant.}
\subsection{A direct proof of Theorem~\ref{thm:GF-Maps}}
In Section \ref{sec:results} we obtained Theorem~\ref{thm:GF-Maps} as a consequence of Theorem~\ref{thm:GF}. Here we explain how to obtain it directly
First of all, by a reasoning identical to the one used to derive~\eqref{eq:colsepmap} one gets
\begin{equation}\label{eq:proof-GF-Maps}
H_{N}^{\alpha}(t+k) = \sum_{r=0}^{2N-m} \binom{t}{r}\,\# \mathcal{U}^{\alpha}(r),
\end{equation}
where $\mathcal{U}^{\alpha}(r)$ is the set of triples $(\pi,A,c_2)$ where $\pi$ is a fixed-point free involution of $[2N]$, $A$ is in $\mA^{\alpha}_n$ and $c_2$ is a a cycle coloring of the product $\pi \circ (1,2,\ldots,2N)$ in $[k+r]$ such that every color in $[k+r]$ is used and for all $i$ in $[k]$ the elements in the subset $A_i$ are colored $i$.
In order to enumerate $\mathcal{U}^{\alpha}(r)$ one considers for each composition $\ga=(\ga_1,\ldots,\ga_\ell)$ the set $\mathcal{M}_{\gamma}$ of pairs $(\pi,c_2)$, where $\pi$ is a fixed-point-free involution of $[2N]$ and $c_2$ is a cycle coloring of the permutation $\pi \circ (1,2,\ldots,2N)$ such that $\gamma_i$ elements are colored $i$ for all $i\in [\ell]$.
One then uses the following analogue of~\eqref{eq:colored-factorizations}:
\begin{equation}\label{eq:HZrefined}
\#\mathcal{M}_{\gamma} = \frac{N(2N-\ell)!}{(N-\ell+1)!} 2^{\ell-N}.
\end{equation}
Using this result in conjunction with Lemma~\ref{lem:nb-markings}, one then obtains the following analogue of~\eqref{eq:cardT}:
\begin{equation}\nonumber
\#\mathcal{U}^{\alpha}(r) = \frac{N(2N-k-r)!}{(N-k-r+1)!}\binom{2N+k-1}{2N-m-r}.
\end{equation}
Plugging this result in \eqref{eq:proof-GF-Maps} completes the proof of Theorem \ref{thm:GF-Maps}.\\
Similarly as \eqref{eq:colored-factorizations}, Equation \eqref{eq:HZrefined} can be obtained bijectively. Indeed by a classical encoding, the set $\mathcal{M}_{\gamma}$ is in bijection with the set of rooted one-face maps with vertices colored in $[\ell]$ in such a way that for all $i\in[\ell]$, there are exactly $\ga_i$ half-edges incident to vertices of color $i$. Using this interpretation, it was proved in~\cite{OB:Harer-Zagier-non-orientable} that the set $\mathcal{M}_{\ga}$ is in bijection with the set of tree-rooted maps with $\ell$ vertices labelled with distinct labels in $[\ell]$ such that the vertex labelled $i$ has degree $\gamma_i$. The latter set is easy to enumerate (using symmetry as in Remark \ref{rmk:symmetry}) and one gets \eqref{eq:HZrefined}.
\section{Concluding remarks: strong separation and connection coefficients}\label{sec:conclusion}
Given a tuple $A=(A_1,\ldots,A_k)$ of disjoint subsets of $[n]$, a permutation $\pi$ is said to be \emph{strongly $A$-separated} if each of the subsets $A_i$, for $i\in[k]$ is included in a distinct cycle of $\pi$. Given a partition $\la$ of $n$ and a composition $\al$ of $m\leq n$, we denote by $\ppi_{\la}^\al$ the probability that the product $\om\circ \rho$ is \emph{strongly} $A$-separated, where $\om$ (resp. $\rho$) is a uniformly random permutation of cycle type $\la$ (resp. $(n)$) and $A$ is a fixed tuple in $\mA_n^\al$. In particular, for a composition $\al$ of size $m=n$, one gets
$$\ppi_{\la}^\al=\frac{K_{\la,(n)}^\al\prod_{i=1}^k (\alpha_i-1)!}{(n-1)!\,\,\#\mC_\la},$$
where $K_{\la,(n)}^\al$ is the \emph{connection coefficient of the symmetric group} counting the number of solutions $(\om,\rho)\in \mC_\la\times \mC_{(n)}$, of the equation $\om\circ \rho=\phi$ where $\phi$ is a fixed permutation of cycle type $\al$.
We now argue that the separation probabilities $\{\psig_{\la}^\al\}_{\al \models m}$ computed in this paper are enough to determine the probabilities $\{\ppi_{\la}^\al\}_{\al \models m}$. Indeed, it is easy to prove that
\begin{equation}\label{eq:weak-to-strong}
\psig_{\la}^\al = \sum_{\be \preceq \al} R_{\al,\be}\ppi_{\la}^\be,
\end{equation}
where the sum is over the compositions $\be=(\be_1,\ldots,\be_\ell)$ of size $m=|\al|$ such that there exists $0=j_0<j_1<j_2<\cdots<j_k=\ell$ such that $(\be_{j_{i-1}+1},\be_{j_{i-1}+1},\ldots,\be_{j_{i}})$ is a composition of $\al_i$ for all $i\in[k]$, and $R_{\al,\be}=\prod_{i=1}^kR_i$ where $R_i$ is the number of ways of partitioning a set of size $\al_i$ into blocks of respective sizes $\be_{j_{i-1}+1},\be_{j_{i-1}+1},\ldots,\be_{j_{i}}$.
Moreover, the matrix $(R_{\al,\be})_{\al,\be\models m}$ is invertible (since the matrix is upper triangular for the lexicographic ordering of compositions). Thus, from the separation probabilities $\{\psig_{\la}^\al\}_{\al \models m}$ one can deduce the strong separation probabilities $\{\ppi_{\la}^\al\}_{\al \models m}$ and in particular, for $m=n$, the connection coefficients $K_{\la,(n)}^\al$ of the symmetric group.\\
\noindent \textbf{Acknowledgment:} We thank Taedong Yun for several stimulating discussions.
|
1,116,691,500,720 | arxiv | \section{Weakly associative algebras}
Let $\K$ be a field of characteristic $0$ and $(A,\ast)$ a $\K$-nonassociative algebra. Recall that an algebra $(A,\ast)$ is a nonassociative algebra if it is a $\K$-vector space with a bilinear binary multiplication $X \ast Y$ which may or not be associative.
The associator of the algebra $(A,\ast)$ is the trilinear map $\mathcal{A}_\ast$ defined by
$$\mathcal{A}_\ast (X,Y,Z)=X \ast (Y \ast Z)- (X \ast Y )\ast Z$$
for any $X,Y,Z \in A.$
\subsection{Definition and examples of weakly-associative algebras}
\begin{definition}\cite{R-Wass}
A $\K$-algebra $(A,\ast)$ is called weakly associative if its associator $\mathcal{A}_\ast$ satisfies
$$\mathcal{A}_\ast (X,Y,Z)+\mathcal{A}_\ast (Y,Z,X)-\mathcal{A}_\ast (Y,X,Z)=0$$
for any $X,Y,Z \in A.$
\end{definition}
To simplify, we shall denote by $\mathcal{WA}_\ast (X,Y,Z)$ the trilinear map
$$\mathcal{WA}_\ast (X,Y,Z)=\mathcal{A}_\ast (X,Y,Z)+\mathcal{A}_\ast (Y,Z,X)-\mathcal{A}_\ast (Y,X,Z)$$
and a $\K$-algebra $(A,\ast)$ is weakly associative if and only if
$$\mathcal{WA}_\ast (X,Y,Z)=0$$
for any $X,Y,Z \in A.$
\medskip
\noindent{\bf Examples.}
\begin{enumerate}
\item Any associative algebra is weakly associative.
\item Any abelian (i.e. commutative) algebra is weakly associative.
\item Any Lie algebra is weakly associative. In fact, if $(\g, \{,\})$ is a $\K$-Lie algebra, the associator of the Lie bracket satisfies
$$\mathcal{A}_{[\, , \, ]} (X,Y,Z)=[X,[Y,Z]\,]-[\, [X,Y],Z]$$
and from the Jacobi identity
$$\mathcal{A}_{[\, , \, ]} (X,Y,Z)=[\, [Z,X],Y].$$
Then
$$\mathcal{WA}_{[\, ,\, ]} (X,Y,Z)=[\, [Z,X],Y]+[\, [X,Y],Z]+[\, [Y,Z],X]$$
and, from the Jacobi identity
$$\mathcal{WA}_{[\, ,\, ]} (X,Y,Z)=0$$
\end{enumerate}
In the next section, we will show another important example of weakly associative algebras given by the class of {\it symmetric Leibniz algebras}. As said in the introduction, the notion of weakly associative algebra has been defined in relation with the classical notion of Poisson algebra. This relation is based on the concept of polarization-depolarization presented in \cite{MRPoisson} which presents Poisson algebras in term of algebras with one nonassociative multiplication so in term of nonassociative algebras.
We can also recall a characterization of weakly associative algebra
\begin{proposition}\label{der}\cite{R-Wass}
Let $(A,\ast)$ be a nonassociative $\K$-algebra. Then for any $X \in A$, the endomorphism
$L_X-R_X$ defined by
$$(L_X-R_X)(Y)=X \ast Y - Y \ast X$$
is a derivation of $(A,\ast)$ if and only if $(A,\ast)$ is weakly associative.
\end{proposition}
\subsection{Polarization-Depolarization of $\K$-algebras} The polarization technique consists in representing a given one-operation $\K$-algebra $(A, \ast)$ without particular symmetry as an algebra with two operations, one commutative
and the other skew symmetric. Explicitly, in the following, we will decompose the multiplication $\ast$ of the $\K$-algebra $(A, \ast)$ using:
\begin{enumerate}
\item $X\bullet Y= \frac{1}{2}(X\ast Y + Y \ast X)$ its symmetric part,
\item $[X,Y]= \frac{1}{2}(X\ast Y - Y \ast X)$ its skewsymmetric part,
\end{enumerate}
for $X,Y \in A$.
The triplet $(A, \bullet , [ \, \, , \, ])$ will be referred to as the {\it polarization} of $(A, \ast)$. Conversely, starting with an algebra $(A, \bullet , [ \, \, , \, ])$ with a commutative product $\bullet$ and a skew symmetric multiplication $[ \, \, , \, ]$, we obtain an algebra $(A, \ast)$ with only one nonassociative multiplication defined by $X \ast Y= X\bullet Y + [X ,Y]$. The algebra $(A, \ast)$ is called the {\it depolarization} of the algebra $(A, \bullet , [ \, \, , \, ])$.
The polarization-depolarization technique of \cite{MRPoisson} then makes a link between Poisson algebras and some nonassociative algebras and permits to present Poisson algebras as a nonassociative algebras $(A,\ast)$ satisfying
$$3(x\ast y)\ast z-3x\ast (y\ast z) = (x\ast z)\star y + (y\ast z)\ast x − (y\ast x)\ast z − (z\ast x)\ast y.$$
Another applications of this technique are found in \cite{Ben}.
The polarization also permits to study weakly associative algebras as nonassociative Poisson algebras.
\begin{definition}
Let $(A,\bullet,[\, ,\, ])$ be a triple where $A$ is a $\K$-vector space with two multiplications $\bullet$ and $ [\, ,\, ]$. We say that $(A,\bullet,[\, ,\, ])$ is a nonassociative Poisson algebra if
\begin{enumerate}
\item $(A,\bullet)$ is a nonassociative commutative algebra,
\item $(A,[\, ,\, ] )$ is a Lie algebra,
\item the Leibniz identity between $[\, ,\, ]$ and $\bullet$ is satisfied, that is
$$[X\bullet Y,Z]=X\bullet[Y,Z]+[X,Z]\bullet Y$$
fo all $X,Y,Z \in A.$
\end{enumerate}
\end{definition}
\noindent{\bf Remark.} As the terminology suggests, nonassociative Poisson algebras generalize Poisson algebras by relaxing the
associative condition on the underlying commutative algebra.
The link between Poisson algebras and weakly associative algebra is summarized in the following result:
\begin{theorem}\cite{R-Wass}
Let $(A,\bullet,[ \; , \, ])$ be a nonassociative Poisson algebra and consider on $A$ the third multiplication $$X \ast Y= X\bullet Y + [X,Y].$$
Then the algebra $(A, \ast)$, that is its depolarization, is weakly associative.
Conversely,
if $(A,\ast)$ is a weakly associative algebra, then its polarization $A(\bullet,[\; ,\, ])$ is a nonassociative Poisson algebra.
\end{theorem}
\noindent{\bf Remark.}
Let $( A,\ast)$ a Lie-admissible algebra and $(A,\bullet, [ \; , \, ])$ its polarization. Then the following relations are equivalent
\begin{enumerate}
\item Leibniz($[ \; ,\, ],\ast$) : $[X\ast Y,Z]-X \ast [Y,Z]-[X,Z] \ast Y=0,$
\item Leibniz($[ \; ,\, ],\bullet$) : $[X\bullet Y,Z]-X\bullet[Y,Z]-[X,Z]\bullet Y=0,$
\end{enumerate}
for any $X,Y,Z \in A.$
Remark also that in order to obtain the classical notion of Poisson algebra, the multiplication $\bullet$ has to be associative. We have
$$\begin{array}{lll}
4\mathcal{A}_\bullet(X,Y,Z)&=& X \ast (Y \ast Z + Z \ast Y)+(Y \ast Z + Z \ast Y) \ast X-(X\ast Y + Y \ast X) \ast Z \\
&&-Z \ast (X \ast Y + Y \ast X) \\
\end{array}$$If we denote by $\mathcal{B}_\ast$ the trilinear map associated to the multiplication $\ast$ defined by
$$\mathcal{B}_\ast(X,Y,Z)=X\ast(Y \ Z)+(Y \ast Z) \ast X$$
then
$$4\mathcal{A}_\bullet(X,Y,Z)= \mathcal{B}_\ast(X,Y,Z)+\mathcal{B}_\ast(X,Z,Y)-\mathcal{B}_\ast(Z,X,Y)-\mathcal{B}_\ast(Z,Y,X).$$
Then a sufficient condition for the associativity of $\bullet$ is that $\mathcal{B}_\ast(X,Y,Z)=0$.
\section{Symmetric Leibniz algebras}
A symmetric Leibniz algebra is an algebra $(A, \ast)$ such that for any $X,Y ,Z \in
A$, we have
$$
\left\{
\begin{array}{l}
X\ast(Y\ast Z) = (X\ast Y)\ast Z + Y\ast(X\ast Z),\\
(Y\ast Z)\ast X = (Y\ast X)\ast Z + Y\ast (Z\ast X).
\end{array}
\right.
$$
If we denote by $\mathcal{A}_\ast$ the associator of the multiplication $ \ast$, that is
$$\mathcal{A}_\ast (X,Y,Z)=X \ast (Y \ast Z)- (X \ast Y ) \ast Z$$
for any $X,Y,Z \in A$, the first identity corresponds to
$$\mathcal{A}_\ast (X,Y,Z)=Y \ast (X \ast Z)$$
and the second to
$$\mathcal{A}_\ast (Y,Z,X)=-(Y\ast X)\ast Z.$$
We deduce that $(A, \ast)$ is a symmetric Leibniz algebra if and only if
$$
\left\{
\begin{array}{l}
\mathcal{A}_\ast (X,Y,Z)=Y \ast (X \ast Z)\\
\mathcal{A}_\ast (X,Y,Z)=-(X\ast Z)\ast Y
\end{array}
\right.
$$
or equivalently
$$
\left\{
\begin{array}{l}
\mathcal{A}_\ast (X,Y,Z)=Y \ast (X \ast Z)\\
\mathcal{B}_\ast(X,Y,Z)=0.
\end{array}
\right.
$$
In particular we deduce
$$\mathcal{A}_\ast (X,Y,Z)+\mathcal{A}_\ast (Y,Z,X)=Y \ast (X \ast Z)
-(Y\ast X)\ast Z=\mathcal{A}_\ast(Y,X,Z)$$
that is
$$\mathcal{A}_\ast (X,Y,Z)+\mathcal{A}_\ast (Y,Z,X)-\mathcal{A}_\ast(Y,X,Z)$$
and $A$ is a weakly associative algebra.
\begin{proposition}
Any symmetric Leibniz algebra is weakly associative.
\end{proposition}
Since any symmetric Leibniz algebra $(A,\ast)$ is weakly associative, it is also Lie admissible \cite{GRLieadm} and its polarized algebra $(A,\bullet,[\; , \, ])$ is a nonassociative Poisson algebra. But we also have that $\mathcal{B}_\ast(X,Y,Z)=0$ for any $X,Y,Z \in A$ so we obtain the following result:
\medskip
\begin{proposition}
Let $(A,\ast)$ be a symmetric Leibniz algebra. Then it is a weakly associative algebra and the commutative multiplication $\bullet$ of its polarization $A(\bullet,[\;,\,])$ is also associative. Then the polarization $A(\bullet,[\;,\, ])$ is a Poisson algebra.
\end{proposition}
So a Poisson algebra can be naturally obtained from each symmetric Leibniz algebra by polarization. We find a result of \cite{A.B}.
\medskip
\noindent Remark. In \cite{GRnonass}, we have studied some classes of nonassociative algebras in terms of action of the symmetric group $\Sigma_3$. The class of symmetric Leibniz algebras are a part of these, more precisely are $v-w$-algebras where $v,w $ are in the algebra group $\K[\Sigma_3]$.
\section{Structure of symmetric Leibniz algebras}
Recall that, if $(A, \ast)$ is a symmetric Leibniz algebra, we have in particular the relation
$$\mathcal{B}_\ast(Y,X,Z)=Y\ast (X \ast Z)+(X \ast Z) \ast Y = 0.$$
This is equivalent, by polarization, to
$$\begin{array}{l}
[Y,[X,Z]]+[Y,X \bullet Z]+Y \bullet [X,Z]+Y \bullet (X \bullet Z)+[[X,Z],Y]+[X \bullet Z,Y]+[X,Z] \bullet Y\\
+(X \bullet Z) \bullet Y=0
\end{array}
$$
which is equivalent to
$$
Y \bullet [X,Z]+Y \bullet (X \bullet Z)+[X,Z] \bullet Y
+(X \bullet Z) \bullet Y=0
$$
or
$$Y \bullet [X,Z]+Y \bullet (X \bullet Z)=0.$$
We deduce that we have also
$$Y \bullet [Z,X]+Y \bullet (Z \bullet X)=0.$$
If we add these two identities we obtain:
\begin{proposition}
Let $(A,\ast)$ be a symmetric Leibniz algebra and $(A,\bullet,[\; ,\, ])$ its polarization. Then $\bullet$ is an associative commutative multiplication which satisfies
$$X \bullet (Y \bullet Z)=0$$
for any $X,Y,Z \in A$.
\end{proposition}
\begin{corollary}
The commutative associative algebra $(A,\bullet)$ is two-step nilpotent.
\end{corollary}
If we denote by $A^2_\bullet$ the subalgebra of $(A,\bullet)$ generated by the products $X\bullet Y$, $X,Y \in A$ and by $A^1_\bullet$ a vectorial subspace of $A$ isomorphic to $A/A^2_\bullet$ then we have the grading
$$A= A^1_\bullet \oplus A^2_\bullet$$
with $$A^1_\bullet \bullet A^1_\bullet = A^2_\bullet, \ A^1_\bullet \bullet A^2_\bullet = A^2_\bullet \bullet A^2_\bullet =0.$$
\medskip
Now lets look at the properties of the Lie bracket associated with the polarization process of the symmetric Leibniz multiplication. We have seen that
$Y \bullet [X,Z]+Y \bullet (X \bullet Z)=0$ for any $X,Y,Z \in A$. Since
$Y \bullet (X \bullet Z)=0$, we deduce
\begin{proposition}
Let $(A, \ast)$ be a symmetric Leibniz algebra and $(A, \bullet, [\; , \, ])$ its polarization.
The derived Lie subalgebra of the Lie algebra $(A,[\; ,\, ])$, denoted by
$[A,A],$ is contained in the center of the associative algebra $(A,\bullet)$.
\end{proposition}
Since for any $X,Y,Z \in A$ we have $X \bullet (Y \bullet Z)=0$ and $X \bullet [Y,Z]=0$, the relation $\mathcal{A}_\ast(X,Y,Z)-Y\ast (X \ast Z)=0$ is equivalent to
$$[X,Y \bullet Z] -[X \bullet Y,Z]-[Y,X\bullet Z]=0.$$
Then we have also
$$[X,Z \bullet Y] -[X \bullet Z,Y]-[Z,X\bullet Y]=0.$$
Since $\bullet$ is commutative, by adding these relations we obtain that
$$[X,Y \bullet Z]=0$$
for any $X,Y,Z \in A$.
\begin{theorem}
Let $(A,\ast)$ be a $\K$-algebra and $(A,\bullet,[\;,\,])$ its polarized algebra. Then $(A,\ast)$ is a symmetric Leibniz algebra if and only if the following conditions are satisfied:
\begin{enumerate}
\item $X \bullet (Y \bullet Z)=0$ for any $X,Y,Z \in A$
(implying that $(A,\bullet)$ is a commutative associative $2$-step nilpotent algebra).
\item $[\; ,\,]$ is a Lie bracket with the property that
$$[X,Y \bullet Z]=0 \ { and} \ X \bullet [Y,Z]=0$$
for any $X,Y,Z \in A$.
\end{enumerate}
\end{theorem}
\pf We have to prove only the converse. If $\bullet$ and $[\; ,\,]$ satisfy
$$X \bullet (Y \bullet Z)=0,\ \ [X,Y \bullet Z]=0, \ \ X \bullet [Y,Z]=0$$
then
$$
\begin{array}{lll}
2(\mathcal{A}_\ast(X,Y,Z)-Y\ast (X \ast Z))&=& X \bullet (Y \bullet Z)-(X \bullet Y )\bullet Z+X \bullet [Y,Z]+[X,Y]\bullet Z\\
&&+[X,Y\bullet Z]-[X\bullet Y,Z]+[X,[Y,Z]]+[Z,[X,Y]]\\
&& -Y\bullet (X \bullet Z)-Y \bullet [X ,Z]-[Y,X \bullet Z]+[Y,[Z,X]]\\
&=&0.
\end{array}
$$
The second identity of symmetric Leibniz algebras is shown in the same way.
\medskip
\noindent{\bf Consequence.} Since any symmetric Leibniz algebra $(A,\ast)$ is weakly associative satisfying $\mathcal{B}_\ast(X,Y,Z)=0$ for any $X,Y,Z \in A$, its polarized algebra has a Poisson algebra structure. From the previous theorem, the Leibniz identity
$$[X\bullet Y, Z]-X\bullet [Y,Z]-[X,Z]\bullet Y=0$$
is trivially satisfied because each term is null. Let us also remark that $(A,\ast,[\; ,\, ])$ satisfies also Leibniz($[\; , \, ],\ast$) is also satisfied, that is
$$[X\ast Y,Z]-X \ast [Y,Z]-[X,Z] \ast Y=0.$$
\section{Finite dimensional case}
Let $(A, \ast)$ a symmetric Leibniz $\mathbb{K}$-algebra and $(A,\bullet,[\; ,\,])$ its polarized Poisson algebra. In the finite dimensional case, we can find a basis $\{u_1,\cdots,u_r,v_1,\cdots,v_p,w_1,\cdots,w_q\}$ of $A$ such that $\{w_1,\cdots,w_q\}$ is a basis of $A^2_\bullet$ and $\{v_1,\cdots,v_p,w_1,\cdots,w_q\}$ a basis of the center of the associative algebra $(A,\bullet)$ (recall that $A^2_\bullet$ is contained in the center of $(A,\bullet)$). The nontrivial structure constants of $(A,\bullet)$ are given by
$$u_i \bullet u_j=\sum _{k=1}^q A_{i,j}^k w_k$$
with $A_{i,j}^k=A_{j,i}^k$.
Since $A^2_\bullet$ is contained in the center of $(A,[\; , \, ])$ and $A^2_{[\, ,\, ]}$ is contained in the center of $(A,\bullet)$, where $A^2_{[\, ,\, ]}$ is the derived Lie subalgebra of $(A,[\; ,\, ])$, we have
$$\begin{array}{l}
[u_i,w_j]=0, \ 1 \leq i \leq r, \ 1 \leq j \leq q,\\
\lbrack v_i,w_j]=0, \ 1 \leq i \leq p, \ 1 \leq j \leq q\\
\end{array}
$$
and
$$\begin{array}{l}
\medskip
[u_i,u_j]=\ds\sum_{k=1}^p C_{i,j}^kv_k+\sum_{l=1}^q D_{i,j}^lw_l,\\
\medskip
\lbrack u_i,v_j]=\ds\sum_{k=1}^p E_{i,j}^kv_k+\sum_{l=1}^q F_{i,j}^lw_l,\\
\medskip
\lbrack v_i,v_j]=\ds\sum_{k=1}^p G_{i,j}^kv_k+\sum_{l=1}^q H_{i,j}^lw_l\\
\end{array}
$$
with the Jacobi polynomial identities.
The classifications in small dimensions of the commutative associative $2$-step nilpotent algebras are already established \cite{dG, Rh}. Recall these results:
\begin{enumerate}
\item Dimension $2$
\begin{enumerate}
\item $A^2_\bullet=0$, that is $A=\K\{v_1,v_2\}$
$$v_1 \bullet v_2 =0.$$
\item $A=\K\{u_1,w_1\}$ and
$$u_1 \bullet u_1 = w_1.$$
\end{enumerate}
\item Dimension $3$ (From now on, we don't write the decomposable algebras).
\begin{enumerate}
\item $A=\K\{u_1,u_2,w_1 \}$
$$u_1 \bullet u_1 = {w_1}, \ u_2 \bullet u_2 = w_1$$
\item $\K=\R$, $A=\K\{u_1,u_2,w_1 \}$
$$u_1 \bullet u_1 = {w_1}, \ u_2 \bullet u_2 = -w_1$$
\item $A=\K\{u_1,u_2,w_1 \}$
$$u_1 \bullet u_2 = u_2 \bullet u_1 = w_1.$$
\end{enumerate}
\item Dimension $4$ ($\K$ is algebraically closed)
\begin{enumerate}
\item $A=\K\{u_1,u_2,u_3 \} \oplus \K\{w_1\}$ and
$$u_1 \bullet u_1 = w_1, \ u_2 \bullet u_2 =w_1, \ u_3 \bullet u_3 = w_1$$
\item $A=\K\{u_1,u_2 \} \oplus \K\{w_1,w_2\}$ and
$$u_1 \bullet u_1 = w_1, \ \ u_1 \bullet u_2=u_2 \bullet u_1=w_2$$
\item $A=\K\{u_1,u_2 \} \oplus \K\{w_1,w_2\}$ and
$$u_1 \bullet u_1 = w_1, \ \ u_2 \bullet u_2 = w_1, \ \ u_1 \bullet u_2=u_2 \bullet u_1=w_2$$
\end{enumerate}
\end{enumerate}
\medskip
Using this list we describe all the small dimensional symmetric Leibniz algebras.
\noindent{\bf 1. $\dim A=2.$}
\begin{enumerate}
\item $A^2_\bullet=0$, that is $A=\K\{v_1,v_2\}$ and
$v_1 \bullet v_2 =0.$
In this case $[v_1,v_2]=av_1+bv_2$ and we obtain two classes of symmetric Leibniz algebras
$$
\left\{
\begin{array}{l}
v_i\ast v_j = 0, \ \forall i,j \in \{i,j\}
\end{array}
\right.
\ \ \ \
\left\{
\begin{array}{l}
v_1 \ast v_2=-v_2 \ast v_1 =v_2.
\end{array}
\right.
$$
\item $A=\K\{u_1,w_1\}$ and
$u_1 \bullet u_1 = w_1.$
Its center is $1$-dimensional and coincides with $A_\bullet ^2=\K\{w_1\}$. We deduce that
$$[u_1,w_1]=0$$ and the associated symmetric Leibniz algebra is given by
$$
\left\{
\begin{array}{l}
u_1 \ast u_1=w_1, \\
u_1 \ast w_1= w_1 \ast u_1=0.
\end{array}
\right.$$
\medskip
\noindent{\bf Remark. Generalization in any dimension.} We can generalize this case, considering $\dim A=n, \dim A_\bullet ^2= 1.$ We consider a basis $\{u_1,v_1,\cdots,v_{n-2},w_1\}$ such that $\{v_1,\cdots,v_{n-2},w_1\}$ is a basis of the center of $(A,\bullet)$ and $\{w_1\}$ a basis of $A_\bullet^2$. The non trivial product of $(A,\bullet)$ is
$$u_1 \bullet u_1= w_1.$$
We define the Lie algebra structures $(A,[\;,\,])$ on the basis $\{u_1,v_1,\cdots,v_{n-2},w_1\}$ as follow:
\begin{itemize}
\item We consider on the $\mathbb{K}$-vector space $\g=\mathbb{K}\{v_1,\cdots,v_{n-2}\}$ any $(n-2)$-dimensional Lie algebra structure with a nontrivial $H^2(\g,\g)$ where $H^*(\g,\g)$ is the Chevalley-Eilenberg cohomology of $\g$.
\item Let $\theta$ be a nontrivial $2$-form on $\g$ (which exists because $H^2(\g,\g) \neq 0)$. We consider a central extension $\g_1=\g \oplus \K\{w_1\}$ of $\g$ associated with $\theta$, that is the bracket of $\g_1$ is defined from the bracket of $\g$ and by the relation
$$[v_i,v_j]_{\g_1}=[v_i,v_j]_{\g}+\theta(v_i,v_j)w_1$$ for $i=1,\cdots,n-2$.
\item Let $g$ be a derivation of $\g_1$ satisfying $g(w_1)=0.$
We consider an extension of $\g_1$ by the derivation $g$, that is if $A=\K\{u_1\}\oplus \g_1$, the bracket of $A$ is that of $\g_1$ and $[u_1,X]=g(X)$ for any $X \in \g_1$.
\end{itemize}
Thus the Lie structure obtained on $A$ satisfies the required conditions.
\noindent{Examples.}
\begin{enumerate}
\item If $\dim \g=1$ that is $\g=\K\{v_1\}$, the Lie algebra $\g_1=\{v_1,w_1\}$ is abelian. Thus the Lie bracket on $(A,[\;,\,])$ is given by
$$[u_1,v_1]=av_1+bw_1.$$
In this case we obtain the $3$-dimensional symmetric Leibniz algebras whose multiplication satisfies
$$
\left\{
\begin{array}{l}
u_1 \ast u_1= w_1,\\
u_1 \ast v_1=-v_1 \ast u_1=av_1+bw_1,
\end{array}
\right.$$
and other products are equal to zero.
\item If $\dim \g=2$ and $\g=\K\{v_1,v_2\}$ is abelian. Then $\g_1=\K\{v_1,v_2,w_1\}$ is the Heisenberg algebra:
$$[v_1,v_2]=w_1$$
and considering a general derivation $g$ of $\g_1$ we obtain the Lie bracket of $(A,[\; , \,])$:
$$[u_1,v_1]=a_1v_1+b_1v_2+c_1w_1, \ [u_1,v_2]=a_2v_1+b_2v_2+c_2w_1, \ [v_1,v_2]=w_1.$$
We obtain the following $4$-dimensional symmetric Leibniz algebras
$$
\left\{
\begin{array}{l}
u_1 \ast u_1 =w_1,\\ u_1\ast v_1=-v_1\ast u_1=a_1v_1+b_1v_2+c_1w_1,\\
u_1\ast v_2=-v_2\ast u_1=a_2v_1+b_2v_2+c_2w_1, \\
v_1 \ast v_2=-v_2 \ast v_1 =w_1.
\end{array}
\right.$$
\noindent{\bf Particular case: the four-dimensional oscillator Lie algebra.} This case corresponds to $a_2=-b_1=1$ and the other parameters equal to zero. Then the Lie bracket of $(A,[\; , \,])$ is given by
$$[u_1,v_1]=v_2, \ [u_1,v_2]=-v_1, \ [v_1,v_2]=w_1.$$
This Lie algebra is usually called the oscillator Lie algebra. It is a linear Lie algebra whose elements are the matrices
$$
\left(
\begin{array}{cccc}
0 & -z & y & 2t \\
0&0&-x & y\\
0 & x & 0 &z\\
0&0&0&0
\end{array}
\right)
$$
The corresponding symmetric Leibniz algebra is given by
$$
\left\{
\begin{array}{l}u_1 \ast u_1 =w_1,\\ u_1\ast v_1=-v_1\ast u_1=-v_2,\\ u_1\ast v_2=-v_2\ast u_1=v_1, \\
v_1 \ast v_2=-v_2 \ast v_1 =w_1.
\end{array}
\right.$$
\end{enumerate}
We find again the result proved in\cite{A.B} in which one shows that the oscillator Lie algebra can be endowed with a symmetric Leibniz algebra structure and with a Poisson algebra structure.
\end{enumerate}
\medskip
\noindent{\bf 2. $\dim A=3.$}
\begin{enumerate}
\item Assume that $\dim A=3$ and the multiplication $\bullet$ is given by
$$u_1 \bullet u_1= u_2 \bullet u_2= w_1.$$
The corresponding Lie bracket satisfies $[u_i,w_1]=0$. We put $[u_1,u_2]=\alpha w_1$. We obtain the symmetric Leibniz algebra
$$
\left\{
\begin{array}{l}
u_1 \ast u_1= u_2\ast u_2=w_1\\
u_1 \ast u_2=-u_2 \ast u_1= \alpha w_1.$$
\end{array}
\right.
$$
\item
Assume that the multiplication $\bullet$ is given by
$$u_1 \bullet u_2= u_2 \bullet u_1= w_1.$$
The corresponding Lie bracket satisfies $[u_i,w_1]=0$. We put $[u_1,u_2]=\alpha w_1$. We obtain the symmetric Leibniz algebra
$$
\left\{
\begin{array}{l}
u_1 \ast u_1= u_2\ast u_2=0\\
u_1 \ast u_2=(1+\alpha) w_1\\
u_2 \ast u_1= (1-\alpha) w_1.$$
\end{array}
\right.
$$
\item If we assume that $(A,\bullet)$ is decomposable with a non trivial product $\bullet$, then $\{u_1,v_1,w_1\}$ is the basis of $A$ and the corresponding symmetric Leibniz algebra is given by
$$
\left\{
\begin{array}{l}
u_1 \ast u_1= w_1,\\
u_1 \ast v_1=-v_1 \ast u_1=av_1+bw_1.
\end{array}
\right.$$
We find the result given in the example (a) in the previous case.
\end{enumerate}
\medskip
\noindent{\bf 3. $\dim A=4.$}
\begin{enumerate}
\item $A=\K\{u_1,u_2,u_3 \} \oplus \K\{w_1\}$ and
$$u_1 \bullet u_1 = w_1, \ u_2 \bullet u_2 =w_1, \ u_3 \bullet u_3 = w_1.$$
In this case we have
$$[u_i,u_j]=\alpha_{i,j}w_1, \ \ 1 \leq i,j \leq 3.$$
The Lie algebra $(A,[\;,\,])$ is a nilpotent Lie algebra isomorphic to the direct sum of the $3$-dimensional Heisenberg algebra with a $1$-dimensional abelian Lie algebra. The corresponding symmetric Leibniz algebras are given by
$$
\left\{
\begin{array}{l}
u_1 \ast u_1= u_2 \ast u_2=u_3 \ast u_3=w_1,\\
u_1 \ast u_2=-u_2 \ast u_1=\alpha_{1,2}w_1,\\
u_1 \ast u_3=-u_3 \ast u_1=\alpha_{1,3}w_1,\\
u_2 \ast u_3=-u_3 \ast u_2=\alpha_{2,3}w_1.\\
\end{array}
\right.$$
\item $A=\K\{u_1,u_2 \} \oplus \K\{w_1,w_2\}$ and
$$u_1 \bullet u_1 = w_1, \ \ u_1 \bullet u_2=u_2 \bullet u_1=w_2.$$
In this case, the non trivial Lie bracket is
$$[u_1,u_2]=\alpha w_1+\beta w_2$$
that is $(A,[\;,\,])$ is the Heisenberg algebra or the abelian Lie algebra. We deduce the corresponding symmetric Leibniz algebra:
$$
\left\{
\begin{array}{l}
u_1 \ast u_1= w_1,\\
u_1 \ast u_2=\alpha w_1 + (\beta+1)w_2,\\
u_2 \ast u_1=-\alpha w_1 - (\beta-1)w_2.\\
\end{array}
\right.$$
\item $A=\K\{u_1,u_2 \} \oplus \K\{w_1,w_2\}$ and
$$u_1 \bullet u_1 = w_1, \ \ u_2 \bullet u_2 = w_1, \ \ u_1 \bullet u_2=u_2 \bullet u_1=w_2$$
In this case also, we have
$$[u_1,u_2]=\alpha w_1+\beta w_2$$
and the corresponding symmetric Leibniz algebra is
$$
\left\{
\begin{array}{l}
u_1 \ast u_1=u_2 \ast u_2= w_1,\\
u_1 \ast u_2=\alpha w_1 + (\beta+1)w_2,\\
u_2 \ast u_1=-\alpha w_1 - (\beta-1)w_2.\\
\end{array}
\right.$$
\item Now we consider that $(A,\bullet)$ is decomposable. The first case is when $(A,\bullet)$ is trivial that is $A=\K\{v_1,v_2,v_3,v_4\}$ that is $v_i \bullet v_j=0.$ In this case $(A,[\;,\,])$ is any $4$-dimensional Lie algebra and the symmetric Leibniz algebra coincides with this Lie algebra, the product $\ast$ is then skew-symmetric.
\item $A=\K\{u_1\} \oplus \K\{v_1,v_2\} \oplus \K\{w_1\}$. This is equivalent to write
$$u_1 \bullet u_1 = w_1.$$
In this case, we can have
$$\begin{array}{l}
\medskip
\lbrack u_1,v_j]=\ds\sum_{k=1}^2 E_{1,j}^kv_k+ F_{1,j}w_1, \ \ j=1,2\\
\medskip
\lbrack v_1,v_2]=\ds \sum_{k=1}^2 G_{1,2}^kv_k+ H_{1,2}w_1.\\
\end{array}
$$
This case has also been studied in a previous example. In particular, we find the $4$-dimensional oscillator Lie algebra.
\item $A=\K\{u_1,u_2\} \oplus \K\{v_1\} \oplus \K\{w_1\}$. This is equivalent to write
$$u_1 \bullet u_1 =u_2 \bullet u_2 = w_1.$$
In this case, the Lie bracket writes :
$$\begin{array}{l}
\medskip
\lbrack u_1,u_2]=\ds C_{1,2}v_1+ D_{1,2}w_1, \\
\medskip
\lbrack u_i,v_1]=\ds E_{i,1}v_1+ F_{i,1}w_1, \ \ i=1,2\\
\end{array}
$$
with the Jacobi condition
$$F_{1,1}E_{2,1}-F_{2,1}E_{1,1}=0.$$
We deduce the following symmetric Leibniz algebras
$$
\left\{
\begin{array}{l}
u_1 \ast u_1=u_2 \ast u_2= w_1,\\
u_1 \ast u_2=-u_2 \ast u_1= C_{1,2}v_1+ D_{1,2}w_1\\
u_1 \ast v_1=-v_1 \ast u_1=E_{1,1}v_1+ F_{1,1}w_1,\\
u_2 \ast v_1=-v_1 \ast u_2=E_{2,1}v_1+ F_{2,1}w_1.\\
\end{array}
\right.$$
\end{enumerate}
\section{Deformation quantization of Poisson algebras in a symmetric Leibniz formal deformation}
As it was recalled in the introduction, the formal deformations of commutative weakly associative algebras gives a construction of nonassociative Poisson algebras. An interesting case corresponds to a formal deformation of an associative commutative algebra in the category of weakly associative algebras which gives a classical Poisson algebra and enlarges the spectrum of deformation quantization. Moreover the class of weakly associative is the only one which permits to construct such a Poisson algebra.
In this paragraph we investigate the formal deformation of commutative symmetric Leibniz algebras in the class of symmetric Leibniz algebras.
Let
$(A, \ast)$ be a commutative symmetric Leibniz algebra. A formal deformation $(A[[t]], \ast_t)$ is given by a symmetric Leibniz formal product which can be represented by a formal series
$$X \ast_t Y=X \ast Y+\sum_{i}t^i\varphi_i(X,Y)$$
for any $X,Y \in A,$ provides $A[[t]]$ with a symmetric Leibniz algebra structure. We then have
$$
\left\{
\begin{array}{l}
X\ast_t(Y\ast_t Z) - (X\ast_t Y)\ast_t Z - Y\ast_t(X\ast_t Z)=0,\\
Y\ast_t (Z\ast_t X) - (Y\ast_t Z)\ast_t X + (Y\ast_t X)\ast_t Z =0
\end{array}
\right.
$$
This system implies
\begin{itemize}
\item at the order 0 that $\ast$ is a symmetric Leibniz product,
\item at the order 1 that the linear bilinear map $\varphi_1$ satisfies
$$\left\{
\begin{array}{ll}
\delta \varphi_1^{(1)}
(X,Y,Z)&=\varphi_1(X, Y\ast Z)-\varphi_1(X\ast Y, Z)-\varphi_1(Y,X\ast Z)\\
&+ X\ast \varphi_1(Y,Z)-\varphi_1(X,Y)\ast Z -Y \ast \varphi_1(X,Z)=0 \\
\delta\varphi_1^{(2)}
(X,Y,Z)&=\varphi_1(Y, Z\ast X)-\varphi_1(Y\ast Z, X)+\varphi_1(Y\ast X, Z)\\
& +Y\ast \varphi_1(Z,X)-\varphi_1(Y,Z)\ast X + \varphi_1(Y,X)\ast Z=0
\end{array}
\right.$$
\item at the order 2 that the linear bilinear map $\varphi_2$ satisfies
$$\left\{
\begin{array}{ll}
\mathcal{A}_{\varphi_1}
(X,Y,Z)-\varphi_1( Y,\varphi_1(X, Z))+\delta\varphi_2^{(1)}
(X,Y,Z)=0\\
\mathcal{A}_{\varphi_1}
(Y,Z,X)+\varphi_1( \varphi_1(Y,X), Z)+\delta \varphi_2^{(2)}
(X,Y,Z)=0
\end{array}
\right.
$$
which is equivalent, since $\ast$ is supposed to be commutative, to
$$\left\{
\begin{array}{ll}
\mathcal{A}_{\varphi_1}
(X,Y,Z)+\mathcal{A}_{\varphi_1}(Y,Z,X)-\mathcal{A}_{\varphi_1}(Y,X,Z)+
\delta \varphi_2^{(1)}(X,Y,Z)+\delta \varphi_2^{(2)}
(X,Y,Z)=0\\
\varphi_1(\varphi_1(X, Z),Y)+\varphi_1( Y,\varphi_1(X, Z))+\delta \varphi_2^{(2)}(Z,X,Y)-\delta \varphi_2^{(1)} (X,Y,Z)=0.
\end{array}
\right.
$$
If we put $\delta \varphi_2=\delta \varphi_2^{(1)}+\delta \varphi_2^{(2)}$ then
$$\left\{
\begin{array}{rl}
\delta \varphi_2
(X,Y,Z)=& \varphi_2(X,Y\ast Z)-\varphi_2(X,Y)\ast Z-
Y \ast \varphi_2(X,Z)-\varphi_2
(Y\ast Z,X)\\
& +Y
\varphi_2( Z,X)+\varphi_2( Y,X) \ast Z\\
= &2 \psi_{\varphi_2}(X,Y\ast Z)-2 \psi_{\varphi_2}(X,Y) \ast Z-2Y\ast \psi_{\varphi_2}(X,Z)
\end{array}
\right.
$$
where $2\psi_{\varphi_2}(U,V)=\varphi_2(U,V)-\varphi_2(V,U)$ for $U,V \in A$.
Then $$\delta \varphi_2(X,Y,Z)-\delta \varphi_2(X,Z,Y)=0.$$
We deduce that
$$\begin{array}{c}
\mathcal{A}_{\varphi_1}
(X,Y,Z)+\mathcal{A}_{\varphi_1}
(Y,Z,X)-\mathcal{A}_{\varphi_1}
(Y,X,Z)-
\mathcal{A}_{\varphi_1}
(X,Z,Y)\\-\mathcal{A}_{\varphi_1}
(Z,Y,X)+\mathcal{A}_{\varphi_1}
(Z,X,Y)=0
\end{array}$$
which is equivalent to say that $\varphi_1$ is Lie-admissible or equivalently that $\psi_{\varphi_1}$ is a Lie bracket.
Since $\delta \varphi_1^{(1)}=\delta \varphi_1^{(2)}=0$, from this calculus we deduce also that
$$
\begin{array}{rl}
\ds \frac{ \delta \varphi_1
(X,Y,Z)}{2}= & \psi_{\varphi_1}(X,Y\ast Z)- \psi_{\varphi_1}(X,Y) \ast Z-Y\ast \psi_{\varphi_1}(X,Z)=0
\end{array}
$$
then $(A,\ast, \psi_{\varphi_1})$ is a nonassociative Poisson algebra defined by the commutative symmetric Leibniz algebra $(A,\ast).$ So we find the same result concerning weakly associative algebra in the symmetric Leibniz context.
\end{itemize}
The Leibniz identity between $\psi_{\varphi_1}$ and $\ast$ is equivalent to $\delta \varphi_1^{(1)}+\delta \varphi_1^{(2)}=0$. Since we have $\delta \varphi_1^{(1)}=\delta \varphi_1^{(2)}=0$ we have to look the consequences of $\tilde{\delta} \varphi_1=\delta \varphi_1^{(1)}-\delta \varphi_1^{(2)}=0$. We have
$$
\begin{array}{ll}
\tilde{\delta} \varphi_1
(X,Y,Z)&=\varphi_1(X, Y\ast Z)-\varphi_1(X\ast Y, Z)-\varphi_1(Y,X\ast Z)\\
&+ X\ast \varphi_1(Y,Z)-\varphi_1(X,Y)\ast Z -Y \ast \varphi_1(X,Z) \\
&-\varphi_1(Y, Z\ast X)+\varphi_1(Y\ast Z, X)-\varphi_1(Y\ast X, Z)\\
& -Y\ast \varphi_1(Z,X)+\varphi_1(Y,Z)\ast X - \varphi_1(Y,X)\ast Z\\
&=2\rho_{\varphi_1}(X, Y\ast Z)-2\varphi_1(X\ast Y, Z)-2\varphi_1(Y,X\ast Z)\\
&+2 X\ast \varphi_1(Y,Z)-2\rho_{\varphi_1}(Y,X)\ast Z-2Y\ast\rho_{\varphi_1}(Z,X)
\end{array}
$$
Since $\varphi_1=\rho_{\varphi_1}+\psi_{\varphi_1}$, we obtain
$$
\begin{array}{ll}
\tilde{\delta} \varphi_1
(X,Y,Z)&=2\rho_{\varphi_1}(X, Y\ast Z)-2\rho_{\varphi_1}(Y,X)\ast Z-2Y\ast\rho_{\varphi_1}(Z,X)-2\rho_{\varphi_1}(X\ast Y, Z)\\
&-2\rho_{\varphi_1}(Y,X\ast Z)+
2 X\ast \rho_{\varphi_1}(Y,Z)-2\psi_{\varphi_1}(X\ast Y, Z)
-2\psi_{\varphi_1}(Y,X\ast Z)\\
&+2 X\ast \psi_{\varphi_1}(Y,Z)
\end{array}
$$
To simplify the writing, because any confusion is possible, we write $XY$ instead $X \ast Y$ and also $\rho,\psi$ in place of $\rho_{\varphi_1},\psi_{\varphi_1}$. The previous identity writes
\begin{eqnarray}\label{rho}
0=&\rho(X, YZ)-\rho(Y,X)Z-Y\rho(Z,X)-\rho(X Y, Z)
-\rho(Y,XZ)+
X \rho(Y,Z)-\psi(XY, Z) \nonumber\\
&-\psi(Y,X Z)
+ X\psi(Y,Z)
\end{eqnarray}
Permuting $X$ and $Z$ we obtain
$$\begin{array}{ll}
0=&\rho(Z, YX)-\rho(Y,Z)X-Y\rho(X,Z)-\rho(Z Y, X)
-\rho(Y,ZX)+
Z \rho(Y,X)-\psi(ZY, X)\\
&-\psi(Y,ZX)
+ Z\psi(Y,X)
\end{array}
$$
Adding these two relations
$$\begin{array}{ll}
0=&-2Y\rho(X,Z)
-2\rho(Y,ZX)-\psi(ZY, X)-2\psi(Y,ZX)
+ Z\psi(Y,X)-\psi(XY, Z)\\&+ X\psi(Y,Z)
\end{array}
$$
We have seen that $\psi_{\varphi_1}$ is a Lie bracket satisfying the Leibniz identity with respect the associative commutative multiplication $\bullet$ but also with respect the multiplication $\ast$. We deduce that
$$-\psi(ZY, X)-2\psi(Y,ZX)
+ Z\psi(Y,X)-\psi(XY, Z)+ X\psi(Y,Z)=-2\psi(Y,XZ).$$
So we obtain
$$Y\rho(X,Z)
+\rho(Y,ZX)+\psi(Y,XZ)=0.$$
\medskip
The identity (\ref{rho}) becomes
$$
-\psi(X,YZ)
+\psi(Z,XY)
+\psi(Y,XZ)
-\psi(XY, Z)-\psi(Y,X Z)
+ X\psi(Y,Z)$$
that is
$$
-\psi(X,YZ)
+2\psi(Z,XY)
+ X\psi(Y,Z)=0.
$$
From the Leibniz identity between $\psi$ and $\ast$, this is equivalent to
$$\psi(Y,Z)X+\psi(X,Y)Z-3\psi(Z,X)Y=0.$$
Then we have also
$$\psi(Z,Y)X+\psi(X,Z)Y-3\psi(Y,X)Z=0.$$
If we add these two last identities we obtain
$$4\psi(X,Y)Z-4\psi(Z,X)Y=0.$$
\begin{theorem}
Let $(A,\ast)$ be a commutative symmetric Leibniz algebra and $\ast_t=\ast +\sum _{i\geq 1} t^i \varphi_i$ a symmetric Leibniz formal deformation of $\ast$ (that is $(A[[t]],\ast_t)$ is a symmetric Leibniz algebra). Then if $\psi_{\varphi_1}$ is the skew-symmetric map attached with $\varphi_1$, the algebra $(A,\ast, \psi_{\varphi_1})$ is a nonassociative Poisson algebra and the Lie bracket $\psi_{\varphi_1}$ satisfies
$$\psi_{\varphi_1}(X,Y)\ast Z-\psi_{\varphi_1}(Z,X)\ast Y=0$$
for any $X,Y,Z \in A$.
\end{theorem}
\medskip
\noindent Remark. A commutative associative algebra is always weakly associative. But it is a symmetric Leibniz algebra if and only if it is $2$-step nilpotent. Then the previous theorem permits to construct from a commutative associative $2$-step nilpotent algebra a classical Poisson algebra via a formal symmetric Leibniz deformation process.
|
1,116,691,500,721 | arxiv | \section{Introduction}
With the surge of computational resources, face recognition using deep representation has been widely applied to many fields such as surveillance, marketing and biometrics\cite{arcface,sphereface}. However, it is still a challenging task to implement face recognition on limited computational cost system such as mobile and embedded systems because of the large scale identities needed to be classified.
Many work propose lightweight networks for common computer vision tasks such as \textit{SqueezeNet}\cite{iandola2016squeezenet}, \textit{MobileNet} \cite{mobilenet}, \textit{MobileNetV2} \cite{mobilenetv2}, \textit{ShuffleNet} \cite{shufflenet}. \textit{SqueezeNet}\cite{iandola2016squeezenet} extensively uses $1\times1$ convolution, achieving $50\times$ fewer parameters than \textit{AlexNet}\cite{alexnet} while maintains AlexNet-level accuracy on ImageNet. \textit{MobileNet}\cite{mobilenet} utilizes depthwise separable convolution to achieve a trade off between latency and accuracy. Based on this work, \textit{MobileNetV2}\cite{mobilenetv2} proposes an inverted bottleneck structure to enhance discriminative ability of network. \textit{ShuffleNet}\cite{shufflenet} and \textit{ShuffleNetV2}\cite{shufflenetv2} uses pointwise group convolution and channel shuffle operations to further reduce computation cost. Even though they cost small computation during inference and achieve good performance on various applications, optimization problems on embedded system still remain on embedded hardware and corresponding compilers \cite{vargnet}. To handle this conflict, \textit{VarGNet} \cite{vargnet} proposes a variable group convolution which can efficiently solve the unbalance of computational intensity inside a block. Meanwhile, we explore that variable group convolution has larger capacity than depthwise convolution with the same kernel size, which helps network to extract more essential information. However, \textit{VarGNet} is designed for general tasks such as image classificaiton and object detection. It decreases spatial area to the half in the head setting to save memory and computational cost, while this setting is not suitable for face recognition task since detailed information of face is necessary. And there is only an average pooling layer between last conv and fully connected layer of the embedding, which may not extract enough discriminative information.
\begin{figure*}
\centering
\subfigure[Normal block]{\label{fig:normal_block}\includegraphics[width=0.5\linewidth]{new1.pdf}}
\subfigure[Down sampling block]{\label{fig:d_block}\includegraphics[width=0.48\linewidth]{new2.pdf}}
\subfigure[Head setting]{\label{fig:head}\includegraphics[width=0.5\linewidth]{new3_1.pdf}}
\subfigure[Embedding setting ]{\label{fig:emb} \includegraphics[width=0.45\linewidth]{new4.pdf}}\\
\caption{Settings of VarGFaceNet. a) is the normal block of VarGFaceNet. We add SE block on normal block of VarGNet. b) is the down sampling block. c) is head setting of VarGFaceNet. We do not use downsample in first conv in order to keep enough information. c) is the embedding setting of VarGFaceNet. We first expand channels from 320 to 1024. Then we employ variable group convolution and pointwise convolution to reduce the parameters and computational cost while remain essential information. }
\label{ROC}
\end{figure*}
Based on \textit{VarGNet}, we propose an efficient variable group convolutional network for lightweight face recognition, shorted as VarGFaceNet. In order to enhance the discriminative ability of \textit{VarGNet} for large scale face recognition task, we first add SE block \cite{se} and PReLU \cite{prelu} on blocks of \textit{VarGNet}. Then we remove the downsample process at the start of network to preserve more information. To decrease parameters of network, we apply variable group convolution to shrink the feature tensor to $1\times 1\times 512$ before fc layer. The performance of VarGFaceNet demonstrates that this embedding setting can preserve discriminative ability while reduce parameters of the network.
To enhance the interpretation ability of lightweight network, we apply knowledge distillation during the training. There are several approaches aim at making the deep network smaller and cost-efficient, such as model pruning, model quantization and knowledge distillation. Among them, knowledge distillation is being actively investigated due to its architectural flexibility. Hinton\cite{hinton2015distilling} introduces the concept of knowledge distillation and proposes to use the softmax output of teacher network to achieve knowledge distillation. To take better advantage of information from teacher network, FitNets\cite{romero2014fitnets} adopts the idea of feature distillation and encourages student network to mimic the hidden feature values of teacher network. After FitNets, there are variant methods attempt to exploit the knowledge of teacher network, such as transferring the feature activation map\cite{heo2018knowledge}, activation-based and gradient-based Attention Maps\cite{yim2017gift}. Recently \textit{ShrinkTeaNet} \cite{duong2019shrinkteanet} introduces an angular distillation loss to focus on angular information of teacher model. Inspired by angular distillation loss we employ an equivalent loss with better implementation efficiency as the guide of VarGFaceNet. Moreover, to relieve the complexity of optimization caused by the discrepancy between teacher model and student model, we introduce recursive knowledge distillation which treats the model of student trained in one generation as pretrained model for the next generation.
We evaluate our model and approach on LFR challenge \cite{LFR}. LFR challenge is a lightweight face recognition challenge which requires networks whose FLOPs is under 1G and memory footprint is under 20M. VarGFaceNet achieves the state-of-the-art performance on this challenge which is shown in Section \ref{Experiments}. Our contributions are summarized as follows:
\begin{itemize}
\item To improve the discriminative ability of \textit{VarGNet} \cite{vargnet} in large-scale face recognition we employ a different head setting and propose a new embedding block. In embedding block, we first expand channels to 1024 by $1\times1$ convolution layer to reserve essential information, then we use variable group conv and pointwise conv to shrink the spatial area to $1\times1$ while saving computational cost. These settings improve the performance on face recognition tasks which shown in Section \ref{Experiments}.
\item To imporve the generalization ability of lightweight models, we propose recursive knowledge distillation which relieves the generalization gap between teacher models and student models in one generation.
\item We analyse the efficiency of variable group convolution and employ an equivalence of angular distillation loss during training. Experiments conducted to show the effectiveness of our approach.
\end{itemize}
\section{Approach}
\subsection{Variable Group Convolution}
Group Convolution was first introduced in \textit{AlexNet} \cite{alexnet} for computational cost reduction on GPUs. Then, the cardinality of group convolution demonstrated a better performance than the dimensions of depth and width in \textit{ResNext} \cite{xie2017aggregated}. Designed for mobile device, \textit{MobileNet} \cite{mobilenet} and \textit{MobileNetV2} \cite{mobilenetv2} proposed depthwise separable convolution inspired by group convolution to save computational cost while keep discriminative ability of convolution. However, depthwise separable convolution spends 95\% computation time in Conv $1\times1$, which causes a large MAdds gap between two consecutive laysers (Conv $1\times1$ and Conv DW $3\times3$) \cite{mobilenet}. This gap is unfriendly to embedded systems who load all weights of the network to perform convolution\cite{xing2019dnnvm}: embedded systems need extra buffers for Conv $1\times1$.
To keep the balance of computational intensity inside a block, \textit{VarGNet} \cite{vargnet} sets the channel numbers in a group as a constant $S$. The constant channel numbers in a group lead to the variable number of groups $n_i$ in a convolution, named variable group convolution. The computational cost of a variable group convolution is:
\begin{equation}
k^2\times h_i \times w_i \times S \times c_{i+1}
\end{equation}
\begin{equation}
S = \frac{c_i}{n_i}
\end{equation}
The input of this layer is $h_i \times w_i \times c_i$ and the output of that is $h_i \times w_i \times c_{i+1}$. $k$ is the kernel size. When variable group convolution is used to replace depthwise convolution in \textit{MobileNet} \cite{mobilenet}, the computational cost of pointwise convolution is:
\begin{equation}
1^2\times h_i \times w_i \times c_{i+1} \times c_{i+2}
\end{equation}
The ratio of computational cost between variable group convolution and pointwise convolution is $\frac{k^2 S}{c_{i+2}}$ while that between depthwise convolution and pointwise convolution is $\frac{k^2}{c_{i+2}}$. In practice, $c_{i+2} \gg k^2$, $S>1$, so $\frac{k^2 S}{c_{i+2}} > \frac{k^2}{c_{i+2}}$. Hence, it will be more computational balanced inside a block when employs variable group convolution on the bottom of pointwise convolution instead of depthwise convolution.
Moreover, $S>1$ means variable group convolution has higher MAdds and larger network capacity than depthwise convoluiton (with the same kernel size), which is capable of extracting more information.
\subsection{Blocks of Variable Group Network}
Communication between off-chip memory and on-chip memory only happens on the start and the end of block computing when a block is grouped and computed together on embedded systems \cite{xing2019dnnvm}. To limit the communication cost, \textit{VarGNet} sets the number of output channels to be same as the number of input channels in the normal block. Meanwhile, \textit{VarGNet} expands the $C$ channels at the start of the block to $2C$ channels using variable group convolution to keep the generalization ability of the block. The normal block we used is shown in Fig. \ref{fig:normal_block}, and down sampling block is shown in Fig. \ref{fig:d_block}. Different from the blocks in \textit{VarGNet} \cite{vargnet}, we add SE block in normal block and employ PReLU instead of ReLU to increase the discriminative ability of the block.
\begin{table*}[htb]
\begin{center}
\setlength{\tabcolsep}{7mm}
\begin{tabular}{l|c|c|c|c|c}
\hline
\hline
Layer & Output Size & KSize & Stride & Repeat & Output Channels \\
\hline
Image & 112x112 & {} & {} & {} & 3 \\
\hline
Conv 1 & 112x112 & 3x3 & 1 & 1 & 40 \\
\hline
Head Block & 56x56 & {} & 2 & 1 & 40 \\
\hline
\multirow{2}{*}{Stage2} & 28x28 & {} & 2 & 1 & \multirow{2}{*}{80} \\
\cline{2-5}
& 28x28 & {} & 1 & 2 & {} \\
\hline
\multirow{2}{*}{Stage3} & 14x14 & {} & 2 & 1 & \multirow{2}{*}{160} \\
\cline{2-5}
& 14x14 & {} & 1 & 6 & {} \\
\hline
\multirow{2}{*}{Stage4} & 7x7 & {} & 2 & 1 & \multirow{2}{*}{320} \\
\cline{2-5}
& 7x7 & {} & 1 & 3 & {} \\
\hline
Conv 5 & 7x7 & 1x1 & 1 & 1 & 1024 \\
\hline
Group Conv & 1x1 & 7x7 & 1 & 1 & 1024 \\
\hline
Pointwise Conv & 1x1 & 1x1 & 1 & 1 & 512 \\
\hline
FC & {} & {} & {} & {} & 512 \\
\hline
\hline
\end{tabular}
\end{center}
\caption{Overall architecture of VarGFaceNet. It only has 1G FLOPs and 5M parameters (memory footprint is 20M saved as float32).}
\label{tab:vargfacenet}
\end{table*}
\subsection{Lightweight Network for Face Recognition}
\subsubsection{Head setting}
The main challenge of face recognition is the large scale identities involved in testing/training phase. It requires discriminative ability as much as possible to support distinguishing millions of identities. In order to reserve this ability in lightweight networks, we use $3\times3$ Conv with stride 1 at the start of network instead of $3\times3$ Conv with stride 2 in \textit{VarGNet}. It is similar to the input setting of \cite{arcface}. The output feature size of first conv in \textit{VarGNet} will be downsampled while ours remains the same as input size, shown in Fig. \ref{fig:head}.
\subsubsection{Embedding setting}
To obtain the embedding of faces, many work \cite{arcface,sphereface} employ a fully-connected layer directly on the top of last convolution. However, the parameters of this fully-connected layer will be huge when output features from last convoluiton are relatively large. For instance, in ResNet 100 \cite{arcface} the output of last conv is $7 \times 7 \times 512$, and the parameters of fc layer (embedding size is 512) are $7 \times 7 \times 512 \times 512$. The overall parameters of fc layer for embedding are 12.25M, and the memory footprint is 49M (float32)!
In order to design a lightweight network (memory footprint is less than 20M, FLOPs is less than 1G), we employ variable group convolution after last conv to shrink the feature maps to $1\times 1 \times 512$ before fc layer. Consequently, the memory footprint of fc layer for embedding is only 1M. Fig.\ref{fig:emb} shows the setting of embedding block. Shrinking the feature tensor to $1\times1\times512$ before fc layer for embedding is risky since information contains by this feature tensor is limited. To avoid the derease of essential information, we expand channels after last conv to retain as much information as possible. Then we employ variable group convolution and pointwise convolution to decrease the parameters and computational cost while keep information.
Specifically, we first use a $1\times1$ Conv to expand the channels from 320 to 1024. Then we employ a $7\times7$ variable group convolution layer (8 channels in a group) to shrink the feature tensors from $7\times7\times1024$ to $1\times1\times1024$. Finally, pointwise convolution is used to connect the channels and output the feature tensors to $1\times1\times 512$. The new embedding block setting only takes up 5.78M while the original fc layer takes up 30M ($7\times7\times320\times512$) on the disk.
Experiments of comparison between our network and \textit{VarGNet} in Section \ref{VgR} demonstrate the efficiency of our network on face recognition tasks.
\subsubsection{Overall architecture}
The overall architecture of our lightweight network (VarGFaceNet) is illustrated in Table \ref{tab:vargfacenet}. The memory footprint of our VarGFaceNet is 20M and FLOPs is 1G. We set $S=8$ in a group empirically. Benefit from variable group convolution, head settings and particular embedding settings, VarGFaceNet can achieve good performance on face recognition task with limited computational cost and parameters. In Section \ref{Experiments}, we will demonstrate the effectiveness of our network on a million distractors face recognition task.
\subsection{Angular Distillation Loss}
Knowledge distillation has been widely used in lightweight network training since it can transfer the interpretation ability of a big network to a smaller network \cite{mobilenet}. Majority tasks that used knowledge distillation are close set tasks \cite{romero2014fitnets, hinton2015distilling}. They apply scores/logits or embeddings/feature magnitude to compute $l2$ distance or cross entropy as loss. However, for open set tasks, scores/logits of training set contain limited information of testing set and the exact match of featuers maybe over-regularized in some situations. To extract useful information and avoid over-regularization, \cite{duong2019shrinkteanet} proposes an angular distillation loss for knowledge distillation:
\begin{equation}
L_a(F_t^i,F_s^i) =\frac{1}{N}\sum^{N}_{i=1} || 1- \frac{F_t^i}{||F_t^i||} * \frac{F_s^i}{||F_s^i||} ||^2_2
\label{eq4}
\end{equation}
$F_t^i$ is the $ith$ feature of teacher model, $F_s^i$ is $ith$ features of student model. $m$ is the number of samples in a batch. Eq. \ref{eq4} first computes cosine similarity between features of teacher and student, then minimizes the $l2$ distance between this similarity and 1. Inspired by \cite{duong2019shrinkteanet}, we propose to use Eq. \ref{eq5} to enhance the implementation efficiency. Since cosine similarity is less than 1, minimize Eq. \ref{eq4} is equivalent to minimize Eq. \ref{eq5}.
\begin{equation}
L_s(F_t^i,F_s^i) = \frac{1}{N}\sum^{N}_{i=1} || \frac{F_t^i}{||F_t^i||} - \frac{F_s^i}{||F_s^i||} ||^2_2
\label{eq5}
\end{equation}
Compared with previous $l2$ loss of exact features, Eq. \ref{eq4} and Eq. \ref{eq5} focus on angular information and the distribution of embeddings.
In addition, we employ arcface \cite{arcface} as our classification loss which also pays attention to angular information:
\begin{equation}
L_{Arc}=-\frac{1}{N}\sum^{N}_{i=1}\log\frac{e^{s(cos(\Theta_{y_i}+m))}}{e^{s(cos(\Theta_{y_i}+m))}+\sum^n_{j=1,j\neq y_i}e^{s cos\Theta_j}}
\label{arc}
\end{equation}
To sum up, the objective function we used in training is:
\begin{equation}
L= L_{Arc} + \alpha L_s
\label{loss}
\end{equation}
We empirically set $\alpha = 7$ in our implementation.
\begin{figure*}[htb]
\begin{center}
\includegraphics[width=1\linewidth]{kd_new.pdf}
\end{center}
\caption{The process of recursive knowledge distillation. We apply the first generation of student to initialize the second generation of student while the teacher model is remained. Angular distillation loss and arcface loss are used to guide training. }
\label{kd_f}
\end{figure*}
\begin{table*}
\begin{center}
\setlength{\tabcolsep}{5mm}
\begin{tabular}{l|c|c|c|c|c}
\hline
\hline
Network & LFW & CFP-FP & AgeDB-30 & deepglint-light (TPR@FPR=1e-8) & Flops \\
\hline
y2 & 0.99700 & 0.97829 & 0.97517 & 0.803 & 933M \\
\hline
VarGFaceNet & 0.99683 & 0.98086 & 0.98100 & 0.855 & 1022M \\
\hline
\hline
\end{tabular}
\end{center}
\caption{VarGFaceNet vs. y2. Performance is recorded within the same epoch. The validation performance of VarGFaceNet is 0.6\% and 0.2\% higher than y2 on AgeDB-30 and CFP-FP respectively. Testing result of VarGFaceNet is 5\% higher than y2.}
\label{scratch}
\end{table*}
\subsection{Recursive Knowledge Distillation}
\label{sec:kd}
Knowledge distillation with one generation is sometimes difficult to transfer enough knowledge when large discrepancy exists between teacher models and student models. For instance, in our implementation, the FLOPs of teacher model is 24G while that of student model is 1G. And the number of parameters of teacher model is 108M while that of student model is 5M. Moreover, the different architecture and block settings between teacher model and student model increase the complexity of training as well. To improve the discriminative and generalization ability of our student network, we use recursive knowledge distillation, which employs the first generation of student to initialize the second generation of student, as shown in Fig. \ref{kd_f}.
In recursive knowledge distillation, we employ the same teacher model in all generations. That means the angular information of samples which guides the student model is invariable. There are two merits if we use recursive knowledge distillation:
\begin{itemize}
\item[1] It will be easier to approach guided direction of teacher when a good initialization is applied.
\item[2] The conflicts between margin of classification loss and guided angular information in the first generation will be relieved in the next generation.
\end{itemize}
The results of our experiments in Section \ref{Experiments} illustrate the performance of recursive knowledge distillation.
\begin{table*}[htb]
\begin{center}
\setlength{\tabcolsep}{7mm}
\begin{tabular}{l|c|c|c|c}
\hline
\hline
Method & LFW & CFP-FP & AgeDB-30 & deepglint-light (TPR@FPR=1e-8) \\
\hline
teacher & 0.99683 & 0.98414 & 0.98083 & 0.86846 \\
student & 0.99683 & 0.98171 & 0.97550 & 0.84341 \\
\hline
teacher & 0.99817 & 0.98729 & 0.98133 & 0.90231 \\
student & 0.99733 & 0.98200 & 0.98100 & 0.85461 \\
\hline
teacher & 0.99833 & 0.99057 & 0.98250 & 0.93315 \\
student & 0.99783 & 0.98400 & 0.98067 & 0.88334 \\
\hline
\hline
\end{tabular}
\end{center}
\caption{Performance of VarGFaceNet with the guide of different teacher models. Performance is recorded within the same epoch. Results of CFP-FP(validation set) and deepglint-light(TPR@FPR=1e-8) (testing set) show that the higher performance of teacher model leads to the better results of student model. }
\label{kd}
\end{table*}
\begin{table*}[h]
\begin{center}
\setlength{\tabcolsep}{5mm}
\begin{tabular}{l|c|c|c|c}
\hline
\hline
Network & LFW & CFP-FP & AgeDB-30 & Flops \\
\hline
r100(teacher) & 0.9987 & 0.9917 & 0.9852 & 24G \\
\hline
VarGNet(student) & 0.9977 & 0.9810 & 0.9810 & 1029M \\
\hline
VarGFaceNet(student) & 0.9985 & 0.9850 & 0.9815 & 1022M \\
\hline
\hline
\end{tabular}
\end{center}
\caption{VarGFaceNet vs. VarGNet. We show the highest performance of every validation dataset. The performance of VarGFaceNet is higher than VarGNet on LFW, AgeDB-30 and CFP-FP.}
\label{VvV}
\end{table*}
\begin{table*}[h]
\begin{center}
\setlength{\tabcolsep}{7mm}
\begin{tabular}{l|c|c|c|c}
\hline
\hline
Method & LFW & CFP-FP & AgeDB-30 & deepglint-light (TPR@FPR=1e-8) \\
\hline
recursive=1 & 0.99783 & 0.98400 & 0.98067 & 0.88334 \\
\hline
recursive=2 & 0.99833 & 0.98271 & 0.98050 & 0.88784 \\
\hline
\hline
\end{tabular}
\end{center}
\caption{Performance of recursive knowledge distillation. Performance is recorded within the same epoch.} Verification results of LFW, AgeDB-30 are increased in the second generation. Performance of testing set deepglint-light(TPR@FPR=1e-8) is increased by 0.4\% the same time.
\label{rkd}
\end{table*}
\section{Experiments}
\label{Experiments}
In this section, we first introduce the datasets and evaluation metric. Then, to demonstrate the effectiveness of our VarGFaceNet, we compare our network with y2 network(a deeper mobilefacenet\cite{chen2018mobilefacenets,arcface}). After that, the investigation for the effect of different teacher models in knowledge distillation is revealed. Finally, we show the competitive performance of VarGFaceNet using recursive knowledge distillation on LFR2019 Challenge.
\subsection{Datasets and Evaluation Metric}
We employ the dataset(clean from MS1M\cite{guo2016ms}) provided by LFR2019 for training. All face images in this dataset are aligned by five facial landmarks predicted from RetinaFace\cite{deng2019retinaface} then resized to $112\times112$. There are 5.1M images collected from 93K identities. For test set, Trillion-pairs dataset \cite{trillionpairs} is used. It contains two parts: 1) ELFW: Face images of celebrities in the LFW name list. There are 274K images from 5.7K identities; 2) DELFW: Distractors for ELFW. There are 1.58 M face images from Flickr. All test images are preprocessed and resized to $112\times112$. We refer deepglint-light to trillionpairs testing set in the following. During the training, we utilize face verification datasets (e.g. LFW\cite{huang2008labeled}, CFP-FP\cite{sengupta2016frontal}, AgeDB-30\cite{moschoglou2017agedb}) to validate different settings using 1:1 verification protocol. Moreover, we employ the TPR@FPR=1e-8 as evaluation metric for identification.
\subsection{VarGFaceNet train from scratch}
To validate the efficiency and effectiveness of VarGFaceNet, we first train our network from scratch, and compare the performance with mobilefacenet(y2) \cite{chen2018mobilefacenets,arcface}. We employ arcface loss as the objective function of classification during training. Tabel \ref{scratch} presents the comparison results of VarGFaceNet and y2. It can be observed that under the limitation of 1G FLOPs, VarGFaceNet is able to reach better face recognition performance on validation sets. Compared with y2, our verification results of AgeDB-30 , CFP-FP have increased 0.6\% and 0.2\% respectively, testing result of deepglint-light (TPR@FPR=1e-8) has increased 5\%. There are two intuitions for the better performance: 1. our network can contain more parameters than y2 when limit FLOPs because of variable group convolution. The biggest number of channels is 256 in y2 while ours is 320 before last conv. 2. Our embedding setting can extract more essential information. y2 expands the number of channels from 256 to 512 then use $7\times7$ depthwise convolution to get the feature tensor before fc layer. We expand the number of channels from 320 to 1024 then use variable group convolution and pointwise convolution which have larger network capacity.
\subsection{VarGFaceNet guided by ResNet}
\label{VgR}
In order to achieve higher performance than train from scratch, bigger networks are applied to perform knowledge distillation using angular distillation loss. Moreover, we conduct experiments to investigate the effect of different teacher models on VarGFaceNet. We employ ResNet 100 \cite{resnet} with SE as our teacher model. The teacher model has 24G FLOPs and 108M parameters. The results are illustrated in Tabel \ref{kd}. It can be observed that 1. even though the architectures of teacher and student are quite different, VarGFaceNet still approaches the performance of ResNet; 2. the performance of VarGFaceNet is highly correlated with teacher model. The higher performance teacher model has, the better interpretation ability VarGFaceNet will learn.
To validate the efficiency of our settings, we conduct comparison experiments between our network and \textit{VarGNet}. Using the same teacher network, we change the head setting of \textit{VarGNet} to our head setting for fair comparison and use the same loss function as above. In Tabel \ref{VvV}, the plain \textit{VarGNet} has lower accuracy in LFW, CFP-FP, AgeDB-30. There is only an average pooling between last conv and fc layer in \textit{VarGNet}. The results illustrate that our embedding setting is more suitable for face recognition task since it can extract more essential information.
\subsection{Recursive Knowledge Distillation}
As we discuss in Section \ref{sec:kd}, when there is a large discrepancy between teacher model and student, knowledge distillation for one generation may not enough for knowledge transfer. To validate it, we use ResNet 100 model as our teacher model, and conduct recursive knowledge distillation on VarGFaceNet. A performance improvement shown in Table \ref{rkd} when we train the model in next generation. The varification result of LFW and CFP-FP is increased by 0.1\% while testing result of deepglint-light(TPR@FPR=1e-8) is 0.4\% higher than pervious generation. Furthermore, we believe that it will lead to better performance if we continue to conduct training in more generations.
\section{Conclusion}
In this paper, we propose an efficient lightweight network called VarGFaceNet for large scale face recognition. Benefit from variable group convolution, VarGFaceNet is capable of finding a better trade-off between efficiency and performance. The head setting and embedding setting specific to face recogniton help preserve information while reduce parametes. Moreover, to improve the interpretation ability of lightweight network, we employ an equivalence of angular distillation loss as our objective function and present a recursive knowledge distillation strategy. The state-of-the-art performance on LFR challenge demonstrates the superiority of our method.
\textbf{Acknowledgments} We would like to thank Xin Wang, Helong Zhou, Zhichao Li, Xiao Jiang, Yuxiang Tuo for their helpful discussion, especially Helong for his advice and discussion on recursive knowledge distillation.
{\small
\bibliographystyle{ieee}
|
1,116,691,500,722 | arxiv | \section{Introduction}
Radon-type transforms that assign to a given function its integrals over various sets of ellipses/ellipsoids arise in migration imaging under an assumption that the medium is acoustic and homogeneous.
The aim of migration is to construct an image of the inside of the earth from seismic reflections recorded at its surface~\cite{gazdags84,robinson83}.
A graphical approach called classical migration was developed systematically by Hagedoorn \cite{hagedoorn54}.
Classical migration had been abandoned after the wave-equation method was introduced by Claerbout~\cite{claerbout71} in 1971.
Gazdag and Sguazzero pointed out that the classical migration procedures that existed at that time were not based on a completely sound theory~\cite{gazdags84}.
However, a correct construction for the wave-equation method was often difficult to find because the experiment data did not fit into a single wave equation.
To adapt the diffraction stack to borehole seismic experiments, a new approach to seismic migration was found.
This approach gave classical migration a sound theory.
After that discovery, classical migration has attracted many researchers in the field.
The underlying idea is that seismic data in the far field can be regarded as if the data are coming from integrals of the earth's acoustic scattering potential over surfaces determined by the velocity model~\cite{millerob87}.
These Radon-type transforms relate to migration imaging as well as Bistatic Synthetic Aperture Radar (BiSAR) \cite{ambartsoumianfknq11,cokert07,krishnanq11,krishnanlq12,yarmanyc08}, Ultrasound Reflection Tomography (URT) \cite{ambartsoumiankq11,gouiaa12,roykcm14}, and radio tomography \cite{wilsonp09,wilsonp10,wilsonpv09}.
Because of these applications, there have been several papers devoted to the topic of elliptical Radon transforms.
The family of ellipses with one focus fixed at the origin and the other one moving along a given line was
considered in~\cite{krishnanlq12}. In the same paper, the family of ellipses with a fixed focal distance was
also studied.
The authors of~\cite{ambartsoumiankq11,gouiaa12} dealt with the case of circular acquisition, when the two foci of ellipses
with a given focal distance are located on a given circle. A family of ellipses with two moving foci was
also handled in~\cite{cokert07}.
Radio tomography, which uses a wireless network of radio transmitters and receivers to image the distribution of
attenuation within the network, was discussed in~\cite{wilsonp09,wilsonp10,wilsonpv09}.
They approximated the obtained signal by the volume integral of the attenuation over this ellipsoid.
One work \cite{smoon,moonert14} derived two inversion formulas of this volume integral of the attenuation over this ellipsoid and studied its properties.
Many works found an approximate inversion for elliptical Radon transforms.
Here we consider migration imaging and introduce a new type of an elliptical Radon transform obtained by restricting the position of the source and receiver in migration imaging.
We find an explicit inversion formula for this elliptical Radon transform arising in migration imaging which is the line/area integral of the function over the ellipse/ellipsoid with foci restricted to a hyperplane.
The rest of this paper is organized as follows. The problem of interest is stated precisely and the elliptical Radon transform is formulated in section~\ref{formulation}. In section 3, we show how to reduce the elliptical Radon transform to the regular Radon transform.
The numerical simulation to demonstrate the suggested 2-dimensional algorithm is presented in section 4.
\section{Formulation of the problem}\label{formulation}
Let $\mathbf s\in\RR^3$ and $\mathbf r\in\RR^3$ represent 3-dimensional source and receiver positions, respectively.
For fixed points $(\mathbf s,\mathbf r)$, an isochron surface $I_{(\mathbf s,\mathbf r,t)}$ \footnote{Hagedoorn called this set a surface of maximum concavity (see~\cite{hagedoorn54}).} is a surface consisting of image points $\xx=(x_1,x_2,x_3)\in\RR^3$ associated by the travel time function $\tau(\xx,\yy)$, which gives the travel time $t\in[0,\infty)$ between two points $\xx\in\RR^3$ and $\yy\in\RR^3$ with a known velocity $v(\xx)$ (see \cite{millerob87}).
Mathematically, $I_{(\mathbf s,\mathbf r,t)}$ can be described as the set of the image points $\xx$ satisfying the constraint that the total travel time from the source $\mathbf s$ though the image point $\xx$ to the receiver $\mathbf r$ is constant and equal to $t$.
The isochron surface $I_{(\mathbf s,\mathbf r,t)}$ can be represented a
$$
I_{(\mathbf s,\mathbf r,t)}=\{\xx\in\RR^3:t=\tau(\xx,\mathbf s)+\tau(\xx,\mathbf r)\}.
$$
Seismic experiments yield data $g(\mathbf s,\mathbf r,t)$ which are functions of the source position $\mathbf s$, receiver position $\mathbf r$, and time $t$.
Assuming an object function $f(\xx)$ on $\RR^3$, the data $g$ is modelled by the integral of $f$ over $I_{(\mathbf s,\mathbf r,t)}$, i.e.,
$$
g(\mathbf s,\mathbf r,t)=\int\limits _{I_{(\mathbf s,\mathbf r,t)}}f(\xx)d\xx\quad(\mbox{see \cite{millerob87}}).
$$
The two detectors have nonzero sizes and time is also passing, so it is reasonable to assume that what we measure is the ``average''-concentrated near the location of the detectors and nearly zero sufficiently far away from the detectors-over a small region of space and a small time interval preceding the time $t$.
In mathematical terms, our data can be written as
$$
g(\mathbf s,\mathbf r,t)=\int\limits _{\RR^3}f(\xx)\delta(\tau(\xx,\mathbf s)+\tau(\xx,\mathbf r)-t)d\xx,
$$
where $\delta$ is the Dirac delta function.
(Most works~\cite{ambartsoumiankq11,ambartsoumianfknq11,cokert07,gouiaa12,krishnanq11,krishnanlq12} dealing with an elliptical Radon transform consider the arc length measure instead.)
Miller, Oristaglio, and Beylkin suggested this model and approximately recovered the object function $f$ by appropriately weighted back projection of the data in \cite{millerob87}.
However, we present the explicit formula for reconstructing $f$ using the specific minimum data of $g$ by changing the variables of positions of the two detectors.
The total dimension of the data $g$ is 7. To reduce the overdeterminacy, we assume that $\mathbf s$ and $\mathbf r$ are located on a hyperplane, say $x_3=0$ and also that the difference vector $\mathbf {s-r}$ is parallel to a given line, say the $x_1$-axis. (In 2 dimensions, the difference vector $\mathbf {s-r}$ is automatically parallel to the $x_1$-axis because the hyperplane is the line.)
Hence, for any $u_2\in\RR$, we write $\mathbf s=(s,u_2,0)$ and $\mathbf r=(r,u_2,0)$.
Also, assuming a velocity $v(\xx)$ is set to a constant 1, $\tau(\xx,\yy)$ becomes the distance $|\xx-\yy|$ between $\xx$ and $\yy$.
Since $\xx\in I_{(\mathbf s,\mathbf r,t)}$ satisfies $t=|\xx-(s,u_2,0)|+|\xx-(r,u_2,0)|$, a point $\xx\in I_{(\mathbf s,\mathbf r,t)}$ can be described by the ellipsoid equation
$$
\frac{(x_1-(s+r)/2)^2}{t^2/4}+\frac{(x_2-u_2)^2}{t^2/4-(r-s)^2/4}+\frac{x_3^2}{t^2/4-(r-s)^2/4}=1.
$$
Let us choose a constant $a$ between 0 and 1.
{\color{black}We consider the fixed value of $r$ satisfying the equation $r=s+ta$, which is a new additional requirement.}
Then a point $\xx\in I_{(\mathbf s,\mathbf r,t)}$ can be also described by the equation
$$
\frac{(x_1-{\color{black}(s+r)/2})^2}{1/4}+\frac{(x_2-u_2)^2}{(1-a^2)/4}+\frac{x_3^2}{(1-a^2)/4}=t^2.
$$
If we set $u_1=(s+r)/2$, then our data become
$$
\begin{array}{ll}
&g((u_1-ta,u_2,0),(u_1+ta,u_2,0),t)\\
&\displaystyle=\int\limits _{\RR^3}f(\xx) \delta\left(\sqrt{\frac{(x_1-u_1)^2}{1/4}+\frac{(x_2-u_2)^2}{(1-a^2)/4}+\frac{x_3^2}{(1-a^2)/4}}-t\right)d\xx\\
&:={\color{black}R_{E,a}}f(\uu,t),
\end{array}
$$
where $\uu=(u_1,u_2)\in\RR^2$.
{\color{black}Here, we restrict the positions of the source and receiver in migration imaging, say, $(u_1-ta,u_2,0)$ and $(u_1+ta,u_2,0)$ for $\uu\in\RR^2$.
In general, reducing the space of positions for two devices is more useful and economical.}
If the function is odd with respect to $x_3$, then ${\color{black}R_{E,a}}f$ is equal to zero.
We thus assume the function $f$ to be even with respect to $x_3$: $f(\xx',x_3)=f(\xx',-x_3)$ where $\xx=(\xx',x_3)\in\RR^{2}\times\RR$.
We call ${\color{black}R_{E,a}}f$ an elliptical Radon transform, since this is the surface area integral of $f$ over the set of these ellipsoids.
We generalize this $3$-dimension setup to $n$-dimension and define a more general form of an elliptical Radon transform.
\begin{defi}
Let $a_1,a_2,\cdots,a_n>0$ be given numbers, let $A:=diag(a_1,\cdots,a_n)$ denote the $n\times n$ diagonal matrix with diagonal entries $a_i$, and let $f$ be a locally integrable function on $\RR^n$, even with respect to $x_n$.
The elliptical Radon transforms of $f$ is defined by
$$
{\color{black}R_{E,A}}f(\uu,t)=\int\limits _{\RR^n}f(\xx) \delta\left(\left|A^{-1}(\xx-(\uu,0))\right|-t\right)d\xx, \quad\mbox{for } (\uu,t)\in\RR^{n-1}\times[0,\infty).
$$
\end{defi}
Note that we do not need the condition $a_i<1$ any more.
If all $a_i$, $i=1,2,\cdots,n$, are equal to 1, the elliptical Radon transform ${\color{black}R_{E,A}}f$ becomes the spherical Radon transform with the centers of the sphere of integration located on the hyperplane, a well-studied transform (see~\cite{andersson88,beltukov09,couranth62,fawcett85,lavrentievrv70,klein03,narayananr10,nattererw01,nilsson97,norton80,palamodov04,reddingn01,yarmany07,yarmany11}).
\section{Inversion formula}\label{defpre}
Here we assume that the object function $f$ does not touch the detectors; that is, the support of $f$ does not intersect the hyperplane $x_n=0$ where the two detectors are located.
To obtain an inversion formula for the elliptical Radon transform, we manipulate ${\color{black}R_{E,A}}f$.
By changing the variables $A^{-1}(\xx-(\uu,0))\to\bar{\xx}$, we can write
\begin{equation}\label{eq:defiref}
\begin{array}{ll}
{\color{black}R_{E,A}}f(\uu,t)&\displaystyle=|\mathbf a|_1\int\limits _{\RR^n}f(A\bar\xx +(\uu,0))\delta(|\bar\xx|-t)d\bar\xx\\
&\displaystyle=|\mathbf a|_1t^{n-1}\int\limits _{|\yy|=1}f(A \yy t+(\uu,0))dS(\yy)\\
&=\displaystyle2|\mathbf a|_1t^{n-1}\int\limits _{|\yy'|\leq 1}f(A (\yy',\sqrt{1-|\yy'|^2})t+(\uu,0))\frac{d\yy'}{\sqrt{1-|\yy'|^2}},
\end{array}
\end{equation}
where $\yy=(\yy',y_n)\in\RR^n$ and $|\mathbf a|_1=a_1a_2\cdots a_n$.
Here $dS(\yy)$ is the area measure of the unit sphere.
Let $\mathfrak A=\{\zz=(\zz',z_n)=(z_1,z_2,\cdots,z_{n-1},z_n)\in\RR^n:0\leq |\zz'|^2\leq z_n\}$ and $\mathfrak B=\{\xx=(\xx',x_n)=(x_1,x_2,\cdots,x_{n-1},x_n)\in\RR^n:x_n\geq 0\}$.
We define a map ${\color{black}m_{n,A}}:\mathfrak A\to\mathfrak B$ by
$$
{\color{black}m_{n,A}}(\zz)=(\bar A\zz',a_n\sqrt{z_n-|\zz'|^2})
$$
where $\bar A:=diag(a_1,\cdots a_{n-1})$ is the $n-1\times n-1$ diagonal matrix.
\begin{prop}\label{prop:property}\indent
\begin{itemize}
\item The map ${\color{black}m_{n,A}}:\mathfrak A\to \mathfrak B$ is a bijection with the inverse map ${\color{black}m_{n,A}}^{-1}:\mathfrak B\to \mathfrak A$ defined by
$$
{\color{black}m_{n,A}}^{-1}(\xx)=(\bar A^{-1}\xx',|A^{-1}\xx|^2).
$$
\item We have
$$
\begin{array}{ll}
\{{\color{black}m_{n,A}}^{-1}(\xx)\in \RR^n:|A^{-1}(\xx-(\uu,0))|=t,\xx\in \mathfrak B\}\\
\displaystyle=\left\{\zz\in \mathfrak A:\zz\cdot\frac{(-2\bar A^{-1}\uu,1)}{\nu_{\bar A}(\uu)}=\frac{t^2-|\bar A^{-1}\uu|^2}{\nu_{\bar A}(\uu)}\right\},
\end{array}
$$
where $\nu_{\bar A}(\uu)=\sqrt{1+4|\bar A^{-1}\uu|^2}$.
\end{itemize}
\end{prop}
The map ${\color{black}m_{n,A}}^{-1}$ transforms an ellipsoid into a hyperplane.
Changing variables using this map ${\color{black}m_{n,A}}$ plays a critical role in reducing the elliptical Radon transform to the regular Radon transform.
\begin{proof}
We can easily check that ${\color{black}m_{n,A}}^{-1}\circ {\color{black}m_{n,A}}(\zz)=\zz$ for $\zz\in \mathfrak A$ and ${\color{black}m_{n,A}}\circ {\color{black}m_{n,A}}^{-1}(\xx)=\xx$ for $\xx\in \mathfrak B$, so ${\color{black}m_{n,A}}:A\to \mathfrak B$ is a bijection.
Consider ${\color{black}m_{n,A}}^{-1}(\xx)\cdot(-2\bar A^{-1}\uu,1)/\nu_{\bar A}(\uu)$ for $\xx\in \mathfrak B$ with $|A^{-1}(\xx-(\uu,0))|=t$:
\begin{equation}\label{eq:mn}
\begin{array}{ll}
\displaystyle {\color{black}m_{n,A}}^{-1}(\xx)\cdot\frac{(-2\bar A^{-1}\uu,1)}{\nu_{\bar A}(\uu)}=(\bar A^{-1}\xx',|A^{-1}\xx|^2)\cdot\frac{(-2\bar A^{-1}\uu,1)}{\nu_{\bar A}(\uu)}\\
\displaystyle =\frac{-2\bar A^{-1}\xx'\cdot\bar A^{-1}\uu+|A^{-1}\xx|^2}{\nu_{\bar A}(\uu)}\\
\displaystyle =\frac{|A^{-1}(\xx-(\uu,0))|^2-|\bar A^{-1}\uu|^2}{\nu_{\bar A}(\uu)}=\frac{t^2-|\bar A^{-1}\uu|^2}{\nu_{\bar A}(\uu)}.
\end{array}
\end{equation}
\end{proof}
We define the function $k(\zz)$ on $\RR^n$ by
$$
k(\zz)=\left\{\begin{array}{ll}\displaystyle\frac{f\circ {\color{black}m_{n,A}}(\zz)}{\sqrt{z_n-|\zz'|^2 }}&\mbox{ if }0\leq|\zz'|^2<z_n,\\
0&\mbox{ otherwise,}\end{array}\right.
$$
where $\zz=(\zz',z_n)\in\RR^n$.
This is equivalent for $x_n>0$, to
$$
f(\xx)=x_na_n^{-1} k\circ {\color{black}m_{n,A}}^{-1}(\xx)=x_na_n^{-1} k(\bar A^{-1} \xx',|A^{-1}\xx|^2),
$$
where $\xx=(\xx',x_n)\in\RR^n$.
By the evenness of $f$, we have
\begin{equation}\label{eq:ffromk}
f(\xx)=|x_n|a_n^{-1} k(\bar A^{-1} \xx',|A^{-1}\xx|^2).
\end{equation}
Let the regular Radon transform $Rk(\boldsymbol{e_\theta},s)$ be defined by
$$
Rk(\boldsymbol{e_\theta},s)=\int\limits _{\boldsymbol{e_\theta}^\perp}k(s\boldsymbol{e_\theta}+\boldsymbol\eta)d\boldsymbol\eta, \qquad (\boldsymbol{e_\theta},s)\in S^{n-1}\times\RR,
$$
where $s\in\RR$ and for $\boldsymbol\theta=(\theta_1,\theta_2,\cdots,\theta_{n-1})\in[0,2\pi)\times[0,\pi]^{n-2},$
$$
\boldsymbol{e_\theta}
\left(\begin{array}c\sin\theta_1\sin\theta_2\cdots\sin\theta_{n-1}\\
\cos\theta_1\sin\theta_2\cdots\sin\theta_{n-1}\\
\cos\theta_2\sin\theta_3\cdots\sin\theta_{n-1}\\
\vdots\\
\cos\theta_{n-2}\sin\theta_{n-1}\\
\cos\theta_{n-1}\end{array}\right)\in S^{n-1}.
$$
This can be represented by
\begin{equation}\label{eq:radondefi}
Rk(\boldsymbol{e_\theta},s)
\frac1{|\cos\theta_{n-1}|}\int\limits _{\RR^{n-1}}k\left(\zz',-\frac{\boldsymbol{e_\theta'}\cdot\zz'}{\cos\theta_{n-1}}+\frac s{\cos\theta_{n-1}}\right)d\zz'
\end{equation}
for $\theta_{n-1}\neq\pi/2$ and $\boldsymbol{e_\theta}=(\boldsymbol{e_\theta'},\cos\theta_{n-1})\in S^{n-1}$.
The following theorem shows the relation between ${\color{black}R_{E,A}}f$ and $Rk$.
\begin{thm}\label{thm:mainrelation}
Let $f\in C(\RR^n)$ be even in $x_n$ and have compact support in $\RR^{n-1}\times\RR\setminus\{0\}$.
Then we have
\begin{equation}\label{eq:mainrel}
{\color{black}R_{E,A}}f(\uu,t)=\frac{2|\mathbf a|_1t}{\nu_{\bar A}(\uu)}Rk\left(\frac{(-2\bar A^{-1}\uu,1)}{\nu_{\bar A}(\uu)},\frac{t^2-|\bar A^{-1}\uu|^2}{\nu_{\bar A}(\uu)}\right).
\end{equation}
Again, $\nu_{\bar A}(\uu)=\sqrt{1+4|\bar A^{-1}\uu|^2}$.
\end{thm}
\begin{proof}
Combining two equations (\ref{eq:defiref}) and (\ref{eq:ffromk}), we have
$$
{\color{black}R_{E,A}}f(\uu,t)=\displaystyle2t^{n}|\mathbf a|_1\int\limits _{|\yy'|\leq 1}k(\yy' t+\bar A^{-1} \uu,|\yy' t+\bar A^{-1} \uu |^2+(1-|\yy'|^2)t^2)d\yy'.
$$
Let us consider the variable change
$$
\zz'=\yy't+\bar A^{-1} \uu\qquad\left(\Longleftrightarrow \yy'=\frac{\zz'-\bar A^{-1} \uu}t\right).
$$
The Jacobian of this transformation is $t^{1-n}$, so
$$
{\color{black}R_{E,A}}f(\uu,t)=\displaystyle2|\mathbf a|_1t\int\limits _{|\zz'-\bar A^{-1}\uu|\leq t}k(\zz',|\zz'|^2+t^2-|\zz'-\bar A^{-1}\uu|^2)d\zz'.
$$
Since $k$ has compact support in $\{\zz\in\RR^2:|\zz'|^2<z_n\}$, we have
\begin{equation}\label{eq:refrk}
\begin{array}{ll}
{\color{black}R_{E,A}}f(\uu,t)&\displaystyle=2|\mathbf a|_1t\int\limits _{\RR^{n-1}}k(\zz',|\zz'|^2+t^2-|\zz'-\bar A^{-1}\uu|^2)d\zz'\\
&=\displaystyle2|\mathbf a|_1t\int\limits _{\RR^{n-1}}k(\zz',t^2-|\bar A^{-1}\uu|^2+2(\bar A^{-1}\uu)\cdot\zz')d\zz'.
\end{array}
\end{equation}
We recognize the right hand side as the integral along the hyperplane perpendicular to
$$
(-2\bar A^{-1}\uu,1)/\nu_{\bar A}(\uu)
$$
with (signed) distance
$$
(t^2-|\bar A^{-1}\uu|^2)/\nu_{\bar A}(\uu)
$$
from the origin.
In this case, the measure for the hyperplane becomes $\nu_{\bar A}(\uu)d\zz'.$
Setting
$$
\boldsymbol{e_\theta}=(-2\bar A^{-1}\uu,1)/\nu_{\bar A}(\uu),\qquad\mbox{and}\qquad s=(t^2-|\bar A^{-1}\uu|^2)/\nu_{\bar A}(\uu)
$$
in equation~\eqref{eq:radondefi} we have the desired formula from equation~\eqref{eq:refrk}.
\end{proof}
Using the projection slice theorem for the regular Radon transform, we obtain an analog of the projection slice theorem for the elliptical Radon transform ${\color{black}R_{E,A}}f$:
\begin{thm}\label{thm:projectionslice}
Let $f\in C^\infty(\RR^n)$ be even in $x_n$ and have compact support in $\RR^{n-1}\times\RR\setminus\{0\}$.
Then we have for $(\boldsymbol\alpha,\beta)\in\RR^{n-1}\times\RR$,
$$
\hat k(\boldsymbol\alpha,\beta)=|\mathbf a|_1^{-1}e^{i\frac{|\boldsymbol\alpha|^2}{4\beta}}\int\limits ^\infty_0{\color{black}R_{E,A}} f\left(-\frac{\bar A \boldsymbol\alpha}{2\beta},t\right)e^{-i\beta t^2}dt
$$
where $\hat k$ is the $n$-dimensional Fourier transform of $k$.
\end{thm}
\begin{proof}
The projection slice theorem implies
$$
\hat k(\sigma\boldsymbol{e_\theta})=\int\limits_{\mathbb{R}} Rk(\boldsymbol{e_\theta},s)e^{-is\sigma}ds.
$$
To use this theorem, we multiply equation~\eqref{eq:mainrel} by $e^{-i\frac{t^2-|\bar A^{-1}\uu|^2}{\nu_{\bar A}(\uu)}\sigma}$ and integrate $\frac{t^2-|\bar A^{-1}\uu|^2}{\nu_{\bar A}(\uu)}$ from $-|\bar A^{-1}\uu|^2/\nu_{\bar A}(\uu)$ to infinity:
$$
\begin{array}{ll}
\displaystyle\int\limits ^\infty_{-|\bar A^{-1}\uu|^2/\nu_{\bar A}(\uu)} Rk\left(\frac{(-2\bar A^{-1}\uu,1)}{\nu_{\bar A}(\uu)},\frac{t^2-|\bar A^{-1}\uu|^2}{\nu_{\bar A}(\uu)}\right)e^{-i\frac{t^2-|\bar A^{-1}\uu|^2}{\nu_{\bar A}(\uu)}\sigma}d\left(\frac{t^2-|\bar A^{-1}\uu|^2}{\nu_{\bar A}(\uu)}\right).
\end{array}
$$
Since $k$ has compact support in $\{\zz\in\RR^n:|\zz'|^2<z_n\}$, the plane perpendicular to $\frac{(-2\bar A^{-1}\uu,1)}{\nu_{\bar A}(\uu)}$ with distance from the origin less than $\frac{-|\bar A^{-1}\uu|^2}{\nu_{\bar A}(\uu)} $ does not intersect the compact support of $k$.
Hence, $\hat k(\sigma/\nu_{\bar A}(\uu)(-2\bar A^{-1}\uu,1))$ is equal to
$$
\begin{array}{ll}
&\displaystyle\int\limits ^\infty_{-|\bar A^{-1}\uu|^2/\nu_{\bar A}(\uu)} Rk\left(\frac{(-2\bar A^{-1}\uu,1)}{\nu_{\bar A}(\uu)},\frac{t^2-|\bar A^{-1}\uu|^2}{\nu_{\bar A}(\uu)}\right)e^{-i\frac{t^2-|\bar A^{-1}\uu|^2}{\nu_{\bar A}(\uu)}\sigma}d\left(\frac{t^2-|\bar A^{-1}\uu|^2}{\nu_{\bar A}(\uu)}\right)\\
&=\displaystyle\int\limits ^\infty_{-|\bar A^{-1}\uu|^2/\nu_{\bar A}(\uu)} \frac{\nu_{\bar A}(\uu)}{2|\mathbf a|_1t} {\color{black}R_{E,A}}f(\uu,t)e^{-i\frac{t^2-|\bar A^{-1}\uu|^2}{\nu_{\bar A}(\uu)}\sigma}d\left(\frac{t^2-|\bar A^{-1}\uu|^2}{\nu_{\bar A}(\uu)}\right)\\
&=\displaystyle\frac{1}{|\mathbf a|_1}\int\limits ^\infty_{0} {\color{black}R_{E,A}}f(\uu,t)e^{-i\frac{t^2-|\bar A^{-1}\uu|^2}{\nu_{\bar A}(\uu)}\sigma}dt
\end{array}
$$
where in the second line, we used Theorem~\ref{thm:mainrelation}.
Setting $\boldsymbol\alpha=-2\sigma\bar A^{-1}\uu/\nu_{\bar A}(\uu)\in\RR^{n-1}$ and $\beta=\sigma/\nu_{\bar A}(\uu)\in\RR$ completes our proof.
\end{proof}
Taking the inverse Fourier transform of a function $k$ and using equation (\ref{eq:ffromk}), we have the following inversion formula:
\begin{cor}
Let $f\in C^\infty(\RR^n)$ be even in $x_n$ and have compact support in $\RR^{n-1}\times\RR\setminus\{0\}$.
Then $f(\xx)$ can be reconstructed as follows:
\begin{equation}\label{eq:kinversion}
\displaystyle\frac{(-1)^{n-1}|x_n| }{2\pi^{n} |\mathbf a|_1^2}\int\limits _{\RR^{n-1}}\left.\Lambda_t ({\color{black}R_{E,A}}f)( \boldsymbol\alpha, t)\right|_{t=|\bar A^{-1}(\boldsymbol\alpha-\xx')|^2+a_n^{-2}x_n^2}d\boldsymbol\alpha,
\end{equation}
where
$$
\Lambda_t h(t)=\int\limits_{\mathbb{R}} \int\limits ^\infty_0 e^{i(t^2-\tau^2)\beta}|\beta|^{n-1}h(\tau)d\tau d\beta.
$$
\end{cor}
\begin{proof}
We get for $x_n>0$,
$
\begin{array}{ll}
f(\xx)&=\displaystyle\frac{x_n }{a_n} k(\bar A^{-1}\xx',|A^{-1}\xx|^2)\\
&=\displaystyle\displaystyle\frac{x_n }{(2\pi)^n a_n}\int\limits _{\RR^{n-1}} \int\limits_{\mathbb{R}} e^{i\bar A^{-1}\xx' \cdot \boldsymbol\alpha}e^{i\beta |A^{-1}\xx|^2}\hat k(\boldsymbol\alpha,\beta) d\beta d\boldsymbol\alpha\\
&=\displaystyle\displaystyle\frac{x_n }{(2\pi)^n a_n|\mathbf a|_1}\int\limits _{\RR^{n-1}}\int\limits_{\mathbb{R}} e^{i\frac{|\boldsymbol\alpha|^2}{4\beta}}e^{i\bar A^{-1}\xx' \cdot \boldsymbol\alpha}e^{i\beta |A^{-1}\xx|^2}G\left(-\frac{\bar A \boldsymbol\alpha}{2\beta}, \beta\right) d\beta d\boldsymbol\alpha,
\end{array}
$$
where in the first line, we used equation (\ref{eq:ffromk}) and in the last line, we used Theorem~\ref{thm:projectionslice} and
$$
G(\uu,w)=\int\limits ^\infty_0{\color{black}R_{E,A}} f(\uu,t)e^{-iwt^2}dt.
$$
The evenness of $f$ in $x_n$ gives us
$$
f(\xx)= \displaystyle\frac{|x_n| }{(2\pi)^n a_n|\mathbf a|_1}\int\limits _{\RR^{n-1}}\int\limits_{\mathbb{R}} e^{i\frac{|\boldsymbol\alpha|^2}{4\beta}} e^{i\boldsymbol\alpha\cdot(\bar A^{-1}\xx')}e^{i\beta |A^{-1}\xx|^2}G\left(-\frac{\bar A \boldsymbol\alpha}{2\beta}, \beta\right) d\beta d\boldsymbol\alpha .
$$
Changing the variables $-\bar A\aalpha/(2\beta)\to\aalpha$, $f(\xx)$ is equal to
$$
\displaystyle\frac{(-2)^{n-1}|x_n| }{(2\pi)^n |\mathbf a|_1^2}\int\limits _{\RR^{n-1}}\int\limits_{\mathbb{R}} e^{i\beta(|\bar A^{-1}\boldsymbol\alpha|^2-2(\bar A^{-1}\boldsymbol\alpha)\cdot(\bar A^{-1}\xx')+|A^{-1}\xx|^2)}G\left( \boldsymbol\alpha, \beta\right) |\beta|^{n-1} d\beta d\boldsymbol\alpha .
$$
By definition of $G$, $f(\xx)$ can be determined through
$$
\displaystyle\frac{(-1)^{n-1}|x_n| }{2\pi^{n} |\mathbf a|_1^2}\int\limits _{\RR^{n-1}}\int\limits_{\mathbb{R}} \int\limits ^\infty_0 e^{i\beta(|\bar A^{-1}\boldsymbol\alpha-\bar A^{-1}\xx'|^2+a_n^{-2}x_n^2-t^2)}{\color{black}R_{E,A}}f( \boldsymbol\alpha, t) |\beta|^{n-1}dtd\beta d\boldsymbol\alpha .
$$
\end{proof}
When $n=2$, equation \eqref{eq:kinversion} becomes
$$
\begin{array}{ll}
f(\xx)&=\displaystyle\frac{|x_2| }{2\pi^{2} a_1^2a_2^2}\int\limits_{\mathbb{R}} \intR\int\limits ^\infty_0 e^{i\beta(|a_1^{-1}(\alpha-x_1)|^2+a_2^{-2}x_2^2-t^2)}{\color{black}R_{E,A}}f(\alpha, t) |\beta|dtd\beta d\alpha\\
&=-\displaystyle\frac{|x_2| }{\pi^{2} a_1^2a_2^2}\int\limits_{\mathbb{R}} \int\limits ^\infty_0 \frac{{\color{black}R_{E,A}}f(\alpha, t)}{(|a_1^{-1}(\alpha-x_1)|^2+a_2^{-2}x_2^2-t^2)^2}dt d\alpha,
\end{array}
$$
since in the distribution sense,
$$
\int\limits_{\mathbb{R}} e^{ix\beta}|\beta|d\beta=-\frac{2}{x^2}.
$$
\section{2-dimensional numerical implementation}
Here we discuss the results of 2-dimensional numerical implementations
In 2-dimension, equation~\eqref{eq:mainrel} becomes
\begin{equation}\label{eq:refrk2}
{\color{black}R_{E,A}}f(u,t)=\frac{2 a_1a_2t}{\sqrt{1+4|a_1^{-1}u|^2}}Rk\left(\frac{(-2a_1^{-1}u,1)}{\sqrt{1+4|a_1^{-1}u|^2}},\frac{t^2-|a_1^{-1}u|^2}{\sqrt{1+4|a_1^{-1}u|^2}}\right).
\end{equation}
Setting $(\cos\theta,\sin\theta)=(-2a_1^{-1}u,1)/\sqrt{1+4|a_1^{-1}u|^2}\in S^{1}$ and
$$
s=\frac{t^2-|a_1^{-1}u|^2}{\sqrt{1+4|a_1^{-1}u|^2}}
$$
in equation~(\ref{eq:refrk2}), we have
\begin{equation*
2R k(\cos\theta,\sin\theta,s)=\displaystyle\frac{|\csc\theta|}{a_1a_2h(s,\theta)}{\color{black}R_{E,A}}f\left(-\frac{a_1\cot\theta}2,h(s,\theta)\right)\mbox{ if }s\csc\theta>-\frac{\cot^2\theta}4
\end{equation*}
where
$$
h(s,\theta)=\sqrt{s\csc\theta+\frac{\cot^2\theta}4}.
$$
(While $Rk(\sin\theta,\cos\theta,s)$ is used in Theorem~\ref{thm:mainrelation}, $Rk(\cos\theta,\sin\theta,s)$ is used to utilize the built-in function in \textsc{Matlab}.
Hence a small change is required.)
Again, since $k$ has compact support in $\{\zz\in\RR^2:|z_1|^2<z_2\}$, the plane perpendicular to $\boldsymbol{e_\theta}=(\cos\theta,\sin\theta)$ with distance from the origin less than $-\frac{\cot^2\theta}{4\csc\theta}$ does not intersect the compact support of $k$.
Therefore, we have
\begin{equation}\label{eq:mainrelation}
2R k(\boldsymbol{e_\theta},s)=\left\{\begin{array}{ll}\displaystyle\frac{|\csc\theta|}{a_1a_2h(s,\theta)}{\color{black}R_{E,A}}f\left(-\frac{a_1\cot\theta}2,h(s,\theta)\right)&\mbox{ if }s\csc\theta>-\dfrac{\cot^2\theta}4,\\
0&\mbox{otherwise,}\end{array}\right.
\end{equation}
First of all, $a_1$ and $a_2$ were set to be 0.8 and 1, respectively.
In the experiments presented here we use the phantom shown in Figure~\ref{fig:spherical} (a).
The phantom, supported within the rectangle $[-1,1]\times[-1,1]$, is the sum of eight characteristic functions of disks.
Notice that it has to be even with respect to the $x_2$-axis and there are four characteristic functions of disks centered at $(0.2,0.4)$, $(0,0.5)$, $(-0.3,0.3)$, and $(-0.5,0.2)$ with radii 0.2 0.15 0.05 and 0.05, whose values are 1, 0.5, 1.5, and 2 above the $x_1$-axis.
Hence we also include their reflection below the $x_1$-axis.
(Actually, our phantom has support in
$$
\{\xx\in\RR^2:(x_1/a_1)^2+((x_1/a_1)^2+x_2^2)^2<1\}.
$$
This implies that $k$ has support in the unit ball and this makes it sufficient to consider the range $[-1,1]$ in $s$.)
The $256\times256$ images are used in Figure~\ref{fig:spherical}.
To reconstruct the image in Figure~\ref{fig:spherical} (b), we have $256\times256$ projections for $\theta\in[0,2\pi]$ and $s\in[-1,1]$ in equation~(\ref{eq:mainrelation}).
All projections are computed by numerical integration.
After finding the function $k$ using the usual inversion code for the regular Radon transform, we obtain the function $f$ using equation~(\ref{eq:ffromk}).
(When using the inversion code for the regular Radon transform, the built-in function ``iradon'' in \textsc{Matlab} was used.
The function ``iradon'' is the inversion of the built-in function ``radon'' in \textsc{Matlab} which considers the number of the pixels where the line passes through.
Thus when computing ${\color{black}R_{E,A}}f$, we also considers the number of the pixels where the ellipse passes through.
We used the default version of the function in which the filter, whose aim is to deemphasize high frequencies, is set to the Ram-Lak filter and the interpolation is set to be linear.)
While Figure~\ref{fig:spherical} (b) demonstrates the image reconstructed from the exact data, Figure~\ref{fig:spherical} (c) shows the absolute value of the reconstruction from noisy data.
The noise is modeled by normally distributed random numbers and this is scaled so that its norm was equal to 5$\%$ of the norm of the exact data.
In Figure~\ref{fig:spherical} (c) the noisy data is modeled by adding the noise values scaled to 5$\%$ of the norm of the exact data to the exact data.
In Figure~\ref{fig:sphericalsurface}, the surface plots of Figure~\ref{fig:spherical} (a) and (b) are provided.
Another phantom and reconstruction are shown in Figure~\ref{fig:sphericalphantom}.
The reconstructed two dimensional data sets consisting of 256$\times$256 projections using the implemented inversion formula have been computed in less than one second (around 0.3 second) on a Intel(R) Cor(TM) i5-3470 CPU @ 3.20 GHz.
\begin{figure}[here]
\begin{center}
\subfigure[]{
\includegraphics[width=0.3\textwidth]{phantomnov.eps} }%
\subfigure[]{
\includegraphics[width=0.3\textwidth]{resultnov.eps}}
\subfigure[]{
\includegraphics[width=0.3\textwidth]{resultnoise5nov.eps}}
\end{center}
\caption{Reconstructions in 2 dimensions: (a) the phantom, (b) the reconstruction form exact data, and (c) the reconstruction from noisy data
\label{fig:spherical}
\end{figure}
\begin{figure}[here]
\begin{center}
\subfigure[]{
\includegraphics[width=0.45\textwidth]{phantomsurfacenov.eps} }%
\subfigure[]{
\includegraphics[width=0.45\textwidth]{resultsurfacenov.eps}}
\end{center}
\caption{Surface plots (a) the phantom and (b) the reconstruction from exact data
\label{fig:sphericalsurface}
\end{figure}
\begin{figure}[here]
\begin{center}
\subfigure[]{
\includegraphics[width=0.45\textwidth]{phantom1nov.eps} }%
\subfigure[]{
\includegraphics[width=0.45\textwidth]{result1nov.eps}}
\end{center}
\caption{Reconstruction in 2D: (a) the phantom and (b) the reconstruction from exact data }
\label{fig:sphericalphantom}
\end{figure}
\section{Conclusion}
This paper is devoted to the study of the elliptical Radon transform arising in seismic imaging.
We provide an inversion formula for the elliptical Radon transform by reducing this transform to the regular Radon transform.
Also, we demonstrate our algorithm by providing numerical simulations.
\section*{Acknowledgements}
The first author thanks C. Jung for fruitful discussions.
The first author was supported in part by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of science, ICT and future planning (2015R1C1A1A02037662)).
|
1,116,691,500,723 | arxiv | \section{Introduction}
The precise morphology of a gravitational waveform encodes a wealth of information about the binary that produced it.
Merging stellar-mass binaries are typically characterized by fifteen parameters: two mass parameters, six spin parameters, and seven extrinsic parameters, which describe the location and orientation of the binary with respect to the detector~\footnote{Binary neutron stars can be characterized by additional tidal parameters and all compact binaries can be characterized with additional eccentricity parameters, depending on their formation scenario.}.
Extracting binary parameters from gravitational-wave measurements enables tremendous science including sky maps for electromagnetic follow-up~\cite{GW170817_properties,GW170817_mma}, measurement of the neutron star equation of state~\cite{GW170817_tidal}, measurement of cosmological parameters~\cite{Schutz,GW170817_Hubble}, probing the fate of massive stars~\cite{mass_uc,mass,o2pop}, understanding the formation mechanisms of compact binaries~\cite{salvo,Stevenson,spin,GerosaBerti,FarrNature,Wysocki18,eccentricity}, and testing general relativity~\cite{GW150914_gr,GRB170817A}.
The parameters of compact binaries are estimated using Bayesian inference software~\cite{lalinference,PyCBCInference,bilby,RapidPE1,RapidPE2}.
The software employs nested sampling~\cite{Skilling2004}, Markov Chain Monte Carlo~\cite{Metropolis1953,Hastings1970,Hogg}, or adaptive mesh refinement~\cite{RapidPE1,RapidPE2} in order to construct posterior distributions for binary parameters and/or to calculate the Bayesian evidence.
Bayesian inference calculations in gravitational-wave astronomy are computationally demanding, and so significant research has been carried out in order to bring down the wall time of calculations, thereby enabling new science; see, e.g.,~\cite{smith,purrer,canizares,gpu_inference,gpu_rit}.
The computational demands of inference have created a premium for fast approximate gravitational waveforms or ``approximants''~\cite{IMRPhenomPv2,SEOBNRv2}.
Fast approximants have enabled breakthrough science.
However, the speed can come at a cost.
The current approximants most commonly used in gravitational-wave inference are constructed using only the leading order, $\ell=2$ modes in the spin-weighted spherical harmonic decomposition, although see~\cite{IMRPhenomHM}.
While these leading-order approximants provide reasonably good estimates of binary parameters, they do not incorporate all of the information in a gravitational waveform, and therefore provide an incomplete picture.
Inference with higher-order modes can provide tighter constraints than those obtained with leading-order waveforms alone.
In particular, higher-order modes are useful breaking degeneracy between binary parameters.
For example, the $\ell=|m|=2$ waveform is totally invariant under a transformation in which the polarization angle and phase of coalescence advance by $\pi/2$.
The ability to break this degeneracy is key to detecting gravitational-wave memory~\cite{memory}.
This is just one example highlighting the scientific potential of inference with higher-order modes.
Astrophysical inference with higher-order mode waveforms was first demonstrated in~\cite{Kumar,RapidPE2}, which applied a numerical relativity surrogate model~\cite{NRSur7dq2} to produce posterior distributions for GW150914, GW170104, GW170608, and GW170814.
More recently, adaptive mesh methods have been employed in order to derive posterior distributions and Bayesian evidence for GW170729 and other events in GWTC-1~\cite{gwtc-1} using a variety of higher-order-mode waveforms~\cite{Chatziioannou}.
In this paper we demonstrate a fast and effective method to calculate posterior distributions and Bayesian evidence for gravitational-wave signals with higher-order-modes using a technique known as ``importance sampling''; see, e.g.,~\cite{Robert&Casella,Liu}.
First, we carry out Bayesian parameter estimation using a low-cost, $\ell=|m|=2$ waveform, which yields an approximate answer on which we can improve.
In the second step, we calculate a weight factor for each posterior sample, which incorporates information from the more expensive higher-order-mode waveform.
Using the weights, we obtain the posterior and evidence, which we would have obtained if we had carried out the entire calculation using the more expensive higher-order mode waveform.
The remainder of this paper is organized as follows.
In Section~\ref{method}, we describe the reweighting formalism.
Then, in Section~\ref{results}, we show results obtained for both simulated data and for events in the LIGO/Virgo catalog, GWTC-1~\cite{gwtc-1,losc}.
Posterior samples and Bayesian evidence for every event in GWTC-1 are available on a companion web page~\cite{hom-git}, along with the code used in our analysis.
We provide a table summarizing the evidence of higher-order modes in GWTC-1.
We show that, while higher-order modes produce tighter constraints than $\ell=2$ analyses, there is not yet a strong signature of higher-order modes in published LIGO/Virgo detections.
In Section~\ref{conclusions}, we discuss other possible directions for future research including novel applications of this reweighting technique, which may be useful for a variety of problems in astrophysics.
Finally, we include an appendix, Section~\ref{catalog_posteriors}, with the rest of the results from GWTC-1.
\section{Methodology}\label{method}
We show how to carry out Bayesian inference using an approximate ``proposal'' likelihood ${\cal L}_{\O}(d|\theta)$ to obtain initial posterior samples, which are then reweighted using a more computationally expensive ``target'' likelihood ${\cal L}(d|\theta)$.
Here, $d$ is the data and $\theta$ represents the parameters.
The proposal\ likelihood is an approximation for the target\ likelihood.
In order for reweighting to be efficient, the proposal\ likelihood should be similar to the target\ likelihood, so that the two likelihoods overlap significantly.
For demonstration purposes, we use as our proposal\ waveform {\sc IMRPhenomD}~\cite{IMRPhenomD}, an aligned-spin, $\ell=|m|=2$ approximant, which is widely used in astrophysical inference thanks to its reliability and speed.
For our target\ waveform, we use {\sc NRHybSur3dq8}\xspace~\cite{NRHybSur3dq8}, a numerical relativity surrogate model, which includes higher-order modes up to $\ell=4$ excepting $(4,\pm1)$ and $(4,0)$, but including $(5,\pm5)$, aligned spin, and mass ratios $m_2/m_1\geq 0.125$.
Our goal is to derive expressions for the target\ posterior
\begin{align}
p(\theta|d) = \frac{{\cal L}(d|\theta) \pi(\theta)}{\cal Z} ,
\end{align}
and the target\ Bayesian evidence
\begin{align}
{\cal Z} = \int d\theta {\cal L}(d|\theta) \pi(\theta) ,
\end{align}
written in terms of a fast-to-calculate, proposal\ likelihood.
The proposal\ quantities are linked to the target\ quantities by a weight factor.
Multiplying the target\ posterior by unity, we obtain
\begin{align}
p(\theta|d) = & \frac{{\cal L}_{\O}(d|\theta)}{{\cal L}_{\O}(d|\theta)}
\frac{{\cal L}(d|\theta) \pi(\theta)}{\cal Z} \nonumber\\
= & w(d|\theta) \frac{{\cal L}_{\O}(d|\theta) \pi(\theta)}{\cal Z} .
\end{align}
Here,
\begin{align}
w(d|\theta) \equiv \frac{{\cal L}(d|\theta)}{{\cal L}_{\O}(d|\theta)} ,
\end{align}
is the weight function.
Multiplying by unity again, we obtain the following expression for the evidence
\begin{align}
{\cal Z} = & {\cal Z}_{\O} \int d\theta \, p_{\O}(\theta|d)
\left(\frac{{\cal L}(d|\theta)}{{\cal L}_{\O}(d|\theta)}\right) \nonumber\\
= & \frac{{\cal Z}_{\O}}{n}\sum_k^n w(d|\theta_k) .
\end{align}
The second line replaces the integral with a discrete sum over $n$ proposal\ posterior samples; see~\cite{intro}.
Carrying out Bayesian inference with the proposal\ likelihood, we obtain ``proposal\ posterior samples'' for the distribution
\begin{align}
p_{\O}(\theta|d) = \frac{{\cal L}_{\O}(d|\theta) \pi(\theta)}{{\cal Z}_{\O}} ,
\end{align}
where ${\cal Z_{\O}}$ is the proposal\ evidence.
We generate our proposal\ samples using the {\sc Bilby}~\cite{bilby} implementation of {\sc CPNest}~\cite{cpnest}.
We compute the weights using likelihoods marginalized over the coalescence time and reference phase (see, e.g.,~\cite{intro}) in order to avoid differences in the definition of these parameters between waveform models.
In a previous iteration of this work~\cite{arxiv}, we employed a slightly different approach.
Instead of marginalizing over time and phase, we chose the values of these parameters that maximized the overlap ${\cal O}$ of the proposal waveform and target waveform:
\begin{align}
{\cal O} \equiv \max_{t,\phi} \frac{\langle h_{\O}^+, h^+ \rangle + \langle h_{\O}^\times, h^\times \rangle}{\sqrt{\left(\langle h_{\O}^+,h_{\O}^+\rangle + \langle h_{\O}^\times,h_{\O}^\times\rangle\right) \left(\langle h^+,h^+\rangle + \langle h^\times,h^\times\rangle\right)}} .
\end{align}
Here, $h_{\O}^{+,\times}$ are the plus and cross components of the proposal\ waveform while $h^{+,\times}$ are the plus and cross components of the target\ waveform.
The angled brackets denote noise-weighted inner products.
This approach, used subsequently in other publications, e.g.,~\cite{isobel}, produces qualitatively similar results to marginalizing over ($t$,$\phi$), albeit with reduced computational efficiency.
Moreover, the marginalization approach is better justified from a statistical point of view.
Weighting each sample by $w(d|\theta)$, and renormalizing, we convert the proposal\ posterior samples into target\ posterior samples.
The application of weight factors has the effect of reducing the effective number of samples~\cite{Kish,Elvira}
\begin{align}
n_\text{eff} = \frac{\left(\sum_k w_k\right)^2}{\sum_k w_k^2} .
\end{align}
After reweighting, it is therefore prudent to calculate $n_\text{eff}$ in order to determine that there are a suitably large number of samples.
It is straightforward to generate more weighted posterior samples by simply combining the results from multiple proposal\ analyses run in parallel.
The method of likelihood reweighting outlined here is similar to the procedure of ``recycling'' commonly used to study the population properties of compact objects; see, e.g.,~\cite{o2pop,intro}.
Previous applications of recycling have, in effect, carried out reweighting to change the {\em prior} in post-processing.
The principle here is the same, except we change the likelihood.
Our formalism can be straightforwardly extended to simultaneously alter the prior (informed by astrophysics) and likelihood (using more sophisticated waveforms).
Reweighting is an application of importance sampling.
For additional details, readers are referred to~\cite{Robert&Casella,Liu}.
\section{Results}\label{results}
We demonstrate likelihood reweighting using a simulated binary black hole merger signal injected into Gaussian noise.
We assume a two-detector LIGO network operating at design sensitivity~\cite{aligo}.
Using {\sc NRHybSur3dq8}\xspace, we inject a binary black hole waveform.
The binary, located at a luminosity distance $d_L=\unit[400]{Mpc}$, has chirp mass $\mathcal{M}=30 M_\odot$ and mass ratio $q=m_2/m_1=0.8$.
The dimensionless aligned spins are $\chi_1=0.4,\chi_2=0.3$.
The signal has a network matched-filter signal-to-noise ratio of $\rho_\text{mf}=55$.
In Fig.~\ref{fig:comparison_corner_plot} we provide a corner plot showing the posterior distribution and credible intervals obtained for this simulated event.
The blue shades indicate the posterior derived using our proposal\ {\sc IMRPhenomD} waveform while the green shades indicates the posterior obtained after reweighting with our target\ {\sc NRHybSur3dq8}\xspace waveform.
The darkness of the contours indicate credible intervals at $1\sigma,2\sigma,3\sigma$.
The true values of each parameter are indicated by orange markers.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.56\linewidth]{qlow_full_marg_intrinsic.pdf}
\includegraphics[width=0.405\linewidth]{qlow_full_marg_extrinsic.pdf}
\caption{
Posterior distributions for a simulated binary black hole waveform.
The blue distributions show the posteriors obtained using the approximant {\sc IMRPhenomD}~\cite{IMRPhenomD}, which includes only the dominant $\ell=|m|=2$ modes.
The green distribution shows the posteriors obtained using the approximant {\sc NRHybSur3dq8}\xspace~\cite{NRHybSur3dq8}, which includes higher-order modes.
The true parameters are indicated with orange lines.
Left: intrinsic binary parameters (total mass, mass ratio, and effective aligned spin). Right: extrinsic parameters (luminosity distance and binary inclination).
}
\label{fig:comparison_corner_plot}
\end{figure*}
While both posteriors includes the true parameter values, the blue {\sc IMRPhenomD} posterior is broad in comparison to the green {\sc NRHybSur3dq8}\xspace posterior.
For some parameters, the posterior shrinks dramatically when we add information from higher-order modes.
Higher-order modes break the degeneracy between distance and inclination while improving our ability to measure the mass ratio.
This, in turn, improves our estimation of the spins.
We calculate the Bayes factor comparing the hypothesis that the data are best fit by {\sc NRHybSur3dq8}\xspace to the hypothesis that they are best fit by {\sc IMRPhenomD}.
This is a measure of degree to which the data prefer a model including higher-order modes \footnote{To be clear, we are not proposing $\ell|m|=22$ gravitational waves as a serious alternative to general relativity. Rather, we are using this Bayes factor in order to ascertain if we can resolve the higher-order mode content predicted by general relativity.}.
The log Bayes factor for our simulated event are reported in the first row of Tab.~\ref{tab:BF}.
We include also the ``efficiency,'' the number of effective samples used in each calculation, normalized by the number of proposal\ samples.
\begin{center}
\begin{table}
\begin{tabular}{ |c|c|c| }
\hline
event & $\ln\text{BF}$ & $n_\text{eff}/n$ \\\hline
simulated & 9.01 & $2.2\times10^{-3}$\\\hline
GW150914 & -0.20 & 0.31\\
GW151012 & 0.40 & 0.36 \\
GW151226 & 0.01 & 0.82\\
GW170104 & 0.01 & 0.57 \\
GW170608 & -0.16 & 0.73\\
GW170729 & 1.23 & 0.13\\
GW170809 & -0.09 & 0.67\\
GW170814 & 0.19 & 0.48\\
GW170818 & 0.31 & 0.74 \\
GW170823 & -0.21 & 0.69 \\\hline
GWTC-1 & 1.49 & N/A\\\hline
\end{tabular}
\caption{
The log Bayes factor and the ``efficiency,'' equal to the number of effective samples $n_\text{eff}$ divided by the initial number of samples $n$.
}
\label{tab:BF}
\end{table}
\end{center}
For the simulated event, the signal-to-noise ratio is sufficiently large to ``detect'' the presence of higher-order modes with a high significance $\ln\text{BF}=9.01$.
Since this simulated event has a high signal-to-noise ratio, the ratio of the effective number of samples to the number of proposal\ samples is small, $\approx0.2\%$.
Thus, a large number of proposal\ runs ($\approx1000$) are required in order to produce a well-sampled target\ posterior.
Fortunately, these proposal\ runs are embarrassingly parallel, which means that the wall time is no longer than a single proposal\ inference run, provided sufficient computational resources are available.
It should also be noted that we are able to get posterior samples and evidence for all of the events in GWTC-1 with just a few parallel runs.
We further note that it may be possible to improve efficiency by restricting the proposal\ prior based on early returns from the target\ posterior.
The events in GWTC-1 have lower signal-to-noise ratio, and so the reweighting procedure is much more efficient.
We now turn toward real data in the GWTC-1 catalog.
For each event, we produce: a set of posterior samples with weights, the proposal\ Bayesian evidence (obtained with {\sc IMRPhenomD}), the target\ Bayesian evidence (obtained with {\sc NRHybSur3dq8}\xspace), the {\sc NRHybSur3dq8}\xspace/{\sc IMRPhenomD} Bayes factor, and corner plots with credible intervals.
The full results are available here:~\cite{hom-git}.
Key summary statistics are provided in Tab.~\ref{tab:BF}.
For illustrative purposes, we also include in Fig.~\ref{fig:GW170729} the corner plot for GW170729, the event with the greatest support for higher-order modes ($\ln\text{BF}=1.23$).
Our target\ posterior for GW170729 is qualitatively similar to previous results from~\cite{Chatziioannou}.
We observe increased support for non-unity mass ratio $q=m_2/m_1$, slightly more support for zero-spin, and changes to the posterior distributions of the extrinsic parameters.
An additional validation of the results for GW170729 are included in Appendix~\ref{validation}.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.56\linewidth]{GW170729_marg_intrinsic.pdf}
\includegraphics[width=0.405\linewidth]{GW170729_marg_extrinsic.pdf}
\caption{
Posterior distributions for GW170729.
The blue distributions show the posteriors obtained using the $\ell=|m|=2$ approximant {\sc IMRPhenomD}~\cite{IMRPhenomD}.
The green distribution shows the posteriors obtained using the approximant {\sc NRHybSur3dq8}\xspace~\cite{NRHybSur3dq8}, which includes higher-order modes.
Left: intrinsic binary parameters (total mass, mass ratio, and effective aligned spin). Right: the extrinsic parameters, luminosity distance and inclination angle.
}
\label{fig:GW170729}
\end{figure*}
\section{Discussion}\label{conclusions}
As the gravitational-wave catalog grows, higher-order modes will provide a more precise view of the properties of compact objects.
Higher-order modes become increasingly important for systems: with large mass ratios $q\lesssim0.5$, with significant spin, and with large total mass~\cite{vijay}.
The method described here will be particularly useful for exploring such events.
In order to make use of likelihood reweighting, it is necessary to employ a suitable proposal\ model, capable of producing posterior samples, which provide a reasonable approximation of the true distribution.
For inference with higher-order modes, this appears to be achievable using quadrupolar approximants.
One must exercise a degree of caution when choosing the proposal\ likelihood to make sure it is suitable for the problem at hand.
For example, binary mergers with very low mass ratios $q\lesssim 1/6$ (not yet observed by LIGO/Virgo) can produce so much higher-order mode content that the quadrupolar approximants yield biased parameter estimates~\cite{bustillo,varma}.
There are two telltale signs, which indicate that the proposal\ likelihood is poorly suited to the problem.
First, the efficiency $n_\text{eff}/n$ will be low even for modest signal-to-noise ratio because the proposal\ samples are all in the tail of the target\ likelihood distribution.
The sample(s) closest to the target\ likelihood peak will be weighted as far more important than the other samples.
Second, the reweighted posterior distributions will be peak at the edges of the sampled region of parameter space.
By comparing the target\ posterior to the proposal, it is straightforward to determine if the target\ posterior is railing against the edge of the proposal\ samples.
If these two tests indicate a poorly chosen proposal\ likelihood, one should devise a better one.
We foresee a number of useful applications for likelihood reweighting including: inferences about gravitational-wave memory~\cite{memory}, noise models that add complexity beyond the usual Gaussian assumption, treatment of calibration errors, and inference with computationally expensive waveforms that include tidal effects or eccentricity~\footnote{After this paper appeared on the arxiv, our method was used in~\cite{isobel}.}.
\section{Acknowledgements}
We thank Rory Smith and Greg Ashton for sharing a beta version of their parallelized nested sampling algorithm, used for benchmarking in the appendix.
We thank Rory Smith for helping to debug our parallel inference runs.
We thank Katerina Chatziioannou, Richard O'Shaughnessy, and Vijay Varma for helpful comments.
We thank Moritz H{\"u}bner, and the {\sc Bilby} team for support.
This is document LIGO-P1900128.
EP, ET, and CT are supported by ARC CE170100004.
ET is supported by ARC FT150100281.
This research has made use of data, software and/or web tools obtained from the Gravitational Wave Open Science Center (https://www.gw-openscience.org), a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration. LIGO is funded by the U.S. National Science Foundation. Virgo is funded by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale della Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by Polish and Hungarian institutes.
\section{Introduction}
The precise morphology of a gravitational waveform encodes a wealth of information about the binary that produced it.
Merging stellar-mass binaries are typically characterized by fifteen parameters: two mass parameters, six spin parameters, and seven extrinsic parameters, which describe the location and orientation of the binary with respect to the detector~\footnote{Binary neutron stars can be characterized by additional tidal parameters and all compact binaries can be characterized with additional eccentricity parameters, depending on their formation scenario.}.
Extracting binary parameters from gravitational-wave measurements enables tremendous science including sky maps for electromagnetic follow-up~\cite{GW170817_properties,GW170817_mma}, measurement of the neutron star equation of state~\cite{GW170817_tidal}, measurement of cosmological parameters~\cite{Schutz,GW170817_Hubble}, probing the fate of massive stars~\cite{mass_uc,mass,o2pop}, understanding the formation mechanisms of compact binaries~\cite{salvo,Stevenson,spin,GerosaBerti,FarrNature,Wysocki18,eccentricity}, and testing general relativity~\cite{GW150914_gr,GRB170817A}.
The parameters of compact binaries are estimated using Bayesian inference software~\cite{lalinference,PyCBCInference,bilby,RapidPE1,RapidPE2}.
The software employs nested sampling~\cite{Skilling2004}, Markov Chain Monte Carlo~\cite{Metropolis1953,Hastings1970,Hogg}, or adaptive mesh refinement~\cite{RapidPE1,RapidPE2} in order to construct posterior distributions for binary parameters and/or to calculate the Bayesian evidence.
Bayesian inference calculations in gravitational-wave astronomy are computationally demanding, and so significant research has been carried out in order to bring down the wall time of calculations, thereby enabling new science; see, e.g.,~\cite{smith,purrer,canizares,gpu_inference,gpu_rit}.
The computational demands of inference have created a premium for fast approximate gravitational waveforms or ``approximants''~\cite{IMRPhenomPv2,SEOBNRv2}.
Fast approximants have enabled breakthrough science.
However, the speed can come at a cost.
The current approximants most commonly used in gravitational-wave inference are constructed using only the leading order, $\ell=2$ modes in the spin-weighted spherical harmonic decomposition, although see~\cite{IMRPhenomHM}.
While these leading-order approximants provide reasonably good estimates of binary parameters, they do not incorporate all of the information in a gravitational waveform, and therefore provide an incomplete picture.
Inference with higher-order modes can provide tighter constraints than those obtained with leading-order waveforms alone.
In particular, higher-order modes are useful breaking degeneracy between binary parameters.
For example, the $\ell=|m|=2$ waveform is totally invariant under a transformation in which the polarization angle and phase of coalescence advance by $\pi/2$.
The ability to break this degeneracy is key to detecting gravitational-wave memory~\cite{memory}.
This is just one example highlighting the scientific potential of inference with higher-order modes.
Astrophysical inference with higher-order mode waveforms was first demonstrated in~\cite{Kumar,RapidPE2}, which applied a numerical relativity surrogate model~\cite{NRSur7dq2} to produce posterior distributions for GW150914, GW170104, GW170608, and GW170814.
More recently, adaptive mesh methods have been employed in order to derive posterior distributions and Bayesian evidence for GW170729 and other events in GWTC-1~\cite{gwtc-1} using a variety of higher-order-mode waveforms~\cite{Chatziioannou}.
In this paper we demonstrate a fast and effective method to calculate posterior distributions and Bayesian evidence for gravitational-wave signals with higher-order-modes using a technique known as ``importance sampling''; see, e.g.,~\cite{Robert&Casella,Liu}.
First, we carry out Bayesian parameter estimation using a low-cost, $\ell=|m|=2$ waveform, which yields an approximate answer on which we can improve.
In the second step, we calculate a weight factor for each posterior sample, which incorporates information from the more expensive higher-order-mode waveform.
Using the weights, we obtain the posterior and evidence, which we would have obtained if we had carried out the entire calculation using the more expensive higher-order mode waveform.
The remainder of this paper is organized as follows.
In Section~\ref{method}, we describe the reweighting formalism.
Then, in Section~\ref{results}, we show results obtained for both simulated data and for events in the LIGO/Virgo catalog, GWTC-1~\cite{gwtc-1,losc}.
Posterior samples and Bayesian evidence for every event in GWTC-1 are available on a companion web page~\cite{hom-git}, along with the code used in our analysis.
We provide a table summarizing the evidence of higher-order modes in GWTC-1.
We show that, while higher-order modes produce tighter constraints than $\ell=2$ analyses, there is not yet a strong signature of higher-order modes in published LIGO/Virgo detections.
In Section~\ref{conclusions}, we discuss other possible directions for future research including novel applications of this reweighting technique, which may be useful for a variety of problems in astrophysics.
Finally, we include an appendix, Section~\ref{catalog_posteriors}, with the rest of the results from GWTC-1.
\section{Methodology}\label{method}
We show how to carry out Bayesian inference using an approximate ``proposal'' likelihood ${\cal L}_{\O}(d|\theta)$ to obtain initial posterior samples, which are then reweighted using a more computationally expensive ``target'' likelihood ${\cal L}(d|\theta)$.
Here, $d$ is the data and $\theta$ represents the parameters.
The proposal\ likelihood is an approximation for the target\ likelihood.
In order for reweighting to be efficient, the proposal\ likelihood should be similar to the target\ likelihood, so that the two likelihoods overlap significantly.
For demonstration purposes, we use as our proposal\ waveform {\sc IMRPhenomD}~\cite{IMRPhenomD}, an aligned-spin, $\ell=|m|=2$ approximant, which is widely used in astrophysical inference thanks to its reliability and speed.
For our target\ waveform, we use {\sc NRHybSur3dq8}\xspace~\cite{NRHybSur3dq8}, a numerical relativity surrogate model, which includes higher-order modes up to $\ell=4$ excepting $(4,\pm1)$ and $(4,0)$, but including $(5,\pm5)$, aligned spin, and mass ratios $m_2/m_1\geq 0.125$.
Our goal is to derive expressions for the target\ posterior
\begin{align}
p(\theta|d) = \frac{{\cal L}(d|\theta) \pi(\theta)}{\cal Z} ,
\end{align}
and the target\ Bayesian evidence
\begin{align}
{\cal Z} = \int d\theta {\cal L}(d|\theta) \pi(\theta) ,
\end{align}
written in terms of a fast-to-calculate, proposal\ likelihood.
The proposal\ quantities are linked to the target\ quantities by a weight factor.
Multiplying the target\ posterior by unity, we obtain
\begin{align}
p(\theta|d) = & \frac{{\cal L}_{\O}(d|\theta)}{{\cal L}_{\O}(d|\theta)}
\frac{{\cal L}(d|\theta) \pi(\theta)}{\cal Z} \nonumber\\
= & w(d|\theta) \frac{{\cal L}_{\O}(d|\theta) \pi(\theta)}{\cal Z} .
\end{align}
Here,
\begin{align}
w(d|\theta) \equiv \frac{{\cal L}(d|\theta)}{{\cal L}_{\O}(d|\theta)} ,
\end{align}
is the weight function.
Multiplying by unity again, we obtain the following expression for the evidence
\begin{align}
{\cal Z} = & {\cal Z}_{\O} \int d\theta \, p_{\O}(\theta|d)
\left(\frac{{\cal L}(d|\theta)}{{\cal L}_{\O}(d|\theta)}\right) \nonumber\\
= & \frac{{\cal Z}_{\O}}{n}\sum_k^n w(d|\theta_k) .
\end{align}
The second line replaces the integral with a discrete sum over $n$ proposal\ posterior samples; see~\cite{intro}.
Carrying out Bayesian inference with the proposal\ likelihood, we obtain ``proposal\ posterior samples'' for the distribution
\begin{align}
p_{\O}(\theta|d) = \frac{{\cal L}_{\O}(d|\theta) \pi(\theta)}{{\cal Z}_{\O}} ,
\end{align}
where ${\cal Z_{\O}}$ is the proposal\ evidence.
We generate our proposal\ samples using the {\sc Bilby}~\cite{bilby} implementation of {\sc CPNest}~\cite{cpnest}.
We compute the weights using likelihoods marginalized over the coalescence time and reference phase (see, e.g.,~\cite{intro}) in order to avoid differences in the definition of these parameters between waveform models.
In a previous iteration of this work~\cite{arxiv}, we employed a slightly different approach.
Instead of marginalizing over time and phase, we chose the values of these parameters that maximized the overlap ${\cal O}$ of the proposal waveform and target waveform:
\begin{align}
{\cal O} \equiv \max_{t,\phi} \frac{\langle h_{\O}^+, h^+ \rangle + \langle h_{\O}^\times, h^\times \rangle}{\sqrt{\left(\langle h_{\O}^+,h_{\O}^+\rangle + \langle h_{\O}^\times,h_{\O}^\times\rangle\right) \left(\langle h^+,h^+\rangle + \langle h^\times,h^\times\rangle\right)}} .
\end{align}
Here, $h_{\O}^{+,\times}$ are the plus and cross components of the proposal\ waveform while $h^{+,\times}$ are the plus and cross components of the target\ waveform.
The angled brackets denote noise-weighted inner products.
This approach, used subsequently in other publications, e.g.,~\cite{isobel}, produces qualitatively similar results to marginalizing over ($t$,$\phi$), albeit with reduced computational efficiency.
Moreover, the marginalization approach is better justified from a statistical point of view.
Weighting each sample by $w(d|\theta)$, and renormalizing, we convert the proposal\ posterior samples into target\ posterior samples.
The application of weight factors has the effect of reducing the effective number of samples~\cite{Kish,Elvira}
\begin{align}
n_\text{eff} = \frac{\left(\sum_k w_k\right)^2}{\sum_k w_k^2} .
\end{align}
After reweighting, it is therefore prudent to calculate $n_\text{eff}$ in order to determine that there are a suitably large number of samples.
It is straightforward to generate more weighted posterior samples by simply combining the results from multiple proposal\ analyses run in parallel.
The method of likelihood reweighting outlined here is similar to the procedure of ``recycling'' commonly used to study the population properties of compact objects; see, e.g.,~\cite{o2pop,intro}.
Previous applications of recycling have, in effect, carried out reweighting to change the {\em prior} in post-processing.
The principle here is the same, except we change the likelihood.
Our formalism can be straightforwardly extended to simultaneously alter the prior (informed by astrophysics) and likelihood (using more sophisticated waveforms).
Reweighting is an application of importance sampling.
For additional details, readers are referred to~\cite{Robert&Casella,Liu}.
\section{Results}\label{results}
We demonstrate likelihood reweighting using a simulated binary black hole merger signal injected into Gaussian noise.
We assume a two-detector LIGO network operating at design sensitivity~\cite{aligo}.
Using {\sc NRHybSur3dq8}\xspace, we inject a binary black hole waveform.
The binary, located at a luminosity distance $d_L=\unit[400]{Mpc}$, has chirp mass $\mathcal{M}=30 M_\odot$ and mass ratio $q=m_2/m_1=0.8$.
The dimensionless aligned spins are $\chi_1=0.4,\chi_2=0.3$.
The signal has a network matched-filter signal-to-noise ratio of $\rho_\text{mf}=55$.
In Fig.~\ref{fig:comparison_corner_plot} we provide a corner plot showing the posterior distribution and credible intervals obtained for this simulated event.
The blue shades indicate the posterior derived using our proposal\ {\sc IMRPhenomD} waveform while the green shades indicates the posterior obtained after reweighting with our target\ {\sc NRHybSur3dq8}\xspace waveform.
The darkness of the contours indicate credible intervals at $1\sigma,2\sigma,3\sigma$.
The true values of each parameter are indicated by orange markers.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.56\linewidth]{qlow_full_marg_intrinsic.pdf}
\includegraphics[width=0.405\linewidth]{qlow_full_marg_extrinsic.pdf}
\caption{
Posterior distributions for a simulated binary black hole waveform.
The blue distributions show the posteriors obtained using the approximant {\sc IMRPhenomD}~\cite{IMRPhenomD}, which includes only the dominant $\ell=|m|=2$ modes.
The green distribution shows the posteriors obtained using the approximant {\sc NRHybSur3dq8}\xspace~\cite{NRHybSur3dq8}, which includes higher-order modes.
The true parameters are indicated with orange lines.
Left: intrinsic binary parameters (total mass, mass ratio, and effective aligned spin). Right: extrinsic parameters (luminosity distance and binary inclination).
}
\label{fig:comparison_corner_plot}
\end{figure*}
While both posteriors includes the true parameter values, the blue {\sc IMRPhenomD} posterior is broad in comparison to the green {\sc NRHybSur3dq8}\xspace posterior.
For some parameters, the posterior shrinks dramatically when we add information from higher-order modes.
Higher-order modes break the degeneracy between distance and inclination while improving our ability to measure the mass ratio.
This, in turn, improves our estimation of the spins.
We calculate the Bayes factor comparing the hypothesis that the data are best fit by {\sc NRHybSur3dq8}\xspace to the hypothesis that they are best fit by {\sc IMRPhenomD}.
This is a measure of degree to which the data prefer a model including higher-order modes \footnote{To be clear, we are not proposing $\ell|m|=22$ gravitational waves as a serious alternative to general relativity. Rather, we are using this Bayes factor in order to ascertain if we can resolve the higher-order mode content predicted by general relativity.}.
The log Bayes factor for our simulated event are reported in the first row of Tab.~\ref{tab:BF}.
We include also the ``efficiency,'' the number of effective samples used in each calculation, normalized by the number of proposal\ samples.
\begin{center}
\begin{table}
\begin{tabular}{ |c|c|c| }
\hline
event & $\ln\text{BF}$ & $n_\text{eff}/n$ \\\hline
simulated & 9.01 & $2.2\times10^{-3}$\\\hline
GW150914 & -0.20 & 0.31\\
GW151012 & 0.40 & 0.36 \\
GW151226 & 0.01 & 0.82\\
GW170104 & 0.01 & 0.57 \\
GW170608 & -0.16 & 0.73\\
GW170729 & 1.23 & 0.13\\
GW170809 & -0.09 & 0.67\\
GW170814 & 0.19 & 0.48\\
GW170818 & 0.31 & 0.74 \\
GW170823 & -0.21 & 0.69 \\\hline
GWTC-1 & 1.49 & N/A\\\hline
\end{tabular}
\caption{
The log Bayes factor and the ``efficiency,'' equal to the number of effective samples $n_\text{eff}$ divided by the initial number of samples $n$.
}
\label{tab:BF}
\end{table}
\end{center}
For the simulated event, the signal-to-noise ratio is sufficiently large to ``detect'' the presence of higher-order modes with a high significance $\ln\text{BF}=9.01$.
Since this simulated event has a high signal-to-noise ratio, the ratio of the effective number of samples to the number of proposal\ samples is small, $\approx0.2\%$.
Thus, a large number of proposal\ runs ($\approx1000$) are required in order to produce a well-sampled target\ posterior.
Fortunately, these proposal\ runs are embarrassingly parallel, which means that the wall time is no longer than a single proposal\ inference run, provided sufficient computational resources are available.
It should also be noted that we are able to get posterior samples and evidence for all of the events in GWTC-1 with just a few parallel runs.
We further note that it may be possible to improve efficiency by restricting the proposal\ prior based on early returns from the target\ posterior.
The events in GWTC-1 have lower signal-to-noise ratio, and so the reweighting procedure is much more efficient.
We now turn toward real data in the GWTC-1 catalog.
For each event, we produce: a set of posterior samples with weights, the proposal\ Bayesian evidence (obtained with {\sc IMRPhenomD}), the target\ Bayesian evidence (obtained with {\sc NRHybSur3dq8}\xspace), the {\sc NRHybSur3dq8}\xspace/{\sc IMRPhenomD} Bayes factor, and corner plots with credible intervals.
The full results are available here:~\cite{hom-git}.
Key summary statistics are provided in Tab.~\ref{tab:BF}.
For illustrative purposes, we also include in Fig.~\ref{fig:GW170729} the corner plot for GW170729, the event with the greatest support for higher-order modes ($\ln\text{BF}=1.23$).
Our target\ posterior for GW170729 is qualitatively similar to previous results from~\cite{Chatziioannou}.
We observe increased support for non-unity mass ratio $q=m_2/m_1$, slightly more support for zero-spin, and changes to the posterior distributions of the extrinsic parameters.
An additional validation of the results for GW170729 are included in Appendix~\ref{validation}.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.56\linewidth]{GW170729_marg_intrinsic.pdf}
\includegraphics[width=0.405\linewidth]{GW170729_marg_extrinsic.pdf}
\caption{
Posterior distributions for GW170729.
The blue distributions show the posteriors obtained using the $\ell=|m|=2$ approximant {\sc IMRPhenomD}~\cite{IMRPhenomD}.
The green distribution shows the posteriors obtained using the approximant {\sc NRHybSur3dq8}\xspace~\cite{NRHybSur3dq8}, which includes higher-order modes.
Left: intrinsic binary parameters (total mass, mass ratio, and effective aligned spin). Right: the extrinsic parameters, luminosity distance and inclination angle.
}
\label{fig:GW170729}
\end{figure*}
\section{Discussion}\label{conclusions}
As the gravitational-wave catalog grows, higher-order modes will provide a more precise view of the properties of compact objects.
Higher-order modes become increasingly important for systems: with large mass ratios $q\lesssim0.5$, with significant spin, and with large total mass~\cite{vijay}.
The method described here will be particularly useful for exploring such events.
In order to make use of likelihood reweighting, it is necessary to employ a suitable proposal\ model, capable of producing posterior samples, which provide a reasonable approximation of the true distribution.
For inference with higher-order modes, this appears to be achievable using quadrupolar approximants.
One must exercise a degree of caution when choosing the proposal\ likelihood to make sure it is suitable for the problem at hand.
For example, binary mergers with very low mass ratios $q\lesssim 1/6$ (not yet observed by LIGO/Virgo) can produce so much higher-order mode content that the quadrupolar approximants yield biased parameter estimates~\cite{bustillo,varma}.
There are two telltale signs, which indicate that the proposal\ likelihood is poorly suited to the problem.
First, the efficiency $n_\text{eff}/n$ will be low even for modest signal-to-noise ratio because the proposal\ samples are all in the tail of the target\ likelihood distribution.
The sample(s) closest to the target\ likelihood peak will be weighted as far more important than the other samples.
Second, the reweighted posterior distributions will be peak at the edges of the sampled region of parameter space.
By comparing the target\ posterior to the proposal, it is straightforward to determine if the target\ posterior is railing against the edge of the proposal\ samples~\footnote{Of course, if the target\ posterior is multi-modal, there is an additional risk that the proposal\ posterior might miss a mode, which would not be apparent through visual inspection. We have not observed such multi-modal behavior in the examples considered in this manuscript, but one can guard against this failure mode through $pp$ tests as described in the appendix.}.
If these two tests indicate a poorly chosen proposal\ likelihood, one should devise a better one.
We foresee a number of useful applications for likelihood reweighting including: inferences about gravitational-wave memory~\cite{memory}, noise models that add complexity beyond the usual Gaussian assumption, treatment of calibration errors, and inference with computationally expensive waveforms that include tidal effects or eccentricity~\footnote{After this paper appeared on the arxiv, our method was used in~\cite{isobel}.}.
\section{Acknowledgements}
We thank Rory Smith and Greg Ashton for sharing a beta version of their parallelized nested sampling algorithm, used for benchmarking in the appendix.
We thank Rory Smith for helping to debug our parallel inference runs.
We thank Katerina Chatziioannou, Richard O'Shaughnessy, and Vijay Varma for helpful comments.
We thank Moritz H{\"u}bner, and the {\sc Bilby} team for support.
This is document LIGO-P1900128.
EP, ET, and CT are supported by ARC CE170100004.
ET is supported by ARC FT150100281.
This research has made use of data, software and/or web tools obtained from the Gravitational Wave Open Science Center (https://www.gw-openscience.org), a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration. LIGO is funded by the U.S. National Science Foundation. Virgo is funded by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale della Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by Polish and Hungarian institutes.
|
1,116,691,500,724 | arxiv | \section{introduction}
The properties of conventional bulk superconductors are well described by the Bardeen-Cooper-Schrieffer (BCS) mean field theory.~\cite{BCS} The BCS theory is valid in the limit when the pairing gap $\Delta$ is much larger than the single-particle mean-level spacing $\delta$. However, in small metallic grains, the discreteness of the spectrum is important and the mean-level spacing can be comparable or larger than the pairing gap. This is the fluctuation-dominated regime, in which BCS theory is no longer a good approximation.
The reduced BCS Hamiltonian was used extensively to study the properties of small metallic grains.~\cite{Muhlschlegel72, Vondelft96, Matveev97, Mastellone98, Braun98,Dukelsky00, Vondelft01} It was found that pairing correlations in the crossover between the bulk BCS limit and the fluctuation-dominated regime manifest through the number-parity dependence of thermodynamic quantities such as the spin susceptibility~\cite{Dilorenzo00,Schechter01,VanHoucke06b,Alhassid07} and the heat capacity.~\cite{Falci02,VanHoucke06b,Alhassid07}
However, the effective low-energy interaction between electrons in a metallic grain contains additional terms beyond the reduced BCS Hamiltonian. Such residual interactions could have significant effects on the signatures of pairing correlations in a finite-size grain. Finding this effective interaction is, in general, a difficult task. However, a remarkably simple effective Hamiltonian emerges in grains whose single-particle dynamics are chaotic or weakly diffusive (in the presence of disorder) in the limit of a large Thouless conductance $g_T$.~\cite{Kurland00, Aleiner02} In such grains the single-particle Hamiltonian of $\sim g_T$ levels around the Fermi energy is described by random matrix theory.~\cite{Mehta91,Alhassid00} The randomness of the single-particle wave functions induce randomness into the corresponding electron-electron interaction matrix elements. These matrix elements can then be decomposed into their average and fluctuating parts. The average interaction is determined by symmetry considerations,~\cite{Kurland00, Alhassid05} and includes, in addition to the classical charging energy, a Cooper-channel BCS-like interaction and an exchange interaction that is proportional to the square of the total spin $\hat{\bf S}$ of the grain. This average interaction together with the one-body Hamiltonian describe the so-called universal Hamiltonian.~\cite{Kurland00,Aleiner02} Residual interaction terms are of the order $1/g_T$ and can be ignored in the limit of large $g_T$.
Much work has been devoted to the understanding of pairing correlations in finite-size systems and, in particular, in small metallic grains.~\cite{Vondelft01} Exchange correlations were also studied extensively in semiconductor quantum dots,~\cite{Alhassid02,Alhassid03,Kiselev06,Burmistrov10} where the pairing interaction is repulsive and can thus be ignored. Much less is known about the properties of a superconducting grain in the presence of both pairing and exchange correlations.
In Ref.~\onlinecite{Schmidt07} we studied the phase diagram of the ground-state spin of a metallic grain that is described by the universal Hamiltonian. The competition between superconductivity and ferromagnetism leads to a narrow coexistence regime in the $J_s/\delta-\Delta/\delta$ plane. This regime can be broadened and tuned by an external Zeeman field. Signatures of this coexistence were identified in the mesoscopic fluctuations of the conductance peak spacings and conductance peak heights in a metallic grain that is weakly coupled to leads.~\cite{Schmidt08} Here we study the competition between pairing and exchange correlations in thermodynamic properties of the grain. In particular, we determine how the signatures of pairing correlations are affected by the spin-exchange interaction. Our studies cover the crossover from the fluctuation-dominated regime to the BCS regime. They are based on a quantum Monte Carlo method that is particularly suitable for the universal Hamiltonian.
The outline of this paper is as follows: the model we use to describe the metallic grain (i.e., the universal Hamiltonian) is discussed in Sec.~\ref{sec:model}, while the quantum Monte Carlo method and in particular its application to the universal Hamiltonian is explained in Sec.~\ref{sec:numapp}. Various thermodynamic properties are calculated in Sec.~\ref{sec:thermprop}. In
particular, we discuss the thermal spin distributions (Sec.~\ref{subsec:spin}), the number of $S=0$ electron pairs (Sec.~\ref{subsec:np}), the canonical pair gap (Sec.~\ref{subsec:cangap}),
the heat capacity (Sec.~\ref{subsec:sh}) and the spin susceptibility
(Sec.~\ref{subsec:sus}). Our conclusions are given in Sec.~\ref{conclusion}.
\section{The model}\label{sec:model}
The universal Hamiltonian of a metallic grain is given by~\cite{Kurland00, Aleiner02}
\begin{equation}
\hat{H} = \sum_{k \sigma} \epsilon_k \hat{c}^{\dag}_{k\sigma}
\hat{c}^{\phantom{\dag}}_{k \sigma} + E_C \hat{N}^2 - G \hat{P}^{\dag}\hat{P} - J_{s} \hat{\bf S}^2~,
\label{eq:universham}
\end{equation}
where $\hat{c}^{\dag}_{k\sigma}$ are creation operators of electrons in spin-degenerate ($\sigma=\pm$) single-particle states with energy $\epsilon_k$, $\hat{N} = \sum_{k \sigma}\hat{c}^{\dag}_{k \sigma}\hat{c}^{\phantom{\dag}}_{k \sigma}$ is the particle-number operator, $\hat{\bf S} = \frac{1}{2} \sum_{k \sigma \sigma'}
\hat{c}^{\dag}_{k \sigma} {\bf \sigma}_{\sigma,\sigma'}
\hat{c}^{\phantom{\dag}}_{k \sigma'}$ is the total spin operator of the grain (${\bf \sigma}$ are Pauli matrices), and $\hat{P}^{\dag} = \sum_k \hat{c}^{\dag}_{k,+}\hat{c}^{\dag}_{k,-}$ is the pair creation operator in time-reversed (spin up/spin down) orbitals. $E_C$ is the charging energy of the grain, while the parameters $G$
and $J_{s}$ are the coupling constants in the Cooper channel and in the exchange
channel, respectively. The universal Hamiltonian describes an isolated mesoscopic grain whose single-particle dynamics are chaotic (or weakly diffusive in a disordered grain) in the limit where the Thouless conductance $g_T \to \infty$. It can be derived from general symmetry considerations.~\cite{Kurland00,Aleiner02,Alhassid05}
Although the form of the universal Hamiltonian in Eq.~(\ref{eq:universham})
is based on the chaotic (or diffusive) nature of the single-particle states,
we do not study here the mesoscopic fluctuations, but assume a generic equidistant
single-particle spectrum (i.e., a picket-fence spectrum) as our benchmark model.
We consider a half-filled band of $2N_o+1$ doubly degenerate levels. The even grain contains $N=2 N_o$ electrons while the odd grain contains $N=2N_o+1$ electrons. The single-particle energies are given by $\epsilon_k = k \delta$ with $k=-N_o,\ldots, N_o$. All energy scales in this work are measured in units of the single-particle mean level spacing $\delta$, and for simplicity we take $\delta=1$.
The reduced BCS model with an attractive pairing force is characterized by two regimes: the fluctuation-dominated regime or perturbative regime $\Delta/\delta \ll 1$ ($\Delta$ is the zero-temperature BCS gap), and the BCS superconducting regime or non-perturbative regime $\Delta/\delta \gg 1$. Here we study the thermodynamics of
the universal Hamiltonian for three different values of $\Delta/\delta$ in the crossover between the fluctuation-dominated regime and the BCS regime: $\Delta/\delta = 0.5, 1$ and $5$. The effective pairing strengths $G$ that correspond to these BCS gaps depend on the band width and are calculated using the appropriate renormalization method.~\cite{Berger98,Alhassid07}
The thermodynamic properties of the reduced BCS model (in the absence of exchange interaction) are universal functions of $T/\delta$ that depend only on $\Delta/\delta$, i.e., changing the model-space size for a fixed $\Delta/\delta$ and renormalizing $G$ leaves the thermodynamic quantities invariant.~\cite{Alhassid07} Of course, choosing a smaller model space restricts the temperature range in which the model is physically meaningful because of truncation effects. In this work we calculate thermodynamic properties for even (odd) grains with $N=50$ ($N=51$) electrons in a half-filled band (of width $N_o=25$) around the Fermi energy in the presence of both pairing and exchange correlations. As long as the number of blocked levels (i.e., singly occupied levels) is much smaller than the total number of levels in the band, the thermodynamic quantities are still universal function of $T/\delta$, but now they depend on two parameters: $\Delta/\delta$ and $J_{s}/\delta$. As
the bandwidth is truncated, $J_s$ remains invariant while the renormalization
of $G$ is approximately independent of $J_s$. We have tested this numerically;
thermodynamic functions for a band width of $N_o=50$ were reproduced by
considering a grain with a band width of $N_o=25$ and an appropriately renormalized coupling strength $G$. As
an example, we show in Fig.~\ref{fig:renorm} the heat capacity and spin
susceptibility for even and odd grains with $N_o=50$ and $N_o=25$. The
pairing strength is renormalized to keep the BCS pairing gap fixed at $\Delta/\delta = 5$, while the exchange coupling is fixed at $J_s = 0.6\,\delta$.
\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0, width=8.5cm] {renorm.eps}
\caption{The heat capacity $C$ and the spin susceptibility $\chi$ [normalized
to its bulk high temperature limit $\chi_P$ (Pauli susceptibility) for $J_s=0$] as a function of temperature $T$ for an even and an odd grain. Results are shown at half filling for band widths of $N_o=25$ and $N_o=50$. The pairing strength $G$ is renormalized such that the BCS pairing gap is kept fixed at $\Delta/\delta =5$, while the exchange constant is fixed at $J_s = 0.6\,\delta$. The coincidence of the corresponding thermodynamic quantities for both band widths demonstrates that the renormalization of $G$ is approximately independent of $J_s$.
\label{fig:renorm}}
\end{center}
\end{figure}
Throughout this work we consider values of the spin coupling constant $J_s$ ranging from $0$ to
$0.91\,\delta$. Values for $J_{s}$ ranging from
$J_s/\delta \approx -0.03 - 0.09$ for copper to $J_{s}/\delta \approx 0.84-0.89$ for palladium were reported in Ref.~\onlinecite{Gorokhov04} (extracted from both experiment and theory).
Since all thermal averages we calculate are canonical (i.e., for a fixed number of electrons), the charging energy term $E_C \hat{N}^2$ is just an overall constant. We therefore put $E_C = 0$ in our calculations without loss of generality.
\section{Quantum Monte Carlo approach}\label{sec:numapp}
To compute thermodynamic properties of a metallic grain, we use a quantum Monte Carlo (QMC) method that is based on the canonical loop updates of Refs.~\onlinecite{Rombouts06, VanHoucke06a}. There, it was shown that this method can be used to simulate the reduced BCS
Hamiltonian [i.e., Eq.~(\ref{eq:universham}) with $J_{s} = 0$] in the canonical ensemble at finite temperature. The QMC method starts from a perturbative expansion of the partition function at inverse temperature $\beta$
\begin{eqnarray}
{\rm{Tr}} \big( e^{-\beta \hat{H}} \big)
& = & \sum_{m=0}^{\infty}
\int_{0}^{\beta} d \tau_m \int_0^{\tau_m} d \tau_{m-1} \cdots \int_0^{\tau_2} d \tau_1
\nonumber\\
& & {\rm{Tr}} \big[\hat{V}(\tau_1) \hat{V}(\tau_2)
\cdots \hat{V}(\tau_m)
e^{-\beta \hat{H}_D}\big],
\label{eq:decompos}
\end{eqnarray}
where $\hat{V}(\tau) = $exp$(-\tau\hat{H}_D) \hat{V} $exp$(\tau\hat{H}_D)$. The Hamiltonian in Eq.~(\ref{eq:decompos}) is assumed to consist of two non-commuting parts, $\hat{H}_D$ and $\hat{V}$.
In case of the reduced BCS Hamiltonian, these were chosen to be
\begin{eqnarray}
\hat{H} & = & \hat{H}_D - \hat{V}, \\
\hat{H}_D & = & \sum_{k \sigma} \epsilon_k \hat{c}^{\dag}_{k \sigma}
\hat{c}^{\phantom{\dag}}_{k \sigma} - G \sum_k \hat{c}^{\dag}_{k,+}\hat{c}^{\dag}_{k,-}
\hat{c}^{\phantom{\dag}}_{k,-}\hat{c}^{\phantom{\dag}}_{k,+}, \label{eq:H0} \\
\hat{V} & = & G \sum_{k \neq l } \hat{c}^{\dag}_{k,+}\hat{c}^{\dag}_{k,-}
\hat{c}^{\phantom{\dag}}_{l,-}\hat{c}^{\phantom{\dag}}_{l,+} \;.
\label{eq:Vpairing}
\end{eqnarray}
The basic idea of the QMC method is to insert a so-called worm operator $\hat{A}$ in the partition function, obtaining an extended partition function ${\rm{Tr}}
\big(\hat{A}e^{-\beta\hat{H}} \big)$.
By propagating this worm operator through imaginary
time according to the rules explained in Refs.~\onlinecite{Rombouts06,VanHoucke06a},
one generates configurations that are
distributed according to the weights occurring in the canonical partition function ${\rm{Tr}}_N
\big( e^{-\beta\hat{H}}\big)$ through a Markov process. The worm propagation
rules are constructed such that the detailed balance condition is satisfied.
In case of the reduced BCS Hamiltonian, the worm operator consists of two parts: one that enables scattering of $S=0$ pairs, and another that enables the breakup of an $S=0$ pair (thus creating two blocked levels).
To study the universal Hamiltonian, it is necessary to include the exchange
interaction term. In general, terms that commute with $\hat{H}_D$ can be incorporated in the current
algorithm by adding them to $\hat{H}_D$. The exchange term $-J_{s} \hat{\bf S}^2$ commutes with $\hat{H}_D$ in Eq.~(\ref{eq:H0}), and
only unpaired electrons (that block levels) contribute to the
total spin $S$. For a given number $b$ of blocked levels, the
degeneracy of many-particle levels in the total spin $S$ is given by
\begin{equation}
d_b(S) = \binom{b}{S+\frac{b}{2}} - \binom{b}{S+1+\frac{b}{2}} \;.
\label{eq:Sweight}
\end{equation}
Since the number $b$ of blocked levels
is known at each step of the Markov process, one can simply take the spin exchange term into
account by adding it to $\hat{H}_D$ and choosing the total
spin of the configuration with a probability proportional to the degeneracy $d_b(S)$.
The non-diagonal part $\hat{V}$ remains the same as for the reduced BCS
model [see Eq.~(\ref{eq:Vpairing})] and there is no change in the canonical loop updates.
\section{Thermodynamic properties}\label{sec:thermprop}
In the following we use the QMC method to study various thermodynamic properties of the grain.
\subsection{Thermal spin distributions}\label{subsec:spin}
We first study the spin distribution at fixed temperature. For that purpose, we consider the ratio of the spin-projected partition function $Z_S$ (at spin $S$) to the
total partition function for a fixed number of electrons
\begin{equation}
\frac{Z_S}{Z} = \frac{{\rm{Tr}}_{N,S}
e^{-\beta\hat{H}}}{{\rm{Tr}}_{N}e^{-\beta\hat{H}}}.
\label{eq:zsz}
\end{equation}
$Z_S$ is normalized such that $\sum_S (2S+1) Z_S/Z = 1$ [i.e., the $(2S+1)$-fold
degeneracy in the spin-projection quantum number $M$ {\bf is} not included in $Z_S$].
We first discuss the case of a pure exchange interaction ($G=0$), for which the ratios (\ref{eq:zsz}) can be expressed in closed form in terms of canonical quantities of non-interacting spinless fermions {\bf ($G=J_{s}=0$)} using the method of Ref.~\onlinecite{Alhassid03}. For $G=0$, we can rewrite (\ref{eq:zsz}) as
\begin{equation}
\frac{Z_S}{Z} = \frac{e^{\beta J_{s}S(S+1)} {\rm{Tr}}_{N,S}
e^{-\beta\hat{H}_0} }{\sum_{S} (2S+1) e^{\beta J_{s}S(S+1)} {\rm{Tr}}_{N,S}
e^{-\beta\hat{H}_0} }\;,
\end{equation}
where $\hat{H}_0$ is the non-interacting Hamiltonian.
The spin-projected quantities can be calculated from the corresponding $M$-projected quantities using
\begin{equation}\label{spin_projection}
{\rm{Tr}}_{N,S} e^{-\beta\hat{H}_0} = {\rm{Tr}}_{N,M=S} e^{-\beta\hat{H}_0} - {\rm{Tr}}_{N,M=S+1} e^{-\beta\hat{H}_0}\;.
\end{equation}
The traces on the r.h.s. of Eq.~(\ref{spin_projection}) can be evaluated in terms of two particle-number projections that correspond to the number of spin-up and the number of spin-down electrons. This leads to~\cite{Alhassid03}
\begin{equation}
{\rm{Tr}}_{N,S} e^{-\beta\hat{H}_0} = e^{-\beta \tilde F_{N/2+S}}e^{-\beta \tilde F_{N/2-S}} - e^{-\beta \tilde F_{N/2+S+1}}e^{-\beta \tilde F_{N/2-S-1}} \;,
\end{equation}
where $\tilde F_q$ is the canonical free energy of $q$ non-interacting spinless fermions with a single-particle spectrum $\epsilon_k$.
In the presence of both exchange and pairing correlations, we evaluate the
ratio $Z_S/Z$ using the QMC method outlined in Sec.~\ref{sec:numapp}. The configurations
generated in the Markov process are distributed according to the weights
appearing in the partition
function ${\rm{Tr}} \big( e^{-\beta \hat{H}} \big)$. Since the degeneracy in
$S$ is known for each configuration [see Eq. (\ref{eq:Sweight})], the ratio $Z_S/Z$ can be evaluated directly through
\begin{equation}
\frac{Z_S}{Z} = \left\langle \frac{d_b(S)}{2^{b}} \right\rangle_{MC}\;,
\end{equation}
where $\langle \ldots \rangle_{MC}$ denotes averaging over all
the configurations generated by the Monte Carlo method, $d_b(S)$ is the degeneracy defined in
Eq. (\ref{eq:Sweight}), and $b$ is the number of blocked levels in the
configuration.
\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0, width=8.5cm] {zse.eps}
\caption{
Thermal ratios $f_S$ of the spin projected partition function to
the full partition function as a function of temperature for an even number
of electrons ($N=50$). The left column
shows the fractions $f_S$ in absence of the pairing interaction ($\Delta/\delta=0$)
for different spin couplings $J_{s}$ (shown in units of $\delta$). The middle (right) column corresponds to
to a gap of $\Delta/\delta = 0.5$ ($\Delta/\delta = 5$). The
different spin values $S$ are indicated by different symbols: $\circ$ ($S=0$),
$\square$ ($S=1$) and $\vartriangle$ ($S=2$).
\label{fig:zse}}
\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0, width=8.5cm] {zso.eps}
\caption{ As in Fig.~\ref{fig:zse} but for a grain with an odd number of electrons
($N=51$). The half-integer spin values $S$ are now indicated by the following symbols: $\circ$ ($S=1/2$),
$\square$ ($S=3/2$) and $\vartriangle$ ($S=5/2$).
\label{fig:zso}}
\end{center}
\end{figure}
We define the thermal fraction $f_S$ of a given spin $S$ by
\begin{equation}
f_S = (2S+1)\frac{Z_S}{Z} \;.
\end{equation}
Figure \ref{fig:zse} shows $f_s$ as a function of temperature for a grain with an even number of electrons. Results for a few lowest spin values are shown for various values of the exchange coupling $J_{s}$ (measured in units of $\delta$) and pairing gap $\Delta/\delta$.
The left column of Fig.~\ref{fig:zse} corresponds to electrons interacting only
through spin exchange ($G=0$ and thus $\Delta/\delta=0$). In the absence of both pairing and exchange interactions {\bf ($G=J_{s}=0$)}, the ground state for an even number of electrons is found by filling the lowest
single-particle energy levels by spin up/spin down electrons resulting in an $S=0$ ground state. At low temperatures ($T\lesssim 0.75\,\delta$), the $S=0$ states give
the largest contribution to the partition function. For higher temperatures, the contribution of higher spin states increases, and in the temperature region $0.75 \,\delta \lesssim T \lesssim 3.5\,\delta$ the largest contribution arises from the $S=1$ states.
The exchange interaction shifts down in energy states with $S \neq 0$; thus less thermal energy is required to excite these states and the fractions $f_S$ of non-zero spin values increase with $J_{s}$. For
$J_{s}=0.4\,\delta$, the $S=0$ states dominate only below $T=0.2\,\delta$. When $J_{s}\geq 0.5\,\delta$, the ground state acquires a finite non-zero spin, and has spin $S=3$ for $J_{s} = 0.845\,\delta$.
Table \ref{table:spinjumps} lists the values of $J_{s}$ at which the ground-state spin changes to a higher value (denoted by $S$) for $\Delta/\delta = 0, 0.5$ and
$1$.
For a strong pairing interaction ($\Delta/\delta = 5$), the system remains fully paired up to an exchange coupling of $J_{s}/\delta \approx 1$, at which it makes a transition to a fully polarized state.
These values of $J_s$ were obtained using Richardson's solution to the pairing
Hamiltonian via the method of Ref.~\onlinecite{Rombouts04}.
The ground-state spin diagram in the presence of pairing correlations and ferromagnetism at zero temperature was discussed in Ref.~\onlinecite{Schmidt07}.
In the presence of pairing correlations, low spin states are favored because the scattering of spin zero pairs lowers the free
energy. The middle column of Fig.~\ref{fig:zse} shows the fraction $f_S$
for a weak pairing force ($\Delta/\delta = 0.5$).
Comparing the fractions $f_S$ at $J_{s}= 0$ and $J_{s}= 0.4\,\delta$ with their corresponding values in the absence of pairing, we observe that the pairing interaction makes the $S=0$ channel more dominant at low temperatures. At higher temperatures pairing correlations are destroyed by thermal excitations, and there is almost no difference between the $\Delta/\delta=0$ and $\Delta/\delta=0.5$ cases. The results of a strong pairing force ($\Delta/\delta = 5$) are shown in the right column of Fig.~\ref{fig:zse}. $S=0$ states remain dominant up to higher temperatures and the spin fractions are less affected by the exchange interaction.
As we increase the pairing strength at fixed $J_s$,
the $S=0$ channel becomes more dominant at low temperatures, and
higher values of $J_s$ are required to make the transition to a higher spin ground state. This in turn affects the finite temperature behavior of the grain.
At a fixed pairing gap $\Delta/\delta$ and for increasing
$J_s$, the crossing point where the $S=1$ channel
becomes dominant shifts to lower temperatures.
Figure \ref{fig:zso} shows results analogous to Fig.~\ref{fig:zse}, but for a grain with an odd number of electrons and thus half-integer spin. In the absence of pairing, the odd grain has an $S=1/2$ ground state for $J_s=0$ and acquires higher spin for a sufficiently strong exchange interaction. As compared with the even case, higher values of $J_{s}$ are required to make the respective transitions to higher spin states (see Table I).
\begin{table}
\begin{tabular}{|c|c|c|c||c|c|c|c|}
\hline
\hline
\multicolumn{4}{|c||}{Even grains} & \multicolumn{4}{c|}{Odd grains} \\
\hline
\multicolumn{2}{|r|}{\phantom{SS} $\Delta/\delta=0$} & $\Delta/\delta=0.5$ & $\Delta/\delta=1$ &
\multicolumn{2}{r|}{\phantom{SSS} $\Delta/\delta=0$} & $\Delta/\delta=0.5$ & $\Delta/\delta=1$ \\
\hline
S & \multicolumn{3}{c||}{$J_{s}$} & S & \multicolumn{3}{c|}{$J_{s}$} \\
\hline
1 & 0.5 & 0.8379 & / & 3/2 & 0.6667 & 0.8320 & / \\
2 & 0.75 & 0.8554 & / & 5/2 & 0.8 & 0.8760 & 0.9079 \\
3 & 0.8333 & 0.8921 & / & 7/2 & 0.8571 & 0.9047 & 0.9193 \\
4 & 0.875 & 0.9147 & 0.9295 & 9/2 & 0.8889 & 0.9229 & 0.9323 \\
\hline
\hline
\end{tabular}
\caption{The $J_{s}$ values (in units of $\delta$) at which the ground state of an even grain (left panel) and an odd grain (right panel) acquires a higher spin value $S$, as $J_s$ is increased at fixed $\Delta/\delta$. Three values of $\Delta/\delta$ are considered (0, 0.5 and 1). For $\Delta/\delta=1$ and $N$
even, the ground-state spin makes a transition from $S=0$ to $S=4$
at $J_{s} = 0.9295\,\delta$.
\label{table:spinjumps}}
\end{table}
\subsection{The number of $S=0$ electron pairs}\label{subsec:np}
Since our QMC method works directly in the space where spin is a good quantum
number, we can evaluate the number of $S=0$ pairs for each
sampled configuration.
Hence we can calculate the average fraction of $S=0$ pairs
\begin{equation}
f_P=\frac{N-\langle b \rangle}{N-p},
\label{eq:np}
\end{equation}
where $N$ the total number of electrons, $b$ the number of blocked (i.e., singly occupied) levels and $p$ the parity of the grain, i.e., $p=0$ ($p=1$) for $N$ even (odd). The normalization in Eq.~(\ref{eq:np}) is chosen to give $f_P=1$ at zero temperature for both the even and odd grain in the absence of spin exchange. Figure \ref{fig:np} shows $f_P$ as a function of temperature for even (open circles) and odd (solid triangles) grains with BCS gaps of $\Delta/\delta=0.5, 1$ and $5$.
In general, the number of pairs decreases with temperature, reflecting the weakening of pairing effects with increased thermal energy.
The temperature at which pairs start to break up depends strongly on $\Delta/\delta$. In the case of strong pairing with $\Delta/\delta=5$ this
temperature is roughly $\sim 1.5\,\delta$ (for $J_{s} =0$) and reduces to a value
of $\sim 0.4\,\delta$ in the weak pairing case with $\Delta/\delta=0.5$.
Even as pairs start to break, their number decreases only slowly with increasing temperature and for $T \sim 3.5\,\delta$ most of the electrons are still paired to $S=0$ (e.g., about $84\%$ for $\Delta/\delta=1$ and $J_{s} = 0.6$). Once pairs start to break up, the fraction of pairs $f_P$ is always larger for the odd grain. This is because the extra electron blocks a level, deferring the transition to higher temperatures. Consequently more thermal energy is required to break up the same number of pairs in the odd grain as compared with the even grain.
The overall effect of the exchange interaction is to reduce the average number
of pairs. Exchange also reduces the threshold temperature at which pairs start to
break. For a pairing gap of $\Delta/\delta=1$ this temperature is
about $\sim 0.5\,\delta$ for $J_s = 0$, and it reduces to $\sim 0.2\,\delta$ for $J_{s} = 0.8\,\delta$. Indeed, in an even grain the exchange interaction
decreases the gap between the $S=0$ ground state and the first $S\neq 0$ excited
state, thereby reducing the thermal energy required to break up a pair. For $\Delta/\delta=0.5$ and $J_s =0.845\,\delta$, the ground state of the even grain is $S=1$ so there are two blocked levels at $T=0$.
\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0, width=8.5cm] {np.eps}
\caption{The fraction $f_P$ of $S=0$ pairs as a function of
temperature for pairing gaps of $\Delta/\delta =0.5$ (left column), $\Delta/\delta
=1$ (middle column) and $\Delta/\delta =5$ (right column), and for different values of the exchange coupling $J_s$ (shown in units of $\delta$). Results for the even grain are shown by circles ($\circ$), while results for the odd grains are denoted by solid triangles ($\blacktriangle$).
\label{fig:np}}
\end{center}
\end{figure}
Pairing correlation effects can be more clearly observed in the increase of the number of $S=0$ pairs as we turn on the pairing interaction at a fixed exchange interaction, i.e., $\langle n_p \rangle - \langle n_p \rangle_{G=0}$ at fixed $J_s$. This pair number excess is simply related to the deficiency of the average number of blocked levels
\begin{equation}
\langle n_p \rangle - \langle n_p \rangle_{G=0} =
\frac{1}{2} \big(\langle b \rangle_{G=0} - \langle b \rangle \big) \;.
\label{eq:npdif}
\end{equation}
Figure \ref{fig:npdif} shows the
excess number of $S=0$ pairs for both even and odd grains.
A clear odd-even effect is observed in this quantity. For a stronger pairing interaction (larger $\Delta/\delta$), the odd-even effect survives up to higher temperatures. For not too large $J_s$, the excess number is zero at $T=0$ because the ground state has the minimal spin and therefore the largest possible number of pairs even in the absence of pairing interaction.
For the largest values of $J_{s}$ (third row in Fig.~\ref{fig:npdif}), the excess number of pairs is non-zero at $T=0$ since the grain is polarized.
\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0, width=8.5cm] {npdif.eps}
\caption{The excess number of $S=0$ pairs [see Eq.~(\ref{eq:npdif})] as a function of temperature. We show results for both even ($\circ$) and odd ($\blacktriangle$)
grains with $\Delta/\delta =0.5$ (left column), $\Delta/\delta =1$ (middle column) and
$\Delta/\delta =5$ (right column).
\label{fig:npdif}}
\end{center}
\end{figure}
\subsection{The canonical pair gap}\label{subsec:cangap}
The canonical pair gap $\Delta_{\rm can}$, defined by~\cite{Vondelft01}
\begin{equation}
\Delta^2_{\rm can}(T,G,J_{s}) = G \bigg( \langle \hat{P}^{\dag}\hat{P} \rangle_{G,J_s}
- \langle \hat{P}^{\dag}\hat{P} \rangle_{G=0,J_s} \bigg) \
\label{eq:cangap}
\end{equation}
measures the pairing correlation energy, namely the increase of pairing energy when the pairing interaction is turned on in the presence of a fixed exchange interaction.
For $J_s=0$ and in the thermodynamic limit, the canonical pair gap $\Delta_{\rm can}$ becomes the familiar BCS gap $\Delta$. For a finite system, the BCS gap $\Delta$ is recovered from $\Delta_{\rm can}$
by applying the mean-field approximation and taking the grand-canonical
averages in Eq.~(\ref{eq:cangap}).
We first discuss the behavior of the canonical pair gap for a weak
pairing interaction $\Delta /\delta=0.5$ (left column of Fig.~\ref{fig:cangap}).
We observe that the pairing correlation energy decreases with increasing temperature. The behavior of $\Delta_{\rm can}$
versus temperature is completely smooth because of the finite size of the grain. The exchange interaction quenches the pairing correlation energy
further since this interaction tends to break up pairs.
At low temperatures and $J_{s}$ not too large, an
odd-even difference is visible in the canonical pair gap. This odd-even effect
is a unique signature of pairing correlations and is reduced by the exchange interaction. For $J_{s} = 0.845\,\delta$ the ground-state spin
of the even (odd) grain is $S=1$ ($S=3/2$) and the odd-even effect in
$\Delta_{\rm can}$ is completely destroyed.
For larger pairing strengths, the exchange interaction does not affect much the canonical pair gap, as can be seen from the middle and right columns of Fig.~\ref{fig:cangap},
corresponding to $\Delta/\delta=1$ and $\Delta/\delta=5$,
respectively.
\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0, width=8.5cm] {cangap.eps}
\caption{The canonical pair gap $\Delta_{\rm can}$ defined in
Eq.~(\ref{eq:cangap}) as a function of temperature. We show results for both even ($\circ$) and odd ($\blacktriangle$)
grains with $\Delta/\delta =0.5$ (left column), $\Delta/\delta =1$ (middle column) and
$\Delta/\delta =5$ (right column). Visible effects of the exchange interaction are limited to the weak pairing case ($\Delta/\delta =0.5$).
\label{fig:cangap}}
\end{center}
\end{figure}
\subsection{The heat capacity}\label{subsec:sh}
Another interesting thermodynamic observable is the heat capacity of the grain
\begin{equation}
C = \frac{d\langle \hat{H} \rangle}{dT},
\label{eq:shdef}
\end{equation}
with $\langle \ldots \rangle$ denoting thermal averaging. Figure \ref{fig:sh} shows the heat capacity in grains with BCS gaps of $\Delta /\delta = 0.5$, $1$ and $5$.
We first discuss the smaller grains with a BCS gap of $\Delta/\delta=0.5$
(left column of Fig.~\ref{fig:sh}). Previous studies have shown that in the absence of exchange interaction ($J_{s} = 0$) the even-grain heat capacity exceeds the odd-grain heat capacity in a temperature range $0.4\,\delta \lesssim T \lesssim 1.3\,\delta$.~\cite{VanHoucke06b,Alhassid07} In this temperature range, $S=0$ pairs start to break up and pairing correlations are quenched, as can be seen from Fig.~\ref{fig:cangap}. The bump in the heat capacity of the even grain reflects a signature of the pairing transition of the finite-size grain. No such effect is observed in the odd case because of the blocking effect of the unpaired electron. This odd-even effect
in the heat capacity is a unique signature of pairing correlations in a finite-size system.
When the exchange interaction is turned on, the odd-even
effect in the heat capacity disappears gradually (see left column of Fig.~\ref{fig:sh}). Since the exchange interaction brings down in energy high spin states while
leaving the $S=0$ states unchanged, it increases the number of unpaired
electrons at finite temperature (even when the even ground state still has $S=0$). These unpaired electrons block levels (in the same way as the single electron blocks a level in the odd grain) and suppress the bump in the heat capacity. For $J_{s}=0.845\,\delta$, the ground state has $S=1$
($S=3/2$) in the even (odd) grain. For this value of $J_s$, the odd-even effect has completely disappeared.
\begin{figure}[ht]
\begin{center}
\includegraphics[angle=0, width=8.5cm] {sh.eps}
\caption{The heat capacity for even ($\circ$) and odd ($\blacktriangle$) grains as a function of temperature. Shown are results for grains with $\Delta /\delta = 0.5$ (left column), $\Delta/\delta = 1$ (middle column) and $\Delta /\delta = 5$ (right column).
\label{fig:sh}}
\end{center}
\end{figure}
A similar behavior is found for grains in the crossover region $\Delta/\delta =
1$ (middle column of Fig.~\ref{fig:sh}),
where the signature of the pairing transition is destroyed by the
exchange interaction. Compared to the smaller grains (with $\Delta/\delta=0.5$),
the odd-even effect is larger, and a larger critical exchange strength
is required to destroy it.
The right column of Fig.~\ref{fig:sh} shows the heat capacity of a grain in the
BCS regime ($\Delta/\delta=5$). The qualitative difference with
the behavior of the heat capacity in the fluctuation-dominated regime ($\Delta/\delta=0.5$) is striking: the signature of
the pairing transition is much stronger and it cannot be destroyed even in the
presence of a strong exchange interaction.
We can understand this effect by comparing the
fraction of $S=0$ pairs in both cases (see left and right column of Fig.~\ref{fig:np}). When the exchange interaction is increased, it is clear that $S \neq 0$ states
are pushed down in energy. At some critical value of the exchange strength,
the ground state eventually acquires a finite spin. Before this happens, the
gap between the $S=0$ ground state and the first $S\neq 0$ excited state
decreases with $J_s$. However, due to strong pairing correlations, this gap
is much larger in the BCS regime. Thus, in the fluctuation-dominated regime,
little thermal energy is needed to excite the system to $S\neq 0$ states,
whereas in the BCS limit $S=0$ states dominate up to
considerably higher temperatures. This effect is also reflected in the
number of $S=0$ pairs (see Fig.~\ref{fig:np}).
For $\Delta/\delta=5$, the excitation gap is still large enough
and a clear finite temperature transition
occurs at a considerably higher temperature ($\sim 1.5\,\delta$) from a $S=0$ state to a state with broken pairs due to thermal excitations. For this $\Delta/\delta=5$ case, the (even) system makes a sudden transition at $J_{s} = 1.0029 \,\delta$ from a $S=0$
ground state to a state where all the electrons in the model space are unpaired. This is known as the Stoner instability.~\cite{Stoner47}
\subsection{The spin susceptibility}\label{subsec:sus}
The spin susceptibility is a measure of the grain's response to an external
magnetic field. Here we discuss the spin susceptibility in the zero-field limit,
defined by
\begin{eqnarray}
\chi (T) = - \frac{\partial^2 \mathcal{F} (T,h)}{\partial h^2}
\bigg|_{h=0}
= \frac{\mu_B^2}{T} \big(\langle \hat{M}^2 \rangle - \langle
\hat{M} \rangle^2 \big),
\end{eqnarray}
where $\mathcal{F}(T,h)= - T \ln [{\rm{Tr}}e^{-\beta (\hat{H}- g \mu_B
\hat{M} h})]$ is
the free energy of the grain in the presence of an
external Zeeman field $h$ and $g$ is the spin g-factor.
The operator $\hat{M}$ is the ``magnetization'' defined as $\hat{M} =
\sum_{i,\sigma} \sigma
\hat{c}^{\dag}_{i,\sigma}\hat{c}^{\phantom{\dag}}_{i,\sigma}$.
For $h=0$, we have $\langle \hat{M}\rangle = 0$ because of spherical symmetry.
Within the reduced BCS model, it was found that pairing
correlations affect the temperature dependence of the spin susceptibility of a grain.~\cite{Dilorenzo00,VanHoucke06b,Alhassid07} In
particular, for an odd number of electrons, the spin
susceptibility shows a re-entrant behavior as a function of $T$ for any value
of the ratio $\Delta/\delta$. This behavior persists in
ultra-small grains, in which the level spacing is larger than the
BCS gap. Since this re-entrant behavior is absent in normal metallic grains,
it was suggested by Di Lorenzo {\em et al.}\cite{Dilorenzo00} that this behavior could be used as a unique signature of pairing correlations in small grains.
Here we study how the exchange interaction affects this re-entrant behavior.
It is straightforward to evaluate the spin susceptibility in the QMC approach since the value of $\hat M$ is known at each step of the Markov
process. To quantify the effects of pairing correlations, we compare our results with the limiting case of spin exchange correlations but no pairing interaction. In this limiting case, the spin susceptibility can be calculated directly using spin projection methods.~\cite{Alhassid03} We find
\begin{equation}
\chi (T) = \frac{\sum_{S} (2S+1) x_{N,S} e^{-\beta F_{N,S}} e^{\beta J_{s} S(S+1)}}{\sum_S (2S+1) e^{-\beta F_{N,S}} e^{\beta J_{s} S(S+1)}},
\end{equation}
where
\begin{eqnarray}
x_{N,S} e^{-\beta F_{N,S}} & = & 4 S^2 e^{-\beta (\tilde{F}_{N/2+S} + \tilde{F}_{N/2-S})}
\nonumber \\
& & - 4 (S+1)^2 e^{-\beta
(\tilde{F}_{N/2+S+1} + \tilde{F}_{N/2-S-1})},
\end{eqnarray}
and
\begin{equation}
e^{-\beta F_{N,S}} = e^{-\beta (\tilde{F}_{N/2+S} + \tilde{F}_{N/2-S})} - e^{-\beta
(\tilde{F}_{N/2+S+1} + \tilde{F}_{N/2-S-1})}.
\end{equation}
The quantity $\tilde{F}_{q}$ is the canonical free energy of $q$ non-interacting
spinless fermions in $2N_o+1$ single-particle levels, which can be evaluated using a particle number projection formula that involves $2N_o+1$ quadrature points.~\cite{Alhassid00}
\begin{figure}[t]
\begin{center}
\includegraphics[angle=0, width=8.5cm] {sus.eps}
\caption{The spin susceptibility, normalized to its $J_{s}=0$ bulk high
temperature limit $\chi_P = 2 \mu_B^2 / \delta$, as function of temperature for an even ($\circ$) and odd ($\blacktriangle$) grain with $\Delta/\delta=0.5$ (left column), $\Delta/\delta=1$ (middle column) and $\Delta/\delta=5$ (right column). The ground-state spin is $S=0$ or $S=1/2$ for all shown values of the spin coupling $J_{s}$ (measured in units of $\delta$).
The dotted lines are the even and odd spin susceptibility for electrons interacting only through an exchange interaction with the indicated strength $J_s$ (in units of $\delta$).
\label{fig:susa}}
\end{center}
\end{figure}
The left column of Fig.~\ref{fig:susa} shows the even and odd spin susceptibility for weak pairing ($\Delta/\delta=0.5$) and for spin exchange couplings
$J_{s}\leq 0.6\,\delta$. For these values of $J_{s}$, the even (odd) ground state has
$S=0$ ($S=1/2$) (see Table \ref{table:spinjumps}). The susceptibility is measured in units of the Pauli susceptibility $\chi_P = 2 \mu_B^2 / \delta$ (the high-temperature value of $\chi$ at $J_s=0$).
In particular, the top left panel of Fig.~\ref{fig:susa} shows the spin susceptibility in the absence of exchange interaction ($J_s=0$). At low temperatures, the spin susceptibility is exponentially suppressed for the even grain, but exhibits the familiar re-entrant effect for the odd grain. This re-entrant behavior is seen for all
cases with $\Delta/\delta=0.5$ and $J_{s} \leq 0.6\,\delta$, including the case $J_{s} = 0.6\,\delta$ for which the signature of pairing correlations is no longer visible in the heat capacity.
This re-entrant behavior originates in the paramagnetic contribution of the spin of the unpaired electron. This contribution is given by $\chi(T)/\chi_P = \delta/2T$ (not shown in the figure), and coincides with the odd-grain QMC results at sufficiently low temperatures ($T \lesssim 0.4\,\delta$). At higher temperatures, the QMC results deviate from this simple behavior since several unpaired electrons contribute to the odd-grain susceptibility. These deviations are correlated with the breakup of $S=0$ pairs (see Fig.~\ref{fig:np}). The stronger the pairing strength, the higher the temperature at which the spin susceptibility deviates from the $\delta/2T$ behavior, as more thermal energy is required to break up pairs.
For comparison, we also show in Fig.~\ref{fig:susa} the even and odd spin susceptibilities when the electrons interact only through the exchange channel (dotted lines). In general, we observe that exchange correlations enhance the spin susceptibility. For $J_s=0.4\,\delta$
and $\Delta/\delta=0.5$, we observe a peak in the even spin susceptibility (in the absence of pairing) around $T \approx 0.1\,\delta$. For this value of $J_s$, the excited triplet state ($S=1$) lies close to the ground-state singlet ($S=0$) and the system could be easily polarized at low temperatures (the $S=0 \to S=1$ ground-state spin transition occurs at $J_s=0.5\,\delta$). At temperatures $T < 0.1\,\delta$, the even spin susceptibility is exponentially suppressed, while at slightly higher temperatures the susceptibility tends to follow the odd spin
susceptibility. At $J_s = 0.6\,\delta$, both the even and odd spin susceptibility diverge at $T=0$, since the ground state has already acquired a finite spin (for $\Delta/\delta=0$).
We also observe from Fig.~\ref{fig:susa} that at high temperatures the spin susceptibility in the presence of pairing correlations approaches its value in the absence of pairing. This behavior is expected since pairing correlations are suppressed at high temperatures.
The middle column of Fig.~\ref{fig:susa} shows the spin susceptibilities for even and odd grains with BCS gap of $\Delta/\delta = 1$.
For the largest exchange value shown ($J_{s}=0.8\,\delta$) the ground state is still
$S=0$ (or $S=1/2$). At low temperatures we observe (for $J_s=0.8\,\delta$) a clear
minimum in the odd spin susceptibility (a signature of pairing correlations),
and the $\delta/T$ behavior of a single unpaired spin is observed now only for $T \lesssim 0.1\,\delta$. For $T \gtrsim \delta$, we note the decrease of the spin susceptibility with temperature.
The case of strong pairing with $\Delta/\delta=5$ is shown in the right column of Fig.~\ref{fig:susa}. $S=0$ pairs start to break up only at higher temperatures and the behavior of a single spin
susceptibility of $\sim \delta/T$ for the odd grain remains valid up to $T \approx \delta$. At high temperatures, the spin susceptibility increases with the exchange
coupling. At this large pairing strength, the first spin jump occurs at $J_{s} = 1.0029\,\delta$, and it immediately polarizes the entire system.
\begin{figure}[h]
\begin{center}
\includegraphics[angle=0, width=8.5cm] {susD05b.eps}
\caption{The spin susceptibility as function of temperature for an even ($\circ$) and odd ($\blacktriangle$) grain with $\Delta/\delta=0.5$. For spin coupling values $J_s$ of $0.8\,\delta$, $0.845\,\delta$, $0.885\,\delta$ and $0.91\,\delta$, the even (odd) ground state has spins $S=0$ ($S=1/2$),
$S=1$ ($S=3/2$), $S=2$ ($S=5/2$) and $S=3$ ($S=7/2$), respectively.
\label{fig:susD05b}}
\end{center}
\end{figure}
Figure \ref{fig:susD05b} shows the spin susceptibility for $\Delta/\delta=0.5$
and spin exchange values of $J_{s}\geq 0.8\,\delta$. At $J_{s}=0.8\,\delta$, the system is close to
its first ground-state spin jump (which occurs at $J_{s} = 0.8379\,\delta$ for an even grain or at $J_{s} = 0.8320\,\delta$ for an odd grain), and the system is easily polarized. The exponential suppression of the
even susceptibility can only be observed at very low temperatures ($T\lesssim \delta/40$). The susceptibility peaks at $T\approx 0.05\,\delta$.
At higher temperature, there is a large number of broken pairs and the even spin susceptibility
coalesces with the odd susceptibility. The odd spin susceptibility is a monotonic function and no re-entrant behavior is observed.
Once the ground-state spin transition has occurred ($J_{s} \geq 0.845\,\delta$), both the even and odd susceptibilities diverge at $T=0$ with the even curve lying slightly below the odd curve. For $J_{s} = 0.845\,\delta$, the odd-grain ground state has $S=3/2$, and we expect a low temperature behavior of $\chi(T)/\chi_P = 5\delta/2T$ (dashed line). For $J_{s} = 0.885\,\delta$ ($J_{s} = 0.91\,\delta$), the ground state has $S=5/2$ ($S=7/2$), leading to a low temperature behavior of $\chi(T)/\chi_P = 35\delta/6T$ ($\chi(T)/\chi_P = 21\delta/2T$) of the odd spin susceptibility.
We conclude that once the exchange strength gets close to its value where the
first ground-state spin jump occurs, the re-entrant behavior in the odd spin
susceptibility disappears. We emphasize, however, that pairing correlations still exist since the canonical pair gap does not vanish.
\section{Conclusion}\label{conclusion}
We have used a quantum Monte Carlo method to calculate the thermodynamic properties of a small superconducting metallic grain that is described by the universal Hamiltonian. These thermodynamic properties have been studied as a function of the BCS gap $\Delta/\delta$ and the exchange interaction strength $J_s/\delta$ (measured in units of the mean-level spacing). The spin exchange interaction competes with the BCS-like pairing interaction, and, in general, we find that number-parity signatures of pairing correlations are suppressed in the presence of a finite exchange interaction. We also find qualitative differences between the superconducting BCS regime and the fluctuation-dominated regime of pairing correlations.
We thank K. Heyde for interesting suggestions and discussions. K. Van Houcke acknowledges financial support of the Fund for Scientific Research - Flanders (Belgium), and the hospitality of the Center for Theoretical Physics at Yale University where part of this work was completed. S. M.A. Rombouts acknowledges support from Grant 220335 of the Seventh Framework Programme of the European Community. This work was supported in part by U.S. DOE grant No.\ DE-FG-0291-ER-40608.
|
1,116,691,500,725 | arxiv | \section{Introduction}
Cryptosystems based on curves of genus $2$ offer per-bit security
and efficiency comparable with elliptic curve cryptosystems.
However, many of the computational problems
related to creating secure instances of genus~$2$ cryptosystems
are considerably more difficult
than their elliptic curve analogues.
Point counting---or, from a cryptographic point of view,
computing the cardinality of a cryptographic group---offers
a good example of this disparity,
at least for curves defined over large prime fields.
Indeed, while computing the order of a cryptographic-sized
elliptic curve with the Schoof--Elkies--Atkin algorithm
is now routine,
computing the order of a comparable genus $2$ Jacobian requires a significant
computational effort~\cite{GaSc04,GaSc10}.
In this article
we describe a number of improvements to the classical Schoof--Pila algorithm
for genus $2$ curves with explicit and efficient real multiplication (RM).
For explicit RM curves over $\mathbb{F}_p$,
we reduce the complexity of Schoof--Pila from ${\widetilde{O}}(\log^8 p)$
to ${\widetilde{O}}(\log^5 p)$.
We applied a first implementation of our algorithms
to find prime-order Jacobians over 128-bit fields
(comparable to prime-order elliptic curves over 256-bit fields,
and therefore suitable for contemporary cryptographic applications).
Going further,
we were able to compute the order of an RM Jacobian defined over a 512-bit prime field,
far beyond the cryptographic range.
(For comparison, the previous record computation in genus $2$
was over a 128-bit field.)
While these RM curves are special, they are not ``too special'':
Every ordinary genus $2$ Jacobian over a finite field
has RM; our special requirement is that this RM be known in advance
and be efficiently computable.
The moduli of curves with RM by
a fixed ring form $2$-dimensional subvarieties (Humbert surfaces)
in the $3$-dimensional moduli space of all genus~$2$ curves.
We can generate random curves with the specified RM by choosing
random points on an explicit model of the corresponding Humbert
surface~\cite{Gruenewald}.
In comparison with elliptic curves,
for which the moduli space is one-dimensional, this
still gives an additional degree of freedom in the random curve selection.
To generate random curves with efficiently computable RM, we choose
random curves from some known one and two-parameter families
(see~\S\ref{sec:families}).
Curves with efficiently computable RM
have an additional benefit in cryptography:
the efficient endomorphism can be used to accelerate scalar multiplication
on the Jacobian, yielding faster encryption and
decryption~\cite{Kohel--Smith,Parketal,Takashima}.
The RM formul\ae{} are also compatible with
fast arithmetic based on theta functions~\cite{Gaudry-theta}.
\section{Conventional Point Counting for Genus $2$ Curves}
Let $\mathcal{C}$ be a curve of genus 2 over a finite field $\mathbb{F}_q$, of odd
characteristic, defined by an affine model $y^2 = f(x)$,
where $f$ is a squarefree polynomial of degree $5$ or $6$ over $\mathbb{F}_q$.
Let $\Jac{\mathcal{C}}$ be the Jacobian of $\mathcal{C}$; we assume $\Jac{\mathcal{C}}$ is ordinary and
absolutely simple. Points on $\Jac{\mathcal{C}}$ correspond to
degree-$0$ divisor classes on $\mathcal{C}$;
we use the Mumford representation for divisor classes
together with the usual Cantor-style composition and reduction algorithms
for divisor class arithmetic~\cite{Galbraith--Harrison--Mireles-Morales,Cantor-reduction}.
Multiplication by $\ell$ on $\Jac{\mathcal{C}}$ is denoted by $[\ell]$, and
its kernel by $\Jac{\mathcal{C}}[\ell]$. More generally, if $\phi$ is an
endomorphism of $\Jac{\mathcal{C}}$ then $\Jac{\mathcal{C}}[\phi] = \ker(\phi)$,
and if $S$ is a set of endomorphisms then $\Jac{\mathcal{C}}[S]$ denotes
the intersection of $\ker(\phi)$ for $\phi$ in $S$.
\subsection{The Characteristic Polynomial of Frobenius}
\label{sec:Frobenius-charpoly}
We let $\pi$ denote the Frobenius endomorphism of $\Jac{\mathcal{C}}$,
with Rosati dual $\dualof{\pi}$ (so $\pi\dualof{\pi} = [q]$).
The characteristic polynomial of $\pi$
has the form
\begin{equation}
\label{eq:Frobenius-charpoly}
\chi(T) = T^4 - s_1 T^3 + (s_2 + 2q)\, T^2 - q s_1 T + q^2,
\end{equation}
where $s_1$ and $s_2$ are integers, and $s_2$ is a translation
of the standard definition.
The polynomial $\chi(T)$ determines the cardinality of $\Jac{\mathcal{C}}(\mathbb{F}_{q^k})$
for all $k$: in particular, $\#\Jac{\mathcal{C}}(\mathbb{F}_{q}) = \chi(1)$.
We refer to the determination of $\chi(T)$ as the {\it point counting problem}.
The polynomial $\chi(T)$ is a {\it Weil polynomial}:
all of its complex roots lie on the circle $|z| = \sqrt{q}$.
This implies the Weil bounds
\begin{equation}
\label{eq:s_1s_2-Weil-bounds}
|s_1| \le 4\sqrt{q}\quad \text{ and }\quad |s_2| \le 4q.
\end{equation}
However, the possible values of $(s_1,s_2)$
do not fill the whole rectangle specified by the Weil bounds.
R\"uck~\cite[Theorem 1.1]{Ruck90}
shows that in fact $s_1$ and $s_2$ satisfy
\[
s_1^2 - 4 s_2 \ge 0 \text{ and } s_2 + 4q \ge 2|s_1| ,
\]
so the possible values of $(s_1,s_2)$ are in the following domain:
\begin{center}
\scalebox{0.75}{
\begin{tikzpicture}[yscale=0.5]
\draw[very thick,->] (-4,0) -- (4,0) node[right] {$s_1/\sqrt{q}$};
\draw[very thick,->] (0,-4) -- (0,4) node[above] {$s_2/q$};
\path[thick,draw=blue, fill=blue!20, opacity=0.6]
(0,-4) -- (-4,4) --
plot[domain=-4:4] (\x,\x*\x/4) -- cycle;
\foreach \x in {-4,...,4} \draw[yscale=2] (\x,-0.1) -- (\x,0.1);
\foreach \x in {-4,...,4} \draw (-0.1,\x) -- (0.1,\x);
\end{tikzpicture}
}
\end{center}
\subsection{The Classical Schoof--Pila Algorithm for Genus $2$ Curves}
\label{sec:Schoof--Pila}
The objective of point counting is to compute $\chi(T)$, or equivalently
the tuple of integers $(s_1,s_2)$. When the characteristic of $\mathbb{F}_{q}$
is large, the conventional approach
is to apply the Schoof--Pila algorithm
as far as is practical, before passing to a baby-step giant-step
algorithm if necessary (see~\S\ref{sec:BSGS}).
The strategy of Schoof's algorithm
and its generalizations is to compute the polynomials
$\chi_\ell(T) = \chi(T) \bmod (\ell)$ for sufficiently many primes (or prime
powers) $\ell$ to reconstruct $\chi(T)$ using the Chinese Remainder Theorem.
Since $\chi_\ell(T)$ is the characteristic polynomial of $\pi$
restricted to $\Jac{\mathcal{C}}[\ell]$ (see~\cite[Proposition 2.1]{Pila}),
we have
$$
\chi_\ell(\pi)(D) = 0 \text{ for all } D \text{ in } \Jac{\mathcal{C}}[\ell].
$$
Conversely, to compute $\chi_\ell(T)$ we let $D$ be a generic
element of $\Jac{\mathcal{C}}[\ell]$ (as in~\S\ref{sec:kernel-ideal} below),
compute the three points
\[
(\pi^2 + [\bar{q}])^2(D),\ \
(\pi^2 + [\bar{q}])\pi(D),\ \text{and}\ \
\pi^2(D),
\]
and then search for the coefficients $(\bar{s}_1,\bar{s}_2)$ of
$\chi_\ell(T)$ in $(\mathbb{Z}/\ell\mathbb{Z})^2$, for which the linear relation
\begin{equation}
\label{eq:conventional-ell-test}
(\pi^2 + [\bar{q}])^2(D)
- [\bar{s}_1]\,(\pi^2 + [\bar{q}])\pi(D)
+ [\bar{s}_2]\,\pi^2(D) = 0
\end{equation}
holds.
If the minimal polynomial of $\pi$ on $\Jac{\mathcal{C}}[\ell]$ is
a proper divisor of $\chi_\ell(T)$---which occurs for at most
a finite number of $\ell$ dividing $\mathrm{disc}(\chi)$---then
the polynomial so determined is not unique, but
$\chi_\ell(T)$ can be determined by deducing the correct
multiplicities of its factors.
Once we have computed $\chi_\ell(T)$ for sufficiently many $\ell$,
we reconstruct~$\chi(T)$ using the Chinese Remainder Theorem.
The Weil and R\"uck bounds together with
a weak version of the prime number theorem tell us how many
$\ell$ are required: Pila notes in~\cite[\S1]{Pila}
that the set of $O(\log q)$ primes $\ell < 21\log q$ will suffice.
We analyse the complexity of the classical Schoof--Pila algorithm
in~\S\ref{sec:Schoof--Pila-complexity}.
\subsection{Endomorphisms and Generic Kernel Elements}
\label{sec:kernel-ideal}
We now recall how to contruct an effective version of a generic
$\ell$-torsion element. We present it in a slightly more general setting,
so that we can use this ingredient in the subsequent RM-specific
algorithm. Therefore, we show how to compute with a generic element
of the kernel of some endomorphism $\phi$ of $\Jac{\mathcal{C}}$, whereas $\phi$ is
just $[\ell]$ in the classical algorithm.
\begin{definition}\label{def:explicit}
Fix an embedding $P \mapsto D_P$ of $\mathcal{C}$ in $\Jac{\mathcal{C}}$.
We say that an endomorphism $\phi$ of~$\Jac{\mathcal{C}}$
is \emph{explicit} if we can effectively compute polynomials
$d_0,d_1,d_2,e_0,e_1,$ and $e_2$ such that
if $P = (x_P,y_P)$ is a generic point of $\mathcal{C}$, then
the Mumford representation of $\phi(D_P)$ is given by
\begin{equation}
\label{eq:endomorphism-image}
\phi(D_P)
=
\Mumfordbig{
x^2 + \frac{d_1(x_P)}{d_2(x_P)}x + \frac{d_0(x_P)}{d_2(x_P)} ,
y - y_P\left(
\frac{e_1(x_P)}{e_2(x_P)}x + \frac{e_0(x_P)}{e_2(x_P)}
\right)
}
.
\end{equation}
The $d_0$, $d_1$, $d_2$, $e_0$, $e_1$, and $e_2$
are called the \emph{$\phi$-division polynomials}.
\end{definition}
If $\phi$ is an explicit endomorphism, then
we can use \eqn{eq:endomorphism-image} (extending $\mathbb{Z}$-linearly)
to evaluate $\phi(D)$ for general divisor classes $D$ in $\Jac{\mathcal{C}}$.
In the case $\phi = [\ell]$, the $[\ell]$-division polynomials
are the $\ell$-division polynomials of Cantor~\cite{Cantor-division}.
The $\phi$-division polynomials depend on the choice of embedding $P
\mapsto D_P$;
we will make this choice explicit when computing the $\phi$-division
polynomials for each of our families
in~\S\ref{sec:families}.
To compute a generic element of $\Jac{\mathcal{C}}[\phi]$,
we generalize the approach of~\cite{GaSc04}
(which computes generic elements of $\Jac{\mathcal{C}}[\ell]$).
The resulting algorithm is essentially the same as in~\cite[\S3]{GaSc04}
(except for the parasite computation step, which we omit)
with $\phi$-division polynomials replacing $\ell$-division polynomials,
so we will only briefly sketch it here.
Let $D = \Mumford{x^2 + a_1x + a_0, y - (b_1x + b_0)}$
be (the Mumford representation of) a generic point of $\Jac{\mathcal{C}}$.
We want to compute a triangular ideal $I_\phi$
in $\mathbb{F}_{q}[a_1,a_0,b_1,b_0]$
vanishing on the nonzero elements of $\Jac{\mathcal{C}}[\phi]$.
The element~$D$ equals $D_{(x_1,y_1)} + D_{(x_2,y_2)}$,
where $(x_1,y_1)$ and $(x_2,y_2)$ are generic points of $\mathcal{C}$.
To find a triangular system
of relations on the $a_i$ and $b_i$
such that $D$ is in $\Jac{\mathcal{C}}[\phi]$
we solve for $x_1$, $y_1$, $x_2$, and $y_2$ in
\[
\phi(D_{(x_1,y_1)}) = -\phi(D_{(x_2,y_2)}) ,
\]
applying \eqn{eq:endomorphism-image} and using resultants
computed with the evaluation--interpolation technique
of~\cite[\S 3.1]{GaSc04}.
We then resymmetrize as in~\cite[\S 3.2]{GaSc04}
to express the result in terms of the~$a_i$ and~$b_i$.
We can now compute with a ``generic'' element
$\Mumford{x^2 + a_1x + a_0, y - (b_1x + b_0)}$
of~$\Jac{\mathcal{C}}[\phi]$
by reducing the coefficients modulo~$I_\phi$
after each operation.
Following the complexity analysis of~\cite[\S3.5]{GaSc04},
we can compute a triangular representation for $I_\phi$
in $O(\delta^2\mathsf{M}(\delta)\log\delta + \mathsf{M}(\delta^2)\log\delta)$
field operations,
where $\delta$ is the maximum among the degrees of
the $\phi$-division polynomials,
and $\mathsf{M}(d)$ is the number of operations
required to multiply polynomials of degree $d$ over $\mathbb{F}_{q}$.
Using asymptotically fast multiplication algorithms,
we can therefore compute~$I_\phi$ in~${\widetilde{O}}(\delta^3)$ field operations.
The degree of $I_\phi$ is in $O(\delta^2)$;
with this triangular representation,
each multiplication modulo $I_\phi$ costs
${\widetilde{O}}(\delta^2)$ field operations.
\subsection{Complexity of Classical Schoof--Pila Point Counting}
\label{sec:Schoof--Pila-complexity}
\begin{proposition}
The complexity of the classical Schoof--Pila algorithm
for a curve of genus $2$ over $\mathbb{F}_{q}$ is in ${\widetilde{O}}((\log q)^8)$.
\end{proposition}
\begin{proof}
To determine $\chi(T)$,
we need to compute $\chi_\ell(T)$ for $O(\log q)$ primes $\ell$
in $O(\log q)$.
To compute $\chi_\ell(T)$,
we must first compute the $\ell$-division polynomials,
which have degrees in $O(\ell^2)$.
We then compute the kernel ideal $I_\ell$;
according to the previous subsection,
the total cost is in ${\widetilde{O}}(\ell^6)$ field operations.
The cost of checking \eqn{eq:conventional-ell-test}
against a generic element of $\Jac{\mathcal{C}}[\ell]$
decomposes into the cost of computing Frobenius images of the generic
element in ${\widetilde{O}}(\ell^4\log q)$ and of finding the matching pair $(\bar s_1,
\bar s_2)$ in ${\widetilde{O}}(\ell^5)$ field operations.
So the total complexity for computing $\chi_\ell(T)$ is
in ${\widetilde{O}}(\ell^4(\ell^2+\log q))$ field operations.
In terms of bit operations, for each $\ell$ bounded by $O(\log q)$,
we compute $\chi_\ell(T)$ in time ${\widetilde{O}}((\log q)^7)$, and the result
follows from the addition of these costs for all the different
$\ell$'s.
\qed
\end{proof}
\subsection{Baby-Step Giant-Step Algorithms}
\label{sec:BSGS}
In practice, computing $\chi_\ell(T)$ with classical Schoof--Pila becomes
impractical for large values of $\ell$. The usual approach is to carry out
the Schoof--Pila algorithm to the extent possible, obtaining congruences
for $s_1$ and $s_2$ modulo some integer $M$, before completing the calculation
using a generic group algorithm such as baby-step giant-step (BSGS).
Our BSGS algorithm of choice is the low-memory parallelized variant of the
Matsuo--Chao--Tsuji algorithm~\cite{GaScXX,Matsuo--Chao--Tsuji}.
The bounds in \eqn{eq:s_1s_2-Weil-bounds} imply that the
search space of candidates for $(s_1,s_2)$ is in $O(q^{3/2})$, and
a pure BSGS approach finds $(s_1,s_2)$ in time and space ${\widetilde{O}}(q^{3/4})$.
However, when we apply BSGS after a partial Schoof--Pila computation,
we obtain a congruence for $(s_1,s_2)$ modulo~$M$.
If $M < 8q$, then the size of the search space
is reduced to $O(q^{3/2}/M^2)$, and the complexity for
finding $(s_1,s_2)$ is reduced to ${\widetilde{O}}(q^{3/4}/M)$.
For larger $M$, the value of $s_1$ is fully determined, and the problem
is reduced to a one-dimensional search space of size $O(q/M)$ for
which the complexity becomes ${\widetilde{O}}(\sqrt{q/M})$.
\section{Point Counting in Genus $2$ with Real Multiplication}
\label{sec:RM}
By assumption, $\Jac{\mathcal{C}}$ is ordinary and simple, so $\chi(T)$
is an irreducible polynomial defining a quartic CM-field with
real quadratic subfield $\mathbb{Q}(\sqrt{\Delta})$. We say that $\Jac{\mathcal{C}}$
(and~$\mathcal{C}$) has \emph{real multiplication} (RM) by $\mathbb{Q}(\sqrt{\Delta}\,)$.
For a randomly selected curve, $\Delta$ is in $O(q)$; but in the sequel
we consider families of curves with RM by $\mathbb{Q}(\sqrt{\Delta})$ for small
$\Delta$ (= $5$ or $8$), admitting an explicit (in the sense of
Definition~\ref{def:explicit}) endomorphism
$\phi$ such that
\begin{equation}
\label{eq:real-subring}
\mathbb{Z}[\phi] = \mathbb{Q}(\sqrt{\Delta}) \cap \ensuremath{\mathrm{End}}(\Jac{\mathcal{C}})
\end{equation}
(that is, $\mathbb{Z}[\phi]$ is the full real subring of $\ensuremath{\mathrm{End}}(\Jac{\mathcal{C}})$), and
\[
\mathrm{disc}\left(\mathbb{Z}[\phi]\right) = \Delta .
\]
We presume that
the trace $\mathrm{Tr}(\phi)$ and norm $\mathrm{N}(\phi)$, such that
$\phi^2 - \mathrm{Tr}(\phi)\phi + \mathrm{N}(\phi) = 0$, are known.
We also suppose that $\phi$ is \emph{efficient}, in the following sense:
\begin{definition}
We say that an explicit endomorphism $\phi$ is
\emph{efficiently computable}
if the cost of evaluating $\phi$ at points of $\Jac{\mathcal{C}}(\mathbb{F}_{q})$
requires only $O(1)$
field operations
(comparable to a few group operations in $\Jac{\mathcal{C}}$).
In practice,
this means that
the $\phi$-division polynomials have small degree.
\end{definition}
The existence of an efficiently computable $\phi$ and knowledge
of $\Delta$ allows us to make significant improvements to each stage
of the Schoof--Pila algorithm.
Briefly:
in~\S\ref{sec:RM-char-poly} we use~$\phi$ to simplify the testing
procedure for each~$\ell$;
in~\S\ref{sec:split-prime} we show that when~$\ell$ splits in~$\mathbb{Z}[\phi]$,
we can use $\phi$ to obtain a radical reduction in complexity for
computing $\chi_\ell(T)$; and
in~\S\ref{sec:search-space} we show that knowing an effective $\phi$ allows
us to use many fewer primes $\ell$.
\subsection{The RM Characteristic Polynomial}
\label{sec:RM-Frobenius-charpoly}
Let $\psi = \pi+\pi^\dagger$;
we consider $\mathbb{Z}[\psi]$,
a subring of the real quadratic subring of $\ensuremath{\mathrm{End}}(\Jac{\mathcal{C}})$.
The characteristic polynomial of $\psi$ is the {\it real Weil polynomial}
\begin{equation}
\label{eq:real-charpoly}
\xi(T) = T^2 - s_1 T + s_2;
\end{equation}
the discriminant of $\mathbb{Z}[\psi]$ is $\Delta_0 = s_1^2 - 4 s_2$.
The analogue for $(s_1,\Delta_0)$ of R\"uck's bounds is
\begin{equation}
\label{eq:s_1d_0-Ruck-bounds}
(|s_1| - 4\sqrt{q})^2 \ge \Delta_0 = s_1^2 - 4 s_2 \ge 0 .
\end{equation}
Equation \eqn{eq:real-subring} implies that
$\mathbb{Z}[\psi]$ is contained in $\mathbb{Z}[\phi]$,
so there exist integers $m$ and $n$ such that
\begin{equation}
\label{eq:RM-psi-relation}
\psi = m + n\phi .
\end{equation}
Both $s_1$ and $s_2$ are determined by $m$ and $n$: we have
\begin{equation}
\label{eq:s_1s_2-from-nm}
s_1 = \mathrm{Tr}(\psi) = 2m + n\mathrm{Tr}(\phi)
\quad \text{and} \quad
s_2 = \mathrm{N}(\psi) = (s_1^2 - n^2\Delta)/4.
\end{equation}
In fact \(n\) is the conductor of \(\mathbb{Z}[\psi]\) in \(\mathbb{Z}[\phi]\)
up to sign: \( |n| = [\mathbb{Z}[\phi]:\mathbb{Z}[\psi]] \),
and hence
\[
\Delta_0 = \mathrm{disc}(\mathbb{Z}[\psi]) = s_1^2 - 4 s_2 = n^2 \Delta .
\]
The square root of the bounds
in \eqn{eq:s_1d_0-Ruck-bounds} gives
bounds on $s_1$ and~$n$:
\[
4\sqrt{q} - |s_1| \ge \sqrt{\Delta_0} = |n| \sqrt{\Delta} \ge 0 ;
\]
In particular,
\( |s_1| \le 4\sqrt{q} \)
and
\( |n| \le 4\sqrt{q/\Delta} \).
Applying the relation in~\eqn{eq:s_1s_2-from-nm},
we have the bounds
\begin{equation}
\label{eq:m-n-bounds}
|m| \le 2(|\mathrm{Tr}(\phi)| + \sqrt{\Delta})\sqrt{q/\Delta}
\quad \text{and}\quad
|n| \le 4\sqrt{q/\Delta}
.
\end{equation}
Both $|m|$ and $|n|$ are in $O(\sqrt{q})$.
\subsection{An Efficiently Computable RM Relation}
\label{sec:RM-char-poly}
We can use our efficiently computable endomorphism $\phi$
to replace the relation
of \eqn{eq:conventional-ell-test}
with a more efficiently computable alternative.
Multiplying \eqn{eq:RM-psi-relation} through by $\pi$,
we have
\[
\psi\pi = \pi^2 + [q] = m\pi + n\phi\pi.
\]
We can therefore compute $\bar{m} = m\bmod\ell$ and $\bar{n} = n\bmod\ell$
by letting $D$ be a generic $\ell$-torsion point,
computing the three points
\[
(\pi^2 + [\bar{q}])(D) ,
\ \
\pi(D) ,
\ \ \text{and} \ \
\phi\pi(D) ,
\]
and then
searching for the $\bar{m}$ and $\bar{n}$ in $\mathbb{Z}/\ell\mathbb{Z}$
such that
\begin{equation}
\label{eq:RM-ell-test}
(\pi^2 + [\bar{q}])(D) - [\bar{m}]\pi(D) - [\bar{n}]\phi\pi(D) = 0
\end{equation}
holds; we can find such an $\bar{m}$ and $\bar{n}$ in $O(\ell)$ group
operations.
Solving \eqn{eq:RM-ell-test} rather than
\eqn{eq:conventional-ell-test} has several advantages.
First, computing
\((\pi^2 + [\bar{q}])(D)\), \(\pi(D)\), and \(\phi\pi(D)\)
requires only two applications of Frobenius,
instead of the four required to compute
\((\pi^2 + [\bar{q}])^2(D)\),
\((\pi^2 + [\bar{q}])\pi(D)\),
and \(\pi^2(D)\)
(and Frobenius applications are costly in practice).
Moreover, either $s_2$ needs to be determined in $O(q)$, or else
the value of $n$ in \eqn{eq:conventional-ell-test}
leaves a sign ambiguity for each prime $\ell$, because only
$n^2 \bmod \ell$ can be deduced from $(\bar{s}_1, \bar{s}_2)$.
In contrast, \eqn{eq:RM-ell-test} determines $n$ directly.
\subsection{Exploiting Split Primes in $\mathbb{Q}(\sqrt{\Delta})$}
\label{sec:split-prime}
Let $\mathbb{Z}[\phi] \subset \ensuremath{\mathrm{End}}(\Jac{\mathcal{C}})$ be an RM order in $\mathbb{Q}(\phi) \cong \mathbb{Q}(\sqrt{\Delta})$.
Asymptotically, half of all primes $\ell$ split: $(\ell) = \mathfrak{p}_1 \mathfrak{p}_2$
in $\mathbb{Z}[\phi]$, where $\mathfrak{p}_1 + \mathfrak{p}_2 = (1)$ (and this carries over to
prime powers $\ell$).
This factorization gives a decomposition of the $\ell$-torsion
\[
\Jac{\mathcal{C}}[\ell] = \Jac{\mathcal{C}}[\mathfrak{p}_1] \oplus \Jac{\mathcal{C}}[\mathfrak{p}_2].
\]
In particular, any $\ell$-torsion point $D$ can be uniquely expressed
as a sum $D = D_1 + D_2$ where $D_i$ is in $\Jac{\mathcal{C}}[\mathfrak{p}_i]$.
According to the Cohen--Lenstra heuristics~\cite{CohenLenstra}, more
than 75\% of RM fields have class number 1;
in each of the explicit RM families in \S\ref{sec:families},
the order $\mathbb{Z}[\phi]$ has class number 1.
All ideals are principal in such an order,
so we may find a generator
for each of the ideals $\mathfrak{p}_i$. Furthermore, the following lemma shows
that we can find a generator which is not too large.
\begin{lemma}
\label{lemma:generator-reduction}
If $\mathfrak{p}$ is a principal ideal of norm $\ell$ in a real quadratic order $\mathbb{Z}[\phi]$,
then there exists an effectively computable generator of $\mathfrak{p}$
with coefficients in $O(\sqrt{\ell})$.
\end{lemma}
\begin{proof}
Let $\alpha$ be a generator of $\mathfrak{p}$,
and $\varepsilon$ a fundamental unit of $\mathbb{Z}[\phi]$.
Let $\gamma \mapsto \gamma_1$
and $\gamma \mapsto \gamma_2$
be the two embeddings of $\mathbb{Z}[\phi]$ in $\mathbb{R}$,
indexed so that
$|\alpha_1| \ge |\alpha_2|$ and $|\varepsilon_1| > 1$
(replacing $\varepsilon$ with $\varepsilon^{-1}$ if necessary).
Then $R = \log(|\varepsilon_1|)$ is the regulator of $\mathbb{Z}[\phi]$.
Set $\beta = \varepsilon^{-k}\alpha$,
where $k = [\log(|\alpha_1/\sqrt{\ell}|)/R]$;
then $\beta = a + b\phi$ is a new generator for $\mathfrak{p}$ such that
\[
-\frac{1}{2}
\le
\frac{\log(|\beta_i/\sqrt{\ell}|)}{R}
\le
\frac{1}{2}\cdot
\]
From the preceding bounds,
\( |\beta_1+\beta_2| = |2a + b\mathrm{Tr}(\phi)| \)
and
\( |\beta_1-\beta_2| = |b\sqrt{\Delta}| \)
are bounded by $2e^{R/2}\sqrt{\ell}$.
Since $\mathrm{Tr}(\phi)$, $\Delta$ and $R$ are
fixed constants, $|a|$ and $|b|$ are in $O(\sqrt{\ell})$.
The ``effective'' part of the result follows from classical
algorithms for quadratic fields.
\qed
\end{proof}
\begin{lemma}
\label{lemma:alpha-division-polynomials}
Let $\Jac{\mathcal{C}}$ be the Jacobian of a genus $2$ curve
over a finite field $\mathbb{F}_q$
with an efficiently computable RM endomorphism $\phi$.
There exists an algorithm which, given a principal ideal $\mathfrak{p}$ of
norm $\ell$ in $\mathbb{Z}[\phi]$,
computes an explicit generator $\alpha$ of $\mathfrak{p}$
and the $\alpha$-division polynomials in $O(\ell)$ field operations.
\end{lemma}
\begin{proof}
By Lemma~\ref{lemma:generator-reduction}, we can compute a generator
$\alpha = [a] + [b]\phi$ with $a$ and $b$ in $O(\sqrt{\ell})$.
The $[a]$- and $[b]$-division polynomials have degrees in $O(\ell)$,
and can be determined in $O(\ell)$ field operations. The division
polynomials for the sum $\alpha = [a] + [b]\phi$ require one sum
and one application of $\phi$; and since $\phi$ is efficiently
computable, this increases the division polynomial degrees and
computing time by at most a constant factor.
\qed
\end{proof}
We can now state the main theorem for RM point counting.
\begin{theorem}
\label{theorem:RM-ptc-complexity}
There exists an algorithm for the point counting problem in a family
of genus~2 curves with efficiently computable RM of class number 1,
whose complexity is in ${\widetilde{O}}((\log q)^5)$.
\end{theorem}
\begin{proof}
Let $\Jac{\mathcal{C}}$ be a Jacobian in a family with efficiently computable
RM by $\mathbb{Z}[\phi]$.
Suppose that $\ell$ is prime, $(\ell) = \mathfrak{p}_1 \mathfrak{p}_2$ in $\mathbb{Z}[\phi]$,
and that the $\mathfrak{p}_i$ are principal.
By Lemma~\ref{lemma:alpha-division-polynomials} we can compute
representative $\alpha$-division polynomials for $\Jac{\mathcal{C}}$ for
each $\mathfrak{p}$ in $\{\mathfrak{p}_1,\mathfrak{p}_2\}$ in time ${\widetilde{O}}(\ell)$, hence
generic points $D_i$ in $\Jac{\mathcal{C}}[\mathfrak{p}_i]$.
We recall that \eqn{eq:RM-ell-test} is the homomorphic
image under $\pi$ of the equation
$$
\psi(D) - [\bar{m}](D) - [\bar{n}]\phi(D) = 0.
$$
When applied to $D_i$ in $\Jac{\mathcal{C}}[\mathfrak{p}_i]$, both $\psi$ and $\phi$
act as elements of $\mathbb{Z}[\phi]/\mathfrak{p}_i \cong \mathbb{Z}/\ell\mathbb{Z}$. Moreover
$\bar{x}_i = \phi \bmod \mathfrak{p}_i$ is known, and it remains to determine
$\bar{y}_i = \psi \bmod \mathfrak{p}_i$ by means of
the discrete logarithm
$$
\psi(D_i) = [\bar{y}_i](D_i) = [\bar{m}+\bar{n}\bar{x}_i](D_i)
$$
in the cyclic group $\langle D_i \rangle \cong \mathbb{Z}/\ell\mathbb{Z}$.
The application of $\pi$ transports this discrete logarithm
problem to that of solving for $\bar{y}_i$ in
$$
D_i'' = [\bar{y}_i]D',
$$
where $D_i' = \pi(D_i)$ and $D_i'' = (\pi^2+[\bar{q}])(D_i)$.
By the CRT, from $(\bar{y}_1,\bar{y}_2)$ in $(\mathbb{Z}/\ell\mathbb{Z})^2$
we recover $\bar{y}$ in $\mathbb{Z}[\phi]/(\ell)$, from which we solve
for $(\bar{m},\bar{n})$ in $(\mathbb{Z}/\ell\mathbb{Z})^2$ such that
$$
\bar{y} = \bar{m} + \bar{n}\phi \in \mathbb{Z}[\phi]/(\ell).
$$
The values of $(\bar{s}_1,\bar{s}_2)$ are then recovered from
\eqn{eq:s_1s_2-from-nm}.
The ring $\mathbb{Z}[\phi]$ is fixed, so as $\log q$ goes to infinity
we find that 50\% of all primes $\ell$ split in $\mathbb{Z}[\phi]$
by the Chebotarev density theorem.
It therefore suffices to consider split
primes in $O(\log q)$.
In comparison with the conventional algorithm
presented in \S\ref{sec:Schoof--Pila}, we reduce from
computation modulo the ideal for $\Jac{\mathcal{C}}[\ell]$ of degree in
$O(\ell^4)$, to computation
modulo the ideals for $\Jac{\mathcal{C}}[\mathfrak{p}_i]$ of degree in $O(\ell^2)$.
This means a reduction from ${\widetilde{O}}(\ell^4(\ell^2+\log q))$
to ${\widetilde{O}}(\ell^2(\ell+\log q))$ field operations for the determination of
each $\chi_{\ell}(T)$,
giving the stated reduction in
total complexity from ${\widetilde{O}}((\log q)^8)$ to ${\widetilde{O}}((\log q)^5)$.
\qed
\end{proof}
\begin{remark}
\label{remark:RM-Schoof-Pila-reduction}
Computing $(m,n)$ instead of $(s_1,s_2)$ allows us to reduce the number
of primes~$\ell$ to be considered by about a half, since
by~\eqn{eq:m-n-bounds} their product
needs to be in $O(\sqrt{q})$ instead of $O(q)$.
While this changes only the constant in the asymptotic complexity
of the algorithm, it yields a significant improvement in practice.
\end{remark}
\begin{remark}
\label{remark:RM-class-number-not-1}
If $\mathbb{Z}[\phi]$ does not have class number $1$, and if $(\ell) = \mathfrak{p}_1\mathfrak{p}_2$
where the $\mathfrak{p}_i$ are not principal, then we may
use a small complementary ideal $(c) = \mathfrak{c}_1\mathfrak{c}_2$ such that $\mathfrak{c}_i\mathfrak{p}_i$
are principal in order to apply Lemma~\ref{lemma:alpha-division-polynomials}
to a larger proportion of small ideals.
Moreover, if $(\bar{m},\bar{n})$ is known modulo $c$, this can be used
to reduce the discrete log problem modulo $\ell$.
Again, since a fixed positive density $1/2h$ of primes are both split
and principal, where $h$ is the class number of $\mathbb{Z}[\phi]$, this does
not affect the asymptotic complexity.
Moreover, the first occurrence of a nontrivial class group is for
$\Delta = 65$, beyond the current range for which an explicit RM
construction is currently known.
\end{remark}
\subsection{Shrinking the BSGS Search Space}
\label{sec:search-space}
In the context of the conventional Schoof-Pila algorithm, we need to
find $s_1$ in $O(\sqrt{q})$ and $s_2$ in $O(q)$.
However, \eqn{eq:RM-psi-relation}, and the effective form of
\eqn{eq:RM-ell-test} (valid for all points $D$ of $\Jac{\mathcal{C}}$),
replaces the determination of $(s_1,s_2)$ with the tuple $(m,n)$ of
integers in $O(\sqrt{q})$.
As a result, the search space is reduced from $O(q^{3/2})$ to $O(q)$.
Thus the BSGS strategy can find $(m,n)$ (which determines $(s_1,s_2)$)
in time and space $O(\sqrt{q})$, compared with $O(q^{3/4})$ when searching
directly for $(s_1,s_2)$.
As in the general case, if one knows $(m,n)$ modulo an integer $M$, then
the area of the search rectangle is reduced by a factor of $M^2$, so we
find the tuple $(m,n)$ in $O(\sqrt{q}/M)$ group operations.
Contrary to the general case of \S\ref{sec:BSGS}, since $m$ and $n$ have
the same order of magnitude, the speed-up is always by a factor of $M$.
\section{Examples of Families of Curves with Explicit RM}
\label{sec:families}
We now exhibit some families of curves and efficient RM endomorphisms
that can be used as sources of inputs to our algorithm.
\subsection{Correspondences and Endomorphisms}
\label{sec:correspondences}
To give a concrete representation for endomorphisms of $\Jac{\mathcal{C}}$,
we use \emph{correspondences}:
that is, divisors on the surface $\XxX{\family{C}}$.
Suppose that $\family{R}$ is a curve on $\XxX{\family{C}}$,
and
let $\pi_1: \family{R} \to \family{C}$ and $\pi_2: \family{R} \to \family{C}$
be the restrictions to $\family{R}$
of the natural projections from $\XxX{\family{C}}$
onto its first and second factors.
We have a pullback homomorphism
\(
(\pi_1)^*: \Pic{\family{C}} \to \Pic{\family{R}}
\),
defined by
\[
(\pi_1)^*\Big(\Big[
\sum_{P \in \family{C}(\overline{\mathbb{F}}_{q})}\!\!\! n_P P\
\Big]\Big)
=
\Big[
\sum_{P \in \family{C}(\overline{\mathbb{F}}_{q})}\!\!\! n_P\!\!\!
\sum_{Q \in \pi_{1}^{-1}(P)}\!\!\! Q\
\Big]
,
\]
where the preimages $Q$ are counted with the appropriate multiplicities.
(A standard moving lemma shows that
we can always choose divisor class representatives
so that each $\pi^{-1}(P)$ is zero-dimensional.)
We also have a pushforward homomorphism
\(
(\pi_2)_*: \Pic{\family{R}} \to \Pic{\family{C}}
\),
defined by
\[
(\pi_2)_*\Big(\Big[
\sum_{Q \in \family{R}(\overline{\mathbb{F}}_{q})}\!\!\! n_Q Q\
\Big]\Big)
=
\Big[ \sum_{Q \in \family{R}(\overline{\mathbb{F}}_{q})}\!\!\! n_Q \pi_2(Q)\ \Big]
.
\]
Note that $(\pi_1)^*$ maps $\Pic[n]{\family{C}}$
into $\Pic[(n\deg\pi_1)]{\family{R}}$
and $(\pi_2)_*$ maps $\Pic[n]{\family{R}}$
into $\Pic[n]{\family{C}}$
for all $n$.
Hence $(\pi_2)_*\circ(\pi_1)^*$
maps $\Pic[0]{\family{C}}$ into
$\Pic[0]{\family{C}}$,
so we have an \emph{induced endomorphism}
\[
\phi = (\pi_2)_*\circ(\pi_1)^* : \Jac{\family{C}} \to \Jac{\family{C}}
.
\]
We write $x_1,y_1$ and $x_2,y_2$ for the coordinates on the first and second
factors of $\XxX{\family{C}}$, respectively
(so $\pi_i(x_1,y_1,x_2,y_2) = (x_i,y_i)$).
In our examples,
the correspondence
$\family{R}$
will be defined by two equations:
\[
\family{R}
=
\variety{A(x_1,x_2),B(x_1,y_1,x_2,y_2)}
.
\]
On the level of divisors,
the image of a generic point $P = (x_P,y_P)$ of $\family{C}$
(that is, a generic prime divisor)
under the endomorphism
\(\phi\)
is given by
\[
\phi:
(x_P,y_P)
\longmapsto
\variety{A(x_P,x),B(x_P,y_P,x,y)}
.
\]
Using
the relations $y_P^2 = f(x_P)$ and $y^2 = f(x)$
(and the fact that correspondences cut out by principal ideals
induce the zero homomorphism),
we can easily replace $A$ and $B$ with
Cantor-reducible generators
to
derive the Mumford representation of $\phi(P)$,
and thus the $\phi$-division polynomials.
\subsection{A $1$-dimensional Family with RM by $\mathbb{Z}[\big(1+\sqrt{5}\big)/2]$}
\label{sec:CTfamily}
\newcommand{\TTV}[1]{\ensuremath{{#1}_{\mathrm{T}}}}
Let $t$ be a free parameter,
and suppose that $q$ is not a power of $5$.
Let $\TTV{\family{C}}$ be the family of curves of genus $2$ over $\mathbb{F}_{q}$
considered by Tautz, Top, and Verberkmoes
in~\cite[Example 3.5]{Tautz--Top--Verberkmoes},
defined by
\[
\TTV{\family{C}}: y^2 = x^5 - 5x^3 + 5x + t .
\]
Let $\tau_5 = \zeta_5 + \zeta_5^{-1}$,
where $\zeta_5$ is a $5$th root of unity in $\overline{\mathbb{F}}_{q}$.
Let
$\TTV{\phi}$
be the endomorphism induced by
the (constant) family of
correspondences
\[
\TTV{\family{R}}
=
\variety{ x_1^2 + x_2^2 - \tau_5 x_1x_2 + \tau_5^2 - 4 , y_1 - y_2 }
\subset \XxX{\TTV{\family{C}}}
.
\]
(Note that \(\TTV{\family{R}}\) and $\TTV{\phi}$
are defined over $\mathbb{F}_{q}(\tau_5)$,
which is equal to $\mathbb{F}_{q}$
if and only if $q \not\equiv\pm2\bmod5$.)
The family $\TTV{\family{C}}$ has an unique point $P_\infty$ at
infinity,
which we can use to define an embedding
\[
P = (x_P,y_P)
\longmapsto
D_P := [(P) - (P_\infty)]
\leftrightarrow
(x - x_P, y - y_P)
\]
of $\TTV{\family{C}}$ in $\Jac{\TTV{\family{C}}}$.
With respect to this embedding,
the $\TTV{\phi}$-division polynomials are
\[
d_2 = 1 ,
\quad
d_1 = -\tau_5 x ,
\quad
d_0 = x^2 + \tau_5^2 - 4 ,
\quad
e_2 = 1,
\quad
e_1 = 0,
\quad
e_0 = 1.
\]
\begin{proposition}
\label{proposition:TTV}
The minimal polynomial of \(\TTV{\phi}\)
is \(T^2 + T - 1\):
that is, \(\TTV{\phi}\)
acts as multiplication by $-(1 + \sqrt{5})/2$
on~\(\Jac{\TTV{\family{C}}}\).
A prime \(\ell\)
splits into two principal ideals in \(\mathbb{Z}[\TTV{\phi}]\)
if and only if \(\ell \equiv \pm1\bmod 5\).
\end{proposition}
\begin{proof}
The first claim is proven
in~\cite[\S3.5]{Tautz--Top--Verberkmoes}.
More directly,
if \(P\) and \(Q\)
are generic points of \(\TTV{\family{C}}\),
then on the level of divisors we find
\[
(\TTV{\phi}^2 + \TTV{\phi})((P) - (Q))
=
(P) - (Q) + \mathrm{div}\left(\frac{y - y(P)}{y - y(Q)}\right).
\]
Hence \(\mathbb{Z}[\TTV{\phi}]\)
is isomorphic to the ring of integers of \(\mathbb{Q}(\sqrt{5})\).
The primes~\(\ell\) splitting in \(\mathbb{Q}(\sqrt{5})\)
are precisely those congruent to \(\pm 1\) modulo \(5\);
and \(\mathbb{Q}(\sqrt{5})\) has class number \(1\),
so the primes over \(\ell\) are principal.
\qed
\end{proof}
The Igusa invariants of $\TTV{\family{C}}$,
viewed as a point in weighted projective space, are
\( ( 140 : 550 : 20(32t^2 - 3) : 25(896t^2 - 3109) : 64(t^2-4)^2) \);
in particular,
$\TTV{\family{C}}$
has a one-dimensional image in the moduli space of curves of genus~$2$.
The Jacobian of the curve with the same defining equation over $\mathbb{Q}(t)$
is absolutely simple (cf.~\cite[Remark 15]{Kohel--Smith}).
\subsection{A $2$-dimensional Family with RM by $\mathbb{Z}[\big(1+\sqrt{5}\big)/2]$}
\label{sec:MestreFive}
\newcommand{\MestreFive}[1]{\ensuremath{{#1}_{\mathrm{H}}}}
Let $s$ and $t$ be free parameters,
and consider the family of curves
\( \MestreFive{\family{C}} : y^2 = \MestreFive{F}(x) \),
where
\[
\MestreFive{F}(x)
=
sx^5 - (2s + t)x^4 + (s^2 + 3s + 2t - 1)x^3 - (3s + t - 3)x^2 + (s - 3)x + 1
.
\]
This family is essentially due to Humbert;
it is equal to the family of Mestre~\cite[\S2.1]{Mestre}
with \( (U,T) = (s,t) \),
and the family of Wilson~\cite[Proposition 3.4.1]{Wilson}
with \( (A,B) = (s,-t-3s+3) \).
The family has a full $2$-dimensional image
in the moduli space of genus $2$ curves.
Let $\MestreFive{\family{R}}$ be the family of correspondences
on \( \XxX{\MestreFive{\family{C}}} \)
defined by
\[
\MestreFive{\family{R}}
=
\variety{ x_1^2x_2^2 + s(s-1)x_1x_2 - s^2(x_1 - x_2) + s^2 , y_1 - y_2 }
;
\]
let \(\MestreFive{\phi}\)
be the induced endomorphism.
The family $\MestreFive{\family{C}}$ has a
unique point $P_\infty$ at infinity,
which we can use to
define an embedding
\[
P = (x_P,y_P)
\longmapsto
D_P := [(P) - (P_\infty)]
\leftrightarrow
(x - x_P, y - y_P)
\]
of $\MestreFive{\family{C}}$ in $\Jac{\MestreFive{\family{C}}}$.
With respect to this embedding,
the \(\MestreFive{\phi}\)-division polynomials
are
\[
d_2 = x^2,
\ \
d_1 = (s^2-s)x + s^2,
\ \
d_0 = -s^2x + s^2,
\ \
e_2 = 1,
\ \
e_1 = 0,
\ \
e_0 = 1.
\]
\begin{proposition}
\label{proposition:MestreFive}
The minimal polynomial of \(\MestreFive{\phi}\)
is \(T^2 + T - 1\):
that is, \(\MestreFive{\phi}\)
acts as multipliction by $-(1 + \sqrt{5})/2$
on~$\Jac{\MestreFive{\family{C}}}$.
A prime \(\ell\)
splits into two principal ideals in \(\mathbb{Z}[\MestreFive{\phi}]\)
if and only if \(\ell \equiv \pm 1 \bmod 5\).
\end{proposition}
\begin{proof}
The first assertion is~\cite[Proposition 2]{Mestre}
with \(n = 5\);
the rest of the proof is exactly as in
Proposition~\ref{proposition:TTV}.
\qed
\end{proof}
\subsection{A $2$-dimensional Family with RM by $\mathbb{Z}[\sqrt{2}]$}
\label{sec:MestreTwo}
\newcommand{\MestreTwo}[1]{\ensuremath{{#1}_{\mathrm{M}}}}
For an example with $\Delta = 8$,
we present a twisted and reparametrized version
of a construction due to Mestre~\cite{Mestre-09}.
Let $s$ and $t$ be free parameters,
let $v(s)$ and $n(s)$ be the rational functions
\[
v = v(s) := \frac{s^2+2}{s^2-2}
\quad \text{and}\quad
n = n(s) := \frac{4s(s^4+4)}{(s^2-2)^3} ,
\]
and let $\MestreTwo{\family{C}}$ be the family of curves
defined by
\[
\MestreTwo{\family{C}}
:
y^2
=
\MestreTwo{F}(x)
:=
(vx - 1)(x-v)(x^4 - tx^2 + vt - 1)
.
\]
The family
of correspondences
on \(\XxX{\MestreTwo{\family{C}}}\)
defined by
\[
\MestreTwo{\family{R}}
=
V\left(\begin{array}{l}
x_1^2x_2^2 - v^2(x_1^2+x_2^2) + 1,
\\
y_1y_2-n(x_1^2+x_2^2-t)(x_1x_2-v(x_1+x_2)+1)
\end{array}\right)
\]
induces an endomorphism
\( \MestreTwo{\phi} \)
of $\Jac{\MestreTwo{\family{C}}}$.
The family $\MestreTwo{\family{C}}$
has two points at infinity, $P_\infty^+$ and $P_\infty^-$.
which are generically only defined over a quadratic extension of
$\mathbb{F}_{q}(s,t)$.
Let $D_\infty = (P_\infty^+) + (P_\infty^-)$
denote the divisor at infinity.
We can use the rational Weierstrass point $P_v = (v,0)$
on $\MestreTwo{\family{C}}$
to define an embedding
\[
P = (x_P,y_P)
\longmapsto
D_P :=
[ (P) + (P_v) - D_\infty ]
\leftrightarrow
\Big((x-x_P)(x - v), y - \frac{y_P}{x_P-v}(x-v)\Big)
\]
of $\MestreTwo{\family{C}}$ in $\Jac{\MestreTwo{\family{C}}}$
(the appropriate composition and reduction algorithms for
divisor class arithmetic on genus 2 curves with an even-degree model,
such as $\Jac{\MestreTwo{\family{C}}}$,
appear in~\cite{Galbraith--Harrison--Mireles-Morales}.)
With respect to this embedding,
the $\MestreTwo{\phi}$-division polynomials are
\[
\begin{array}{r@{\;=\;}l@{\qquad}r@{\;=\;}l}
d_2 & x^2 - v^2 ,
&
e_2 & (x^2 - v^2)\MestreTwo{F}(x) ,
\\
d_1 & 0 ,
&
e_1 & n(x-v)(x^4 - tx^2 + tv^2 - 1) ,
\\
d_0 & -v^2x^2 + 1 ,
&
e_0 & n(vx - 1)(x^4 - tx^2 + tv^2 - 1) .
\end{array}
\]
\begin{proposition}
\label{proposition:MestreTwo}
The minimal polynomial of \(\MestreTwo{\phi}\)
is \(T^2 - 2\):
that is,
\(\MestreTwo{\phi}\) acts as multiplication by $\sqrt{2}$
on $\Jac{\MestreTwo{\family{C}}}$.
A prime \(\ell\)
splits into two principal ideals in \(\mathbb{Z}[\MestreTwo{\phi}]\)
if and only if \(\ell\equiv\pm1\bmod{8}\).
\end{proposition}
\begin{proof}
Let \(P\) and \(Q\)
be generic points of \MestreTwo{\family{C}}.
An elementary but lengthy calculation shows that
on the level of divisors,
\[
\MestreTwo{\phi}^2((P) - (Q))
=
2(P)-2(Q) + \mathrm{div}\left(\frac{x+x(P)}{x+x(Q)}\right)
,
\]
so
\(\MestreTwo{\phi}^2([D]) = 2[D]\) for all \([D]\) in
\(\Pic[0]{\MestreTwo{\family{C}}}\).
Hence \(\MestreTwo{\phi}^2 = [2]\),
and $\mathbb{Z}[\MestreTwo{\phi}]$ is isomorphic to
the maximal order of $\mathbb{Q}(\sqrt{2})$.
The primes \(\ell\) splitting in $\mathbb{Q}(\sqrt{2})$
are precisely those congruent to \(\pm 1\) modulo \(8\);
further, $\mathbb{Q}(\sqrt{2})$ has class number 1,
so the primes over \(\ell\) are principal.
\qed
\end{proof}
\begin{remark}
As noted above,
this construction is a twisted reparametrization
of a family of isogenies described by
Mestre in~\cite[\S2.1]{Mestre-09}.
Let \(a_1\) and \(a_2\)
be the roots of $T^2 - tT + v^2t-1$ in $\overline{\mathbb{F}_{q}(v,t)}$.
Mestre's curves \(C'\) and \(C\)
are equal
(over \(\mathbb{F}_{q}(v,a_1,a_2)\))
to
our \(\MestreTwo{\family{C}}\)
and its quadratic twist by
\(A = 2(v^2-1)(v^2+1)^2 = (2n)^2 \),
respectively.
We may specialize the proofs in~\cite{Mestre-09}
to show that $\MestreTwo{\family{C}}$
has a two-dimensional image in the moduli space of curves of genus $2$,
and that the Jacobian of the curve with the same defining equation over $\mathbb{Q}(s,t)$
is absolutely simple.
Constructions of curves with RM by \(\mathbb{Z}[\sqrt{2}]\)
are further investigated in Bending's thesis~\cite{Bending}.
\end{remark}
\begin{remark}
The algorithms described here should be readily adaptable
to work with Kummer surfaces instead of Jacobians.
In the notation of~\cite{Gaudry-theta},
the Kummers with parameters $(a,b,c,d)$
satisfying $b^2 = a^2-c^2-d^2$
have RM by $\mathbb{Z}[\sqrt{2}]$,
which can be made explicit as follows:
the doubling algorithm decomposes into two identical steps,
since $(A:B:C:D) = (a:b:c:d)$,
and the components after one step are
the coordinates of a Kummer point.
The step therefore defines an efficiently computable endomorphism
which squares to give multiplication by 2.
\end{remark}
\section{Numerical Experiments}
We implemented our algorithm in C++ using the NTL library.
For non-critical steps, including computations in quadratic fields, we
used Magma for simplicity. With this implementation,
the determination of $\chi(T)$ for a curve over a 128-bit prime field
takes approximately 3 hours on one core of a Core2 processor at 2.83 GHz.
This provides a proof of concept rather than an optimized implementation.
\subsection{Cryptographic Curve Generation}
When looking for a cryptographic curve, we used an early-abort
strategy, where we switch to another curve as soon as either the
order of the Jacobian order or its twist can not be prime.
Using our adapted version of Schoof algorithm, we guarantee that
the group orders are not divisible by any prime that splits in the real
field up to the CRT bound used.
In fact, any prime that divides the group order of a curve having RM
by the maximal order of $\mathbb{Q}(\sqrt{\Delta})$ must either be a split (or
ramified) prime, or divide it with multiplicity 2.
As a consequence, the early abort strategy works much better than in
the classical Schoof algorithm, because, it suffices to test half the
number of primes up to our CRT bound.
We ran a search for a secure curve over a prime field of 128 bits, using
a CRT bound of 131.
Our series of computations frequently aborted early, and resulted in 245
curves for which $\chi(T)$ was fully determined, and for which neither
the group order nor its twist was divisible by a prime less than 131.
Considering these twists, this provided 490 group orders, of which 27 were
prime, and therefore
suitable for cryptographic use.
We give here the data for one of these curves, that was furthermore
twist-secure: both the Jacobian and the twist Jacobian order are prime.
Let $\mathcal{C}/\mathbb{F}_q$, where $q=2^{128}+573$, be the curve in the family
$\TTV{\family{C}}$ of \S\ref{sec:CTfamily} specialized to
$t=75146620714142230387068843744286456025$. The characteristic
polynomial $\chi(T)$ is determined by
$$
(s_1,s_2) = (-26279773936397091867,\;
-90827064182152428161138708787412643439),
$$
giving prime group orders for the Jacobian:
$$
115792089237316195432513528685912298808995809621534164533135283195301868637471,
$$
and for its twist:
$$
115792089237316195414628441331463517678650820031857370801365706066289379517451.
$$
We note that correctness of the orders is easily verified on
random points in the Jacobians.
\subsection{A Kilobit Jacobian}
Let $q$ be the prime $2^{512}+1273$,
and consider the curve over $\mathbb{F}_{q}$ from the family
$\TTV{\family{C}}$ of \S\ref{sec:CTfamily}
specialized at
\[
\begin{array}{rl}
t = &
2908566633378727243799826112991980174977453300368095776223\\ &
2569868073752702720144714779198828456042697008202708167215\\ &
32434975921085316560590832659122351278 .
\end{array}
\]
This value of $t$ was randomly chosen, and carries no special structure.
We computed the values of the pair $(s_1\bmod\ell,n\bmod\ell)$ for this curve
for each split prime $\ell$ up to $419$; this was enough to
uniquely determine the true value of $(s_1, n)$ using the Chinese Remainder
Theorem.
The numerical data for the curve follows:
\[
\begin{array}{r@{\;}c@{\;}l}
\Delta & = & 5 \\
s_1 & = &
-10535684568225216385772683270554282199378670073368228748\\ & &
7810402851346035223080\\
n & = &
-37786020778198256317368570028183842800473749792142072230\\ & &
993549001035093288492\\
s_2 & = & (s_1^2 - n^2\Delta)/4 \\
& = &
990287025215436155679872249605061232893936642355960654938\\ & &
008045777052233348340624693986425546428828954551752076384\\ & &
428888704295617466043679591527916629020\\
\end{array}
\]
The order of the Jacobian is therefore
\[
\begin{array}{r@{\;}c@{\;}l}
N & = & (1+q)^2 - s_1(1+q) + s_2\\
& = &
179769313486231590772930519078902473361797697894230657273\\ & &
430081157732675805502375737059489561441845417204171807809\\ & &
294449627634528012273648053238189262589020748518180898888\\ & &
687577372373289203253158846463934629657544938945248034686\\ & &
681123456817063106485440844869387396665859422186636442258\\ & &
712684177900105119005520.
\end{array}
\]
The total runtime for this computation was about 80 days on a single core
of a Core 2 clocked at 2.83 GHz. In practice, we use the inherent
parallelism of the algorithm, running one prime~$\ell$ on each available core.
We did not
compute the characteristic polynomial modulo small prime powers
(as in~\cite{GaSc10}),
nor did we use BSGS to
deduce the result from partial modular information
as in \S\ref{sec:search-space}
(indeed, we were more
interested in measuring the behaviour of our algorithm for large values
of~$\ell$).
These improvements with an exponential-complexity
nature bring much less than in the classical point
counting algorithms, since they have to be balanced with a
polynomial-time algorithm with a lower degree.
For this example, we
estimate that BSGS and small prime powers could have saved a factor
of about 2 in the total runtime.
\subsection{Degrees of Division Polynomials}
For each prime $\ell$ splitting in $\mathbb{Z}[\TTV{\phi}]$,
we report the degree of the $\alpha$-division polynomial $d_2$
(where $\alpha$ is the endomorphism of norm $\ell$ that was used).
By Lemma~\ref{lemma:generator-reduction},
$\deg(d_2)$ is in $O(\ell)$;
the table below gives the ratio $\deg(d_2)/\ell$,
thus measuring the hidden constant in the $O()$ notation.
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c||}
\hline $\ell$ &
11 & 19 & 29 & 31 & 41 & 59 & 61 & 71 & 79 & 89 & 101 & 109 & 131 \\
\hline $\deg d_2 / \ell$ &
1.82 & 2.05 & 2.07 & 1.94 & 2.05 & 2.10 & 1.97 & 2.03 & 2.01 & 2.02 &
1.98 & 2.02 & 2.02 \\
\hline
\hline $\ell$ &
139 & 149 & 151 & 179 & 181 & 191 & 199 & 211 & 229 & 239 & 241 & 251 &
269 \\
\hline $\deg d_2 / \ell$ &
2.12 & 2.04 & 1.99 & 2.00 & 2.01 & 2.09 & 2.21 & 1.99 & 2.18 & 2.01 &
2.05 & 2.07 & 2.17 \\
\hline
\hline $\ell$ &
271 & 281 & 311 & 331 & 349 & 359 & 379 & 389 & 401 & 409 & 419 & & \\
\hline $\deg d_2 / \ell$ &
2.01 & 1.99 & 2.11 & 2.12 & 2.13 & 2.02 & 2.00 & 2.16 & 2.03 & 2.10 &
2.00 & & \\
\hline
\end{tabular}
\end{center}
We have
$\deg(d_1) = \deg(d_2)+1$ and $\deg(d_0) = \deg(d_2)+2$.
All of these degrees depend only on the curve family $\TTV{\family{C}}$,
and not on the individual curve chosen.
|
1,116,691,500,726 | arxiv | \section{Introduction}
The holographic model of superconductors, which is constructed by a
gravitational theory of a Maxwell field coupled to a charged complex
scalar field via anti-de Sitter/conformal field theory (AdS/CFT)
correspondence \cite{Witten,Maldacena-1,Maldacena-2}, has been
investigated extensively in recent years (for reviews, see Refs.
\cite{HartnollRev,HerzogRev,HorowitzRev}). According to the AdS/CFT
dictionary, the emergence of the scalar hair in the bulk AdS black
hole corresponds to the formation of a charged condensation in the
boundary dual CFTs. This brings a remarkable connection between the
condensed matter and the gravitational physics which attracts
considerable interest for its potential applications to the
condensed matter physics \cite{HorowitzPRD78}-\cite{Brihaye}. At the
moment when the condensation occurs in the boundary CFT and in the
gravitational counterpart a non-trivial hair for the black hole is
triggered, there appears a phase transition \cite{gubser,Hart}. The
phenomenological signature of this phase transition was recently
disclosed in the perturbation around such AdS black holes
\cite{HeXi,ZhangCai}.
Motivated by the application of the Mermin-Wagner theorem to the
holographic superconductors there were studies of the effects of the
curvature corrections on the (3 + 1)-dimensional superconductor
\cite{Gregory,Pan-Wang,Ge-Wang,Pan2,Brihaye}. It was found that
higher curvature corrections make condensation harder. In addition,
the large Gauss-Bonnet factor gives the correction to the disclosed
universal value for the conductivity $\omega_g/T_c\approx 8$
\cite{HorowitzRev} in the probe limit
\cite{Gregory,Pan-Wang,Ge-Wang}. Furthermore Brihaye \emph{et al.}
observed that the decrease of the critical temperature at which
condensation sets in is stronger as the Gauss-Bonnet coupling
increases, which happens even beyond the probe approximation
\cite{Brihaye}. In order to get more objective picture on the
influence given by the high curvature on the condensation, in this
work we are going to study the perturbation in the $5$-dimensional
Gauss-Bonnet-AdS black hole backgrounds. We will concentrate on the
bulk high temperature AdS black holes and pay more attention on how
the Gauss-Bonnet term influences the perturbation in the bulk
background spacetime when the temperature of the black hole drops.
Further we are going to study the critical phenomenon once the AdS
black hole approaches marginally stable mode and the charged scalar
field starts to condensate. We will examine how the Gauss-Bonnet
term affects the critical behavior. Recently, Maeda \emph{et al.}
\cite{maeda} studied most of the static critical exponents of
holographic superconductors for a Reissner-Nordstr$\ddot{o}$m (RN)
AdS black hole with planar horizon and found that they take the
standard mean-field values. We will generalize their work to the
$5$-dimensional Gauss-Bonnet-AdS black hole configurations and
examine the effect of Gauss-Bonnet term on the critical behavior. We
will focus our attention on the high-temperature phase for
simplicity and study the linear perturbations of the bulk equations
of motion in the probe approximation.
The outline of this work is as follows. In section II, we deal with
the perturbation equations of the charged scalar field in the
$5$-dimensional Gauss-Bonnet-AdS black hole spacetime. In section
III, we investigate the perturbation in the bulk and examine the
critical phenomenon of the holographic superconductors. We will
conclude in the last section with our main results.
\section{Perturbation Equations in the Gauss-Bonnet AdS Black Hole}
Let us begin with the $D=p+2$ dimensional charged Gauss-Bonnet black
hole described by the metric~\cite{BD,Cai,Cai2,Cvet,ge}
\begin{eqnarray}\label{metric}
ds^{2}&=&-H(r)dt^{2}+H^{-1}(r)dr^{2}+\frac{r^{2}}{l^{2}}dx_{p}^{2}~,
\end{eqnarray}
where the $U(1)$ gauge field reads
\begin{eqnarray}\label{potential}
A_{t}&=&\frac{Q}{4\pi(D-3)}~(r_{H}^{D-3}-r^{D-3}).
\end{eqnarray}
Here
\begin{eqnarray}
H(r)&=&\frac{r^{2}}{2\alpha}\left[1-\sqrt{1-\frac{4\alpha}{l^{2}}\left(1-\frac{m
l^{2}}{r^{D-1}}+\frac{Q_{0}^{2}l^{2}}{r^{2D-4}}\right)}\right]~,
\end{eqnarray}
where $\alpha$ is the Gauss-Bonnet coefficient, $r_{H}$ is the
horizon radius and $l$ corresponds to the AdS radius. The
gravitational mass $M$ and the charge $Q$ are expressed as
\begin{eqnarray}
M&=&\frac{(D-2)mV_p}{16\pi G_{D}}~,\nonumber\\
Q^{2}&=&\frac{2\pi (D-2)(D-3)Q_{0}^{2}}{G_{D}}~,\nonumber
\end{eqnarray}
where $V_p$ is the volume of the $p$-dimensional Euclidean space
$dx_p^2$ and $G_D$ is the $D$-dimensional Newton constant. Note that
in the asymptotic region (r$\rightarrow\infty$), we have
\begin{equation}
H(r)=\frac{r^{2}}{2\alpha}\left(1-\sqrt{1-\frac{4\alpha}{l^{2}}}\right)~.\nonumber
\end{equation}
We can define the effective AdS radius
\begin{eqnarray}
\quad\quad\quad\quad\quad
l_{eff}^{2}=\frac{2\alpha}{1-\sqrt{1-\frac{4\alpha}{l^2}}}\sim\left\{
\begin{array}{ll} l^2~, &
~~~~\alpha\rightarrow 0~, \\
\frac{l^2}{2}~, & ~~~~ \alpha\rightarrow \frac{l^2}{4}~.
\end{array} \right.
\end{eqnarray}
The upper bound of the Gauss-Bonnet coefficient $\alpha\leq l^2/4$
is known as the Chern-Simons limit. Besides there also exists a
lower bound $\alpha\geq-\frac{(3D-1)(D-3)}{4(D+1)^{2}}$ by
considering the causality of dual field theory on the boundary
\cite{ge,Buchel,Boer}. Using a coordinate transformation, the metric
(\ref{metric}) and the potential (\ref{potential}) can be rewritten
as
\begin{equation}\label{new metric}
ds^{2}=\frac{l^{2}}{u^{2}}\left[-\frac{r_{H}^{2}(1+c)^{\frac{2}{1-p}}f(u)}{J^{2}(u)}dt^{2}+\frac{J^{\frac{2}{p-1}}(u)}{f(u)}du^{2}
+r_{H}^{2}(1+c)^{\frac{2}{1-p}}J^{\frac{2}{p-1}}(u)dx_{p}^{2}\right]~,
\end{equation}
\begin{eqnarray}
A_{t}&=&\mu \left[1-\frac{1+c}{J(u)}u^{p-1}\right]~,
\end{eqnarray}
with
\begin{eqnarray}
Q_{0}&=&c^{\frac{1}{2}}r_{H}^{p-2}~,\nonumber\\
u&=&\frac{r_{H}}{r(1+c-c~r_{H}/r)^{p-1}}~,\nonumber\\
J(u)&=&1+cu^{p-1}~,\nonumber\\
g(u)&=&J^{\frac{2p}{p-1}}(u)-(1+c)^{\frac{2p}{p-1}}u^{p+1}~,\nonumber\\
f(u)&=&\frac{J(u)}{2\alpha}\left[J(u)-\sqrt{J^{2}(u)-4\alpha
g(u)}\right]~,
\end{eqnarray}
where $\mu$ is the chemical potential and $c$ is related to the
parameter $Q_{0}$. Obviously $u=0$ is the AdS boundary and $u=1$ is
the location of the horizon. The metric (\ref{new metric}) will be
reduced to the $D$-dimensional RN-AdS black hole if we take the
limit $\alpha\rightarrow0$. On the other hand, it becomes the
neutral Gauss-Bonnet-AdS black hole~\cite{Cai} if $Q_0 = 0$. There
are four parameters $\alpha,~c,~r_{H}~$and $\mu$ which parameterize
the background (\ref{new metric}). In fact, not all of them are
independent, they are related by
\begin{equation}
\rho=l^{p-2}r_{H}^{p-1}~\mu~,
\end{equation}
where $\rho$ is the charge density. The temperature $T$ and the
chemical potential $\mu$ of the black hole are given by
\begin{eqnarray}
T=\frac{p+1-(p-1)c~r_{H}}{4\pi}~,
\end{eqnarray}
\begin{eqnarray}\label{chemical potential}
\mu=\sqrt{\frac{2p}{p-1}}c^{1/2}l~r_{H}~.
\end{eqnarray}
In order to investigate the perturbation in the bulk spacetime and
the critical phenomena when the black hole approaches marginally
stable from the high temperature phase, we will consider the
minimally coupled, charged scalar perturbation,
$\psi_{\varpi,~q}(u)e^{-i(\omega t+kx)}$, with mass $m$, which obeys
the wave equation
\begin{equation}\label{perturbation equation}
\left[u^{p}\frac{d}{du}(\frac{f}{u^{p}}\frac{d}{du})+\frac{J^{2p/(p-1)}(\varpi+\aleph)^{2}}{f(1+c)^{2/(1-p)}}
-\frac{q^{2}}{(1+c)^{2/(1-p)}}-\frac{l^{2}m^{2}J^{2/(p-1)}}{u^{2}}\right]\psi_{\varpi,~q}(u)=0~,
\end{equation}
where we have defined four dimensionless quantities:
\begin{eqnarray}
\varpi&:=&\omega/r_{H}~,\nonumber\\
q&:=&|k|/r_{H}\nonumber~,\nonumber\\
\aleph&:=&\frac{eA_{t}}{r_{H}}=\sigma\left(\frac{1}{1+c}-\frac{u^{p-1}}{J}\right)~,\nonumber\\
\sigma&:=&(1+c)^{p/(p-1)}\frac{e\mu}{r_{H}}.
\end{eqnarray}
Using Eq. (\ref{chemical potential}), we can rewrite $\sigma$ as
\begin{equation}\label{sigma}
\sigma=\sqrt{\frac{2p}{p-1}}~(le)~c^{1/2}(1+c)^{p/(p-1)}~.
\end{equation}
Taking $e\rightarrow\infty$ and keeping $\sigma$ fixed, we can
employ the probe approximation following \cite{maeda}. Equation
(\ref{sigma}) tells that in the limit
$c\propto(le)^{-2}\rightarrow0$ the background (\ref{new metric})
becomes a neutral Gauss-Bonnet-AdS black hole in $D$
dimensions~\cite{Cai}.
Near the AdS boundary $u\sim0$, Eq. (\ref{perturbation equation})
becomes
\begin{equation}
\left[u^p\partial_{u}\left(\frac{1-\sqrt{1-4\alpha}}{2\alpha}u^{-p}~\partial_{u}\right)-l^{2}m^{2}u^{-2}\right]\psi_{\varpi,~q}(u)=0,
\end{equation}
and $\psi_{\varpi,~q}(u)$ has a fall-off behavior as
\begin{equation}
\psi_{\varpi,~q}(u)\sim\psi_{\varpi,~q}^{-}u^{\lambda_{-}}+\psi_{\varpi,~q}^{+}u^{\lambda_{+}},
\end{equation}
where
\begin{eqnarray}
\lambda_{\pm}:=\frac{1}{2}\left[p+1\pm\sqrt{(p+1)^{2}+4m^{2}l_{eff}^{2}/l^{2}}\right]~.
\end{eqnarray}
In the AdS/CFT duality, the order parameter expectation value
$\langle\mathcal{O}_{\varpi,~q}\rangle$ corresponds to
$\psi_{\varpi,~q}^{+}$ while the source term is
$\psi_{\varpi,~q}^{-}$, so the response function can be defined
by~\cite{maeda}
\begin{equation}\label{response function}
\chi_{\varpi,~q}:=\left.\frac{\delta\langle\mathcal{O}_{\varpi,~q}\rangle}{\delta\psi_{\varpi,~q}^{-}}\right
|_{\psi_{\varpi,~q}^{-}\rightarrow0}\propto\frac{\psi_{\varpi,~q}^{+}}{\psi_{\varpi,~q}^{-}}~.
\end{equation}
We aim to investigate the perturbation and the critical phenomenon,
so that we have to solve the equation of motion of the scalar field,
Eq. (\ref{perturbation equation}), based on the boundary conditions
at the horizon and at the boundary. After obtaining the coefficients
$\psi_{\varpi,~q}^{\pm}$, we can study the behavior of the response
function.
Near the horizon $u\sim1$, Eq. (\ref{perturbation equation}) becomes
\begin{equation}\label{new perturbation equation}
f\frac{\partial}{\partial u}\left[f\frac{\partial}{\partial
u}\psi_{\varpi,~q}(u)\right]+(1+c)^{\frac{2p+2}{p-1}}\varpi^{2}\psi_{\varpi,~q}(u)=0~,
\end{equation}
and its solution is given by $\psi_{\varpi,~q}(u)\sim(1-u)^{\pm
i\frac{\omega}{4\pi T}}$. We impose the ``incoming wave" boundary
condition at the horizon, so $\psi_{\varpi,~q}(u)\sim(1-u)^{-
i\frac{\omega}{4\pi T}}$. Introducing a new variable $\varphi$ as
$\psi_{\varpi,~q}(u)=\mathcal{\Re}(u)\varphi_{\varpi,~q}(u)$ and
choosing
$\mathcal{\Re}(u)=exp[i(1+c)^{1/(p-1)}\int^{u}_{0}du\frac{J^{p/((p-1)}}{f}(\varpi+\aleph)]$,
we can express the boundary condition at the horizon as
$\varphi_{\varpi,~q}(~u=1)=const.$, and Eq. (\ref{perturbation
equation}) becomes
\begin{eqnarray} \label{main equation}
&&\left\{\frac{d^{2}}{du^{2}}+\left[(\frac{d}{du}\ln\frac{f}{u^{p}})+2i\frac{(\varpi+\aleph)
J^{p/(p-1)}}{f(1+c)^{1/(1-p)}}\right]\frac{d}{du}\right.\nonumber\\&&\left.-\frac{q^{2}(1+c)^{2/(p-1)}u^{2}+l^2m^2J^{2/(p-1)}}{u^{2}f}
+i\frac{u^p}{f}\frac{d}{du}\left[\frac{J^{p/(p-1)}(\varpi+\aleph)}{u^{p}(1+c)^{1/(1-p)}}\right]\right\}\varphi_{\varpi,~q}(u)=0~.
\end{eqnarray}
Near the AdS boundary $u\sim0$, $\varphi_{\varpi,~q}$ behaves as
\begin{equation}\label{asymptotic behavior}
\varphi_{\varpi,~q}(u)\sim\varphi_{\varpi,~q}^{-}u^{\lambda_{-}}+\varphi_{\varpi,~q}^{+}u^{\lambda_{+}}~.
\end{equation}
The boundary conditions at the horizon are now given by
\begin{eqnarray}\label{boundary}
\varphi_{\varpi,~q}|_{~u=1}&=&1~,\nonumber\\
\left.\frac{\varphi^{\prime}_{\varpi,~q}}{\varphi_{\varpi,~q}}\right|_{~u=1}
&=&\left.\frac{q^{2}(1+c)^{2/(p-1)}+l^2m^2J^{2/(p-1)}u^{-2} -i
u^p\frac{d}{du}[\frac{J^{p/(p-1)}(\varpi+\aleph)}{u^p(1+c)^{1/(1-p)}}]}{\partial_{u}f-p~
u^{-1}f+2i(\varpi+\aleph)(1+c)^{1/(p-1)}J^{p/(p-1)}}\right|_{~u=1}~.
\end{eqnarray}
Eq. (\ref{main equation}) is a linear equation and
$\varphi_{\varpi,~q}(u)$ must be regular at the horizon. Since we do
not concentrate on the amplitude of $\varphi_{\varpi,~q}(u)$, we can
set $\varphi_{\varpi,~q}(~u=1)=1$.
\section{Numerical Results}
In this section, we will numerically solve Eq. (\ref{main equation})
under the boundary conditions (\ref{boundary}) in the probe
approximation. We will first examine the behavior of the charged
scalar field perturbation, which can present us an objective picture
on how the black hole approaches the marginally stable mode when the
temperature drops. In addition we will determine the coefficients
$\varphi_{\varpi,~q}^{\pm}$ from the asymptotic behavior
(\ref{asymptotic behavior}), and then study the response function
$\chi_{\varpi,~q}\propto\varphi_{\varpi,~q}^{+}/\varphi_{\varpi,~q}^{-}$.
This can tell us the critical behavior of some physical quantities
when the system approaches the critical point. Without loss of
generality, hereafter we will set $e=1$ and AdS radius $l=1$ in our
calculation.
To disclose the high curvature influence on the perturbation and the critical phenomenon, we will concentrate on the 5-dimensional (p=3)
Gauss-Bonnet AdS black holes with the scalar mass $m^2=-3$ and the Gauss-Bonnet coupling parameter within the range
$-\frac{7}{36}\leq\alpha\leq\frac{1}{4}$.
\begin{figure}[ht]
\includegraphics[width=300pt]{fig1.eps}
\caption{\label{fig1}(Color online) The trajectories of the
imaginary parts of the lowest quasinormal frequency for different
values of $\alpha$. While the temperature \textbf{drops to the
critical} point, the system approaches the marginally stable mode.}
\end{figure}
At first we report the influence of the Gauss-Bonnet term on the
scalar perturbation behavior. We concentrate on the lowest
quasinormal frequency which gives the relaxation time \cite{wang}.
We can obtain the quasinormal frequencies by solving Eq. (\ref{main
equation}) based on the boundary conditions (\ref{boundary}) at the
horizon and $\varphi_{\varpi,~q=0}^{-}=0$ at the AdS boundary. The
objective influence of the Gauss-Bonnet term on the imaginary parts
of the lowest quasinormal frequency of the perturbation is shown in
Fig. 1. We see that all the imaginary parts of the quasinormal
frequencies are negative, which shows that the black hole spacetime
is stable. For the larger Gauss-Bonnet coefficient, the imaginary
part of the lowest quasinormal frequency has larger deviation from
zero. This implies that the higher curvature correction can ensure
the system to be more stable and can slow down the process to make
the high temperature black hole phase become marginally stable. This
objective picture of studying the quasinormal modes is consistent
with the observation in \cite{Gregory,Pan-Wang} that the higher
curvature correction can hinder the condensation of the scalar hair
on the boundary. With the decrease of the black hole temperature, we
observe that the lowest quasinormal frequency approaches the origin
and vanishes when the temperature of the system reaches the critical
value, which indicates that the system approaches marginally stable.
The lowest quasinormal frequency approaches the origin with equal
spacing and we fit the results for different Gauss-Bonnet
coefficient in polynomials as below
\begin{eqnarray}
\alpha&=&-0.19,~\varpi_{QNM} \sim (2.57-0.69i)\times10^{-12}
+(1.74-0.42i)~\varepsilon_{\sigma} -
(0.27+0.49i)~\varepsilon_{\sigma}^{2},\nonumber\\
\alpha&=&-0.1,~~~\varpi_{QNM} \sim (1.21-1.50i)\times10^{-13} +
(1.96-0.46i)~\varepsilon_{\sigma} -
(0.37+0.60i)~\varepsilon_{\sigma}^{2},\nonumber\\
\alpha&=&0,~~~~~~~~\varpi_{QNM} \sim (-2.92-0.81i)\times10^{-13} +
(2.23-0.50i)~\varepsilon_{\sigma} -
(0.52+0.76i)~\varepsilon_{\sigma}^{2},\nonumber\\
\alpha&=&0.1,~~~~~\varpi_{QNM} \sim (-1.05+0.23i)\times10^{-9} +
(2.56-0.57i)~\varepsilon_{\sigma} -
(0.72+1.00i)~\varepsilon_{\sigma}^{2},\nonumber\\
\alpha&=&0.2,~~~~~\varpi_{QNM} \sim (1.01-3.40i)\times10^{-13}
+(2.97-0.69i)~\varepsilon_{\sigma} -
(1.28+1.37i)~\varepsilon_{\sigma}^{2},
\end{eqnarray}
where $\varepsilon_{\sigma}=1-\sigma/\sigma_{c}$.
\begin{figure}[ht]
\includegraphics[width=300pt]{fig2.eps}
\caption{\label{fig2}(Color online) The values of
$(\sigma_{c}+2)/40$ (black line) and the critical temperature
$T_{c}$ (blue line) as a function of the Gauss-Bonnet coefficient
$\alpha$. It is shown that the critical temperature decreases with
the increase of the Gauss-Bonnet coefficient.}
\end{figure}
Let's turn to discussing the thermodynamic susceptibility, which can be obtained numerically by solving Eq. (\ref{main equation}) with
$\varpi=q=0$ under the boundary condition (\ref{boundary}) at the horizon and $\varphi_{\varpi=0,~q=0}^{-}=0$ at the AdS boundary using the
shooting method. The dimensionless parameter $\sigma$ determines the phase structure and its critical value $\sigma_{c}$ can be calculated
numerically. In Fig. 2 we exhibit the critical point $\sigma_{c}$ and the critical temperature $T_{c}$ for the background system with different
Gauss-Bonnet coefficient. In the $5$-dimensional spacetime $T_{c}\propto\rho^{1/3}$. The behavior of $T_c$ is consistent with the result obtained
from the analysis of the condensation in \cite{Gregory, Pan-Wang}, which decreases with the increase of the Gauss-Bonnet coefficient. When the
Gauss-Bonnet term disappears, our result goes back to that got in \cite{maeda} for the $5$-dimensional RN-AdS background.
To examine the critical behavior of the thermodynamical
susceptibility $\chi$, we deviate $\sigma$ away from the critical
value $\sigma_c$ and denote the deviation by $\varepsilon_{\sigma}$.
After examining $\varphi^{\pm}_{\varpi=0,~q=0}$ as the function of
$\sigma$ near the critical point for different Gauss-Bonnet
coefficient, we find as expected that the critical behavior
$\varphi^{-}_{\varpi=0,~q=0}$ vanishes while
$\varphi^{+}_{\varpi=0,~q=0}$ approaches to a constant when
$\sigma=\sigma_{c}$. The Gauss-Bonnet term affects the tendency to
the critical behavior which can be observed from the thermodynamical
susceptibility for the stationary homogeneous source
$\chi=\chi_{\varpi=0,~q=0}\propto\frac{\varphi^{+}_{\varpi=0,~q=0}}{\varphi^{-}_{\varpi=0,~q=0}}$
as shown in Fig. 3. The results are fitted by polynomials as below
\begin{eqnarray}
\alpha&=&-0.19,~~~~~~~1/\chi \sim 1.04\times10^{-9} +
0.44~\varepsilon_{\sigma} - 0.59~\varepsilon_{\sigma}^{2},\nonumber\\
\alpha&=&-0.1,~~~~~~~~1/\chi \sim 3.44\times10^{-9} + 0.33
~\varepsilon_{\sigma} -1.19~\varepsilon_{\sigma}^{2},\nonumber\\
\alpha&=&0,~~~~~~~~~~~~~1/\chi \sim 1.74\times10^{-8} +
0.25~\varepsilon_{\sigma} - 3.36~\varepsilon_{\sigma}^{2},\nonumber\\
\alpha&=&0.1,~~~~~~~~~~1/\chi \sim 3.20\times10^{-10} + 0.19
~\varepsilon_{\sigma} - 7.30~\varepsilon_{\sigma}^{2},\nonumber\\
\alpha&=&0.2,~~~~~~~~~~1/\chi \sim 6.20\times10^{-11} +
0.15~\varepsilon_{\sigma} +0.75 ~\varepsilon_{\sigma}^{2}.
\end{eqnarray}
At the critical point $\sigma_c$, $\chi$ diverges as $\chi\propto
1/\varepsilon_{\sigma}$ regardless of the values of the Gauss-Bonnet
coefficient. Defining $\chi\propto|\varepsilon_{\sigma}|^{-\gamma}$,
we find the critical exponent of the thermodynamic susceptibility
$\gamma=1$ at the critical point. Although the Gauss-Bonnet term
cannot modify the critical exponent of the thermodynamic
susceptibility, it does influence the tendency of the thermodynamic
susceptibility when the critical point is approached. In the
vicinity of the critical point, we see that higher curvature
correction has bigger thermodynamical susceptibility.
\begin{figure}[h]
\includegraphics[width=300pt]{fig3.eps}
\caption{\label{fig3}(Color online) The thermodynamic susceptibility
$\chi$ as a function of $\varepsilon_{\sigma}$ for different
$\alpha$. Plotted are $1/\chi$ for the deviation
$\varepsilon_{\sigma}=10^{-5}n$, $n=1,2,\cdots,20$.}
\end{figure}
\begin{figure}[h]
\includegraphics[width=300pt]{fig4.eps}
\caption{\label{fig4}(Color online) The correlation length $\xi$ as
a function of $\varepsilon_{\sigma}$ for different $\alpha$. Plotted
are $\xi^{-2}$ for the deviation $\varepsilon_{\sigma}=10^{-5}n$.}
\end{figure}
Now we shift our gear to discuss the correlation length and static
susceptibility. The critical behavior of the system near the
critical point is determined by the large-scale fluctuations. The
correlation length is the scale parameter that exists in the system
near the phase transition point. It increases while the temperature
approaches its critical value and becomes infinite at the moment of
the phase transition. Now we check the influence imposed by the
Gauss-Bonnet term on the correlation length $\xi$, which is defined
by $\xi^{2}:=-q^{-2}$ \cite{maeda}. We consider a perturbation with
$\varpi=0$ for different $\alpha$, and solve Eq. (\ref{main
equation}) with $\varpi=0$ under boundary conditions: Eq.
(\ref{boundary}) at the horizon and $\varphi_{\varpi=0,~q}^{-}=0$ at
the AdS boundary. Fig. \ref{fig4} shows $\xi^{2}$ with the interval
$\Delta\varepsilon_{\sigma}=10^{-5}$ towards the critical value
$\sigma_{c}$. The results can be fitted by polynomials as
\begin{eqnarray}
\alpha&=&-0.19,~~~~~~\xi^{-2} \sim 1.40\times10^{-11} +
9.07~\varepsilon_{\sigma} -
6.90~\varepsilon_{\sigma}^{2},\nonumber\\
\alpha&=&-0.1,~~~~~~~~\xi^{-2} \sim 8.36\times10^{-12} + 10.99
~\varepsilon_{\sigma} -
8.69~\varepsilon_{\sigma}^{2},\nonumber\\
\alpha&=&0,~~~~~~~~~~~~~\xi^{-2} \sim 1.53\times10^{-12} +
13.55~\varepsilon_{\sigma} -
11.16~\varepsilon_{\sigma}^{2},\nonumber\\
\alpha&=&0.1,~~~~~~~~~~\xi^{-2} \sim 2.05\times10^{-12} + 16.97
~\varepsilon_{\sigma} -
14.50~\varepsilon_{\sigma}^{2},\nonumber\\
\alpha&=&0.2,~~~~~~~~~~\xi^{-2} \sim 2.79\times10^{-12} +
22.32~\varepsilon_{\sigma} - 19.63 ~\varepsilon_{\sigma}^{2}.
\end{eqnarray}
This shows that the correlation length $\xi$ depends on the
Gauss-Bonnet coefficient $\alpha$. For the smaller $\alpha$, the
correlation length is bigger for the same deviation from the
critical point of the system, which means that it is easier for the
system to approach the phase transition point when the Gauss-Bonnet
coefficient is smaller. However at the critical point
$\xi^{-2}\propto\varepsilon_{\sigma}$, i.e.,
$\xi\propto\varepsilon_{\sigma}^{-1/2}$ is always true for all
chosen Gauss-Bonnet coefficients $\alpha$, which shows that the
critical exponent $\nu=1/2$ is independent of the Gauss-Bonnet term.
\begin{figure}[h]
\includegraphics[width=300pt]{fig5.eps}
\caption{\label{fig5}(Color online) The static susceptibility at the
critical point $\chi_{\varpi=0,~q}\mid_{T_{c}}$ as a function of
$q^{2}$ for various $\alpha$. Plotted are
$1/\chi_{\varpi=0,~q}\mid_{T_{c}}$ for $q^{2}=10^{-5}n$.}
\end{figure}
The static critical exponent $\eta$ is determined by the static
susceptibility at the critical point
$\chi_{\varpi=0,~q}\mid_{~T_{c}}\propto q^{\eta-2}$ \cite{maeda} and
can be obtained from its $q$-dependence. Solving Eq. (\ref{main
equation}) with $\varpi=0$ under the boundary condition
(\ref{boundary}) at the horizon, we can get
$\left.\chi_{\varpi=0,~q}\right|_{~T_{c}}\propto~\frac{\varphi_{\varpi=0,~q}^{+}}{\varphi_{\varpi=0,~q}^{-}}$
from the behavior at the AdS boundary. Fig. \ref{fig5} shows
$\left.\chi_{\varpi=0,~q}\right|_{~T_{c}}$ as a function of $q$ and
the fitting results are listed below
\begin{eqnarray}
\alpha&=&-0.19,~~~~~~\left.1/\chi_{\varpi=0,~q}\right|_{~T_{c}} \sim
-1.41\times10^{-11} + 0.048~q^{2} - 0.27~q^{4},\nonumber\\
\alpha&=&-0.1,~~~~~~~~\left.1/\chi_{\varpi=0,~q}\right|_{~T_{c}}
\sim
-2.38\times10^{-10} + 0.030 ~q^{2} -0.65~q^{4},\nonumber\\
\alpha&=&0,~~~~~~~~~~~~\left.1/\chi_{\varpi=0,~q}\right|_{~T_{c}}
\sim 1.92\times10^{-14} + 0.018~q^{2} +0.016~q^{4},\nonumber\\
\alpha&=&0.1,~~~~~~~~~~\left.1/\chi_{\varpi=0,~q}\right|_{~T_{c}}
\sim 2.66\times10^{-14} + 0.011 ~q^{2} +0.027~q^{4},\nonumber\\
\alpha&=&0.2,~~~~~~~~~~\left.1/\chi_{\varpi=0,~q}\right|_{~T_{c}}
\sim 8.96\times10^{-14} + 0.0068~q^{2} -0.0010 ~q^{4}.
\end{eqnarray}
We see that the Gauss-Bonnet term affects the slope of the inverse
static susceptibility. From the fitting result we obtain
$\left.1/\chi_{\varpi=0,~q}\right|_{~T_{c}}\propto q^{-2}$, which
suggests that the exponent $\eta=0$ within numerical errors for
various $\alpha$.
\section{Conclusions and discussions}
We investigated the perturbations of charged scalar field in a $5$-dimensional Gauss-Bonnet-AdS black hole background and paid attention to the
effect of the Gauss-Bonnet term on the critical behavior of the system. From the perturbation behavior we obtained the objective picture on how
the high curvature influences the spacetime perturbation and the formation of the scalar hair. Our results from the dynamical perturbation
support that observed in the study of the condensation phenomena \cite{Gregory,Pan-Wang}. The high curvature will slow down the process for the
system with high temperature to approach the marginally stable state and hinder the condensation of the scalar hair. These effects can also be
read from the susceptibility and the correlation length in the process when the system approaches the marginally stable moment.
We also calculated the critical exponents for holographic superconductors when the critical point of the system is approached from the high
temperature phase. We observed that although the Gauss-Bonnet term affects the processes of the systems to approach the critical moments, they do
not change the static critical exponents, namely the static critical exponents still take the mean-field values. This confirmed the conjecture
that the critical exponents are determined by the matter fields in the system and are independent of the gravity sector of the
system~\cite{maeda}. This is mainly due to the fact that the gravity sector just simply provides a background in the high temperature phase
analysis or in the probe approximation, while the matter fields undergo a second-order phase transition (from zero to nonzero
condensation)~\cite{maeda}. Note that without the charged scalar field, the system does not have any critical phenomenon. This shows the role of
the charged scalar field in the phase transition.
\begin{acknowledgments}
RGC and BW thank the organizers and participants for various
discussions during the workshop on ``Dark Energy and Fundamental
Theory" held at Xidi, Anhui, China, May 28-June 6, 2010, supported
by the Special Fund for Theoretical Physics from the National
Natural Science Foundation of China under grant no: 10947203. This
work was partially supported by the National Natural Science
Foundation of China.
\end{acknowledgments}
|
1,116,691,500,727 | arxiv | \section{Introduction}\label{sec:introduction}
In a series of recent papers
\cite{attal12open,attal2012open,sinayskiy2013open,sweke2013dissipative,sinayskiy2012properties,pawela2014generalized}
various aspects of open quantum walks have been discussed. This is a novel and
very promising approach to the quantum walks.
Quantum walks have long been studied~\cite{Reitzner,
Ambainis2008quantum, Ampadu, Spatiotemporal1201.4839, percolation} and have
numerous applications, such as: search algorithms \cite{grover1997search,
kempe2003walksearch, portugal2013quantum, childs2004spatialsearch,
sadowski04efficient}, quantum agents~\cite{miszczak2014magnus} and quantum
games~\cite{flitney2002introduction,piotrowski2003invitation,pawela2013cooperative,pawela2013quantum}.
Open walks generalize this well studied model and in particular allow one to
incorporate decoherence that is an always present factor when considering
quantum systems.
The importance of managing decoherence has motivated the study of this problem
in numerous fields, such as quantum
control~\cite{dahleh1990optimal,viola1999universal,viola2003robust,james2004risk,d2006quantum,dong2009sliding,pawela2014quantum,pawela2014quantum2,gawron2014decoherence,pawela2013various},
quantum
games~\cite{du2002experimental,flitney2005quantum,flitney2007multiplayer,pawela2013enhancing,gawron2014relativistic}and
quantum walks~\cite{grover_fail, Franco1303.5319, Kendon, kendon0209005v3,
ampadu0localization, Chandrashekar_few_ie2d }.
In this work we analyze the asymptotic behavior of open quantum walks.
Especially we consider the possibility to determine the time limit properties
of walks with non-homogeneous structure. The theorems for the homogeneous case
are proven~\cite{attal2012central}. In this work we consider two different
approaches: the possibility to reduce the walk to the homogeneous one and
provide walk's asymptotic properties as it is. In the first case, we construct
a set of rules and methods that allows to determine when it is possible to
reduce a walk. In the second case, we state a new central limit theorem that
allows us to derive asymptotic distribution under certain conditions. We
illustrate this approaches with appropriate numerical examples.
\section{Preliminaries}\label{sec:oqw}
\subsection{Quantum states and channels}
\begin{definition}
We call an operator $\rho \in L(\mathcal{X})$ for some Hilbert space $\mathcal{X}$ a density
operator iff $\rho \geq 0$ and $\Tr \rho = 1$. We denote the set of all density
operators on $\mathcal{X}$ by $\Omega(\mathcal{X})$.
\end{definition}
\begin{definition}
A superoperator $\Phi$ is a linear mapping acting on linear operators $L(\mathcal{X})$
on a finite dimensional Hilbert space $\mathcal{X}$ and transforming them into
operators on another finite dimensional Hilbert space $\mathcal{Y}$ i. e.
\begin{equation}
\Phi: L(\mathcal{X}) \rightarrow L(\mathcal{Y}).
\end{equation}
\end{definition}
\begin{definition}
Given superoperators
\begin{equation}
\Phi_1: L(\mathcal{X}_1) \rightarrow L(\mathcal{Y}_1), \ \Phi_2: L(\mathcal{X}_2)
\rightarrow
L(\mathcal{Y}_2),
\end{equation}
we define the product superoperator
\begin{equation}
\Phi_1 \otimes \Phi_2: L(\mathcal{X}_1 \otimes\mathcal{X}_2)
\rightarrow L(\mathcal{Y}_1\otimes \mathcal{Y}_2),
\end{equation}
to be the unique linear mapping that satisfies:
\begin{equation}
(\Phi_1 \otimes \Phi_2)(A_2 \otimes A_2) =
\Phi_1(A_1) \otimes \Phi_2(A_2),
\end{equation}
for all operators $A_1\in L(\mathcal{X}_1), A_2 \in L(\mathcal{X}_2)$. The extension for
operators not in the tensor product form follows from linearity.
\end{definition}
\begin{definition}\label{def:channel}
A quantum channel is a superoperator $\Phi: L(\mathcal{X}) \rightarrow L(\mathcal{Y})$ that
satisfies the following restrictions:
\begin{enumerate}
\item $\Phi$ is trace-preserving, i.e. $\forall {A \in L(\mathcal{X})} \
\Tr(\Phi(A))=\Tr(A)$,
\item \label{item:CP}$\Phi$ is completely positive, that is for every
finite-dimensional Hilbert space $\mathcal{Z}$ the product of $\Phi$ and an identity
mapping on $L(\mathcal{Z})$ is a non-negativity preserving operation, i.e.
\begin{equation}
\forall {\mathcal{Z}} \ \forall {A \in L(\mathcal{X} \otimes \mathcal{Z})}, \ {A \geq 0} \
(\Phi\otimes {\rm 1\hspace{-0.9mm}l}_{L(\mathcal{Z})})(A) \geq 0.
\end{equation}
\end{enumerate}
\end{definition}
Note that quantum channels map density operators to density operators.
\begin{definition}
The Kraus representation of a quantum channel $\Phi: L(\mathcal{X}) \rightarrow L(\mathcal{Y})$
is given by a set of operators $K_i \in L(\mathcal{X}, \mathcal{Y})$. The action of the
superoperator $\Phi$ on $A \in L(\mathcal{X})$ is given by:
\begin{equation}
\Phi(A)=\sum_i K_i A K_i^\dagger,
\end{equation}
with the restriction that
\begin{equation}
\sum_i K_i^\dagger K_i={\rm 1\hspace{-0.9mm}l}_{\mathcal{X}}.
\end{equation}
\end{definition}
\begin{definition}
Given a superoperator $\Phi: L(\mathcal{X}) \rightarrow L(\mathcal{Y})$, for every operator $A
\in L(\mathcal{X}), B \in L(\mathcal{Y})$ we define the conjugate superoperator $\Phi^\dagger:
L(\mathcal{Y}) \rightarrow L(\mathcal{X})$ as the mapping satisfying
\begin{equation}
\forall A \in L(\mathcal{X}) \; \forall B \in L(\mathcal{Y}) \quad \mathrm{Tr} (\Phi(A)B) = \mathrm{Tr} (A
\Phi^\dagger(B)). \label{eq:conjugate-channel}
\end{equation}
\end{definition}
Note, that the conjugate to a completely positive superoperator is completely
positive, but is not necessarily trace-preserving.
\subsection{Open quantum walks}
The model of the open quantum walk was introduced by Attal \emph{et al.}\
\cite{attal12open} (see also \cite{sinayskiy12open}). To introduce the open
quantum walk (OQW) model, we consider a random walk on a graph with the set
of vertices $V$ and directed edges $\{(i, j): \; i, j \in V\}$. The
dynamics on the graph is described in the space of states $\mathcal{V} =
\mathbb{C}^V$ with an orthonormal basis $\{ \ket{i} \}_{i \in V}$. We model
an internal degree of freedom of the walker by attaching a Hilbert space
$\mathcal{X}$ to each vertex of the graph. Thus, the state of the quantum walker is
described by an element of the space $\Omega(\mathcal{X} \otimes \mathcal{V})$.
To describe the dynamics of the quantum walk, for each directed edge $(i, j)$
we introduce a set of operators $\{K_{ijk} \in L(\mathcal{X})\}$. These
operators
describe the change in the internal degree of freedom of the walker due to the
transition from vertex $j$ to vertex $i$. Choosing the operators $K_{ijk}$ such
that
\begin{equation}
\sum_{ik} K_{ijk}^\dagger K_{ijk} = {\rm 1\hspace{-0.9mm}l}_{\mathcal{X}},
\end{equation}
we get a Kraus representation of a quantum channel for each vertex $j \in V$ of
the graph. As the operators $K_{ijk}$ act only on $\mathcal{X}$, we introduce the
operators $M_{ijk} \in L(\mathcal{X} \otimes \mathcal{V})$
\begin{equation}
M_{ijk} = K_{ijk} \otimes \ketbra{i}{j},\label{eq:kraus-channel}
\end{equation}
where$\ket{i}, \ket{j} \in \mathcal{V}$ which perform the transition from vertex $j$ to
vertex $i$ and internal state evolution. It is straightforward to check that
$\sum_{ijk} M_{ijk}^\dagger
M_{ijk} = {\rm 1\hspace{-0.9mm}l}_{\mathcal{X} \otimes \mathcal{V}}$.
\begin{definition}
A discrete-time open quantum walk is given by a quantum channel $\Phi: L(\mathcal{X}
\otimes \mathcal{V}) \rightarrow L(\mathcal{X} \otimes \mathcal{V})$ with the Kraus representation
\begin{equation}
\forall A \in L(\mathcal{X} \otimes \mathcal{V}) \quad \Phi(A) = \sum_{ijk} M_{ijk} A
M_{ijk}^\dagger,
\end{equation}
where operators $M_{ijk} \in L(\mathcal{X} \otimes \mathcal{V})$ are defined in
Eq.~\eqref{eq:kraus-channel}.
\end{definition}
\subsection{Asymptotic behavior of open quantum walks}
Recently Attal \emph{et al.} \cite{attal2012central} provided a description of
asymptotic behavior of open quantum walks in the case when the behavior of
every vertex is the same i. e. all vertices belong to one class. We call such
networks homogeneous.
In order to describe asymptotic properties of an open quantum walk we will
use the notion of quantum trajectory process associated with this open quantum
walk.
\begin{definition}
We define the quantum trajectory process as a classical Markov chain assigned
to an open quantum walk constructed as a simulation of the walk with
measurement at each step. The initial state is $(\rho_0, X_0)\in \Omega(\mathcal{X})
\times \mathbb{Z}^d$ with probability 1. The state $(\rho_n, X_n)$ at step $n$
evolves into one of the $2d$ states corresponding to possible directions
$\Delta_j$, $j=\pm1,\ldots,\pm d$:
\begin{equation}
\left(\frac{1}{p_j}K_j \rho K_j^\dagger, X_j+\Delta_j\right),
\end{equation}
with probability $p(j)=\mathrm{Tr}(K_j\rho K_j^\dagger)$. We also define a transition
operator for a Markov chain $(\rho, \Delta X)$ associated with this trajectory
process
\begin{equation}\label{eq:definition:P}
P[(\rho, \Delta_i), (\rho', \Delta_j)]=
\left\{\begin{array}{ll}
\mathrm{Tr}(K_j\rho K_j^\dagger)
& \mathrm{if} \rho'=\frac{K_j \rho K_j^\dagger}{Tr(K_j \rho
K_j^\dagger)}, \\
0 & \mathrm{else}.
\end{array}\right.
\end{equation}
\end{definition}
We define an auxiliary channel $\Phi: L(\mathcal{X}) \rightarrow L(\mathcal{X})$ that mimics
the behavior of the walk when all the internal states are the same as
\begin{equation}
\Phi(\rho)=\sum_{j=1}^{2d} K_j \rho K_j^\dagger.\label{eq:aux-channel}
\end{equation}
We assume that the channel has a unique invariant state $\rho_\infty
\in
\Omega(\mathcal{X})$.
Additionally we define a vector that approximates the estimated asymptotic
transition for the channel $\Phi$:
\begin{equation}
\ket{m}=\sum_{j=1}^{2d}\mathrm{Tr}(K_j\rho_{\infty}K_j^\dagger)\ket{j},\label{eq:aux-vector}
\end{equation}
where $\ket{j} \in \mathbb{R}^d$ and for $j>d$ we put $\ket{j}=-\ket{j-d}$.
Let us recall the theorem by Attal \emph{et al.}~\cite{attal2012central}.
\begin{theorem}\label{th:original}
Consider a quantum open walk on $\mathbb{Z}^d$ associated with transition
operators $\{K_1, \ldots, K_{2d}\}$. We assume that a channel $\Phi$ admits a
unique invariant state. Let $(\rho_n, X_n)_{n\ge 0}$ be the quantum trajectory
process associated with this open quantum walk, then
\begin{equation}
\lim\limits_{n \rightarrow \infty} \frac{\mathbb{E}(\ket{X_n})}{n} = \ket{m},
\end{equation}
and probability distribution of normalized random variable $X_n$
\begin{equation}
\frac{\ket{X_n}-n \ket{m}}{\sqrt{n}},
\end{equation}
converges in law to the Gaussian distribution in ${\mathbb{R}}^d$.
\end{theorem}
\section{Results}
We are mainly interested in open quantum walks that are defined on networks
with many classes of vertices. In this paper we assume that there is a finite
number of vertex classes $\Gamma=\{C_1, \ldots, C_n\}$. The transitions in each
vertex is given by Kraus operators defined for each class separately
$\{K_1^{c(X)}, \ldots, K_{2d}^{c(X)}\}_{X\in \mathbb{Z}^d} \subset L(\mathcal{X})$, where
$c(X)\in\Gamma$ is class of the vertex $X\in\mathbb{Z}^d$, Such that $\sum_j
\left(K_j^C \right)^\dagger K_j^C={\rm 1\hspace{-0.9mm}l}_{\mathcal{X}}$. We define a transition operator of
the Markov chain as in Eq.~\eqref{eq:definition:P}:
\begin{equation}\label{eq:pc}
P_C[(\rho, \Delta_i), (\rho', \Delta_j)]=
\left\{\begin{array}{ll}
\mathrm{Tr}\left(K_j^C\rho \left( K_j^C \right)^\dagger\right)
& \mathrm{if} \rho'=\frac{K_j^C \left( K_j^C \right)^\dagger}{Tr\left(K_j^C
\rho
\left( K_j^C \right)^\dagger\right)}, \\
0 & \mathrm{else}.
\end{array}\right.
\end{equation}
Next, we define a channel $\Phi^C$ for each class $C$ as in
Eq.~(\ref{eq:aux-channel})
\begin{equation}
\Phi^C(\rho)=\sum_{j=-d, j\ne 0}^{d} K_j^C \rho (K_j^C)^\dagger.\label{eq:phic}
\end{equation}
Again, we assume that $\Phi^C$ has a unique invariant state $\rho_\infty^C \in
\Omega(\mathcal{X})$. Additionally for each class $C$ we define a vector as in
Eq.~(\ref{eq:aux-vector})
\begin{equation}
\ket{m_C}=\sum_{j=1}^{2d}
\mathrm{Tr}(K_j^C\rho_{\infty}^C(K_j^C)^\dagger)\ket{j}\label{eq:mc},
\end{equation}
where $\ket{j} \in \mathbb{R}^d$ and for $j>d$ we put $\ket{j}=-\ket{j-d}$.
In order to provide a description of distribution evolution of open quantum
walks on non-homogeneous networks we analyze two cases. First in
Section~\ref{sec:reducible}, we model a walk with vertices defined in such a
way that it is possible to reduce the network to the homogeneous case. Secondly
in Section~\ref{sec:irreducible}, we study a network which is irreducible in
the above sense but satisfies some basic properties that allow us to develop
other techniques.
\subsection{Reducible open quantum walks}\label{sec:reducible}
Let us consider an open quantum walk with several classes of vertices. We aim
to analyze the possibility to construct a new walk that that behaves the same
way in the asymptotic limit.
\begin{definition}
We call an open quantum walk reducible if there is a class $A$ that for some
integer $l$ each $l$-step path from a vertex of type $A$ always leads to a
vertex of type $A$.
\end{definition}
When considering a reducible OQW we can consider these paths as edges and
reduce the network to the homogeneous case.
\begin{definition}\label{def:abstract-class}
For a reducible quantum walk with $N$ possible paths we construct a new set of
Kraus operators $\{K_1^R, .., K_N^R\} \subset L(\mathcal{X})$ such that each operator
is a composition of all the operators corresponding to the consecutive steps
composing one of the paths from vertex $A$ to another vertex $A$, i. e. for a
path $q$ consisting of vertices $X_1, \dots, X_l$ and direction changes
$\Delta_1, \ldots, \Delta_l$ the corresponding operator is
\begin{equation}
K_q^R = K_{\Delta_l}^{C(X_l)}\cdot\ldots\cdot K_{\Delta_1}^{C(X_1)},
q=1,\ldots,N.
\end{equation}
We call the OQW based on these operators a reduced open quantum walk.
\end{definition}
The simplest example of a reducible open quantum walk is a walk on
$\mathbb{Z}^2$ presented in Fig.~\ref{fig:2d-reducible-network}. Starting in a
vertex of class $A$, after two steps we always end up in a vertex of class $A$.
We use that property to construct a new walk with only one vertex type and
exactly the same asymptotic behavior. In
Fig.~\ref{fig:2d-reducible-network-big} we present a more complex example of a
network with these properties.
\begin{figure*}[!ht]
\centering\includegraphics{fig-1}
\caption{An example of a 2D reducible OQW. The operators are defined in the
text. The dashed lines show possible paths from one vertex of type $A$ to
another
vertex of this type.}\label{fig:2d-reducible-network}
\end{figure*}
\begin{figure*}[!ht]
\centering\includegraphics{fig-2}
\caption{An example of a 2D reducible OQW. The arrows show possible
transitions. Each path from one vertex of type $A$ leads to another vertex of
this
type with exactly 4 steps.}
\label{fig:2d-reducible-network-big}
\end{figure*}
\subsubsection{Central limit theorem and its proof}
\begin{theorem}\label{th:reducible}
Consider a reducible quantum open walk on $\mathbb{Z}^d$. By $P$ we denote the
abstract class of vertices constructed as described in
Definition~\ref{def:abstract-class}. We assume that a channel constructed with
these paths $\Phi^P$ has a unique invariant state $\rho_\infty \in \Omega(\mathcal{X})$
with
average transition vector $\ket{m_P}$. Let $(\rho_n, X_n)_{n\ge 0}$ be the
quantum trajectory process associated to this open quantum walk, then
\begin{equation}
\lim_{n\rightarrow\infty} \frac{\mathbb{E}( \ket{X_n})}{n}=\ket{m_P},
\end{equation}
and probability distribution of normalized random variable $X_n$
\begin{equation}
\frac{\ket{X_n}-n \ket{m_P}}{\sqrt{n}},
\end{equation}
converges in law to the Gaussian distribution in ${\mathbb{R}}^d$.
\end{theorem}
\begin{proof}
We apply the Theorem~\ref{th:original} to the reduced OQW as in Def.
\ref{def:abstract-class}. As all the path's lengths are equal and describe all
possible paths starting from a vertex of type $A$ we have that $\sum_{q=1}^N
K_q^{R\dagger} K_q^R = \sum_{q=1}^{N} \left( K_1^q\ldots K_l^q \right)^\dagger
K_1^q \ldots K_l^q = {\rm 1\hspace{-0.9mm}l}_{\mathcal{X}}$. Thus the new walk satisfies assumptions of the
Theorem~\ref{th:original}. One step of this walk corresponds exactly to $l$
steps of the original walk. The one-to-one correspondence assures that the
asymptotic behavior is the same.
\end{proof}
\subsubsection{Example}\label{sec:example-reducible}
We show the application of Theorem~\ref{th:reducible} by considering a walk on
a network presented in the Fig. \ref{fig:2d-reducible-network}. The Kraus
operators for vertices of type $A$ are defined as follows:
\begin{equation}
\begin{split}
A_U(X) = & \alpha \ket{0}\bra{0}X\ket{0}\bra{0} + (1 - \alpha) \ket{1}\bra{0}
X \ket{0}\bra{1}, \\
A_R(X) = & \frac12 \ket{1}\bra{1}X\ket{1}\bra{1} + \frac12 \ket{3}\bra{1} X
\ket{1}\bra{3} \\
A_D(X) = & \alpha \ket{3}\bra{2}X\ket{3}\bra{2} + (1 - \alpha) \ket{2}\bra{2}
X \ket{2}\bra{2}, \\
A_L(X) = & \frac12 \ket{3}\bra{3}X\ket{3}\bra{3} + \frac12 \ket{0}\bra{3}
X \ket{3}\bra{0}.
\end{split}
\end{equation}
The operators for vertices of type $B$ are:
\begin{equation}
\begin{split}
B_U(X) = & \alpha \ket{1}\bra{0}X\ket{0}\bra{1} + (1 - \alpha) \ket{3}\bra{0}
X \ket{0}\bra{3}, \\
B_R(X) = & \frac12 \ket{0}\bra{1}X\ket{1}\bra{0} + \frac12 \ket{2}\bra{1} X
\ket{1}\bra{2}, \\
B_D(X) = & \alpha \ket{1}\bra{2}X\ket{2}\bra{1} + (1 - \alpha) \ket{3}\bra{2}
X \ket{2}\bra{3}, \\
B_L(X) = & \frac12 \ket{0}\bra{3}X\ket{3}\bra{0} + \frac12 \ket{2}\bra{3}
X \ket{3}\bra{2}.
\end{split}
\end{equation}
In our example we set $\alpha=0.81$. The behavior of this particular walk is
presented in Fig.~\ref{fig:2d-reducible-walk}. As expected, after a
sufficiently large
number of steps, the distribution is Gaussian and moves towards the left and
down.
\begin{figure}[!h]
\centering
\subfloat[\label{fig:reducible-a}]{\includegraphics[width=0.49\textwidth]{pattern-layout-010}}
\subfloat[\label{fig:reducible-b}]{\includegraphics[width=0.49\textwidth]{pattern-layout-050}}\\
\subfloat[\label{fig:reducible-c}]{\includegraphics[width=0.49\textwidth]{pattern-layout-100}}
\subfloat[\label{fig:reducible-d}]{\includegraphics[width=0.49\textwidth]{pattern-layout-200}}
\caption{An example of a reducible OQW on a 2D lattice shown in
Fig.~\ref{fig:2d-reducible-network}. The plots show the distribution of the
walks for various time steps and a cross section through the center of the
distribution. Panel~\protect\subref{fig:reducible-a} $n=10$,
panel~\protect\subref{fig:reducible-b} $n=50$,
panel~\protect\subref{fig:reducible-c} $n=100$,
panel~\protect\subref{fig:reducible-d} $n=200$.}\label{fig:2d-reducible-walk}
\end{figure}
\subsection{Irreducible OQWs}\label{sec:irreducible}
The assumptions introduced in Theorem~\ref{th:reducible}
allow us to analyze some non-homogeneous OQW, but the class of such walks is
still very limited. In this section we aim to provide a way to determine
asymptotic behavior of less restricted family of OQWs.
\subsubsection{Theorem and proof}\label{sec:irreducible-theorem}
Lets consider an OQW on a network composed with several types of vertices on an
infinite lattice. The main assumption of the following theorem is that the
distribution of vertices is regular over the lattice. In other words, the
probability of finding a vertex of particular type is transition invariant.
\begin{definition}\label{def:regular-network}
A regular network is a network where density of every vertex class $C\in\Gamma$
is transition invariant. In other words for any $\epsilon\in\mathbb{R}$ there is fixed
neighborhood size (ball radius) $\delta\in\mathbb{R}$ that for any two neighborhoods
the distance of vertex class probability distribution is bounded by $\epsilon$.
Thus, when restricted to any finite neighborhood of the considered hypercube,
there is fixed probability $p_{C}$ of finding a vertex of class $C$ in
any random direction from any randomly chosen vertex.
\end{definition}
\begin{theorem}\label{th:irreducible}
Given an open quantum walk on $\mathbb{Z}^d$ with vertex classes
$c(X)\in\Gamma$ for $X
\in \mathbb{Z}^d$ and associated transition operators $\{K_1^{c(X)}, \ldots,
K_{2d}^{c(X)}\}_{X\in \mathbb{Z}^d} \subset L(\mathcal{X})$ we construct for each class of
vertices $C\in\Gamma$ a quantum channel $\Phi^C$ as in Eq.~\eqref{eq:phic} with
a unique
invariant state $\rho_\infty^C \in \Omega(\mathcal{X})$ and an average position vector
$\ket{m} = \sum_{C\in\Gamma} p_C \ket{m_C}$, where $\ket{m_C}$ is
obtained from Eq.~\eqref{eq:mc} and $p_C$ from Def. \ref{def:regular-network}.
Let $(\rho_n, X_n)_{n\ge 0}$ be the quantum
trajectory process associated with this open quantum walk, then
\begin{equation}
\lim_{n\rightarrow\infty} \frac{\mathbb{E}(\ket{X_n})}{n}= \ket{m},
\end{equation}
and probability distribution of normalized random variable $X_n$
\begin{equation}
\frac{\ket{X_n}- n \ket{m}}{\sqrt{n}},
\end{equation}
converges in law to the Gaussian distribution.
\end{theorem}
Before we prove Theorem~\ref{th:irreducible}, let us introduce three technical
lemmas.
\begin{lemma}\label{lem:ker-im-sum}
For every superoperator $\Phi:L(\mathcal{X})\rightarrow L(\mathcal{X})$ the space
$L(\mathcal{X}) = \mathrm{Ker}(\Phi) \oplus \mathrm{Im}(\Phi^\dagger)$.
\end{lemma}
\begin{proof}
First we show that if $A\perp \mathrm{Im}(\Phi^\dagger)$ then $A\in\mathrm{Ker}(\Phi)$ for $A\in L(\mathcal{X})$.
Let us assume $A\perp \mathrm{Im}(\Phi^\dagger)$. Then for every $B\in L(\mathcal{X})$ it holds that
$\Tr(A\Phi^\dagger (B))=0$. Then $\Tr(\Phi(A)B)=0$. Thus $\Phi(A)=0$ and
$A\in\mathrm{Ker}(\Phi)$.
Now we show that if $A\in\mathrm{Ker}(\Phi)$ then $A\perp\mathrm{Im}(\Phi^\dagger)$. We assume $\Phi(A)=0$.
Then for any chosen $B\in L(\mathcal{X})$ it holds that $\Tr(\Phi(A)B)=0$, thus
$\Tr(A\Phi^\dagger(B)=0)$, hence $A\perp\mathrm{Im}(\Phi^\dagger)$.
\end{proof}
\begin{lemma}\label{lem:Lexists}
Given a channel $\Phi^C$ corresponding to vertex class $C$ with associated
Kraus $\{K_1^C \ldots K_{2d}^C\} \subset L(\mathcal{X})$ which has a unique invariant
state $\rho_\infty$, for every $l\in\mathbb{R}^d$ there exists $L_l^C \in
L(\mathcal{X})$ such that
\begin{equation}
\left(\1_{L(\mathcal{X})} -\left( \Phi^C \right)^\dagger \right)\left(L_l^C\right) =
\sum_{j=1}^{2d} \left(\left(K_j^C \right)^\dagger K_j^C \braket{j}{l} \right) -
\braket{m_C}{l} \1_{\mathcal{X}}.
\end{equation}
\end{lemma}
\begin{proof}
First we compute $\braket{m_C}{l}$. We get
\begin{equation}
\braket{m_C}{l} = \sum_i^{2d} \mathrm{Tr}\left(K_i^C\rho_z \left( K_i^C
\right)^\dagger\braket{i}{l}\right).
\end{equation}
Next, we move all the terms to one side of the equation and write all terms
under the trace
\begin{equation}
\sum_{i=1}^{2d} \mathrm{Tr}\left(K_i^C\rho_z \left(K_i^C\right)^\dagger\braket{i}{l} -
\braket{m_C}{l} \rho_z\1_{\mathcal{X}} \right)=0,
\end{equation}
where we multiplied $\braket{m_C}{l}$ by $\rho_\infty^C {\rm 1\hspace{-0.9mm}l}_{\mathcal{X}}$.
Finally, we use the fact that trace is cyclic and linear and get:
\begin{equation}\label{eq:lemmaL}
\mathrm{Tr} \rho_z \left(\sum_{i=1}^{2d} \left(K_i^C\right)^\dagger K_i^C\braket{i}{l}
- \braket{m_C}{l}\1_{\mathcal{X}} \right)=0.
\end{equation}
Thus we obtain that the term under the bracket in Eq. (\ref{eq:lemmaL}) is
orthogonal to $\rho_\infty^C$ and as it is the only invariant state of $\Phi^C$
we get that $\mathrm{ker}({\rm 1\hspace{-0.9mm}l}_{L(\mathcal{X})}-\Phi^C)=\rho_\infty^C$. Then, from Lemma
\ref{lem:ker-im-sum}, the states orthogonal to the kernel are in the image of
the conjugated superoperator, hence we get:
\begin{equation}
\sum_{i=1}^{2d} \left(K_i^C\right)^\dagger K_i^C \braket{i}{l} -
\braket{m_C}{l} \1_\mathcal{X} \in
\mathrm{ker}\left({\rm 1\hspace{-0.9mm}l}_{L(\mathcal{X})} -
\Phi^C\right)^\perp=\mathrm{im}\left(\1_{L(\mathcal{X})}-\left(\Phi^C\right)^\dagger\right).
\end{equation}
Hence, we have shown that $L_l^C$ exists.
\end{proof}
\begin{lemma}\label{lem:f}
For each class $C$ and a vector $l \in \mathbb{R}^d$ a function
\begin{equation}
f_C: \Omega(\mathcal{X}) \times \mathbb{R}^d \rightarrow \mathbb{R},
\end{equation}
given by the explicit formula
\begin{equation}
f_C(\rho, x)= \mathrm{Tr}(\rho L_l^C) + \braket{x}{l}\label{eq:fc},
\end{equation}
satisfies
\begin{equation}
({\rm 1\hspace{-0.9mm}l}-P_C)f_C(\rho, x)=\braket{x}{l} - \braket{m_C}{l},
\end{equation}
where $P_C$ is given by Eq.~\eqref{eq:pc}.
\end{lemma}
\begin{proof}
We apply the $P_C$ operator as defined in Eq.~\ref{eq:pc}.
Let us note that $(P_C f_C)(\rho, x) = \sum_{\rho', x'}P_C[(\rho, x), (\rho',
x')]f_z(\rho', x')$.
Applying the definition of $P_C$ to \eqref{eq:fc} we get:
\begin{equation}
\begin{split}
(\mathrm{Id} - P_C)f_C(\rho, x) & = \mathrm{Tr}(\rho L_l^C) + \braket{x}{l} -
\left[\mathrm{Tr}\left(\sum_{i=1}^{2d} K_i^C\rho \left(K_i^C\right)^\dagger L_l^C
\right) \right. \\
& + \left. \sum_{i=1}^{2d} \mathrm{Tr} \left(K_i^C\rho
\left(K_i^C\right)^\dagger\right)\braket{i}{l}\right],
\end{split}
\end{equation}
where $\mathrm{Id}$ is an identity operation on the space of functions
$\mathbb{R}^d\rightarrow \mathbb{R}$.
Now, using Lemma~\ref{lem:Lexists} we get
\begin{equation}
\mathrm{Tr}\left[\rho
\left(\1_{L(\mathcal{X})}-\left(\Phi^C\right)^\dagger\right)\left(L_l^C\right)-
\sum_{i=1}^{2d} \left(K_i^C\right)^\dagger K_i^C \braket{i}{l} \right] +x\cdot l
=\braket{x}{l} - \braket{m_C}{l}.
\end{equation}
which completes the proof.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{th:irreducible}]
For a random variable $X_n$ we expand the formula
$F_l=\braket{X_n}{l}-n\braket{m}{l}$:
\begin{equation}
F_l =\braket{X_n}{l} -
n\braket{m}{l}=\braket{X_0}{l}+\sum_{k=1}^n(\bra{X_k}-\bra{X_{k-1}})-\bra{m})\ket{l}.
\end{equation}
Recall that $\sum_{C \in \Gamma} p_C=1$, $m=\sum_{C \in \Gamma} p_C m_C$ and we
denote $\bra{X_k} - \bra{X_{k-1}} = \bra{\Delta X_k}$, we get:
\begin{equation}
F_l =\braket{X_0}{l}+\sum_{k=1}^n\sum_{C \in \Gamma} p_C(\bra{\Delta X_k}-
\bra{m_C})\ket{l}.
\end{equation}
From Lemma~\ref{lem:f} we get
$(\ket{X}-\ket{m_C})\ket{l}=(\mathrm{Id}-P_C)f_C(\rho, \ket{X})$ for some $\rho
\in \Omega(\mathcal{X})$, hence:
\begin{equation}
\begin{split}
F_l & =\braket{X_0}{l} + \sum_{k=1}^n\sum_{C \in \Gamma} p_C (\mathrm{Id}-P_C)
f_C (
\rho, \ket{\Delta X_k})= \\
& = \braket{X_0}{l} + \sum_{k=1}^n\sum_{C \in \Gamma} p_C (f_C(\rho,
\ket{\Delta X_k}) - P_Cf_C(\rho, \ket{\Delta X_k})).
\end{split}
\end{equation}
After rearranging the sum in the formula for $F_l$ we get:
\begin{equation}
\begin{split}
F_l = &\braket{X_0}{l} + \sum_{k=2}^n\sum_{C \in \Gamma} [p_C(f_C(\rho,
\ket{\Delta X_k})-P_C f_C(\rho, \ket{\Delta X_{k-1}}))] + \\
&+ \sum_{C \in \Gamma} p_C f_C(\rho, \ket{\Delta X_1})-\sum_{C \in \Gamma} p_C
P_Cf_C(\rho, \ket{\Delta X_n})=M_n + R_n.
\end{split}
\end{equation}
Now we consider $M_n$ and $R_n$ separately. First we discuss $M_n$:
\begin{equation}
M_n = \sum_{C \in \Gamma} \sum_{k=2}^{n} p_C(f_C(\rho, \ket{\Delta X_k}) -
P_Cf_C(\rho, \ket{\Delta X_{k-1}})).
\end{equation}
We notice that $M_n$ is a centered martingale i.e.
\begin{equation}
\mathbb{E}[\Delta M_n|\mathcal{F}_{n-1}] = 0,
\end{equation}
where $\Delta M_n = M_n - M_{n-1}$ and $\mathcal{F}$ denotes filtering for stochastic
process $M_n$~\cite{brown1971, hall1980martingale}. This follows from the
definition of $P_C$. Additionally $|\Delta M_n|$ is bounded from above i.e.
$|\Delta M_n| < M_{\max}$ as $\Delta M_n$ includes terms corresponding to one
step of the walk.
In the case of $R_n$ we have:
\begin{equation}
R_n=\braket{X_0}{l} + \sum_{C \in \Gamma} f_C(\rho, \ket{\Delta X_1}) - \sum_{C
\in \Gamma} P_C f_C(\rho, \ket{\Delta X_n}).
\end{equation}
From the definition of $f_C$ we notice that $R_n$ is bounded as the first two
terms are constant and the last one $P_Cf_C(\rho, \ket{\Delta X_n})=Tr(\rho
L_l^C)+\braket{\Delta X_n}{l}$ is clearly bounded, hence$|R_n| < R_{\max}$ and
$R_n$ does not influence the asymptotic behavior.
Now it suffices to show that the following two equalities hold (for proof see
Theorem 3.2 and Corollary 3.1 in~\cite{hall1980martingale}):
\begin{equation}
\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{k=1}^n \mathbb{E}[(\Delta
M_k)^2{{\rm 1\hspace{-0.9mm}l}}_{|\Delta M_k|\ge \epsilon \sqrt{n}}| \mathcal{F}_{k-1}] = 0
\label{eq:matingale-zero},
\end{equation}
and
\begin{equation}
\lim_{n\rightarrow\infty} \frac{1}{n} \sum_{k=1}^n \mathbb{E}[(\Delta M_k)^2| \mathcal{F}_{k-1}]
= \sigma^2
\label{eq:matingale-sigma},
\end{equation}
to obtain that $M_n/\sqrt{n}$ converges in distribution to $\mathcal{N}(0, \sigma^2)$,
where
\begin{equation}
{{\rm 1\hspace{-0.9mm}l}}_{|\Delta M_k|\ge \epsilon \sqrt{n}}=\left\{
\begin{matrix}
{\rm 1\hspace{-0.9mm}l}, {|\Delta M_k| \ge \epsilon \sqrt{n}},\\
0, {|\Delta M_k| < \epsilon \sqrt{n}},
\end{matrix}
\right.
\end{equation}
introduces restricted expectation values.
We prove Eq.~\eqref{eq:matingale-zero} using the fact that $|\Delta M_k|$ is
bounded, hence the sum in Eq.~\eqref{eq:matingale-zero} terminates for some $N
\in \mathbb{N}$.
In order to prove the equality in Eq.~\eqref{eq:matingale-sigma} we expand
$(\Delta M_k)^2$:
\begin{equation}
\begin{split}
(\Delta M_k)^2 & = \left( \sum_{C \in \Gamma}
p_C (\mathrm{Tr}\rho_kL_l^C - \mathrm{Tr} \rho_{k-1}L_l^C+
(\bra{\Delta X_k}-\bra{m_C})\ket{l} ) \right)^2= \\
& \left(\sum_{C \in \Gamma} p_C \Delta M_k^C \right)^2= \sum_{C,C'\in \Gamma}
p_C p_{C'} \Delta M_k^C \Delta M_k^{C'},
\end{split}
\end{equation}
where $\Delta M_k^C=( \mathrm{Tr}(\rho_kL_l^C)-\mathrm{Tr}(\rho_{k-1}L_l^C)+(\bra{\Delta
X_k}-\bra{m_C})\ket{l})$. Next, we expand the product
\begin{equation}
\begin{split}
\Delta M_k^C \Delta M_k^{C'}&=
[\mathrm{Tr}(\rho_kL_l^C)-\mathrm{Tr}(\rho_{k-1}L_l^C)+(\bra{\Delta
X_k}-\bra{m_C})\ket{l}] \times \\
&\times [\mathrm{Tr}(\rho_kL_l^{C'})-\mathrm{Tr}(\rho_{k-1}L_l^{C'})+(\bra{\Delta
X_k}-\bra{m_C'})\ket{l}].
\end{split}
\end{equation}
We divide this expression into three terms $\Delta M_k^{C'} \Delta M_k^C =
T_{C,C'}^{(1,k)} + T_{C,C'}^{(2,k)} + T_{C,C'}^{(3,k)}$. Henceforth, we will
drop indexes $C,C', k$ when unambiguous. The term $T^{(1)}$ is equal to:
\begin{equation}
T^{(1)}=\mathrm{Tr}(\rho_kL_l^{C'})\mathrm{Tr}(\rho_kL_l^{C})-\mathrm{Tr}(\rho_{k-1}L_l^{C'})\mathrm{Tr}(\rho_{k-1}L_l^C).
\end{equation}
We compute $\mathbb{E}(T^{(1)}|\mathcal{F}_{k-1})$ by adding the term $\pm \mathrm{Tr}(\rho_kL_l^{C'})
\mathrm{Tr}(\rho_kL_l^C)$
\begin{equation}
\begin{split}
\mathbb{E}[T1|\mathcal{F}_{k-1}] & =\mathbb{E}(\mathrm{Tr}(\rho_{k}L_l^{C'})\mathrm{Tr}(\rho_{k}L_l^C)|\mathcal{F}_{l-1})
-\mathrm{Tr}(\rho_{k}L_l^{C'})\mathrm{Tr}(\rho_{k}L_l^C)+ \\
& +\mathrm{Tr}(\rho_{k} L_l^{C'}) \mathrm{Tr}(\rho_{k} L_l^C)- \mathrm{Tr}(\rho_{k-1} L_l^{C'})
\mathrm{Tr}(\rho_{k-1} L_l^C),
\end{split}
\end{equation}
we obtain a sum of two terms that can be interpreted as an increment part of a
martingale and an increment part of a sum respectively. Thus after a summation
over $k$ both terms
are bounded and we get the equality
\begin{equation}
\lim_{n\rightarrow\infty} \frac{1}{n} \sum_{k=1}^n \sum_{C,C' \in \Gamma} p_C
p_{C'}
\mathbb{E}[T^{(1,k)}_{C,C'} | \mathcal{F}_{k-1}]=0.
\end{equation}
The term $T^{(2)}$ is given by:
\begin{equation}
T^{(2)} = -\mathrm{Tr}(\rho_{k-1} L_l^{C'})\Delta M_k^C -\mathrm{Tr}(\rho_{k-1} L_l^C)\Delta
M_k^{C'}.
\end{equation}
We note that $\mathbb{E}(\Delta M_k|\mathcal{F}_{k-1})=0$. Thus after summation over $C$ and
${C'}$ we get the the expectation value of the whole term $T^{(2),k}_{C,C'}$:
\begin{equation}
\lim_{n\rightarrow\infty} \frac{1}{n} \sum_{k=1}^n \sum_{C,C' \in \Gamma } p_C
p_{C'}
\mathbb{E}[T2_{C,C'}^k|\mathcal{F}_{k-1}] = 0.
\end{equation}
We will calculate the term $T^{(3)}$ using the definition of the expectation
value. We write the probability of $\ket{\Delta X}$ being equal to $\ket{j}$
and $\rho_k$ being $K_j \rho_{k-1} K_j^\dagger / \mathrm{Tr}(A_j \rho_{k-1}
A_j^\dagger)$ as $\mathrm{Tr}(K_j \rho_{k-1} K_j^\dagger)$. This can be expressed in a
nice trace form:
\begin{equation}
\begin{split}
\mathbb{E}[ T3^{C, {C'}}|\mathcal{F}_{k-1}] & = \mathbb{E}[(\bra{\Delta
X_k}-\bra{m_C})\ket{l}(\bra{\Delta
X_k}-\bra{m_{C'}})\ket{l} +
\\&
+ \mathrm{Tr}(\rho_kL_l^{C'})(\braket{\Delta X_k}{l}-\braket{m_C}{l})+
\\&
+ \mathrm{Tr}(\rho_kL_l^C)(\braket{\Delta X_k}{l}-\braket{m_{C'}}{l})|\mathcal{F}_{k-1}]=
\\&
= \sum_{i=1}^{2d} \mathrm{Tr}(K_i^{c(k-1)}\rho_{k-1} {K_i^{c(k-1)\dagger}}) \times \\&
\times [(\bra{i}-\bra{m_C}) \ket{l} (\bra{i} - \bra{m_{C'}}) \ket{l}+
\\&
+ \mathrm{Tr}(K_i^{c(k-1)}\rho_{k-1} K_i^{c(k-1)\dagger} L_l^{C'}) (\braket{i}{l} -
\braket{m_C}{l})\times
\\&
\times \mathrm{Tr}(K_i^{c(k-1)}\rho_{k-1} K_i^{c(k-1)\dagger} L_l^{C}) (\braket{i}{l} -
\braket{m_{C'}}{l})],
\end{split}
\end{equation}
where $c(k-1)$ is the class of $X_{k-1}$. Thus we can define
$\Xi_{C,C'}^{c(k-1)}$ so that
\begin{equation}
\mathbb{E}[ T3^{C,C'}|\mathcal{F}_{k-1}] =\mathrm{Tr}(\rho_{k-1}\Xi_{C,C'}^{c(k-1)}).
\end{equation}
After summation over $C$ and $C'$ the value is equal to:
\begin{equation}
\lim_{n\rightarrow\infty} \frac{1}{n}
\sum_{k=1}^n \sum_{C,C' \in \Gamma}p_C p_{C'} \mathbb{E}[T3_{C,C'}^k|\mathcal{F}_{k-1}]
=\lim_{n\rightarrow\infty} \frac{1}{n} \sum_{k=1}^n \mathrm{Tr}(\rho_{k-1}\Xi^{c(k-1)}),
\end{equation}
where $\Xi^{c(k-1)}=\sum_{C,C'\in \Gamma}p_C p_{C'} \Xi{C,C'}^{c(k-1)}$. By the
ergodic theorem (Th. 4.2 in \cite{attal2012central}) this
converges to:
\begin{equation}
\lim_{n\rightarrow\infty} \frac{1}{n} \sum_{k=1}^n \mathrm{Tr}(\rho_{k-1}\Xi^{c(k-1)})
= \mathrm{Tr}(\rho_{\infty} \Xi) = \sigma^2_l,
\end{equation}
with $\Xi=\sum_{c} p_c \Xi^{c}$.
Finally, after summing of all of the terms we get:
\begin{equation}
\lim_{n\rightarrow\infty} \frac{1}{n} \sum_k \mathbb{E}[(\Delta M_k)^2|\mathcal{F}_{k-1}] =
\sigma_l^2,
\end{equation}
which completes the proof.
\end{proof}
\subsubsection{Example}\label{sec:example-irreducible}
As an example of a walk consistent with description in
Section~\ref{sec:irreducible-theorem} we consider a walk with the same vertex
types as in the reducible case, that is:
\begin{equation}
\begin{split}
A_U(X) = & \alpha \ket{0}\bra{0}X\ket{0}\bra{0} + (1 - \alpha) \ket{1}\bra{0}
X \ket{0}\bra{1}, \\
A_R(X) = & \frac12 \ket{1}\bra{1}X\ket{1}\bra{1} + \frac12 \ket{3}\bra{1} X
\ket{1}\bra{3} \\
A_D(X) = & \alpha \ket{3}\bra{2}X\ket{3}\bra{2} + (1 - \alpha) \ket{2}\bra{2}
X \ket{2}\bra{2}, \\
A_L(X) = & \frac12 \ket{3}\bra{3}X\ket{3}\bra{3} + \frac12 \ket{0}\bra{3}
X \ket{3}\bra{0}.
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
B_U(X) = & \alpha \ket{1}\bra{0}X\ket{0}\bra{1} + (1 - \alpha) \ket{3}\bra{0}
X \ket{0}\bra{3}, \\
B_R(X) = & \frac12 \ket{0}\bra{1}X\ket{1}\bra{0} + \frac12 \ket{2}\bra{1} X
\ket{1}\bra{2}, \\
B_D(X) = & \alpha \ket{1}\bra{2}X\ket{2}\bra{1} + (1 - \alpha) \ket{3}\bra{2}
X \ket{2}\bra{3}, \\
B_L(X) = & \frac12 \ket{0}\bra{3}X\ket{3}\bra{0} + \frac12 \ket{2}\bra{3}
X \ket{3}\bra{2}.
\end{split}
\end{equation}
Although, in this case we assign the type to a vertex randomly with a uniform
distribution.
The channels formed from Kraus operators $A_x$ and $B_x$ where $x \in {U, R, L,
D}$ both have a unique invariant state. The behavior
of the network is presented in the Fig.~\ref{fig:2d-irreducible-walk}. We
obtain a similar behavior as in the reducible case, although the convergence to
a Gaussian distribution is slower.
\begin{figure}[!h]
\centering
\subfloat[\label{fig:irreducible-a}]{\includegraphics[width=0.49\textwidth]{random-layout-010}}
\subfloat[\label{fig:irreducible-b}]{\includegraphics[width=0.49\textwidth]{random-layout-100}}\\
\subfloat[\label{fig:irreducible-c}]{\includegraphics[width=0.49\textwidth]{random-layout-200}}
\subfloat[\label{fig:irreducible-d}]{\includegraphics[width=0.49\textwidth]{random-layout-500}}
\caption{An example of realization of OQW with a random uniform distribution of
vertex types. The figures show the distribution of the walk and cross section
through the center for various time steps:
panel~\protect\subref{fig:reducible-a} $n=10$,
panel~\protect\subref{fig:reducible-b} $n=100$,
panel~\protect\subref{fig:reducible-c} $n=200$,
panel~\protect\subref{fig:reducible-d} $n=500$.}\label{fig:2d-irreducible-walk}
\end{figure}
\section{Conclusions}
The aim of this paper was to provide formulas describing the behavior of the
open quantum walk in the asymptotic limit. We described two cases: networks
that are reducible to the 1-type case and networks with random, uniformely
distributed vertex types. This result allows one to analyze behavior of walks
with a more complex structure compared to the known results. We have illustrated
our claims with numerical examples that show possible applications and
correctness of our theorems. The networks are still restricted to vertices that
exhibits invariant states.
We provided examples showing that the theorems are valid in the case of a 2D
regular lattice with two vertex types. In Section~\ref{sec:example-reducible}
we shown application to the reducible case, when the assignment of vertex
types is regular and translation invariant. Next, in
Section~\ref{sec:example-irreducible} we turned to a random, uniformly
distributed assignment of vertex types.
These theorems can also be applied to the non-lattice graphs. Different types
of vertices allow also to apply this in the case of graphs with non-constant
degrees. This may be very useful in modeling complex structures, especially of
regular definition as in the case of Apollonian networks.
These possibilities are important as open quantum walks with different vertex
classes have application in quantum biology and dissipative quantum computing.
\section*{Acknowledgments}
Work by {\L}P was supported by the Polish Ministry of Science and Higher
Education under the project number IP2012 051272. PS was supported by the Polish
Ministry of Science and Higher Education within ``Diamond Grant'' Programme
under the project number 0064/DIA/2013/42.
|
1,116,691,500,728 | arxiv | \section{Introduction}
The AdS/CFT duality \cite{dual} has been widely used to discuss the
meson spectra and dense and hot quark matter. The string description
of realistic QCD has not been successfully formulated yet. Many
efforts are invested in searching for such a realistic description
by using the "top-down" approach, \textit{i.e.} by deriving
holographic QCD from string theory, as well as by using the
"bottom-up" approach, \textit{i.e.} by examining possible \textit{holographic%
} QCD models from experimental data and lattice results.
In the "bottom-up" approach, the most economic way is to search for
a deformed ${\rm AdS}_5$ metric
\cite{KKSS2006,Andreev:2006ct,Andreev:2006vy,Shock-2006,Ghoroku-Tachibana,
Csaki:2006ji,Gursoy,Zeng:2008sx,Pirner:2009gr,Brodsky:2010ur}, which
can describe the known experimental data and lattice results of QCD,
e.g. hadron spectra and the heavy quark potential. The simplest
holographic QCD model is the hard-wall ${\rm AdS}_5$ model
\cite{Hardwall-Polchinski}, which can describe the lightest meson
spectra in $80-90\%$ agreement with the experimental data. However,
the hard-wall model cannot produce the Regge behavior for higher
excitations. It is regarded that the Regge behavior is related to
the linear confinement. It has been suggested in Ref.
\cite{KKSS2006} that a negative quadratic dilaton term $-z^2$ in the
action is needed to produce the right linear Regge behavior of
$\rho$ mesons or the linear confinement. The most direct physical
quantity related to the confinement is the heavy-quark potential.
The lattice result which is consistent with the so called Cornell
potential \cite{Cornell} has the form of $ V_{Q{\bar
Q}}(R)=-\frac{\kappa}{R}+\sigma_{str}R+V_0$. Where $\kappa\approx
0.48$, $\sigma_{str}\approx 0.183 {\rm GeV}^{2}$ and $V_0=-0.25 {\rm
GeV}$, the first two parameters can be interpreted as
$\frac{4\alpha_s}{3}$ and QCD "string" tension, respectively.
In order to produce linear behavior of heavy flavor potential,
Andreev and Zakharov in Ref.\cite{Andreev:2006ct} suggested a
positive quadratic term modification \cite{Andreev:2006vy} in the
deformed warp factor of the metric, which is different from the
soft-wall model in \cite{KKSS2006}. In Ref. \cite{White:2007tu}, the
authors found that the heavy quark potential from the positive
quadratic model is closer to the Cornell potential than that from
the backreaction model \cite{Shock-2006}, which contains higher
order corrections.
It is clearly seen from the Cornell potential that the Coulomb
potential dominates in the ultraviolet (UV) region and the linear
potential dominates in the infrared (IR) region. It motivates people
to take into account the QCD running coupling effect into the
modified metric
\cite{Csaki:2006ji,Gursoy,Zeng:2008sx,Pirner:2009gr,Brodsky:2010ur}.
In Ref.\cite{Pirner:2009gr}, Pirner and Galow have proposed a
deformed metric which resembles the QCD running coupling, and the
Pirner-Galow metric can fit the Cornell potential reasonably well.
However, as shown in Ref. \cite{Galow:2009kw} the corresponding
dilaton potential solved from the Einstein equation is unstable, and
the corresponding beta function does not agree with the QCD beta
function.
The motivation of this work \cite{He-Huang-Yan} is to show that a
deformed ${\rm AdS}_5$ metric with an explicit infrared cutoff
included in the logarithmic correction $-c_0\log[(z_{IR}-z)/z_{IR}]$
can describe the heavy quark potential as well as the QCD $\beta$
function very well, at the same time it can have a stable dilaton
potential from the gravity side.
\section{The deformed $AdS_5$ model}
To search for the possible \textit{holographic} QCD models, the most
economic way of breaking conformal invariance is to add a deformed
warp factor $h(z)$ in the metric background, and the general metric
$\mathcal {A}_s(z)$ in the string frame and in the Euclidean space
has the following form:
\begin{eqnarray}
ds^2=G_{\mu\nu}^s dX^\mu dX^\nu &=& \frac{h(z)L^2}{z^2}\left(
dt^2+d\vec{x}^2+dz^2\right) \label{h-general} \\
&=& e^{2\mathcal {A}_s(z)}\left( dt^2+d\vec{x}^2+dz^2\right).
\label{metric-general}
\end{eqnarray}
As pointed in \cite{Gursoy}, that the logarithmic term $c_0\log z$
itself cannot produce confinement, while a logarithmic correction
with an infrared cut-off in the form of $c_0\log (z_{IR}-z)$ can
have confinement at IR. Therefore, we propose the following form for
the deformed warp factor \cite{He-Huang-Yan} as
\begin{equation}
h(z)=\exp\left( -\frac{\sigma
z^2}{2}-c_0\ln(\frac{z_{IR}-z}{z_{IR}})\right).
\label{metric-firstmodel}
\end{equation}
The coefficients $\sigma$ and $c_0$ can be either positive or
negative. An IR cut-off $z_{IR}$ explicitly sets in the metric,
which has the same effect as the hard-wall model. When $c_0=0$,
$\sigma>0$ and $\sigma<0$ corresponds to the soft-wall model
\cite{KKSS2006} and Andreev-Zakharov model, respectively. In
Ref.\cite{Pirner:2009gr}, in order to mimic the QCD running coupling
behavior, Pirner and Galow proposed the deformed warp factor
\begin{equation}
h_{PG}(z)=\frac{\log\left (\frac{1}{\epsilon} \right )}{\log\left
[\frac{1}{(\Lambda z)^2+\epsilon}\right ]}. \label{metric-PG}
\end{equation}
This metric with asymptotically conformal symmetry in the UV and
infrared slavery in the IR region yields a good fit to the heavy
$Q\bar Q$-potential with $\Lambda=264\,\text{MeV}$ and
$\epsilon=\Lambda^2 l_s^2=0.48$. It is worthy of mentioning that the
deformed warp factor $h_{PG}(z)$ is dominated by a quadratic term $
\sigma z^2$ in the UV regime and a logarithmic term
$-\log(z_{IR}-z)$ in the IR regime, respectively. The deformed
metric in Eq.(\ref{metric-firstmodel}) when taking the parameter of
$\sigma=0.08, c_0=1, z_{IR}=2.73 {\rm GeV}^{-1}$ can mimic the
Pirner-Galow deformed metric in Eq.(\ref{metric-PG}).
Following the standard procedure, one can derive the interquark
distance $R$ as a function of $z$
\begin{eqnarray}
R(z) &=&2 z \int_0^1 d\nu \frac{e^{2\mathcal {A}_s(z)}}{e^{2\mathcal
{A}_s(\nu z)}}\frac{1}{\sqrt{1- \left (\frac{e^{2\mathcal
{A}_s(z)}}{e^{2\mathcal {A}_s(\nu z)}}\right )^2}}.
\label{distance1}
\end{eqnarray}
The heavy quark potential can be worked out from the Nambu-Goto
string action:
\begin{eqnarray} V_{Q\bar
Q}(z)&=&\frac{1}{\pi\sigma_s}\int_0^1 d\nu e^{2\mathcal {A}_s(\nu
z)}z\frac{1}{\sqrt{1-\left ( \frac{e^{2\mathcal
{A}_s(z)}}{e^{2\mathcal {A}_s(\nu z)}}\right )^2}}.
\label{VQQ-general}
\end{eqnarray}
It is noticed that the integral in Eq.(\ref{VQQ-general}) in
principle include some poles, which induces $ V_{Q\bar
Q}(z)\rightarrow \infty$. The infinite energy should be extracted
through certain regularization procedure. The divergence of
$V_{Q\bar Q}(z)$ is related to the vacuum energy for two static
quarks.
According to the G{\"u}rsoy -Kiritsis-Nitti(GKN) framework
\cite{Gursoy}, the noncritical string background dual to the
QCD-like gauge theories can be described by the following action in
the Einstein frame:
\begin{equation}
S_{5D-Gravity}=\frac{1}{2\kappa_5^2}\int d^5x \sqrt{-G^E} \left (R
-\frac{4}{3}\partial_{\mu}\phi\partial^{\mu}\phi-V_B(\phi)\right)\,.
\label{5D_action}
\end{equation}
Where $R$ is the Ricci scalar, $\phi$ is the dilaton field, and
$V_B(\phi)$ the dilaton potential. The metric in the Einstein frame
is denoted by $G_{\mu\nu}^E$. Replacing
$A(z)=\mathcal{A}_s(z)-\frac{2}{3}\phi$, we obtain the following two
independent Einstein's equations:
\begin{eqnarray}
V_B(\phi(z))&=& -4
e^{\frac{4}{3}\phi-2\mathcal{A}_s}[(\phi')^2+3(\mathcal{A}_s')^2-4\phi'\mathcal{A}_s'
], \nonumber \\
\phi''&=& \frac{3}{2}\mathcal{A}_s'' + 2
\mathcal{A}_s'\phi'-\frac{3}{2} (\mathcal{A}_s')^2. \label{EQ-As}
\end{eqnarray}
Different from the original GKN paper, we will determine the metric
structure $\mathcal{A}_s$ from heavy quark potential, then solve the
dilaton field $\phi$ and the dilaton potential $V_B(\phi)$ from
Eq.~(\ref{EQ-As}). The resulting second order differential equation
for $\phi(z)$ needs two boundary conditions.
In the GKN framework, the scalar filed or dilaton field $\phi$
encodes the running of the Yang-Mills gauge theory's coupling
$\alpha$. For convenience, the renormalized dilaton field $\phi$ has
been defined as $\alpha=\frac{g_{YM}^2}{4 \pi}=e^{\phi}$. For a 5D
holographic model, its $\beta$ function is related to the deformed
warp factor $A(z)$ by
\begin{equation}
\beta\,\equiv\,E\frac{d\alpha }{d
E}=\frac{e^\phi d\phi}{d A}=\frac{e^{\phi(z)}\cdot \phi'(z)}{A'(z)}.
\label{calculatebeta}
\end{equation}
The QCD $\beta$-function at 2-loop level has the following form:
\begin{equation}
\beta(\alpha)=-b_0 \alpha^2 - b_1 \alpha^3,
\label{QCDbeta}
\end{equation}
with $b_0=\frac{1}{2\pi}(\frac{11}{3}N_c-\frac{2}{3}N_f)$, and
$b_1=\frac{1}{8\pi^2}(\frac{34}{3} N_c^2-(\frac{13}{3}
N_c-\frac{1}{N_c})N_f)$. By choosing $N_c=3$ and $N_f=4$, one has
$b_0=\frac{25}{6\pi}$ and $b_1=\frac{77}{12\pi^2}$.
\section{Heavy quark potential, QCD beta function and dilaton
potential}
We consider two cases: 1) with only quadratic correction when
$c_0=0$; 2) with only logarithmic correction when $\sigma=0$. In the
numerical calculations, we choose the ${\rm AdS}_5$ radius $L=1{\rm
GeV}^{-1}$, and the Coulomb part is fixed by choosing the string
tension $\sigma_s=0.38$.
The heavy quark potential as functions of quark anti-quark distance
$R$ for the case with only quadratic correction when $c_0=0$ is
shown in Fig. \ref{Vqq-Rz} (a), and for the case with only
logarithmic correction is shown in Fig.\ref{Vqq-Rz} (b). For the
first case, the best fit of the heavy quark potential gives
$\sigma=-0.22 {\rm GeV}^2 $, which is negative and corresponds to
the Andreev-Zakharov model. For the case with only logarithmic
correction when $\sigma=0$, the best fitted heavy quark potential
(the black solid line in Fig.\ref{Vqq-Rz} (b)) gives $c_0=0.272 {\rm
GeV}^2$ and $z_{IR}=2.1 {\rm GeV}^{-1}$. The results are compared
with that from the Pirner-Galow model (short dashed line) and the
experimental data (the long dashed line) and the UV analytical
result in Fig.\ref{Vqq-Rz} (b).
\begin{figure}[h]
\begin{center}
\epsfxsize=6.0 cm \epsfysize=6.0 cm \epsfbox{VQQ-c0zero}
\hspace*{0.1cm} \epsfxsize=6.0 cm \epsfysize=6.0 cm
\epsfbox{VQQ-sigma0-c00272} \vskip -0.05cm \hskip 0.15 cm
\textbf{( a ) } \hskip 6.5 cm \textbf{( b )} \\
\end{center}
\caption{ (a) The heavy quark potential as functions of $R$ in the
case of $c_0=0$, and $\sigma=0.1,0.01,-0.22,-0.4 {\rm GeV}^2$. (b)
The heavy quark potential as functions of the distance $R$ in the
case of $\sigma=0$ and $c_0=0.272$ and $z_{IR}=2.1 {\rm GeV}^{-1}$.
} \label{Vqq-Rz}
\end{figure}
The $\beta$ function as a function of $\alpha$ and the dilaton
potential as a function of $\phi$ are shown in Fig. \ref{beta-Vphi}
for the case of quadratic correction and for the case of logarithmic
correction, respectively. For the case with only quadratic
correction, the used two types of boundary conditions are:
\begin{eqnarray}
& & {\rm 1stBC}: \phi(z=0.87)={\rm log}(0.25), ~
\phi'(z=0.87)=0.9, \nonumber \\
& & {\rm 2ndBC}: \phi(z=0.87)={\rm log}(0.25), ~ \phi(z=0.38)={\rm
log}(0.18). \label{BC-c0zero}
\end{eqnarray}
For the case with only logarithmic correction, the used two types of
boundary conditions are:
\begin{eqnarray}
& & {\rm 1st BC}: \phi(z=0.9)={\rm log}(0.25), ~
\phi'(z=0.9)=1.7, \nonumber \\
& & {\rm 2nd BC}: \phi(z=0.9)={\rm log}(0.25), ~ \phi(z=0.39)={\rm
log}(0.185). \label{BC-sigmazero}
\end{eqnarray}
\begin{figure}[h]
\begin{center}
\epsfxsize=6.0 cm \epsfysize=6.0 cm \epsfbox{Beta-c0zero}
\hspace*{0.1cm} \epsfxsize=6.0 cm \epsfysize=6.0 cm
\epsfbox{Beta-sigma0-c00272} \vskip -0.05cm \hskip 0.15 cm
\textbf{( a ) } \hskip 6.5 cm \textbf{( b )} \\
\epsfxsize=6.0 cm \epsfysize=6.0 cm \epsfbox{Vphi-c0zero}
\hspace*{0.1cm} \epsfxsize=6.0 cm \epsfysize=6.0 cm
\epsfbox{Vphi-sigma0-c00272} \vskip -0.05cm \hskip 0.15 cm
\textbf{( c ) } \hskip 6.5 cm \textbf{( d )} \\
\end{center}
\caption[]{The beta function as a function of coupling constant
$\alpha$ and the dilaton potential as a function of $\phi$ in the
case with only quadratic correction ($c_0=0$, $\sigma=-0.22$) and in
the case with only logarithmic correction ($\sigma=0$, $c_0=0.272$
and $z_{IR}=2.1 {\rm GeV}^{-1}$). The boundary conditions are
described in Eqs.(\ref{BC-c0zero}) and (\ref{BC-sigmazero}).}
\label{beta-Vphi}
\end{figure}
For both cases with only quadratic correction and with only
logarithmic correction, it is found that by using the second type
boundary condition, i.e, the boundary condition used in
\cite{Galow:2009kw}, the produced $\beta$ function is not a
monotonic function of coupling $\alpha$. This behavior is due to the
fixing running coupling constant at two points. By using the first
type of boundary condition, the produced $\beta$ function is
monotonically decreasing with the coupling constant $\alpha$, and it
agrees reasonably well with the QCD $\beta$ function at 2-loop
level, which is shown by dashed line in Fig. \ref{beta-Vphi}.
For the case with only quadratic correction, it is found that for
both types of boundary conditions, $V_B(\phi)$ decreases with
$\phi$, the dilaton potential in the IR regime is not bounded from
below, which might indicate an unstable vacuum. This behavior is
also shown in the Pirner-Galow model in Ref. \cite{Galow:2009kw}.
However, in the model with only logarithmic correction, it is found
that for both types of boundary condition, the dilaton potential
$V_B(\phi)$ is stable which is bounded from below in the IR.
\section{Conclusion}
We found that in the deformed $AdS_5$ model with only logarithmic
correction in the deformed warp factor, the heavy quark potential
can be fitted very well and the beta function of the running
coupling agrees well with QCD $\beta$ function at 2-loop level.
Comparing with the Andreev-Zakharov model and the Pirner-Galow
model, the corresponding dual dilaton potential is stable in the
model with only logarithmic correction.
\section*{Acknowledgements}
M.H. thanks the Yukawa Institute for Theoretical Physics at Kyoto
University and the organizers of NFQCD2010 for their hospitality.
The work of M.H. is supported by CAS program "Outstanding young
scientists abroad brought-in", CAS key project KJCX3-SYW-N2,
NSFC10735040, NSFC10875134, and K.C.Wong Education Foundation, Hong
Kong.
|
1,116,691,500,729 | arxiv | \section{Introduction}
The Gauss map $G$ of a submanifold $M$ into $G(n,m)$ in $\wedge ^{n}\mathbb{
}_{s}^{m},$ where $G(n,m)$ is the Grassmannian manifold consisting of all
oriented $n-$planes through the origin of $\mathbb{E}_{s}^{m}$ and $\wedge
^{n}\mathbb{E}_{s}^{m}$ is the vector space obtained by the exterior product
of $n$ vectors in $\mathbb{E}_{s}^{m}$ is a smooth map which carries a point
$p$ in $M$ into the oriented $n-$plane in $\mathbb{E}_{s}^{m}$ obtained from
parallel translation of the tangent space of $M$ at $p$ in $\mathbb{E
_{s}^{m}.$ Since the vector space $\wedge ^{n}\mathbb{E}_{s}^{m}$ identify
with a semi-Euclidean space $\mathbb{E}_{t}^{N}$ for some positive integer
t,$ where $N=\left(
\begin{array}{c}
m \\
\end{array
\right) ,$ the Gauss map is defined by $G:M\rightarrow G(n,m)\subset \mathbb
E}_{t}^{N},$ $G(p)=\left( e_{1}\wedge ...\wedge e_{n}\right) \left( p\right)
$. The notion of submanifolds with finite type Gauss map was introduced by
B. Y.Chen and P.Piccinni in 1987 \cite{chen} and after then many works were
done about this topic, especially 1-type Gauss map and 2- type Gauss map.
If a submanifold $M$ of a Euclidean space or pseudo-Euclidean space has
1-type Gauss map $G$, then $G$ satisfies
\begin{equation*}
\Delta G=\lambda \left( G+C\right)
\end{equation*}
for some $\lambda \in \mathbb{R}$ and some constant vector $C.$
On the other hand the Laplacian of the Gauss map of some typical well-known
surfaces satisfy the form o
\begin{equation}
\Delta G=f\left( G+C\right)
\end{equation
for some smooth function $f$ on $M$ and some constant vector $C.$ A
submanifold of a Euclidean space or pseudo-Euclidean space is said to have
pointwise 1-type Gauss map, if its Gauss map satisfies (1) for some smooth
function $f$ on $M$ and some constant vector $C.$ If the vector $C$\ in (1)
is zero, a submanifold with pointwise 1-type Gauss map is said to be of the
first kind, otherwise it is said to be of the second kind.
A lot of papers were recently published about rotational surfaces with
pointwise 1-type Gauss map in four dimensional Euclidean and pseudo
Euclidean space in \cite{aksoyak 1},\cite{arslan 1},\cite{arslan 2}, \cit
{dursun 1}, \cite{dursun 2} \cite{kim}.Timelike and spacelike rotational
surfaces of elliptic, hyperbolic and parabolic types in Minkowski space
\mathbb{E}_{1}^{4}$ with pointwise 1-type Gauss map were studied in \cit
{bektas 1, bektas 2}. Aksoyak and Yayl\i \ in \cite{aksoyak 2} studied boost
invariant surfaces (rotational surfaces of hyperbolic type) with pointwise
1-type Gauss map in Minkowski space $\mathbb{E}_{1}^{4}$. They gave a
characterization for flat boost invariant surfaces with pointwise 1-type
Gauss map. Also they obtain some results for boost invariant marginally
trapped surfaces with pointwise 1-type Gauss map. Ganchev and Milousheva in
\cite{milo} defined three types of rotational surfaces with two dimensional
axis rotational surfaces of elliptic, hyperbolic and parabolic type in
pseudo Euclidean space $\mathbb{E}_{2}^{4}$. They classify all rotational
marginally trapped surfaces of elliptic, hyperbolic and parabolic type,
respectively.
In this paper, we study rotational surfaces of elliptic, hyperbolic and
parabolic type with pointwise 1-type Gauss map which have spacelike profile
curve in four dimensional pseudo Euclidean space and give all
classifications of flat rotational surfaces of elliptic, hyperbolic and
parabolic type with pointwise 1-type Gauss map.
\section{Preliminaries}
Let $\mathbb{E}_{s}^{m}$ be the $m-$dimensional pseudo-Euclidean space with
signature $(s,m-s)$. Then the metric tensor $g$ in $\mathbb{E}_{s}^{m}$ has
the form
\begin{equation*}
g=\sum \limits_{i=1}^{m-s}\left( dx_{i}\right) ^{2}-\sum
\limits_{i=m-s+1}^{m}\left( dx_{i}\right) ^{2}
\end{equation*
where $(x_{1},...,x_{m})$ is a standard rectangular coordinate system in
\mathbb{E}_{s}^{m}.$
A vector $v$ is called spacelike (resp., timelike) if $\left \langle
v,v\right \rangle >0$ (resp., $\left \langle v,v\right \rangle <0$). Avector
$v$ is called lightlike if it $v\neq 0$ and $\left \langle v,v\right \rangle
=0,$ where $\left \langle ,\right \rangle $ is indefinite inner scalar
product with respect to $g.$
Let $M$ be an $n-$dimensional pseudo-Riemannian submanifold of a $m-
dimensional pseudo-Euclidean space $\mathbb{E}_{s}^{m}$ and denote by
\tilde{\nabla}$ and $\nabla $ Levi-Civita connections of $\mathbb{E}_{s}^{m}$
and $M$ $,$ respectively. We choose local orthonormal frame $\left \{
e_{1},...,e_{n},e_{n+1},...,e_{m}\right \} $ on $M$ with $\varepsilon
_{A}=\left \langle e_{A},e_{A}\right \rangle =\pm 1$ such that $e_{1},$...,
e_{n}$ are tangent to $M$\ and $e_{n+1},$...,$e_{m}$ are normal to $M.$ We
use the following convention on the ranges of indices: $1\leq i,j,k,$...
\leq n$, $n+1\leq r,s,t,$...$\leq m$, $1\leq A,B,C,$...$\leq m.$
Denote by $\omega _{A}$ the dual-1 form of $e_{A}$ such that $\omega
_{A}\left( X\right) =\left \langle e_{A},X\right \rangle $ and $\omega _{AB}$
the connection forms defined b
\begin{equation*}
de_{A}=\sum \limits_{B}\varepsilon _{B}\omega _{AB}e_{B},\text{ \ \ }\omega
_{AB}+\omega _{BA}=0.
\end{equation*
Then the formulas of Gauss and Weingarten are given b
\begin{equation*}
\tilde{\nabla}_{e_{k}}^{e_{i}}=\sum \limits_{j=1}^{n}\varepsilon _{j}\omega
_{ij}\left( e_{k}\right) e_{j}+\sum \limits_{r=n+1}^{m}\varepsilon
_{r}h_{ik}^{r}e_{r}
\end{equation*
an
\begin{equation*}
\tilde{\nabla}_{e_{k}}^{e_{s}}=-\sum \limits_{j=1}^{n}\varepsilon
_{j}h_{kj}^{s}e_{j}+D_{e_{k}}^{e_{s}},\text{ \ }D_{e_{k}}^{e_{s}}=\su
\limits_{r=n+1}^{m}\varepsilon _{r}\omega _{sr}\left( e_{k}\right) e_{r},
\end{equation*
where $D$ is the normal connection, $h_{ik}^{r}$ the coefficients of the
second fundamental form $h.$
For any real function $f$ on $M,$ the Laplacian operator of $M$ with respect
to induced metric is given by
\begin{equation}
\Delta f=-\varepsilon _{i}\sum \limits_{i}\left( \tilde{\nabla}_{e_{i}
\tilde{\nabla}_{e_{i}}f-\tilde{\nabla}_{\nabla _{e_{i}}^{e_{i}}}f\right) .
\end{equation
The mean curvature vector $H$ and the Gaussian curvature $K$of $M$ in
\mathbb{E}_{s}^{m}$ are defined by
\begin{equation}
H=\frac{1}{n}\sum \limits_{s=n+1}^{m}\sum \limits_{i=1}^{n}\varepsilon
_{i}\varepsilon _{s}h_{ii}^{s}e_{s}
\end{equation
an
\begin{equation}
K=\sum \limits_{s=n+1}^{m}\varepsilon _{s}\left(
h_{11}^{s}h_{22}^{s}-h_{12}^{s}h_{21}^{s}\right) ,
\end{equation
respectively. We recall that a surface $M$ is called minimal if its mean
curvature vector vanishes identically, i.e. $H=0.$ If the mean curvature
vector satisfies $DH=0,$ then the surface $M$ is said to have parallel mean
curvature vector. Also if Gaussian curvature of $M$ vanishes identically,
i.e. $K=0,$ the surface $M$ is called flat.
\section{Rotational Surfaces with Pointwise 1-Type Gauss Map in $\mathbb{E
_{2}^{4}$}
In this section, we consider rotational surfaces of elliptic, hyperbolic and
parabolic type in four dimensional pseudo-Euclidean space $\mathbb{E
_{2}^{4} $ which are defined by Ganchev and Milousheva in \cite{milo} and
investigate these rotational surfaces with pointwise 1-type Gauss map.
Denote by $\left \{ \epsilon _{1},\epsilon _{2},\epsilon _{3},\epsilon
_{4}\right \} $ the standart orthonormal basis of $\mathbb{E}_{2}^{4},$
i.e., $\epsilon _{1}=(1,0,0,0),$ $\epsilon _{2}=(0,1,0,0),$ $\epsilon
_{3}=(0,0,1,0)$ and $\epsilon _{4}=(0,0,0,1),$ where $\left \langle \epsilon
_{1},\epsilon _{1}\right \rangle =\left \langle \epsilon _{2},\epsilon
_{2}\right \rangle =1,$ $\left \langle \epsilon _{3},\epsilon
_{3}\right
\rangle =\left \langle \epsilon _{4},\epsilon _{4}\right \rangle
=-1. $
\subsection{Rotational surfaces of elliptic type with pointwise 1-type Gauss
map in $\mathbb{E}_{2}^{4}$}
In this subsection, firstly we consider the rotational surfaces of elliptic
type with harmonic Gauss map. Further we give a characterization of the flat
rotational surfaces of elliptic type with pointwise 1-type Gauss map and
obtain a relationship for non-minimal these surfaces with parallel mean
curvature vector and pointwise 1-type Gauss map of the first kind.
Rotational surface of elliptic type $M_{1}$ is defined b
\begin{equation*}
\varphi \left( t,s\right)
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & \cos t & -\sin t \\
0 & 0 & \sin t & \cos
\end{pmatrix
\left(
\begin{array}{c}
x_{1}(s) \\
x_{2}(s) \\
x_{3}(s) \\
\end{array
\right)
\end{equation*
\begin{equation}
M_{1}:\text{ }\varphi \left( t,s\right) =\left(
x_{1}(s),x_{2}(s),x_{3}(s)\cos t,x_{3}(s)\sin t\right) ,
\end{equation
where the surface $M_{1}$ is obtained by the rotation of the curve
x(s)=(x_{1}(s),x_{2}(s),x_{3}(s),0)$ about the two dimensional Euclidean
plane span$\left \{ \epsilon _{1},\epsilon _{2}\right \} .$ Let the profile
curve of $M_{1}$ be unit speed spacelike curve. In that case $\left(
x_{1}{}^{\prime }(s)\right) ^{2}+\left( x_{2}{}^{\prime }(s)\right)
^{2}-\left( x_{3}{}^{\prime }(s)\right) ^{2}=1$. We suppose that
x_{3}(s)>0. $ The moving frame field $\left \{
e_{1},e_{2},e_{3},e_{4}\right
\} $ on $M_{1} $\ is determined as follows:
\begin{eqnarray*}
e_{1} &=&\left( x_{1}{}^{\prime }(s),x_{2}{}^{\prime }(s),x_{3}{}^{\prime
}(s)\cos t,x_{3}{}^{\prime }(s)\sin t\right) , \\
e_{2} &=&\left( 0,0,-\sin t,\cos t\right) , \\
e_{3} &=&\frac{1}{\sqrt{1+\left( x_{3}{}^{\prime }\right) ^{2}}}\left(
-x_{2}{}^{\prime }(s),x_{1}{}^{\prime }(s),0,0\right) , \\
e_{4} &=&\frac{1}{\sqrt{1+\left( x_{3}{}^{\prime }\right) ^{2}}}\left(
x_{3}{}^{\prime }(s)x_{1}{}^{\prime }(s),x_{3}{}^{\prime }(s)x_{2}{}^{\prime
}(s),(1+\left( x_{3}{}^{\prime }\right) ^{2})\cos t,(1+\left(
x_{3}{}^{\prime }\right) ^{2})\sin t\right) ,
\end{eqnarray*
where $e_{1},e_{2}$ and $e_{3},e_{4}$ are tangent vector fields and normal
vector fields to $M_{1},$ respectively.Then it is easily seen that
\begin{equation*}
\left \langle e_{1},e_{1}\right \rangle =\left \langle e_{3},e_{3}\right
\rangle =1,\text{ }\left \langle e_{2},e_{2}\right \rangle =\left \langle
e_{4},e_{4}\right \rangle =-1.
\end{equation*
We have the dual 1-forms as:
\begin{equation*}
\omega _{1}=ds\text{ \ \ \ \ and \ \ \ \ }\omega _{2}=-x_{3}(s)dt.
\end{equation*
After some computations, the components of the second fundamental form and
the connection forms are given as follows
\begin{eqnarray}
h_{11}^{3} &=&-d(s),\ h_{12}^{3}=0,\ h_{22}^{3}=0, \\
h_{11}^{4} &=&-c(s),\text{ \ }h_{12}^{4}=0,\text{ \ }h_{22}^{4}=b(s) \notag
\end{eqnarray
an
\begin{eqnarray*}
\omega _{12} &=&a(s)b(s)\omega _{2},\text{ \ \ }\omega _{13}=-d(s)\omega
_{1},\text{ \ \ }\omega _{14}=-c(s)\omega _{1}, \\
\omega _{23} &=&0,\text{ \ \ }\omega _{24}=-b(s)\omega _{2},\text{ \ \
\omega _{34}=a(s)d(s)\omega _{1}.
\end{eqnarray*}
The covariant differentiations with respect to $e_{1}$ and $e_{2}$ are
computed as:
\begin{eqnarray}
\tilde{\nabla}_{e_{1}}e_{1} &=&-d(s)e_{3}+c(s)e_{4}, \\
\tilde{\nabla}_{e_{2}}e_{1} &=&a(s)b(s)e_{2}, \notag \\
\tilde{\nabla}_{e_{1}}e_{2} &=&0, \notag \\
\tilde{\nabla}_{e_{2}}e_{2} &=&a(s)b(s)e_{1}-b(s)e_{4}, \notag \\
\tilde{\nabla}_{e_{1}}e_{3} &=&d(s)e_{1}-a(s)d(s)e_{4}, \notag \\
\tilde{\nabla}_{e_{2}}e_{3} &=&0, \notag \\
\tilde{\nabla}_{e_{1}}e_{4} &=&c(s)e_{1}-a(s)d(s)e_{3}, \notag \\
\tilde{\nabla}_{e_{2}}e_{4} &=&b(s)e_{2}, \notag
\end{eqnarray
where
\begin{eqnarray}
a(s) &=&\frac{x_{3}^{\prime }(s)}{\sqrt{1+\left( x_{3}{}^{\prime }\right)
^{2}}},\text{ } \\
b(s) &=&\frac{\sqrt{1+\left( x_{3}{}^{\prime }\right) ^{2}}}{x_{3}(s)},\text{
} \\
c(s) &=&\frac{x_{3}^{\prime \prime }(s)}{\sqrt{1+\left( x_{3}{}^{\prime
}\right) ^{2}}}, \\
d(s) &=&\frac{x_{1}^{\prime \prime }(s)x_{2}^{\prime }(s)-x_{2}^{\prime
\prime }(s)x_{1}^{\prime }(s)}{\sqrt{1+\left( x_{3}{}^{\prime }\right) ^{2}}
.
\end{eqnarray
By using (3), (4) and (6), the mean curvature vector and Gaussian curvature
of the surface $M_{1}$ are obtained as
\begin{equation}
H=\frac{1}{2}\left( -d(s)e_{3}+\left( c(s)+b\left( s\right) \right)
e_{4}\right)
\end{equation
and
\begin{equation}
K=c(s)b\left( s\right) ,
\end{equation
respectively.
By using (2) and (7) the Laplacian of the Gauss map of $M_{1}$\ is computed
as
\begin{equation}
\Delta G=L(s)\left( e_{1}\wedge e_{2}\right) +M(s)\left( e_{2}\wedge
e_{3}\right) +N(s)\left( e_{2}\wedge e_{4}\right) ,
\end{equation
where
\begin{equation}
L(s)=d^{2}(s)-b^{2}\left( s\right) -c^{2}\left( s\right) ,
\end{equation
\begin{equation}
M(s)=d^{\prime }\left( s\right) +a(s)d(s)(b(s)+c(s)),
\end{equation
\begin{equation}
N(s)=b^{\prime }(s)+c^{\prime }(s)+a(s)d^{2}(s).
\end{equation}
\begin{theorem}
\label{teo 1}Let $M_{1}$ be rotation surface of elliptic type given by the
parametrization (5). If $M_{1}$ has harmonic Gauss map then it has constant
Gaussian curvature.
\end{theorem}
\begin{proof}
Let the Gauss map of $M_{1}$be harmonic , i.e., $\Delta G=0.$ In that case
from (14), (15), (16) and (17) we have
\begin{eqnarray}
d^{2}(s)-b^{2}\left( s\right) -c^{2}\left( s\right) &=&0, \\
d^{\prime }\left( s\right) +a(s)d(s)(b(s)+c(s)) &=&0, \notag \\
b^{\prime }(s)+c^{\prime }(s)+a(s)d^{2}(s) &=&0. \notag
\end{eqnarray
By multiplying both sides of second equation of (18) with $d(s)$ and using
the third equation of (18) we have tha
\begin{equation}
d(s)d^{\prime }\left( s\right) -b(s)b^{\prime }\left( s\right)
-c(s)c^{\prime }\left( s\right) =(b(s)c(s))^{\prime }.
\end{equation}
By evaluating the derivative of the first equation of (18) with respect to
s $ and using (19), we have that $b(s)c(s)=$constant and from (13) it
implies that $K=K_{0}=$constant.
\end{proof}
\begin{theorem}
\label{teo 2}Let $M_{1}$ be the flat rotation surface of elliptic type given
by the parameterization (5). Then $M_{1}$ has pointwise 1-type Gauss map if
and only if the profile curve of $M_{1}$ is characterized in one of the
following way:
i
\begin{eqnarray}
x_{1}(s) &=&-\frac{1}{\delta _{1}}\sin \left( -\delta _{1}s+\delta
_{2}\right) +\delta _{4}, \\
x_{2}(s) &=&\frac{1}{\delta _{1}}\cos \left( -\delta _{1}s+\delta
_{2}\right) +\delta _{4}, \notag \\
x_{3}(s) &=&\delta _{3}, \notag
\end{eqnarray
where $\delta _{1},$\ $\delta _{2},$\ $\delta _{3}$\ and $\delta _{4}$ are
real constants and the Gauss map of $M_{1}$\ holds (1) for $f=\delta
_{1}^{2}-\frac{1}{\delta _{3}^{2}}$ and $C=0.$ If $\delta _{1}\delta
_{3}=\pm 1$ then the function $f$ becomes zero and it implies that the Gauss
map is harmonic.
ii
\begin{eqnarray}
x_{1}(s) &=&\int \left( 1+\lambda _{1}^{2}\right) ^{\frac{1}{2}}\cos \left(
\frac{\lambda _{3}}{\lambda _{1}\left( 1+\lambda _{1}^{2}\right) ^{\frac{1}{
}}}\ln (\lambda _{1}s+\lambda _{2})+\lambda _{4}\right) ds, \notag \\
x_{2}(s) &=&\int \left( 1+\lambda _{1}^{2}\right) ^{\frac{1}{2}}\sin \left(
\frac{\lambda _{3}}{\lambda _{1}\left( 1+\lambda _{1}^{2}\right) ^{\frac{1}{
}}}\ln (\lambda _{1}s+\lambda _{2})+\lambda _{4}\right) ds, \notag \\
x_{3}(s) &=&\lambda _{1}s+\lambda _{2},
\end{eqnarray
where $\lambda _{1},$ $\lambda _{2},$ $\lambda _{3}$ and $\lambda _{4}$ are
real constants and the Gauss map of $M_{1}$holds (1) for $f(s)=\frac{1}
\left( \lambda _{1}s+\lambda _{2}\right) ^{2}}\left( \frac{\lambda _{3}^{2}}
1+\lambda _{1}^{2}}-1\right) $ and $C=\lambda _{1}^{2}e_{1}\wedge
e_{2}+\lambda _{1}\left( 1+\lambda _{1}^{2}\right) ^{\frac{1}{2}}e_{2}\wedge
e_{4}.$
\end{theorem}
\begin{proof}
We suppose that $M_{1}$ has pointwise 1-type Gauss map. By using (1) and
(14), we get
\begin{eqnarray}
-f+f\left \langle C,e_{1}\wedge e_{2}\right \rangle &=&-L(s), \\
f\left \langle C,e_{2}\wedge e_{3}\right \rangle &=&-M(s), \notag \\
f\left \langle C,e_{2}\wedge e_{4}\right \rangle &=&N(s) \notag
\end{eqnarray
and
\begin{equation}
\left \langle C,e_{1}\wedge e_{3}\right \rangle =\left \langle C,e_{1}\wedge
e_{4}\right \rangle =\left \langle C,e_{3}\wedge e_{4}\right \rangle =0.
\end{equation
By taking the derivatives of all equations in (23) with respect to $e_{2}$
and using (22) we obtain
\begin{eqnarray}
a(s)N(s)-L(s)+f &=&0, \\
a(s)M(s) &=&0, \notag \\
M(s) &=&0, \notag
\end{eqnarray
respectively. From above equations, we have two cases. One of them is $a(s)=0
$, $M(s)=0$ and the other is $a(s)\neq 0,$ $M(s)=0.$ Firstly, we suppose
that $a(s)=0$ and $M(s)=0.$ By using (8), we have that $x_{3}(s)=\delta _{3}
=constant. It implies that $c(s)=0,$ $b\left( s\right) =\frac{1}{\delta _{3}}
$ and $M_{1}$ is flat. Since the profile curve $x$\ is spacelike curve which
is parameterized by arc-length, we can pu
\begin{equation}
x_{1}^{\prime }(s)=\cos \delta \left( s\right) \text{ (or resp. }\sin \delta
\left( s\right) \text{ )and }x_{2}^{\prime }(s)=\sin \delta \left( s\right)
\text{ (or resp. }\cos \delta \left( s\right) \text{)},
\end{equation
where $\delta $ is smooth angle function. Without loss of generality we
assume that
\begin{equation*}
x_{1}^{\prime }(s)=\cos \delta \left( s\right) \text{ and }x_{2}^{\prime
}(s)=\sin \delta \left( s\right)
\end{equation*}
We can do similar computations for the another case, too. By using third
equation of (24)\ and (16) we obtain that
\begin{equation}
d\left( s\right) =\delta _{1},\text{\ }\delta _{1}\text{\ is non zero
constant.}
\end{equation
On the other hand by using (11), (25) and (26) we ge
\begin{equation}
\delta \left( s\right) =-\delta _{1}s+\delta _{2},
\end{equation
where $\delta _{1},$ $\delta _{2}$ are real constants. Then by substituting
(27) into (25) and taking the integral we have the equation (20). Also the
Laplacian of the Gauss map of $M_{1}$ with the equations $a(s)=0,$ $b\left(
s\right) =\frac{1}{\delta _{3}}$, $c(s)=0$ and $d\left( s\right) =\delta _{1}
$ is found as $\Delta G=\left( \delta _{1}^{2}-\frac{1}{\delta _{3}^{2}
\right) G$
Now we suppose that $a(s)\neq 0$ and $M(s)=0.$ Since the surface $M_{1}$ is
flat, i.e., $K=0.$ By using (13) we have that $c(s)=0.$ From (10) we get
\begin{equation}
x_{3}(s)=\lambda _{1}s+\lambda _{2}
\end{equation
for some constants $\lambda _{1}\neq 0$ and $\lambda _{2}.$ In that case by
using (8), (9) and (28) we have
\begin{equation}
a(s)=\frac{\lambda _{1}}{\left( 1+\lambda _{1}^{2}\right) ^{\frac{1}{2}}}
\end{equation
and
\begin{equation}
b(s)=\frac{\left( 1+\lambda _{1}^{2}\right) ^{\frac{1}{2}}}{\lambda
_{1}s+\lambda _{2}}.
\end{equation
Let consider that $M(s)=0$ with $c(s)=0$. In that case from (16), we obtain
tha
\begin{equation}
d^{\prime }\left( s\right) +a(s)b(s)d(s)=0
\end{equation
By using (29), (30) and (31) we have
\begin{equation}
d(s)=\frac{\lambda _{3}}{\lambda _{1}s+\lambda _{2}},
\end{equation
where $\lambda _{3}$\ is constant of integration. On the other hand, Since
the profile curve $x$\ is spacelike curve which is parameterized by
arc-length, we can pu
\begin{eqnarray}
x_{1}^{\prime }(s) &=&\left( 1+\lambda _{1}^{2}\right) ^{\frac{1}{2}}\cos
\lambda \left( s\right) , \\
x_{2}^{\prime }(s) &=&\left( 1+\lambda _{1}^{2}\right) ^{\frac{1}{2}}\sin
\lambda \left( s\right) , \notag
\end{eqnarray
where $\lambda $ is smooth angle function. By differentiating (33) we obtain
\begin{eqnarray}
x_{1}^{\prime \prime }(s) &=&-\left( 1+\lambda _{1}^{2}\right) ^{\frac{1}{2
}\sin \lambda \left( s\right) \lambda ^{\prime }\left( s\right) , \\
x_{2}^{\prime \prime }(s) &=&\left( 1+\lambda _{1}^{2}\right) ^{\frac{1}{2
}\cos \lambda \left( s\right) \lambda ^{\prime }\left( s\right) . \notag
\end{eqnarray
By using (11), (28), (33) and (34)$,$ we ge
\begin{equation}
d(s)=-\left( 1+\lambda _{1}^{2}\right) ^{\frac{1}{2}}\lambda ^{\prime
}\left( s\right) .
\end{equation
By combining (32) and (35) we obtain
\begin{equation}
\lambda \left( s\right) =-\frac{\lambda _{3}}{\lambda _{1}\left( 1+\lambda
_{1}^{2}\right) ^{\frac{1}{2}}}\ln (\lambda _{1}s+\lambda _{2})+\lambda _{4}.
\end{equation
So by substituting (36) into (33), we get
\begin{eqnarray}
x_{1}(s) &=&\int \left( 1+\lambda _{1}^{2}\right) ^{\frac{1}{2}}\cos \left(
\frac{\lambda _{3}}{\lambda _{1}\left( 1+\lambda _{1}^{2}\right) ^{\frac{1}{
}}}\ln (\lambda _{1}s+\lambda _{2})+\lambda _{4}\right) ds, \notag \\
x_{2}(s) &=&\int \left( 1+\lambda _{1}^{2}\right) ^{\frac{1}{2}}\sin \left(
\frac{\lambda _{3}}{\lambda _{1}\left( 1+\lambda _{1}^{2}\right) ^{\frac{1}{
}}}\ln (\lambda _{1}s+\lambda _{2})+\lambda _{4}\right) ds, \notag
\end{eqnarray}
Conversely, the surface $M_{1}$ whose the profil curve given by (21) is
pointwise 1-type Gauss map fo
\begin{equation*}
f(s)=\frac{1}{\left( \lambda _{1}s+\lambda _{2}\right) ^{2}}\left( \frac
\lambda _{3}^{2}}{1+\lambda _{1}^{2}}-1\right)
\end{equation*
and
\begin{equation*}
C=\lambda _{1}^{2}e_{1}\wedge e_{2}+\lambda _{1}\left( 1+\lambda
_{1}^{2}\right) ^{\frac{1}{2}}e_{2}\wedge e_{4}.
\end{equation*}
\end{proof}
\begin{theorem}
\label{teo 3} A non- minimal rotational surfaces of elliptic type $M_{1}$
defined by (5) has pointwise 1-type Gauss map of the first kind if and only
if the mean curvature vector of $M_{1}$ is parallel .
\end{theorem}
\begin{proof}
From (12) we have that $H=\frac{1}{2}\left( -d(s)e_{3}+\left( c(s)+b\left(
s\right) \right) e_{4}\right) .$ Let the mean curvature vector of $M_{1}$ be
parallel , i.e., $DH=0.$ Then we get
\begin{equation*}
D_{e_{1}}H=\frac{1}{2}\left( -M(s)e_{3}+N(s)e_{4}\right) =0.
\end{equation*
In this case we obtain that $M(s)=N(s)=0.$ From (14), we have that $\Delta
G=L(s)e_{1}\wedge e_{2}.$
Conversely, if $M_{1}$ has pointwise 1-type Gauss map of the first kind then
from (14) we get $M(s)=N(s)=0$ and it implies that $M_{1}$ has parallel mean
curvature vector.
\end{proof}
\begin{corollary}
\label{cor 1}If rotational surfaces of elliptic type $M_{1}$ given by (5) is
minimal then it has pointwise 1-type Gauss map of the first kind.
\end{corollary}
\subsection{Rotational surfaces of hyperbolic type with pointwise 1-type
Gauss map in $\mathbb{E}_{2}^{4}$}
In this subsection, firstly we consider rotational surfaces of hyperbolic
type with harmonic Gauss map. Further we obtain a characterization of flat
rotational surfaces of hyperbolic type with pointwise 1-type Gauss map and
give a relationship for non-minimal these surfaces with parallel mean
curvature vector and pointwise 1-type Gauss map of the first kind. The
proofs of theorems in this subsection are similar the proofs of theorems in
previous section so we give the theorems as without proof.
Rotational surface of hyperbolic type $M_{2}$ is defined b
\begin{equation*}
\varphi \left( t,s\right)
\begin{pmatrix}
\cosh t & 0 & \sinh t & 0 \\
0 & 1 & 0 & 0 \\
\sinh t & 0 & \cosh t & 0 \\
0 & 0 & 0 &
\end{pmatrix
\left(
\begin{array}{c}
x_{1}(s) \\
x_{2}(s) \\
0 \\
x_{4}(s
\end{array
\right)
\end{equation*
\begin{equation}
M_{2}:\text{ }\varphi \left( t,s\right) =\left( x_{1}(s)\cosh
t,x_{2}(s),x_{1}(s)\sinh t,x_{4}(s)\right) ,
\end{equation
where the surface $M_{2}$ is obtained by the rotation of the curve
x(s)=(x_{1}(s),x_{2}(s),0,x_{4}(s))$ about the two dimensional Euclidean
plane span$\left \{ \epsilon _{2},\epsilon _{4}\right \} .$ Let the profile
curve of $M_{2}$ be unit speed spacelike curve. In that case $\left(
x_{1}{}^{\prime }(s)\right) ^{2}+\left( x_{2}{}^{\prime }(s)\right)
^{2}-\left( x_{4}{}^{\prime }(s)\right) ^{2}=1$. We assume that $x_{1}(s)>0.$
The moving frame field $\left \{ e_{1},e_{2},e_{3},e_{4}\right \} $ on
M_{2} $\ is choosen as follows
\begin{eqnarray*}
e_{1} &=&\left( x_{1}{}^{\prime }(s)\cosh t,x_{2}{}^{\prime
}(s),x_{1}{}^{\prime }(s)\sinh t,x_{4}{}^{\prime }(s)\right) , \\
e_{2} &=&\left( \sinh t,0,\cosh t,0\right) , \\
e_{3} &=&\frac{1}{\sqrt{\varepsilon \left( \left( x_{1}{}^{\prime }\right)
^{2}-1\right) }}\left( 0,x_{4}{}^{\prime }(s),0,x_{2}{}^{\prime }(s)\right) ,
\\
e_{4} &=&\frac{1}{\sqrt{\varepsilon \left( \left( x_{1}{}^{\prime }\right)
^{2}-1\right) }}\left( \left( 1-\left( x_{1}{}^{\prime }\right) ^{2}\right)
\cosh t,-x_{1}{}^{\prime }(s)x_{2}{}^{\prime }(s),\left( 1-\left(
x_{1}{}^{\prime }\right) ^{2}\right) \sinh t,-x_{1}{}^{\prime
}(s)x_{4}{}^{\prime }(s)\right) ,
\end{eqnarray*
where $e_{1},e_{2}$ and $e_{3},e_{4}$ are tangent vector fields and normal
vector fields to $M_{2},$ respectively and $\varepsilon $\ is signature of
\left( x_{1}{}^{\prime }\right) ^{2}-1.$ If $\left( x_{1}{}^{\prime }\right)
^{2}-1$ is positive (resp. negative) then $\varepsilon =1$ (resp.
\varepsilon =-1$). It is easily seen that
\begin{equation*}
\left \langle e_{1},e_{1}\right \rangle =-\left \langle e_{2},e_{2}\right
\rangle =1,\text{ }\left \langle e_{3},e_{3}\right \rangle =-\left \langle
e_{4},e_{4}\right \rangle =\varepsilon .
\end{equation*
we have the dual 1-forms as:
\begin{equation*}
\omega _{1}=ds\text{ \ \ \ \ and \ \ \ \ }\omega _{2}=-x_{1}(s)dt.
\end{equation*
After some computations, components of the second fundamental form and the
connection forms are obtained by
\begin{eqnarray}
h_{11}^{3} &=&d(s),\ h_{12}^{3}=0,\ h_{22}^{3}=0, \\
h_{11}^{4} &=&c(s),\text{ \ }h_{12}^{4}=0,\text{ \ }h_{22}^{4}=-\varepsilon
b(s) \notag
\end{eqnarray
an
\begin{eqnarray*}
\omega _{12} &=&a(s)b(s)\omega _{2},\text{ \ \ }\omega _{13}=d(s)\omega _{1}
\text{ \ \ }\omega _{14}=c(s)\omega _{1}, \\
\omega _{23} &=&0,\text{ \ \ }\omega _{24}=\varepsilon b(s)\omega _{2},\text{
\ \ }\omega _{34}=a(s)d(s)\omega _{1}.
\end{eqnarray*
The covariant differentiations with respect to $e_{1}$ and $e_{2}$ are
computed as:
\begin{eqnarray}
\tilde{\nabla}_{e_{1}}e_{1} &=&\varepsilon d(s)e_{3}-\varepsilon c(s)e_{4} \\
\tilde{\nabla}_{e_{2}}e_{1} &=&a(s)b(s)e_{2} \notag \\
\tilde{\nabla}_{e_{1}}e_{2} &=&0 \notag \\
\tilde{\nabla}_{e_{2}}e_{2} &=&a(s)b(s)e_{1}+b(s)e_{4} \notag \\
\tilde{\nabla}_{e_{1}}e_{3} &=&-d(s)e_{1}-\varepsilon a(s)d(s)e_{4} \notag
\\
\tilde{\nabla}_{e_{2}}e_{3} &=&0 \notag \\
\tilde{\nabla}_{e_{1}}e_{4} &=&-c(s)e_{1}-\varepsilon a(s)d(s)e_{3} \notag
\\
\tilde{\nabla}_{e_{2}}e_{4} &=&-\varepsilon b(s)e_{2} \notag
\end{eqnarray
where
\begin{equation*}
a(s)=\frac{x_{1}^{\prime }(s)}{\sqrt{\varepsilon \left( \left(
x_{1}{}^{\prime }\right) ^{2}-1\right) }},
\end{equation*
\begin{equation*}
b(s)=\frac{\sqrt{\varepsilon \left( \left( x_{1}{}^{\prime }\right)
^{2}-1\right) }}{x_{1}(s)},
\end{equation*
\begin{equation*}
c(s)=\frac{x_{1}^{\prime \prime }(s)}{\sqrt{\varepsilon \left( \left(
x_{1}{}^{\prime }\right) ^{2}-1\right) }},
\end{equation*
\begin{equation*}
d(s)=\frac{x_{2}^{\prime \prime }(s)x_{4}^{\prime }(s)-x_{4}^{\prime \prime
}(s)x_{2}^{\prime }(s)}{\sqrt{\varepsilon \left( \left( x_{1}{}^{\prime
}\right) ^{2}-1\right) }}.
\end{equation*
By using (3), (4) and (38), the mean curvature vector and Gaussian curvature
of the surface $M_{2}$ are obtained as follows
\begin{equation*}
H=\frac{1}{2}\left( \varepsilon d(s)e_{3}-\varepsilon \left(
c(s)+\varepsilon b\left( s\right) \right) e_{4}\right)
\end{equation*
and
\begin{equation*}
K=c(s)b\left( s\right) ,
\end{equation*
respectively.
By using (2) and (39) the Laplacian of the Gauss map of $M_{2}$\ is computed
as
\begin{equation*}
\Delta G=L(s)\left( e_{1}\wedge e_{2}\right) +M(s)\left( e_{2}\wedge
e_{3}\right) +N(s)\left( e_{2}\wedge e_{4}\right) ,
\end{equation*
where
\begin{equation*}
L(s)=\varepsilon \left( d^{2}(s)-c^{2}\left( s\right) -b^{2}\left( s\right)
\right) ,
\end{equation*
\begin{equation*}
M(s)=\varepsilon \left( d^{\prime }\left( s\right) +\varepsilon
a(s)d(s)\left( c(s)+\varepsilon b\left( s\right) \right) \right) ,
\end{equation*
\begin{equation*}
N(s)=-\varepsilon \left( c^{\prime }(s)+\varepsilon b^{\prime
}(s)+\varepsilon a(s)d^{2}(s)\right) .
\end{equation*}
\begin{theorem}
\label{teo 4}Let $M_{2}$ be rotation surface of hyperbolic type given by the
parameterization (37). If $M_{2}$\ has Gauss map harmonic then it has
constant Gaussian curvatrure.
\end{theorem}
\begin{theorem}
\label{teo 5}Let $M_{2}$ be flat rotation surface of hyperbolic type given
by the parameterization (37). Then $M_{2}$ has pointwise 1-type Gauss map if
and only if the profile curve of $M_{2}$ is characterized in one of the
following way:
i
\begin{eqnarray*}
x_{1}(s) &=&\delta _{1}, \\
x_{2}(s) &=&-\frac{1}{\delta _{2}}\sinh \left( -\delta _{2}s+\delta
_{3}\right) +\delta _{4}, \\
x_{4}(s) &=&-\frac{1}{\delta _{2}}\cosh \left( -\delta _{2}s+\delta
_{3}\right) +\delta _{4},
\end{eqnarray*}
where $\delta _{1},$\ $\delta _{2},$\ $\delta _{3}$\ and $\delta _{4}$ are
real constants and the Gauss map $G$ holds (1) for $f=\frac{1}{\delta
_{1}^{2}}-\delta _{2}^{2}$ and $C=0.$ If $\delta _{1}\delta _{2}=\pm 1$ then
the function $f$ becomes zero and it implies that the Gauss map is harmonic.
ii
\begin{eqnarray*}
x_{1}(s) &=&\lambda _{1}s+\lambda _{2}, \\
x_{2}(s) &=&\int \left( \lambda _{1}^{2}-1\right) ^{\frac{1}{2}}\sinh \left(
\frac{\lambda _{3}}{\lambda _{1}\left( \lambda _{1}^{2}-1\right) ^{\frac{1}{
}}}\ln (\lambda _{1}s+\lambda _{2})+\lambda _{4}\right) ds, \\
x_{4}(s) &=&\int \left( \lambda _{1}^{2}-1\right) ^{\frac{1}{2}}\cosh \left(
\frac{\lambda _{3}}{\lambda _{1}\left( \lambda _{1}^{2}-1\right) ^{\frac{1}{
}}}\ln (\lambda _{1}s+\lambda _{2})+\lambda _{4}\right) ds,
\end{eqnarray*
where $\lambda _{1},$ $\lambda _{2},$ $\lambda _{3}$ and $\lambda _{4}$ are
real constants and without loss of generality we suppose that $\lambda
_{1}^{2}-1>0.$ Morever the Gauss map $G$ holds (1) for the function $f(s)
\frac{1}{\left( \lambda _{1}s+\lambda _{2}\right) ^{2}}\left( 1-\frac
\lambda _{3}^{2}}{\lambda _{1}^{2}-1}\right) $ and $C=-\lambda
_{1}^{2}e_{1}\wedge e_{2}+\lambda _{1}\left( \lambda _{1}^{2}-1\right) ^
\frac{1}{2}}e_{2}\wedge e_{4}.$
\end{theorem}
\begin{theorem}
\label{teo 6} A non- minimal rotational surfaces of hyperbolic type $M_{2}$
defined by (37) has pointwise 1-type Gauss map of the first kind if and only
if $M_{2}$ has parallel mean curvature vector
\end{theorem}
\begin{corollary}
\label{cor 2}If rotational surfaces of hyperbolic type $M_{2}$ given by (37)
is minimal then it has pointwise 1-type Gauss map of the first kind.
\end{corollary}
\subsection{Rotational surfaces of parabolic type with pointwise 1-type
Gauss map in $\mathbb{E}_{2}^{4}$}
In this subsection, we study rotational surfaces of parabolic type with
pointwise 1-type Gauss map. We show that flat rotational surface of
parabolic type has pointwise 1-type Gauss map if and only if its Gauss map
is harmonic. Also we conclude that flat rotational surface of parabolic type
has harmonic Gauss map if and only if it has parallel mean curvature vector.
We consider the pseudo-orthonormal base $\left \{ \epsilon _{1},\xi _{2},\xi
_{3},\epsilon _{4}\right \} $ of $\mathbb{E}_{2}^{4}$\ such that $\xi _{2}
\frac{\epsilon _{2}+\epsilon _{3}}{\sqrt{2}},$\ $\xi _{3}=\frac{-\epsilon
_{2}+\epsilon _{3}}{\sqrt{2}}$ $\left \langle \xi _{2},\xi _{2}\right \rangle
=\left \langle \xi _{3},\xi _{3}\right \rangle =0$ and $\left \langle \xi
_{2},\xi _{3}\right \rangle =-1.$ Let consider $\alpha $ spacelike curve is
given b
\begin{equation*}
x\left( s\right) =x_{1}(s)\epsilon _{1}+x_{2}(s)\epsilon
_{2}+x_{3}(s)\epsilon _{3}
\end{equation*
or we can express $x$\ according to pseudo-orthonormal base $\left \{
\epsilon _{1},\xi _{2},\xi _{3},\epsilon _{4}\right \} $ as follows
\begin{equation*}
x\left( s\right) =x_{1}(s)\epsilon _{1}+p(s)\xi _{2}+q(s)\xi _{3},
\end{equation*
where $p(s)=\frac{x_{2}(s)+x_{3}(s)}{\sqrt{2}}$ and $q(s)=\frac
-x_{2}(s)+x_{3}(s)}{\sqrt{2}}.$\ The rotational surface of parabolic type
M_{3}$ is defined b
\begin{equation}
M_{3}:\varphi \left( t,s\right) =x_{1}(s)\epsilon _{1}+p(s)\xi
_{2}+(-t^{2}p(s)+q(s))\xi _{3}+\sqrt{2}tp(s)\epsilon _{4},
\end{equation
We suppose that $x$\ is parameterized by arc-length, that is, $\left(
x_{1}{}^{\prime }(s)\right) ^{2}-2p^{\prime }(s)q^{\prime }(s)=1.$ Now we
can give a moving orthonormal frame $\left \{ e_{1},e_{2},e_{3},e_{4}\right \}
$\ for $M_{3}$ as follows
\begin{eqnarray*}
e_{1} &=&x_{1}{}^{\prime }(s)\epsilon _{1}+p^{\prime }(s)\xi
_{2}+(-t^{2}p^{\prime }(s)+q^{\prime }(s))\xi _{3}+\sqrt{2}tp^{\prime
}(s)\epsilon _{4}, \\
e_{2} &=&-\sqrt{2}t\xi _{3}+\epsilon _{4}, \\
e_{3} &=&\epsilon _{1}+\frac{x_{1}{}^{\prime }(s)}{p^{\prime }(s)}\xi _{3},
\\
e_{4} &=&x_{1}{}^{\prime }(s)\epsilon _{1}+p^{\prime }(s)\xi _{2}+(\frac{1}
p^{\prime }(s)}+q^{\prime }(s)-t^{2}p^{\prime }(s))\xi _{3}+\sqrt{2
tp^{\prime }(s)\epsilon _{4},
\end{eqnarray*
where $p^{\prime }(s)$ is non zero. Then it is easily seen that
\begin{equation*}
\left \langle e_{1},e_{1}\right \rangle =\left \langle e_{3},e_{3}\right \rangle
=1,\text{ }\left \langle e_{2},e_{2}\right \rangle =\left \langle
e_{4},e_{4}\right \rangle =-1.
\end{equation*
We have the dual 1-forms as:
\begin{equation*}
\omega _{1}=ds\text{ \ \ \ \ and \ \ \ \ }\omega _{2}=-\sqrt{2}p\left(
s\right) dt.
\end{equation*
Also we obtain components of the second fundamental form and the connection
forms as
\begin{eqnarray}
h_{11}^{3} &=&c(s),\ h_{12}^{3}=0,\ h_{22}^{3}=0, \\
h_{11}^{4} &=&-b(s),\text{ \ }h_{12}^{4}=0,\text{ \ }h_{22}^{4}=a(s) \notag
\end{eqnarray
an
\begin{eqnarray*}
\omega _{12} &=&a(s)\omega _{2},\text{ \ \ }\omega _{13}=c(s)\omega _{1}
\text{ \ \ }\omega _{14}=-b(s)\omega _{1}, \\
\omega _{23} &=&0,\text{ \ \ }\omega _{24}=-a(s)\omega _{2},\text{ \ \
\omega _{34}=-c(s)\omega _{1}.
\end{eqnarray*
Then we get the covariant differentiations with respect to $e_{1}$ and $e_{2}
$ as follows
\begin{eqnarray}
\tilde{\nabla}_{e_{1}}e_{1} &=&c(s)e_{3}+b(s)e_{4}, \\
\tilde{\nabla}_{e_{2}}e_{1} &=&a(s)e_{2}, \notag \\
\tilde{\nabla}_{e_{1}}e_{2} &=&0, \notag \\
\tilde{\nabla}_{e_{2}}e_{2} &=&a(s)e_{1}-a(s)e_{4}, \notag \\
\tilde{\nabla}_{e_{1}}e_{3} &=&-c(s)e_{1}+c(s)e_{4}, \notag \\
\tilde{\nabla}_{e_{2}}e_{3} &=&0, \notag \\
\tilde{\nabla}_{e_{1}}e_{4} &=&b(s)e_{1}+c(s)e_{3}, \notag \\
\tilde{\nabla}_{e_{2}}e_{4} &=&a(s)e_{2}, \notag
\end{eqnarray
where
\begin{equation}
a(s)=\frac{p^{\prime }(s)}{p(s)},
\end{equation
\begin{equation}
b(s)=\frac{p^{\prime \prime }(s)}{p^{\prime }(s)},
\end{equation
\begin{equation}
c(s)=\frac{x_{1}^{\prime \prime }(s)p^{\prime }(s)-p^{\prime \prime
}(s)x_{1}^{\prime }(s)}{p^{\prime }(s)}.
\end{equation
By using (3), (4) and (41), the mean curvature vector and Gaussian curvature
of the surface $M_{3}$ are obtained as follows
\begin{equation}
H=\frac{1}{2}\left( c(s)e_{3}+\left( a(s)+b\left( s\right) \right)
e_{4}\right)
\end{equation
and
\begin{equation}
K=a(s)b\left( s\right) ,
\end{equation
respectively.
By using (2) and (42) the Laplacian of the Gauss map of $M_{3}$\ is computed
as
\begin{equation}
\Delta G=L(s)\left( e_{1}\wedge e_{2}\right) +M(s)\left( e_{2}\wedge
e_{3}\right) +N(s)\left( e_{2}\wedge e_{4}\right) ,
\end{equation
where
\begin{equation}
L(s)=c^{2}(s)-a^{2}\left( s\right) -b^{2}\left( s\right) ,
\end{equation
\begin{equation}
M(s)=c^{\prime }\left( s\right) +c(s)(a(s)+b(s)),
\end{equation
\begin{equation}
N(s)=c^{2}(s)+a^{\prime }\left( s\right) +b^{\prime }\left( s\right) .
\end{equation}
\begin{theorem}
\label{teo 7}Let $M_{3}$ be flat rotation surface of parabolic type given by
the parameterization (40). Then $M_{3}$ has pointwise 1-type Gauss map if
and only if the profile curve of $M_{3}$ is given b
\begin{eqnarray*}
x_{1}(s) &=&\frac{\varepsilon }{\mu _{1}}\left( \ln (\mu _{1}s+\mu _{2})(\mu
_{1}s+\mu _{2})\right) +\left( \mu _{4}-\varepsilon \right) s+\mu _{5}, \\
p(s) &=&\mu _{1}s+\mu _{2}, \\
q(s) &=&\frac{1}{2\mu _{1}}\int \left( \left( \varepsilon \ln \left( \mu
_{1}s+\mu _{2}\right) +\mu _{4}\right) ^{2}-1\right) ds,
\end{eqnarray*
where $\mu _{1},$ $\mu _{2},$ $\mu _{4},$ $\mu _{5}$ real constants. Morever
the surface $M_{3}$ has harmonic Gauss map for $f=0.$
\end{theorem}
\begin{proof}
We suppose that $M_{3}$ has pointwise 1-type Gauss map. In that case the
Gauss map of $M_{3}$\ holds (1). By using (1) and (48), we get
\begin{eqnarray}
-f+f\left \langle C,e_{1}\wedge e_{2}\right \rangle &=&-L(s), \\
f\left \langle C,e_{2}\wedge e_{3}\right \rangle &=&-M(s), \notag \\
f\left \langle C,e_{2}\wedge e_{4}\right \rangle &=&N(s) \notag
\end{eqnarray
and
\begin{equation}
\left \langle C,e_{1}\wedge e_{3}\right \rangle =\left \langle C,e_{1}\wedge
e_{4}\right \rangle =\left \langle C,e_{3}\wedge e_{4}\right \rangle =0.
\end{equation
By taking the derivatives of all equations in (53) with respect to $e_{2}$
and using (52) we obtain
\begin{eqnarray}
L(s)-N(s) &=&f, \\
M(s) &=&0, \notag
\end{eqnarray
respectively$.$ Since the surface $M_{3}$ is flat, i.e., $K=0$ from (47) we
have that $b(s)=0$. From (44) we obtain that
\begin{equation}
p(s)=\mu _{1}s+\mu _{2}
\end{equation
for some constants $\mu _{1}\neq 0$ and $\mu _{2}.$ By using (43) and (55)
we have that
\begin{equation}
a(s)=\frac{\mu _{1}}{\mu _{1}s+\mu _{2}}.
\end{equation
If we consider $M(s)=0$ with the equations $b(s)=0$ and $a(s)=\frac{\mu _{1
}{\mu _{1}s+\mu _{2}},$ from (50) we get
\begin{equation}
c(s)=\frac{\mu _{3}}{\mu _{1}s+\mu _{2}}.
\end{equation
On the other hand, by using the first equation of (54), (49), (51), (56) and
(57) we obtain that $f=0$. It means that $L(s)=N(s)=0$ and we hav
\begin{equation*}
\mu _{3}=\varepsilon \mu _{1},\ \varepsilon =\pm 1.
\end{equation*
If we consider (45), (55) and (57) we get
\begin{equation}
x_{1}(s)=\frac{\varepsilon }{\mu _{1}}\left( \ln (\mu _{1}s+\mu _{2})(\mu
_{1}s+\mu _{2})\right) +\left( \mu _{4}-\varepsilon \right) s+\mu _{5},
\end{equation
where $\mu _{4},$ $\mu _{5}$ are constants of integration. Since $x$ is unit
speed spacelike curve we ge
\begin{equation}
q^{\prime }(s)=\frac{\left( x_{1}{}^{\prime }(s)\right) ^{2}-1}{2p^{\prime
}(s)}.
\end{equation
By substituting (55) and (58) into (59) we obtai
\begin{equation*}
q(s)=\frac{1}{2\mu _{1}}\int \left( \left( \varepsilon \ln \left( \mu
_{1}s+\mu _{2}\right) +\mu _{4}\right) ^{2}-1\right) ds.
\end{equation*
This completes the proof.
\end{proof}
\begin{theorem}
\label{teo 8} Let $M_{3}$ be flat rotational surfaces of parabolic type
given by (40). $M_{3}$ has harmonic\ Gauss map if and only if its mean
curvature vector is parallel.
\end{theorem}
\begin{proof}
We suppose that $M_{3}$ has parallel mean curvature vector, i.e., $DH=0.$
From (46) we have that
\begin{equation*}
D_{e_{1}}H=\frac{1}{2}\left( M(s)e_{3}+N(s)e_{4}\right) =0.
\end{equation*
In this case we obtain that $M(s)=N(s)=0.$ Since $M_{3}$ is a flat surface,
from the previous theorem we have
\begin{equation*}
b(s)=0\text{ and }a(s)=\frac{\mu _{1}}{\mu _{1}s+\mu _{2}}.
\end{equation*
By considering the equation $M(s)=0$ with above equations and using (50) we
get
\begin{equation*}
c(s)=\frac{\mu _{3}}{\mu _{1}s+\mu _{2}},
\end{equation*
where $\mu _{3}$\ is the constant of integration. It implies that $L(s)=0.$
Hence we obtain that Gauss map of $M_{3}$ is harmonic .
Conversely, if $M_{3}$ is harmonic then it is easily seen that $DH=0.$
\end{proof}
The first author is supported by Ahi Evran University :PYO-EGF.4001.15.002.
|
1,116,691,500,730 | arxiv | \section{Introduction}
In this work, we provide a generally applicable method for solving inverse problems, and for obtaining statistical bounds on the error of the prediction. The bounds are proven to hold in probability via methods of conformal prediction \cite{romano2019conformalized}, without making a-priori assumptions on the underlying distributions and by only using a finite data set.
In inverse problems, the goal is to recover the true source signal $x \in \mathcal{X}$ from a corrupted observation $z \in \mathcal{Z}$ obtained from the so-called \emph{measurement} or \emph{forward} operator $\mathcal{F}:\mathcal{X}\rightarrow \mathcal{Z}$
\begin{equation}\label{eq:inverse_problem}
z = \mathcal{F}(x)+\nu,
\end{equation}
where $\nu$ denotes the measurement noise. Since in most applications of interest the forward operator $\mathcal{F}$ is not invertible or at least severly ill-conditioned, \eqref{eq:inverse_problem} is not well-posed in general. In a probabilistic framework, $x$, $z$, and $\nu$ are replaced by the random variables $X$, $Z$, and $N$ and, consequently, \eqref{eq:inverse_problem} is modelled by a common distribution $\mathrm{p}(x,z)$. In this setting, solving the inverse problem amounts to estimating the random variable $X$ based on an observation of $Z$, which is frequently done by invoking the posterior distribution computed using Bayes theorem
\begin{equation}\label{eq:bayes_thm}
\mathrm{p}(x|z) = \frac{\mathrm{p}(z|x)\mathrm{p}(x)}{\mathrm{p}(z)}
\end{equation}
respectively its negative logarithm
\[-\log(\mathrm{p}(x|z))=-\log(\mathrm{p}(z|x))-\log(\mathrm{p}(x))+const.\]
While a model for the data likelihood $\mathrm{p}(z|x)$ is easily found if the noise distribution is fixed, a lot of research has gone into modelling the prior distribution $\mathrm{p}(x)$, see \Cref{sec:related_works_reg}. The present work, however, is directed at a different aim: Given any (approximate) model of the prior distribution, and a resulting method to obtain point estimates, we want to obtain error bounds with coverage guarantees for the point estimate that hold true even if the model of the prior distribution and/or the method to obtain point estimates are not 100\% accurate.
To explain our approach, let us assume for simplicity $\mathcal{X} = \mathbb{R}$. Given a prediction $\hat{x}(Z)$ for $X$ and an i.i.d. sample $(X_i,Z_i)_{i=1}^m$, we want to obtain a method for predicting quantiles $\hat{S}_q = \hat{s}_q((X_i,Z_i)_{i=1}^m,Z)$ of the squared error $S=s(X,Z)=(X-\hat{x}(Z))^2$ satisfying
\begin{equation}\label{eq:intro_coverage}
\mathrm{P}[S\leq \hat{S}_q] \geq q
\end{equation}
where $q\in (0,1)$ can be defined arbitrarily by the user. Here, the probability is computed over $(X_i,Z_i)_{i=1}^m$ and $(X,Z)$, and \eqref{eq:intro_coverage} shall be satisfied without assumptions on the underlying distributions. Our approach to obtain the quantiles $\hat{S}_q$ builds on the relation between error and posterior variance. More precisely, given an estimator $\hat{t}(z)\approx \Var[X|Z=z]$, $\hat{T}=\hat{t}(Z)$ for the posterior variance, we first approximate the distribution of the $(S,\hat{T})$ on the sample $(X_i,Z_i)_{i=1}^m$, where we split the range of $\hat{T}$ into different bins $\tau_k = [t_k,t_{k+1})$ for $k=1,2,\ldots$ For each bin $\tau_k$, we then set $\hat{S}^{\tau_k}_q$ to be an approximation of the $q$-quantile of $\{S_i\;|\; \hat{t}(Z_i) \in \tau_k\}$. For this, we make use of conformal prediction techniques to ensure coverage (see \cite{romano2019conformalized} for the original work that motivated this part of our approach). In other words, $\hat{S}^{\tau_k}_q$ is an estimate of the $q$-quantile of $S|\hat{T}\in\tau_k$ based on a finite sample. For a new data point $Z$, the estimated $q$-quantile $\hat{S}_q$ such that \eqref{eq:intro_coverage} holds is then obtained as $\hat{S}_q = \hat{S}^{\tau_k}$, where $k$ is such that $\hat{t}(Z) \in \tau_k$. With this, our method acts as a regression from posterior variance to the squared error, where the regression is carried out pointwise on different variance bins (thus it is non-linear). Our coverage guarantees hold independently of the true distribution of $(X,Z)$ and even in the case that both $\hat{x}(Z)$ and $\hat{t}(Z)$ are sub-optimal estimators (in fact, they can even be arbitrary). The reason why this is possible is because we make \emph{no guarantees on the magnitude of the error bounds} (i.e., on the size of the interval of uncertainty). Naturally, the error bounds will be smaller, the better the prior $\mathrm{p}(x)$ and the estimators $\hat{x}(Z)$ and $\hat{t}(Z)$. While our coverage guarantees hold in any case, it can only be observed after carrying out our method on a new datum, if the error bounds obtained for this datum are sufficiently small for practical use. Our experiments show that this is the case with different, reasonable priors.
For the estimation of statistical quantities, such as the posterior variance or the expected posterior, we rely on Langevin based Markov chain Monte Carlo sampling \cite{laumont2022bayesian}, where we introduce a novel variation of the Langevin algorithm applicable to non-smooth log-distributions. The contributions of our work can be summarized as follows.
\paragraph{Contributions}
\begin{itemize}
\item We propose a novel method for error estimation for inverse problems in imaging. Based on a finite i.i.d. sample we estimate quantiles of the reconstruction error conditioned on an estimate of the posterior variance.
\item Coverage of the estimated quantiles is proven without making assumptions on the underlying distributions and by only invoking a finite data set by means of conformal prediction.
\item Numerical experiments supporting our claims are conducted on multiple inverse problems as well as with different regularization methods. In particular we show that the error guarantees are satisfied in practice and provide additional empirical evidence demonstrating the strong relation between posterior variance and error.
\item Within our numerical experiments, we propose a new approach for posterior sampling in the presence of a non-smooth convex functional based on the primal-dual algorithm.
\item Furthermore, a novel method for evaluation of Langevin-like algorithms by means of a Markov Random Field is presented.
\end{itemize}
\section{Related Works}\label{sec:related_works}
\subsection{Regularization and Posterior Sampling}\label{sec:related_works_reg}
As the framework for solving inverse problems is well-established, most research nowadays revolves around the choice of the prior respectively regularizer $\mathrm{p}(x)$ in \eqref{eq:bayes_thm}.
Two categories can be distinguished: In the past hand-crafted priors with certain assumptions on regularity such as sparsity \cite{rof92, sa01, chpo11, da04, ho20_ip_review} were commonly the preferred choice, while nowadays the field of inverse problems is mainly dominated by data driven approaches that either directly learn an inverse mapping \cite{ja08, le17, zb18} or attempt to learn a suitable prior.
The latter is frequently done by means of unrolling \cite{ham18, koef21, aga18} or as in recent works by the use of generative approaches \cite{bor17, na19, son19,habring2022generative}.
Priors, rather than penalty functionals, are of particular interest, since they offer the possibility of sampling due to their access to the posterior \cite{laumont2022bayesian}.
Although the sampling method itself is subject to ongoing research \cite{do21, pe16,ro99}, Langevin algorithms for Markov chain Monte Carlo sampling \cite{roberts1996exponential,durmus2019high,durmus2018efficient} are widely used either for inference \cite{laumont2022bayesian} or also in training \cite{zach2022computed, son19}.
\subsection{Uncertainty Quantification}
Uncertainty quantification is the task of quantifying the certainty or confidence of a prediction. Note, however, that there is a conceptual difference between uncertainty and prediction error since a very confident prediction can still be erroneous. While in the literature it has, e.g., been proposed to use the posterior variance as a measure of uncertainty \cite{zach2022computed}, a link to the reconstruction error is rarely found in existing works. It is precisely this link between uncertainty and error that we investigate in this work. Most works about uncertainty quantification in the literature either utilize Bayesian neural networks or posterior sampling using Markov chain Monte Carlo methods. In \cite{blundell2015weight} an algorithm for learning a distribution over the weights of a neural network is proposed. In \cite{kega17,narnhofer2021bayesian} the authors use Bayesian neural networks to investigate different types of uncertainty (aleatoric vs epistemic) and in \cite{rep19} the authors use hypothesis tests in a Bayesian framework to quantify the probability of the presence of certain structures in an image reconstruction. For a review of Bayesian uncertainty quantification of deep learning methods see \cite{abdar2021review}. In \cite{gal2016dropout} it is shown that dropout training in deep neural networks is equivalent to approximate Bayesian inference in deep Gaussian processes yielding tools to model uncertainty. In \cite{sc18} Markov chain dropout is used for deep learning in MRI reconstruction to investigate uncertainty and the authors also qualitatively relate the uncertainty to the reconstruction error. In \cite{ra20} the authors sample from the posterior using Monte Carlo sampling and modelling the gradient of the log prior distribution by a denoising auto encoder. In \cite{lu22} the authors propose to learn the reverse noise process in the context of MRI reconstruction allowing for sampling from the posterior. Closely related to our approach, in \cite{zach2022computed} the authors propose posterior sampling using Langevin dynamics combined with a generative prior to estimate the posterior variance.
\subsection{Conformal Prediction}
The proposed work also builds on ideas of conformal prediction and risk control, methods of obtaining distribution free confidence bounds for model predictions based on a finite training sample. Risk control describes methods, where an i.i.d. data set is used to bound the expected loss. The procedure builds on concentration inequalities, such as Hoeffding's inequality, which allow to bound an expected value based on a finite sample mean \cite{angelopoulos2021learn,angelopoulos2022image,medarametla2021distribution,angelopoulos2022conformal,bates2021distribution}. On the other hand, in conformal prediction, introduced in \cite{vovk2005algorithmic,shafer2008tutorial}, experience on a data set is used to estimate quantiles and/or confidence regions for a random variable, typically under the assumption of exchangeability of the sample \cite{tibshirani2019conformal,lei2018distribution,lei2015conformal,lei2013distribution,sadinle2019least,romano2019conformalized}. The present work can, indeed, be put into this framework: If the proposed method outputs a point prediction $\hat{x}$ and an upper bound for the squared error $\hat{s}_q$, then $[\hat{x}-\sqrt{\hat{s}_q}, \hat{x}+\sqrt{\hat{s}_q}]$ is a $q$ confidence interval for the ground truth. In terms of applications, our work is related most closely to \cite{angelopoulos2022image} where the authors propose a method predicting per-pixel confidence intervals for image regression. Contrary to our work, however, the authors guarantee a bounded expected rate of wrongly predicted pixel intervals in an image using risk control based on concentration inequalities. In the present work we bound the error probability for each pixel individually. From a theoretic viewpoint, our work is in the same spirit as \cite{romano2019conformalized}. There authors use conformal prediction to calibrate confidence intervals obtained by quantile regression. Distinct from both of the mentioned works we do not assume a given method for interval predictions, but only require access to a point predictor. Using data to estimate the distribution of this method's error, we implicitly obtain a set predictor. Moreover, we account for heteroscedasticity by conditioning the conformal prediction procedure on the given observation.
\section{Theoretical Results}
\label{sec:main}
In this section we first explain the proposed method in an abstract setting before elaborating on the application in imaging.
\subsection{Preliminaries}
We denote random variables as upper case letters and deterministic quantities or realizations of random variables as lower case letters. Random variables obtained as functions of other random variables are consequently denoted as the upper case letter of the function, e.g., we denote the posterior variance as $T=t(Z)$ with the function $t(z)=\Var[X\;|\;Z=z]$ evaluated at the random variable $Z$. Moreover, approximations/estimators are denoted with a hat, e.g., the estimator of the posterior variance $t$ is denoted as $\hat{t}$ and analogously for the random variables $T,\hat{T}$. Independence of two random variables $X,Y$ is denoted as $X\rotatebox[origin=c]{90}{$\models$} Y$. For a real valued random variable $Y$ with cumulative distribution function $F(y) = \mathrm{P}[Y\leq y]$, we denote for $q\in (0,1)$ the $q$-quantile as
\[y_q = \inf\{ y\in\mathbb{R}\; | \; F(y)\geq q\}.\]
Further for i.i.d. random variables $Y_1,\dots,Y_n$, we define the empirical quantile $\hat{Y}_q$ as the quantile of the empirical distribution function $\hat{F}(y) = \frac{1}{n}\sum_{i=1}^n \mathbbm{1}_{Y_i\leq y}$,
\[\hat{Y}_q = \inf\{ y\in\mathbb{R}\; | \; \hat{F}(y)\geq q\}\]
which can be computed explicitly as $\hat{Y}_q = Y_{(\lceil qn \rceil)}$ with $Y_{(k)}$ being the k-th smallest value in $Y_1,\dots,Y_n$. We present a slight modification of a key result from conformal prediction, which can be found in different versions in \cite{vovk2005algorithmic,lei2018distribution,tibshirani2019conformal,romano2019conformalized}. In particular, the presented proofs are a generalization of \cite[Lemmas 1,2, Appendix]{romano2019conformalized} where we additionally allow for a random sample size.
\begin{lemma}\label{lem:cqr1}
Let $Y_1, Y_2,\dots,Y_{N+1}$ real valued random variables with $N\in\mathbb{N}$ a random sample size. Assume that $Y_1, Y_2,\dots,Y_{N+1}$ are i.i.d. with respect to the probability measure $\mathrm{P}[\;.\;|\; N=n]$ for every $n\in\mathbb{N}$, $n\geq 1$. Then for any $q\in (0,1)$ and $n\geq 1$,
\[\mathrm{P}[Y_{N+1}\leq \hat{Y}_q\;|\; N=n]\geq q\]
where $\hat{Y}_q = Y_{(\lceil(N+1)q\rceil)}$ is the empirical q-quantile of $Y_1, Y_2,\dots,Y_{N+1}$.
\end{lemma}
\begin{remark}
The sample size is set to $N+1$ instead of $N$ in order to enable a consistent notation for later results where the first $N$ random variables will be the estimation data and the $(N+1)$-th sample a new data point.
\end{remark}
\begin{proof}
By definition of the empirical quantile it is always true that
\[q\leq \hat{F}(\hat{Y}_q).\]
As a consequence we find for fixed $n\geq 1$
\begin{equation}
\begin{aligned}
q\leq\mathbb{E}[ \hat{F}(\hat{Y}_q)\;|\;N=n]
=& \mathbb{E}[ \frac{1}{N+1}\sum\limits_{i=1}^{N+1}\mathbbm{1}_{Y_i\leq\hat{Y}_q}\;|\;N=n]
=\mathbb{E}[ \frac{1}{n+1}\sum\limits_{i=1}^{n+1}\mathbbm{1}_{Y_i\leq\hat{Y}_q}\;|\;N=n] \\
=&\frac{1}{n+1}\sum\limits_{i=1}^{n+1}\mathrm{P}[Y_i\leq\hat{Y}_q\;|\;N=n]
=\mathrm{P}[Y_{n+1}\leq\hat{Y}_q\;|\;N=n]\\
=&\mathrm{P}[Y_{N+1}\leq\hat{Y}_q\;|\;N=n].
\end{aligned}
\end{equation}
%
%
%
%
%
%
\end{proof}
In the preceding Lemma we have proven the intuitive assertion that the empirical quantile of $Y_1,\dots Y_{N+1}$ has guaranteed coverage for any of the $Y_i$. In the subsequent result, on the other hand, we conclude, that by correcting the quantile by a factor $(1+\frac{1}{N})$ coverage is also guaranteed for a new unseen i.i.d random variable.
\begin{lemma}\label{lem:cqr}
Let $Y_1, Y_2,\dots,Y_{N},Y$ be real valued random variables with $N\in\mathbb{N}$ random such that $Y_1, Y_2,\dots,Y_{N},Y$ are i.i.d. with respect to the probability measure $\mathrm{P}[\;.\;|\; N=n]$ for every $n\in\mathbb{N}$, $n\geq 1$. Assume additionally that $(1+\frac{1}{n})q\leq 1$, then
\[\mathrm{P}[Y\leq \hat{Y}_{(1+\frac{1}{N})q}\;|\;N=n]\geq q\]
where now $\hat{Y}_{(1+\frac{1}{N})q} = Y_{(\lceil(N+1)q\rceil)}$ is the empirical quantile of $Y_1, Y_2,\dots,Y_{N}$.
We call $\hat{Y}_{(1+\frac{1}{N})q}$ the \emph{conformalized} q-quantile.
\end{lemma}
\begin{proof}
Let us denote $Y_{N+1}=Y$ and $Y_{(k,m)}$ the k-th largest value in $Y_1,\dots, Y_m$. Note that for $k$ such that $1\leq k\leq n$, $Y_{n+1}\leq Y_{(k,n)}$ if and only if $Y_{n+1}\leq Y_{(k,n+1)}$. Thus, since by assumption $\lceil(n+1)q\rceil\leq n$,
\begin{equation}
\begin{aligned}
\mathrm{P}[Y_{N+1}\leq Y_{(\lceil(N+1)q\rceil,N)}\;|\;N=n] = \mathrm{P}[Y_{N+1}\leq Y_{(\lceil(N+1)q\rceil,N+1)}\;|\;N=n]
\end{aligned}
\end{equation}
The result follows by applying \Cref{lem:cqr1} to the right-hand side of the equation.
\end{proof}
\subsection{The Proposed Method}
Let us assume for now that the signal space $\mathcal{X}=\mathbb{R}$. Let $(X_i,Z_i)_{i=1}^m\subset \mathcal{X} \times \mathcal{Z}$ be an i.i.d. sample and $(X,Z)\sim (X_i,Z_i)$. Further, let $\hat{x}:\mathcal{Z}\rightarrow \mathcal{X}$ be a prediction function for the ground truth $X$, e.g., $\hat{x}(z)\approx \mathbb{E}[X\;|\;Z=z]$, an approximation of the expected posterior. Define the squared error of the prediction $S = s(X,Z) = (\hat{x}(Z)-X)^2$ and accordingly $S_i = s(X_i,Z_i)$. A naive approach to obtain guaranteed error bounds would be to simply apply \Cref{lem:cqr} to the random variables $S_i, S$. Note, however, that this would yield an error bound which is independent of the given observation $Z$. In particular for heteroscedastic data this will result in inefficient error estimates. While in \cite{romano2019conformalized} the authors propose to account for heteroscedasticity by applying \Cref{lem:cqr} to an interval predictor which incorporates heteroscedasticity already, we propose a different approach, namely integrating information about $Z$ in the form of conditional probabilities. Denote an approximation of the posterior variance as $\hat{t}(z) \approx \Var[X|Z=z]$, $\hat{T} = \hat{t}(Z)$ and $\hat{T}_i = \hat{t}(Z_i)$. Ideally, we would apply \Cref{lem:cqr} to the random variable $S|\hat{T}=\tau$ for any fixed value of $\tau \in \mathbb{R}$ yielding guarantees on the error for any fixed value of the approximated variance. This way we would make use of all the information contained in $\hat{T}$. But since the estimation of a quantile of the random variable $S$ conditioned on any point-value of $\hat{T}$ is unfeasible in practice, where only a finite amount of data is available, a relaxation is needed. This is achieved by partitioning $[0,\infty)$ into disjoint intervals $\tau_k = [t_k,t_{k+1})$, $k=0,1,2,\dots$, and, for any given interval $\tau_k$, considering the error conditioned on $\hat{T}\in\tau_k$. To realize this, for any given interval $\tau_k$, first define the estimated conditional error quantile
\begin{equation}\label{eq:quantile_estimator}
\hat{S}^{\tau_k}_q \coloneqq (1+\frac{1}{N_{\tau_k}})q-\text{empirical quantile of } \{S_i\;|\; \hat{T}_i \in \tau_k\}.
\end{equation}
where $N_{\tau_k}=|\{S_i\;|\; \hat{T}_i \in \tau_k\}|$. In the case that $(1+N_{\tau_k})q>N_{\tau_k}$, we set $\hat{S}^{\tau_k}_q$ to be the essential supremum of the random variable $S$. Given a new observation $Z$, the final method can then be described as follows:
\begin{enumerate}
\item Compute $\hat{x}(Z)$ and $\hat{t}(Z)$.
\item Pick $\tau_k$, such that $\hat{t}(Z)\in \tau_k$.
\item Compute the error quantile estimator $\hat{S}_q$ as $\hat{S}^{\tau_k}_q$.
\end{enumerate}
\begin{remark}\label{rmk:choice_T_x_bar}
The following results do not rely on exact knowledge of the posterior expectation or variance, which is why we already introduced $\hat{X}$ and $\hat{T}$ as approximations above. As a matter of fact, \Cref{prop:coverage,cor:coverage} below are entirely independent of the particular choices of the functions $\hat{x}(z)$ and $\hat{t}(z)$. In particular, all results remain true despite using heuristics within the computations of the approximations of posterior expectation and variance, such as the choice of the thinning parameter explained in \Cref{rmk:thinning} below. While the theory is not affected by the choices of $\hat{x}$ and $\hat{t}$, the tightness, and thus quality, of the error estimates will crucially depend on the predictive capability of $\hat{T}$ with respect to the error $S$. Our choice is motivated by the high similarity between the squared error to the expected posterior $(X-\mathbb{E}[X\;|\;Z])^2$ and the posterior variance $\Var[X\;|\;Z]=\mathbb{E}[(X-\mathbb{E}[X\;|\;Z])^2]\;|\;Z]$, which is, in fact, the conditional expectation of the former.
\end{remark}
As a direct consequence of \Cref{lem:cqr}, we obtain the following coverage guarantee.
\begin{proposition}\label{prop:coverage}
Let $(X_i,Z_i)_{i=1}^m\subset \mathcal{X}\times \mathcal{Z}$ and $(X,Z) \in \mathcal{X}\times \mathcal{Z}$ be i.i.d. Let $S=s(X,Z) = (X-\hat{x}(Z))^2$ and $\hat{T}=\hat{t}(Z)$ and assume that $S$ is bounded from above almost surely. Assume further that $\mathrm{P}[\hat{T}\in\tau_k]>0$, then the estimated conditional error quantile satisfies
\[\mathrm{P}[S\leq \hat{S}^{\tau_k}_q\;|\; \hat{T}\in \tau_k]\geq q\]
\end{proposition}
\begin{proof}
Fix $\tau_k$ and let $(S_{i_j})_{j=1}^{N_{\tau_k}}$ be the set of all $S_i$ for which $\hat{T}_i\in \tau_k$. That is,
\begin{align}
i_1 = \min\{1\leq i\leq m\;|\;\hat{T}_i\in\tau_k\},\quad i_{j+1} = \min\{i_j< i \leq m\;|\;\hat{T}_i\in\tau_k\}.
\end{align}
In a sequence $(S_i,\hat{T}_i)_{i>m}$, of i.i.d. copies of $(S,\hat{T})$ let $S_{i_{N_{\tau_k}+1}}$ be defined as the next sample with variance $\hat{T}\in\tau_k$ in this sequence, i.e.,
\[i_{N_{\tau_k}+1} = \min\{m< i\;|\;\hat{T}_i\in\tau_k\}.\]
Note that by the Borel-Cantelli Lemma, $i_{N_{\tau_k}+1}<\infty$ a.s., since $\mathrm{P}[\hat{T}\in\tau_k]>0$. For the distributions of these random variables we find for any $n\geq 1$, $j\leq n$ and $s\in\mathbb{R}$ by the law of total probability
\begin{equation}
\begin{aligned} \label{eq:intex_to_bin_change}
\mathrm{P}[S_{i_j}\leq s\;|\; N_{\tau_k}=n] = \sum_{l=1}^m \mathrm{P}[S_{i_j}\leq s\;|\; i_j=l, N_{\tau_k}=n]\;\mathrm{P}[i_j=l\;|\; N_{\tau_k}=n]\\
=\sum_{l=1}^m \mathrm{P}[S_l\leq s\;|\; \hat{T}_i\in \tau_k \text{ for $i=l$ and for $(n-1)$ different $i\in\{1,\dots,m\}\setminus\{l\}$} ]\\ \;\mathrm{P}[i_j=l\;|\; N_{\tau_k}=n]\\
=\sum_{l=1}^m \mathrm{P}[S_l\leq s\;|\; \hat{T}_l\in \tau_k ]\;\mathrm{P}[i_j=l\;|\; N_{\tau_k}=n]\\
=\sum_{l=1}^m \mathrm{P}[S\leq s\;|\; \hat{T}\in \tau_k ]\;\mathrm{P}[i_j=l\;|\; N_{\tau_k}=n]\\
= \mathrm{P}[S\leq s\;|\; \hat{T}\in \tau_k ]
\end{aligned}
\end{equation}
and the same result can be obtained for $j=n+1$ via an analogous computation. Further we can deduce independence as follows. Let $s_1,\dots,s_{n+1}\in\mathbb{R}$,
\begin{equation}
\begin{aligned}
\mathrm{P}[\forall j=1,2,\ldots,n+1: S_{i_j}\leq s_j\;|\; N_{\tau_k}=n] \\
= \sum_{l_1<l_2<\dots<l_n\leq m<l_{n+1}}\mathrm{P}[\forall j:\;S_{i_j}\leq s_j\;|\; \forall j:\;i_j=l_j,\;N_{\tau_k}=n]\;\mathrm{P}[\forall j:\;i_j=l_j, \;|\;N_{\tau_k}=n]\\
= \sum_{l_1<l_2<\dots<l_n\leq m<l_{n+1}} \mathrm{P}[\forall j:\;S_{l_j}\leq s_j\;|\; \hat{T}_i\in\tau_k \Leftrightarrow i \in \{l_1,\ldots,l_n\}]\;\mathrm{P}[\forall j:\;i_j=l_j\;|\;N_{\tau_k}=n]\\
= \sum_{l_1<l_2<\dots<l_n\leq m<l_{n+1}} \mathrm{P}[\forall j:\;S_{l_j}\leq s_j\;|\; \forall j:\;\hat{T}_{l_j}\in\tau_k]\;\mathrm{P}[\forall j:\;i_j=l_j\;|\;N_{\tau_k}=n]\\
= \sum_{l_1<l_2<\dots<l_n\leq m<l_{n+1}} \mathrm{P}[\forall j:\;S_{j}\leq s_j\;|\; \forall j:\;\hat{T}_{j}\in\tau_k]\;\mathrm{P}[\forall j:\;i_j=l_j\;|\;N_{\tau_k}=n]\\
= \mathrm{P}[\forall j:\;S_{j}\leq s_j\;|\; \forall j:\;\hat{T}_{j}\in\tau_k]
= \prod\limits_{j=1}^{n+1}\mathrm{P}[S_{j}\leq s_j\;|\; \hat{T}_{j}\in\tau_k]
= \prod\limits_{j=1}^{n+1}\mathrm{P}[S_{i_j}\leq s_j\;|\; N_{\tau_k}=n]\\
\end{aligned}
\end{equation}
where the last step follows from \eqref{eq:intex_to_bin_change}.
Thus, we can apply \Cref{lem:cqr} to $(S_{i_j})_{j=1}^{N_{\tau_k}}$, $S_{i_{N_{\tau_k}+1}}$ and find for $(1+n)q<n$
\begin{equation}
\begin{aligned}
q\leq \mathrm{P}[S_{i_{N_{\tau_k}+1}}\leq \hat{S}^{\tau_k}_q\;|\; N_{\tau_k}=n]\\
= \sum\limits_{l>m} \mathrm{P}[S_{i_{N_{\tau_k}+1}}\leq \hat{S}^{\tau_k}_q\;|\; i_{N_{\tau_k}+1}=l,\;N_{\tau_k}=n]\;\mathrm{P}[i_{N_{\tau_k}+1}=l\;|\; N_{\tau_k}=n]\\
= \sum\limits_{l>m} \mathrm{P}[S_l\leq \hat{S}^{\tau_k}_q\;|\; i_{N_{\tau_k}+1}=l,\;N_{\tau_k}=n]\;\mathrm{P}[i_{N_{\tau_k}+1}=l\;|\; N_{\tau_k}=n]\\
= \sum\limits_{l>m} \mathrm{P}[S\leq \hat{S}^{\tau_k}_q\;|\; \hat{T}\in\tau_k,\;N_{\tau_k}=n]\;\mathrm{P}[i_{N_{\tau_k}+1}=l\;|\; N_{\tau_k}=n]\\
= \mathrm{P}[S\leq \hat{S}^{\tau_k}_q\;|\; \hat{T}\in\tau_k,\;N_{\tau_k}=n].
\end{aligned}
\end{equation}
In the case that $(1+n)q\geq n$ (in particular $n=0$), we find
\[q\leq 1=\mathrm{P}[S\leq \hat{S}^{\tau_k}_q\;|\;\hat{T}\in\tau_k,\;N_{\tau_k}=n]\]
as well by definition of $\hat{S}^{\tau_k}$ as the essential supremum of $S$. As a result, the law of total probability yields
\begin{equation}
\begin{aligned}
\mathrm{P}[S\leq \hat{S}^{\tau_k}_q\;|\;\hat{T}\in\tau_k] = \sum\limits_{n=0}^\infty \underbrace{\mathrm{P}[S\leq \hat{S}^{\tau_k}_q\;|\;\hat{T}\in\tau_k,\;N_{\tau_k}=n]}_{\geq q}\mathrm{P}[N_{\tau_k}=n\;|\;\hat{T}\in\tau_k]\geq q.
\end{aligned}
\end{equation}
\end{proof}
\begin{remark}
Assuming $S$ to be bounded a.s. in \Cref{prop:coverage} is only necessary for the definition of $\hat{S}^{\tau_k}_q$ in the case that $(1+N_{\tau_k})q>N_{\tau_k}$ as the essential supremum of $S$. This condition is trivially satisfied for the application in imaging where a pixel's color or gray scale value is in $[0,1]$ or $[0,255]$.
\end{remark}
We conclude coverage without the conditioning on a specific $\tau_k$.
\begin{corollary}\label{cor:coverage}
With the conditions of \Cref{prop:coverage}, define $\hat{S}_q = \hat{S}^{\tau_k}_q$ where $\tau_k$ is such that $\hat{T}\in\tau_k$, then
\[\mathrm{P}[S\leq \hat{S}_q]\geq q\]
\end{corollary}
\begin{proof}
Applying \Cref{prop:coverage} and the law of total probability we can compute
\begin{align*}
\mathrm{P}[S\leq \hat{S}_q] &= \sum_{k=0}^\infty \mathrm{P}[S\leq \hat{S}_q\;|\;\hat{T}\in\tau_k]\;\mathrm{P}[\hat{T}\in\tau_k]\\
&= \sum_{k=0}^\infty \mathrm{P}[S\leq \hat{S}^{\tau_k}_q\;|\; \hat{T}\in\tau_k]\;\mathrm{P}[\hat{T}\in\tau_k]
\geq \sum_{k=0}^\infty q\;\mathrm{P}[\hat{T}\in\tau_k] = q.
\end{align*}
\end{proof}
\subsection{Application to Bayesian Imaging}
In the previous section we introduced the proposed method for estimating the prediction error in a general setting. In this section we explain the specific application to inverse problems in imaging.
In order to apply our method to high dimensional image data, we model the samples $X_i$ as the gray scale values of individual image pixels and $Z_i$ as the given observation of the image. That is, our method is applied to each image pixel separately. The method consists of two parts. First for a given a sample $(x_i,z_i)_{i}$, we compute $(s_i,\hat{t}_i)$ and afterwards, for new data $z$, $\hat{t}(z)$ is computed and, for $\tau_k$ such that $\hat{t}(z)\in\tau_k$, the empirical quantile $\hat{s}^{\tau_k}_q$ is determined. The procedure is depicted in \Cref{algo:error_estimation}.
\begin{algorithm}
\setstretch{1.15}
\caption{Error estimation.}\label{algo:error_estimation}
\textbf{Input:} Observation $z$, random sample $(x_i,z_i)_{i=1}^m$, variance bins $(\tau_k)_k$, confidence level $q\in(0,1)$.\\
\textbf{Output:} Point prediction $\hat{x}$, error estimate $\hat{s}_q$.
\begin{algorithmic}[1]
\For{$i=1,2,\dots,m$}
\State $\hat{x}_{i} = \hat{x}(z_i)$
\State $\hat{t}_i = \hat{t}(z_i)$
\State $s_i = (\hat{x}_i-x_i)^2$
\EndFor
\State $\hat{x}=\hat{x}(z)$
\State $\hat{t}=\hat{t}(z)$
\State Pick bin $\tau_k$ with $\hat{t}\in \tau_k$.
\State $n_{\tau_k} = \left|\left\{i\;\middle|\; \hat{t}_i\in\tau_k\right\}\right|$
\State Compute $\hat{s}_q$ according to \eqref{eq:quantile_estimator}.
\end{algorithmic}
\end{algorithm}
\begin{figure}[htb]
\includegraphics[width= \linewidth]{ures_PDF.pdf}
\caption{Left: Joint distribution $\mathrm{p}(x,z)$ with marginal distributions $\mathrm{p}(x)$, $\mathrm{p}(z)$ and posterior distributions $\mathrm{p}(x|z)$. Right: Posterior expectation $\mathbb{E}[X|Z=z]$ and posterior variance $t=\Var[X|Z=z]$ as functions of $z$ as well as the conditional distribution of the error $p(s|t)$ for specific instances of $t$. As in later usage, posterior variance and error are already presented in logarithmic scaling.}
\label{fig:PDF}
\end{figure}
\subsection{Toy example in 1D}
Let us consider a toy example with known, continuous probability distributions. In this case we can work with exact quantiles instead of empirical estimates using conformal prediction.%
Let $X$ be a uni-variate mixture of Gaussian distributions. More precisely, define the density of $X$ as $\mathrm{p}(x) = \sum_{k=1}^K\alpha_k \mathcal{N}(c_k,\sigma^2_{x_k})$ with $\alpha_k>0$ and $\sum_{k=1}^K\alpha_k=1$ and let $Z = X + N$, with $N \sim \mathcal{N}(0,\sigma^2_z)$.
The joint density of $(X,Z)$ reads as
\begin{equation}
\mathrm{p}(x,z) = \sum_{k=1}^K\alpha_k \mathcal{N}(\mu_k,\Sigma_k),
\label{eq:joint}
\end{equation}
with
\begin{align*}
\mu_k &=
\begin{pmatrix}
c_k \\
c_k
\end{pmatrix},\quad
\Sigma_k=
\begin{pmatrix}
\sigma^2_{x_k} & \sigma^2_{x_k} \\
\sigma^2_{x_k} & \sigma^2_{x_k} +\sigma^2_z
\end{pmatrix},
\end{align*}
and is visualized in \Cref{fig:PDF} for $(c_1,c_2,c_3) = (-1,0,1)$, $\sigma^2_{x_k} = 0.05^2$, $\alpha_k = \frac{1}{3}$ for all $k$ and $\sigma^2_{z} = 0.3^2$. There we show the joint distribution of $(X,Z)$ as well as the corresponding marginal distributions. We additionally plot the resulting posterior distribution for observations of $z=0.55$ and $z=0.0$, the posterior expected value, and the posterior variance as well as the error. The latter two are presented in logarithmic scale.
In this toy example we can explicitly compute the joint distribution of error and variance $(S,T)$ using a change of variables. The result can be found in \Cref{fig:cum1D}, where on the left we plot the joint distribution and on the right the cumulative conditional distribution with respect to $S$, $(s,t)\mapsto \mathrm{P}[S\leq s | T=t]$ is shown. The dashed line indicates the 0.9 quantile of the conditional error distribution. An application of the proposed method in this ideal scenario reads as follows: Given an observation $Z=z$, compute the point prediction $\hat{x}(z)$ and posterior variance $t(z)$. Then in \Cref{fig:cum1D} the $0.9$ quantile of the error is obtained as the value of $s$ where the dashed quantile line intersects the vertical line at $t=t(z)$. While this example is mathematically well-behaved, in imaging applications, neither are statistics of the posterior distribution mathematically tractable, nor do we have access to an unlimited amount of data. The former forces us to make use of sampling techniques while the latter demands the use of conformal prediction in order to retain theoretical guarantees. In \Cref{fig:cum1Dempiric} we illustrate the conformal prediction strategy for the toy example. We estimate the distribution empirically from $2\cdot 10^6$ synthetically generated samples. In the right figure we added the exact conditional quantiles as well as the estimation based on conformal prediction, demonstrating the accuracy of the approximation.
\begin{figure}[htb]
\centering
\includegraphics[width= 0.7\linewidth]{ures_Cumulative_analytic.pdf}
\caption{Joint log-distribution $\log\mathrm{p}(s,t)$ (left) and cumulative conditional distribution over error (right). The gray dashed line indicates the conditional 0.9 quantile.}
\label{fig:cum1D}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width= 0.7\linewidth]{ures_Cumulative_empiric.pdf}
\caption{Empirical joint log-distribution $\log\mathrm{p}(s,t)$ (left) and empirical cumulative conditional distribution over error (right). The dashed gray line indicates the exact conditional 0.9 quantile of the distribution, the red lines indicates the respective estimated conformalized quantile.}
\label{fig:cum1Dempiric}
\end{figure}
\section{Algorithms for the Computation of Posterior Expectation and Variance}
Within our methods we need to compute approximations of posterior expectation and variance. For the typical models of image distributions, an analytical computation is not feasible. Hence, we are forced to compute statistics based on samples. In the following we introduce algorithms for sampling from distributions $\pi(x)$ from the exponential family $\pi(x) \propto \exp{\left(-E(x) \right)}$, on $\mathbb{R}^d$, $d\geq 1$ with a potential (or energy) $E(x)$. In particular in the case of posterior sampling from a Bayesian model
\[E(x) = -\log(\mathrm{p}(z|x)) - \log(\mathrm{p}(x)) = -\log(\mathrm{p}(x|z)) + const.\]
which is equal to the log-posterior up to an unknown normalization constant. The presented algorithms yield samples $(x_k)_{k=1}^K$ from to the distribution $\pi(x) = -\log(\mathrm{p}(x|z))$ based on the observation $z$. Using this samples, as point prediction and variance approximation we will use the sample mean and sample variance, respectively, i.e.,
\begin{equation}\label{eq:used_estimators}
\hat{x}(z)=\frac{1}{K}\sum\limits_{k=1}^K x_k,\quad \hat{t}(z)=\frac{1}{K}\sum\limits_{k=1}^K (x_k-\hat{x}(z))^2.
\end{equation}
After dealing with differentiable energies in \Cref{ssec:ULA}, in \Cref{ssec:PDLV}, we propose a novel sampling method called unadjusted Langevin primal-dual algorithm (ULPDA), which combines Langevin dynamics with a primal-dual algorithm. Further, in \Cref{sec:markov_random_fields}, we introduce Belief Propagation (BP) on Markov random fields (MRF) which is later used as a comparison benchmark for the Langevin algorithm.
\subsection{Langevin Algorithms for Differentiable Energies}\label{ssec:ULA}
Assume that the energy $E(x)$ is differentiable with Lipschitz continuous gradient, i.e., there exists a constant $L \geq 0$ such that
\[
\norm[2]{\nabla E(x) - \nabla E(y)} \leq L \norm[2]{x-y},
\]
for all $x,y \in \mathbb{R}^d$. Langevin based sampling algorithms are discretizations of the over-damped Langevin diffusion equation~\cite{legy97,roberts1996exponential},
\begin{equation}
\rm{d}X_t = -\nabla E(X_t) + \sqrt{2} \rm{d}B_t,
\label{eq:LVcont}
\end{equation}
where $\rm{d}B_t$ is a $d$-dimensional Brownian motion. The stationary distribution of $(X_t)_t$ is the distribution $\pi$ itself \cite{roberts1996exponential}. The striking advantage of the Langevin diffusion equation over classical density-based sampling algorithms is that only information about gradients is required and, hence, no normalization constant is needed, that is, it is sufficient to have knowledge of the distribution $\pi$ up to a multiplicative scalar factor. Intuitively, in \eqref{eq:LVcont} a gradient ascent towards a mode of $\pi$ is performed but the declination injected by the Brownian motion ensures that generated samples cover the entire space exactly with density $\pi(x)$. The scaling factor $\sqrt{2}$ balances the deterministic force induced by the gradient of the log distribution and the stochastic force induced by the Brownian motion.
For a numerical simulation of the stochastic Langevin diffusion equation, the simplest approach is the Euler-Maruyama discretization~\cite{klo13} which is rooted in the classical forward-Euler method. This yields an iterative algorithm, known as the Unadjusted Langevin algorithm, which essentially resembles a gradient descent on the energy $E(x)$ combined with noise (ULA)~\cite{durmus2018efficient}, see \Cref{algo:unaj_langevin}, where
\begin{algorithm}
\setstretch{1.15}
\caption{Unadjusted Langevin Algorithm (ULA)}\label{algo:unaj_langevin}
\textbf{Input:} Initial values $x_0 \in \mathbb{R}^d$, and step sizes $\tau_k > 0 $.\\
\textbf{Output:} Sequence $\{x_k\}_{k\geq 0}$.
\begin{algorithmic}[1]
\For{$k=0,\ldots,K-1$}
\State $x_{k+1} = x_k - \tau_k \nabla E(x_k) + \sqrt{2 \tau_k} \xi_k $
\EndFor
\end{algorithmic}
\end{algorithm}
$\xi_k \sim \mathcal{N}(0,\mathrm{Id}_d)$ is i.i.d. from a $d$-dimensional Gaussian distribution. It has been shown that the Markov chain $\{x_k\}_{k\geq 0}$ generated by the ULA converges to a stationary distribution $\pi_\tau(x)$ and the target distribution $\pi(x)$ is obtained for $\tau_k \to 0$ and $\sum_k \tau_k = \infty$ \cite{lamberton2002recursive,lamberton2003recursive}. Conversely, for $\tau_k > 0$, the stationary distribution of $\{x_k\}_{k\geq 0}$ contains some sort of bias and hence remains \emph{unadjusted}. By the inclusion of an additional Metropolis test, the ULA can be turned into the Metropolis adjusted Langevin algorithm (MALA)~\cite{gar96}, whose stationary distribution is $\pi$, even in case that $\tau_k$ remains strictly bounded away from $0$.
\subsection{Primal-Dual Langevin Algorithm for Non-differentiable Energies}\label{ssec:PDLV}
We will now investigate the case of a non-differentiable potential with the aim of developing a Langevin based sampling algorithm which is also applicable, e.g., for potentials
such as those incorporating TV-regularization, see \Cref{sec:tvl2_denoising}. A remedy making ULA applicable nonetheless is smoothing the potential $E(x)$, either by directly replacing any non-smooth function by a smooth approximation or by considering other model-based smoothness approaches such as the Moreau envelope~\cite{durmus2018efficient}, whose gradient can be computed efficiently whenever the proximal map can be computed efficiently. However, we propose a different approach. Let us consider the general case of an energy of the form $E:\mathbb{R}^d\rightarrow\mathbb{R}$,
\begin{equation}\label{eq:pd_general_energy}
E(x) = \sup_{p\in\mathbb{R}^{d'}} \scp{D x}{p} + g(x) - f^*(p).
\end{equation}
with $D: \mathbb{R}^d \to \mathbb{R}^{d'}$ a linear operator with operator norm $L = \norm{D}$, $f,g$ proper, convex, and lower-semicontinuous and $f^*$ the convex conjugate of $f$. Potentials of this type can be efficiently minimized by first-order primal-dual algorithms~\cite{chpo11}. Interestingly, the flexibility of the step size condition in the primal-dual algorithm opens up the possibility to turn the primal-dual algorithm into a primal algorithm ($\tau_k \to 0, \; \sigma_k \to \infty$), minimizing a particular Moreau-envelope of the primal problem, or into a dual algorithm ($\tau_k \to \infty, \; \sigma_k \to 0$), maximizing a particular Moreau-envelope of the dual problem. Therefore, it is quite natural to propose an Unadjusted Langevin Primal-Dual Algorithm (ULPDA), which is a primal-dual algorithm with small primal step size and the Gaussian noise injected into the primal variable. The algorithm is summarized in \Cref{algo:pd_langevin}.
\begin{algorithm}
\setstretch{1.15}
\caption{Unadjusted Langevin Primal-Dual Algorithm (ULPDA)}\label{algo:pd_langevin}
\textbf{Input:} Initial values $x_0 \in \mathbb{R}^d$, $p_0 \in \mathbb{R}^{d'}$, and step sizes $\sigma_k \tau_k L^2 \le 1 $, $\theta_k \in [0,1]$, $L = \|D\|$\\
\textbf{Output:} Sequence $\{x_k\}_{k\geq 0}$.
\begin{algorithmic}[1]
\For{$k=0,\ldots,K-1$}
\State \begin{equation}\label{eq:algo_pb}
\begin{cases}
\bar p_{k} & = p_{k} + \theta_k( p_{k} - p_{k-1}) , \\
x_{k+1}& = \prox_{\tau_k g} (x_k - \tau_k D^* \bar p_k )+ \sqrt{2 \tau_k} \xi_k, \\
p_{k+1} &= \prox_{\sigma_k f^\ast} (p_k + \sigma_k D x_{k+1} ).
\end{cases}
\end{equation}
\EndFor
\end{algorithmic}
\end{algorithm}
\begin{remark}\label{rmk:thinning}\
\begin{itemize}
\item Consecutive iterates $(x_k,x_{k+1})$ from Langevin algorithms are not independent in general, which is why it is recommended to compute the approximations in \eqref{eq:used_estimators} based on the sample $(x_{kH})_{k}$, instead of $(x_k)_k$, where $H\in\mathbb{N}$, $H>>1$ defines the number of skipped samples in order to obtain a thinned version of the Markov chain. For sufficiently large $H$, it can be assumed that after thinning $(x_{kH},x_{kH+H})$ are approximately i.i.d. This, however, leads to a vast increase of computational effort. Throughout our experiments we set the thinning length to $H=1$. As explained in \Cref{rmk:choice_T_x_bar} the theoretical guarantees are not affected by heuristics in this part of the method. Moreover, an empirical justification of this choice can be found in \Cref{sec:appendix_thinning}, where we show that the effect of thinning regarding the desired statistical quantities is negligible.
\item In order to reduce memory consumption, in practice $\hat{x}$ and $\hat{t}$ are computed from the sample according to Welford's algorithm \cite{wel62}.
\end{itemize}
\end{remark}
\subsection{Belief Propagation}\label{sec:markov_random_fields}
Conversely to the sampling based methods introduced in the previous sections, Belief Propagation (BP) \cite{pea82,tapfre03} is a sampling free algorithm for computing marginals of a given discrete multivariate distribution. The method is designed for graphical models and, in particular, we consider the application to Markov random fields (MRF) \cite{lau1996}. Let $\mathcal{G} = (\mathcal{V},\mathcal{E})$ be a graph with nodes $\mathcal{V}$ and edges $\mathcal{E}$, where in the case of imaging each node $\nu \in \mathcal{V}$ corresponds to an image pixel, and edges $(\nu,\nu') \in \mathcal{E} \subseteq \mathcal{V}^2$ define a neighborhood system of pixels. In a MRF we are given a random vector $X$ corresponding to the graph $\mathcal{G}$, consisting of discrete random variables $X_\nu$ for $\nu\in\mathcal{V}$ which take values in a label set $\mathcal{L} = \{l_1,...,l_{L}\}$. The Markov property is encoded in the model by requiring that for $I,J\subset\mathcal{V}$ not intersecting or adjacent,
$X_I \rotatebox[origin=c]{90}{$\models$} X_J|X_{\mathcal{V}\setminus (I\cup J)}$, that is, $X_I$ independent of $X_J$ conditioned on $X_{\mathcal{V}\setminus (I\cup J)}$.
Using BP, the marginal distributions
\[
\mathrm{P}[X_\nu=x_\nu] = \sum_{\substack{x'\in \mathcal{L}^{|\mathcal{V}|}:\\x'_\nu=x_\nu}} \mathrm{P}[X=x'].
\]
can be computed efficiently with results being exact in case of tree-like graphs and highly accurate approximations otherwise. In turn the pixel-wise posterior expectation $\hat{x}$ and variance $\hat{t}$ are computed as
\[
\hat{x}_\nu = \sum_{x_\nu \in \mathcal{L}} x_\nu\mathrm{P}[X_\nu=x_\nu], \hspace{2em} \hat{t}_\nu = \sum_{x_\nu \in \mathcal{L}} (x_\nu - \hat{x}_\nu)^2\mathrm{P}[X_\nu=x_\nu],
\]
Due to its high accuracy, the result of the BP algorithm can be used as a benchmark for the sampling based methods.
\begin{figure}[htb]
\includegraphics[width=\linewidth]{ures_bp_comp_abs.pdf}
\caption{Mean absolute difference of MMSE and posterior variance estimated using ULPDA for different values of $\tau$ and ULA with Huber TV ($\TV_h$) compared to BP as function of the number of samples with discretization threshold $\Delta=\frac{1}{1024}$.}
\label{fig:bp_comp_abs}
\end{figure}
\section{Numerical experiments}
\label{sec:num}
In this section, we present numerical results of various experiments with the proposed method, verifying our theoretical predictions. For the source code to reproduce the presented results see \cite{error_estimation_source_code}. Throughout the experiments, we compute the error quantiles $\hat{s}_q$ of the reconstruction according to \Cref{algo:error_estimation} in different settings. In our first experiment, presented in \Cref{sec:tvl2_denoising}, we consider total variation based image denoising where we also present a comparison of the ULPDA results to the highly accurate BP results. Afterwards we replace the total variation regularization with a fields of experts \cite{ro09} regularizer in \Cref{sec:foe_denoising} and with total deep variation \cite{koef21} in \Cref{sec:tdv_denoising}. In \Cref{sec:mri_recon} we consider the inverse problem of accelerated magnetic resonance imaging showing the flexibility of our approach with respect to the forward operator of the inverse problem. In \Cref{sec:comp} we compare the proposed method to the one from \cite{angelopoulos2022image}. While the theory and the algorithms were presented for $X$ being one-dimensional, we consider high-dimensional image data in the following. This is done by applying our methods to each pixel separately, that is, the error quantile for an image is computed for each pixel separately based on the approximate posterior variance of this pixel's gray scale value. Doing so, each image sample provides as many samples of the relation between posterior variance and error as its number of pixels. However, strictly speaking, only one of these pixel-wise samples could be used in the quantile estimation, since different pixels of the same image are not independent in general. Our empirical investigation of this effect, see \Cref{sec:neighbouring_pixels}, however, has shown that using all pixels of a single image has a negligible effect on the obtained coverage in practice. For this reason, we use all pixels of each image sample in the subsequent experiments, and refer to \Cref{sec:neighbouring_pixels} for further details.
To evaluate our method quantitatively, we use two different metrics, the coverage and the magnitude of the estimated quantiles, where the former is a verification of the theoretical results as well as a measure for tightness of the quantiles and the latter a measure of the quality of the estimate. We define the coverage as the rate of correct predictions where the true reconstruction error is smaller or equal than the estimated quantile. Precisely, for an image with true error $s\in \mathbb{R}^{M\times N}$ and predicted pixel-wise quantile $\hat{s}_q\in \mathbb{R}^{M\times N}$, the coverage is defined as
\[
\text{coverage} = \frac{\left| \left\{(i,j)\mid s_{i,j}\leq(\hat{s}_q)_{i,j}\right\} \right|}{NM}.
\]
and for a data set containing multiple images we compute the mean of the coverage over all test images. The best method is the one yielding the smallest estimated quantiles satisfying the prescribed coverage.
For our experiment we use gray scale images with values in $[0,1]$ from the BSDS training set corrupted with zero mean Gaussian noise with $\sigma=15/255$. For the estimation of the distribution of $(S,\hat{T})$ 400 images are used and we evaluate our method on a test set of 68 unseen images, referred to as BSDS 68 in the following.
\begin{figure}[htb]
\centering
\resizebox{0.8\textwidth}{!}{%
\includegraphics[scale=1]{article-figure0.pdf}
}
\caption{Comparison of ULPDA-sampling results and BP reconstruction results for denoising with $\sigma=15/255$. The ULPDA reconstructions were obtained for a primal step size of $\tau=\expnumber{5}{-5}$ and 50k iterations ($\expnumber{8}{-4}$~\protect\includegraphics[width=1.5cm,height=.2cm]{ures_cividis.png}~$\expnumber{3.5}{-2}$, $0$~\protect\includegraphics[width=1.5cm,height=.2cm]{ures_hot.png}~$0.01/\expnumber{1}{-3}$ for MMSE and variance difference respectively).}
\label{fig:bp_comp_abs_vis}
\end{figure}
\subsection{TV-$\ell_2$ Denoising}\label{sec:tvl2_denoising}
In denoising the aim is to recover an unknown noise-free image $x \in \mathbb{R}^{M\times N}$ from a noisy observation $z\in \mathbb{R}^{M\times N}$, where the relationship between $x$ and $z$ is described by the following linear forward model
\[
z = x + \nu,
\]
where each noisy pixel $\nu_{i,j}$ is independently sampled from a Gaussian distribution with zero mean and variance $\sigma^2$. Based on the forward model and the i.i.d. Gaussian assumption, the likelihood is simply given by the Gaussian distribution
\[
\mathrm{p}(z|x) \propto \exp{\left( -\frac{1}{2\sigma^2} \norm[2]{x-z}^2\right)}.
\]
In this particular experiment, as in ~\cite{ch97,rof92}, we assume a prior~\footnote{Note that due to the lack of coercivity of $\TV(x)$ its associated prior is not well defined. However, by combining the prior with the likelihood function, the posterior is well-defined.} based on the total variation,
\[
\mathrm{p}(x) \propto \exp{\left( -\frac{\TV(x)}{\lambda}\right)},
\]
where $\lambda$ plays the role of a variance parameter of the prior. Combining the likelihood with the prior, the negative log-posterior respectively the variational model is given by
\begin{equation}\label{eq:tvl2}
-\log(\mathrm{p}(x|z)) = \frac{1}{2\sigma^2} \norm[2]{x-z}^2 + \frac{1}{\lambda} \TV(x) + const,
\end{equation}
where we recall that an additive constant term is irrelevant for sampling via Langevin algorithms as only evaluations of the gradient of $-\log(\mathrm{p}(x|z))$ are used. In what follows, we consider the anisotropic total variation
\[
\TV(x) = \sum_{i,j} |x_{i+1,j} - x_{i,j}| + |x_{i,j+1} - x_{i,j}| = \norm[1]{\mathrm{D} x},
\]
where $\mathrm{D} : \mathbb{R}^{M\times N} \to
\mathbb{R}^{M\times N\times 2}$ is a suitable finite differences operator, defined by
\begin{equation} \label{eq:defDx}
\begin{split}
(\mathrm{D} x)_{i,j,1} &= \begin{cases}
x_{i+1,j} - x_{i,j} & \text{if } 1 \leq i < M,\\
0 & \text{else}, \end{cases} \\*
(\mathrm{D} x)_{i,j,2} &= \begin{cases}
x_{i,j+1} - x_{i,j} & \text{if } 1 \leq j < N,\\
0 & \text{else}.
\end{cases}
\end{split}
\end{equation}
See for example~\cite{chpo16} for further information. The anisotropic total variation is chosen in order to render the problem amenable for the BP algorithm presented in \Cref{sec:markov_random_fields}. While we make use of the novel ULPDA, \Cref{algo:pd_langevin}, for the error estimation, in the following paragraph we first confirm functionality and estimate hyperparameters of this novel algorithm by a comparison to the highly accurate results obtained with BP.
\begin{figure}[htb]
\centering
\includegraphics[width=0.7\linewidth]{ures_denoising_1e-4_BSDS_stats_TV.pdf}
\label{fig:empirical_dist_TV_l2}
\caption{TV-$\ell_2$ Denoising $\sigma = 15/255$. The left figure shows the joint log density of error S and estimated variance $\hat{T}$, while the right figure shows the conditional cumulative distribution for the error.
The black line indicates the conformalized 0.9 quantile.}
\end{figure}
\paragraph{A Comparison of ULPDA and Belief Propagation}
For BP to be applicable we have to discretize the TV denoising problem. Thus, we replace the continuous gray scale \emph{label} space $[0,1]$ by the discretization $l_k = k/L$, $k=0,...,L$. The node set $\mathcal{V}=\{(i,j)\;|\;i=1,\dots,M,\; j=1,\dots,N\}$ is the set of all image pixels and the edge set $\mathcal{E}$ the set of $4$-nearest neighbors on the pixel grid. In particular, in this setting the the log posterior probability factors nicely
\begin{equation}\label{eq:mrf_logdist}
\begin{aligned}
- \log \mathrm{P}(X=x|Z=z)= &- \log \mathrm{P}(Z=z|X=x)- \log \mathrm{P}(X=x) +const.\\
=&\sum_{\nu \in \mathcal{V}} g_{\nu}(x_\nu)+\sum_{(\nu, \nu') \in \mathcal{E}} f_{\nu, \nu'}(x_\nu,x_\nu') + const,
\end{aligned}
\end{equation}
where $g_\nu:\mathcal{L}\rightarrow \mathbb{R}$ are unary terms representing the data likelihood and the pairwise terms $f_{\nu, \nu'}:\mathcal{L}^2\rightarrow \mathbb{R}$ correspond to the prior. These are defined as
\[g_\nu(x_\nu) = \frac{1}{2\sigma^2}(x_\nu-z_\nu)^2,\quad f_{\nu, \nu'}(x_\nu, x_\nu') = \frac{1}{\lambda}|x_\nu-x_\nu'|.\]
In the following experiment we set the number of labels to $L=1024$ and use the Sweep Belief Propagation algorithm with 10 iterations to obtain the respective results.
In \Cref{fig:bp_comp_abs}, for a noisy observation $z$ with $\sigma=15/255$, on the left side we show the mean absolute difference $MD(\hat{x}_{BP},\hat{x}_{LV,k})=\frac{1}{MN} \sum_{i,j} |\hat{x}_{LV,k,ij}-\hat{x}_{BP,ij}|$ of the MMSE computed with BP ($\hat{x}_{BP}$) to the one computed with ULPDA ($\hat{x}_{LV,k}$) for different values of $\tau$ in dependence on the number of samples $K$ of ULPDA. On the right hand side we compare the posterior variances $\hat{t}_{BP}$ and $\hat{t}_{LV,k}$ in the same manner.
Additionally, we compare with the results of a smooth approximation of the total variation by Huber TV~\cite{chpo16} where we use the ULA scheme \Cref{algo:unaj_langevin} for sampling.
As \Cref{fig:bp_comp_abs} indicates, the UPDLA results converge faster for primal step sizes of $\tau >\expnumber{1}{-5}$.
Based in this results, for subsequent experiments with UPDLA, a primal step size $\tau=\expnumber{5}{-5}$ and $K = 50.000$ iterations is used since this yields a good tradeoff between approximation accuracy and computational cost. For the task of denoising, we further found that no burn in phase is necessary since a steady state is acquired within a few iterations.
The qualitative visual comparison between the BP and ULPDA results for the favoured hyperparameters, depicted in \Cref{fig:bp_comp_abs_vis}, show that no visual deviation can be observed between $\hat{x}_{LV}$ and $\hat{x}_{BP}$.
The same holds for the estimated posterior variances $\hat{t}_{LV}$ and $\hat{t}_{BP}$. A visualization of the pixel marginals obtained with the two methods can be found in \Cref{fig:margs}.
\paragraph{Results for Error Estimation}
In \Cref{fig:empirical_dist_TV_l2} we show the resulting estimated distribution of $(S,\hat{T})$ obtained from posterior sampling with ULPDA. Exemplary reconstruction results alongside the pixelwise estimated quantile are shown in \Cref{fig:rec_1e-4}. The quantitative results for this experiment can be found in \Cref{tab:quantquant}.
\begin{figure}[htb]
\centering
\resizebox{0.8\textwidth}{!}{%
\includegraphics[scale=1]{article-figure1.pdf}
}
\caption{TV-$\ell_2$ Denoising. Reconstruction results for samples from the BSDS dataset and a noise level of $\sigma = 15/255$.
From left to right: ground truth image~$x$, observation~$z$, reconstruction~$\hat{x}_{LV}$, true error~$s$, and predicted $0.9$ quantile $\hat{s}_{0.9}$ ($0$ \protect\includegraphics[width=1.5cm,height=.2cm]{ures_viridis.png} $0.02$).}
\label{fig:rec_1e-4}
\end{figure}
\subsection{Fields-of-Experts-$\ell_2$ Denoising}\label{sec:foe_denoising}
In this section we consider the task of denoising again, but replace the total variation regularization by a more complex, non-convex data driven prior, the so-called Fields of Experts (FoE) \cite{ro09} which reads as
$\mathrm{p}(x)~\propto~\exp{\left( -\frac{FoE(x)}{\lambda}\right)}$ with
\[
FoE(x) = \sum_{c=1}^C\sum_{i=1}^{M}\sum_{j=1}^{N} \phi_c((k_c * x)_{i,j}),
\]
$C \in \mathbb{N}$ experts of the form $\phi_c(t) =\alpha_c \log (1+t^2_i)$ with $\alpha_c$ and 2D convolution kernels $k_c$ being the trainable parameters of the method. The respective negative log-posterior reads as
\begin{equation}\label{eq:FoEl2}
-\log(\mathrm{p}(x|z)) = \frac{1}{2\sigma^2} \norm[2]{x-z}^2 + \frac{1}{\lambda} FoE(x) + const.
\end{equation}
Since the described prior is non-convex, the ULPDA algorithm is not applicable. However, sampling from the posterior is still possible using the ULA algorithm described \cref{algo:unaj_langevin}
\begin{figure}[htb]
\centering
\includegraphics[width=0.7\linewidth]{ures_denoising_FoE_1e-4_BSDS_stats_FoE.pdf}
\label{fig:foe_estimated_distribution}
\caption{FoE-$\ell_2$ Denoising $\sigma = 15/255$. The left figure shows the joint log density of error S and estimated variance $\hat{T}$, while the right figure shows the conditional cumulative distribution for the error.
The black line indicates the conformalized 0.9 quantile.}
\end{figure}
The FoE prior used for this experiment was trained on the BSDS dataset in a bi-level optimization scheme as presented in~\cite{ch14}.
The ULA algorithm is performed with 50k iterations. A step size of $\tau=\expnumber{1}{-4}$ and a regularization parameter of $\lambda = 0.125$ were determined empirically.
As before we use 400 images from the BSDS data set to estimate the distribution of $(S,\hat{T})$ and evaluate on a test set of 68 unseen images.
The estimated error-variance distribution can be found in \Cref{fig:foe_estimated_distribution}. Qualitative results are shown in \Cref{fig:rec_Foe_1e-4} and quantitative results in \Cref{tab:quantquant} again.
\begin{figure}[htb]
\centering
\resizebox{0.8\textwidth}{!}{%
\includegraphics[scale=1]{article-figure2.pdf}
}
\caption{FoE-$\ell_2$ Denoising. Reconstruction results for samples from the BSDS dataset and a noise level of $\sigma = 15/255$.
From left to right: ground truth image~$x$, corrupted image~$z$, reconstruction~$\hat{x}_{LV}$ , true error~$s$, and predicted $0.9$ quantile $\hat{s}_{0.9}$ ($0$ \protect\includegraphics[width=1.5cm,height=.2cm]{ures_viridis.png} $0.02$).}
\label{fig:rec_Foe_1e-4}
\end{figure}
\subsection{Total Deep Variation-$\ell_2$ Denoising}\label{sec:tdv_denoising}
The last experiment in the additive Gaussian denosing setting is performed using the Total Deep Variation (TDV)~\cite{koef21} regularizer.
As in the previous experiment, this regularizer is a non-convex data driven prior, however, orders of magnitude more powerful in terms of its approximation capability due to the greater amount of trainable parameters.
Formally, the prior is defined as $\mathrm{p}(x) \propto \exp{\left( -\frac{TDV(x)}{\lambda}\right)}$ with
\[
TDV(x) = \sum_{i=1}^{M}\sum_{j=1}^{N} \Psi(x)_{i,j},
\]
where $\Psi:\mathbb{R}^{M\times N}\to\mathbb{R}^{M\times N}$ is a U-Net inspired convolutional neural network. We refer the reader to~\cite{koef21} for a more detailed description of the TDV. As in the prevoius experiment, we use the ULA for sampling. Values of $\tau =\expnumber{1}{-3}$, $\lambda =\frac{1}{5.7}$ and $\beta = 150$ have empirically proven to yield satisfactory results.
Note that the natural choice of $\beta = \frac{1}{\sigma^2}$ was neglected, since $\beta$ was subject to optimization in the training of the TDV and must therefore be chosen accordingly.
As before, we use 50k samples of the respective Langevin algorithm.
The data setup is the same as in the previous experiment.
The estimated error-variance distribution is shown in \Cref{fig:TDV_estimated_distribution}, qualitative results in \Cref{fig:rec_TDV}, and quantitative ones again in \Cref{tab:quantquant}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.7\linewidth]{ures_denoising_TDV_BSDS_stats_TDV.pdf}
\label{fig:TDV_estimated_distribution}
\caption{TDV-$\ell_2$ Denoising $\sigma = 15/255$. The left figure shows the joint log density of error S and estimated variance $\hat{T}$, while the right figure shows the conditional cumulative distribution for the error.
The black line indicates the conformalized 0.9 quantile.}
\end{figure}
\begin{figure}[htb]
\centering
\resizebox{0.8\textwidth}{!}{%
\includegraphics[scale=1]{article-figure3.pdf}
}
\caption{TDV-$\ell_2$ Denoising. Reconstruction results for samples from the BSDS dataset and a noise level of $\sigma = 15/255$.
From left to right: ground truth image~$x$, corrupted image~$z$, reconstruction~$\hat{x}_{LV}$ , true error~$s$, and predicted $0.9$ quantile $\hat{s}_{0.9}$ ($0$ \protect\includegraphics[width=1.5cm,height=.2cm]{ures_viridis.png} $0.02$).}
\label{fig:rec_TDV}
\end{figure}
\subsection{MRI Reconstruction}\label{sec:mri_recon}
We now consider an inverse problem with a non-trivial forward operator and reuse the same TDV as in the previous section as a prior. The aim of accelerated MRI reconstruction is to recover an image $x \in \mathbb{R}^{M\times N}$ from a complex-valued observation $z \in \mathbb{C}^{M\times N}$ representing the undersampled k-space data. The relation of image and k-space data is given by the linear operator ${A:\mathbb{C}^{M\times N} \rightarrow\mathbb{C}^{M\times N}}$ composed of the two-dimensional discrete Fourier transform $\mathcal{F}:\mathbb{C}^{M\times N}\rightarrow \mathbb{C}^{M\times N}$ and an undersampling operator $M:\mathbb{C}^{M\times N}\rightarrow \mathbb{C}^{M\times N}$, together with additive measurement noise $\nu$, formally expressed as
\begin{equation}
z = Ax + \nu = M\mathcal{F}x + \nu.
\label{eq:forward}
\end{equation}
Since the gradient step on the negative log-likelihood can be efficiently solved as proximal mapping, we use a proximal version of the ULA namely P-ULA~\cite{pe16} in this experiment. %
The proximal map of the negative log-likelihood reads as
\[
\prox_{\tau_k\mathcal{D}}(\Tilde{x})=\text{real}(\mathcal{F}^{-1}((\mathrm{Id}+\tau_k\beta M^\ast M)^{-1}(\mathcal{F}\Tilde{x}+\tau_k \beta M^\ast z))).
\]
Note that the TDV used in this setup is identical to the one used in the previous section and is therefore not specifically trained for the purpose of accelerated MRI reconstruction.
In contrast to the denoising experiments, the choice of the data-likelihood weighting is not clear, due to the linear operator.
As in \cite{narnhofer2021bayesian} we use a high value for the weighting parameter $\beta=\expnumber{1}{7}$ which almost leads to a projection on the acquired k-space lines due to the proximal map.
The regularization parameter was set to a value of $\lambda = \frac{1}{15}$.
Another difference to the denoising experiments is that we conduct a burn in phase of 500 iterations to acquire a steady state in this setup.
We used 420 central sclices from the fastmri knee multicoil training dataset \cite{zb18} with CORPD contrast to generate real valued groundtruth images from the root-sum-of-squares solution, which is further used to estimate the distribution of $(S, \hat{T})$.
Evaluation was performed on a test set of 100 images from the respective validation dataset. The estimated joint distribution of error and variance is shown in \Cref{fig:joint_distribution_mri}, qualitative results of the method in \Cref{fig:CORPD41}, and quantitative ones in \Cref{tab:quantquantMRI}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.7\linewidth]{ures_MRI_TDV_MRI_stats_TDV.pdf}
\label{fig:joint_distribution_mri}
\caption{TDV-$\ell_2$ 4-fold MRI reconstruction. The left figure shows the joint log density of error S and estimated variance $\hat{T}$, while the right figure shows the conditional cumulative distribution for the error.
The black line indicates the conformalized 0.9 quantile.}
\end{figure}
\begin{figure}[htb]
\centering
\resizebox{0.8\textwidth}{!}{%
\includegraphics[scale=1]{article-figure4.pdf}
}
\caption{MRI reconstruction results for CORPD data and $R=4$.
From left to right: ground truth image~$x$, zero filling~$z$, reconstruction~$\hat{x}_{LV}$ , true error~$s$, and predicted $0.9$ quantile $\hat{s}_{0.9}$~($0$~\protect\includegraphics[width=1.5cm,height=.2cm]{ures_viridis.png}~$0.06$)}
\label{fig:CORPD41}
\end{figure}
\subsection{Comparison to the State of the Art}\label{sec:comp}
In this section we compare the proposed method to an existing method for error estimation in imaging. While there are several methods available for uncertainty quantification, a fair comparison can only be made to other methods also estimating the prediction error.
\paragraph{The Comparison Method} Given a heuristic algorithm predicting pixel wise confidence intervals, in \cite{angelopoulos2022image} the authors propose a method of calibrating said algorithm in order to obtain guaranteed coverage. More precisely, assume we are given a heuristic predictor $\mathcal{T}:\mathcal{Z}\rightarrow \mathbb{R}^{3\times N\times M}$, $\mathcal{T}(z)=(\hat{l}(z),\hat{x}(z),\hat{u}(z))$, where $\hat{x}$ is a point estimate of the ground truth image and $\hat{l}$, $\hat{u}$ denoting lower and upper bounds of the heuristic pixel-wise confidence intervals, satisfying $\hat{l}_{i,j}(z)<\hat{x}_{i,j}(z)<\hat{u}_{i,j}(z)$ for all pixels $(i,j)$. Define a scaled interval predictor for $\lambda>0$ as $\mathcal{T}_\lambda:\mathcal{Z}\rightarrow \mathbb{R}^{3\times N\times M}$,
\begin{equation}
\begin{aligned}
&\mathcal{T}_\lambda(z)=(\hat{l}_\lambda(z),\hat{x}(z),\hat{u}_\lambda(z)),\\
&\hat{l}_\lambda(z) = \hat{x}(z)+\lambda(\hat{l}(z)-\hat{x}(z)),\\
&\hat{u}_\lambda(z) = \hat{x}(z)+\lambda(\hat{u}(z)-\hat{x}(z)).
\end{aligned}
\end{equation}
Using a \emph{calibration} data set $(X_i,Z_i)_{i\in\mathcal{I}_\text{cal}}$, the parameter $\Lambda = \lambda((X_i,Z_i)_{i\in\mathcal{I}_\text{cal}})$ is chosen to ensure
\begin{equation}\label{eq:im2imuq}
\mathrm{P}\Bigg[\underbrace{\mathbb{E}\left[\frac{1}{NM}\left|\{(i,j)\;|\;X_{i,j}\in [l_\Lambda(Z)_{i,j},u_\Lambda(Z)_{i,j}]\}\right|\;\middle|\; (X_i,Z_i)_{i\in\mathcal{I}_\text{cal}}\right]}_\text{expected coverage conditioned on calibration data}\geq 1-\alpha\Bigg]\geq 1-\delta
\end{equation}
for $\alpha,\delta\in (0,1)$ defined arbitrarily by the user. That is, the method from \cite{angelopoulos2022image} ensures a bounded expected coverage on a new sample in probability over the calibration data. Two main differences to the proposed method are i) that in \cite{angelopoulos2022image} a method yielding a point estimate, a lower and an upper interval bound is needed already before the calibration, whereas for the proposed method a prior on the image data is sufficient, and ii) while the proposed method yields a direct probability on the coverage, in \eqref{eq:im2imuq} we obtain nesting of coverage and expectation. The benefit of the result in \eqref{eq:im2imuq} is that it allows for a prediction over the entire image, which comes at the cost of not knowing which pixels are the ones not correctly covered.
\paragraph{Experimental Setup}
For the comparison we consider again TV-$\ell_2$ denoising as in \Cref{sec:tvl2_denoising}. We apply the method from \eqref{eq:im2imuq} with the heuristic interval predictor defined via
\begin{equation}\label{eq:im2imheur}
\hat{l}(z) = \hat{x}(z)-\sqrt{\hat{t}(z)},\quad \hat{u}(z) = \hat{x}(z)+\sqrt{\hat{t}(z)}.
\end{equation}
with $\hat{t}(z)$ the estimated posterior variance. While in \eqref{eq:im2imuq} the authors present several different trained methods yielding heuristic interval predictors we decided to use \eqref{eq:im2imheur} since we think it allows for a reasonable comparison, focused on the calibration of the predictor ensuring coverage and rather than on the performance of the heuristic. Moreover, since the heuristic estimator would need to be trained on a distinct data set, additional sources of volatility with respect to the performance would be introduced via the training process. In order to obtain results comparable to our method, for the q-quantile we pick $\alpha$, $\delta$ performing a grid search minimizing the average estimated interval size while requiring that $(1-\alpha)(1-\delta)=q$ which is the most sensible choice according \eqref{eq:im2imuq} in order to obtain a coverage of approximately $q$ over all image pixels. The coverage results can be found in \Cref{tab:quant_comp} with a box plot comparing the interval sizes to the proposed method in \Cref{fig:boxcomp}. In this case the interval size of the proposed method is computed as $2\sqrt{\hat{s}_q}$, twice the estimated error, and the interval size of \cite{angelopoulos2022image} as $\hat{u}_\lambda-\hat{l}_\lambda$.
\begin{table}[htb]
\caption{Quantitative comparison of the proposed method and \cite{angelopoulos2022image} for BSDS 68 test data and the task of denoising with TV-$\ell_2$. Average coverage for different quantiles.}
\centering
\begin{tabular}{l|cc|cc|cc|cc}
{} & \multicolumn{2}{c|}{Coverage} & \multicolumn{2}{c|}{Discrepancy to quantile in \% } \\
\noalign{\smallskip}
\textbf{Quantile} &{Reference} & {Ours} & {Reference} & {Ours} \\ \hline\hline
0.85 & 0.8973 & 0.8444 & 4.73 & 0.6\\
0.90 & 0.9406 & 0.8951 & 4.06 & 0.49\\
0.95 & 0.9805 & 0.9465 & 3.05 & 0.35\\\hline
\end{tabular}
\label{tab:quant_comp}
\end{table}
\begin{figure}[htb]
\label{fig:boxcomp}
\centering
\includegraphics[width=0.7\linewidth]{ures_boxcomp_small.pdf}
\caption{Box plots of the estimated interval size on test data for different values of $q$. Comparison of the proposed method and the reference \cite{angelopoulos2022image}. Each individual box shows the respective estimated interval size, with the box representing the inter quantile range and the orange bar being the median.}
\end{figure}
\subsection{Evaluation of the Numerical Experiments}
In this section, we evaluate and discuss the results of the performed experiments. For this, we stress that the aim of the proposed work is not to obtain highly accurate reconstructions of the unknown ground truth, but rather to obtain tight bounds of the reconstruction error satisfying the desired coverage. First of all, note that quantitatively in all experiments coverage is obtained accurately, as can be seen in \Cref{tab:quantquant,tab:quantquantMRI}. The slight deviation from the predicted coverage despite the theoretical results only requiring a finite data set can be explained since, while our method only relies on a finite data set, the evaluation of probabilites, indeed, requires an infinite amount of data in order to be exact. Coverage, however, is guaranteed by the theoretical results for all methods anyway. Differences in performance should, thus, be measured rather in terms of the magnitude of the predicted error quantiles. In \Cref{fig:boxlog_denoising} we therefore show box plots of the estimated error quantiles for our denosing experiments. We can observe that all methods yield practically useful error estimates with magnitudes of the order $10^{-3}$ to $10^{-2}$ for images with gray scale values in $[0,1]$. Over all values of $q$, we can clearly note that the TDV prior yields the tightest estimates, followed by the FoE prior and lastly the TV prior. This trend is not surprising, as TDV is the most complex of the three models and TV clearly the simplest one. Let us focus now on a qualitative analysis of the results. In \Cref{fig:empirical_dist_TV_l2,fig:foe_estimated_distribution,fig:TDV_estimated_distribution,fig:joint_distribution_mri} we find that for all conducted experiments, the cumulative distribution of the reconstruction error conditioned on the posterior variance is mostly monotonically increasing, empirically supporting the predictive capability of the posterior variance for the error. Exceptions of monotonicity are observed only for extreme values of the variance in the form of oscillations and are most likely caused by a lack of data in these regions. Throughout all experiments, we observe that the estimated error quantiles structurally resemble the true reconstruction errors, see \Cref{fig:rec_1e-4,fig:rec_Foe_1e-4,fig:rec_TDV,fig:CORPD41}. In general, errors are concentrated at image edges which can be expected as most priors, especially TV type, penalize precisely image edges. The main differences between different priors are found in the heatmaps' sharpness around edges and overall smoothness and are a result of locality of the respective prior. The TV prior acts only on three neighbouring pixels resulting in sharp edges with the disadvantage of yielding noisy estimates with non-trivial error quantiles also in constant regions of the image, see \Cref{fig:rec_1e-4}. The FoE prior yields smoother error estimates where noise is only visible slightly in constant regions, \Cref{fig:rec_Foe_1e-4}. Again the best results are obtained with TDV, \Cref{fig:rec_TDV}, where error estimates are practically zero in constant regions of the image while still providing more detailed estimates than FoE (see the details of the castle) at the cost of some oversmoothing, see for instance the predicted error at the bush in the lower image.
Furthermore, the reconstruction in the second row of \Cref{fig:rec_TDV} shows a pixel wit a high error which was not detected by our method, which can be explained by the fact that we only guarantee a 90$\%$ chance of a respective error being below the estimated quantile.
The overall tendency of TDV yielding the best and FoE the second best results is also reflected in \Cref{tab:quantquant} in the respective values of the mutual information of $S$ and $\hat{T}$, which is a quantitative measure for the predictive capabilities of $\hat{T}$ wrt. $S$, see \Cref{sec:MI}. We observe a significant increase from TV to FoE and a further, smaller increase to TDV. For the sake of completeness we also added the PSNR and SSIM values in the tables. Regarding a comparison to the state of the art, our method outperforms the one proposed in \cite{angelopoulos2022image} in terms of interval sizes as shown in \Cref{fig:boxcomp}. Among all considered quantiles, the interval sizes with the proposed method tend to be statistically smaller with tighter coverage as shown in \Cref{tab:quant_comp}. It should, however, be mentioned that the method from \cite{angelopoulos2022image} might yield improved results using a more sophisticated heuristic interval predictor as explained in \Cref{sec:comp}. Moreover, coverage on a per pixel base cannot be prescribed for the method in \cite{angelopoulos2022image} and the applied choice of $\alpha,\delta$ from \eqref{eq:im2imuq} satisfying $(1-\alpha)(1-\delta)=q$ might not be optimal.
\begin{table}[htb]
\caption{Quantitative Results for different regularization approaches for denoising with $\sigma=15/255$ on the BSDS 68 dataset. Left: PSNR, SSIM, and mutual information (\Cref{sec:MI}) of $(S;T)$ for different experiments. Right: Coverage for different estimated quantiles and experiments.}
\centering
\begin{tabular}{l|c|c|c|c||c|c|c|c}
\noalign{\smallskip}
\hline
\backslashbox[25mm]{\footnotesize{\textbf{Method}}}{\footnotesize{\textbf{Metric}}} & PSNR & SSIM & I(S;T) & \backslashbox[25mm]{\footnotesize{\textbf{Method}}}{\footnotesize{\textbf{Quantile}}} & 0.85 & 0.9 & 0.95 & 0.99 \\ \hline\hline
Corrupted & 24.78 & 0.5820 & - & & & & & \\
TV-$\ell_2$ & 29.34 & 0.7913 & 0.1231 & TV-$\ell_2$ & 0.8444 & 0.8951 & 0.9465 & 0.9888 \\
FoE-$\ell_2$ & 30.29 & 0.8370 & 0.1609 & FoE-$\ell_2$ & 0.8447 & 0.8948 & 0.9459 & 0.9883 \\
TDV-$\ell_2$ & 30.79 & 0.8484 & 0.1650 & TDV-$\ell_2$ & 0.8527 & 0.9023 & 0.9516 & 0.9906 \\\hline
\end{tabular}
\label{tab:quantquant}
\end{table}
\begin{figure}[htb]
\label{fig:boxlog_denoising}
\includegraphics[width=\linewidth]{ures_boxlog_small.pdf}
\caption{Box plots of the estimated error quantiles for denoising on test data for different values of $q$ and different priors. Each individual box shows the respective distribution of errors, with the box representing the inter quantile range and the orange bar being the median.}
\end{figure}
\begin{table}[htb]
\caption{Quantitative Results for 4-fold accelerated MRI reconstruction with the TDV on the fastmri validation dataset.}
\centering
\begin{tabular}{l|c|c|c|c||c|c|c|c}
\noalign{\smallskip}
\hline
\backslashbox[25mm]{\footnotesize{\textbf{Method}}}{\footnotesize{\textbf{Metric}}} & PSNR & SSIM & I(S;T) & \backslashbox[25mm]{\footnotesize{\textbf{Method}}}{\footnotesize{\textbf{Quantile}}} & 0.85 & 0.9 & 0.95 & 0.99 \\ \hline\hline
Corrupted & 10.66 & 0.0145 & - &&&&&\\
TDV-$\ell_2$ & 34.79 & 0.8812 & 0.1554 & TDV-$\ell_2$ & 0.8461 & 0.8966 & 0.9478 & 0.9895 \\\hline
\end{tabular}
\label{tab:quantquantMRI}
\end{table}
\section{Conclusions}
\label{sec:conclusions}
In this work we propose a general framework for error estimation for inverse problems in imaging that is flexible with respect to the forward operator as well as the chosen reconstruction approach as long as posterior sampling is possible. Given any point estimate for the inverse problem and an i.i.d. sample, the method enables to estimate pixel-wise quantiles of the reconstruction error for an unseen sample. Coverage of the estimated error quantiles is guaranteed without assumptions on the underlying distributions. In various experiments we show that the proposed method works accurately for different regularization approaches as well as inverse problems. Accurate knowledge about possible reconstruction errors might be especially important for critical tasks such as medical imaging where medical experts base the patients treatment on reconstructed images and therefore have an option to revise diagnoses based on structures that clearly show a high error potential. In future work the mutual information $I(S;T)$ between posterior variance and error might become an interesting metric to quantify the predictive capability of the posterior variance for the reconstruction error of a specific method. Furthermore future approaches could consider a different choice of statistical entities, such as substituting the MMSE by the median and the variance by the median absolute deviation and canonically the mean squared error by the mean absolute difference.
\paragraph{Limitations}
From a theoretical point of view, the presented results guarantee the desired coverage for each pixel individually. However, they do not directly carry over to an assertion regarding coverage of all pixels in an image. In order to circumvent this issue, in \cite{angelopoulos2022image} the authors bound the expected value of the rate of correctly covered pixels in an image using methods of risk control. A side effect of this approach, however, is that it is unknown which pixels are correctly covered and which are not. Requiring all pixels of an image to be covered correctly, on the other hand, was observed to lead to inefficient quantile estimates in our experiments, which is why we decided to use the pixel wise approach. Moreover, from a practical point of view the amount of data is a limiting factor. For numerical experiments we used all pixels of an image for estimation despite the fact that in this setting, the sample might not be i.i.d. While theoretically not favourable, this heuristic did not show to be disadvantageous in empirical experiments.
\section*{#1}%
\else
\vspace{.05in}\footnotesize
\parindent .2in
{\upshape\bfseries #1. }\ignorespaces
\fi}
{\if@twocolumn\else\par\vspace{.1in}\fi}
\newenvironment{keywords}{\begin{@abssec}{\keywordsname}}{\end{@abssec}}
\newenvironment{keyword}{\begin{@abssec}{\keywordname}}{\end{@abssec}}
\newenvironment{MSCcodes}{\begin{@abssec}{\MSCname}}{\end{@abssec}}
|
1,116,691,500,731 | arxiv | \section*{Appendix}
The density of states $g(E)$ is calculated as
\begin{align} \label{TotalElectrons}
g(E) = \frac{1}{V} \sum_n \int_{V} \delta(E - \epsilon_{n,\bf{k}}) d\bf{k}
\end{align}
where $\epsilon_{n,\bf{k}}$ is the energy of band $n$ at k-point $\bf{k}$, and $V$ is the volume of the reciprocal primitive cell. The Gaussian and Fermi smearing methods approximate the $\delta(E - \epsilon_{n,\bf{k}})$ function in the following ways:
\noindent Gaussian smearing:
\begin{align}
\delta(E - \epsilon_{n,\bf{k}}) \approx \frac{1}{\sigma \sqrt{\pi}} e^{-\left( \frac{E - \epsilon_{n,\bf{k}}}{\sigma} \right)^2}
\end{align}
\noindent Fermi smearing:
\begin{align}
\delta(E - \epsilon_{n,\bf{k}}) \approx \frac{e^{-\frac{E - \epsilon_{n,\bf{k}}}{\sigma}} }{\sigma \left( 1 + e^{-\frac{E - \epsilon_{n,\bf{k}}}{\sigma}} \right)^2}
\end{align}
\section*{Acknowledgements}
We acknowledge support from NSF DMREF award \#1729487. MYT acknowledges support from the United States Department of Energy through the Computational Science Graduate Fellowship (DOE CSGF) under Grant Number DE-SC0020347. This research was supported in part through the computational resources and staff contributions provided for the Quest high performance computing facility at Northwestern University which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology. Work by AG, JP, and AJ was supported by the U.S Department of Energy, Office of Basic Energy Sciences, and the Early Career Research Program. JM and KAP acknowledge support from the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division under contract no. DE-AC02-05-CH11231 (Materials Project program KC23MP). This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DE-AC02-05CH11231.
\section{}
\subsection{}
\subsubsection{}
|
1,116,691,500,732 | arxiv | \section{Introduction}
The {\it Herschel Space Observatory} (Pilbratt et al.\ 2010) has provided us with high sensitivity and angular resolution maps of nearby galaxies in the far-infrared, which has allowed us to spatially resolve their cold dust emission. Dust plays a key role in the chemistry of the interstellar medium, acting as a catalyst for the formation of molecular gas, the fuel for star formation. However, it complicates our view of galaxies by obscuring UV and optical photons from stars and then re-radiating this light in the infrared. Thus, over a third of a galaxy's bolometric luminosity comes to us at these longer wavelengths (e.g. Draine et al.\ 2003;
Bernstein et al.\ 2002). \let\thefootnote\relax\footnote{$\dagger$ The first two lead authors have been co-equal contributors to the majority of the work presented in this paper.}
In the past it has been very difficult to detect the dust emission in galaxies between 200 and 850$\mu$m. Ground-based telescopes have lacked the sensitivity and, prior to {\it Herschel}, space telescopes could not make detections at these wavelengths.
Apart from the measurement of dust luminosities and masses, the maps from {\it Herschel}, spanning 70 to 500~$\mu$m, can be used to estimate two
important quantities:
the gas mass and the average intensity of the radiation field heating dust within galaxies. In the first case, since dust is generally well-mixed with gas,
we can use dust mass estimates inferred by SED fitting as a proxy for the total gas mass, assuming a gas-to-dust ratio (e.g. Hildebrand 1983, Boselli et al.\ 2002, Eales et al.\ 2010, Eales et al.\ 2012). While atomic gas measurements are relatively well-known due to 21 cm line measurements, molecular gas measures are more challenging since we are forced to use an alternative tracer like CO rather than directly measure molecular hydrogen.
This requires the calibration of this tracer, which is known to vary with environment, metallicity and density (e.g. Shetty et al.\ 2011).
{\it Herschel} maps have been used to help spatially resolve this calibration factor (i.e. the X-factor) improving our view of the molecular gas
component in galaxies (Sandstrom et al.\, sub.\ ). Typically dust emission maps also have higher resolutions, at least at the shorter wavelengths, and superior sensitivity to CO maps.
Dust emission SED fitting also provides a measure of the average radiation field energy density
heating the dust or, alternatively, the average dust temperature. These latter quantities are connected to the luminosity of the heating sources and are
therefore, in principle, useful to understand which radiation sources are heating the dust. In particular,
in order to use dust emission to infer star formation rates, one would like to quantify the fraction of dust heating due to recent star formation.
However, the dust can also be powered by an older stellar population (e.g.\ Popescu \& Tuffs 2002, Groves et al.\ 2012, Bendo et al.\ 2012, Smith et al.\ 2012).
{\it Herschel} has improved our view of the morphological structure of the dust emission and has allowed us to better separate compact sources from diffuse emission in some of the far-infrared bands.
We find both clumpy, compact regions associated with sites of recent star formation, detected also in H$\alpha$ emission, and a smoother,
more diffuse component (e.g.\ Verley et al.\ 2010). Although {\it Herschel} images have a resolution of 6$''$ at the shortest wavelengths (70$\mu$m), at
longer wavelengths it increases to 36$''$ at 500~$\mu$m. Therefore, to date most studies have been forced to degrade the high-resolution PACS (Photodetector Array Camera; Poglitsch et al.\ 2010) maps (70 and 160~$\mu$m) in order to match them to the longer
wavelength SPIRE (Spectral Photometric Imaging REceiver; Griffin et al.\ 2010) data (i.e.\ Bendo et al.\ 2010, Smith et al.\ 2012, Bendo et al.\ 2012, Boquien et al.\ 2012, Aniano et al.\ 2012). This has meant that these studies have averaged over large physical areas (i.e. pixel-by-pixel analysis at 36$''$ resolution or
azimuthal averages). Thus, the structure evident in the PACS maps is mostly lost. In this work, we attempt to capitalize on the
highest resolution PACS maps, while still making use of the lower resolution SPIRE maps, in order to study the compact, clumpy regions in the FIR continuum emission. Our method relies on the
multi-wavelength extraction tool, \textsc{getsources} (Men'shchikov et al.\ 2012), which allows us to preserve the native resolution of the images.
This paper uses this new technique on M83, a galaxy from the Very Nearby Galaxies Survey (PI: C.D. Wilson). The close
proximity of M83 (4.5 Mpc; Thim et al.\ 2003) affords us high spatial resolution (130 pc at 70~$\mu$m). M83 also has a strong spiral structure and
prominent bar, allowing us to investigate different regions within the galaxy.
The main goal of this work is the investigation of the properties of the compact sources detected in the FIR, in terms of gas masses, star formation
and dust heating. Specifically, we address the following questions: 1) Which range of giant molecular cloud masses are associated with the FIR compact
sources in M83?; 2) How efficient is the star formation within them?; and 3) Is the local star formation the predominant radiation source powering the dust
emission?
To this purpose, we developed a procedure consisting of the following steps. We use \textsc{getsources} to detect and extract compact regions from the FIR (70-350~$\mu$m),
which we call ``clumps''. We also measure the flux in the MIR (8 and 24~$\mu$m) and H$\alpha$ using the clump area and position. In all cases a local
background emission component is subtracted. For each well-detected source, the corresponding dust luminosity, dust mass, average radiation field energy density
and dust temperature are determined using a
two-component dust emission SED fitting method of the MIR and FIR. Using the dust mass we infer a gas mass for each source, using a constant gas-to-dust
ratio. We also derive the clump SFRs by applying the Calzetti et al.\ (2007) calibration on the measured H$\alpha$ and 24~$\mu$m fluxes.
We structure the paper in the following sections: background and
motivation (\S 2); observations and data reduction (\S3); compact
source extraction (\S 4); dust emission SED fitting (\S 5); star
formation rates and gas mass estimations; presentation of results (\S 7) and discussion (\S 8).
We conclude with a summary of the main findings (\S 9).
\section{Background \& Motivations}
\subsection{Gas \& Star Formation on sub-kpc scales}
During the last few years, high-resolution maps of nearby galaxies in several wavelengths including UV, IR, emission line and radio have become gradually available and opened the possibility to study the relation between gas mass surface density and star formation, known as Schmidt-Kennicutt law (hereafter S-K relation;
Kennicutt 1998; Schmidt 1959) on scales of kpc/sub-kpc within galaxies. The S-K relation is commonly expressed as
\begin{equation}
\Sigma_{\text{gas}} \text{ = A}\Sigma_{\text{SFR}}^{\text{N}}\text{.}
\end{equation}
The value of the exponent N is particularly important and remains debated. N=1 implies that the star formation efficiency (SFE) or gas depletion timescale
is constant, while N$>$1 implies that the SFE is greater for high-density regions, or that the timescale is shorter for high-density regions.
While global averages across galaxies have shown that the total gas mass (HI + H$_{2}$) surface density has a power law relation with the SFR surface density
(i.e. N=1.4; Kennicutt et al.\ 1989), spatially resolved studies of star formation have shown that the SFR surface density is more tightly coupled to the
molecular gas surface density (e.g.\ Wong \& Blitz 2002 and Bigiel et al.\ 2008).
Reliable measurements of SFR within galaxies can be performed by combining optical/UV and infrared flux measurements, according to calibrations such as Calzetti et al.\ (2007) or
Zhu et al.\ (2008) for H$\alpha$+24$\mu$m and Bigiel et al.\ (2008) for UV+24$\mu$m. In these empirically based calibrations, dust emission is used to estimate the fraction of optical/UV light attenuated by dust. These calibrations are now considered the most accurate way to determine SFRs within galaxies, at least when used on galaxy regions which are bright enough (SFR $>=$ 0.001 M$_{\odot}$ yr$^{-1}$, Kennicutt \& Evans 2012).
Spatially resolved studies of the SFR typically involve aperture photometry of the SFR tracers. The size of the aperture has been primarily dictated by the beam sizes. Typically background emission is subtracted from the SFR tracers either locally, in annuli directly surrounding the apertures, or, in the case of crowded regions, more extended zones are used (Calzetti et al.\ 2005).
Background subtraction is typically performed because the star formation tracers have contamination from other sources not directly
associated with star formation$^1$\footnote{ $^1$The relative amount of background emission that needs to be removed is still the subject of much debate. See e.g. Leroy et al.\ (2012) for a discussion on the nature of the diffuse MIR emission.}.
The 24$\mu$m emission includes cirrus emission from an older stellar population (Popescu \& Tuffs 2002), which may contribute as much
as 30\% of the emission (Kennicutt et al.\ 2007). H$\alpha$ emission may also include diffuse ionized gas (Ferguson et al.\ 1996). Background subtraction has proven to have effects on the S-K relation. Subtracting a diffuse component more readily suppresses the fainter star forming regions (Liu et al.\ 2011). Thus, studies that have used aperture photometry on star forming regions and background subtraction have found a super-linear relation (e.g. Kennicutt et al.\ 2007). Meanwhile pixel-by-pixel analyses that have not subtracted a diffuse component have recovered a linear relation (e.g. Bigiel et al.\ 2008).
The measures of star formation tracers are quite advanced due to the relatively high resolutions available but tracing gas in extragalactic studies has
proven to be more complicated. While measurements of the atomic gas component are available by using the 21 cm line emission (e.g.\ THINGS; Walter et al.\ 2008),
cold molecular hydrogen has proven more challenging.
Due to the low mass of the molecule, it requires high temperatures to excite the rotational transitions of molecular hydrogen. Thus, it is virtually
impossible to measure the total amount of molecular hydrogen directly.
Instead, an alternative tracer like CO is commonly used (Bolatto et al.\ 2013). Typically conversions from CO are done using a constant X-factor. However, this factor can vary considerably depending upon the metallicity, density and temperature (Kennicutt \& Evans 2012). Accurate measures of the molecular component associated with star formation are particularly important, because it seems that star formation is more directly coupled to molecular gas than to total gas.
Due to the comparatively low resolution and sensitivity of the molecular gas maps, only few studies (e.g. Rahman et al.\ 2011) have subtracted diffuse emission from the gas.
However, diffuse gas unrelated to star formation sites is most likely present. Low mass clouds that are unresolved will present themselves as a diffuse
component and may not host star formation. Ideally, one should also account for a diffuse gas component.
Furthermore, there is growing evidence that low mass GMCs may not have a corresponding star forming region (e.g. Hirota et al.\ 2011).
In light of the issues surrounding the molecular gas measures we are
motivated to capitalize on the high resolution dust emission maps from
{\it Herschel}.
By employing a gas-to-dust ratio we present an alternative way to probe the star formation efficiency on spatially resolved scales.
Inspired by the aperture photometry studies of star forming regions, we employ the dust maps to isolate and extract clumpy regions which are likely associated with
individual or groups of molecular clouds.
For both the star formation tracers and the dust we can also subtract a diffuse background component, thus treating both the star formation and gas tracers in a
similar fashion.
\subsection{Dust heating within galaxies}
In general, dust emission within galaxies without AGNs is powered by
radiation coming from both sites of recent star formation and from
more evolved stellar populations. However, there is a longstanding
debate about the exact fraction of dust heating contributed by each
stellar population (e.g. Law et al.\ 2011, Boquien
et al.\ 2011, Bendo et al.\ 2012, for recent references), which depends on several factors: the intrinsic emission spectral energy distribution
of the stellar populations, the dust mass and optical properties,
the relative dust-star geometry.
Recent observational works investigating the origin of the radiation
heating the dust in nearby galaxies have looked for correlations
between the source dust temperature (or, alternatively, FIR colour) and
source stellar
populations luminosities, as traced by SFR for the young stars and NIR
luminosity for the old stars (e.g. Bendo et al.\ 2012, Boquien et al.\
2011, Smith et al.\ 2012, Foyle et al.\ 2012).$^2$\footnote{ {$^2$ The quantities typically considered are \textit{surface densities}
of SFR and NIR luminosity rather than total values. For pixel-by-pixel analyses this amounts to multiplication with a constant,
thus it does not affect the nature of the correlation with dust temperature.}}
The presence of an observed correlation of this kind has often been used as evidence that a
particular stellar population is the dominant source heating the
dust. Although it might seem intuitive that a higher dust temperature
should correspond to a stellar population with a higher intrinsic luminosity,
deducing the origin of the radiation heating the dust in this way can
be potentially misleading. The reason is that the intrinsic luminosity
of a stellar population and the average dust temperature are not
necessarily proportional to each other, even in the case where the
stellar population in question is the dominant source of dust heating.
The luminosity of a stellar population gives the amount of total radiative energy per unit time injected by a stellar population locally in the
interstellar medium. Therefore, one can expect at most a
proportionality between the observed dust luminosity and the intrinsic stellar
population luminosity for a set of sources, provided all the sources
are mainly heated locally by the same kind of stellar population and heating from radiation
sources external to the areas considered is negligible$^3$\footnote{$^3$The latter assumption can easily break down when considering arbitrary
galactic regions. For the compact FIR sources associated with star formation regions, the heating is usually thought to be dominated by the local young
stellar populations but see further discussion in \S8.3.}.
Specifically, one can express the
total dust luminosity L$_{\rm{dust}}$ for a source as:
\begin{equation}
{L}_{\rm{dust}}={L}_{\rm{stars}}^{\rm{int}}\left(1-e^{-\tau}\right)
\label{ldusteq}
\end{equation}
where $L_{\rm{stars}}^{\rm{int}}$ is the total intrinsic luminosity
from a certain stellar population and $\tau$ is the source luminosity-weighted
optical depth. The optical depth can be expressed as:
\begin{equation}
e^{-\tau}=\frac{L_{\rm{stars}}^{\rm{out}}}{L_{\rm{stars}}^{\rm{int}}}
\end{equation}
where $L_{\rm{stars}}^{\rm{out}}$ is the escaping unabsorbed source stellar
population luminosity. All the aforementioned luminosities are integrated over
wavelength spanning the entire emission spectral range.
From Eq. \ref{ldusteq} one can see that, if $\tau$ has similar values for a sample of sources, a correlation between L$_{\rm{dust}}$
and $L_{\rm{stars}}^{\rm{int}}$ will be found. Note also that
Eq.\ref{ldusteq} implies that $L_{\rm{dust}}$ cannot exceed
$L_{\rm{stars}}^{\rm{int}}$. If this is
observed for a sample of sources, it would mean that the
dust is significantly heated by radiation coming from another stellar
population beyond that associated with $L_{\rm{stars}}^{\rm{int}}$.
In order to have a proportionality between the source intrinsic stellar luminosity and the source average dust temperature, an additional
assumption is required: each of the detected sources should be
associated with a similar amount of dust mass, $M_{\rm{dust}}$. Assuming that the dust emission can be described by a modified blackbody function such that $\kappa_{\nu}B_{\nu}(T_{\rm{dust}})$,
where $\kappa_{\nu}\propto \nu^\beta$ is
the absorption coefficient, $\beta$ is the emissivity, $B_{\nu}$ the Planck function and $T_{\rm{dust}}$ the dust temperature, it can be shown that
$L_{\rm{dust}} \propto M_{\rm{dust}}\int{\kappa_{\nu}B_{\nu}(T_{\rm{dust}})d\nu} \propto M_{\rm{dust}}T_{\rm{dust}}^{4+\beta}$.
By combining the latter relation with Eq.~\ref{ldusteq}, it follows that:
\begin{equation}
\frac{L_{\rm{stars}}^{\rm{int}}}{M_{\rm{dust}}} \propto \frac{T_{\rm{dust}}^{4+\beta}}{\left(1-e^{-\tau}\right)}
\label{lstarequ}
\end{equation}
If $M_{\rm{dust}}$ and $\tau$ have similar values for all the sources considered, the average intensity of the radiation heating the dust approximately
scales only with the
stellar luminosity and, as a consequence, the dust temperature $T_{\rm{dust}}$ is directly related only to $L_{\rm{stars}}^{\rm{int}}$ through
Eq.~\ref{lstarequ}.
In this case, it is likely that a specific stellar population is heating the dust, if one finds that the intrinsic
luminosity of that stellar population correlates with the observed dust temperature.
However, the amount of dust mass, the dust-star geometry and, therefore, the value of $\tau$, can be substantially different for each source.
Thus, in general, it cannot be expected that the luminosity of a stellar population is correlated with the dust temperature, even if that stellar
population is responsible for the dust heating.
Eq.~\ref{ldusteq} and \ref{lstarequ} are almost equivalent to each
other and they both express the relation between the dust luminosity and stellar
population luminosity. The only difference is that $L_{\rm{dust}}$ in Eq.~\ref{ldusteq}
includes all the dust and PAH emission throughout the entire infrared range. In Eq.~\ref{lstarequ}, we assumed that the dust emission can be
modelled by a single modified blackbody curve. This is a good approximation if one considers only the FIR region of the emission spectra. However, usually
the FIR luminosity is the dominant contribution to $L_{\rm{dust}}$.
Thus, Eq.~\ref{lstarequ} can be used to probe the extent to which a given stellar population is powering the dust emission.
We can do so by comparing the local, cold dust temperature with a measure of the local stellar population luminosity divided by the corresponding dust mass.
As before, if the value of $\tau$ is varying in a relatively small range, a correlation between $L_{\rm{stars}}^{\rm{int}}/{M_{\rm{dust}}}$ and $T_{\rm{dust}}$
will be
observed if the stellar population considered is responsible for
heating the dust.
For example, one can use the SFR value as a tracer of the UV luminosity of the young stellar populations associated with each source.
Therefore, if the radiation from young stars dominates the dust heating, the SFR/$M_{\rm{dust}}$
ratio is expected to be more tightly coupled to the dust temperature than the SFR alone.
Thus, an observed correlation between SFR/$M_{\rm{dust}}$ and $T_{\rm{dust}}$ would suggest that star formation is powering the observed
dust emission.
Given this, we are motivated to include a comparison of the measured SFR/$M_{\rm{dust}}$ (or, equivalently, SFE if one assumes a constant dust-to-gas mass ratio)
with $T_{\rm{dust}}$ for the sources
we detect, in addition to simply a comparison of the SFR and dust temperature, which is typically seen in the literature.
The results of these comparisons are shown in Sect. \S7 and discussed in Sect. \S8.
\section{Observations}
In this work we use the {\it Herschel} far-infrared maps of M83 from the Very Nearby Galaxies Survey (PI: C. D. Wilson) as well as ancillary mid-infrared
and H$\alpha$ maps to trace dust/PAH emission and star formation rate. Fig.~\ref{maps} shows M83 at each of the wavebands considered.
\subsection{Far-Infrared Images}
We use far-infrared (FIR) images from the {\it {\it Herschel} Space Observatory} to trace cold dust emission.
We use 70 and 160~$\mu$m maps taken with the Photodetector Array Camera (PACS; Poglitsch et al.\ 2010) and 250 and 350~$\mu$m maps taken with the
Spectral Photometric Imaging REceiver (SPIRE; Griffin et al.\ 2010)$^{4}$\footnote[4]{$^{4}$We performed an initial test including the 500~$\mu$m map as well, but we found
that the uncertainties on the source fluxes were so large that they were not useful to constrain the source SEDs. This was caused by the
low resolution of the 500~$\mu$m map (PSF FWHM=36$''$). Thus, we decided to not include the 500~$\mu$m map in our analysis.}.
The PACS images are processed using both \textsc{hipe} v5 and \textsc{scanamorphos} v8
(Roussel 2012) and the SPIRE images are processed with \textsc{hipe} and \textsc{brigade} (Smith et al.\ 2012). The PACS images were corrected from the v5 photometric calibration files to v6 with corrective factors of 1.119 and 1.174 for the 70- and 160-~$\mu$m maps, respectively. The SPIRE images are
multiplied by 0.9828 and
0.9839 for the 250 and 350~$\mu$m maps respectively in order to convert from monochromatic intensities of point sources to monochromatic extended sources. The images are kept in their native
resolution with a FWHM of the PSF of 6.0$''$, 12.0$''$, 18.2$''$ and 24.5$''$ for the 70, 160, 250 and 350~$\mu$m maps, respectively. For more details on how the images were processed we refer to
Bendo et al.\ (2012) and Foyle et al.\ (2012, hereafter F12).
\subsection{Mid-Infrared Images}
We trace the warm dust and PAH emission using mid-infrared maps (MIR) taken from the {\it Spitzer} Local Volume Legacy Survey (Dale et al.\ 2009).
Specifically, we use the 8 and 24~$\mu$m maps from IRAC and MIPS instruments. We subtract the stellar component of the emission in the 8~$\mu$m map
using a scaling of the IRAC 3.6~$\mu$m map, according to the relation provided by Helou et al.\ (2004): $F_{\nu}$(8 $\mu$m, dust) = $F_{\nu}$(8 $\mu$m)-0.232$F_{\nu}$(3.6 $\mu$m).
In the 24$\mu$m map the nuclear region is saturated, so this region is excluded from our analysis. Because of reasons explained in \S4, we degrade the resolution of the MIR maps to 6$''$ in order to match the resolution of the 70~$\mu$m map.
\subsection{H$\alpha$ Images}
We use continuum subtracted H$\alpha$ maps from the Survey for Ionization in Neutral Gas Galaxies (SINGG; Meurer et al.\ 2006).
We correct the H$\alpha$ maps for Galactic extinction using a factor 1.167 from the NASA/IPAC Extragalactic Database that is based on Schlegel et al.\ (1998).
As for the MIR maps, we degrade the H$\alpha$ map to a resolution of 6$''$ matching that of the 70~$\mu$m map.
We use the H$\alpha$ maps in conjunction with the 24~$\mu$m MIR map, in order to measure the star formation rate (SFR) of the extracted compact sources
(discussed in greater detail in \S6).
\begin{figure*}
\centering
\includegraphics[trim=0mm 0mm 0mm 0mm,width=180mm]{fig_allmaps_m83.eps}
\caption{M83 at different wavelengths at the resolution used in this study denoted by the white circle in the bottom right of each panel. Upper row: H$\alpha$ from SINGG survey and {\it Spitzer} IRAC 8 and MIPS 24~$\mu$m maps. Lower row: {\it Herschel} 70, 160, 250 and 350~$\mu$m maps.}
\label{maps}
\end{figure*}
\section{Compact Source Extraction}
In order to extract compact sources in the FIR maps (70-350~$\mu$m), we use the new
multi-scale, multi-wavelength tool, \textsc{getsources} (Men'shchikov
et al.\ 2012). \textsc{getsources} is specifically designed to work
with the FIR images of {\it Herschel}. The data from {\it
Herschel} spans a range of angular resolutions from 6.0$''$ to 36.0$''$ and, thus, any source extraction code must be able to handle these extremes, which poses concerns for source blending at longer wavelengths.
Rather than extracting sources directly from observed images (i.e. \textsc{gaussclump}; Stutzki \& Gusten 1990, \textsc{clumpfind}; Williams et al.\ 1994 or \textsc{sextractor}; Bertin \& Arnouts 1996), \textsc{getsources} analyzes spatial decompositions of the images across different scales and different wavelengths. Wavelength independent images are generated to detect sources at each spatial scale. The original images are then used to perform photometry on the detected sources.
This procedure takes into account the different angular resolutions of the maps, background subtraction and source blending.
We briefly describe the key steps of the process here, but refer the reader to Men'shchikov et al.\ (2012) for a detailed description on
how \textsc{getsources} extracts and measures the properties of compact sources, including a comparison with other similar codes.
After the images are aligned to the same spatial grid, \textsc{getsources} decomposes of the original maps into single scale detection images.
This is done by using a
process of successive unsharp masking, where the original images are convolved with Gaussians and subtracted successively, in order to enhance
the visibility of emission on different scales. The FWHM of the Gaussians
varies between twice the pixel size to a maximum of 18 times the resolution of the image or the image size. The image resolution is the only information
the user needs to provide. The background and noise in the single scale
detection images is then removed by intensity thresholding. The clean single-scale detection images at each wavelength are then combined into single-scale
wavelength-independent detection images, allowing for the use of all the information across all bands simultaneously for the source detection. Many detection codes rely on independent catalogues at each waveband which are then matched using an
association radius, which can introduce large unknown errors. By dividing the images into single scales, this process can be avoided.
On the combined single-scale detection images, a given source will appear at a small scale and gradually get brighter until it is seen at a scale roughly the true
size of the source. Beyond this, the source begins to vanish again. \textsc{getsources} tracks the evolution of the source through the spatial scales and
creates source masks to identify the sources. The scale where the source is brightest provides an initial estimate of the source {\it footprint}. A source must have a signal-to-noise ratio of at least 3$\sigma$ in at least two bands in order to be considered detected.
Coordinates of the sources are determined using the moments of the intensities. Once the sources have been detected in the combined images over all spatial
scales, \textsc{getsources} performs source flux measurements on the observed images at each waveband and simultaneously subtracts a background by interpolating
under the sources {\it footprints}. Partially overlapping sources are ``deblended'' using an iterative process.
Upon completing the extraction and measurements, the user is supplied with a table giving the location and size of each source at each waveband.
Due to the fact that the resolution decreases with increasing wavelength, the apparent source size increases with wavelength. The measured properties also
include total flux, peak flux, degree of source blending, monochromatic and global detection significance. The detection significance at each wavelength is
essentially a signal-to-noise ratio, which is determined by the ratio of the peak flux of the source to the standard deviation in an annulus surrounding the
source on the detection maps. The global detection significance is determined by the square root of the sum of the
squares of the detection significance at each wavelength. As discussed in the following sections, we apply a minimum threshold for the global detection
significance value to remove some sources not well detected.
We now briefly describe the parameters we defined in the source extraction process.
\subsection{Preparing Images}
We supply all images (H$\alpha$, MIR and FIR) to \textsc{getsources}, but only use the FIR images to detect the compact sources, since we seek to trace the location of cold compact clouds.
The images are aligned to the same grid and converted to MJy sr$^{-1}$, with a pixel size of 1.4$''$.
For the SPIRE images, \textsc{getsources} uses the beam areas in order to convert the units from Jy/beam (423 $\pm$ 3, 751 $\pm$ 4, 1587 $\pm$ 9
arcsec$^{2}$, for the 250, 350 and 500~$\mu$m images respectively). The images are all aligned to the WCS of the 70$\mu$m maps. Observational masks are
created, which denote the image area over which \textsc{getsources} is meant to look for sources, which speeds the detection process. The alignment and masks are visually checked.
Since \textsc{getsources} has been designed to work with FIR {\it Herschel} maps, we decided to degrade the MIR and H$\alpha$ maps to the angular resolution of the 70~$\mu$m map.
This should reduce possible systematic effects due to the use of the program on a larger range of spatial resolutions than the one on which it has been designed and tested.
Furthermore, we performed a test to examine the efficiency of \textsc{getsources} in recovering source fluxes at different resolutions, by running the program
on a set of convolved 70$\mu$m maps. In this test, source fluxes are expected to be the same at each resolution. We found the accuracy
of the source flux measurement decreases with poorer resolution, but the average systematic effect does not seem to be large enough to affect our
results substantially (see Appendix A for more details).
\subsection{Compact Source Measurements}
The source detection has been performed using only the FIR maps, since we aimed to select sources due to their FIR brightness and not necessarily MIR and H$\alpha$ counterparts.
However, MIR and H$\alpha$ maps have been provided to \textsc{getsources} for source photometry.
\textsc{getsources} detects 186 compact sources across M83. However, in the analysis described in the following sections,
we only include those sources with a detection significance above 20. Furthermore, since a high global detection
significance does not necessarily imply that the photometry
is accurate for each single waveband, we further removed all the sources with a photometric signal-to-noise ratio lower than 1$\sigma$ in at least one band
between 8 and 250~$\mu$m (that is, we occasionally retained sources detected in all bands but the 350~$\mu$m band). After this source selection,
our sample is reduced to 121 sources.
Fig.~\ref{sources_on_map} shows the location of all the detected sources overlaid on the 70~$\mu$m map, with colours denoting the detection significance.
Sources that are retained in the study are denoted with triangles, while sources that are excluded are denoted with squares.
We note that the sources lie almost exclusively on the bar ends and the spiral arms. Very few sources are detected in the interarm regions and along
the bar. This does not mean that compact sources do not exist in
these regions, but rather, the sources there are not bright enough to be detected.
Thus, it seems that only the bar ends and spiral arms harbour bright compact sources that we can detect.
Our spatial resolution at 70~$\mu$m allows us to measure intrinsic sources sizes only for sources having radii larger than 130 pc.
We find a median radius of 150 pc at this waveband. This implies that
the majority of our sources are unresolved or barely resolved.
Thus, it is not possible to determine a physical radius for all the sources.
At each waveband we obtain the total flux of each of the extracted sources (hereafter, F$_{\text{source, HB}}$) and of the corresponding
background emission, which has been interpolated and subtracted by \textsc{getsources}.
Because \textsc{getsources} interpolates the background from regions very close to the sources, this background measurement is actually determined in the
vicinity of the spiral arms. We find that the background makes up
more than 50\% of the flux in the footprint. Fig.~\ref{comp_bk} shows
the relative flux of the source to the total flux in the footprint.
The relative fraction decreases with wavelength mainly because, due to the
poorer resolution at longer wavelengths, the footprint size increases
and thus a relatively larger amount of background flux is included in
the footprint. However, part of this relative flux variation might be also due to an intrinsic higher fraction of diffuse background
emission at longer wavelengths.
We find that at all wavelengths the source makes up less than 50\% of the emission in the footprint. For this reason, we
refer to this way of performing the source photometry as ``high-background'' source flux measurement.
In order to identify possible systematic uncertainties due to the way the background is
subtracted, we also consider an alternative measure for the background, estimated in the interarm regions, as shown in the next subsection.
\begin{figure*}
\centering
\includegraphics[trim=0mm 0mm 0mm 0mm,width=100mm]{sources_on_map_m83.eps}
\caption{M83 at 70~$\mu$m with the compact source locations marked with colors corresponding to the detection significance (Log 10 scale). A detection significance of 20 is used as a cut-off for retaining sources for further analysis.
Sources that are used in the study are denoted with triangles and excluded sources are denoted with squares (see text for criteria). Apertures showing the measurement locations of interarm emission are displayed in white.}
\label{sources_on_map}
\end{figure*}
\begin{figure}
\centering
\includegraphics[trim=0mm 0mm 0mm 0mm,angle=-90,width=80mm]{COMP_bk.ps}
\caption{Median value of the ratio of the measured source flux to the total flux in
the source footprint for both the high (full circles) and low (open
squares) source background subtractions. At
longer wavelengths the background is greater due to the increasing
size of the footprint.}
\label{comp_bk}
\end{figure}
\subsection{Low-Background Source Flux Measurement}
The background estimated by \textsc{getsources} is
basically a measure of the smooth emission component associated mainly with the spiral arms.
However, typically the photometry of star forming regions is done by
subtracting an
average brightness measured in the interarm regions close to the sources (i.e. Calzetti et al.\ 2007 and Liu et al.\ 2011) and, as said before, it is
important to check if there are substantial differences on the final results if the background measurement is performed differently.
For these reasons, we decided to perform aperture photometry in several interarm regions selected by eye (see Fig. \ref{sources_on_map}).
Specifically, we selected 18 interarm regions,
which are all sufficiently far away from the detected sources at all wavelengths. The diameter size of the apertures is 25$''$ which is large enough to obtain
a good estimate of the rather uniform interarm emission. For each aperture we measured the average surface brightness with an uncertainty derived from the standard deviation of the surface brightness within each aperture and the flux calibration uncertainty.
Once we obtained the flux measurements for the interarm apertures, we performed a ``low background'' source flux measurement in the following way:
for each source, we consider the flux in the
full footprint determined by \textsc{getsources} and subtract the interarm surface brightness, estimated from the aperture that is
closest to the source, multiplied by the area of the source footprint (typically, the distance between the centres of each source and the closest interarm
region is about 1 kpc or less)$^5$\footnote{ $^5$We note that there are variety of possible approaches for subtracting background emission. Here we contrast two techniques, one which utilizes an interpolation scheme in the vicinity of the source and one that relies on a more distant estimate in the interarm regions.}. Since the \textsc{getsources} background is usually larger than that determined in
the interarm region, in general the ``low background'' (LB) source fluxes are higher than the ``high-background'' (HB) ones.
Fig.~\ref{comp_bk} shows that the LB measurements have higher fluxes relative to their background at all wavelengths than HB flux measurements.
\section{SED Fitting}
In the absence of a strong contribution from AGNs, the galaxy dust/PAH emission SED on sub-kpc/kpc scales can be modelled as the sum of two
emission components:
a warm component largely emitting in the MIR, produced by dust in photo-dissociation regions (PDRs) and heated predominantly by young stellar populations;
and a diffuse emitting component, emitting mostly in the FIR and MIR PAH line emission, which can be
powered both
by older stellar populations and by the fraction of UV photons escaping from PDRs.
This concept is at the base of the SED fitting method developed by
Natale et al.\ (2010, NA10), which we used in this work to fit all the well-detected source emission SEDs derived in \S4.
In the following we explain the main features of this SED fitting
method, together with minor updates, and its application to our data set. We refer the reader to NA10
for additional details.
Using the mentioned fitting method, each observed source dust emission SED is fit by
combining two infrared SED components (see Fig.\ref{sedfit}). The first component is a PDR SED template which has been selected by Popescu et al.\ (2011)
among the models of Groves et al.\ (2008) because it provides a good fit to the dust
emission from Milky Way star formation regions (specifically the chosen model is for compactness parameter $log(C)=6.5$, solar metallicity and hydrogen
column density $log(N)=22$, see \S 2.8 of Popescu et al.\ 2011 for more details). The second component, suitable to fit the diffuse dust emission,
is taken from a grid of
uniformly heated dust emission templates, obtained by using the dust emission code of Fischera \& Dopita (2008). The diffuse dust emission is calculated
assuming
a Milky Way
dust+PAH composition (exact dust model parameters can be found in Table 2 of Fischera \& Dopita 2008) and taking into account the stochastic heating of grains following the
method of Guhathakurta \& Draine (1989), combined with the step
wise analytical solution of Voit (1991). Since we assumed a
fixed dust/PAH composition, each element of the grid is determined only by the parameters of the radiation field heating the dust.
The adopted spectral shape of the radiation field is the classical Mathis et al.\ (1983) profile, which was derived for the local interstellar radiation field,
scaled by two linear factors: $\chi_{\rm{UV}}$, which multiplies the whole curve, and $\chi_{\rm{col}}$ which multiplies only the optical part of the Mathis spectra
(see appendix B.2 of NA10). Therefore, $\chi_{\rm{UV}}$ can be seen as the intensity of the UV radiation field and
$\chi_{\rm{col}}$ as the optical to UV ratio in the units of the Mathis et al.\ (1983) profile.
Compared to NA10, we extended the size of the
grid of the diffuse dust templates, in order to cover a larger range of possible radiation field parameters. Specifically, $\chi_{\rm{UV}}$ and $\chi_{\rm{col}}$ are both allowed to vary
between 0.1 and 10, a range which is reasonably large to include
all the plausible values of the diffuse radiation field intensity and colour within galaxies (note that even higher radiation fields and, therefore,
warmer dust are associated with the PDR component in our SED fitting procedure). We also point out that there is a degeneracy
between $\chi_{\rm{col}}$ and the relative abundance of PAH and solid dust grains, in the sense that they both affect the 8~$\mu$m/FIR ratio (see \S B.\ 2 of NA10).
One of the main differences between our dust emission models and those of Draine \& Li (2007) is
that in their models the PAH abundance is varied but the optical/UV intensity ratio is fixed to the Mathis et al.\ (1983) value.
Therefore, this should be taken into account when considering results involving the $\chi_{\rm{col}}$ parameter.
We performed a $\chi^2$ minimization fitting of the data with the two dust emission components by varying four free parameters: $\chi_{\rm{UV}}$ and $\chi_{\rm{col}}$, the
radiation
field parameters defined above; $M_{d}$ the dust mass of the
diffuse dust component; F$_{24}$, the fraction of 24~$\mu$m emission provided by the PDR component. For each pair of $\chi_{\rm{UV}}$
and $\chi_{\rm{col}}$ values, the parameters $F_{\rm{24}}$ and $M_{\rm dust}$ correspond to linear scaling factors for the amplitudes of the PDR template and the
diffuse dust component respectively. The fit takes into account colour corrections calculated for each SED template following the conventions adopted
for each
instrument (See also Sect. 5 of NA10 for details and the {\it Spitzer} and {\it Herschel} observer manuals for the colour correction definitions).
The uncertainties on each parameter are calculated by analyzing the variation of
$\chi^{2}$ around the minimum found by the fitting procedure. The one sigma interval is defined as the minimum variation of a given fitting parameter
around its best fit value,
which produces a variation $\Delta \chi^{2}=\chi^2-\chi^2_{min}$ always higher than 1, independently of all the possible values of the other fitting parameters.
The program checks also if ``islands'' of $\chi^{2}$ low values (such that $\Delta \chi^{2}<1$) are present, which are detached from the region where the minimum of $\chi^{2}$ has been found.
In that case,
a conservative
uncertainty covering all the regions of low $\chi^{2}$ values is provided for the fitting parameter.
Apart from the fitting parameters, our SED fitting procedure provides the total dust emission luminosity and the luminosities of each SED component.
However, in contrast to blackbody fits, our method does not provide a single average dust temperature, since the
output total spectra of the
dust emission code is determined by probability distributions of dust temperatures, which are different for each dust grain size and composition.
Furthermore, in principle
different combinations of UV and optical radiation field energy densities can cause the dust to have approximately the same average cold dust temperature (that is, similar
FIR peak wavelength),
which would not be
immediately evident by comparing different pairs of $\chi_{\rm{UV}}$ and $\chi_{\rm{col}}$ values.
However, for the purposes of comparisons with other works that have relied on blackbody fits, it is useful to quote an average dust temperature. Taking advantage of the fact
that the cold
FIR part of the diffuse dust emission component ($\lambda$ $>$70~$\mu$m) can be well fit with a modified blackbody curve with a dust emissivity index of
$\beta$=2 (consistent with observational results, i.e. F12, Davies et al.\ 2012, Auld et al.\ 2013), it is straightforward to associate to each
diffuse dust template the dust temperature of the modified blackbody which
best reproduces its FIR part. We will refer to this cold dust
temperature as T$_{\rm dust}$. In Appendix B we compare the dust
masses and temperatures found using the SED fitting method outlined
here to those determined directly from a modified blackbody function
fit to the FIR wavelengths.
\begin{figure*}
\centering
\includegraphics[trim=0mm 0mm 0mm 0mm,width=150mm]{source_visual_sed_49_green.eps}
\caption{Example of source flux measurement and SED fitting. The maps on the left show a small area of the disk of M83 around one of the detected sources.
The ellipses overplotted on each map delineate the source footprint determined by \textsc{getsources}. On the right, we show the two-component SED fit of the
same source with best fit parameter and SFR values listed aside. The dashed and the dotted line represent the diffuse dust emission component and the PDR emission component
respectively.}
\label{sedfit}
\end{figure*}
\section{Star Formation Rates and Gas Masses}
Although the sources are detected in the FIR bands, measurements of the flux in the source footprint
in the H$\alpha$ and MIR maps are also made. H$\alpha$ and 24$\mu$m emission can be used in conjunction to trace recent star formation - both unobscured and obscured (Calzetti et al.\ 2007).
\textsc{getsources} also interpolates and subtracts a background for these images as well. A visual comparison of the source measurements in H$\alpha$ confirmed that the source footprints are centred on bright emission peaks. In this way, we feel confident that the flux measured in the footprints is directly related to an active star forming region.
For each source, we combined the fluxes measured in H$\alpha$ and 24 $\mu$m emission to derive a star formation rate by using the calibration of
Calzetti et al.\ (2007):
\begin{equation}
\text{SFR} [M_\odot \text{ yr}^{-1}] = 5.3 \times 10^{-42} (L(\text{H}_{\alpha}) + 0.031L(\text{24} \mu\text{m}))\text{.}
\end{equation}
There are two key differences between our SFR and that derived by Calzetti et al.\ (2007). We detect the compact sources in the FIR maps rather than
the H$\alpha$. This means we essentially estimate the SFR in the region of a FIR compact source. However, a visual check shows that almost all sources in
the FIR are also present in H$\alpha$. A second difference is related to how the background is treated. Here, we rely on the background subtraction
performed by \textsc{getsources}. This background is interpolated in the region surrounding the sources and takes into consideration neighbouring sources
and deblending. In Calzetti et al.\ (2007), the background is determined in 12 rectangular regions surrounding sources. These regions cover a large number of
pixels and extend well beyond the local neighbourhood of the sources. The mode of these regions is then used as a measure of the background. Our method,
produces a more `local' background, which is greater than that which would be found by using the same method of Calzetti et al.\ (2007).
In order to check the effect of choosing a different background level, we also determine
source star formation rates by using the H$\alpha$ and 24~$\mu$m emission in the total source footprint, after subtraction of the background estimated
from the nearest interarm aperture. This is exactly the same approach as for the LB measure for the compact source dust emission.
We note a potential problem with the SFR as outlined above. During the source extraction, it is not known which sizes the sources will potentially have. The size of the region plays an important role in deciding whether the calibration described above can be used. Regions which are too small may have SFRs which are too low.
It is known that SFRs below $\approx$ 0.001 M$_{\odot}$ yr$^{-1}$ can be problematic because there may be incomplete
sampling of the IMF and the assumption of continuous star formation in the last few Myr may not be valid (Kennicutt \& Evans 2012; Leroy et al.\ 2012).
Furthermore, the adopted calibration assumes that all the ionizing photons are absorbed by the gas in the HII regions.
However, a fraction of ionizing photons could escape the star formation regions before ionizing the gas or, instead, could be
absorbed by dust (see e.g. Boselli et al.\ 2009,
Calzetti 2012, Relano et al.\ 2012). Due to all these concerns, it is useful to consider the SFR measured in this process as a ``corrected'' H$\alpha$
luminosity. Whether this luminosity can accurately trace the SFR will depend on the source size and the value of the SFR.
We estimate the source gas masses $M_{\text gas}$ by making use of the fact that the dust mass can be used as a proxy for the total gas mass (both molecular and atomic),
provided one can translate the dust mass with a constant gas-to-dust ratio (GDR) to a gas mass. Dust mass is known to be better correlated with the total gas mass than with the atomic or molecular component alone (i.e.\ Corbelli et al.\ 2012).
We assume a constant GDR of 100 for the compact regions. While, there is some evidence that the GDR may vary within galaxies (e.\ g.\ Sandstrom et al. 2013), particularly with metallicity, given that our sources lie almost exclusively along the spiral arms, it is not obvious that a varying GDR should be used for these sources. For example, the metallicity values for M83 are based on azimuthal averages, which include interarm regions. Thus, employing a varying GDR based on metallicity values would require introducing another assumption, namely that the GDR depends more on radius than arm and interarm regions. In the pixel-by-pixel analysis of F12, the GDR was found to be relatively constant
on the spiral arms ($\approx$ 100) and, in contrast to the interarm regions, declined only slightly with radius with the inner regions of the spiral arm having a GDR of 130 and the outer tips of the spiral arm having a GDR of 120. Given the current uncertainties and possible systematic effects, we simply adopt a value of 100, which is in the range found by F12 (mean value of 84 and standard deviation of 40).
By combining the SFR and gas mass measurement for each source, we estimate the star formation efficiency defined as SFE=SFR/$M_{\text gas}$ for both types of compact source flux measurement.
We stress that this SFE is different from that presented in other works (Leroy et al.\ 2008; Bigiel et al.\ 2008 etc.) which describe the SFE using only the molecular gas component. However, it is likely that the dust is mostly tracing molecular gas in these regions. M83 is known to be a galaxy with a dominant molecular component and, over the region we consider, the molecular gas constitutes 80\% of the total gas (Crosthwaite et al.\ 2002). F12 found that on the spiral arms, where most sources lie, the molecular gas component has column densities on the order of 100 M$_{\odot}$ pc$^{-2}$, whereas the atomic gas had column densities of less than 10 M$_{\odot}$ pc$^{-2}$.
\section{Results}
Upon completing the SED fitting, as described in \S5, we further remove sources that have high (total) $\chi^{2}$ values, that is $\chi^{2}>10$, or
have a dust mass uncertainty greater than a factor of 2. In the first case we remove all the sources which are not well-fit
by our model (mainly because the observed SED shape is too irregular to be fitted with our two-component model), while in the second we remove those
sources with flux uncertainties that are too large at some wavebands and, therefore, cause the fitting parameters to not be constrained within small ranges.
After applying this selection criteria, we are left with 90 sources. Appendix D presents tables of the extracted flux and SED fitting parameters for both the
HB and LB source measurements. In this section we present the results for this set of sources. Specifically we describe
the distributions of the inferred source parameters in\S7.1, their radial variation in \S7.2 and their interdependence in \S7.3.
\begin{figure*}
\centering
\includegraphics[trim=0mm 0mm 0mm 0mm,width=100mm,angle=-90]{BigBox.ps}
\caption{Box-and-whisker plots showing the distribution of values for
the compact source dust mass (top left), temperature (top middle),
$F_{24}$ (top left)), $\chi_{\rm{UV}}$ (bottom right) and $\chi_{\rm{col}}$
(bottom middle). In each panel the source measurements with high
background (left) and low background (right) subtractions are shown. The solid horizontal line denotes the median value with the box delineating the 25th and 75th quartile range. The lines extend to the maxima and minima with open circles marking outliers (see text). }
\label{bigbox}
\end{figure*}
\subsection{Inferred Source Parameter distributions}
In order to show the distributions of inferred parameters and compare them for the two types of source flux measurements, we created box-and-whisker plots
which are shown in Fig.~\ref{bigbox}, \ref{lumbox} and \ref{bigbox_sfr_sfe}. The solid line in the boxes shows the median value and the box delineates the 25th
and 75th quartiles. The lines
extend to either the maximum and minimum values or to 1.5 times the
75th and 25th quartiles. If there are values beyond the later range,
they are denoted with open circles.
In the following we describe in detail the distributions for each set of parameters, that is, the SED fitting parameters, the source luminosities and the
star formation rates and efficiencies.
\subsubsection{SED fitting parameters}
As explained in \S5, the SED fitting has four free parameters $\chi_{\rm{UV}}$, $\chi_{\rm{col}}$, $M_{\rm{dust}}$ and $F_{24}$. The box-and-whisker plots of these
parameters and of the dust temperature are shown in Fig.\ref{bigbox}.
The $\chi_{\rm{UV}}$ values (see bottom left panel), representing the UV radiation energy density in the units of the Mathis et al.\ (1983) profile (hereafter, MMP profile),
show a
similar distribution for both types of source measurements with most values between 0.1 and 2. Therefore, the average UV radiation field energy density of
the sources seems to be typically from a few tenth up to a factor 2 the intensity of the local Milky Way interstellar radiation field,
as described by the MMP curve.
The bottom right panel shows the distribution of $\chi_{\rm{col}}$. As explained in \S5, $\chi_{\rm{col}}$ expresses the ratio of the optical to UV radiation field
energy density (relative to the standard MMP curve). Here the two measurements show some differences with the HB source measurement showing
higher values and a greater range. This could mean that the average optical/UV ratio needed to fit the source emission is higher when one subtracts the
higher \textsc{getsources} background. Alternatively, this could mean
that the PAH abundance required for the HB fit is lower than for the LB fit (because of the $\chi_{\rm{col}}$-PAH abundance degeneracy described
in \S5). In other words, a lower fraction of 8~$\mu$m emission relative to the FIR emission is removed by the background estimated in the interarm apertures.
As explained in \S5, the cold dust temperature is not fit in our SED
fitting procedure but is determined by the values of both $\chi_{\rm{UV}}$ and $\chi_{\rm{col}}$.
Both source measurements show similar temperature ranges (see top middle panel of Fig.~\ref{bigbox}), with the HB measurement only slightly higher
(about 1 K higher).
The source dust masses are shown in the upper left panel. In both cases, the compact sources have a quite narrow range in dust masses peaking close to
10$^{5}$ M$_{\odot}$ and with most values between 10$^4$ and 10$^6$M$_\odot$. The small range of masses is likely due to our resolving power, which prevents us from detecting fainter sources which
tend to also have smaller dust masses. The LB measurements for the sources have higher masses, because the background determined in the interarm aperture
subtracts less flux than the \textsc{getsources} background.
The distribution of $F_{24}$, the relative amount of 24~$\mu$m emission associated with PDR, is shown in the top right panel. Both source measurements show relatively
similar values, mainly in the range 0.25-0.6, which suggests that a substantial part of the 24~$\mu$m emission is contributed by stochastically heated small
grains in the diffuse dust component. However, we note that PDR SED dust emission spectra can vary substantially (Groves et al. 2008) and might not
be accurately reproduced by our PDR template for each individual case.
\subsubsection{Source Luminosities}
The SED fitting procedure also provides us with a measure of the total infrared luminosity of the sources as well as the luminosities due to the PDR
and the diffuse dust emission component.
Fig.~\ref{lumbox} shows the distribution of the inferred luminosity values for each component. The total luminosity values are found in a rather
small interval around $10^{41}~\rm{ergs/s}$. This is likely due to the detection technique.
We find that \textsc{getsources} does not detect sources below a minimum dust infrared luminosity of $\approx 0.5\times10^{41}~\rm{ergs/s}$. Since there are not many sources with dust
luminosities higher than a few times $10^{41}~\rm{ergs/s}$, the inferred dust luminosity range covered is rather small and this has important consequences
on the inferred range of SED fitted parameters and SFR values (see discussion in \S8). Fig.~\ref{lumbox} also shows that the diffuse dust emission component
dominates the dust luminosity. We point out that this does not
necessarily mean that the dust heating from star forming regions is not responsible for most of the dust
emission, since the diffuse dust emission component can be powered by both radiation from older stellar populations and the fraction of UV photons escaping
from PDRs. The origin of the dust heating will be further discussed in $\S8.3$.
\subsubsection{Star formation rates and efficiencies}
Fig.~\ref{bigbox_sfr_sfe} shows the distribution of the SFR and SFE inferred values. Due to the small areas considered here (i.e. average radius of sources
is 150 pc), the SFRs are quite low. As outlined in \S6 if the SFRs are too low (i.e. below 0.001 M$_{\odot}$ yr$^{-1}$) then the calibration of H$\alpha$ and 24~$\mu$m fluxes may not adequately trace the SFR. However, the majority of
the sources have SFRs above this value and furthermore, we have restricted ourselves to regions associated with recent star formation.
These regions should have a good correspondence between gas and star formation. We note that the H$\alpha$ contribution represents half or slightly more than half of the SFR for the sources. The median ratio of the H$\alpha$ luminosity, $L(\text{H}_{\alpha})$, to the scaled 24~$\mu$m luminosity, 0.031$L(\text{24} \mu\text{m}$), is 1.4.
The SFEs of the source measurements have median values with median deviation values of 3.57 $\pm$ 3.45 $\times$ 10$^{-10}$ yr$^{-1}$ and 3.1 $\pm$ 3.1 $\times$ 10$^{-10}$ yr$^{-1}$
in the case of the HB and LB measurement, respectively. This is consistent with the
findings of Leroy et al.\ (2008) who found that in terms of molecular gas, the SFE has a mean value of
5.25 $\pm$ 2.5 $\times$ 10$^{-10}$ yr$^{-1}$.
The LB and HB measurements show considerable differences between the inferred SFR and source gas masses, with the LB measurements having higher values in
both cases. However, the LB and HB measurements show roughly the same range of SFE values. This means that roughly the same relative amount of background is being subtracted for both the SFR tracers and dust emission. This suggests the amount of background subtraction does not affect the results of the SFE, provided the same technique is used for both the gas and SFR tracers.
\begin{figure}
\centering
\includegraphics[trim=0mm 0mm 0mm 0mm,width=50mm,angle=-90]{lumBox.ps}
\caption{Box-and-whisker plots showing the distribution of
luminosities for the sources in the high background (left) and low
background (right) subtractions. We show the distributions of
the total luminosity, PDR luminosity component and total diffuse
component. See Fig.~\ref{bigbox} for details.}
\label{lumbox}
\end{figure}
\begin{figure}
\centering
\includegraphics[trim=0mm 0mm 0mm 0mm,width=45mm,angle=-90]{BigBox_sfr_sfe.ps}
\caption{Box-and-whisker plots showing the distribution of values for
the SFR (left) and SFE (right). In each panel the source
measurements with high background (left) and low background (right)
subtractions are shown. See Fig.~\ref{bigbox} for details.}
\label{bigbox_sfr_sfe}
\end{figure}
\begin{figure}
\centering
\includegraphics[trim=0mm 0mm 0mm 0mm,width=70mm,angle=-90]{regions_luminosity.ps}
\caption{The radial variation of the luminosity components including
the total (black circles), PDR component (blue triangles) and
diffuse component (red squares), for the compact sources with high
background subtraction (left) and low background subtraction (right). In the upper left corner we list the Spearman rank correlation coefficient for each component and radius. The relative fraction of the PDR component to the total (blue triangles) and diffuse component to the total (red squares) are shown on the bottom panel for both cases. }
\label{lumrad}
\end{figure}
\subsection{Radial Variations}
We now turn to the radial variations of the SED fitting parameters, the SFR and SFE and of the dust luminosities.
Previous works which have
used azimuthal averages (e.g. Mu\~noz-Mateos et al.\ 2009, Pohlen et al.\ 2010, Engelbracht et al.\ 2010, Boquien
et al.\ 2011) have found that the parameters
describing the dust,
including temperature and surface mass density, and the SFR tend to decrease with radius particularly for late-type spirals. These studies have averaged over both the compact regions and the
more diffuse emission. By separating the compact regions, we can better disentangle how and if the radial position affects the dust properties
of these sources.
In order to determine the deprojected radial position, we assume an inclination and position angle for M83 of 24$^{\circ}$ and 225$^{\circ}$ respectively (Tilanus \& Allen 1993).
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[trim=0mm 0mm 0mm 70mm,width=80mm,angle=-90]{radial_correlations_sfr1.ps}&
\end{tabular}
\caption{The radial variation of the mass, SFR and SFE for the two compact source measures. The upper right hand corner lists the Spearman rank correlation coefficient. The error bars in red denote the median uncertainties of the points in three bins between 0-2 kpc, 2-4 kpc and 4-6 kpc.}
\label{rad1}
\end{figure}
\begin{figure}
\centering
\includegraphics[trim= 0mm 0mm 0mm 20mm,width=80mm,angle=-90]{radial_correlations3.ps}
\caption{The radial variation of $\chi_{\rm{col}}$, $\chi_{\rm{UV}}$, $F_{24}$
and temperature for the two compact source measures. The upper right hand corner lists the Spearman rank correlation coefficient. The error bars in red denote the median uncertainties of the individual measurements in three bins between 0-2 kpc, 2-4 kpc and 4-6 kpc.}
\label{rad2}
\end{figure}
\subsubsection{Source infrared luminosities}
Fig.~\ref{lumrad} shows how the luminosity of the sources (upper panels) and the relative fraction of PDR and diffuse dust emission luminosity
(lower panels)
varies with radius. In the upper left corner we list the Spearman rank correlation coefficient for the correlation between the luminosity components and
radius. Values of the correlation coefficient approaching $+$1 or $-$1 reflect stronger correlations or anti-correlations respectively. Values close or
less than 0.5 represent mild or weak correlations. In both the HB and LB measurements, we find only a mild correlation between the source
luminosities and radius. The source luminosities show a slight decrease with radial position with scatter.
We also note that the regions with the highest PDR luminosity fraction are found in the inner regions, at a radial distance of 2-3~kpc
from the galaxy centre (roughly the end of the bar). However, the diffuse dust component dominates the emission for all the sources, except in a few cases.
\subsubsection{Radial Variations of the SED Fitting Parameters, SFR and SFE}
Fig.~\ref{rad1} and \ref{rad2} show the SED fitting parameters and the SFR and SFE values plotted versus the source deprojected radial positions for both the flux
measurements of the compact sources. The median uncertainties of the individual measurements are shown by the red error bars and the Spearman rank coefficient is listed in the upper right of each panel.
For both the HB and LB measurements, we find little to no variations with radius for any of the SED parameters including dust mass, temperature,
F$_{24}$, $\chi_{\rm{UV}}$, $\chi_{\rm{col}}$. There is a slight trend showing that the SFR decreases with radius, but this is far less than what is typically
seen in studies on late-type spiral galaxies that employ azimuthal averages. It seems that the properties of the compact regions are quite uniform
and do not vary much with
location in the galaxy. This is perhaps not too surprising, since our source sample covers only a relatively small range of dust luminosities,
as shown in \S7.1. Therefore, the sources we consider likely represent similar types of objects. Although the low spatial
resolution did not allow us to detect a larger range of source luminosities, it is striking that the detectable bright sources lie almost exclusively on
the spiral arms or at the ends of the bar in M83, which means that the local environment may in fact be quite similar as well.
This suggests that azimuthal averages of dust properties could well mask important differences in environment within an azimuthal bin. The radial decreases
that are typically seen in other galaxies and for M83 in F12 using azimuthal averages might simply reflect the increasing contribution of the interarm regions at larger radii.
We note, however, that the SFE is quite constant with radius. This finding is consistent with studies that use azimuthal averages and pixel-by-pixel analyses
(e.g.\ Leroy et al.\ 2008, Blanc et al.\ 2009, Foyle et al.\ 2010, Bigiel et al.\ 2011).
\subsection{Correlations between source properties}
Having seen how the dust and star formation parameters of the compact sources vary with radius, we now turn to how they are inter-related and what
types of correlations may exist between some of these parameters. In particular, we examine how the source dust mass and temperature are related
to the radiation field parameters $\chi_{\text UV}$ and $\chi_{\text col}$ and with the star formation rate and efficiency.
Fig.~\ref{masscorr}
shows the dust temperature, $\chi_{\text UV}$, SFR and SFE plotted versus the dust mass, while Fig.~\ref{tempcorr} shows $\chi_{\text UV}$, $\chi_{\text col}$, SFR and SFE values plotted against the dust temperature$^{6}$\footnote[6]{$^{6}$In Appendix C we present how these parameters vary as a function of colour.}. As before, the red bars show the median uncertainties in equally spaced bins
and the Spearman correlation coefficients are shown in the upper-left of each panel. In general, we find that both LB and HB measurements of the compact sources
present quite similar results.
We observe a strong
correlation of $\chi_{\text UV}$ with the dust temperature, while the optical-UV intensity ratio $\chi_{\text col}$ is not as strongly correlated to the dust temperature measured for the sources. These findings reflect the fact the UV radiation field intensity
is the fundamental parameter that determines the average dust temperature, while the optical-UV intensity ratio varies randomly among sources
having same average dust temperatures. We point out that this does not mean that optical photons are not an important source of dust heating.
Actually, the optical part of the radiation field accounts for more than 50\% of the dust heating for the diffuse dust SED component when
$\chi_{\text col} > 1$.
\begin{figure}
\centering
\includegraphics[trim=0mm 0mm 0mm 50mm,width=100mm,angle=-90]{mass_correlations1.ps}
\caption{Correlation plots showing dust temperature, $\chi_{\rm{UV}}$, SFR and SFE versus dust mass for the HB (left) and LB source flux measurements (right).
The Spearman rank coefficient is shown in the upper right of each panel. The red uncertainties show the median uncertainties in bins of 10$^{0.4}$ M$_{\odot}$. }
\label{masscorr}
\end{figure}
\begin{figure}
\centering
\includegraphics[trim=0mm 0mm 0mm 100mm,width=100mm,angle=-90]{temp_correlations1.ps}
\caption{Correlation plots showing $\chi_{\rm{col}}$, $\chi_{\rm{UV}}$, the SFR and SFE versus dust temperature for the HB (left) and LB source flux measurement (right).
The Spearman rank coefficient is shown in the upper right of each panel. The red uncertainties show the median uncertainties in bins of 2 K. }
\label{tempcorr}
\end{figure}
The upper left panel of Fig.~\ref{masscorr} reveals a mild anti-correlation between the dust temperature and mass for the sources. The typical error bars are much smaller than the inferred range of dust mass and dust temperature. Thus,
this anti-correlation cannot be explained by the uncertain determination of dust temperature and mass, which are connected in the SED fitting since
approximately $L_{\text dust}\propto M_{\text dust}T_{\text dust}^{4+\beta}$. Therefore, we are confident that the sources with higher
dust masses tend to have lower dust temperatures in the sample we considered. Given the strong correlation between $\chi_{\text UV}$ and temperature,
we also find an anti-correlation between $\chi_{\text UV}$ and dust mass in the second row from the top of Fig.~\ref{masscorr}.
We will discuss the significance of the dust mass - dust temperature anti-correlation in \S8.3.
Fig.~\ref{masscorr} (third row, left) shows that for both measures of the compact sources the dust mass is only weakly correlated with the SFR.
If the dust traces gas, this plot should be similar to the S-K relation (Kennicutt 1998), except that typically surface densities are
plotted rather than total values. The S-K relation for nearby galaxies typically shows a tight correlation between gas surface densities
and the SFR surface densities, at least when the scales considered are large enough (e.g.\ Schruba et al.\ 2010, Bigiel et al.\ 2008, see other references in \S 2.1).
The relatively constant SFR with mass suggests that more massive sources are less efficient at forming stars and will be further discuss in \S8.2.
We also find that for both measures of the compact sources, the SFR is only mildly correlated with the dust temperature (see third row of Fig.~\ref{tempcorr}).
This is not unexpected since, as explained in Sect. \S2.2, the SFR is connected to the dust temperature through the dust mass and the total luminosity-weighted
optical-depth.
In contrast to the lack of correlations for the SFR, we find a mild and strong correlation respectively of dust mass and dust
temperature with the SFE.
While the SFE was roughly constant with radius,
we find that the SFE is anti-correlated with dust mass and correlated with dust temperature. These findings have not been seen before, as most studies
have found that the SFE does not vary much with other properties in terms of pixel-by-pixel and azimuthal averages (i.e.\ Leroy et al.\ 2008). This will be further discussed in \S8.2 and \S8.3. We note that that the anti-correlation between the SFE and mass is primarily due to the fact that the SFR does not vary much with the inferred gas mass. Thus, the correlated axes produce the anti-correlation. To check this, we performed Monte-Carlo simulation assuming the SFR was roughly constant with a Gaussian distribution and a mean and standard deviation equivalent to that found in our measurements. We assumed a similar distribution for the simulated source masses. The simulation naturally produced a linear plot of SFE versus mass with a slope of -1.05 $\pm$ 0.15. The compact sources in our analysis produced a relation with a slope of -0.71 $\pm$ 0.14. The slopes while not identical within the uncertainties, are similar, suggesting that the correlated axes are the primary reason for the relation.
We also examined the relation between dust luminosity and the SFR, as shown in Fig.\ref{lumsfr}. We see that not only is the total dust luminosity
but also the PDR and diffuse dust emission luminosities are well-correlated with the SFR. As explained in section $\S2.2$, this is what is expected
if a young stellar population is responsible for the dust heating and the total luminosity-weighted opacity of the sources, $\tau$, does not vary arbitrarily among the sources.
\begin{figure}
\centering
\includegraphics[trim=20mm 20mm 0mm 20mm,width=40mm,angle=-90]{sfr_luminosity.ps}
\caption{Dust luminosity versus the SFR for both measures of compact
sources including the total luminosity (black circles), PDR
component (blue triangles) and diffuse component (red squares). The
correlation coefficients for each component are shown in the upper
right. The solid line denotes the relation of
SFR=2.8$\times$10$^{-44}\times$ $L_{\text{dust}}$ (Calzetti 2012), which assumes that all the young stellar luminosity is absorbed and re-emitted by dust.}
\label{lumsfr}
\end{figure}
\section{Discussion}
In this section we discuss the properties of the detected compact sources of M83 in terms of gas masses and locations (\S8.1), star formation (\S8.2)
and dust heating (\S8.3).
\subsection{Source Gas Masses and Locations}
The inferred dust masses of the compact sources lie within the range of 10$^{4}$-10$^{6}$ M$_{\odot}$. These dust masses can be used to estimate the
source gas masses as discussed in \S6. We use a gas-to-dust ratio of 100, which means the gas masses of the sources have values in the range
10$^{6}$-10$^{8}$ M$_{\odot}$. These values correspond to the high end of the molecular cloud mass distribution function derived both observationally (see e.g.\ Solomon et al.\ 1987, Rosolowski et al.\ 2005, Gratier et al.\ 2012) and by numerical simulations (e.g. Nimori et al.\ 2012). Furthermore, previous CO studies of nearby galaxies have highlighted the presence of very massive clouds with gas masses of order of 10$^{7}$ M$_{\odot}$ or higher, which are usually referred to as Giant Molecular Associations (GMA) (e.g. Vogel et al.\ 1988, Koda et al.\ 2009, Muraoka et al.\ 2009). The high masses of our compact sources suggests that they are GMAs rather than less massive GMCs.
However, because dust continuum emission does not provide kinematical information, it is not possible to check whether the detected sources are gravitationally bound clouds. The smallest cloud radius we can measure is 130 pc due the 70~$\mu$m beam size (6$''$). The majority of the detected sources are found on the spiral arms. Studies of the molecular gas in M83 have shown that most of the bright compact
sources on the spiral arms are bound clouds in GMAs. Muraoka et al.\ (2009) examined CO in M83 and measured the virial parameter, $\alpha$, defined as the ratio of the virial mass to the CO luminosity mass. They found that $\alpha$ is almost equal to unity in on-arm clouds, suggesting that they are in a gravitationally bound state (see also Rand et al.\ 1999, Lundgren et al.\ 2004). The resolution of their study was 7$''$.5, which is comparable to our resolution. Thus, it is quite likely that our sources are bound GMAs.
The spiral arms are the natural location where ISM gas is brought to higher densities, thus leading to the formation of massive clouds.
Therefore, it is not surprising, given our resolution and sensitivity, that we only detect compact sources on the spiral structure. We note that the bar-spiral arm
transition region harboured the most sources and that just few were detected along the length of the bar. Due to shear motions in bars, GMCs can be
easily pulled apart (Downes et al.\ 1996). In contrast to spiral arms, bars exhibit lower star formation rates and in some cases lower SFE
(Momose et al.\ 2010). Meanwhile, studies of atomic and molecular gas have shown that the most massive complexes are found in the bar-spiral arm
transition region. This can be explained by orbit crowding when gas on highly elliptical orbits in the bar converges with the gas orbiting in the spiral
structure (e.g.\ Kenney \& Lord 1991). The location of the compact sources detected in the FIR is consistent with these findings in the gas.
\subsection{Star Formation Rates and Efficiencies}
One of the goals of this work has been to infer cloud gas masses using the dust emission of compact sources. Combined with a measure of the SFR, we can then examine the SFE in these regions.
There have been many recent studies that have examined star formation on spatially-resolved scales. Typically these studies have employed one of
three methods: pixel-by-pixel analyses, where the SFR tracers and gas tracers are compared in individual pixels; azimuthal average analyses, where average
values in radial bins are compared; or aperture photometry on compact sources bright in some SFR tracer such as H$\alpha$ emission.
Our work differs in several key ways and, before discussing the results, we summarize these differences here:
1) We use the dust emission to detect the compact sources. While most studies have focused on aperture photometry of active star forming regions
(e.g.\ Calzetti et al.\ 2005, Kennicutt et al.\ 2007, etc.), few have selected compact sources as seen in dust emission maps, which may be more akin to
locating peaks in the gas (e.g. Schruba et al.\ 2010). However, there is a good correspondence between the detected FIR sources and star formation regions, which is demonstrated by the tight correlation between total infrared luminosities of the sources and their SFRs (see Fig. 13); 2)
we perform a diffuse background subtraction of the dust emission using both a local background defined by \textsc{getsources} and one determined from
nearby apertures in the interarm regions. Previous studies have performed a background subtraction on the star formation tracers but not on the gas mass
(though see Rahman et al.\ 2011); 3) it is important to keep in mind, when comparing this work with others, that we analysed the total SFR and
dust masses for individual sources while most studies consider the surface densities of the gas mass and SFR when examining trends between the two. The reason why we did not
consider surface densities is that it is not clear what the source area should be, since the apparent size of the sources
varies on each map depending on the map resolution. In addition, since most of the sources are not well-resolved, it is not possible to determine the
physical source area which is typically smaller than the PSF beam area.
As discussed in \S2, previous works have found that, on relatively large spatial scales ($>$ 500 pc) clear correlations are found between the gas, SFR
and dust as well as a tendency for these parameters to decline with radius. However, as we saw in \S7, the compact regions detected in this study show a relatively constant
SFR and dust mass with radius, albeit with large scatter and there is little to no correlation between the SFR and dust mass. Due to the small spatial
scales of these sources ($<$ 300 pc), one does not expect to recover the relations found by averaging over much larger areas. Recently, it has become clear
that on small spatial scales (less than 400 pc ) the S-K relation between the gas surface density and SFR surface density breaks down (Calzetti et al.\ 2012, Feldmann \& Gnedin 2011 and Schruba et al.\ 2010). This has been found to be even more prominent in the case where
gas peaks are selected (Schruba et al.\ 2010). The scatter is due to the fact that individual GMCs and HII regions will show varying gas mass to star
formation rates depending on their evolutionary state. Averaging over large areas means that many objects in different states are averaged and the
relation between SFR and gas mass is recovered.
The scales on which the SFR and gas mass surface density are measured play an important role in defining the relation (Liu et al.\ 2011). Typical SFR tracers suitable for relatively large galaxy regions are not straightforwardly applicable to small regions of sub-kpc scales. In fact, in this case, the intrinsic assumption that the stochastic characteristics of star formation are averaged out from the integration on large areas can easily break down (i.e.\ Calzetti et al.\ 2012). This happens because, one ideally needs a complete sampling of the stellar IMF. Small regions may not encompass a large enough stellar population to do so.
Furthermore, beyond the issue of sampling the stellar IMF, there is the problem of time-averaging. The SFR is calibrated based on the assumption of a constant SFR over 100 Myr (Calzetti et al.\ 2007). While this is true for entire galaxies, on small scales this assumption may no longer hold and can create a large scatter in the relation.
Our sources have sizes of roughly 300 pc and thus, we find large scatter between the inferred gas mass and the SFR. We also note that our sources populate
only a narrow range in FIR luminosity and dust mass (see Fig.\ref{bigbox} and \ref{lumbox}).
Thus, we see a relatively constant SFR for sources with such similar characteristics. If we could populate the plot with lower mass regions,
an S-K relation with considerable scatter might be recovered. Our results reflect stochasticity and the fact that individual star forming regions can exhibit a range of SFRs for a given dust or gas mass. However, we should note a few other caveats that might introduce scatter into the relation.
First, it is possible that dust emission does a poor job of tracing molecular gas especially in light of our use of a constant GDR. There may be large variations in the GDR (i.e. due to metallicity variations), which means that our gas mass estimates might not be reliable. However, the compact regions all lie on the spiral arms and beyond the nuclear region of the galaxy. In F12, the largest variations in the GDR were seen in the central regions and, beyond, the GDR was quite constant. It is also possible that some of the peaks found in the dust emission are not associated with active star forming regions. However, we found a correlation between the dust luminosity and the SFR and studies like Verley et al.\ (2010) have found a tight correlation between compact regions in M33 and star formation rate tracers. We should also note that all of the sources were detected in the 70$\mu$m map, which is also known as a good tracer of the SFR (Li et al.\ 2010).
A second possibility is that our background emission subtraction has introduced scatter. Rahman et al.\ (2011) showed that as one increases the amount of diffuse emission subtracted from the gas, the scatter in the S-K relation increases. However, particularly given that we are using dust emission, which may trace gas of different phases, it is important to account for a diffuse component.
Despite the large scatter, we find that the SFR varies little with the inferred gas mass and is roughly constant. This implies that the more massive clouds are less efficient at forming stars. A physical reason for this is that the more massive clumps might be more extended and thus have lower gas surface densities. Studies of GMCs have revealed that only the densest gas is directly associated with star formation. In this way, the more massive and thus extended regions may have roughly comparable SFRs (or lower SFEs). We note that we also found that there is a strong anti-correlation between the SFE and the inferred gas mass. However, as discussed in \S 7.3, this is is largely a product of the correlated axis since mass appears both in the y and x-axes (see Murray (2011) who examined galactic GMCs and found an anti-correlation which was attributed to a similar effect). We also note that we find no radial variation of the SFE of the compact regions. Previous works, which have examined kpc-sized regions, have found that the SFE is relatively constant regardless of the variable considered (i.e. Leroy et al.\ 2008). This is consistent with our findings with radius.
\subsection{Dust Heating}
We showed in \S2.2 that the simultaneous presence of a correlation between the
SFR and source dust luminosity as well as a correlation between the SFE and the dust temperature suggests that the radiation impinging on the dust
mass is mainly produced by a local, young stellar population.
Contrary to common lore, a strong correlation between SFR and dust temperature should not be necessarily expected in this case.
Indeed we saw in \S7 (see Fig.~\ref{lumsfr} and \ref{tempcorr} ) that while the dust temperature was only mildly correlated with SFR, it was strongly
correlated with the SFE. We also saw that the dust luminosity was correlated with the SFR (see Fig.~\ref{lumsfr}). This supports a scenario where a young,
local stellar population is powering the dust emission of the FIR bright sources of M83.
In Fig.~\ref{lumsfr} the solid line illustrates the maximum level of dust luminosity that can be powered by the young stellar population by assuming that all
the luminosity from this population is absorbed and re-emitted by dust (optically thick case) using the following relation by Calzetti (2012):
\begin{equation}
SFR=2.8\times10^{-44}L_{\text dust}
\label{sfr_ltir}
\end{equation}
This relation has been derived using the same assumptions on the star formation
history, metallicity and IMF as the H$\alpha$ based calibration we used to infer SFR of the sources.
This line should delineate the maximum value, yet we find many sources lie close to or above this limit. Specifically we find that the percentage of
sources that lie above or only within 2$\sigma$ below this limit is 60\% or 20\% in the HB or LB case respectively.
While it is surprising to find sources that lie above or close to this maximum limit, these findings can be explained in one of two ways.
Firstly, it is possible that some regions are indeed very close to being completely optically thick and
therefore the total dust luminosity will be approximately equal to the intrinsic young stellar population luminosity in those cases. However, this is
unlikely because these regions are also H$\alpha$ emitters, which means, by definition, that at least some fraction of the young stellar population
luminosity is able to escape
unabsorbed (at least through re-emission in H$\alpha$). Furthermore, we checked that the differences between the observed dust luminosities and the dust
luminosities for an optically thick case, predicted by the relation in Eq.~\ref{sfr_ltir}, do not depend on the ratio F(H$\alpha$)/SFR[H$\alpha-24\mu m$].
This suggests that the scatter of the observed points is not driven by differences in attenuation.
Secondly, it is possible that there is a local older stellar population or stellar populations outside the projected source area contributing
substantially to the heating. In this
case, the approximate equality between dust and young stellar population luminosity would be reached through this extra-heating from other radiation sources.
Thus, the infrared total luminosity could still be used as a star formation rate indicator in those cases despite the fact that the dust heating
is not powered exclusively by a local, young stellar population (see Bendo et al.\ 2012 also). This might explain why the recent work of Verley et al.\ (2010) and Boquien et al.\ (2010)
showed that the single {\it Herschel} 100, 160 and 250 $\mu$m band luminosities of individual sources in M33 are good tracers of SFR, since most of the dust
luminosity is emitted at FIR wavelengths.
We should note that the SFE vs $T_{\rm{dust}}$ correlation is at least partially driven by the $M_{\rm{dust}}$ vs $T_{\rm{dust}}$ anti-correlation
(more evident for the LB measurement), since the SFR alone does not
correlate with $M_{\rm{dust}}$ and only mildly with $T_{\rm{dust}}$. There are multiple potential explanations for an anti-correlation between
mass and temperature.
One possible explanation is that it is due to the different
evolutionary stages of the sources, with the more massive sources having a relatively lower number of young, recently formed stars.
In this way, the radiation field would be less intense for massive sources and the dust would be cooler. Alternatively, it could be that the more massive
sources are more efficient at shielding radiation originating from outside the local region. As outlined in the previous paragraph, there is likely some
heating due to other stellar populations other than just the local, young population, and thus such shielding could keep more massive sources cooler.
The anti-correlation between dust temperature and mass could be in part due to the lower limit in dust luminosity for the sources.
Fig.\ref{mass_vs_temp} shows the inferred dust masses versus dust temperatures with the plotted points colour coded depending
on the source total dust luminosity. Since most of the dust luminosity is emitted at FIR wavelengths where $L_{\rm{dust}} \propto M_{\rm{dust}}T_{\rm{dust}}^{4+\beta}$,
the points correspond to higher total dust luminosities while moving from the bottom-left to the upper-right region of the plot. From this figure one
can see that there is a lower limit in luminosity ($\approx 5-6\times10^{40}$erg/s), determined by the minimum fluxes for source detection.
This explains why there are no sources populating the lower-left part of the diagram. The steepness of the observed anti-correlation is consistent with
the $4+\beta$ exponent (with $\beta$=2),
as shown by the line in Fig.~\ref{mass_vs_temp}. However, higher mass sources do not populate uniformly the entire range of temperatures in the
upper-right part of the diagram, where source detection would easily be possible. Therefore, while the low-luminosity limit may contribute to the
anti-correlation between dust temperature and mass, it cannot fully account for it. Thus, there should be at least some physical restrictions at play.
We note that the lower limit to the luminosity
range explored in our analysis does introduce a bias that should be taken into account. Essentially, our sources are the brightest FIR compact sources in the
disk of M83 and they are associated with the high-end of the molecular cloud mass distribution. As discussed in \S8.1, the sources are associated with gas
masses in the range $\approx$10$^{6}$-10$^{8}$~M$_{\odot}$, while their total dust luminosities are in the range $\approx$0.5-5$\times$10$^{41}$erg/s.
The results of this work should be considered valid only within these parameter ranges.
We found
a mild correlation between SFR and $T_{\rm{dust}}$. However, as shown in Appendix C, a better correlation is found when plotting the inferred
$70/160~\rm{\mu m}$ flux ratio versus the SFR. The difference between the two cases is due to the fact that $T_{\rm{dust}}$ in our SED fitting
procedure refers to the cold dust temperature of the diffuse dust component. Meanwhile the observed 70 and 160$\mu$m fluxes have contributions from both the diffuse component and the PDR component. Nonetheless, in Appendix C we also show that the correlation between SFE and $70/160~\rm{\mu m}$ flux ratio is clearly tighter
and this is consistent with the arguments presented in \S2.2 and very similar to the strong correlation found by plotting SFE versus $T_{\rm{dust}}$.
Pixel-by-pixel analyses have tended to find varying results with no strong consensus
for a correlation between SFR and modified black body dust temperatures or color temperatures. We should note that the pixel-by-pixel analyses have
considered the star formation rate surface density (hereafter SFRSD) as
opposed to the SFR used here. For M83, F12
found a correlation between the SFRSD and $T_{\rm{dust}}$. A similar result for M83 has been found by Bendo et al.\ (2012)
when comparing the $70/160~\rm{\mu m}$ flux ratio with the observed H$\alpha$
brightness, uncorrected for internal dust attenuation. However, Bendo et al.\ (2012) found a weaker correlation for M81 and NGC 2403.$^{7}$\footnote[7]{$^{7}$The weaker correlation is attributed to the dust heating by bulge stars in
M81 and artefacts in the images of NGC 2403; see the
discussion in that paper.}
Boquien et al.\ (2011) found a mild correlation between SFRSD
and the $70/160\rm{\mu m}$ flux ratio for M33. In addition, Smith et al.\ (2012) for the Andromeda galaxy and
Skibba et al.\ (2012) for the Large and Small Magellanic Clouds found no or only a weak correlation between SFRSD and $T_{\rm{dust}}$. Therefore,
it seems that in general the correlation between SFR vs tracers of dust temperature is not observed to be very strong when one considers either
total SFRs for individual FIR sources or SFR averages on areas equivalent to pixel sizes. As explained in \S2.2, this is expected even for the case where
young stellar populations are responsible for the dust heating because the dust temperature also depends on the dust mass and the total luminosity-weighted
optical depth (see Eq.~\ref{lstarequ}), which are not uniform throughout a galaxy disk.
As we have seen, while isolating FIR bright sources has highlighted the presence of interesting correlations between the inferred source parameters,
the narrow range of luminosities of the sources has introduced a potentially important bias. In order to elucidate the influence of this bias on the
inferred relationships, we need to expand this work to include sources within a
wider range of luminosities, particularly fainter sources. To do so, we plan a follow-up study which will include other spiral galaxies that are even
closer (i.e. NGC 2403 and M33). The closer proximity will allow for an even better spatial resolution, which is necessary to detect sources in a wider range
of luminosities and determine if the results for the GMAs in M83 are also valid for GMCs in other galaxies.
\begin{figure}
\centering
\includegraphics[trim=0mm 0mm 0mm 0mm,width=80mm]{dust_mass_vs_dust_temp_col.eps}
\caption{Source dust masses versus dust temperature for the HB (left) and LB (right) measurements. The points are colour-coded based on their total dust luminosity. The solid line represents the relation $L_{\text dust}$ $\propto$ $M_{\text dust}$ $T_{\text dust}$$^{6}$ for $L_{\text dust}$=6$\times$10$^{40}$ergs/s.}
\label{mass_vs_temp}
\end{figure}
\section{Conclusions}
The main aim of this paper has been the investigation of the star formation and dust heating properties of the compact FIR bright sources as observed
on the {\it Herschel} maps of the nearby spiral galaxy M83. By combining the source detection and photometry algorithm \textsc{getsources}, the dust emission SED fitting method of Natale et al.\ (2010)
and the [H$\alpha$ - 24~$\mu$m] star formation rate calibration by Calzetti et al.\ (2007), we have developed a new procedure to determine gas masses, radiation field intensities,
cold dust temperatures, dust luminosities, star formation rates and star formation efficiencies associated with those sources.\\
The main results of our analysis are the following:
\begin{itemize}
\item We have found that the well-detected compact FIR sources are mostly associated with giant molecular associations, with gas masses in the range
10$^{6}$-10$^{8}$ M$_{\odot}$ and dust total infrared luminosities in the range 0.5$\times 10^{41}$-$10^{42}$ergs/s. The majority of the sources are
located on the spiral arms of M83 with only few sources found in the interarm and within the bar in the central region of M83.
\item None of the inferred physical quantities for the sources shows a strong variation with radius, including SFR, gas masses and dust temperature.
Previous studies have usually found radial variation for these quantities, although only after averaging on larger areas including interarm regions and
without subtracting any local background.
\item The SFR does not seem to correlate strongly with the gas mass of the sources. The lack of correlation is most likely due to the small spatial
scales considered ($\approx$200-300 pc) and/or the relatively small range of inferred gas masses.
\item The star formation efficiency SFE, defined as SFR/M$_{\text{gas}}$,
shows an anti-correlation with source gas mass. This finding suggests that the more massive GMAs are less efficient
in forming stars in the last few Myr. However, we note that this anti-correlation is a consequence of the roughly constant SFR with inferred gas mass.
\item We found that the SFR correlates well with total dust luminosity, which is consistent with a scenario where dust is predominantly heated by the local
young stellar population. However, between 20-60\% of the sources show dust luminosities which are greater than or only 2$\sigma$ below those predicted when
the heating is due only to a local young stellar population, embedded in a optically thick dust distribution. It is unlikely that the
sources are completely optically thick, as they are also bright in H$\alpha$. Thus, it seems that there must be some extra-heating by either a local older
stellar population or by an external radiation field.
\item We found a correlation between the SFE and $T_{\rm{dust}}$ which is tighter than the mild correlation we found between SFR and $T_{\rm{dust}}$.
This is expected if the dust is heated primarily by recent star formation.
\item We found a mild anti-correlation between dust and mass temperature. While our sources have a low-luminosity limit, which may contribute to this
anti-correlation, we find that this can not fully account for it. Thus, we speculate that the more massive sources are more efficient at shielding from an
impinging radiation field or that more massive sources are in an earlier stage of star formation.
\end{itemize}
We plan to use the same procedure presented in this pilot work on a set of nearby galaxies observed by {\it Herschel}. This will help to further clarify
the origin of the observed correlations and their implications for star formation and dust heating associated with FIR bright sources on galactic scales.
\\
\\
We are grateful to the anonymous referee for their careful reading of this work and their comments. K. Foyle acknowledges helpful conversations with S.\ J.\ Kiss, V. K\"onyves, P.\ G.\ Martin, A.\ Men'shchikov, M.\ Reid and R.\ Skibba. This research was supported by grants from the Canadian Space Agency and the Natural Science and Engineering Research Council of Canada (PI: C. D. Wilson). MPS has been funded by the Agenzia Spaziale Italiana (ASI) under contract I/005/11/0. PACS has been developed by
a consortium of institutes led by MPE (Germany) and including
UVIE (Austria); KU Leuven, CSL, IMEC (Belgium); CEA, LAM
(France); MPIA (Germany); INAF-IFSI/OAA/OAP/OAT, LENS,
SISSA (Italy); IAC (Spain). This development has been supported
by the funding agencies BMVIT (Austria), ESA-PRODEX (Belgium), CEA/CNES (France), DLR (Germany), ASI/INAF (Italy),
and CICYT/MCYT (Spain). SPIRE has been developed by a consortium of institutes led by Cardiff University (UK) and including
Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France);
IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory
(Sweden); Imperial College London, RAL, UCL-MSSL, UKATC,
Univ. Sussex (UK); and Caltech, JPL, NHSC, Univ. Colorado
(USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS
(France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC
(UK); and NASA (USA). HIPE is a joint development by the Herschel Science Ground Segment Consortium, consisting of ESA, the
NASA Herschel Science Center, and the HIFI, PACS and SPIRE
consortia. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion
Laboratory, California Institute of Technology, under contract with
the National Aeronautics and Space Administration.
|
1,116,691,500,733 | arxiv | \section{Introduction}
\aj{This paper continues the theme of the book ``Extending the Theory of Composites to Other Areas of Science'' \cite{Milton:2016:ETC},
(reviewed in \cite{Sharma:2017:BRE,Grabovsky:2018:BRE})
using tools in the theory of composites \cite{Cherkaev:2000:VMS,Torquato:2002:RHM,Milton:2002:TOC,Allaire:2002:SOH,Tartar:2009:GTH}
to obtain results pertinant to other problems. Specifically in this paper we use ideas from the Theory of Composites to
derive bounds on the generalized spectrum of the operator whose inverse, when it exists, gives the Green's function.}
Here we modify the definition of the Green's function to avoid unnecessary complexities of having to deal with complicated
function spaces, and to simplify the analysis. We believe this is a much better approach to operator theory than conventionally taken.
The approach builds upon that developed in \cite{Milton:2018:ERG} where the infinite body Green's function, defined in the appropriate way,
was found to satisfy certain exact identities for wide classes of inhomogeneous media. \aj{These exact identities generalize the theory
of exact relations for composites \cite{Grabovsky:1998:EREa,Grabovsky:1998:EREb,Grabovsky:2000:ERE} (see also Chapter 17 of \cite{Milton:2002:TOC} and the book \cite{Grabovsky:2016:CMM}) that, like our analysis, derives from the splitting of the relevant Hilbert space into orthogonal subspaces.
Such exact relations identify tensor manifolds such that the effective tensor lies on the manifold whenever the local tensor takes values
in the manifold.}
There is, of course, an enormous body of literature establishing bounds on the spectra of operators, that reflects the importance of this problem. For compact Hermitian operators
on Hilbert spaces the Courant-Fischer-Weyl min-max principle gives a variational characterization of the eigenvalues, and can be used to bound the eigenvalues through the Cauchy
interlacing principle. Horn's theorem \cite{Horn:1954:DSM} bounds the eigenvalues of a Hermitian matrix in terms of its diagonal elements.
Various inequalities
\cite{Weyl:1912:AVE,Lidskii:1950:PVS,Wielandt:1955:OSE,Horn:1962:BPI,Thompson:1971:ESH,Helmke:1995:EIS,Klyachko:1954:DSM} relate the eigenvalues
of a sum of Hermitian matrices to the eigenvalues of the matrices in the sum: an excellent summary can be found in the paper of Fulton \cite{Fulton:2000:EIF}.
Such eigenvalue problems can be regarded as determining when certain linear pencils of Hermitian matrices have a nontrivial nullspace, when one of the matrices is the identity matrix.
Here we are concerned with such questions for linear operators rather than matrices (without the restriction that one of the operators is the identity operator)
and the bounds we obtain incorporate information about the underlying
partial differential equation, using Null-Lagrangians, quasiconvexity and its generalizations
\cite{Tonelli:1920:SCV,Terpstra:1938:DBF,Morrey:1952:QSM,Meyers:1965:ASA,Morrey:1966:MIC,Murat:1978:CPC,Tartar:1979:CCA,
Tartar:1979:ECH,Ball:1981:NLW,Murat:1981:CPC,Murat:1985:CVH,Tartar:1985:EFC,Benesova:2017:WLS},
notably using the notion of $Q^*$-convexity \cite{Milton:2013:SIG,Milton:2015:ATS}. Interestingly, there
are other connections between the spectrum of matrices, quasiconvexity, and rank-one convexity aside from those developed here: see \cite{Buliga:208:FAM} and references therein.
\aj{It is well known that the classic equation of electrical conductivity in the absence of current sources $\nabla \cdot\bfm\sigma\nabla V=0$ in which $V({\bf x})$ is the electric potential and $\bfm\sigma({\bf x})$ is the conductity tensor, can be alternatively
formulated as
\begin{equation} {\bf j}({\bf x})=\bfm\sigma({\bf x}){\bf e}({\bf x}),\quad\nabla \cdot{\bf j}({\bf x})=0,\quad{\bf e}({\bf x})=-\nabla V, \eeq{0.0}
where ${\bf j}$ and ${\bf e}$ is the electrical current and field. This alternative formulation, involving fields satisfying differential constraints
linked by a constitutive law involving a tensor that contains information about the material moduli, and its generalizations, forms the basis for much analysis in the theory
of composites (see, e.g. \cite{Milton:2002:TOC}), and has been advocated, for example, by Strang \cite{Strang:1988:FEE}. As shown in \cite{Milton:2009:MVP} and in Chapter 1 of \cite{Milton:2016:ETC}, and as will
be reviewed below (see also the appendix), the formulation can be extended to wave equations in the frequency domain
and to the Schr{\"o}dinger equation. It provides a route to obtaining results that may not be directly evident from standard formulations.}
\aj{The formulation naturally extends to the} large variety of linear equations in physics that can be written as a system of second-order linear partial differential equations:
\begin{equation} \sum_{i=1}^d\frac{\partial}{\partial x_i}\left(\sum_{j=1}^d\sum_{\beta=1}^sL_{i\alpha j\beta}({\bf x})\frac{\partial u_\beta({\bf x})}{\partial x_j}\right)=f_\alpha({\bf x}), \quad\alpha=1,2,\ldots,s,\eeq{0}
for the $s$-component potential ${\bf u}({\bf x})$. If the integral of ${\bf f}({\bf x})$ over ${\mathbb R}^d$ is zero, these can be reexpressed as
\begin{equation} J_{i\alpha}({\bf x})=\sum_{j=1}^d\sum_{\beta=1}^sL_{i\alpha j\beta}({\bf x})E_{j\beta}({\bf x})-h_{i\alpha}({\bf x}),\quad E_{j\beta}({\bf x})=\frac{\partial u_\beta({\bf x})}{\partial x_j},\quad \sum_{i=1}^d\frac{\partial J_{i\alpha}({\bf x})}{\partial x_i}=0,
\eeq{1}
where, counter to the usual convention, we find it convenient to let the divergence act on the first index of ${\bf J}$, and to let the gradient in ${\bf E}=\nabla{\bf u}$ be associated
with the first index of ${\bf E}$, and ${\bf h}({\bf x})$ is chosen so
\begin{equation} \sum_{i=1}^d\frac{\partial h_{i\alpha}({\bf x})}{\partial x_i}=f_\alpha({\bf x}). \eeq{2}
Examples include, for instance, the electrostatics equation, linear elastostatics equation, piezoelectric equation, the quasistatic acoustic, Maxwell, and elastodynamic, equations
(where the fields and moduli are generally complex): see Chapter 2 in \cite{Milton:2002:TOC} for numerous examples.
If we restrict attention to the space of square integrable fields, so that ${\bf E}({\bf x})$ and ${\bf J}({\bf x})$ are square-integrable, integration by parts shows that
\begin{equation} ({\bf J},{\bf E})\equiv\int_{{\mathbb R}^d} ({\bf J}({\bf x}),{\bf E}({\bf x}))_{\cal T}\,d{\bf x}=0,\quad ({\bf J}({\bf x}),{\bf E}({\bf x}))_{\cal T}=\sum_{i=1}^d\sum_{\alpha=1}^s J_{i\alpha}({\bf x})[E_{i\alpha}({\bf x})]^*,
\eeq{3}
where the asterisk denotes complex conjugation. Thus ${\bf E}({\bf x})$ and ${\bf J}({\bf x})$ belong to orthogonal spaces: ${\cal E}$ the set of square-integrable fields ${\bf E}({\bf x})$ such that ${\bf E}=\nabla{\bf u}$
for some $m$ component potential ${\bf u}$, and ${\cal J}$ the set of square-integrable fields ${\bf J}({\bf x})$ such that $\nabla \cdot{\bf J}=0$. With these definitions, the equations \eq{1}
take the equivalent, more abstract, form
\begin{equation} {\bf J}({\bf x})={\bf L}({\bf x}){\bf E}({\bf x})-{\bf h}({\bf x}),\quad {\bf J}\in{\cal J},\quad {\bf E}\in{\cal E},\quad {\bf h}\in{\cal H}.
\eeq{4}
where ${\cal H}={\cal E}\oplus{\cal J}$ consists of square integrable $d\times m$ matrix-valued fields. \aj{The benefit of this more abstract formulation is that it brings under the
one umbrella a wide variety of physical equations, and enables one to develop results, such as exact identities satisfied by the Green's function \cite{Milton:2018:ERG}, that
may be obscured in more conventional approaches. Also the orthogonality of the spaces ${\cal E}$ and ${\cal J}$ naturally leads to minimization variational principles (even
for wave equations in lossy media) and the norms of relevant operators can often be easily estimated, thanks to the fact that the projection operators onto
${\cal E}$ and ${\cal J}$, like any projection, have norm one. Additionally, the formalism allows one to see that the ``Dirichlet-to-Neumann'' map that governs the response
of inhomogeneous bodies has Herglotz-type analytic properties not only as a function of frequency, but also as a function of the component moduli, and consequently there are
associated integral representation formulas for this map (see Chapters 3 and 4 in \cite{Milton:2016:ETC}, also arXiv:1512.05838 [math.AP]).}
Given any simply-connected region $\Omega$ with smooth
boundary $\partial\Omega$, the fields in ${\cal E}$ and ${\cal J}$ also satisfy the key identity
\begin{equation} \int_\Omega ({\bf J}({\bf x}),{\bf E}({\bf x}))_{\cal T}\,d{\bf x}=\int_{\partial\Omega}B(\partial{\bf J},\partial{\bf E})\,dS\text{ for any }{\bf E}\in{\cal E},\quad{\bf J}\in{\cal J},
\eeq{5}
where $\partial{\bf E}$ and $\partial{\bf J}$ denote the boundary fields associated with ${\bf E}$ and ${\bf J}$ respectively,
and $B(\partial{\bf J},\partial{\bf E})$ is linear in $\partial{\bf J}$ and antilinear $\partial{\bf E}$, i.e., for any $c_1,c_2\in\mathbb{C}$,
\begin{equation} B(c_1\partial{\bf J},c_2\partial{\bf E})=c_1 c_2^*B(\partial{\bf J},\partial{\bf E}), \eeq{5.aa}
in which $c_2^*$ denotes the complex conjugate of $c_2$.
In the context of the equations \eq{1}, the boundary field $\partial{\bf E}$ can be identified with the
potential ${\bf u}({\bf x})$ for ${\bf x}\in\partial\Omega$, and $\partial{\bf J}$ can be identified with the boundary fluxes ${\bf n}({\bf x})\cdot{\bf J}({\bf x})$ where ${\bf n}({\bf x})$
is the outwards normal to $\partial\Omega$. \aj{ We are avoiding a precise definition of $\partial{\bf J}$ and $\partial{\bf E}$ as these boundary fields are not central to the paper,
they depend on the problem of interest, and are not so easily defined in
cases (such as plate equations) where the differential constraints on the fields are higher than first order. Roughly speaking,
they are associated with physical problems for which the response of a body is governed by some ``Dirichlet-to-Neumann'' map and $\partial{\bf J}$ and $\partial{\bf E}$ are the boundary fields
associated with this map. Various examples of the boundary fields $\partial{\bf J}$ and $\partial{\bf E}$ will be given below, and in the appendix.}
More generally, other equations of physics, including heat and wave equations, can be expressed in the form \eq{4} with an identity
like \eq{5} holding. For example, at fixed frequency $\omega$ with a $e^{-i\omega t}$ time dependence, as recognized in \cite{Milton:2009:MVP}
the acoustic equations, with $P({\bf x})$ the pressure, ${\bf v}({\bf x})$ the velocity, $\bfm\rho({\bf x},\omega)$ the effective mass density matrix, and $\kappa({\bf x},\omega)$ the bulk modulus, take the form
\begin{equation} \underbrace{\begin{pmatrix}-i{\bf v} \\ -i\nabla \cdot{\bf v} \end{pmatrix}}_{{\bf J}({\bf x})}
=\underbrace{\begin{pmatrix}-(\omega\bfm\rho)^{-1} & 0 \\ 0 & \omega/\kappa\end{pmatrix}}_{{\bf L}({\bf x})}\underbrace{\begin{pmatrix}\nabla P \\ P\end{pmatrix}}_{{\bf E}({\bf x})},
\eeq{5.A}
(and $\partial{\bf E}$ and $\partial{\bf J}$ can be identified with the boundary values of $P({\bf x})$ and ${\bf n}\cdot{\bf v}({\bf x})$ at $\partial\Omega$, respectively).
Here we allow for effective mass density matrices that, at a given frequency, can be anisotropic and complex valued as may be the case in metamaterials
\cite{Schoenberg:1983:PPS,Willis:1985:NID,Milton:2006:CEM,Milton:2007:MNS}.
Maxwell's equations, with ${\bf e}({\bf x})$ the electric field, ${\bf h}({\bf x})$ the magnetizing field, $\bfm\mu({\bf x},\omega)$ the magnetic permeability,
$\bfm\varepsilon({\bf x})$ the electric permittivity, take the form
\begin{equation} \underbrace{\begin{pmatrix}-i{\bf h} \cr i\nabla \times{\bf h}\end{pmatrix}}_{{\bf J}({\bf x})}
=\underbrace{\begin{pmatrix}-{[\omega\bfm\mu]}^{-1} & 0 \\ 0 & \omega\bfm\varepsilon \end{pmatrix}}_{{\bf L}({\bf x})}
\underbrace{\begin{pmatrix}\nabla \times{\bf e} \\ {\bf e}\end{pmatrix}}_{{\bf E}({\bf x})},
\eeq{5.B}
(and $\partial{\bf E}$ and $\partial{\bf J}$ can be identified with the tangential values of ${\bf e}({\bf x})$ and ${\bf h}({\bf x})$ at $\partial\Omega$, respectively).
The linear elastodynamic equations, with ${\bf u}({\bf x})$ the displacement, $\bfm\sigma({\bf x})$ the stress, ${\bfm{\cal C}}({\bf x},\omega)$ the elasticity tensor,
$\bfm\rho({\bf x},\omega)$ the effective mass density matrix, take the form
\begin{equation} \underbrace{\begin{pmatrix} -\bfm\sigma/\omega \\ -\nabla \cdot\bfm\sigma/\omega\end{pmatrix}}_{{\bf J}({\bf x})}
=\underbrace{\begin{pmatrix}-{\bfm{\cal C}}/\omega & 0 \\ 0 & \omega\bfm\rho\end{pmatrix}}_{{\bf L}({\bf x})}\underbrace{\begin{pmatrix}\nabla {\bf u} \\ {\bf u}\end{pmatrix}}_{{\bf E}({\bf x})},
\eeq{5.C}
(and $\partial{\bf E}$ and $\partial{\bf J}$ can be identified with the values of ${\bf u}({\bf x})$ and the traction ${\bf n}\cdot\bfm\sigma({\bf x})$ at $\partial\Omega$, respectively)
where we note that it is not necessary to introduce the strain field (symmetrized gradient of the displacement ${\bf u}$) as the elasticity tensor
${\bfm{\cal C}}$ annihilates the antisymmetric part of $\nabla{\bf u}$. The preceeding three equations have been written in this form so $\mathop{\rm Im}\nolimits{\bf L}({\bf x})\geq 0$
when $\mathop{\rm Im}\nolimits\omega\geq 0$, where complex frequencies have the physical meaning of the solution increasing exponentially in time.
Using an approach of Gibiansky and Cherkaev \cite{Cherkaev:1994:VPC} this allows one to express the solution as the minimum of
some appropriately defined functional \cite{Milton:2009:MVP,Milton:2010:MVP}. The Schr{\"o}dinger equation for the wavefunction $\psi({\bf x})$
of a single electron in a magnetic field, in the time domain, with ${\bf b}=\nabla \times\bfm\Phi$ the magnetic induction,
$V({\bf x},t)$ the time-independent electric potential, $e$ is the charge on the electron, and $m$ its mass,
takes the form \cite{Milton:2016:ETC}:
\begin{equation}
\underbrace{\begin{pmatrix}
{\bf q}_x\\
q_t\\
\nabla\cdot{\bf q}_x+\frac{\partial q_t}{\partial t}
\end{pmatrix}}_{{\bf J}({\bf x})}
=
\underbrace{\begin{pmatrix}
\frac{-{\bf I}}{2m} & 0 & \frac{{\rm i} e \bfm\Phi}{2m}\\
0 & 0 & -\frac{{\rm i}}{2}\\
\frac{-{\rm i} e \bfm\Phi}{2m} & +\frac{{\rm i}}{2} & -e V
\end{pmatrix}}_{{\bf L}({\bf x})}
\underbrace{\begin{pmatrix}
\nabla\psi\\
\frac{\partial\psi}{\partial t}\\
\psi
\end{pmatrix}}_{{\bf E}({\bf x})},
\eeq{5.D}
in which dimensions have been chosen so that $\hbar=1$, where $\hbar$ is Planck's constant divided by $2\pi$
(and $\partial{\bf E}$ and $\partial{\bf J}$ can be identified with the values of $\psi({\bf x})$ and the flux ${\bf n}_x\cdot{\bf q}_x({\bf x})+n_tq_t$ at $\partial\Omega$, respectively,
in which $({\bf n}_x,n_t)$ is the outwards normal to the region $\Omega$ in space-time).
The multielectron Schr{\"o}dinger equation with a time dependence $e^{-iEt/\hbar}$ and with $\hbar=1$ takes the form \cite{Milton:2016:ETC}:
\begin{equation}
\underbrace{\begin{pmatrix}{\bf q}({\bf x}) \\ \nabla \cdot{\bf q}({\bf x})\end{pmatrix}}_{{\bf J}({\bf x})}
=\underbrace{\begin{pmatrix}-{\bf A} & 0 \\ 0 & E-V({\bf x})\end{pmatrix}}_{{\bf L}({\bf x})}\underbrace{\begin{pmatrix}\nabla \psi({\bf x}) \\ \psi({\bf x})\end{pmatrix}}_{{\bf E}({\bf x})},
\eeq{5.E}
where $V({\bf x})$ is the potential and ${\bf A}$ in the simplest approximation is ${\bf I}/(2m)$ in which $m$ is the
mass of the electron, but it may take other forms to take into account the reduced mass of the
electron, or mass polarization terms due to the motion of the atomic nuclei. Here ${\bf x}$ lies in a multidimensional space ${\bf x}=({\bf x}_1,{\bf x}_2,\ldots,{\bf x}_N)$ where following,
for example, \cite{Parr:1994:DFT}, each ${\bf x}_i$ represents a pair $({\bf r}_i,s_i)$ where ${\bf r}_i$ is a three dimensional vector
associated with the position of electron $i$ and
$s_i$ denotes its spin (taking discrete values $+1/2$ for spin up or $-1/2$ for spin down). Accordingly, $\nabla$ represents the operator
\begin{equation}
\nabla = (\nabla_1,\nabla_2,\ldots,\nabla_N),\quad\text{where}~~
\nabla_j = \left(\frac{\partial}{\partial
r_1^{(j)}},\frac{\partial}{\partial r_2^{(j)}},\frac{\partial}{\partial
r_3^{(j)}}\right).
\eeq{5.F}
When the energy $E$ is complex, $E=E'+iE"$, then minimization variational principles for the multielectron Schr{\"o}dinger equation exist (see Chapter 13 in \cite{Milton:2016:ETC}).
\aj{Specifically, consider the functional
\begin{equation} W(\psi')=\sum_s\int_{\Omega^N}[p({\bf x},\psi')]^2+(E'')^2|\psi'({\bf x})|^2~d{\bf r},\quad p({\bf x},\psi')=\nabla \cdot{\bf A}\nabla\psi'+(E'-V({\bf x}))\psi',
\eeq{5.Fa}
where the sum is over all $2^N$ spin configurations $s=(s_1,s_2,\ldots,s_N)$ as each $s_j$ takes values $+1/2$ or $-1/2$. Then in any body $\Omega$ with appropriate boundary
conditions on $\psi'$ on $\partial\Omega^N$, $W(\psi')$ is minimized when $\psi'$ is the real part of the wave function $\psi$ which satisfies the Schr{\"o}dinger equation
\begin{equation} \nabla \cdot{\bf A}\nabla\psi+(E-V({\bf x}))\psi=0, \eeq{5.Fb}
in which the potential $V({\bf x})$ is assumed to be real. Following the prescription outlined in \cite{Milton:2010:MVP}, a variety of different boundary conditions on $\psi'$ can be handled by suitably adjusting the functional $W(\psi')$: see Section 13.3 in \cite{Milton:2016:ETC}.}
With only pair potentials the multielectron Schr{\"o}dinger equation is also equivalent to the desymmetrized multielectron Schr{\"o}dinger equation
which takes the form
\begin{equation} {\bf J}({\bf x})=\underbrace{\begin{pmatrix}-{\bf A} & 0 \\ 0 & E-g({\bf x}_1,{\bf x}_2)\end{pmatrix}}_{{\bf L}({\bf x})}\underbrace{\begin{pmatrix}\nabla \psi({\bf x}) \\ \psi({\bf x})\end{pmatrix}}_{{\bf E}({\bf x})},
\quad \bfm\Lambda{\bf J}({\bf x})=\begin{pmatrix}{\bf q}({\bf x}) \\ \nabla \cdot{\bf q}({\bf x})\end{pmatrix},\quad \bfm\Lambda{\bf E}({\bf x})={\bf E}({\bf x}),
\eeq{5.G}
where $\bfm\Lambda$ is an appropriate symmetrization operator defined in Chapter 12 of \cite{Milton:2016:ETC}. The desymmetrized multielectron Schr{\"o}dinger equation has the advantage
(with complex values of $E$ and source terms) that it can be solved iteratively by going back and forth between real and Fourier space using Fast Fourier
transforms where the Fourier transforms only need to be done on the two variables ${\bf x}_1$ and ${\bf x}_2$, not on all variables. Many additional equations
can be expressed in the canonical form: these include, at least, the thermoacoustic equations at constant frequency, and in the time domain
(without assuming a $e^{-i\omega t}$ time dependence) the acoustic, Maxwell, elastodynamic, piezoelectric, and plate equations,
Biot poroelastic equations, thermal conduction and diffusion equations,
thermoelastic equation, and the Dirac equation for the electron. The interested reader is referred to Chapter 1 of \cite{Milton:2016:ETC} for more details.
The formalism also extends to scattering problems, where one needs to introduce an auxiliary space ``at infinity'' to ensure orthogonality
of appropriately defined spaces ${\cal E}$ and ${\cal J}$ \cite{Milton:2017:BCP}.
Of course, in all these equations one can allow for an additional source term ${\bf h}({\bf x})$, as is needed to define the Green's function. For wave equations in the
time domain, rather than the frequency domain, one has to impose boundary conditions appropriate to selecting the causal Green's function. To avoid such complications
our focus will be on static and quasistatic equations, and on wave equations in the frequency domain. \aj{Beyond the forms \eq{5.A}, \eq{5.B}, and \eq{5.C} for acoustics,
electromagnetism, and elastodynamics, canonical forms in the frequency domain for thermoacoustics, thermoelasticity, and the plate equations, are given in the appendix}.
The classical Poincare inequality seeks bounds on the lowest eigenvalue of the Dirichlet Laplacian operator that is associated with the equation
\begin{equation} \underbrace{\begin{pmatrix}{\bf Q} \\ \nabla \cdot{\bf Q} \end{pmatrix}}_{{\bf J}({\bf x})}
= \underbrace{\begin{pmatrix} 1 & 0 \\ 0 & -z\end{pmatrix}}_{{\bf L}}\underbrace{\begin{pmatrix}\nabla {\bf u} \\ {\bf u}\end{pmatrix}}_{{\bf E}({\bf x})},
\eeq{5.Ga}
on a domain $\Omega$ with ${\bf u}=0$ on $\partial\Omega$. Bounds on higher order eigenvalues have been obtained too: see, for example, \cite{Ilyin:2006:LBS} and references
therein.
In general, the fields ${\bf J}({\bf x})$, ${\bf E}({\bf x})$, and ${\bf h}({\bf x})$ take values in some tensor space ${\cal T}$. The tensor space has the characteristic feature that
there is a inner product $(\cdot,\cdot)_{\cal T}$ on ${\cal T}$ such that for every rotation ${\bf R}$ in $\mathbb{R}^d$ there is a corresponding operator ${\cal R}({\bf R}){\cal T}\to{\cal T}$
such that ${\cal R}({\bf I})={\bf I}$
\begin{equation} ({\cal R}({\bf R}){\bf A},{\cal R}({\bf R}){\bf B})_{\cal T}=({\bf A},{\bf B})_{\cal T}\text{ for all }{\bf A},{\bf B}\in{\cal T}.
\eeq{5b}
This inner product assigned to ${\cal T}$ is useful when one is considering a body containing a $m$-phase polycrystalline material with moduli
\begin{eqnarray} &~& {\bf L}({\bf x})=[{\cal R}({\bf R}({\bf x}))]^\dagger\widetilde{{\bf L}}({\bf x}){\cal R}({\bf R}({\bf x})),\quad \widetilde{{\bf L}}({\bf x})=\sum_{i=1}^m\chi_i({\bf x}){\bf L}_i,\nonumber \\
&~& [{\cal R}({\bf R}({\bf x}))]^\dagger{\cal R}({\bf R}({\bf x}))={\bf I},\quad \sum_{i=1}^m\chi_i({\bf x})=1,
\eeqa{10}
where the ${\bf L}_i$ are the tensors of the individual phases, $\chi_i({\bf x})$ is the characteristic functions of phase $i$, taking the value $1$ in phase $i$ and zero
outside of it, and ${\bf R}({\bf x})$ is a rotation field giving the orientation of the material at the point ${\bf x}$. A particular, but important, case is a multicomponent body
where ${\bf R}({\bf x})={\bf I}$ and ${\cal R}({\bf R}({\bf x}))={\bf I}$ for all ${\bf x}$ so that \eq{10} reduces to
\begin{equation} {\bf L}({\bf x})=\sum_{i=1}^m\chi_i({\bf x}){\bf L}_i, \quad \sum_{i=1}^m\chi_i({\bf x})=1.
\eeq{10A}
We will not generally require
that the inner product assigned to ${\cal T}$ satisfy the property \eq{5b}. There is good reason for removing this constraint. Indeed, in say three dimensions, with $m=3$ the potentials
$u_1({\bf x})$, $u_2({\bf x})$, and $u_3({\bf x})$ could represent three different scalar potentials such as voltage, temperature, and pressure. However it is mathematically equivalent
to the problem of linear elasticity where these represent components of the displacement field ${\bf u}({\bf x})$. Despite the mathematical equivalence, the potentials they behave quite differently
under rotations, and thus ${\cal R}({\bf R})$ is different.
In most applications the spaces ${\cal H}$, ${\cal E}$ and ${\cal J}$ are real-symmetric, in the sense that
if a field belongs to them, then so does the complex conjugate field. Our analysis goes through without this assumption:
it will only be used just below equation \eq{F.3a} to simplify the criterion \eq{F.3}.
\aj{Our focus is on the generalized spectrum rather than the spectrum, and there is good reason for that. The spectrum typically consists of the range of values of the
energy, frequency (or frequency squared), or wave number away from which the Green's function exists. However, in many applications the component moduli also
depend upon frequency, and this greatly complicates the picture. Thus it makes sense to separate the dependence on the component moduli, as
embodied in the generalized spectrum. Varying the frequency then corresponds to following a trajectory in the space of component moduli
that may intersect the generalized spectrum, and at frequencies
at these intersections the Green's function does not exist. }
\section{Showing the Green's function exists when suitable boundedness and coercivity conditions are satisfied}
For fields ${\bf P}({\bf x})$ and ${\bf P}'({\bf x})$ defined within $\Omega$ we define the inner product and norm
\begin{equation} ({\bf P},{\bf P}')_\Omega=\frac{1}{V(\Omega)}\int_\Omega ({\bf P}({\bf x}),{\bf P}'({\bf x}))_{\cal T}\,d{\bf x},\quad |{\bf P}|_\Omega=\sqrt{({\bf P},{\bf P})_\Omega},
\eeq{6}
where $V(\Omega)$ is the volume of $\Omega$.
Define ${\cal H}_{\Omega}$ to consist of those fields ${\bf P}$ taking values in ${\cal T}$ that are square integrable over ${\bf Q}$ in the sense that the norm $|{\bf P}|_\Omega$ is finite.
Define ${\cal E}_\Omega^0\subset {\cal H}_{\Omega}$ as the restriction to $\Omega$ of those fields in ${\cal E}$ satisfying the boundary constraint $\partial{\bf E}=0$.
Alternatively ${\cal E}_\Omega^0$ can be viewed as consisting
of those fields ${\bf E}({\bf x})$ in ${\cal H}_{\Omega}$ that when extended to ${\cal H}$ by defining ${\bf E}({\bf x})=0$ outside $\Omega$, have the property that this extended field
lies in ${\cal E}$. We define ${\cal J}_\Omega\subset {\cal H}_{\Omega}$ as the restriction to $\Omega$ of those fields in ${\cal J}$
(with no boundary constraint), and we let $\overline{\cal J}_\Omega$ denote its closure. The subspaces ${\cal E}_\Omega^0$ and $\overline{\cal J}_\Omega$ are orthogonal and span ${\cal H}_{\Omega}$
as proved in Lemma 2.3 of \cite{Grabovsky:2016:CMM}.
To define the Green's operator, with the boundary condition $\partial{\bf E}=0$,
the equations of interest now take the form
\begin{equation} {\bf J}({\bf x})={\bf L}({\bf x}){\bf E}({\bf x})-{\bf h}({\bf x}),\quad {\bf J}\in\overline{\cal J}_\Omega,\quad {\bf E}\in{\cal E}_\Omega^0,\quad {\bf h}\in{\cal H}_\Omega.
\eeq{6a}
We assume that within $\Omega$, ${\bf L}({\bf x})$ takes the form of a pencil of linear operators:
\begin{equation} {\bf L}({\bf x})=\sum_{i=1}^nz_i{\bf L}^{(i)}({\bf x}).
\eeq{7}
For example, in the setting of the polycrystalline body \eq{10}, the $z_1$, $z_2$, $\ldots$, $z_n$ could be taken as the matrix elements (in some representation)
of the $m$ tensors ${\bf L}_1$, ${\bf L}_2$,$\ldots$, ${\bf L}_m$. Alternatively we may just write ${\bf L}({\bf x})={\bf L}({\bf x},{\bf L}_1,{\bf L}_2,\ldots,{\bf L}_m)$
where it is implicitly understood that this is being regarded as a function of the elements of all the matrices ${\bf L}_1$, ${\bf L}_2$,$\ldots$,${\bf L}_m$ using
a basis of ${\cal T}$ to represent these as matrices.
Let $\bfm\Gamma_1^{\Omega}$ denote the self-adjoint projection into ${\cal E}_\Omega^0$, which then annihilates any field in $\overline{\cal J}_\Omega$. Applying it to both sides of
\eq{6a} we get
\begin{equation} 0={\bf M}{\bf E}-\bfm\Gamma_1^{\Omega}{\bf h},\text{ where } {\bf M}=\bfm\Gamma_1^{\Omega}{\bf L}\bfm\Gamma_1^{\Omega}=\sum_{i=1}^nz_i{\bf M}_i,\quad {\bf M}^{(i)}=\bfm\Gamma_1^{\Omega}{\bf L}^{(i)}\bfm\Gamma_1^{\Omega}. \eeq{7.a}
The generalized spectrum of ${\bf M}$ can then be defined as the set of those values ${\bf z}=(z_1,z_2,\ldots,z_n)$ in $\mathbb{C}^n$ where the
operator ${\bf M}$ does not have an inverse on the space ${\cal E}_\Omega^0$. Outside the generalized spectrum the inverse exists and defines the Green's function ${\bf G}_\Omega$:
\begin{equation} {\bf E}={\bf G}_\Omega{\bf h},\quad {\bf G}_\Omega=\bfm\Gamma_1^{\Omega}{\bf M}^{-1}\bfm\Gamma_1^{\Omega}=\bfm\Gamma_1^{\Omega}(\bfm\Gamma_1^{\Omega}{\bf L}\bfm\Gamma_1^{\Omega})^{-1}\bfm\Gamma_1^{\Omega}.
\eeq{7.b}
Note that there is an equivalence class of sources ${\bf h}$ that yield the same field ${\bf E}$: we can add to ${\bf h}$ any field in $\overline{\cal J}_\Omega$ without disturbing
${\bf E}$. In the context of the equations \eq{0} there is still only a unique ${\bf E}({\bf x})$ associated with a given source ${\bf f}({\bf x})$, since if
$\nabla \cdot{\bf h}'=\nabla \cdot{\bf h}={\bf f}$ then ${\bf h}'-{\bf h}\in\overline{\cal J}_\Omega$.
We first show that ${\bf M}$ has an inverse on ${\cal E}_\Omega^0$ if ${\bf z}=(z_1,z_2,\ldots,z_n)$ takes values in a domain ${\cal Z}(\alpha,\beta,\theta)$ of $\mathbb{C}^n$, defined as
that domain where ${\bf L}$ satisfies the following boundedness and coercivity conditions:
\begin{equation} \beta >\sup_{\substack{{\bf P}\in{\cal H} \\ |{\bf P}|=1}}|{\bf L}{\bf P}|_\Omega, \text{ (Boundedness)}
\eeq{8}
\begin{equation} \mathop{\rm Re}\nolimits (e^{i\theta}{\bf L}{\bf E},{\bf E})_\Omega\geq \alpha|{\bf E}|_\Omega^2 \text{ for all } {\bf E}\in{\cal E}_{\Omega}. \text{ (Coercivity)}
\eeq{9}
The latter is the usual coercivity condition and makes sense: if ${\bf E}\in{\cal E}_\Omega^0$ is in the point spectrum of ${\bf M}$ in the sense that ${\bf M}{\bf E}=0$ for some $z_1,z_2,\ldots,z_n$ then clearly
$({\bf L}{\bf E},{\bf E})_\Omega=0$ and \eq{9} cannot hold. Under these boundedness and coercivity conditions, we will see that given any ${\bf h}$ there is a unique solution to \eq{6a} for ${\bf E}$,
which thus defines the Green's operator ${\bf G}_\Omega$ in \eq{7.b}.
To show uniqueness, suppose for some given ${\bf h}$ that there is another solution ${\bf E}'\in{\cal E}_\Omega^0$
and ${\bf J}'\in\overline{\cal J}_\Omega$. Subtracting solutions we get
\begin{equation} {\bf J}-{\bf J}'={\bf L}({\bf E}-{\bf E}'). \eeq{11a}
The coercivity condition \eq{9} with ${\bf P}={\bf E}-{\bf E}'$ implies
\begin{equation} \alpha|{\bf E}-{\bf E}'|^2=\alpha|{\bf P}|^2\leq \mathop{\rm Re}\nolimits ({\bf P},e^{i\theta}{\bf L}{\bf P}) \leq \mathop{\rm Re}\nolimits({\bf E}-{\bf E}',e^{i\theta}({\bf J}-{\bf J}')).
\eeq{11b}
By the orthogonality of ${\cal E}_\Omega^0$ and $\overline{\cal J}_\Omega$ the expression on the right is zero which forces ${\bf E}={\bf E}'$, thus establishing uniqueness.
To establish existence, we let $\bfm\Gamma_1^{\Omega}$ denote the projection into ${\cal E}_\Omega^0$,
which then annihilates any field in $\overline{\cal J}_\Omega$. Rewrite the constitutive law in \eq{4} as
\begin{equation} {\bf J}({\bf x})-c{\bf E}({\bf x})=\delta{\bf L}({\bf x}){\bf E}({\bf x})-{\bf h}({\bf x}),\quad \delta{\bf L}({\bf x})={\bf L}({\bf x})-c{\bf I},
\eeq{14}
where $c=|c|e^{-i\theta}$, and $|c|$ is a free constant that ultimately will be chosen very large to ensure convergence of a series expansion for ${\bf G}_\Omega$.
Applying the operator $\bfm\Gamma_1^{\Omega}/c$ to both sides, we get
\begin{equation} -{\bf E}=(\bfm\Gamma_1^{\Omega}\delta{\bf L}/c){\bf E}-\bfm\Gamma_1^{\Omega}{\bf h}/c
\eeq{15}
giving \eq{7.b} with
\begin{equation} {\bf G}_\Omega=[{\bf I}+\bfm\Gamma_1^{\Omega}\delta{\bf L}/c]^{-1}\bfm\Gamma_1^{\Omega}/c
\eeq{18}
as our Green's function operator. Expanding this formulae, and using the fact that $(\bfm\Gamma_1^{\Omega})^2=\bfm\Gamma_1^{\Omega}$ we obtain the series expansion
\begin{equation} {\bf G}_\Omega=-(\bfm\Gamma_1^{\Omega}/c)\sum_{j=0}^\infty [\bfm\Gamma_1^{\Omega}(-\delta{\bf L}/c)\bfm\Gamma_1^{\Omega}]^j.
\eeq{19}
We want to show this converges to ${\bf G}_\Omega$ when $|c|$ is sufficiently large, and thus establishes the existence of the solution ${\bf E}$ to \eq{6a} when
${\bf h}$ is given.
Now, for any field ${\bf E}\in{\cal E}_\Omega^0$, introduce ${\bf E}'=-\bfm\Gamma_1^{\Omega}(\delta{\bf L}/c){\bf E}=\bfm\Gamma_1^{\Omega}({\bf I}-{\bf L}/c){\bf E}$ giving $\bfm\Gamma_1^{\Omega}{\bf L}{\bf E}/c={\bf E}-{\bf E}'$. The boundedness of ${\bf L}$ and implies
\begin{equation} \beta^2|{\bf E}|^2/|c|^2\geq |{\bf E}-{\bf E}'|^2,
\eeq{20}
and by expanding $|{\bf E}-{\bf E}'|^2$ we get
\begin{equation} 2\mathop{\rm Re}\nolimits ({\bf E},{\bf E}')\geq |{\bf E}'|^2+ [1-(\beta/|c|)^2]|{\bf E}|^2.
\eeq{21}
Now recall that $c$ has been chosen so $c/|c|=e^{-i\theta}$. Then the coercivity \eq{9} implies
\begin{equation}
\mathop{\rm Re}\nolimits ({\bf E},{\bf E}-{\bf E}') \geq \alpha|{\bf E}|^2/|c|.
\eeq{22}
By combining this with \eq{21} we obtain
\begin{equation} [1+(\beta/|c|)^2-2\alpha/|c|]|{\bf E}|^2\geq |{\bf E}'|^2.
\eeq{23}
Let
\begin{equation} \|{\bf S}\|_\Omega= \sup_{\substack{{\bf P}\in{\cal H} \\ |{\bf P}|_\Omega=1}}|{\bf S}{\bf P}|_\Omega \eeq{23.a}
denote the standard norm of an operator ${\bf S}$. Then we get
\begin{eqnarray} &~&\|\bfm\Gamma_1^{\Omega}(\delta{\bf L}/c)\bfm\Gamma_1^{\Omega}\|_\Omega^2 = \sup_{\substack{{\bf E}\in{{\cal E}_\Omega^0} \\ |{\bf E}|_\Omega=1}}(\bfm\Gamma_1^{\Omega}(\delta{\bf L}/c){\bf E},\bfm\Gamma_1^{\Omega}(\delta{\bf L}/c){\bf E})_\Omega \nonumber \\
&~& \quad\quad\quad \leq 1+(\beta/|c|)^2-2\alpha/|c|=1-(\alpha/\beta)^2 \text{ with }|c|=\beta^2/\alpha.
\eeqa{23.b}
Clearly this is less than $1$ and the series \eq{19} converges.
Incidentally, if we truncate the series expansion \eq{19}, the truncated expansion will be a polynomial
in the variables $z_1$, $z_2$, $\ldots$, $z_n$. Using the result that a sequence of analytic functions that converges uniformly on any compact set
of a domain is analytic in that domain [see theorem 10.28 of Rudin \cite{Rudin:1987:RCA}] we see
that ${\bf G}_\Omega$ is an operator valued analytic function of ${\bf z}$ in the domain ${\cal Z}(\alpha,\beta,\theta)$ for any value of $\theta$ and for arbitrarily small value of $\alpha>0$ and any arbitrarily large value of
$\beta$, such that \eq{8} and \eq{9} hold. This is also a corollary of the well known result that the Green's function is analytic
away from its spectrum. Clearly the domain of analyticity of ${\bf G}_\Omega(z_1,z_2,\ldots,z_n)$ is at least
\begin{equation} \bigcup_{\theta,\alpha,\beta}{\cal Z}(\alpha,\beta,\theta).
\eeq{25}
Consequently the generalized spectrum must be confined to the set of $\mathbb{C}^n$ outside the region \eq{25}.
\section{Herglotz type analytic properties of the Green's operator}
In the theory of composites, the Herglotz type analytic property of the effective tensor
as a function of the moduli or tensors of the component materials has played an important role.
Particularly, it has served as a useful tool for deriving bounds on the effective tensor given the component
moduli, and possibly some information about the geometry such as the volume fractions
of the individual phases: see, for example,
\cite{Bergman:1978:DCC,Milton:1979:TST,Milton:1981:BCP,Milton:1981:BTO,Bergman:1980:ESM,Bergman:1982:RBC,DellAntonio:1986:ATO,Golden:1983:BEP,Golden:1985:BEP,Clark:1994:MEC,Clark:1995:OBC,Clark:1997:CFR}
and Chapters 18, 27, and 28 of \cite{Milton:2002:TOC}. Recently, in Chapter 6 of \cite{Milton:2016:ETC} (available as https://arxiv.org/abs/1602.03383) Mattei and Milton found these analytic properties
useful for deriving bounds on the response in the time domain. Also, it has been established by Cassier, Milton, and Welters that the Dirichlet to Neumann map governing the response of bodies shares
these Herglotz type properties: see Chapters 3 and 4 of \cite{Milton:2016:ETC}, the latter of which is also available as https://arxiv.org/abs/1512.05838.
Here we show that these Herglotz type properties extend to the Green's operator. Of course, analytic and Herglotz type properties of the Green's operator have an
extensive history: what is new here is the Herglotz type analytic properties as functions of the tensors ${\bf L}_1$, ${\bf L}_2$,$\dots$, ${\bf L}_m$ of the phases.
Note that we are free to rewrite \eq{6a} as
\begin{equation} {\bf J}'({\bf x})={\bf L}'({\bf x}){\bf E}({\bf x})-{\bf h}'({\bf x}),\quad {\bf J}'\in{\cal J},\quad {\bf E}\in{\cal E},\quad {\bf h}'\in{\cal H},
\eeq{26}
with
\begin{equation} {\bf J}'=e^{i\psi}{\bf J},\quad {\bf L}'=e^{i\psi}{\bf L},\quad {\bf h}'=e^{i\psi}{\bf h}.
\eeq{27}
So by appropriately redefining ${\bf L}$ we may assume the coercivity condition \eq{9} is satisfied with $e^{i\theta}=-i$, i.e. $\theta=-\pi/2$. Then the condition implies
\begin{equation}
\mathop{\rm Im}\nolimits ({\bf P},{\bf L}{\bf P})_\Omega \geq ({\bf T}{\bf P},{\bf P})_\Omega+\alpha|{\bf P}|_\Omega^2\quad \text{ for all } {\bf P}\in{\cal H}_{\Omega},
\eeq{28}
and since
\begin{equation} ({\bf G}_\Omega{\bf h},{\bf h})_\Omega=({\bf E},{\bf h})_\Omega=({\bf E},{\bf L}{\bf E}-{\bf J})_\Omega=({\bf E},{\bf L}{\bf E})_\Omega=({\bf L}{\bf E},{\bf L}{\bf E})_\Omega^*,
\eeq{29}
we conclude that for all ${\bf h}$ with support in $\Omega$,
\begin{equation} \mathop{\rm Im}\nolimits ({\bf h},{\bf G}_\Omega{\bf h})_\Omega=-\mathop{\rm Im}\nolimits({\bf E},{\bf L}{\bf E})_\Omega\leq 0.
\eeq{30}
In the context of the $m$-phase polycrystalline material \eq{10} we may consider ${\bf G}_\Omega$ to be a function ${\bf G}_\Omega({\bf L}_1, {\bf L}_2, \ldots,{\bf L}_m)$
that is analytic in the domain where ${\bf L}_1, {\bf L}_2, \ldots,{\bf L}_m$, and has the reverse Herglotz property
\begin{equation} \mathop{\rm Im}\nolimits ({\bf h},{\bf G}_\Omega{\bf h})_\Omega\leq 0 \text{ when } \mathop{\rm Im}\nolimits{\bf L}_i>0 \text{ for all }i,
\eeq{31}
the homogeneity property that for all $\lambda\in\mathbb{C}$,
\begin{equation} {\bf G}_\Omega(\lambda{\bf L}_1, \lambda{\bf L}_2, \ldots,\lambda{\bf L}_m)=\lambda^{-1}{\bf G}_\Omega({\bf L}_1, {\bf L}_2, \ldots,{\bf L}_m),
\eeq{32}
and the normalization property that
\begin{equation} {\bf G}_\Omega({\bf I}, {\bf I}, \ldots,{\bf I})=\bfm\Gamma_1^{\Omega}.
\eeq{33}
Taking $\lambda=i$ in \eq{32}, and using \eq{30} we get
\begin{equation} \mathop{\rm Re}\nolimits ({\bf h},{\bf G}_\Omega{\bf h})\geq 0 \text{ when } \mathop{\rm Re}\nolimits{\bf L}_i>0 \text{ for all }i.
\eeq{34}
\aj{Of course if the tensors ${\bf L}_1$, ${\bf L}_2$,\ldots,${\bf L}_m$ themselves have Herglotz properties as a function of frequency $\omega$, in the sense that
their imaginary parts are positive definite when $\omega$ lies in the upper half plane $\mathop{\rm Im}\nolimits \omega>0$, then ${\bf G}_\Omega(\omega)$ will inherit the reverse
Herglotz property
\begin{equation} \mathop{\rm Im}\nolimits ({\bf h},{\bf G}_\Omega{\bf h})_\Omega\leq 0 \text{ when } \mathop{\rm Im}\nolimits\omega>0, \eeq{34a}
and there will be associated integral representation formulas for the operator valued Herglotz function $-{\bf G}_\Omega(\omega)$, involving a positive semidefinite
operator valued measure derived from the values that $-{\bf G}_\Omega(\omega)$ takes as $\omega$ approaches the real axis.}
\section{Using $Q_\Omega$-convex functions to establish coercivity}
The coercivity condition \eq{9} is generally hard to verify for a given operator ${\bf L}$ as it requires one to test it for all fields ${\bf E}\in{\cal E}_\Omega^0$.
However, suppose we are given a real valued quadratic form, $Q_\Omega({\bf P})$ defined for all ${\bf P}\in{\cal H}_\Omega$, such that
\begin{equation} Q_\Omega({\bf E})\geq 0\quad\text{ for all }{\bf E}\in{\cal E}_\Omega^0.
\eeq{Q.1}
If we can find a constant $t\geq 0$ such that the $Q_\Omega$-coercivity condition defined by
\begin{equation} \mathop{\rm Re}\nolimits (e^{i\theta}{\bf L}{\bf P},{\bf P})_\Omega \geq tQ_\Omega({\bf E})+\alpha|{\bf P}|_\Omega^2 \quad\text{ for all } {\bf P}\in{\cal H}_{\Omega},
\eeq{Q.2}
holds, then by taking ${\bf P}={\bf E}$, with ${\bf E}\in{\cal E}_\Omega^0$ it is clear that the coercivity condition \eq{9} will be satisfied.
Following ideas of Murat and Tartar \cite{Murat:1978:CPC,Tartar:1979:CCA,Tartar:1979:ECH,Murat:1981:CPC,Murat:1985:CVH,Tartar:1985:EFC} and Milton \cite{Milton:2013:SIG,Milton:2015:ATS}
one can take this idea further. Given an integer $\ell\geq 1$, define
\begin{eqnarray}
\mathfrak{H}_\Omega& = & \{\mathbb{P}~|~\mathbb{P}=({\bf P}_1,{\bf P}_2,\ldots,{\bf P}_\ell),~{\bf P}_i\in{\cal H}_\Omega\text{ for }i=1,2,\ldots\ell\}, \nonumber \\
\mathfrak{E}_\Omega^0& = & \{\mathbb{E}~|~\mathbb{E}=({\bf E}_1,{\bf E}_2,\ldots,{\bf E}_\ell),~{\bf E}_i\in{\cal E}_\Omega^0\text{ for }i=1,2,\ldots\ell\}.
\eeqa{Q.3}
Fields in these spaces take values in a tensor space $\mathfrak{T}$ consisting of tensors
\begin{equation} \mathfrak{T} = \{\mathbb{A}~|~\mathbb{A}=({\bf A}_1,{\bf A}_2,\ldots,{\bf A}_\ell),~{\bf A}_i\in{\cal T} \text{ for }i=1,2,\ldots\ell\},
\eeq{Q.3aa}
and on $\mathfrak{T}$ we define the inner product
\begin{equation} (\mathbb{A},\mathbb{A'})_{\mathfrak{T}}=\sum_{i=1}^\ell({\bf A}_i,{\bf A}'_i)_{{\cal T}}.
\eeq{Q.3ab}
Similarly, given any pair of fields $\mathbb{P},\mathbb{P}'\in \mathfrak{H}_\Omega$ we define the inner product and norm:
\begin{equation} (\mathbb{P},\mathbb{P}')_{\mathfrak{H}_\Omega}=\sum_{i=1}^\ell({\bf P}_i,{\bf P}'_i)_{\Omega},\quad |\mathbb{P}|_{\mathfrak{H}_\Omega}=\sqrt{(\mathbb{P},\mathbb{P})_{\mathfrak{H}_\Omega}}.
\eeq{Q.3a}
Introduce an operator $\mathbb{L}:\mathfrak{H}_\Omega\to\mathfrak{H}_\Omega$ defined by
\begin{equation} \mathbb{L}\mathbb{E}=\mathbb{L}({\bf E}_1,{\bf E}_2,\ldots,{\bf E}_\ell) = ({\bf L}{\bf E}_1,{\bf L}{\bf E}_2,\ldots,{\bf L}{\bf E}_\ell).
\eeq{Q.4}
Now suppose we are given a real valued quadratic form, $Q_\Omega(\mathbb{P})$ defined for all $\mathbb{P}\in\mathfrak{H}_\Omega$, such that
\begin{equation} Q_\Omega(\mathbb{E})\geq 0\text{ for all }\mathbb{E}\in\mathfrak{E}_\Omega^0.
\eeq{Q.5}
If we can find a constant $t\geq 0$ such that the generalized $Q_\Omega$-coercivity condition,
\begin{equation} \mathop{\rm Re}\nolimits (e^{i\theta}\mathbb{L}\mathbb{P},\mathbb{P})_{\mathfrak{H}_\Omega} \geq t Q_\Omega(\mathbb{P})+\alpha|\mathbb{P}|_{\mathfrak{H}_\Omega}^2
\text{ for all } \mathbb{P}\in\mathfrak{H}_{\Omega}
\eeq{Q.6}
holds, then by taking $\mathbb{P}=\mathbb{E}$ where $\mathbb{E}\in\mathfrak{E}_\Omega^0$ we get
\begin{equation} \mathop{\rm Re}\nolimits (e^{i\theta}\mathbb{L}\mathbb{E},\mathbb{E})_{\mathfrak{H}_\Omega} \geq \alpha|\mathbb{E}|_{\mathfrak{H}_\Omega}^2.
\eeq{Q.7}
This can be rewritten as
\begin{equation} \sum_{i=1}^\ell\mathop{\rm Re}\nolimits(e^{i\theta}{\bf L}{\bf E}_i,{\bf E}_i)_{\Omega}\geq \alpha\sum_{i=1}^\ell|{\bf E}_i|^2_{\Omega},
\eeq{Q.8}
and clearly implies that the coercivity condition \eq{9} holds.
This choice of quadratic functions $Q_\Omega$ enables one to get convergent series expansions for ${\bf G}_\Omega$ on a
domain ${\cal Z}(\alpha,\beta,\theta,\ell,Q_\Omega)$ defined as that region of $\mathbb{C}^n$ where the boundedness and $Q_\Omega$-coercivity conditions
\eq{8} and \eq{Q.6} hold, in which, by rescaling $Q_\Omega$, one may assume that $t=1$. One should get tighter bounds on the generalized spectrum, as it lies outside the domain
\begin{equation} \bigcup_{\theta,\alpha,\beta,\ell,Q_\Omega}{\cal Z}(\alpha,\beta,\theta,\ell,Q_\Omega).
\eeq{Q.11}
Obviously the $Q_\Omega$-coercivity condition \eq{Q.2} is just a special case of the $Q_\Omega$-coercivity condition \eq{Q.6} (obtained by taking $\ell=1$ or by
taking $\mathbb{T}$ to be block diagonal). So from now on we will concentrate on finding quadratic functions $Q_\Omega(\mathbb{P})$ satisfying \eq{Q.5}.
\section{Finding appropriate quadratic $Q_\Omega$-convex functions}
\subsection{Generating position independent translations}
Identifying quadratic forms such that $Q_\Omega$-convexity condition \eq{Q.5} holds on $\mathfrak{E}_\Omega^0$, and then determining
when the $Q_\Omega$-coercivity condition \eq{Q.6} holds is still a difficult task. Progress can be made by limiting attention
to quadratic functions $Q_\Omega(\mathbb{P})$ generated by a self adjoint operator that acts locally in space, with action given by an operator $\mathbb{T}({\bf x})$,
called the translation operator, so that
\begin{eqnarray} Q_\Omega(\mathbb{P}) & = & \frac{1}{V(\Omega)}\int_\Omega(\mathbb{T}({\bf x})\mathbb{P}({\bf x}),\mathbb{P}({\bf x}))_{\mathfrak{T}}\,d{\bf x} \nonumber \\
& = & \frac{1}{V(\Omega)}\int_\Omega\sum_{i=1}^\ell\sum_{j=1}^\ell({\bf T}_{ij}({\bf x}){\bf P}_j({\bf x}),{\bf P}_i({\bf x}))_{\cal T}\,d{\bf x},
\eeqa{F.0}
where the ${\bf T}_{ij}({\bf x})$ represent the individual blocks of $\mathbb{T}({\bf x})$, with ${\bf T}_{ji}({\bf x})={\bf T}_{ij}^\dagger({\bf x})$ to ensure that $\mathbb{T}$ is self-adjoint.
The key point is that $\mathbb{T}$ need not act separately on the individual components of $\mathbb{P}$, but can couple them. Then the $Q_\Omega$-coercivity
condition \eq{Q.6} becomes
\begin{equation} \int_\Omega\sum_{i=1}^\ell\mathop{\rm Re}\nolimits(e^{i\theta}{\bf L}({\bf x}){\bf P}_i({\bf x}),{\bf P}_i({\bf x}))_{{\cal T}}\,d{\bf x}\geq t\int_\Omega\sum_{i=1}^\ell\sum_{j=1}^\ell({\bf T}_{ij}({\bf x}){\bf P}_j({\bf x}),{\bf P}_i({\bf x}))_{{\cal T}}\,d{\bf x}
+\alpha\int_\Omega\sum_{i=1}^\ell|{\bf P}_i|^2_{{\cal T}}\,d{\bf x}.
\eeq{F.0a}
and will clearly be satisfied if and only if for all ${\bf x}\in\Omega$, and for all $\mathbb{A}=({\bf A}_1,{\bf A}_2,\ldots,{\bf A}_\ell)\in\mathfrak{T}$,
\begin{equation} \sum_{i=1}^\ell\mathop{\rm Re}\nolimits(e^{i\theta}{\bf L}({\bf x}){\bf A}_i,{\bf A}_i)_{{\cal T}}\geq t\sum_{i=1}^\ell\sum_{j=1}^\ell({\bf T}_{ij}({\bf x}){\bf A}_j,{\bf A}_i)_{{\cal T}}
+\alpha\sum_{i=1}^\ell|{\bf A}_i|^2_{{\cal T}}.
\eeq{F.0b}
This can be written as a block matrix inequality:
\begin{equation} \begin{pmatrix} \mathop{\rm Re}\nolimits(e^{i\theta}{\bf L}({\bf x}))-\alpha{\bf I} & 0 & \ldots & 0 \\
0 & \mathop{\rm Re}\nolimits(e^{i\theta}{\bf L}({\bf x}))-\alpha{\bf I} & \ldots &0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \ldots & \mathop{\rm Re}\nolimits(e^{i\theta}{\bf L}({\bf x}))-\alpha{\bf I}\end{pmatrix}
-t\begin{pmatrix} {\bf T}_{11}({\bf x}) & {\bf T}_{12}({\bf x}) & \ldots & {\bf T}_{1\ell}({\bf x}) \\
{\bf T}_{21}({\bf x}) & {\bf T}_{22}({\bf x}) & \ldots & {\bf T}_{2\ell}({\bf x}) \\
\vdots & \vdots & \ddots & \vdots \\
{\bf T}_{\ell 1}({\bf x}) & {\bf T}_{\ell 2}({\bf x}) & \ldots & {\bf T}_{\ell\ell}({\bf x})\end{pmatrix}\geq 0,
\eeq{F.0c}
where the inequality holds in the sense that the matrix on the left is a positive definite matrix. Thus, when the moduli are shifted (translated) by $\mathbb{T}({\bf x})$,
they satisfy a local coercivity condition on $\mathfrak{T}$.
For a multicomponent medium with ${\bf L}({\bf x})$ given
by \eq{10A} this inequality clearly becomes
\begin{equation} \begin{pmatrix} \mathop{\rm Re}\nolimits(e^{i\theta}{\bf L}_i)-\alpha{\bf I} & 0 & \ldots & 0 \\
0 & \mathop{\rm Re}\nolimits(e^{i\theta}{\bf L}_i)-\alpha{\bf I} & \ldots &0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \ldots & \mathop{\rm Re}\nolimits(e^{i\theta}{\bf L}_i)-\alpha{\bf I}\end{pmatrix}
-t\begin{pmatrix} {\bf T}_{11}({\bf x}) & {\bf T}_{12}({\bf x}) & \ldots & {\bf T}_{1\ell}({\bf x}) \\
{\bf T}_{21}({\bf x}) & {\bf T}_{22}({\bf x}) & \ldots & {\bf T}_{2\ell}({\bf x}) \\
\vdots & \vdots & \ddots & \vdots \\
{\bf T}_{\ell 1}({\bf x}) & {\bf T}_{\ell 2}({\bf x}) & \ldots & {\bf T}_{\ell\ell}({\bf x})\end{pmatrix}\geq 0,
\eeq{F.0d}
for all ${\bf x}$ in phase $i$ (i.e. such that $\chi_i({\bf x})=1$), and for all
$i=1,2,\ldots,n$.
In the case where $\mathbb{T}$ is a constant operator, independent of ${\bf x}$, there is a simple algebraic route to finding $\mathbb{T}$ for which
the $Q_\Omega$-convexity condition \eq{Q.5} is satisfied. Then the inequalities in \eq{Q.5} will surely hold if they hold on the larger space
$\mathfrak{E}$, where $\mathfrak{E}$ is comprised of fields $\mathbb{E}({\bf x})=({\bf E}_1({\bf x}),{\bf E}_2({\bf x}),\ldots,{\bf E}_\ell({\bf x}))$
defined for ${\bf x}\in\mathbb{R}^d$ where all the component fields ${\bf E}_i$ lie in ${\cal E}$. Thus we need to identify quadratic functions $Q_*(\mathbb{A})$ defined for $\mathbb{A}\in\mathfrak{T}$
that satisfy
\begin{equation} \int_{\mathbb{R}^d} Q_*(\mathbb{E}({\bf x}))\,d{\bf x}\geq 0 \text{ for all }\mathbb{E}\in\mathfrak{E}.
\eeq{F.1}
We call this the $Q_*$-convexity condition, and functions that satisfy it will be called $Q_*$-convex. The terminology arises because of similarities with
the definition of $Q_*$-convexity given in \cite{Milton:2013:SIG,Milton:2015:ATS}, where it was used to obtain boundary field inequalities.
Then we may simply take
\begin{equation} Q_\Omega(\mathbb{E})=\int_\Omega~ Q_*(\mathbb{E}({\bf x}))\,d{\bf x}.
\eeq{F.2}
The field $\mathbb{E}({\bf x})$, being square integrable, will have some Fourier expansion
\begin{equation} \mathbb{E}({\bf x})=\int_\mathbb{R}^d\widehat{\mathbb{E}}({\bf k})e^{i{\bf k}\cdot{\bf x}}\,d{\bf k},
\eeq{F-3}
and the differential constraints on $\mathbb{E}({\bf x})$ imply that $\widehat{\mathbb{E}}({\bf k})$ takes values in
some subspace $\mathfrak{E}_{\bf k}$. Substituting \eq{F-3} in \eq{F.1}, and doing the integration over ${\bf x}$ we see that
\eq{F.1} holds if
\begin{equation} Q_*(\mathbb{E}({\bf k}))=(\mathbb{E}({\bf k})\mathbb{T},\mathbb{E}({\bf k}))_{\mathfrak{T}}\geq 0\text{ for all }{\bf k}\in\mathbb{R}^d.
\eeq{F-4}
Thus the $Q_*$-convexity condition \eq{F.1} is satisfied if the algebraic condition that
\begin{equation} Q_*(\mathbb{A})\geq 0 \text{ for all } \mathbb{A}\in\mathfrak{E}_{\bf k}
\eeq{F.3}
holds for all ${\bf k}\in\mathbb{R}^d$. As $\mathbb{T}$ is selfadjoint, if we make the substitution $\mathbb{A}= \mathbb{A}'+i\mathbb{A}''$,
where $\mathbb{A}'$ and $\mathbb{A}''$ are the real and imaginary parts of $\mathbb{A}$, we obtain
\begin{equation} Q_*(\mathbb{A})=Q_*(\mathbb{A}')+Q_*(\mathbb{A}'').
\eeq{F.3a}
Now suppose the spaces ${\cal H}$, ${\cal E}$ and ${\cal J}$ are real-symmetric, in the sense that
if a field belongs to them, so does the complex conjugate field. Then if $\mathbb{A}$ lies in $\mathfrak{E}_{\bf k}$,
so too does its complex conjugate, and hence $\mathbb{A}'\in\mathfrak{E}_{\bf k}$ and $\mathbb{A}''\in\mathfrak{E}_{\bf k}$.
With this assumption on the subspaces we see from \eq{F.3a} that it is only necessary to test \eq{F.3}
for real values of $\mathbb{A}\in\mathfrak{E}_{\bf k}$.
The condition \eq{F.3} for $Q_*$-convexity reduces to
the normal quasiconvexity condition when the subspace ${\cal E}$ is scale invariant, i.e. if ${\bf E}\in{\cal E}$, then
the field ${\bf E}'({\bf x})$ defined by ${\bf E}'({\bf x})={\bf E}(\lambda{\bf x})$ also lies in ${\cal E}$ for all real nonzero choices of the scale factor
$\lambda$. In that event, $\mathfrak{E}_{\bf k}$ remains invariant under the scale change ${\bf k}\to{\bf k}/\lambda$.
When $\mathbb{T}$ does not depend on ${\bf x}$ then the inequalities \eq{F.0d} do not involve the characteristic
functions of the phases and thus we obtain geometry independent bounds on the generalized spectrum (in the sense that these
bounds hold for all choices of the $\chi_i({\bf x})$). For polycrystalline multiphase bodies where ${\bf L}({\bf x})$
takes the form \eq{10}, one has to made additional assumptions about $\mathbb{T}$ to obtain geometry independent
bounds that are independent both of the choice of the $\chi_i({\bf x})$ and the rotation field ${\bf R}({\bf x})$. Here we
follow the ideas developed in \cite{Avellaneda:1988:ECP}: see also sections 24.2 and 24.8 in \cite{Milton:2002:TOC}.
It is convenient to
take $\ell$ as the dimension of ${\cal T}$. Then, taking an orthonormal basis ${\bf e}_1$, ${\bf e}_2$, $\ldots$, ${\bf e}_\ell$ of ${\cal T}$ we can regard
the elements of $\mathfrak{T}$ as linear maps from ${\cal T}$ to ${\cal T}$. Thus if $\mathbb{A}=({\bf A}_1,{\bf A}_2,\ldots,{\bf A}_\ell)\in\mathfrak{T}$ is
given, we define its action on ${\cal T}$ through its action on the basis elements according to the prescription
\begin{equation} \mathbb{A}{\bf e}_i={\bf A}_i. \eeq{FF.1}
Now given a rotation ${\bf R}$ we can define an associated operator $\mathbb{R}({\bf R})$ acting on linear maps from ${\cal T}$ to ${\cal T}$, defined
by
\begin{equation} [\mathbb{R}({\bf R})\mathbb{A}]{\bf e}_j={\cal R}({\bf R})^\dagger\mathbb{A}[{\cal R}({\bf R}){\bf e}_j], \eeq{FF.2}
where ${\cal R}({\bf R}):{\cal T}\to{\cal T}$ is that operator on the tensor space which corresponds to a rotation ${\bf R}$: see \eq{5b}.
Since ${\cal R}({\bf R}){\bf e}_j$ is a linear combination of the ${\bf e}_i$, the action of $\mathbb{A}$ on ${\cal R}({\bf R}){\bf e}_j$ can be computed via
\eq{FF.1}. Now $\mathbb{T}$ acts on a map $\mathbb{A}$ from ${\cal T}$ to ${\cal T}$ to produce a new map $\mathbb{T}\mathbb{A}$ from ${\cal T}$ to ${\cal T}$.
We seek $\mathbb{T}$ that are rotationally invariant in the sense that for all rotations ${\bf R}$,
\begin{equation} \mathbb{T}[\mathbb{R}({\bf R})\mathbb{A}]=\mathbb{R}({\bf R})[\mathbb{T}\mathbb{A}].
\eeq{FF.3}
Now $\mathbb{L}$ defined in \eq{Q.4}, acts locally and we can write
\begin{equation} [\mathbb{L}\mathbb{P}]({\bf x})=[\mathbb{L}({\bf x})]\mathbb{P}({\bf x})={\bf L}({\bf x})\mathbb{P}({\bf x}),
\eeq{FF.4}
which thus defines $\mathbb{L}({\bf x})$ where the expression on the far right can be regarded as the composition of the two
operators ${\bf L}({\bf x}):{\cal T}\to{\cal T}$ and $\mathbb{P}({\bf x}):{\cal T}\to{\cal T}$. Similarly, we can define the operator $\widetilde{\mathbb{L}}$ that
acts locally, given by
\begin{equation} [\widetilde{\mathbb{L}}\mathbb{P}]({\bf x})=[\widetilde{\mathbb{L}}({\bf x})]\mathbb{P}({\bf x})=\widetilde{{\bf L}}({\bf x})\mathbb{P}({\bf x})
\eeq{FF.4a}
in which $\widetilde{{\bf L}}({\bf x})$ is defined by \eq{10}. Then we have
\begin{eqnarray} [\mathbb{L}\mathbb{R}({\bf R})\mathbb{A}]{\bf e}_j & = & {\bf L}{\cal R}({\bf R})^\dagger\mathbb{A}[{\cal R}({\bf R}){\bf e}_i] \nonumber \\
& = & {\cal R}({\bf R})^\dagger\widetilde{{\bf L}}\mathbb{A}[{\cal R}({\bf R}){\bf e}_i] \nonumber \\
& = & \mathbb{R}({\bf R})[\widetilde{\mathbb{L}}]{\bf e}_j.
\eeqa{FF.4b}
The condition \eq{Q.6} then holds if and only if
\begin{equation} \mathop{\rm Re}\nolimits (e^{i\theta}\mathbb{L}\mathbb{A},\mathbb{A})_{\mathfrak{T}} \geq t (\mathbb{T}\mathbb{A},\mathbb{A})_{\mathfrak{T}}+\alpha(\mathbb{A},\mathbb{A})_{\mathfrak{T}}
\text{ for all } \mathbb{A}:{\cal T}\to{\cal T}.
\eeq{FF.5}
We are free to replace $\mathbb{A}$ by $\mathbb{R}({\bf R})\mathbb{A}$ and then using the rotational invariance of $\mathbb{T}$, and the rotational
properties \eq{FF.4b} of $\mathbb{L}$, the condition becomes
\begin{equation} \mathop{\rm Re}\nolimits (e^{i\theta}\widetilde{\mathbb{L}}\mathbb{A},\mathbb{A})_{\mathfrak{T}} \geq t (\mathbb{T}\mathbb{A},\mathbb{A})_{\mathfrak{T}}+\alpha(\mathbb{A},\mathbb{A})_{\mathfrak{T}}
\text{ for all } \mathbb{A}:{\cal T}\to{\cal T}.
\eeq{FF.6}
With $\widetilde{{\bf L}}({\bf x})$ given by \eq{10} we see \eq{FF.6} holds if and only if \eq{F.0d} is satisfied.
\subsection{Generating position dependent translations using Null-Lagrangians or coordinate transformations}
More general $Q_\Omega$ functions, with $\mathbb{T}({\bf x})$ depending on ${\bf x}$, can also be obtained. The simplest case is when $\mathbb{E}({\bf x})$
is a linear function of a potential ${\bf u}({\bf x})$, its gradient $\nabla{\bf u}({\bf x})$. Then we may look for quadratic functions $L({\bf x},{\bf u},\nabla{\bf u})$ that are Null-Lagrangians
in the sense that
\begin{equation} \int_{\Omega}L({\bf x},{\bf u}({\bf x}),\nabla{\bf u}({\bf x}))\,d{\bf x} =0.
\eeq{F.3aa}
Such functions have been completely characterized, even if they are not quadratic \cite{Olver:1988:SNL}.
The integrand here may not be expressible simply as a quadratic function of $\mathbb{E}({\bf x})$ but we could look for operators $\mathbb{T}({\bf x})$
such that for all ${\bf x}\in\Omega$,
\begin{equation} (\mathbb{T}({\bf x})\mathbb{E}({\bf x}),\mathbb{E}({\bf x}))_{\mathfrak{T}}\geq L({\bf x},{\bf u}({\bf x}),\nabla{\bf u}({\bf x})),
\eeq{F.3b}
where here ${\bf u}({\bf x})$ and $\nabla{\bf u}({\bf x})$ are those arguments that appear implicitly in the expression for $\mathbb{E}({\bf x})$.
Another approach is to make coordinate transformations.
Thus, for example, suppose we replace ${\bf x}$ by ${\bf y}$, and suppose the differential constraints on
$\mathbb{E}({\bf y})$ imply that is a linear function of a potential ${\bf w}({\bf y})$, its gradient $\nabla_{{\bf y}}{\bf w}({\bf y})$, a field ${\bf V}({\bf y})$ and
its divergence $\nabla_{{\bf y}}\cdot{\bf V}({\bf y})$ (that is possibly constrained to be zero). Here $\nabla_{{\bf y}}$ and $\nabla_{{\bf y}}\cdot$ denote the gradient
and divergence with respect to the ${\bf y}$ variables. Furthermore, suppose we have identified
a quadratic function $Q_*(\mathbb{E}({\bf y}))=F({\bf w}({\bf y}),\nabla_{{\bf y}}{\bf w}({\bf y}),{\bf V}({\bf y}),\nabla_{{\bf y}}\cdot{\bf V}({\bf y}))$ such that
\begin{equation} \int_{\Omega}F({\bf w}({\bf y}),\nabla_{{\bf y}}{\bf w}({\bf y}),{\bf V}({\bf y}),\nabla_{{\bf y}}\cdot{\bf V}({\bf y}))\,d{\bf y}=\int_{\Omega} Q_*(\mathbb{E}({\bf y}))\,d{\bf y}\geq 0 \text{ for all }\mathbb{E}\in\mathfrak{E}.
\eeq{F.4}
Now consider a coordinate transformation from ${\bf y}$ to a variable ${\bf x}$ such that ${\bf y}=\bfm\psi({\bf x})$, i.e., ${\bf x}=\bfm\psi^{-1}({\bf y})$
that leaves points outside and on the boundary of $\Omega$ unmoved, i.e.
\begin{equation} \psi({\bf y})={\bf y}\text{ for }{\bf y}\in\partial\Omega \text{ or }{\bf y}\text{ outside } \Omega.
\eeq{F.5}
Then making the change of variables in the integral \eq{F.5} we obtain
\begin{equation} \int_{\Omega}\widetilde{F}({\bf x},{\bf u}({\bf x}),\nabla{\bf u}({\bf x}),{\bf Q}({\bf x}),\nabla \cdot{\bf Q}({\bf x}))\,d{\bf y} \geq 0,
\eeq{F.6}
where with ${\bf y}=\bfm\psi({\bf x})$,
\begin{eqnarray} &~& {\bf u}({\bf x})={\bf w}({\bf y}), \quad {\bf Q}({\bf x})=\bfm\Lambda({\bf y}){\bf V}({\bf y})/\det[\bfm\Lambda({\bf y})], \nonumber \\
& ~& \nabla{\bf u}({\bf x})=[\bfm\Lambda({\bf y})]^{-1}\nabla_{{\bf y}}{\bf w}({\bf y}),\quad \nabla \cdot{\bf Q}({\bf x})=[\nabla_{{\bf y}}\cdot{\bf V}({\bf y})]/\det[\bfm\Lambda({\bf y})],\nonumber \\ \nonumber \\
&~& \Lambda_{ij}({\bf y})=\frac{\partial x_j({\bf y})}{\partial y_i}=\frac{\partial \psi^{-1}_j({\bf y})}{\partial y_i},
\eeqa{F.6a}
and
\begin{eqnarray}
&~& \widetilde{F}({\bf x},{\bf u}({\bf x}),\nabla{\bf u}({\bf x}),{\bf Q}({\bf x}),\nabla \cdot{\bf Q}({\bf x})) \nonumber \\
&~&\quad\quad =\frac{F({\bf u}({\bf x}),\bfm\Lambda(\bfm\psi({\bf x}))\nabla{\bf u}({\bf x}),\det[\bfm\Lambda(\bfm\psi({\bf x}))][\bfm\Lambda(\bfm\psi({\bf x}))]^{-1}{\bf Q}({\bf x}),\det[\bfm\Lambda(\bfm\psi({\bf x}))]\nabla \cdot{\bf Q}({\bf x}))}{\det[\bfm\Lambda(\bfm\psi({\bf x}))]}, \nonumber \\
&~&
\eeqa{F.7}
in which $\nabla \cdot{\bf Q}=0$ if $\nabla_{{\bf y}}\cdot{\bf V}({\bf y})=0$. We now can take
\begin{equation} Q_\Omega(\widetilde{\mathbb{E}})=\int_{\Omega}\widetilde{F}({\bf x},{\bf u}({\bf x}),\nabla{\bf u}({\bf x}),{\bf Q}({\bf x}),\nabla \cdot{\bf Q}({\bf x}))\,d{\bf x}, \eeq{F.7a}
where we suppose that $\widetilde{\mathbb{E}}$ is a linear function of the potential ${\bf u}({\bf y})$, its gradient $\nabla_{{\bf y}}{\bf u}({\bf y})$, and a divergence free field ${\bf Q}({\bf y})$,
and involves them in such a way that the integrand in \eq{F.7a} can be expressed simply as a function of $\widetilde{\mathbb{E}}({\bf x})$ (otherwise, one would
look for a quadratic function of $\widetilde{\mathbb{E}}({\bf x})$ that bounds $\widetilde{F}({\bf x},{\bf u}({\bf x}),\nabla{\bf u}({\bf x}),{\bf Q}({\bf x}),\nabla \cdot{\bf Q}({\bf x}))$ from above for all ${\bf x}\in\Omega$,
in a similar fashion as was done in \eq{F.3b}).
\subsection{Generating position dependent translations by making substitutions}
Yet another approach is to make substitutions. As suggested by the analysis of \cite{Olver:1988:SNL}
having identified a quadratic function $Q_*(\mathbb{E}({\bf x}))=F({\bf w}({\bf x}),\nabla{\bf w}({\bf x}))$ such that
\begin{equation} \int_{\Omega}F({\bf w}({\bf x}),\nabla{\bf w}({\bf x}))\,d{\bf x}=\int_{\Omega} Q_*(\mathbb{E}({\bf x}))\,d{\bf x}\geq 0 \text{ for all }\mathbb{E}\in\mathfrak{E},
\eeq{F.8}
we make the substitution
\begin{equation} {\bf w}({\bf x})={\bf S}({\bf x}){\bf u}({\bf x})+{\bf s}({\bf x}),
\eeq{F.9}
where we require that ${\bf s}({\bf x})=0$ and ${\bf u}({\bf x})=0$ for ${\bf x}\in\partial\Omega$ to ensure that ${\bf w}({\bf x})=0$ on $\partial\Omega$. Here we suppose that at each ${\bf x}\in\Omega$, ${\bf w}({\bf x})$
and ${\bf s}({\bf x})$ are $p$-component vectors, while ${\bf u}({\bf x})$ is an $s$-component vector, so ${\bf S}({\bf x})$ is a $p\times s$ matrix.
Since $F$ is quadratic we may write
\begin{equation} F({\bf w},{\bf W})=w_\alpha A_{\alpha\beta}w_\beta^*+2\mathop{\rm Re}\nolimits\left(w _\alpha B_{\alpha j\beta}W_{j\beta}^*\right)+W_{i\alpha}C_{i\alpha j\beta}W_{j\beta}^* ,
\eeq{F.10}
where sums over repeated indices are assumed (Einstein summation convention) and
${\bf A}$ and ${\bf C}$ are Hermitian ($A_{\alpha\beta}^*=A_{\beta\alpha}$ and $C_{i\alpha j\beta}^*=C_{j\beta i\alpha}$).
Making the substitution \eq{F.9} we get
\begin{eqnarray} &~& F({\bf S}({\bf x}){\bf u}({\bf x})+{\bf s}({\bf x}),[\nabla{\bf u}({\bf x})]{\bf S}({\bf x})^T+[\nabla{\bf S}({\bf x})]{\bf u}({\bf x})+\nabla{\bf s}({\bf x})) \nonumber \\
&~&\quad\quad= F({\bf S}({\bf x}){\bf u}({\bf x}),[\nabla{\bf u}({\bf x})]{\bf S}({\bf x})^T+[\nabla{\bf S}({\bf x})]{\bf u}({\bf x}))+F({\bf s}({\bf x}),\nabla{\bf s}({\bf x}))\nonumber \\
&~&\quad\quad\quad +2\mathop{\rm Re}\nolimits[u_\gamma(S_{\beta\gamma}A_{\beta\alpha}s_\alpha^*+S_{\alpha\gamma}B_{\alpha j \beta}s_{\beta,j}^*+s_\alpha^*B_{\alpha j \beta}^*S_{\beta\gamma,j}+S_{\alpha\gamma,i}C_{i\alpha j\beta}s_{\beta,j}^*)] \nonumber \\
&~&\quad\quad\quad +2\mathop{\rm Re}\nolimits[u_{\gamma,i}(S_{\beta\gamma}B_{\alpha i \beta}^*s_{\alpha}^*+S_{\alpha\gamma} C_{i\alpha j\beta} s_{\beta,j}^*)].
\eeqa{F.11}
Let us require that for all $\gamma=1,2,\ldots,s$,
\begin{eqnarray} &~& S_{\beta\gamma}A_{\beta\alpha}s_\alpha^*+S_{\alpha\gamma}B_{\alpha j \beta}s_{\beta,j}^*+s_\alpha^*B_{\alpha j \beta}^*S_{\beta\gamma,j}+S_{\alpha\gamma,i}C_{i\alpha j\beta}s_{\beta,j}^*=0,\nonumber \\
&~& (S_{\beta\gamma}B_{\alpha i \beta}^*s_{\alpha}^*+S_{\alpha\gamma} C_{i\alpha j\beta} s_{\beta,j}^*)_{,i}=0.
\eeqa{F.13}
Given ${\bf s}({\bf x})$ this places $2s$ linear restrictions on the $ps$ components of ${\bf S}({\bf x})$. So one would expect to find solutions, at least when $p\geq 2$.
Then the function
\begin{eqnarray} \widetilde{F}({\bf x},{\bf u}({\bf x}),\nabla{\bf u}({\bf x}))& = &
F({\bf S}({\bf x}){\bf u}({\bf x}),[\nabla{\bf u}({\bf x})]{\bf S}({\bf x})^T+[\nabla{\bf S}({\bf x})]{\bf u}({\bf x})) \nonumber \\
& = &F({\bf S}({\bf x}){\bf u}({\bf x})+{\bf s}({\bf x}),[\nabla{\bf u}({\bf x})]{\bf S}({\bf x})^T+[\nabla{\bf S}({\bf x})]{\bf u}({\bf x})+\nabla{\bf s}({\bf x})) \nonumber \\
&~&-F({\bf s}({\bf x}),\nabla{\bf s}({\bf x}))-2\mathop{\rm Re}\nolimits[u_{\gamma,i}(S_{\beta\gamma}B_{\alpha i \beta}^*s_{\alpha}^*+S_{\alpha\gamma} C_{i\alpha j\beta} s_{\beta,j}^*)]
\eeqa{F.14}
is a quadratic function of ${\bf u}$ and $\nabla{\bf u}$. In general the identity \eq{F.8} holds as an inequality. However there can be special
fields ${\bf w}({\bf x})$ such that it holds as an equality \cite{Kang:2013:BVF3d,Milton:2013:SIG,Milton:2015:ATS}. Special fields
are more likely to exist if $Q_*(\mathbb{E}({\bf x}))$ is chosen to be extremal in the sense that it loses its $Q_*$-convexity property
if any strictly convex quadratic function is subtracted from it. In the case where $\mathbb{T}({\bf x})$ does not depend on ${\bf x}$,
an algorithm for numerically constructing extremal $Q_*$-convex functions (based on earlier work in \cite{Kohn:1988:OBE,Milton:1990:CSP})
is given in \cite{Milton:2013:SIG}.
For functions of gradients, an explicit example of an extremal $Q_*$-convex function that is not a Null-Lagrangian is given
in \cite{Harutyunyan:2015:EEE} and curiously there is a connection between such functions and extremal polynomials \cite{Harutyunyan:2015:REE,Harutyunyan:2016:TCE}.
Let ${\bf s}({\bf x})$ be one of these special fields. Then we have
\begin{equation} \int_\Omega F({\bf s}({\bf x}),\nabla{\bf s}({\bf x}))\,d{\bf x}=0,
\eeq{F.15}
and, using the fact that ${\bf u}$ vanishes on $\partial\Omega$, we see that
\begin{eqnarray} &~& \int_\Omega \widetilde{F}({\bf x},{\bf u}({\bf x}),\nabla{\bf u}({\bf x}))\,d{\bf x} \nonumber \\
&~&\quad\quad =\int_\Omega F({\bf w}({\bf x}),\nabla{\bf w}({\bf x}))\,d{\bf x}
-\int_\GO2\mathop{\rm Re}\nolimits[u_{\gamma,i}(S_{\beta\gamma}B_{\alpha i \beta}^*s_{\alpha}^*+S_{\alpha\gamma} C_{i\alpha j\beta} s_{\beta,j}^*)] \nonumber \\
&~&\quad\quad \geq -\mathop{\rm Re}\nolimits\int_{\partial\Omega}u_\gamma(S_{\beta\gamma}B_{\alpha i \beta}^*s_{\alpha}^*+S_{\alpha\gamma} C_{i\alpha j\beta} s_{\beta,j}^*)n_i \nonumber \\
&~&\quad\quad \geq 0,
\eeqa{F.16}
in which the $n_i({\bf x})$ are the elements of ${\bf n}({\bf x})$, the outwards normal to $\partial\Omega$.
Substitutions can also be made if $\mathbb{E}({\bf x})$ involves a field ${\bf V}({\bf x})$ and its divergence ${\bf v}({\bf x})=\nabla \cdot{\bf V}({\bf x})$. Having identified a quadratic function
$Q_*(\mathbb{E}({\bf y}))=F({\bf V}({\bf y}),\nabla \cdot{\bf V}({\bf y}))$ such that
\begin{equation} \int_{\Omega}F({\bf V}({\bf x}),\nabla \cdot{\bf V}({\bf x}))\,d{\bf x}=\int_{\Omega} Q_*(\mathbb{E}({\bf x}))\,d{\bf x}\geq 0 \text{ for all }\mathbb{E}\in\mathfrak{E},
\eeq{F.17}
we make the substitution
\begin{equation} {\bf V}({\bf x})={\bf Q}({\bf x}){\bf S}({\bf x})+{\bf s}({\bf x}).
\eeq{F.18}
Since $F$ is quadratic we may write
\begin{equation} F({\bf V},{\bf v})=V_{i\alpha} A_{i\alpha j\beta}V_{j\beta}^*+2\mathop{\rm Re}\nolimits\left(v_\alpha B_{\alpha j\beta}V_{j\beta}^*\right)+v_{\alpha}C_{\alpha\beta}v_{\beta}^* ,
\eeq{F.19}
where ${\bf A}$ and ${\bf C}$ are Hermitian ($A_{i\alpha j\beta}^*=A_{j\beta i\alpha}$ and $C_{\alpha\beta}^*=C_{\beta\alpha}$).
Making the substitution \eq{F.9} we get
\begin{eqnarray} &~& F({\bf S}({\bf x}){\bf Q}({\bf x})+{\bf s}({\bf x}),[\nabla \cdot{\bf Q}({\bf x})]{\bf S}({\bf x})+[{\bf Q}({\bf x})]^T\nabla{\bf S}({\bf x})+\nabla \cdot{\bf s}({\bf x})) \nonumber \\
&~&\quad\quad= F({\bf S}({\bf x}){\bf Q}({\bf x}),[\nabla \cdot{\bf Q}({\bf x})]{\bf S}({\bf x})+[{\bf Q}({\bf x})]^T\nabla{\bf S}({\bf x})) +F({\bf s}({\bf x}),\nabla \cdot{\bf s}({\bf x}))\nonumber \\
&~&\quad\quad\quad +2\mathop{\rm Re}\nolimits[Q_{i\gamma}(S_{\gamma\alpha}A_{i\alpha j\beta}s_{j\beta}^*+S_{\gamma\alpha,i}B_{\alpha j \beta}^*s_{j\beta}^*+S_{\gamma\beta}B_{\alpha i \beta}^*s_{j\alpha,j}^*+S_{\gamma\alpha,i}C_{\alpha\beta}s_{j\beta,j}^*)] \nonumber \\
&~&\quad\quad\quad +2\mathop{\rm Re}\nolimits[Q_{i\gamma,i}(S_{\gamma\alpha}B_{\alpha j \beta}^*s_{j\beta}^*+S_{\gamma\alpha} C_{\alpha\beta} s_{j\beta,j}^*)].
\eeqa{F.20}
We require that
\begin{eqnarray} &~& S_{\gamma\alpha}A_{i\alpha j\beta}s_{j\beta}^*+S_{\gamma\alpha,i}B_{\alpha j \beta}^*s_{j\beta}^*+S_{\gamma\beta}B_{\alpha i \beta}^*s_{j\alpha,j}^*+S_{\gamma\alpha,i}C_{\alpha\beta}s_{j\beta,j}^*)=Y_{\gamma,i}, \nonumber \\
&~& S_{\gamma\alpha}B_{\alpha j \beta}^*s_{j\beta}^*+S_{\gamma\alpha} C_{\alpha\beta} s_{j\beta,j}^*=0,
\eeqa{F.21}
for some potential ${\bf Y}$. Then the function
\begin{eqnarray} \widetilde{F}({\bf Q}({\bf x}),\nabla \cdot{\bf Q}({\bf x})) & = & F({\bf S}({\bf x}){\bf Q}({\bf x}),[\nabla \cdot{\bf Q}({\bf x})]{\bf S}({\bf x})+[{\bf Q}({\bf x})]^T\nabla{\bf S}({\bf x})) \nonumber \\
& = & F({\bf S}({\bf x}){\bf Q}({\bf x})+{\bf s}({\bf x}),[\nabla \cdot{\bf Q}({\bf x})]{\bf S}({\bf x})+[{\bf Q}({\bf x})]^T\nabla{\bf S}({\bf x})+\nabla \cdot{\bf s}({\bf x})) \nonumber \\
&~&\quad-F({\bf s}({\bf x}),\nabla \cdot{\bf s}({\bf x})) -2\mathop{\rm Re}\nolimits[Q_{i\gamma}Y_{\gamma,i}]
\eeqa{F.22}
is a quadratic function of ${\bf Q}({\bf x})$ and $\nabla \cdot{\bf Q}({\bf x})$. Again we look for special fields ${\bf s}({\bf x})$ such that
\begin{equation} \int_\Omega F({\bf s}({\bf x}),\nabla \cdot{\bf s}({\bf x}))\,d{\bf x}=0.
\eeq{F.23}
We get
\begin{eqnarray} \int_\Omega\widetilde{F}({\bf Q}({\bf x}),\nabla \cdot{\bf Q}({\bf x}))\,d{\bf x} & = &\int_\Omega F({\bf V}({\bf x}),\nabla \cdot{\bf V}({\bf x}))\,d{\bf x}-2\mathop{\rm Re}\nolimits[Q_{i\gamma}Y_{\gamma,i}] \nonumber \\
&\geq & \int_{\partial\Omega}Y_{\gamma}Q_{i\gamma}n_i\,dS=0,
\eeqa{F.24}
where we have used the fact that ${\bf n}\cdot{\bf Q}=0$ on $\Omega$, as implied by the boundary constraint $\partial\mathbb{E}=0$.
\section{Periodic boundary conditions}
The theory we have developed can easily be extended to allow for other boundary conditions. The extension
to Green's functions where we impose the constraint that $\partial{\bf J}=0$ on the boundary of $\Omega$, rather than
$\partial{\bf E}=0$, is obvious: just swap the roles of the spaces ${\cal E}$ and ${\cal J}$.
Periodic boundary conditions, with $\Omega$ as the unit cell of periodicity, have the advantage that one has an explicit expression in Fourier space for the projection operator $\bfm\Gamma_1^\Omega$
onto ${\cal E}_\Omega$ which is taken as the space of all $\Omega$-periodic fields, that are square integrable in the unit cell of periodicity and satisfy the appropriate differential constraints.
${\cal J}_\Omega$ is then taken as the orthogonal complement of ${\cal E}_\Omega$ in the space ${\cal H}_\Omega$ of $\Omega$-periodic fields that are square integrable within the unit cell of periodicity.
Thus we also have explicit expression in Fourier space for the projection operator $\bfm\Gamma_2^\Omega={\bf I}-\bfm\Gamma_1^\Omega$ onto the space ${\cal J}_\Omega$. We are then interested in solutions
of \eq{6a} as before, with ${\bf L}({\bf x})$ being $\Omega$-periodic. The analysis proceeds as before, and with $\mathfrak{E}_\Omega^0$ defined by \eq{Q.3},
we now can directly identify functions $Q_\Omega(\mathbb{E})$ satisfying \eq{Q.5} when they are of the form \eq{F.0}
with $\mathbb{T}({\bf x})$ independent of ${\bf x}$.
Further bounds on the generalized spectrum of the Green's operator can be obtained when ${\bf L}({\bf x})$ is non-singular for all ${\bf x}\in\Omega$. Then \eq{6a} can be rewritten as
\begin{equation} {\bf E}({\bf x})=[{\bf L}({\bf x})]^{-1}{\bf J}({\bf x})-\widetilde{{\bf h}},\quad {\bf E}\in{\cal E}_\Omega,\quad{\bf J}\in{\cal J}_\Omega,\quad \widetilde{{\bf h}}=-[{\bf L}({\bf x})]^{-1}{\bf h}({\bf x})\in{\cal H}_\Omega.
\eeq{P.1}
This is exactly the same form as before but with the roles of ${\cal E}_\Omega$ and ${\cal J}_\Omega$, and ${\bf h}$ and $\widetilde{{\bf h}}$ interchanged, and with ${\bf L}({\bf x})$ replaced by
$[{\bf L}({\bf x})]^{-1}$. If $\widetilde{{\bf G}}_\Omega$ is the Green's operator for this problem, so that ${\bf J}=\widetilde{{\bf G}}_\Omega\widetilde{{\bf h}}$, we clearly have ${\bf E}={\bf G}_\Omega{\bf h}$ with
\begin{equation} {\bf G}_\Omega=[{\bf L}({\bf x})]^{-1}-[{\bf L}({\bf x})]^{-1}\widetilde{{\bf G}}_\Omega[{\bf L}({\bf x})]^{-1}.
\eeq{P.2}
So if, for some $\widetilde{\beta}>\widetilde{\alpha}>0$,
the boundedness and coercivity conditions
\begin{eqnarray} &~& \widetilde{\beta} >\sup_{\substack{{\bf P}\in{\cal H} \\ |{\bf P}|=1}}|{\bf L}^{-1}{\bf P}|_\Omega, \nonumber \\
&~& \mathop{\rm Re}\nolimits (e^{i\theta}{\bf L}^{-1}{\bf J},{\bf J})_\Omega\geq \widetilde{\alpha}|{\bf J}|_\Omega^2 \text{ for all } {\bf J}\in{\cal J}_{\Omega},
\eeqa{P.3}
are met (and ${\bf L}({\bf x})$ is non-singular for all ${\bf x}\in\Omega$), then $\widetilde{{\bf G}}_\Omega$ and hence ${\bf G}_\Omega$ exists.
More generally, following ideas developed in \cite{Cherkaev:1994:VPC}, section 18 of \cite{Milton:1990:CSP}, \cite{Milton:2009:MVP,Milton:2010:MVP}
and Chapter 14 of \cite{Milton:2016:ETC}, we may take a splitting of the operator ${\bf L}$:
\begin{equation} {\bf L}={\bf L}_A+{\bf L}_B, \eeq{P.4}
in which ${\bf L}_A$ is non-singular. One may, for example, take ${\bf L}_A$ as the Hermitian part of ${\bf L}$ and ${\bf L}_B$
as the anti-Hermitian part, but other choices are possible too.
Then we consider the pair of equations
\begin{eqnarray} {\bf J} & = & [{\bf L}_A+{\bf L}_B]{\bf E}-{\bf h},\quad {\bf J}\in{\cal J}_\Omega,\quad {\bf E}\in{\cal E}_\Omega,\quad{\bf h}\in{\cal H}_\Omega,\nonumber \\
{\bf J}' & = & [{\bf L}_A-{\bf L}_B]{\bf E}'-{\bf h}',\quad {\bf J}'\in{\cal J}_\Omega,\quad {\bf E}'\in{\cal E}_\Omega,\quad{\bf h}'\in{\cal H}_\Omega,
\eeqa{P.5}
the first of which is of course equivalent to \eq{6a}. Adding and subtracting these gives
\begin{eqnarray} {\bf J}+{\bf J}'& = & {\bf L}_A({\bf E}+{\bf E}')+{\bf L}_B({\bf E}-{\bf E}')-{\bf h}-{\bf h}', \nonumber \\
{\bf J}-{\bf J}'& = & {\bf L}_A({\bf E}-{\bf E}')+{\bf L}_B({\bf E}+{\bf E}')-{\bf h}+{\bf h}', \eeqa{P.6}
or equivalently, in block matrix form,
\begin{equation} \begin{pmatrix} {\bf J}+{\bf J}' \\ {\bf J}-{\bf J}'\end{pmatrix}
=\begin{pmatrix} {\bf L}_A & ~& {\bf L}_B \\ {\bf L}_B &~& {\bf L}_A \end{pmatrix}
\begin{pmatrix} {\bf E}+{\bf E}' \\ {\bf E}-{\bf E}'\end{pmatrix}-\begin{pmatrix} {\bf h}+{\bf h}' \\ {\bf h}-{\bf h}'\end{pmatrix}.
\eeq{P.7}
Going one step further, we can solve the first equation in \eq{P.6} for ${\bf E}+{\bf E}'$ in terms of
${\bf J}+{\bf J}'$ and ${\bf E}-{\bf E}'$:
\begin{equation} {\bf E}+{\bf E}'={\bf L}_A^{-1}[{\bf J}+{\bf J}'-{\bf L}_B({\bf E}-{\bf E}')+{\bf h}+{\bf h}']. \eeq{P.8}
Then substituting this in the second equation in \eq{P.6} gives
\begin{equation} {\bf J}-{\bf J}'={\bf L}_A({\bf E}-{\bf E}')+{\bf L}_B{\bf L}_A^{-1}[{\bf J}+{\bf J}'-{\bf L}_B({\bf E}-{\bf E}')+{\bf h}+{\bf h}']-{\bf h}+{\bf h}'.
\eeq{P.9}
These two equations take the block matrix form
\begin{eqnarray} \underbrace{\begin{pmatrix} c_E({\bf E}+{\bf E}') \\ c_J({\bf J}-{\bf J}') \end{pmatrix}}_{\underline{{\bf J}}}
& = & \underbrace{\begin{pmatrix} c_Ed_J{\bf L}_A^{-1} & ~ & -c_Ed_E{\bf L}_A^{-1}{\bf L}_B \\ c_Jd_J{\bf L}_B{\bf L}_A^{-1} & ~& c_Jd_E({\bf L}_A-{\bf L}_B{\bf L}_A^{-1}{\bf L}_B) \end{pmatrix}}_{\underline{{\bf L}}}
\underbrace{\begin{pmatrix} ({\bf J}+{\bf J}')/d_J \\ ({\bf E}-{\bf E}')/d_E \end{pmatrix}}_{\underline{{\bf E}}} \nonumber \\
&~&\quad\quad -\underbrace{\begin{pmatrix} -c_E{\bf L}_A^{-1}({\bf h}+{\bf h}') \\ c_J({\bf h}-{\bf h}') -c_J{\bf L}_B{\bf L}_A^{-1}({\bf h}+{\bf h}') \end{pmatrix}}_{\underline{{\bf h}}}.
\eeqa{P.10}
in which we have introduced the additional nonzero complex factors $c_E$, $c_J$, $d_E$, and $d_J$ as they may help in establishing coercivity.
We now define
\begin{eqnarray} \underline{{\cal H}}_\Omega & = &\left\{\underline{{\bf P}}=\left.\begin{pmatrix} {\bf P}_1 \\ {\bf P}_2 \end{pmatrix}\quad \right| \quad{\bf P}_1,{\bf P}_2\in {\cal H}_\Omega\right\}, \nonumber \\
\underline{{\cal E}}_\Omega & = & \left\{\underline{{\bf E}}=\left.\begin{pmatrix} {\bf J}_0 \\ {\bf E}_0 \end{pmatrix} \quad \right| \quad{\bf J}_0 \in {\cal J}_\Omega,\quad{\bf E}_0\in {\cal E}_\Omega\right\}, \nonumber \\
\underline{{\cal J}}_\Omega & = &\left\{\underline{{\bf E}}=\left.\begin{pmatrix} {\bf E}_0 \\ {\bf J}_0 \end{pmatrix} \quad \right| \quad{\bf E}_0 \in {\cal E}_\Omega,\quad{\bf J}_0\in {\cal J}_\Omega\right\},
\eeqa{P.11}
and on $\underline{{\cal H}}_\Omega$ we introduce the inner product and norm
\begin{equation} \left(\begin{pmatrix} {\bf P}_1 \\ {\bf P}_2 \end{pmatrix},\begin{pmatrix} {\bf P}_3 \\ {\bf P}_4 \end{pmatrix}\right)_{\underline{{\cal H}}_\Omega}=({\bf P}_1,{\bf P}_3)_\Omega+({\bf P}_2,{\bf P}_4)_\Omega,
\quad |\underline{{\bf P}}|_{\underline{{\cal H}}_\Omega}=\sqrt{(\underline{{\bf P}},\underline{{\bf P}})_{\underline{{\cal H}}_\Omega}},
\eeq{P.11a}
so that the subspaces $\underline{{\cal E}}_\Omega$ and $\underline{{\cal J}}_\Omega$ are orthogonal complements.
If, for some $\underline{\beta}> \underline{\alpha}>0$, $\underline{{\bf L}}$ satisfies the boundedness and coercivity conditions,
\begin{eqnarray} &~& \underline{\beta} >\sup_{\substack{\underline{{\bf P}}\in\underline{{\cal H}}_\Omega \\ |\underline{{\bf P}}|=1}}|\underline{{\bf L}}\,\underline{{\bf P}}|_{\underline{{\cal H}}_\Omega}, \nonumber \\
&~& \mathop{\rm Re}\nolimits (\underline{{\bf L}}\,\underline{{\bf E}},\underline{{\bf E}})_{\underline{{\cal H}}_\Omega}\geq\underline{\alpha}|\underline{{\bf E}}|_{\underline{{\cal H}}_\Omega}^2 \text{ for all } \underline{{\bf E}}\in\underline{{\cal E}}_{\Omega},
\eeqa{P.12}
then, say with ${\bf h}'=-{\bf h}$, \eq{P.10} will have a unique solution
for the fields $\underline{{\bf E}}$ and $\underline{{\bf J}}$. From these fields one extracts the fields ${\bf E}$, ${\bf J}$, ${\bf E}'$ and ${\bf J}'$ that solve \eq{P.5}. Conversely,
if the equations \eq{P.5} did not have a unique solution, there would not be a unique solution to \eq{P.10} in contradiction to what the coercivity of $\underline{{\bf L}}$ implies.
Thus if the boundedness and coercivity conditions \eq{P.12} hold, then the Green's operator ${\bf G}_\Omega$ for the equations \eq{6a} exists. As previously, one can look for appropriate
$Q_\Omega$-convex functions that guarantee that the coercivity condition in \eq{P.12} holds. At this stage it is unclear
if sharper bounds on the generalized spectrum can be obtained by making this splitting -- this needs further exploration.
\section{Quasiperiodic Periodic boundary conditions and bounds on the Floquet-Bloch spectrum}
Quasiperiodic boundary conditions, as appropriate to wave equations, can easily be dealt with too within the existing framework,
and the associated spectrum is the Floquet-Bloch spectrum. Bounds on this spectrum for periodic operators are of wide interest. Following the seminal papers of
John \cite{John:1987:SLP} and Yablonovitch \cite{Yablonovitch:1987:ISE} there was tremendous excitement in obtaining
acoustic, electromagnetic and elastic band-gap materials for which the relevant operator had no spectrum in a frequency window
for any value of ${\bf k}$. A rigorous proofs of the existence of band-gaps was given by Figotin and Kuchment \cite{Figotin:1996:BGSI,Figotin:1996:BGSII}.
See also the book of Joannopoulos et.al. \cite{Joannopoulos:2008:ISS}. Upper bounds on photonic band gaps have been given by
Rechtsman and Torquato \cite{Rechtsman:2009:MOU}. More recently Lipton and Viator \cite{Lipton:2017:BWC} obtained convergent power series
for the Bloch wave spectrum in periodic media. For general theory related to Floquet theory
for partial differential equations, and Bloch waves, see Wilcox's paper \cite{Wilcox:1978:TBW} and Kuchment's book \cite{Kuchment:1993:FTP}.
Here we concentrate on the acoustic equations \eq{5.A} as
the extension to other wave equations is obvious. We assume that the effective mass density matrix $\bfm\rho({\bf x},\omega)$, and the bulk modulus
$\kappa({\bf x},\omega)$ are periodic with now $\Omega$ denoting the unit cell of periodicity. We look for solutions where
the pressure ${\bf P}({\bf x})$ and velocity ${\bf v}({\bf x})$ are quasiperiodic, i.e.
\begin{equation} {\bf P}({\bf x})=e^{i{\bf k}\cdot{\bf x}}\widetilde{{\bf P}}({\bf x}),\quad {\bf v}({\bf x})=e^{i{\bf k}\cdot{\bf x}}\widetilde{{\bf v}}({\bf x}),
\eeq{B-1}
in which $\widetilde{{\bf P}}({\bf x})$ and $\widetilde{{\bf v}}({\bf x})$ are periodic functions of ${\bf x}$ with period cell $\Omega$. It follows that
\begin{eqnarray} \nabla{\bf P}({\bf x})& = & e^{i{\bf k}\cdot{\bf x}}\nabla\widetilde{{\bf P}}({\bf x})+i{\bf k} e^{i{\bf k}\cdot{\bf x}}{\bf k}\widetilde{{\bf P}}({\bf x}), \nonumber \\
\nabla \cdot{\bf v}({\bf x})& = & e^{i{\bf k}\cdot{\bf x}}\nabla \cdot\widetilde{{\bf v}}({\bf x})+ie^{i{\bf k}\cdot{\bf x}}{\bf k}\cdot\widetilde{{\bf v}}({\bf x}).
\eeqa{B-2}
Making this substitution in \eq{5.A} we obtain
\begin{equation} \begin{pmatrix}-i\widetilde{{\bf v}} \\ {\bf k}\cdot\widetilde{{\bf v}}-i\nabla \cdot\widetilde{{\bf v}} \end{pmatrix}
=\begin{pmatrix}-(\omega\bfm\rho)^{-1} & 0 \\ 0 & \omega/\kappa\end{pmatrix}\begin{pmatrix}i{\bf k}\widetilde{P}+\nabla\widetilde{P} \\ \widetilde{P}\end{pmatrix},
\eeq{B-3}
or equivalently
\begin{equation} \underbrace{\begin{pmatrix}-i\widetilde{{\bf v}} \\-i\nabla \cdot\widetilde{{\bf v}} \end{pmatrix}}_{\widetilde{{\bf J}}({\bf x})}
=\underbrace{\begin{pmatrix}-(\omega\bfm\rho)^{-1} & -i(\omega\bfm\rho)^{-1}{\bf k} \\ i{\bf k}(\omega\bfm\rho)^{-1} & z+(\omega/\kappa)\end{pmatrix}}_{\widetilde{{\bf L}}({\bf x})}\underbrace{\begin{pmatrix} \nabla\widetilde{P} \\ \widetilde{P}\end{pmatrix}}_{\widetilde{{\bf E}}({\bf x})},
\eeq{B-4}
where $z={\bf k}\cdot{\bf k}$. Written in this form we see that $\widetilde{{\bf L}}$ is a linear function of $z$ and the elements of ${\bf k}$, and hence we recover
the known result \cite{Wilcox:1978:TBW} that the Green's function will be analytic in these variables away from the spectrum.
All the analysis applies as before. We take ${\cal E}_{\Omega}$ to consist of $\Omega$-periodic fields $\widetilde{{\bf E}}({\bf x})$ having the same form
as that appearing on the left hand side of \eq{B-4} and we take ${\cal J}_{\Omega}$ to consist of $\Omega$-periodic fields $\widetilde{{\bf J}}({\bf x})$ having the same form
as that appearing on the right hand side of \eq{B-4}. Integration shows that these spaces are orthogonal.
\acknowledgements{\aj{The author is grateful to Yury Grabovsky for pointing out that in the formulation of \eq{6a} one needs the closure
$\overline{\cal J}_\Omega$ of ${\cal J}_\Omega$ rather than just ${\cal J}_\Omega$. Also the referee is thanked for helpful comments.} The author is thankful to the National Science Foundation for support through the Research Grant DMS-1211359.}
|
1,116,691,500,734 | arxiv | \section*{RabbitMQ current design - Not part of the paper structure}
\begin{figure}
\centering
\resizebox{\textwidth}{!}{
\input{Diagrams/rbmq-seq-diag_ENTERINITMODE}
}
\caption{Sequence Diagram for the function: $EnterInitializationMode$}
\label{fig:my_label}
\end{figure}
\begin{figure}
\centering
\resizebox{.9\textwidth}{!}{
\input{Diagrams/rbmq-seq-diag_DOSTEP}
}
\caption{Sequence Diagram for the function: $DoStep$}
\label{fig:my_label}
\end{figure}
\begin{figure}
\centering
\resizebox{.9\textwidth}{!}{
\input{Diagrams/rbmq-seq_CHECKSHD}
}
\caption{Sequence Diagram for the function: $checkSHD$, that deals with consumed system health data.}
\label{fig:my_label}
\end{figure}
\begin{figure}
\centering
\resizebox{.6\textwidth}{!}{
\input{Listings/msg2Send_SHD}
}
\caption{Message on system health data published by the rabbitMQFMU.}
\label{fig:my_label}
\end{figure}
\begin{figure}
\centering
\resizebox{.6\textwidth}{!}{
\input{Listings/msg2Consume_SHD}
}
\caption{Message on system health data consumed by the rabbitMQFMU}
\label{fig:my_label}
\end{figure}
\section{Background}\label{sec:background}
In this Section we provide a brief summary on the relevant concepts for this work, such as co-simulation, FMI standard, master algorithms, and the tools we use for their realisation.
Thereafter, we present the first released version of the RMQFMU (RMQFMU\textsubscript{0}), which we have extended as presented in this paper.
\subsection{Concepts and Tools}
The realisation of CPSs and constituent systems is a cross-disciplinary process, where the different components are modelled using different formalisms and modelling tools~\cite{gomes2018co}.
In order to evaluate the behaviour of such systems as a whole, co-simulation techniques are used, which require the integration of the separate models into what are called multi-models~\cite{7496424}.
For the latter to be possible, the tools used to produce the individual models need to adhere to some standard.
One such standard is the Functional Mock-up Interface 2.0 for Co-simulation (FMI)~\cite{FMIStandard2.0.1} that defines the C-interfaces to be exposed by each model, as well as interaction constraints, packaging, and a static description format.
An individual component that implements the FMI standard is called Functional Mock-up Unit (FMU).
A co-simulation is executed by an orchestration engine that employs a given master algorithm. The master algorithm defines the progression of a co-simulation in terms of getting outputs via \texttt{getXXX} function calls, setting inputs via \texttt{setXXX} function calls, and stepping the individual FMUs in time via the \texttt{doStep} function calls.
Common master algorithms are the Gauss-Seidel and Jacobi (the curious reader is referred to~\cite{gomes2018co} for an overview of these algorithms and a survey on co-simulation).
In this work, we use the open-source INTO-CPS tool-chain~\cite{Fitzgerald&15}, for the design and execution of co-simulation multi-models, with Maestro~\cite{Thule&19} as orchestration engine employing an FMI-based Jacobi master algorithm.
\subsection{Overview of the RMQFMU\textsubscript{0}}
RMQFMU\textsubscript{0} was implemented to enable getting external/historical data into the co-simulation environment, for either replaying such data in simulation, or for performing different kinds of analysis, e.g. checking its difference from expected data (the reader is referred to the Water-Tank Case-Study for an example\footnote{Available at \url{https://github.com/INTO-CPS-Association/example-single_watertank_rabbitmq}, visited May 6, 2021}).
Essentially, RMQFMU\textsubscript{0} subscribes to messages with a specified routing key and outputs messages as regular FMU outputs at specific points in time based on the timestamp of the messages.
At every call of the \texttt{doStep} function, the RMQFMU\textsubscript{0} attempts to consume a message from the server.
A retrieved message is placed in an incoming queue, from which it is thereafter processed, according to the quality constraints (\textit{maxage} and \textit{lookahead}), described shortly.
In case there is no data available, the RMQFMU\textsubscript{0} will wait for a configurable timeout before exiting.
Note that, the \texttt{doStep} is tightly bound with the consuming operation, i.e. they are both contained within the main thread of execution.
If on the other hand there is data, its validity with respect to time is checked.
The \textit{maxage}, main parameter of the RMQFMU\textsubscript{0}, allows to configure the age of data within which it is considered valid at a given time-step.
Additionally, the \textit{lookahead} parameter specifies how many messages will be processed at a step of RMQFMU and thereby at the given point in time\footnote{In this version, the functionality based on the \textit{lookahead} is not fully implemented.}.
The behaviour of the RMQFMU\textsubscript{0} has been formally verified in previous work~\cite{thule2020formally}.
\section{RMQFMU\textsubscript{2} Configuration Guidelines}\label{sec:guidelines}
The parameters of the RMQFMU\textsubscript{2} should be configured based on the use-case in order to result in desired behaviour.
If possible, the simulation step-size should match the frequency of sending data $f_{data}$.
However, depending on the time granularity needed by different FMUs, achieving such alignment might not be possible.
The \textit{maxage} value has to be big enough to make up for any time gap between data, but also low enough to ensure consistency in domain of use.
E.g. for some applications, it might the case that data should not be older than $200$ms to ensure correct operation.
The \textit{lookahead} can be set to counter-balance the effect of the \textit{maxage}, as it allows to jump to the latest data that is available in the queue (within the limits of the \textit{lookahead}).
Note however, larger values of the \textit{lookahead} will result in bigger time jumps between the messages outputted.
If more intermittent values are required, then a lower value should be adopted.
The frequency of sending data influences the meaningful range of values for the \textit{lookahead}, e.g. sending data every $2$ms, will result in $500$ messages per second.
For a simulation step-size of $100$ms, a reasonable value of the \textit{lookahead} would be circa $50$.
Finally, the speed of the simulation also affects what is present in the queue at any time, as a result impacting RMQFMU\textsubscript{2} outputs.
\section{Introduction}\label{sec:intropap}
Cyber-physical systems (CPSs) refer to systems that combine computational and physical processes, and play an important role in the development of intelligent systems.
Using real world operation data could possibly facilitate smart decision-making~\cite{banerjee2012ensuring}.
Harnessing this potential is by no means trivial, with challenges including safety~\cite{baheti2011cyber}, reliability, and security~\cite{ecsel2016multi}, among others, that need to be tackled.
Additionally, CPS development is a multi-disciplinary process, where different components will be modelled and validated by different tools used in the different involved disciplines.
The evaluation of such complex systems can be performed through co-simulation~\cite{Gomes&18}.
This is made possible by developing the components according to some standard, e.g. the Functional Mock-up Interface (FMI) adopted in our work, with components referred to as functional mock-up units (FMUs).
However, this is not sufficient, as CPSs together with their environment(s) are subject to continuous change, and evolve through time, possibly diverging significantly from the initial co-simulated results~\cite{fitzgerald2019multi}.
Digital twins (DTs), defined as digital replicas of the physical components, also known as physical twins (PTs), can be used to follow the behaviour of the PTs and CPSs during operation, potentially through learning~\cite{Fitzgerald&14f}, by adapting the models, as well as performing different tasks such as monitoring or predictions.
The DT and the aforementioned (or more) operations can be developed via the co-simulation of a modelled system that corresponds to the deployed CPS.
In order to connect the deployed CPS to the DT, the implementation of data brokering between them becomes a necessity.
Such data brokering can be useful in different scenarios; in this paper we identify four such:
\begin{enumerate}
\item For data analysis and visualisation of system behaviour, where the user is interested in feeding back to the co-simulation data recorded from the operation of the hardware, i.e. historical data, to perform some analysis on the data, potentially supporting the visualisation of said data.
\item For co-simulation, where the user is interested in coupling systems that are simulated in different environments, {in those cases for which it is easier to use data-brokering than to implement the FMI interface for a non FMI-compliant component, e.g. connecting to a Gazebo simulation}~\footnote{In other cases, the generation of an FMU might be preferable. Tools like UNIFMU~\cite{legaard2020rapid} for quick FMU prototyping, or PVSio-web based tool-kits~\cite{palmieri2018flexible}, are designed to help users prototype FMUs for their co-simulations.}
\item For the realisation of the digital shadow (DS), where the digital models of the physical twin are contained within the co-simulation environment.
The user is interested in getting the live data from the PT into the co-simulation environment in order to estimate the difference between actual and simulated data along with visualising the operation of the physical twin.
This is similar to the first usage, however the digital shadow would operate with live data.
\item For the realisation of the digital twin, where the user is interested in enabling the communication link from the DS to the PT, thus fully realising the implementation of the DT concept.
Indeed, components within the co-simulation environment, based on the live data, could send live feedback to the PT.
This can be useful for monitoring the behaviour of the PT, or during CPS development, to lower the cost of testing with Hardware in the Loop
\end{enumerate}
In this publication we extend an existing Data-Broker called RabbitMQ FMU (RMQFMU~\footnote{\url{https://github.com/INTO-CPS-Association/fmu-rabbitmq.git}})~\cite{thule2020formally}, initially suitable for the first scenario, to enable data brokering both to and from an FMI2-enabled co-simulation, applicable for all the aforementioned scenarios.
We present an experience report on applying the RMQFMU to two actual cases undergoing development to become digital twin systems: a tabletop robot arm and an autonomous agriculture robot.
As part of this experience report we present guidelines on how to configure the RMQFMU parameters by presenting various experiments that show their effects and relation to one another.
Through the application of RMQFMU to the aforementioned cases and experiments several needs were discovered such as:
\begin{enumerate}
\item Data Platform - Get all available data instead of minimally-needed
data to enable decision making, possibly allowing to jump ahead to
future data.
\item Performance - RMQFMU shall be as fast as possible.
\item Data Delay - RMQFMU shall output the newest data if available.
\end{enumerate}
To address need (1) and (2), the existing RMQFMU, henceforth referred to as
RMQFMU\textsubscript{0}, was realised with multi-threading instead of
single-threading. This new multi-threaded version is referred to as
RMQFMU\textsubscript{1}. Experimentation related to detailing the
configuration of RMQFMU\textsubscript{1} and its related effects led to the
discovery of need (3).
In order to mitigate (3), yet another version of RMQFMU was realised
referred to as RMQFMU\textsubscript{2}, which represents the latest version
of the RMQFMU.
The rest of this paper is organised as follows.
The next section provides an overview of the FMI standard, the first version of the RMQFMU (RMQFMU\textsubscript{0}), as well as some of the technologies employed in the realisation of the case-studies.
Afterwards, we informally derive the requirements for RMQFMU, based on the usage scenarios.
Section~\ref{sec:rabbitmq_fmu} presents RMQFMU\textsubscript{1,2}, how it mitigates some inadequacies of RMQFMU\textsubscript{0}, and how it compares to RMQFMU\textsubscript{0}.
Thereafter, in Section~\ref{sec:experiments} the individual case-studies are described, and the results gained with RMQFMU\textsubscript{1,2} are presented.
Moreover, a detailed comparison between RMQFMU\textsubscript{1} and RMQFMU\textsubscript{2} is made.
Section~\ref{sec:guidelines} provides a set of guidelines meant to aid the use of RMQFMU\textsubscript{2}, whereas Section~\ref{sec:conclusion} concludes the paper.
\section{Concluding Remarks}\label{sec:conclusion}
In this paper we have described an extended RMQFMU data-broker that enables coupling an FMI-based co-simulation environment to a non-FMI external component, via the AMQP protocol.
The RMQFMU supports communication in both directions, and can be used in different contexts ranging from replaying historical data into a co-simulation, to enabling the realisation of the DT concept by connecting the DT to its physical counterpart.
We evaluate this component in terms of its performance, i.e. the real-time duration of a simulation step.
Our results show the benefit of implementing a threaded solution, that effectively decouples the \texttt{doStep} logic from the consumption from the rabbitMQ server.
Moreover, we explore different values of its configurable parameters, and provide usage guidelines for practitioners.
There are {five potential} directions for future work.
First, we are interested in enabling the RMQFMU to take bigger step-sizes if necessary.
In some monitoring applications, where only the latest data-points are relevant, such jump could be of use to mitigation components that deal with out of synchronisation situations.
Second, we plan to formally verify the presented RMQFMU, thus extending previous work that tackled its initial version~\cite{thule2020formally}.
Third, we intend to profile the overhead of RMQFMU as data-broker compared to an FMU that directly queries a database, for example in the context of large amounts of data.
Fourth, we will enable the master-algorithm to learn the optimal step-sizes automatically and deal with recurring variation at run-time.
{Fifth, we are interested in the implementation of a more general solution, that can be configured to use different libraries and protocols, e.g. ZeroMQ.}
\section{The RMQFMU}\label{sec:rabbitmq_fmu}
The design of the RMQFMU is based on a set of informally derived requirements, covering those functionalities needed for the adoption of the RMQFMU in the scenarios described in
Section~\ref{sec:intropap}.
These requirements are given as follows:
\begin{enumerate}
\item The RMQFMU is able to get the data published by an external system to the RabbitMQ server.
This is relevant for all scenarios.
\item The RMQFMU is able to publish data to the RabbitMQ server, thus closing the communication link.
This is relevant for the second and fourth scenario, i.e. for co-simulation and DT realisation
\item The FMU steps as fast as possible. Indeed the role of the RMQFMU is that of a data broker, as such it is not part of the system being simulated, rather a facilitating entity.
Therefore, the delays it causes should be minimal and impact the overall co-simulation environment as little as possible.
\item Provide quality constraints, e.g. with respect to age of data (\textit{maxage}).
\item Provide performance constraints, e.g. number of messages processed per step (\textit{lookahead}).
\end{enumerate}
RMQFMU\textsubscript{0} already fulfils requirements 1 and 4.
Requirement 5 is partially implemented in this version as well, in that while the \textit{lookahead} determines how many messages to retrieve from the incoming queue at every time-step, there will not be more than one message at a time in this queue, given that a consume call is performed at the \texttt{doStep} call until a valid message is retrieved.
\begin{comment}
Given these purposes we will outline the requirements for the rabbitmq fmu, among which:
\begin{itemize}
\item Main requirement: Bridge real-world data into a co-simulation.
\item Secondary requirement: Be opaque! It is VITAL to know to which degree it can keep up, such that measures can be taken.
\item As Fast As Possible Stepping - The broker is not part of the system to be simulated, it is a facilitating entity. Thus, it can ONLY be too slow.
\item Quality constraints - when is data too old?
\item Performance constraints - Lookahead? Amount of messages to process? Which parameter can have the biggest influence on performance? \casper{Do we need a timing setup. Perhaps via logging and scalarvariables? Perhaps some preprocessing macros, I am sure that Henrik has some ideas here?}
\item A platform on which to extend for particular scenarios. (i.e. by getting the data out of the socket layer)
\item Previous implementation - good enough - lacking when applied?
\item New requirements based on DT lead to need for new platform
\item New platform in terms of: Performance
\item New platform in terms of: system health
\item New platform in terms of: enabling future work
\item Guarantees? Max age guarantee, do we need others?
\item Threaded vs non-threaded configuration
\end{itemize}
\end{comment}
In order to tackle requirement 2, we enable the configuration of the inputs of the RMQFMU\textsubscript{1,2} as needed, i.e., a user can define as many inputs as desired, of one the following types: integer, double, boolean, and string.
These inputs will be sent to the RabbitMQ server, on change; in other words, the values of the inputs will be forwarded if they have changed as compared to the previous step taken by the RMQFMU\textsubscript{1,2}.
This check is performed within the \texttt{doStep}.
In order to tackle requirement 3, we investigate the potential benefit of a threaded configuration of the RMQFMU\textsubscript{1,2}.
Indeed, the FMU allows for a build-time option to enable a multi-threaded implementation, with a separate thread to interface and consume data from the RabbitMQ server, parsing incoming data and placing it in the FMU incoming queue.
The main \texttt{doStep} function of the FMU still executes in the main thread context of the calling simulation orchestration application, and it reads and processes data from the FMU incoming queue and produces outputs. If the multi-threaded option is disabled (default), the \texttt{doStep} function will consume data directly from the RabbitMQ server in the context of the calling main thread.
In general the potential performance benefits of the threaded implementation will depend on the amount of data being consumed, parsed, and provided in the internal incoming FMU queue when the \texttt{doStep} function is called.
Hence, the benefits will depend largely on the co-simulation environment.
If the co-simulation is fast, in vague terms if the delay between each call of the FMU \texttt{doStep} function is short compared to the rate of the incoming data, then the FMU will be mostly blocking for I/O.
In this situation, the separate consumer thread may not provide much benefit as the \texttt{doStep} function will still need to wait/block for incoming data.
On the other hand, if the co-simulation is slower compared to the rate of data, then the separate thread may be able to consume, parse, and provide data to the internal incoming FMU queue, in parallel to the main co-simulation thread running its orchestration engine and executing other potential FMUs or monitors.
In addition, any performance benefits of a threaded implementation also depends largely on the execution platform having a multi-core processor and the OS/thread environment to be able to take advantage.
The threaded implementation allows for the full realisation of the \textit{lookahead} functionality, thus tackling requirement 5.
The consumer thread is continuously retrieving data -- when available -- and placing these messages in the incoming queue, from where said data is processed in chunks of \textit{lookahead} size.
\section{Experiments}~\label{sec:experiments}
The RMQFMU\textsubscript{1,2} has been evaluated with a series of experiments, across different combinations of parameters such as the \textit{maxage} and \textit{lookahead}, and in two different case-studies.
The purpose of these setups is to provide an understanding on the performance of the RMQFMU\textsubscript{1,2}, its parameter tuning, as well as provide some insight into the effect of external factors.
The first case-study presents a scenario based on data from a deployed industrial robot, thus adequately representing an industrial case.
As a result, we perform the performance evaluation in this case-study.
Additionally, we add mockup components to the co-simulation structure, to mimic a more realistic co-simulation with different components for different purposes, such as data analysis, prediction among others.
We also consider the impact of external factors, such as the frequency of sending data to the RMQFMU\textsubscript{1,2}.
Whereas, the second case-study presents a setup with data from the Gazebo simulation of the Robotti agricultural robot~\cite{foldager2018design}, rather simple both in terms of the size of the messages being sent and the internal structure of the co-simulation.
The parameters of interest are the \textit{lookahead}, \textit{maxage}, and the frequency with which we send data into the RMQFMU\textsubscript{1,2}.
The experimentation is divided in two phases.
In the first, we evaluate the impact of the multi-threaded implementation on the performance of the RMQFMU.
Specifically, we compare RMQFMU\textsubscript{1} to RMQFMU\textsubscript{0}, and argue for the suitability of adopting the multi-threaded configuration as the primary implementation for the RMQFMU.
Additionally we consider the effects of the \textit{maxage} and \textit{lookahead} for RMQFMU\textsubscript{1}.
In the second phase, we evaluate the alterations to RMQFMU\textsubscript{1}, i.e. RMQFMU\textsubscript{2}, in order to tackle the side effects observed in phase 1, providing both performance and behaviour results.
\subsection{Case-study 1: Universal robot 5e (UR)}
This case study concerns an industrial robot called UR5e\footnote{Available at \url{https://www.universal-robots.com/products/ur5-robot/}, visited April 7, 2021} (Figure~\ref{fig:ur5e}).
The robot has a reach of 85 mm, payload of 5kg and 6 rotating joints.
It is chosen because the UR5e is in production, thus represents a realistic component in a digital twin setting, and thereby a realistic amount of data.
The data set is generated from a series of movements used for calibrating the digital twin to a related physical twin.
Therefore, all joints are exercised.
The test data from the UR robot used is present in csv format (35123 lines including the header) and consists of messages (excluding the header) with 107 float values (one being time) and 10 integer values.
The messages have a sample rate (frequency $f_{data}$) of $2$ms and will be replayed/produced into the RabbitMQ broker with this rate.
The test data contains a substantial number of time gaps, i.e.\ places where two consecutive messages in the input data are spaced with more than $2$ms.
Data is replayed at a constant frequency of $2$ms and as such some messages are produced faster than their timestamps indicate. These effects are discussed in the test results.
Tests are also performed on a \textit{cleaned} test data with a constant message spacing of $2$ms between all pairs of messages - i.e.\ no additional time gaps.
To investigate the behaviour of the FMU, we perform tests
with different variations and relations between the simulation step size and the delay imposed by the simulation environment. {The tests are run on a 2.6GHz Intel Core i7 (6 cores), with 32GB RAM.}
We perform the tests with and without the threading option to investigate its effect, $t_{\mathit{on}}$ and $t_{\mathit{off}}$ respectively.
We first consider simulation step duration and environment delays that simulates a setup where data is expected to be consumed as fast as its produced - i.e.\ every $2$ms and the overhead of the simulation delay is minimal.
In this case we consider both the simulation step and the simulation delay to match the input data rate of $2$ms.
Then we consider situations with larger simulation step duration and larger simulation delays - to mimic co-simulation environments with additional FMUs or monitors.
The full list of test data used is provided in Table~\ref{tab:exp1}.
In short, the results will provide an insight into how these different parameters influence the performance of the RMQFMU.
Note, that for these tests we use a fixed \textit{lookahead} of $1$ and a fixed \textit{maxage} of $300$ms. The max age is chosen to cover for time gaps in the input data up to this range.
\begin{minipage}{\textwidth}
\begin{minipage}[t]{0.4\textwidth}
\centering
\raisebox{-15ex-\height}{\includegraphics[scale=0.15]{Figures/ur5e.png}}
\captionof{figure}{UR5e robo
}
\label{fig:ur5e}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}[t]{0.4\textwidth}
\centering
%
\input{Tables/experiments_table_cas1}
\end{minipage}
\end{minipage}
\footnotetext{Due to space constraints, we present in the paper results for some selected cases marked with $*$. The results as a whole can be found in the technical report~\cite{frasheri2021rmqfmu}.}
\subsection{Case-study 2: Gazebo Simulation of Robotti}
The purpose of this case-study is to provide a basic example of a co-simulation receiving {data recorded~\footnote{We record the data to ease the testing process across different runs.} from }the Gazebo simulation of the Robotti agricultural robot~\cite{foldager2018design}, through the RMQFMU, in order to gain insight into how the configurable parameters of the RMQFMU affect its behaviour.
{Tests are run on a 2GHz Quad-Core Intel Core i5, with 16GB RAM.}
In this case the co-simulation environment consists of the RMQFMU and a monitor FMU.
The data of interest in this scenario, i.e. the data sent through the RMQFMU consists of: the $x$ and $y$ positions of the robot, and the $x$ and $y$ position of the nearest obstacle.
Additionally, a sequence number is attached to each message to keep track of the outputted messages during the processing of the results.
The monitor takes as input such data, and computes the distance between the robot and obstacle for every time-step.
In case such distance is below a predefined safety threshold, the monitor will issue an emergency stop to the Gazebo simulation of the robot.
\input{Tables/experiments_table_cas2b}
For this case-study we are interested in the following parameters (see Table~\ref{tab:exp2}), the \textit{maxage} $ma$ of the messages, and the \textit{lookahead} $la$, and how they affect the behaviour of the RMQFMU, i.e. in terms of the sequence of messages outputted at every time-step.
Note that for all the cases the threaded option is used ($t_{\mathit{on}}$).
Parameters such as the time-step size $t_s$, the frequency of sending data $f_{data}$ to the co-simulation are fixed for cases $1-4$, such that they align.
We argue such decision with the simplicity of the case-study, which could allow a user in reality to align these values.
The time-step of the simulation is fixed to $0.1$s for adequate granularity.
Whereas for cases $5-12$ we consider $f_{data}$ that doesn't match the size of the simulation step, and observe the effects of the \textit{maxage} and \textit{lookahead}.
\subsection{Phase 1 Results}
\begin{figure}[t]
\centerline{\includegraphics[scale=1]{Figures/rbmqv1/urdata/case3and4.pdf}}
\caption{Cases 3 and 4: step=100ms, delay=100ms,
thread=$t_{\mathit{off}}$/$t_{\mathit{on}}$ \vspace{-2mm}}
\label{fig:log_f2_s100_z100_orig}
\vspace{-2mm}
\end{figure}
\subsubsection{Case-study 1}
In this paper we will present only the results from cases $3$ and $4$ (Table~\ref{tab:exp1}) due to space constraints, and refer the reader to the technical report~\cite{frasheri2021rmqfmu} that contains the results over all cases. The results not included in this paper are coherent with the ones included.
In the included scenarios, the simulation step and the simulation delay are set each to $100$ms, in order to mimic a co-simulation environment that imposes a step size of $100$ms and has a simulation delay also of $100$ms, caused e.g.\ by other FMUs or monitors.
The results of these two tests are shown in Figure~\ref{fig:log_f2_s100_z100_orig}.
The number of messages to process in each step is $50$, given the input frequency of $2$ms.
Additionally, given the simulation delay equal to $100$ms, for the threaded configuration all $50$ messages will be available in the RMQFMU\textsubscript{1} queue when the step function is called. The artificial simulation delay of $100$ms covers enough time for the separate consumer thread to have consumed approximately $50$ messages. While for the non-threaded configuration, the step function itself needs to consume approximately $50$ messages off the socket interface inside a single step.
From Figure~\ref{fig:log_f2_s100_z100_orig} left-hand side graph it can be observed that for this test, the threaded configuration (red) shows an improvement in average step duration of approximately $10$ms compared to the unthreaded configuration (blue).
This difference corresponds to the overhead of consuming a single message from the socket interface via the rabbitmq client library, parsing the message, and finally adding it to the internal RMQFMU\textsubscript{1} queue.
As approximately $50$ messages need to be consumed in a single step, this accounts to an overhead of approximately $200\mu$s per message ($10$ms $/$ $50$ = $200\mu$s).
The right-hand side graph in Figure~\ref{fig:log_f2_s100_z100_orig} shows the internal FMU queue size at the exit of each simulation time step. The increase of queue size in the threaded configuration (red) is mostly an effect of gaps occurring in the input data.
Large gaps in the input will cause the RMQFMU\textsubscript{1} to stay at its current output for a predefined \textit{maxage} time period ideally covering the input gap.
This effect can also be observed in the left-hand side graph, when the step duration occasionally lowers to around $0$ when the RMQFMU\textsubscript{1} stays at its current output to cover a gap.
For this period, data will still be consumed by the separate RMQFMU\textsubscript{1} consumer thread and added to the internal queue.
The RMQFMU\textsubscript{1} implementation must therefore include guards to respect internal queue size limitations.
The internal RMQFMU\textsubscript{1} queue size in the unthreaded configuration (blue) is always $0$ at simulation step exit, since that configuration consumes only a single message off the socket interface per step.
In this configuration, a queue size build up will occur in the socket layer rather than the FMU layer.
\subsubsection{Case-study 2}
Similarly to Case-study 1, we will present only the results from cases $9-12$ (Table~\ref{tab:exp2}) due to space constraints, and refer the reader to the technical report~\cite{frasheri2021rmqfmu} that contains the results over all cases.
Note that the results not included remain coherent with the ones presented in the current paper.
The results for cases $9-12$ are displayed in Figure~\ref{fig:fig4}.
For a \textit{maxage} equal to $400$ms, we can see that there is not much an effect of the value set for the \textit{lookahead}.
This is due to the fact that with lower \textit{maxage}, the data becomes invalid sooner than for higher \textit{maxage}, thus triggering the RMQFMU\textsubscript{1} to fetch newer data from the incoming queue.
Nonetheless, for a $ma=2$s, it is possible to note a difference for the two \textit{lookahead} values, where for $la=5$, there is a bigger jump in the sequences of data for specific time-steps, when a new value is needed.
Additionally, sequence numbers remain fixed for consecutive number of time-steps, whereas for $la=2$ more in-between value are outputted, as expected. In the latter however, there is more delay in the output values.
Beside the \textit{maxage} value, the frequency of sending data also impacts the effect of the \textit{lookahead}, as it directly affects the number of messages found in the incoming queue.
\begin{figure
\vspace{-3mm}
\centering
\includegraphics[scale=1]{Figures/rbmqv1/gazebo/f200ms.pdf}
\caption{Messages outputted by the RMQFMU\textsubscript{1}, for varying \textit{ma}, \textit{la}, and $f_{data}=200$ms.\vspace{-2mm}
}
\label{fig:fig4}
\end{figure}
The number of skipped messages is not exactly the same from run to run, as observed in Figure~\ref{fig:fig2}, with a \textit{lookahead} equal to $50$ and \textit{maxage} fixed to $2000$ms, over 5 independent runs.
One factor that impacts the state of the queue is the speed of the execution of a simulation step.
In Figure~\ref{fig:fig2}, one can observe the wall-clock time for two of the simulation runs.
The first run is executed faster than real-time, at least until time step $2.0$, whereas the second is executed slower than real-time.
This depends on the underlying execution environment, as well as the load on said environment.
As such, a specific \textit{lookahead} will not necessarily output the same sequence numbers at each time step over different independent runs.
This effect is also visible in Figure~\ref{fig:fig4}, where near the end, for both \textit{lookahead} values, the sequence numbers that are outputted are similar.
This means that, there were no newer messages in the queue at those particular times for \textit{lookahead}$=5$, as the simulation was running faster than real-time.
\begin{figure}[tbh!]
\centering
\includegraphics[scale=1]{Figures/rbmqv1/gazebo/realtime.pdf}
\caption{Messages outputted by RMQFMU\textsubscript{1}, over 5 runs, for $ma = 2$s and $la = 50$.}
\label{fig:fig2}
\end{figure}
\subsection{Phase 2 Results}
As discussed in the previous results, the RMQFMU\textsubscript{0,1} has a rather conservative interpretation of the \textit{maxage} parameter.
The FMU will \textit{always} stay at its current output, if the timestamp of the output is within \textit{maxage} range from the current simulation step timestamp.
This means that, the timestamp of the output plus the \textit{maxage} is greater than or equal to the current simulation timestamp.
However, this interpretation is rather strict, in the sense that it doesn't take into account if newer data is present in the incoming FMU queue.
Thus, we want to relax this condition such that the FMU will only stay at its current output, if the output is within the \textit{maxage} and in addition there is no newer data available.
As such, the FMU should always move ahead to the latest data valid within the current step, and the \textit{maxage} is only considered when there is no new input.
We have implemented this change of \textit{maxage} semantics in the RMQFMU\textsubscript{2} version of the FMU.
The change itself had some side-effects impacting the performance of the FMU step function.
In short, the change means that the FMU has to consider if data is present before considering the \textit{maxage}.
However, situations may occur where data is present in the incoming queue, but it may have time-stamps in the future of the current simulation time. This can occur e.g.\ when replaying with a fixed interval historical data including gaps.
If this situation occurs, the RMQFMU\textsubscript{0,1} versions of the FMU would move the future data from the incoming queue into an internal processing queue, that would gradually keep increasing in size and thereby impacting the step performance.
To remedy this issue, RMQFMU\textsubscript{2} has an additional change to the internal queue structure.
In the following paragraphs we present results of experiments with our two case-studies using RMQFMU\textsubscript{2}.
\subsubsection{Case-study 1}
The effect of the RMQFMU\textsubscript{2} changes to the case-study 1 can be observed in Figure~\ref{fig:fig5}. The test configuration for this test is identical to Case 4 of Table~\ref{tab:exp1}. The results of this case with RMQFMU\textsubscript{1} are present in Figure~\ref{fig:log_f2_s100_z100_orig}.
To demonstrate the effect of moving to the latest input data available, we have specified a \textit{lookahead} size of $100$ in this test.
This covers more messages than being produced on average within a simulation time step of $100$ms and a data frequency of $2$ms, which is approximately $100 / 2 = 50$ messages.
Practically, the RMQFMU\textsubscript{2} consumes as many messages as possible within a time step.
It is possible to observe from Figure~\ref{fig:fig5},
graph on the left,
that in contrast to Figure~\ref{fig:log_f2_s100_z100_orig}, the step duration now has decreased from around $6000$us to around $1000$us.
This is due to the fact that we now use the larger (100 = move as close to step time as possible) \textit{lookahead} size as opposed to the original size 1, additionally to the improvement in the internal queue implementation. Also, the larger \textit{lookahead} size has the effect, that the queue size (right-hand sideL graph) stays low for a longer simulation time compared to earlier test results.
However, the queue size still increases due to both data being produced more frequently than consumed and to gaps in the input data. Thus, as mentioned before, the FMU needs to guard any internal queue size limitations.
\begin{figure}[tbh!]
\centering
\includegraphics[scale=1]{Figures/rbmqv2/urdata/combo.pdf}
\caption{Step duration and sequence numbers outputted by RMQFMU\textsubscript{2} at every step, for step-size of $100$ms, and delay of $100$ms.}
\label{fig:fig5}
\end{figure}
\vspace{-4mm}
\subsubsection{Case-study 2}
The new version of the RMQFMU\textsubscript{2} removes the initial delay in the outputted sequence numbers noticeable for large values of the \textit{maxage}, e.g. $2000$ms (Figure~\ref{fig:fig6}, for Cases $1-4$), as compared to RMQFMU\textsubscript{1} (Figure~\ref{fig:fig4})~\footnote{Note, the initial delay is present in all cases for RMQFMU\textsubscript{1}.}.
The utility of the \textit{maxage} parameter becomes evident, once gaps are introduced in the data, in these experiments equal to $500$ms, Figure~\ref{fig:fig6b}, left graph, and $1000$ms, right graph.
It can be observed how when there is no input, the sequence value that is outputted is the last value that is valid from a time perspective.
Naturally, for larger gaps, the sequence numbers outputted are more delayed, e.g. for $t=5$s, the RMQFMU\textsubscript{2} with $500$ms gaps is at number $10$, whereas for $1000$ms is at $5$.
Note that, with RMQFMU\textsubscript{1}, given the initial delay, we would expect a lower sequence number outputted at $t=5$s.
\begin{figure}[bth!]
\centering
\includegraphics[scale=1]{Figures/rbmqv2/gazebo/f100ms.pdf}
\caption{Messages outputted by the RMQFMU\textsubscript{2} at every step, for $f_{data}=100$ms.}
\label{fig:fig6}
\end{figure}
\begin{figure}[tbh!]
\centering
\includegraphics[scale=1]{Figures/rbmqv2/gazebo/f100ms_gaps.pdf}
\caption{Messages outputted by the RMQFMU\textsubscript{2} at every step, for $f_{data}=100$ms and $la=1$, with gaps of $500$ms (left), and $1000$ms (right).}
\label{fig:fig6b}
\end{figure}
In order to show the utility of the \textit{lookahead}, we ran the experiments with simulation step-size equal to $100$ms, for values of \textit{maxage} in \{200ms, 2000ms\}, with frequency of sending data equal to $2$ms (Figure~\ref{fig:fig7}).
For smaller values of the \textit{maxage}, the delay of the RMQFMU\textsubscript{2} for \textit{lookahead} equal to $1$, while consistently present, is rather low.
However, for larger values of the \textit{maxage}, the delay for low \textit{lookahead} values is non-negligible.
It is possible to observe that for the duration of the \textit{maxage} the sequence numbers gradually increase by $1$, as expected for \textit{lookahead} equal to $1$.
After the initial \textit{maxage} duration has passed, there is a jump in the sequence numbers, that continue to follow the results for \textit{lookahead} equal to $50$, albeit with a more or less constant delay until the end.
A similar effect was observed in Figure~\ref{fig:fig4}, with the difference that in the current version, the RMQFMU\textsubscript{2} checks if there are newer messages in the queue.
This is due to the fact that after $2$s, older in-between values become invalid, and thus are not outputted, with RMQFMU\textsubscript{2} going back to the queue for more recent messages.
\begin{figure}[tbh!]
\centering
{ \includegraphics[scale=1]{Figures/rbmqv2/gazebo/f2ms_pap.pdf}}
\caption{RMQFMU\textsubscript{2} output at every step, for $f_{data}=2$ms, over $2$s.}
\label{fig:fig7}
\end{figure}
\section{Related Work}
\input{Discussion}
\input{Conclusion}
\section{Background}
\label{sec:back}
This section provides background information on the INTO-CPS toll chain and its current Design Space Exploration (DSE) support, and introduces the concept of Multi-Objective Optimisation (MOO) and existing MOO libraries.
\subsection{INTO-CPS and Design Space Exploration (DSE) Support}
\label{sec:back:into}
INTO-CPS is an integrated tool chain for comprehensive Model-Based Design (MBD) of Cyber-Physical Systems (CPSs)''~\cite{Larsen&17a}. INTO-CPS is based around the Functional Mock-up Interface (FMI)\footnote{\url{https://fmi-standard.org/}} that allows collaborative models of CPS to be defined, configured, and co-simulated. INTO-CPS is comprised of two main components:
\begin{enumerate}
\item A front-end called the INTO-CPS Application that allows for configuration and execution of co-simulations.
\item A back-end program called the Co-simulation Orchestration Engine (COE) called Maestro~\cite{Thule&17}, which provides functionality for, and oversees, the actual execution of co-simulations.
\end{enumerate}
Around the core of co-simulation, INTO-CPS provides other useful functionalities such as automatic Design Space Exploration (DSE), in which multiple co-simulations are executes automatically and the results compared. This functionality is provided by a set of Python scripts which can be downloaded from within the INTO-CPS Application. The INTO-CPS Application and DSE scripts provide the ability to configure DSE using the following:
\begin{description}
\item[Search Algorithm] The ability to choose between which optimisation algorithm should be used to perform the search. The currently available choices are `Exhaustive Search' and `Genetic'.
\item[Experiment Parameters] The ability to define which parameters of the model should be considered part of the design space and their bounds.
\item[Parameter Constraints] The ability to define constraints on the varying parameters e.g., ``parameter A should always have a value greater than parameter B, to be considered a feasible solution.''
\item[External Script Objectives] The ability to point to a script that uses the results of a simulation run to calculate the model's fitness according to a custom objective.
\item[Results Output] Once a run of DSE has completed, the current implementation outputs a `Pareto front' plot showing the optimal solutions regarding the objectives defined, and the set of input parameters required to recreate these optimal configurations.
\end{description}
The work reported in this paper aims to replicate these features using a MOO library, with the aim of providing the user with a seamless option to apply further algorithms in their exploration of design space.
\subsection{Multi-Objective Optimisation (MOO)}
\label{sec:back:moo}
Multi-Objective Optimisation (MOO) is the act of optimising a system as according to multiple specific objective criteria~\cite{gunantara2018review}. A solution is considered Pareto optimal if no aspect of that solution can be improved without making another aspect of the solution worse off. Due to the nature of this, any optimisation problem that has more than one objective will most likely have more than one Pareto-optimum solution. MOO algorithms may attempt to find a representative set of optimisations, to quantify the trade off, or pick one solution based on the preferences of the Decision Maker (DM).
The work reported in this paper links a MOO library to the INTO-CPS tool chain, thus we do not go into details regarding how various MOO algorithms work. The core concept is the same as the existing genetic DSE offered by INTO-CPS --- there is a set of possible parameters and a method which proposes solutions, and selects and refines these based on the results of co-simulation. Some MOO algorithms are based on genetic algorithms, such as the popular Non-dominated Sorting Genetic Algorithm-II (NSGA-II)~\cite{deb2002fast} and Strength Pareto Evolutionary Algorithm 2 (SPEA-2)~\cite{zitzler2001spea2} algorithms. Other MOO algorithms are not genetic, such as Particle Swarm Optimization and Simulated Annealing~\cite{suman2006survey}.
When selecting a library for the implementation, a number of options were considered. The three main options were MOEA, jMetal and pymoo. MOEA (Multi-Objective Evolutionary Algorithms) is a free and open-source framework for Java\footnote{\url{http://moeaframework.org/}}. It has an extensive list of algorithms and high-quality documentation, however is only two-years old and many of the guides are behind a paywall. jMetal\footnote{\url{http://jmetal.sourceforge.net/}} stands for Metaheuristic Algorithms in Java and is similar to MOEA. It has been established for much longer, with its fifth major release in 2015. Documentation is also freely available, although somewhat lacking for the newest version. The pymoo\footnote{\url{https://pymoo.org/}} framework offers state of the art single- and multi-objective optimization algorithms in Python. Although it has excellent documentation, it offers fewer algorithms and has not yet reached a major release. Based on this search, jMetal was selected as the most sensible option.
\section{Conclusions and Future Work}
\label{sec:conc}
This paper presented work on integrating support for a library of Multi-Objective Optimisations (MOO) algorithms into the INTO-CPS tool chain to provide additional options for Design Space Exploration (DSE) of models through co-simulation. It demonstrated a working solution and initial integration into the INTO-CPS Application. The solution was demonstrated by using two common MOO algorithms (NSGA-II and SMPSO) to explore the design space of the Robotti case study, comparing against results presented at a previous workshop~\cite{bogomolov2021tuning}.
The results show that the MOO algorithms were able to find similar trends to the exhaustive search presented previously, however when limited to the same 125 evaluations by co-simulation, performance was worse overall. The MOO library also introduces an overhead in execution time, which can be mitigated through parallel execution support. The case study chosen was selected given the previous work on using DSE to explore the parameter space, and provided a useful benchmark to check the correct functioning of the new MOO features.
The results show that MOO is not suitable for every use case, thus would not replace existing DSE solutions for INTO-CPS, but rather complement them. It is likely that the MOO solutions would offer an improvement to the time needed to perform DSE on CPS models when the design space in question has a high number of dimensions, in particular where multiple objectives compete and require trading off. In addition, of particular importance is to also apply this approach to case studies with a larger parameter spaces (both in the range of each parameter and the number of parameters).
One key area of future work is to develop case studies within the INTO-CPS examples repository that truly demonstrate multi-objective characteristics. This will allow a better comparison of existing DSE and MOO performance, particularly given the new multi-threading support in the existing DSE scripts. In this way, we can begin to identify guidelines for users about when and where each approach is likely to be most beneficial.
\section{Introduction}
\label{sec:intro}
Modelling and Simulation are important and powerful tools when developing systems of any kind, especially within the field of engineering, and increasingly so in the field of software development. Simulating models of the systems we design allows us to gain a solid understanding of how they might behave once deployed in the real world. This is of particular importance for Cyber-Physical Systems (CPSs)~\cite{Lee10}, where software must interact with hardware and it is difficult to test exhaustively and in a non-destructive way.
One of the key benefits of using simulation in this way is the ability to rapidly test different design parameters of a system without having to first make the financial and time commitments of creating or changing a physical prototype. This places simulation as an invaluable tool for performing Design Space Exploration (DSE). The `design space' of a system can be defined as the set of all configurations of its `design parameters', and therefore DSE is the act of exploring and assessing the range of configurations present in this design space, with the goal of discovering ones in which the system meets certain performance criteria most optimally.
DSE is one feature of INTO-CPS~\cite{Larsen&17a}, a tool chain for model-based design of CPS, built around a co-simulation engine called Maestro~\cite{Thule&192}. INTO-CPS currently offers exhaustive search (i.e. all combinations of a given set of design parameters), and genetic search (solving through natural selection of the fittest designs). Depending on the complexity of the model in question, the number of simulation runs needed to perform these searches accurately enough can make this process increasingly time consuming.
In industry, it is often paramount that tasks take the minimum amount of time needed to be performed properly, to keep costs down and to ensure that crucial deadlines are met. There are a wide variety of Multi-Objective Optimisation (MOO) that attempt to efficiently search for optimal solutions in complex design spaces. Many of these are implemented in existing, well-maintained libraries (see Section~\ref{sec:back:moo}). This paper describes work in extending the DSE capabilities of INTO-CPS with features that attempt to widen the range of DSE options for users of INTO-CPS.
By incorporating state-of-the-art MOO library to INTO-CPS, the user can now choose from the most appropriate and up-to-date optimisation algorithm for their CPS model, allowing them to discover optimal parameter configurations in the most efficient way possible. Furthermore, integrating an open-source, third-party library would allow the application to be easily updated to stay up to date with the most cutting-edge algorithms, eliminating the need to manually implement newly released ones from the ground up.
This paper also applies the MOO support to the Robotti optimisation case study, which was reported as a previous workshop~\cite{bogomolov2021tuning}. This case study demonstrates that the MOO integration works, but does not demonstrate the full benefits that MOO could have on co-simulations with larger design spaces and more competing objectives that require trading-off.
This remainder of the paper is structured as follows. Section~\ref{sec:back} provides some background detail on INTO-CPS and MOO. Section~\ref{sec:what} describes integration of a MOO library into the INTO-CPS framework. Section~\ref{sec:study} provides a refresher on the Robotti optimisation case study. Section~\ref{sec:results} compare the results from the previous Robotti DSE with those form the new MOO feature. Finally, Section~\ref{sec:conc} presents from conclusions and future work.
\section{Results}
\label{sec:results}
This section presents results of applying the MOO support to the Robotti case study described above, using the four scenarios presented in Figure~\ref{fig:res_scenarios} (\emph{sin1}, \emph{turn\_ramp1}, \emph{speed\_ramp1}, \emph{speed\_step1}). In order to ensure comparisons with exhaustive search are valid, and any comparisons of speed were made on the same machine, the exhaustive search was re-run for the four scenarios. These results are shown in Figure~\ref{fig:res_scenarios2}.
Note that the trends found are the same except for \emph{turn\_ramp1} which found a completely different trend. To try and account for this discrepancy, two manual co-simulations were run with values of 3000 for \emph{m\_robot} and 20000 for \emph{cAlphaF}, and \emph{mu} values of 0.3 and 0.7, respectively. After running these two co-simulations and evaluating their cross-track error using the objective script, the configuration with a mu value of 0.3 returned a considerably lower error value (in line with Figure~\ref{fig:res2_turn_ramp1}). It is therefore possible that Bogomolov et al.~\cite{bogomolov2021tuning} used some different setup that affected these results. In any case, the results from the re-run are used here to ensure comparisons are made on a consistent setup.
\subsection{Optimisation of Robotti using MOO Algorithms}
\label{sec:results:moo}
The MOO functionality was then used to optimise these scenarios using two different algorithms, and compare against exhaustive search. Given its popularity, the NSGA-II (Non-dominated Sorting Genetic Algorithm-II) algorithm was chosen, as well as the SMPSO (Speed-constrained Multi-objective Particle-Swarm Optimisation) since it is the most different from the evolutionary approach of NSGA-II. To ensure fairness, the MOO algorithms were initially limited to the same 125 co-simulations as the exhaustive search. Table~\ref{tab:initial} shows the mean cross track error found by the two algorithms and the exhaustive search.
\begin{figure}[tb]
\centering
\subfloat[$\mathsf{sin1}$ Scenario\label{fig:res2_sin1}]
{\includegraphics[width=0.45\textwidth]{source/Diagrams/SinEx}}
\subfloat[$\mathsf{turn\_ramp1}$ Scenario\label{fig:res2_turn_ramp1}]
{\includegraphics[width=0.45\textwidth]{source/Diagrams/TurnRampEx}}\\
\subfloat[$\mathsf{speed\_ramp1}$ Scenario\label{fig:res2_speed_ramp1}]
{\includegraphics[width=0.45\textwidth]{source/Diagrams/SpeedRampEx}}
\subfloat[$\mathsf{speed\_step1}$ Scenario\label{fig:res2_speed_step1}]
{\includegraphics[width=0.45\textwidth]{source/Diagrams/SpeedStepEx}}
\caption{Results of a re-run exhaustive search of the four scenarios}
\label{fig:res_scenarios2}
\end{figure}
\begin{table}
\centering
\caption{Mean cross track error found by algorithm and scenario}\label{tab:initial}
\begin{tabular}{p{2.5cm}p{2cm}c} \hline
\textbf{Scenario} & \textbf{Algorithm} & \textbf{Mean Cross Track Error} \\ \hline
\multirow{3}{*}{\emph{sin1}} & NSGA-II & 1.226 \\
& SMPSO & 1.225 \\
& Exhaustive & 1.188 \\ \hline
\multirow{3}{*}{\emph{turn\_ramp1}} & NSGA-II & 5.348 \\
& SMPSO & 5.348 \\
& Exhaustive & 3.640 \\ \hline
\multirow{3}{*}{\emph{speed\_ramp1}} & NSGA-II & 1.508 \\
& SMPSO & 1.498 \\
& Exhaustive & 1.491 \\ \hline
\multirow{3}{*}{\emph{speed\_step1}} & NSGA-II & 3.653 \\
& SMPSO & 3.655 \\
& Exhaustive & 5.348 \\ \hline
\end{tabular}
\end{table}
As can be seen, it appears that the MOO algorithms are unable to find optimal solutions as well as the previous study's exhaustive search, when completing only 125 evaluations. In the \emph{sin1} scenario, both algorithms came somewhat close to the error values found in that study, but in every other scenario, both algorithms calculated a much worse optimal amount of error. Plots of the intermediate solutions are given in Figure~\ref{fig:res_scenarios3}. By inspection, it appears as though the trends are the same as those in Figure~\ref{fig:res_scenarios2}; \emph{sin1} is in the opposite corner to \emph{turn\_ramp1} and \emph{speed\_ramp1}, and \emph{speed\_step1} shows no overall trend.
To investigate whether further co-simulations would find as optimal a solution as the exhaustive search, the optimisations for \emph{sin1} were repeated using NSGA-II and incrementally increasing the maximum number of evaluations. These are shown in Figure~\ref{fig:sin1}. It was not until around 750 evaluations that the NSGA-II consistently produces results as well as the exhaustive search.
This was somewhat surprising, as we expected that using MOO algorithms would find the most optimal configuration much faster in terms of the number of evaluations. Or failing that, in the same number of evaluations, the MOO algorithm would find a more optimal solution. Since such a small set of values on the range of each parameter was tested during the exhaustive study, values between those specified would not be tested. Because of this, we assumed more optimal solutions would be missed, and that the MOO algorithms would find them since they would have the ability to test the full range of values possible.
\begin{figure}[tb]
\centering
\subfloat[$\mathsf{sin1}$ Scenario\label{fig:res3_sin1}]
{\includegraphics[width=0.48\textwidth]{source/Diagrams/Sin1NSGAII125}}
\subfloat[$\mathsf{turn\_ramp1}$ Scenario\label{fig:res3_turn_ramp1}]
{\includegraphics[width=0.48\textwidth]{source/Diagrams/TurnRamp1NSGAII125}}\\
\subfloat[$\mathsf{speed\_ramp1}$ Scenario\label{fig:res3_speed_ramp1}]
{\includegraphics[width=0.48\textwidth]{source/Diagrams/SpeedRamp1NSGAII125}}
\subfloat[$\mathsf{speed\_step1}$ Scenario\label{fig:res3_speed_step1}]
{\includegraphics[width=0.48\textwidth]{source/Diagrams/SpeedStep1NSGAII125}}
\caption{Results of a NSGA-II results for the four scenarios limited to 125 co-simulations}
\label{fig:res_scenarios3}
\end{figure}
\begin{figure}[tb]
\centering
\subfloat[$\mathsf{sin1}$ (125 co-simulations)\label{fig:sin1_125}]
{\includegraphics[width=0.48\textwidth]{source/Diagrams/Sin125}}
\subfloat[$\mathsf{sin1}$ (250 co-simulations)\label{fig:fig:sin1_250}]
{\includegraphics[width=0.48\textwidth]{source/Diagrams/Sin250}}\\
\subfloat[$\mathsf{sin1}$ (500 co-simulations)\label{fig:fig:sin1_500}]
{\includegraphics[width=0.48\textwidth]{source/Diagrams/Sin500}}
\subfloat[$\mathsf{sin1}$ (750 co-simulations)\label{fig:fig:sin1_750}]
{\includegraphics[width=0.48\textwidth]{source/Diagrams/Sin750}}
\caption{Results of NSGA-II on \emph{sin1} with increasing numbers of co-simulations}
\label{fig:sin1}
\end{figure}
These findings suggested that, in terms of number of evaluations, the MOO solution was much slower at finding optimal solutions. However, we hypothesise that for models with many more dimensions, MOO algorithms would converge on optimal configurations in fewer evaluations than in an exhaustive search. Due to the nature of the exhaustive search, as the number of dimensions increase, the number of evaluations increases exponentially. This is because the number of evaluations is given by:
\[ | D_1 | \cdot | D_2 | \cdot | D_3 | \ldots \cdot | D_n | \]
\noindent where $N$ and $D_n$ is the set of values that makes up the $n$th dimension. If the cardinality of each set tested is uniform, this would give the exhaustive search a time complexity of $O(c^n)$ in terms of the number of evaluations, where $n$ is the number of dimensions. The time complexity of NSGA-II on the other hand is not affected by the number of dimensions directly, since the time complexity of NSGA-II is given by $O(GMN^2)$, where $G$ is the number of generations, $M$ is the number of objectives and $N$ is the population size~\cite{jensen2003reducing}. While some or all of these three variables would likely need to increase with the number of dimensions to continue providing accurate optimisation configurations, this would not correspond to as much as an exponential increase in evaluations.
Using equation derived above, doubling the number of dimensions of the Robotti design space to six dimensions would increase the number of evaluations performed by the exhaustive search from 125 to 31,250. Even if this increase in dimensions required by NSGA-II to double the number of generations completed, this search would still only comprise of 1,500 evaluations. Because of this, we believe that problems with larger numbers of dimensions, the MOO solution could be much more efficient at DSE when compared with exhaustive search. This however requires further study through development of co-simulation examples that exhibit multi-objective behaviour.
\subsection{Evaluation of Execution Time}
\label{sec:results:time}
Table~\ref{tab:speed} compares the execution speed of the exhaustive DSE, along with the NSGA-II algorithm running on one and six threads. It shows that when running each method for the same number of evaluations, using NSGA-II is slightly slower than using an exhaustive search, likely due to the overhead of using the jMetal library. However, because the MOO solution supports parallel execution, the ability to run multiple simulations at once decreases optimisation time significantly. In this case, utilising six parallel cores decreased optimisation by a factor of three when compared to single thread evaluation. Note that the most recent version of the existing DSE scripts now support parallel execution, so further benchmarks are required to truly compare approaches, using a range of case studies to show the strengths and weaknesses of the two approaches.
\begin{table}
\centering
\caption{Execution time for exhaustive and NSGA-II algorithms}\label{tab:speed}
\begin{tabular}{p{2.5cm}p{3.5cm}cc} \hline
\textbf{Scenario} & \textbf{Algorithm} & \textbf{Mean Time / Evaluation (s)} & \textbf{Mean Total (s)}\\ \hline
\multirow{3}{*}{\emph{sin1}}
& NSGA-II (6 threads) & 2.2 & 280 \\
& NSGA-II (1 thread) & 6.1 & 771 \\
& Exhaustive (1 thread) & 5.3 & 664 \\ \hline
\multirow{3}{*}{\emph{turn\_ramp1}}
& NSGA-II (6 threads) & 3.8 & 480 \\
& NSGA-II (1 thread) & 10.9 & 1365 \\
& Exhaustive (1 thread) & 9.1 & 1141 \\ \hline
\multirow{3}{*}{\emph{speed\_ramp1}}
& NSGA-II (6 threads) & 3.9 & 488 \\
& NSGA-II (1 thread) & 11.8 & 1485 \\
& Exhaustive (1 thread) & 10.4 & 1296 \\ \hline
\multirow{3}{*}{\emph{speed\_step1}}
& NSGA-II (6 threads) & 4.6 & 584 \\
& NSGA-II (1 thread) & 12.3 & 1540 \\
& Exhaustive (1 thread) & 10.8 & 1350 \\ \hline
\end{tabular}
\end{table}
\section{Case Study}
\label{sec:study}
To demonstrate the work reported in this paper against existing functionality, we use the same case study as Bogomolov et al.~\cite{bogomolov2021tuning} presented at a previous workshop. The paper reports on a pilot study to assess the viability of DSE as a tool in improving the fidelity of a co-simulation model, based on real-world data observed in field trials of the real system. The study is based on the Robotti unmanned platform developed by Agro Intelligence (Agrointelli) for~(see Figure~\ref{fig:RealRobotti}). In the remainder of this section, the case study and previous results are discussed. Section~\ref{sec:results:time} discusses the performance of the current solution and the selection of the MOO algorithms.
The authors note that this study does not exhibit true multi-objective behaviours that require trade off to best demonstrate . We suggest in Section~\ref{sec:conc} that new case studies are required in the INTO-CPS examples repository that do show multi-objective characteristics and would better demonstrate the strengths and weaknesses of both the DSE and MOO approaches. We selected this case study because it was readily available and was previously reported at this venue, which meant that the results could be directly compared to demonstrate the functionality of the new MOO add on, if not demonstrate its full utility.
In this study, the real Robotti system carried out field trials under four scenarios with a total of 27 runs. Each scenario is a pattern of control outputs designed to assess how Robotti moves under different conditions. The control outputs and responses of the Robotti are then recorded for use in model optimisation. This data is then used to optimise the co-simulation by finding which physical parameters in the model most closely replicate those in the real Robotti. In this way, the optimisation should deliver a co-simulation that better models the actual system. The four scenarios were as follows~\cite{bogomolov2021tuning}:
\begin{description}
\item[Speed step:] The speed of each wheel is increased incrementally in lock step, resulting in the Robotti driving straight.
\item[Speed ramp:] The speed of each wheel is increased smoothly. This also results in a straight path as in the speed step tests.
\item[Turn ramp:] The speed of the right wheel is increased smoothly, while the left wheel is kept constant. This results in the Robotti turning left.
\item[Sin:] The speed of the left and right wheel is increased and decreased to produce the sinusoidal motion.
\end{description}
\begin{figure}[tb]
\centering
\includegraphics[width=0.7\columnwidth,trim=0 150 0 130, clip]{Figures/robotti_3_infield2b}
\caption{\label{fig:RealRobotti}The Robotti unmanned platform}
\end{figure}
The measure of accuracy in this study is the mean cross-track error. Given the known path and the simulated path under a given set of parameters, the mean difference between then provides a measure of how well the parameters match the (unknown) attributes of the real Robotti. A lower mean difference therefore means a closer representation of reality.
The design space involves three parameters: \emph{m\_robot}, the mass of the Robotti; \emph{cAlphaF}, the steering stiffness; and \emph{mu}, the friction between the wheel and ground. In the study, exhaustive search is used for the DSE, with five values for each parameter. This results in 125 co-simulation per scenario, and 1250 for the ten scenarios used. Results for one each of the four scenarios are shown in Figure~\ref{fig:res_scenarios}. These show that the Sine and Turn ramp scenarios strongly suggest the same specific parameter values (in the lower right of the plots). Speed ramp shows a region of optimal solutions, and a secondary weaker region of good solutions, while Speed step suggests no overall optimal solution.
\begin{figure}[tb]
\centering
\subfloat[$\mathsf{sin1}$ Scenario\label{fig:res_sin1}]
{\includegraphics[width=0.48\textwidth]{Figures/Sin1.png}}
\subfloat[$\mathsf{turn\_ramp1}$ Scenario\label{fig:res_turn_ramp1}]
{\includegraphics[width=0.48\textwidth]{Figures/TurnRamp1.png}}\\
\subfloat[$\mathsf{speed\_ramp1}$ Scenario\label{fig:res_speed_ramp1}]
{\includegraphics[width=0.48\textwidth]{Figures/SpeedRamp1.png}}
\subfloat[$\mathsf{speed\_step1}$ Scenario\label{fig:res_speed_step1}]
{\includegraphics[width=0.48\textwidth]{Figures/SpeedStep1.png}}
\caption{Mean cross track error of four scenarios against different parameter values~\protect\cite{bogomolov2021tuning}}
\label{fig:res_scenarios}
\end{figure}
\section{Multi-Objective Optimisation (MOO) Support}
\label{sec:what}
\begin{figure}[tb]
\includegraphics[width=0.99\textwidth]{Figures/Picture1}
\caption{Class diagram of the inheritance hierarchy of \texttt{Algorithm}}
\label{fig:algo}
\end{figure}
The support for Multi-Objective Optimisation (MOO) involved creating an interface between the jMetal~\cite{durillo2006jmetal} library and the Co-Simulation Orchestration Engine (COE). Figure~\ref{fig:algo} shows a diagram of the INTO-CPS Application interacting with the new MOO support tool and the COE. Here the frontend can configure a MOO problem, selecting the algorithm, parameter space, objective scripts and maximum iterations. The jMetal library then executes the selected algorithm. When testing solutions proposed by the algorithm, a co-simulation configuration is generated that configures the COE to run one or more co-simulations. The selected objective scripts are used to feed back into the jMetal library for the algorithm to generate a new set of solutions. This process repeats until a given number of iterations is reached.
The jMetal library is object-oriented. The key classes called \texttt{Problem}, representing the MOO problem to be solved, and \texttt{Solution}, representing a possible solution to the optimisation. In this case, the values of parameters of the model. The \texttt{Problem} class has a primarily method called \texttt{evaluate} which determines the fitness of a proposed \texttt{Solution}. Hence the primary effort of this implementation is to generate co-simulation configurations for \texttt{Solution} instances proposed by the MOO algorithm(s) and evaluate the \texttt{Problem} using the COE and objective scripts.
Each algorithm in JMetal comes with its own builder, which allows you to configure and initialise an algorithm. The MOO functionality was built in an object-oriented approach to wrap these algorithms in an interface. Each algorithm to be imported from jMetal is represented as its own class, which inherits from an abstract superclass called \texttt{Algorithm}. The superclass also contain an abstract method \texttt{runAlgorithm} that subclasses are required to implement. Calling this function should run the algorithm and output the results. This superclass contains empty fields for:
\begin{itemize}
\item Each operator kind of operator, i.e.:
\begin{itemize}
\item Crossover
\item Mutation
\item Selection
\end{itemize}
\item Every variant of algorithm parameter, e.g.
\begin{itemize}
\item Crossover probability and distribution index
\item Mutation probability and distribution index
\item Mutation probability and distribution index
\item Population size
\item Maximum number of evaluations
\item Etc.
\end{itemize}
\end{itemize}
To allow the user or INTO-CPS Application to select an algorithm by name, all possible algorithms are enumerated in a type called \texttt{AlgorithmVariant}. A class called \texttt{RunHandler} takes a string input and attempts to resolve which subclass of \texttt{Algorithm} should construct and run. This is shown in the class diagram in Figure~\ref{fig:runhandler}.
\begin{figure}[tb]
\includegraphics[width=0.99\textwidth]{source/Diagrams/Picture1}
\caption{Class diagram to showing the role of \texttt{RunHandler}}
\label{fig:runhandler}
\end{figure}
To make the integration with the INTO-CPS Application as simple as possible, the MOO implementation takes a single JSON (Javascript Object Notation) file as its only command line argument. This replicates the style of both the COE and existing DSE scripts. This means that tool integration only requires the INTO-CPS Application to fetch the required information from the user, store that information in a JSON file, then execute the MOO solution with this JSON as an argument.
\clearpage
The input JSON used should contain the following fields (based on the existing DSE JSON file), with an example is presented in Figure~\ref{fig:json}:
\begin{itemize}
\item The path of the COE
\item The path of the JSON configuring the base co-simulation
\item The length of the simulation, in seconds
\item The algorithm to use
\item An array of the model's parameters that make up the design space, each with the following attributes:
\begin{itemize}
\item Parameter identifier
\item Upper bound
\item Lower bound
\end{itemize}
\item An array of the objectives, each with the following attributes:
\begin{itemize}
\item Type (``SCRIPT'' or ``PARAMETER'')
\item Identifier (path to script, or parameter identifier)
\item A Boolean value for whether the objective should be maximised (false means minimised)
\end{itemize}
\end{itemize}
\begin{figure}[tb]
\centering
\includegraphics[width=0.75\textwidth]{source/Diagrams/Picture3}
\caption{An example of the input JSON used to configure an MOO search}
\label{fig:json}
\end{figure}
In addition to basic functionality of running each algorithm offered by jMetal, the solution also includes concurrent execution of solution evaluation and custom constraint handling through Python scripts. At the time of writing, the INTO-CPS Application is currently being updated to provide a MOO option in the existing DSE interface. A screenshot showing this is given in Figure~\ref{fig:into}. It is expected that this functionality will be available in the next release of the INTO-CPS Application expected in Q3 2021.
\begin{figure}
\centering
\includegraphics[width=0.95\textwidth]{source/MOOAlgorithmHighlited}
\caption{A screenshot of the INTO-CPS Application showing MOO algorithm choices}
\label{fig:into}
\end{figure}
\subsection{Specification Language Server Protocol}
\label{sec:slspIntroduction}
The purpose of the \gls{slsp} protocol is to facilitate support for specification language specific features using the same architecture as for \gls{lsp} and \gls{dap}.
The \gls{slsp} protocol is an extension to the \gls{lsp} protocol.
This means that it uses the same base protocol as \gls{lsp} and relies on functionality from \gls{lsp} such as the synchronisation between client and server.
The protocol is developed with the intention that it should also be usable by other languages than \gls{vdm} and possibly be included in the \gls{lsp} specification.
Hence, the \gls{slsp} protocol uses language neutral data types, which allows the client to be language-agnostic enabling it to be used with any server that supports the \gls{slsp} protocol.
At the time of writing the \gls{slsp} protocol defines messages to support \gls{pog} and \gls{ct}, and it is used in the \gls{vscode} extension for those features.
\subsection{Visual Studio Code Extension}
\Gls{vscode} operates with a rich extensibility model that enables users to include support for different programming languages, debuggers and various tools to support the development workflow, by downloading extensions from the \gls{vscode} extensions marketplace\footnote{See \url{https://marketplace.visualstudio.com/vscode}}.
This also enables developers to make extensions that they find missing from the marketplace, which is how \gls{vdm} is supported in \gls{vscode}; by creating an extension for each \gls{vdm} dialect.
The extensions include a \gls{slsp} client and a \gls{dap} client.
Besides communicating with the server the extension also has the responsibility of doing syntax highlighting as this is not supported in any of the protocols.
As illustrated in \Cref{fig:ExtensionArchitecture}, \gls{vscode} includes generic support for both the \gls{lsp} and \gls{dap} protocols which allows the client implementation to be carried out with little effort.
The generic protocol support provides the extension with handlers for all the \gls{lsp} and \gls{dap} messages which include code to synchronise the editor and the server, handling of the editor navigation and displaying messages from the server.
Thus, the extension provides configuration of the generic protocol support, launching and connecting to the server, syntax highlighting, and the \gls{gui} to support the \gls{slsp} features.
\Cref{fig:VSCodeSnippet_pog} depicts the \gls{vscode} environment with the VDM-SL extension active.
\begin{figure}[htb]
\centering
\includegraphics[width=1\textwidth]{figures/VSCodeSnippet_pog.pdf}
\caption{VDM-SL VS Code Extension: (1) Main Editor; (2) Project Explorer; (3) File Outline; (4) Proof Obligation View}
\label{fig:VSCodeSnippet_pog}
\end{figure}
\subsubsection{Syntax Highlighting:}
Syntax highlighting in \gls{vscode} is handled on the client side as it is not supported by the protocols.
Instead syntax highlighting is performed using TextMate grammars \cite{Gray07}.
The TextMate grammars are structured collections of Oniguruma regular expressions\footnote{See \url{https://github.com/kkos/oniguruma}}, implemented using a JSON schema.
The grammar file specifies a set of rules, that is used with pattern matching to transform the visible text into a list of tokens and colour these according to a colour scheme.
\subsubsection{Specification Language Server Protocol Client:}
As custom for language extensions the VDM extensions are activated when a file with a matching file format is opened, \eg opening a `\texttt{.vdmsl}' file actives the VDM-SL extension.
On activation the client launches the server and connects to it.
When connected, the client forwards all relevant information to the server using the \gls{lsp} protocol.
Furthermore, any responses or notifications from the server triggers feedback to the user, which is handled by the generic \gls{lsp} client integration in \gls{vscode}.
In addition to setting up the support for \gls{lsp} the client also provides customised \gls{gui} views to support \gls{pog} and \gls{ct}. This is not directly supported by \gls{vscode} and \gls{lsp} since these features are not commonly used for traditional programming languages.
The \gls{pog} view, illustrated in \Cref{fig:VSCodeSnippet_pog}, provides a list of the proof obligations for a specification, where expanding an element displays the actual proof obligation.
The view is implemented using the \gls{vscode} Webview API\footnote{See \url{https://code.visualstudio.com/api/extension-guides/webview}.}, which is customised using CSS, HTML and JavaScript.
The \gls{ct} view, illustrated in \Cref{fig:VSCodeSnippet_ct}, shows the tests that are generated based on the traces in the specification, these are structured in a tree view.
Traces can be fully or partially executed using either an execution filter or executing a test group at a time.
A tests execution sequence and result is displayed in a separate tree view.
Tests can also be debugged, which is facilitated using the \gls{dap} protocol.
The \gls{ct} view is implemented using the \gls{vscode} Tree View API\footnote{See \url{https://code.visualstudio.com/api/extension-guides/tree-view}.}. which reduces the complexity of implementing a view.
\begin{figure}[htb]
\centering
\includegraphics[width=1\textwidth]{figures/VSCodeSnippet_ct.pdf}
\caption{VDM-SL VS Code Extension: (1) Combinatorial Testing View; (2) Test Result View; (3) Debug Console}
\label{fig:VSCodeSnippet_ct}
\end{figure}
\subsubsection{Debug Adapter Protocol Client:}
The \gls{dap} protocol support is added to the client by specifying a \texttt{DebugAdapterDescriptorFactory} which establishes the connection between the generic debugger client and the server upon starting a debug session.
This would normally include launching a \gls{dap} server and connect to it.
However, as the \gls{lsp} and \gls{dap} server is combined into one we simply connect to the port specified for \gls{dap} communication.
\subsection{Enabling the use of LSP and DAP in VDMJ}
Internally, VDMJ provides a set of language services to enable VDM specifications to be processed, and by default the coordination of these services is handled by a command-line processor. So for example, a Parser service is used to analyse a set of VDM source files to produce an Abstract Syntax Tree (AST); if there are no syntax errors, that AST is further processed by a Type Checker; and if there are no type errors, the tree is further processed by an Interpreter (in a read-eval loop) or by a Proof Obligation Generator. If the Interpreter encounters a runtime problem or a debug breakpoint, it stops and interacts with the command-line to allow the user to examine the stack and variables, for example.
Although the command-line is the default way to interact with VDMJ, this is not assumed in the design. An alternative means to interact is provided using the DBGP protocol\footnote{See \url{https://xdebug.org/docs/dbgp}}. This allows VDMJ to interact with a more sophisticated IDE, such as Overture. DBGP messages are exchanged over a local TCP socket, allowing the IDE to coordinate the execution of expressions within the specification, and to debug using a richer user interface. The VDMJ end of the DBGP socket coordinates the same services as the command-line does, but with requests and responses being exchanged via the socket, using DBGP messages.
So adapting VDMJ to use the LSP and DAP protocols is a matter of creating a new handler to accept socket connections from a client, and process the JSON/RPC messages defined in the respective LSP and DAP standards. Since LSP and DAP are expressed in general language terms, there is generally a natural mapping from abstract concepts in these protocols to the specific case of a VDM specification.
Creating the LSP part of the VDMJ handler was relatively simple. The biggest difference to the existing connection handlers is that LSP is an ``editing'' protocol. That is, rather than being asked to process a fixed set of unchanging specification files, LSP requires the server to accept the creation and editing of files on the fly. So whereas normally, the Parser would only be called once to process a set of fixed specification files, in the LSP handler it can be called repeatedly as edits are passed from the client. Since the VDMJ parser is relatively efficient, it is responsive enough to re-parse the affected file as edits are made. A decision was taken to only type-check the specification when the client deliberately saves the work (\ie writes changes to disk). This is the same behaviour as Overture.
Execution and debugging via the DAP protocol required the creation of a new ``DebugLink'' subclass in VDMJ. The existing command-line and DBGP protocols each extend an abstract DebugLink class, which allows the Interpreter service to interact with an arbitrary client to coordinate a debugging session. In this case, the client is the DAP client. Since the DebugLink was designed to be extended, this was relatively straightforward -- in fact the DAPDebugLink is an extension of the command-line class with the protocol aspects changed from textual read/writes to JSON message exchanges. It is about 200 lines of Java. The only complexity in this area is in the case of multi-threaded VDM++ specifications. Special care must be taken to be sure that all of the separate threads and their data are coordinated correctly, with appropriate mutex protection of the (single) communication channel to the DAP client.
In addition to parsing, type-checking and executing a specification, the LSP protocol also enables the server to offer various language services that allow the client to build a more intelligent user interface. For example, the outline of a file (its contents, in terms of the VDM definitions within and their location) can be queried; any name symbol in a file can be used to navigate to that symbol's definition (\eg moving from a function application to its definition, possibly in a different file); and name-completion services are offered, where the first few characters of a name can be typed, and the server will offer possible completions. These services do not exist in the base VDMJ, and were therefore added as part of the LSP development. They are mostly implemented via visitors (using the VDMJ visitor framework) which process the AST.
The Proof Obligation Generation and Combinatorial Testing features of VDMJ are maked available with the SLSP protocol by adding new ``slsp'' RPC methods. Internally, the server routes these new requests to the VDMJ components that are able to perform the analyses.
Proof Obligations (POs) can be generated relatively quickly (usually in a fraction of a second, even for hundreds of obligations), so the ``slsp/POG/generate'' method takes a one-shot approach, generating and returning all the POs in the specification by default, optionally allowing the UI to limit the scope to a sub-folder of specification files or a single file. The returned obligations are represented as VDM source, as a stack of contexts with a primitive obligation at the end. This allows the UI to decide how to represent the indentation of the full obligation without requiring it to understand the VDM AST.
Combinatorial Testing is a more complex problem, requiring multiple interactions between the UI and the server as new ``slsp/CT'' methods. Unlike Proof Obligations, Combinatorial Tests can expand to millions of test cases and their execution can take many hours. This in turn means that the UI must have some indication of the progress of the execution, and it must also have the ability to cleanly terminate the execution in the event that the test run is taking too long. The base LSP protocol allows for the asynchronous cancellation of an action, and this is implemented in the LSP Server by running CT in a separate thread, allowing the main thread to listen for more interactions from the UI. This complication means that a check has to be added to prevent the user from trying to launch more than one test execution at the same time (which the runtime could not easily support). The server also has to guard against changes to the specification between the expansion of the tests and their subsequent execution. Once a test run is complete, individual tests can be sent to the Interpreter for debugging. This is enabled via the standard DAP protocol, with the addition of a new ``runtrace'' launch command. So the runtime thinks that the user has started a normal interactive debugging session, but instead of executing something like ``print fac(10)'', they are executing ``runtrace A'Test 1234'', which will debug test number 1234 of the expansion of the A'Test trace.
In terms of packaging, the SLSP/DAP combination of server functionality is in its own jar, separate from (and dependent on) the standard VDMJ jar. The SLSP/DAP jar is about 174Kb. VDMJ itself is about 2.4Mb.
\subsection{Assessing the Implementation Effort}
The implementation effort is quantified by counting the Lines of Code (LoC) used for implementing support for a given protocol and the features it enables.
Providing support the protocols for VDMJ requires 7855 LoC, whereas support for \gls{vdm} in the \gls{vscode} extension only requires 1880 LoC where most are used for the new \gls{slsp} features.
The few lines of code for the extension can mainly be attributed to the generic \gls{lsp} and \gls{dap} protocol implementations exposed by the \gls{vscode} API\footnote{See \url{https://code.visualstudio.com/api/references/vscode-api}}.
A detailed overview of the lines needed to implement the language support is shown in \Cref{tab:LoCOverview}.
In the table the `json' and `rpc' directory contain a basic JSON/RPC system; `dap' and `lsp' are the protocol handlers for each; `lsp/lspx' is the handler for SLSP; `vdmj' is the DebugLink and visitors to implement the various features (i.e.\ these link to the VDMJ jar); and `workspace' contains the code to maintain the collection of files.
For the \gls{vscode} extension `LSP client' and `DAP client' is the setup of each client; `POG' and `CT' is the protocol handlers and GUI support for each feature; `syntax highlighting' contains the TextMate grammars for \gls{vdm}; and `SLSP Protocol' is the definition of the \gls{slsp} protocol and expansion of the LSP client.
\input{tables/LoCOverview}
A comparison between LoC for the \gls{vscode} extension, migrating Overture to Emacs (as in \cite{Tran&19}) and the Overture IDE (version 3.0.1) not counting the core is seen in \Cref{tab:LoCCompare}.
From the comparison we find that the \gls{vscode} extension requires more code than the Emacs migration but Overture consists of far more LoC than both.
However, if we only look at the \gls{lsp} and \gls{dap} client support that covers all the normal programming language features they only require 210 LoC.
The \gls{slsp} parts of the extension are made almost generic, hence these parts can be reused for other specification languages to enable support for the same features in \gls{vscode}.
For VDMJ to support the protocols it requires considerably more LoC compared to the Emacs approach from \cite{Tran&19}, which weights against the protocol approach for enabling \gls{vdm} feature support.
However, the protocol solution is a standardisation of the decoupling, hence the server supporting the language features (in this case VDMJ) only has to be implemented once to be used with any development tool supporting the protocols.
In comparison, the \gls{vscode} extension and migration of the Overture language core to another development tool, as done with Emacs, must be performed for each \gls{ide} that is to be supported.
Furthermore, the Overture migration depends on Emacs packages, which may not exist in other \glspl{ide} or have a different interface.
Thus, the analysis of finding suitable packages for another \gls{ide} will have to be repeated, possibly making the server solution faster to implement in a new \gls{ide}.
\input{tables/LoCCompare}
To provide a fair comparison between the three implementation efforts, we compare the features that are available in each.
The comparison is done by the same set of features as in \cite{Tran&19}, this is found in \Cref{tab:featureCompare}.
As illustrated, the Overture IDE and the Emacs migration both support more features than the \gls{vscode} extension.
However, some of the features provided by the Emacs migration is only available through a command-line interface and not by a \gls{gui}.
\gls{vscode} does support implementation of integrated command-line interfaces like the Emacs migration, however the implementation efforts related to this are unknown.
Providing this kind of command-line interface has intentionally been left out of the \gls{vscode} extension, as we wanted to replicate the workflow from Overture.
\input{tables/featureCompare}
\subsection{Protocol Coverage of Language Features}
\label{sec:ProtocolCoverage}
To get an overview of the specification language features that are covered by the protocols, we have examined and grouped the features that are sought after to support specification languages such as \gls{vdm}, B and Z.
This is illustrated in \Cref{fig:ProtocolCoverage}, where the language features have been divided into four categories:
\begin{enumerate}
\item \textbf{Editor}: features that support writing a given specification, e.g.\ type- and syntax-checking.
\item \textbf{Translation}: features that support translating a specification to other formats, e.g.\ executable code and LaTeX.
\item \textbf{Validation}: features that support validation of a specification.
\item \textbf{Verification}: features that support verification of a specification.
\end{enumerate}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.53\textwidth]{figures/SLSP_Protocol_coverage.pdf}
\caption{Specification language features covered by existing protocols}
\label{fig:ProtocolCoverage}
\end{figure}
As a result of the initial implementation and by comparing the features supported by the \gls{lsp} protocol with the features from Overture, it is found that the protocol is able to support all of the editor related features except syntax highlighting.
Thus, specification languages that benefits from the editor features found in \Cref{fig:ProtocolCoverage}, can use a direct implementation of the \gls{lsp} protocol to support this feature set.
The \gls{dap} protocol can be used for decoupling the debugging feature as it supports common debug functionality such as different types of breakpoints, variable values and more.
This is useful for specification languages that allow execution of specifications.
Additionally, the \gls{slsp} extension to \gls{lsp} facilitates support for both \gls{pog} and \gls{ct}.
As found in \Cref{fig:ProtocolCoverage} there are still many features related to specification languages that are not supported by protocols.
Thus, in order to achieve fully decoupled language server support for \gls{vdm} one or more protocols must be developed that support the remaining features.
From our development of \gls{slsp} we believe that it is possible to develop a language-agnostic protocol to support all of these features, which should make it possible to apply the protocol to multiple specification languages and potentially increase the industrial uptake of specification languages.
\subsection{VDMJ}
VDMJ \cite{Battle09} is a command-line tool written in Java, that provides basic language support for the \gls{vdm} dialects VDM-SL, VDM++ and VDM-RT.
It includes a parser, a type checker, an interpreter, a debugger, a proof obligation generator and a combinatorial test generator with coverage recording, as well as VDMUnit support for automatic testing and user definable annotations.
These are implemented using an extensible \gls{ast} analysed using a visitor framework\cite{Gamma&95}.
Using VDMJ for the language server allows parts of the language support to be reused.
However, many features of VDMJ are not supported by standardised protocols, thus only a subset of the functionality is used in the language server.
\subsection{Visual Studio Code}
\gls{vscode}\footnote{See \url{https://code.visualstudio.com/}} is a free source code editor.
It has built-in support for the programming languages JavaScript and TypeScript and further enables support for other languages through a rich ecosystem of extensions.
\gls{vscode} uses a folder or workspace system for interacting with a project and a document system for handling the source code files in the project.
This allows \gls{vscode} to be language-agnostic and delegate language specific functionality to an extension.
Given this design reasoning, a need for the standardisation of decoupling between editor and extension has been identified.
At the time of writing, three different protocols have been developed for this purpose.
Namely \gls{lsp}\footnote{See \url{https://microsoft.github.io/language-server-protocol/}} (described in \Cref{subsec:lsp}), \gls{dap}\footnote{See \url{https://microsoft.github.io/debug-adapter-protocol/}} (described in \Cref{subsec:dap}) and Language Server Index Format (LSIF)\footnote{See \url{https://microsoft.github.io/language-server-protocol/overviews/lsif/overview/}}.
\subsection{Language Server Protocol}
\label{subsec:lsp}
Since most \glspl{ide} support a variety of programming languages, \gls{ide} developers face the problem of keeping up with ongoing programming language evolution \cite{Bunder19a}.
At the same time language providers are interested in providing as many \gls{ide} integrations as possible to serve a broad audience.
Consequently, integrating every language, $m$, in every \gls{ide}, $n$, leads to a $m \times n$ complexity.
\begin{figure}[htb]
\centering
\includegraphics[width=0.75\textwidth]{figures/LSPApproach.pdf}
\caption[\gls{lsp} approach to language support.]
{\gls{lsp} approach to language support. Borrowed from \cite{Rodriguez&18}.}
\label{fig:LSPApproach}
\end{figure}
The \gls{lsp} protocol defines a standardised protocol to be used to decouple a language-agnostic development tool (client) and a language-specific server that provides language features like syntax-checking, hover information and code completion.
This is illustrated in \Cref{fig:LSPApproach}.
The client is responsible for managing editing actions without any knowledge of the language and the server validates the correctness of the source code and reports issues and language-specific information to the client.
To facilitate this the \gls{lsp} protocol communicates using language neutral data types such as document references and document positions.
Many tools support the \gls{lsp} protocol which reduces the time needed to create a client implementation\footnote{See \url{https://microsoft.github.io/language-server-protocol/implementors/tools/}}.
Furthermore, new development tools only have to support the protocol, which can be done with little effort compared to native integration \cite{Bunder19b}.
Additionally, as the server is separated from the development tool it can be used for multiple tools.
This allows tools to easily support multiple languages and features.
Thus, by decoupling the language implementation from the editor integration the complexity is reduced to $m + n$.
\subsection{Debug Adapter Protocol}
\label{subsec:dap}
The \gls{dap} protocol is a standardised protocol for decoupling \glspl{ide}, editors and other development tools from the implementation of a language-specific debugger.
The \gls{dap} protocol uses language neutral data types, which makes the protocol possible to use for any text-based language.
The debug features supported by the protocol includes: different types of breakpoints, variable values, multi-process and thread support, navigation through data structures and more.
To be compatible with existing debugger components, the protocol relies on an intermediary debug adapter component.
It is used to wrap one or multiple debuggers, to allow communication using the \gls{dap} protocol.
The adapter is then part of a two-way communication with a generic debugger component, which is integrated in a given development environment as illustrated in \Cref{fig:DAPArchitecture}.
Thus, the protocol reduces a $m \times n$ problem of implementing each language debugger for each development tool into a $m + n$ problem.
\begin{figure}[htb]
\centering
\includegraphics[width=0.8\textwidth]{figures/DAP_architecture.pdf}
\caption[The decoupled architecture where the \gls{dap} protocol is used.]
{The decoupled architecture where the \gls{dap} protocol is used. }
\label{fig:DAPArchitecture}
\end{figure}
\section{Introduction}
\label{sec:Introduction}
Formal specifications have been reported to increase productivity and reliability by describing system functionality with rigour, enabling early detection and correction of specification-induced problems in software development.
VDM-SL is a formal specification language with an executable subset \cite{Fitzgerald&09}.
Interpreted executions of the specification, or specification animation, can be used to simulate and try out the behaviour of the system before implementation takes place.
Specification animation is a powerful tool for deepening understanding of the problem domain.
Software development is typically an ill-defined problem where the problem to be solved is not defined in advance.
Although a development project starts with objectives and a tentative goal, the developers do not always have a full understanding of the problem domain.
During the development, the developers learn and understand the problems to be solved, and specify, design, and implement the software as a means to solve them.
Especially in the early stages, it is important to learn about the problem domain by defining and animating the formal specification, and also feedback from stakeholders.
Exploration involves frequent modification to the specification to identify associated concepts and find their definition with appropriate abstraction and concise presentation.
Refactoring is a technique to modify the program source while keeping the behaviour of the program code \cite{Mens&04}.
Refactoring has mainly been used to improve the maintainability of code.
In particular, in agile development, refactoring plays an important role in removing design distortions caused by added features and in preparing for further feature additions.
The tool support for refactoring originated from the programming community of Smalltalk \cite{Roberts&97}.
Smalltalk is by its nature bundled with powerful IDEs featuring class browsers that strongly support prototyping.
It is not a coincidence that Smalltalk also delivered a unit testing framework to the programming community which later derived JUnit \cite{Louridas&05}.
Both refactoring tools and unit testing framework support rapid modification to the program code in prototyping.
Exploratory specification and prototyping share rapid modification to the source as well as learning aspects through trial and error.
Refactoring in agile development support the maintenance aspect of prototyping.
Rapid trials and errors followed by modifications often make the source code less readable and thus make further modifications harder.
On the other hand, the exploratory specification values its learning aspect.
We expect refactoring techniques to help the engineers widen and deepen their understandings of the problem domain.
Section \ref{sec:RefactoringForExploration} will discuss the role of refactoring in the exploratory specification phase and show families of semi-automated refactoring operations for exploratory specification.
In Section \ref{sec:RefactoringSupportInViennaTalk}, refactoring functionalities implemented in ViennaTalk will be described with an example.
Afterwards, Section \ref{sec:RelatedWork} explains existing research on refactoring of VDM-SL.
Finally, Section \ref{sec:DiscussionAndConcludingRemarks} discusses the result of implementation and concludes the paper.
\section{Refactoring for exploration}
\label{sec:RefactoringForExploration}
Refactoring is technically a collection of transformations on abstract syntax trees (ASTs) that preserve the behaviour of the existing code.
Refactoring of program code often benefits the maintainability of the code including readability and extensibility by reorganising the structure of the code.
Refactoring is often applied to code fragments that typically contain magic numbers, frequent expressions that can be abstracted, definitions located in inappropriate modules, misleading identifiers, and so on.
In the context of software maintenance, such code fragments are often called {\it code smell}.
The exploratory specification is the stage of writing a specification with a partial understanding of the application domain.
Software development typically goes parallel with acquiring new concepts in the target domain and exploring how they are associated with other known concepts.
Since a specification engineer who writes a specification does not have an accurate grasp of the entire domain, many parts of the specification remain tentative.
The specification at this stage is not always comprehensive and concise due to limited understanding of the domain and thus contains {\it code smell}.
In the exploratory specification, they are signs of learning opportunities.
While code smell in software maintenance is considered a threat to the maintainability of the code, it is an opportunity for learning in the exploratory specification.
Refactoring on a formal specification does not add new functionality to the system to be developed, but adds more senses to the model.
Our approach to refactoring operations focuses on names.
In programming, refactoring is sometimes carried out to improve performance by modifying to faster code and data structures without changing the functionality.
Although VDM-SL has an executable subset, performance is less emphasised in general.
We value the readability and maintainability brought by appropriate naming and abstraction.
In this research, we identified and implemented a set of semi-automated refactoring operations for naming, renaming, and unnaming a fragment of the specification source.
The goal is to provide a safe way to modify the specification without unintended change of the behaviour and also to reduce the workload of the editing task.
\subsection{Refactoring for naming}
The first group of refactoring operations is to give a name to a fragment of the specification.
In VDM-SL, there are three kinds of scopes of names: (1) global names, e.g. module name, (2) module-wide names defined in types sections, values sections, state sections, functions sections and operations sections, and (3) local names such as local definitions in let expressions/statements, and pattern identifiers for the pattern matching in parameters of functions/operations, local binding in set/sequence/map comprehensions, lambdas, quantifiers, loop constructs and so on.
The {\it Extract} family of refactoring operations generates the language constructs enumerated above from a fragment of the specification.
The {\it Use} family of refactoring operations replaces fragments in the same pattern with another that uses names.
\subsubsection{Extracting module-wide names}
In VDM-SL, type definitions, value definitions, state variables, function definitions, and operation definitions have the module-wide scope.
{\it Extract type} is a refactoring operation to define a new type definition for the specified type expression.
The newly defined type name will be used In the original place of the specified expression.
{\it Extract value} defines a new value definition for the specified expression.
Please note that a refactoring operation should be safe; the resulting specification should not break the well-formedness of the specification.
If the specified expression contains a reference to a state variable or an operation call, the tool should not apply the operation.
\begin{figure}
\begin{vdmsl}
functions
priceIncludingTax : real -> real
priceIncludingTax(displayPrice) ==
displayPrice + displayPrice * 0.1
\end{vdmsl}
\vspace*{-2em}\begin{center} $\Downarrow$ {\it Extract value on }{\tt 0.1}\end{center}\vspace*{-2em}
\begin{vdmsl}
values
TAX_RATE = 0.1
functions
priceIncludingTax : real -> real
priceIncludingTax(displayPrice) ==
displayPrice + displayPrice * TAX_RATE
\end{vdmsl}
\vspace*{-1em}
\caption{An example application of the {\it Extract value} refactoring}
\label{fig:spec-extract-value}
\end{figure}
Figure \ref{fig:spec-extract-value} illustrates {\it Extract value} operation that transforms the definition above the arrow into the one below by adding a value definition.
The original definition specifies the computation of price with 10\% sales tax using a magic number {\tt 0.1}.
By extracting {\tt 0.1} as a constant value named {\tt TAX\_RATE}, the definition becomes more expressive to the readers.
{\it Extract state variable} adds a new state variable.
The expression specified by the user will be used in the initialiser of the state definition.
The specified expression will be used as the right-hand side of the type definition.
If the specified expression contains a stateful subexpression, the tool must block the refactoring.
Also, the tool must guarantee that all the record constructor expressions with the state must be given an appropriate element for the newly added field.
A conservative approach is to prohibit the {\it Extract state variable} refactoring when there are one or more record constructors with the state.
{\it Extract function} and {\it Extract operation} define a new definition of function and operation accordingly.
To define a new function, a let expression is a language construct of VDM-SL that has corresponding functionality with function application: parameters, arguments and a body.
If there is no dependency among local definitions in a let expression, pairs of a local name and its value can be interpreted as pairs of a parameter and an argument as the local bindings to evaluate the body expression.
The tool must check whether the body expression contains a stateful subexpression and also a reference to other local names that are invisible in the top-level scope.
The let statement can similarly be converted into an operation definition.
\begin{figure}
\begin{vdmsl}
functions
priceIncludingTax : real -> real
priceIncludingTax(displayPrice) ==
displayPrice +
let rate:real = 0.1, taxable = displayPrice in taxable * rate
\end{vdmsl}
\vspace*{-2em}\begin{center} $\Downarrow$ {\it Extract function}\end{center}\vspace*{-2em}
\begin{vdmsl}
functions
priceIncludingTax : real -> real
priceIncludingTax(displayPrice) ==
displayPrice + tax(0.1, displayPrice);
tax : real * ? -> ?
tax(rate, taxable) == taxable * rate
\end{vdmsl}
\vspace*{-1em}
\caption{An example application of {\it Extract function} refactoring}
\label{fig:spec-extract-function}
\end{figure}
Figure \ref{fig:spec-extract-function} shows an example application of {\it Extract function}.
In the original specification, a 10\% sales tax computation is specified using let expression.
{\it Extract function} generates the definition of the function {\tt tax} from the let expression.
Please note that the type name {\tt ?} means so-called {\it any} type.
Any-type is not a legitimate type in the language specification of VDM-SL\cite{Larsen&13b}, but is a feature supported by many VDM-SL interpreters including VDMJ, Overture tool, VDMTools and ViennaTalk.
To turn the specification into a legitimate one, a concrete type should be given for the any-type, i.e. {\tt real} in this particular case.
It is in general difficult to infer a static type of an arbitrary expression though this case can be trivially inferred.
This issue will be discussed in Section \ref{sec:DiscussionAndConcludingRemarks}.
\subsubsection{Extracting local names}
The let expression is a language construct to make local definitions.
Parameters of function or operation also have local scopes.
Defining those local names for expressions add contextual meaning to the expressions and thus are expected to gain readability.
Those names also help abstraction.
{\it Extract let expression} turns an expression {\it exp} into {\tt let} {\it localname} {\tt =} {\it exp} {\tt in} {\it localname}.
Although the resulting form is trivial, this refactoring can be effectively applied along with refactoring operations for scoping described later.
{\it Extract local definition} adds a local definition with the specified subexpression in a let expression.
{\it Extract parameter in function} adds a parameter to the function definition and also adds the specified subexpression as an argument to the function for all applications of the function.
The subexpression ported as the argument must be valid in the function applications.
Also, the subexpression must be side-effect free because the argument is evaluated before the body of the let expression and the change of order of evaluation may produce different results than the original.
{\it Extract parameter in operation} similarly adds a parameter to an operation definition and the extra argument to the callers.
\begin{figure}
\begin{vdmsl}
functions
tax : real -> real
tax(taxable) == taxable * 0.1
\end{vdmsl}
\vspace*{-2em}\begin{center} $\Downarrow$ {\it Extract parameter in function}\end{center}\vspace*{-2em}
\begin{vdmsl}
functions
tax : real * ? -> real
tax(taxable, rate) == taxable * rate
\end{vdmsl}
\vspace*{-1em}
\caption{An example application of {\it Extract parameter in function} refactoring}
\label{fig:spec-extract-parameter-in-function}
\end{figure}
Figure \ref{fig:spec-extract-parameter-in-function} illustrates an example application of {\it Extract parameter in function}.
The magic number {\tt 0.1} that appears in the original specification is turned into a parameter of the function so that the function gains it generality.
Please note that the {\it any}-type appears also in this refactoring.
Although the literal value {\tt 0.1} is apparently of the {\tt real} type, it is not trivial in general to find an appropriate name of the type among possibly many alias names of the {\tt real} type.
\subsubsection{Using names}
Once a name is given to an expression, the occurrences of the same expression may better use the name.
The {\it Use} family of refactoring operations supports the user to deal with fragments that frequently appear in the specification.
{\it Use type} replaces an occurrence of the right-hand side of a type definition with the type name.
{\it Use value} does the same on a value definition.
{\it Use function} replaces an expression that matches with the definition body of a function with a function application that results in the same expression.
If a function has a precondition, {\it Use function} is not applicable to the function because the resulting function application may possibly violate the precondition.
\subsection{Refactoring for renaming}
In the exploratory stage of specification, the specifier has limited knowledge of the system and the domain, often resulting in misleading names.
Modification to the name, its location and its scope is also expected to be frequently performed to improve the quality of the specification as the knowledge is updated.
\subsubsection{Changing the name}
Renaming is frequently applied among refactoring operations in programming languages \cite{Negara&13}.
Renaming is not mere string replacement because the lexically same name can be semantically different names.
A semi-automated renaming operation that takes account of syntax and semantics of the language is desirable to conduct safe and efficient renaming.
The {\it Rename} family of refactorings supports rename types, quote types, values, functions, state variables, operations, local definitions and parameters.
\subsubsection{Changing the scope}
\begin{figure}
\begin{vdmsl}
functions
priceIncludingTax : real -> real
priceIncludingTax(displayPrice) ==
let rate = 0.1, taxable = displayPrice
in displayPrice + taxable * rate
\end{vdmsl}
\vspace*{-2em}\begin{center} {\it Narrow let expression} $\Downarrow$ \hspace{1cm} $\Uparrow$ {\it Widen let expression}\end{center}\vspace*{-2em}
\begin{vdmsl}
functions
priceIncludingTax : real -> real
priceIncludingTax(displayPrice) ==
displayPrice +
let rate = 0.1, taxable = displayPrice in taxable * rate
\end{vdmsl}
\vspace*{-1em}
\caption{An example application of {\it Widen/Narrow let expression} refactoring}
\label{fig:spec-widen-narrow-let}
\end{figure}
Every name has its scope and choosing the right scope of a local name is considered generally good for readability and maintainability.
The {\it Widen} family of refactorings widens the body of the {\tt in}-clause in a let expression or a let statement into its enclosing expression or statement.
The {\it Split} family of refactoring separate one local definition from an inner let expression to the outer let expression.
If the let expression is not nested, the {\it Split} refactoring will create another let expression that encloses the original let expression.
The {\it Narrow} family of refactorings narrows the body into its subexpression if and only if the locally defined names appear only in the subexpression.
Figure \ref{fig:spec-widen-narrow-let} shows an example of widening and narrowing a let expression.
\subsubsection{Changing the location}
VDM-SL supports a modular construction of a model.
The choice of the module to define a type, value or function is important but sometimes not trivial.
The {\it Move} family of refactoring operation is to move a definition from one module to another and update the import/export declarations of each module.
\subsection{Refactoring for unnaming}
In the exploration to a better presentation of the model, a function definition or a value definition may have little use or be found confusing.
A let expression may introduce unnecessary syntactical complexity to the presentation.
Removal of a name needs not only just deleting its definition but also replace the name with its flat presentation.
Semi-automated refactoring operation for these tasks is expected to save the user's cognitive workload.
The {\it Inline} family of refactoring operations supports inlining both local and module-wide names.
{\it Inline type}, {\it Inline value} and {\it Inline function application} are inverse of their counterparts in the {\it Use} family, which replace the expression with the definition body.
{\it Inline let expression} and {\it Inline let statement} replace the occurrences of the locally defined names with their definition bodies and remove the local definition in the let expression or statement.
The {\it Remove} family of refactorings deletes the declaration of names when no reference to the names is left in the specification.
A {\it Remove} refactoring can be either automatically performed after an {\it Inline} refactoring or triggered independently.
\section{Refactoring Support in ViennaTalk}
\label{sec:RefactoringSupportInViennaTalk}
ViennaTalk\cite{Oda&17a} is an IDE for the exploratory stage of VDM-SL specification built on top of the Pharo Smalltalk environment.
ViennaTalk has a browser called VDM Browser designed to empower the use of animation in writing a specification, evaluating expressions, unit testing, UI prototyping, web API prototyping, and executable documentation.
While text-based source editors in conventional IDEs provide text-editing operation at the source file tree, VDM Browser holds an interpreting process that contains the source text and the binding information of state variables.
We chose to implement another browser named Refactoring Browser to implement semi-automated refactoring operations.
The design of the new Refactoring Browser is revised to hold an abstract syntax tree (AST) because refactoring can more naturally be defined on AST rather than the source text.
The source text displayed on the Refactoring Browser is promptly generated from the AST when the user selects the definition to edit.
The source text edited by the user is parsed and merged into the AST, and the source text will be disposed.
It is widely advised that refactoring should be performed with unit testing to ensure that the functionality is not changed.
Unit testing functionality was ported from VDM Browser to Refactoring Browser with a modification to UI.
While VDM Browser has a Unit Testing tab to display the result of unit tests behind the source pane, Refactoring Browser places the unit testing pane aside the source pane so that the result of unit testing is always visible and has more chance to catch the user's eyes.
History functions to record versions of the specification are added to Refactoring Browser.
Semi-automated refactoring may cause unexpected modifications to the specification, and the ability to roll back to previous versions is desirable.
Refactoring Browser records a mirror copy of AST into history at every execution of refactoring operations and manual edits.
\begin{figure}
\begin{center}
\includegraphics[width=1\textwidth]{screenshot-RefactoringBrowser.png}
\end{center}
\caption{Screenshot of Refactoring Browser}
\label{fig:screenshot-refactoring-browser}
\end{figure}
Figure \ref{fig:screenshot-refactoring-browser} is a screenshot of the Refactoring Browser.
The left top dropbox that reads ``load'' is a history selector that lists all versions of the AST with the names of refactoring operations, and let the user choose a version to roll back.
The list widget on the left is called module list that shows all modules defined in the specification, e.g. {\tt Luhn}, {\tt LuhnTest} and {\tt UnitTesting} in the Figure \ref{fig:screenshot-refactoring-browser}.
The up centre list widget shows sections in the selected module, and the right list enumerates top-level definitions in the sections.
The source text of the selected top-level definition is generated from the AST and shown in the lower text pane.
In the screenshot, the function {\tt total} in the functions section of the {\tt Luhn} module is edited.
A refactoring operation is a semi-automated manipulation of AST.
Each refactoring is defined as a subclass of ViennaRefactoring class.
When a user opens a context menu, each subclass of ViennaRefactoring is created and tested executability to the specified AST node.
All refactoring instances that returned true to the executability test will be listed on the context menu.
In Figure \ref{fig:screenshot-refactoring-browser}, the keyword {\tt let} highlighted in the source is selected by the user, and the context menu shows all applicable refactoring operations on the function.
\subsection{Example: LUHN}
\begin{figure}
\begin{vdmsl}
module Luhn
exports all
definitions
types
Digit = nat inv d == d < 10;
functions
luhn : seq1 of Digit -> Digit
luhn(data) == total(data) * 9 mod 10;
total : seq of Digit -> nat
total(data) ==
if data = []
then 0
else
(let
multipler = len data mod 2 + 1,
product = hd data * multipler
in
total(tl data) +
(if product < 10
then product
else product mod 10 + 1))
measure slen;
slen : seq of Digit -> nat
slen(data) == len data;
end Luhn
\end{vdmsl}
\caption{The original source of the {\tt Luhn} module}
\label{fig:luhn-original}
\end{figure}
Luhn algorithm, also known as mod-10, is a checksum algorithm mainly used to verify a series of numbers and/or letters against accidental errors such as typos.
Figure \ref{fig:luhn-original} shows a part of the specification of the Luhn algorithm published at the Overture website\footnote{\url{https://www.overturetool.org/download/examples/VDMSL/LUHNSL/index.html}} with modification.
The {\tt lune} function computes the check digit for the given sequence of digits.
Although the algorithm is well-known and specification is already mature, the specification can be modified more informative by applying a series of semi-automated refactoring operations and a few manual edits on ViennaTalk's Refactoring Browser.
Two improvements will be illustrated and explained in the rest of this section.
\subsubsection{Improve the definition of {\tt total}}
The first improvement is on the function {\tt total}.
A let expression appears in the definition body of {\tt total}.
The subexpression {\tt total(tl data)} is included in the let expression, but does not refer to any local definition of the let expression.
The {\it Narrow let expression} refactoring can be applied to adjust the scope of the local definitions.
The let expression will be refactored to the below.
\begin{vdmsl}
total(tl data) +
(let
multipler = len data mod 2 + 1,
product = hd data * multipler
in
(if product < 10 then product else product mod 10 + 1))
\end{vdmsl}
The expression is now separated into the recursion part and the computation on each element of {\tt data}.
Extracting a parameter {\tt datum} from {\tt hd data} in the let expression and extract a function definition from the let expression will simplify the definition of the {\tt total} function.
The result of {\it Extract local definition in let expression} and {\it Split let} in order is shown below.
\begin{vdmsl}
total(tl data) +
(let datum = hd data in
let
multipler = len data mod 2 + 1,
product = datum * multipler
in
(if product < 10 then product else product mod 10 + 1))
\end{vdmsl}
This time, the Refactoring Browser does not enable {\it Extract function from let expression} because the reference to the local parameter {\tt data} is left in the body of the let expression.
The local definition of {\tt multipler} should be moved to the outer let expression by {\it Split let}.
\begin{vdmsl}
total(tl data) +
(let
datum = hd data,
multipler = len data mod 2 + 1,
in
let product = datum * multipler in
(if product < 10 then product else product mod 10 + 1))
\end{vdmsl}
The Refactoring Browser enables the user to apply {\it Extract function from let expression} to the outer let expression.
The following shows the resulting definition of the {\tt total} function and the newly defined {\tt single} function.
\begin{vdmsl}
total : seq of Digit -> nat
total(data) ==
if data = []
then 0
else
(total(tl data) + single(hd data, len data mod 2 + 1));
single : ? * ? -> ?
single(datum, multipler) ==
let product = datum * multipler in
(if product < 10 then product else product mod 10 + 1)
\end{vdmsl}
The type of the {\tt single} function is not specified well, and the parameter {\tt multipler} should be constrained by adding a precondition.
The definition of {\tt single} will be manually edited into the below.
\begin{vdmsl}
single : Digit * nat -> Digit
single(datum, multipler) ==
let product = datum * multipler in
(if product < 10 then product else product mod 10 + 1)
pre multipler in set {1, 2}
\end{vdmsl}
The precondition of {\tt single} is important.
Many explanation of this formula writes that each digit in the 1st, 3rd, 5th, ... place from the right should be doubled.
{\tt multipler} should be either 1, meaning not doubled, or 2 meaning doubled.
The extraction of the function {\tt single} created a place to formally document the constraint while the constraint can be inferred from {\tt len data mod 2 + 1} in the original specification.
The refactored specification works the same as the original one, and asserts two constraints explicit: (1) the resulting value to be added is in the range of {\tt Digit} type, and (2) the {\tt multipler} should be either {\tt 1} or {\tt 2}.
The refactoring involved five semi-automated operations and one manual edit.
The automated operations do not only save the keyboard typing efforts but also prevent the user from breaking the semantics of the original specification.
\subsubsection{Generalising {\tt 10} as a base}
A literal value {\tt 10} appears in multiple places in the specification.
A constant value can be defined by {\it Extract value} refactoring.
\begin{vdmsl}
values
base = 10;
types
Digit = nat inv d == d < base;
functions
luhn : seq1 of Digit -> Digit
luhn(data) == total(data) * 9 mod base;
total : seq of Digit -> nat
total(data) ==
if data = []
then 0
else
total(tl data) + single(hd data, len data mod 2 + 1)
measure slen;
slen : seq of Digit -> nat
slen(data) == len data;
single : Digit * nat -> Digit
single(datum, multipler) ==
let product = datum * multipler
in
(if product < base
then product
else product mod base + 1)
pre multipler in set {1, 2};
end Luhn
\end{vdmsl}
A literal number {\tt 9} appears in the definition of {\tt luhn}.
The original expression {\tt total(data) * 9 mod 10} is a short form of more explanatory expression {\tt (10 - total(data) mod 10) mod 10}, intending a single-digit value that will be summed up with the last digit of {\tt total(data)} to make a natural number whose last digit is {\tt 0}.
The below is the result of manual editing to {\tt (10 - total(data) mod 10) mod 10} and apply the {\it Use value in all occurrences} operation.
\begin{vdmsl}
luhn : seq1 of Digit -> Digit
luhn(data) == (base - total(data) mod base) mod base;
\end{vdmsl}
\begin{figure}
\begin{vdmsl}
module Luhn
exports all
definitions
values
base = 10;
types
Digit = nat inv d == d < base;
functions
luhn : seq1 of Digit -> Digit
luhn(data) == (base - total(data) mod base) mod base;
total : seq of Digit -> nat
total(data) ==
if data = []
then 0
else
total(tl data) + single(hd data, len data mod 2 + 1)
measure slen;
slen : seq of Digit -> nat
slen(data) == len data;
single : Digit * nat -> Digit
single(datum, multipler) ==
let product = datum * multipler
in
(if product < base
then product
else product mod base + 1)
pre multipler in set {1, 2};
end Luhn
\end{vdmsl}
\caption{The refactored source of the {\tt Luhn} module}
\label{fig:luhn-refactored}
\end{figure}
The resulting specification is shown in Figure \ref{fig:luhn-refactored}.
The specification gained generality so that it can be easily extended to other variations of the Luhn algorithm to validate a sequence of letters other than just digits.
\section{Related Work}
\label{sec:RelatedWork}
Pedersen and Mathiesen pioneered the application of refactoring to VDM-SL~\cite{Petersen&17a}.
They identified refactoring operations applicable to VDM-SL, including rename, add parameter, convert function to operation, extract definition and remove operations, and built a Proof-of-Concept implementation on the Overture tool.
Refactoring is discussed as means to address {\it model smells} and improve the maintainability of the model while its counterpart in the programming languages is to address {\it code smell} in the field of software maintenance.
This paper pays attention to the same technique, refactoring, for a different purpose.
This paper shed light on refactoring as guidance for the specifier to learn the system and domain, which is more performed at the earlier phase of the specification than the phase to improve the rigour and conciseness of the model.
\section{Discussion and Concluding Remarks}
\label{sec:DiscussionAndConcludingRemarks}
Refactoring, in general, is often performed to improve readability and maintainability in both programming languages and specification languages.
Refactoring in programming is widely practised in the maintenance of program sources to keep them readable and extendable.
The objective of this research is to design a set of semi-automated refactoring operations that support the learning process in the exploratory stage of the specification rather than the maintenance phase of the specification.
The question is whether the automated refactoring techniques widely used in programming is also useful in the exploratory stage of the specification.
The authors used the Refactoring Browser to modify several existing models including the Luhn algorithm and found the semi-automated refactoring operations safe and efficient.
The semi-automated refactoring operations make coarse-grained interaction with the user comparing the conventional text editing operations.
Each refactoring has a different scope and makes a different amount of changes to the source text.
The user's expectation of the changes to be made by a refactoring operation may differ from the actual one.
The automated history mechanism implemented in the Refactoring Browser encourages the use of refactoring operations.
While the use of refactoring tools in the exploratory stage look promising, we observed difficulties to apply the semi-automated refactoring operations to VDM-SL.
One difficulty is assertions, especially invariants and preconditions.
Judging the applicability of a refactoring operation that replaces two expressions often involves the satisfiability of assertions.
Even though an expression matches with the definition body of a function, it is not trivial to confirm that the arguments in the function application always satisfy the precondition of the function and the type invariants of the parameter types.
Automated inference of types on extracted identifiers is also problematic.
For example, the type of constant value in the values section can be omitted while the export signatures of values require explicit typing.
The refactoring operations implemented in this research avoid this issue by using the {\it any}-type ({\tt ?}) which is not supported by the language specification.
To generate a formally valid fragment of specification, the user's manual edit is required.
Multiple refactoring operations are often required to make an apparent improvement as seen in the Luhn example.
Support for planning a series of refactoring operations to achieve an expected achievement is required.
History is a promising tool to show the strategic view of the modification to the specification with a series of fine-grained refactoring operations applied so far.
Further study to design a better interaction model with the combination of refactoring operations and history is needed.
\section*{Acknowledgments}
The authors thank Nick Battle for providing the original specification of Luhn algorithm and supplementary documentation.
We would thank the anonymous reviewers for their valuable comments and suggestions.
\input{3-Oda.bbl}
\subsection{Client}
\label{subsec:client}
The client is an entity responsible for making calls to the sandbox. It could be understood as an application that acts upon a behalf of a sandbox user. The HSM is a multi-user system, hence multiple clients can access the different sandboxes and system features. Since the system can handle multiple clients, each client carries a unique identity as \lstinline[style=VDM]{ClientId=nat}. In a current iteration of the HSM, the client can select multiple tools and models from a repository, i.e. a library of preexisting items hosted on the system and launch a new sandbox that distributes these tools across as many newly spawned virtual machines as there are selected tools. In the current HSM analysis model, the models are not associated with a specific tool, what is sufficient as the functionality expressed within the table of roles and profiles does not define interaction between a model and a tool. The selection of tools and models and launching of a new sandbox with these entities is shown in Listing~\ref{lst:selections}. The client can further destroy an existing sandbox, connect and disconnect from a sandbox and upload repository items.
\begin{vdmsl}[style=VDM, label={lst:selections}, caption={Selection of sandbox entities and launching of a sandbox}]
SelectToolsFromRepository: ClientId * SelectedTools ==> ()
SelectToolsFromRepository(cId, tIds) ==
clientst.selectedTools := SelectTools(tIds, cId)
pre cId in set validClients;
SelectModelsFromRepository: ClientId * SelectedModels ==> ()
SelectModelsFromRepository(cId, mIds) ==
clientst.selectedModels := SelectModels(mIds, cId)
pre cId in set validClients;
LaunchNewSandbox: ClientId ==> ()
LaunchNewSandbox(cId) ==
StartNewSandbox(cId, clientst.selectedTools, clientst.selectedModels)
pre cId in set validClients;
\end{vdmsl}
In this iteration of the HSM the client can not only upload a tool once to the repository, but can keep the tool up to date by updating the tool with potential new versions. Another option is to upload an archive of files to a specific sandbox and download data generated by a tool running on a specific virtual machine within a sandbox that the client has access to. The uploading of a tool, updating of a tool, uploading of an archive and downloading of an archive is shown in Listing~\ref{lst:repositoryOperations}. It is important to note that the pre-condition \lstinline[style=VDM]{pre cId in set validClients} checks that the client is recognized as valid by the system. After the data download the post-condition \lstinline[style=VDM]{post card clientst.downloadedData = card clientst~.downloadedData + 1} checks that the client has received the downloaded data. While this post-condition would not allow for a repeated download of the same data, it is sufficient within our analysis.
\begin{vdmsl}[style=VDM, label={lst:repositoryOperations}, caption={Upload, update and download of items}]
UploadTool : ClientId * OSId * Version * Private * OsOnly ==> ()
UploadTool(cId, oId, v, p, oo) == SaveTool(cId, oId, v, p, oo)
pre cId in set validClients and oId in set brokerst.validOSs;
UpdateTool : ClientId * ToolId ==> ()
UpdateTool(cId, tId) == UploadNewToolVersion(cId, tId)
pre cId in set validClients;
UploadArchive : ClientId * SandboxId * token ==> ()
UploadArchive(cId, sbId, arch) ==
UploadArchiveToSandbox(cId, sbId, arch)
pre cId in set validClients;
DownloadArchive : ClientId * ServerId * SandboxId ==> ()
DownloadArchive(cId, sId, sbId) ==
let d = DownloadArchiveFromServer(cId, sId, sbId) in
clientst.downloadedData := clientst.downloadedData union {d}
pre cId in set validClients
post card clientst.downloadedData =
card clientst~.downloadedData + 1;
\end{vdmsl}
\subsection{Broker}
\label{subsec:broker}
The broker is the part of the HSM that handles the calls from the client and provides replies from the system to the client. The broker is a single entity hence it does not need a specific identity. The broker state is responsible for handling of client/sandbox ownership relation, for example a relation \lstinline[style=VDM]{Guests = map ClientId to set of SandboxId} specifies that the specific clients are guests to specific sandboxes. The broker further specifies other relations such as ownership of tools, assignments of tools to sandboxes and others. One very important function of the broker is to start new sandboxes. This requires starting of new virtual machines under the sandbox. The operation for launching of new sandboxes is shown in Listing~\ref{lst:launch}. This operation also calls \textit{BuildNewServers} operation responsible for spinning out new virtual machines under the system. Another operation responsible for getting the operating systems out of a specific tool used when launching a sandbox is \textit{GetToolOSs}. This is necessary in order to keep track of which operating systems are used under a specific sandbox. These operations are described in more detail in Section~\ref{subsec:system}. The pre-condition on the operation states that the sandbox can only contain valid tools and valid models, i.e. items that are recognized by the sandboxing system. The post-condition states that new sandbox is added upon completion of the operation, the calling client is added as an owner of the newly spawned sandbox.
\begin{vdmsl}[style=VDM, label={lst:launch}, caption={Launching of a new sandbox}]
StartNewSandbox : ClientId * set of ToolId * set of ModelId ==> SandboxId
StartNewSandbox(cId, tId, mId) ==
let sId = GenerateNewSandboxId()
in
(brokerst.sandboxOSs := brokerst.sandboxOSs munion {sId |-> GetToolOSs(GetToolsByToolIds(tId, brokerst.validTools))};
brokerst.sandboxModels := brokerst.sandboxModels munion {sId |-> mId};
(brokerst.sandboxTools := brokerst.sandboxTools munion {sId |-> tId};
systemSandboxes := systemSandboxes munion {sId |-> mk_Sandbox(sId, BuildNewServers(tId,mk_token(nil)), {})});
if cId in set dom brokerst.owners then
brokerst.owners(cId) := brokerst.owners(cId) union {sId}
else
brokerst.owners := brokerst.owners munion {cId |-> {sId}};
return sId)
pre tId <> {} and tId subset dom brokerst.validTools
and (mId <> {} => mId subset brokerst.validModels)
post card dom systemSandboxes = card dom systemSandboxes~ + 1
and cId in set dom brokerst.owners
and RESULT in set dom systemSandboxes;
\end{vdmsl}
The broker is an integral part of the system often responsible for handling operation calls according to the table of roles and profiles, encoded as pre and post conditions.
\subsection{Gateway}
The gateway is a connectivity component. It provides connections between clients and virtual machines within the sandbox. The gateway holds a state of connections, while it is the broker that assigns these connections. The gateway is defined as shown in Listing~\ref{lst:gateway} with the connections handling as shown in Listing~\ref{lst:gatewayConnections}. All of the sandbox connections and disconnections are recorded within the gateway state. The operation called during connections and disconnections is the \textit{UpdateConnections} that is defined within system and is described in detail in Section~\ref{subsec:system}. Within the Listing~\ref{lst:gatewayConnections} the precondition checks that the client actually has rights as defined within the table of roles and profiles and the client is either an owner of the sandbox it is connecting to, or a guest that was invited towards this sandbox.
\begin{vdmsl}[style=VDM, label={lst:gateway}, caption={Gateway state}]
GateWaySt ::
connectedClients : ConnectedClients
connectedServers : ConnectedServers;
ConnectedClients = set of ClientId;
ConnectedServers = set of ServerId;
\end{vdmsl}
\begin{vdmsl}[style=VDM, label={lst:gatewayConnections}, caption={Sandbox connection initiation}]
BrokerInitiateSandboxAccess: ClientId * SandboxId ==> bool
BrokerInitiateSandboxAccess(cId, sId) ==
let sandboxes = GetSystemSandboxes(),
servers = dunion {dom s.sandboxServers
| s in set rng sandboxes
& s.sandboxId = sId}
in
(for all x in set servers do
UpdateConnections(cId, x, sId, true);
return true)-- used to simulate error handling
pre not ClientIsNull(cId,brokerst.providers,brokerst.consumers,
brokerst.owners, brokerst.guests)
and ((cId in set dom brokerst.owners and sId in set brokerst.owners(cId))
or (cId in set dom brokerst.guests and sId in set brokerst.guests(cId)));
\end{vdmsl}
\subsection{Sandbox}
\label{subsec:sandbox}
The sandbox is considered to be an isolated entity, a container that contains several virtual machines hosting tools and models that user can interact with remotely. Each sandbox isolates its data from other sandboxes and access to it is governed by the table of roles and profiles. The sandbox itself can receive data remotely via a client once it is launched, this data is only present within a selected sandbox and cannot be shared to other sandboxes. An important notion within the sandboxing system is the ownership model, i.e. the user could be an owner of a sandbox and be able to invite other as guests and also destroy the sandbox. The sandbox is defined as shown in Listing~\ref{lst:sandbox}.
\begin{vdmsl}[style=VDM, label={lst:sandbox}, caption={Sandbox definition}]
SandboxId = nat;
SandboxServers = map ServerId to Server;
UploadedData = set of token;
Sandbox::
sandboxId : SandboxId
sandboxServers : SandboxServers
uploadedData : UploadedData
\end{vdmsl}
Each of the sandboxes has a unique identity, defines virtual machines that belong to this sandbox and has a set of data that has been uploaded from a client (starting as an empty set). Since each sandbox must have an unique identity it is necessary to generate new identity each time a new sandbox is launched. This is handled by functions and operations shown in Listing~\ref{lst:sandboxId}. This shows that the identity of sandboxes is incremental.
\begin{vdmsl}[style=VDM, label={lst:sandboxId}, caption={Sandbox identity}]
Max: SystemSandboxes -> nat
Max(ss) ==
if ss = {|->} then 0
else let max in set dom ss be st
forall d in set dom ss & d <= max
in
max;
GenerateNewSandboxId: () ==> nat
GenerateNewSandboxId() == return Max(systemSandboxes) + 1;
\end{vdmsl}
The current iteration of the model allows for sandboxes consisting of multiple virtual machines, increasing the complexity from all previous attempts to model the system.
\subsection{Server}
\label{subsec:server}
The server is considered to be a virtual machine within the system used to host a specific tool. Each server within the system has a unique identity where the identity is generated similar to the sandbox identities, i.e. they are incremental. Within the system each server needs to be a part of a sandbox in order for a connection to be established via a client. The server definition is shown in Listing~\ref{lst:server}. Each server defines what tool it carries and potentially there could be a piece of abstract data that has been generated by the interaction of a tool and a model. This data can then be obtained by a client and when the server is started is starts empty as \lstinline[style=VDM]{BuildNewServers(tId,mk_token(nil))}. Within the model this is used for analysis of clients ability to download data from a server according to the table of roles and profiles.
\begin{vdmsl}[style=VDM, label={lst:server}, caption={Server definition}]
ServerId = nat;
Data = token;
Server::
serverId : ServerId
toolId : ToolId
data : Data
\end{vdmsl}
\subsection{Tool}
\label{subsec:tool}
The tool is the newest addition to the modeling effort. The tool is considered to be an application that is used to interact with specific models and is deployed within a specific virtual machine under a sandbox. In some cases the tool does not contain an application but only an operating system. The tool can further be set as private, meaning that only the user that provided it to the repository of tools could select it and deploy it within a sandbox. The tool further defines a version as in this iteration of the system it is possible to update existing tools rather than keep all of the different tool versions within the platform. The tool is defined as shown in Listing~\ref{lst:tool}. The operating system of the tool is specified by the identity of the operating system.
\begin{vdmsl}[style=VDM, label={lst:tool}, caption={Tool definition}]
Version = nat;
Private = bool;
OsOnly = bool;
Tool::
osId : OSId
version : Version
private : Private
osOnly : OsOnly
\end{vdmsl}
\subsection{System}
\label{subsec:system}
Finally the system could be understood as an abstraction encompassing the core system functionality. For example the system defines operations for generation of new identities, starting of new virtual machines and assigning connections to the gateway. The operation utilized to create new servers is shown in Listing~\ref{lst:newserv}. Here a new virtual machine is created with its properties when required due to a new sandbox being launched. This operation creates a new server for every tool selected by the client.
\begin{vdmsl}[style=VDM, label={lst:newserv}, caption={New server (virtual machine) creation}]
BuildNewServers: set of ToolId * Data ==> map ServerId to Server
BuildNewServers(tId, d) ==
(dcl servers : map ServerId to Server := {|->};
for all t in set tId do
let newServerId = GenerateNewServerId()
in(
systemServers:= systemServers union {newServerId};
servers(newServerId) := mk_Server(newServerId, t, d);
);
return servers;);
\end{vdmsl}
The system furthermore reads out the operating system identities out of the specific tools, information that is used to setup relation between sandboxes and operating systems. This function is shown in Listing~\ref{lst:getos}.
\begin{vdmsl}[style=VDM, label={lst:getos}, caption={Get operating system identities}]
GetToolOSs: set of Tool -> set of OSId
GetToolOSs(tools) == {t.osId | t in set tools};
\end{vdmsl}
Finally an important operation is an operation responsible for changes in connections, updating the state of the gateway. This operation is shown in Listing~\ref{lst:updatecon}. Both connections and disconnections are handled by this operation.
\begin{vdmsl}[style=VDM, label={lst:updatecon}, caption={Update the gateway connections}]
UpdateConnections: ClientId * ServerId * SandboxId * bool ==> ()
UpdateConnections(cId, sId, sbId, connect)==
if connect
then (if cId in set dom gatewayConnections
then atomic
(gatewayConnections(cId):= gatewayConnections(cId) union {sId};
gatewayConnectionsSandbox(cId) := gatewayConnectionsSandbox(cId) union {sbId};
brokerst.activeSandboxes := brokerst.activeSandboxes union {sbId})
else atomic
(gatewayConnections := gatewayConnections munion {cId|-> {sId}};
gatewayConnectionsSandbox := gatewayConnectionsSandbox munion {cId |-> {sbId}};
brokerst.activeSandboxes := brokerst.activeSandboxes union {sbId}))
else atomic
(gatewayConnections(cId) := gatewayConnections(cId) \ {sId};
gatewayConnectionsSandbox(cId) := gatewayConnectionsSandbox(cId) \ {sbId};
brokerst.activeSandboxes := brokerst.activeSandboxes \ {sbId})
pre (connect=false) => cId in set dom gatewayConnections;
\end{vdmsl}
The file System.vdmsl also holds the initialization of the model building up initial states of different entities and holds an invariant \lstinline[style=VDM]{dom ss.gatewayConnections = dom ss.gatewayConnectionsSandbox}, specifying that all of the sandbox connections and server connections are established using the same clients.
\section{Introduction}
\input{introduction}
\section{Architecture}\label{sec:arch}
\input{architecture}
\section{Access rights and hierarchy}\label{sec:rights}
\input{accesRights}
\section{Formal access model}\label{sec:VDM}
\input{model}
\section{Formal analysis}\label{sec:analyse}
\input{analysis}
\section{Related work}\label{sec:related}
\input{related-work}
\section{Conclusion}\label{sec:conclude}
\input{conclusion}
\subsubsection*{Acknowledgements.}
The work presented here is partially supported by the HUBCAP Innovation Action funded by the European Commission's Horizon 2020 Programme under Grant Agreement 872698. We would also like to express our thanks to the anonymous reviewers.
\input{4-Kulik.bbl}
\subsection{What does BDS involve?}\label{subsec:what-is-bds?}
The BDS process is much like BDD, and it too requires a DSL in which to write scenarios.
However, unlike BDD, where the inputs are ACs and the output is code, with BDS we look to establish ACs and use those to drive the creation of the specification, with both being valuable artifacts of the process.
For each behaviour, the first step is to write a natural language description of the AC.
A scenario is then created to exemplify the AC; this may involve the extension of the DSL if new language is needed to express the criterion.
This scenario now forms an executable acceptance criterion (EAC) -- we shall discuss how these are run in section~\ref{subsec:executable-acceptance-criteria} -- that we can use to validate the behaviour of both the specification and the target implementation.
When first run this EAC should fail, otherwise it is likely that the AC is redundant or that the scenario does not correctly capture the unicity of the AC.
The specifier then makes just enough changes to the specification to satisfy the EAC without invalidating any others.
This process is repeated until all of the ACs of the behaviour have been established and agreed with the stakeholder.
At this point the EACs can be handed to a developer to implement the behaviour.
The developer can now start their work with a set of ACs, including examples scenarios that fail against the target implementation --- an ideal place to start BDD.
We can rapidly create scenarios to exemplify behaviour questions that arise during implementation, and we can run them against the specification to determine the expected outcome.
If the developer is not satisfied, we can take that example and the specified results, and present to the stakeholder.
If the stakeholder believes this scenario is distinct from our existing EACs and wants a change of behaviour, we make the scenario a new EAC and change just enough of the specification to satisfy it.
At this point, the specifier may discover that it is not possible to make a change that is consistent with the existing EACs.
If so, the conflicting criteria are presented to the stakeholder and a resolution agreed.
The use of the DSL allows us to do this in the language of the stakeholder, but without the ambiguity of natural language.
Similarly, the developer may discover efficiency or performance issues with implementing a particular EAC, or in fact, any member of the team may spot some behaviour they disagree with.
Once again, these can be discussed with the stakeholder, the outcome of the EAC changed, and the requirements of the behaviour, the specification, and the implementation iteratively refined.
The use of EACs not only encourages a more agile approach to specification, but enables its continuous validation through DSL-based requirements which are accessible to everyone involved in the project.
\subsection{Running executable acceptance criteria}\label{subsec:executable-acceptance-criteria}
Running scenarios requires a DSL in which to express the actions and checks of the system in the stakeholder's language, and means by which to translate that language into commands and queries that the implementation can execute.
Cucumber~\cite{wynne2017cucumber} is synonymous with BDD, but we found it unsuitable for use with BDS.
Although Cucumber accepts scenarios in the given-when-then format, the distinction between the blocks is lost at runtime and the scenario is reduced to a single sequence of steps.
When executing a program it is perhaps unimportant that certain steps are declarative and others are imperative\footnote{We found it useful to make the distinction with the implementation too. Anaplan is a transactional system and it was useful for us to be able to create the declarative, initial context in a single transaction.}, but when animating a specification we wanted to transform the declarative steps into a single declaration rather than calling a series of functions.
After some time trying to adapt Cucumber to our needs, we found that we were better able to make progress with a custom Kotlin DSL~\cite{subramaniam2021programming}.
In practice, we found the majority of our effort in this area was in creating the DSL and adapting to the specification and the various implementations, rather than in the mechanics of execution.
We briefly describe our approach to these mechanics for completeness, and it is likely that another team employing BDS would benefit from replicating the approach rather than using our tooling directly.
Example of some trivial EACs are given in listings~\ref{lst:create} and ~\ref{lst:add}\footnote{Note that the keyword \texttt{whenever} is used in the scenario as \texttt{when} is a Kotlin keyword.}.
\noindent
\begin{minipage}[t]{.5\textwidth}
\begin{lstlisting}[caption=EAC for creating a list, label=lst:create]
@Eac("A created list is empty")
fun create() {
whenever {
createAList("list")
}
then {
listContains("list")
}
}
\end{lstlisting}
\end{minipage}
\noindent
\begin{minipage}[t]{.5\textwidth}
\begin{lstlisting}[caption=EAC for adding to a list, label=lst:add]
@Eac("When an entity is added,
it is contained by the list")
fun addToAList() {
given {
thereIsAList("list", "a")
}
whenever {
addEntityToList("list", "b")
}
then {
listContains("list", "a", "b")
}
}
\end{lstlisting}
\end{minipage}
\noindent
A single EAC is run using a custom JUnit runner;
this means that all the usual JUnit tools are available for running from an IDE, a build tool, or a continuous integration environment.
To run an EAC, the runner first identifies what implementations are available on the classpath and whether any constraints have been placed on the EAC.
For example, we might want to skip a particular implementation due to a known bug and use an annotation to indicate this.
Once the list of available implementations has been determined, the runner will attempt to run the EAC against each of them.
The DSL is backed by an API that is implemented by an adapter for each implementation.
The JUnit runner will use this API to convert the DSL into a series of implementation specific commands.
There is not a one-to-one correspondence here as, particularly with the declaration, an adapter may choose to maintain a state and create a single command corresponding to several calls to the API.
The adapter is also allowed to return an `unsupported command' from any API method -- indicating that there is piece of functionality that has not yet been implemented, or perhaps which may never be implemented.
For this reason, during conversion, the runner does not instantiate an implementation and none of the commands are executed.
Only if all commands are supported does the runner actually execute them against the implementation, otherwise it can quickly skip the EAC without having to partially execute the scenario.
This is extremely valuable for BDS where -- unlike BDD -- we are usually working with more than one adapter and development may not immediately follow specification.
When the specification of a behaviour is complete, we can mark that behaviour as unsupported in the implementation's adapter until development can begin.
This capability is even more useful when working with multiple implementations --- as in our Anaplan project.
Consider the example in~\ref{subsec:the-need-for-formal-methods}: we want to be able to specify both the dense and sparse behaviour of calculating $0^0$, but clearly each implementation will only support one outcome.
We can extend our DSL to be able to disambiguate the two methodologies\footnote{To prevent cluttering, we would only specify the arithmetic used in the EAC when it mattered and we would expect identical behaviour otherwise.} and write two EACs as shown in listings~\ref{lst:dense} and~\ref{lst:sparse}.
Our two sparse platforms would return an `unsupported command' for the EAC requiring dense arithmetic and would be skipped, but would be able to run the EAC requiring sparse arithmetic (and vice versa for the two dense platforms).
Our specification would be able to support both arithmetic methodologies and evaluate both EACs.
From a documentation perspective, this is ideal as we can place the EACs side by side, to compare and contrast the two behaviours.
\noindent
\begin{minipage}{.5\textwidth}
\begin{lstlisting}[caption=Dense EAC,label={lst:dense}]
given {
thereIsAFormula("f",
"POWER(0,0)",
arithmetic = DENSE)
}
then {
formulaEvaluatesTo("f", 1)
}
\end{lstlisting}
\end{minipage}
\begin{minipage}{.5\textwidth}
\begin{lstlisting}[caption=Sparse EAC,label={lst:sparse}]
given {
thereIsAFormula("f",
"POWER(0,0)",
arithmetic = SPARSE)
}
then {
formulaEvaluatesTo("f", 0)
}
\end{lstlisting}
\end{minipage}
\subsection{Adapting to a specification}\label{subsec:adapting-to-a-specification}
A requirement of BDS is the capability to create an adapter from the DSL to the specification, which requires a specification framework where animation is feasible.
We considered a number of formal specification systems with suitable tooling, but -- to reaffirm the arguments presented in section 3 of~\cite{larsen2011formal} -- as engineers in industry we preferred the flexibility and familiarity of VDM~\cite{larsen2010overture} to more alien, proof-oriented methods such as Event-B~\cite{abrial2010modeling}.
VDM's Overture ecosystem provides a choice of IDEs that can animate a specification from the UI, while the published JARs enabled us to perform animation programmatically.
Specifically, the capability to animate any function or operation from the specification was a significant boon as it enabled us to construct focused EACs rather than requiring a full system definition in every instance.
This flexibility, and the maturity of the tooling, enabled us to build the specification adapter iteratively, doing `just enough' to run new EACs.
Having this capability enabled us to be fully agile in our process, and we believe that it made VDM an ideal choice for our project, and for BDS generally.
In \cite{oda2015vdm}, Oda illustrates how animation in VDM can be used to bridge the language gap between that of a project's stakeholders and that of a formal specification.
The EACs used in BDS provide an extra layer of abstraction to provide a single language that can also be used by developers.
Sections 4, 5, and 6 of~\cite{oda2015vdm} demonstrate how animation can be driven interactively through various interfaces, but for our EACs we took an even simpler approach.
Our VDM adapter creates a module that represents the EAC, animates it, and then checks the final returned value to determine success.
The module consists of a single operation that begins with all the declarations made in the {\em given} section of the EAC, followed by the commands corresponding to the {\em whenever} section.
Each check in the {\em then} section is translated to a block that returns \texttt{false} if the check is failed.
If the animation proceeds through all checks successfully then the EAC has passed and \texttt{true} is returned.
\begin{vdmsl}[caption=A translated EAC specification,label=lst:animation,basicstyle=\scriptsize\ttfamily]
AnimateCheck: () ==> bool
AnimateCheck() ==
(
(
dcl formulaA: Formula`Formula := mk_Formula`UnsafeFormula([
DataTypeLiterals`CreateNumberLiteral(0.0),
DataTypeLiterals`CreateNumberLiteral(0.0),
AnaplanFormulaFunctions`POWER_SPARSE_2
], {1 |-> [], 2 |-> [], 3 |-> [1, 2]});
(
dcl f: Formula`Formula := formulaA;
let
expected = ValueOption`Create[Number`Number](0)
in
if not TestEquality`Equals(FormulaModelling`Evaluate(f, { |-> }), expected)
then (
IO`println("* Check 'FormulaEvaluationCheck' failed (tolerant = true)");
IO`print(" Expected: ");
IO`println(expected);
IO`print(" Actual: ");
IO`println(FormulaModelling`Evaluate(f, { |-> }));
return false;
)
else IO`println("* Check 'FormulaEvaluationCheck' passed");
);
);
return true
)
\end{vdmsl}
\noindent
For example, the EAC in listing~\ref{lst:sparse} generates the VDM operation shown in listing~\ref{lst:animation}.
If an EAC fails, it is trivial to open the workspace in Overture and debug the animation of the specification using its rich tooling.
\section{First Section}
\subsection{A Subsection Sample}
Please note that the first paragraph of a section or subsection is
not indented. The first paragraph that follows a table, figure,
equation etc. does not need an indent, either.
Subsequent paragraphs, however, are indented.
\subsubsection{Sample Heading (Third Level)} Only two levels of
headings should be numbered. Lower level headings remain unnumbered;
they are formatted as run-in headings.
\paragraph{Sample Heading (Fourth Level)}
The contribution should contain no more than four levels of
headings. Table~\ref{tab1} gives a summary of all heading levels.
\begin{table}
\caption{Table captions should be placed above the
tables.}\label{tab1}
\begin{tabular}{|l|l|l|}
\hline
Heading level & Example & Font size and style\\
\hline
Title (centered) & {\Large\bfseries Lecture Notes} & 14 point, bold\\
1st-level heading & {\large\bfseries 1 Introduction} & 12 point, bold\\
2nd-level heading & {\bfseries 2.1 Printing Area} & 10 point, bold\\
3rd-level heading & {\bfseries Run-in Heading in Bold.} Text follows & 10 point, bold\\
4th-level heading & {\itshape Lowest Level Heading.} Text follows & 10 point, italic\\
\hline
\end{tabular}
\end{table}
\noindent Displayed equations are centered and set on a separate
line.
\begin{equation}
x + y = z
\end{equation}
Please try to avoid rasterized images for line-art diagrams and
schemas. Whenever possible, use vector graphics instead (see
Fig.~\ref{fig1}).
\begin{figure}
\includegraphics[width=\textwidth]{fig1.eps}
\caption{A figure caption is always placed below the illustration.
Please note that short captions are centered, while long ones are
justified by the macro package automatically.} \label{fig1}
\end{figure}
\begin{theorem}
This is a sample theorem. The run-in heading is set in bold, while
the following text appears in italics. Definitions, lemmas,
propositions, and corollaries are styled the same way.
\end{theorem}
\begin{proof}
Proofs, examples, and remarks have the initial word in italics,
while the following text appears in normal font.
\end{proof}
For citations of references, we prefer the use of square brackets
and consecutive numbers. Citations using labels or the author/year
convention are also acceptable. The following bibliography provides
a sample reference list with entries for journal
articles~\cite{ref_article1}, an LNCS chapter~\cite{ref_lncs1}, a
book~\cite{ref_book1}, proceedings without editors~\cite{ref_proc1},
and a homepage~\cite{ref_url1}. Multiple citations are grouped
\cite{ref_article1,ref_lncs1,ref_book1},
\cite{ref_article1,ref_book1,ref_proc1,ref_url1}.
\section{Future work}\label{subsec:future-work}
We have begun to use the tools developed for BDS to assist with verifying our implementations.
We are constructing a verification framework that generates semi-random scenarios from the instructions available in the DSL and runs them against target systems.
These scenarios are slightly more expressive than those used for validation.
For example, it is possible to use a query to determine the valid points in a cube and then check that the value at each point in the target system satisfies the specification.
Briefly, the verification process proceeds as follows:
\begin{enumerate}
\item Generate a scenario.
\item Check that the {\em given} and {\em whenever} blocks satisfy the specification -- that is, that the scenario satisfies the precondition and that the command make sense.
\item Run the scenario against the implementation in a mode that queries rather than checks.
\item Use the results of the query to generate an equivalent scenario containing a {\em then} block, and run that scenario against the specification.
\end{enumerate}
\noindent
At the time of writing, we are continuing to develop the verification framework, and we are also scaling up its use in checking the correctness of the alternative calculation engine.
Currently, we have only preliminary results, which are not yet ready for publication.
\subsection{A matter of scale}\label{subsec:a-matter-of-scale}
A customer's Anaplan estate will typically consist of a number of connected models.
Each model will usually focus on a specific business problem, allowing detailed analysis, but will also export summarized data to be used in other models.
As mentioned previously, when any change is made within a model the impact of that change will immediately be reflected throughout the model -- akin to making a change in a spreadsheet.
Furthermore, unlike spreadsheets -- which are still commonly used for business planning -- Anaplan models will frequently have billions of cells.
To efficiently maintain the consistency of the Anaplan model, all cube data is held in memory to enable rapid retrieval and calculation.
Larger instances of models can consume more than 700GB of memory to store the cube data alone.
When the memory required to process the data, operate an API, and run the OS is added, it can take a server with 1TB of memory to operate a model.
Although such amounts of memory can be installed in your own data centres, it is at or near the limit of instances offered by cloud hosting solutions.
This is a concern as it can limit options for disaster failover.
Despite the scale of model offered by Anaplan, we are sometimes asked to provide for cubes with trillions of addressable cells.
Often this is due to practices carried over from less flexible, legacy OLAP tools, but sometimes they are due to inherent properties of the scenario being modelled and, upon closer inspection, we find that in the vast majority of these scenarios not all of these addressable cells are `valid'.
For example, an aviation company may wish to measure the time taken to fly between airports and compare it with their competitors.
According to IATA~\cite{iata}, there are approximately 9100 airports and 1100 airlines.
To track the duration of flights between any two airports by day and airline for the period of a year would require \num{3.3e13} cells.
IATA also reports~\cite{iata2} that there are less than 40 million flights made a year\footnote{In fact, many of these flights will be between the same two airports on the same day.}.
In practice, therefore, less than a thousandth of one percent of the addressable cells will ever be populated.
\subsection{An alternative calculation engine}\label{subsec:an-alternative-calculation-engine}
The Anaplan calculation engine was optimised for densely populated models.
Using large chunks of contiguous memory for cube data enables efficient calculation, aggregation, and dependency management through mechanical sympathy with the CPU caches \cite{drepper2007every}.
Although memory may not always be assigned for every addressable cell, the requirement to use contiguous memory would be inefficient for cubes like those in our example.
We decided to begin work on an alternative calculation engine that was optimised for the sparse population of a model.
Here we would only assign memory for populated cells -- enabling larger models -- and make use of sparse computation strategies to retain efficiency.
This calculation engine will be offered alongside the existing engine, enabling a user to choose appropriately for the business case they are modelling.
We would generally expect the user to be able to create the same models with either engine, and for both engines to give the same calculation results.
The calculation engine is an integral component of Anaplan, and in order to introduce another one we needed to refine the interface between the engine and the rest of the platform.
This was a task with risk, as it was critical to neither impact the functionality nor performance of the existing calculation engine.
The approach taken was to completely fork the platform, quickly make changes to the fork to determine best practice and, once established, apply production-quality changes to the mainline.
The new sparse calculation engine would then be developed in parallel and continuously integrated.
This led to having four alternative platforms available for the duration of the project: the fork and mainline platforms, each paired with either the dense or sparse engine.
\subsection{The need for formal methods}\label{subsec:the-need-for-formal-methods}
The aim of the project was to have all four platforms eventually exhibit the same behaviour.
However, there would naturally be some variation throughout the lifetime of the project.
We would be working using the agile methodology~\cite{shore2007art} and the features available in all four alternatives would vary over time and would not be equivalent at most points.
Furthermore, in some instances we wanted to alter the existing behaviour since, like most organically evolving software products, Anaplan is the culmination of design decisions made over many years.
If we were to revisit some of these decisions today, we might choose other options, as now we have more or better data that has invalidated our previous assumptions.
This is particularly true for design decisions that were influenced by the implementation method at the time or which would be made differently for a sparse paradigm.
Typically, these decisions impact performance or efficiency.
For example, consider the value of $0^0$: in Mathematics this is either left
undefined or set by convention to $0^0 = 1$, and indeed the original definition
of the $\operatorname{POWER}$ function in Anaplan follows this convention,
giving $\operatorname{POWER}(0,0) = 1$. This makes sense in a dense context,
where we can expect that the vast majority of points will have a meaningful
user-defined value. On the other hand, in a sparse context most points will not
have a user-defined value and will instead have a default value, which for
numbers is $0$. As a result, the memory consumption would be dramatically
increased if $\operatorname{POWER}(0,0)$ returned $1$, leading to performance
degradation, so for our sparse engine we chose a different convention:
$\operatorname{POWER}(0,0) = 0$.
We needed the means to not only ensure that each system exhibited the same base behaviour, but also to rigorously capture any differences between them.
We had a difficulty here as the platform has been developed using agile principles, and one of the tenets of this methodology is to prefer `working software over comprehensive documentation'~\cite{beck2001agile}.
Like most teams~\cite{wagenaar2018working}, we still possess and value documentation, but the working software is the `source of truth', and we did not have an independent means of benchmarking its correctness.
The decision was made that a specification was needed.
It was necessary that we were able to capture both the existing behaviour and any differences between the alternative platforms.
It was essential that we could do so rigorously and without ambiguity, making the use of formal methods an obvious choice.
However, we also needed the specification to fit into our agile way or working;
it would not be possible to complete the specification before starting work on the new calculation engine, and the platforms would change as the specification was written.
Furthermore, we needed to make use of the specification as soon as possible to enable quick identification and resolution of any discrepancy between the platforms' behaviour.
In summary, we needed to combine the rigour of formal methods and the flexibility of agile methods.
\subsection{Existing work}\label{subsec:related-work}
In the same way that none of the leading relational database systems satisfies Codd's twelve rules~\cite{codd1990relational}, commercial OLAP systems are inconsistent in their mathematical foundations.
Anaplan encourages the use of many small, dense, and distinct cubes that are connected through formulae to form a model.
Some of the cubes explicitly represent inputs to the model and others its outputs, such that dashboards, charts and reports are constructed automatically from those outputs.
Thus, a user does not use the typical OLAP operations, such as roll-up, drill-down, or slice and dice to {\em interact with the cube}.
Instead once a model has been constructed, the user simply makes a change and observes its immediate impact on the output artifacts.
Formalisms have been proposed for a `pure' OLAP approach~\cite{macedo2015linear,lenz2009formal}, but these have focused on the use of operations -- particularly roll-up -- within a single cube;
our specification needed to guarantee consistency across a directed graph of cubes.
Furthermore, Anaplan's connected planning methodology enables the use of multiple models -- where each typically corresponds to an area of a business -- and our specification was also required to define the asynchronous relationships between models.
These differences, and a desire to use the same methods throughout, discouraged us from pursuing the use of an existing OLAP specification.
\subsection{Specifying a behaviour}\label{subsec:specifying-a-behaviour}
The general procedure of BDS has been detailed in section~\ref{subsec:what-is-bds?}, while here we elaborate a little more on the process followed in our project.
The first step of BDS is to identify an AC.
For example, the behaviour that allows adding an entity to the end of a list could be described as `when an entity is added, it is contained by the list'.
We would then write an EAC for this -- as previously seen in listing~\ref{lst:add}.
Here, we had the advantage that we could immediately run the EAC against the dense platforms to validate that we had captured the behaviour correctly.
We would then try to run the EAC against the sparse platforms.
If the required functionality had not yet been implemented, we could ensure that the adapter returned the \emph{unsupported} command.
If the EAC ran and succeeded we could conclude that it has been implemented correctly, but if it ran and failed our next step would be to determine whether the difference was believed to be correct or incorrect.
If the difference was acknowledged to be incorrect, we could raise a bug referencing the EAC and feed it into the appropriate team's backlog.
On the other hand, if the development team believed that they had implemented the correct behaviour, we would raise a ticket in a formal difference backlog.
At this point we would mark the EAC as `under review' until the difference was discussed by a working group at a fortnightly meeting, where all the stakeholders would discuss differences and make an informed decision on whether the proposals should be accepted or rejected\footnote{Note that, as this was an agile process, there was always a possibility that a future discovery could invalidate the decision, and members of the team were free to reopen the discussion and present new evidence to the working group before a new decision was made.}.
Once the difference was resolved, it would be necessary to return to the EAC and ensure if reflected the decision made.
The specifier then runs the EAC against the specification to ensure it fails, the EAC is marked as unsupported by the specification, and the author pushes the work to a branch and raises a pull request (PR) for review by another team member.
Once the review of the PR is complete -- and all necessary builds have passed -- the EAC is merged to the mainline, and specification can proceed.
There is not too much to say about writing the specification itself, other than to reinforce that the only changes should be those required to make the EAC pass, and that the presence of EACs does not divest the author of the responsibility to write units tests within the VDM, which should focus on testing the correctness of each individual function, as well as its pre and post conditions.
Having the EACs in place should discourage the specifier from writing end-to-end tests directly in the VDM, and instead ensure the focus is on good coverage of the specification.
\subsection{Continuous validation}\label{subsec:continuous-validation}
An important part of agility is continually integrating the pieces of work from all parts of the team to ensure that changes do not cause problems elsewhere.
The Gradle plugin introduced in~\cite{fraser2018integrating} enables us to build the specification, run all the VDM unit tests in Jenkins, and publish the specification to Artifactory.
Change to the DSL and its associated adapters are managed like any other agile development work -- with frequent commits, PRs, and adoption upstream.
The implementation adapters leverage Java's service loading mechanism to use whichever version of the implementation they find on their classpath, while the VDM adapter was necessarily tied to a specific version of the specification, since part of its function was to package the text-based specification into a Java library that could be `run'.
The EACs are all placed into a single validation repository, and whenever the mainline of this project builds successfully, it also checks for newer versions of the DSL, the adapters and the implementations.
If a new version is found, the system automatically raises a PR to update the version numbers of the dependencies, and if this PR builds successfully, it is considered a team priority to merge it quickly.
On occasion, a version update PR will fail, typically due to either a regression of behaviour in an implementation, or because some previously unsupported action has been implemented, causing an EAC that was previously being skipped to fail.
At this point we follow the same process that we use when a new EAC fails, as described previously.
Again, resolving this difference is considered a team priority.
As with any agile development team, team members will work on a failing mainline or version update build -- rather than regular assigned work -- until it is fixed.
\section{Introduction}\label{sec:introduction}
\input{introduction}
\section{Background}\label{sec:background}
\input{background}
\section{Behaviour driven specification}\label{sec:behaviourDrivenSpec}
\input{specification}
\section{Integrating into the Agile Process}
\input{agile-process}
\section{Results}
\input{results}
\section{Concluding remarks}\label{sec:concludingRemarks}
\input{concluding-remarks}
\section{Injection Moulding Machine Model in VDM-RT}
This document describes the model of an injection moulding machine. The purpose of the injection moulding machine is to produce plastic parts within specifications. The injection moulding machine contains plastic granulate that is heated, injected into a given shape, and solidified to yield the final part. For obtaining high quality plastic parts, it is critical to ensure that the right amount of melted plastic is injected.
For this, temperature, pressure and screw position are machine parameters to control during the injection process. The execution of the injection cycle needs to be carried out in a specific timely manner.
The injection moulding system consists of:
\begin{itemize}
\item an motor to move the screw forward and backward
\item heaterbands to add additional energy if the temperature in the barrel is too low
\item a pressure sensor to determine the pressure in front of the screw
\item a position sensor to determine the position of the screw
\item a temperature sensor to determine the temperature
\item a mould that needs to be closed during injection and open to eject the solidified plastic part
\item a software controller for safe execution of the injection cycle and quality elements
\end{itemize}
Screw Position (real), 0 to some maximum number
Heaterband <ON> | <OFF>
Mould <OPEN> | <CLOSE>
The controller must ensure that the injection cycle is executed in a timely manner while maintaining the correct temperature, pressure and screw position.
The controller is connected to sensors and actuators that enable the controller to:
\begin{itemize}
\item monitor temperature
\item monitor pressure
\item monitor screw position
\item monitor the status of the mould
\item position the screw (which increases/decreases pressure)
\item
\end{itemize}
The controller has three modes:
\begin{itemize}
\item Initialisation Mode
\item Production Mode
\item Emergency stop mode
\end{itemize}
In Initialisation Mode the controller increases the temperature, opens the mould, and moves the screw backwards.
In Production Mode the controller runs through a cyclic process:
\begin{itemize}
\item Filling: screw moves forward to inject the plastic and fill the mould up to 95\% (position controlled)
\item Packing: screw moves forward to fill the remaining 5\% and achieve a specific pressure (pressure controlled)
\item Cooling: wait for plastic to solidify
\item Dosing: screw moves backward to melt material for the next filling
\item Ejecting: mould opens, element is pushed out of the mould, mould closes
\end{itemize}
In Emergency Stop Mode the controller terminates and hands over responsibility to the physical environment.
Before beginning with the actual modelling process, it is advised to have a clear purpose for the model. Thereafter, abstraction at an appropriate level can be applied to capture relevant characteristics. For defining the purpose of the model, we describe use cases identifying all actors interacting with the system.
Differentiate between closed loop process control and open loop phase control
Machine components and parameter
\textbf{Temperature control of the barrel and at the nozzel}
The temperature control monitors the actual temperature in the different heating zones and sends control signal to actuators. The electrical resistance heater bands are turned on and off using semi conductor relays for a specific amount of time. Commonly the heating power of the injection machine is given in percentage e.g. 50\% heating power equal to power supply to the heating bands for half the time.
$P_{heat} = U \cdot I = R \cdot I^{2}$
$W_{heat} = \frac{U^2}{R} \cdot t$
\textbf{Injection actuator}
To begin the injection phase the open loop phase control sents a signal to the frequency converter which in turn controls the servomotor. The frequency converter controls the motor based on the set injection speed. The pressure control behind the screw has only a monitoring function in this phase. Position is feedback from the servomotor. Pressure is feedback between the gear box and the screw.
$T = \frac{P}{\omega}$ where $\omega = \frac{2 \pi n}{60}$
\textbf{Plastification actuator}
During plastification the screw rotation is controlled by feedback of the position through a frequency converter, controlling a servo motor which is connected to the screw through a gear box. The set value given from the operator to the machine is screw rotation speed. Based on the screw rotation speed the torque of the motor is controlled. Plastification stops as soon as the shot volume is achieved.
\textbf{Pressure and force measurement}
Usually injection pressure and force are measured between the screw shaft and the actuation unit of the screw. Also, the pressure during plastification is measured in front of the screw. The process parameters that are measured include injection pressure, pack and holding pressure, and back pressure.
\subsection{DT model}
The overall purpose of this VDM(++/rt) model is to describe the (high-level) control of the injection moulding process.
R1: The model to be developed should describe the 6 states of the injection moulding cycle.
R2:
\subsubsection{The Environment Class}
The \textit{Environment} class contains elements of the IMM that may be replaced by CT models. It is used to mimic the behaviour of the CT world in DE domain. External inputs to the IMM include user settings for injection job, button press to exchange of mould, button press for initialization i.e. preheat, button press for start of injection cycle, emergency stop key, on/off switch.
The environment class defines a set of reserved names for the external inputs, and an instance variable to map the input names to a sequence of readings related to points in the approximated curve (see pg. 173, Co-sim book).
A \textit{Step} operation computes the new state of the environment in periodic intervals dt.
Further, the environment class contains a \textit{mainLoop} operation. This operation determines the system time and updates the environment model reflecting the causal relationship between the external inputs and elements of the IMM.
\subsection{Sensors and Actuators}
\subsection{Controller Class}
\textbf{Controller States}
The different states of the controller:
PowerOff
PowerOn
Initialize
Idle
Manual vs Operate
And EmergencyStop
\subsection{Safety Unit Class}
e.g. endstop protection, detect if plastic elements stuck in mould
\subsection{CT model}
To transition to a Co-model, we replace the environment class by the CT model.
The CT model for the IMM describes the motion of the screw using a 1st order ordinary differential equation during filling phase (velocity controlled):
\begin{equation}
\tau_1 \diff{v}{t} + v = K_1 v_f
\end{equation}
where $v_f$ describes the velocity trajectory, and $tau_1$, $K_1$ are constants.
Using Newton's Second Law the hydraulic pressure is found:
\begin{equation}
\diff{v}{t} = \frac{\sum F}{M} = \frac{P_{hyd}A_{hyd} - P_{brl}A_{brl} - C_f sign(v) - B_f v}{M}
\end{equation}
Polymer flows due to the displacement of the screw and the change in pressure in the antechamber (in front of the screw). The pressure in the antechamber $P_{noz}$ equals the barrel pressure $P_{brl}$ during fill and is found using the continuity equation:
\begin{equation}
\diff{P_{noz}}{V_{brl}} = \frac{K_p}{V_{brl}}(A_{brl}\diff{x}{t}-Q_1)
\end{equation}
The volume of polymer in the barrel $V_{brl}$ is equal to the initial volume $V_{brl,0}$ subtracted the volume displaced by the screw motion:
\begin{equation}
V_{brl} = V_{brl,0} - A_{brl}x
\end{equation}
The bulk modulus $K_p$, describes the resistance of the polymer to compression and is expressed using Tait equation of state (assuming constant temperature):
\begin{equation}
K_p = \frac{P_{noz} + B_T}{C}(1 - C \log(1+\frac{P_{noz}}{B_T}))
\end{equation}
To derive the polymer flow from the nozzle into the mould cavity the momentum equation is used assuming laminar flow and incompressible fluid in a cylindrical conduit:
\begin{equation}
\diff{Q_1}{t} \frac{P_{noz} - P_{flowfront} - \sum \frac{F_{s,i}}{A_{c,i}}}{\rho_{avg} \sum \frac{H_{L,i}}{A_{c,i}}}
\end{equation}
where $F_S,i$ is the resistiv shear force in section i, and $A_{c,i}$ is the surface area in section i. $\rho_{avg}$ is found by evaluating Tait equation of state at $T_0$ and mean nozzle pressure $P_{noz}/2$.
To find the resistiv shear force, Cross equation is used.
Switch-over from velocity controlled filling to pressure controlled packing happens when the total volume of the mould equals to the injection plastic volume:
\begin{equation}
V_{tot} = \sum V_i
\end{equation}
\textbf{Packing}
During packing the hydraulic pressure equals:
\begin{equation}
tau_2 \diff{P_{hyd}}{t} + P_{hyd} = K_2 P_{hold}
\end{equation}
During packing the assumption that the polymer is incompressible does not hold true. Thus, continuity and momentum equation need to be evaluated individually for each section.
The polymer density for section i:
\begin{equation}
\diff{\rho_i}{t} = \frac{\rho{i-1}Q_i - \rho_i Q_{i+1}}{V_i}
\end{equation}
and the flow:
\begin{equation}
\diff{Q_i}{t} = (P_{i-1} - P_i - \frac{1}{2} \sum \frac{F_{s,i}}{A_{c,i}}) ( \frac{\rho_{i-1}}{2} \sum \frac{H_{L,i}}{A_ {c,i}})^{-1}
\end{equation}
\textbf{Cooling}
The melt temperature during filling is assumed to be constant.
During packing the temperature of the melt starts cooling assuming perpendicular heat loss to the mould surface and no shear heating the temperature change in section i equals:
\begin{equation}
\diff{T_i}{t} = - \frac{U_{o,i}A_{s,i}(T_i - T_{mold})}{\rho_i V_i C_{p,i}}
\end{equation}
with the heat transfer coefficient $U_{o,i}$
\begin{equation}
U_{o,i} = (\frac{1}{h_c} + \frac{D_{h,i}/2}{k})^{-1}
\end{equation}
The heat capacitance $C_{p,i}$ is assumed variable:
\begin{equation}
C_{p,i} =
\begin{cases}
c_{1,u} + c_{2,u} \bar{T} + c_{3,u} e^{-c_{4,u}\bar{T_i}^2} & \text{if $\bar{T_i} \geq 0$} \\
c_{1,l} + c_{2,l} \bar{T} + c_{3,l} e^{-c_{4,l}\bar{T_i}^2} &\text{if $\bar{T_i} < 0$}
\end{cases}
\end{equation}
where $\bar{T_i} = T_i - c_5$
The fraction of solidified polymer is estimated as
\begin{equation}
\phi_{s,i} = \frac{\int C_{p,i} d\bar{T}}{\int C_{p,i}d\bar{T}}
\end{equation}
with the integral limits from $\bar{T_i}$ to $T_m = 10K$.
The solidification of the polymer changes the hydraulic diameter, the cross-sectional area, and the surface area in section i as:
\begin{equation}
D_{H,i} = D_{H0,i} \sqrt{1-\phi_{s,i}}
\end{equation}
\begin{equation}
A_{c,i} = \frac{\pi D_{H,i}^2}{4}
\end{equation}
\begin{equation}
A_{s,i} = \pi D_{H,i} H _{L,i}
\end{equation}
The pressure in the ith conduit equals:
\begin{equation}
P_i = P_{i,Tait}(1- \phi{s,i})
\end{equation}
\subsection{Co-simulation approach}
\section{Injection Moulding Process}
\label{sec:immp}
The typical aspect of a horizontal IMM used in a manufacturing setting is shown in Fig.~\ref{fig:my_label}. The machine consists of a workbench enclosed by panels and a gate used by operators to access the clamping unit, which alongside the injection unit sits at the top of the workbench. The two units are the main components involved in the production of plastic parts that are ejected from the mould inside the clamping unit to a side bucket below the gate. Modern machines may use a conveyor belt situated under the clamping unit to collect and route the piece to a quality control check and automatically scrap non-compliant pieces.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{Content/Figures/IMM_door.png}
\caption{3D representation of an injection moulding machine with the gate highlighted in blue. Below the gate, the part is ejected from the machine and passed onto the belt.}
\label{fig:my_label}
\end{figure}
The machine controller and other components are packaged under the workbench with sensors and actuators connected to different parts of the machine. For example, given that the gate near the clamping unit must be locked during operation to avoid damage and safety hazards to the operators, it is common to have a sensor that gauges the state of the gate (open/closed) and feed it to the controller among other interlocking mechanisms. This is one of the first properties (a safety property) that any IMM controller must satisfy:
\begin{prop}
The machine must be idle when the gate is open.
\end{prop}
\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{Content/Figures/Injection_moulding.png}
\caption{Schematics of the components of an IMM}
\label{fig:IMM}
\end{figure}
In Fig.~\ref{fig:IMM} we consider the main components of a single screw, hybrid IMM composed of a clamping unit and an injection unit. The units are further split into actuators.
The clamping unit consists of an actuator and a control system to close the mould that provides the clamping pressure and ejects the part.
The injection unit consists of various components - the barrel, the screw, the hopper, heat bands, the nozzle - and an electrical and hydraulic actuator, for plastification and forcing the polymer into the mould cavities respectively.
\subsection{The Injection Moulding Process}
The IM process is cyclic and consists of four states which are associated with the activation of the different machine tool components and amenable to an automaton representation as shown in Figure \ref{fig:automaton}. At the beginning of the process, the screw tip is assumed to be at a position of $x=0$. At this point, the barrel is filled with molten polymer material, and the mould is closed in the clamping unit. At the start of the process, the machine is in the Filling state and is controlled based on the screw velocity. Hydraulic pressure is applied to the screw to achieve the set screw velocity. After the screw reaches a certain position, $x_3$, or the expected time, ($tfill$), the filling phase is complete. Subsequently, the machine transitions to the Packing phase and switches from velocity to pressure control. For the Packing, Cooling/Classification, and Ejection phases, the state transition is time controlled. After the Ejection phase, the machine becomes Idle and a new cycle may start. In the following paragraphs, we describe the different phases of the IM process and some of the properties that one can check or assure in the development of a VDM model.
\begin{figure}
\centering
\includegraphics[width=0.95\textwidth]{Content/Figures/automaton.pdf}
\caption{States and phase transitions of an IM process depicted as an automaton}
\label{fig:automaton}
\end{figure}
\paragraph{Injection/Filling Phase:} The material is filled into the mould cavity (form) under high pressure and speed. During injection, the material starts to solidify. In this phase, the screw is position/velocity controlled, and deviations from the final position reached by the screw are used as an input to the quality control.
For example, an undershoot of the screw position means a lack of material in the mould cavity, thus a defect in the part being produced. This is the second property (the first on quality control) we illustrate about our IMM controller model:
\begin{prop}
The screw must reach EndPositionFill. For the scenario in this paper, EndPositionFill equals 20.3 mm.
\end{prop}
\paragraph{Packing Phase:} Based on a pre-defined position, the machine control switches to the packing phase. The material inside the cavity is still flowable along the cross section and it is possible to influence the final part weight e.g. to compensate for shrinkage. During packing, the remaining material is further compressed into the cavity at constant pressure. At the end of the packing phase, the gate freezes off such that no more material can flow into or out of the mould cavity.
Excessive and insufficient packing pressure leads to various defects of the moulded part such as mould sticking and empty holes, respectively.
Therefore, we monitor the values achieved and check if they are inside a target interval. This is the third property (the second on quality control) we illustrate about our IMM controller model:
\begin{prop}
The hydraulic pressure peak must occur at PeakTimeIntervall and be PeakPressureBand. For the scenario this paper, these are set to $0.9 < t < 1.1$ s and $9 \pm 1$ MPa.
\end{prop}
\paragraph{Cooling Phase \& Plastification Phase:}
After the packing and holding phases, cooling is applied through the coolant channels of the mould to lower the temperature and completely harden the material. In parallel, a new charge of material for the next shot is heated, mixed (homogenised) and transported by screw backward rotation. The transition to the ejection phase is specified by a defined cooling time.
\paragraph{Ejection phase:} The mould opens, the solidified part is ejected, and the mould is closed. The final part is transferred via a belt to a storage box. A quality controller automatically rejects parts that do not meet defined standards based on monitoring relevant process variables.
This concludes our brief presentation of an IMM and the IM process. For more details about this application domain, we refer the readers to the work of \cite{KULKARNI2017,Woll&97}. In terms of the process, we will provide more details on the continuous behaviour of the machine in Section \ref{sec:validation}, where we introduce the continuous time model of screw motion and injection process.
\subsubsection*{Appendix:}
Our work in progress (Overture Project) is available at: \url{https://tinyurl.com/VDMIMM}.
\section{An Injection Moulding Machine Model in VDM-RT}
\label{sec:model}
In our idealised IMM, we conceive a computer/controller receiving readings from an open/close sensor on the state of the gate, a position sensor receiving the $x$ position of the screw tip, and a pressure sensor receiving the nozzle pressure readings. The controller affects the system by issuing screw tip velocity target setpoints to the injection unit local controller, issuing commands to the belt controller on whether to accept or scrap a part, and performs PD control of an electric drive returning the screw tip to the 0-position during the Cooling phase.
Fig. \ref{fig:UML} visualises the design of the IMM which includes (1) \texttt{Actuator}, (2) \texttt{Belt}, (3) \texttt{Controller}, \texttt{(4) Environment}, (5) \texttt{GateSensor}, (6) \texttt{IMM\_System}, (7) \texttt {InjectionMachine}, (8) \texttt{MotorActuator}, (9) \texttt{PositionSensor},
\noindent
(10) \texttt{PressureSensor} and (11) \texttt{Timer}. All components are represented as VDM-RT files. Besides, the model receives a scenario input and returns the simulation output as CSV files. Below, we highlight various relevant aspects of the model components.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{Content/Figures/model-structure.pdf}
\caption{UML class diagram of the VDM-RT model for an Injection moulding machine controller. The red dotted rectangle indicates physical components of the machine tool (sensors, actuators and environment). In addition, the VDM model includes World, System and Controller classes.}
\label{fig:UML}
\end{figure}
\subsection{World, InjectionMachine and Environment Class}
Today, injection moulding machines integrate complex control and software systems including closed-loop process, open-loop phase,
safety and quality control. We developed an abstract model of the control architecture of an injection moulding machine.
This model was developed in Overture version 3.0.0. We selected the VDM-RT dialect to implement the process/machine controller,
quality controller and data acquisition system as a distributed control architecture.
The \texttt{World} is the top-level entry point to the model. A scenario is input from a CSV file containing time steps, screw position,
hydraulic pressure and the order to start production of a new plastic element. The logger operation returns a CSV file after each step of the simulation.
In our current model, the \texttt{InjectionMachine} models the continuous behaviour of the injection moulding machine. The \texttt{World} controls the simulation. Whenever time advances by calling the corresponding world functionality, the current screw position and pressure are set in the \texttt{InjectionMachine}. The Step functionality in the \texttt{InjectionMachine} sets the sensor readings for both the Position and Pressure sensor. In turn, the \texttt{Controller} can only access the sensor readings to update the current believed state of the system. This separation allows investigating problems arising from a controller being out of synchronization, i.e. sampling slower than needed.
The \texttt{Environment} describes the interaction of the injection moulding machine with the surrounding environment by receiving the element's output from the injection moulding process.
\subsection{Example of Safety Analysis}
One feature of our controller is a safety switch coupled to the gate. Our model uses an invariant to check whether the gate is closed during production. Inside the Controller class, we defined
the StateType and LockState reflecting the state of the machine and state of the gate respectively.
\begin{vdmpp}[style=VDM, label={lst:safety}, caption={Highlights from the Controller class}]
types
StateType = <Filling> | <Packing> | <Cooling> | <Ejecting> | <Idle>;
LockState = <Open> | <Close>;
\end{vdmpp}
The current state of the machine operation is kept track of by updating the $ctl\_state$ variable, and the state of the gate is fed to the lock variable. Accordingly, safety is verified by invoking a function that checks at no point the door is open and the machine is not idle.
\begin{vdmpp}[style=VDM, label={lst:instancevars}, caption={Instance variables used to assert safety}]
instance variables
ctl_state : StateType;
lock : LockState;
inv InterlockInvariant(lock,ctl_state)
functions
InterlockInvariant : LockState * StateType -> bool
InterlockInvariant (ls,s) == (ls = <Open>) => (s = <Idle>)
\end{vdmpp}
\subsection{Example of Process Control}
The execution of the sequential open-loop process control in injection moulding is modelled by the Step function in the \texttt{Controller}. The ctl\_state instance variable represents the current process stage. For each phase, the controller logic is unique and specified in the relevant operation - FillStep, PackStep, CoolStep and EjectStep.
\begin{vdmpp}[style=VDM, label={lst:stages}, caption={Different IMM operation stages}]
public Step: () ==> ()
Step() == (
cases ctl_state:
<Idle> -> return,
<Filling> -> FillStep(),
<Packing> -> PackStep(),
<Cooling> -> CoolStep(),
<Ejecting> -> EjectStep() end;);
\end{vdmpp}
The FillStep operation implements the behaviour of the controller during the filling phase. With the molten plastic in the barrel and the screw tip at position 0, the screw starts rotating to push the melt into the mould. The rotation is controlled according to the desired velocity (setpoint) of the horizontal displacement of the screw tip. The velocity setpoints, v1, v2 and v3, were obtained from the work of \cite{Woll&97} and they are set according to positions also defined in that work for x1, x2 and x3. In Listing \ref{lst:fill}, we depict the specification of the setpoint control switching based on a reading of the screw tip position sensor, and after achieving the desired position, the machine switches to PackStep by using a variable assignment function Delta.
\begin{vdmpp}[style=VDM, label={lst:fill}, caption={Screw velocity setpoint assignment based on current position}]
public FillStep : () ==> ()
FillStep() == (
-- Fetch sensor reading
ctl_x := IMM_System`ps.getReading();
-- Run control flow
if (ctl_x < x1)
then (IMM_System`imm.v_t := v1)
elseif (ctl_x < x2)
then (IMM_System`imm.v_t := v2)
elseif (ctl_x < x3)
then (IMM_System`imm.v_t := v3)
-- Transition to Packing
else Delta() )
pre ctl_state = <Filling>;
\end{vdmpp}
The CoolStep operation implements the behaviour of the controller during the cooling phase. With the molten plastic solidified, the screw starts rotating backwards to prepare for the next injection cycle. The screw should reach a specified position to prepare exactly the amount of plastic required to fill the mould completely. This behaviour is modelled as a PD controller using the position error to set the target velocity of the screw.
\begin{vdmpp}[style=VDM, label={lst:cool}, caption={Return to initial state of the screw using PD control}]
public CoolStep : () ==> ()
CoolStep() == (
dcl t : real := IMM_System`timer.GetTime();
dcl dt : real := (t - last_time);
last_time := t;
ctl_x_last := ctl_x;
ctl_x := IMM_System`ps.getReading();
e_ti_last := e_ti;
e_ti := 0 - ctl_x_last;
e'_ti := (e_ti - e_ti_last)/dt;
IMM_System`imm.v_t := K_p * e_ti + K_d * e'_ti;
if (IMM_System`timer.GetTime()-tphase >= t_cool)
then Delta(); )
pre ctl_state = <Cooling>
post if (ctl_state = <Ejecting>) then abs (e_ti - 0) < 0.1;
\end{vdmpp}
\subsection{Example of Quality Control}
During the filling phase, constant screw velocity and reaching the screw setpoints are key to achieving a stable process. During the packing phase, the peak injection pressure dictates quality defects of the injection moulded part. Quality control monitors relevant machine parameters and automatically differentiates between good and bad quality parts.
\begin{vdmpp}[style=VDM, label={lst:eject}, caption={Return to initial state of the screw using PD control}]
public EjectStep : () ==> ()
EjectStep() == (
let success : bool = PartIsOK()
in if (success) then
ctl_bin := <OK>
else
ctl_bin := <Scrap>;
pre ctl_state = <Ejecting>;
\end{vdmpp}
In VDM it is easy to define the quality control by expressing a logical formula over the historical values of the x position of the screw and the hydraulic peak value inside the interval. The Boolean variable, \verb'hyd_peak_ok', is set to false in cases where the pressure is above the threshold outside the right interval.
\begin{vdmpp}[style=VDM, label={lst:PartIsOK}, caption={Part of the quality control logic defined in the VDM specification}]
pure PartIsOK : () ==> bool
PartIsOK () == return (ctl_x_max >= 20.3) and hyd_peak_ok and (hyd_peak_max >= 9.1);
\end{vdmpp}
\section{Related Work}
\label{sec:related}
In this section, we present a brief review of related work. We follow a top-down approach, where we list the works about Digital Twins followed by models of IMM. Finally, we delve into
modelling examples in VDM and cover relevant works.
The authors of \cite{Bibow&20} use a Domain Specific Language for specifying events, connecting to an injection moulding machine, and automate the otherwise time-consuming setup of new production jobs. Our work uses standard VDM to specify the behaviour of a controller for an IMM and validates it with a continuous-time model developed in MATLAB. Although our work towards a Digital Twin setup is in progress, we are still far from Hardware In the Loop (HIL) development.
Autonomous vehicles are also an area where Digital Twins bring added value. The work in \cite{LumerKlabbers&21} presents a framework resourcing to some
common technologies that we use, as the standard VDM supporting IDEs \cite{Rask&21} are part of the INTO-CPS Application \cite{Macedo&19b} ecosystem. Our work has more emphasis on the specification of the controller and is less oriented towards HIL, as both a desktop model of a commercial IMM or the usage of one is less feasible than the kind of research platform we are comparing ourselves with.
In the following, we provide a small account of scientific works on models of injection moulding and IMMs. The works are listed chronologically.
The only executable model we found is a MATLAB\footnote{\url{https://se.mathworks.com/help/physmod/hydro/ug/injection-molding-actuation-system.html}}
example based on a machine design from 1989. Contrary to ours, the model relies on purely hydraulics actuation designs. New machines are now full Cyber-Physical Systems with several computation units inside and the actuation is now done using electric pumps and drives.
The model in \cite{Woll&97} is used as a key reference in our work. In their work, the authors lay a precise mathematical formulation of the behaviour of an IMM, yet the authors do not model the controller logic, nor repeated iterations. In our work, we have developed a MATLAB implementation and use it to derive test cases/scenarios to our VDM model.
A step towards a co-simulation model of parts of an injection moulding machine is taken in \cite{Hostert&05}. The dynamic model of the clamping unit combines a multi-body simulation of the mechanical structure (ADAMS/ANSYS), a hydraulic model (DSHplus), and a control model (MATLAB/Simulink). The model allows testing of interactions between sub-components and analysis with varying design parameters, but overlooks the injection unit and interactions of machine and process parameters. Our work focuses on the injection unit and the influence of machine parameters on the process flow.
The authors of \cite{Steinegger&13} present standardised (according to IEC 61499) function blocks for typical faults in manufacturing systems that decouple fault handling and control code and reduce the complexity of the control architecture. The methodology is evaluated on an injection moulding machine.
The usage of VDM modelling to develop incremental models of CPS systems span inceptional cases such as \cite{Macedo&08} or \cite{MacedoNL2019}, to extensive guidelines in the form of complete books \cite{Fitzgerald&13a}.
VDM has been enabled with recent tooling support in Visual Studio Code \cite{Rask&20,Rask&21} and together with the INTO-CPS application and the Functional Mock-up Interface-based co-simulation, it may become a valuable tool to build Digital Twins \cite{Fitzgerald&19}.
\section{Conclusion and Future Work}
\label{sec:conclusion}
As of today, various models of selected IMM components, associated processes, and controllers have been developed. Nevertheless, none of the models covers all aspects of an injection moulding process. Also, models are seldom open-source or available in the public domain.
This paper develops a high-level description of the control architecture of an injection moulding machine in VDM-RT. We selected VDM as it is traditionally used to specify the discrete part of a CPS. The developed functionality includes aspects of the process, quality and safety control.
We also set up a machine-process model based on \cite{Woll&97} and conduct a first validation of the control functionality.
To validate the model, we generated three scenarios that were used as CSV inputs to the model. Our validation of the model shows the discrete event controller model behaves as expected in the scenarios. The model intends to provide controls to the CT model and not match its behaviour, so we decided to not use the results immediately as a measure of compliance. Our goal is to test the VDM model using data generated from the CT model. We cross-check the results by observing the response as a plot of pressure and position/velocity and the outcome is following our expectation.
The resulting model is a first step towards a co-simulation model that is applicable to control an injection moulding machine tool in a Digital Twin setting. Currently, we are unable to accurately point towards the effective benefits beyond the theoretical model-based design advantages. The modelling activity is in its initial stages and more research is needed to couple the model with a physical system.
Next steps include enriching the models, refining the VDM model and adding testing (e.g., combinatorial tests) and at a future stage towards a Digital Twin for an IMM, we plan to interface the VDM model in a co-simulation environment.
\section{Model Validation}
\label{sec:validation}
In this section, we describe the steps that were followed to perform a first check of the suitability of the model. We cross-check the output behaviour of the model execution while running it against scenarios simulating the events/inputs to the system sensors at each time instant. To generate the scenarios, we have implemented a continuous model of the expected behaviour of an IMM in MATLAB and exported the events to a CSV file that is in turn read by the VDM model.
\subsection{Continuous Time Model}
We follow the work by \cite{Woll&97}, where the authors develop a first-principles model of an IMM linking the controllable machine parameters (e.g., the motion of the screw) to measurable process variables (e.g. injection pressure). Accordingly, we only introduce the equations containing the parameters used to generate the scenarios for validating our VDM model. For a comprehensive overview of the first-principles model, we refer the reader to \cite{Woll&97}.
The model describes the injection process based on a set of selected equations for the Filling, Packing and Cooling phases. The set of equations consists of Newtons Second Law and thermodynamic relationships such as the equations of continuity, momentum and energy. In each phase the equations are solved for hydraulic pressure, nozzle pressure, ram velocity, polymer flow, and cavity pressure.
During the Filling phase, Newton's Second Law is used for describing the velocity of the screw using first order dynamic equations. As shown in Fig. \ref{fig:ramvel}, the screw velocity is changed at positions $x_1, x_2, x_3$ to the associated velocity setpoints $v_1, v_2, v_3$.
The screw velocity $v$ is described by
\begin{equation}\label{6-Bottjer:eq:1order_fill}
\tau_1 \diff{v}{t} + v = K_1 v_f
\end{equation}
where $v_f$ describes the velocity trajectory, and $\tau_1$, $K_1$ are constants.
Using Newton's Second Law the hydraulic pressure is found (compare Fig. \ref{fig:hp})
\begin{equation}\label{6-Bottjer:eq:Newton}
\diff{v}{t} = \frac{\sum F}{M} = \frac{P_{hyd}A_{hyd} - P_{brl}A_{brl} - C_f sign(v) - B_f v}{M}
\end{equation}
where hydraulic pressure is applied to the screw, and barrel pressure, $P_{brl}A_{brl}$, Coulomb friction, $C_f sign(v)$, and viscous damping, $B_f v$, resist the motion of the screw. $M$ is the combined mass of the actuators.
During the Packing phase, the process switches to pressure-based control to reach holding pressure, $P_{hold}$. Eq. \ref{eq:1order_fill} is replaced by
\begin{equation}
\tau_2 \diff{P_{hyd}}{t} + P_{hyd} = K_2 P_{hold}
\end{equation}
describing the transition of hydraulic pressure to the holding pressure, $P_{hold}$. Eq. \ref{eq:Newton} is solved for describing the deceleration of the screw. Fig. \ref{fig:ct_out} shows the pressure response and screw velocity as outputted from the continuous model.
\begin{figure}
\centering
\subfloat[Combined nozzle pressure response (MPa) for Packing, Filling, and Cooling Phase\label{fig:hp}.]{\includegraphics[width=5.75cm]{Content/Figures/RAMVel_upd.png}}
\hfill
\subfloat[Ram velocity (mm/s) with the three velocity set points $v_1, v_2, v_3$ for the transition positions $x_1, x_2, x_3$\label{fig:ramvel}.]{\includegraphics[width=5.75cm]{Content/Figures/HP.png}}
\hfill
\caption{Output generated by the continuous time model. These plots were obtained from our MATLAB implementation of \cite{Woll&97}'s model}
\label{fig:ct_out}
\end{figure}
\begin{comment}
\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{Content/Figures/RAMVel_upd.png}
\caption{Ram velocity (mm/s) with the three velocity set points $v_1, v_2, v_3$ for the transition positions $x_1, x_2, x_3$\label{fig:ramvel}. These plots were obtained from our MATLAB implementation of \cite{Woll&97}'s model}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{Content/Figures/HP.png}
\caption{Combined nozzle pressure response (MPa) for Packing, Filling, and Cooling Phase\label{fig:hp}. These plots were obtained from our MATLAB implementation of \cite{Woll&97}'s model}
\end{figure}
\end{comment}
\begin{comment}
Polymer flows due to the displacement of the screw and the change in pressure in the antechamber (in front of the screw). The pressure in the antechamber $P_{noz}$ equals the barrel pressure $P_{brl}$ during Filling and is found using the continuity equation
\begin{equation}
\diff{P_{noz}}{V_{brl}} = \frac{K_p}{V_{brl}}(A_{brl}\diff{x}{t}-Q_1)
\end{equation}
Further, the volume of polymer in the barrel $V_{brl}$ is equal to the initial volume $V_{brl,0}$ subtracted the volume displaced by the screw motion
\begin{equation}
V_{brl} = V_{brl,0} - A_{brl}x
\end{equation}
\end{comment}
\subsection{Extracting Scenarios from the Continuous Time Model}
We validated the model using three scenarios generated using the physical model of an injection moulding machine developed in MATLAB.
Fig. \ref{fig:profiles} shows the hydraulic pressure during actuation of the injection ram for three different scenarios. Scenario A) uses the original parameters introduced in \cite{Woll&97}, while Scenario B) and Scenario C) simulate an injection process where the position setpoint, $x_2$, is not reached or overshoot, respectively.
Slight variations in the process conditions during the injection cycle can lead to bad quality parts. Variation in hydraulic pressure can cause quality issues with the polymer part. Hydraulic pressure above the pre-set value lead excessive polymer flow into the mould. This causes a lid surrounding the injection part at the parting line of the mould halves. Lower pressure than the pre-set value leads to underfilling of the mould cavities and causes incomplete parts. We varied the hydraulic pressure, a controllable variable actuating the injection screw during the Filling phase, for validating the main functionality of our VDM model that are:
\begin{enumerate}
\item switching of the injection phases at defined conditions
\item setting of the velocity set points during filling
\item returning of the screw in the cooling phase, i.e. plasticising new polymer for the next injection cycle
\item the auto scrap control responsible for detecting quality issues from process data and automatically reject corresponding elements
\end{enumerate}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{Content/Figures/profiles.png}
\caption{Three inputs to the VDM model that were generated using a deterministic injection moulding machine model. The hydraulic pressure is shown for A) optimal operating conditions (x2 = 15.2 mm, the same behaviour as in Fig. \ref{fig:ct_out}), B) the second position set point is not reached (x2 = 9 mm), and C) second position is overshoot (x2 = 20 mm).}
\label{fig:profiles}
\end{figure}
\section{Introduction}
Since its invention in the late 19th century, the Injection Moulding Machine (IMM) has changed the plastics manufacturing industry. At the same time, the IMM design has evolved from a purely mechanical process to a full cyber-physical system \cite{Bibow&20} that utilise a high degree of computerisation to add smart features to the process.
The development of digital models is an essential step in the process of adding smart features to a cyber-physical system. The models may serve various purposes, e.g., models may be exercised in sync with the machine operations to detect operation anomalies, a typical Digital Twin setting or used to specify the behaviour of its components and estimate the system response by simulation. One of the first modelling tasks in a new project is to elicit the requirements and properties about the physical behaviour of the machine components and the computational processes controlling them.
In this article, we report on a new model of an idealised IMM derived from our first-hand experience obtained while working with a leading company within the sector of automated plastic manufacture.
Our work has so far materialised in a model of a controller for an injection moulding machine based on the application of the Vienna Development Method (VDM) \cite{Fitzgerald&05}. The model intends to serve as a specification of the IMM safety, process, and quality control requirements for a typical machine that can be found on a factory floor. Given that a subset of the VDM models are executable, it is possible to validate the requirements and understanding expressed in the specification by executing
tests, running the controller in parallel with a model of the machine components behaviours, or generating implementations of the controller to deploy in the physical machine using co-simulation \cite{Gomes&182}. In addition, it is possible to use the executable model as a basis for discussion, expertise sharing, and use it as a test bench in an academic setting.
Our work achieved a first-principles model (i.e., a model based on the governing physics) of the discrete event controller using VDM-RT \cite{Verhoef&06b} of the IMM and validated it against scenarios obtained from a continuous time model of the IMM behaviour as described in \cite{Woll&97}. We implemented the continuous time model in MATLAB and used it to generate tests for the VDM model.
The next validation step is to connect the VDM model with the continuous time model by use of co-simulation. In addition, it is possible to use the executable model as a basis for discussion, expertise sharing, and use it as a test bench in an academic setting.
Both the VDM model and the co-simulation model are to be made publicly available \footnote{The VDM model is available at: \url{https://www.overturetool.org/download/examples/VDM++/IMMPP}.}, so they can be used in the future by engineers that are interested in modelling and simulation of IMM as well as the application of Digital Twins in a manufacturing setting. We plan to add the VDM model to the Overture examples repository and the co-simulation in the INTO-CPS Association repository \cite{Larsen&16a}.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Content/Figures/ModelConnectionsDiagram.pdf}
\caption{The connection between the discrete event (DE) model and the continuous time (CT) model. The DE models change between a number of states at events occurring in time, while in the CT models the system state variables change continuously over time.} \label{fig:connection}
\end{figure}
The remainder of the paper is organised as follows.
Section~\ref{sec:related} provides an overview of related work, and a general overview of the injection moulding machine in Section~\ref{sec:immp}.
Then, Section~\ref{sec:model} contains an overview of the VDM model.
In Section~\ref{sec:validation}, we provide an account of the continuous time model and current validation of the VDM model.
Lastly, Section~\ref{sec:conclusion} provides a conclusion, discussion of the results, and an outline of the future work.
\subsubsection*{Acknowledgements.}
The work presented here is partially supported by the the Poul Due Jensen Foundation for funding the project Digital Twins for Cyber-Physical Systems (DiT4CPS). This work is also supported by the Manufacturing Academy
of Denmark (MADE) in the MADE FAST project (see \url{http://www.made.dk/}). We would also like to thank Daniella Tola for feedback on the first draft of this paper.
\bibliographystyle{splncs04}
\input{6-Bottjer.bbl}
\section{Implementation of Genetic Algorithms in INTO-CPS DSE}
\label{sec:implement}
Several operations have been implemented in order to allow the straightforward construction of GAs for DSE. We here describe their functionality in terms of the GA elements described above in Section~\ref{sec:background}. These have been selected mainly on the grounds of their extensive use in the literature. We do not yet have sufficient experience with DSE in INTO-CPS to have developed heuristics on which combinations of operators might be used in different design spaces.
\subsection{Selection Operators}
\label{sec:selection}
To simulate the selection of mating pairs in a meaningful way a fitness function must be created and then applied to each organism in the generation. In the case of DSE with INTO-CPS, the fitness function is expressed in terms of the results obtained from each co-simulation.
\subsubsection{Random}
\label{sec:selection-random}
This is the simplest operation and does not actually require the ranking form the previous generation. This algorithm is included for completeness; it may be useful to start the algorithm with in situations in which it may not be practical to create a ranking for organisms in generation 0 in some scenarios.
\subsubsection{Roulette Wheel}
\label{sec:selection-roulette}
Initially proposed by Holland~\cite{Holland92}, this operation can be thought of in terms of a pie chart with the more fit individuals taking up a larger portion of the pie. From this the probability of a specific organism being chosen is controlled by Eqn.~\ref{eqn:roulette} where $p_i$ is the probability of the $i^{th}$ organism being selected relative to its fitness, $f_i$.
Eqn.~\ref{eqn:roulette} does assume that a higher fitness value is always better. This may not be the case for the diverse applications of INTO-CPS. Thus a reversed version of this was also implemented that offers a higher probability of selection for organisms with a lower fitness value.
\begin{equation}
p_i = \frac{f_i}{\sum^{n}_{j=0}{f_j}}
\label{7-Rose:eqn:roulette}
\end{equation}
\subsubsection{Ranked}
\label{sec:selection-ranked}
This is a variation of Roulette Wheel that, instead of using the raw fitness value for ranking, will instead normalise the values first~\cite{Obikto98}. This solves an issue that when using the raw fitness value it is possible for a single organism to dominate the process by having an extremely high fitness.
The ranking is a total ordering from $1$ to $n$ where $n$ is the organism with the worst raw fitness value. Organisms having the same fitness are ranked in the order they are processed. As can be seen in Fig.~\ref{fig:ranking-tab}, `a' has an extremely high fitness value compared to the other organisms and thus dominates the selection being selected $74.2\%$ of the time. When normalised, `a' is still chosen $33.6\%$ of the time.
\begin{figure}[h!]
\begin{center}
\begin{tabular}{c c c c c}
\multirow{2}{*}{Organism} & \multirow{2}{*}{Fitness} & \multirow{2}{*}{Normalised Fitness} & \multicolumn{2}{c}{Times Selected} \\
& & & Roulette Wheel & Ranked \\
\hline
a & $300$ & $1$ & $742$ ($74.2\%$) & $336$ ($33.6\%$) \\
b & $50$ & $2$ & $118$ ($11.8\%$) & $257$ ($25.7\%$) \\
c & $30$ & $3$ & $52$ ($5.2\%$) & $196$ ($19.6\%$) \\
d & $35$ & $4$ & $78$ ($7.8\%$) & $127$ ($12.7\%$) \\
e & $5$ & $5$ & $10$ ($1\%$) & $84$ ($8.4\%$) \\
\end{tabular}
\end{center}
\caption{Difference in selection bias between using raw fitness values and normalised values.}
\label{fig:ranking-tab}
\end{figure}
\subsubsection{Tournament}
\label{sed:selection-tournament}
This operator is based on the bracketing systems that are seen in sports where $x$ individuals are chosen to compete between each other from all $n$ individuals in the population.
The size, $x$, of each tournament should be a small number else the fittest individual in the population will dominate, leading to premature convergence. This value will be left to be user defined in the implementation and taken as an argument but with a default value of $2$.
To simplify implementation, this operator will accept only the normalised rankings. This is because using the normalised ranking, an organism with a ranking of $1$ will always be more fit than an organism with ranking $2$, ranking $2$ is always more fit than $3$, etc. Thus, we do not need to account for if the DSE is looking to minimise or maximise the fitness in the implementation of the operator as it has already been dealt with by the normalisation operation.
\subsection{Mating Operators}
\label{sec:mating}
A traditional GA would select from the gene pool created by the previous selection operators to create mating pairs. However, due to the implementation in this paper, this step is combined into the selection operation.
\subsection{Crossover Operators}
\label{sec:crossover}
In our DSE setting, we think of the parameter values that characterise a given design as a sequence, analogous to chromosomes. Each parameter value in the sequence is a key to a specific design characteristic. Crossover operators emulate the recombination stage in natural reproduction. During this stage in natural systems two homologous chromosomes come together touching chromatids. Where these chromatids touch, they break and re-join exchanging DNA between the two chromatids~\cite{Pearson15}. In computational crossing the process does not need to simulate chromosomes moving around and touching each other. Instead we can select contiguous keys (which in our setting are design parameter values) from each parent to add into the child. Note that this process does not change the values of the inherited parameters: each inherited value comes from a parent.
\subsubsection{$n$-Point}
\label{sec:crossover-npoint}
One of the simplest crossing functions that mimics biology is n-point crossover. The simple case, 1-point crossover, is the traditional operation for a GA to implement~\cite{Sivanandam&08-2}. To give the user more options when using the GA library, it was decided to implement the more general n-point version.
This operator selects $n$ keys from the parent, takes contiguous keys from the parent and adds them to the child. This process is illustrated in Fig.~\ref{fig:npointcrossing}. In the diagram each coloured rectangle represents a key in the parent. In this example 3-point crossing is selected so three points to cross are selected, shown in yellow, blue and purple. The child is created with contiguous keys from A until the first crossing point is reached, when it is reached, we switch to taking keys from B, this process repeats until there are no keys left.
\begin{figure}[h!]
\centering
\includegraphics[width=.3\paperwidth]{figures/nPoint.png}
\caption{Example of $n$-point crossover.}
\label{fig:npointcrossing}
\end{figure}
\subsubsection{Uniform}
\label{sec:crossover-uniform}
This crossover strategy differs from n-point in that it uses a bitmask to decide which genes are inherited by the child. Fig.~\ref{fig:uniformcrossing} illustrates this strategy. Again, the coloured rectangles represent genes. The crossing mask would be a randomly generated bit string with the value 0 (white rectangles) meaning take a gene value from A and value 1 (black rectangles) meaning take a gene from B.
\begin{figure}[h!]
\centering
\includegraphics[width=.3\paperwidth]{figures/uniform.png}
\caption{Example of uniform crossover.}
\label{fig:uniformcrossing}
\end{figure}
\subsubsection{BLX}
\label{sec:crossover-blx}
Blend Crossover \cite{Eshelman&93,Takahashi&01} is a more advanced operator that also incorporates elements of mutation into the crossover phase. This incorporation of changing gene values might argue for this being a mutation operation rather than a crossover operation. However, due to the need for two separate individuals, we will treat it as a crossover operation~(the same argument can also be made for the SBX operator discussed below).
This operation takes the two parent gene values and uses them as a range from which the child's value can be selected~(See Fig.~\ref{fig:blxcrossing}). The extents of the ranges past the parent values are controlled by $\alpha$. The child value, $g$, is selected from this range as shown by Eqn.~\ref{eqn:blx}. Common values of $\alpha$ for this equation are $0.5$ \cite{Eshelman&93} and $0.366$ \cite{Takahashi&01}, $0.5$ is chosen as the default value.
\begin{figure}[h!]
\centering
\includegraphics[width=.3\paperwidth]{figures/blxcrossing.png}
\caption{BLX crossover.}
\label{fig:blxcrossing}
\end{figure}
\begin{equation}
g \in{(\min{(p_{1g}, p_{2g})} - \alpha |p_{1g} - p_{2g}|, \max{(p_{1g}, p_{2g})} + \alpha |p_{1g} + p_{2g}|)}
\label{7-Rose:eqn:blx}
\end{equation}
\subsubsection{SBX}
\label{sec:crossover-sbx}
This is similar to BLX crossover however biases the child gene value to be closer to the parent gene values as depicted in Fig.~\ref{fig:sbxcorssing}. This operation is designed for real value genes proposed in~\cite{Deb&00}. The distribution is achieved using Eqns.~\ref{eqn:sbx1} and~\ref{eqn:sbx2}. The value $\beta$ is used to control the spread of the resulting gene with $\eta$ called the distribution index, with larger values meaning the resulting offspring values should be closer to the parents.
\begin{figure}[h!]
\centering
\includegraphics[width=.3\paperwidth]{figures/sbxcrossing.png}
\caption{SBX crossover.}
\label{fig:sbxcorssing}
\end{figure}
\begin{equation}
\beta =
\left\{
\begin{array}{@{}lr@{}}
\mu \leq 0.5, & (2\mu) \\
\mu > 0.5, & {\frac{1}{2(1-\mu)}}
\end{array}
\right\} ^\frac{1}{\eta+1}
\label{7-Rose:eqn:sbx1}
\end{equation}
\begin{equation}
\beta = 0.5 \times
\left\{
\begin{array}{@{}l}
(1 + \beta)p_{1g} + (1 - \beta)p_{2g} \\
(1 - \beta)p_{1g} + (1 + \beta)p_{2g}
\end{array}
\right.
\label{7-Rose:eqn:sbx2}
\end{equation}
\subsection{Mutation Operators}
\label{sec:mutation}
Aside from the BLX and SBX operations that some may consider mutation operations, the only realistic option is to select values from a normal distribution in the case of a real value algorithm, as a bit flipping operation would effectively result in a random search. Selection from a normal distribution can be done in two ways: either absolute selection as in Eqn.~\ref{eqn:mutationabs} or a relative distribution as in Eqn.~\ref{eqn:mutationrel}.
It is recommended that small values of $\sigma$ are normally used, as larger values will effectively turn the algorithm into a random search. Larger values of $\sigma$ are do not accurately represent the reality of evolution since large changes do not happen suddenly. Using a larger $\sigma$ value can be used to prime the search space in the initial generation to help ensure that the full search space is explored.
As the fitness of organisms converges, it is possible to decrease the value of $\sigma$ slowly to yield a process like simulated annealing.
\begin{equation}
g' = g + \Phi(0, \sigma)
\label{7-Rose:eqn:mutationabs}
\end{equation}
\begin{equation}
g' = g \times (1 + \Phi(0, \sigma))
\label{7-Rose:eqn:mutationrel}
\end{equation}
\subsection{Elitism}
\label{sec:elitism}
This operator ensures that a certain percentage of the best individuals from the previous generation will be carried over into the next generation. This is to ensure that the best individual will always be produced by the algorithm no matter what stage it was generated at. Organisms can be chosen at random for removal as they have not yet been ranked. Whilst this does introduce the possibility of removing the new best organism it is probable that it will be recreated in the generation after and the probability of it being removed a second time is low especially with large generation sizes and a low number of organisms being carried over from the previous generation.
\subsection{Diversity Control}
\label{sec:divcontrol}
Diversity control is the mechanism that attempts to prevent premature convergence by applying penalties to organisms based on how similar they are to the rest of the population. This paper implements the first operator from the GOSET manual~\cite{USNA&07}. The diversity operator is based on calculating a mean distance between all organisms, using that to calculate a threshold and then using that threshold to count the number of neighbours an organism has. Although its $O(n^2)$ complexity makes it possibly the most expensive part of the entire GA implementation, it is highly accurate.
\begin{figure}[h!]
\centering
\includegraphics[width=.5\paperwidth]{figures/divcont.png}
\caption{Diversity control.}
\label{fig:divcon}
\end{figure}
\subsection{Controller}
\label{sec:controller}
There are several possible solutions to controlling the iterative process. We can stop after a set number of generations, or after a fitness threshold is reached by $n$ organisms, or after there has been little change is fitness for $n$ generations (convergence stopping), or after the algorithm has run for a certain amount of time. The final option does not appear to be realistic in the context of DSE for INTO-CPS, but the other options have been implemented:
\begin{description}
\item[$n$ Generation.]
\label{sec:controller-ngen}
This is the simplest of the controllers to implement is stopping after n generations. This approach whilst guaranteed to stop at some point is not guaranteed to produce the fittest individuals.
\item[Fitness Threshold.]
\label{sec:controller-firthres}
This requires a certain percentage of organisms to pass a threshold before stopping. This controller is guaranteed to produce organisms of at least some minimum quality, however, this alone does not guarantee the algorithm to stop. To guarantee that the algorithm will eventually stop we have implemented a limit to the number of generations that is configurable by the user.
\item[Convergence.]
\label{sec:controller-convergence}
To find if the search has GA has converged, we need two things: firstly, the threshold at which we consider things to have converged, and secondly the number of previous generations to look at to check for convergence. As convergence is when the average fitness of $n$ sequential generations is relatively similar we keep a list of the previous $n$ generations in the algorithm and then check the standard deviation of that list to check for convergence.
\end{description}
\section{Introduction}
\label{sec:intro}
In the model-based design of Cyber-Physical Systems~(CPSs), developers work with semantically heterogeneous collections of models (which we term \emph{multi-models}) describing computational and physical processes. Co-simulation is thus of particular interest as methods for evaluating design alternatives in order to find those that best meet combinations of requirements (e.g., for performance or cost).
In this context, a \emph{design space} is a collection of multi-models that represent alternative designs for a proposed CPS. These designs differ from one another in terms of the values of \emph{design parameters}. The process of \emph{Design Space Exploration~(DSE)} aims to identify the combinations of values for design parameters that specify designs which are optimal with respect to a defined criterion. In the DESTECS project DSE is seen as a way of managing the complex embedded systems design process~\cite{Gamble&14}. More recently, DSE has been implemented in the Integrated Toolchain for Cyber-Physical Systems~(INTO-CPS\footnote{\url{https://into-cps.org/}})~\cite{Larsen&17a2}.
Exhaustive evaluation of all designs in a design space is infeasibly costly for all but the simplest multi-models. For even moderately complex multi-models, it is necessary to hunt for optimal designs by iteratively selecting subsets of designs, evaluating these and using the results to suggest where an optimal design might lie. Genetic Algorithms~(GAs), which implement optimisation methods inspired by Darwinian principles of natural selection, may provide significant performance improvements over exhaustive methods. There are, for example, suggestions that GAs are feasible for DSE in the hardware/software codesign domain~\cite{Panerati&17}. However, an implementation of GA-based DSE has not yet been integrated in the INTO-CPS toolchain to permit the evaluation of this approach.
This paper reports an implementation and initial evaluation of GAs in DSE over multi-models in the INTO-CPS toolchain\footnote{The implementation code is available at \url{https://github.com/INTO-CPS-Association/dse_scripts}}. In order to provide this implementation, it was necessary to lay groundwork by updating the INTO-CPS DSE infrastructure, identify GAs relevant to DSE, implement such algorithms in a general setting and integrate them with the toolchain. An evaluation was undertaken, comparing a GA-based approach to the exhaustive DSE for the Robotti agricultural robot previously described by Bogomolov et al.~\cite{Bogomolov&21}.
We assume that the reader is familiar with the principle of collaborative modelling and co-simulation based on the Functional Mockup Interface~(FMI) standard employed in INTO-CPS. Section~\ref{sec:background} provides further background on DSE and GAs in INTO-CPS. Section~\ref{sec:update} briefly describes technical updates made to INTO-CPS in order to accommodate our GA-based approach. We then are in a position to describe the GA approaches implemented in INTO-CPS~(Section~\ref{sec:implement}) and the evaluation on the Robotti data~(Section~\ref{sec:case}). We discuss the study so far and potential directions for future work in Section~\ref{sec:discussion}.
\section{Background}
\label{sec:background}
\subsection{DSE support in the INTO-CPS Toolchain}
\label{sec:into}
For a brief account of related work on DSE, we direct the reader to \cite{Fitzgerald&17a} and more recently \cite{Bogomolov&21}. In the DESTECS project on embedded systems, which was largely focussed on binary multi-models (\emph{co-models}) the need for a systematic approach to DSE was identified, using the Automated Co-model Analysis features of the DESTECS tool. A range of techniques for design parameter selection were considered, including the use of screenign experiments, fractional experiments using orthogonal matrices (Taguchi methods), space-filling searches as well as exhaustive parameter sweeps~\cite{Gamble&14}.
INTO-CPS emerged as an open toolchain permitting the co-simulation of multi-models of CPSs that conform to the Functional Mockup Interface~(FMI) standard\footnote{\url{https://fmi-standard.org/}}. The toolchain supports a workflow in which a multi-model architecture represented in a SysML profile supported by the Modelio tool which can then be used to generate the FMI specifications for the interfaces of constituent models. Co-simulation is managed through a Co-simulation Orchestration Engine~(COE) known as Maestro~\cite{Thule&193}. A web-based application can be used to launch the COE.
The purpose of DSE is to evaluate multi-models under a series of co-simulations, in order to arrive at a \emph{ranking} against an \emph{objective (or cost) function} which expresses the selection criteria that are important to the developer~\cite{Fitzgerald&17a}. One or more multi-models selected on the basis of a ranking can then be taken forward for further design. In this way, DSE can be used to reduce the range of prototypes that may have to be produced and on which expensive tests may have to be performed.
INTO-CPS supports DSE by running co-simulations on selected points in the design space. Scripts implemented in a combination of Java and Python allow the analysis to be performed from the INTO-CPS Application, and the ranking to be displayed.
Fitzgerald et al.~\cite{Fitzgerald&17a} offer a systematic approach to DSE in INTO-CPS through a specific SysML profile and demonstrate feasibility of a simple GA on a line follower robot as a first closed-loop algorithm for INTO-CPS. In this paper, we describe a recent and more comprehensive integration of GA techniques in INTO-CPS.
\subsection{Genetic Algorithms}
\label{sec:ga}
GAs are proposed by Holland~\cite{Holland92} as an adaptive heuristic approach analogous to the biological process of natural selection~\cite{Darwin&59} whereby the selection of genes occurs through internal competition leading to unfavourable genes eventually being removed from the gene pool and convergence to an optimum with some variation~\cite{Sivanandam&08}. Fig.~\ref{fig:mainloop} shows the main loop of a typical GA that aims to select a set of optimal candidate solutions. Each iteration produces a new generation of candidate solutions. These are evaluated and a selection is made by a process that typically ensures the fitter candidates are have a higher chance of selection. In order to create the next generation, these are mated, and genetic operators such as crossover and mutation are applied. Elimination may remove infeasible solutions before a decision is made on termination. This may be based on a range of criteria such as having reached specified number of generations or when improvements between iterations are below a threshold.
\begin{figure}
\centering
\includegraphics[width=0.75\columnwidth]{figures/mainloop}
\caption{\label{fig:mainloop} Outline of a Genetic Algorithm.}
\end{figure}
In our DSE context, a solution is a candidate design represented in terms of the values of the design parameters of interest. The genetic operators that have been implemented are described in Section~\ref{sec:implement} below.
\section{Discussion and Future Work}
\label{sec:discussion}
We have provided an implementation of GAs as a basis for DSE in INTO-CPS, and evaluated this using an established case study. Some groundwork had to be laid in order to enable GA-based DSE, including the updating of existing scripts and the introduction of threading. However, the implementation appears to be viable.
Aside from increasing the sophistication of the GA operators now implemented for INTO-CPS DSE, this opens up a range of other areas of future work.
First, CPSs such as Robotti may develop faults over time and it may be useful for the INTO-CPS toolchain to integrate Fault Space Exploration~(FSE) in addition to DSE. This could be used to find the level of fault tolerance that a system can withstand before resulting in undefined, possibly unsafe, behaviour. It is reasonable to think of this in the context of resilience modelling~\cite{Jackson&17}. Second, GAs are part of the broader class of Evolutionary Algorithms (EAs), suggesting that a wider examination of the orle of EAs in DSE would be worthwhile. Third, as a CPS may be deployed in many different environments it is not unreasonable to find that a set of parameters may be optimal for one scenario but not another, as shown in the DSE results for Robotti. Thus, it may be useful to implement an approach that can automatically produce a `compromise' set of parameter values that are optimal across all scenarios for a given CPS. This could take the results of DSEs as input and produce an `average organism' from this data.
\section{Updating INTO-CPS to Support GA-based DSE}
\label{sec:update}
\label{sec:update-threading}
\label{sec:update-misc}
Prior to the work reported here, the main means of performing DSE in INTO-CPS was by using outdated Python~2 scripts and an exhaustive search method. Indeed, the authors of~\cite{Bogomolov&21} have noted the overhead caused by the reliance on Python~2. In addition, Python~2 support ceased in 2020. The goal of the work reported here was to update the DSE scripts to use modern Python~3, to improve usability, and to provide INTO-CPS users with more options based on GAs when performing DSE.
The code that is provided with INTO-CPS for DSE consists of 8 scripts that interact with each other as shown in Fig.~\ref{fig:DSEscripts}. Algorithm Selector is used to select the desired DSE algorithm to use. The selected algorithm script then calls helper functions from the Common script to run and evaluate a specific configuration using the COE. To produce a human readable output in both CSV and HTML the Output HTML script is run.
\begin{figure}
\centering
\includegraphics[width=0.75\columnwidth]{figures/DSEscripts}
\caption{\label{fig:DSEscripts} Outline of DSE Scripts in INTO-CPS.}
\end{figure}
\paragraph{Conversion to Python 3.} It was decided that, to maintain compatibility, the overall structure of the scripts would remain the same with only minor changes to improve flexibility. Any new options that would be added would also default to the original behaviour of the scripts. The largest change in respect to Python 2 and 3 was the requirement for brackets on print statements and also the http library used to connect to the COE had also changed its name. As such the upgrade process was not difficult and is fully compatible with anything that may be reliant on the way the original Python 2 scripts executed.
\paragraph{Multi-Threading} The theoretical speed improvements associated with multi-threading are exponential with increasing thread counts to a threshold, specifically the time taken for a single simulation. Although the original scripts and COE were not designed with multi-threading in mind, they are mostly thread safe, and those parts that were not (mainly to do with writing simulation results to a file ready for ranking) required some updates to properly implement the multi-threading.
From Fig.~\ref{fig:threadedtestperf} indicates that the threading performs as expected with a test controller that simply echoes its input and with the case study described in Section~\ref{sec:case-oldVnew}.
\begin{figure}[h!]
\centering
\includegraphics[width=.4\paperwidth]{figures/threadedtestvtheoretical.png}
\caption{Actual Multi-Threaded Performance}
\label{fig:threadedtestperf}
\end{figure}
More options and flexility were also added to the upgraded DSE scripts:
\begin{description}
\item[Splitting Results Generation.] Previously there was no option to prevent the generation of the HTML \& CSV results files. This has been changed so that the user may specify how they would like their results to be output.
\item[Arguments.] As it is possible to run the DSE scripts easily via a command line, it is desirable that they produce useful errors. With the old Python 2 scripts the error outputs were simply from Python telling the user that a command was not found. With the updated scripts this has been changed so that the script will notify the user not only that a parameter is missing but also which parameter is missing. A \texttt{--help} command has also been added so that the user can see all the options available to them.
\item[Progress Bar.] A progress bar was added to the console output of the scripts to give the user some sense that the scripts are running and things are happening.
\end{description}
\section{Evaluation on the Robotti Case}
\label{sec:case}
\subsection{Existing DSE Algorithms against Upgraded Versions}
\label{sec:case-oldVnew}
Fig.~\ref{fig:oldvnew-results} shows the results of speed testing done using Robotti data, which suggest that the upgrades described in Section~\ref{sec:update} have had a positive effect on runtime. Each scenario was run $5$ times to find the average runtime. As no functionality was changed the results of the ranking, HTML and CSV output are also consistent with the original version, apart from a change in the order of the columns in the outputs, which we suggest may be caused by a change in how sort operates between Python~2 and~3, although we cannot find evidence to support this.
\begin{figure}[h!]
\begin{center}
\begin{tabular}{c c c c c c c}
Scenario & Original & \multicolumn{2}{c}{Python 3} & \multicolumn{3}{c}{Python 3 Threaded} \\
& Average Time / s & Average Time / s & Diff / \% & Threads & Average Time / s & Diff / \% \\
\hline
Sin 2 & $2977.611$ & $2988.454$ & $0$ & $10$ & $496.001$ & $-83$ \\
Speed Ramp 1 & $2054.087$ & $1756.910$ & $-14$ & $10$ & $329.819$ & $-84$ \\
Speed Step 1 & $1960.502$ & $1360.295$ & $-31$ & $5$ & $356.744$ & $-82$ \\
Turn Ramp 1 & $2389.692$ & $1604.571$ & $-33$ & $8$ & $359.412$ & $-85$ \\
\end{tabular}
\end{center}
\caption{Average Run Time Comparisons}
\label{fig:oldvnew-results}
\end{figure}
\subsection{GA-based DSE of Robotti}
\label{sec:case-robottiGA}
Now we consider the use of GA-based DSE on Robotti. The GA setup was initially the same as for the test functions used to validate the implementations of the GA operations. However, as can be seen in \ref{fig:initialresults} this did not produce the expected results. The issue here is that the diversity control operation was too aggressive, preventing convergence by essentially turning the algorithm into an exhaustive search. This is suggested by the mean line in Fig.~\ref{fig:initialresultsC}. At around $35$ generations the mean fitness begins to rise and maximum fitness remains relatively constant, suggesting that the search is not converging. This highlights the need for parameter tuning to ensure GAs run effectively. To resolve this, the diversity control was disabled as it was found it was not needed for this simulation.
\begin{figure}
\centering
\subfloat[Gene values across all generations.\label{fig:initialresultsA}]
{\includegraphics[width=0.32\textwidth]{figures/initialspeedrampa.png}}
\subfloat[Fitness against gene values.\label{fig:initialresultsB}]
{\includegraphics[width=0.32\textwidth]{figures/initialspeedrampb.png}}
\subfloat[Fitness trends.\label{fig:initialresultsC}]
{\includegraphics[width=0.32\textwidth]{figures/initialspeedrampc.png}}
\caption{Initial Speed Ramp results.}
\label{fig:initialresults}
\end{figure}
Fig.~\ref{fig:robottiGA-exhaustiveVga} summarises the results of DSE in each of the Robotti scenarios compared against the best found by the exhaustive search. As can be seen in Fig.~\ref{fig:robottiGA-exhaustiveVga} the GA-based DSE performs as expected, finding the best parameter combinations within acceptable margins. As we can see, although in the order of $\times10^{-6}$, the genetic algorithm-based search is able to find slightly better parameter combinations for both the Sin and Turn Ramp scenarios. This is a consequence of the ability to select from a continuous range of values, so the GA found a combination that could not have been evaluated in the exhaustive DSE. The results also highlight that DSE scenarios must be selected with care, as if DSE was only performed on Speed Step then we would come to the false conclusion that the chosen parameters do not affect the running of the robot.
\begin{figure}[h!]
\begin{center}
\begin{tabular}{c c c c c c c c c}
\multirow{2}{*}{Scenario} & \multicolumn{3}{c}{Exhaustive Best Combination} & \multicolumn{3}{c}{GA Best Combination} & \multirow{2}{*}{Cross Track Error Difference} \\
& cAlphaF & $\mu$ & Mass & cAlphaF & $\mu$ & Mass & \\
\hline
Sin 2 & $20000$ & $0.7$ & $3000$ & $20000$ & $0.70$ & $3000$ & $-1.46\times10^{-6}$ \\
Speed Ramp 1 & $38000$ & $0.4$ & $1000$ & $37019$ & $0.45$ & $1038$ & $8.40\times10^{-2}$ \\
Speed Step 1 & $20000$ & $0.3$ & $1000$ & $34725$ & $0.45$ & $1027$ & $0.00$ \\
Turn Ramp 1 & $20000$ & $0.3$ & $3000$ & $20000$ & $0.30$ & $3000$ & $-9.21\times10^{-7}$ \\
\end{tabular}
\end{center}
\caption{Exhaustive Best Combination vs GA Best Combination. Results are rounded appropriately. Cross track error difference calculated as Exhaustive$-$GA.}
\label{fig:robottiGA-exhaustiveVga}
\end{figure}
\begin{figure}
\centering
\subfloat[Gene values across all generations.\label{fig:sinresultsA}]
{\includegraphics[width=0.32\textwidth]{figures/sinresultsa.png}}
\subfloat[Fitness against gene values.\label{fig:sinresultsB}]
{\includegraphics[width=0.32\textwidth]{figures/sinresultsb.png}}
\subfloat[Fitness trends.\label{fig:sinresultsC}]
{\includegraphics[width=0.32\textwidth]{figures/sinresultsc.png}}
\caption{Sin Results.}
\label{fig:sinresults}
\end{figure}
\begin{figure}
\centering
\subfloat[Gene values across all generations.\label{fig:speedrampA}]
{\includegraphics[width=0.32\textwidth]{figures/speedrampa.png}}
\subfloat[Fitness against gene values.\label{fig:speedrampB}]
{\includegraphics[width=0.32\textwidth]{figures/speedrampb.png}}
\subfloat[Fitness trends.\label{fig:speedrampC}]
{\includegraphics[width=0.32\textwidth]{figures/speedrampc.png}}
\caption{Speed Ramp results}
\label{fig:speedrampresults}
\end{figure}
\begin{figure}
\centering
\subfloat[Gene values across all generations.\label{fig:speedstepA}]
{\includegraphics[width=0.32\textwidth]{figures/speedstepa.png}}
\subfloat[Fitness against gene values.\label{fig:speedstepB}]
{\includegraphics[width=0.32\textwidth]{figures/speedstepb.png}}
\subfloat[Fitness trends.\label{fig:speedstepC}]
{\includegraphics[width=0.32\textwidth]{figures/speedstepc.png}}
\caption{Speed Step results.}
\label{fig:speedstepresults}
\end{figure}
\begin{figure}
\centering
\subfloat[Gene values across all generations.\label{fig:turnrampA}]
{\includegraphics[width=0.32\textwidth]{figures/turnrampa.png}}
\subfloat[Fitness against gene values.\label{fig:turnrampB}]
{\includegraphics[width=0.32\textwidth]{figures/turnrampb.png}}
\subfloat[Fitness trends.\label{fig:turnrampC}]
{\includegraphics[width=0.32\textwidth]{figures/turnrampc.png}}
\caption{Turn Ramp results.}
\label{fig:turnrampresults}
\end{figure}
\chapter*{Preface}
\markboth{Preface}{Preface}
The 19th in the ''Overture'' series of workshops on the Vienna Development Method (VDM), associated tools and applications was held as a hybrid event both online at in person at Aarhus University on October 22, 2021.
VDM is one of the longest established formal methods, and yet has a lively community
of researchers and practitioners in academia and industry grown around the modelling
languages (VDM-SL, VDM++, VDM-RT) and tools (VDM VSCode, VDMTools, VDMJ, ViennaTalk,
Overture, Crescendo, Symphony, and the INTO-CPS chain). Together, these provide a
platform for work on modelling and analysis technology that includes static and dynamic
analysis, test generation, execution support, and model checking.
Research in VDM is driven by the need to precisely describe systems. In order to do so, it is also necessary for the associated tooling to serve the current needs of researchers and practitioners and therefore remain up to date. The 19th Workshop reflected the breadth and depth of work supporting and applying VDM. This technical report includes first a paper on industrial usage. This is followed by a session on VDM related papers concerning techniques for developing models and applying VDM for modelling systems. The last sessions is related to Cyber-Physical Systems and simulation.
As the pandemic is still in effect across the world this workshop was held both online and in person. We applaud the possibility of meeting some collegues in person and still remaining in contact with others online. It is our sincere hope that the next Overture Workshop will see an increase in personal attendance.
We would like to thank the authors, PC members, reviewers and participants for
their help in making this a valuable and successful workshop, and we look forward
together to meeting once more in 2022.
\medskip
\begin{flushright}\noindent
Hugo Daniel Macedo, Aarhus
Casper Thule, Aarhus\\
Ken Pierce, Newcastle\\
\end{flushright}
\tableofcontents
\chapter*{Organization}
\section*{Programme Committee}
\begin{longtable}{p{0.3\textwidth}p{0.7\textwidth}}
Tomo Oda & Software Research Associate Incorporated, Japan\\[12pt]
Marcel Verhoef & European Space Agency, The Netherlands\\[12pt]
Paolo Masci & National Institute of Aerospace (NIA), USA\\[12pt]
Peter Gorm Larsen & Aarhus University, Denmark\\[12pt]
Nick Battle & Newcastle University, UK\\[12pt]
Fuyuki Ishikawa & National Institute of Informatics, Japan\\[12pt]
Keijiro Araki & National Institute of Technology, Kumamoto College, Japan\\[12pt]
Sam Hall & Newcastle University, UK\\[12pt]
Hugo Daniel Macedo & Aarhus University, Denmark\\[12pt]
Ken Pierce & Newcastle University, UK\\[12pt]
Marcel Verhoef & European Space Agency, The Netherlands\\[12pt]
Kenneth G. Lausdahl & AGROCorp International and Aarhus University, Denmark
\end{longtable}
\mainmatter
\titlerunning{19th Overture Workshop, 2021}
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|
1,116,691,500,735 | arxiv | \section{Introduction}
\label{intro}
Homogenization problems appear in various contexts of science and engineering and involve
the interaction of two or more oscillatory scales. In this work we focus on the simplest possible
mathematical paradigms of periodic homogenization.
Our objective is to develop an analytical
tool that is capable of understanding the shapes of periodic oscillatory functions when the scales
of oscillations are a-priori known (or expected), and use it in order to transform the homogenization
problem into a limit problem for a kinetic equation. The calculation of an effective equation
becomes then an issue of studying a hyperbolic (or diffusive) limit for the kinetic equation. The
procedure is well adapted in identifying the specific characteristics of the underlying homogenization
problem and provides an efficient tool for the rigorous justification of multiscale asymptotic
expansions.
The main idea is motivated from considerations of kinetic theory. When the statistics of interacting particles
is studied it is customary to introduce an empirical measure and to study its statistical properties in the
(weak) limit when the number of particles gets large. Likewise, for an oscillating family of
functions $\{ u_{{\varepsilon}} \}$ if we want to study the shape of periodic oscillations at a predetermined scale
we may introduce an inner variable that counts the content of oscillation at such scale.
For instance, to count oscillations at the scale $\frac{x}{{\varepsilon}}$ one can introduce
\begin{equation}
\label{intro1}
f_{{\varepsilon}} (x, v) = u_{{\varepsilon}}(x) \delta_{p} (v - \frac{x}{{\varepsilon}})
\end{equation}
where $\delta_{p}$ is the periodic delta function, and study the family $\{ f_{{\varepsilon}} \}$. A-priori bounds
for $\{ u_{{\varepsilon}} \}$ translate to uniform bounds for $\{ f_{{\varepsilon}} \}$: if for example
$u_{{\varepsilon}}$ is uniformly bounded in $L^{2}$, $u_{{\varepsilon}} \in_{b} L^{2}$, then
$f_{{\varepsilon}} \in_{b} L^{2} (M({\mathbb T}^{d}))$ and, along a subsequence,
\begin{equation}
\label{intro2}
f_{{\varepsilon}} \rightharpoonup f \quad \text{weak$\star$ in $L^{2} (M({\mathbb T}^{d}))$} \, ,
\end{equation}
where $M({\mathbb T}^{d})$ stands for the periodic measures.
In addition, the resulting $f$ is better: $f \in L^{2}(L^{2}({\mathbb T}^{d}))$.
The above object should be compared to the concept of double-scale limit
introduced in the influential work of Nguetseng \cite{Nguetseng}
and applied to a variety of homogenization problems \cite{Allaire,E,HX,FP}.
In the double-scale limit one tests the family
$\{ u_{{\varepsilon}} \}$ against oscillating test functions and develops a
representation theory for the resulting weak-limits.
It turns out, \cite{Nguetseng}, that for a uniformly bounded family
$u_{{\varepsilon}} \in_{b} L^{2}$ and test functions $\varphi$ periodic in $v$
\begin{equation}
\label{intro3}
\int u_{{\varepsilon}} (x) \varphi(x,\frac{x}{{\varepsilon}} )dx
\to \int \int f(x,v) \varphi(x,v) dx dv
\end{equation}
where $f \in L^{2}(L^{2}({\mathbb T}^{d}))$. The reader should note that
this is precisely the content of \eqref{intro1}, \eqref{intro2},
which thus provide an alternative interpretation to the double scale limit.
However, what seems to have been missed, perhaps because
Nguetseng's analysis \cite{Nguetseng} proceeds without writing down
\eqref{intro1} but rather by establishing directly \eqref{intro3},
is that the measures $f_{{\varepsilon}}$ satisfy in their own right very
interesting equations.
This is a consequence of additional properties, like
\begin{equation}
\label{intro4}
\Big ( \nabla_{x} + \frac{1}{{\varepsilon}} \nabla_{v} \Big ) f_{{\varepsilon}} (x, v) =
\nabla_{x} u_{{\varepsilon}}(x) \delta_{p} (v - \frac{x}{{\varepsilon}}) \, ,
\end{equation}
obtained by applying differential operators that annihilate
the singular measure.
Properties like \eqref{intro4}, in turn, suggest a procedure for embedding
homogenization problems into limit problems for kinetic equations.
In the sequel we develop this perspective,
using as paradigms the problem of
hyperbolic homogenization, and the problem of enhanced diffusion.
The double-scale limit \cite{Nguetseng} along with
the technique of multiscale asymptotic expansions \cite{BLL} have been quite
effective in the development of homogenization theory
with considerable progress in several contexts ({\it e.g.}
\cite{MPP}, \cite{Allaire}, \cite{AM}, \cite{E},
\cite{FP}, \cite{HX}).
Other tools have also been used for the
homogenization of linear hyperbolic problems:
Among them are of course Young measures, developed by Tartar
and used for the homogenization of some particular linear transport
equations in two dimensions (see \cite{Tartar1} and
\cite{Tartar2}). Wigner measures (see \cite{GMMP}) may also be mentioned.
As our first example we consider the hyperbolic homogenization problem
\begin{equation}
\label{intro5}
\begin{aligned}
\dt {u_{{\varepsilon}}} + a \big (x, \frac{x}{{\varepsilon}} \big) \cdot \nabla_{x} u_{{\varepsilon}} &= 0
\\
u_{{\varepsilon}} (0, x) &= U^{0}(x, \frac{x}{{\varepsilon}}) \, ,
\end{aligned}
\end{equation}
with $a(x,v)$ a divergence free field periodic in $v$, is transformed to the problem of
identifying the hyperbolic limit ${\varepsilon} \to 0$ of the kinetic initial-value problem
\begin{equation}
\label{intro6}
\begin{aligned}
\dt {f_{\varepsilon}} +a(x,v)\cdot\nabla_x f_{\varepsilon} +\frac{1}{{\varepsilon}}a(x,v)\cdot
\nabla_v f_{\varepsilon} &=0,
\\
f_{\varepsilon}(t=0,x,v)
&=U^0(x,v)\;\delta_{p} (v-\frac{x}{{\varepsilon}})
\end{aligned}
\end{equation}
Homogenization for \eqref{intro5} has been studied by Brenier
\cite{Brenier},
E \cite{E}, Hou and Xin \cite{HX} and, in fact,
the effective equation is sought - motivated by the double-scale limit
- in a class of kinetic equations. Eq. \eqref{intro5} is by no means
the only interesting hyperbolic problem for homogenization; we refer
to \cite{AHZ}, \cite{Hamdache},
\cite{GouPou} (where a kinetic equation itself is homogenized), and
to
\cite{AlVa} for an example concerning a Schr\"odinger equation (the list is
of course not exhaustive).
For \eqref{intro5}, our analysis proceeds by studying
the hyperbolic limit for the kinetic equation \eqref{intro6}.
We find that if the kernel of the cell-problem
\begin{equation}
\label{introh1}
K_x=\Big\{g\in L^2( {\mathbb R} ^d\times{\mathbb T}^d)\quad \Big|\ a(x, v)\cdot
\nabla_v g=0\ in\ {\mathcal D}'\Big\}
\end{equation}
is {\it independent} of $x$, then it is possible to identify
the effective equation.
Namely, when the vector fielfd $a = a(v)$ is independent of $x$
the effective equation for $f$ reads
\begin{equation}
\label{intro7}
\begin{aligned}
\dt f + (P a) \cdot \nabla_{x} f &= 0
\\
f(t=0, x,v) &= P U^{0} (x,v) \, ,
\end{aligned}
\end{equation}
where $P$ is the projection operator on the kernel $K$, and in turn
$u = \int_{{\mathbb T}^d} f dv$ (see Theorem \ref{hyphomthm}).
By contrast, when $a = a(x)$ and $K_{x}$ depends on $x$, a counterexample
is constructed that shows that the effective equation can not be
a pure transport equation (see section \ref{secce}).
In section \ref{sectr}, this analysis is extended for homogenization
problems where a periodic fine-scale structure is transported
by a divergence-free vector field (see equations \eqref{tos} and \eqref{kintos})
analogous results to the case of \eqref{intro5} are found.
Such kinetic equations might turn very useful for devising computational
algorithms for the computation of homogenization problems.
A second paradigm is the problem of enhanced diffusion
\begin{equation}
\label{intro8}
\begin{aligned}
\partial_{t} u_{{\varepsilon}} + \frac{1}{{\varepsilon}} a(x, \frac{x}{{\varepsilon}}) \cdot \nabla_{x} u_{{\varepsilon}}
&= \alpha \triangle_{x} u_{{\varepsilon}}
\\
u_{{\varepsilon}} (0, x) = U^{0} (x, \frac{x}{{\varepsilon}})
\end{aligned}
\end{equation}
with $a(x,v)$ periodic, divergence-free and with mean $\int_{{\mathbb T}^{d}} a = 0$. The results formally
obtained by multiscale asymptotics have been validated for this problem by
McLaughlin, Papanicolaou and Pironneau \cite{MPP},
Avellaneda and Majda \cite{AM},
and Fannjiang and Papanicolaou \cite{FP}.
We revisit this problem from the perspective of the
kinetic decomposition and transform it to the problem of
identifying the ${\varepsilon} \to 0$ limit
\begin{equation}
\label{intro9}
\begin{aligned}
\dt {f_{\varepsilon}} + \frac{1}{{\varepsilon}} a(x, v)\cdot\nabla_x f_{\varepsilon}
&+ \frac{1}{{\varepsilon}^{2}}
\big ( a(x,v) \cdot \nabla_{v} f_{{\varepsilon}} - \alpha \triangle_{v} f_{{\varepsilon}} \big )
\\
&= \alpha \triangle_{x} f_{{\varepsilon}}
+ \frac{2\alpha}{{\varepsilon}} \nabla_{x} \cdot \nabla_{v} f_{{\varepsilon}}
\, ,
\\
f_{\varepsilon}(t=0,x,v)
&=U^0(x,v)\;\delta_{p} (v-\frac{x}{{\varepsilon}}).
\end{aligned}
\end{equation}
The latter is a limit for the transport-diffusion equation \eqref{intro9}
in the so-called diffusive scale, and its analysis provides the
effective equation \eqref{basic}-\eqref{cellp}
of enhanced diffusion (see Theorem \ref{enhdiffthm}). This example indicates the efficiency of
this approach in the rigorous validation of multi-scale asymptotic expansions.
Finally, we note that the scales of the drift and of the diffusion
in \eqref{intro8} may be chosen differently from $1/{\varepsilon}$ and $\alpha$,
yielding other interesting homogenization problems, see for instance
Capdeboscq \cite{Capdeboscq1,Capdeboscq2}.
The article is organized as follows. Analytical considerations like the proper definition of \eqref{intro1},
the characterization of the weak limit points of $f_{{\varepsilon}}$ under various uniform bounds,
the differential relations such as \eqref{intro4}, and the identification of asymptotics for $f_{{\varepsilon}}$
are developed in section \ref{msde} and in appendix I. In section \ref{homohyp}, we study the
hyperbolic homogenization problem \eqref{intro5}, derive the effective equation, and
produce the counterexample mentioned before.
Some material from ergodic theory needed in the derivation is outlined in the appendix II.
In section \ref{sectr}, we study the transport via a divergence-free field depending
on an oscillating fine-scale, we derive the associated kinetic equation, and discuss
the connection of the two formulations via characteristics and the derivation of an
effective equation. Finally, in section \ref{endiff} we study the parabolic homogenization
problem \eqref{intro8} and derive the enhanced diffusion equation via the kinetic decomposition.
\section{Multi-scale decomposition}
\label{msde}
Let $\{ u_{{\varepsilon}}(x) \}$ be a family of functions defined on a open set
$\Omega \subset {\mathbb R} ^{d}$
that contains periodic oscillations and suppose that the scales of oscillations
are either a-priori known (or anticipated).
Our goal is to introduce an analytical object that
will prompt the anticipated scale(s) of oscillations and quantify the structure
of oscillations in the family at the preselected scale(s).
Suppose that periodic oscillations of length ${\varepsilon}$ are anticipated in
the family $\{ u_{{\varepsilon}} \}$.
To focus on them we consider a periodic grid with sides of length ${\varepsilon}$
in each coordinate direction. The grid splits the Euclidean space into distinct cubic cells
of volume ${\varepsilon}^{d}$, and it is arranged so that the centers of the cells
occupy the lattice ${\varepsilon} {\mathbb Z}^{d}$. Let $\Omega$ be placed on that grid,
and define a function $\chi_{{\varepsilon}} : \Omega \to {\varepsilon} {\mathbb Z}^{d}$ that maps the generic
$x \in \Omega$ to the center $\chi_{{\varepsilon}}(x)$ of the cell containing $x$.
To each point $x \in \Omega$ there is associated a decomposition $(\chi_{{\varepsilon}}(x), v)$
where $\chi_{{\varepsilon}}(x) \in {\varepsilon} {\mathbb Z}^{d}$ stands for the center of the cell that $x$ occupies,
and $v \in {\mathbb T}^{d}$ is the vector difference $x-\chi_{{\varepsilon}}(x) $ as measured in units
of distance ${\varepsilon}$, that is $x = \chi_{{\varepsilon}}(x) + {\varepsilon} v$.
We introduce the quantity
\begin{equation}
\label{decompose}
f_{{\varepsilon}}(x, v) = u_{{\varepsilon}} (x) \delta_{p} \big (v - \frac{x - \chi_{{\varepsilon}}(x)}{{\varepsilon}} \big )
\, , \quad x \in {\mathbb R} ^{d} \, , \; v \in {\mathbb T}^{d} \, ,
\end{equation}
where $\delta_{p}$ stands for a periodization of the usual delta function
with period $1$ in each coordinate direction, and ${\mathbb T}^{d}$ stands for the d-dimensional torus,
the quotient of $ {\mathbb R} ^{d}$ by the subgroup ${\mathbb Z}^{d}$.
We note that the map $x \mapsto (\chi_{{\varepsilon}}(x), v) $ is single valued for points that
fall into a single cell, but multi-valued for points that fall onto the boundaries between
adjacent cells. For the latter points there would be two different decompositions
$(\chi_{{\varepsilon}}, v)$ and $(\chi'_{{\varepsilon}}, v')$ associated to the same point $x \in \Omega$.
Nevertheless, in that case $x = \chi_{{\varepsilon}} + {\varepsilon} v = \chi'_{{\varepsilon}} + {\varepsilon} v'$ and,
due to the use of a periodic delta function,
$$
\delta_{p} \big (v - \frac{x - \chi_{{\varepsilon}}}{{\varepsilon}} \big ) =
\delta_{p} \big (v - \frac{x }{{\varepsilon}} \big ) =
\delta_{p} \big (v - \frac{x - \chi'_{{\varepsilon}}}{{\varepsilon}} \big )
$$
Hence, both decompositions provide
the same outcome in \eqref{decompose} with $f_{{\varepsilon}}$ defined
for $x \in \Omega$ and $v \in {\mathbb T}^{d}$.
The operator $\nabla_{x} + \frac{1}{{\varepsilon}} \nabla_{v}$ annihilates
the form $v - \frac{x }{{\varepsilon}}$ and that - at least formally - yields the formula
\begin{equation}
\label{formder}
\big ( \nabla_{x} + \frac{1}{{\varepsilon}} \nabla_{v} \big ) f_{{\varepsilon}} =
(\nabla_{x} u_{{\varepsilon}}) \delta_{p} \big (v - \frac{x }{{\varepsilon}} \big )
\end{equation}
In the sequel, we provide formal definitions
for the decomposition \eqref{decompose} and extensions as well as
differentiation properties like \eqref{formder}
that are helpful in later sections for validating multiscale expansions.
\subsection{Definitions}
We make extensive use of distributions defined on the torus ${\mathbb T}^{d}$.
Such distributions are in one-to-one correspondence with periodic
distributions
$T$ on $ {\mathbb R} ^{d}$ of period 1 in each coordinate direction,
that is distributions satisfying for $i = (i_{1}, ... , i_{d}) \in
{\mathbb Z}^{d}$ the
property
$\tau_{i}T = T$ where $\tau_{i}$ is the shift operator,
see \cite[p. 229]{Schwartz}.
The same notation is used for both interpretations of periodic distributions.
Let $\delta_{p}$ be the periodic delta function of period 1, defined
by its action $< \delta_{p} , \psi > = \psi(0)$
on continuous periodic test functions $\psi \in C({\mathbb T}^{d})$.
We use the notation $C_{p} = C ({\mathbb T}^{d})$ for the continuous periodic functions,
$C^{\infty}_{p}= C^{\infty}({\mathbb T}^{d})$ for periodic test functions
and $M_{p} = M^{1} ({\mathbb T}^{d})$ for the periodic measures,
with period $1$ in each coordinate direction.
Recall that $C_{p}$ is separable and that bounded sets in
$M_{p} = \big (C_{p} \big )^{*}$ are sequentially precompact in the weak-$\star$
topology of $M_{p}$.
\subsubsection{The double-scale kinetic decomposition}
\label{ss11}
Our objective is to define the product
\begin{equation}
\label{decomp}
f_{{\varepsilon}}(x, v) = u_{{\varepsilon}} (x) \delta_{p} \big (v - \frac{x }{{\varepsilon}} \big )
\end{equation}
which, in view of the periodicity of $\delta_{p}$ and
$\frac{\chi_{{\varepsilon}}}{{\varepsilon}} \in {\mathbb Z}^{d}$,
coincides with \eqref{decompose}. Products of distributions are not in general well defined.
However, due to the tensor-product-like structure the product in \eqref{decomp} is well
defined by using the Schwartz kernel theorem \cite[Ch V]{Hormander}. We defer the details
for the example in section \ref{ss12}, and note that
the definition of \eqref{decomp}
is effected for $u_{{\varepsilon}} \in L^{1}_{loc}(\Omega)$ by setting
\begin{equation}
<f_{{\varepsilon}}, \theta> = \int_{ {\mathbb R} ^{d}} u_{{\varepsilon}}(x)
\theta \big (x , \frac{x }{{\varepsilon}} \big ) \, dx
\end{equation}
for $\theta (x,v) \in C^{\infty}_{c} (\Omega ; C^{\infty} { ({\mathbb T}^{d})})$.
Moreover, we have the differentiation formula:
\begin{lemma}
\label{lem1}
For $u_{{\varepsilon}} \in W^{1,1}_{loc}(\Omega)$,
$$
\big ( \nabla_{x} + \frac{1}{{\varepsilon}} \nabla_{v} \big )
\left (u_{{\varepsilon}} \delta_{p} \big (v - \frac{x }{{\varepsilon}} \big ) \right ) =
(\nabla_{x} u_{{\varepsilon}}) \delta_{p} \big (v - \frac{x }{{\varepsilon}} \big )
$$
\end{lemma}
\begin{proof}
For $\theta (x,v) \in C^{\infty}_{c} (\Omega ; C^{\infty}_{p})$, we
have
$$
\begin{aligned}
< \big ( \nabla_{x} + \frac{1}{{\varepsilon}} \nabla_{v} \big ) f_{{\varepsilon}} , \theta>
&=
- < f_{{\varepsilon}} , \big ( \nabla_{x} + \frac{1}{{\varepsilon}} \nabla_{v} \big ) \theta>
\\
&= - \int_{ {\mathbb R} ^{d}} u_{{\varepsilon}}(x)
\Big (\nabla_{x} \theta + \frac{1}{{\varepsilon}} \nabla_{v} \theta \Big)
\big (x , \frac{x }{{\varepsilon}} \big ) \, dx
\\
&= - \int_{ {\mathbb R} ^{d}} u_{{\varepsilon}}(x) \nabla_{x}
\big ( \theta \big (x , \frac{x }{{\varepsilon}} \big ) \big ) \, dx
\\
&= \int_{ {\mathbb R} ^{d}} (\nabla_{x} u_{{\varepsilon}} )(x)
\theta \big (x , \frac{x }{{\varepsilon}} \big ) \, dx
\\
&= < (\nabla_{x} u_{{\varepsilon}} ) \delta_{p} \big (v - \frac{x }{{\varepsilon}} \big ) , \theta >
\end{aligned}
$$
\end{proof}
\subsubsection{A generalization}
\label{ss12}
Let $\alpha (x) $ be a smooth vector field and
$u_{{\varepsilon}} \in L^{1}_{loc}(\Omega)$. We proceed to define the product
\begin{equation}
\label{vardecomp}
f_{{\varepsilon}} = u_{{\varepsilon}} (x) \delta_{p} \big (v - \frac{\alpha (x) }{{\varepsilon}} \big ) \, .
\end{equation}
Naturally it should act on tensor products $\varphi \otimes \psi$ of test functions via
the formula
\begin{equation}
\label{deftensor}
< f_{{\varepsilon}}, \varphi \otimes \psi > =
\int_{ {\mathbb R} ^{d}} u_{{\varepsilon}}(x) \varphi (x) \psi \big( \frac{\alpha (x) }{{\varepsilon}} \big )
\, dx \, .
\end{equation}
To define \eqref{vardecomp}, we employ the Schwartz kernel theorem
\cite[Thm 5.2.1]{Hormander}.
Consider the linear map
$$
\mathcal{K} : C^{\infty}({\mathbb T}^{d}) \to \mathcal{D}'(\Omega)
\quad \text{defined by } \quad \mathcal{K} \psi = u_{{\varepsilon}}(x)
\psi \big ( \frac{\alpha(x)}{{\varepsilon}} \big )
$$
If $\psi_{n} \to 0$ in $C^{\infty}({\mathbb T}^{d})$ then $\mathcal{K} \psi_{n} \to 0$
in $\mathcal{D}'(\Omega)$. The kernel theorem implies that there exists a {\it unique}
distribution $K$ such that $< K , \varphi \otimes \psi > = ( \mathcal{K} \psi , \varphi)$,
that is $K$ acts on tensor products via \eqref{deftensor} and is the desired product.
It satisfies, for $\theta \in C^{\infty}_{c}(\Omega ; C^{\infty}({\mathbb T}^{d}))$,
\begin{equation}
\label{defn2}
< u_{{\varepsilon}} \delta_{p} \big (v - \frac{\alpha (x) }{{\varepsilon}} \big ), \theta > =
\int_{ {\mathbb R} ^{d}} u_{{\varepsilon}}(x) \theta \big(x, \frac{\alpha (x) }{{\varepsilon}} \big ) \, dx
\, ,
\end{equation}
which can also serve as a direct definition of $f_{{\varepsilon}}$.
Of course smoothness of $a(x)$ is required for the above definition: at
least $a \in C(\Omega ; {\mathbb R} ^d)$ if $f_{\varepsilon}$ is interpreted as a measure, and
more if $f_{\varepsilon}$ is interpreted as a distribution and we need to take derivatives.
We now prove.
\begin{lemma}
\label{lem2}
Let $u_{{\varepsilon}} \in W^{1,1}_{loc}(\Omega)$ and
$\alpha \in C^1 (\Omega ; {\mathbb T}^{d})$. Then
$$
\big ( \nabla_{x} + \frac{1}{{\varepsilon}} (\nabla \alpha)^{T} \nabla_{v} \big )
\left (u_{{\varepsilon}} \delta_{p} \big (v - \frac{\alpha(x)}{{\varepsilon}} \big ) \right ) =
(\nabla_{x} u_{{\varepsilon}}) \delta_{p} \big (v - \frac{\alpha(x) }{{\varepsilon}} \big )
$$
\end{lemma}
\begin{proof}
For the $k$-th coordinate, we have
$$
\begin{aligned}
<
\Big ( \partial_{x_{k}}
&+ \frac{1}{{\varepsilon}} \sum_{j} \frac{\partial \alpha_{j}}{\partial x_{k}} \partial_{v_{j}}
\Big ) u_{{\varepsilon}} \delta_{p} \big (v - \frac{\alpha(x)}{{\varepsilon}} \big ) , \theta
>
\\
&=
- < u_{{\varepsilon}} \delta_{p} \big (v - \frac{\alpha(x)}{{\varepsilon}} \big ) ,
\partial_{x_{k}} \theta + \frac{1}{{\varepsilon}} \sum_{j} \partial_{v_{j}}
\Big ( \frac{\partial \alpha_{j}}{\partial x_{k}} \theta \Big )
>
\\
&= - \int_{ {\mathbb R} ^{d}} u_{{\varepsilon}} (x)
\partial_{x_{k}} \Big ( \theta(x, \frac{\alpha(x)}{{\varepsilon}} ) \Big ) \, dx
\\
&= \int_{ {\mathbb R} ^{d}} (\partial_{x_{k}}u_{{\varepsilon}}) (x)
\ \theta(x, \frac{\alpha(x)}{{\varepsilon}} ) \, dx
\\
&= < (\partial_{x_{k}}u_{{\varepsilon}}) \delta_{p} \big (v - \frac{\alpha(x)}{{\varepsilon}} \big )
, \theta>
\end{aligned}
$$
\end{proof}
\subsubsection{A multiscale kinetic decomposition}
\label{ss13}
We pursue next the construction of decompositions in cases when more than
two scales are involved. Suppose that for an oscillating family $\{ u_{{\varepsilon}} \}$ we wish to
focus on oscillations at the scales $1$, $\frac{x}{{\varepsilon}}$ and $\frac{x}{{\varepsilon}^{2}}$.
We define
\begin{equation}
\label{defn3sc}
f_{{\varepsilon}} (x,v,w) =
u_{{\varepsilon}} (x)
\delta_{p} \big( v - \frac{x }{{\varepsilon}} \big)
\delta_{p} \big( w - \frac{v }{{\varepsilon}} \big) \, ,
\qquad x \in \Omega, \, v\in {\mathbb T}^{d}, w\in{\mathbb T}^{d} \, ,
\end{equation}
or, in terms of the action on test functions,
$\theta (x,v,w) \in
C^{\infty}_{c}(\Omega ; C^{\infty} ({\mathbb T}^{d}\times {\mathbb T}^{d}))$ via the formula
\begin{equation}
\label{defnpro3sc}
<f_{{\varepsilon}} , \theta> = \int_{ {\mathbb R} ^{d}}
\theta \big(x, \frac{x}{{\varepsilon}} , \frac{x}{{\varepsilon}^{2}}\big) \, dx
\, .
\end{equation}
In a straightforward generalization of Lemma \ref{lem1}, $f_{{\varepsilon}}$ satisfies, for
$u_{{\varepsilon}} \in W^{1,1}_{loc}(\Omega)$, the
differentiation formula
\begin{equation}
\begin{aligned}
\big ( \nabla_{x} + \frac{1}{{\varepsilon}} \nabla_{v}
+ \frac{1}{{\varepsilon}^{2}} \nabla_{w} \big)
f_{{\varepsilon}}
=
(\nabla_{x} u_{{\varepsilon} })(x)
\delta_{p} \big( v - \frac{x}{{\varepsilon}} \big)
\delta_{p} \big( w - \frac{v}{{\varepsilon}} \big)
\, .
\end{aligned}
\end{equation}
To motivate the definition \eqref{defn3sc} consider for simplicity the
case that $1/{\varepsilon}$ is an integer. Fix a first grid of size ${\varepsilon}$ and introduce
the quantities $\chi_{{\varepsilon}}(x)$ and
$v = \frac{x - \chi_{{\varepsilon}}}{{\varepsilon}} \in {\mathbb T}^{d}$ as before.
To focus on the scale $\frac{x}{{\varepsilon}^{2}}$ we consider a second grid of length ${\varepsilon}^{2}$
embedded in the first grid. When $1/ {\varepsilon}$ is an integer,
the grids fit perfectly onto one-another.
Define the function $\psi_{{\varepsilon}} : {\mathbb T}^{d} \to {\varepsilon} {\mathbb Z}^{d}$ that takes the generic point $v$
to the center of the inner cell containing $v$, and introduce a second inner variable
$w = \frac{v - \psi_{{\varepsilon}}(v)}{{\varepsilon}} \in {\mathbb T}^{d}$ describing the vector distance between
$v$ and the center of the inner cell containing $v$ in units of length ${\varepsilon}$. The process defines
a decomposition of the physical space $x \mapsto (\chi_{{\varepsilon}}(x), v, \psi_{{\varepsilon}}(v), w)$,
and allows to define a kinetic function representing three scales by
$$
\begin{aligned}
f_{{\varepsilon}} (x,v,w) &= u_{{\varepsilon}} (x)
\delta_{p} \big( v - \frac{x - \chi_{{\varepsilon}}}{{\varepsilon}} \big)
\delta_{p} \big( w - \frac{v - \psi_{{\varepsilon}}}{{\varepsilon}} \big)
\\
&=
u_{{\varepsilon}} (x)
\delta_{p} \big( v - \frac{x }{{\varepsilon}} \big)
\delta_{p} \big( w - \frac{v }{{\varepsilon}} \big) \, ,
\qquad x \in \Omega, \, v\in {\mathbb T}^{d}, w\in{\mathbb T}^{d} \, ,
\end{aligned}
$$
This definition is also good when $1/ {\varepsilon}$ is not an integer as can be seen
by the formula \eqref{techn2} in the appendix.
\subsection{Multiscale analysis of uniformly bounded families of functions}
Nguetseng \cite{Nguetseng} introduced the notion of double scale limit,
which has been a very effective technical tool in the development of periodic homogenization
theory.
His approach does not use the kinetic decomposition \eqref{decompose}, but
the double-scale limit is precisely the weak limit of the measures introduced
in \eqref{decompose}.
For this reason, we will review the results of Nguetseng \cite{Nguetseng}
from the perspective of the theory presented here, and produce some further asymptotic analysis
of kinetic decompositions for uniformly bounded families of functions.
In the sequel, the notation $u_{{\varepsilon}} \in_{b} X$ means that the family $\{u_{{\varepsilon}}\}$
belongs in a bounded set of the Banach space $X$.
\subsubsection{Uniform $L^{2}$-bounds}
Suppose first that $\{u_{{\varepsilon}}\}$ satisfies $u_{{\varepsilon}} \in_{b} L^{2}(\Omega)$.
We define $f_{{\varepsilon}}$ by \eqref{decomp} and note that
\begin{equation}
\label{l2bound}
f_{{\varepsilon}} \in_{b} L^{2} (\Omega ; M_{p}) \, .
\end{equation}
The Riesz representation theory asserts that there is an isometric isomorhism between the
dual of $C_{p}=C({\mathbb T}^{d})$ and the Banach space of periodic Radon measures
$M_{p} = M^{1} ({\mathbb T}^{d})$ on the torus. Since $C_{p}$ is separable, bounded sets in $M_{p}$
are sequentially precompact in the weak-$\star$ topology of $M_{p}$. Also, since $C_{p}$
is separable, so is $L^{2} (\Omega ; C_{p} )$ and thus bounded sets in $L^{2} (\Omega ; M_{p})$
are sequentially precompact in the weak-$\star$ topology of $L^{2} (\Omega ; M_{p})$.
As a consequence \eqref{l2bound} implies that, along a subsequence,
\begin{equation}
\label{weakst}
f_{{\varepsilon}} \rightharpoonup f \quad \text{ weak-$\star$ in $L^{2} (\Omega ; M_{p})$}
\end{equation}
with $f \in L^{2} (\Omega ; M_{p})$, that is
\begin{equation}
\label{weakstdet}
\begin{aligned}
< f_{{\varepsilon}} , \theta > &= \int u_{{\varepsilon}}(x) \theta (x, \frac{x}{{\varepsilon}}) \, dx
\\
&\to <f, \theta> = \iint f(x,v) \theta(x,v) \, dx dv
\qquad \text{ for $\theta \in L^{2} (\Omega ; C_{p})$}
\end{aligned}
\end{equation}
\bigskip
\noindent
{\sc Examples.}
A few examples will illustrate the properties of this convergence.
1.
Note first that
\begin{equation}
\label{ex1}
\delta_{p} \big( v - \frac{x}{{\varepsilon}} \big) \rightharpoonup 1
\qquad \text{weak-$\star$ in $L^{\infty}(\Omega ; M_{p})$} \, ,
\end{equation}
that is
\begin{equation}
\label{ex1det}
< \delta_{p} \big( v - \frac{x}{{\varepsilon}} \big) , \theta > =
\int_{\Omega} \theta (x, \frac{x}{{\varepsilon}}) dx \to
\int_{\Omega} \int_{{\mathbb T}^{d}} \theta (x,v) dx dv \, ,
\end{equation}
for $\theta \in L^{1}(\Omega ; C_{p})$. This is a classical result, see \cite{BLL},
and a proof is provided for completeness in the appendix.
2. If $u_{{\varepsilon}} \to u$ strongly in $L^{2}(\Omega)$, then
\begin{equation}
\label{ex3}
u_{{\varepsilon}} (x) \delta_{p} \big( v - \frac{x}{{\varepsilon}} \big) \rightharpoonup u(x)
\qquad \text{weak-$\star$ in $L^{2} (\Omega ; M_{p})$}.
\end{equation}
Indeed, since $u_{{\varepsilon}} \to u$ in $L^{2}(\Omega)$ and
$\psi \big( \frac{x}{{\varepsilon}} \big) \rightharpoonup \int_{T^{d}} \psi (v) dv$
weakly in $L^{2} (\Omega)$, for $\theta = \varphi \otimes \psi$ a tensor
product
$$
\begin{aligned}
< u_{{\varepsilon}} \delta_{p} \big( v - \frac{x}{{\varepsilon}}\big ) , \varphi \otimes \psi >
&=
\int_{ {\mathbb R} ^{d}} u_{{\varepsilon}} (x) \psi \big( \frac{x}{{\varepsilon}} \big) \varphi(x) dx
\\
&\to \int_{ {\mathbb R} ^{d}} \int_{{\mathbb T}^{d}} u(x) \varphi(x) \psi(v) dx dv
\end{aligned}
$$
Finite sums of tensor products $\sum_{j} \varphi_{j} \otimes \psi_{j}$ are dense
in $L^{2}(\Omega ; C_{p} )$ and
\eqref{ex3} follows by a density argument.
3. For $u_{{\varepsilon}} = a(\frac{x}{{\varepsilon}})$. where $a(v)$ is a periodic function, we obtain
\begin{equation}
\label{weakex4}
a(\frac{x}{{\varepsilon}}) \rightharpoonup \int_{{\mathbb T}^{d}} a(v) dv
\end{equation}
and, by \eqref{techn1},
\begin{equation}
\label{ex4}
a(\frac{x}{{\varepsilon}}) \delta_{p} \big( v - \frac{x}{{\varepsilon}}\big ) \rightharpoonup
a(v)
\end{equation}
Observe that the weak limit \eqref{weakex4} retains only the information of the average of $a$
while the double scale kinetic limit \eqref{ex4} also retains the information of the shape of $a$.
Equation \eqref{techn2} in the appendix
indicates that if we were to prompt $u_{{\varepsilon}} = a(\frac{x}{{\varepsilon}})$ with test functions oscillating
on a scale different than ${\varepsilon}$ then the information on the shape of $a$ is lost and only the
average is perceived.
Therefore, the double scale decomposition works well when the scale of oscillations
are {\it a-priori} known, so that the right oscillatory scale is prompted.
4. Finally, let $u_{{\varepsilon}} = a(\frac{x}{{\varepsilon}}) b(\frac{x}{{\varepsilon}^{2}})$ where
$a$ and $b$ are periodic functions. Then \eqref{techn1} and \eqref{techn2} give
\begin{align*}
a(\frac{x}{{\varepsilon}}) b(\frac{x}{{\varepsilon}^{2}}) &\rightharpoonup \int_{{\mathbb T}^{d}} a(y)dy \;
\int_{{\mathbb T}^{d}} b(z)dz
\\
a(\frac{x}{{\varepsilon}}) b(\frac{x}{{\varepsilon}^{2}}) \delta_{p} \big( v - \frac{x}{{\varepsilon}}\big )
&\rightharpoonup a(v) \int_{{\mathbb T}^{d}} b(z)dz
\\
a(\frac{x}{{\varepsilon}}) b(\frac{x}{{\varepsilon}^{2}}) \delta_{p} \big( v - \frac{x}{{\varepsilon}}\big )
\delta_{p} \big( w - \frac{v}{{\varepsilon}}\big )
&\rightharpoonup a(v) \, b(w)
\end{align*}
\bigskip
In the following proposition, we give a simplified proof of \cite[Thm 1]{Nguetseng}
concerning the double-scale limit for sequences that are uniformly bounded in $L^{2}(\Omega)$.
\begin{proposition}
\label{prop1}
Let $u_{{\varepsilon}} \in_{b} L^{2}(\Omega)$. Then, along a subsequence,
$$
f_{{\varepsilon}} \rightharpoonup f \qquad \text{ weak-$\star$ in $L^{2} (\Omega \, ; M_{p})$}
$$
with $f \in L^{2} (\Omega \times {\mathbb T}^{d})$.
\end{proposition}
\begin{proof}
Let $\theta \in C^{\infty}_{c} (\Omega ; C^{\infty}_{p} )$ be a test function. Then
$$
<f_{{\varepsilon}} , \theta> = \int_{\Omega} u_{{\varepsilon}} (x) \theta(x, \frac{x}{{\varepsilon}}) dx
$$
and
$$
\begin{aligned}
|<f_{{\varepsilon}} , \theta>| &\le \| u_{{\varepsilon}} \|_{L^{2}(\Omega)}
\left ( \int_{\Omega} |\theta (x, \frac{x}{{\varepsilon}})|^{2} dx \right )^{1/2}
\\
&\le C \left ( \int_{\Omega} |\theta (x, \frac{x}{{\varepsilon}})|^{2} dx \right )^{1/2}
\\
&\stackrel{\eqref{ex3}}{\to} C \left ( \int_{\Omega} \int_{{\mathbb T}^{d}}
|\theta (x, v)|^{2} dx dv \right )^{1/2}
\end{aligned}
$$
Hence, $f_{{\varepsilon}} \in_{b} \left ( L^{2} (\Omega ; C_{p}) \right )^{*}$,
$f_{{\varepsilon}} \rightharpoonup f$ weak-$\star$ in $L^{2} (\Omega ; M_{p})$
and $f \in L^{2} (\Omega ; M_{p})$. Moreover,
$$
\frac{ |<f , \theta>| }{ \| \theta \|_{ L^{2} (\Omega \times {\mathbb T}^{d}) } }
=
\lim \frac{ |<f_{{\varepsilon}} , \theta>| }{ \| \theta \|_{ L^{2} (\Omega \times {\mathbb T}^{d}) } }
\le C
$$
and $f \in L^{2} (\Omega \times {\mathbb T}^{d})$.
\end{proof}
\subsubsection{ Uniform $H^{1}$-bounds}
Next consider the case of families $\{u_{{\varepsilon}}\}$ that are uniformly bounded
in $H^{1} (\Omega)$. The first proposition is essentially
a rephrasing of \cite[Thm 3]{Nguetseng}.
\begin{proposition}
\label{prop2}
Let $u_{{\varepsilon}} \in_{b} H^{1} (\Omega)$. Then, there exist $u \in H^{1} (\Omega)$,
$\pi \in L^{2} (\Omega ; H^{1}({\mathbb T}^{d}))$ such that, along a subsequence,
$$
\begin{aligned}
f_{{\varepsilon}} = u_{{\varepsilon}} \delta_{p} \big( v - \frac{x}{{\varepsilon}} \big)
&\rightharpoonup u(x)
\qquad \text{ weak-$\star$ in $L^{2} (\Omega ; M_{p} )$ }
\\
(\nabla_{x} u_{{\varepsilon}} - \nabla_{x} u ) \delta_{p} \big( v - \frac{x}{{\varepsilon}} \big)
&\rightharpoonup \nabla_{v} \pi (x,v)
\qquad \text{ weak-$\star$ in $L^{2} (\Omega ; M_{p} )$ }
\end{aligned}
$$
\end{proposition}
\begin{proof}
Along subsequences (whenever necessary)
$u_{{\varepsilon}} \rightharpoonup u$ weakly in $H^{1}(\Omega)$,
$u_{{\varepsilon}} \to u$ in $L^{2}(\Omega)$ and
$u_{{\varepsilon}} \delta_{p} \big( v - \frac{x}{{\varepsilon}} \big) \rightharpoonup u$
weak-$\star$ in $L^{2}(\Omega ; M_{p} )$.
Moreover, Proposition \ref{prop1} implies
$$
g_{i, {\varepsilon}} : =
\big( \frac{\partial u_{{\varepsilon}}}{\partial x_{i}} - \frac{\partial u}{\partial x_{i}} \big)
\delta_{p} \big( v - \frac{x}{{\varepsilon}} \big)
\rightharpoonup g_{i} (x,v)
$$
weak-$\star$ in $L^{2}(\Omega ; M_{p})$ with $g_{i} \in L^{2} (\Omega \times {\mathbb T}^{d}) $.
For $\varphi \in C_{c}^{\infty}(\Omega)$, $\psi_{i} \in C^{\infty}_{p}$, we have
$$
\begin{aligned}
- \int_{\Omega} (u_{{\varepsilon}} - u)
&\left [
\varphi (x) \frac{1}{{\varepsilon}} \frac{\partial \psi_{i}}{\partial v_{i}} \big( \frac{x}{{\varepsilon}} \big)
+ \frac{\partial \varphi}{\partial x_{i}}(x) \psi_{i} \big( \frac{x}{{\varepsilon}} \big)
\right ] \, dx dv
\\
&= \int_{\Omega} \frac{\partial(u_{{\varepsilon}} - u) }{\partial x_{i}}
\varphi (x) \psi_{i} \big( \frac{x}{{\varepsilon}} \big) \; dx dv
\\
&\to
\int_{ {\mathbb R} ^{d}} \int_{{\mathbb T}^{d}} g_{i}(x,v) \varphi (x) \psi_{i} (v) dx dv
\end{aligned}
$$
We apply the above formula to a test function $\Psi = (\psi_{1}, ... , \psi_{d})$
that satisfies $\div_{v} \Psi = 0$.
Then
$$
- \int_{ {\mathbb R} ^{d}} (u_{{\varepsilon}} - u)
\left [
\varphi (x) \frac{1}{{\varepsilon}} \frac{\partial \psi_{i}}{\partial v_{i}} \big( \frac{x}{{\varepsilon}} \big)
+ \frac{\partial \varphi}{\partial x_{i}}(x) \psi_{i} \big( \frac{x}{{\varepsilon}} \big)
\right ] \, dx dv \to 0
$$
and we conclude that for a.e. $x \in \Omega$
\begin{equation}
\int_{{\mathbb T}^{d}} \sum_{i} g_{i}(x,v) \psi_{i}(v) \, dv = 0
\quad \text{for any $\Psi$ with $\div_{v} \Psi = 0$}.
\end{equation}
A lemma from \cite[Lemma 4]{Nguetseng} then implies
there exists $\pi \in L^{2}(\Omega ; H^{1}({\mathbb T}^{d}))$ such that
$G = (g_{1}, ... , g_{d}) = \nabla_{v} \pi $.
\end{proof}
The next proposition is novel and establishes the asymptotics of $f_{{\varepsilon}}$
when the family $\{ u_{{\varepsilon}}\}$ is uniformly bounded in $H^{1}(\Omega)$.
\begin{proposition}
\label{prop3}
Let $u_{{\varepsilon}} \in_{b} H^{1} (\Omega)$. Then
\begin{equation}
\label{propasym}
\begin{aligned}
\frac{1}{{\varepsilon}}(\delta_{p}(v - \frac{x}{{\varepsilon}}) - 1)
&\to 0 \quad \text{in ${\mathcal D}'$}
\\
g_{{\varepsilon}} := u_{{\varepsilon}}(x) \frac{1}{{\varepsilon}}(\delta_{p}(v - \frac{x}{{\varepsilon}}) - 1)
&\in_{b} H^{-1} (\Omega ; M_{p} )
\\
g_{{\varepsilon}} &\rightharpoonup g \quad
\text{ weak-$\star$ in $H^{-1} (\Omega ; M_{p})$}
\\
g &\in L^{2} (\Omega ; H^{1} ({\mathbb T}^{d})) \, ,
\end{aligned}
\end{equation}
and $f_{{\varepsilon}}$ enjoys the asymptotic expansion
$$
f_{{\varepsilon}} = u_{{\varepsilon}} + {\varepsilon} g + o({\varepsilon}) \quad \text{in ${\mathcal D}'$} \, .
$$
\end{proposition}
\begin{proof}
It is instructive to first give a quick proof for the case of one space dimension. Consider
the function $H(v) = v$, $v \in [0,1]$ and let $H_{p}$ denote its periodic extension
of period $1$. $H_{p} \in L^{\infty}( {\mathbb R} )$ satisfies
$\partial_{v} H_{p} (v) = 1 - \delta_{p} (v)$ and
\begin{equation}
\label{diden}
\frac{1}{{\varepsilon}} \big ( \delta_{p} (v - \frac{x}{{\varepsilon}}) - 1 \big )
=
\partial_{x} H_{p} ( v - \frac{x}{{\varepsilon}} )
\end{equation}
Using standard properties of weak
convergence we obtain for $\theta \in C^{1}_{c}(\Omega , C_{p})$
$$
< H_{p} (v - \frac{x}{{\varepsilon}}) , \theta >
\to
\int_{\Omega}\int_{{\mathbb T}^{d}} \int_{{\mathbb T}^{d}} H_{p} (v-w) \theta(x,v) dw dv dx
$$
and
$$
\begin{aligned}
< \frac{1}{{\varepsilon}} \big ( \delta_{p} (v - \frac{x}{{\varepsilon}}) - 1 \big ) , \theta >
&=
< \partial_{x} H_{p} ( v - \frac{x}{{\varepsilon}} ) , \theta >
\\
&=
- < H_{p} ( v - \frac{x}{{\varepsilon}} ) , \partial_x \theta >
\\
&\rightharpoonup
- \int_{\Omega} \int_{{\mathbb T}^{d}} \int_{{\mathbb T}^{d}} H_{p} (v-w) \partial_{x} \theta(x, v) dw dv dx
\\
&= 0 \, .
\end{aligned}
$$
Using lemma \ref{lem1} and \eqref{diden}
we have, for $u_{{\varepsilon}} \in H^{1}(\Omega)$, the identities
$$
\big ( \partial_{x} + \frac{1}{{\varepsilon}} \partial_{v} \big )
\big ( u_{{\varepsilon}} H_{p} (v - \frac{x}{{\varepsilon}}) \big )
= (\partial_{x} u_{{\varepsilon}}) H_{p} (v - \frac{x}{{\varepsilon}})
$$
and
$$
u_{{\varepsilon}} \frac{1}{{\varepsilon}} \big ( \delta_{p} (v - \frac{x}{{\varepsilon}}) - 1 \big )
=
\partial_{x} \Big ( u_{{\varepsilon}} H_{p} (v - \frac{x}{{\varepsilon}}) \Big )
- (\partial_{x} u_{{\varepsilon}}) H_{p} (v - \frac{x}{{\varepsilon}}) \, .
$$
The remaining three properties are direct consequences of the last formula.
Consider next the multi-dimensional case.
For $\theta \in C^{1}_{c}(\Omega ; C_{p} )$,
let ${\rm supp \,} \theta$ denote the support (in $x$) of $\theta$, and fix
${\varepsilon} < \frac{1}{\sqrt{d}} \, {\rm dist \,} ( {\rm supp \,} \theta , \partial \Omega)$.
We cover ${\rm supp \,} \theta$ by cubes $C_{k}$ centered at points $\chi_{k} \in {\varepsilon} {\mathbb Z}^{d}$
of latteral size ${\varepsilon}$.
The number of the cubes covering ${\rm supp \,} \theta$ is of the order
${\varepsilon}^{-d} O(|{\rm supp \,} \theta|)$, and the covering is arranged so that
${\rm supp \,} \theta \subset \cup_{k=1}^{N} C_{k} \subset \Omega$. We observe that
by construction $\frac{1}{{\varepsilon}} \chi_{k} \in {\mathbb Z}^{d}$ and compute
$$
\begin{aligned}
<g_{{\varepsilon}} & , \theta >
= \int_{\Omega} u_{{\varepsilon}} (x) \frac{1}{{\varepsilon}}
\Big (
\theta(x, \frac{x}{{\varepsilon}}) - \int_{{\mathbb T}^{d}} \theta(x,v) dv
\Big )
dx
\\
&= \sum_{k \in {\mathbb Z}^{d}} \int_{C_{k}} u_{{\varepsilon}}(x) \frac{1}{{\varepsilon}}
\Big (
\theta(x, \frac{x}{{\varepsilon}}) - \int_{{\mathbb T}^{d}} \theta(x,v) dv
\Big )
dx
\\
&= \sum_{k \in {\mathbb Z}^{d}} {\varepsilon}^{d} \int_{{\mathbb T}^{d}}
u_{{\varepsilon}} (\chi_{k} + {\varepsilon} w)
\frac{1}{{\varepsilon}}
\Big (
\theta(\chi_{k} + {\varepsilon} w , w) - \int_{{\mathbb T}^{d}} \theta(\chi_{k} + {\varepsilon} w , v) dv
\Big ) dw
\\
&= \sum_{k \in {\mathbb Z}^{d}} {\varepsilon}^{d} \int_{{\mathbb T}^{d}}
\frac{1}{{\varepsilon}}
\Big (
(u_{{\varepsilon}} \theta)(\chi_{k} + {\varepsilon} w , w)
- \int_{{\mathbb T}^{d}} (u_{{\varepsilon}} \theta)(\chi_{k} + {\varepsilon} \rho , w) d\rho
\Big )
dw
\end{aligned}
$$
We employ the Poincar\'e inequality
$$
\left | v(z) - \int_{{\mathbb T}^{d}} v(z') dz' \right | \le
\int_{{\mathbb T}^{d}} | \nabla v| (z') dz'
$$
for $v(z) = (u_{{\varepsilon}} \theta)(\chi_{k} + {\varepsilon} z , v)$ to obtain
$$
\begin{aligned}
\frac{1}{{\varepsilon}} \Big |
(u_{{\varepsilon}} \theta)(\chi_{k} + {\varepsilon} z , v)
&- \int_{{\mathbb T}^{d}} (u_{{\varepsilon}} \theta)(\chi_{k} + {\varepsilon} \rho , v) d\rho
\Big |
\\
&\le
\int_{{\mathbb T}^{d}} |\nabla_{x} (u_{{\varepsilon}} \theta)| (\chi_{k} +{\varepsilon} \rho , v) d\rho
\end{aligned}
$$
and
$$
\begin{aligned}
| <g_{{\varepsilon}} , \theta > |
&\le
\sum_{k \in {\mathbb Z}^{d}} {\varepsilon}^{d} \int_{{\mathbb T}^{d}} \int_{{\mathbb T}^{d}}
| \nabla_{x} (u_{{\varepsilon}} \theta) | (\chi_{k} + {\varepsilon} \rho , w) \, d\rho dw
\\
&= \int_{\Omega} \int_{{\mathbb T}^{d}} |\nabla_{x} (u_{{\varepsilon}} \theta)|(x,v) \, dxdv
\\
&\le \| u_{{\varepsilon}} \|_{H^{1}(\Omega)}
\left ( \|\theta \|_{L^{2}(\Omega \times {\mathbb T}^{d})}
+ \|\nabla_{x} \theta \|_{L^{2}(\Omega \times {\mathbb T}^{d})} \right )
\end{aligned}
$$
From here we obtain \eqref{propasym}$_{2}$, \eqref{propasym}$_{3}$ (along a subsequence),
and that
$$
g \in H^{-1} (\Omega ; L^{2} ({\mathbb T}^{d}))
$$
In addition, we have
$$
\begin{aligned}
\nabla_{v} g_{{\varepsilon}} &= {\varepsilon} \big ( \nabla_{x} + \frac{1}{{\varepsilon}} \nabla_{v} \big ) g_{{\varepsilon}}
- {\varepsilon}\nabla_{x} g_{{\varepsilon}}
\\
&= (\nabla_{x} u_{{\varepsilon}}) \big ( \delta_{p}(v - \frac{x}{{\varepsilon}}) - 1 \big )
- {\varepsilon} \nabla_{x} g_{{\varepsilon}}
\\
&\in_{b} L^{2}(\Omega ; M_{p} ) + H^{-2} (\Omega ; M_{p} )
\end{aligned}
$$
Therefore,
$$
\begin{aligned}
< \nabla_{v} g_{{\varepsilon}} , \theta > &= \int_{\Omega} \nabla_{x} u_{{\varepsilon}}(x)
\Big ( \theta(x, \frac{x}{{\varepsilon}}) - \int_{{\mathbb T}^{d}} \theta(x,v)dv \Big ) dx
+ {\varepsilon} <g_{{\varepsilon}}, \nabla_{x} \theta >
\\
| <\nabla_{v} g , \theta >| &= \lim | <\nabla_{v} g_{{\varepsilon}} , \theta >|
\\
&\le {\overline \lim} \left [
\int_{\Omega} |\nabla_{x}u_{{\varepsilon}}(x) \theta (x, \frac{x}{{\varepsilon}}) | dx
+
\int_{\Omega} \int_{{\mathbb T}^{d}} |\nabla_{x}u_{{\varepsilon}}| | \theta | dv dx
\right ]
\\
&\le C \| \theta \|_{L^{2}(\Omega \times {\mathbb T}^{d})}
\end{aligned}
$$
which implies
$$
\nabla_{v} g \in L^{2} (\Omega \times {\mathbb T}^{d})
$$
and gives using the Poincar\'e inequality the desired \eqref{propasym}$_{4}$.
To see the first property, consider a test function $\theta = \varphi \otimes \psi$
which is a tensor product of $\varphi \in C^{\infty}_{c}(\Omega)$
and $\psi \in C^{\infty} ({\mathbb T}^{d})$. Then
$$
\begin{aligned}
&<\frac{1}{{\varepsilon}} \big (\delta_{p} (v - \frac{x}{{\varepsilon}}) - 1 \big ) ,
\varphi \otimes \psi >
\\
&\quad
= \sum_{k \in {\mathbb Z}^{d}} {\varepsilon}^{d} \int_{{\mathbb T}^{d}}\frac{1}{{\varepsilon}}
\varphi (\chi_{k} + {\varepsilon} w )
\Big ( \psi(w) - \int_{{\mathbb T}^{d}} \psi \Big )
dw
\\
&\quad
=
\sum_{k \in {\mathbb Z}^{d}} {\varepsilon}^{d} \int_{{\mathbb T}^{d}}
\frac{ \varphi(\chi_{k} + {\varepsilon} w ) - \varphi(\chi_{k})
- \nabla \varphi (\chi_{k}) \cdot {\varepsilon} w}{{\varepsilon}}
\big ( \psi(w) - \int_{{\mathbb T}^{d}} \psi \big ) dw
\\
&\qquad
+ \sum_{k \in {\mathbb Z}^{d}} {\varepsilon}^{d} \nabla \varphi (\chi_{k}) \cdot
\int_{{\mathbb T}^{d}} w \Big ( \psi(w) - \int_{{\mathbb T}^{d}} \psi \Big ) dw
\\
&\quad = O({\varepsilon}) + \int_{\Omega} \nabla \varphi (x) dx \cdot
\int_{{\mathbb T}^{d}} w \Big ( \psi(w) - \int_{{\mathbb T}^{d}} \psi \Big ) dw
\\
&\quad
\to 0
\end{aligned}
$$
as $\varphi$ is of compact support. Since
$$
\frac{1}{{\varepsilon}} \big (\delta_{p} (v - \frac{x}{{\varepsilon}}) - 1 \big )
\in_{b} H^{-1} (\Omega ; M_{p}) = \left ( H_{0}^{1} (\Omega ; C_{p}) \right )^{*}
$$
and finite sums of tensor products $\sum_{j} \varphi_{j} \otimes \psi_{j}$ are dense
in $H_{0}^{1} (\Omega ; C_{p})$ we obtain \eqref{propasym}$_{1}$.
\end{proof}
\section{Homogenization of hyperbolic equations}
\label{homohyp}
In this section we consider certain homogenization problems for transport equations.
First we develop an example where the effective equation can be calculated with the help
of the double scale kinetic decomposition. Then we provide a counter-example where the
double scale limit is not the right object to treat the effective equation.
\subsection{Effective equation}
\label{subseffeq}
Consider the transport equation
\begin{equation}
\label{transport}
\begin{aligned}
\dt {u_{{\varepsilon}}} + a \big (\frac{x}{{\varepsilon}} \big) \cdot \nabla_{x} u_{{\varepsilon}} &= 0
\\
u_{{\varepsilon}} (0, x) &= U^{0}(x, \frac{x}{{\varepsilon}})
\end{aligned}
\end{equation}
We assume that $a(v)$ is a $C^{1}$ vector field, periodic with period $1$,
and satisfying $\div\, a = 0$, and that the initial data
$U^{0} \in L^{2}( {\mathbb R} ^{d} \times {\mathbb T}^{d})$ is 1-periodic in $v$
and satisfy the uniform bounds
\begin{equation}
\label{hypdata}
\int_{ {\mathbb R} ^{d}} | U^{0} (x, \frac{x}{{\varepsilon}}) |^{2} dx \le C \, .
\tag {h$_{d}$}
\end{equation}
Under this hypothesis standard energy estimates for \eqref{transport} imply the
uniform bound on solutions
\begin{equation}
\label{unifl2}
\int_{ {\mathbb R} ^{d}} | u_{{\varepsilon}} (t, x) |^{2} dx
\le \int_{ {\mathbb R} ^{d}} | U^{0} (x, \frac{x}{{\varepsilon}}) |^{2} dx
\le C \quad \forall t > 0 \, .
\end{equation}
We introduce
\begin{equation}
\label{dsdef}
f_{{\varepsilon}} (t,x,v) = u_{{\varepsilon}}(t,x) \delta_{p} \big( v - \frac{x}{{\varepsilon}} \big)
\quad t \in {\mathbb R} _{+}, \, x\in {\mathbb R} ^{d} , \, v\in{\mathbb T}^{d} \, ,
\end{equation}
and use Lemma \ref{lem1} to check that $f_{{\varepsilon}}$ satisfies
\begin{align}
\dt {f_{\varepsilon}} +a(v)\cdot\nabla_x f_{\varepsilon} +\frac{1}{{\varepsilon}}a(v)\cdot
\nabla_v f_{\varepsilon} &=0,\quad \text{in ${\mathcal D}'$}
\label{maineq}
\\
f_{\varepsilon}(t=0,x,v)
&=U^0(x,v)\;\delta_{p} (v-\frac{x}{{\varepsilon}})
\label{newinit}
\end{align}
with periodic boundary conditions ($v \in {\mathbb T}^d$).
The uniform bound \eqref{unifl2} implies
\begin{equation}
f_{\varepsilon} \in_{b}
L^\infty \big ( [0,\ \infty) ,\ L^2( {\mathbb R} ^d, M_{p}) \big ) \, ,
\label{bounds}
\end{equation}
and thus by Proposition \ref{prop1}, along a subsequence if necessary,
\begin{equation}
\label{weakl}
f_{{\varepsilon}} \rightharpoonup f \quad \text{weak-$\star$ in
$L^\infty \big ( [0,\ \infty) ,\ L^2( {\mathbb R} ^d, M_{p}) \big )$}
\end{equation}
with $f$ enjoying the improved regularity
\begin{equation}
\label{wlreg}
f \in L^\infty([0,\ \infty) ,\ L^2( {\mathbb R} ^d \times {\mathbb T}^{d})) \, .
\end{equation}
Our objective is to calculate the effective limit of \eqref{transport} by computing the
hydrodynamic limit problem for the kinetic equation \eqref{maineq}-\eqref{newinit}.
Note that if $f$ satisfies a well-posed problem then this provides a complete determination
of the weak limit of $u_{{\varepsilon}}$ since
$$
u_{{\varepsilon}} = \int_{{\mathbb T}^{d}} f_{{\varepsilon}} dv \rightharpoonup \int_{{\mathbb T}^{d}} f dv = u
$$
We introduce
\begin{equation}
\label{projkernel}
K=\Big\{g\in L^2({\mathbb T}^d)\quad
\Big|\ a(v)\cdot
\nabla_v g=0\ in \ {\mathcal D}'\Big\}.
\end{equation}
and remark that $K$ is the space of solutions of the cell-problem obtained by the method
of multiscale asymptotic expansion \cite{BLL} for the homogenization problem
\eqref{transport} (see \cite{HX}, \cite{E}).
Let $P$ denote the $L^{2}$-projection operator on the kernel $K$.
We prove
\begin{theorem}
\label{hyphomthm}
Under hypothesis \eqref{hypdata} the effective limit of problem \eqref{transport}
is obtained as $u = \int_{{\mathbb T}^{d}} f dv$ where
$f \in L^\infty([0,\ \infty) \, , \, L^2( {\mathbb R} ^d \times {\mathbb T}^{d}))$,
$f(t,x,\cdot) \in K$ for a.e. $(t,x)$, and $f$
is the unique solution of the kinetic problem
\begin{equation}
\begin{aligned}
\dt f + (P a) \cdot \nabla_{x} f &= 0
\\
f(t=0, x,v) &= P U^{0} (x,v) \, ,
\end{aligned}
\end{equation}
where $P$ is the projection operator on the kernel $K$.
\end{theorem}
\begin{proof} Let $f_{{\varepsilon}}$ and $f$ be as in \eqref{bounds}, \eqref{weakl}
and \eqref{wlreg}. The proof is split in three steps:
{\em Step 1~: The limit $f$ belongs to $K$.}
The kernel $K$ is defined in \eqref{projkernel}.
We may consider elements of $K$ as functions of $t,\,x$ and $v$ instead of
only $v$, as $t$ and $x$ play the role of parameter in the
definition of $K$. Thus we have
\begin{equation}
K_x=\Big\{g\in L^2( {\mathbb R} ^d\times{\mathbb T}^d)\quad
\Big|\ a(v)\cdot
\nabla_v g=0\ in\ {\mathcal D}'\Big\},
\end{equation}
and
\begin{equation}
K_{t,x}=\Big\{g\in L^\infty([0,\ \infty),\ L^2( {\mathbb R} ^d\times{\mathbb T}^d))\quad
\Big|\ a(v)\cdot
\nabla_v g=0\ in\ {\mathcal D}'\Big\}.
\end{equation}
We may also define all the
\[
K^p=\Big\{g\in L^p({\mathbb T}^d))\quad
\Big|\ a(v)\cdot
\nabla_v g=0\ in\ {\mathcal D}'\Big\},
\]
and their extensions $K_x^p$ and $K_{t,x}^p$.
The convergence \eqref{weakl} states that for $\phi$ in
$L^1([0,\ \infty),\ L^2( {\mathbb R} ^d,\ C_{p}))$ we have
\[
\int_0^\infty\int_{ {\mathbb R} ^d\times{\mathbb T}^d} \phi(t,x,v)\,df_{\varepsilon} \longrightarrow
\int_0^\infty\int_{ {\mathbb R} ^d\times{\mathbb T}^d} \phi(t,x,v)\,df.
\]
Take $\phi\in C^\infty_c([0,\ \infty)\times {\mathbb R} ^d\times{\mathbb T}^d)$ and compute
\[
\int_0^\infty\int_{ {\mathbb R} ^d\times{\mathbb T}^d} a(v)\cdot\nabla_v \phi\,df_{\varepsilon}=
-{\varepsilon}\int_0^\infty\int_{ {\mathbb R} ^d\times{\mathbb T}^d}(\partial_t\phi+a(v)\cdot
\nabla_x \phi)\,df_{\varepsilon}.
\]
Passing to the limit, we conclude that
\[
\int_0^\infty\int_{ {\mathbb R} ^d\times{\mathbb T}^d} a(v)\cdot\nabla_v \phi\,df=0.
\]
On the other hand $f \in L^\infty([0,\ \infty), L^2( {\mathbb R} ^d\times{\mathbb T}^d))$, so
$f\in K_{t,x}$.
\medskip
{\em Step 2~: The limit equation.}
Consider a function $\phi \in K_x$. We wish to mollify $\phi$ and use it
as a test function in the weak form of \eqref{maineq}. Since $K_x$ depends
only parametrically in $x$, we may select $\phi$ to be compactly
supported in $x$.
Take $H(x) \in C^\infty_c( {\mathbb R} ^d) $ with $\int_{ {\mathbb R} ^d} H(x)\,dx=1$, and
$\bar H(v) \in C^\infty_c ((0,\ 1)^d)$ with
$\int_{{\mathbb T}^d} \bar H(v)\,dv=1$.
For any $\phi$ define
\[\begin{split}
&\phi_{n}=\int_{ {\mathbb R} ^d} n^d\,H\left(n(x-y)\right)\;\phi(y,v)\,dy,\\
&\phi_{n,m}=\int_{{\mathbb T}^d}
\bar H_m(v-\eta)\;\phi_n(x,\eta)\,d\eta,\\
\end{split}
\]
with $\bar H_m(v)=m^d\sum_{k\in {\mathbb Z}^d} \bar H(m(v+k))$, periodic
and well
defined for all $m$ as $\bar H(m v)$ is compactly supported in
$(0,\ 1/m)^d$.
Then for any $\phi\in K_x$ we have
$$
\int_{ {\mathbb R} ^{d}\times {\mathbb T}^{d}} \phi(y,\eta) a(\eta) \cdot
\nabla \bar H_{m} (v-\eta) H_{n} (x-y) \, dy d\eta =0 \, .
$$
Thus, for $a$ Lipshitz continuous,
$$
\begin{aligned}
\int_{ {\mathbb R} ^d\times{\mathbb T}^d} &a(v)\cdot\nabla_v\phi_{n,m}\,df_{\varepsilon}
\\
&=
\int_{ {\mathbb R} ^d\times{\mathbb T}^d}\!\!\int_{{\mathbb T}^d} (a(v)-a(\eta))\cdot \nabla \bar H_m
\left(v-\eta\right) \phi_n(x,\eta)\,d\eta\,df_{\varepsilon}
\\
&= \int_{ {\mathbb R} ^d\times{\mathbb T}^d}\!\!\int_{{\mathbb T}^d}\int_0^1
\big(\zeta\cdot\nabla a(v-(1-t)\zeta)\big)\,
\\
&\qquad\qquad\qquad\qquad
\cdot \sum_{k\in{\mathbb Z}^{d}} m^{d+1}\nabla \bar H\left(m(\zeta +k)\right)
\phi_n(x,v-\zeta)\,dt\,d\zeta\,df_{\varepsilon}.\\
\end{aligned}
$$
Notice that $m^{d+1}\zeta\otimes \nabla\bar H(m\zeta)$ converges in
the sense of distributions toward $C\,(Id)\,\delta$ with $C$ a numerical constant.
Moreover thanks to \eqref{bounds}, and to the fact that $\phi_n\in
L^2({\mathbb T}^d,\ C_c( {\mathbb R} ^d))$ and $\nabla a\in C ({\mathbb T}^d)$, we may pass to
the limit in $m$ in the previous equality and find
\[\begin{split}
\lim_{m\rightarrow\infty}
\int_{ {\mathbb R} ^d\times{\mathbb T}^d} &a(v)\cdot\nabla_v\phi_{n,m}\,df_{\varepsilon}=C\,
\int_{ {\mathbb R} ^d\times{\mathbb T}^d} {\rm div}\,
a(v)\,\phi_n(x,v)\,df_{\varepsilon}=0,
\end{split}
\]
as $a$ is divergence free.
Multiplying \eqref{maineq} by $\phi_{n,m}$ and taking first
the limit $m\rightarrow \infty$ and then the limit ${\varepsilon}\rightarrow 0$,
we find that for any $\phi\in K_x$ compactly supported in $x$ we have
\[
\partial_t \int_{ {\mathbb R} ^d\times {\mathbb T}^d} \phi_n f\,dx\,dv -\int_{ {\mathbb R} ^d\times{\mathbb T}^d}
a(v)\cdot\nabla_x\phi_n\,f\,dx\,dv=0.
\]
This relation can now easily be extended by approximation to any $\phi \in K_x$.
Let us denote $\bar a$ the orthogonal projection on $K$ of $a$. The new
function $\bar a$ belongs to $L^\infty({\mathbb T}^d)$ as the projection
operator $P$ is continuous on every $L^p({\mathbb T}^d)$ for all $1\leq p\leq\infty$,
but does not necessarily
have any further
regularity, Lipschitz for instance (see the appendix where we recall
the basic properties of $P$). Now as $f\in K_{t,x}$ and
$\nabla_x\phi_n\in K_x$,
then $\nabla_x\phi_n \,f\in K^1_{t,x}$ and consequently
\[
\frac{d}{dt} \int_{ {\mathbb R} ^d\times {\mathbb T}^d} \phi_n f\,dx\,dv -\int_{ {\mathbb R} ^d\times{\mathbb T}^d}
\bar a(v)\cdot\nabla_x\phi_n\,f\,dx\,dv=0.
\]
On the other hand, the projection operator $P$ may be trivially extended on
$K_{t,x}$ from $K$ as $t$ and $x$ are only parameters and
of course it commutes
with derivatives in $t$ or $x$. Now, for any $\phi\in L^2( {\mathbb R} ^d\times{\mathbb T}^d)$
\[
\int_{ {\mathbb R} ^d\times{\mathbb T}^d} (\phi-P\phi)\,f\,dx\,dv=0,
\int_{ {\mathbb R} ^d\times{\mathbb T}^d} \bar a\cdot\nabla(\phi_n-P\phi_n)\,f\,dx\,dv=0.
\]
Finally for any $\phi\in L^2( {\mathbb R} ^d\times{\mathbb T}^d)$, we have that
\[
\partial_t \int_{ {\mathbb R} ^d\times {\mathbb T}^d} \phi_n f\,dx\,dv -\int_{ {\mathbb R} ^d\times{\mathbb T}^d}
\bar a(v)\cdot\nabla_x\phi_n\,f\,dx\,dv=0.
\]
This implies that $f \in K_{t,x}$ is a solution in the sense of distribution to
\begin{equation}
\partial_t f+\bar a(v)\cdot\nabla_x f=0.\label{limiteq}
\end{equation}
\medskip
{\em Step 3~: Conclusion.} Let us begin with the identification of the
initial value $f(t=0)$ which has a sense since $\partial_t f\in L^\infty([0,\
\infty),\ H^{-1}( {\mathbb R} ^d,\ L^2({\mathbb T}^d)))$ because of \eqref{limiteq} and as
$f\in L^\infty([0,\
\infty),\ L^2( {\mathbb R} ^d\times {\mathbb T}^d)))$. For every $\phi\in K_x$, as
\begin{equation}
\frac{d}{dt} \int_{ {\mathbb R} ^d\times {\mathbb T}^d} \phi_n df_{\varepsilon} =\int_{ {\mathbb R} ^d\times{\mathbb T}^d}
a(v)\cdot\nabla_x\phi_n\,df_{\varepsilon},
\label{inter}
\end{equation}
then $\int_{ {\mathbb R} ^d\times {\mathbb T}^d} \phi_n df_{\varepsilon}(t,.,.)$ has a limit as
$t\rightarrow 0$ and this limit is, thanks to \eqref{newinit}
\[
\int_{ {\mathbb R} ^d} \phi_n(x,x/{\varepsilon})\,U^0(x,x/{\varepsilon})\,dx.
\]
Moreover because of \eqref{inter}, we may pass to the limit in ${\varepsilon}$
and deduce that
\[
\int_{ {\mathbb R} ^d\times {\mathbb T}^d} \phi_n f(t,x,v)\,dxdv
\mathop{ \; \; \longrightarrow \; \;}_{t\to 0}
\int_{ {\mathbb R} ^d\times {\mathbb T}^d} \phi_n f(0,x,v)=
\int_{ {\mathbb R} ^d\times {\mathbb T}^d} \phi_n U^0(x,v).
\]
On the other hand we of course have for any $\phi\in L^2$ as $f\in K_{t,x}$
$$
0=\int_{ {\mathbb R} ^d\times {\mathbb T}^d} (\phi_n-P\phi_n) f(t,x,v)\,dxdv
\mathop{ \; \; \longrightarrow \; \;}_{t\to 0}
\int_{ {\mathbb R} ^d\times {\mathbb T}^d} (\phi_n-P\phi_n) f(0,x,v).
$$
Combining the last two equalities we get that
\begin{equation}
f(t=0,x,v)=P\,U^0(x,v).
\label{initf}
\end{equation}
Finally, we notice that Eq. \eqref{limiteq} combined with \eqref{initf} has
a unique solution in the space of distribution, through standard arguments of
kinetic theory and as, even though $\bar a$ is only bounded, it does not
depend on $x$. Therefore any extracted subsequence of
$f_{\varepsilon}$ has only one possible limit and the whole sequence $f_{\varepsilon}$ converges
toward the solution of \eqref{limiteq} with \eqref{initf}.
\end{proof}
\bigskip
\noindent
{\sc Examples.}
We calculate the equation for the double scale limit $f$ and the associated
effective equation for certain examples, always within the framework of
\eqref{transport}.
1. First consider the case that $a(v)$ is ergodic.
Then
$$
\begin{aligned}
K &= \{ g \in L^2({\mathbb T}^d) : g = const. \}
\\
P_K g &= \int_{{\mathbb T}^d} g dv = : \overline{ g }
\end{aligned}
$$
The equation for $f$ becomes
$$
\partial_t f + \overline{ a} \cdot \nabla_x f = 0
$$
and of course $u = \int_{{\mathbb T}^d} f dv$.
2. Consider next the homogenization problem
$$
\begin{aligned}
\partial_t u_{\varepsilon} + b \big ( \frac{x_2}{{\varepsilon}} \big ) \partial_{x_1} u_{\varepsilon} &= 0
\\
u_{\varepsilon} (0, x_1 , x_2) &= U^0 (x_1, x_2, \frac{x_1}{{\varepsilon}}, \frac{x_2}{{\varepsilon}})
\end{aligned}
$$
where $u_{\varepsilon} = u_{\varepsilon} (t,x)$, $x = (x_1, x_2) \in {\mathbb R} ^2$,
and the vector field $a(x_1, x_2) = (b(x_2), 0)$ corresponds
to a shear flow with $b(x_2)\neq 0$ for $a.e.\ x_2$. We compute
$$
\begin{aligned}
K &= \{ g \in L^2({\mathbb T}^2) : b(v_2) \partial_{v_1} g = 0 \}
= \{ g = \psi (v_2) \; \big | \; \forall \psi \in L^2({\mathbb T}^1) \}
\\
P_K g &= \int_0^1 g(v_1, v_2) dv_2
\end{aligned}
$$
Since $f \in K$ we conclude that $f = f(t, x_1, x_2, v_2)$ and
satisfies the problem
$$
\begin{aligned}
\partial_t f + b(v_2) \partial_{x_1} f &= 0
\\
f (0, x_1, x_2 , v_2) &= P_K U^0 = \int_0^1 U^0 (x_1, x_2, v_1, v_2) dv_1
\end{aligned}
$$
The weak limit $u = \int_{{\mathbb T}^2} f $ satisfies the
integrated equation.
3. It is possible to give a more general framework for the situation
of the previous example.
Suppose that the divergence free vector field $a$ is such that the
following description of $K$ is true: There exist
functions $\xi_1,\ldots \xi_N$, $N\leq d$ from ${\mathbb T}^d$ to
$ {\mathbb R} $. These functions are
local coordinates in the sense that they may be
completed by $\xi_{N+1},\ldots \xi_d$ and that the change of
coordinates $v$ to $(\xi_1(v),\ldots, \xi_d(v))$ is a $C^1$
diffeomorphism from ${\mathbb T}^d$ to some domain $O\subset {\mathbb R} ^d$. And finally
\[
K=\{\psi(\xi_1(v),\ldots,\xi_N(v))\;|\ \forall \psi\in L^2(O) \}.
\]
For instance in dimension $d=2$, as ${\rm div}\;a=0$, there
is always $\xi:\ {\mathbb R} ^d\rightarrow {\mathbb R} $ such that $a=\nabla^\perp
\xi$. Now if in addition
$\xi$ is a periodic regular function with
$\nabla \xi(v)\neq 0$ for all $v$, which is a non trivial assumption,
then $K$ is exactly the set of functions $\psi(\xi)$.
In that case, we may define $g(t,x,\xi)=f(t,x,V(\xi_1,\ldots,\xi_d))$
with $V$ the inverse change of variables. Then $g$ does not depend on
$\xi_{N+1},\ldots , \xi_d$ and it simply satisfies
\[
\partial_t g+b(\xi_1,\ldots,\xi_N)\cdot\nabla_x g=0,
\]
with $b(\xi)=\bar a(V(\xi))=\int a(V(\xi))d\xi_{N+1}\ldots d\xi_d$.
4. Notice now that the kernel $K$, endowed with the usual $L^2$ scalar
product, is a Hilbert space and so
that the kernel $K$ admits an orhonormal basis $\{ \psi_k (v) \}$,
possibly countable. Since $f \in K$ it will be given in a Fourier
expansion
$$
f = \sum_{k=1}^\infty m_k (t,x) \psi_k (v)
\quad \text{where $m_k = < f , \psi_k >$}.
$$
Moreover, we see that
$$
<P_K (a) f , \psi_k > = < P_K (a f) , \psi_k > = < af , \psi_k >
$$
and one computes that the set of moments $m_k$ satisfies the initial
value problem
$$
\begin{aligned}
&\partial_t m_k + \sum_{j=1}^d \Big ( \sum_{n=1}^\infty
<a_j \psi_n , \psi_k > \frac{ \partial m_n}{\partial x_j} \Big ) = 0
\\
&m_k (0, x) = <P_K U^0 (x, \cdot ) , \psi_k >
= \int_{{\mathbb T}^d} U^0(x,v)
\overline{ \psi_k (v) } dv
\end{aligned}
$$
As the wave speed $a$ is real,
$<a_j \psi_n , \psi_k > = <\psi_n , a_j \psi_k >$, and the system of moments
is an infinite symmetric hyperbolic system.
\subsection{The multiscale case: A counterexample}
\label{secce}
A natural extension of the previous analysis is to deal with transport
coefficients depending on more than one scale. Consider for example
the equation
\begin{equation}
\partial_t u_{\varepsilon}+a_{\varepsilon}\cdot\nabla_x u_{\varepsilon}=0,
\label{multiscaletransport}
\end{equation}
with $a_{\varepsilon}=a(x,x/{\varepsilon})$ and $a(x,v)$ a Lipschitz function,
or even with $a_{\varepsilon}=a(x,x/{\varepsilon},x/{\varepsilon}^2)$ (or with as many
scales as one cares to introduce). Assume again that
$\hbox{div}_v a(x,v)=0$ and $\hbox{div}_w a(x,v,w)=0$.
Is it possible to derive an equation for the double scale limit (or
for the triple scale limit when $a(x,x/{\varepsilon},x/{\varepsilon}^2)$) in the case of
\eqref{multiscaletransport}?
In fact, it is relatively easy to show that the previous approach does not
work! Everything goes as before in the beginning; upon defining
\[
f_{\varepsilon}(t,x,v,w)=u_{\varepsilon}(t,x)\,{\delta_{p}}(v-x/{\varepsilon})\,{\delta_{p}}(w-v/{\varepsilon}),
\]
as in paragraph \ref{ss13}, one simply obtains the generalized kinetic
equation
\[
\partial_t
f_{\varepsilon}+\nabla_x\cdot(a(x,v,w)f_{\varepsilon})+\frac{1}{{\varepsilon}}a\cdot\nabla_v
f_{\varepsilon}+\frac{1}{{\varepsilon}^2}a\cdot\nabla_w f_{\varepsilon}=0.
\]
However it is not always possible to derive a well posed problem for the
hydrodynamic limit, even in the simple setting where $a$ depends only on
$x,v$ and the equation for $f_{\varepsilon}(t,x,v)$ is
\[
\partial_t
f_{\varepsilon}+\nabla_x\cdot(a(x,v)f_{\varepsilon})+\frac{1}{{\varepsilon}}a(x,v)\cdot\nabla_v
f_{\varepsilon}=0.
\]
Indeed the only information that we have is that any limit $f$ belongs
to the kernel which now depends on $v$ and $x$
\[
K=\{f\in L^2( {\mathbb R} ^d\times{\mathbb T}^d)\;|\; a(x,v)\cdot\nabla_v f=0\}.
\]
On the other hand, when projecting the equation on $K$, it is not possible to
handle the term with the $x$ derivative as projection on $K$ and
differentiation in $x$ no longer commute. This is associated to the
possibility that the dimensionality of $K$ may vary with $x$.
In addition, at the level of the double scale limit, this is not a mere
technical problem, rather the double scale limit is in general not unique and
depends on the choice of the extracted subsequence in ${\varepsilon}$.
This can be simply seen for the problem
\begin{equation}
\begin{split}
&\partial_t u_{\varepsilon} +a(x)\cdot\nabla_x u_{\varepsilon}=0,\quad t\in {\mathbb R} _+,\
x\in {\mathbb R} ^2,\\
&u_{\varepsilon}(t=0,x)=U^0(x,x/{\varepsilon}).
\end{split}\label{c-ex}
\end{equation}
The oscillations are due only to the initial data as the transport
coefficient no longer depends on ${\varepsilon}$, and $u$ the weak limit
of $u_{\varepsilon}$ satisfies the same equation. Take now
\[
a_1(x)=1,\quad a_2(x)=x_1\,\mbox{{\text 1}\kern-.24em {\text l}}_{0\leq x_1\leq 1}+1\,\mbox{{\text 1}\kern-.24em {\text l}}_{x_1>1},
\]
so that $a$ is Lipschitz and divergence free, and select the initial data
\[
U^0(x,v)=K(x)\,L(v_2),
\]
with $K$ and $L$ two $C^\infty$ functions, $L$ periodic on $ {\mathbb R} $ of
period $1$ and with zero average, and $K$ compactly supported with support in $x_1$ in $[-1,\
-1/2]$.
As the average of $U^0(x,v)$ in $v$ vanishes for all $x$, the weak
limit $u$ of $u_{\varepsilon}$ is uniformly $0$.
For $1\leq t\leq 3/2$, the support in $x_1$ of the solution $u_{\varepsilon}$
is entirely in the interval $[0,\ 1]$. Therefore any double scale
limit $f$ should satisfy
\[
\partial_{v_1} f+x_1\partial_{v_2}f=0.
\]
It is easy to check that the only $L^2$ solutions to this last equations are the functions
which depend only on $x$ and not on $v$. Therefore for $1\leq t\leq
3/2$ the double scale limit is equal to $u$, {\rm i.e.} uniformly
vanishes.
Let us finally compute the double scale limit for $t>2$ and check that
it does not vanish. For that introduce the characteristics $X(t,x)$
\[
\partial_t X(t,x)=a(X(t,x)),\quad X(0,x)=x.
\]
We need the
characteristics only for those $x$ which belong to the support of
$U^0$, that is for $-1<x_1<-1/2$.
As $a_1=1$, we simply have
\[
X_1(t,x)=x_1+t.
\]
As to $X_2$, as long as $X_1<0$ or $t<t_0=-x_1$ (remember $x_1\in[-1,\
-1/2]$) it is equal to $x_2$.
For $t_0<t<t_1=1-x_1$ (corresponding to $X_1$ in $[0,\ 1]$), we have
\[
\partial_t X_2=X_1=x_1+t.
\]
As a consequence
\[
X_2(t_1)=x_2+x_1\,(t_1-t_0)+\frac{t_1^2}{2}-\frac{t_0^2}{2}=x_2+x_1+\frac{(1-x_1)^2}{2}-
\frac{x_1^2}{2}=x_2+1/2.
\]
After $t>t_1$, $\partial_t X_2=1$ and so
\[
X_2(t)=x_2+1/2+t-t_1=x_2+1/2+t-1+x_1=x_2+X_1-1/2.
\]
With this, the solution $u_{\varepsilon}$ is given for $t>2$ by
\[\begin{split}
u_{\varepsilon}(t,x)&=u_{\varepsilon}(0,x_1-t,x_2-x_1+1/2)\\
&=K(x_1-t,x_2-x_1+1/2)\, L(x_2/{\varepsilon}-x_1/{\varepsilon}+1/2{\varepsilon}).
\end{split}\]
For every $\alpha\in [0,\ 1]$, choose a subsequence ${\varepsilon}_n$ such that
$1/2{\varepsilon}_n$ converges to $\alpha$ modulo $1$. Then the double scale
limit associated to this subsequence is the function
\[
f(t,x,v)=K(x_1-t,x_2-x_1+1/2)\, L(v_2-v_1+\alpha).
\]
Instead of one unique limit, we obtain a whole family which clearly
indicates the ill-posedness of the problem at the level of the double
scale limit.
\section{Transport of oscillating fine-scale}
\label{sectr}
An interesting question that can be studied using the techniques
developped in section \ref{homohyp} is the problem of transport of an oscillatory fine-scale structure
under a divergence-free vector field. Consider the homogenization problem
\begin{equation}
\label{tos}
\begin{aligned}
\partial_{t} u_{{\varepsilon}} + \nabla_{x} \cdot A(t, x, \frac{\varphi(t,x)}{{\varepsilon}}) u_{{\varepsilon}} &= 0
\\
u_{{\varepsilon}} (0, x) = U^{0} (x, \frac{x}{{\varepsilon}})
\end{aligned}
\end{equation}
where $\varphi : {\mathbb R} ^{d} \times {\mathbb R} \to {\mathbb R} ^{d}$ is a $C^{2}$ map describing the fine scale of
oscillations that satisfies for some $c >0$
\begin{equation}
\label{hypso}
\begin{aligned}
\text{$\varphi (\cdot, t)$ is surjective and invertible for $t$ fixed}
\\
\det \nabla (\varphi^{-1} (\cdot , t)) \ge c > 0
\\
\varphi (\cdot , 0 ) = id
\end{aligned}
\tag {h$_{so}$}
\end{equation}
and $a_{{\varepsilon}} = A(t, x, \frac{\varphi(t,x)}{{\varepsilon}})$ is a divergence free field.
The latter is guaranteed provided $A(t,x,v)$ is a $C^{1}$ vector field $1$-periodic in $v$
such that
\begin{equation}
\label{hyptvf}
\begin{aligned}
\nabla_{x} \cdot A(t,x,v) &= 0
\\
\mathop{\mathrm{tr}} \big ( \nabla_{v} A \nabla_{x} \varphi \big )
= \sum_{i,j = 1}^{d} \frac{\partial A_{i}}{\partial v_{j}} \frac{\partial \varphi_{j}}{\partial x_{i}}
&=0
\end{aligned}
\tag {h$_{tvf}$}
\end{equation}
The initial data
$U^{0} \in L^{2}( {\mathbb R} ^{d} \times {\mathbb T}^{d})$ are 1-periodic in $v$
and satisfy the uniform bounds
\begin{equation}
\label{todata}
\int_{ {\mathbb R} ^{d}} | U^{0} (x, \frac{x}{{\varepsilon}}) |^{2} dx \le C \, .
\tag {h$_{d}$}
\end{equation}
Then standard energy estimates imply that solutions of \eqref{tos}
satisfy the uniform bound
\begin{equation}
\label{uniforml2}
\int_{ {\mathbb R} ^{d}} | u_{{\varepsilon}} (t, x) |^{2} dx
\le \int_{ {\mathbb R} ^{d}} | U^{0} (x, \frac{x}{{\varepsilon}}) |^{2} dx
\le C \, .
\end{equation}
Our objective is to calculate an effective equation for the weak limits of $\{u_{{\varepsilon}}\}$.
The counterexample of section \ref{secce} indicates that we can not expect to do that in full
generality. A more precise statement of what will be achieved is that we will identify
conditions on the vector field $A$
and the structure function $\varphi$ under which an effective equation is calculated.
\subsection{Reformulation via a kinetic problem}
We introduce the "kinetic function"
\begin{equation}
\label{kinf}
f_{{\varepsilon}} = u_{{\varepsilon}} (t,x) \delta_{p} \big (v - \frac{\varphi (t,x) }{{\varepsilon}} \big ) \, ,
\end{equation}
which is well defined (see section \ref{ss12}) as a measure. Due to the identities
in lemma \ref{lem2} of section \ref{ss12}, it is possible to transform
the homogenization problem \eqref{tos} into a hyperbolic limit for a kinetic initial value problem:
\begin{lemma}
If $u_{{\varepsilon}}$ a weak solution of \eqref{tos} then
$f_{{\varepsilon}}$ in \eqref{kinf} verifies in ${\mathcal D}'$ the kinetic problem
\begin{equation}
\label{kintos}
\begin{aligned}
\partial_{t} f_{{\varepsilon}} + \nabla_{x} \cdot (A f_{{\varepsilon}}) + \nabla_{v} ( \frac{1}{{\varepsilon}} B f_{{\varepsilon}})
&= 0 \, ,
\\
f_{{\varepsilon}}(0,x,v) &= U^{0} (x,v) \delta_{p} (v - \frac{x}{{\varepsilon}}) \, ,
\end{aligned}
\end{equation}
where the vector field $B(t,x,v)$, defined by
\begin{equation}
\label{trvf}
B_{i} = \big (\partial_{t} + A \cdot \nabla_{x} \big ) \varphi_{i}
\, , \quad i = 1 , ... , d \, ,
\end{equation}
is $1$-periodic in $v$ and divergence-free,
$\nabla_{v} \cdot B = 0$.
\end{lemma}
\begin{proof}
Let $u_{{\varepsilon}}$ be a weak solution of \eqref{tos}. By \eqref{hyptvf}, $B$ defined in \eqref{trvf}
is $1$-periodic in $v$ and divergence free.
We consider a test function
$\theta \in C_{c}^{1} \big ( [0, \infty ) \times {\mathbb R} ^{d} ; C^{1} ({\mathbb T}^{d}) \big)$
and compute
$$
\begin{aligned}
\int_{0}^{\infty} \int_{ {\mathbb R} ^{d}} \int_{{\mathbb T}^{d}}
&\big [ \partial_{t} \theta + \nabla_{x} (A \cdot \theta )
+ \frac{1}{{\varepsilon}} \nabla_{v} ( B \cdot\theta) \big ] df_{{\varepsilon}}
\\
&= \int_{0}^{\infty} \int_{ {\mathbb R} ^{d}}
u_{{\varepsilon}}(x,t) \Big ( \partial_{t} \theta + A \cdot \nabla_{x} \theta
+ \frac{1}{{\varepsilon}} B \cdot \nabla_{v} \theta \Big ) (t,x,\frac{\varphi}{{\varepsilon}})
dxdt
\\
&= \int_{0}^{\infty} \int_{ {\mathbb R} ^{d}}
u_{{\varepsilon}} \left [ \partial_{t} \big (\theta (t,x,\frac{\varphi}{{\varepsilon}}) \big )
+ A (t,x,\frac{\varphi}{{\varepsilon}}) \cdot \nabla_{x} \big ( \theta (t,x,\frac{\varphi}{{\varepsilon}}) \big )
\right ]
dxdt
\\
&= - \int_{ {\mathbb R} ^{d}} U^{0} (x, \frac{x}{{\varepsilon}}) \theta(0,x,\frac{x}{{\varepsilon}}) dx
\\
&= - \int_{ {\mathbb R} ^{d}} \int_{{\mathbb T}^{d}} \theta(0,x,v) df_{{\varepsilon}} (0,x,v) \, ;
\end{aligned}
$$
that is, $f_{{\varepsilon}}$ is a weak solution of \eqref{kintos}.
\end{proof}
An alternative, albeit formal, derivation of \eqref{kintos} may be obtained by studying
characteristics. The characteristic curve of \eqref{tos} emanating from the point $y$
is defined by
$$
\begin{cases}
\frac{dx}{dt} = A(x,t, \frac{\varphi}{{\varepsilon}}) & \\
x(0,y) = y & \\
\end{cases}
$$
and is denoted by $x = X (t ; y)$.
Along such curves we have
$$
\frac{d}{dt} \Big (\frac{\varphi}{{\varepsilon}} \Big ) = \frac{1}{{\varepsilon}}
\Big ( \varphi_{t} + A(x,t,\frac{\varphi}{{\varepsilon}})\cdot \nabla_{x} \varphi \Big )
$$
The two equations together can be embedded into the system of ordinary
differential equations
\begin{equation}
\label{charsys}
\begin{cases}
\frac{dx}{dt} = A(t,x,v) & \\
\frac{dv}{dt} = \frac{1}{{\varepsilon}}
\Big ( \varphi_{t} + A(x,t,v)\cdot \nabla_{x} \varphi\Big )
\end{cases}
\end{equation}
in the following sense: If $(Y(t; y,u), U(t; y, u))$ is the solution of \eqref{charsys}
emanating from the point $(y,u)$ then
$$
\begin{aligned}
X(t ; y) &= Y ( t ; y , \varphi (y,0))
\\
\frac{\varphi( X(t;y), t)}{{\varepsilon}} &= U (t ; y , \varphi (y,0))
\end{aligned}
$$
Note that \eqref{kintos}$_{1}$ is precisely the Liouville equation associated to the characteristic
system \eqref{charsys}.
\subsection{Conditions leading to an effective equation}
Our next goal is to derive an effective equation for the hydrodynamic limit
of \eqref{kintos}. We first show that under hypothesis \eqref{hypso}
the definition \eqref{kinf} still induces
good properties for the weak limit points of $\{ f_{{\varepsilon}} \}$.
\begin{lemma}
\label{lem42}
Under hypotheses \eqref{hypso} and \eqref{uniforml2},
\begin{align}
\label{pro1}
\delta_{p} \big ( v - \frac{\varphi(x,t))}{{\varepsilon}} \big ) \rightharpoonup 1
\qquad \text{in ${\mathcal D}'$}
\\
\label{pro2}
f_{{\varepsilon}} \in_{b} L^{\infty} \big ( (0,\infty) ; L^{2} ( {\mathbb R} ^{d} ; M_{p}) \big )
\end{align}
and, along a subsequence (if necessary),
\begin{align}
\label{pro3}
f_{{\varepsilon}} &\rightharpoonup f
\qquad \text{weak-$\star$ in
$ L^{\infty} \big ( (0,\infty) ; L^{2} ( {\mathbb R} ^{d} ; M_{p}) \big )$}
\\
\label{pro4}
&\text{with} \;
f \in L^{\infty} \big ( (0,\infty) ; L^{2} ( {\mathbb R} ^{d} \times {\mathbb T}^{d}) \big )
\end{align}
\end{lemma}
\begin{proof}
For $\theta \in C_{c}^{\infty} \big ( (0,\infty) \times {\mathbb R} ^{d} ; C^{\infty} ({\mathbb T}^{d}) \big )$
we have
$$
\begin{aligned}
< \delta_{p} \big ( v - \frac{\varphi(x,t))}{{\varepsilon}} \big ) , \theta >
&=
\int_{0}^{\infty} \int_{ {\mathbb R} ^{d}} \theta (x,t, \frac{\varphi(x,t))}{{\varepsilon}}) dx dt
\\
&=
\int_{0}^{\infty} \int_{ {\mathbb R} ^{d}} \theta ( \varphi^{-1}(y,t), t, \frac{y}{{\varepsilon}})
|\det \nabla_{y}(\varphi^{-1})| dy dt
\\
&\to
\int_{0}^{\infty} \int_{ {\mathbb R} ^{d}}\int_{{\mathbb T}^{d}} \theta(t,x,v) dv dx dt
\end{aligned}
$$
Note next that for $(x,t)$ fixed
$$
\| f_{{\varepsilon}} (x,t,\cdot)\|_{M_{p}} = |u_{{\varepsilon}} (x,t)|
$$
and thus \eqref{pro2} and \eqref{pro3} follow from \eqref{uniforml2}. Finally,
$$
\begin{aligned}
|<f_{{\varepsilon}}, \theta>| &\le
\| u_{{\varepsilon}}\|_{L^{\infty}(L^{2})} \,
\int_{0}^{\infty} \left (
\int_{ {\mathbb R} ^{d}} \big | \theta (x,t, \frac{\varphi(x,t))}{{\varepsilon}}) \big |^{2} dx
\right)^{\frac{1}{2}} dt
\\
&\le
C \int_{0}^{\infty} \left (
\int_{ {\mathbb R} ^{d}} \big | \theta (x,t, \frac{\varphi(x,t))}{{\varepsilon}}) \big |^{2} dx
\right)^{\frac{1}{2}} dt
\\
&\to C \int_{0}^{\infty} \left (
\int_{ {\mathbb R} ^{d}} \int_{{\mathbb T}^{d}} \big | \theta (x,t, v) \big |^{2} dx dv
\right)^{\frac{1}{2}} dt
\end{aligned}
$$
and \eqref{pro4} follows.
\end{proof}
\begin{remark}
A hypothesis of the type of \eqref{hypso} is essential for the validity
of \eqref{pro1} and accordingly for \eqref{pro4}. For instance, in the extreme case that $\varphi$
is a constant map, $\varphi(x,t) \equiv c$, it is possible by choosing appropriate sequences
${\varepsilon}_{n} \to 0$ to achieve any weak limit
$$
\delta_{p} \big ( v - \frac{c}{{\varepsilon}_{n}}) \rightharpoonup \delta_{p} (v - v_{o})
\quad \text{with any $0 < v_{o} < 1$}.
$$
The regularity of $f$ is then no better than the regularity of $\{ f_{{\varepsilon}} \}$ and
\eqref{pro4} is of course violated.
\end{remark}
We conclude by providing a formal derivation of an effective equation. Consider the
${\varepsilon} \to 0$ limit of \eqref{kintos}-\eqref{trvf} and recall that, by \eqref{uniforml2}
and lemma \ref{lem42}, we have $f_{{\varepsilon}} \rightharpoonup f$ as in \eqref{pro3} and \eqref{pro4}.
Define the set
$$
K_{t,x} = \big \{ g \in L^{\infty} ( (0, \infty) ; L^{2}( {\mathbb R} ^{d} \times {\mathbb T}^{d})) \; \Big | \;
B(t,x,v) \cdot \nabla_{v} g = 0 \big \} \, .
$$
The set $K = N({\mathcal B})$ is the null space of the operator ${\mathcal B} := B\cdot \nabla_{v}$ and in general it will
depend on $(t,x)$. We will derive the effective equation under the hypothesis
\begin{equation}
\label{hypmain}
N({\mathcal B}) \; \text{ is independent of $(x,t)$}
\tag {H}
\end{equation}
Then we have the decomposition
$$
L^{2} = N({\mathcal B}) \oplus \overline{R({\mathcal B}^{T})} = K \oplus K^{\perp}
$$
and the spaces remain the same for any point $(x,t)$. Let $P$ denote the $L^{2}$-projection on the
set $K$. Any $\theta \in L^{2}({\mathbb T}^{d})$ can be decomposed as
$$
\theta = P\theta + (I-P)\theta =: \psi + \phi
$$
Moreover the differentiation operators $\partial_{t}$ and $\nabla_{x}$ commute with the
projector $P$.
For $\psi \in N({\mathcal B})$, using $\nabla_{x} \cdot A = \nabla_{v} \cdot B = 0$,
we derive from \eqref{kintos} that
\begin{equation}
\label{interm}
\partial_{t} < f , \psi > + \nabla_{x} <Af , \psi> = 0
\end{equation}
where the brackets denote the usual inner product in $L^{2}({\mathbb T}^{d})$. One
easily sees that
$f(t,x, \cdot) \in K$ for a.e. $(t,x)$. Given $\theta \in L^{2}({\mathbb T}^{d})$
let $\psi = P \theta$. Then
$$
<f , \theta > = <f , \psi >
$$
and
$$
<A_{i} f, \psi> = < P(A_{i} f) , \psi> = <P(A_{i} f) , \theta>
$$
Since $f \in K$ we have $P(A_{i} f) = P(A_{i}) f$ and we conclude that \eqref{interm}
can be expressed in the form
$$
\partial_{t} < f , \theta > + \nabla_{x} \cdot <P(A) f , \theta > = 0 \, ,
\quad \theta \in L^{2}({\mathbb T}^{d}) \, .
$$
The effective equation thus takes the form
\begin{equation}
\label{effto}
\partial_{t} f + \nabla_{x} \cdot ( PA) f = 0 \, .
\end{equation}
The above derivation of \eqref{effto} is formal and is based on hypothesis \eqref{hypmain},
which is quite restrictive especially when viewed together with \eqref{hyptvf} that has to be
satisfied simultaneously. We view this equation as a theoretical framework of when an effective
equation can be computed. To derive it rigorously one needs an analysis as in Theorem \ref{hyphomthm}
and we will not pursue the details here. The counterexample in section \ref{secce} indicates
that the hypothesis \eqref{hypmain} is essential.
We list two examples that can be viewed under the above framework. First,
the homogenization problem \eqref{transport} is a special case of \eqref{tos}
with the obvious identifications. A second example is given by the problem
\begin{equation}
\label{special}
\begin{aligned}
\partial_{t} u_{{\varepsilon}} + a(x) \cdot \nabla_{x} u_{{\varepsilon}} &= 0
\\
u_{{\varepsilon}} (x, 0) &= U (x, \frac{x}{{\varepsilon}})
\end{aligned}
\end{equation}
where $a$ is a divergence free field, $\nabla_{x} \cdot a = 0$.
Define $\varphi(t,x)$ to be the backward characteristic emanating
from the point $x$.
Then $\varphi = (\varphi_{1}, ... , \varphi_{n})$ satisfies
$$
\begin{aligned}
\partial_{t} \varphi_{i} + a(x) \cdot \nabla_{x} \varphi_{i} &= 0
\\
\varphi_{i} (0,x) = x_{i}
\end{aligned}
$$
The problem \eqref{special} fits under the framework of \eqref{tos} under
the selections
$$
A(t,x, v) = a(x) \, . \quad
B_{i} (t,x,v) = \big ( \partial_{t} + a(x) \cdot \nabla_{x} \big ) \varphi_{i} = 0
\, .
$$
The kinetic equation for
$f_{{\varepsilon}} = u_{{\varepsilon}} \delta_{p} \big ( v - \frac{\varphi}{{\varepsilon}}\big)$
becomes
$$
\begin{aligned}
\partial_{t} f_{{\varepsilon}} + a(x) \cdot \nabla_{x} f_{{\varepsilon} } &=0
\\
f_{{\varepsilon}} (0,x,v) = U (x,v) \delta_{p} \big ( v - \frac{x}{{\varepsilon}}\big)
\end{aligned}
$$
while the limiting $f$ satisfies the same transport equation with initial condition
$f(0,x,v) = U(x,v)$. Hence, it is computed explicitly by
$$
f(t,x,v) = U (\varphi(t,x),v) \, .
$$
\section{Enhanced diffusion}
\label{endiff}
In this section we study the enhanced diffusion problem
\begin{equation}
\label{enhdiff}
\begin{aligned}
\partial_{t} u_{{\varepsilon}} + \frac{1}{{\varepsilon}} a(x, \frac{x}{{\varepsilon}}) \cdot \nabla_{x} u_{{\varepsilon}}
&= \alpha \triangle_{x} u_{{\varepsilon}}
\\
u_{{\varepsilon}} (0, x) = U^{0} (x, \frac{x}{{\varepsilon}})
\end{aligned}
\end{equation}
where $a(x,v)$ is a Lipshitz vector field periodic (with period 1) in $v$ that satisfies
\begin{equation}
\label{vecf}
\nabla_{x}\cdot a = \nabla_{v} \cdot a =0 \, , \quad \int_{{\mathbb T}^{d}} a(x,v) dv = 0 \, ,
\tag {h$_{vf}$}
\end{equation}
$\alpha > 0$ is constant and $U^{0} \in L^{2} ( {\mathbb R} ^{d} \times {\mathbb T}^{d})$.
We use this as an example to develop the methodology of section \ref{msde}.
For previous work and a commentary on the significance of this problem
we refer to Avellaneda-Majda \cite{AM},
Fannjiang-Papanicolaou \cite{FP} and references therein.
It is assumed that the initial data oscillates
at the scale ${\varepsilon}$ and satisfy the uniform bound
\begin{equation}
\label{diffdata}
\int_{ {\mathbb R} ^{d}} | U^{0} (x, \frac{x}{{\varepsilon}}) |^{2} dx \le C \, .
\tag {h$_{d}$}
\end{equation}
Standard energy estimates then imply the
uniform bounds
\begin{equation}
\label{unifdiff}
\int_{ {\mathbb R} ^{d}} | u_{{\varepsilon}} (t, x) |^{2} dx
+ \alpha \int_{0}^{t} \int_{ {\mathbb R} ^{d}} |\nabla u_{{\varepsilon}}|^{2} dxdt
\le \int_{ {\mathbb R} ^{d}} | U^{0} (x, \frac{x}{{\varepsilon}}) |^{2} dx
\le C
\end{equation}
for solutions of \eqref{enhdiff}.
We introduce the kinetic decomposition
\begin{equation}
\label{diffdsdef}
f_{{\varepsilon}} (t,x,v) = u_{{\varepsilon}}(t,x) \delta_{p} \big( v - \frac{x}{{\varepsilon}} \big)
\quad t \in {\mathbb R} _{+}, \, x\in {\mathbb R} ^{d} , \, v\in{\mathbb T}^{d} \, ,
\end{equation}
and use Lemma \ref{lem1} to obtain that $f_{{\varepsilon}}$ satisfies the transport-diffusion equation
\begin{equation}
\label{maindiff}
\begin{aligned}
\dt {f_{\varepsilon}} + \frac{1}{{\varepsilon}} a(x, v)\cdot\nabla_x f_{\varepsilon}
&+ \frac{1}{{\varepsilon}^{2}}
\big ( a(x,v) \cdot \nabla_{v} f_{{\varepsilon}} - \alpha \triangle_{v} f_{{\varepsilon}} \big )
\\
&= \alpha \triangle_{x} f_{{\varepsilon}}
+ \frac{2\alpha}{{\varepsilon}} \nabla_{x} \cdot \nabla_{v} f_{{\varepsilon}}
\, , \qquad \text{in ${\mathcal D}'$}
\\
f_{\varepsilon}(t=0,x,v)
&=U^0(x,v)\;\delta_{p} (v-\frac{x}{{\varepsilon}})
\end{aligned}
\end{equation}
with periodic boundary conditions on the torus in the $v$ variable.
Our objective is to analyze the ${\varepsilon} \to 0$ limit of this problem and through this process
to calculate the effective equation satisfied by the weak limit of $u_{{\varepsilon}}$. We note
that this is a hydrodynamic limit problem in the diffusive scaling for the kinetic
equation \eqref{maindiff}.
We prove.
\begin{theorem}
\label{enhdiffthm}
Under hypothesis \eqref{vecf} and \eqref{diffdata} we have the following
asymptotic behavior
for $f_{{\varepsilon}}$ as ${\varepsilon} \to 0$:
\begin{align}
f_{{\varepsilon}} (t,x,v) &\rightharpoonup u(t,x) \quad \text{weak-$\star$ in
$L^\infty([0,\ \infty) ,\ L^2( {\mathbb R} ^d, M_{p})$}
\\
f_{{\varepsilon}} (t,x,v) &= u_{{\varepsilon}}(t,x) + {\varepsilon} g(t,x,v) + o({\varepsilon}) \, , \quad \text{in ${\mathcal D}'$}
\end{align}
where $u$ of class \eqref{unifdiff} and
$g \in L^{2} ( (0, \infty) \times \Omega ; H^{1} ({\mathbb T}^{d}))$ satisfy
respectively
\begin{align}
\partial_{t} u - \alpha \triangle_{x} u +
\nabla_{x} \cdot \int_{{\mathbb T}^{d}} a(x,v) g (t,x,v) = 0
\label{basic}
\\
\alpha \triangle_{v} g - \nabla_{v} \cdot a g =
a \cdot \nabla_{x} u
\label{gsol}
\end{align}
The weak limit $u$ satisfies the effective diffusion equation
$$
\begin{aligned}
\partial_{t} u &= \alpha \sum_{i, j} \partial_{x_{i}} \left (
\Big (\delta_{i j} + \int_{{\mathbb T}^{d}} \nabla_{v} \chi_{i} \cdot \nabla_{v} \chi_{j} dv
\Big )
\partial_{x_{j}} u \right )
\\
u (0,x) &= \int_{{\mathbb T}^{d}} U^{0} (x,v) \, dv
\end{aligned}
$$
where $\chi_{k}$, $k = 1, ... , d$, is the solution of the cell problem
\begin{equation}
\label{cellp}
\alpha \triangle_{v} \chi_{k} - \nabla_{v} \cdot a \chi_{k} = a \cdot e_{k}
\end{equation}
\end{theorem}
\begin{proof} Let $f_{{\varepsilon}}$ be defined as in \eqref{diffdsdef}. Then $f_{{\varepsilon}}$ satisfies
the problem \eqref{maindiff} and $u_{{\varepsilon}} = \int_{{\mathbb T}^{d}} f_{{\varepsilon}}$.
The proof is split in three steps:
\medskip
{\em Step 1~: Characterization of the weak limit.}
From \eqref{unifdiff} and Lemma \ref{lem1} we obtain uniform bounds
for $f_{{\varepsilon}}$:
\begin{equation}
\label{fbounds}
\begin{aligned}
f_{\varepsilon} &\in_{b} L^\infty([0,\ \infty) \, ; \, L^2( {\mathbb R} ^d, M_{p}) \, ,
\\
\big ( \nabla_{x} + \frac{1}{{\varepsilon}} \nabla_{v} \big ) f_{{\varepsilon}}
&\in_{b} L^{2} ( (0, \infty) \times {\mathbb R} ^{d} \, ; \ M_{p})
\end{aligned}
\end{equation}
Using (a slight variant of) Proposition \ref{prop1} we see that, along a subsequence if necessary,
$f_{{\varepsilon}}$ satisfies
\begin{equation}
\label{wl}
\begin{aligned}
f_{{\varepsilon}} &\rightharpoonup f \quad \text{weak-$\star$ in
$L^\infty([0,\ \infty) ;\ L^2( {\mathbb R} ^d , M_{p})$}
\\
f &\in L^\infty([0,\ \infty) \, ; \, L^2( {\mathbb R} ^d \times {\mathbb T}^{d})) \, ,
\end{aligned}
\end{equation}
that is
$$
\int \theta(t,x,v) df_{{\varepsilon}}(t,x,v) \to \int \theta(t,x,v) f(t,x,v) dt dx dv
$$
for $\theta \in L^{1} ((0,\infty) ; L^{2} ( {\mathbb R} ^{d}, C_{p}))$.
Passing to the limit ${\varepsilon} \to 0$ in \eqref{maindiff} and using \eqref{fbounds}
we see that $f$ satisfies
$$
\alpha \triangle_{v} f - \nabla_{v} \cdot a f = 0 \, ,
\quad \text{in ${\mathcal D}'$},
$$
and for any test function $\theta$
$$
\begin{aligned}
<\nabla_{v} f , \theta > = \lim_{{\varepsilon} \to 0} <\nabla_{v} f_{{\varepsilon}}, \theta>
= 0
\, .
\end{aligned}
$$
Hence, $\nabla_{v}f = 0$ in ${\mathcal D}'$ and
\begin{equation}
\label{cl}
f(t,x,v) = \int_{{\mathbb T}^{d}} f dv = : u(t,x)
\end{equation}
\medskip
{\em Step 2~: Asymptotics of $f_{{\varepsilon}}$.} Define next
\begin{equation}
g_{{\varepsilon}} (t,x,v) = \frac{1}{{\varepsilon}} \big ( f_{{\varepsilon}} (t,x,v) - u_{{\varepsilon}} (t,x) \big )
\, , \quad t \in {\mathbb R} _{+} \, , \, x \in {\mathbb R} ^{d} \, , \, v \in {\mathbb T}^{d} \, ,
\end{equation}
where $u_{{\varepsilon}} = \int_{{\mathbb T}^{d}} f_{{\varepsilon}}$. We proceed along the lines
of Proposition \ref{prop3} replacing the bounds of that proposition by the bound
\eqref{unifdiff} and accounting for the extra dependence in time. After minor
modifications in the proof we obtain for any $T>0$
\begin{equation}
\label{proptimeasym}
\begin{aligned}
g_{{\varepsilon}} &\in_{b} L^{2} \big ( (0,T) \, ; \, H^{-1} (\Omega , M_{p}) \big )
\\
g_{{\varepsilon}} &\rightharpoonup g \quad
\text{ weak-$\star$ in $L^{2} \big ( (0,T) \, ; \, H^{-1} (\Omega , M_{p}) \big )$}
\\
g &\in L^{2} ( (0, \infty) \times \Omega \, ; \, H^{1} ({\mathbb T}^{d})) \, ,
\quad \int_{{\mathbb T}^{d}} g = 0 \, .
\end{aligned}
\end{equation}
Accordingly, $f_{{\varepsilon}}$ enjoys the asymptotic expansion
$$
f_{{\varepsilon}} = u_{{\varepsilon}} + {\varepsilon} g + o({\varepsilon}) \quad \text{in ${\mathcal D}'$} \, .
$$
On the other hand, on account of \eqref{enhdiff}, \eqref{maindiff} and \eqref{vecf},
it follows that $u_{{\varepsilon}}$ and $g_{{\varepsilon}}$ satisfy
\begin{equation}
\begin{aligned}
\partial_{t} u_{{\varepsilon}} - \alpha \triangle_{x} u_{{\varepsilon}} +
\nabla_{x} \cdot \int_{{\mathbb T}^{d}} a(x,v) \frac{f_{{\varepsilon} } - u_{{\varepsilon}} }{{\varepsilon}} = 0
\end{aligned}
\end{equation}
and
\begin{equation}
\begin{aligned}
\alpha \triangle_{v} g_{{\varepsilon}} - \nabla_{v} \cdot a g_{{\varepsilon}} =
{\varepsilon} (\partial_{t} f_{{\varepsilon}} - \alpha \triangle_{x} f_{{\varepsilon}}) -
2 \alpha \nabla_{x} \cdot \nabla_{v} f_{{\varepsilon}}
+ \nabla_{x} \cdot a f_{{\varepsilon}}
\end{aligned}
\end{equation}
Using \eqref{wl}, \eqref{cl} and \eqref{proptimeasym}, we pass to the limit ${\varepsilon} \to 0$
and deduce that $u$, $g$ satisfy \eqref{basic} and \eqref{gsol} respectively.
\medskip
{\em Step 3~: Characterization of the limit problem.} Due to its regularity the solution $g$
of \eqref{gsol} is unique and can be expressed in the form
$$
g = \nabla_{x} u(t,x) \cdot \chi (x,v)
$$
where $\chi = ( \chi_{1}, ... , \chi_{d})$ is the solution of the cell problem
\begin{equation}
\alpha \triangle_{v} \chi_{k} - \nabla_{v} \cdot a \chi_{k} = a \cdot e_{k} = a_{k}
\, .
\end{equation}
A direct computation shows that solutions of \eqref{cellp} satisfy the property
$$
\frac{1}{2} \int_{{\mathbb T}^{d}} (a_{i} \chi_{k} + a_{k} \chi_{i} ) dv =
- \alpha \int_{{\mathbb T}^{d}} \nabla_{v} \chi_{i} \cdot \nabla_{v} \chi_{k} dv
$$
and \eqref{basic} may be written in the
equivalent forms
$$
\begin{aligned}
\partial_{t} u &= \sum_{i, j} \partial_{x_{i}} \left (
\big (\alpha \delta_{i j} - \frac{1}{2} \int_{{\mathbb T}^{d}} (a_{i} \chi_{j} + a_{j} \chi_{i}) \, dv
\big )
\partial_{x_{j}} u \right )
\\
&= \alpha \sum_{i, j} \partial_{x_{i}} \left (
\Big ( \delta_{i j} + \int_{{\mathbb T}^{d}} \nabla_{v} \chi_{i} \cdot \nabla_{v} \chi_{j} dv
\Big ) \partial_{x_{j}} u \right )
\end{aligned}
$$
The latter is a diffusion equation with positive definite diffusion matrix
$$
\begin{aligned}
D_{i j} &= \delta_{i j} + \int_{{\mathbb T}^{d}} \nabla_{v} \chi_{i} \cdot \nabla_{v} \chi_{j} dv
\\
\sum_{i j} D_{i j} \nu_{i} \nu_{j} &=
|\nu|^{2} + \int_{{\mathbb T}^{d}} | \nabla_{v} (\chi \cdot \nu) |^{2} dv \, , \quad
\text{$\nu \in {\mathbb R} ^{d}$},
\end{aligned}
$$
determined through the solution of \eqref{cellp}.
\end{proof}
\section{Appendix I}
\label{appendix}
We prove in the appendix a lemma that is used in the justification of multiscale
decompositions.
\begin{lemma}
\label{technical}
Let $\Omega$ be an open subset of $ {\mathbb R} ^{d}$,
$\theta \in C_{c}(\Omega)$, $\varphi \in C({\mathbb T}^{d})$, $\psi \in C({\mathbb T}^{d})$,
and suppose that $\delta = \delta ({\varepsilon}) \to 0$ as ${\varepsilon} \to 0$. Then, as ${\varepsilon} \to 0$,
\begin{align}
\int_{\Omega} \theta(x) \varphi(\frac{x}{{\varepsilon}}) dx &\to
\int_{{\mathbb T}^{d}}\varphi(z)dz \; \int_{\Omega} \theta (x) dx
\label{techn1}
\\
\int_{\Omega} \theta(x) \varphi(\frac{x}{{\varepsilon}}) \psi (\frac{x}{{\varepsilon} \delta}) dx &\to
\int_{{\mathbb T}^{d}}\psi(y)dy \; \int_{{\mathbb T}^{d}}\varphi(z)dz \; \int_{\Omega} \theta (x) dx
\label{techn2}
\end{align}
\end{lemma}
\begin{proof}
Fix $\theta \in C_{c}(\Omega)$ and let
${\varepsilon} < \frac{1}{\sqrt{d}} \, {\rm dist \,} ( {\rm supp \,} \theta , \partial \Omega)$.
We consider a cover of the support of the function $\theta$
by cubes $C_{k}$ centered at points $\chi_{k} \in {\varepsilon} {\mathbb Z}^{d}$ of latteral size ${\varepsilon}$.
The number of the cubes covering ${\rm supp \,} \theta$ satisfies $N {\varepsilon}^{d} = O(|{\rm supp \,} \theta|)$,
and since ${\varepsilon} \sqrt{d} < {\rm dist \,} ( {\rm supp \,} \theta , \partial \Omega)$ the covering
can be arranged so that
${\rm supp \,} \theta \subset \cup_{k=1}^{N} C_{k} \subset \Omega$ while
$\frac{1}{{\varepsilon}} \chi_{k} \in {\mathbb Z}^{d}$. We have
$$
\begin{aligned}
\int_{\Omega} \theta(x) \varphi(\frac{x}{{\varepsilon}}) dx
& = \sum_{k=1}^{N} \int_{C_{k}} \theta(x) \varphi(\frac{x}{{\varepsilon}}) dx
\\
&= \sum_{k=1}^{N} {\varepsilon}^{d} \int_{{\mathbb T}^{d}} \theta (\chi_{k} + {\varepsilon} z)
\varphi ( \frac{1}{{\varepsilon}} \chi_{k} + z) \, dz
\\
&= \sum_{k=1}^{N} {\varepsilon}^{d} \int_{{\mathbb T}^{d}} \theta (\chi_{k} + {\varepsilon} z)
\varphi ( z) \, dz
\end{aligned}
$$
and thus
$$
\begin{aligned}
I &= \int_{\Omega} \theta(x) \varphi(\frac{x}{{\varepsilon}}) dx -
\int_{{\mathbb T}^{d}}\varphi(y)dy \; \int_{\Omega} \theta (x) dx
\\
&= \sum_{k=1}^{N} {\varepsilon}^{d} \int_{{\mathbb T}^{d}} \theta(\chi_{k} + {\varepsilon} y) \varphi(y) dy
-
\int_{{\mathbb T}^{d}} \varphi(y) dy \sum_{k=1}^{N} {\varepsilon}^{d}
\int_{{\mathbb T}^{d}} \theta(\chi_{k} + {\varepsilon} z) dz
\\
&= \int_{{\mathbb T}^{d}} \varphi(y) \sum_{k=1}^{N} {\varepsilon}^{d} \int_{{\mathbb T}^{d}}
\left ( \theta(\chi_{k} + {\varepsilon} y) - \theta(\chi_{k} + {\varepsilon} z) \right ) dz \, dy
\end{aligned}
$$
Using the uniform continuity of $\theta$ and that $N = O(\frac{1}{{\varepsilon}^{d}})$ we
deduce $I \to 0$ as ${\varepsilon} \to 0$ and \eqref{techn1}.
Next observe that
$$
\begin{aligned}
&\int_{\Omega} \theta(x) \varphi(\frac{x}{{\varepsilon}}) \psi (\frac{x}{{\varepsilon} \delta}) dx -
\int_{{\mathbb T}^{d}}\psi(y)dy \; \int_{{\mathbb T}^{d}}\varphi(z)dz \; \int_{\Omega} \theta (x) dx
\\
&\quad =
\int_{\Omega} \theta(x) \varphi(\frac{x}{{\varepsilon}}) \psi (\frac{x}{{\varepsilon} \delta}) dx
- \int_{{\mathbb T}^{d}} \psi(y) dy \; \int_{\Omega} \theta(x) \varphi(\frac{x}{{\varepsilon}}) dx
\\
&\qquad + \int_{{\mathbb T}^{d}} \psi(y)dy
\left(
\int_{\Omega} \theta(x) \varphi(\frac{x}{{\varepsilon}}) dx - \int_{{\mathbb T}^{d}}\varphi(z) dz
\; \int_{\Omega} \theta(x)dx
\right)
\\
= I_{1} + I_{2}
\end{aligned}
$$
and that $I_{2} \to 0$ as ${\varepsilon} \to 0$.
Consider now a covering of ${\rm supp \,} \theta$ by cubes $\bar C_{k}$ centered at points
$\chi_{k} \in {\varepsilon} \delta {\mathbb Z}^{d}$ of latteral size ${\varepsilon} \delta$.
As in the preceding argument we can arrange the cubes so that
${\rm supp \,} \theta \subset \cup_{k=1}^{\bar N} \bar C_{k} \subset \Omega$ and their number
$\bar N$ satisfies
$\bar N ({\varepsilon} \delta)^{d} = O(|{\rm supp \,} \theta|)$. We have
$$
\begin{aligned}
\int_{\Omega} \theta(x) \varphi(\frac{x}{{\varepsilon}}) \psi (\frac{x}{{\varepsilon} \delta}) dx
&=
\sum_{k=1}^{{\bar N}} \int_{\bar C_{k}}
\theta(x) \varphi(\frac{x}{{\varepsilon}}) \psi (\frac{x}{{\varepsilon} \delta}) dx
\\
&=
\sum_{k=1}^{{\bar N}} ({\varepsilon} \delta)^{d} \int_{{\mathbb T}^{d}}
\theta(\bar \chi_{k} + {\varepsilon} \delta z)
\varphi(\frac{1}{{\varepsilon}} \bar \chi_{k} + \delta z) \psi (z) dz
\end{aligned}
$$
and
$$
\begin{aligned}
I_{1} &=
\sum_{k=1}^{{\bar N}} \int_{\bar C_{k}}
\theta(x) \varphi(\frac{x}{{\varepsilon}}) \psi (\frac{x}{{\varepsilon} \delta}) dx
- \int_{{\mathbb T}^{d}} \psi(z) dz \; \sum_{k=1}^{\bar N}\int_{\bar C_{k}}
\theta (x) \varphi(\frac{x}{{\varepsilon}}) dx
\\
&=
\int_{{\mathbb T}^{d}} \psi(z) \sum_{k=1}^{\bar N} ({\varepsilon} \delta)^{d} \int_{{\mathbb T}^{d}}
\Big ( \theta (\bar \chi_{k} +{\varepsilon} \delta z)
\varphi( \frac{1}{{\varepsilon}}\bar \chi_{k} + \delta z)
\\
&\qquad \qquad \qquad \qquad \qquad \qquad -
\theta (\bar \chi_{k} +{\varepsilon} \delta y)
\varphi( \frac{1}{{\varepsilon}}\bar \chi_{k} + \delta y)
\Big ) \, dy \, dz
\end{aligned}
$$
Again, since $\bar N = O(\frac{1}{({\varepsilon} \delta)^{d}})$ and $\lim_{{\varepsilon}\to 0}\delta = 0$,
we deduce $I_{2} \to 0$ as ${\varepsilon} \to 0$
and \eqref{techn2}.
\end{proof}
\begin{remark} Relation \eqref{techn1} is classical (see \cite{BLL}) and is only proved here
as a precursor to the proof of \eqref{techn2}. Equation \eqref{techn2} indicates that oscillations
of different scales do not correlate and suggests that the definition \eqref{defn3sc}
is meaningful.
Both equations can be extended for test functions $\theta \in C(\bar \Omega)$ provided
that $\Omega \subset {\mathbb R} ^{d}$ is a bounded open set and its boundary $\partial \Omega$
is of finite $d-1$ Hausdorff dimension.
\end{remark}
As an application we prove \eqref{ex1}.
\begin{lemma}
\label{lemapen}
We have
$$
\delta_p \big ( v - \frac{x}{{\varepsilon}} \big ) \rightharpoonup 1
\qquad \text{weak-$\star$ in $L^\infty(\Omega, M_p)$ }
$$
\end{lemma}
\begin{proof}
We need to show that for $\theta \in L^1(\Omega, C_p)$ we have
$$
\int_{\Omega} \theta (x, \frac{x}{{\varepsilon}}) dx \to \int_\Omega \int_{{\mathbb T}^d}
\theta(x,v) dx dv
$$
Equation \eqref{techn1} justifies that for $\theta = \chi(x) \otimes \varphi(v)$
a tensor product with $\chi \in C_c(\Omega)$ and $\varphi \in C_p$ and
by a density argument also for $\chi \in L^1 (\Omega)$, $\varphi \in C_p$.
To complete the proof we need to show that finite sums of tensor products
$\sum_j \chi_j \otimes \varphi_j$ are dense in $L^1 (\Omega, C_p)$. Fix
$\theta \in L^1 (\Omega, C_p)$ and consider a decomposition of the torus
${\mathbb T}^d$ into squares of size $1/n$. Take a partition of unity
$\varphi_i \in C_p$, $i = 1, ..., n^d$, with each $\varphi_i$ supported
in a square of size $2/n$ and $\sum_i \varphi_i =1$. Let $v_i$ be the
center of each square and define
$$
\theta_n (x,v) = \sum_i \theta (x, v_i) \varphi_i (v)
$$
Clearly, $\theta_n$ is a sum of tensor products. Now define
$$
\sup_{v \in {\mathbb T}^d} |\theta(x,v) - \theta_n (x,v) |
\le \sup_{v \in {\mathbb T}^d , |h| < \frac{2}{n}} |\theta(x,v) - \theta (x,v+h) |
=: g_n(x)
$$
and thus
$$
\| \theta - \theta_n \|_{L^1(C_p)} \le \int_\Omega |g_n(x)| dx
$$
Note that $g_n(x) \to 0$ for a.e. $x\in \Omega$ and that
$|g_n(x)| \le 2 \sup_{v\in {\mathbb T}^d} |\theta(x,v)|$. The latter
is an $L^1$ function by the very definition of $\theta$, and the
dominated convergence theorem implies $\int_\Omega |g_n|dx \to 0$.
\end{proof}
\section{Appendix II: Some basic results of ergodic theory}
The purpose of this appendix is to recall some well known properties
of the classical ergodic theory
for the projection on the kernel
\[
K=\{f\in L^2({\mathbb T}^d)\,|\;a(v)\cdot\nabla_v f(v)=0\},
\]
where the last equality is of course in the sense of distributions.
Let us define the characteristics associated with $a$ which are the
solutions on ${\mathbb T}^d$ of the following differential equation
\[
\partial_t T(t,v)=a(T(t,v)),\quad T(0,v)=v.
\]
Then assuming that
\begin{equation}
a\in W^{1,\infty}({\mathbb T}^d),\quad \nabla_v\cdot a=0,\label{hypa}
\end{equation}
the characteristics are well defined and for a fixed $t$, the
transform $v\rightarrow T(t,v)$ is a mesure preserving homeomorphism
of ${\mathbb T}^d$. We then have the well-known theorem (see Sinai \cite{Si}
for more details)
\begin{theorem}
For every $f\in L^p({\mathbb T}^d)$ with $1\leq p<\infty$, there exists a
unique function in $L^p({\mathbb T}^d)$, denoted by $Pf$, such that
\[
\frac{1}{t}\int_0^t f(T(s,v))\,ds\longrightarrow Pf(v),\ \hbox{as}\
t\rightarrow \infty,\
\hbox{strongly\ in}\ L^p({\mathbb T}^d).
\]
Moreover $Pf$ satisfies in the sense of distribution
\[
a(v)\cdot\nabla_v Pf(v)=0,
\]
and if $f\in L^2({\mathbb T}^d)$, then $Pf$ is exactly the orthogonal
projection of $f$ on $K$.
\label{ergodic}
\end{theorem}
This immediately implies the
\begin{corollary}
The orthogonal projection on $K$ may be extended as an operator on
$L^p({\mathbb T}^d)$ for every $1\leq p<\infty$. In addition if $f\in L^2\cap
L^p({\mathbb T}^d)$ with $1\leq p\leq\infty$ ($p=\infty$ allowed), then $P_K
f$ also belongs to $L^2\cap
L^p({\mathbb T}^d)$.
\end{corollary}
{\bf Proof of Theorem \ref{ergodic}.} This proof exactly corresponds
to the one in \cite{Si} in the particular case which we consider.
Notice that if, for $f\in L^p({\mathbb T}^d)$, $\frac{1}{t}\int_0^t
f(T(s,v))\,ds$ converges to $Pf$ then trivially
\[
Pf(T(t,v))=Pf(v)\quad \forall\ t.
\]
Therefore we automatically have in the sense of distribution that
\[
a(v)\cdot\nabla_v Pf=0.
\]
Take now $f$ in $L^p$ and assume
first that there exists $g\in L^p$ with $a\cdot\nabla_v g=0$ and
$h\in W^{1,p}({\mathbb T}^d)$ such that
\[
f=g+a\cdot\nabla_v h.
\]
Then notice that in the sense of distribution
\[
\partial_t\left(g(T(t,v))\right)=a(T(t,v))\cdot\nabla_v g(T(t,v))=0,
\]
and so
\[
g(T(t,v))=g(T(0,v))=g(v).
\]
On the other hand
\[
a(T(t,v))\cdot\nabla_v h(T(t,v))=\partial_t\left(h(T(t,v))\right),
\]
and therefore
\[
\frac{1}{t}\int_0^t f(T(s,v))\,ds=g(v)+\frac{h(T(t,v))-h(v)}{t}.
\]
Consequently in this case $\frac{1}{t}\int_0^t f(T(s,v))\,ds$
converges to $g$ which is unique as a consequence. This proves the
theorem on the set
\[
L_p=\{g+a(v)\cdot\nabla_v h(v)\,|\;h\in W^{1,\infty}({\mathbb T}^d),\ g\in
L^p({\mathbb T}^d)\ \hbox{with}\ a\cdot\nabla_v g=0\}.
\]
Let us first prove that $L_2$ is dense in $L^2({\mathbb T}^d)$.
If $L_2$ is not dense, then there exists $f\in
L^2\setminus \{0\}$ orthogonal to $L_2$. This implies that for all
$h\in W^{1,\infty}({\mathbb T}^d)$
\[
\int_{{\mathbb T}^d} f(v)\,a(v)\cdot\nabla_v h(v)\,dv=0,
\]
or in other words $f\in K$. But $K\subset L_2$ and $f$ should
consequently be
orthogonal to $K$, which is impossible. Notice that $Pf$ necessarily is the orthogonal
projection on $K$ as $a\cdot\nabla h$ belongs to $K^\perp$.
Now for any $f\in L^p$. If $1\leq p<2$, take $g_n+a\cdot\nabla_v h_n=f_n\in L_2$ converging
toward $f$ in $L^p$ (first take ${\hat f}_n \in L^2$
and then select $f_n$ by diagonal extraction). We have that
\[
\begin{aligned}
\|\frac{1}{t}\int_0^t f(T(s,v))\,ds &- \frac{1}{t'}\int_0^{t'
}f(T(s,v))\,ds\|_{L^p}
\\
&\leq
2\|f-f_n\|_{L^p}
+\left(\frac{1}{t}+\frac{1}{t'}\right)\;\|h_n\|_{L^p}.
\end{aligned}
\]
So the sequence $\frac{1}{t}\int_0^t f(T(s,v))\,ds$ is of Cauchy in
$L^p$ and hence converges to a unique limit $Pf$.
Finally if $f\in L^\infty({\mathbb T}^d)$, then $f\in L^2({\mathbb T}^d)$ and
$\frac{1}{t}\int_0^t f(T(s,v))\,ds$
converges to $Pf$ in $L^2$. As the first quantity is uniformly bounded
in $L^\infty$, $Pf\in L^\infty$ and the convergence holds in every
$L^p$, $p<\infty$. By interpolation, one eventually obtains the
desired result for $f\in L^p({\mathbb T}^d)$.\cqfd
\section*{Acknowledgements}
PEJ was partially supported by the HYKE european network.
AET is partially supported by the National Science Foundation.
|
1,116,691,500,736 | arxiv | \section{Introduction}
A classical way of obtaining submodules of the Bergman space $\ber$ is from a zero variety in $\bn$. The corresponding quotient module is then just the closure of linear span of those reproducing kernels in this variety. The Geometric Arveson-Douglas Conjecture asks whether the quotient module is $(p-)$ essential normal. In this paper, we develop a machinery to tell whether the sum of two quotient modules, each related to a zero variety in $\bn$, is closed. In other words, we prove that when two varieties satisfy certain hypotheses, the quotient module related to their union, which is itself a zero variety, is exactly the sum of the two quotient modules. Therefore properties of the union variety is determined by properties of the two original ones. This ``decomposing variety'' technique allows us to obtain complicated examples from simple ones (cf. \cite{Douglas Wang remark}\cite{Sha Ken}\cite{Shalit}). As a consequence, under certain hypotheses, the essential normality results on the union variety follow from those on each variety.
A complex Hilbert space $\mathcal{H}$ is called a Hilbert module (over the polynomial ring $\mathbb{C}[z_1,\cdots,z_n]$) if for every $p\in\mathbb{C}[z_1,\cdots,z_n]$ there is a bounded linear operator $M_p$ on $\mathcal{H}$ and the map $\mathbb{C}[z_1,\cdots,z_n]\to B(\mathcal{H}),~p\to M_p$ is an algebra homomorphism. A submodule $P\subset\mathcal{H}$ is a Hilbert subspace of $\mathcal{H}$ that is closed under the module multiplications $M_p$. Let $Q$ be the orthogonal complement of $P$ in $\mathcal{H}$. Then $Q$ is the quotient Hilbert module with the homomorphism taking $z_i$ to
the compression of $M_{z_i}$ to $Q$. A Hilbert module $\mathcal{H}$ is said to be essentially normal ($p$-essentially normal) if the commutators $[M_{z_i}, M_{z_j}^*]$ belong to the compact operators $\mathcal{K}(\mathcal{H})$ (Schatten $p$ class $\mathcal{S}^p$), for any $1\leq i, j\leq n$.
In the study of spaces of holomorphic functions on the open unit ball $\bn$, for example, the Bergman space, the Hardy space and the Drury-Arveson space, a well known property is that they are essentially normal, i.e., the multiplication operators of analytic polynomials are essentially normal.
In his paper \cite{Arv Dirac}, Arveson made a conjecture about essential normality of submodules and quotient modules of the Drury-Arveson space. The conjecture was then refined by the first author and extended to other spaces \cite{Dou index}. In this paper, we consider the Bergman space.
\begin{arv}
Assume $I$ is a homogeneous ideal of the polynomial ring $\mathbb{C}[z_1,\cdots,z_n]$
and $[I]$ is the closure of $I$ in $\ber$. Then for all $p>\dim Z(I)$, the quotient module $Q=[I]^{\perp}$ is $p$-essentially normal. Here
$$
Z(I)=\{z\in\bn:p(z)=0, \forall p\in I\}
$$
and $\dim Z(I)$ denotes the complex dimension of $Z(I)$.
\end{arv}
Results on the Arveson-Douglas Conjecture include \cite{Arv p summable}\cite{Douglas Sarkar}\cite{Douglas Wang}\cite{Guo}$\sim$\cite{Shalit} and many others.
For a polynomial ideal $I$, the zero variety $Z(I)$ determines another submodule
$$
P=\{f\in\ber: f|_{Z(I)}=0\}.
$$
In many cases, especially when $I$ is radical, one can prove that $[I]=P$. Specializing to this case, we have a geometric version of the conjecture \cite{Sha Ken}.
\begin{geo}
Let $M$ be a homogeneous variety in $\bn$. Let
$$
P=\{f\in\ber: f|_M=0\}
$$
and $Q=P^{\perp}=\overline{span}\{K_{\lambda}:\lambda\in M\}$. Then the quotient module $Q$ is $p$-essentially normal for every $p>\dim M$.
\end{geo}
Although the Geometric Arveson-Douglas Conjecture is about homogenous varieties, it was shown to hold in many non-homogenous cases. Assuming $M$ is smooth on $\pbn$, the conjecture was proved by Engli\v{s} and Eschmeier \cite{Englis}, the first author, Tang and Yu \cite{DYT} and us \cite{our paper} under additional assumptions, using very different techniques. In particular, in \cite{our paper}, we introduced tools in complex harmonic analysis to solve the problem. One approach is to decompose the variety into simpler ones. In \cite{Sha Ken}, Shalit and Kenndy discussed this problem and obtained positive results on unions of linear varieties or homogenous varieties intersecting only at the origin. In this paper, we continue to develop the machinery in \cite{our paper} and discuss a simple case which allow some singularity on $\pbn$. Suppose $M_1$ and $M_2$ are two varieties, each having nice properties, the variety $M=M_1\cup M_2$ might have singular points on $\pbn$. In this paper, we prove the Geometric Arveson-Douglas Conjecture on $M$ assuming the varieties $M_1$ and $M_2$ intersect ``nicely''. Our main result is the following.
\begin{thm}\label{main theorem}
Suppose $\tilde{M_1}$ and $\tilde{M_2}$ are two analytic subsets of an open neighborhood of $\clb$. Let $\tilde{M_3}=\tilde{M_1}\cap\tilde{M_2}$. Assume that
\begin{itemize}
\item[(i)]$\tilde{M_1}$ and $\tilde{M_2}$ intersect transversely with $\pbn$ and have no singular points on $\pbn$.
\item[(ii)]$\tilde{M_3}$ also intersects transversely with $\pbn$ and has no singular points on $\pbn$.
\item[(iii)]$\tilde{M_1}$ and $\tilde{M_2}$ intersect cleanly on $\partial\mathbb{B}_n$.
\end{itemize}
Let $M_i=\tilde{M_i}\cap\mathbb{B}_n$ and $Q_i=\overline{span}\{K_{\lambda}: \lambda\in M_i\}$, $i=1,2,3.$ Then $Q_1\cap Q_2/Q_3$ is finite dimensional and $Q_1+Q_2$ is closed. As a consequence, $Q_1+Q_2$ is $p$-essentially normal for $p>2d$, where $d=\max\{\dim M_1, \dim M_2\}$.
\end{thm}
A key ingredient is that when the varieties $M_1$ and $M_2$ satisfy condition (i), the projections $Q_1$ and $Q_2$ are in the Toeplitz algebra $\toe$ (cf. \cite{our paper}). This allows us to apply a result by Su\'{a}rez (cf. \cite{Suarez07}) about essential norms of Toeplitz operators. Then we show that the angle between the two spaces $Q_1$ and $Q_2$ depend essentially on the angles between $M_1$ and $M_2$ at points in $\pbn$. Therefore under the hypotheses of Theorem \ref{main theorem}, $Q_1$ and $Q_2$ have positive angle, which implies that their sum is closed. As a consequence, an equality in index classes is stated in the Summary.
Our method offers a framework of proving closeness of sum of spaces. This could be useful not only in this paper, but also in the future.
We would like to thank Xiang Tang for discussing with us, reading the draft of this paper and giving valuable suggestions and advises.
\section{Preliminary}
To simplify notation, throughout this paper, we will use the same letter to denote both the space and the projection operator onto it. For example, the letter $Q$ is used to denote both the quotient module and the projection operator onto $Q$.
For $n\in\mathbb{N}$, let $\bn$ be the open unit ball in $\cn$. Let $\ber$ be the Bergman space on $\bn$.
$$
\ber=\{f\mbox{ holomorphic on }\bn: \int_{\bn}|f|^2dv<\infty\}.
$$
Here $v$ is the normalized Lebesgue measure on $\bn$, $v(\bn)=1$. The Bergman space $\ber$ has reproducing kernels
$$
K_z(w)=\frac{1}{(1-\langle w,z\rangle)^{n+1}}, ~~~~~~z,w\in\bn.
$$
It becomes a Hilbert module with module map given by pointwise multiplication.
For $g\in L^{\infty}(\bn)$, the Toeplitz operator is defined by
$$
T_g:\ber\to\ber, f\mapsto P_{\ber}(gf)
$$
where $P_{\ber}$ is the projection operator from $L^2(\bn)$ to $\ber$. The Toeplitz algebra $\toe$ is the $C^*$ subalgebra of $B(\ber)$ generated by $T_g$, $g\in L^{\infty}(\bn)$.
\begin{defn}
Let $\Omega$ be a complex manifold. A set $A\subset\Omega$ is called a \emph{(complex) analytic subset} of $\Omega$ if for each point $a\in\Omega$ there exist a neighborhood $U\ni a$ and functions $f_1,\cdots,f_N$ holomorphic on this neighborhood such that
$$
A\cap U=\{z\in U: f_1(z)=\cdots=f_N(z)=0\}.
$$
A point $a\in A$ is called \emph{regular} if there is a neighborhood $U\ni a$ in $\Omega$ such that $A\cap U$ is a complex submanifold of $\Omega$. A point $a\in A$ is called a \emph{singular point} of $A$ if it is not regular.
\end{defn}
\begin{defn}
Let $Y$ be a manifold and let $X, Z$ be two submanifolds of $Y$. We say that the submanifolds $X$ and $Z$ are \emph{transversal} if $\forall x\in X\cap Z$, $T_x(X)+T_x(Z)=T_x(Y)$.
\end{defn}
The following two theorems are the main result of our paper \cite{our paper}. We will use them in the proof.
\begin{thm}
If there exists a positive, finite, regular, Borel measure~$\mu$ on $M$ such that the $L^2(\mu)$ norm and Bergman norm are equivalent on $Q$, i.e., $\exists C, c>0$ such that $\forall f\in Q$,
$$c\|f\|^2\leq\int_M|f(w)|^2d\mu(w)\leq C\|f\|^2,$$
then the projection operator $Q$ belongs to the Toeplitz algebra $\toe$ and consequently, the quotient module~$Q$~is essentially normal.
\end{thm}
In the sequel, we will say $\mu$ is an ``equivalent measure'' on $Q$ if it satisfies the above hypotheses.
\begin{thm}\label{one variety}
Suppose~$\tilde{M}$~is a complex analytic subset of an open neighborhood of~$\clb$~satisfying the following conditions:
\begin{itemize}
\item[(1)] $\tilde{M}$~intersects~$\pbn$~transversely.
\item[(2)] $\tilde{M}$~has no singular points on~$\pbn$.
\end{itemize}
Let~$M=\tilde{M}\cap\bn$ and let~$P=\{f\in\ber: f|_M=0\}$. Then $Q$ has an ``equivalent measure''. As a consequence, the projection operator $Q\in\toe$ and the quotient module~$Q$~is $p$-essentially normal for any $p>2\dim M$.
\end{thm}
Next, we introduce some tools in complex harmonic analysis that will be used in this paper (see \cite{Zhu Kehe}\cite{Suarez04}\cite{Suarez07} for more details).
\begin{defn}
For~$z\in\bn$, write~$P_z$~for the orthogonal projection onto the complex line~$\mathbb{C}z$ and~$Q_z=I-P_z$. The map
$$
\varphi_z(w)=\frac{z-P_z(w)-(1-|z|^2)^{1/2}Q_z(w)}{1-\langle w,z\rangle}
$$
defines an automorphism of $\mathbb{B}_n$.
\end{defn}
The map $\varphi_z$ has many nice properties, for example, it maps affine spaces to affine spaces. Also, $\varphi_z\circ\varphi_z=id$~and~$\varphi_z(0)=z$.
\begin{lem}\label{basic about varphi}
Suppose~$a$, $z$, $w\in\bn$, then
\begin{itemize}
\item[(1)] $$1-\langle\varphi_a(z),\varphi_a(w)\rangle=\frac{(1-\langle a,a\rangle)(1-\langle z,w\rangle)}{(1-\langle z,a\rangle)(1-\langle a,w\rangle)}.$$
\item[(2)] As a consequence of (1),
$$1-|\varphi_a(z)|^2=\frac{(1-|a|^2)(1-|z|^2)}{|1-\langle z,a\rangle|^2}.$$
\item[(3)] The Jacobian of the automorphism~$\varphi_z$~is
$$(J\varphi_z(w))=\frac{(1-|z|^2)^{n+1}}{|1-\langle w,z\rangle|^{2(n+1)}}.$$
\end{itemize}
\end{lem}
\begin{defn}
For $z, w\in\mathbb{B}_n$, define
$$
\rho(z,w)=|\varphi_z(w)|.
$$
$\rho$ is called the pseudo-hyperbolic metric on $\mathbb{B}_n$. Define
$$
\beta(z,w)=\frac{1}{2}\log\frac{1+\rho(z,w)}{1-\rho(z,w)}.
$$
$\beta$ is called the hyperbolic metric on $\bn$.
\end{defn}
From Lemma \ref{basic about varphi}, one can show that:
\begin{lem}
Fix $r>0$, then $\exists c, C>0$ such that
\begin{itemize}
\item[(1)] $c<\frac{1-|z|^2}{1-|w|^2}<C$, $\forall z, w, \beta(z,w)<r$.
\item[(2)] $c<\frac{|1-\langle z,w\rangle|}{1-|z|^2}<C$, $\forall z,w, \beta(z,w)<r$.
\end{itemize}
\end{lem}
For~$r>0$, $z\in\bn$, write
$$D(z,r)=\{w\in\bn: \beta(w,z)< r\}=\{w\in\bn: \rho(w,z)< s_r\},$$
where~$s_r=\tanh r$.
\begin{lem}{\cite[2.2.7]{Rudin}}\label{description of D(z,R)}
For~$z\in\bn$, $r>0$, the hyperbolic ball~$D(z,r)$~consists of all~$w$~that satisfy:
$$\frac{|Pw-c|^2}{s_r^2\rho^2}+\frac{|Qw|^2}{s_r^2\rho}<1,$$
where~$P=P_z$, $Q=Q_z$, and
$$c=\frac{(1-s_r^2)z}{1-s_r^2|z|^2},~~~\rho=\frac{1-|z|^2}{1-s_r^2|z|^2}.$$
\end{lem}
Thus~$D(z,r)$~is an ellipsoid with center~$c$, radius of~$s_r\rho$~in the~$z$~direction and~$s_r\sqrt{\rho}$~in the directions perpendicular to~$z$. Therefore the Lebesgue measure of~$D(z,r)$~is
$$v_n(D(z,r))=Cs_r^{2n}\rho^{n+1},$$
where~$C>0$~is a constant depending only on~$n$.
Note that when we fix~$r$, $\rho$~is comparable with~$1-|z|^2$. Hence~$v(D(z,r))$~is comparable with~$(1-|z|^2)^{n+1}$.
\begin{lem}\label{4equivalent}
Let~$\nu$~be a positive, finite, regular, Borel measure on~$\bn$~and~$r>0$. Then the following are equivalent.
When one of these conditions holds, $\nu$ is called a Carleson measure (for $\ber$).
\begin{itemize}
\item[(1)]$\sup_{z\in\bn}\int_{\bn}\frac{(1-|z|^2)^{n+1}}{|1-\langle w, z\rangle|^{2(n+1)}}d\nu(w)<\infty,$
\item[(2)]$\exists C>0:\int|f|^2d\nu\leq C\int|f|^2dv~\mbox{for all}~f\in\ber,$
\item[(3)]$\sup_{z\in\bn}\frac{\nu(D(z,~r))}{v_n(D(z,~r))}<\infty,$
\end{itemize}
\end{lem}
Let $A$ be the algebra of bounded functions on $\mathbb{B}_n$ which are uniformly continuous in the hyperbolic metric, equipped with the supreme norm. Then $A$ is a commutative $C^*$ algebra. Let $M_A$ be its maximal ideal space. Then the unit ball $\mathbb{B}_n$ is naturally contained in $M_A$ as evaluations. The algebra $A$ can be used to study the properties of the Toeplitz operators( cf. \cite{Suarez04}\cite{Suarez07}).
\begin{defn}
A sequence $\{z_m\}\subseteq\mathbb{B}_n$ is said to be \emph{separated} if there exists $\delta>0$ such that $\rho(z_k,z_l)\geq\delta$ for $k\neq l$.
If $x, y\in M_A$, define
$$\rho(x,y)=\sup\rho(\mathcal{S},\mathcal{T}),$$
where $\mathcal{S}$, $\mathcal{T}$ run over all separated sequences in $\mathbb{B}_n$ so that $x\in\overline{\mathcal{S}}^{_A}$ and $y\in\overline{\mathcal{T}}^{_A}$. Here $\overline{\mathcal{S}}^{_A}$ denotes the closure in $M_A$. Define
$$
\beta(x,y)=\frac{1}{2}\log\frac{1+\rho(x,y)}{1-\rho(x,y)}.
$$
\end{defn}
For $x\in M_A$ and any net $\{z_{\alpha}\}$ that converges to it, there is a map $\varphi_x:\mathbb{B}_n\to M_A$ such that $a\circ\varphi_x\in A$ and $a\circ\varphi_{z_{\alpha}}\to a\circ\varphi_x $ uniformly on compact sets of $\mathbb{B}_n$, for all $a\in A$ (cf. \cite{Suarez04}).
The following Lemma was proved in \cite{Suarez04}, Section 3.2 for the unit disc and the same proof works for the $\mathbb{B}_n$ case verbatimly.
\begin{lem}\label{extended rho}
Let $x, y\in M_A\backslash\mathbb{B}_n$. Then
\begin{itemize}
\item[(1)] $\rho(x,y)=a<1$ if and only if $y=\varphi_x(w)$ for some $w$ with $|w|=a$.
\item[(2)] $y=\varphi_x(\xi)$ with $\xi\in\mathbb{B}_n$ if and only if every separated sequences $\mathcal{S}$, $\mathcal{T}$ such that $x\in\overline{\mathcal{S}}^{_A}$ and $y\in\overline{\mathcal{T}}^{_A}$ satisfy $\rho(\mathcal{T},\{\varphi_{z_n}(\xi): z_n\in\mathcal{S}\})=0$.
\item[(3)] $\rho(\varphi_x(\xi_1),\varphi_x(\xi_2))=\rho(\xi_1,\xi_2)$ for every $\xi_1$, $\xi_2\in\mathbb{B}_n$.
\item[(4)] $\beta$ is a $[0,+\infty]$-valued metric on $M_A$.
\end{itemize}
\end{lem}
\begin{defn}
For $z\in\mathbb{B}_n$, define $U_z: L_a^2(\mathbb{B}_n)\to L_a^2(\mathbb{B}_n)$ to be
$$
U_z(f)=f\circ\varphi_z\cdot k_z.
$$
Here $k_z$ is the normalized reproducing kernel.
$$
k_z(w)=\frac{K_z(w)}{\|K_z\|}=\frac{(1-|z|^2)^{(n+1)/2}}{(1-\langle w,z\rangle)^{n+1}}.
$$
Then $U_z$ is an unitary operator on $L_a^2(\mathbb{B}_n)$ with $U_z^*=U_z$.
\end{defn}
The following Lemmas can be found in Section 8 and 10 of \cite{Suarez07}.
\begin{lem}
If $S\in\mathcal{T}(L^{\infty})$, then the map $\Psi_S:\mathbb{B}_n\to(\mathcal{B}(L_a^2(\mathbb{B}_n)), SOT)$, $z\mapsto S_z:=U_zSU_z$ extends continuously to $M_A$. We write $S_x$ for the operator $\Psi_S(x)$.
\end{lem}
\begin{lem}
$x\in M_A$, $S, T\in\mathcal{T}(L^{\infty})$, then
$$
(ST)_x=S_xT_x, (S_x)^*=(S^*)_x, \|S_x\|\leq\|S\|.
$$
\end{lem}
From the Lemma, we see that for any normal Toeplitz operator $S$ and any $f\in C(\sigma(S))$, $f(S)_x=f(S_x)$.
\begin{lem}\label{essential norm}
$S\in\mathcal{T}(L^{\infty})$, then
$$
\|S\|_e=\sup_{x\in M_A\backslash\mathbb{B}_n}\|S_x\|.
$$
\end{lem}
Suppose $H$ is a Hilbert space and $H_1$, $H_2$ are subspaces of $H$. When is $H_1+H_2$ closed? Write $H_3=H_1\cap H_2$. Then $H_1+H_2=(H_1\ominus H_3+H_2\ominus H_3)\oplus H_3$. Therefore $H_1+H_2$ is closed if and only if $H_1\ominus H_3+H_2\ominus H_3$ is closed. In the case when $H_1\cap H_2=\{0\}$, by open mapping Theorem we know that $H_1+H_2$ is closed if and only if the norm on $H_1+H_2$ is equivalent to the norm on $H_1\oplus H_2$.
\begin{defn}\label{defnangle}
Suppose $H_1$, $H_2$ are subspaces of a Hilbert space $H$. And write $H_3=H_1\cap H_2$.
We define the \emph{angle} of $H_1$ and $H_2$ to be
$$
\arccos\sup\{\frac{|\langle u,v\rangle|}{\|u\|\|v\|}: u\in H_1\ominus H_3, v\in H_2\ominus H_3\}.
$$
\end{defn}
\begin{lem}\label{angle}
The angle is positively related to the following quantities.
\begin{itemize}
\item[(1)] $\inf\{\frac{\|u-v\|^2}{\|u\|^2+\|v\|^2}:u\in H_1\ominus H_3, v\in H_2\ominus H_3\}$,
\item[(2)] $1-\|H_2H_1-H_3\|$,
\item[(3)] $1-\|H_1H_2H_1-H_3\|$.
\end{itemize}
\end{lem}
\begin{proof}
For the relation between the angle and (1), take $v$ by $-v$ in the previous equality. We get
$$
\frac{\|u-v\|^2}{\|u\|^2+\|v\|^2}=1-\frac{2Re\langle u,v\rangle}{\|u\|^2+\|v\|^2}.
$$
Therefore
\begin{eqnarray*}
&&\inf\{\frac{\|u-v\|^2}{\|u\|^2+\|v\|^2}:u\in H_1\ominus H_3, v\in H_2\ominus H_3\}\\
&=&1-\sup\{\frac{2Re\langle u,v\rangle}{\|u\|^2+\|v\|^2}:u\in H_1\ominus H_3, v\in H_2\ominus H_3\}\\
&=&1-\sup\{\frac{2|\langle u,v\rangle|}{\|u\|^2+\|v\|^2}:u\in H_1\ominus H_3, v\in H_2\ominus H_3\}
\end{eqnarray*}
Since
$$
\frac{|\langle u,v\rangle|}{\|u\|^2+\|v\|^2}\leq\frac{|\langle u,v\rangle|}{2\|u\|\|v\|}=\frac{|\langle au,a^{-1}v\rangle|}{\|au\|^2+\|a^{-1}v\|^2},
$$
where $a=\sqrt{\frac{\|v\|}{\|u\|}}$, we have
\begin{eqnarray*}
&&\inf\{\frac{\|u-v\|^2}{\|u\|^2+\|v\|^2}:u\in H_1\ominus H_3, v\in H_2\ominus H_3\}\\
&=&1-\sup\{\frac{|\langle u,v\rangle|}{\|u\|\|v\|}:u\in H_1\ominus H_3, v\in H_2\ominus H_3\}.
\end{eqnarray*}
Also, since $(H_2H_1-H_3)^*(H_2H_1-H_3)=H_1H_2H_1-H_3$, the quantities (2) and (3) are positively related. Now we show that (1) and (2) are positively related. Fix $u\in H_1\ominus H_3$, for any $v\in H_2\ominus H_3$,
\begin{eqnarray*}
&&\frac{\|u-v\|^2}{\|u\|^2+\|v\|^2}\geq \frac{\|u-v\|^2}{2\|u\|^2+\|u-v\|^2}\\
&\geq&\frac{\|u-H_2u\|^2}{2\|u\|^2+\|u-H_2u\|^2}\geq\frac{\|u-H_2u\|^2}{3\|u\|^2+3\|H_2u\|^2}.
\end{eqnarray*}
Also,
$$
\frac{\|u-H_2u\|^2}{\|u\|^2+\|H_2u\|^2}\approx\frac{\|u-H_2u\|^2}{\|u\|^2}.
$$
The first inequality shows that the infimum in (1) is obtained (modulo a constant) by taking $v=H_2u$. The second inequality shows that (1) and (2) are positively related. This completes the proof.
\end{proof}
From our discussion before Definition \ref{defnangle} and Lemma \ref{angle}(1), the following corollary is immediate.
\begin{cor}
$H_1+H_2$ is closed if and only if their angle is non-zero.
\end{cor}
As a consequence of Lemma \ref{angle}, when the projection operators to subspaces $H_1$, $H_2$ and $H_3$ all change continuously, the quantities in (3) change continuously, therefore the angles have a uniform lower bound on any compact set of parameters. This fact will be used in the proof of our main theorem.
\section{Decomposition of Varieties}
We begin this section with an example.
\begin{exam}\label{example hyperplane}
Suppose $\tilde{M}_1$ and $\tilde{M}_2$ are two linear subspaces of $\cn$. $\tilde{M}_3=\tilde{M}_1\cap \tilde{M}_2$. Let $M_i=\tilde{M}_i\cap\bn$ and $Q_i=\overline{span}\{K_{\lambda}| \lambda\in M_i\}\subseteq\ber$, $i=1,2,3$. Then
$$
\|Q_2Q_1Q_2-Q_3\|\leq a<1
$$
where the number $a$ depends on the angle between $\tilde{M}_1$ and $\tilde{M}_2$. As a consequence, $Q_1+Q_2$ is closed and $Q_1\cap Q_2=Q_3$.
\begin{proof}
First, to simplify notation, we use $Q_i$ and $M_i$ to denote both the spaces and the projection operators.
Let $\epsilon>0$ be determined later. Choose $k\in\mathbb{N}$ (depending on $\epsilon$) so that $\forall v\in M_2\ominus M_3$,
$$
|(M_2M_1)^kv|\leq\epsilon|v|.
$$
Clearly the operator $Q_2Q_1Q_2-Q_3$ vanishes on $Q_2^{\perp}$ and $Q_3$. For any $f\in Q_2\ominus Q_3$ with $\|f\|=1$, since
$$
(Q_2Q_1Q_2-Q_3)^kf=(Q_2Q_1)^kf,
$$
it suffices to prove that
$$
\|(Q_2Q_1)^kf\|\leq a^k.
$$
Let $d=\dim M_2$, by Example 3.3 in \cite{our paper}, the measure $\mu=c(1-|z|^2)^{n-d}dv_{M_2}$ with a suitable normalizing constant $c$ has the property that
$$\|Q_2 g\|^2=\int_{M_2}|g|^2d\mu, ~~~\forall g\in\ber.$$
Here $v_{M_2}$ is the volume measure on $M_2$.
It is easy to see that $Q_if(z)=f(M_i(z)), i=1,2,3$. Here we use $M_i$ to denote the projection operators to $\tilde{M}_i$. Now for any $z\in M_2$,
\begin{eqnarray*}
(Q_2Q_1)^kf(z)&=&Q_1(Q_2Q_1)^{k-1}f(z)\\
&=&(Q_2Q_1)^{k-1}f(M_1z)\\
&=&(Q_2Q_1)^{k-1}f(M_2M_1z)\\
&=&\cdots\\
&=&f((M_2M_1)^kz).
\end{eqnarray*}
\begin{eqnarray*}
(M_2M_1)^kz&=&(M_2M_1)^kM_3z+(M_2M_1)^k(1-M_3)z\\
&=&M_3z+(M_2M_1)^k(1-M_3)z.
\end{eqnarray*}
By the choice of $k$,
$$
|(M_2M_1)^k(1-M_3)z|^2\leq\epsilon^2|(1-M_3)z|^2\leq\epsilon^2(1-|M_3z|^2).
$$
Therefore the pseudo-hyperbolic metric
$$\rho((M_2M_1)^kz, M_3z)\leq r_{\epsilon},$$
where $r_{\epsilon}\to0$ when $\epsilon\to0$.
Before continuing, we need the following lemma.
\begin{lem}\label{continuity of function}
$\exists C>0$, $\forall g\in Hol(\mathbb{B}_d)$, $\forall z, w\in\mathbb{B}_d$, $\beta(z, w)<1/2$
$$
|g(z)-g(w)|^2\leq C\frac{\rho(z, w)^2}{(1-|w|^2)^{d+1}}\int_{D(w)}|g(\eta)|^2dv(\eta),
$$
where $D(w)=\{z | \beta(z,w)<1\}$.
\end{lem}
\begin{proof}
Using a reproducing kernel argument, it is easy to show that for $g\in Hol(\mathbb{B}_d)$ and $|\lambda|\in D(0,1/2)$,
$$
|g(\lambda)-g(0)|^2\leq C|\lambda|^2\int_{D(0)}|g(\eta)|^2dv(\eta).
$$
So if $\beta(z,w)<1/2$,
\begin{eqnarray*}
|g(z)-g(w)|^2&=&|g\varphi_w(\varphi_w(z))-g\varphi_w(0)|^2\\
&\leq&C|\varphi_w(z)|^2\int_{D(0)}|g\varphi_w(\eta)|^2dv(\eta)\\
&=&C\rho(z,w)^2\int_{D(w)}|g(\lambda)|^2\frac{(1-|w|^2)^{d+1}}{|1-\langle \lambda,w\rangle|^{2(d+1)}}dv(\lambda)\\
&\leq&C\frac{\rho(z,w)^2}{(1-|w|^2)^{d+1}}\int_{D(w)}|g(\lambda)|^2dv(\lambda)
\end{eqnarray*}
This completes the proof of lemma.
\end{proof}
From the Lemma and previous argument,
\begin{eqnarray*}
&&|f((M_2M_1)^kz)|\\
&=&|f((M_2M_1)^kz)-f(M_3z)|\\
&\leq&Cr_{\epsilon}\frac{1}{(1-|M_3z|^2)^{d+1}}\int_{D(M_3z)}|f(\eta)|^2dv_{M_2}(\eta)(1-|z|^2)^{n-d}dv_{M_2}(z).
\end{eqnarray*}
Therefore
\begin{eqnarray*}
&&\|(Q_2Q_1)^kf\|^2=\int_{M_2}|(Q_2Q_1)^kf(z)|^2(1-|z|^2)^{n-d}dv_{M_2}(z)\\
&=&\int_{M_2}|f((M_2M_1)^kz)|^2(1-|z|^2)^{n-d}dv_{M_2}(z)\\
&\leq&Cr_{\epsilon}^2\int_{M_2}\frac{1}{(1-|M_3z|^2)^{d+1}}\int_{D(M_3z)}|f(\eta)|^2dv_{M_2}(\eta)(1-|z|^2)^{n-d}dv_{M_2}(z)\\
&=&Cr_{\epsilon}^2\int_{M_2}|f(\eta)|^2\int_{\{z\in M_2: M_3z\in D(\eta)\}}\frac{(1-|z|^2)^{n-d}}{(1-|M_3z|^2)^{d+1}}dv_{M_2}(z)dv_{M_2}(\eta).
\end{eqnarray*}
If $ M_3z\in D(\eta)$, then $\beta(M_3z,M_3\eta)\leq\beta(M_3z,\eta)<1$ and therefore
$$
\beta(M_3\eta,\eta)\leq\beta(M_3z,M_3\eta)+\beta(M_3z,\eta)<2.
$$
Thus $1-|M_3z|^2\approx 1-|M_3\eta|^2\approx 1-|\eta|^2$. We claim that
$$
\int_{\{z\in M_2: M_3z\in D(\eta)\}}(1-|z|^2)^{n-d}dv_{M_2}(z)\leq C(1-|\eta|^2)^{n+1}.
$$
For $z\in M_2$, write temporarily $z=(z',z'')$ where $z'$ corresponds to the coordinates in $M_3$.
\begin{eqnarray*}
&&\int_{\{z\in M_2: M_3z\in D(\eta)\}}(1-|z|^2)^{n-d}dv_{M_2}(z)\\
&=&C\int_{z'\in D(\eta)}(1-|z'|^2)^{n-d+d-d_3}\int_{\lambda\in\mathbb{B}_{d-d_3}}(1-|\lambda|^2)^{n-d}dv(\lambda)dv(z')\\
&\leq&C(1-|\eta|^2)^{n-d_3}(1-|\eta|^2)^{d_3+1}\\
&=&C(1-|\eta|^2)^{n+1}.
\end{eqnarray*}
This proves the claim.
Therefore
$$
\|(Q_2Q_1)^kf\|^2\leq Cr_{\epsilon}^2\|f\|^2.
$$
Take $\epsilon>0$ such that $Cr_{\epsilon}^2<1$ in the beginning, then our proof is complete.
\end{proof}
\end{exam}
\begin{rem}\label{inverse of exam 1}
From Lemma \ref{continuity of function}, it is easy to see that
$$
\|k_z-k_w\|\leq C\rho(z,w)
$$
when $\rho(z,w)$ is small. This tells us that the inverse of Example \ref{example hyperplane} is also true: if the angle between $M_1$ and $M_2$ is small, then so is the angle between $Q_1$ and $Q_2$. Take $z_1\in M_1$, $z_2\in M_2$ such that $z_i\perp M_3$, then $Q_3k_{z_i}=0, i=1,2$ and $\|k_{z_1}-k_{z_2}\|\leq C\rho(z_1,z_2)$. When the angle of $M_1$ and $M_2$ is small we can take such $z_i$ so that $\rho(z_1,z_2)$ is small. Therefore by Lemma \ref{angle}, the angle between $Q_1$ and $Q_2$ is small.
\end{rem}
\begin{exam}\label{counter example}
Suppose $\tilde{M}_1$ and $\tilde{M}_2$ are two affine spaces, $\emptyset\neq \tilde{M}_1\cap \tilde{M}_2\cap\overline{\mathbb{B}_n}\subseteq\partial\mathbb{B}_n$. Let $M_i=\tilde{M}_i\cap\bn$, $Q_i=\overline{span}\{K_{\lambda}:\lambda\in M_i\}$. Then $Q_1\cap Q_2=\{0\}$ and $Q_1+Q_2$ is not closed.
\end{exam}
\begin{proof}
Since $Q_1\cap Q_2$ is the orthogonal space of a polynomial ideal with generators of degree one, and has no zero points inside $\bn$ and only one on $\partial\mathbb{B}_n$, $Q_1\cap Q_2=\{0\}$, we have $Q_1\cap Q_2=\{0\}$.
For $z\in\mathbb{B}_n$, it is easy to prove that $Q_{iz}$ is the projection to the space $\overline{span}\{K_{\lambda}:\lambda\in\varphi_z(M_i)\}$. Therefore, without loss of generality, we assume that $M_1$ is a linear subspace.
We claim that $\rho(M_1,M_2)=0$. To prove this, take $z\in M_1\cap M_2\cap\partial\mathbb{B}_n$, then $rz\in M_1$, $\forall 0<r<1$. Change coordinates so that $z=(z_1,0,\ldots,0)$. Since $M_2$ is an affine space that intersects $\mathbb{B}_n$, after possibly changing the order of basis, $M_2$ has expression
$$
w=(w',L(w'))+(z_1,0,\ldots,0), w\in M_2,
$$
where $L$ is a linear function of $w'=(w_1,\ldots,w_d)$, $d=\dim M_2$. Take $w_r=((r-1)z',L((r-1)z'))+z$, then
$$
\varphi_{rz}(w_r)=(\varphi_{rz'}(rz'),\frac{(1-r^2)^{1/2}}{1-r^2}\cdot(r-1)L(z'))=(0,O((1-r^2)^{1/2})).
$$
Therefore $\rho(rz,w_r)\to0, r\to1$. This proves the claim. The rest of the proof is same as in Remark \ref{inverse of exam 1}.
\end{proof}
Next we discuss the more general case: suppose $M_1$ and $M_2$ are two varieties and $M_3=M_1\cap M_2$. Let $Q_i=\overline{span}\{K_{\lambda}:\lambda\in M_i\}$, $1=1,2,3$. When do we know that $Q_1+Q_2$ is closed and $Q_1\cap Q_2/Q_3$ is finite dimensional? This question is important because when it holds, the essential normality of $Q=Q_1+Q_2$ follows from the essential normality of each $Q_1$ and $Q_2$. Moreover, we can obtain the index class of $Q_1+Q_2$ from the index classes of $Q_1$, $Q_2$ and $Q_3$. This would allow us to obtain index results for complicated varieties by decomposing it into nice pieces.
As preparation for our main result, we establish a few lemmas.
\begin{lem}\label{rhoxy}
Suppose $x$, $y\in M_A\backslash \mathbb{B}_n$ and $\rho(x,y)<1$, then there exists a unitary operator $U$ such that for any $S\in\mathcal{T}(L^{\infty})$,
$$
S_y=U^*S_xU.
$$
\end{lem}
\begin{proof}
Suppose $z_{\alpha}\to x$, $\{z_{\alpha}\}\subseteq\bn$. Since $\rho(x,y)<1$, by Lemma \ref{extended rho}, $\exists\lambda\in\mathbb{B}_n$ such that $w_{\alpha}:=\varphi_{z_{\alpha}}(\lambda)\to y$.
\begin{eqnarray*}
U_{w_{\alpha}}f(z)&=&f\circ\varphi_{w_{\alpha}}(z)\frac{(1-|w_{\alpha}|^2)^{(n+1)/2}}{(1-\langle z,w_{\alpha}\rangle)^{n+1}}\\
&=&f\circ U_{\alpha}\circ\varphi_{\lambda}\circ\varphi_{z_{\alpha}}(z)a_{\alpha}\frac{(1-|z_{\alpha}|^2)^{(n+1)/2}}{(1-\langle z,z_{\alpha}\rangle)^{n+1}}\frac{(1-|\lambda|^2)^{(n+1)/2}}{(1-\langle\varphi_{z_{\alpha}}(z),\lambda\rangle)^{n+1}}\\
&=&a_{\alpha}U_{z_{\alpha}}U_{\lambda}(f\circ U_{\alpha})(z).
\end{eqnarray*}
Here
$$
a_{\alpha}=\frac{(1-\langle z_{\alpha},\lambda\rangle)^{n+1}}{|1-\langle z_{\alpha},\lambda\rangle|^{n+1}}
$$
is a number of absolute value $1$ and $U_{\alpha}$ is a unitary operator on $\mathbb{C}^n$ such that $\varphi_{w_{\alpha}}=U_{\alpha}\circ\varphi_{\lambda}\circ\varphi_{z_{\alpha}}$(see the proof of Lemma 6.2 in \cite{Suarez07} for existence of such $U_{\alpha}$).
Now we can take a subnet such that $U_{\alpha}\to U'$. Here $U'$ is a unitary operator on $\mathbb{C}^n$.
Therefore, for $f, g\in\mathbb{C}[z_1,\ldots,z_n]$,
\begin{eqnarray*}
\langle S_{w_{\alpha}}f,g\rangle&=&\langle SU_{w_{\alpha}}f,U_{w_{\alpha}}g\rangle\\
&=&\langle SU_{z_{\alpha}}U_{\lambda}(f\circ U_{\alpha}),U_{z_{\alpha}}U_{\lambda}(g\circ U_{\alpha})\rangle\\
&=&\langle S_{z_{\alpha}}U_{\lambda}(f\circ U_{\alpha}),U_{\lambda}(g\circ U_{\alpha})\rangle\\
&=&\langle S_{z_{\alpha}}U_{\lambda}(f\circ U_{\alpha}-f\circ U'),U_{\lambda}(g\circ U_{\alpha}))\rangle\\
& &+\langle S_{z_{\alpha}}U_{\lambda}(f\circ U'),U_{\lambda}(g\circ U_{\alpha}-g\circ U')\rangle\\
& &+\langle S_{z_{\alpha}}U_{\lambda}(f\circ U'),U_{\lambda}(g\circ U')\rangle.
\end{eqnarray*}
Note that $f\circ U_{\alpha}$ tends to $f\circ U'$ in norm, and that $\|S_{\alpha}\|\leq\|S\|$, by standard argument, the first two terms converge to $0$. Define $Uf=U_{\lambda}(f\circ U')$. Taking limit, we see that for any polynomial $f, g$,
$$
\langle S_yf,g\rangle=\langle S_xUf,Ug\rangle.
$$
Therefore $S_y=U^*S_xU$. This completes the proof.
\end{proof}
\begin{lem}\label{3equivalent}
Suppose $Q_1$, $Q_2$, $Q_3$ are closed linear subspaces of $L_{a}^2(\mathbb{B}_n)$, $Q_3\subseteq Q_1\cap Q_2$, the projection operators $Q_i\in\mathcal{T}(L^{\infty})$. Then the following are equivalent.
\begin{itemize}
\item[(1)] $Q_1+Q_2$ is closed and $Q_1\cap Q_2/Q_3$ is finite dimensional.
\item[(2)] $\|Q_2Q_1Q_2-Q_3\|_e<1$.
\item[(3)] $\exists 0<a<1$ such that $\forall x\in M_A\backslash \mathbb{B}_n$,
$$
\|Q_{2x}Q_{1x}Q_{2x}-Q_{3x}\|<a.
$$
\end{itemize}
\end{lem}
\begin{proof}
The equivalence of (1) and (2) can be obtained by analysing the spectral decomposition of the operator $Q_2Q_1Q_2-Q_3$. And the equivalence of (3) and (2) is by Lemma \ref{essential norm}.
\end{proof}
\begin{lem}\label{Mx}
Suppose $\tilde{M}$ is an analytic subset of an open neighborhood of $\overline{\mathbb{B}_n}$. $\tilde{M}$ is smooth on $\partial\mathbb{B}_n$ and transversal with $\partial\mathbb{B}_n$. Let $M=\tilde{M}\cap\mathbb{B}_n$ and
$$Q=\overline{span}\{K_{\lambda}|\lambda\in M\}.$$
Then for $x\in M_A\backslash \mathbb{B}_n$.
If $\rho(x,\overline{M}^{_A})=1$, then $Q_x=0$;
If $x\in\overline{M}^{_A}$, then $Q_x=\overline{span}\{K_{\lambda}|\lambda\in M_x\}$, where $M_x=\tilde{M}_x\cap\bn$ and
$$
\tilde{M}_x=\{v\in T\tilde{M}|_{\hat{x}}: v\perp\hat{x}\}+\mathbb{C}\hat{x}.
$$
Where $\hat{x}\in\pbn$ is obtained by evaluating $x$ at each index function $z_i$.
If $\rho(x,\overline{M}^{_A})<1$, then $\exists y\in\overline{M}^{_A}$ such that $Q_x$ is unitary equivalent to $Q_y$.
Here $\overline{M}^{_A}$ denotes the closure of $M$ in $M_A$.
\end{lem}
\begin{proof}
By the proof in \cite{our paper}, there exists an ``equivalent measure'' $\mu$ on $M$ such that $0$ is isolated in $\sigma(T_{\mu})$ and $Q=Ran T_{\mu}$. In other words, if we take a continuous function $f$ on $\mathbb{R}$ such that $f(0)=0$ and $f$ takes value $1$ on $\sigma(T_{\mu}\backslash\{0\})$, then the projection operator $Q=f(T_{\mu})$. Therefore $Q_x=f((T_{\mu})_x)$. Suppose $z_{\alpha}\to x$, $z_{\alpha}\in\mathbb{B}_n$. The operators $(T_{\mu})_{z_{\alpha}}$ tend to $(T_{\mu})_x$ in the strong operator topology. Since
$$
(T_{\mu})_{z_{\alpha}}=T_{\mu_{z_{\alpha}}},
$$
where the positive measure $\mu_{z_{\alpha}}$ is defined by
$$
\int gd\mu_{z_{\alpha}}=\int g\circ\varphi_{z_{\alpha}}|k_{z_{\alpha}}|^2d\mu.
$$
From the definition,
$$
\|\mu_{z_{\alpha}}\|=\int d\mu_{z_{\alpha}}=\int |k_{z_{\alpha}}|^2d\mu\leq C\|k_{z_{\alpha}}\|=C
$$
since $\mu$ is a Carleson measure. Therefore
$$
\|\mu_{z_{\alpha}}\|\leq C.
$$
Therefore the net $\{\mu_{z_{\alpha}}\}$ has a subnet that converges to some measure $\mu_x$ in the $weak^*$ topology. Then
$$
\langle(T_{\mu})_xg,h\rangle=\int g\bar{h}d\mu_x, \forall g,h\in \mathbb{C}[z_1,\ldots,z_n].
$$
So $\mu_x$ is a Carleson measure and $(T_{\mu})_x=T_{\mu_x}$.
From our construction, we see that $Q_x$ is the projection operator onto $Range T_{\mu_x}=\overline{span}\{K_{\lambda}|\lambda\in supp\mu_x\}$.
Next we discuss about $supp\mu_x$. We claim that
\begin{eqnarray*}
supp\mu_x&=&M_x:=\{w\in\mathbb{B}_n| \rho(\varphi_{z_{\alpha}}(w), M)\to0\}\\
&=&\{w\in\mathbb{B}_n| \rho(w,\varphi_{z_{\alpha}}(M))\to0\}.
\end{eqnarray*}
For any $w\in\mathbb{B}_n$ and $0<r<1$,
\begin{eqnarray*}
\mu_{z_{\alpha}}(D(w,r))&=&\int\chi_{D(w,r)}d\mu_{z_{\alpha}}\\
&=&\int\chi_{D(\varphi_{z_{\alpha}}(w),r)}|k_{z_{\alpha}}|^2d\mu\\
&\approx&\mu(D(\varphi_{z_{\alpha}}(w),r))\frac{|1-\langle w,z_{\alpha}\rangle|^{2(n+1)}}{(1-|z_{\alpha}|^2)^{n+1}},
\end{eqnarray*}
where the last inequality is because for $\lambda\in D(\varphi_{z_{\alpha}}(w),r)$,
\begin{eqnarray*}
|k_{z_{\alpha}}(\lambda)|^2&=&\frac{(1-|z_{\alpha}|^2)^{n+1}}{|1-\langle\lambda,z_{\alpha}\rangle|^{2(n+1)}}\\
&\approx&\frac{(1-|z_{\alpha}|^2)^{n+1}}{|1-\langle\varphi_{z_{\alpha}}(w),z_{\alpha}\rangle|^{2(n+1)}}\\
&=&\frac{|1-\langle w,z_{\alpha}\rangle|^{2(n+1)}}{(1-|z_{\alpha}|^2)^{n+1}}.
\end{eqnarray*}
Then if $\rho(\varphi_{z_{\alpha}}(w),M)\to0$, for any $0<r<1$, we have $$\mu(D(\varphi_{z_{\alpha}}(w),r))\approx(1-|\varphi_{z_{\alpha}}(w)|^2)^{n+1}.$$ So
\begin{eqnarray*}
\mu_{z_{\alpha}}(D(w,r))&\approx&(1-|\varphi_{z_{\alpha}}(w)|^2)^{n+1}\frac{|1-\langle w,z_{\alpha}\rangle|^{2(n+1)}}{(1-|z_{\alpha}|^2)^{n+1}}\\
&=&(1-|w|^2)^{n+1}.
\end{eqnarray*}
Therefore $\mu_x(D(w,r))>0$ for any $0<r<1$, i.e., $w\in supp\mu_x$.
On the other hand, if $\rho(\varphi_{z_{\alpha}}(w), M)\nrightarrow0$, by taking a subnet we can assume that $\rho(\varphi_{z_{\alpha}}(w),M)>\epsilon>0$. Take $r<\epsilon$ in the proof and it is easy to see that $\mu_x(D(w,r))=0$. Therefore $w$ is not in the support. This completes the proof of our claim.
Now we study the set $M_x$.
First, suppose $\rho(x,\overline{M}^{_A})=1$. By definition, this means that for any $0<r<1$, there exists a net $z_{\alpha}\to x$, such that $\rho(\{z_{\alpha}\}, M)>r$. Therefore for any $0<r'<1$, choose $r>r'$, then from the proof above it is easy to see that $\mu_x(D(0,r'))=0$, which implies $\mu_x=0$. Therefore $Q_x=0$.
Second, the case $\rho(x,\overline{M}^{_A})<1$ is by Lemma \ref{rhoxy}.
Finally, when $x\in\overline{M}^{_A}$, suppose $\tilde{M}$ has local expression $$
w=(w_1,\ldots,w_d,F_{d+1}(w'),\ldots,F_n(w')), w\in\tilde{M}\cap B(z,\delta),
$$
where $w'=(w_1,\ldots,w_d)$. Note that we are using the same kind of expression as in \cite{our paper}, where the basis and functions change continuous with $z$. And the point $z$ always has expression $(z_1,0,\ldots,0)$.
Suppose $w\in M_x$ and suppose $z_{\alpha}\in M$, $z_{\alpha}\to x$. Then by definition $\exists\lambda_{\alpha}\in M$ such that $$\rho(w,\varphi_{z_{\alpha}}(\lambda_{\alpha}))\to0.$$
which is equivalent to $|w-\varphi_{z_{\alpha}}(\lambda_{\alpha})|\to0$.
Take any $\epsilon>0$ such that $|w|+\epsilon<1$, then for some subnet we have $\rho(\lambda_{\alpha},z_{\alpha})=|\varphi_{z_{\alpha}}(\lambda_{\alpha})|<|w|+\epsilon<1$. Therefore $$1-\langle\lambda_{\alpha},z_{\alpha}\rangle\approx1-|z_{\alpha}|^2.$$
Since $\lambda_{\alpha}=(\lambda_{\alpha}',F_{\alpha}(\lambda_{\alpha}'))$
under the basis determined by $z_{\alpha}$.
$$
\varphi_{z_{\alpha}}(\lambda_{\alpha})=(\eta_{\alpha,1},\eta_{\alpha,2},\ldots,\eta_{\alpha,d},\ldots,\eta_{\alpha,n})
$$
where
$$
\eta_{\alpha,1}=\frac{z_{\alpha,1}-\lambda_{\alpha,1}}{1-\langle\lambda_{\alpha},z_{\alpha}\rangle},~~\eta_{\alpha,i}=-\frac{(1-|z_{\alpha}|^2)^{1/2}}{1-\langle\lambda_{\alpha},z_{\alpha}\rangle}\lambda_{\alpha,i},~~i=2,\ldots,d.
$$
and
$$
\eta_{\alpha,i}=-\frac{(1-|z_{\alpha}|^2)^{1/2}}{1-\langle\lambda_{\alpha},z_{\alpha}\rangle}F_{\alpha,i}(\lambda_{\alpha}'),~~i=d+1,\ldots,n.
$$
For simplicity we omit the subscript $\alpha$.
Now $$F_i(\lambda')=L_i(\lambda')+O(|z-\lambda|^2)=L_i(\lambda')+O(1-|z|^2).$$
Here $L_i$ is the linear part of $F_i$:
$$
L_i(\lambda')=\sum_{j=1}^dA_j(\lambda_j-z_j)=A_1(\lambda_1-z_1)+\sum_{j=2}^dA_j\lambda_j.
$$
Then
$$
\eta_i=(1-|z|^2)^{1/2}A_1\eta_1+\sum_{j=2}^dA_j\eta_j+O((1-|z|^2)^{1/2}),~~~j=d+1,\ldots,n.
$$
Since $\eta\to w$ as $z\to x$ and the coefficients $A_i$ converges to the corresponding value at $\hat{x}$. We see that
$$
w_i=\sum_{j=2}^dA_jw_j, i=d+1,\ldots,n.
$$
Also if $w$ is of this form, the argument above also implies that $w\in M_x$.
To write more explicitly, $M_x=\tilde{M}_x\cap\bn$ and
$$
\tilde{M}_x=\{v\in T\tilde{M}|_{\hat{x}}: v\perp\hat{x}\}+\mathbb{C}\hat{x}.
$$
\end{proof}
Now suppose $M_1$ and $M_2$ are as in Lemma \ref{Mx}. Let $M_3=M_1\cap M_2$ and let $Q_i=\overline{span}\{K_{\lambda}|\lambda\in M_i\}$. We want to find a suitable condition to ensure that $Q_1+Q_2$ is closed and $Q_1\cap Q_2/Q_3$ is finite dimensional. An equivalent condition is that $\|Q_2Q_1Q_2-Q_3\|_e<1$. From Theorem \ref{one variety}, the projections $Q_1$ and $Q_2$ are already in $\mathcal{T}(L^{\infty})$. Assume $M_3$ is also as in Lemma \ref{Mx}, then by Lemma \ref{3equivalent}, we only need to look at the operators $Q_{2x}Q_{1x}Q_{2x}-Q_{3x}$, $x\in M_A\backslash\mathbb{B}_n$. From Lemma \ref{Mx} we know $Q_{1x}$, $Q_{2x}$ and $Q_{3x}$ are projections to quotient modules corresponding to linear varieties $M_{ix}$. We list the cases that are possible:
\begin{itemize}
\item[(1)]$\rho(x,\overline{M_1}^{_A})<1$, $\rho(x, \overline{M_2}^{_A})<1$. In this case, there are two possibilities for $M_3$:
\begin{itemize}
\item[(1-a)] $\rho(x,\overline{M_3}^{_A})<1$ or
\item[(1-b)]$\rho(x,\overline{M_3}^{_A})=1$.
\end{itemize}
\item[(2)]$\rho(x,\overline{M_1}^{_A})=1$, $\rho(x,\overline{M_2}^{_A})<1$, then $\rho(x,\overline{M_3}^{_A})=1$.
\item[(3)]$\rho(x,\overline{M_1}^{_A})<1$, $\rho(x,\overline{M_2}^{_A})=1$, then $\rho(x,\overline{M_3}^{_A})=1$.
\item[(4)]$\rho(x,\overline{M_1}^{_A})=1$, $\rho(x,\overline{M_2}^{_A})=1$, then $\rho(x,\overline{M_3}^{_A})=1$.
\end{itemize}
The case (1-b) corresponds to that of Example \ref{counter example}, which we want to avoid.
Under the cases (2)(3)(4), the operator $Q_{2x}Q_{1x}Q_{2x}-Q_{3x}=0$. In the case (1-a), we can assume $x\in\overline{M_3}^{_A}$, then it becomes Example \ref{example hyperplane}. The only thing that matters is the angle between the two linear subspaces $M_{1x}$ and $M_{2x}$ which are defined in Lemma \ref{Mx}.
To summarize, we need the following conditions:
\begin{itemize}
\item[(1)] $M_1$ and $M_2$ are transverse with $\pbn$, smooth on $\partial\mathbb{B}_n$.
\item[(2)] $M_3$ is also transverse with $\pbn$, smooth on $\partial\mathbb{B}_n$.
\item[(3)] If $x\in M_A\backslash \mathbb{B}_n$ and $\rho(x, \overline{M_1}^{_A})<1$, $\rho(x,\overline{M_2}^{_A})<1$, then $\rho(x,\overline{M_3}^{_A})<1$.
\item[(4)] $\forall x\in\overline{M_3}^{_A}\backslash\mathbb{B}_n$, $M_{1x}\cap M_{2x}=M_{3x}$.
\item[(5)] For all $x\in\overline{M_3}^{_A}\backslash\mathbb{B}_n$, the angles between $M_{1x}$ and $M_{2x}$ have a lower bound.
\end{itemize}
Next we seek properties that would ensure condition (1)-(5). Let us begin with a definition.
\begin{defn}
Let $K$ and $L$ be embedded submanifolds of a manifold $M$ and suppose that their intersection $K\cap L$ is also an embedded submanifold of $M$. Then $K\cap L$ are said to have \emph{clean intersection} if for each $p\in K\cap L$ we have
$$
T_p(K\cap L)= T_pK\cap T_pL.
$$
\end{defn}
Condition (1) and (2) must be assumed so that each of the three quotient modules alone are essentially normal. Assume further that $M_1$ and $M_2$ intersect cleanly at each point of intersection in $\partial\mathbb{B}_n$, then it is not hard to verify (1)(2)(4)(5). Condition (4) follows from the definition of clean intersection and the expression we obtained for $M_x$ in Lemma \ref{Mx}. The fact that $M_i$ is smooth on $\pbn$ tells us that the projection operator to $M_{ix}$ depend continuously on $x$, $i=1,2,3$. Therefore condition (5) follows from Lemma \ref{angle} and compactness. The fact that the assumptions above also ensure condition (3) takes some effort to prove. Assuming this, we have reached our main theorem of this paper.
\begin{thm}\label{mainthm}
Suppose $\tilde{M_1}$ and $\tilde{M_2}$ are two analytic subsets of an open neighborhood of $\clb$. Let $\tilde{M_3}=\tilde{M_1}\cap\tilde{M_2}$. Assume that
\begin{itemize}
\item[(i)]$\tilde{M_1}$ and $\tilde{M_2}$ intersect transversely with $\pbn$ and have no singular points on $\pbn$.
\item[(ii)]$\tilde{M_3}$ also intersects transversely with $\pbn$ and has no singular points on $\pbn$.
\item[(iii)]$\tilde{M_1}$ and $\tilde{M_2}$ intersect cleanly on $\partial\mathbb{B}_n$.
\end{itemize}
Let $M_i=\tilde{M_i}\cap\mathbb{B}_n$ and $Q_i=\overline{span}\{K_{\lambda}: \lambda\in M_i\}$, $i=1,2,3.$ Then $Q_1\cap Q_2/Q_3$ is finite dimensional and $Q_1+Q_2$ is closed. As a consequence, $Q_1+Q_2$ is $p$-essentially normal for $p>2d$, where $d=\max\{\dim M_1, \dim M_2\}$.
\end{thm}
As stated above the theorem, the only thing left for us to verify is the following lemma.
\begin{lem}\label{condition 3}
Assume the same conditions as Theorem \ref{mainthm}, then the technical condition (3) holds.
\end{lem}
We break the proof into several lemmas.
\begin{lem}\label{condition 3 lemma 1}
Suppose $z=(z_1,0,\cdots,0)\in\bn$, then
$$
\frac{\partial}{\partial w_1}|\varphi_z(w)|^2(0)=\bar{z_1}(|z_1|^2-1)
$$
and
$$
\frac{\partial}{\partial w_i}|\varphi_z(w)|^2(0)=0,~~~i=2,3,\cdots,n.
$$
As a consequence, if $M$ is any complex manifold passing through $0$ and obtains its minimal hyperbolic distant to $z$ at the point $0$, then $z$ must be orthogonal to the tangent space $TM|_0$.
\end{lem}
\begin{proof}
The two formulas are obtained by direct computation. To prove the last statement, one only need to observe that the derivative of $|\varphi_z(w)|^2$ in $u$ direction is
$$
\frac{\partial|\varphi_z(w)|^2}{\partial u}(0)=\sum_{i=1}^nu_i\frac{\partial|\varphi_z(w)|^2}{\partial w_i}(0)=\langle u,z\rangle(|z_1|^2-1).
$$
Since the minimal value of $|\varphi_z(w)|^2$ is obtained at $0$, the derivative of $|\varphi_z(w)|^2$ along all directions in $TM|_0$ must be $0$. Therefore $z$ is orthogonal to $TM|_0$. This completes the proof.
\end{proof}
\begin{lem}\label{condition 3 lemma 2}
Suppose $M$ satisfies the hypotheses of Theorem \ref{one variety}, and suppose $\{z_{\alpha}\}, \{w_{\alpha}\}\subseteq M$ are two separated nets such that , viewed as points in $M_A$, $z_{\alpha}$ tends to a point $x\in M_A\backslash\bn$, viewed as points in $\clb$, $w_{\alpha}$ tends to $\hat{x}$. Then any limit point of the net $\{\varphi_{z_{\alpha}}(w_{\alpha})\}$ is in $\overline{M_x}\subseteq\clb$.
\end{lem}
\begin{proof}
For convenience we omit the subscript $\alpha$. Using the same convention as before, we take the basis at each $z$, so
$$
z=(z_1,0,\cdots,0),~~~w=(w',F(w')),
$$
where $w'=(w_1,\cdots,w_d)$ and $F=(f_{d+1},\cdots,F_n)$ is the expression of $\tilde{M}$ depending continuously on $z$. Same as in the proof of Lemma \ref{Mx}, we have
$$
\varphi_z(w)=(\eta_1,\cdots,\eta_n),
$$
where
$$
\eta_1=\frac{z_1-w_1}{1-\langle w,z\rangle},~~~\eta_i=-\frac{(1-|z|^2)^{1/2}w_i}{1-\langle w,z\rangle},i=2,\cdots,d
$$
and
$$
\eta_j=-\frac{(1-|z|^2)^{1/2}F_j(w')}{1-\langle w,z\rangle},j=d+1,\cdots,n.
$$
We write $F_j(w')=L_j(w')+O(|w-z|^2)$, where $L$ is the linear part of $F$.
Since
$$
|w-z|^2=|w|^2+|z|^2-2Re\langle w,z\rangle\leq2(1-Re\langle w,z\rangle)\leq2|1-\langle w,z\rangle|,
$$
for $j=d+1,\cdots,n$,
$$
\eta_j+\frac{(1-|z|^2)^{1/2}L_j(w')}{1-\langle w,z\rangle}=\frac{(1-|z|^2)^{1/2}}{1-\langle z,w\rangle}O(|1-\langle w,z\rangle|)\to0,~~z\to\hat{x}.
$$
The rest of the proof is as in Lemma \ref{Mx}(3).
\end{proof}
\begin{proof}[\textbf{Proof of Lemma \ref{condition 3}}]
Suppose $x\in M_A\backslash\bn$ and $\rho(x,\overline{M_1}^{_A})<1$, $\rho(x,\overline{M_2}^{_A})<1$, we will show that $\rho(x,\overline{M_3}^{_A})<1$. Clearly, $\hat{x}\in\tilde{M}_1\cap\tilde{M}_2=\tilde{M}_3$. By Lemma \ref{Mx}, without loss of generality, we assume $x\in\overline{M_1}^{_A}$.
Let $\{z_{\alpha}\}\subseteq M_1$ such that $z_{\alpha}\to x$. Let $w_{\alpha}\in M_2$ and $\lambda_{\alpha}\in M_3$ such that $\rho(z_{\alpha},w_{\alpha})=\rho(z_{\alpha},M_2)$ and $\rho(z_{\alpha},\lambda_{\alpha})=\rho(z_{\alpha},M_3)$. Take subnets (using the same notation) such that both nets converge in $M_A$. Suppose $w_{\alpha}\to y\in M_A$ and $\lambda_{\alpha}\to \xi\in M_A$. Clearly $\hat{y}=\hat{\xi}=\hat{x}$. For convenience we omit the subscript $\alpha$ in the sequel.
Since $\rho(\varphi_{\lambda}(z),0)=\rho(\varphi_{\lambda}(z),\varphi_{\lambda}(M_3))$, by Lemma \ref{condition 3 lemma 1}, $\varphi_{\lambda}(z)\perp\varphi_{\lambda}(M_3)$. The latter tends uniformly to $M_{3\xi}$ while the first has a subnet that converges to some point $a$ in $\pbn$ by compactness. Therefore $a\perp M_{3\xi}$.
On the other hand, $\rho(\varphi_{\lambda}(z),\varphi_{\lambda}(w))=\rho(z,w)\to\rho(x,\overline{M_2}^{_A})<1$. Since $|\varphi_{\lambda}(z)|=\rho(\lambda,z)\to1$, we have the Euclidean distance $|\varphi_{\lambda}(z)-\varphi_{\lambda}(w)|\to0$. Therefore $\varphi_{\lambda}(w)\to a$. By Lemma \ref{condition 3 lemma 2}, $a\in M_{1\xi}\cap M_{2\xi}$ which equals $M_{3\xi}$ by the clean intersection condition and the experession of $M_{i\xi}$ in Lemma \ref{Mx}. So $a$ is a vector of length $1$ which both belong to $M_{3\xi}$ and is perpendicular to $M_{3\xi}$. A contradiction. Therefore such $x$ does not exist. This completes the proof.
\end{proof}
\section{Summary}
A classical way of proving Geometric Arveson-Douglas Conjecture is by ``decomposing the variety'' (cf. \cite{Sha Ken}). In this paper, we introduce ideas in complex harmonic analysis to solve this problem. This approach has the advantage of ``localizing'' the problem, which allows us to reduce the problem to simpler cases. Su\'{a}rez's results (\cite{Suarez04}\cite{Suarez07}) play an important role here.
The ideas in our last paper \cite{our paper} and this paper should be considered as two continuous steps towards analysing the varieties. First, we approximate the variety at points close to $\pbn$, using simpler varieties (in our case, their linearizations at the points). Then we use results on these simpler varieties to obtain essential normality results of the original variety. After the first step, we are able to ``localize'' the variety at points in $M_A\backslash \bn$. We then obtain results on relation between the angle of two quotient modules and the relative positions of ``localizations'' of the corresponding varieties. Finally, we give sufficient conditions for the sum of two quotient modules to be closed, i.e., the angle to be positive. This gives us results on unions of varieties. In the future, when we proved more results in the first step, we can use similar techniques in this paper to generate more complicated examples.
Another consequence of our result is an index result. Given $Q_i$ as in Theorem \ref{mainthm}, consider the exact sequence
$$
0\to Q_1\cap Q_2\to Q_1\oplus Q_2\to Q_1+Q_2\to0.
$$
Here we define the first map to be the embedding and second map to be the difference of two entries. In general, given such a short exact sequence and given that the sum on the right side is closed, then the essential normality of the two modules imply the essential normality of both their sum and their intersection (cf. \cite{Douglas Wang remark}).
Also, By BDF theory, the essentially normal quotient modules $Q_i$, $i=1,2,3$ and $Q=Q_1+Q_2$ define index classes, or elements in $K_1(\tilde{M_i}\cap\pbn)$ and $K_1((\tilde{M_1}\cup\tilde{M_2})\cap\pbn)$, respectively. Since $Q_1\cap Q_2/Q_3$ is finite dimensional, the index class $[Q_1\cap Q_2]=[Q_3]$. Therefore we have an equation of index classes
$$[Q_3]+[Q]=[Q_1]+[Q_2].$$
In particular, if we assume further that the varieties $M_i$, $i=1,2,3$ satisfy the assumptions of \cite{DYT}, then the index results in \cite{DYT} apply to $[Q_i]$ and we get a formula for $[Q]$ from the above equation.
|
1,116,691,500,737 | arxiv | \section{Introduction}
In some application domains, for example legal reasoning, knowing that something
holds (or it is presumed to hold) is not enough to draw further conclusions from it.
One has to determine to what degree one can assert that it holds. In other words
statements in rules (here we use the term `rule' to indicate a mechanism/principle
to assert conclusions from already established assertions) have an associated proof
standard. Accordingly, a party wanting to assert a particular assertion has the burden
to prove that assertion with the appropriate standard (or a stronger one). Consider the
following rule:
\[
\mathit{IllegalBehaviour}, \neg\mathit{Justification} \Rightarrow \mathit{Liability}
\]
Suppose there is factual evidence about the illegal behaviour. The information in the rule
is not enough, since it does not prescribe the burden needed to assess whether the behaviour
was justified or not. According to \cite{PrakkenSartor2007,jurix10burden}, in a civil case the lack of
justification is subject to the so-called \emph{burden of production}, i.e., there is a
credible argument for it, while in a criminal case the \emph{burden of persuasion} applies (i.e.,
more sceptical reasoning must be used).
Let us consider a concrete scenario. Party A
caused some injuries to B.
Party A was much stronger than Party B, and thus the action causing injury is not justified.
On the other hand, Party A claims that they acted in self defence since they were
under threat from Party B. The scenario can now be modelled by the rules:
\[
\begin{array}{lrclclrcl}
& \mi{Injury}, \neg \mi{Justification} & \Rightarrow & \phantom{\neg} \mi{Liability}\\
& \mi{Threat} & \Rightarrow & \phantom{\neg} \mi{Justification}\\
& \mi{Stronger} & \Rightarrow &\neg \mi{Justification} \\
\end{array}
\]
Here, in case we are not able to assess whether the threat was real,
we have a credible argument for $\neg\mi{Justification}$ (because Party A is stronger), but
we do not have a sceptical argument for it (because it might be that the threat was real, and then the outcome
from the two conflicting rules is undetermined).
Thus, we can establish liability in a civil case,
but Party A is not criminally liable. Accordingly, we can reformulate the initial rule
in the following two principles:
\[
\begin{array}{lrclclrcl}
& \mi{Tort}, \mi{BurdenProduction}(\neg \mi{Justification}) & \Rightarrow & \mi{CivilCaseLiability}\\
& \mi{Crime}, \mi{BurdenPersuasion}(\neg \mi{Justification}) & \Rightarrow & \mi{CriminalCaseLiability} \\
\end{array}
\]
where $\mi{BurdenProduction}$ and $\mi{BurdenPersuasion}$ are annotations describing the mode
in which we have to prove the lack of justification for the illegal behaviour.
Legal reasoning has developed so-called \emph{proof standards} (e.g., scintilla of evidence,
substantial evidence, preponderance of evidence, beyond reasonable doubt) according to which
assertions have to be justified. \cite{GordonWalton:proof} proposed to encode proof standards
using rule-based argumentation with salience, and \cite{icail2011carneades} shows how to
represent the proof standards of \cite{GordonWalton:proof} where, essentially, each
proof standard corresponds to a different degree of provability in some defeasible logic
variant. In particular,
\cite{icail2011carneades} argues that the proof standard of beyond reasonable doubt
corresponds to provability in the ambiguity propagating variant of defeasible logic.
However, as the following example illustrates there are examples where more than one such proof
standards must be used. This means that incompatible variants of defeasible
logic have to work side-by-side.
Suppose that a piece of evidence $A$ suggests that the defendant in a legal case
is not responsible while a second piece of evidence $B$ indicates that he/she is
responsible; moreover, the sources are equally reliable. According to the underlying
legal system a defendant is presumed innocent (i.e., not guilty) unless responsibility
has been proved (beyond reasonable doubt).
The above scenario is encoded by the following rules:
\[
\begin{array}{lrclclrcl}
r_1: & \mathit{EvidenceA} &\Rightarrow & \neg\mathit{Responsible}\\
r_2: & \mathit{EvidenceB} &\Rightarrow & \phantom{\neg} \mathit{Responsible}& ~~~~~ &
r_3: & \mathit{Responsible} &\Rightarrow & \phantom{\neg} \mathit{Guilty}\\
&&&&& r_4: &{} & \Rightarrow & \neg \mathit{Guilty}
\end{array}
\]
where $r_{3}$ is stronger than $r_{4}$.
Given both $\mathit{EvidenceA}$ and $\mathit{EvidenceB}$, the literal
$\mathit{Responsible}$ is ambiguous. There are applicable rules ($r_1$ and
$r_2$) for and against the literal, with no way to adjudicate between them. As a
consequence $r_3$ is not applicable, and so there is no applicable rule
arguing against the presumption of innocence (rule $r_{4}$).
In an ambiguity blocking setting we obtain a $\neg\mathit{Guilty}$ verdict;
the ambiguity about responsibility is blocked from applying to $\mathit{Guilty}$.
In contrast, in an ambiguity propagating setting,
the ambiguity of $\mi{Responsible}$ propagates to $\mathit{Guilty}$, and thus the literals
$\mathit{Guilty}$ and $\neg \mathit{Guilty}$ are ambiguous too; hence an undisputed
conclusion cannot be drawn.
When we look at the example above, is it appropriate to say that we have reached
a not guilty verdict without any reasonable doubt?
The evidence supporting that the defendant was responsible has not been refuted.
This example supports the contention of \cite{icail2011carneades} that
ambiguity propagating inference is a more appropriate representation of
proof beyond a reasonable doubt.
Let us extend the scenario. Suppose that the legal system allows for compensation
for wrongly accused people. A person (defendant) has been wrongly accused if
the defendant is found innocent, where innocent is defined as $\neg\mi{Guilty}$.
In addition, by default, people are not entitled to compensation. The additional
elements of this scenario are modelled by the rules:
\[
\begin{array}{lrclclrcl}
r_5:& \neg\mi{Guilty} & \Rightarrow &\phantom{\neg} \mi{Compensation}\\
r_6:& {} & \Rightarrow & \neg\mi{Compensation} \\
\end{array}
\]
where $r_5$ is stronger than $r_6$.
In the full scenario, the defendant is not found innocent, and so is not entitled to compensation.
If we take a purely ambiguity blocking stance then, since we are not able to determine
whether there was responsibility, the defendant is not guilty, and then the
defendant is entitled to compensation. On the other hand, in a purely ambiguity
propagating setting, $\mi{Guilty}$ and $\neg\mi{Guilty}$ are ambiguous,
and this makes $\mi{Compensation}$ and $\neg\mi{Compensation}$ ambiguous;
we are in a position where we cannot decide whether the defendant is
entitled or not to compensation.
Thus, both choices are unsatisfactory:
either the defendant receives compensation despite not being found innocent
or no decision is made about compensation.
What we want is a regime where we can reason about guilt in an ambiguity propagating way,
but then reason about compensation in an ambiguity blocking way.
This can be achieve by replacing rule $r_{5}$ with
\[
r'_{5}: \mi{BeyondReasonableDoubt}(\neg\mi{Guitly}) \Rightarrow \mi{Compensation}
\]
where, similarly to what we have done in the previous example, $BeyondReasonableDoubt$
is an annotation to the literal $\neg\mi{Guilty}$ that holds in case the literal is
provable under ambiguity propagation,
and the proof standard for $\mi{Compensation}$ can be chosen to be ambiguity blocking.
The purpose of this paper is to provide a formalism -- \emph{annotated defeasible logic} --
in which such distinctions can be expressed,
define its semantics, and investigate properties of the formalism.
This paper is organised as follows.
In the next section we provide brief background on defeasible logics.
We then introduce annotated defeasible logic,
and define its behaviour with a meta-program.
In the following section we establish some properties of annotated defeasible logic,
including its relationship to existing defeasible logics and
the relative inference strength of the additional inference rules we introduce.
Finally, we show that annotated defeasible logic has the flexibility to deal with different
notions of failure, corresponding to different semantics of negation-as-failure in logic programs.
Due to space limitations, parts of the paper -- including proof sketches -- are presented
in the supplementary material accompanying the paper at the TPLP archive.
\section{Defeasible Logics}
In this section we can only present an outline of defeasible logics.
Further details can be obtained from \cite{TOCL10} and the references therein.
We address propositional defeasible logics,
but many results should extend to a first-order language.
A defeasible theory is built from a language $\Sigma$ of literals (which we assume is closed under negation)
and a language $\Lambda$ of labels.
A \emph{defeasible theory} $D = (F, R, >)$ consists of a set of facts $F$, a finite set of rules $R$,
each rule with a distinct label from $\Lambda$,
and an acyclic relation $>$ on $\Lambda$ called the \emph{superiority relation}.
This syntax is uniform for all the logics considered here.
Facts are individual literals expressing indisputable truths.
Rules relate a set of literals (the body), via an arrow, to a literal (the head), and are one of three types:
a strict rule, with arrow $\rightarrow$;
a defeasible rule, with arrow $\Rightarrow$;
or
a defeater, with arrow $\leadsto$.
Strict rules represent inferences that are unequivocally sound if based on definite knowledge;
defeasible rules represent inferences that are generally sound.
Inferences suggested by a defeasible rule may fail, due to the presence in the theory
of other rules.
Defeaters do not support inferences, but may impede inferences suggested by other rules.
The superiority relation provides a local priority on rules with conflicting heads.
Strict or defeasible rules whose bodies are established defeasibly represent claims
for the head of the rule to be concluded.
When both a literal and its negation are claimed,
the superiority relation contributes to the adjudication of these conflicting claims by an inference rule,
leading (possibly) to a conclusion.
Defeasible logics derive conclusions that are outside the syntax of the theories.
Conclusions may have the form
${+}d q$, which denotes that under the inference rule $d$ the literal $q$ can be concluded,
or
$-d q$, which denotes that the logic can establish that under the inference rule $d$ the literal $q$ cannot be concluded.
The syntactic element $d$ is called a proof tag.
In general, neither conclusion may be derivable:
$q$ cannot be concluded under $d$, but the logic is unable to establish that.
Tags ${+}\Delta$ and $-\Delta$ represent monotonic provability (and unprovability)
where inference is based on facts, strict rules, and modus ponens.
We assume these tags and their inference rules are present in every defeasible logic.
What distinguishes a logic is the inference rules for defeasible reasoning.
The four logics discussed in \cite{TOCL10} correspond to four different pairs of inference rules,
tagged $\partial$, $\delta$, $\partial^*$, and $\delta^*$;
they produce conclusions of the form (respectively) ${+}\partial q$, $-\partial q$, ${+}\delta q$, $-\delta q$, etc.,
where $q$ is a literal.
These logics all abide by the Principle of Strong Negation \cite{flexf}, which asserts that the condition
for applying a $-d$ inference rule should be the strong negation of the condition for applying ${+}d$.
The inference rules $\delta$ and $\delta^*$ require auxiliary tags and inference rules,
denoted by $\supp_\delta$ and $\supp_{\delta^*}$, respectively\footnote{
Note that in previous works these have been denoted by $\supp$ and $\supp^*$ or $\int$ and $\int^*$.
This change of notation is made to accommodate new forms of support introduced in this paper.
},
expressing that there is at least (weak) support for the conclusion.
These inference rules are available in the supplementary material.
For each of the four principal defeasible tags $d$, the corresponding logic is denoted by ${\bf DL}(d)$.
We write $D \vdash +d q$ (respectively, $D \vdash -d q$) if $+d q$ ($-d q$) can be proved by ${\bf DL}(d)$.
The four principal tags and corresponding inference rules represent different intuitions about defeasible reasoning,
that is, define different forms of defeasibility:
in $\partial$ and $\partial^*$ ambiguity is blocked, while in $\delta$ and $\delta^*$ ambiguity is propagated;
in $\partial$ and $\delta$ rules for a literal act as a team to overcome competing rules,
while in $\partial^*$ and $\delta^*$ an individual rule must overcome all competing rules.
The scenario in the introduction with rules $r_1 - r_4$ exemplifies the treatments of ambiguity.
For an example of team defeat, consider rules $s_1$ and $s_2$ for $q$ and rules $s_3$ and $s_4$ for $\neg q$,
with $s_1 > s_3$ and $s_2 > s_4$;
then no individual rule for $q$ can overcome the rules for $\neg q$,
but $s_1$ and $s_2$ -- as a team -- can,
because every rule for $\neg q$ is overridden by some rule in the team.
A more detailed discussion of ambiguity and team defeat in the ${\bf DL}$ framework is given in \cite{TOCL10,Maher12}.
In \cite{MG99,flexf}, the inference rules in ${\bf DL}(d)$ were reformulated as a meta-program ${\cal M}_d$:
a logic program that takes a representation of a defeasible theory $D$ as input and specifies
what conclusions can be drawn from the theory according to the $d$ inference rules.
(The combined meta-program and theory is denoted by ${\cal M}_d(D)$.)
We will take this meta-programming formulation as our starting point,
rather than the inference rules as presented in \cite{TOCL10},
for example.
This meta-program formulation is given in the supplementary material.
We assume, initially, that the logic programming semantics in use is
Kunen's semantics \cite{Kunen}, which expresses the 3-valued logical consequences of
the Clark completion of a logic program.
Equivalently,
Kunen's semantics is the set of all consequences of $\Phi \uparrow n$ for any finite $n$,
where $\Phi$ is Fitting's semantic function for logic programs \cite{Fitting}.
(Fitting's semantics, which is the least fixedpoint of $\Phi$, expresses the
logical consequences of 3-valued Herbrand models of
the Clark completion of a logic program.)
Although defeasible logics are usually founded on proofs, there are alternative semantics for these logics:
a model-theoretic semantics was defined in \cite{Maher02},
a denotational semantics for ${\bf DL}(\partial)$ was presented in \cite{densem},
and an argumentation semantics for ${\bf DL}(\partial)$ was given in \cite{JLC04}.
Each of these approaches provides an alternative characterization of the
conclusions derivable by proofs in the logic.
However, in this paper we only use the meta-programming formulation of the proof systems.
In the following, annotated defeasible logic will be defined as an integration of the four defeasible logics discussed above.
However, it should be clear that the same approach can be applied to any set of defeasible logics
employing the same logic programming semantics.
\section{Annotated Defeasible Logic} \label{sect:ADL}
Annotated defeasible logic is the formalism we propose, motivated by the discussion in the introduction.
We begin by addressing its syntax, which is an extension of the syntax of defeasible logics.
A tag is any one of the proof tags, or the additional tag $\mt{free}$.
An annotated literal has the form $t \: q$, where $t$ is a tag and $q$ is a literal.
An \emph{annotated defeasible rule} has the form
\[
r: ~~ L_1, \ldots, L_n \Rightarrow q
\]
where $r$ is a label, $q$ is a literal and each $L_i$ is either an annotated literal or a fail-expression,
where a \emph{fail-expression} has the form $\mt{fail} ~ L$, where $L$ is an annotated literal.
An annotated defeater is defined similarly; strict rules are not annotated.
Roughly, the meaning of a rule
\[
r: ~~ t_1 \: q_1, \ldots, t _n\: q_n, \mt{fail}\: t _{n+1}\: q_{n+1}, \ldots, \mt{fail}\: t_m \: q_m \Rightarrow q
\]
is that if $q_i$ can be proved using inference rule $t_i$, for $1 \leq i \leq n$,
and proof of $q_i$ can be demonstrated to fail using inference rule $t_i$, for $n+1 \leq i \leq m$,
then we have a \emph{prima facie} reason to infer $q$.
As with all defeasible logics, such an inference can be overridden by another rule.
A proof tag only indicates which inference rule should be applied to resolve conflict concerning that literal.
Thus, an annotated literal $t \: q$ is asking, roughly, for ${+}t \: q$ to be proved.
A fail-expression $\mt{fail} \: t \: q$ is asking, roughly, for ${-}t \: q$ to be proved.
The $\mt{free}$ tag has a different meaning than the proof tags.
A free literal $\mt{free} q_i$ must be proved by the same inference rule that is intended to prove $q$.
This provides a mechanism by which defeasible rules can be agnostic as to inference rule,
which can be determined later,
just as defeasible rules in current defeasible logics are.
An \emph{annotated defeasible theory} is a defeasible theory where the defeasible rules are annotated
and fail-expressions are allowed.
Alternatively, we can think of an annotated defeasible theory as consisting of
an unannotated defeasible theory (the \emph{underlying theory}) $D$ that allows fail-expressions, and
an \emph{annotation function} $\alpha$ that maps each body literal occurrence to its annotation.
In this case we denote the annotated defeasible theory by $\alpha(D)$.
We can consider $\alpha$ a total function, or consider it a partial function mapping literal occurrences to proof tags.
The unmapped literals are $\mt{free}$.
We now turn to expressing the meaning of annotated defeasible theories using the meta-programming approach.
The semantics of a theory is parameterized by a logic programming semantics, which is applied to a meta-program.
Given an annotated defeasible theory $D=(F,R,>)$, the theory is represented by facts as follows:
\begin{enumerate}
\item $\mt{fact}(p)$. \hfill if $p\in F$
\item $\mt{strict}(r_i,p,[L_1,\dots,L_n])$.
\hspace*{\fill}
if
$r_i:L_1,\dots,L_n\to p\in R$
\item $\mt{defeasible}(r_i,p,[L_1,\dots,L_n])$.
\hspace*{\fill}
if
$r_i:L_1,\dots,L_n\Rightarrow p\in R$
\item $\mt{defeater}(r_i,p,[L_1,\dots,L_n])$.
\hspace*{\fill}
if
$r_i:L_1,\dots,L_n\leadsto p\in R$
\item $\mt{sup}(r_i,r_j)$.
\hspace*{\fill}
for each pair of rules such that
$r_i > r_j$
\end{enumerate}
where the $L_i$ are annotated literals or fail-expressions.
The meta-program to which these facts are input is denoted by ${\cal M}$, while
the combination of ${\cal M}$ and the representation of $D$ is denoted by ${\cal M}(D)$.
In what follows, we permit ourselves some syntactic flexibility in presenting
the meta-program. (For example, we enumerate a list instead of explicitly iterating over it,
and express the complementation operation ${\sim\!\:\!}$ as a function\footnote{
The complement of $p$ is $\neg p$ and the complement of $\neg p$ is $p$.
${\sim\!\:\!}$ is unrelated to $\mt{fail}$, since it is the complement of classical negation.}.
Furthermore, tags and $\mt{fail}$ are unary functors.)
However, there is no technical difficulty in using
conventional logic programming syntax to represent this program.
Before we get to the predicates that define the meaning of theories, we define some auxiliary predicates.
As discussed in the introduction to defeasible logics,
the different proof tags represent different forms of defeasibility.
In particular, some forms block ambiguity, while others propagate ambiguity;
some use team defeat, while others require an individual rule to overcome all conflicting rules.
The following facts are used to specify, for each proof tag:
that it is a proof tag,
whether it expresses team defeat or individual defeat,
and whether the inference rule blocks or propagates ambiguity.
Strictly speaking, we should distinguish the proof tags appearing syntactically in ${\cal M}$
from the tags appearing in conclusions (which are not part of the syntax of defeasible logics, but part of its meta-theory).
However, because there is a clear correspondence between the two, we find it clearer to use the same symbol for both. \\
{\small
\begin{minipage}{0.3\textwidth}
\begin{Fact}\label{f:teamB}
$\mt{team}( \partial ).$
\end{Fact}
\begin{Fact}\label{f:teamP}
$\mt{team}( \delta ).$
\end{Fact}
\begin{Fact}\label{f:indivB}
$\mt{indiv}( \partial^* ).$
\end{Fact}
\begin{Fact}\label{f:indivP}
$\mt{indiv}( \delta^* ).$
\end{Fact}
\end{minipage} \hfill
\begin{minipage}{0.3\textwidth}
\begin{Fact}\label{f:Iblock}
$\mt{ambiguity\_blocking}( \partial^* ).$
\end{Fact}
\begin{Fact}\label{f:Tblock}
$\mt{ambiguity\_blocking}( \partial ).$
\end{Fact}
\begin{Fact}\label{f:Iprop}
$\mt{ambiguity\_propagating}( \delta^* ).$
\end{Fact}
\begin{Fact}\label{f:Tprop}
$\mt{ambiguity\_propagating}( \delta ).$
\end{Fact}
\end{minipage} \hfill
\begin{minipage}{0.3\textwidth}
\begin{Fact}\label{f:tagTB}
$\mt{proof\_tag}( \partial^* ).$
\end{Fact}
\begin{Fact}\label{f:tagIB}
$\mt{proof\_tag}( \partial ).$
\end{Fact}
\begin{Fact}\label{f:tagTP}
$\mt{proof\_tag}( \delta^* ).$
\end{Fact}
\begin{Fact}\label{f:tagIP}
$\mt{proof\_tag}( \delta ).$
\end{Fact}
\end{minipage}
} \\\ \\
The following clauses define the class of all rules and the class of supportive rules.
Defeaters are not supportive rules because they can only be used to prevent other conclusions;
they cannot support any conclusion.
{\small
\begin{clause}
$\mt{supportive\_rule}(Label,Head,Body)$:-\\
\> $\mt{strict}(Label,Head,Body)$.\\
\\[-.5\baselineskip]
$\mt{supportive\_rule}(Label,Head,Body)$:-\\
\> $\mt{defeasible}(Label,Head,Body)$.
\end{clause}
\begin{clause}
$\mt{rule}(Label,Head,Body)$:-\\
\> $\mt{supportive\_rule}(Label,Head,Body)$.\\
\\[-.5\baselineskip]
$\mt{rule}(Label,Head,Body)$:-\\
\> $\mt{defeater}(Label,Head,Body)$.
\end{clause}
}
The next clauses express monotonic provability. \\
{\small
\begin{Clause}\label{strictly1}
$\mt{definitely}(X)$ :-\\
\> $\mt{fact}(X)$.
\end{Clause}
\begin{Clause}\label{strictly2}
$\mt{definitely}(X)$ :-\\
\> $\mt{strict}(R,X,[\seq{Y}])$,\\
\> $\mt{definitely}(Y_1)$,\dots,$\mt{definitely}(Y_n)$.
\end{Clause}
}
In the predicate expressing defeasible inference, $\mt{defeasibly}$,
one argument is written as a subscript $\mt{Z}$ in the following clauses.
That argument takes as its value one of the four proof tags and represents
the inference rule that should be applied to resolve conflict for the literal in the other argument,
unless the literal has a proof annotation.
All clauses for predicates with a subscript $\mt{Z}$
implicitly contain $\mt{proof\_tag}(\mt{Z})$ in their body.
In clause \ref{defeasibly_free} we see that $\mt{free}$-annotated literals are to be proved according to $\mt{Z}$.
In clause \ref{defeasibly_fail}, fail-expressions are defined: failure is implemented by negation.
This is valid because the logics involved satisfy the Principle of Strong Negation.
For such logics, the conditions for $-d$ inference rules are a negation of the conditions for $+d$ inference rules.
In both defeasible logics and logic programming, failure-to-prove is a primitive notion,
available in defeasible logics through negative tags and in logic programming through negation.
Hence, it is not surprising that failure is implemented by negation in the meta-program.
The remaining two clauses are reflective of the basic structure of defeasible reasoning.
Clause \ref{defeasibly1} expresses that any literally that is definitely true
(proved monotonically from facts and strict rules)
is also defeasibly true.
Clause \ref{defeasibly_notfree} handles an annotated literal by using the tag $Y$
as the subscript argument in subsidiary computations.
This clause says that a literal $X$, annotated by $Y$, is proved if
the negation of $X$ is not proved monotonically
and there is a supportive rule $R$ that is not overruled,
each of whose body literals are proved defeasibly according to $Y$.
{\small
\begin{Clause}\label{defeasibly_free}
$\mt{defeasibly_Z}(\mt{free}\ X)$ :-\\
\> $\mt{proof\_tag}(Z)$,\\
\> $\mt{defeasibly_Z}(Z\ X)$.
\end{Clause}
\begin{Clause}\label{defeasibly_fail}
$\mt{defeasibly_Z}(\mt{fail}\ X)$ :-\\
\> $\mt{not\ defeasibly_Z}(X)$.
\end{Clause}
\begin{Clause}\label{defeasibly1}
$\mt{defeasibly_Z}(X)$ :-\\
\> $\mt{definitely}(X)$.
\end{Clause}
\begin{Clause}\label{defeasibly_notfree}
$\mt{defeasibly_Z}(Y\ X)$ :-\\
\> $\mt{proof\_tag}(Y)$,\\
\> $\mt{not\ definitely}({\sim\!\:\!} X)$,\\
\> $\mt{supportive\_rule}(R,X,[\seq{W}])$,\\
\> $\mt{defeasibly_Y}(W_1)$,\dots,$\mt{defeasibly_Y}(W_n)$,\\
\> $\mt{not\ overruled_Y}(R,X)$.
\end{Clause}
}
The basic structure of overruling a rule is similar for all defeasible logics:
the body of the overruling rule must be proved and the rule not ``defeated''.
However,
it varies depending on whether the logic blocks or propagates ambiguity.
In an ambiguity blocking logic, the body of the overruling rule must be established defeasibly
whereas,
in an ambiguity propagating logic, the body of the overruling rule need only be supported.
{\small
\begin{Clause}\label{overruled_AB}
$\mt{overruled_Z}(R,X)$ :-\\
\> $\mt{ambiguity\_blocking}(Z)$, \\
\> $\mt{rule}(S,{\sim\!\:\!} X,[\seq{U}])$,\\
\> $\mt{defeasibly_Z}(U_1)$,\dots,$\mt{defeasibly_Z}(U_n)$,\\
\> $\mt{not\ defeated_Z}(R,S,{\sim\!\:\!} X)$.
\end{Clause}
\begin{Clause}\label{overruled_AP}
$\mt{overruled_Z}(R,X)$ :-\\
\> $\mt{ambiguity\_propagating}(Z)$, \\
\> $\mt{rule}(S,{\sim\!\:\!} X,[\seq{U}])$,\\
\> $\mt{supported_Z}(U_1)$,\dots,$\mt{supported_Z}(U_n)$,\\
\> $\mt{not\ defeated_Z}(R,S,{\sim\!\:\!} X)$.
\end{Clause}
}
The notion of defeat varies, depending on whether a logic involves team defeat or individual defeat.
In individual defeat, the overruling rule $S$ is defeated if the rule $R$ it tries to overrule is superior to $S$.
In team defeat, $S$ is defeated if there is a rule $T$ (possibly the same as $R$) that is superior to $S$ and
whose body can be proved.
{\small
\begin{Clause}\label{defeated_team}
$\mt{defeated_Z}(R,S,{\sim\!\:\!} X)$ :-\\
\> $\mt{team}(Z)$, \\
\> $\mt{sup}(T,S)$, \\
\> $\mt{supportive\_rule}(T,X,[\seq{V}])$,\\
\> $\mt{defeasibly_Z}(V_1)$,\dots,$\mt{defeasibly_Z}(V_n)$.
\end{Clause}
\begin{Clause}\label{defeated_indiv}
$\mt{defeated_Z}(R,S,{\sim\!\:\!} X)$ :-\\
\> $\mt{indiv}(Z)$, \\
\> $\mt{sup}(R,S)$.
\end{Clause}
}
The structure of this meta-program makes one point clear that was less readily apparent in \cite{flexf} or \cite{TOCL10}:
treatment of ambiguity concerns how the body of an overruling rule is proved,
while the choice of team/individual defeat concerns how an overruling rule can be defeated.
For the ambiguity propagating logics we must define the notion of ``supported''.
The intuition is that a literal is supported if there is a chain of supportive rules that form a proof tree
for the literal, and each supportive rule is not beaten (i.e. overruled) by a rule that is proved defeasbily.
In ordinary defeasible logics support is only needed for the ambiguity propagating logics
but, for annotated defeasible theories, we also need to have support for ambiguity blocking logics.
This is because we might wish to use, as part of the support,
a rule that contains an annotated literal such as $\partial q$.
Hence the $\mt{supported}$ predicate is defined uniformly,
with a parameter $\mt{Z}$ specifying the form of defeasibility underlying the support.
Thus we are introducing new forms of support: $\supp_\partial$ and $\supp_{\partial^*}$.
As with $\mt{defeasibly}$, the clauses for $\mt{supported}$ address
free literals,
fail-expressions,
literals that are proved definitely,
and proof-annotated literals.
Note how the parameter $\mt{Z}$ to $\mt{supported}$ is used by $\mt{beaten}$ to select the
form of defeasibility for which the body of an overruling rule must be proved.
{\small
\begin{Clause}\label{support_free}
$\mt{supported_Z}(\mt{free}\ X)$ :-\\
\> $\mt{supported_Z}(Z\ X)$.
\end{Clause}
\begin{Clause}\label{support_fail}
$\mt{supported_Z}(\mt{fail}\ X)$ :-\\
\> $\mt{not\ supported_Z}(X)$.
\end{Clause}
\begin{Clause}\label{support_definite}
$\mt{supported_Z}(X)$ :-\\
\> $\mt{definitely}(X)$.
\end{Clause}
\begin{Clause}\label{support_notfree}
$\mt{supported_Z}(Y\ X)$ :-\\
\> $\mt{proof\_tag}(Y)$,\\
\> $\mt{supportive\_rule}(R, X, [\seq{W}])$,\\
\> $\mt{supported_Y}(W_1)$,\dots,$\mt{supported_Y}(W_n)$,\\
\> $\mt{not\ beaten_Y}(R,X)$.
\end{Clause}
\begin{Clause}\label{beaten1}
$\mt{beaten_Z}(R,X)$ :-\\
\>$\mt{rule}(S,{\sim\!\:\!} X,[\seq{W}])$,\\
\>$\mt{defeasibly_Z}(W_1)$,\dots,$\mt{defeasibly_Z}(W_n)$,\\
\>$\mt{sup}(S,R)$.
\end{Clause}
}
Let us now examine how to put annotated defeasible logic to work by
revisiting the compensation example presented in the introduction.
As we have already discussed, $\mi{Guilty}$ must be proven
with the ``beyond reasonable doubt'' proof standard to derive that
the defendant is entitled to receive a compensation.
As we have alluded to in the introduction, \cite{GordonWalton:proof}
proposed to model proof standards such as scintilla of evidence,
preponderance of evidence, clear and convincing case, beyond reasonable
doubts and dialectical validity using rule based argumentation. For
example, they define that the proof standard of preponderance of
evident for a literal $p$ is satisfied if and only if the maximum weight
of applicable arguments for $p$ exceeds some threshold $\alpha$, and the
difference between the maximum weight of the applicable arguments for $p$
and the maximum weight of the applicable arguments against $p$ exceeds some
threshold $\beta$. \cite{icail2011carneades} shows how the weights and
thresholds can be modelled by a preference relation (superiority) over
arguments (rules) and it establishes the following relationships between the
proof standards and proof tags:
\begin{center}
$\begin{array}{lc}
\text{Proof standard(s)} & \text{Proof tag}\\
\hline
\text{scintilla of evidence} & \sigma\\
\text{preponderance of evidence, clear and convincing case} & \partial^{*}\\
\text{beyond reasonable doubt, dialectic validity} & \delta^{*}
\end{array}$
\end{center}
where the distinction between preponderance of evidence and clear and convincing case, and
beyond reasonable doubt and dialectic validity depends on how the weights associated
to the arguments and thresholds are translated in instances of the superiority relation
in the resulting theories. Furthermore, \cite{icail2011carneades} provides examples
where the definitions of proof standards given in \cite{GordonWalton:proof} exhibit
some counter-intuitive conclusions. To obviate such limitations he proposes an
alternative correspondence between proof tags in defeasible logic variants
and proof standards, including the following:
\begin{center}
$\begin{array}{lll}
\text{Proof standard(s)} & ~~~ & \text{Proof tag}\\
\hline
\text{substantial evidence} & & \sigma\\
\text{preponderance of evidence} & & \partial\\
\text{beyond reasonable doubt} & & \delta\\
\text{dialectic validity} & & \delta \text{ (when the superiority relation is ignored)}
\end{array}$
\end{center}
Thus, the proof standard of beyond reasonable doubt corresponds to
defeasible provability using ambiguity propagation.
Accordingly, we can replace $\mi{BeyondReasonableDoubt}$ in rule $r'_5$ with $+\delta$.
All the other literals appearing in the body of the rules do not
require special proof standards, and thus we can annotate them with
$\mt{free}$. Consequently, the formalization of this scenario in
annotated defeasible logic is:
\[
\begin{array}{lrcl}
r_1\colon & \mt{free}\:\mi{EvidenceA} & \Rightarrow &\neg\mi{Responsible}\\
r_2\colon & \mt{free}\:\mi{EvidenceB} & \Rightarrow &\phantom{\neg}\mi{Responsible}\\
r_3\colon & \mt{free}\:\mi{Responsible} & \Rightarrow & \phantom{\neg}\mi{Guilty}\\
r_4\colon && \Rightarrow & \neg\mi{Guilty} \\
r_5\colon & +\delta\neg\:{Guilty} &\Rightarrow &\phantom{\neg}\mi{Compensation}\\
r_6\colon & & \Rightarrow & \neg\mi{Compensation}
\end{array}
\]
It is easy to verify that we now derive $+\partial \neg\mi{Compensation}$, that the defendant
is not entitled to compensation, as the scenario requires.
\section{Properties of Annotated Defeasible Theories}
We now investigate properties of annotated defeasible logic,
exploiting its logic programming underpinnings.
The first theorem relates the meta-program for annotated defeasible logic
to the meta-programs for existing defeasible logics ${\bf DL}(d)$.
Those logics do not contain fail-expressions.
We write $\models_K$ for logical consequence under Kunen's semantics \cite{Kunen}.
Recall that ${\cal M}_d(D)$ is the meta-programming representation for $D$ in ${\bf DL}(d)$,
while ${\cal M}(\alpha(D))$ is the meta-programming representation for $D$ annotated by $\alpha$.
\begin{theorem} \label{thm:correct}
Let $D = (F, R, >)$ be a defeasible theory, and $\alpha$ be an annotation function for that theory.
Let $d \in \{ \delta^*, \delta, \partial^*, \partial \}$.
Suppose $\alpha(R)$ contains only annotations $\mt{free}$ and $d$, and there is no fail-expression in $R$.
Then, for every literal $q$
\begin{itemize}
\item
${\cal M}(\alpha(D)) \models_K \mt{defeasibly}_d(d~q)$ iff
${\cal M}_d(D) \models_K \mt{defeasibly}(q)$
\item
${\cal M}(\alpha(D)) \models_K \neg \mt{defeasibly}_d(d~q)$ iff
${\cal M}_d(D) \models_K \neg \mt{defeasibly}(q)$
\end{itemize}
\noindent
Furthermore, if $d \in \{ \delta^*, \delta \}$,
\begin{itemize}
\item
${\cal M}(\alpha(D)) \models_K \mt{supported}_d(d~q)$ iff
${\cal M}_d(D) \models_K \mt{supported}(q)$
\item
${\cal M}(\alpha(D)) \models_K \neg \mt{supported}_d(d~q)$ iff
${\cal M}_d(D) \models_K \neg \mt{supported}(q)$
\end{itemize}
\end{theorem}
The proof is based on separately unfolding ${\cal M}(\alpha(D))$ and ${\cal M}_d(D)$
until they have essentially the same form.
As an immediate corollary to this theorem, we see that annotated defeasible theories are a conservative extension of defeasible theories.
Let the \emph{free annotation function} be the annotation function that maps every body literal occurrence in $D$ to $\mt{free}$.
For any defeasible theory $D$,
the unannotated theory behaves exactly the same as the theory annotated by the free annotation function.
\begin{corollary} \label{cor:consext}
Suppose that $\alpha_F$ is the free annotation function for $D$.
Let $d \in \{ \delta^*, \delta, \partial^*, \partial \}$.
Then,
for every literal $q$,
\begin{itemize}
\item
${\cal M}(\alpha_F(D)) \models_K \mt{defeasibly_d}(q)$ iff $D \vdash +d q$
\item
${\cal M}(\alpha_F(D)) \models_K \neg \mt{defeasibly_d}(q)$ iff $D \vdash -d q$
\end{itemize}
\noindent
Furthermore, if $d \in \{ \delta^*, \delta \}$,
\begin{itemize}
\item
${\cal M}(\alpha_F(D)) \models_K \mt{supported_d}(q)$ iff $D \vdash +\supp_d q$
\item
${\cal M}(\alpha_F(D)) \models_K \neg \mt{supported_d}(q)$ iff $D \vdash -\supp_d q$
\end{itemize}
\end{corollary}
For any tag $d$ and an annotated defeasible theory $D$ we define
${+}d(D) = \{ q ~|~ D \vdash {+}d q\} = \{ q ~|~ {\cal M}(D) \models_K \mt{defeasibly_d}(q) \}$ and
${-}d(D) = \{ q ~|~ D \vdash {-}d q\} = \{ q ~|~ {\cal M}(D) \models_K \neg \mt{defeasibly_d}(q) \}$.
Similarly, we define
${+}\supp_d(D)$ as $\{ q ~|~ {\cal M}(D) \models_K \mt{supported_d}(q) \}$ and
${-}\supp_d(D)$ as $\{ q ~|~ {\cal M}(D) \models_K \neg \mt{supported_d}(q) \}$.
We can now extend the inclusion theorem of \cite{TOCL10} to the new tags and annotated defeasible logic.
This theorem shows the relative inference strength of the different forms of defeasibility.
\begin{theorem}[Inclusion Theorem] \label{thm:inclusion}
Let $D$ be an annotated defeasible theory.
\begin{itemize}
\item[(a)] ${+}\Delta(D) \subseteq {+}\delta^*(D) \subseteq {+}\delta(D) \subseteq {+}\partial(D)
\subseteq {+}\supp_\delta(D) \subseteq {+}\supp_{\delta^*}(D)$
\item[(b)] $-\supp_{\delta^*}(D) \subseteq -\supp_\delta(D) \subseteq -\partial(D) \subseteq -\delta(D)
\subseteq -\delta^*(D) \subseteq -\Delta(D)$
\item[(c)] ${+}\partial(D) \subseteq {+}\supp_\partial(D) \subseteq {+}\supp_\delta(D)$
\item[(d)] $-\supp_\delta(D) \subseteq {-}\supp_\partial(D) \subseteq {-}\partial(D)$
\item[(e)] ${+}\delta^*(D) \subseteq {+}\partial^*(D) \subseteq {+}\supp_{\partial^*}(D) \subseteq {+}\supp_{\delta^*}(D)$
\item[(f)] $-\supp_{\delta^*}(D) \subseteq {-}\supp_{\partial^*}(D) \subseteq {-}\partial^*(D) \subseteq -\delta^*(D)$
\end{itemize}
\end{theorem}
The proof is by induction on the iteration stages of Fitting's $\Phi_{{\cal M}(D)}$ function.
The inclusions in this theorem are presented graphically in Figure \ref{fig:inclusion}.
The relation $t_1 \subset t_2$ expresses that,
for all defeasible theories $D$, $+t_1(D) \subseteq +t_2(D)$ and $-t_1(D) \supseteq -t_2(D)$,
and, for some defeasible theory $D$, $+t_1(D) \subset +t_2(D)$.
The containments come from the theorem, while their strictness is demonstrated by simple examples.
Examples also show that there are no containments that can be added to the figure.
\begin{figure*}
\[
\begin{array}{rcccl}
\Delta \subset \delta^* & \subset & \delta \subset \partial \subset \supp_\partial \subset \supp_\delta & \subset & \supp_{\delta^*} \\
\\
& \rotatebox[origin=c]{-45}{$\mathbf{\subset}$} & & \rotatebox[origin=c]{45}{$\mathbf{\subset}$} & \\
\\
& & \partial^* ~~~~ \subset ~~~~ \supp_{\partial^*} &\\
\end{array}
\]
\caption{Ordering of inference rules by relative inference strength.}
\label{fig:inclusion}
\end{figure*}
This ordering on tags can be extended to annotation functions.
Let $\alpha_1$ and $\alpha_2$ be annotation functions for a defeasible theory $D$.
We define $\alpha_1 \sqsubseteq \alpha_2$ iff for every body occurrence $o$ of every literal in $D$,
$\alpha_1(o) \subset \alpha_2(o)$.
If such an ordering had implications for the conclusions of the annotated theories, it would provide
a useful basis from which to reason about annotated defeasible theories.
Unfortunately, the most obvious possibility -- a kind of monotonicity --
does not hold, as the following example shows.
\begin{example}
Let $D$ consist of the rules
\[
\begin{array}{lrclclrcl}
r_1: & & \Rightarrow & \phantom{\neg} p & ~~~~~~~~~~~~~ &
r_5: & q & \Rightarrow & \phantom{\neg} s \\
r_2: & & \Rightarrow & \neg p & &
r_6: & & \Rightarrow & \neg s \\
r_3: & & \Rightarrow & \phantom{\neg} q \\
r_4: & \neg p & \Rightarrow & \neg q \\
\end{array}
\]
with $r_5 > r_6$.
Let $\alpha_1$ map $q$ in $r_5$ to $\delta$, and $\alpha_2$ map $q$ in $r_5$ to $\partial$
(with all other occurrences mapped to $\mt{free}$).
Then $\alpha_1 \sqsubseteq \alpha_2$.
Rules $r_1$ - $r_4$ are a standard example distinguishing ambiguity blocking and propagating behaviours.
$+\partial q$ and $-\delta q$ can be concluded.
Consequently, in $\alpha_1(D)$ we conclude $+\partial \neg s$ and $-\partial s$
while in $\alpha_2(D)$ we conclude $-\partial \neg s$ and $+\partial s$.
Thus we see that a strengthening of the annotation function (in the $\sqsubseteq$ ordering)
does not necessarily lead to a strengthening of the conclusions
of the annotated defeasible theory.
\end{example}
For the defeasible logics we address, the consequences of a defeasible theory
can be computed in linear time, with respect to the size of the theory \cite{Maher2001,TOCL10},
but these logics only support one form of defeasibility.
Annotated defeasible logic allows the interaction between the different inference rules
but, nevertheless, we expect its consequences can also be computed in linear time,
although with a larger constant factor.
(Certainly, it is straightforward to show we can compute consequences in quadratic time.
See the supplementary material.)
Let
\[
\begin{array}{rl}
{\cal C}(D) =
& \{ {+}d q ~|~ {\cal M}(D) \models_K \mt{defeasibly}_d(q), d \in T \} ~\cup \\
& \{ {-}d q ~|~ {\cal M}(D) \models_K \neg \mt{defeasibly}_d(q), d \in T \} ~\cup \\
& \{ {+}\supp_d q ~|~ {\cal M}(D) \models_K \mt{supported}_d(q), d \in T \} ~\cup \\
& \{ {-}\supp_d q ~|~ {\cal M}(D) \models_K \neg \mt{supported}_d(q), d \in T \} \\
\end{array}
\]
where $D$ is an annotated defeasible theory,
$T = \{\partial, \partial^*, \delta, \delta^*\}$ refers to the four main forms of defeasibility,
and $q$ ranges over annotated literals.
\begin{conjecture} \label{thm:linear}
Let $D$ be an annotated defeasible theory, and $|D|$ be the number of symbols in $D$.
Then the set of consequences ${\cal C}(D)$ can be computed in time O($|D|$).
\end{conjecture}
\section{Different Forms of Failure}
One advantage of the framework of \cite{MG99,flexf} is that different notions of failure can be obtained
by different semantics for logic programs.
In this section we demonstrate that annotated defeasible logic is a conservative extension of those logics for many such semantics.
Many of the logic programming semantics we will focus on can be seen to be derived from the
3-valued stable models \cite{Pmodels} (also known as \emph{partial stable models},
but distinct from partial stable models in \cite{SaccaZaniolo}).
In addition to the semantics based on all partial stable models,
there is the well-founded model \cite{WF91},
which is the least partial stable model under the information ordering \cite{Pmodels} (called $F$-least in \cite{Pmodels});
the (2-valued) stable models \cite{stable};
the regular models \cite{regular},
which are the maximal partial stable models under set inclusion on the positive literals;
and
the L-stable models \cite{Lstable},
which are the maximal partial stable models under set inclusion on positive and negative literals or,
equivalently,
the minimal partial stable models under set inclusion on the undefined literals.
The interest in these semantics derives from the use of their counterparts
in abstract argumentation \cite{LPeqArg}.
Let ${\cal S}$ denote the collection of semantics mentioned above, with the exception of the stable semantics.
That is, ${\cal S} = \{ \mi{partial~stable, well\mbox{--}founded, regular, L\mbox{--}stable, Kunen, Fitting} \}$.
These semantics (and the stable semantics) are preserved by unfolding
(see \cite{Aravindan,cdr}).
Consequently, Theorem \ref{thm:correct} extends to the semantics in ${\cal S}$:
\begin{theorem} \label{thm:correct2}
Let $D = (F, R, >)$ be a defeasible theory, and $\alpha$ be an annotation for that theory.
Let $d \in \{ \delta^*, \delta, \partial^*, \partial \}$.
Suppose $\alpha(R)$ contains only annotations $\mt{free}$ and $d$, and there is no fail-expression in $R$.
Let $S \in {\cal S}$.
Then
\begin{itemize}
\item
${\cal M}(\alpha(D)) \models_S \mt{defeasibly_d}(q)$ iff
${\cal M}_d(D) \models_S \mt{defeasibly}(q)$
\item
${\cal M}(\alpha(D)) \models_S \neg \mt{defeasibly_d}(q)$ iff
${\cal M}_d(D) \models_S \neg \mt{defeasibly}(q)$
\end{itemize}
\noindent
and, if $d \in \{ \delta^*, \delta \}$,
\begin{itemize}
\item
${\cal M}(\alpha(D)) \models_S \mt{supported_d}(q)$ iff
${\cal M}_d(D) \models_S \mt{supported}(q)$
\item
${\cal M}(\alpha(D)) \models_S \neg \mt{supported_d}(q)$ iff
${\cal M}_d(D) \models_S \neg \mt{supported}(q)$
\end{itemize}
More generally,
the S-models of ${\cal M}(\alpha(D))$ restricted to $\mt{defeasibly_d}$ are identical (up to predicate renaming)
to the S-models of ${\cal M}_d(D)$ restricted to $\mt{defeasibly}$.
\end{theorem}
In particular, annotated defeasible logic under the well-founded semantics
extends the well-founded defeasible logics \cite{MG99,WFDL}.
This theorem does \emph{not} apply to the stable model semantics, because of the possibility that
${\cal M}_d(D)$ has stable models but ${\cal M}(D)$ does not.
This, in turn, occurs because ${\cal M}(D)$ represents all the inference rules, while ${\cal M}_d(D)$ does not.
Technically, the proof fails because the deletion of irrelevant clauses is not sound under the stable model semantics.
To see what can go wrong, consider the following example.
\begin{example}
Let $D$ consist of the rules
\[
\begin{array}{lrclclrcl}
r_1: & & \Rightarrow & \phantom{\neg} p & ~~~~~~~~~~~&
r_3: & & \Rightarrow & \phantom{\neg} q \\
r_2: & p, q & \Rightarrow & \neg p & &
r_4: & & \Rightarrow & \phantom{\neg} q \\
& & & & &
r_5: & & \Rightarrow & \neg q \\
& & & & &
r_6: & & \Rightarrow & \neg q \\
\end{array}
\]
with $r_3 > r_5$ and $r_4 > r_6$.
After unfoldings and simplifications,
${\cal M}(D)$ contains
{\small
\begin{Clause}
$\mt{defeasibly_\partial}(\partial\ p)$ :-\\
\> $\mt{not\ overruled_\partial}(r_1, p)$.
\end{Clause}
\begin{Clause}
$\mt{overruled_\partial}(r_1, p)$ :-\\
\> $\mt{defeasibly_\partial}(\partial\ p)$, \\
\> $\mt{defeasibly_\partial}(\partial\ q)$.
\end{Clause}
}
\noindent
and similar clauses for $\partial^*$
(as well as other clauses).
It is clear that if $\mt{defeasibly_\partial}(\partial\ q)$ holds then the structure of these two clauses
prevents the existence of a stable model,
while if $\neg \mt{defeasibly_\partial}(\partial\ q)$ then $\mt{defeasibly_\partial}(\partial\ p)$ holds in every stable model,
assuming there is nothing else preventing the formation of stable models.
The same applies for $\partial^*$.
Now, $\mt{defeasibly_\partial}(\partial\ q)$ holds, but $\mt{defeasibly_{\partial^*}}(\partial^*\ q)$ does not.
It follows, from the proof of Theorem \ref{thm:correct},
that ${\cal M}_{\partial^*}(D)$ has stable models but ${\cal M}(D)$ does not.
\end{example}
Thus Theorem \ref{thm:correct2} holds for stable models only when \emph{all} forms of defeasibility
and supportedness have stable models.
\section{Related Work}
Among the features of annotated defeasible theories are:
(1)
the language supports multiple forms of defeasibility within a single defeasible theory,
indeed within a single rule;
(2) the language provides explicit fail-expressions;
(3) the framework has the ability to incorporate different notions of failure-to-prove,
corresponding to different semantics of negation-as-failure.
No other formalism for defeasible reasoning has all these features.
Courteous logic programs \cite{Grosof97} (and later developments \cite{LPDA,ASPDA})
permit negation-as-failure expressions in defeasible rules, which are essentially the same as fail-expressions.
\cite{LPwN} discussed a specific transformation for eliminating these expressions from courteous logic programs;
that transformation is not sound for ambiguity propagating logics.
Our meta-programming approach to fail-expressions was discussed in \cite{MG99}, for a language with a single form of defeasibility, and our Theorem \ref{thm:correct} extends to languages with such fail-expressions.
Within proof-theoretic treatments of defeasible logics
(see, for example \cite{MN10} and \cite{TOCL10})
the logics can incorporate multiple forms of defeasibility, but they don't interact.
For example, the proof of $+\partial q$ cannot depend on the proof of $+\delta p$:
it can only depend on proofs of $\partial$ conclusions.
Within the meta-programming framework of \cite{MG99,flexf}
a logic has only a single form of defeasibility,
although this can be easily remedied by the use of multiple variants of the $\mt{defeasibly}$ predicate.
Still, the multiple forms don't interact.
Structured argumentation approaches, such as ASPIC+ \cite{ASPIC+},
use unannotated rules without an inference rule (in the sense above)
and hence define a single form of defeasibility.
A meta-program component of the languages LPDA and ASPDA \cite{LPDA,ASPDA},
called an argumentation theory,
is capable of specifying a different inference rule for each literal,
but not for each \emph{occurrence} of each literal.
Thus, although they provide more interaction than the defeasible logics,
they do not provide the ability to apply different inference rules to the same atom.
It should be noted that the logics of \cite{TOCL10}
are able to simulate each other \cite{Maher12,Maher13}
(and ASPIC+ appears expressive enough to simulate these logics),
but such an approach to incorporating multiple forms of defeasibility leads to an unnatural representation
and has computational penalties.
It also fails to represent free-expressions,
since the top level form of defeasibility must be fixed before simulations can be coded.
Annotated logic programs \cite{GALP}
are an extension of logic programs to multi-valued logics,
where the truth values are assumed to form an upper semi-lattice.
Atoms in the body are annotated by truth values
and the head is annotated by a function of those truth values.
Thus there are some similarities to annotated defeasible logic,
in the use of annotations, including a similarity of
variable annotations and free-expressions.
However, annotated defeasible logic uses proof tags -- not truth values -- as annotations,
and does not assume any ordering on the annotations.
Further, the semantics of annotated logic programs is essentially a disjunction of the conclusions of rules,
so this formalism is unable to represent the overriding of a rule by a competing rule.
Most defeasible logics support a single semantics of failure:
Kunen's \cite{TOCL10}, well-founded \cite{MG99,MN10,WFDL,Grosof97,LPDA}, stable \cite{DefLog,Maier13,ASPDA}.
Apart from the framework of \cite{flexf}, the only defeasible formalisms supporting multiple semantics
are structured argumentation languages like ASPIC+ \cite{ASPIC+}.
But such languages do not support multiple forms of defeasibility.
The annotation mechanism we presented is closely related to the introduction
of modal literals in modal defeasible logic \cite{tplp:goal}, where each rule is labelled with the mode ($\Box$)
its conclusion can be proved and the literals $\Box q$ and $\Diamond q$ correspond to
$+\partial_{\Box} q$ and $-\partial_{\Box}\neg q$. While each modality has its own inference
rule, each supports a single form of defeasibility. This raised the question whether different
forms of defeasibility could be combined: the present paper offers a positive answer.
\section{Conclusion}
We have argued that we need a formalism that supports different kinds of defeasible reasoning,
and introduced annotated defeasible logic to fulfil that requirement.
The semantics of the annotated logic is defined through a logic program,
and we are able to exploit that medium to prove properties of the logic.
\bibliographystyle{acmtrans}
|
1,116,691,500,738 | arxiv | \section{Introduction}
Testing the Kerr hypothesis is a central goal of current strong gravity research~\cite{Berti:2015itd,Barack:2018yly}. This is the hypothesis that astrophysical black holes (BHs), when near equilibrium, are well described by the Kerr metric \cite{Kerr:1963ud}. This working assumption is intimately connected with the no-hair conjecture \cite{Ruffini:1971bza}, which states that the dynamically formed equilibrium BHs have no other macroscopic degrees of freedom beyond those associated with Gauss laws (and hence gauge symmetries) -- see $e.g.$ \cite{Herdeiro:2015waa,Sotiriou:2015pka,Volkov:2016ehx,Cardoso:2016ryw} for reviews.
Over the last few years it has been realized that the Kerr hypothesis can be challenged \textit{even} within General Relativity (GR) and \textit{even} with physically simple and reasonable energy-matter contents, due to the discovery that free, massive complex scalar or vector fields can endow the Kerr BH with synchronised bosonic ``hair"~\cite{Herdeiro:2014goa,Herdeiro:2015gia,Herdeiro:2016tmi,Santos:2020pmh}, see, $e.g.$~\cite{Herdeiro:2015tia,Delgado:2016jxq,Herdeiro:2018daq,Wang:2018xhw,Herdeiro:2018djx,Kunz:2019bhm,Kunz:2019sgn,Delgado:2019prc,Collodel:2020gyp} for generalizations. Moreover, if these fields are sufficiently ultra-light, this ``hair" could occur in the mass range of astrophysical BH candidates, spanning the interval from a few solar masses, $\sim M_\odot$ (stellar mass BHs), to $\sim 10^{10} \ M_\odot$ (supermassive BHs).
The existence of such hairy BHs circumvents various no-scalar (and no-Proca) hair theorems - see $e.g.$~\cite{Bekenstein:1972ny,Bekenstein:1996pn,Herdeiro:2015waa}. Notwithstanding, the key test to the no-hair conjecture is a dynamical one. Are these hairy BHs dynamically robust? That is, can they form dynamically and be sufficiently stable? The current understanding is that: 1) there are dynamical formation channels, namely: $a)$ the superrading instability~\cite{Brito:2015oca} of the Kerr solution~\cite{East:2017ovw,Herdeiro:2017phl} and $b)$ mergers of bosonic stars~\cite{Sanchis-Gual:2020mzb}; 2) these hairy BHs may, themselves, still be afflicted by superradiant instabilities~\cite{Herdeiro:2014jaa,Ganchev:2017uuo} but these can be very long-lived, with a time scale that can exceed a Hubble time~\cite{Degollado:2018ypf}. The evidence so far, therefore, indicates BHs with synchronised ultralight bosonic hair are an interesting challenge to the Kerr hypothesis even in GR. This has led to different phenomenological studies, including the study of their shadows \cite{Cunha:2015yba,Cunha:2016bpi,Cunha:2019ikd} and their $X$-ray phenomenology, namely the iron K$\alpha$ line \cite{Ni:2016rhz,Zhou:2017glv} and QPOs~\cite{Franchini:2016yvq}.
An important outstanding issue is the fundamental physics nature of the putative ultralight bosonic field that could endow BHs with ``hair". Possible embeddings in high energy physics have been suggested, such as the string axiverse~\cite{Arvanitaki:2009fg}, which proposes a landscape of ultralight axion-like particles emerging from string compactifications. If these particles have sufficiently small couplings with standard model particles, they are dark matter, only observable via their gravitational effects, one of which would be endowing spinning BHs with synchronised hair. The QCD axion, in which these axion-like particles are inspired, is a pseudo Nambu-Goldstone boson suggested by the Peccei-Quinn mechanism \cite{Peccei:1977hh} to solve the strong CP problem in QCD \cite{tHooft:1976snw,Jackiw:1976pf}. It possesses a self-interactions potential~\cite{Weinberg:1977ma,Wilczek:1977pj}, characterise by two parameters: the mass of the scalar field $m_a$ and the decay constant $f_a$. Thus, it would be interesting to assess the gravitational effects of such a potential for the axion-like particles that could endow spinning BHs with synchronised hair. This is the goal of the present paper.
Kerr BHs with synchronised bosonic hair have a solitonic limit, obtained when taking the BH horizon to zero size, wherein they reduce to spinning boson stars. In the case of the original model with a free, complex scalar field, the solitonic limit yields the so-called, spinning \textit{mini} boson stars~\cite{Schunck:1996he,Yoshida:1997qf,Schunck:2003kk}. Recently, axion boson stars have been constructed, wherein the complex scalar field is under the action of a QCD axion-like potential~\cite{Guerra:2019srj,Delgado:2020udb}, with the two aforementioned parameters.\footnote{Axionic stars with real fields have been considered, $e.g.$ in~\cite{Hertzberg:2018lmt,Visinelli:2017ooc}.} In the limit when $f_a \rightarrow \infty$, the potential reduces to a simple mass term, and the axion boson stars reduce to mini boson stars. Thus, in the same way, that the hairy BHs in~\cite{Herdeiro:2014goa} are the BH generalisation of mini boson stars, we shall construct here the BH generalisations of the rotating axion boson stars in~\cite{Delgado:2020udb}. Moreover, we shall study the basic physical properties and phenomenology of these BHs, hereafter dubbed \textit{Kerr BHs with synchronised axionic hair} (KBHsAH), or simply ``axionic BHs".
This work is organised as follows. In Section 2 we introduce the model together with the equations of motion and the \textit{ansatz} we used to solve them; we also present the QCD potential. In Section 3 we display the numerical framework to tackle the field equations and to obtain the BH solutions, discuss the boundary conditions of the problem and how to extract physical quantities from the data. In Section 4 we show the numerical results, presenting the domain of existence of the numerical solutions together with an analysis of some of their physical properties and phenomenology. In Section 5 we present conclusions and some final remarks.
\section{The model}
The theory for which we shall obtain hairy BH solutions is the Einstein-Klein-Gordon model, that describes a massive complex scalar field, $\Psi$, minimally coupled to Einstein's gravity. The action of the theory is, using units such that $G = c = \hbar = 1$,
\begin{equation}\label{Eq:Action}
\mathcal{S} = \int d^4 x \sqrt{-g} \left[ \frac{R}{16\pi} - g^{\mu\nu} \partial_{\mu} \Psi^* \partial_{\nu} \Psi - V(|\Psi|^2) \right] \ ,
\end{equation}
where $R$ is the Ricci scalar, $\Psi$ is the complex scalar field (`*' denotes complex conjugation) and $V$ is the scalar self-interaction potential. The equations of motion resulting from the variation of the action with respect to the metric, $g_{\mu\nu}$, and scalar field, are,
\begin{eqnarray}
&E_{\mu\nu} \equiv R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R - 8\pi \ T_{\mu\nu} = 0 \ ,& \label{Eq:FieldEquationsMetric} \\
&\Box \Psi - \dfrac{d V}{d |\Psi|^2} \Psi = 0 \ ,& \label{Eq:FieldEquationsScalarField}
\end{eqnarray}
where
\begin{equation}
T_{\mu\nu} = 2 \partial_{(\mu} \Psi^* \partial_{\nu)} \Psi - g_{\mu\nu} \left( \partial^\alpha \Psi^* \partial_\alpha \Psi + V\right) \ ,
\end{equation}
is the energy-momentum tensor associated with the scalar field.
A global $U(1)$ transformation $\Psi \rightarrow e^{i\chi} \Psi$, where $\chi$ is a constant, leaves the above action invariant; thus it is possible to write a scalar 4-current \cite{Herdeiro:2014goa}, $j^\mu = -i \left( \Psi^* \partial^\mu \Psi - \Psi \partial^\mu \Psi^* \right)$ which is conserved: $D_\mu j^\mu = 0$. The existence of this symmetry and conserved current implies the existence of a conserved quantity -- the \textit{Noether charge} -- that can be computed by integrating the timelike component of the 4-current,
\begin{equation}
Q = \int_\Sigma j^t \ .
\end{equation}
This quantity is interpreted as the number of scalar particles in a given solution, albeit this relation only becomes rigorous after field quantisation. For solitonic solutions (without an event horizon), moreover, $Q$ is related with the total angular momentum as~\cite{Yoshida:1997qf,Schunck:1996he}
\begin{equation}
J = m Q \ .
\end{equation}
This is a generic relation for rotating boson stars, already observed in other models with a self-interactions potential - see, $e.g.$~\cite{Kleihaus:2007vk}.
As BH generalisations of the work done on rotating axion boson stars~\cite{Delgado:2020udb}, we are interested in stationary, regular on and outside the event horizon, axi-symmetric and asymptotically flat solutions of the above equations of motion. The spacetime generated by these solutions possesses two commuting Killing vector fields, $\xi$ and $\eta$, which, in a suitable coordinate system, can be written as $\xi = \partial_t$ and $\eta = \partial_\varphi$ corresponding to stationarity and axi-symmetry, respectively. With these Killing vector fields, we can define the following \textit{ansatz} for the metric and the scalar field, written in terms of the spheroidal coordinates, $\{t, r, \theta, \varphi\}$,
\begin{eqnarray}
&ds^2 = -e^{2F_0} N dt^2 + e^{2F_1} \left( \frac{dr^2}{N} + r^2 d\theta^2 \right) + e^{2F_2} r^2 \sin^2 \theta \left( d\varphi - W dt \right)^2 \hspace{2pt} , \hspace{10pt} N \equiv 1 - \frac{r_H}{r} \hspace{2pt} , \label{Eq:AnsatzMetric} \\
&\Psi = \phi\ e^{i\left(\omega t - m \varphi \right)} \label{Eq:AnsatzScalarField}~,
\end{eqnarray}
where $ \{F_1, F_2, F_0, W; \phi \}$ are \textit{ansatz} functions that depend exclusively on the $(r,\theta)$ coordinates; $r_H$ is the radial coordinate of the event horizon; $\omega$ is the angular frequency of the scalar field; and $m = \pm 1, \pm 2, \dots$ is the azimuthal harmonic index of the scalar field.
In this work, we aim to study BHs with a surrounding axion-like scalar field; thus we specify the scalar self-interaction potential as describing axion interactions. Following~\cite{Guerra:2019srj, Delgado:2020udb}, we will use the QCD axion potential~\cite{diCortona:2015ldu} to describe the axion interactions, added to a constant term in order to obtain asymptotically flat solutions. The potential in question has the form,
\begin{equation}\label{Eq:AxionPotential}
V(\phi) = \frac{2 \mu_a^2 f_a^2}{B} \left[ 1 - \sqrt{1 - 4 B \sin^2 \left( \frac{\phi}{2 f_a} \right)} \right]~,
\end{equation}
where $B = \frac{z}{(1+z)^2} \approx 0.22$ is a constant, in which $z \equiv m_u/m_d \approx 0.48$ is the ratio between the up and down masses, $m_u$ and $m_d$, respectively; $\mu_a$ and $f_a$ define the axion-like particle (ALP) mass and quartic self-interaction coupling, respectively, and are two free parameters. This can be understood by performing an expansion of the potential around $\phi = 0$,
\begin{equation}\label{Eq:AxionPotentialExpansion}
V(\phi) = \mu_a^2 \phi^2 - \left( \frac{3B-1}{12} \right) \frac{\mu_a^2}{f_a^2} \phi^4 + \frac{1+15 B(3B-1)}{360 f_a^4}\mu_a^2 \phi^6+\dots ~.
\end{equation}
In this way, we confirm $\mu_a$ is the ALP mass and $f_a$ is the quartic self-interaction coupling,
\begin{equation}
m_a = \mu_a \hspace{2pt} , \hspace{10pt} \lambda_a = - \left( \frac{3B - 1}{12} \right) \frac{\mu_a^2}{f_a^2} \hspace{2pt}.
\end{equation}
In this work, we shall refer to $\mu_a$ and $f_a$ as the ALP mass and decay constant, respectively.
We note that the above expansion is only valid on the regime where $ \phi \ll f_a $.
In fact, to leading order, only the mass term remains. Therefore, as already noted in \cite{Delgado:2020udb} for the case of boson stars, for sufficiently large decay constant $f_a$, we expect the BHs solutions to become very similar to the original Kerr BHs with synchronised scalar hair, obtained in \cite{Herdeiro:2014goa}.
In a similar fashion to the family of Kerr BHs with synchronised scalar hair, the solutions obtained in the work are possible due to the so-called \textit{synchronisation condition}. This condition can be interpreted as a synchronisation between the angular velocity of the event horizon, $\Omega_H$, corresponding to the value of the metric function $W$ at the horizon, $\Omega_H=W(r_H)$, and the phase angular velocity of the scalar field, $\omega/m$, $cf.$~\eqref{Eq:AnsatzScalarField}:
\begin{equation}
\label{conds}
\omega = m \Omega_H \hspace{2pt} .
\end{equation}
Finally, let us mention that, as remarked in
\cite{Delgado:2020udb},
the potential
(\ref{Eq:AxionPotential})
allows
for the existence
of solutions even in the absence of the gravity term in the action (\ref{Eq:Action}).
The simplest case corresponds to (non-gravitating) Q-ball-like solitons in a flat spacetime background.
As expected, these solutions possess generalizations on a Kerr BH background.
These bound states are in synchronous rotation with the BH horizon, $i.e.$ they still obey the condition
(\ref{conds}) and share most of the properties of the non-linear Q-clouds in \cite{Herdeiro:2014pka}.
A discussion of these aspects will be
reported elsewhere.
\section{Framework}
\subsection{Boundary conditions}
To obtain numerical BH solutions appropriate boundary conditions must be imposed that enforce the sough physical behaviours. Such boundary conditions will now be summarised.
\begin{itemize}
\item \textit{Asymptotically boundary conditions:} Asymptotic flatness implies that all \textit{ansatz} functions must go to zero asymptotically,
\begin{equation}
\lim_{r \rightarrow \infty} F_i = \lim_{r \rightarrow \infty} W = \lim_{r \rightarrow \infty} \phi = 0 \hspace{2pt} .
\end{equation}
\item \textit{Axial boundary conditions:} Axial symmetry, together with regularity on the symmetry axis, implies that,
\begin{equation}
\partial_\theta F_i = \partial_\theta W = \partial_\theta \phi = 0 \hspace{2pt}, \hspace{10pt} \text{at} \hspace{10pt} \theta = \{0, \pi \} \hspace{2pt} .
\end{equation}
Furthermore, we require the absence of conical singularities, thus, $F_1 = F_2$ on the axis. Since we focus on even parity solutions, which typically correspond to the fundamental solutions, they are symmetric \textit{w.r.t.} the equatorial plane. Hence, one only needs to solve the equations of motion in the range $0 \leqslant \theta \leqslant \pi/2$ and impose the following boundary conditions,
\begin{equation}
\partial_\theta F_i = \partial_\theta W = \partial_\theta \phi = 0 \hspace{2pt} , \hspace{10pt} \text{at} \hspace{10pt} \theta = \frac{\pi}{2} \hspace{2pt} .
\end{equation}
\item \textit{Event horizon boundary conditions:} To simplify the study of these boundary conditions, let us introduce a radial coordinate transformation, $x = \sqrt{r^2 - r_H^2}$. With this coordinate transformation, we can perform a series expansion of the \textit{ansatz} functions at the horizon, $x = 0$, and find that,
\begin{eqnarray}
&&F_i = F_i^{(0)} + x^2 F_i^{(2)} + \mathcal{O}(x^4) \hspace{2pt} , \\
&&W = \Omega_H + \mathcal{O}(x^2) \hspace{2pt} , \\
&&\phi = \phi^{(0)} + \mathcal{O}(x^2) \hspace{2pt} ,
\end{eqnarray}
where $F_i^{(0)}, F_i^{(2)},\phi^{(0)}$ are functions of $\theta$.
With these series expansions, we can naturally impose the following boundary conditions,
\begin{equation}
\partial_x F_i = \partial_x \phi = 0 \hspace{2pt} , \hspace{5pt} W = \Omega_H \hspace{2pt} , \hspace{10pt} \text{at} \hspace{10pt} r = r_H
\end{equation}
\end{itemize}
\subsection{Extracting physical quantities}
The main physical quantities of interest for our analysis are encoded in the metric functions evaluated either at the horizon or at spacial infinity. For the former, we obtain the horizon angular velocity by using the horizon boundary condition mentioned in the previous sections, $\Omega_H = W|_{r_H}$, together with Hawking temperature, $T_H$, as well as the horizon area, $A_H$, through the following expressions,
\begin{equation}
T_H = \frac{1}{4\pi r_H} e^{(F_0 - F_1)|_{r_H}} \hspace{2pt} , \hspace{10pt} A_H = 2\pi r_H^2 \int_0^\pi d\theta\ \sin \theta\ e^{(F_1 + F_2)|_{r_H}} \ .
\end{equation}
The entropy of the computed BH follows from the Bekenstein-Hawking formula, $S = A_H/4$.
For the latter, we can compute the ADM mass, $M$, and angular momentum, $J$, through the asymptotic behaviour of $g_{tt}$ and $g_{t\varphi}$,
\begin{equation}
g_{tt} = -e^{2F_0} N + e^{2F_2} W^2 r^2 \sin^2 \theta \rightarrow -1 + \frac{2M}{r} + \dots \hspace{2pt}, \hspace{10pt} g_{t\varphi} = -e^{2F_2} W r^2 \sin^2 \theta \rightarrow - \frac{2J}{r} \sin^2 \theta + \dots
\end{equation}
The above quantities are mutually related through a Smarr-type formula \cite{Smarr:1972kt},
\begin{equation}\label{Eq:SmarrRelation}
M = 2 T_H S + 2 \Omega_H \left( J - J^\Psi \right) + M^\Psi \hspace{2pt},
\end{equation}
where we have introduced two new quantities, which can not be computed directly from the horizon or spacial infinity data,
\begin{equation}
M^\Psi = -2 \int_\Sigma dS_\mu \left( T^\mu_\nu \xi^\nu - \frac{1}{2} T \xi^\mu \right) \hspace{2pt} , \hspace{10pt} J^\Psi = \int_\Sigma d S_\mu \left( T^\mu_\nu \eta - \frac{1}{2} T \eta^\mu \right) \hspace{2pt} ,
\end{equation}
corresponding to the scalar field mass and angular momentum, respectively\footnote{The scalar field potential $V(|\Psi|^2)$ enters (\ref{Eq:SmarrRelation}) via the $M^\Psi$-term.}.
In their definition, $\Sigma$ is a spacelike surface, bounded by the 2-sphere at infinity, $S^2_\infty$, and the spatial section of the horizon, $H$. Moreover, the angular momentum of the scalar field is related to the Noether charge which arises from the global $U(1)$ symmetry of the scalar field, $Q$, as $J^\Psi = m Q$. For a solution composed entirely of axionic scalar hair (no horizon), the ADM angular momentum must be equal to the angular momentum of the scalar field; likewise, for a solution without axionic scalar hair, the ADM angular momentum of the solution equals the horizon angular momentum. Thus, in order to evaluate how hairy a given BH solution is, we define the following dimensionless parameter:
\begin{equation}
q \equiv \frac{J^\Psi}{J} = \frac{m Q}{J} \hspace{2pt}.
\end{equation}
One easily sees that, when $q = 0$, we have a bald BH, corresponding to a Kerr BH. On the other end, when $q=1$, we have a solution entirely compose of axionic hair, which corresponds to a rotation axion boson star \cite{Delgado:2020udb}. For any other value $0 < q < 1$, we have a rotating BH surrounded by a non-trivial, backreacting, massive rotating axion scalar field.
\subsection{Numerical approach}
To perform the numerical integration of the equations of motion resulting from Eqs. \eqref{Eq:FieldEquationsMetric} and \eqref{Eq:FieldEquationsScalarField} with the \textit{ansatz} Eqs. \eqref{Eq:AnsatzMetric} and \eqref{Eq:AnsatzScalarField}, it is useful to rescale key quantities by $\mu_a$,
\begin{equation}
r \rightarrow r \mu_a \hspace{2pt}, \hspace{10pt} \phi \rightarrow \phi \sqrt{4\pi} \hspace{2pt} , \hspace{10pt} \omega \rightarrow \omega/\mu_a~,
\end{equation}
together with $ f_a \rightarrow f_a \sqrt{4\pi}$.
This leads to the disappearance of the ALP mass constant
from the equations of motion numerically solved, but all global quantities will be express in terms of $\mu_a$.
In our approach, by expanding the equations of motion, we get a set of five coupled, non-linear, elliptic partial differential equations for the \textit{ansatz} functions, $\mathcal{F}_a = \{ F_0, F_1, F_2, W; \phi \}$. They are compose of the Klein-Gordon equation, Eq. \eqref{Eq:FieldEquationsScalarField}, together with the following combination of the Einstein equations, Eq. \eqref{Eq:FieldEquationsMetric},
\begin{eqnarray}
&&E^r_r + E^\theta_\theta - E^\varphi_\varphi - E^t_t = 0 \hspace{2pt} , \\
&&E^r_r + E^\theta_\theta - E^\varphi_\varphi + E^t_t + 2W E^t_\varphi = 0 \hspace{2pt} , \\
&&E^r_r + E^\theta_\theta + E^\varphi_\varphi - E^t_t - 2W E^t_\varphi = 0 \hspace{2pt} , \\
&&E^t_\varphi = 0 \hspace{2pt} .
\end{eqnarray}
The remaining equations, $E^r_\theta = 0$ and $E^r_r - E^\theta_\theta = 0$ are not solved directly, instead, they are used as constraint equations to evaluate the accuracy of the numerical solution. Typically they are satisfied at the level of the overall numerical accuracy.
Our numerical treatment can be summarised as follows. We restrict the domain of integration to the region outside the horizon. Using the aforementioned radial coordinate transformation $x = \sqrt{r^2 - r_H^2}$, we transform the radial region of integration from $[r_H, \infty)$ to $[0,\infty)$. Then, we introduce a new radial coordinate that maps the semi-infinite region $[0,\infty)$ to the finite region $[0,1]$. Such map can be defined in several ways, but in this work we choose the new coordinate, $\bar{x} = x/(x+1)$. After this, the equations $\mathcal{F}_a$ are discretised on a grid in $\bar{x}$ and $\theta$. Most of the results presented here were obtained on an equidistant grid with $251 \times 30$ points. The grid covers the integration region $0 \leqslant \bar{x} \leq 1$ and $0 \leqslant \theta \leq \pi/2$.
The equations of motion have been solved subject to the boundary conditions introduced above by using a professional package, entitled \textsf{FIDISOL/CADSOL}~\cite{schoen}, which employs a Newton-Raphson method with an arbitrary grid and consistency order. This code uses the finite difference method, providing also an error
estimate for each unknown function. For the solutions in this work, the maximal numerical error for the functions is estimated to be on the order of $10^{-3}$. The Smarr relation, Eq. \eqref{Eq:SmarrRelation}, provides a further test of the numerical accuracy, leading to error estimates of the same order.
In our scheme, there are four input parameters: \textbf{i)} the decay constant $f_a$ in the potential, Eq. \eqref{Eq:AxionPotential}; \textbf{ii)} the angular frequency of the scalar field $\omega$; \textbf{iii)} the azimuthal harmonic index $m$; and \textbf{iv)} the radial coordinate of the event horizon $r_H$. The number of nodes $n$ of the scalar field, as well as all other quantities of interest mentioned before, are computed from the numerical solution. For simplicity, we have restricted our study to the fundamental configurations, \textit{i.e.} with a nodeless scalar field, $n = 0$ and with $m = 1$. Also, from the results presented in \cite{Delgado:2020udb}, we shall illustrate the effect\footnote{ We have confirmed the existence of KBHsAH for various $f_a$ ranging from $0.02$ to $10$.}
of the axion potential on the hairy BHs by performing a thorough study of the solutions with the specific decay constant $f_a = 0.05$.
\section{Numerical Results}
\subsection{The domain of existence}
At the end of the previous section, we fixed two of the four input parameters of the problem ($f_a,m$). Thus, the full domain of existence is obtained by varying the remaining two input parameters: the angular frequency of the scalar field, $\omega$; and the radial coordinate of the event horizon, $r_H$. Since it is impossible to obtain all possible BH solutions, we obtained a very large number of them ($\sim$ 30000) and we have extrapolated this large discrete set of solutions into the continuum, which defines the region where one can find the BH solutions with axionic hair.
Such a region can be expressed and plotted in various ways. In the left panel of Fig. \ref{Fig:MassAngularMomentum} (main panel), we show it in an ADM mass $M\mu_a$ \textit{vs.} angular frequency $\omega/\mu_a$ plane. We can observe that \textit{most} (but not all) of the numerical solutions region is bounded by two specific lines\footnote{In fact, there is a third line corresponding to extremal hairy BHs, which possess vanishing Hawking temperature. In this work we have only studied the neighbouring solutions of these extremal BHs, but not the latter \textit{per se}, which would require a different metric ansatz.},
\begin{itemize}
\item The \textit{axion boson star line} - corresponding to the solitonic limit, in which both the event horizon radius and area vanish, $r_H = 0$ and $A_H = 0$, and the solutions have no BH horizon; therefore $q = 1$. Such line is represented in both panels of Fig. \ref{Fig:MassAngularMomentum} as a red solid line.
\item The \textit{existence line} - corresponding to specific subset of vacuum Kerr BHs which can support stationary scalar clouds (with an infinitesimally small $\phi$), first discussed by Hod~ \cite{Hod:2012px,Hod:2013zza}, thus having $q = 0$. These solutions are obtained by linearising the theory, and since, on that regime, the axion self-interacting potential reduces to the mass potential -- \textit{cf.} Eq. \eqref{Eq:AxionPotentialExpansion} -- the existence line will be the same as the one obtained for the family of Kerr BHs with scalar hair. Such line is represented in both panels of Fig. \ref{Fig:MassAngularMomentum} as a blue dotted line.
\end{itemize}
\begin{figure}[ht!]
\begin{center}
\includegraphics[height=.22\textheight, angle =0]{Graphics_w_vs_M.pdf}
\includegraphics[height=.22\textheight, angle =0]{Graphics_w_vs_J.pdf}
\end{center}
\caption{Domain of existence of KBHsAH for $f_a = 0.05$ in the $M\mu_a$ \textit{vs.} $\omega/\mu_a$ plane (left panel) and in the $J\mu_a^2$ \textit{vs.} $\omega/\mu_a$ plane (right panel). In both panels the insets correspond to the analogous domain of existence for the original Kerr BHs with synchronised hair (no scalar self-interactions), corresponding to $f_a \rightarrow \infty$. For the latter family of solutions we also present the extremal line, composed of BHs with vanishing Hawking temperature.}
\label{Fig:MassAngularMomentum}
\end{figure}
Fig.~\ref{Fig:MassAngularMomentum} (left panel) exhibits a novel property of this class of BHs: the solutions' region is no longer totally bounded by the boson star line in this particular representation. For large decay constant, $f_a$, we recover the family of Kerr BHs with synchronised scalar hair~\cite{Herdeiro:2014goa} (inset), for which the solutions' region \textit{is} totally bounded by the boson star line (together with an existence line -- scalar clouds --, and extremal line -- zero temperature BHs). The consequence of this observation is that for certain frequencies, the ADM mass is not maximised by a boson star, but rather by a hairy BH.
By changing the representation, however, and plotting the domain of existence in the ADM angular momentum $J\mu_a^2$ \textit{vs.} angular frequency $\omega/\mu_a$ plane -- right panel of Fig. \ref{Fig:MassAngularMomentum} --, we see that, for the angular momentum, the boson star line bounds all axionic BHs with decay constant $f_a = 0.05$. Thus for all frequencies, the angular momentum is maximised by a boson star.
Another distinctive feature of the domain of solutions of the axionic BHs is the existence of a local maximum for the mass and angular momentum at $\omega \sim 0.9$, which is not the global maximum. For solutions with smaller angular frequency, it is possible to have BHs with more mass and angular momentum than the ones near the local maximum; in fact, these quantities are maximised for the solutions with the smallest possible value of angular frequency. Such is not the case in the absence of the axionic potential (inset of both panels in Fig. \ref{Fig:MassAngularMomentum}).
In Fig. \ref{Fig:PhaseSpaceSpin} both the ADM angular momentum, $J$ and the dimensionless spin, $j$, defined as $j \equiv J/M^2$, are exhibited $vs.$ the ADM mass, $M$, on the left and right panels, respectively.
\begin{figure}[ht!]
\begin{center}
\includegraphics[height=.22\textheight, angle =0]{Graphics_M_vs_J.pdf}
\includegraphics[height=.22\textheight, angle =0]{Graphics_M_vs_spin.pdf}
\end{center}
\caption{ADM angular momentum, $J$, (left panel) and dimensionless spin, $j = J/M^2$, (right panel) as a function of the ADM mass $M$. As in Fig.~\ref{Fig:MassAngularMomentum} the insets show the free scalar field limit $f_a \rightarrow \infty$, where the extremal line is also shown.}
\label{Fig:PhaseSpaceSpin}
\end{figure}
In the former, we can see that the axionic BHs can have a higher mass and angular momentum than their $f_a \rightarrow \infty$ counterparts (inset). We can also see a zigzag of the boson star line. This behaviour is explained by the sudden drop on the ADM mass and angular momentum around $\omega/\mu_a \approx 0.84$ -- \textit{cf.} Fig. \ref{Fig:MassAngularMomentum}.
In the right panel, we see a considerable violation of the Kerr bound, $j \leqslant 1$ in part of solution space. This already occurred for the $f_a \rightarrow \infty$ limit (inset). Again, it is possible to visualise the zigzag behaviour of the boson star line.
Let us now study the horizon geometry of the axionic BHs. Their event horizon has a spherical topology but a spheroidal geometry, similarly to Kerr BHs. This can be seen by studying the spatial cross-section of the horizon, through the induced metric,
\begin{equation}
d\Sigma^2 = r_H^2 \left[ e^{2 F_1(r_H,\theta)} d\theta^2 + e^{2F_2(r_H,\theta)} \sin^2 \theta d \varphi^2 \right] \hspace{2pt} .
\end{equation}
Due to the rotation of the solutions, the horizon is squashed at the poles. To show this, we compute the horizon circumference along the equator, $L_e$, and along the poles, $L_p$,
\begin{equation}
L_e = 2\pi r_H e^{F_2 (r_H, \pi/2)} \hspace{2pt}, \hspace{10pt} L_p = 2 r_H \int_0^\pi d\theta e^{F_1(r_H,\theta)} \hspace{2pt}.
\end{equation}
We define the sphericity as the ratio of both circumferences above \cite{Delgado:2018khf},
\begin{equation}
\mathfrak{s} = \frac{L_e}{L_p}
\end{equation}
For values in which the sphericity is greater (lower) than 1, the horizon will be squashed (elongated) at the poles, leading to an oblate (prolate) spheroid. From the left panel in Fig. \ref{Fig:SphevH} we see that all solutions have a sphericity larger than the unity; thus all solutions have an oblate horizon, as expected.
\begin{figure}[ht!]
\begin{center}
\includegraphics[height=.22\textheight, angle =0]{Graphics_rH_vs_Sphe.pdf}
\includegraphics[height=.22\textheight, angle =0]{Graphics_rH_vs_vH.pdf}
\end{center}
\caption{The sphericity, $\mathfrak{s}$ (left panel), and the horizon linear velocity, $v_H$ (right panel), as a function of the event horizon radial coordinate, $r_H \mu_a$. The black solid line corresponds to the Smarr line, which, in contrast with the $f_a \rightarrow \infty$ limit, is no longer constant for the sphericity. Below (above) it, we have (do not have) embeddable BH horizons in Euclidean 3-space. The white region in the far left region of each panel corresponds to solutions outside our scan (challenging due to the small $r_H$). }
\label{Fig:SphevH}
\end{figure}
Another physical quantity of interest associated with the horizon is its linear velocity $v_H$ \cite{Delgado:2018khf,Delgado:2019prc,Herdeiro:2015moa}. Such quantity measures how fast the null geodesics generators of the horizon rotate relatively to a static observer at spatial infinity. Its definition is quite simple, only taking into account the perimetral radius of the circumference located at the equator, $R_e \equiv L_e/2\pi$, and the horizon angular velocity, $\Omega_H$,
\begin{equation}
v_H = \frac{L_e}{2\pi} \Omega_H \hspace{2pt} .
\end{equation}
The horizon linear velocity is presented in the right panel of Fig. \ref{Fig:SphevH}. The central feature in this plot is the fact that all solutions have horizon linear velocity smaller than the unity, which, in the units we are using, corresponds to the speed of light. Therefore, null geodesics generators of the horizon never rotate relatively to the asymptotic observer at superluminal speeds, even though some solutions strongly violate the Kerr bound $j \leqslant 1$.
A final insight about the horizon geometry of the axionic BHs is obtained from investigating whether an isometric embedding of the spatial sections of the horizon is possible in Euclidean 3-space $\mathbb{E}^3$. For a Kerr BH, such embedding is possible iff its dimensionless spin obeys $j \leqslant j^{(S)}$~\cite{Smarr:1973zz}, where $ j^{(S)} \equiv \sqrt{3}/2$ was dubbed the \textit{Smarr point}~\cite{Delgado:2018khf}. For $j > j^{(S)}$ the Gaussian curvature of the horizon becomes negative in the vicinity of the poles \cite{Smarr:1973zz}, which, prevents the embedding (due to occurring at a fixed point of the axi-symmetry).
As expected, this feature also occurs for the axionic BHs. Due to the existence of scalar hair around the BH, we have an extra degree of freedom, which converts the Smarr point into a \textit{Smarr line}. Such line is represented in both panels of Fig. \ref{Fig:SphevH} as a solid black line. One observes that, for both the sphericity and the horizon linear velocity, the Smarr line is not constant. This contrasts with the behaviour for $f_a \rightarrow \infty$; in that case, the sphericity of the Smarr line was constant and equal to the value of the Smarr point in Kerr \cite{Delgado:2018khf}. Thus, the axion potential destroys the constancy of the sphericity along the Smarr line.
\subsection{Other properties}
\subsubsection{Ergo-regions}
An ergo-region is a part of a spacetime, outside the event horizon, wherein the norm of the asymptotically timelike Killing vector $\xi = \partial_t$ becomes positive and thus the vector becomes spacelike. Ergoregions are associated with the possibility of energy extraction from a spinning BH, via the Penrose process \cite{Penrose:1969pc,Penrose:1971uk}, or superradiant scattering~\cite{Brito:2015oca}. In the context of BHs with synchronised hair, the superradiant instability of vacuum Kerr BHs in the presence of ultralight scalar fields is one of the possible channels of formation of these hairy BHs~\cite{East:2017ovw,Herdeiro:2017phl}. Thus, it is of relevance to analyse ergo-regions for the axionic BHs.
Kerr BHs possess an ergo-region whose boundary has a spherical topology and touches the BH horizon at the poles -- such surface is called an \textit{ergo-sphere}. For the axionic BHs in the $f_a \rightarrow \infty$ limit, the ergo-regions can be more complicated~\cite{Herdeiro:2014jaa}. Some solutions have a Kerr-like ergo-region; but others have a more elaborate ergo-region topology, with two disjoint parts, one Kerr-like and another of toroidal topology. The latter were dubbed \textit{ergo-Saturns}. The toroidal ergo-region is inherited from the mini boson star environment around the horizon, since these stars develop such ergo-regions, when sufficiently compact. Such ergo-torii also occur for spinning axion boson stars~\cite{Delgado:2020udb}. Thus, we expect some axionic BHs to develop an ergo-Saturn. This is confirmed in Fig. \ref{Fig:Ergoregions}.
We have found that the ergo-region of axionic BHs follows a qualitatively similar distribution to that of their $f_a \rightarrow \infty$ limit: there are solutions which possess an ergo-sphere and others that develop an ergo-Saturn. The latter occur on the far left of the domain of existence in Fig. \ref{Fig:Ergoregions}, where the most massive BHs exist, having the lowest possible values of the angular frequency of the scalar field.
\begin{figure}[ht!]
\begin{center}
\includegraphics[height=.28\textheight, angle =0]{Graphics_Ergo_w_vs_M.pdf}
\end{center}
\caption{Ergo-regions. BHs with axionic hair have an ergo-sphere in the light blue region and an ergo-Saturn in the dark blue region. The inset shows the free scalar field case ($f_a \rightarrow \infty$), for comparison.}
\label{Fig:Ergoregions}
\end{figure}
\subsubsection{Light rings and timelike innermost stable circular orbits}
A phenomenological aspect of importance is the structure of circular orbits (COs) of both massless and massive particles around a BH. In particular, the (timelike) innermost stable circular orbit (ISCO) and the (null) light rings (LRs) are of special relevance. The former is associated with a cut-off frequency of the emitted synchrotron radiation generated from accelerated charges in accretion disks; the latter is related to the real part of the frequency of BH quasi-normal modes \cite{Cardoso:2008bp}, as well as to the BH shadow \cite{Cunha:2018acu}.
The structure of COs can be obtained as follows. Given the geometry in Eq. \eqref{Eq:AnsatzMetric}, we can compute the effective Lagrangian for equatorial, $\theta = \pi/2$, geodesic motion as,
\begin{equation}\label{Eq:EffectiveLagrangian}
2\mathcal{L} = \frac{e^{2 F_1}}{N}\dot{r}^2 + e^{2 F_2}r^2 \left( \dot{\varphi} - W \dot{t} \right)^2 - e^{2F_0} N \dot{t} = \epsilon \ ,
\end{equation}
where all \textit{ansatz} functions depend only on the radial coordinate $r$, the dot, $\dot{ }$, denotes the derivative \textit{w.r.t} the proper time, and $\epsilon = \{-1,0\}$ for massive (timelike) particles and massless (lightlike) particles, respectively. Due to the existence of two Killing vector fields, we can write $\dot{t}$ and $\dot{\varphi}$ in terms of the energy $E$ and angular momentum $L$ of the particle,
\begin{eqnarray}
&&E = \left( e^{2F_0} N - e^{2F_2} r^2 W^2 \right) \dot{t} + e^{2F_2} r^2 W \dot{\varphi} \ , \\
&&L = e^{2 F_2} r^2 \left( \dot{\varphi} - W \dot{t} \right) \ .
\end{eqnarray}
Inverting the above system of equations and replacing the result into the effective Lagrangian, Eq. \eqref{Eq:EffectiveLagrangian}, we can obtain an equation for $\dot{r}$, which defines a potential $V(r)$,
\begin{equation}
\dot{r}^2 = V(r) \equiv e^{-2 F_1} N \left[ \epsilon + \frac{e^{2F_0}}{N} \left( E - L W \right)^2 - e^{2F_2} \frac{L^2}{r^2} \right] \ .
\end{equation}
In order to obtain COs, both the potential and its derivative must be zero, \textit{i.e.} $V(r) = V'(r) = 0$. Depending on whether we are considering massless or massive particles, these two equations will yield different results.
For massless particles, the first equation, $V(r) = 0$, will give two algebraic equation for the impact parameter of the particle, $b_+ \equiv L_+/E_+$ and $b_- \equiv L_-/E_-$, corresponding to co- and counter-rotating orbits, respectively. The second equation, $V'(r) = 0$, together with the impact parameters, will give the radial coordinate of the co- and counter-rotating LRs. Whenever it is possible to obtain a real solution for the radial coordinate, the BH possesses LRs.
In Fig. \ref{Fig:LRs}, we show the distribution of hairy BHs with different number of LRs in the angular frequency, $\omega/\mu_a$ \textit{vs.} angular momentum, $J \mu^2$ plane. The left panel shows the counter-rotating case, whereas the right panel shows the co-rotating case.
In both cases, it is always possible to have at least one LR, as for the Kerr BH. In the counter-rotating case, however, if the surrounding scalar field is compact enough, an extra pair of LRs emerge, leading to a hairy BH with 3 LRs. This is the case of a large set of hairy BHs with low values of $\omega/\mu_a$. In fact, we see the same behaviour for the free scalar field case (inset plot); even the solid black line separating the two regions has a qualitatively similar shape.
The radial coordinate of the several LRs is shown in Figs. \ref{Fig:CounterCOs} and \ref{Fig:CoCOs} as a blue dashed line.
\begin{figure}[ht!]
\begin{center}
\includegraphics[height=.22\textheight, angle =0]{Graphics_LR_w_vs_J_counter.pdf}
\includegraphics[height=.22\textheight, angle =0]{Graphics_LR_w_vs_J_co.pdf}
\end{center}
\caption{Number of LRs. The left (right) panel shows the counter-rotating (co-rotating) case. Hairy BHs can have 1 LR, as for the case of Kerr, in the light blue region, or 3 LRs, in the dark blue region. The inset follows the same description but for $f_a \rightarrow \infty$.}
\label{Fig:LRs}
\end{figure}
\begin{figure}[ht!]
\begin{center}
\includegraphics[height=.22\textheight, angle =0]{Graphics_MaxPhi_vs_r_Counter_fa_0999.pdf}
\includegraphics[height=.22\textheight, angle =0]{Graphics_MaxPhi_vs_r_Counter_fa_0900.pdf} \\
\includegraphics[height=.22\textheight, angle =0]{Graphics_MaxPhi_vs_r_Counter_fa_0800.pdf}
\includegraphics[height=.22\textheight, angle =0]{Graphics_MaxPhi_vs_r_Counter_fa_0500.pdf}
\end{center}
\caption{Structure of counter-rotating COs for four sets of axionic BHs with constant $q = \{0.5, 0.8, 0.9, 0.999\}$. The maximal value of the scalar field $\phi_\text{max}$ varies with $q$; thus the vertical scale changes for the four plots.}
\label{Fig:CounterCOs}
\end{figure}
For massive particles, $V(r) = V'(r) = 0$ will yield two algebraic equations for the energy and angular momentum of the particle, $\{E_+,L_+\}$ and $\{E_-,L_-\}$ corresponding to co- and counter-rotating orbits, respectively. Their stability can be verified by analysing the sign of the second derivative of the potential $V(r)$. Given a BH solution it will have three distinct regions concerning (timelike) COs:
\begin{itemize}
\item \textit{No circular orbits (No COs)} - Whenever we obtain an imaginary solution for the energy and angular momentum of the massive particle. This is a region in which no COs exist. Such region is shaded in light red, in Figs. \ref{Fig:CounterCOs} and \ref{Fig:CoCOs}.
\item \textit{Unstable circular orbits (UCOs)} - In this region, it is possible to obtain real solutions for the energy and angular momentum, but the sign of the second derivative of the potential $V(r)$ is positive, implying that COs are unstable. Such region is shaded in light yellow, in Figs. \ref{Fig:CounterCOs} and \ref{Fig:CoCOs}.
\item \textit{Stable circular orbits (SCOs)} - In this region, it is also possible to obtain real solutions for the energy and angular momentum, but now the second derivative of the potential $V(r)$ has a negative sign, thus implying that COs are stable. Such region is shaded in light green, in Figs. \ref{Fig:CounterCOs} and \ref{Fig:CoCOs}.
\end{itemize}
The ISCO, as the name entails, is the innermost stable circular orbit, \textit{i.e.}, the stable CO with the smallest radial coordinate. At this orbit, the second derivative of the potential vanishes, as it corresponds to the transition between the SCOs region and the UCOs region. The ISCO is represented as a solid red line in Figs. \ref{Fig:CounterCOs} and \ref{Fig:CoCOs}.
In Fig.~\ref{Fig:CounterCOs} we present the structure of counter-rotating COs for four sets of hairy BHs with constant $q = \{0.5, 0.8, 0.9, 0.999\}$. For each set of BHs, we have constructed a plot showing the maximal value of the scalar field, $\phi_\text{max}$, as a function of the radial coordinate $x = \sqrt{r^2 - r_H^2}$. This ensures that every horizontal line (lines with constant $\phi_\text{max}$) corresponds to a unique BH solution making it easier to analyse the structure of COs.
The first observation from Fig.~\ref{Fig:CounterCOs} is that, by increasing $q$, it is possible to obtain solutions with larger $\phi_\text{max}$. This is reasonable since by increasing $q$ the BHs become hairier. Let us now analyse each individual plot. For the case where $q = 0.5$ (right bottom panel), the structure of the counter-rotating COs for all BHs is identical to the one for Kerr. There is only one LR and one ISCO, which separates the No COs region from the UCOs region, and separates the UCOs region from the SCOs region, respectively. At large enough $x$ we only have SCOs, but as we approach the horizon, we reach the ISCO and consequently enter the UCOs region. If we continue towards the horizon, we eventually cross the LR and enter the No COs region.
The radial coordinate of the LR and ISCO increases monotonically with the increase of the maximal value of the scalar field.
For the case where $q = 0.8$ (left bottom panel) most of the structure is similar, but a new feature emerges. Starting at the regime where there are BHs with a dilute scalar field ($\phi_\text{max} < 0.11$), their structure is the same as for Kerr, but for BHs with $\phi_\text{max} > 0.11$ and $\phi_\text{max} < 0.15$ an extra region of UCOs appears besides the one already present between the LR and the ISCO, and disconnected from the latter. The new region appears as a single point around $x\mu_a \approx 4.8$ and increases in size as $\phi_\text{max}$ increases until its inner boundary merges with the ISCO. For $\phi_\text{max} > 0.15$, the structure is again the same as that of Kerr, but now the radial coordinate of the ISCO is significantly larger than for BHs with a more dilute scalar field. In fact, there is a discontinuity when we study the evolution of the radial coordinate of the ISCO as it changes with the maximum value of the scalar field, as one can see in the left bottom panel of Fig. \ref{Fig:CounterCOs}.
In the $q = 0.9$ case (right top panel), we have a structure similar as for the $q = 0.8$ when we consider BHs with $\phi_\text{max} < 0.21$, but, for the remaining BHs, the scalar field environment is compact enough to develop a pair of extra LRs. Such pairs give rise to a new region of No COs, disconnected from the already existing one between the event horizon and innermost LR. It starts as a single point for a BH with $\phi_\text{max} \approx 0.21$; then it grows in size until its inner boundary (one of the LRs) merges with the innermost LR, connecting both regions of No COs. At the same time, the ISCO also merges with the LRs, meaning that such BH has a degenerate point in which two LRs and the ISCO converge. For BHs with a larger value of $\phi_\text{max}$, a Kerr-like structure again emerges, but now both the LR and the ISCO appear at a larger radial coordinate than for small $\phi_\text{max}$.
Lastly, in the $q = 0.999$ case (left top panel), the most complex structure is observed. This case inherits the several features discussed in the previous cases, such as the existence of two different and disconnected UCOs and No COs regions, but also presents a new one. For BHs with $\phi_\text{max} > 0.18$, a third region of UCOs between the two already existing ones. This third region starts as a single point and increases slowly in size as $\phi_\text{max}$ increases until its inner boundary connects with the ISCO, and the innermost region of UCOs merges with this new region. Although it is not visible in the left top panel of Fig. \ref{Fig:CounterCOs}, one can draw the conclusion that there is a BH with larger $\phi_\text{max}$ for which the innermost two LRs together with the ISCO converge to the same single point, akin to what happened in the $q=0.9$ case.
Now we turn to the co-rotating COs. In Fig.~\ref{Fig:CoCOs} we followed the same idea that we used to study the counter-rotating COs and plotted four sets of BHs solutions with constant $q = \{0.5, 0.8, 0.9, 0.999\}$ in the $\phi_\text{max}$ \textit{vs.} $x$ plane.
Fig. \ref{Fig:CoCOs} manifests that the structure of co-rotating COs is much simpler than the counter-rotating one; only the $q=0.999$ case presents significant differences from the other ones; the structure of the latter is the same as for a Kerr BH. Moreover, the radial coordinate of the co-rotating LR and ISCO is always smaller than the one of the counter-rotating LR and ISCO, as it is for the Kerr BH~\cite{Bardeen:1972fi}.
\begin{figure}[ht!]
\begin{center}
\includegraphics[height=.26\textheight, angle =0]{Graphics_MaxPhi_vs_r_Co_fa_0999.pdf}
\includegraphics[height=.26\textheight, angle =0]{Graphics_MaxPhi_vs_r_Co_fa_0900.pdf}
\includegraphics[height=.26\textheight, angle =0]{Graphics_MaxPhi_vs_r_Co_fa_0800.pdf}
\includegraphics[height=.26\textheight, angle =0]{Graphics_MaxPhi_vs_r_Co_fa_0500.pdf}
\end{center}
\caption{Structure of co-rotating COs for four sets of axionic BHs with constant $q = \{0.5, 0.8, 0.9, 0.999\}$. Note that the maximal value of the scalar field $\phi_\text{max}$ varies with $q$, thus the vertical scale changes for the four plots.}
\label{Fig:CoCOs}
\end{figure}
Let us comment on the qualitatively different $q=0.999$ case (first panel of Fig. \ref{Fig:CoCOs}). For BHs with a dilute scalar field, a Kerr-like structure is observed; but above $\phi_\text{max} \approx 0.14$ a second region of UCOs emerges. This region exists until we reach a BH with $\phi_\text{max} \approx 0.30$, where both regions of UCOs merge together. For BHs with slighter larger $\phi_\text{max}$ a second region of UCOs again emerges, but now this region occurs at large radii, quickly decreasing to smaller radii as $\phi_\text{max}$ increases. Although not shown in this panel, we can predict that this new region of UCOs will converge, as the previous one, with the already existing and closer to the event horizon UCOs region.
\section{Conclusions and Remarks}
In this work, we have constructed and analysed BHs with synchronised axionic hair, which are BH generalisation of the rotating axion boson stars recently found in \cite{Delgado:2020udb} - see also~\cite{Guerra:2019srj}. These are stationary, axi-symmetric, asymptotically flat and regular on and outside the event horizon solutions of the Einstein-Klein-Gordon equations of motion with a QCD axion-like potential, Eq.~\eqref{Eq:AxionPotential}. This family of axionic BHs is described by three parameters: the radial coordinate of the event horizon, $r_H$, the angular frequency of the scalar field, $\omega$ and the decay constant of the QCD potential, $f_a$. In this work, we have thoroughly scanned the space of solutions with decay constant $f_a = 0.05$, since, from the results found in \cite{Delgado:2020udb}, this yields a case with considerable impact of the axion potential, and hence considerable differences from the free scalar field case, obtained as the $f_a \rightarrow \infty$ limit. The latter yields the original example of Kerr BHs with synchronised scalar hair \cite{Herdeiro:2014goa}. Even larger deviations from this original example may occur for even smaller $f_a$, but then the numerics to obtain such solutions becomes more challenging.
When comparing the axionic BHs with their free scalar field counterparts~\cite{Herdeiro:2014goa}, there are both differences and similarities. Some key differences are: (i) axionic BHs can have more mass and angular momentum and lower values of the angular frequency of the scalar field; (ii) the existence of a local maximum for the mass and angular momentum, that does not exists for the $f_a \rightarrow \infty$ case; (iii) the presence of a small region of frequencies where the mass of axionic BHs is no longer bounded by the axion boson stars. In such region, we have a degeneracy of solutions with the same angular frequency and ADM mass, but such degeneracy is easily lifted by specifying any other physical quantity; (iv) the variation of the sphericity along the Smarr line, which, in the free scalar field case was constant and equal to the sphericity of the Smarr point for Kerr BHs, but varies for axionic BHs.
Concerning the similarities, we may emphasise: (i) the clear violation of the Kerr bound, $j \leqslant 1$; (ii) the sphericity and horizon linear velocity are bounded by the same existence line. Thus, both families can only have the same values of sphericity and horizon linear velocity as the Kerr ones, which implies that all BHs have a horizon which is an oblate spheroid and the rotation of its null generators (relatively to a static observer at spatial infinity) never exceeds the speed of light; (iii) the topology of the ergo-regions is either an ergo-sphere or an ergo-Saturn. In both families, the ergo-Saturn only appears for the solutions with the lower values of angular frequency.
In this work, we also presented a study of the structure of COs. Counter-rotating COs show a more complex structure than the co-rotating COs counterpart. Concerning the former, we saw that solutions with low or moderate amounts of hair (\textit{e.g.} $q = 0.5$) will have a structure similar to the Kerr one. For very hairy solutions ($q \geqslant 0.8$), a new region of UCOs can emerge, as well as a new region of No COs when the axionic hair outside the event horizon is compact enough to yield an extra pair of LRs. In the extreme case, where most of the BH solution is composed of axionic hair ($q = 0.999$), it possible to have a third region of UCOs, leading to an intercalation of SCOs and UCOs regions.
Let us briefly comment on energy conditions. In \cite{Delgado:2020udb}, it was shown that the weak energy condition (WEC) and the dominant energy condition (DEC) always hold for the case of axion boson stars, whereas the strong energy condition (SEC) could be violated, and in fact, it is violated for a zero angular momentum observer (ZAMO). For the BH generalisation we can prove, following~\cite{Delgado:2020udb}, that the WEC and DEC are never violated, but the SEC can be violated, for instance, for a ZAMO.
Due to the groundbreaking results presented by the LIGO-Virgo collaboration detecting several gravitational waves events, starting with~\cite{Abbott:2016blz}, as well as the results by the EHT collaboration on the first image of a shadow of the M87 supermassive BH~\cite{Akiyama:2019cqa}, two possible and interesting directions to follow up on this work are: (1) to study the possible gravitational waves generated by the collision of two axionic hairy BHs, both in the head-on scenario, as well as, in the more realistic scenario of an in-spiral binary system. Recently~\cite{CalderonBustillo:2020srq} such collisions were made for Proca stars~\cite{Brito:2015pxa} obtaining a suggestive agreement with the particular gravitational wave event GW190521~\cite{Abbott:2020tfl}; (2) and to study the shadow of axionic hairy BHs, to analyse the impact of the QCD axion-like potential in the BH shadows. This would be a generalisation of the analysis in~\cite{Cunha:2015yba}.
\section*{Acknowledgements}
J. D. is supported by the FCT grant SFRH/BD/130784/2017. This work is supported by the Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT - Funda\c c\~ao para a Ci\^encia e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020 and by national funds (OE), through FCT, I.P., in the scope of the framework contract foreseen in the numbers 4, 5 and 6 of the article 23, of the Decree-Law 57/2016, of August 29, changed by Law 57/2017, of July 19. We acknowledge support from the projects PTDC/FIS-OUT/28407/2017, CERN/FIS-PAR/0027/2019 and PTDC/FIS-AST/3041/2020. This work has further been supported by the European Union's Horizon 2020 research and innovation (RISE) programme H2020-MSCA-RISE-2017 Grant No.~FunFiCO-777740. The authors would like to acknowledge networking support by the COST Action CA16104.
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1,116,691,500,739 | arxiv | \section{Introduction}
Pulsars are fast-rotating neutron stars. Most of them spin down steadily, converting their rotational energy into electromagnetic radiation and particle outflows. This class of pulsars is thus known as rotation-powered pulsars (RPPs). They typically have spin periods of $\sim$0.1--10\,s and period derivatives of $\sim10^{-12}$--$10^{-17}$\,s\,s$^{-1}$, implying magnetic fields of $B\sim10^{11}$--$10^{13}$\,G. Thanks to the revolution of astronomical instrumentation in the past two decades, several subtypes of pulsars have been discovered, and they occupy different regions in the $P$-$\dot{P}$ diagram. Magnetars are an extreme group of pulsars with long spin periods of 2--12\,s and strong magnetic fields of $10^{14}$--$10^{15}$\,G inferred from the spin-down rate. They are remarkable for their energetic burst activities and high X-ray luminosities \citep[see review by][]{Mereghetti2008, MereghettiPM2015}. The theoretical interpretation is that the energetic features are powered by the dissipation and decay of a strong magnetic field \citep{ThompsonD1995, ThompsonD1996}, although the mechanism that converts the magnetic energy into the surface heating remains an open question \citep[see][]{TurollaZW2015, BeloborodovL2016}. Moreover, recent discoveries blurred the boundary between magnetars and RPPs. The low field magnetars SGR 0418+5729 and Swift J1822.3$-$1606 have spin-down-inferred dipole magnetic fields of $6\times10^{12}$\,G and $1.4\times10^{13}$\,G, respectively, comparable to typical RPPs \citep{ReaET2010, ReaIP2013, ScholzNL2012}. Furthermore, the discovery of magnetar-like bursts of high magnetic field RPPs J1846$-$0258 with $B=4.9\times10^{13}$\,G \citep{GavriilGG2008, NgSG2008}, and J1119$-$6127 with $B=4.1\times10^{13}$\,G \citep{ArchibaldKT2016, GogusLK2016} also challenges the division between these two classes of neutron stars. To interpret the fact that magnetar and high magnetic field RPPs share similar behaviors, a unified model has been proposed \citep{PernaP2011, ViganoRP2013} based on the magneto-thermal evolution \citep[e.g.,][]{PonsG2007}.
Because pulsars are born as hot objects, young pulsars provide good targets for measuring the thermal emission and testing the above theory. There are only three high magnetic field ($B>10^{13}$\,G) RPPs younger than 2000 years: PSRs\,B1509$-$58, J1846$-$0258, and J1119$-$6127. \object{PSR\,B1509$-$58} (hereafter B1509) is the only one for which the thermal emission has not been studied. To complete the sample, we carry out a measurement of the surface temperature of {B1509}\ with the \emph{Chandra X-ray Observatory} in this study.
\begin{figure}[ht]
\epsscale{1.15}
\plotone{B1509_image.eps} \caption{{\it Chandra}\ X-ray image of {B1509}\ and its PWN in MSH 15-5{\it 2} in the 0.5--7\,keV band. All the X-ray events are projected onto the red dashed line in the CC mode observation to form a 1D image. This image is created by accumulating all the archival observations with {\it Chandra}\ ACIS-I and operated in the TE mode. \label{chandra_image}}
\vspace{-0.4cm}
\end{figure}
{B1509}\ has a spin period of 150\,ms, which was first discovered in X-rays with the {\it Einstein Observatory} \citep{SeawardH1982}. It is embedded in a bright supernova remnant MSH 15$-$5{\it 2} \citep{MillsSH1960}. Radio pulsations were detected subsequently, and a large period derivative of $\sim 1.5\times 10^{-12}$ s\,s$^{-1}$ was also confirmed \citep{ManchesterTD1982, WeisskopfED1983}. This implies a dipole field of $B=1.5\times10^{13}$\,G, which is higher than that of SGR 0418+5729. The radio pulse profile shows a sharp peak, while the X-ray profile has a single, broad, and asymmetric peak and lags the radio peak by about one quarter of the cycle \citep{KawaiOB1991}. The broad X-ray pulse profile was conventionally described by two Gaussian components \citep{CusumanoMM2001, GeLQ2012}. Using 11 years of radio timing observations, the detailed spin-down evolution suggests a characteristic age of 1700 years \citep{KaspiMS1994}. The distance is $5.2\pm1.4$\,kpc estimated from the H{\sc i} measurement \citep{GaenslerBM1999}.
{B1509}\ is associated with a bright pulsar wind nebula (PWN). The high-resolution {\it Chandra}\ image in Figure~\ref{chandra_image} shows a bright jet in the southeast and a semicircular arc 30\arcsec\ north of the pulsar \citep{GaenslerAK2002}. A few bright clumps were found in the northwest and are occasionally seen in the southeast, indicating the turbulent nature of the flows near the pulsar \citep{DeLaneyGA2006}. Moreover, an inner ring-like structure with a 10\arcsec\ radius surrounding the pulsar was found, which may correspond to the wind termination shock \citep{YatsuKS2009}. Although {\it Chandra}\ can resolve {B1509}\ from the surrounding PWN, its surface temperature has never been determined because the emission is too bright, such that the spectrum was heavily distorted by the pile-up effect. \citep{GaenslerAK2002}.
Using the {\it EXOSAT}, {\it Ginga}, and SIGMA data, \citet{GreiveldingerCM1995} determined a hydrogen column density of the PWN of $N_{\rm H}\approx9.5\times10^{21}$\,cm$^{-2}$ and concluded that the pulsar had a harder emission than its surrounding nebula. The pulsed spectrum obtained with the \emph{RXTE} can be well fit by a power law (PL) with a photon index $\Gamma=1.36\pm0.01$. The off-pulse spectrum, which was dominated by the surrounding nebular emission due to the poor spatial resolution, shows a much softer PL with $\Gamma=2.215\pm0.005$ \citep{MarsdenBG1997}. A pulse phase-resolved spectral analysis showed that $\Gamma$ remains stable during the pulsed phase, such that the spectra of the two components of the pulse shape are indistinguishable \citep{RotsJM1998}. \citet{GeLQ2012} presented a more comprehensive phase-resolved spectral analysis using 15 years of {\it RXTE}\ data with a total exposure time of $\sim$578\,ks. They treated the 0.2 cycles off-pulse emission as the background and found that $\Gamma$ increases from 1.33 to 1.47 as the flux decreases. In addition, the photon indices observed with LECS, MECS, and PDS onboard {\it BeppoSAX}\ are significantly different and the 0.1--300\,keV broadband spectrum is better fit by a curved log-parabolic model than by a single PL \citep{CusumanoMM2001}. The recent {\it NuSTAR}\ observation provides supporting evidence for the log-parabolic model \citep{ChenAK2015}, although the parameters slightly deviate from those determined from {\it BeppoSAX}. However, both the PL and the log-parabolic models are equally good in a detailed phase-resolved analysis.
In this paper, we present a detailed phase-resolved analysis of the X-ray emission of {B1509}, using high temporal resolution data taken with {\it Chandra}. We describe the observations in Section \ref{observation}, including a new observation made with the ACIS-S in the continuous-clocking (CC) mode and an archival observation made with the High Resolution Camera (HRC). Section \ref{analysis} presents the data analysis and results, including the detection of the off-pulse emission from the neutron star, the on- and off-pulse spectra, and the phase-resolved analysis. We then discuss the results in Section \ref{discussion} and give a summary in Section~\ref{summary}.
\section{Observations and Data Reduction}\label{observation}
A new observation of {B1509}\ was made with {\it Chandra}\ on 2013 March 29 with a total exposure of 60.1\,ks \dataset[ ADS/Sa.CXO#obs/14805]{ObsID 14805}. The ACIS-S CCD array was operated in the CC mode with a high timing resolution of 2.85\,ms, such that the pile-up is negligible. Note that the CC mode observation only has one-dimensional spatial information since the events are collapsed along CCD columns into a line. The roll angle of the telescope was carefully chosen to avoid the bright features of the PWN, including the jet and the clumps, from falling onto the same CCD column as the pulsar (see Figure~\ref{chandra_image}). We also used the archival observation \dataset[ ADS/Sa.CXO#obs/5515]{ObsID 5515} made with the HRC-I on 2015 June 13 to verify the detection of the off-pulse emission.
We reprocessed both the ACIS-S and HRC data using the task {\tt chandra\_repro} in the Chandra Interactive Analysis of Observations (CIAO) v4.8 with the most recent calibration database (CALDB) 4.7.0. To perform an accurate timing analysis, we corrected all photon arrival times to the barycenter of the solar system with the CIAO tool {\tt axbary}, based on the JPL solar system ephemeris DE405. We examined the background light curve and found no significant background flares in either observations. Therefore, all the time spans with exposures of 59.9 ks for ACIS-S CC mode data and 44.9 ks for HRC data were used in this analysis.
\begin{figure}[ht]
\centering
\includegraphics[bb=0 0 290 270, width=0.42\textwidth, clip]{profile_0.5_7.0keV_semilog_matlab5.eps}
\caption{Projected photon distribution of {B1509}\ obtained with the {\it Chandra}\ CC mode observation in the 0.5--7\,keV energy range (see Figure~\ref{chandra_image}). Data were collected from (a) the phase-averaged and (b) off-pulse phases. Positive offset is defined as the northwest direction from the pulsar. The orange shaded area indicates the source region and the gray shaded area indicates the background region. The blue dashed line is the mean background value. \label{projected_counts}}
\end{figure}
\section{Data Analysis and Results}\label{analysis}
\setcounter{footnote}{0}
\subsection{Radio Ephemeris}
We used the pulsar ephemeris obtained with the Parkes Observatory, which is a part of the timing program for {\it Fermi}\ \citep{WeltevredeJM2010}. A total of 88 times of pulse arrivals (TOAs) between MJD 54220 and MJD 56512 were used to generate the short time-baseline ephemeris. We then used the TEMPO2 package to fit the TOAs with the rotational frequency and its first two time-derivatives. Because of the significant timing noise of {B1509}, we furthermore used the FITWAVES algorithm \citep{HobbsLK2004}, which employs the sinusoidal curves instead of higher-order polynomials, to fit the residuals. We used five harmonics to whiten the timing noise, and the final rms residual is 1.4\,ms. The dispersion measure was set to $252.5\pm0.3$\,pc\,cm$^{-3}$.
\subsection{Source and Background Selection}\label{off_pulse}
Unlike previous studies that treated the off-pulse X-ray emission as the background, the superb angular resolution of {\it Chandra}\ can minimize the contamination from the surrounding PWN, enabling us to investigate the off-pulse emission. The source selection criterion for the CC mode observation, which is indicated by the orange shaded area in Figure~\ref{projected_counts}, is defined as a 4\arcsec\ wide box centered on {B1509}. The point spread function (PSF) generated by the raytrace simulation tool {\tt ChaRT}\footnote{\url{http://cxc.harvard.edu/ciao/PSFs/chart2/}} suggested that our source selection region contains more than 95\% of the source flux. We obtained a total of $\sim$82,000 counts in 0.5--7\,keV.
In the projected count profile in Figure~\ref{projected_counts}(a), a bump is clearly seen on the northwest side of the pulsar. This is likely related to the bright clumps in the PWN \citep{GaenslerAK2002, DeLaneyGA2006}. To avoid overestimating the background level, we chose a 4\arcsec\ wide region with 1\arcsec\ separation from the source region in the southeast as the background (indicated as the gray shaded area in Figure~\ref{projected_counts}) in the analysis. We also tried using the average of both sides as the background in the spectral analysis and found that the results are similar.
\begin{figure}[t]
\includegraphics[bb=-10 -10 290 280, width=0.42\textwidth, clip]{pulse_profile_0.5_7.0keV_matlab7.eps}
\caption{(a) Folded pulse profile of {B1509}\ in 0.5--7\,keV. We divided the photons into 128 bins per spin period and plotted two cycles. The green shaded region defines the off-pulse phase. The blue dashed line denotes the background level. The red solid line is the best-fit two-Gaussian model, with the two components shown by the red dashed lines. (b) The radio pulse profile in an arbitrary unit. The two profiles are aligned in phase, with phase zero corresponding to the radio peak. \label{fold_lc}}
\end{figure}
\begin{figure}[h]
\centering
\epsscale{1.1}
\plotone{phase_average_off_pulse.eps}
\caption{The HRC phase-averaged and off-pulse images of {B1509}\ (see Figure~\ref{fold_lc} for the definition of the phase range). The off-pulse emission is obviously a point source consistent with the phase-averaged emission. \label{hrc_image}}
\vspace{-0.5cm}
\end{figure}
\subsection{Timing Analysis}\label{timing}
We folded the photon arrival times of the CC mode observation according to the radio timing ephemeris. The X-ray and radio profiles are shown in Figure~\ref{fold_lc} and the radio peak is defined as phase zero. The X-ray profile shows a single, broad, and asymmetric peak, with a fast rise and a relatively slower decay, and it lags the radio peak by more than 0.2 cycles. To investigate the off-pulse emission, we collected all X-ray photons at the pulse minimum, defined as phase 0.8--1.0 (see Figure~\ref{fold_lc}), and plot the count distribution along the projected direction in Figure~\ref{projected_counts}(b). An excess of a width comparable to the model PSF is clearly seen at the pulsar position, indicating that the pulsar is still much brighter than the surrounding during the off-pulse phase. We furthermore confirmed the detection of the off-pulse emission by examining the archival HRC off-pulse image (see Figure~\ref{hrc_image}). We searched the period of HRC data with the $H$-test algorithm \citep{deJagerRS1989} since the observation is not covered by the radio ephemeris, and then folded the photon arrival times according to the best-determined period. The pulse profile is consistent with the CC mode profile in Figure~\ref{fold_lc}. The off-pulse HRC image is fully consistent with the pulsed image and the model PSF.
\begin{deluxetable*}{cccccc}
\tablecaption{Best-fit Parameters for the Pulse profile of B1509 Observed with {\it Chandra}\ in Different Energy Bands. \label{profile_parameters}}
\tablehead{\colhead{Parameter} & \colhead{0.5--7\,keV} & \colhead{0.5--1.7\,keV} & \colhead{1.7--2.7\,keV} &\colhead{2.7--4.0\,keV} &\colhead{4.0--7.0\,keV}}
\startdata
$\mu_1$ & 0.257(1) & 0.252(3) & 0.260(3) & 0.261(3) & 0.260(3) \\
$\sigma_1$ & 0.062(2) & 0.063(5) & 0.058(3) & 0.068(4) & 0.062(2)\\
$\mu_2$ & 0.399(3) & 0.398(9) & 0.395(4) & 0.409(8) & 0.398(5)\\
$\sigma_2$ & 0.137(2) & 0.136(5) & 0.140(3) & 0.137(4) & 0.137(2) \\
F$(\rm{A}_1)$ & 0.25(1) & 0.25(4) & 0.22(2) & 0.30(3) & 0.25(2) \\
PF$_{\rm{area}}$ & 0.87(2) & 0.80(5) & 0.87(3) & 0.90(5) & 0.90(4)\\
$\chi^{2}/dof$ & 183.3/121 & 38.6/43 & 44.1/43 & 71.1/43 & 47.6/43
\enddata
\end{deluxetable*}
Previous studies found that the pulse profile can be fit with two Gaussian components \citep{KuiperHK1999, CusumanoMM2001, GeLQ2012}. Moreover, the leading narrower peak contributes less emission at higher energy bands \citep{GeLQ2012}, and even vanishes in the 30\,MeV--1\,GeV band observed with the {\it Fermi}\ Large Area Telescope \citep{KuiperH2015}. The two-Gaussian function is expressed as
\begin{equation}
f(\phi)=C+\frac{A_1}{\sqrt{2\pi}\sigma_1}e^{-\frac{1}{2}\left( \frac{\phi-\mu_1}{\sigma_1} \right)^2}+\frac{A_2}{\sqrt{2\pi}\sigma_2}e^{-\frac{1}{2}\left( \frac{\phi-\mu_2}{\sigma_2} \right)^2}
\end{equation}
where $f(\phi)$ is the X-ray counts at phase $\phi$, $C$ is a constant term representing the off-pulse emission, and $\mu$ and $\sigma$ are the peak location and the width of individual Gaussian functions. The normalization factor for each Gaussian function is $A/\sqrt{2\pi}\sigma$, where $A$ is the area below the individual Gaussian function. We fit the {\it Chandra}\ 0.5--7\,keV\,profile with the above function and obtained an acceptable result with a reduced $\chi^2$ value of 1.5. The two Gaussian components peak at phase $0.257\pm0.001$ and $0.399\pm0.003$, respectively. Their phase difference is consistent with those measured with {\it RXTE}\ in 2--5\,keV \citep{GeLQ2012} and {\it BeppoSAX}\ MECS in 1.6--10\,keV \citep{CusumanoMM2001}. The best-fit model and the individual components are shown in the second cycle of Figure~\ref{fold_lc} (a). The narrower component contributes $25.1\pm1.3$\,\% of the total pulsed emission, which is in an agreement with the previous results \citep{KuiperHK1999,CusumanoMM2001}. The best-fit parameters are listed in Table~\ref{profile_parameters}.
To investigate the energy dependence of the pulse profile, we divided the X-ray photons into four energy bands: 0.5--1.7, 1.7--2.7, 2.7--4, and 4--7\,keV, with a similar number of counts in each band. We then folded the X-ray light curves with 50 phase bins in each energy band, and fit the profiles with the double Gaussian function. The background-subtracted pulse profiles, the best-fit two-Gaussian functions, and the profile of each Gaussian component are shown in Figure~\ref{fold_lc_hardness}. The best-fit parameters, including $\mu_1$, $\mu_2$, $\sigma_1$, and $\sigma_2$ are listed in Table~\ref{profile_parameters}.
We noted that all fits have $\chi^2_\nu \sim 1.0$ except for fit at 2.7--4\,keV, which has $\chi^2_\nu \sim 1.6$. This could be contributed by the bump between the two Gaussian peaks at phase 0.4--0.45. We further divided the bands into more subbands and performed the same analysis, and found that the profile occasionally cannot be fit well. In these cases, the second peak jitters slightly and results in the bump structure around the best-fit second peak. Therefore, we concluded that the local structure is likely a statistical fluctuation. The parameters $\mu$ and $\sigma$ for each component do not significantly vary with respect to energy and are consistent with the results obtained in the similar energy bands of previous studies \citep{CusumanoMM2001, GeLQ2012}. The flux ratio $F(A_1)=A_1/(A_1+A_2)$ varies between $\sim$0.2 and $\sim$0.3 and also shows no significant energy dependency either. This means that the pulse profile does not change evidently throughout the {\it Chandra}\ energy range, although the strength of the narrower peak decreases as the energy increases beyond the {\it Chandra}\ band \citep{KuiperH2015}.
\begin{figure}
\epsscale{1.0}
\plotone{fold_lc_hardness.eps} \caption{Pulse profiles of {B1509}\ in four energy bands (0.5--1.7, 1.7--2.7, 2.7--4, and 4--7\,keV) and the hardness ratio obtained with {\it Chandra}. The hardness ratio is defined as $(\rm{H}-\rm{S})/(\rm{H}+\rm{S})$, where $H$ is the counts in 2.7--7\,keV, and $S$ is the counts in 0.5--2.7\,keV. The best-fit model is marked by thick red lines, and the individual Gaussian components are shown by blue dashed lines. \label{fold_lc_hardness}}
\vspace{-0.3cm}
\end{figure}
Another important derived parameter is the pulsed fraction. This can be estimated either from the pulsed and total area below the pulse profile (PF$_{\rm{area}}$) or by the root mean square (rms) method based on Fourier decomposition \citep{DibKG2009,ChenAK2015}. The variations in PF obtained with both methods are shown in Figure~\ref{pulsed_fraction}. We found that PF shows an increasing trend with energy, indicating that the off-pulse emission contributes more in the soft X-ray band. This gives us a hint of the softening at the pulse minimum. The flux ratio $F(A_1)$ and PF$_{\rm{area}}$ are listed in Table~\ref{profile_parameters}. The pulsed fraction obtained with {\it Chandra}\ HRC observation is $\sim 0.88$ \citep{ChenAK2015}, which is consistent with the mean value of these four energy bands.
The hardness ratio (HR), which is defined as
\begin{equation}
\rm HR=\frac{H-S}{H+S}\ ,
\end{equation}
where $H$ is the hard X-ray counts and $S$ is the soft X-ray counts, can provide us crude information of the spectral behavior. Here we defined $S$ as the number of counts in 0.5--2.7\,keV, which is the combination of the two softest bands in the analysis above, and $H$ as the number of counts in 2.7--7.0\,keV. The phase variation in HR is shown in the bottom panel of Figure~\ref{fold_lc_hardness}. It is clear that the ratio is stable at HR$\gtrsim 0$ between phase 0.15 and 0.65, indicating that the spectral behaviors of these two Gaussian peaks are the same. Moreover, the HR drops to negative values at the off-pulse phases, which also supports the softening indicated by the PF. This could be caused by a change in PL index with the pulse phase, or by the presence of an additional soft component (see Section \ref{spectral_analysis} below).
\begin{figure}
\epsscale{1.25}
\plotone{pulsed_fraction1.eps} \caption{Pulsed fraction of {B1509}\ observed with {\it Chandra}\ ACIS estimated by the RMS method (black circles) and the area method (red crosses) \label{pulsed_fraction}}
\end{figure}
\subsection{Spectral Analysis}\label{spectral_analysis}
\subsubsection{Phase-averaged Spectrum}
We first extracted the spectrum from the source and background regions described in Section \ref{off_pulse}. The spectral fitting was carried out with the Sherpa package. We performed the spectral fitting over the 0.5--7\,keV range and grouped the photons to at least 50 counts per energy bin. We fit a PL with the absorption model {\tt tbnew}\footnote{\url{http://pulsar.sternwarte.uni-erlangen.de/wilms/research/tbabs/}} and set the solar abundance according to \citet{WilmsAM2000}. The best-fit parameters are $N_{\rm{H}}=(1.43\pm0.04)\times10^{22}$\,cm$^{-2}$, $\Gamma=1.18\pm0.03$, and $\chi_{\nu}^2=0.98$, where all reported uncertainties are in the 90\,\% confidence interval. The 0.5--7\,keV absorbed flux is $(2.0\pm0.1) \times 10^{-11}$\,erg\,cm$^{-2}$\,s$^{-1}$. The $N_{\rm{H}}$ value is significantly higher than those previously reported using {\it Chandra}\ \citep{GaenslerAK2002}, {\it XMM-Newton}\ \citep{SchockBJ2010}, and {\it BeppoSAX}\ \citep{CusumanoMM2001} observations. This could be attributed to the different choice of abundance table and absorption model. We also followed previous studies to use the abundances by \citet{AndersG1989} with the absorption model {\tt phabs}, and obtained $N_{\rm{H}}=(9.7\pm0.3)\times 10^{21}$\,cm$^{-2}$, which agrees well with the {\it Chandra}\ and {\it BeppoSAX}\ values \citep{CusumanoMM2001, GaenslerAK2002}, and is $15\pm4$\,\% smaller than the abundance determined by \citet{SchockBJ2010}.
\begin{figure}
\includegraphics[bb=20 15 410 345, width=0.47\textwidth, clip]{spec_pulsed_pl_leftbg_0.5_7.0keV.eps}
\caption{Pulsed spectrum of {B1509}. The data points in the top panel showed the 0.5--7\,keV spectrum of {B1509}, and the red curve corresponds to the best-fit PL model. The bottom panel shows the fit residual. \label{on_pulse_spectrum}}
\end{figure}
\begin{deluxetable*}{cccccc}
\tablecaption{Best-fit Parameters for the PL and {\it Logpar} Models of the Pulsed X-ray Spectrum of {B1509}. \label{pl_logpar}}
\tablehead{\colhead{Parameter} & \multicolumn{2}{c}{{\it Chandra}} & \multicolumn{2}{c}{{\it NuSTAR}} & \colhead{{\it BeppoSAX}} \\
\colhead{} & \colhead{PL} & \colhead{{\it Logpar}} & \colhead{PL} & \colhead{{\it Logpar}} & \colhead{PL} }
\startdata
$N_{\rm{H}}$ ($10^{22}$ cm$^{-2}$) & 1.57(6) & 1.5(1) & 0.95(fixed) & 0.95(fixed) & 0.91(fixed)\\
$\Gamma$ & 1.18(4) & \nodata & 1.386(7) & \nodata & \nodata\\
$\alpha$ & \nodata & 0.9(3) & \nodata & 1.16(5) & 0.96(8) \\
$\beta$ & \nodata & 0.2(2) & \nodata & 0.11(2) & 0.16(4) \\
$\chi^{2}/dof$ & 352.7/388 & 350.4/387 & 278/254 & 254/253 & 26.6/36
\enddata
\tablecomments{The {\it NuSTAR}\ and {\it BeppoSAX}\ results of pulsed spectra are listed for reference \citep{CusumanoMM2001, ChenAK2015}. }
\end{deluxetable*}
\begin{figure*}[ht]
\includegraphics[bb=20 15 410 330, width=0.45\textwidth, clip]{off_pl_simultaneous_50bin.eps}
\hspace{1.5cm}
\includegraphics[bb=20 15 410 330, width=0.45\textwidth, clip]{off_pl_bb_simultaneous_50bin.eps}
\caption{The {\it Chandra}\ off-pulse (phase 0.8--1.0) spectrum of {B1509}\ fit by (a) a PL model and (b) a PL+BB model. \label{off_pulse_fitting}}
\end{figure*}
\subsubsection{Pulsed Spectrum}
We furthermore extracted the 0.5--7\,keV pulsed spectrum by subtracting the off-pulse emission from the pulsed one (phase 0--0.7) to compare our result with previous studies \citep{CusumanoMM2001, ChenAK2015}. After the background subtraction, we collected $\sim$71,000 photons. Following the same choice of abundance table and absorption model, the pulsed spectrum can be well fit with an absorbed PL, and the best-fit model with residuals is shown in Figure~\ref{on_pulse_spectrum}. The best-fit parameters are $N_{\rm{H}}=(1.57\pm0.06)\times10^{22}$\,cm$^{-2}$, $\Gamma=1.18\pm0.04$, and $\chi^2_{\nu}=0.91$. The absorbed X-ray flux within 0.5--7\,keV is $(2.5\pm0.2) \times 10^{-11}$\,erg\,cm$^{-2}$\,s$^{-1}$, where the phase duration is accounted for hereafter. The parameters are consistent with those of the phase-averaged spectrum above because the X-ray emission is dominated by the pulsed emission.
The photon index we obtained agrees with the value determined with {\it BeppoSAX}\ MECS \citep[$\Gamma=1.19\pm 0.04$,][]{CusumanoMM2001}, but it is significantly lower than the those {\it RXTE}\ ($\Gamma\approx1.35)$) and {\it NuSTAR}\ ($\Gamma=1.386\pm0.07$) values \citep{RotsJM1998, GeLQ2012, ChenAK2015}. Previous studies found that the photon index of {B1509}\ could change with energy, and the log-parabola model ({\it logpar}) gives a better fit to the broadband spectrum \citep{CusumanoMM2001, ChenAK2015}. The mathematical form of the {\it logpar} model is
\begin{equation}
f(x)=A\left(\frac{x}{x_{\rm{ref}}} \right)^{-\alpha-\beta\log(x/x_{\rm{ref}})},
\end{equation}
where $x$ is the energy in units of keV and $x_{\rm{ref}}$ is the reference energy frozen at 1\,keV, $\alpha$ is the photon index at $x_{\rm{ref}}$, $\beta$ is the curvature term, and $A$ is the normalization. We employed this model to fit the pulsed {\it Chandra}\ spectrum, and the best-fit parameters are listed in Table~\ref{pl_logpar}. We note that the {\it logpar} model does not significantly improve the fit, and $\alpha$ is consistent with $\Gamma$ in the PL model. In addition, $\beta$ is not well determined, indicating that the spectral curvature is not obvious over the {\it Chandra}\ energy range. We also show the {\it NuSTAR}\ and {\it BeppoSAX}\ results in Table \ref{pl_logpar} for comparison. It is clear that $\alpha$ and $\Gamma$ determined from {\it Chandra}\ are consistent with the $\alpha$ determined with {\it NuSTAR}\ and {\it BeppoSAX}.
\begin{deluxetable}{lllcc}
\tabletypesize{\scriptsize}
\tablecaption{Best-fit Parameters for the Simultaneous Fits to the Pulsed and Off-pulse Spectra of B1509 with Different Statistics and Models. \label{off_pulse_spectral_table}}
\tablehead{\multicolumn{2}{c}{Model} & \colhead{Parameter} & \colhead{$\chi^2$ Statistic} & \colhead{Cash Statistic} \\
\colhead{Pulsed} & \colhead{Off-pulse} & \colhead{ } & \colhead{ } & \colhead{ } }
\startdata
PL & PL & $N_{\rm{H}}$ ($10^{22}$ cm$^{-2}$) & 1.54(6) & 1.55(6) \\
& & $\Gamma_{\textrm{pulsed}}$ & 1.16(3) & 1.16(3) \\
& & $\Gamma_{\textrm{off-pulse}}$ & 1.85(15) & 1.78(13) \\
& & statistic$/dof$ & 430.9/442 & 1000.7/883 \\
\tableline
PL & PL+BB & $N_{\rm{H}}$ ($10^{22}$ cm$^{-2}$) & 1.57(6) & 1.58(6) \\
& & $\Gamma_{\textrm{pulsed}}$ & 1.18(4) & 1.18(3) \\
& & $\Gamma_{\textrm{off-pulse}}$ & $1.5^{+0.2}_{-0.3}$ & 1.4(2) \\
& & $kT$ (keV) & $0.17_{-0.05}^{+0.06}$ & $0.16_{-0.04}^{+0.05}$ \\
& & $R_{\rm{BB}}$ (km) & $9_{-5}^{+38}$ & $10_{-5}^{+39}$ \\
& & statistic$/dof$ & 413.7/440 & 979.9/881 \\
\tableline
PL & PL+BB & $N_{\rm{H}}$ ($10^{22}$ cm$^{-2}$) & 1.57(6) & 1.58(5) \\
& & $\Gamma_{\textrm{pulsed}}$ & 1.18(4) & 1.18(3) \\
& & $\Gamma_{\textrm{off-pulse}}$ & 1.6(2) & 1.5(2) \\
& & $kT$ (keV) & $0.147_{-0.01}^{+0.007}$ & $0.148_{-0.009}^{+0.007}$ \\
& & $R_{\rm{BB}}$ (km) & 13 (fixed) & 13 (fixed) \\
& & statistic$/dof$ & 414.2/441 & 980.2/882
\enddata
\tablecomments{We did not attempt to add the blackbody component to the pulsed spectrum because it is dominated by the nonthermal emission and the thermal component is subtracted out by the off-pulse emission.}
\end{deluxetable}
\subsubsection{Off-pulse Spectrum}\label{off_pulse_fit}
The off-pulse spectrum, which has never been investigated before, can be studied using our new {\it Chandra}\ observation. Following the same source and background selection procedure, we collected $\sim$1900 net counts in the energy range of 0.5--7\,keV. Because the Galactic absorption should not change with pulse phase, we fit the pulsed and off-pulse spectra simultaneously with tied $N_{\rm{H}}$. We first tried a single PL model, and obtained a best-fit off-pulse $\Gamma=1.85\pm0.15$, which is obviously softer than the pulsed one ($\Gamma=1.16\pm0.03$). Figure~\ref{off_pulse_fitting}(a) shows the off-pulse spectrum and the best-fit PL model. The residuals show a significant curvature, especially in the low-energy end, indicating that an additional component is needed.
\begin{figure}[t]
\includegraphics[bb=0 0 432 390, width=0.49\textwidth, clip]{phase_resolved_parameters_simultaneous.eps}
\caption{Phase variation of the PL flux (upper panel) and the photon index (lower panel) for {B1509}. The black points are obtained with a simple PL fit and the red points are obtained with PL+BB fit. All uncertainties are at 90\,\% confidence level. \label{phase_resolved}}
\end{figure}
We therefore added a blackbody (BB) component to the off-pulse spectrum and tried fitting its temperature and normalization. We obtained $\Gamma=1.5^{+0.2}_{-0.3}$, $kT=0.17_{-0.05}^{+0.06}$\,keV, and $\chi_{\nu}^2=0.94$, which significantly improves the fit with an $F$-test null hypothesis probability of $1.3\times10^{-4}$. Assuming a distance of 5.2 kpc, the BB radius is $R_{\rm{BB}}=9_{-5}^{+38}$\,km, not very well constrained albeit compatible with the canonical neutron star radius. We then fixed the normalization by assuming that the emission is from the entire surface of a neutron star with a radius of 13 km. The best-fit result yielded $\Gamma=1.6\pm0.2$, $kT=0.147^{+0.007}_{-0.01}$\,keV, and the absorbed 0.5--7\,keV fluxes of $2.2^{+0.6}_{-0.5}\times10^{-12}$\,erg\,cm$^{-2}$\,s$^{-1}$ and $8^{+20}_{-5}\times10^{-14}$\,erg\,cm$^{-2}$\,s$^{-1}$ for the PL and BB components, respectively. We note that $\Gamma$ is flatter than that obtained by fitting a single PL because part of the soft X-ray emission is contributed by the BB component. The best-fit parameters are listed in Table~\ref{off_pulse_spectral_table}. We also used the abundances by \citet{AndersG1989} with the absorption model {\tt phabs} since the soft BB component could sensitively depend on the absorption model . The result is consistent with that obtained using {\tt tbnew}. We also attempted to use the Cash statistics \citep{Cash1979} to perform the same simultaneous fits without grouping the photons into energy bins. The result is fully consistent with that obtained from the $\chi^2$ statistics (see Table~\ref{off_pulse_spectral_table}), and the likelihood ratio test suggested that adding a BB component can significantly improve the fit at a null hypothesis probability of $6\times10^{-6}$.
\begin{deluxetable}{lcc}
\tabletypesize{\scriptsize}
\tablecaption{Variation in Photon Index of B1509 Obtained with the PL and the PL+BB Fits to the {\it Chandra}\ Phase-resolved Spectrum. \label{phase_resolved_table}}
\tablehead{\colhead{Pulse Phase} & \colhead{$\Gamma$ (PL)} & \colhead{$\Gamma$ (PL+BB)} }
\startdata
0.0--0.1 & $1.55\pm0.11$ & $1.46\pm0.18$ \\
0.1--0.15 & $1.30^{+0.09}_{-0.06}$ & $1.30\pm0.10$ \\
0.15--0.2 & $1.27_{-0.05}^{+0.06}$ & $1.29_{-0.05}^{+0.08}$ \\
0.2--0.225 & $1.12\pm0.06$ & $1.15_{-0.07}^{+0.06}$ \\
0.225--0.25 & $1.12_{-0.04}^{+0.06}$ & $1.16_{-0.05}^{+0.07}$ \\
0.25--0.275 & $1.15_{-0.04}^{+0.06}$ & $1.20_{-0.05}^{+0.06}$ \\
0.275--0.3 & $1.15_{-0.04}^{+0.06}$ & $1.20_{-0.05}^{+0.06}$ \\
0.3--0.325 & $1.10_{-0.04}^{+0.06}$ & $1.14\pm0.05$ \\
0.325--0.35 & $1.09_{-0.04}^{+0.05}$ & $1.13_{-0.05}^{+0.07}$ \\
0.35--0.375 & $1.14_{-0.04}^{+0.06}$ & $1.18\pm0.06$ \\
0.375--0.4 & $1.12_{-0.04}^{+0.05}$ & $1.16_{-0.05}^{+0.06}$ \\
0.4--0.425 & $1.17_{-0.05}^{+0.06}$ & $1.20_{-0.09}^{+0.07}$ \\
0.425--0.45 & $1.17_{-0.04}^{+0.06}$ & $1.20_{-0.05}^{+0.07}$ \\
0.45--0.475 & $1.10_{-0.07}^{+0.06}$ & $1.13\pm0.07$ \\
0.475--0.5 & $1.20\pm0.07$ & $1.24_{-0.09}^{+0.06}$\\
0.5--0.525 & $1.18_{-0.07}^{+0.06}$ & $1.20_{-0.09}^{+0.08}$\\
0.525--0.55 & $1.16_{-0.05}^{+0.08}$ & $1.19_{-0.07}^{+0.09}$\\
0.55--0.575 & $1.11\pm0.08$ & $1.13_{-0.10}^{+0.09}$ \\
0.575--0.6 & $1.29_{-0.07}^{+0.08}$ & $1.30_{-0.11}^{+0.09}$ \\
0.6--0.65 & $1.27\pm0.06$ & $1.28_{-0.11}^{+0.09}$ \\
0.65--0.7 & $1.38\pm0.08$ & $1.37\pm0.11$ \\
0.7--0.8 & $1.41_{-0.10}^{+0.12}$ & $1.32_{-0.11}^{+0.13}$ \\
0.8--0.9 & $1.57\pm0.15$ & $1.41_{-0.22}^{+0.14}$ \\
0.9--1.0 & $2.05\pm0.19$ & $1.82_{-0.18}^{+0.23}$ \vspace{0.3 mm} \\
\tableline
Pulsed & $1.18\pm0.04$ & \nodata \\
Off-pulse & $1.85\pm0.15$ & $1.5^{+0.3}_{-0.2}$ \\
Phase-averaged & $1.18\pm0.03$ & \nodata \\
\tableline
\tableline
\multicolumn{3}{c}{Tied Parameters and Statistics} \\
\tableline
Parameter & PL & PL+BB\\
\tableline
$N_{\rm{H}}$ ($10^{22}$ cm$^{-2}$) & $1.44\pm0.04$ & $1.55\pm0.05$ \\
$kT$ (keV) & \nodata & $0.142^{+0.007}_{-0.009}$\\
$\chi^2/dof$ & 2016.4/2145 & 1993.8/2144
\enddata
\tablecomments{The phase-averaged, pulsed, and off-pulse results are also listed for reference. We did not attempt to add the blackbody component to the phase-averaged spectrum because it is dominated by the pulsed emission that is well described by a power law.}
\end{deluxetable}
We also tried using a more physical neutron star atmosphere model, {\tt nsa}, to describe the soft component. However, the fitting is not significantly better than a simple BB and the best-fit surface temperature is similar. Finally, as a consistency check, we convolved the best-fit BB+PL model with the HRC response, and this predicts 386 counts from the pulsar, which is fully consistent with the observed 390 counts. Note that the background is negligible at a level of $\lesssim 10$ photons.
\subsubsection{Phase Variation of the Spectrum}
To investigate the phase variation of the spectrum, we divided the X-ray photons into 24 phase bins. The bin length for the low-intensity (phase 0.7--1.1), the transition (phase 0.1--0.2 and 0.6--0.7), and the high-intensity (phase 0.2--0.6) states are 0.1, 0.05, and 0.025, respectively. We first fit all the 24 spectra simultaneously with a simple PL model and a single $N_{\rm{H}}$ value. The fit is acceptable with an $N_{\rm{H}}=(1.44\pm0.04)\times10^{22}$\,cm$^{-2}$. The variation of the PL flux and the best-fit photon index with phase is shown in Figure~\ref{phase_resolved}. $\Gamma$ stays at $\approx1.15$ between phase 0.2--0.6 and increases when the flux decreases after phase $\sim0.6$, until it reaches a maximum value of $2.0\pm0.2$ in the faintest phase bin. We did not find significant spectral variation around phase 0.4, indicating that the bump structure in the profile (see Section \ref{timing}) is possibly a statistical fluctuation.
Similar to the analysis in Section~\ref{off_pulse_fit}, we then added a BB component across all phases and set the BB temperature and radius as free parameters. The fit was significantly improved with an $F$-test null hypothesis probability of $4.6\times10^{-6}$, and the BB temperature is $kT=0.13\pm0.03$\,keV. However, the BB radius is $18_{-10}^{+70}$\,km, not well constrained although consistent with that in the off-pulse fit. Therefore, we fixed the BB normalization by assuming that BB emission arises from the entire surface of a neutron star. The fits yielded a consistent $N_{\rm{H}}=(1.55\pm0.05)\times10^{22}$\,cm$^{-2}$, and the BB temperature was determined as $kT=0.142^{+0.007}_{-0.009}$\,keV. After adding a BB component, the photon indices are systematically lower than those obtained from a single PL fit when the flux is low. The photon index again shows variability, albeit slightly less significant. The results are listed in Table~\ref{phase_resolved_table}, together with the parameters for the pulsed, off-pulse, and phase-averaged spectra.
\section{Discussion}\label{discussion}
\subsection{Thermal Emission}
Our spectral analysis shows evidence of thermal emission from {B1509}, with a BB temperature of $\sim$0.14\,keV. To compare the surface temperature with other high magnetic field RPPs, typical RPPs, and magnetars, we plotted their thermal luminosities and compared with theoretical models in Figure~\ref{cooling_curve}. Neutron stars are born as hot objects and then cool down rapidly by neutrino emission through the Urca process in the core, although different equations of state may result in a variety of cooling timescales \citep[see, e.g., ][]{NomotoT1987, LattimerPP1991, AkmalPR1998}. In general, the surface temperature of a neutron star drops to $T\lesssim 10^6$\,K after $\sim 10^3$ yr. The observational results of typical RPPs \citep[see, e.g.,][]{ShterninYH2011, WeisskopfTY2011} agree with the theoretical predictions, and they occupy the lower part of Figure~\ref{cooling_curve}.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{age_luminosity.eps} \caption{Theoretical cooling curves adopted from \citet{ViganoRP2013} for a 1.4M$_{\odot}$ neutron star with zero magnetic field and an Fe envelope (solid line), zero magnetic field and an H envelope (dotted line), strong magnetic field and an Fe envelope (dashed line), and strong magnetic field and an H envelope (dash-dotted line). The data points show the observed thermal luminosities of magnetars (green diamond), high magnetic field RPPs (blue squares), typical RPPs (black crosses), and {B1509}\ (red filled circle). The magnetar values are adopted from \citet{ViganoRP2013}, with updated ages from the McGill Magnetar Catalog \citep{OlausenK2014}. The RPP measurements and upper limits are from \citet{ShterninYH2011}, \citet{WeisskopfTY2011}, and \citet{ViganoRP2013}. The high magnetic field RPPs are from \citet{GonzalezKL2004}, \citet{KaspiM2005}, \citet{McLaughlinRG2007}, \citet{SafiHarbK2008}, \citet{KaplanEC2009}, \citet{ZhuKG2009}, \citet{LivingstoneNK2011}, \citet{NgKH2012}, \citet{KeaneMK2013}, and \citet{OlausenZV2013}. \label{cooling_curve}}
\end{figure}
The discovery of magnetars breaks the expectation owing to their high surface temperature of $T > 5\times 10^6$\,K and high thermal luminosities $L=10^{34}-10^{36}$\,erg\,s$^{-1}$. They are thought to be powered by the dissipation of a strong magnetic field \citep{ThompsonD1995}. A theory that unifies the magnetars and RPPs is the magneto-thermal evolution model \citep{PonsMG2009, PernaP2011, ViganoRP2013}. In this model, the decay of the crustal magnetic field causes the magnetar activities and the high surface temperature, and it is dominated by the Ohmic dissipation and the Hall drift \citep{GlampedakisJS2011}. The time evolution of the $B$-field can be described by
\begin{equation}
\begin{split}
\frac{\partial\mathbf{B}}{\partial t} =& -\nabla\times\left\{ \frac{c^{2}}{4\pi\sigma}\nabla\times\left(\rm{e}^{\nu}\mathbf{B}\right) \right. \\
& \left. +\frac{c}{4\pi en_{e}}\left[\nabla\times\left(\rm{e}^{\nu}\mathbf{B}\right)\right]\times\mathbf{B}\right\},
\end{split}
\end{equation}
where $\mathbf{B}$ is the surface field measured by the observer at rest, $\sigma$ is the conductivity, $\rm{e}^{\nu}$ is the lapse function that accounts for the relativistic redshift, $e$ is the elementary charge, and $n_e$ is the electron number density. A stronger initial field indicates a higher energy dissipation and results in a higher surface luminosity. We adopted the theoretical cooling curves produced by \citet{ViganoRP2013}, which took all these effects into account. We plotted two cases of the initial magnetic field strengths of 0 and $3\times10^{15}$\,G. Below ages of $\sim 10^4$ yr, magnetars, high magnetic field RPPs, and typical RPPs occupy different regions in Figure~\ref{cooling_curve}. Typical RPPs are consistent with the evolutionary tracks of $B=0$ with luminosities lower than $\sim10^{33}$\,erg\,s$^{-1}$. Magnetars, which have much higher thermal luminosities at $10^{34}$--$10^{36}$\,erg\,s$^{-1}$, are well described by the tracks of high $B$-fields. The thermal luminosity of {B1509}\ is comparable with other high magnetic field RPPs, e.g., PSR J1119$-$6127 with $L\approx2\times10^{33}$\,erg\,s$^{-1}$ \citep{NgKH2012}, which are in between the typical RPPs and magnetars. This result fits the magneto-thermal evolution model well.
Finally, we note that the return current in the magnetosphere could also heat the neutron star surface \citep{ZhangC1997, ZhangC2000}. The hard X-rays from the polar cap are reflected back to the entire stellar surface and cause a temperature of
\begin{equation}
T_s\sim 3.8\times 10^5 f_0^{1/4}P^{-5/12}B_{12}^{1/4}\rm{\,K~,}
\end{equation}
where $f_0\sim0.12$ for {B1509}\ is the average size of the outer gap, $P$ is the spin period, and $B_{12}$ is the surface magnetic field in a unit of $10^{12}$\,G \citep{ZhangC2000}. In the case of {B1509}, this effect gives a surface temperature of only $\sim0.08$\,keV, contributing a thermal luminosity lower than 9\,\% of the measured value. Hence, it is negligible in our discussion.
In the magneto-thermal theory above, the luminosity of a neutron star is related to the magnetic field strength, which consists of the poloidal and the toroidal components. However, only the former can be inferred from the spin down, and the latter is not directly observable. Our detection of thermal emission from {B1509}\ completes the sample of young, high magnetic field RPPs with age $<2000$\,years. It has a similar thermal luminosity and age as other sources (namely PSRs\,J1846$-$0258 and J1119$-$6127) in the group. We therefore speculate that {B1509}\ has a comparable total $B$-field strength, although its dipole field is slightly weaker. High magnetic field RPPs are expected to show magnetar-like bursts as magnetars. This idea was first suggested by \citet{KaspiM2005} and then observed in PSRs\,J1846$-$0258 and J1119$-$6127. By estimating the frequency of starquakes, \citet{PernaP2011} suggested that all types of pulsars could show magnetar-like activities, although the occurrence rate is low for lower field and aged objects. If this is true, we may expect to find magnetar-like activities in {B1509}\ in the sam way as in the other two high magnetic field RPPs since their ages and total $B$-fields are similar.
\subsection{Nonthermal Emission}
Another intriguing feature of {B1509}\ is that the X-ray spectrum becomes softer when the flux decreases from the pulse peak. Similar behavior has been found in other young pulsars, including the Crab \citep{WeisskopfTY2011, GeLQ2012}, PSR\,J1930+1852 \citep{LuWG2007}, and PSR\,B0540$-$69 \citep{HirayamaNE2002,GeLQ2012}. Of these, the Crab pulsar is the best-studied source. Its profile shows a double-peaked feature from radio to $\gamma$-ray bands \citep{ToorS1977, PravdoS1981}. The variation in photon index between the two peaks has been interpreted by both the outer-gap model \citep{ZhangC2002} and a phenomenological two-component model \citep{MassaroCL2000}. A recent {\it Chandra}\ observation discovered a large photon index of the Crab's off-pulse emission \citep{WeisskopfTY2011}, which cannot be explained by any previous models.
Unlike the Crab, the other pulsars show single-peaked X-ray profiles, and the profiles of both PSRs\,B1509$-$58 and B0540$-$69 can be fit with two Gaussians. Taking this feature into account, their X-ray spectra are hardest between the two Gaussian peaks \citep{GeLQ2012}, which qualitatively agrees with the behavior of the Crab if the two Gaussians are considered as the two peaks. Moreover, these two pulsars exhibit significant softening of the off-pulse emission similar to that of the Crab pulsar \citep{HirayamaNE2002}. For PSR\,J1930+1852, this was only inferred from the HR \citep{LuWG2007} and could possibly be due to contamination by the BB emission. Therefore, further deep observation is needed to investigate its spectral behavior.
We can qualitatively explain the phase variation of the photon index of {B1509}\ using the outer-gap model. In this model, the discharged pairs in the pulsar magnetosphere produce gamma-rays, which subsequently create secondary pairs that give synchrotron emission in the X-ray band. The viewing angle of {B1509}\ only allows us to detect the synchrotron radiation of the pairs created by the incoming gamma-rays \citep{WangTC2013}. The created pairs are quickly cooled down to the lowest Landau level via synchrotron emission, with the photon energy given by
\begin{equation}
E_{syn}\sim 3h\gamma^2B/4\pi m_ec\sim 170{\rm MeV}(\gamma/10^3)^2(B/10^{10}{\rm G})\rm{,}
\end{equation}
with $\gamma$ being the Lorentz factor of the pairs and $B$ being the magnetic field strength at the pair-creation site. The spectrum has a high-energy cutoff of $E_{max}\sim170{\rm\,MeV}(B/10^{10}{\rm\,G})$ owing to the maximum energy of the created pair with $\gamma\sim1000$. This is much higher than the {\it Chandra}\ energy range. In addition, a low-energy turnover is present due to the cooled-down pair with a final Lorentz factor of $\gamma\sim1$ as $E_{min}\sim170{\rm\,eV}(B/10^{10}{\rm\,G})$. Below this, the spectrum is described by the low-energy tail of the synchrotron radiation, which has an asymptotic form of a PL with $\Gamma\sim 2/3$ \citep{RybickiL1986}. Between these two critical energies, the spectrum has $\Gamma\sim 1.5$ determined by the energy distribution of the pairs. Hence, the photon index in the X-ray band is controlled by the value of $E_{min}$, which is proportional to the magnetic field strength.
According to this model, the pair-creation process of {B1509}\ has two origins. The first origin is magnetic-photon pair creation, which occurs near the magnetic pole because a stronger magnetic field is needed, thus the synchrotron emission is observed at a narrower viewing angle. As the $B$-field is strong near the pole, $E_{min}$ is lower and hence this emission has a flatter spectrum. The pulsed X-ray emission of {B1509}\ is dominated by this process. The other process is photon-photon pair creation, which is distributed more isotropically in the entire magnetosphere, and $E_{min}$ is located in optical/UV bands owing to the lower magnetic field. The off-pulse emission is dominated by this process, and the X-ray emission is much softer than the pulsed one. Consequently, the variation in spectral index can be qualitatively explained by the various amounts of contribution from these emission processes across the rotational phase.
\section{Summary}\label{summary}
We performed a detailed timing and phase-resolved spectral analysis of the high magnetic field RPP {B1509}\ using high-resolution {\it Chandra}\ observations. The pulse profile can be fit by two Gaussian components, and the pulsed fraction is energy dependent. The 0.5--7\,keV pulsed spectrum is well described by a PL with a photon index of 1.18, lower than the indices at higher energy bands. This result is consistent with the broadband curved {\it logpar} spectrum. The off-pulse spectrum can be fit by a PL plus BB model. The photon index is significantly higher, and the BB has a temperature of $\sim$0.14\,keV. The thermal luminosity of {B1509}\ we obtained is comparable with the luminosities of two other young high magnetic field RPPs, J1846$-$0258 and J1119$-$6127, and it is brighter than the luminosities of normal RPPs. This result supports the magneto-thermal evolution model, indicating that these sources belong to the transition class of neutron stars between magnetars and RPPs and could show magnetar-like activities. We also found that the nonthermal emission in the off-pulse phase has a much softer spectrum. This behavior is similar to other young and energetic pulsars, including the Crab, PSRs\,B0540$-$69 and J1930+1852. We interpreted it qualitatively using the outer-gap model, in which the pulsed emission of {B1509}\ is dominated by the magnetic-photon pair creation near the stellar surface, while the off-pulse nonthermal emission is dominated by the isotropic photon-photon pair creation.
\acknowledgments
We thank the referee for the comments that improved this paper. The scientific results reported in this article are based on observations made by the {\it Chandra X-ray Observatory} and data obtained from the {\it Chandra Data Archive}. This research has made use of software provided by the Chandra X-ray Center (CXC) in the application packages CIAO, ChIPS, and Sherpa. The Parkes radio telescope is part of the Australia Telescope National Facility, which is funded by the Commonwealth of Australia for operation as a National Facility managed by Commonwealth Science and Industrial Research Organization (CSIRO). C.-P.~H. and C.-Y.~N.~are supported by a GRF grant of Hong Kong Government under HKU 17300215P. J.~T. is supported by the NSFC grants of China under 11573010.
\emph{Facilities:} CXO (ACIS, HRC), Parkes
\emph{Software:} CIAO \citep{FruscioneMA2006}, Sherpa \citep{FreemanDS2001}
\input{ms1.bbl}
\end{document}
|
1,116,691,500,740 | arxiv | \section{Introduction}
Feature selection has been one of the most significant issues in machine learning and data mining. Following the success of Lasso~\citep{Tibshirani94},
learning algorithms with sparse regularization~(a.k.a. sparse
learning) have recently received significant attention. A classical problem is to
estimate a signal $\beta^*\in \mathbb{R}^{d}$ from {\lt a feature
matrix $X\in\mathbb{R}^{n\times d}$ and an observation $y=X\beta^* +
\text{noise} \in \mathbb{R}^{n}$,} under the assumption that
$\beta^*$ is sparse~(i.e., $\beta^*$ has $\bar{k} \ll d$ nonzero
elements).
Previous studies have proposed many powerful tools to estimate $\beta^*$.
In addition, in certain applications, reducing the
number of features has a significantly practical value~(e.g., sensor selection in our case).
The general sparse learning problems can be formulated as follows \citep{Jalali11}:
{\lt
\begin{equation}
\begin{aligned}
\bar{\beta}:=arg\min_{\beta}:&~Q(\beta; X, y) \quad \quad s.t.:&~\|\beta\|_0\leq \bar{k}.
\label{eq:intro1}
\end{aligned}
\end{equation}
where $Q(\beta; X, y)$ is a convex smooth function in terms of
$\beta$ such as the least square loss~\citep{TroppTIT04}~(regression),
the Gaussian MLE (or log-determinant
divergence)~\citep{Ravikumar11}~(covariance selection), and the logistic
loss~\citep{Kleinbaum10}~(classification). $\|\beta\|_0$
denotes $\ell_0$-norm, that is, the number of nonzero entries of
$\beta \in \mathbb{R}^d$. Hereinafter, we denote $Q(\beta; X, y)$ simply as $Q(\beta)$.}
From an algorithmic viewpoint, we are mainly interested in three
aspects for the estimator $\hat{\beta}$: (i) estimation error $\|\hat{\beta} - \bar{\beta}\|$; (ii)
objective error $Q(\hat{\beta}) - Q(\bar{\beta})$; and (iii) feature
selection error, that is, the difference between
$\rm{supp}(\hat{\beta})$ and $\bar{F}:=\rm{supp}(\bar{\beta})$, where $\rm{supp}(\beta)$ is a feature
index set corresponding to nonzero elements in $\beta$.
Since the constraint defines a
non-convex feasible region, the problem is non-convex and generally
NP-hard.
There are two types of approaches to solve this problem in the literature.
Convex-relaxation approaches replace $\ell_0$-norm by $\ell_1$-norm
as a sparsity penalty. Such approaches include Lasso
\citep{Tibshirani94}, Danzig selector \citep{Candes05}, and
L1-regularized logistic regression \citep{Kleinbaum10}. Alternative
greedy-optimization approaches include~orthogonal matching pursuit (OMP)~\citep{TroppTIT04, Zhang09}, backward elimination, and forward-backward greedy method~(FoBa)~\citep{Zhang11}, which use
greedy heuristic procedure to estimate sparse signals. Both types
of algorithms have been well studied from both theoretical and empirical
perspectives.
{\color{black}
FoBa has been shown to give better theoretical properties than LASSO and Dantzig selector for the least squared loss function:} $Q(\beta) = {1\over
2} \|X\beta - y\|^2$~\citep{Zhang11}. \citet{Jalali11} has recently
extended it to general convex smooth functions. Their method and
analysis, however, pose computational and theoretical issues. First,
since FoBa solves a large number of single variable optimization
problems in every forward selection step, it is
computationally expensive for general convex functions if the
sub-problems have no closed form solution. Second, though they have
empirically shown that FoBa performs well for general smooth convex
functions, their theoretical results are weaker than those for
the least square case~\citep{Zhang11}. More precisely, their upper
bound for estimation error is looser and their analysis requires more
{\color{black} restricted} conditions for {\lt feature selection and signal recovery
consistency.} The question of whether or not FoBa can achieve {\color{black} the
same theoretical bound in the general case as in the least square case
motivates this work.}
This paper addresses the theoretical and
computational issues associated with the standard FoBa algorithm (hereinafter referred to as FoBa-obj because it solves single variable problems to minimize the objective in each forward selection step).
We study a new algorithm referred to as ``gradient'' FoBa~(FoBa-gdt) which significantly improves the computational efficiency of FoBa-obj. The key difference is that FoBa-gdt only evaluates gradient information in
individual forward selection steps rather than solving a large number of single variable optimization problems.
Our contributions are summarized as follows.
{\bf Theoretical Analysis of FoBa-obj and FoBa-gdt}
This paper presents three main theoretical contributions. First, we derive better theoretical bounds for estimation error, objective error, and
feature selection error than existing analyses for FoBa-obj for general smooth convex functions~\citep{Jalali11} under the same condition: restricted strong convexity condition. Second, we show that FoBa-gdt achieves the same theoretical performance as FoBa-obj. Our new bounds are consistent with the bounds of a {\color{black}} %\newcommand{\bt}{\color{red} special case, i.e., the least square case, and fills in the theoretical gap between the general loss~\citep{Jalali11} and the least squares loss case~\citep{Zhang11}}. Our result also implies an interesting result: when the signal noise ratio is big enough, the NP hard problem \eqref{eq:intro1} can be solved by using FoBa-obj or FoBa-gdt. Third, we show that the restricted strong convexity condition is satisfied for a class of commonly used machine learning objectives, e.g., logistic loss and least square loss, if the number of independent samples is greater than $\bar{k}\log d$ where $\bar{k}$ is the sparsity number and $d$ is the dimension of the variable.
{\bf Application to Sensor Selection}
{\color{black}
We have applied FoBa-gdt with the CRF loss function (referred to as FoBa-gdt-CRF) to sensor selection from time-series binary location signals~(captured by pyroelectric sensors) for human activity recognition at homes, which is a fundamental problem in smart
home systems and home energy management systems. In comparison with forward greedy and L1-regularized CRFs (referred to as L1-CRF), FoBa-gdt-CRF requires the smallest number of sensors for achieving comparable recognition accuracy. Although this paper mainly focuses on the theoretical analysis for FoBa-obj and FoBa-gdt, we conduct additional experiments to study the behaviors of FoBa-obj and FoBa-gdt in Appendix (part~\ref{sec:addexp}).
}
\subsection{Notation}
Denote $e_j\in \mathbb{R}^d$ as the $j^{th}$ natural basis in the
space $\mathbb{R}^d$. The set difference $A-B$ returns the elements
that are in $A$ but outside of $B$. Given any integer $s>0$, the
restricted strong convexity constants (RSCC) $\rho_-(s)$ and
$\rho_+(s)$ are defined as follows: for any {\color{black}} %\newcommand{\bt}{\color{red} $\|t\|_0\leq s$ and
$t=\beta'-\beta$}, we require
\begin{align}
{\rho_-(s)\over 2}\|t\|^2 \leq Q(\beta') - Q(\beta) -\langle \triangledown Q(\beta), t\rangle
\leq {\rho_+(s)\over 2}\|t\|^2. \label{eqn_RSCC}
\end{align}
Similar definitions can be found in \citep{Bahmani11, Jalali11,
Negahban10, Zhang09}. If the objective function takes the quadratic
form $Q(\beta)={1\over 2}\|X\beta-y\|^2$, then the above definition is
equivalent to the restricted isometric property (RIP)
\citep{CandesTIT05}:
\begin{equation}
{\rho_-(s)}\|t\|^2 \leq \|Xt\|^2 \leq {\rho_+(s)}\|t\|^2, \nonumber
\end{equation}
where the well known RIP constant can be defined as $\delta =
\max\{1-\rho_-(s),~\rho_+(s)-1\}$. To give tighter values for $\rho_+(.)$ and $\rho_-(.)$, we only require \eqref{eqn_RSCC} to hold for all $\beta\in\mathcal{D}_s:=\{\|\beta\|_0\leq s~|~Q(\beta)\leq Q(0)\}$ throughout this paper. Finally we define $\hat\beta(F)$ as $ \hat\beta(F):=\arg\min_{\rm{supp}(\beta)\subset
F}:Q(\beta).$ Note that the problem is convex as long as $Q(\beta)$
is a convex function. Denote $\bar{F}:=\rm{\text{supp}}(\bar{\beta})$ and $\bar{k}:=|\bar{F}|$.
We make use of order notation throughout this paper. If $a$ and $b$ are both positive quantities that depend on {\color{black} $n$ or $p$}, we write $a=O(b)$ if $a$ can be bounded by a fixed multiple of $b$ for all sufficiently large dimensions. We write $a=o(b)$ if for {\rm any} positive constant $\phi>0$, we have $a\leq \phi b$ for all sufficiently large dimensions. We write $a=\Omega(b)$ if both $a=O(b)$ and $b=O(a)$ hold.
\begin{algorithm}[h!]
\caption{FoBa (\fbox{FoBa-obj}~\dbox{FoBa-gdt})}
\begin{algorithmic}[1]
\REQUIRE \fbox{$\delta > 0$}~\dashbox{$\epsilon > 0$}
\ENSURE $\beta^{(k)}$
\STATE Let $F^{(0)}=\emptyset$, $\beta^{(0)}=0$, $k=0$,
\WHILE{TRUE} {\label{alg_obj_stop_start}} \STATE \%\% stopping
determination \IF{\fbox{$Q(\beta^{(k)})-\min_{\alpha,j\notin
F^{(k)}}Q(\beta^{(k)}+\alpha e_j)< \delta$}\par\dbox{$\|\triangledown
Q(\beta^{(k)})\|_\infty< \epsilon$}} \STATE break \ENDIF
{\label{alg_obj_stop_end}} \STATE \%\% forward step
{\label{alg_obj_forward_start}}
\STATE
\fbox{$i^{(k)}=\arg\min_{i\notin F^{(k)}}\{\min_\alpha
Q(\beta^{^{(k)}}+\alpha e_i)\}$}\par\dbox{$i^{(k)}=\arg\max_{i\notin F^{(k)}}:~|\nabla Q(\beta^{(k)})_i|$} \STATE $F^{(k+1)}=F^{(k)}\cup
\{i^{(k)}\}$ \STATE $\beta^{(k+1)} = \hat{\beta}(F^{(k+1)})$ \STATE
$\delta^{(k+1)}=Q(\beta^{(k)})-Q(\beta^{(k+1)})$ \STATE $k=k+1$
\label{alg_obj_forward_end} \STATE \%\% backward step
\label{alg_obj_backward_start} \WHILE{TRUE}
\IF{$\min_{i\in F^{(k+1)}}Q(\beta^{(k)}-\beta_i^{(k)}e_i)-Q(\beta^{(k)}) \geq
\delta^{(k)}/2$} \STATE break \ENDIF
\STATE
$i^{(k)}=\arg\min_i Q(\beta^{(k)}-\beta^{(k)}_ie_i)$ \STATE $k=k-1$
\STATE $F^{(k)}=F^{(k+1)}-\{i^{(k+1)}\}$ \STATE $\beta^{(k)} =
\hat{\beta}(F^{(k)})$ \ENDWHILE \label{alg_obj_backward_end}
\ENDWHILE
\end{algorithmic}
\label{alg_FoBaobj}
\end{algorithm}
\section{Related Work}
\citet{TroppTIT04} investigated the behavior of the orthogonal
matching pursuit (OMP) algorithm for the least square case, and
proposed a sufficient condition (an $\ell_\infty$ type condition)
for guaranteed feature selection consistency. \citet{Zhang09}
generalized this analysis to {\color{black} the case of measurement noise}. In
statistics, OMP is known as \rm{boosting} \citep{Buhlmann06} and
similar ideas have been explored in Bayesian network learning
\citep{Chickering02}. \citet{Shalev-ShwartzSZ10}
extended OMP to the general convex smooth function and studied
the relationship between objective value reduction and output
sparsity. Other greedy methods such as ROMP \citep{Needell09} and
CoSaMp \citep{Needell08} were studied and shown to have theoretical
properties similar to those of OMP. \citet{Zhang11} proposed a Forward-backward (FoBa)
greedy algorithm for the least square case, which is an extension of OMP but has stronger theoretical guarantees as well as better empirical performance: feature selection consistency is guaranteed under the sparse eigenvalue condition, which is an $\ell_2$ type condition weaker than the $\ell_\infty$ type condition. Note that if the data matrix is a Gaussian random matrix, the $\ell_2$ type condition requires the measurements $n$ to
be of the order of $O(s\log d)$ where $s$ is the sparsity of the true
solution and $d$ is the number of features, while the
$\ell_\infty$ type condition requires $n=O(s^2\log d)$; see~\citep{zhangzhang12, LiuJMLR12}. \citet{Jalali11} and \citet{JohnsonJR12} extended the FoBa algorithm to general convex functions and applied it to
sparse inverse covariance estimation problems.
Convex methods, such as LASSO \citep{ZhaoY06} and Dantzig selector
\citep{Candes05}, were proposed for sparse learning. The basic idea
behind these methods is to use the $\ell_1$-norm to approximate the
$\ell_0$-norm in order to transform problem~\eqref{eq:intro1} into a
convex optimization problem. They usually require restricted
conditions referred to as irrepresentable conditions (stronger than
the RIP condition) for guaranteed feature selection consistency
\citep{Zhang11}. A multi-stage procedure on LASSO and Dantzig
selector \citep{LiuJMLR12} relaxes such condition, but it is
still stronger than RIP.
\section{The Gradient FoBa Algorithm} \label{sec:alg}
{\color{black}
This section introduces the standard FoBa algorithm, that is, FoBa-obj, and its variant FoBa-gdt. Both algorithms start from an empty feature pool $F$ and follow the same procedure in every iteration consisting of two steps: a forward step and a backward step. The forward step evaluates the ``goodness'' of all features outside of the current feature set $F$, selects the best feature to add to the current feature pool $F$, and then optimizes the corresponding coefficients of all features in the current feature pool $F$ to obtain a new $\beta$. The elements of $\beta$ in $F$ are nonzero and the rest are zeros. The backward step \emph{iteratively} evaluates the ``badness'' of all features outside of the current feature set $F$, removes ``bad'' features from the current feature pool $F$, and recomputes the optimal $\beta$ over the current feature set $F$. Both algorithms use the same definition of ``badness'' for a feature: the increment of the objective after removing this feature. Specifically, for any features $i$ in the current feature pool $F$, the ``badness'' is defined as $Q(\beta-\beta_i e_i) - Q(\beta)$, which is a positive number. It is worth to note that the forward step selects one and only one feature while the backward step may remove zero, one, or more features. Finally, both algorithms terminate when no ``good'' feature can be identified in the forward step, that is, the ``goodness'' of all features outside of $F$ is smaller than a threshold.
The main difference between FoBa-obj and FoBa-gdt lies in the definition of ``goodness'' in the forward step and their respective stopping criterion. FoBa-obj evaluates the goodness of a feature by its maximal reduction of the objective function. Specifically, the ``goodness'' of feature $i$ is defined as $Q(\beta) - \min_{\alpha}Q(\beta+\alpha e_i)$ (a larger value indicates a better feature). This is a direct way to evaluate the ``goodness'' since our goal is to decrease the objective as much as possible under the cardinality condition. However, it may be computationally expensive since it requires solving a large number of one-dimensional optimization problems, which may or may not be solved in a closed form. To improve computational efficiency in such situations, FoBa-gdt uses the partial derivative of $Q$ with respect to individual coordinates (features) as its ``goodness' measure: specifically, the ``goodness'' of feature $i$ is defined as $|\nabla Q(\beta)_i|$. Note that the two measures of ``goodness'' are always nonnegative. If feature $i$ is already in the current feature set $F$, its ``goodness'' score is always zero, no matter which measure to use. We summarize the details of FoBa-obj and FoBa-gdt in Algorithm~\ref{alg_FoBaobj}: the plain texts correspond to the common part of both algorithms, and the ones with solid boxes and dash boxes correspond to their individual parts. The superscript $(k)$ denotes the $k^{th}$ iteration incremented/decremented in the forward/backward steps.
Gradient-based feature selection has been used in a forward greedy method \citep{Zhang11a}. FoBa-gdt extends it to a Forward-backward procedure (we present a detailed theoretical analysis of it in the next section). The main workload in the forward step for FoBa-obj is on Step 4, whose complexity is $O(TD)$, where $T$ represents the iterations needed to solve $\min_{\alpha}:~Q(\beta^{(k)}+\alpha e_j)$ and $D$ is the number of features outside of the current feature pool set $F^{(k)}$. In comparison, the complexity of Step 4 in FoBa is just $O(D)$.
When $T$ is large, we expect FoBa-gdt to be much more computationally efficient. The backward steps of both algorithms are identical. The computational costs of the backward step and the forward step are comparable in FoBa-gdt (but not FoBa-obj), because their main work loads are on Step 10 and Step 21 (both are solving $\hat{\beta}(.)$) respectively and the times of running Step 21 is always less than that of Step 10.
}
\section{Theoretical Analysis} \label{sec:result}
This section first gives the termination condition of Algorithms~\ref{alg_FoBaobj} with FoBa-obj and FoBa-gdt because the number of iterations directly affect the values of RSCC ($\rho_+(.)$, $\rho_-(.)$, and their ratio), which are the key factors in our main results. Then we discuss the values of RSCC in a class of commonly used machine learning objectives. Next we present the main results of this paper, including upper bounds on objective, estimation, and feature selection errors for both FoBa-obj and FoBa-gdt. We compare our results to those of existing analyses of FoBa-obj and show that our results fill the theoretical gap between the least square loss case and the general case.
\subsection{Upper Bounds on Objective, Estimation, and Feature Selection Errors} \label{sec:result:1}
{\color{black}
We first study the termination conditions of FoBa-obj and FoBa-gdt, as summarized in Theorems~\ref{thm_main2_obj} and \ref{thm_main2} respectively.
}
\begin{theorem}\label{thm_main2_obj}
Take $\delta > {4\rho_+(1)\over
\rho_-(s)^2}\|\triangledown Q(\bar\beta)\|_\infty^2$ in Algorithm~\ref{alg_FoBaobj} with FoBa-obj where $s$ can be any positive integer satisfying
$s\leq n$ and
\begin{equation}
(s-\bar{k})>(\bar{k}+1)\left[\left(\sqrt{\rho_+(s)\over \rho_-(s)}+1\right){2\rho_+(1)\over \rho_-(s)}\right]^2.
\label{eq:thm_1}
\end{equation}
Then the algorithm terminates at some $k\leq s-\bar{k}$.
\end{theorem}
\begin{theorem}\label{thm_main2}
Take $\epsilon > {2\sqrt{2}\rho_+(1)\over
\rho_-(s)}\|\triangledown Q(\bar\beta)\|_\infty$ in Algorithm~\ref{alg_FoBaobj} with FoBa-gdt, where $s$ can be any positive integer satisfying $s\leq n$ and Eq.(\ref{eq:thm_1}).
Then the algorithm terminates at some $k\leq s-\bar{k}$.
\end{theorem}
{\color{black}
To simply the results, we first assume that the condition number $\kappa(s):= {\rho_+(s)/\rho_-(s)}$ is bounded (so is ${\rho_+(1) / \rho_-(s)}$ because of $\rho_+(s)\geq \rho_+(1)$). Then both FoBa-obj and FoBa-gdt terminate at some $k$ proportional
to the sparsity $\bar{k}$, {\color{black} similar to OMP
\citep{Zhang11a} and FoBa-obj \citep{Jalali11, Zhang11}.} Note that the value of $k$ in Algorithm~\ref{alg_FoBaobj} is exactly the cardinality of $F^{(k)}$ and the sparsity of $\beta^{(k)}$. Therefore, Theorems~\ref{thm_main2_obj} and \ref{thm_main2} imply that if $\kappa(s)$ is bounded, FoBa-obj and FoBa-gdt will output a solution with sparsity proportional to that of the true solution $\bar{\beta}$.
}
Most existing works simply assume that $\kappa(s)$ is bounded or have similar assumptions. We make our analysis more complete by discussing the values of $\rho_+(s)$, $\rho_-(s)$, and their ratio $\kappa(s)$. Apparently, if $Q(\beta)$ is strongly convex and Lipschitzian, then $\rho_-(s)$ is bounded from below and $\rho_+(s)$ is bounded from above, thus restricting the ratio $\kappa(s)$. To see that $\rho^+(s)$, $\rho^-(s)$, and $\kappa(s)$ may still be bounded under milder conditions, we consider a common structure for $Q(\beta)$ used in many machine learning formulations:
\begin{equation}
Q(\beta) = {1\over n}\sum_{i=1}^n l_i(X_{i.}\beta, y_i) + R(\beta)
\end{equation}
where $(X_{i.}, y_i)$ is the $i^{th}$ training sample with $X_{i.}\in \mathbb{R}^d$ and $y_i\in \mathbb{R}$, $l_i(.,.)$ is convex with respect to the first argument and could be different for different $i$, and both $l_i(.,.)$ and $R(.)$ are twice differentiable functions. $l_i(.,.)$ is typically the loss function, e.g., the quadratic loss $l_i(u, v) = (u-v)^2$ in regression problems and the logistic loss $l_i(u, v) = \log (1+\exp\{-uv\})$ in classification problems. $R(\beta)$ is typically the regularization, e.g., $R(\beta)={\mu\over 2}\|\beta\|^2$.
\begin{theorem} \label{thm_kappa}
Let $s$ be a positive integer less than $n$, and $\lambda^-$, $\lambda^+$, $\lambda^-_R$, and $\lambda^+_R$ be positive numbers satisfying
\[
\lambda^- \leq \nabla_1^2l_i(X_{i.}\beta, y_i) \leq \lambda^+,\quad \lambda^-_RI\preceq\nabla^2 R(\beta) \preceq \lambda^+_RI
\]
($\nabla_1^2l_i(.,.)$ is the second derivative with respect to the first argument) for any $i$ and $\beta\in\mathcal{D}_s$. Assume that $\lambda^-_R+0.5\lambda^- > 0$ and the sample matrix $X\in\mathbb{R}^{n\times d}$ has independent sub-Gaussian isotropic random rows or columns (in the case of columns, all columns should also satisfy $\|X_{.j}\|=\sqrt{n}$).
If the number of samples satisfies $n\ge C s\log d$, then
\begin{subequations}
\begin{align}
\rho_+(s) \leq & \lambda^+_R + 1.5\lambda^+ \label{eqn_thm_kappa_+}\\%\left(1+c\sqrt{s\log d \over n}\right)\\
\rho_-(s) \geq & \lambda^-_R + 0.5\lambda^- \label{eqn_thm_kappa_-}\\%\left(1-c\sqrt{s\log d \over n}\right)
\kappa(s) \leq & \frac{\lambda^+_R + 1.5\lambda^+}{\lambda^-_R + 0.5\lambda^-}=: \kappa \label{eqn_thm_kappa_+-}
\end{align} \label{eqn_thm_kappa_all}
\end{subequations}
hold with high probability\footnote{``With high probability'' means that the probability converges to $1$ with the problem size approaching to infinity.}, where $C$ is a fixed constant. Furthermore, define $\bar{k}$, $\bar{\beta}$, and $\delta$ (or $\epsilon$) in Algorithm~\ref{alg_FoBaobj} with FoBa-obj (or FoBa-gdt) as in Theorem~\ref{thm_main2_obj} (or Theorem~\ref{thm_main2}). Let
\begin{equation}
s = \bar{k} + 4\kappa^2(\sqrt{\kappa}+1)^2(\bar{k}+1)
\label{eqn_thm_kappa_s}
\end{equation}
and $n\geq Cs\log d$. We have that $s$ satisfies \eqref{eq:thm_1} and Algorithm~\ref{alg_FoBaobj} with FoBa-obj (or FoBa-gdt) terminates within at most $4\kappa^2(\sqrt{\kappa}+1)^2(\bar{k}+1)$ iterations with high probability.
\end{theorem}
Roughly speaking, if the number of training samples is large enough, i.e., $n\geq \Omega(\bar{k}\log d)$ (actually it could be much smaller than the dimension $d$ of the train data), we have the following with high probability: Algorithm~\ref{alg_FoBaobj} with FoBa-obj or FoBa-gdt outputs a solution with sparsity at most $\Omega(\bar{k})$ (this result will be improved when the nonzero elements of $\bar{\beta}$ are strong enough, as shown in Theorems~\ref{thm_main1_obj} and \ref{thm_main1}); $s$ is bounded by $\Omega(\bar{k})$; and $\rho_+(s)$, $\rho_-(s)$, and $\kappa(s)$ are bounded by constants. One important assumption is that the sample matrix $X$ has independent sub-Gaussian isotropic random rows or columns. In fact, this assumption is satisfied by many natural examples, including Gaussian and Bernoulli matrices, general bounded random matrices whose entries are independent bounded random variables with zero mean and unit variances. Note that from the definition of ``sub-Gaussian isotropic random vectors'' \citep[Definitions 19 and 22]{Vershynin11}, it even allows the dependence within rows or columns but not both. Another important assumption is $\lambda^-_R+0.5\lambda^- > 0$, which means that either $\lambda^-_R$ or $\lambda^-$ is positive (both of them are nonnegative from the convexity assumption). We can simply verify that (i) for the quadratic case $Q(\beta)={1\over n}\sum_{i=1}^n(X_{i.}\beta-y_i)^2$, we have $\lambda^-=1$ and $\lambda^-_R=0$; (ii) for the logistic case with bounded data matrix $X$, that is $Q(\beta)={1\over n}\sum_{i=1}^n\log (1+\exp\{-X_{i.}\beta y_i\}) + {\mu\over 2} \|\beta\|^2$, we have $\lambda^-_R = \mu>0$ and $\lambda^- > 0$ because $\mathcal{D}_s$ is bounded in this case.
Now we are ready to present the main results: the upper bounds of estimation error, objective
error, and feature selection error for both algorithms. $\rho_+(s)$, $\rho_+(1)$, and $\rho_-(s)$ are involved in all bounds below. One can simply treat them as constants in understanding the following results, since we are mainly interested in the scenario when the number of training samples is large enough. We omit proofs due to space limitations (the proofs are provided in Appendix). The main results for FoBa-obj and FoBa-gdt are presented in Theorems~\ref{thm_main1_obj} and \ref{thm_main1} respectively.
\begin{theorem}\label{thm_main1_obj}
Let $s$ be any number that satisfies \eqref{eq:thm_1} and choose $\delta$ as in Theorem~\ref{thm_main2_obj} for Algorithm~\ref{alg_FoBaobj} with FoBa-obj. Consider the output $\beta^{(k)}$ and its support set $F^{(k)}$. We have
\begin{align*}
\|\beta^{(k)}-\bar\beta\|^2 \leq& {16\rho_+^2(1)\delta\over \rho_-^2(s)}\bar\Delta,\\
Q(\beta^{(k)})-Q(\bar\beta)\leq& {2\rho_+(1)\delta\over \rho_-(s)}\bar\Delta,\\
{\rho_-(s)^2\over 8\rho_+(1)^2}|F^{(k)}-\bar{F}|\leq&
|\bar{F}-F^{(k)}|\leq 2\bar\Delta,
\end{align*}
where $\gamma={4\sqrt{\rho_+(1)\delta}\over \rho_-(s)}$ and
$\bar\Delta := |\{j\in \bar{F}-F^{(k)}: |\bar{\beta}_j|<\gamma\}|$.
\end{theorem}
\begin{theorem}\label{thm_main1}
Let $s$ be any number that satisfies \eqref{eq:thm_1} and choose $\epsilon$ as in Theorem~\ref{thm_main2} for Algorithm~\ref{alg_FoBaobj} with FoBa-gdt. Consider the output $\beta^{(k)}$ and its support
set $F^{(k)}$. We have
\begin{align*}
\|\beta^{(k)}-\bar\beta\|^2 \leq& {8\epsilon^2\over \rho_-^2(s)}\bar\Delta,\\
Q(\beta^{(k)})-Q(\bar\beta)\leq& {\epsilon^2\over \rho_-(s)}\bar\Delta,\\
{\rho_-(s)^2\over 8\rho_+(1)^2}|F^{(k)}-\bar{F}|\leq&
|\bar{F}-F^{(k)}|\leq 2\bar\Delta,
\end{align*}
where $\gamma={2\sqrt{2}\epsilon\over \rho_-(s)}$ and $\bar\Delta :=
|\{j\in \bar{F}-F^{(k)}: |\bar{\beta}_j|<\gamma\}|$.
\end{theorem}
Although FoBa-obj and FoBa-gdt use different criteria to evaluate the
``goodness'' of each feature, they actually guarantee the same
properties. Choose $\epsilon^2$ and $\delta$ in the order of $\Omega(\|\nabla Q(\bar{\beta})\|^2_{\infty})$. For both algorithms, we have that the estimation error $\|\beta^{(k)}-\bar{\beta}\|^2$ and the objective error $Q(\beta^{(k)})-Q(\bar\beta)$ are bounded by $\Omega(\bar\Delta \|\nabla Q(\bar{\beta})\|^2_{\infty})$, and the feature selection errors $|F^{(k)}-\bar{F}|$ and $|\bar{F}-F^{(k)}|$ are bounded by $\Omega(\bar\Delta)$. $\|\nabla Q(\bar\beta)\|_\infty$ and $\bar\Delta$ are two key factors in these bounds. $\|\nabla Q(\bar\beta)\|_\infty$ roughly represents the noise level\footnote{To see this, we can consider the least square case (with standard noise assumption and each column of the measurement matrix $X\in \mathbb{R}^{n\times d}$ is normalized to 1): $\|\nabla Q(\bar\beta)\|_\infty \leq \Omega(\sqrt{n^{-1}\log d}\sigma)$ holds with high probability, where $\sigma$ is the standard derivation.}. $\bar{\Delta}$ defines the number of weak channels of the true solution $\bar\beta$ in $\bar{F}$. One can see that if all channels of $\bar{\beta}$ on $\bar{F}$ are strong enough, that is, $|\bar{\beta}_j| > \Omega(\|\nabla Q(\bar{\beta})\|_\infty)~\forall j\in\bar{F}$, $\bar\Delta$ turns out to be $0$. In other words, all errors (estimation error, objective error, and feature selection error) become $0$, when the signal noise ratio is big enough. Note that under this condition, the original NP hard problem~\eqref{eq:intro1} is solved exactly, which is summarized in the following corollary:
\begin{corollary}
Let $s$ be any number that satisfies \eqref{eq:thm_1} and choose $\delta$ (or $\epsilon$) as in Theorem~\ref{thm_main2_obj} (or \ref{thm_main2}) for Algorithm~\ref{alg_FoBaobj} with FoBa-gdt (or FoBa-obj). If $${|\bar{\beta}_j| \over \|\nabla Q(\bar{\beta})\|_{\infty}} \ge {8\rho_+(1)\over \rho_-^2(s)}\quad \forall j\in \bar{F},$$ then problem~\eqref{eq:intro1} can be solved exactly.
\end{corollary}
One may argue that since it is difficult to set $\delta$ or $\epsilon$, it is still hard to solve \eqref{eq:intro1}. In practice, one does not have to set $\delta$ or $\epsilon$ and only needs to run Algorithm~\ref{alg_FoBaobj} without checking the stopping condition until all features are selected. Then the most recent $\beta^{(\bar{k})}$ gives the solution to \eqref{eq:intro1}.
\subsection{Comparison for the General Convex Case}
\citet{Jalali11} analyzed FoBa-obj for general convex smooth functions and here we compare our results to theirs. They chose the true model
$\beta^*$ as the target rather than the true solution $\bar\beta$.
In order to simplify the comparison, we assume that the distance
between the true solution and the true model is not too
great\footnote{This assumption is not absolutely fair, but holds in
many cases, such as in the least square case, which will be made clear in
Section~\ref{sec:result:3}.}, that is, we have $\beta^*\approx
\bar\beta$, $\rm{supp}(\beta^*)=\rm{supp}(\bar{\beta})$, and $\|\nabla Q(\beta^*)\|_\infty \approx \|\nabla
Q(\bar\beta)\|_\infty$. We compare our results from
Section~\ref{sec:result:1} and the results in \citep{Jalali11}. In
the estimation error comparison, we have from our results:
\begin{align*}
&\|\beta^{(k)}-\beta^*\|\approx \|\beta^{(k)}-\bar\beta\| \\
\leq & \Omega({\bar\Delta}^{1/2}\|\nabla Q(\bar\beta)\|_\infty)
\approx \Omega({\bar\Delta}^{1/2}\|\nabla Q(\beta^*)\|_\infty)
\end{align*}
and from the results in \citep{Jalali11}: $\|\beta^{(k)}-\beta^*\| \leq \Omega(\bar{k}\|\nabla Q(\beta^*)\|_\infty).$
Note that ${\Delta}^{1/2} \leq \bar{k}^{1/2} \ll \bar{k}$.
Therefore, {\color{black} under our assumptions with respect to $\beta^*$} and $\bar\beta$,
our analysis gives a tighter bound. Notably, when there are a large number of
strong channels in $\bar\beta$ (or approximately $\beta^*$), we will have
$\bar{\Delta} \ll \bar{k}$.
Let us next consider the condition {\color{black} required for feature selection
consistency, that is,
$\rm{supp}(F^{(k)})=\rm{supp}(\bar{F})=\rm{supp}(\beta^*)$. We have from our results:
$$\|\bar{\beta}_j\| \geq \Omega(\|\nabla Q(\bar\beta)\|_\infty)~\forall j \in \rm{supp}(\beta^*)$$and from the results in \citep{Jalali11}:
$$\|{\beta}^*_j\| \geq \Omega(\bar{k}\|\nabla Q(\beta^*)\|_\infty)~\forall j \in \rm{supp}(\beta^*).$$
When $\beta^*\approx \bar\beta$ and $\|\nabla Q(\beta^*)\|_\infty
\approx \|\nabla Q(\bar\beta)\|_\infty$, our results guarantee feature selection consistency under a weaker condition.}
\subsection{A Special Case: Least Square Loss} \label{sec:result:3}
We next consider the least square case: $Q(\beta)={1\over
2}\|X\beta-y\|^2$ and shows that our analysis for the two algorithms in
Section~\ref{sec:result:1} fills in a theoretical gap between this
special case and the general convex smooth case.
{\color{black} Following previous studies \citep{Candes05, Zhang11a, ZhaoY06}}, we
assume that $y=X\beta^* +\varepsilon$ where the entries in
$\varepsilon$ are independent random sub-gaussian variables,
$\beta^*$ is the true model with the support set $\bar{F}$ and
the sparsity number $\bar{k}:=|\bar{F}|$, and $X\in
\mathbb{R}^{n\times d}$ is normalized as $\|X_{.i}\|^2=1$ for all
columns $i=1,\cdots,d$. We then have following inequalities with
high probability \citep{Zhang09}:
\begin{align}
&\|\nabla Q(\beta^*)\|_\infty = \|X^T\varepsilon\|_\infty \leq \Omega(\sqrt{n^{-1}\log d}), \label{eqn_gQstar}
\\
&\|\nabla Q(\bar\beta)\|_\infty \leq \Omega(\sqrt{n^{-1}\log d}), \label{eqn_gQbar}
\\
&\|\bar\beta - \beta^*\|_2 \leq \Omega(\sqrt{n^{-1}\bar{k}}), \label{eq:secResult1}
\\
&\|\bar\beta-\beta^*\|_\infty \leq \Omega(\sqrt{n^{-1}\log \bar{k}}), \label{eq:secResult2}
\end{align}
implying that $\bar\beta$ and $\beta^*$ are quite close when
the true model is really sparse, that is, when $\bar{k}\ll n$.
An analysis for FoBa-obj in the least square case \citep{Zhang11} has
indicated that the following estimation error bound holds with high probability:
\begin{align}
&\|\beta^{(k)}-\beta^*\|^2 \leq \Omega(n^{-1}(\bar{k}+ \nonumber
\\
\quad &\log d|\{j\in \bar{F}:~|\beta_j^*|\leq \Omega(\sqrt{n^{-1}\log d})\}|))
\label{eq:secIntro1}
\end{align}
as well as the following condition for feature selection consistency:
if $|{\beta}_j^*|\geq \Omega(\sqrt{n^{-1}\log d})~\forall j\in \bar{F}$, then
\begin{align}
{\rm supp}(\beta^{(k)})={\rm supp}(\beta^*)
\label{eq:secIntro2}
\end{align}
Applying the analysis for general convex smooth
cases in \citep{Jalali11} to the least square case, one obtains the following estimation
error bound from Eq.~\eqref{eqn_gQstar}
\begin{equation*}
\begin{aligned}
\|\beta^{(k)}-\beta^*\|^2 \leq \Omega(\bar{k}^2\|\triangledown
Q(\beta^*)\|_\infty^2) \leq \Omega({n^{-1} \bar{k}^2 \log d})
\end{aligned}
\end{equation*}
and the following condition of feature selection consistency: if $|{\beta}_j^*|\geq
\Omega({\sqrt{\bar{k}n^{-1}\log d}})~\forall~j\in \bar{F}$, then
\begin{equation*}
\begin{aligned}
\rm{supp}(\beta^{(k)})=\rm{supp}(\beta^*).
\end{aligned}
\end{equation*}
One can observe that the general analysis gives a looser bound for
estimation error and requires a stronger condition for feature
selection consistency than the analysis for the special case.
Our results in Theorems~\ref{thm_main1_obj} and~\ref{thm_main1} bridge this gap when combined with Eqs.~\eqref{eq:secResult1} and \eqref{eq:secResult2}.
The first inequalities in Theorems~\ref{thm_main1_obj} and~\ref{thm_main1} indicate that
{\small
\begin{align*}
&\|\beta^{(k)} - \beta^*\|^2 \leq (\|\beta^{(k)}-\bar\beta\| + \|\bar\beta - \beta^*\|)^2 \\
\leq &\Omega\bigg(n^{-1}(\bar{k}+ \log d~|~\{j\in \bar{F}-F^{(k)}:\\
&~|\bar{\beta}_j|<\Omega(n^{-1/2}\sqrt{\log d})\}|)\bigg)~~~~[\text{from Eq.~\eqref{eq:secResult1}}]\\
\leq &\Omega\bigg(n^{-1}(\bar{k}+ \log d~|~\{j\in \bar{F}-F^{(k)}:\\
&~|\beta^*_j|<\Omega(n^{-1/2}\sqrt{\log d})\}|)\bigg)~~~~[\text{from
Eq.~\eqref{eq:secResult2}}]
\end{align*}
}
which is consistent with the results in Eq.~\eqref{eq:secIntro1}. The
last inequality in Theorem~\ref{thm_main1} also implies that feature selection consistency is guaranteed as well, as long as
$|\bar\beta_j|>\Omega(\sqrt{n^{-1}\log d})$ (or $|\beta^*_j|>\Omega(\sqrt{n^{-1}\log d})$) for all $j\in \bar{F}$. This
requirement agrees with the results in
Eq.~\eqref{eq:secIntro2}.
\section{Application: Sensor Selection for Human Activity Recognition}
Machine learning technologies for smart home systems and home energy
management systems have recently attracted much attention. Among the
many promising applications such as optimal energy control,
emergency alerts for elderly persons living alone, and automatic
life-logging, a fundamental challenge for these applications is to
recognize human activity at homes, with the smallest number of
sensors. The data mining task here is to minimize the number of
sensors without significantly worsening recognition accuracy. We
used pyroelectric sensors, which return binary signals in reaction
to human motion.
Fig.~\ref{fig:sensor position} shows our experimental room layout
and sensor locations. The numbers represent sensors, and the
ellipsoid around each represents the area covered by it. We used 40 sensors, i.e., we observe a 40-dimensional binary time series. A single person lives in the room
for roughly one month, and data is collected on the basis of manually tagging his
activities into the pre-determined 14 categories summarized in
Table~\ref{table:activities}. For data preparation
reasons, we use the first 20\%~(roughly one week) samples in the
data, and divide it into 10\% for training and 10\% for testing. The
numbers of training and test samples are given in
Table~\ref{fig:sensor position}.
Pyroelectric sensors are preferable over cameras for two
practical reasons: cameras tend to create a psychological barrier
and pyroelectric sensors are much cheaper and easier to implement at
homes. Such sensors only observe noisy binary location information.
This means that, for high recognition accuracy, history (sequence)
information must be taken into account. The binary time series data follows a linear-chain conditional random field (CRF)~\citep{LaffertyMP01, Sutton06}. Linear-chain CRF gives a smooth and convex loss function; see Appendix~\ref{sec:CRFsyn} for more details of CRF.
Our task then is sensor selection on the basis of noisy binary time
series data, and to do this we apply our FoBa-gdt-CRF (FoBa-gdt with CRF objective function). Since it is very expensive to evaluate the CRF objective value and its gradient, FoBa-obj becomes impractical in this case (a large number of optimization problems in the forward step make it computationally very expensive). Here, we
consider a sensor to have been ``used'' if at least one feature
related to it is used in the CRF. Note that we have 14
activity-signal binary features~(i.e., indicators of sensor/activity
simultaneous activations) for each single sensor, and therefore we
have $40 \times 14 = 560$ such features in total. In addition, we
have {\lt $14 \times 14 = 196$} activity-activity binary
features~(i.e., indicators of the activities at times $t-1$ and $t$).
As explained in Section~\ref{sec:CRFsyn}, we only enforced sparsity
on the first type of features.
\begin{figure}[t]
\centering
\vspace{-3mm}
\subfigure{\includegraphics[width=0.45\textwidth, height=0.45\textwidth]{fig/fig1.eps}}
\vspace{-10mm}
\caption{Room layout and sensor locations.} \label{fig:sensor position}
\vspace{-5mm}
\end{figure}
\begin{table}[t]
{\tiny
\caption{Activities in the sensor data set}
\begin{center}
\begin{tabular}{ c c c }
\\ \toprule
ID & Activity & train / test samples\\ \midrule
1 & Sleeping & 81K / 87K \\
2 & Out of Home (OH) & 66K / 42K \\
3 & Using Computer & 64K / 46K \\
4 & Relaxing & 25K / 65K \\
5 & Eating & 6.4K / 6.0K \\
6 & Cooking & 5.2K / 4.6K \\
7 & Showering (Bathing) & 3.9K / 45.0K \\
8 & No Event& 3.4K / 3.5K \\
9 & Using Toilet & 2.5K / 2.6K \\
10 & Hygiene (brushing teeth, etc.)& 1.6K / 1.6K \\
11 & Dishwashing & 1.5K /1.8K \\
12 & Beverage Preparation & 1.4K / 1.4K \\
13 & Bath Cleaning/Preparation & 0.5K / 0.3K \\
14 & Others & 6.5K / 2.1K \\ \midrule
Total & - & 270K / 270K \\ \bottomrule
\end{tabular}
\end{center}
}
\label{table:activities}
\vspace{-5mm}
\end{table}
First we compare FoBa-gdt-CRF with Forward-gdt-CRF (Forward-gdt
with CRF loss function) and L1-CRF\footnote{L1-CRF solves the
optimization problem with CRF loss + L1 regularization. Since it is
difficult to search the whole space L1 regularization parameter value space, we
investigated a number of discrete values.} in terms of test
recognition error over the number of sensors selected~(see the top
of Fig.~\ref{fig:sensor accuracy}). We can observe that
\begin{itemize}[noitemsep,nolistsep,leftmargin=*]
\item The performance for all methods get improved when the umber of sensors increases.
\item FoBa-gdt-CRF and Forward-gdt-CRF achieve comparable performance. However, FoBa-gdt-CRF reduces the error rate slightly faster, in terms of the number of sensors.
\item FoBa-gdt-CRF achieves its best performance with 14-15 sensors while Forward-gdt-CRF needs 17-18 sensors to achieve the same error level. We obtain sufficient accuracy by using fewer than 40 sensors.
\item FoBa-gdt-CRF consistently requires fewer features than Forward-gdt-CRF to achieve the same error level when using the same number of sensors.
\end{itemize}
\begin{figure}[t]
\centering
\subfigure{\includegraphics[scale=0.32]{fig/SensorSelectionCRF_Error.eps}}
\subfigure{\includegraphics[scale=0.32]{fig/SensorSelectionCRF_Feature.eps}}\\
\subfigure{\includegraphics[scale=0.80]{fig/sensor_individual_accuracy.eps}}
\vspace{-5mm}
\caption{Top: comparisons of FoBa-gdt-CRF, Forward-gdt-CRF and L1-CRF. Bottom: test error rates~(FoBa-gdt-CRF) for individual activities.}
\label{fig:sensor accuracy}
\vspace{-5mm}
\end{figure}
We also analyze the test error rates of FoBa-gdt-CRF for individual
activities. We consider two cases with the number of sensors being 10 and 15, and entered
their test error rates for each individual activity in the bottom of
Fig.~\ref{fig:sensor accuracy}. We observe that:
\begin{itemize}[noitemsep,nolistsep,leftmargin=*]
\item The high frequency activities (e.g., activities \{1,2,3,4\}) are well recognized in both cases. In other words, FoBa-gdt-CRF is likely to select sensors (features) which contribute to the discrimination of high frequency activities.
\item The error rates for activities \{5, 7, 9\} significantly improve when the number of sensors increases from 10 to 15. Activities 7 and 9 are {\it Showering} and {\it Using Toilet}, and the use of additional sensors \{36, 37, 40\} {\lt seems} to have contributed to this improvement.
Also, a dinner table was located near sensor 2, which is why the error rate w.r.t. activity 5 (Eating) significantly decreases from the case \# of sensors=10 to \# of sensors=15 by including sensor 2.
\end{itemize}
\begin{table}[t]
{\tiny
\begin{center}
\caption{Sensor IDs selected by FoBa-gdt-CRF.}
\centering
\begin{tabular}{ c c }
\\ \toprule
\# of sensors=10 & \{1, 4, 5, 9, 10, 13, 19, 28, 34, 38\} \\
\# of sensors=15 & \{\# of sensors=10\} + \{2, 7, 36, 37, 40\} \\ \bottomrule
\end{tabular}
\end{center}
}
\label{table:selected sensors}
\vspace{-5mm}
\end{table}
\section{Conclusion
This paper considers two forward-backward greedy methods, a state-of-the-art greedy method FoBa-obj and its variant FoBa-gdt which is more efficient than FoBa-obj, for solving sparse feature selection problems with general convex smooth functions. We systematically analyze the theoretical properties of both algorithms. Our main contributions include: (i) We derive better theoretical bounds for FoBa-obj and FoBa-gdt than existing analyses regarding FoBa-obj in \citep{Jalali11} for general smooth convex functions. Our result also suggests that the NP hard problem~\eqref{eq:intro1} can be solved by FoBa-obj and FoBa-gdt if the signal noise ratio is big enough; (ii) Our new bounds are consistent with the bounds of a special case (least squares)~\citep{Zhang11} and fills a previously existing theoretical gap for general convex smooth functions \citep{Jalali11}; (iii) We provide the condition to satisfy the restricted strong convexity condition in commonly used machine learning problems;
(iv) We apply FoBa-gdt (with the conditional random field objective) to the sensor selection problem for human indoor activity recognition and our results show that FoBa-gdt can successfully remove unnecessary sensors and is able to select more valuable sensors than other methods (including the ones based on forward greedy selection and L1-regularization). As for the future work, we plan to extend FoBa algorithms to minimize a general convex smooth function over a low rank constraint.
\section{Acknowledgements}
We would like to sincerely thank Professor Masamichi Shimosaka of the
University of Tokyo for providing sensor data collected in
his research and Professor Stephen Wright of the University of
Wisconsin-Madison for constructive comments and helpful advice.
The majority of the work reported here was done during the internship of the first
author at NEC Laboratories America, Cupertino, CA.
\clearpage
{
\bibliographystyle{icml2014}
|
1,116,691,500,741 | arxiv | \section{Introduction}%
\subsection{}
Let $\g$ be a complex semisimple Lie algebra and $\Uhg$ its quantised enveloping algebra.
The starting point of the present paper is the
construction of $\Uhg$ from the
{\it dynamical} Knizhnik--Zamolodchikov (DKZ) equations obtained by the first author \cite{TL}.
Let $(\cdot,\cdot)$ be an invariant inner product on $\g$, $\Omega\in\g\otimes\g$ the corresponding
Casimir element, and $\h\subset\g$ a Cartan subalgebra. Consider the DKZ on $n=2$ points, that
is the $\End(\Ug^{\otimes 2})$--valued connection on $\IC\ni z=z_1-z_2$ given by
\begin{equation}\label{eq:intro DKZ}
d-\left(\sfh\frac{\Omega}{z}+\ad\muone\right) dz
\end{equation}
where $\mu\in\h$, $\muone=\mu\otimes 1$, and $\sfh$ is a formal deformation parameter. Just as
its non--dynamical counterpart which is obtained for $\mu=0$, the connection \eqref{eq:intro DKZ}
has a regular singularity at $z=0$, and admits a canonical fundamental solution $\Upsilon_0$ which
is asymptotic to $z^{\sfh\Omega}$ as $z\to 0$.
\subsection{}\label{eq:intro J}
The dynamical term $\ad\muone$ gives rise to an {\it irregular singularity} at $z=\infty$. Assuming that
$\mu$ is real, so that all Stokes rays lie in $\IR$, and regular, it is proved in \cite{TL} that \eqref{eq:intro DKZ}
admits two canonical fundamental solutions $\Upsilon_\pm$ which are asymptotic to $e^{z\ad\muone}
\cdot z^{\sfh\Omega_0}$ as $z\to\infty$ with $\Im z\gtrless 0$, where $\Omega_0\in\h\otimes\h$ is the
projection of $\Omega$.
Consider now the regularised holonomy of \eqref{eq:intro DKZ} from $\pm\iota\infty$ to $0$ \ie the
element $J_\pm\in\Ug^{\otimes 2}\sfml$ given by $J_\pm=\Upsilon_0(z)^{-1}\cdot\Upsilon_\pm(z)$,
where $\Im z\gtrless 0$. One of the main results of \cite{TL} is that $J_\pm$, regarded as a twist,
kills the KZ associator $\Phi\KKZ$ which arises from the (non--dynamical, reduced) KZ equations
on $n=3$ points
\[d-\sfh\left(\frac{\Omega_{12}}{z}+\frac{\Omega_{23}}{z-1}\right)dz\]
Let $\Delta_\pm=J_\pm^{-1}\Delta(\cdot)J_\pm$ and
$R_\pm=(J_\pm^{21})^{-1}e^{\hbar\Omega/2}J_\pm$ be the corresponding twisted coproduct and
$R$--matrix, where $\hbar=2\pi\iota\sfh$. It follows that $\left(\Ug\fml,\Delta_\pm,R_\pm\right)$ is
a quasitriangular Hopf algebra, which can be shown to be isomorphic to the quantum group $\Uhg$.
\subsection{}
In contrast to earlier constructions of $\Uhg$ from the (non--dynamical) KZ equations
\cite{Dr4,KL,ek-1}, the above construction is entirely transcendental \ie does not rely
on cohomological arguments or the representation theory of $\g$, and perhaps more
naturally explains how $\Uhg$ arises from such equations.
One additional feature is its compatibility with the Casimir equations of $\g$ introduced in \cite{DC,
MTL,TL00,FMTV}. Specifically, the twist $J_\pm$ is a smooth function of $\mu\in\hreg^{\IR}$, and
satisfies the PDE
\[d J_\pm=
\half{\sfh}\sum_{\alpha\in\sfPhi_+}\frac{d\alpha}{\alpha}
\left(\Delta(\Kalpha)J_\pm-J_\pm(\onetwo{\Kalpha})\right)\]
where $\sfPhi_+$ is a chosen system of positive roots, and $\Kalpha$ the Casimir of the $\sl{2}
$--subalgebra of $\g$ corresponding to $\alpha$. This compatibility is a key ingredient in proving
that the monodromy of the Casimir connection of $\g$ is given by Lustzig's quantum Weyl group
operators \cite{TL1,TL,ATL3}.
\subsection{}
Let now $G$ be the connected and simply connected complex Lie group corresponding to $\g$.
Irregular singularities were exploited earlier by Boalch to linearise the Poisson structure on the
\PLg $G^*$ dual to $G$ \cite{Bo1,Bo2}.
Boalch considered connections on the holomorphically trivial $G$--bundle over $\IP^1$ of the form
\begin{equation}\label{eq:intro IODE}
d-\left(\frac{A}{z^2}+\frac{B}{z}\right) dz
\end{equation}
where $A\in\h$ is regular, and $B\in\g$.
Assume that $A$ is real, so that the Stokes rays of \eqref{eq:intro IODE} lie in $\IR$,\footnote
{Contrary to the reality assumption made in \ref{eq:intro J}, the assumption that $A\in\h^\IR$
is inessential, and is only made in the Introduction to simplify the exposition.} and set $\IH_
\pm=\{z\in\IC|\,\Im z\gtrless 0\}$. Then, there are unique holomorphic fundamental solutions
$\gamma_\pm:\IH_\pm\to G$ of \eqref{eq:intro IODE}, which are asymptotic to $e^{-A/z}\cdot
z^{[B]}$ as $z\to 0$ in $\IH_\pm$, where $[B]$ is the projection of $B$ onto $\h$.
Define the {\it Stokes matrices} $S_\pm\in G$ by the analytic continuation identities
\[ \wt{\gamma_-}=\gamma_+\cdot S_+
\aand
\wt{\gamma_+}=
\gamma_-\cdot S_-\cdot e^{2\pi\iota[B]}\]
where $\wt{\cdot}$ denotes counterclockwise analytic continuation, and the identities hold
in $\IH_+$ and $\IH_-$ respectively. The elements $S_\pm\in G$ are unipotent. Specifically,
$A$ determines a partition $\sfPhi=\sfPhi_+\sqcup\sfPhi_-$ of the root system by $\sfPhi_
\pm=\{\alpha\in\sfPhi|\,\alpha(A)\gtrless 0\}$, and $S_\pm$ lies in the unipotent subgroup
$N_\pm\subset G$ with Lie algebra $\n_\pm=\bigoplus_{\alpha\in\sfPhi_\pm}\g_\alpha$.
\Omit{
\subsection{}
Let $\SR=\left\{\alpha(A)\cdot\IR_{>0}\right\}_{\alpha\in\sfPhi}$ be the set of Stokes rays of the
connection \eqref{eq:intro IODE}, and fix a $\log$ cut $\ell=\IR_{>0}\cdot e^{\iota\theta}$. For
any ray
\[r=\IR_{>0}\cdot e^{\iota\sfPhi}\qquad\text{let}\qquad
\IH_r=\bigcup_{\theta\in(-\pi/2,\pi/2)}e^{\iota\theta}r\]
be the open halfplane bisected by $r$. If $r\notin\SR$,
there is a unique holomorphic fundamental solution $\gamma_r:\IH_r\to G$ of \eqref{eq:intro IODE},
which is asymptotic to $e^{-A/z}\cdot z^{[B]}$ on $\IH_r\setminus\ell$, where
$[B]$ is the projection of $B$ onto $\h$.
The solution $\gamma
_r$ is locally constant \wrt the ray $r$, so long as the latter does not cross a Stokes ray or
$\ell$.
\subsection{}
Fix now such an $r$, and assume for simplicity that the log cut does
not lie in the halfplane swept by $e^{\iota\theta}r$, $\theta\in[0,\pi]$.
Define the {\it Stokes matrices} $S_\pm\in G$ relative to the choice
of $r$ by the analytic continuation identities
\[ \left.\wt{\gamma_{r}\phantom{.}}\right|_{-r}=
\gamma_{-r}\cdot S_+
\aand
\left.\wt{\gamma_{-r}}\right|_{r}=
\gamma_{r}\cdot S_-\cdot e^{2\pi\iota[B]}\]
where $\wt{\cdot}$ denotes counterclockwise analytic continuation.
The elements $S_\pm\in G$ are unipotent. Specifically, the pair
$(A,r)$ determines a partition $\sfPhi=\sfPhi_+\sqcup\sfPhi_-$ of the
root system given by
\[\sfPhi_+=
\{\alpha\in\sfPhi|\,\alpha(A)\in e^{\iota\theta}r, \theta\in (0,\pi)\}=
-\sfPhi_-\]
and $S_\pm$ lies in the unipotent subgroup $N_\pm\subset G$ with
Lie algebra $\n_\pm=\bigoplus_{\alpha\in\sfPhi_\pm}\g_\alpha$.
}
\subsection{}
Let $B_\pm\subset G$ be the Borel subgroups corresponding to $\sfPhi_\pm$,
$H=B_+\cap B_-$ the maximal torus with Lie algebra $\h$, and
consider the fibred product
\[B_+\times_H B_-=\{(b_+,b_-)\in B_+\times B_-|\, \pi_+(b_+)\pi_-(b_-)=1\}\]
where $\pi_\pm:B_\pm\to H$ are the quotient maps. Following \cite
{Bo2}, we define the {\it Stokes map} to be the analytic map $\calS:
\g\longrightarrow B_+\times_H B_-$ given by
\begin{equation}\label{eq:intro Stokes}
B\longrightarrow
\left(S_+^{-1}\cdot e^{-\iota\pi[B]},
S_-\cdot e^{\iota\pi[B]}\right)
\end{equation}
\Omit{Note that $B_-\times_H B_+$ maps to $G$ via the map
$\beta:(b_+,b_-)\to b_+\cdot b_-^{-1}$.\comment{$\beta$ is a principal
bundle over its image with structure group the order two elements in
$H$.} Moreover, by Proposition \ref{pr:monodromy reln}, the composition
$\beta\circ\calS_r$ is the map $\g\to G$ given by
\[B\longrightarrow
(S^r_+)^{-1}\cdot e^{-2\pi\iota[B]}\cdot (S^r_-)^{-1}=
C_r\cdot e^{-2\pi\iota B}\cdot C_r^{-1}\]}
\subsection{}
The pair $(B_+,B_-)$
gives rise to a solution $\sfr\in\b_-\otimes
\b_+$ of the classical \YBE given by
\begin{equation}\label{eq:intro standard cybe}
\sfr = x_i\otimes x^i + \half{1} t_a\otimes t^a
\end{equation}
where $\{x_i\},\{x^i\}$ are bases of $\n_-,\n_+$ which are
dual \wrt $(\cdot,\cdot)$, and $\{t_a\},\{t^a\}$ are dual bases of $\h$.
The element $\sfr$ gives $\g$ the structure of a \qt Lie bialgebra, with
cobracket $\delta:\g\to\g\wedge\g$ given by $\delta(x)=[x\otimes 1+1
\otimes x,\sfr]$.
The dual Lie bialgebra $(\g^*,\delta^t,[\cdot,\cdot]^t)$ may be identified, as
a Lie algebra, with
\[\b_+\times_\h\b_-=\{(X_+,X_-)\in\b_+\oplus\b_-|\pi_+(X_+)+\pi_-(X_-)=0\}\]
where $\pi_\pm:\b_\pm\to\h$ is the quotient map.
This endows $G^*=B_+\times_H B_-$ with the structure of a \PL group,
which is dual to $G$.
\subsection{}
Endow $\g^*$ with its standard \KKS Poisson structure
\[ \{f,g\}(x)=\langle[d_x f,d_xg],x\rangle\]
where $d_xh\in T^*_x\g^*=\g$ is the differential of $h$ at $x$,
and $[\cdot, \cdot]$ is the Lie bracket on $\g$.
Let $\nnu:\g^*\to\g$ be the isomorphism induced by the bilinear form
$(\cdot,\cdot)$, and identify $\g$ and $\g^*$ by using $\nnu^\vee=-1/
(2\pi\iota)\nnu$. The following remarkable result is due to Boalch \cite
{Bo1,Bo2}.
\begin{thm
The map $\calS:\g^*\to G^*$ is a Poisson map, and generically a local
complex analytic diffeomorphism. In particular, $\calS$ gives a linearisation
of the Poisson structure on $G^*$.
\end{thm}
\Omit{
Let $\nnu:\g^*\to\g$ be the isomorphism induced by the bilinear form
$(\cdot,\cdot)$, and set $\nnu^\vee=-1/(2\pi\iota)\nnu$. The following
remarkable result is due to Boalch \cite{Bo1,Bo2}.
\begin{thm}\label{th:Boalch}
The map $\calS\circ\nnu^\vee:\g^*\to G^*$ is a Poisson map, and
generically a local complex analytic diffeomorphism.
In particular, $\calS\circ\nnu^\vee$ gives a linearisation of the
Poisson structure on $G^*$.
\end{thm}
}
\subsection{}
One of goals of the present paper is to prove that Boalch's linearisation result, specifically
the fact that $\calS$ is a Poisson map, can be obtained as a semiclassical limit of the
transcendental construction of $\Uhg$.
Our overall strategy is the following. Since $\calS$ is holomorphic, it suffices to show that its
Taylor series $\whS$ at $0\in\g^*$ is a formal Poisson map. This in turn follows if $\whS$ can
be quantised. We therefore seek quantisations $\IC_\hbar\fmls{\g^*}$ and $\IC_\hbar\fmls{G^*}$
of the algebras of functions on the formal \PL groups corresponding to $\g^*$ and $G^*$, together
with an algebra isomorphism $\whS^*_\hbar:\IC_\hbar\fmls{G^*}\to\IC_\hbar\fmls{\g^*}$ such
that the following diagram is commutative
\begin{equation}\label{eq:diagram}
\xymatrix@C=2cm
\IC_\hbar\fmls{\g^*}\ar[d] & \IC_\hbar\fmls{G^*}\ar[l]_{\whS^*_\hbar}\ar[d]\\
\IC\fmls{\g^*} & \IC\fmls{G^*}\ar[l]^{\whS^*}\\
}\end{equation
where the vertical arrows are the specialisations at $\hbar=0$, and the bottom one the
pullback of $\whS$.
\subsection{}\label{ss:intro duality}
A formal quantisation of the dual $P^*$ of a \PLg $P$ can be obtained from Drinfeld's {\it quantum
duality principle} as follows \cite{DrICM,Ga}. Let $\UU$ be a quantised enveloping algebra which
deforms the Lie bialgebra $\p$ of $P$. Thus, $\UU$ is a topologically free Hopf algebra over $
\IC\fml$ such that $\UU/\hbar\UU$ is isomorphic to $U\p$ and, for any $x\in\p$ with cobracket
$\delta(x)\in\p\wedge\p$
\[\delta(x)=\left.\frac{\Delta(\wt{x})-\Delta^{21}(\wt x)}{\hbar}\right|_{\hbar=0}\]
where $\wt{x}\in\UU$ is an arbitrary lift of $x$. Then, $\UU$ admits a canonical Hopf subalgebra
$\UU'$ which is commutative mod $\hbar$, and endowed with a canonical Poisson isomorphism
$\imath_\UU:\UU'/\hbar\UU'\to\IC\fmls{P^*}$.
The simplest example of Drinfeld duality arises when $P$ is the Lie group $G$ endowed with
the trivial Poisson structure. The corresponding Lie bialgebra is $\g$ with the trivial cobracket,
and $P^*$ is the additive abelian group $\g^*$ with cobracket given by the transpose of the bracket
on $\g$. In this case, $\UU$ can be taken to be $\Ug\fml$ with undeformed product and coproduct.
The corresponding subalgebra $\UU'$ is the Rees algebra of formal power series $\sum_{n\geq 0}
x_n\hbar^n$ where the filtration order of $x_n$ is at most $n$, and the specialisation $\imath_
\UU$ is the symbol map $\UU'/\hbar\UU'\to\prod_{n\geq 0}S^n\g=\IC\fmls{\g^*}$.
\subsection{}
To obtain a formal quantisation of $G^*$, we seek a QUE deforming the quasitriangular
Lie bialgebra $(\g,\sfr)$, where $\sfr\in\b_-\otimes\b_+$ is the canonical element \eqref
{eq:intro standard cybe}.
One such quantisation is the
\DJ quantum group $\Uhg$ corresponding to $\g$. That, however, shifts the problem of
filling in the diagram \eqref{eq:diagram} to one of finding an algebra isomorphism $(\Uhg)'
\to\UU'$, where $\UU=\Ug\fml$, and showing that the latter quantises $\whS^*$.
Alternatively, we may resort to a {\it preferred} quantisation of $\g$, that is a QUE
which is equal to $\UU$ as algebras. A class of such quantisations may be obtained
as a {\it twist quantisation}, that is by using an element $J\in 1+\frac{\hbar}{2}\sfj+
\hbar^2\UU^{\otimes 2}$ satisfying $\sfj-\sfj^{21}=\sfr-\sfr^{21}$, together with the
twist equation
\[\Phi\cdot J_{12,3}\cdot J_{1,2}=J_{1,23}\cdot J_{2,3}\]
where $\Phi$ is a given associator. Then,
$\UU_J=
\left(\UU,J^{-1}\Delta_0(\cdot)J,J^{-1}_{21}e^{\hbar\Omega/2}J\right)$
is a QUE which quantises $(\g,\sfr)$, and $(\UU_J)'$ is a formal quantisation
of $G^*$.
\subsection{}
A general result of \EH asserts that if the twist $J$ is {\it admissible}, that is such
that $\hbar\log(J)\in(\UU')^{\otimes 2}$, the Drinfeld algebras $(\UU_J)'$ and $\UU'$
{\it coincide} \cite{EH}. In this case, the equality $e:(\UU_J)'\to\UU'$ clearly is an
algebra isomorphism, and descends to a Poisson isomorphism
$e\claj:\IC\fmls{G^*}\to\IC\fmls{\g^*}$ given by the composition
\[e\claj = \imath_\UU \circ e_0 \circ\imath_{\UU_J}^{-1}\]
where $\imath_\UU:\UU'/\hbar\UU'\to\IC\fmls{\g^*}$ is the symbol map, $\imath_
{\UU_J}:(\UU_J)'/\hbar(\UU_J)'\to\IC\fmls{G^*}$ the canonical identification mentioned
in \ref{ss:intro duality}, and $e_0=\id$ the reduction of $e$ mod $\hbar$.
One of the main results of this paper is that if $J=J_+$ is (one of) the twist(s)
arising from the dynamical KZ equations described in \ref{eq:intro J},
then $J$ is admissible, and the corresponding map $e\claj$ is equal to the
Stokes map $\whS^*$. In particular, the latter is a Poisson map.
\subsection{} \label{ss:EEM
A key ingredient in proving the identity $e\claj=\whS^*$ is a result of \EEM \cite{EEM}
which gives an explicit formula for $e\claj$, under the additional assumptions that
$\Phi$ is a Lie associator and that the admissible twist $J$ lies in $\UU'\otimes\UU
\cap\UU\otimes\UU'$.
Consider to that end the quotient $\UU\otimes\UU'/\hbar\,\UU\otimes\UU'\cong\Ug
\fmls{\g^*}$, where the latter is the algebra of $\Ug$--valued formal power series
on $\g^*$. Let $G\fmls{\g^*}_+\subset\Ug\fmls{\g^*}$ be the prounipotent group of
$\IC\fmls{\g^*}$--points of
$G$ such that their value
at $0\in\g^*$ is equal to $1$. Then, the following holds \cite{EEM}
\begin{enumerate}
\item The {\it semiclassical limit $\jmath=\scl{J}$}, that is the image of $J$ in $\Ug\fmls{\g^*}$,
lies in $G\fmls{\g^*}_+$ and is therefore a formal map $\g^*\to G$.\\
\item
Let
\[\beta:G^*\to G,\qquad (b_+,b_-)\to b_+\cdot b_-^{-1}\]
be the {\it big cell map}. Then, the composition of $e\claj$ with $\beta$
is the formal map $\g^*\to G$ given by the {\it twisted exponential map}
\begin{equation}\label{eq:intro eem}
e_\jmath(\lambda)=
\jmath(\lambda)^{-1}\cdot e^{\nnu(\lambda)}\cdot\jmath(\lambda)
\end{equation}
where $\nu:\g^*\to\g$ is the isomorphism given by the inner product.
\end{enumerate}
\subsection{}
Since $\beta$ is an isomorphism when regarded as a formal map, it suffices to prove
that $e\claj\circ\wh{\beta}^*=\whS^*\circ\wh{\beta}^*=\wh{\beta\circ\calS}^*$ that is, by
\eqref{eq:intro eem} that
\begin{equation}\label{eq:intro to prove}
\wh{\beta\circ\calS}=\jmath(\lambda)^{-1}\cdot e^{\nnu(\lambda)}\cdot\jmath(\lambda)
\end{equation}
By definition of $\calS$ \eqref{eq:intro Stokes}, the composition $\beta\circ\calS$
is the map $B\to S_+^{-1}\cdot e^{-2\pi\iota[B]}\cdot S_-^{-1}$, which is the {\it
clockwise} monodromy around $z=0$ of \eqref{eq:intro IODE} expressed in the
solution $\gamma_-$. By parallel transport to $z=\infty$, where \eqref{eq:intro IODE}
has a regular singularity with residue $-B$, $\beta\circ\calS$ is also equal to
\begin{equation}\label{eq:intro infty}
B\to C^{-1}_-\cdot e^{-2\pi\iota B}\cdot C_-
\end{equation}
where $C_-=C_-(B)\in G$ is the {\it connection matrix},
that is the element relating $\gamma_-$ to the canonical
fundamental solution $\gamma_\infty$ which is asymptotic to $z^
{2\pi\iota B}$ near $z=\infty$.
\Omit{
where $C_-=C_-(B)\in G$ is the {\it connection matrix} of \eqref{eq:intro IODE}
that is the element defined by $\gamma_-(z)=\gamma_\infty(z)\cdot C_-$, $z\in
\IH_-$, where $\gamma_\infty$ is the canonical fundamental solution asymptotic
to $z^{2\pi\iota B}$ near $z=\infty$.
}
Comparing the right--hand sides of \eqref{eq:intro to prove} and \eqref{eq:intro infty},
and recalling that $\g^*$ and $\g$ are identified by $-1/2\pi\iota\cdot\nu$, it therefore
suffices to show that $B\to\wh{C_-}$ is the semiclassical limit of the DKZ twist $J$.\footnote
{The problem of obtaining a quantisation of the connection matrix $C_-$ formulated
in \cite{Xu}, together with our intuition that such a quantisation should be given by
the DKZ twist $J$, were in fact the original impetus of this project.}
\subsection{}
The fact that $J$ is a quantisation of the connection matrix $C_-$ follows from
the uniqueness of canonical fundamental solutions of
\eqref{eq:intro IODE}, when the structure group is an arbitrary affine algebraic group,
specifically the prounipotent group $G\fmls{\g^*}_+$ \cite{BTL2}.
It stems from the basic, but seemingly novel observation that the {\it semiclassical
limit of the DKZ equation \eqref{eq:intro DKZ} is equal to the ODE \eqref
{eq:intro IODE}}.\footnote{This is related to, but different from, the fact that
a different semiclassical limit of the KZ equations are the (non--linear)
Schlesinger equations \cite{Re}.}
More precisely, if $\Upsilon$ is a solution of
\[\frac{d\Upsilon}{dz}=
\left(\ad\muone+\sfh\frac{\Omega}{z}\right)\Upsilon\]
with values in $\UU\otimes\UU'$, the semiclassical limit $\gamma$
of $\Upsilon$, as a formal function of $\lambda\in\g^*$ with values
in $\Ug$, is readily seen to satisfy
\[\frac{d\gamma}{dz}=
\left(\ad\mu+\frac{\nnu(\lambda)}{2\pi\iota z}\right)\gamma\]
where $\nnu(\lambda)=\id\otimes\lambda(\Omega)$ which, after
the change of variable $z\to 1/z$, and the replacement $\ad\mu
\to -\Az, \nnu(\lambda)\to -2\pi\iota B$ is precisely the equation \eqref
{eq:intro IODE}.\footnote{The appearance of the factor $2\pi\iota$
is due to the fact that the identification $\UU'/\hbar\UU'\cong\wh{S\g}$
is given by mapping $x\in\g$ to $\hbar x=2\pi\iota\sfh x\in\UU'$.}
\subsection{Outline of paper}
In Sections \ref{se:G Stokes} and \ref{se:Ug Stokes}, we review the definition of
the Stokes data and map for the connection \eqref{eq:intro IODE}, and the
transcendental construction of $\Uhg$ given in \cite{TL}. In Section \ref{se:R = S},
we show that quantum $R$--matrix of $\Uhg$ is a Stokes matrix of the dynamical
KZ equation. Section \ref{se:quantum duality} reviews Drinfeld's duality principle.
Section \ref{se:scl DKZ} contains the first part of our main results, namely the fact that
the semiclassical limits of the DKZ equations and its canonical solutions at $0$ and
$\infty$ are equal to the connection \eqref{eq:intro IODE} and its canonical solutions,
after a change of variables.
Section \ref{se:EEM} describes the linearisation formula of Enriquez--Etingof--Marshall.
Finally, in Section \ref{se:alt boalch}, we prove that the Stokes map is Poisson and,
in Section \ref{Section:isomono} relate quantum and classical isomonodromic equations.
\subsection*{Acknowledgements}
\noindent
We would like to thank Anton Alekseev and Pavel Etingof for their helpful discussions and useful comments.
\Omit{
\section{Introduction}%
Let $\g$ be a complex, semisimple Lie algebra, $\frak h$ a Cartan subalgebra of $\g$, and
$b_\pm\subset\g$ a pair of opposite Borel subalgebras intersecting along $\frak h$. Let $G$ be the simply-connected Poisson-Lie group corresponding to $(\g,r)$, and $G^*=
B_-\times_H B_+$ its dual. The $G$--valued Stokes phenomena were used in \cite{Bo2}, to give a canonical, analytic linearisation of the \PLg structure on $G^*$, and was used more
recently by the second author \cite{Xu} to study the vertex-IRF gauge transformation of (dynamical) $r$-matrices.
On the other hand, in a recent paper of the first author \cite{TL}, the Stokes phenomena of the dynamical $KZ$ equations (introduced by Felder, Markov, Tarasov and Varchenko in \cite{FMTV}) was used to construct a Drinfeld twist
killing the KZ associator, and therefore give an explicit transcendental construction
of the Drinfeld-Jimbo quantum group $U_h(\g)$. Because the dual Poisson Lie group $G^*$ is the semiclassical limit of the Drinfeld-Jimbo quantum group $U_h(\g)$, the Stokes phenomena used in \cite{Bo1} and \cite{TL} should relate to each other in a similar way as $G^*$ is related to $U_h(\g)$. Another stronger hint is that the vertex-IRF gauge equation satisfied by the connection matrices in \cite{Xu} is the semiclassical limit of Drinfeld's twist equation satisfied by the quantum connection matrix/differential twists in \cite{TL}.
\vspace{3mm}
In this paper, we prove that this is the case, i.e., both Boalch's and the second author's constructions can be obtained
as semiclassical limits of the first author's construction in \cite{TL}.
From a different point of view, we actually show that the quantization problem (quantization of Poisson Lie groups to Quantum groups) can be understood in the frame of the deformation of certain irregular Riemann-Hilbert problem (meromorphic ODE).
Along the way, we introduce the quantum Stokes matrices of the dynamical KZ
equations and prove they are $R$--matrix of $\Uhg$. We study the quantum isomonodromy deformation equation of the $R$-matrix, and prove its semiclassical limit recovers the classical isomonodromy equations of Jimbo--Miwa--Ueno \cite{JMU}. In the end, we unveil the Poisson geometric meaning of the centralizer property satisfied by the twist $J$. As an application, we construct a Ginzburg-Weinstein linearization compatible with the Gelfand-Zeitlin systems. More details are as follows.
\vspace{3mm}
{\bf Quantum Stokes matrices and Yang-Baxter equations}
Following Felder-Markov-Tarasov-Varchenko \cite{FMTV}, the dynamical Knizhnik--Zamolodchikov equation is
\begin{eqnarray}\label{intro:eq}
\nabla_{\DKKZ}F_\hbar:=\frac{dF_\hbar}{dz}-\left(\sfh\frac{\Omega}{z}+\ad\muone\right)F_\hbar=0,
\end{eqnarray}
where $F_\hbar (z)\in U(\g)^{\hat{\otimes}2}\fml$, $\mu\in\eta$ and the Casimir element $\Omega:=\sum e_a\otimes e_a$ for any orthogonal basis $\{e_a\}$ of $\g$.
The canonical solutions at the irregular singularity was discovered by the first author in \cite{TL}. That is for any $\mu\in\hreg^\IR$, let $\IH_\pm=\{z\in\IC|\,\Im(z)\gtrless 0\}$ be the two Stokes sectors, then there are canonical holomorphic solutions $\Upsilon_\pm$ of equation \eqref{intro:eq} on each $\IH_\pm$.
Let us define the quantum Stokes matrices $S_{\hbar\pm}\in\Ug^{\hat{\otimes}2}\fml$ by
\[\Upsilon_+=\Upsilon_-\cdot S_{\hbar+}\aand
\Upsilon_-\cdot e^{\hbar\Omega_0}=\Upsilon_+\cdot S_{\hbar-}\]
where the first identity is understood to hold in $\IH_-$ after $\Upsilon
_+$ has been continued across the ray $\IR_{\geq 0}$, and the second
in $\IH_+$ after $\Upsilon_-$ has been continued across $\IR_{\leq 0}$. Our first main theorem is
\begin{thm}
For any $u\in \frak t_{\rm reg}$, the quantum Stokes matrices $S_{\hbar\pm}$ satisfies the Yang-Baxter equation
$$S^{12}_{\hbar\pm}S^{13}_{\hbar\pm}S^{23}_{\hbar\pm}=S^{23}_{\hbar\pm}S^{13}_{\hbar\pm}S^{12}_{\hbar\pm}\in U(\g)^{\hat{\otimes}3}
\fml.$$
\end{thm}
Here if we write $S=\sum X_a\otimes Y_a$, then $S^{12}:=\sum X_a\otimes Y_a\otimes 1$, and $S^{13}:=\sum X_a\otimes 1\otimes Y_a$.
\vspace{3mm}
{\bf Quantum isomonodromy deformations}
Note that the quantum Stokes matrices $S_{\hbar\pm}$ depends on the irregular data $\mu\in\hreg^\IR$.
The dependence is studied in Section \ref{Section:quantumSmatrix}, where we prove that it is described by a quantum isomonodromic deformation equation.
\begin{prop}\label{pr:Spm}
As functions of $\mu\in\C$, the quantum Stokes matrices $S_\pm$
satisfy
\[d_\h S_{\hbar\pm}=
\frac{\sfh}{2}\sum_{\alpha\in\sfPhi_+}\frac{d\alpha}{\alpha}
\left[\onetwo{\Kalpha},S_{\hbar\pm}\right]\]
\end{prop}
Here $\Kalpha's$ are the Casimir operators. See Section \ref{Section:quantumSmatrix} for the conventions.
In Section \ref{Section:isomono}, we prove the semiclassical limit of the quantum isomonodromy equation recovers the isomonodromy equations of Jimbo--Miwa--Ueno \cite{JMU}, see also Boalch \cite{Bo2}. Along the way, we interpret the symplectic nature (Hamiltonian description) of isomonodromy deformation via the (infinitesimal) gauge transformation of the Casimir operator on quantum $R$-matrices.
\vspace{3mm}
{\bf Semiclassical limit and linearization of Poisson Lie groups}
In Section \ref{section:scl}, we show that the semiclassical limit of the dynamical $KZ$ equation is the meromorphic differential equation
\[\frac{dF}{dz}=\left(\frac{B}{z}+\ad\mu\right)F\]
where $B:\g^*\to\g$ is the linear isomorphism given by $x\to
x\otimes\id(\Omega)$.
The Stokes map, also known as Boalch's dual exponential map, is an analytic map $\calS:\g^*\rightarrow G^*$ associating any $B\in\g^*$ to the Stokes matrices of the above equation. See Section \ref{ss:Stokes mat} for more details. Then we prove that
\begin{thm}
The semiclassical limit of the quantum Stokes matrices $S_{\hbar\pm}$ of the
dynamical KZ equation are the Stokes matrices $S_{\mp}^{-1}$ of the
connection \eqref{eq:dKZ} respectively, thought of as functions
$\g^*\to B_\pm$.
\end{thm}
As an immediate consequence, see Section \ref{sclisPoisson} for more details, we recovery Boalch's remark theorem.
\begin{prop}\cite{Bo1}The Stokes map $\calS$ is a local Poisson isomorphism.
\end{prop}
Therefore we give a new interpretation of Boalch's theorem from the perspective of (quasi-)Hopf algebras.
\vspace{3mm}
{\bf Gelfand-Zeitlin systems and centralizer properties/relative Drinfeld twists}
The Stokes map $\calS$ restricts to a diffeomorphism between
the Poisson manifolds $u(n)^*$ and $U(n)^*$, known as a Ginzburg-Weinstein linearization \cite{GW}. These Poisson manifolds carry more structures: Guillemin-Sternberg \cite{GS}
introduced the Gelfand--Zeitlin integrable system on $u(n)^*$. Later on, Flaschka-Ratiu \cite{FR} described
a multiplicative Gelfand-Zeitlin system for the dual Poisson Lie group $U(n)^*$.
However, the Stokes map $\calS$ in general is not compatible with the Gelfand-Zeitlin systems. As noted by Boalch \cite{Bo1}
\vspace{3mm}
{\em "Note that the hope of \cite{FR}, that the property of fixing a positive Weyl chamber
would pick out a distinguished Ginzburg-Weinstein isomorphism, does
not hold: the dependence of the monodromy map on the irregular type is
highly non-trivial."}
\vspace{3mm}
Thus the task is to pick out a distinguished Ginzburg-Weinstein isomorphism which intertwines the Gelfand-Zeitlin systems.
In Section \ref{GZsystems}, we will do it using the centralizer property of the differential twist constructed by the first author in \cite{TL}. Geometrically, {\bf mention its relation with the DCP/wonderful compactification of $\eta_{\rm reg}$.}
To be more precise, set $\g:={\rm gl}_n$ and let $\g_n\subset\cdot\cdot\cdot \g_1\subset \g_0=\g$ be the Gelfand-Zeitlin chain. The centralizer property of the differential twists allows us to define a relative twist $C_{\g_i}$ with respect to each pair $\g_i\subset \g_{i-1}$ for $1\le i\le n$. Then the semiclassical limit of each $C_{\g_i}$ gives rise to a map $C_i:\g^*\rightarrow G$ (defined on a dense subset of $\g^*$ containing ${\rm Herm}_n\subset \g^*$ which is good enough for our purpose).
\begin{thm}
Define $\Gamma:=C_n\cdot\cdot\cdot C_1$ as the pointwise multiplication of all the map $C_i's$. Then the composition $${\rm Ad}_\Gamma \circ {\rm exp} : {\rm Herm}(n)\cong u(n)^*\rightarrow {\rm Herm}^+(n)\cong U(n)^*$$
is a Poisson
diffeomorphism compatible with the Gelfand-Zeitlin systems.
\end{thm}
Here ${\rm Herm}(n)$ ( ${\rm Herm}^+(n)$) denotes the set of (positive definite) Hermitian n by n matrices, which
is naturally isomorphic to the dual Lie algebra $u(n)^*$
(dual Poisson Lie group $U(n)^*$). See Section \ref{GZsystems} for more details.
The existence of a Ginzburg-Weinstein linearizatioin compatible with the Gelfand-Zeitlin systems was conjectured by Flaschka-Ratiu \cite{FR}, first proved
by Alekseev-Meinrenken \cite{AM} and recently proved by the second author \cite{Xu} using different approaches.
\subsection*{Acknowledgements}
\noindent
We would like to thank Anton Alekseev and Pavel Etingof for their helpful discussions and useful comments.
\subsection{Linearisation of Poisson--Lie groups}
\subsection{Acknowledgements}
We would like to thank Anton Alekseev and Pavel Etingof for their helpful discussions and useful comments.
}
\section{Stokes phenomena and \PL groups} \label{se:G Stokes}
\Omit{
Use a different letter from G: G is ss/reductive, but we will be linearising G* NOT G.
Let $G$ be a Poisson--Lie group, that is a Lie group endowed with a Poisson
structure, such that group multiplication $G\times G\to G$ is a Poisson map.
Induces Lie bialgebra structure on g.
This section reviews various instances of linearisation of PL structures, that is....
Mention first result due to Weinstein: non constructive. Then mention Alekseev
maybe (also non-construvie). Then say what are going to review: 1) Boalch
result for G ss using Stokes phenomena (based on the interpretatoon of g*,
and G* as de Rham/Betti moduli spaces of irregular connections on IP^1
2) EEM: applies to any qt Lie bialgebra. But what exactly are they linearising:
G or G* (not the same thing, the first is a qt PL group, not the second)
4. Xu's result: another linearisation of G using the connection matrix.
So context is Boalch's, but are relyong on the EEM ODE
Note somewhere that the dual of a qt Lie bialgebra is *not* a qt Lie bialgebra
}
\subsection{$G$--valued irregular connections on $\IP^1$}\label{ss:G Stokes}
Let $G$ be an affine algebraic group defined over $\IC$, $H\subset G$
a maximal torus, and $\h\subset\g$ the Lie algebras of $H$ and $G$
respectively. Let $\sfPhi\subset\h^*$ be the set of roots of $\g$ relative
to $\h$, and $\hreg=\h\setminus\bigcup_{\alpha\in\sfPhi}\Ker\alpha$
the set of regular elements in $\h$.
Let $\P$ be the holomorphically trivial, principal $G$--bundle on
$\IP^1$, and consider the meromorphic connection $\nabla$ on
$\P$ given by
\begin{equation}\label{eq:dKZ}
\nabla=d-\left(\frac{\Az}{z^2}+\frac{B}{z}\right)dz.
\end{equation}
where $\Az,B\in\g$. We assume henceforth that $\Az\in\hreg$.
By definition, the {\it Stokes rays} of $\nabla$ are the rays $\IR
_{>0}\cdot\alpha(\Az)\subset\IC^*$, $\alpha\in\sfPhi$, that is the
rays through the non--zero eigenvalues of $\ad(A)$.
A ray $r$ is called {\it admissible} if it is not a Stokes ray.
\subsection{Canonical fundamental solutions}
To each admissible ray $r$, and determination of $\log z$, there
is a canonical fundamental solution $\csol_r$ of $\nabla$ with
prescribed asymptotics in the open half--plane
\[\halfplane_r=\left\{ue^{\iota\phi}|\,u\in r, \phi\in(-\pi/2,\pi/2)\right\}\]
Specifically, the following result is proved in \cite{BJL} for $G=GL
_n(\IC)$, in \cite{Bo2} for $G$ reductive, and in \cite{BTL2} for an
arbitrary affine algebraic group.\footnote{We use the formulation
of \cite{BTL2}, which does not rely on formal power series solutions.}
Denote by $[B]$ the projection of $B$ onto $\h$ corresponding to
the root space decomposition $\g=\h\bigoplus_{\alpha\in\sfPhi}\g_\alpha$.
\begin{thm}\label{jurk}
Let $r=\IR_{>0}\cdot e^{\iota\theta}$ be an admissible ray. Then,
there is a unique holomorphic function $\ch_r:\halfplane_r\to G$
such that
\begin{enumerate}
\item $\ch_r$ tends to 1 as $z\to 0$ in any closed sector of
$\halfplane_r$ of the form
\[|\arg(z\cdot e^{-\iota\theta})|\leq \frac{\pi}{2}-\delta,\qquad\delta>0\]
\item For any determination of $\log z$ with a cut along the ray $c$,
the function
\[\csol_r=\ch_r\cdot e^{-\Az/z}\cdot z^{[B]}\]
where $z^{[B]}=\exp([B]\log z)$, satisfies $\nabla\csol_r=0$ on
$\IH_r\setminus c$.
\end{enumerate}
\end{thm}
\subsection{Stokes phenomena}
For a given determination of $\log z$, with a cut along a ray $\cut$,
the canonical solution $\csol_r$ is locally constant \wrt the choice
of $r$, so long as $r$ does not cross a Stokes ray. More precisely,
the following holds. For any subset $\Sigma\subset\IC$, let $\g_
\Sigma\subseteq\g$ be the direct sum of the eigenspaces of
$\ad(A)$ corresponding to the eigenvalues contained in $\Sigma$,
\[\g_\Sigma=\bigoplus_{\substack{\alpha\in\sfPhi\sqcup\{0\}:\\ \alpha(\Az)\in\Sigma}}\g_\alpha\]
where $\g_0=\h$. Note that $[\g_{\Sigma_1},\g_{\Sigma_2}]
\subseteq\g_{\Sigma_1+\Sigma_2}$. In particular, if $\Sigma$ is
an open convex cone, $\g_\Sigma$ is a nilpotent subalgebra of
$\g$.
\begin{prop}\label{pr:r and r'}
Let $r,r'$ be admissible rays such that $r\neq-r'$, so that $\halfplane
_r\cap\halfplane_{r'}\neq\emptyset$, and denote by $\ol{\Sigma}(r,r')
\subset\IC^\times$ the closed convex cone bounded by $r$ and $r'$.
Let
\[S:\halfplane_r\cap\halfplane_{r'}\setminus\cut\longrightarrow G\]
be the locally constant function defined by $\csol_r=\csol_{r'}\cdot S$.
Then, the following holds.
\begin{enumerate}
\item $S$ takes values in the unipotent elements of $G$,
and $\log S$ in the nilpotent subalgebra $\g_{\ol{\Sigma}
(r,r')}$.
\item In particular, if $\ol{\Sigma}(r,r')$
does not contain any Stokes rays, the solutions $\csol_r$ and
$\csol_{r'}$ coincide on $\halfplane_r\cap\halfplane_{r'}\setminus\cut$.
\end{enumerate}
\end{prop}
\begin{pf}
The asymptotic behaviour of $\csol_r$ and $\csol_{r'}$ implies that
\begin{equation}\label{eq:S asy}
z^{[B]}\cdot e^{-\Az/z}\cdot S\cdot e^{\Az/z}\cdot z^{-[B]}
=
\left(\gamma_{r'}\cdot e^{\Az/z}\cdot z^{-[B]}\right)^{-1}\cdot\gamma_r\cdot e^{\Az/z}\cdot z^{-[B]}
\to 1
\end{equation}
as $z\to 0$ along any ray $\rho$ in $\halfplane_r\cap \halfplane
_{r'}\setminus\cut$.
By \cite[Lemma 6]{Bo2} and \cite[Prop. 6.3]{BTL2}, the restriction
of $S$ to $\rho$ is unipotent, and $\log S$ lies in $\g_{\halfplane
_\rho}$.
Up to a permutation, we may assume that the counterclockwise
angle from $r$ to $r'$ is less than $\pi$, so that $\IH_r\cap\IH_{r'}$
is the open convex cone bound by the rays $r'e^{-i\pi/2}$ and $re^
{i\pi/2}$.
If the cut $\cut$ is not contained in $\halfplane_r\cap\halfplane_{r'}$,
$S$ takes a single value. Since the intersection of the half--planes
$\halfplane_\rho$ as $\rho$ varies in $\halfplane_r\cap\halfplane_{r'}$
is the closed convex cone bounded by $r$ and $r'$, it follows that
$\log S\in\g_{\ol{\Sigma}(r,r')}$.
If, on the other hand, $\cut$ disconnects $\halfplane_r\cap\halfplane
_{r'}$ into two open cones $\Sigma_<,\Sigma_>$, listed in counterclockwise
order, then $\csol_r=\csol_{r'}\cdot S_\lessgtr$ on $\Sigma_\lessgtr$,
for some $S_\lessgtr\in G$.
The previous argument
then shows that $S_\lessgtr$ are unipotent, and that
\[\log S_<\in\g_{\ol{\Sigma}(e^{-\iota\pi/2}c,r')}
\aand
\log S_>\in\g_{\ol{\Sigma}(r,e^{\iota\pi/2}c)}\]
Analytic continuation across $\cut$ implies that $S_>=e^{2\pi\iota[B]}
\cdot S_<\cdot e^{-2\pi\iota[B]}$. Since any $\g_\Sigma$ is stable under
$\Ad(e^{2\pi\iota[B]})$, this implies that
\[\log S_\lessgtr\in
\g_{\ol{\Sigma}(e^{-\iota\pi/2}c,r')}\cap
\g_{\ol{\Sigma}(r,e^{\iota\pi/2}c)}
=\g_{\ol{\Sigma}(r,r')}\]
\end{pf}
\subsection{Stokes data}
For any two rays $r,r'$, let $\sect(r,r')\subset\IC^\times$ be the (not
necessarily convex) closed sector swept by $e^{\iota\theta}\cdot r$,
as $\theta$ ranges from $0$ to the positive angle between $r$ and
$r'$. If $r,r'$ are admissible, and different from the log cut $\cut$,
define an element $S_{r'r}\in G$ by the identity
\[\left.\wt{\left.\csol_{r}\right|_{r}}\right|_{r'}=
\left.\csol_{r'}\right|_{r'}\cdot S_{r',r}\cdot e^{2\pi\iota[B]\epsilon^\cut_{r'r}}\]
where the \lhs is the \cc analytic continuation to $r'$ of the restriction
of $\csol_r$ to $r$, and $\epsilon_{r'r}^\cut$ is $1$ if $\cut$ lies
in $\sect(r,r')$, and $0$ otherwise.
\begin{prop}\label{pr:uni and fact}
The following holds
\begin{enumerate}
\item If the positive angle formed by $r$ and $r'$ is at most $\pi$, $S_{r'r}$
is unipotent, and its logarithm lies in the nilpotent subalgebra $\g_{\sect(r,r')}$.
\item If the admissible ray $r'\neq c$ lies in $\sect(r,r'')$, the following
factorisation holds
\[ S_{r''r}=
S_{r''r'}\cdot e^{2\pi\iota[B]\epsilon^\cut_{r''r'}}\cdot S_{r'r}\cdot e^{-2\pi\iota[B]\epsilon^\cut_{r''r'}}\]
\end{enumerate}
\end{prop}
\begin{pf}
(1) Let $\ell$ be a ray in $\IH_r\cap\IH_{r'}\setminus\cut$. Then
\[\left.\wt{\left.\csol_r\right|_r}\right|_\ell=\left.\csol_r\right|_\ell\cdot e^{2\pi\iota[B]\cdot\epsilon^\cut_{\ell r}}
\aand
\left.\wt{\left.\csol_{r'}\right|_\ell}\right|_{r'}=\left.\csol_{r'}\right|_{r'}\cdot e^{2\pi\iota[B]\cdot\epsilon^\cut_{r'\ell}}\]
By Proposition \ref{pr:r and r'}, $\left.\csol_r\right|_{\ell}=\left.\csol_{r'}
\right|_{\ell}\cdot S$, where $S\in G$ is a unipotent element whose log
lies in $\g_{\ol{\Sigma}(r,r')}$. Computing analytic continuation in stages
yields
\[\begin{split}
\left.\wt{\left.\csol_r\right|_r}\right|_{r'}
&=
\left.\wt{\left.\wt{\left.\csol_r\right|_r}\right|_{\ell}}\right|_{r'}
=
\left.\wt{\left.\csol_r\right|_\ell}\right|_{r'}\cdot e^{2\pi\iota[B]\cdot\epsilon^\cut_{\ell r}}\\
&=
\left.\wt{\left.\csol_{r'}\right|_\ell}\right|_{r'}\cdot S\cdot e^{2\pi\iota[B]\cdot\epsilon^\cut_{\ell r}}
=
\left.\csol_{r'}\right|_{r'}\cdot e^{2\pi\iota[B]\cdot\epsilon^\cut_{r'\ell}}\cdot S\cdot e^{2\pi\iota[B]\cdot\epsilon^\cut_{\ell r}}\\
&=
\left.\csol_{r'}\right|_{r'}\cdot \Ad(e^{2\pi\iota[B]\cdot\epsilon^\cut_{r'\ell}})(S)\cdot e^{2\pi\iota[B]\cdot\epsilon^\cut_{r'r}}
\end{split}\]
so that $S_{r'r}=\Ad(e^{2\pi\iota[B]\cdot\epsilon^\cut_{r'\ell}})(S)$.
(2) Computing analytic continuation from $r$ to $r''$ in stages yields
\[\begin{split}
\left.\wt{\left.\csol_r\right|_r}\right|_{r''}
&=
\left.\wt{\left.\wt{\left.\csol_r\right|_r}\right|_{r'}}\right|_{r''
=
\left.\wt{\left.\csol_{r'}\right|_{r'}}\right|_{r''}\cdot S_{r'r}\cdot e^{2\pi\iota[B]\epsilon^\cut_{r'r}}\\
&=
\left.\csol_{r''}\right|_{r''}\cdot S_{r''r'}\cdot e^{2\pi\iota[B]\epsilon^\cut_{r''r'}}\cdot S_{r'r}\cdot e^{2\pi\iota[B]\epsilon^\cut_{r'r}}
\end{split}\]
Since the result is also equal to $\left.\csol_{r''}\right|_{r''}\cdot S_{r''r}\cdot
e^{2\pi\iota[B]\epsilon^\cut_{r''r}}$, the result follows.
\end{pf}
\subsection{Stokes factors}
Given a Stokes ray $\ell$, the {\it Stokes factor} $S_\ell$ is the
unipotent element of $G$ defined by $S_\ell=S_{r'r}$, where $
r,r'\neq\cut$ are admissible rays such that $\sect(r,r')$ contains
no other Stokes rays than $\ell$, and does not contain the cut
$\cut$ if the latter is different from $\ell$. By Proposition \ref
{pr:uni and fact}, the definition of $S_\ell$ is independent of
the choice of $r,r'$. The following is a direct consequence of
Proposition \ref{pr:uni and fact}.
\begin{prop}\label{pr:clockwise}
The following holds
\begin{enumerate}
\item If $c$ does not lie in $\sect(r,r')$, then
\[S_{r'r}=\ccprod_{\ell}S_\ell\]
where $\ell$ ranges over the Stokes rays contained in $\sect(r,r')$, and
$S_\ell$ is placed to the left of $S_{\ell'}$ if $\ell$ is contained in $\sect
(\ell',r')$.
\item If $c$ lies in $\sect(r,r')$, then
\[S_{r'r}=\ccprod_{\ell}S_\ell\cdot e^{2\pi\iota[B]}\cdot \ccprod_{\ell}S_\ell\cdot e^{-2\pi\iota[B]}\]
where the leftmost product ranges over the Stokes rays contained in
$\sect(c,r')$, and the rightmost one over those contained in $\sect(r,c)$
except for $c$ is the latter is a Stokes ray.
\end{enumerate}
\end{prop}
\subsection{Stokes matrices}\label{ss:Stokes mat class}
Let $r$ be a ray such that both $\pm r$ are admissible, and distinct from
the log cut $\cut$. Assume further that $\cut$ lies in the cone $\sect(-r,r)$.
\comment{This last condition is only necessary so that the monodromy
relation comes out looking nicer.}
By definition, the {\it Stokes matrices} $S^r_\pm$ are the unipotent elements
of $G$ defined by
\[S^r_+=S_{-r\,r}\aand S^r_-=S_{r\,-r}\]
The pair $(A,r)$ determines a partition $\sfPhi=\sfPhi_+\sqcup\sfPhi_-$ of
the root system given by $\sfPhi_\pm=\{\alpha\in\sfPhi|\,\alpha(A)\in\sect
(\pm r,\mp r)\}$. By Proposition \ref{pr:uni and fact}, the Stokes matrices
$S^r_+,S^r_-$ lie in $N_+,N_-$ respectively, where $N_\pm=N_\pm(A,r)
\subset G$ is the unipotent subgroup with Lie algebra $\n_\pm=\bigoplus
_{\alpha\in\sfPhi_\pm}\g_\alpha$. Moreover, if $A$ is fixed, the Stokes
matrices $S^r_\pm$ (and the subgroups $N_\pm$) are locally constant
in $r$, so long as $\pm r$ do not cross a Stokes ray or $\cut$.
\subsection{Connection matrix}\label{ss:connection}
Recall that the connection $\nabla$ is said to be {\it non--resonant} at
$z=\infty$ if none of the eigenvalues of $\ad(B)$ are positive integers.
The following is well--know (see, \eg \cite{Wasow} for $G=GL_n(\IC)$).
\begin{lemma}\label{le:nr dkz}
If $\nabla$ is non--resonant, there is a unique holomorphic function
$\chh_\infty:\IP^1\setminus\{0\}\to G$ such that $\chh_\infty(\infty)=1$ and,
for any determination of $\log z$, the function $\csol_\infty=\chh_\infty
\cdot z^{B}$ is a solution of $\nabla\gamma_\infty=0$.
\end{lemma}
Fix a log cut $\cut$ and, for any admissible ray $r$ distinct from $\cut$,
define the {\it connection matrix} $C_r\in G$ by
\[\csol_\infty=\csol_r\cdot C_r\]
where the identity is understood to hold on $r$. By Proposition \ref
{pr:r and r'}, $C_r$ is locally constant \wrt $r$, so long as $r$ does
not cross a Stokes ray or $\cut$.
\subsection{Monodromy relation}
The connection matrix $C_r$ is related to the Stokes matrices $S_
\pm^r$ by the following {\it monodromy relation}.
\begin{prop}\label{pr:monodromy reln}
The following holds
\[ C_r\cdot e^{2\pi\iota B}\cdot C_r^{-1}=S^r_-\cdot e^{2\pi\iota[B]}\cdot S^r_+\]
\end{prop}
\begin{pf}
By definition of $S^r_\pm$, the monodromy of $\csol_r$ around a
positive loop $p_0$ around $0$ based at a point $z_0\in r$
is the \rhs of the stated identity. On the other hand, the monodromy of
$\csol_\infty$ around $p_0$ is $e^{2\pi\iota B}$. Since $\csol
_r=\csol_\infty\cdot C_r^{-1}$, the former monodromy is
conjugate to the latter by $C_r$.
\end{pf}
\Omit{
Let $\g\nr\subset\g$ be the dense open subset consisting of elements
$B$ such that none of the eigenvalues of $\ad(B)$ are positive integers.
Consider the map $C:\g\nr\to G$ given by
mapping $B$ to the connection matrix of $\nabla=d-\left(\Az/z^2+
B/z\right)$.
\comment{Remove this paragraph? If keep it need to make sure that
make it clear that $A$ is fixed.}
}
\subsection{The Stokes map}\label{ss:stokes map}
Let $N_\pm\subset G$ be the unipotent subgroups corresponding
to $(A,r)$, and $B_\pm=H\ltimes N_\pm\subset G$ the solvable
subgroups with Lie algebras $\b_\pm=\h\ltimes\n_\pm$. Consider
the fibred product
\[B_+\times_H B_-=\{(b_+,b_-)\in B_+\times B_-|\, \pi_+(b_+)\pi_-(b_-)=1\}\]
where $\pi_\pm:B_\pm\to H$ are the quotient maps. Following \cite
{Bo2}, we define the {\it Stokes map} to be the analytic map $\calS_r:
\g\longrightarrow B_+\times_H B_-$ given by
\[B\longrightarrow
\left((S^r_+)^{-1}\cdot e^{-\iota\pi[B]},
S^r_-\cdot e^{\iota\pi[B]}\right)\]
Note that $B_-\times_H B_+$ maps to $G$ via the map
$\beta:(b_+,b_-)\to b_+\cdot b_-^{-1}$.\comment{$\beta$ is a principal
bundle over its image with structure group the order two elements in
$H$.} Moreover, by Proposition \ref{pr:monodromy reln}, the composition
$\beta\circ\calS_r$ is the map $\g\to G$ given by
\[B\longrightarrow
(S^r_+)^{-1}\cdot e^{-2\pi\iota[B]}\cdot (S^r_-)^{-1}=
C_r\cdot e^{-2\pi\iota B}\cdot C_r^{-1}\]
\subsection{Linearisation of $G^*$}\label{ss:linearisation}
Assume now that $G$ is reductive, and fix a symmetric, non--degenerate,
invariant bilinear form $(\cdot,\cdot)$ on $\g$. The pair of opposite Borel
subalgebras $\b_\pm$ of $\g$ then gives rise to a solution $\sfr\in\b_-\otimes
\b_+$ of the classical \YBE given by
\begin{equation}\label{eq:standard cybe}
\sfr = x_i\otimes x^i + \half{1} t_a\otimes t^a
\end{equation}
where $\{x_i\},\{x^i\}$ are bases of $\n_-,\n_+$ respectively which are
dual \wrt $(\cdot,\cdot)$, and $\{t_a\},\{t^a\}$ are dual bases of $\h$.
The element $\sfr$ gives $\g$ the structure of a \qt Lie bialgebra, with
cobracket $\delta:\g\to\g\wedge\g$ given by $\delta(x)=[x\otimes 1+1
\otimes x,\sfr]$.
The dual Lie bialgebra $(\g^*,\delta^t,[\cdot,\cdot]^t)$ may be identified, as
a Lie algebra, with
\[\b_+\times_\h\b_-=\{(X_+,X_-)\in\b_+\oplus\b_-|\pi_+(X_+)+\pi_-(X_-)=0\}\]
where $\pi_\pm:\b_\pm\to\h$ is the quotient map.
This endows $G^*=B_+\times_H B_-$ with the structure of a Poisson--Lie
group, which is dual to $G$.
Endow now $\g^*$ with its standard Kirillov--Kostant--Souriau Poisson
structure given by
\[ \{f,g\}(x)=\langle[d_x f,d_xg],x\rangle\]
where $d_xh\in T^*_x\g^*=\g$ is the differential of $h$ at $x$,
and $[\cdot, \cdot]$ is the Lie bracket on $\g$.
Let $\nnu:\g^*\to\g$ be the identification induced by the bilinear form
$(\cdot,\cdot)$, and set $\nnu^\vee=-1/(2\pi\iota)\nnu$. The following
remarkable result is due to Boalch \cite{Bo1,Bo2}.
\begin{thm}\label{th:Boalch}
The map $\calS\circ\nnu^\vee:\g^*\to G^*$ is a Poisson map, and
generically a local analytic diffeomorphism.
\end{thm}
In particular, $\calS\circ\nnu^\vee$ gives a linearisation of the
\PL structure on $G^*$. We shall give an alternative proof
of the fact that $\calS\circ\nnu^\vee:\g^*\to G^*$ is a Poisson map
in Section \ref{se:alt boalch}.
\section{Stokes phenomena and quantum groups} \label{se:Ug Stokes}%
This section is an exposition of \cite{TL}. We explain in particular
how the dynamical KZ equations give rise to a twist which kills the
KZ associator. Sections \ref{se:filtered V}--\ref{se:filtered A} contain
background material required to do calculus with values in infinite--dimensional
filtered vector spaces and their endomorphisms. Throughout the
paper, $\sfh,\hbar$ are two formal parameters related by $\hbar
=2\pi\iota\sfh$.
\subsection{Filtered vector spaces}\label{se:filtered V}
Let $\V$ be a vector space over a field $\sfk$ endowed with
a decreasing filtration
\[\V=\V_0\supseteq\V_1\supseteq\V_2\supseteq\cdots\]
and $\imath$ the map $\V\rightarrow\ds{\lim_{\longleftarrow}}\,
\V/\V_n$. Recall that $\V$ is said to be {\it separated} if $\imath$
is injective, and {\it complete} if $\imath$ is surjective.
If $\sfk=\IC$, and the quotients $\V/\V_n$ are finite--dimensional, we
shall say that a map $F:X\to\V$, where $X$ is a topological space
(resp. a smooth or complex manifold) is continuous (resp. smooth
or holomorphic) if its truncations $F_n:X\to\V/\V_n$ are. If $\V$ is
separated and complete, giving such an $F$ amounts to giving
continuous (resp. smooth or holomorphic) maps $F_n:X\to\V/\V_n$
such that $F_n=F_m\mod \V_m/\V_n$, for any $n\geq m$.
\subsection{Filtered endomorphisms}\label{se:filtered E}
Let $\V$ be as in \ref{se:filtered V}, and $\E\subset\End_\sfk(\V)$
the subalgebra defined by
\[\E=\left\{T\in\End_\sfk(\V)|\,T(\V_m)\subseteq\V_m,m\geq 0\right\}\]
Consider the decreasing filtration $\E=\E_0\supseteq\E_1\supseteq
\cdots$ where $\E_n\subset\E$ is the two--sided ideal given by $\E_
n=\{T\in\E|\,\Im(T)\subseteq\V_n\}$. Note that if the quotients $\V/\V
_n$ are finite--dimensional, the same holds for
\[\E/\E_n\cong
\{T\in\End_\sfk(\V/\V_n)|\,T(\V_m/\V_n)\subseteq \V_m/\V_n,\,0\leq m\leq n\}\]
In particular, if $\sfk=\IC$, we may speak of a continuous (resp. smooth,
holomorphic) map with values in $\E$.
\begin{lemma}\hfill
\begin{enumerate}
\item If $\V$ is separated, so is $\E$.
\item If $\V$ is complete, so is $\E$.
\end{enumerate}
\end{lemma}
\begin{pf}
(1) holds because $\bigcap_{n\geq 0}\E_n=\{T\in\E|\,\Im(T)\subseteq
\bigcap_{n\geq 0}\V_n\}$.
(2) Let $T_n\in\E/\E_n$ be such that $T_n=T_m\mod\E/\E_m$ for
any $n\geq m$. It suffices to find $T\in\End_\sfk(\V)$ such that
$T=T_n\mod\E_n$ for any $n\geq 0$, for it then follows that $T\in
\E$. Let $\{v_i\}_{i\in I}$ be a basis of $\V$. For any $i\in I$, $\{T_n
v_i\}$ is a well--defined element of $\lim_n\V/\V_n$. By completeness
of $\V$, there exists $u_i\in\V$ such that $u=T_n v_i\mod\V_n$
for any $n$. Setting $Tv_i=u_i$ gives the required $T$.
\end{pf}
\subsection{Filtered algebras}\label{se:filtered A}
Let $A$ be a $\sfk$--algebra endowed with an increasing algebra
filtration $\sfk=A_0\subseteq A_1\subseteq\cdots$, and $A\fml^o$
the (completed Rees) algebra given by
\[A\fml^o=
\{\sum_{k\geq 0}a_k\hbar^k\in A\fml|\,
a_k\in A_{k}\}\
Endow $A\fml^o$ with the decreasing filtration
\begin{equation}\label{eq:filtration}
A\fml^o_n=A\fml^o\cap\hbar^nA\fml
\end{equation}
\wrt which it is easily seen to be separated and complete.
Note that each $A\fml^o_n$ is a two--sided, $\IC\fml$--ideal in $A\fml^o$,
and that the quotients
\[A\fml^o/A\fml^o_n\cong
A_{0}\oplus\hbar A_{1}\oplus \cdots\oplus\hbar^{n-1} A_{n-1}\]
are \fd if $A$ is filtered by \fd subspaces.
\subsection{Example}
We shall be interested in the case when $A=\Ug^{\otimes m}$ is a tensor
power of an enveloping algebra, with filtration given by $A_k=(U\g_{\leq
k})^{\otimes m}$. Then,
\begin{equation}\label{eq:for future}
\Ug^{\otimes m}\fml^o=
\UU'\otimes\UU^{\otimes m-1}
\cap
\UU\otimes\UU'\otimes\UU^{\otimes m-2}
\cap
\cdots
\cap
\UU^{\otimes m-1}\otimes\UU'
\end{equation}
where $\UU=\Ug\fml$ and $\UU'=\Ug\fml^o$.
Note that $\Ug^{\otimes m}\fml^o\cap\Ug^{\otimes m}=\sfk$. However, if $x
\in\Ug_{\leq k}$, $i=1,\ldots,m$, and
\[x^{(i)}=1^{\otimes i-1}\otimes x\otimes 1^{\otimes m-i}\in\Ug^{\otimes m}\]
then $\hbar^{k-1}\negthinspace\ad x^{(i)}$ is a derivation of $\Ug^{\otimes
m}\fml^o$, which preserves the filtration $\Ug^{\otimes m}\fml^o_n$.
\subsection{The dynamical KZ equations}
Let now $\g$ be a complex reductive Lie algebra, $\h\subset\g$ a Cartan
subalgebra, and $(\cdot,\cdot)$ an invariant inner product on $\g$. Let
$\sfPhi=\{\alpha\}\subset\h^*$ be the root system of $\g$ relative to $\h$,
choose $x_\alpha\in\g_\alpha$ for any $\alpha\in\sfPhi$ such that $(x_
\alpha,x_{-\alpha})=1$, and set
\[\Kalpha=x_\alpha x_{-\alpha}+x_{-\alpha}x_\alpha\]
Endow $\A=\Ug^{\otimes 2}\fml^o$ with the filtration $\A_n=\Ug^{\otimes 2}
\fml^o\cap\hbar^n\Ug^{\otimes 2}$ as in \eqref{eq:filtration}, and filter $\E=
\{T\in\End_{\IC}(\A)|\,T(\A_n)\subseteq\A_n\}$ as in \ref{se:filtered E}. Since
the quotients $\A/\A_n$ and $\E/\E_n$ are finite--dimensional, we may
speak of continuous, smooth or holomorphic functions with values in
$\A$ and $\E$.
The dynamical KZ (DKZ) connection is the $\E$--valued
connection on $\IC$ given by
\begin{equation}\label{eq:DKZ}
\nabla\DKKZ=d-\left(\sfh\frac{\Omega}{z}+\ad\muone\right) dz
\end{equation}
where $\mu\in\h$, $\Omega\in\g\otimes\g$ is the invariant tensor corresponding
to $(\cdot,\cdot)$ and, given an element $a\in\A$, we abusively denote by $a$
the corresponding left multiplication operator $L(a)\in\E$.
\subsection{Fundamental solution at $z=0$} \label{ss:Upsilon zero}
\begin{prop}\label{pr:Fuchs 0}
\hfill
\begin{enumerate}
\item
For any $\mu\in\h$, there is a unique holomorphic function
$H_0:\IC\to\A$ such that $H_0(0,\mu)=1$ and, for any
determination of $\log z$, the $\E$--valued function
\[\Upsilon_0(z,\mu)=e^{z\ad\muone}\cdot H_0(z,\mu)\cdot z^{\sfh\Omega}\]
satisfies $\nabla\DKKZ\Upsilon_0=0$.
\item $H_0$ and $\Upsilon_0$ are invariant under the diagonal action of $\h$.
\item $H_0$ and $\Upsilon_0$ are holomorphic functions of $\mu$,
and $\Upsilon_0$ satisfies
\
\left(d_\h-\half{\sfh}\sum_{\alpha\in\Phi_+}\frac{d\alpha}{\alpha}
\Delta(\Kalpha)-z\ad d\muone\right)\Upsilon_0=
\Upsilon_0\left(d_\h-\half{\sfh}\sum_{\alpha\in\Phi_+}\frac{d\alpha}{\alpha}
\Delta(\Kalpha)\right)
\
\end{enumerate}
\end{prop}
\subsection{Fundamental solutions at $z=\infty$} \label{ss:Upsilon infty}
Let $\halfplane_\pm=\{z\in\IC|\,\Im(z)\gtrless 0\}$.
\begin{thm}\label{th:Stokes infty}\hfill
\begin{enumerate}
\item
For any $\mu\in\hreg^\IR$, there are unique holomorphic functions
$H_\pm:\halfplane_\pm\to\A$ such that $H_\pm(z,\mu)$ tends to $1$
as $z\to\infty$ in any sector of the form $|\arg(z)|\in(\delta,\pi-\delta)$,
$\delta>0$ and, for any determination of $\log z$, the $\E$--valued
function
\[\Upsilon_\pm(z,\mu)=H_\pm(z,\mu)\cdot z^{\sfh\Omega_0}\cdot e^{z\ad\muone}\]
satisfies $\nabla\DKKZ\Upsilon_\pm=0$.
\item $H_\pm$ and $\Upsilon_\pm$ are invariant under the diagonal action of $\h$.
\item $H_\pm$ and $\Upsilon_\pm$ are smooth functions of $\mu$,
and $\Upsilon_\pm$ satisfies
\
\left(d_\h-\half{\sfh}\sum_{\alpha\in\Phi_+}\frac{d\alpha}{\alpha}
\Delta(\Kalpha)-z\ad d\muone\right)\Upsilon_\pm=
\Upsilon_\pm\left(d_\h-\half{\sfh}\sum_{\alpha\in\Phi_+}\frac{d\alpha}{\alpha}
(\onetwo{\Kalpha})\right)
\
\end{enumerate}
\end{thm}
\subsection{Remark}
The PDEs (3) in Proposition \ref{pr:Fuchs 0} and Theorem \ref{th:Stokes infty}
do not take values in $\E$, since left multiplication by $\sfh\Delta(\Kalpha),
\sfh\Kalpha^{(1)}$ and $\sfh\Kalpha^{(2)}$ does not preserve $\A$. Let,
however, $\A\subsetneq\wt{\A}\subset\Ug^{\otimes 2}\fml$ be the Rees
algebra \wrt the laxer filtration $(\Ug^{\otimes 2})_k=\sum_{a+b=2k}\Ug_
{\leq a}\otimes\Ug_{\leq b}$,
and $\wt{\E}$ the corresponding algebra of endomorphisms. Then,
$\Upsilon_0,\Upsilon_\pm$, and left multiplication by $\sfh\Delta
(\Kalpha),\sfh\Kalpha^{(1)}$ and $\sfh\Kalpha^{(2)}$ all lie in $\wt{\E}$,
and these PDEs should be understood as holding in $\wt{\E}$.
\subsection{$\IZ_2$--equivariance}\label{ss:Z2}
Let $\U\subset\IC$ be an open subset. For any functions $F:\U\to\A$
and $G:\U\to\E$, define $F^\vee:-\,\U\to\A$ and $G^\vee:-\,\U\to\E$ by
\[F^\vee(z)=e^{z\ad(\onetwo{\mu})}(F(-z)^{21})
\quad\text{and}\quad
G^\vee(z)=e^{z\ad(\onetwo{\mu})}\cdot G(-z)^{21}\]
where $G(-z)^{21}=(1\,2)\cdot G(-z)\cdot(1\,2)$. If $F,G$ are local solutions
of the dynamical KZ equations with values in $\A$ and $\E$ respectively,
then so are $F^\vee,G^\vee$.
\begin{lemma}\label{le:Z2}
The following holds
\begin{enumerate}
\item For $z\in\halfplane_\pm$,
\[\Upsilon_0^\vee(z)=\Upsilon_0(z)\cdot e^{\mp\pi\iota\sfh\Omega}\]
\item For $z\in\halfplane_\mp$,
\[ \Upsilon_\pm^\vee(z)=\Upsilon_\mp(z)\cdot e^{\pm\pi\iota\sfh\Omega_0}\]
\end{enumerate}
\end{lemma}
\begin{pf}
(1) The uniqueness of the holomorphic part $H_0$ of $\Upsilon_0$
implies that $(e^{z\ad\muone}\cdot H_0)^\vee=e^{z\ad\muone}\cdot
H_0$. It follows that $\Upsilon_0^\vee(z)=H_0(z)\cdot (-z)^{\sfh\Omega}
=\Upsilon_0(z)\cdot e^{\mp\iota\pi\sfh\Omega_0}$ since $\log(-z)=
\log z\mp\iota\pi$, depending on whether $\Im z\gtrless 0$.
(2) Similarly, for $z\in\halfplane_\mp$,
\[\begin{split}
\Upsilon_\pm^\vee(z)
&=
e^{z\ad(\onetwo{\mu})}\cdot (1\,2)\cdot H_\pm(-z)\cdot e^{-z\ad\muone}\cdot (-z)^{\sfh\Omega_0}\cdot (1\,2)\\
&=
H_\pm^\vee(z)\cdot e^{z\ad\muone}\cdot (-z)^{\sfh\Omega_0}
\end{split}
\]
The uniqueness of $H_\pm$ implies that $H_\pm^\vee=H_\mp$,
from which the result follows.
\end{pf}
\subsection{Another $\IZ_2$--equivariance}
Let $\U\subset\IC$ be an open subset. For any functions $F:\U\to\A$
and $G:\U\to\E$, define $\wt{F}:\U\to\A$ and $\wt{G}:\U\to\E$ by
\[\wt{F}(z)=e^{-z\ad(\onetwo{\mu})}(F(z)^{21})
\quad\text{and}\quad
\wt{G}(z)=e^{-z\ad(\onetwo{\mu})}\cdot (1\,2)\cdot G(z)\cdot(1\,2)\]
If $F,G$ are local solutions of the dynamical KZ equations with parameter
$\mu\in\h$, then $\wt{F},\wt{G}$ are solutions of the DKZ equations
with parameter $-\mu$.
\begin{lemma}\label{le:another Z2}
The following holds
\[\wt{\Upsilon}_0(z;\mu)=\Upsilon_0(z;-\mu)
\aand
\wt{\Upsilon}_\pm(z;\mu)=\Upsilon_\pm(z;-\mu)\]
\end{lemma}
\begin{pf}
By definition,
\[\wt{\Upsilon}_0(z;\mu)=
e^{-z\ad(\onetwo{\mu})}\cdot e^{z\ad\mu^{(2)}}\cdot H_0^{21}(z;\mu)\cdot z^{\hbar\Omega}
=
e^{-z\ad\mu^{(1)}}\cdot H_0^{21}(z;\mu)\cdot z^{\hbar\Omega}
\]
which coincides with $\Upsilon_0(z;-\mu)$ by uniqueness. Similarly,
\begin{multline*}
\wt{\Upsilon}_\pm(z;\mu)
=
e^{-z\ad(\onetwo{\mu})}\cdot H_\pm^{21}(z;\mu)\cdot e^{z\ad\mu^{(2)}}\cdot z^{\hbar\Omega_0}\\
=
H_\pm^{21}(z;\mu)\cdot e^{-z\ad\mu^{(1)}}\cdot z^{\hbar\Omega_0}
=
\Upsilon_\pm(z;-\mu)
\end{multline*}
where the second equality uses the fact that $H_\pm$ is of weight zero, and
the third follows by uniqueness.
\end{pf}
\subsection{Differential twist}\label{ss:differential twist}
Fix henceforth the standard determination of $\log z$ with a cut
along the negative real axis, and let $\Upsilon_0,\Upsilon_\pm$
be the corresponding fundamental solutions of the dynamical KZ
equations given in \ref{ss:Upsilon zero} and \ref{ss:Upsilon infty}
respectively. We shall consider $\Upsilon_0$ and $\Upsilon_\pm$
as (single--valued) holomorphic functions on $\IC\setminus\IR_
{\leq 0}$.
\begin{defn}
The differential twist is the smooth function $J_\pm:\h\reg^\IR\to\Ug
^{\otimes 2}\fml^o$ defined by
\[J_\pm=\Upsilon_0(z)^{-1}\cdot\Upsilon_\pm(z)\]
where $z\in\IC\setminus\IR_{\leq 0}$.
\end{defn}
\begin{rem}
$J_\pm$ takes in fact values in $\E$. However, the form of
$\Upsilon_0$ and $\Upsilon_\pm$ shows that
\[J_\pm=
z^{-\sfh\Omega}\cdot H_0(z)^{-1}\cdot
\exp(-z\ad\muone)\left(H_\pm\right)\cdot z^{\sfh\Omega_0}
\]
so that it is a left multiplication operator. We therefore
abusively identify $J_\pm$ and $J_\pm(1^{\otimes 2})$.
\end{rem}
\begin{prop}\label{pr:Jpm}
The following holds
\[J_-=e^{\pi\iota\sfh\Omega}\cdot J_+^{21}\cdot e^{-\pi\iota\sfh\Omega_0}\]
\end{prop}
\begin{pf}
Let $G^\vee(z)=e^{z\ad(\onetwo{\mu})}\cdot G(-z)^{(1\,2)}$
be the involution defined in \ref{ss:Z2}. By definition, $J_+^{21}=(\Upsilon_0^\vee)^{-1}\cdot\Upsilon_+^\vee$,
where the \rhs is evaluated for $\Im z<0$. By Lemma \ref{le:Z2}, this is
equal to $e^{-\pi\iota\sfh\Omega}\cdot\Upsilon_0^{-1}\cdot\Upsilon_-
\cdot e^{\pi\iota\sfh\Omega_0}$.
\end{pf}
\subsection{}
For any $\mu\in\hreg^\IR$, set $\sfPhi_+(\mu)=\{\alpha\in\sfPhi|\alpha(\mu)>0\}$.
\begin{thm}\label{th:J}\hfill
\begin{enumerate}
\item $J_\pm$ kills the KZ associator $\Phi\KKZ\in U\g^{\otimes 3}\fml^o$,
that is
\[\Phi\KKZ\cdot\Delta\otimes\id(J_\pm)\cdot J_\pm\otimes 1
=\id\otimes\Delta(J_\pm)\cdot 1\otimes J_\pm\]
\item $J_\pm
=1^{\otimes 2}+\half{\hbar}\sfj_\pm$ mod $\hbar^2$, where
\[\sfj_\pm=
\mp\Omega_-
+\frac{1}{\pi\iota}\sum_{\alpha\in\sfPhi_+(\mu)}
(\log\alpha+\gamma)\left(\Omega_\alpha+\Omega_{-\alpha}\right)\]
with $\Omega_\alpha=x_\alpha\otimes x_{-\alpha}$, $\Omega
_\pm=\sum_{\alpha\in\sfPhi_+(\mu)}\Omega_{\pm\alpha}$, and $\gamma
=\lim_n(\sum_{k=1}^n\frac{1}{k}-\log(n))$ the Euler--Mascheroni constant.
In particular,
\begin{equation}\label{eq:J alt}
\sfj_\pm-\sfj_\pm^{21}
=
\Omega_\pm-\Omega_\mp
\end{equation}
\item As a function of $\mu\in\hreg^\IR$, $J_\pm$ satisfies
\[d_\h J_\pm=
\half{\sfh}\sum_{\alpha\in\Phi_+(\mu)}\frac{d\alpha}{\alpha}
\left(\Delta(\Kalpha)J_\pm-J_\pm(\onetwo{\Kalpha})\right)\]
\end{enumerate}
\end{thm}
\begin{rem}
Note that the PDE satisfied by $J_\pm$ is independent of the chamber
which $\mu$ lies in since $d\log\alpha=d\log(-\alpha)$ and $\Kalpha=
\mathcal K_{-\alpha}$. Note also that this PDE takes values in $\A$.
Indeed, although neither the left multiplication operator $L(\sfh\Delta(
\Kalpha))$ nor the right multiplication $R(\sfh\Kalpha^{(1)}+\sfh\Kalpha
^{(2)})$ leaves $\A$ inavariant, the fact that $\Delta(\Kalpha)=\Kalpha^{(1)}+\Kalpha
^{(2)}+2(\Omega_\alpha+\Omega_{-\alpha})$ implies that
\[L(\sfh\Delta(\Kalpha))-
R(\sfh\Kalpha^{(1)}+\sfh\Kalpha^{(2)})
=
2L(\sfh\Omega_\alpha+\sfh\Omega_{-\alpha})
+
\ad(\sfh\Kalpha^{(1)}+\sfh\Kalpha^{(2)})
\]
which preserves $\A$ since $\sfh\Omega_\alpha
\in\A$, and $\ad(\sfh\Kalpha^{(i)})$ leave $\A$ invariant by \ref{se:filtered A}.
\end{rem}
\Omit{
Give 1--jet of solutions at 0 and infty
Relation btw J_21(mu) and J(-mu)}
\subsection{Quantisation of $(\g,\sfr)$}
Fix a chamber $\C\subset\h\reg^\IR$, and set $\sfPhi_+=\sfPhi_+(\mu)$,
$\mu\in\C$. Let
\[\sfr=
\Omega_++\half{1}\Omega_0=
\sum_{\alpha\in\sfPhi_+}x_\alpha\otimes x_{-\alpha}+\half{1}\Omega_0\]
be the Drinfeld--Sklyanin $r$--matrix corresponding to $\C$, and $(\g,\sfr)$
the corresponding \qt Lie bialgebra.\footnote{Note that the $\sfr$ considered
in \ref{ss:linearisation} corresponds to the {\it anti}fundamental chamber.}
Set $R\KKZ=e^{\hbar\Omega/2}$, and let
\[\left(U\g\fml,\Delta_0,R\KKZ,\Phi\KKZ\right)\]
be the \qtqha structure on $\Ug\fml$ underlying the monodromy of the KZ
equations \cite{Dr4}, where $\Delta_0$ is the standard cocommutative
coproduct on $U\g$. If $\mu\in\C$, the differential twist $J_\pm=J_\pm(\mu)$
allows to twist this structure, and yields a \qt Hopf algebra $(U\g\fml,\Delta
_\pm,R_\pm)$, where\footnote{Note that $\Delta_\pm$ and $R_\pm$
depend on the additional choice of $\mu\in\C$. Specifically, if $\mu_0,
\mu_1\in\C$, $p:[0,1]\to\C$ is a path with $p(0)=\mu_0,p(1)=\mu_1$,
and $a_p\in\Ug\fml_0$ is the holonomy of the Casimir connection
along $p$, then
\[\Delta_\pm(x)(\mu_1)=
a_p^{\otimes 2}\cdot\Delta_\pm(a_p^{-1}xa_p)(\mu_0)\cdot(a_p^{\otimes 2})^{-1}
\aand
R_\pm(\mu_1)=a_p^{\otimes 2}\cdot R_\pm(\mu_0)\cdot(a_p^{\otimes 2})^{-1}\]
In particular, the \qt Hopf algebras corresponding to different values
of $\mu\in\C$ are all isomorphic.}
\[\Delta_\pm(x)=J_\pm^{-1}\cdot\Delta_0(x)\cdot J_\pm
\aand
R_\pm=(J_\pm^{-1})^{21}\cdot R\KKZ\cdot J_\pm\]
\begin{thm}\hfill
\begin{enumerate}
\item $(U\g\fml,\Delta_+,R_+)$ is a quantisation of $(\g,\sfr)$.
\item $(U\g\fml,\Delta_-,R_-)$ is a quantisation of $(\g,\sfr^{21})$.
\item $(U\g\fml,\Delta_\pm,R_\pm)$ is isomorphic, as a \qt Hopf
algebra, to the Drinfeld--Jimbo quantum group corresponding to
$\g$.
\end{enumerate}
\end{thm}
\begin{pf}
(1)--(2) By \eqref{eq:J alt}, the coefficient of $\hbar$ in $R_\pm$ is
$\half{1}(\Omega\pm\Omega_+\mp\Omega_-)$, which is equal to
$\sfr$ for $R_+$ and $\sfr^{21}$ for $R_-$.
(3) This follows, for example, from Drinfeld's uniqueness of the
quantisation of $(\g,\sfr)$ \cite{DrICM} given that the Chevalley
involution of $\g$ clearly lifts to $(U\g\fml,\Delta_\pm,R_\pm)$.
\end{pf}
\begin{rem} It follows from (4) of Theorem \ref{th:J} that
\begin{equation}\label{eq:Rpm}
R_-=R\KKZ^0\cdot R_+^{21}\cdot (R\KKZ^0)^{-1}
\end{equation}
\end{rem}
\section{The $R$--matrix as a quantum Stokes matrix} \label{se:R = S}%
\subsection{Quantum Stokes matrices}\label{ss:quant Stokes mat}
Recall that $\halfplane_\pm=\{z\in\IC|\,\Im(z)\gtrless 0\}$. Define the quantum
Stokes matrices $S_\pm\in\Ug^{\otimes 2}\fml^o$ by
\[\Upsilon_+=\Upsilon_-\cdot S_+\aand
\Upsilon_-\cdot e^{\hbar\Omega_0}=\Upsilon_+\cdot S_-\]
where the first identity is understood to hold in $\halfplane_-$ after $\Upsilon
_+$ has been continued across the ray $\IR_{\geq 0}$, and the second
in $\halfplane_+$ after $\Upsilon_-$ has been continued across $\IR_{\leq 0}$.
\begin{prop}\label{pr:Spm}
The following holds
\begin{enumerate}
\item $S_-=
e^{-\iota\pi\sfh\Omega_0}\cdot
S_+^{21}\cdot
e^{\iota\pi\sfh\Omega_0}$.
\item
$J_+^{-1}\cdot e^{2\pi\iota\sfh\Omega}\cdot J_+=
S_+^{-1}\cdot e^{2\pi\iota\sfh\Omega_0}\cdot S_-^{-1}$
\item As functions of $\mu\in\C$, the quantum Stokes matrices $S_\pm$
satisfy
\[d_\h S_\pm=
\frac{\sfh}{2}\sum_{\alpha\in\sfPhi_+}\frac{d\alpha}{\alpha}
\left[\onetwo{\Kalpha},S_\pm\right]\]
\end{enumerate}
\end{prop}
\begin{pf}
(1) Let $f$ be a holomorphic function on $\halfplane_\pm$, and denote by
$\P_\pm(f)$ the analytic continuation of $f$ to $\halfplane_\mp$ across the
half--axis $\IR_{\gtrless 0}$. By Lemma \ref{le:Z2}, and the definition
of $S_-$,
\[\P_-(\Upsilon_+^\vee)
=\P_-(\Upsilon_-)\cdot e^{\iota\pi\sfh\Omega_0}
=\Upsilon_+\cdot S_-\cdot e^{-\iota\pi\sfh\Omega_0}\]
On the other hand, if $\imath:\IC\to\IC$ is the inversion $z\to -z$,
\[\begin{split}
\P_-(\Upsilon_+^\vee)
&=
e^{z\ad(\onetwo{\mu})}\cdot(1\,2)\cdot \P_-(\Upsilon_+\circ\imath)\cdot (1\,2)\\
&=
e^{z\ad(\onetwo{\mu})}\cdot(1\,2)\cdot \P_+(\Upsilon_+)\circ\imath\cdot (1\,2)\\
&=
e^{z\ad(\onetwo{\mu})}\cdot(1\,2)\cdot \Upsilon_-\circ\imath\cdot S_+\cdot (1\,2)\\
&=
\Upsilon_-^\vee\cdot S^{21}_+\\
&=
\Upsilon_+\cdot e^{-\iota\pi\sfh\Omega_0}\cdot S^{21}_+
\end{split}\]
where the last identity uses Lemma \ref{le:Z2}.
(2) By construction, the monodromy of the fundamental solution $\Upsilon
_0$ around a positively oriented loop $\gamma_0$ around $0$ is $e^{2\pi
\iota\sfh\Omega}$. Let now $\gamma_\infty$ be a clockwise loop around
$\infty$ based at $x_0\in\halfplane_+$. Since such a loop crosses the negative
real axis before the positive one, the monodromy of $\Upsilon_+$ around
$\gamma_+$ is $S_+^{-1}\cdot e^{2\pi\iota\sfh\Omega_0}\cdot S_-^{-1}$.
The result now follows from the fact that $\gamma_\infty$ is homotopic to
$\gamma_0$, and $\Upsilon_+=\Upsilon_0\cdot J_+$.
(3) follows from the PDE satisfies by $\Upsilon_0$ and $\Upsilon_\pm$.
\end{pf}
\subsection{The $R$--matrix as a quantum Stokes matrix}
\begin{thm}
The following holds
\[R_+
=e^{\pi\iota\sfh\Omega_0}\cdot S_-^{-1}
\aand
R_-
=e^{\pi\iota\sfh\Omega_0}\cdot S_+^{-1}
\]
\end{thm}
\begin{pf}
By definition of $S_+$, $\Upsilon_+=\Upsilon_-\cdot S_+$, when both
$\Upsilon_\pm$ are considered as single--valued functions on $\IC
\setminus\IR_{\leq 0}$. On the other hand, by definition of $J_\pm$,
\[\Upsilon_+=\Upsilon_0\cdot J_+=\Upsilon_-\cdot J_-^{-1}\cdot J_+\]
Using Proposition \ref{pr:Jpm} therefore yields
\[S_+=
e^{\iota\pi\sfh\Omega_0}\cdot(J_+^{-1})^{21}\cdot e^{-\iota\pi\sfh\Omega}\cdot J_+=
e^{\iota\pi\sfh\Omega_0}\cdot(R_+^{-1})^{21}\]
where the last equality uses the fact that $R\KKZ=\exp(\pi\iota\sfh\Omega)
=R\KKZ^{21}$. The first stated identity now follows from (1) of Proposition
\ref{pr:Spm}. The second one follows from the first and \eqref{eq:Rpm}.
\end{pf}
\section{Quantum duality principle and semiclassical limits} \label{se:quantum duality}
\subsection{Quantised universal enveloping algebras}
Let $\sfk$ be a field of characteristic zero, and $\UU$ a quantised universal
enveloping algebra (QUE) over $\sfk$, that is a topologically free Hopf algebra
over $\sfk[[\hbar]]$ such that $\UU/\hbar\UU$ is isomorphic to the enveloping
algebra $U\g$ of a Lie algebra $\g$ over $\sfk$. Then, $\UU$ induces a Lie
bialgebra structure on $\g$, with cobracket $\delta:\g\to\g\otimes\g$ given by
\[\delta(x)=\left.\frac{\Delta(\wt{x})-\Delta^{21}(\wt x)}{\hbar}\right|_{\hbar=0}\]
where $\wt{x}\in\UU$ is an arbitrary lift of $x$.
\subsection{The algebra $\UU'$}
Let $\eta:\IC\fml\to\UU$ and $\eps:\UU\to\IC\fml$ be the unit and counit,
respectively. $\UU$ splits as $\Ker(\eps)\oplus\IC\fml\cdot 1$, with projection
onto the first summand given by $\pi=\id-\eta\circ\eps$. Let $\Delta^{(n)}:
\UU\to\UU^{\otimes n}$ be the iterated coproduct recursively defined by
$\Delta^{(0)}=\eps$, $\Delta^{(1)}=\id$, and $\Delta^{(n)}=\Delta\otimes
\id^{\otimes (n-2)}\circ\Delta^{(n-1)}$ for $n\geq 2$.
Following Drinfeld, define the subspace $\UU'\subset\UU$ by \cite{DrICM,
Ga}
\[\UU'=
\left\{x\in\UU\left|\, \pi^{\otimes n}\circ\Delta^{(n)}(x)\in\hbar^n\UU^{\otimes n},\,n\geq 1\right.\right\}\]
The definition of $\UU'$ extends that of the completed Rees algebra of $\Ug$
to an arbitrary QUE. Specifically, the following holds.
\begin{lemma}\label{le:example}
If $\UU=U\g[[\hbar]]$ with undeformed coproduct, then
$x=\sum_{n\geq 0}x_n\hbar^n$ lies in $\UU'$ if, and only if the filtration
order of $x_n$ in $U\g$ is less than or equal to $n$.
\end{lemma}
\begin{pf}
It is easy to see that, for any $x_1,\ldots,x_k\in\g$
\[\pi^{\otimes n}\circ\Delta^{(n)}(x_1\cdots x_k)=
\sum_{\substack{I_1\sqcup\cdots\sqcup I_n=\{1,\ldots,k\}\\ |I_i|\neq 0}}
x_{I_1}\otimes\cdots\otimes x_{I_n}
\]
where, for any $I=\{i_1,\ldots,i_m\}$, with $i_1<\cdots<i_m$, we set $x_I=
x_{i_1}\cdots x_{i_m}$. In particular, $\pi^{\otimes n}\circ\Delta^{(n)}(x_1
\cdots x_k)=0$ if $n\geq k+1$. This implies that $\hbar^\ell x_1\cdots x_k
\in\UU'$ if, and only if $k\leq\ell$.
\end{pf}
\subsection{Quantum duality principle}
Assume now that $\g$ is finite--dimensional, let $\left(\g^*,\delta^t,[\cdot,
\cdot]^t\right)$ be the dual bialgebra, and $G^*$ the formal \PLg with
Lie algebra $\g$. By definition, the algebra of functions on $G^*$ is the
topological Poisson Hopf algebra given by $\sfk[[G^*]]
=(U\g^*)^*$. The following result is due to Drinfeld.
\begin{thm}[\cite{DrICM,Ga}]\label{th:duality}
$\UU'$ is a topologically free $\sfk[[\hbar]]$--module, and a sub Hopf
algebra of $\UU$. Its multiplication is commutative mod $\hbar$, and
$\UU'/\hbar\UU'$ is isomorphic, as a local, complete Poisson Hopf
algebra to $\sfk[[G^*]]$.
\end{thm}
\subsection{The isomorphism $\UU'/\hbar\UU'\cong\sfk[[G^*]]$} \label{ss:iso formula}
If $\UU=\Ug[[\hbar]]$ with undeformed coproduct, then $\delta=0$ and
$\g^*$ has trivial bracket. In this case $G^*$ is the (germ at 0 of the) abelian group $\g^*
$ and, by Lemma \ref{le:example},
$\UU'/\hbar\UU'=\wh{\gr{\Ug}}=\sfk[[\g^*]]$, where $\wh{\cdot}$ is the
graded completion.
More generally, the isomorphism $\UU/\hbar\UU\cong\Ug$ induces
a canonical isomorphism
\[i_\Delta:\UU'/\hbar\UU'\longrightarrow\sfk[[G^*]]\]
as follows \cite[Rem. 3.7]{EH}.
Identify $U\g^*$ as the quotient of the tensor algebra $T\g^*$ endowed
with the standard concatenation product and (cocommutative) shuffle
coproduct, and $(U\g^*)^*$ with a sub Hopf algebra of its dual $(T\g^*)
^*=\wh{T\g}=\prod_{n\geq 0} \g^{\otimes n}$, where
the latter is endowed with the (commutative) shuffle product
and deconcatenation coproduct. Then, the isomorphism $i_\Delta:\UU'/\hbar
\UU'\to\sfk[[G^*]]=(U\g^*)^*\subset\wh{T\g}$ is given by noticing that
if $x\in\UU'$, $\left(\left.\frac{1}{\hbar^n} \pi^{\otimes n}\circ\Delta^{(n)}
(x)\right)\right |_{\hbar=0}$ lies in $\g^{\otimes n}\subset (\Ug)^{\otimes n}$
for any $n$, and setting
\begin{equation}\label{eq:iso formula}
i_\Delta(x)=
\left\{\left.\frac{\pi^{\otimes n}\circ\Delta^{(n)}(x)}{\hbar^n}\right |_{\hbar=0}\right\}_{n\geq 0}
\in
\prod_{n\geq 0} \g^{\otimes n}
\end{equation}
\subsection{Semiclassical limit}
\comment{What are $G/\g$? At the beginning just algebraic groups,
but pretty soon reductive too.}
If $\UU$ is a QUE which deforms $U\g$, and $A\in\UU\otimes\UU'$,
we denote by $\scl{A}$ the {\it semi--classical limit} of $A$, that is its
class in $\UU\otimes\UU'/\hbar\UU\otimes\UU'$. By Theorem \ref
{th:duality}, $\scl{A}$ lies in $U\g\wh{\otimes}\sfk[[G^*]]$, and is
therefore a (formal) function on $G^*$ with values in $U\g$.
\comment{Mention completed tensor products somewhere}
\section{Semiclassical limit of the dynamical KZ equation}\label{se:scl DKZ}
The goal of this section is to prove that the Stokes data of the ODE
\eqref{eq:dKZ} are the semiclassical limits of the Stokes data of the
dynamical KZ equations \eqref{eq:DKZ}. Technicalities aside, this
stems from the observation that if $\Upsilon$ is a solution of
\[\frac{d\Upsilon}{dz}=
\left(\ad\muone+\sfh\frac{\Omega}{z}\right)\Upsilon\]
with values in $\UU\otimes\UU'$, the semiclassical limit $\gamma$
of $\Upsilon$, as a formal function of $\lambda\in\g^*$ with values
in $\Ug$, satisfies
\[\frac{d\gamma}{dz}=
\left(\ad\mu+\frac{\nnu(\lambda)}{2\pi\iota z}\right)\gamma\]
where $\nnu(\lambda)=\id\otimes\lambda(\Omega)$ which, after
the change of variable $z\to 1/z$, and the replacement $\ad\mu
\to -\Az, \nnu(\lambda)\to -2\pi\iota B$ is precisely the equation \eqref
{eq:dKZ}.\footnote{The appearance of the factor $2\pi\iota$
is due to the fact that the identification $\UU'/\hbar\UU'\cong\wh{S\g}$
is given by mapping $x\in\g$ to $\hbar x=2\pi\iota\sfh x\in\UU'$.}
\subsection{Formal Taylor series groups}
Let $G$ be an affine algebraic group over $\IC$. The ring of
regular functions $\IC[G]$ is a Hopf algebra, with coproduct
$\Delta f(g_1,g_2)=f(g_1g_2)$, counit $\eps(f)=f(1)$, and
antipode $Sf(g)=f(g^{-1})$.
If $(R,m_{R},1_{R})$ is a commutative, unital $\IC
$--algebra, the $R$--points of $G$ are, by definition, the
set of $\IC$--algebra morphisms $G(R)=\Alg_\IC(\IC[G],R)$.
$G(R)$ is a group, with multiplication $\phi\cdot\psi=m_
{R}\circ\phi\otimes\psi\circ\Delta$, unit $1_{R}\circ
\epsilon$, and inverse $\phi^{-1}=\phi\circ S$. Let $\mm\subset
R$ be a maximal ideal, and denote by $G(R)_\mm\subset
G(R)$ the normal subgroup consisting of maps $\gamma:\operatorname{Spec}
R\to G(\IC)$ such that $\gamma(\mm)=1$, that is
\[G(R)_\mm=
\left\{\varphi\in\Alg_\IC(\IC[G],R)|\,\varphi(I)\subset\mm\right\}\]
where $I=\Ker\eps$ is the augmentation ideal. We shall need the
following elementary
\begin{lemma}
If $R$ is a complete local ring with unique maximal ideal $\mm$,
then $G(R)_\mm$ may be identified with the set of grouplike
elements of the topological Hopf algebra
$$\Ug\wh{\otimes}R=\lim_p \Ug\otimesR/\mm^p$$
\end{lemma}
\begin{pf}
Let $\IC[[G]]=\lim\IC[G]/I^n$ be the completion of $\IC[G]$ at the
identity, and identify $\Ug$, as a Hopf algebra, with the continuous
dual
\[\IC[[G]]^*=\left\{\varphi\in\Hom_\IC(\IC[G],\IC)|\,\varphi(I^n)=0, n\gg 0\right\}\]
If $\mm^p=0$ for some $p$, and $\phi\in G(R)_\mm$, $\phi$ vanishes
on $I^p$ and therefore lies in $\left(\IC[G]/I^p\right)^*\otimesR\subset
\IC[[G]]^*\otimesR$. In general, $\mm$ is of finite order in $R/\mm^p$
for any $p\geq 1$, so that $G(R)_\mm=\lim_p G(R/\mm^p)_\mm$
embeds into $\lim_p\Ug\otimesR/\mm^p$.
\end{pf}
We shall be interested below in the case when $R=\IC[[V]]$ is
the completion of the algebra of regular functions on the vector
space $V=\g$ or $V=\g^*$ at $0$. We denote in this case $G(R),
G(R)_\mm$ and $\Ug\wh{\otimes}R$ by $G[[V]],G[[V]]_+$
and $\Ug[[V]]$ respectively. As algebraic groups over $\IC$,
$G[[V]]$ and $G[[V]]_+$ are the inverse limits
\[ G[[V]]=\lim_{\longleftarrow} G[[V]]^{(m)}
\aand
G[[V]]_+=\lim_{\longleftarrow} G[[V]]^{(m)}_+\]
where $G[[V]]^{(m)}=G(\IC[[V]]/I^m)$, respectively, and $G[[V]]_+$
is prounipotent.
\subsection{Semiclassical limit of canonical solutions of the DKZ equations}
Consider the ODE
\begin{align}
\frac{d\gamma}{dz}
&=
\left(\frac{A}{z^2}+\frac{B}{z}\right)\gamma
\label{eq:class}\\
\intertext{and the dynamical KZ equation}
\frac{d\Upsilon}{dz}
&=
\left(\ad\muone+\sfh\frac{\Omega}{z}\right)\Upsilon
\label{eq:quant}
\end{align}
where $A,\mu\in\hreg$, and $B\in\g$.
Fix throughout the standard determination of the logarithm, with a cut along
$\IR_{<0}$. The following result shows that the semiclassical limits of the
canonical fundamental solutions of \eqref{eq:quant} at $z=0,\infty$ are the
canonical fundamental solutions of \eqref{eq:class} at $z=\infty,0$, after
the change of variable $z\to 1/z$.
\begin{prop}\label{pr:scl DKZ
Let $\nu:\g^*\to\g$ be the isomorphism given by $\lambda\to\lambda\otimes
\id(\Omega)$, and set $\nu^\vee=-\nu/2\pi\iota$.
\begin{enumerate}
\item
Let $\csol_\infty$ be the canonical solution of \eqref{eq:class} near
$z=\infty$, and write $$\csol_\infty=e^{-\Az/z}\cdot h_\infty\cdot z^B$$
where $h_\infty:\IP^1\setminus 0\to G$ is such that $h_\infty(\infty)=1$.
Regard $h_\infty$ as a holomorphic function of $B\in\g\nr$ such that
$\left.h_\infty(z)\right|_{B=0}\equiv 1$, and let
\[\whh_\infty:\IP^1\setminus 0\longrightarrow G[[\g]]_+\]
be its formal Taylor series at $B=0$.
Let $\qsol_0=e^{z\ad\muone}\cdot H_0\cdot z^{\sfh\Omega}$ be
the canonical solution of \eqref{eq:quant} near $z=0$. Then, the
semiclassical limit of $H_0$ takes values in $G[[\g^*]]
_+\subset\Ug[[\g^*]]$. Moreover, if $\mu=-\Az$, then
\[\scl{H_0(z)}(\lambda)=\whh_\infty(1/z;\nu^\vee(\lambda))\]
\item
Assume now that $\Az\in\hreg^\IR$. Let $$\csol_\pm=h_\pm\cdot
e^{-\Az/z}\cdot z^{[B]}:\halfplane_\pm\to G$$ be the canonical solution
of \eqref{eq:class} at $z=0$ corresponding to the
half--plane $\halfplane_\pm=\{z\in\IC|\,\Im(z)\gtrless 0\}$. Regard $h_\pm$ as a holomorphic
function of $B\in\g$ such that $\left.h_\pm(z)\right|_{B=0}\equiv 1$,
and let
\[\whh_\pm:\halfplane_\pm\longrightarrow G[[\g]]_+\]
be its formal Taylor series at $B=0$.
Let $\qsol_\pm=H_\pm\cdot e^{z\ad\muone}\cdot z^{\sfh\Omega_0}$,
be the canonical solution of \eqref{eq:quant} at $z=\infty$ corresponding
to the half--planes $\halfplane_\pm$. Then, the semiclassical limit of
$H_\pm$ takes values in $G[[\g^*]]_+\subset\Ug[[\g^*]]$. Moreover,
if $\mu=-\Az$, then
\[\scl{H_\pm(z)}(\lambda)=\whh_\mp(1/z;\nu^\vee(\lambda))\]
\Omit{
Let $H_\pm:\halfplane_\pm\to\A$ be the holomorphic component of the
canonical solution of the DKZ equations at $z=\infty$ corresponding
to the half--planes $\halfplane_\pm$. Then, the semiclassical limit $\scl{H_
\pm}$ of $H_\pm$ takes values in $G[[\g^*]]_+\subset\Ug[[\g^*]]$.
Moreover, if $\mu=-\Az$, then
\[\scl{H_\pm(z)}(B,\mu)=\whh_\mp(1/z;-B,\Az)\]
}
\end{enumerate}
\end{prop}
\begin{pf}
(1)
By definition, $H_0$ is a solution of
\[\frac{dH_0}{dz}=\frac{\sfh}{z}\left(\ell(e^{-z\ad\muone}(\Omega))-\rho(\Omega)\right)H_0\]
where $\ell,\rho$ denote left and right multiplication respectively. Thus,
as en element of $\UU\otimes\UU'/\hbar\UU\otimes\UU'=\Ug[[\g^*]]$,
the semiclassical limit $h_0$ of $H_0$ satisfies
\[\frac{dh_0}{dz}=\frac{1}{2\pi\iota z}\left(\ell(e^{-z\ad\mu}(\nnu)-\rho(\nnu)\right)h_0\]
together with the initial condition
$h_0(0)=1$. We claim that $h_0$ takes values in $G[[\g^*]]_+\subset
\Ug[[\g^*]]$. Indeed, both $\Delta\otimes\id(h_0)$ and $h_0^{13}h_0^{23}$
satisfy
\[\frac{dh}{dz}=\frac{1}{2\pi\iota z}\left(\ell(e^{-z\ad\mu}(\nnu^1+\nnu^2))-\rho(\nnu^1+\nnu^2)\right)h\]
and the result follows by uniqueness. The claimed equality now follows
from the uniqueness statement of Lemma \ref{le:nr dkz}, applied to the affine
algebraic groups $G[\IC[\g^*]/I^m]$, $m\geq 1$.
(2) is proved similarly.
\end{pf}
\subsection{Semiclassical limit of the differential twist}\label{ss:scl J}
\begin{thm}\label{th:scl J}
Assume that $\Az\in\hreg^\IR$, and let $C_\pm=\csol_\pm^{-1}\cdot\csol
_\infty$ be the connection matrix of \eqref{eq:class} (see \ref{ss:connection}).
Regard $C_\pm$ as a $G$--valued holomorphic function of $B\in\g\nr$
such that $\left.C_\pm\right|_{B=0}=1$, and let $\wh{C}_\pm\in G[[\g]]_
+$ be its formal Taylor series at $B=0$.
Let $J_\pm=\Upsilon_0^{-1}\cdot\Upsilon_\pm$ be the differential
twist defined in \ref{ss:differential twist}. Then, if $\mu=-A$, the
semiclassical limit of $J_\pm$ is given by
\[\scl{J_\pm}(\lambda)=\wh{C}_\mp(\nu^\vee(\lambda))^{-1}\]
\end{thm}
\begin{pf}
By definition,
$J_\pm=
z^{-\sfh\Omega}\cdot H_0(z)^{-1}\cdot
\exp(-z\ad\muone)\left(H_\pm\right)\cdot z^{\sfh\Omega_0}
$,
where $z\in\halfplane_\pm$. By Proposition \ref{pr:scl DKZ},
\[\scl{J_\pm}=
z^{-\nnu/2\pi\iota}\cdot \wh{h}_\infty(1/z;-\nnu/2\pi\iota)^{-1}\cdot \exp(z\ad(\Az))
\left(\wh{h}_\mp(1/z;-\nnu/2\pi\iota)\right)\cdot z^{[\nnu]/2\pi\iota}
\]
On the other hand,
\[C_\pm(B)=
w^{-[B]}\cdot e^{\Az/w}\cdot h_\pm(w)^{-1}\cdot
e^{-\Az/w}\cdot h_\infty(w)\cdot w^B\]
where $w\in\halfplane_\pm$.
\end{pf}
\subsection{Semiclassical limit of the quantum Stokes matrices}
\label{ss:sc Stokes mat}
\begin{thm}
Let $A\in\hreg^\IR$, and $S_\pm$ the Stokes matrices of
the ODE \eqref{eq:class} relative to the ray $-\iota\IR_{>0}$
(see \ref{ss:Stokes mat class}). Regard $S_\pm$ as a $G$--valued
holomorphic function of $B\in\g$ such that $\left.S_\pm\right|
_{B=0}=1$, and let $\wh{S}_\pm\in G[[\g]]_+$ be its formal
Taylor series at $0$.
Let $\mu\in\hreg^\IR$, and $S^\hbar_\pm$ the Stokes matrices
of the dynamical KZ equation \eqref{eq:quant} (see \ref {ss:quant Stokes mat}).
Then, $S^\hbar_\pm$ take values in $\UU\wh{\otimes}\UU'$,
and its semi--classical limit in $G[[\g^*]]_+\subset\Ug[[\g^*]]$.
Moreover, if $\mu=-A$, then
\[\scl{S^\hbar_\pm}(\lambda)=\wh{S}_\pm(\nu^\vee(\lambda))\]
\end{thm}
\begin{pf}
Let $\qsol_\pm=H_\pm\cdot e^{z\ad\muone}\cdot z^{\sfh\Omega_0}$
be the canonical solutions of the DKZ equations corresponding to
the halfplanes $\halfplane_\pm$, and $\wt{\qsol}_+=\wt{H}_+\cdot e^{z\ad
\muone}\cdot z^{\sfh\Omega_0}$ the analytic continuation of $\qsol_+$
across $\IR_{>0}$. By definition,
\[S_+^\hbar=
\qsol_-^{-1}\cdot\wt{\qsol}_+=
z^{-\sfh\Omega_0}\cdot \exp(-z\ad\muone)\left(H_-^{-1}\cdot
\wt{H}_+\right)\cdot z^{\sfh\Omega_0}\]
for $z\in\halfplane_-$. By Proposition \ref{pr:scl DKZ},
\[\scl{S^\hbar_+}=z^{-[\nnu]/2\pi\iota}\cdot\exp(-z\ad(\mu))\left(\wh{h}_+(1/z;-\nnu/2\pi\iota)^{-1}\cdot\wt{\wh{h}}_-(1/z;-\nnu/2\pi\iota)\right)z^{[\nnu]/2\pi\iota}\]
where $\wt{\wh{h}}_-$ is the analytic continuation of $\wh{h}_-$
across $\IR_{>0}$.
On the other hand, if $\gamma_\pm(w)=h_\pm(w)\cdot e^{-A/w}\cdot w^{[B]}$
are the canonical solutions of \eqref{eq:class} corresponding to $w\in\IH_\pm$,
and $\wt{\gamma}_-$ is the analytic continuation of $\gamma_-$ across
$\IR_+$ then, by definition
\[S_+=
\csol_+^{-1}\cdot\wt{\csol}_-=
w^{-[B]}\cdot e^{A/w}\cdot h_+(w)^{-1}\cdot \wt{h}_-(w)\cdot e^{-A/w}\cdot w^{[B]}
\]
The Taylor series of $S_+$ at $B=0$ therefore coincides with $\scl{S^\hbar_+}$
provided $A=-\mu$, $w=1/z$, and $B$ is replaced by $-\nnu(\lambda)/2\pi\iota$. The proof that $\scl
{S^\hbar_-}=\wh{S}_-(-\nnu/2\pi\iota)$ is identical.
\end{pf}
\section{Formal linearisation via quantisation} \label{se:EEM}%
\subsection{}
Let $(\p,\sfr)$ be a \fd \qt Lie bialgebra over a field $\sfk$ of characteristic
zero. Thus, $\p$ is a Lie algebra, $\sfr\in\p\otimes\p$ satisfies the classical
\YBE (CYBE)
\[[\sfr_{12},\sfr_{23}+\sfr_{13}]+[\sfr_{13},\sfr_{23}]=0\]
and is such that $\Omega=\sfr+\sfr^{21}$ is invariant under $\p$. In particular,
$\p$ is a Lie bialgebra with cobracket $\delta:\p\to\p\wedge\p$ given by
$\delta(x)=[x\otimes 1+1\otimes x,\sfr]$.
Let $\p^*$ be the dual Lie bialgebra to $\p$, and $P,P^*$ the formal
\PL groups with Lie algebras $\p,\p^*$. The CYBE imply that the maps
$\ell,\rho:\p^*\to\p$ given by
\[\ell(\lambda)=\lambda\otimes\id(\sfr)\aand \rho(\lambda)
=-\id\otimes\lambda(\sfr)\]
are morphisms of Lie algebras. We denote the corresponding morphisms
of formal groups $P^*\to P$ by $\LL$ and $\RR$ respectively, and by $\beta:P^*
\to P$ the {\it big cell map}
\[g^* \longrightarrow \LL(g^*)\cdot\RR(g^*)^{-1}\]
The differential of $\beta$ at $1$ is $\ell-\rho:\lambda\to\lambda\otimes\id(\Omega)=
:\nnu(\lambda)$. In particular, $\beta$ is an isomorphism of formal manifolds
if $\sfr$ is {\it non--degenerate}, that is such that $\nnu:\p^*\to\p$ is an isomorphism.
\subsection{}
Set $\UU=U\p[[\hbar]]$ and let $\Phi\in\UU^{\otimes 3}$ be an associator,
that is an element satisfying $\Phi\in 1+\frac{\hbar^2}{24}[\Omega_{12},\Omega
_{23}]+\hbar^3\UU^{\otimes 3}$, and such that $(\UU,\Delta_0,e^{\hbar\Omega/2},
\Phi)$ is a \qt quasi--Hopf algebra. Let $J\in 1+\frac{\hbar}{2}\sfj+\hbar^2\UU^
{\otimes 2}$ be a twist such that $\sfj-\sfj^{21}=\sfr-\sfr^{21}$, and the following
twist equation holds
\begin{equation}\label{eq:twist eq
\Phi\cdot J_{12,3}\cdot J_{1,2}=J_{1,23}\cdot J_{2,3}
\end{equation
Then,
$\UU_J=\left(\UU,\Delta_J=J^{-1}\Delta_0(\cdot)J,R_J=J^{-1}_{21}e^{\hbar\Omega/2}J\right)$
is a \qt Hopf algebra, which is a quantisation of $(\p,\sfr)$. By Theorem \ref{th:duality},
$(\UU_J)'$ is therefore a quantisation of the Poisson algebra $\sfk[[P^*]]$.
\subsection{}
Assume that the twist $J$ is {\it admissible}, that is such that $\hbar\log(J)\in(\UU')
^{\otimes 2}$. The following linearisation result is due to Enriquez--Halbout \cite
[Prop. 4.2]{EH}.
\begin{prop}
The subalgebras $\UU'$ and $(\UU_J)'$ of $\UU$ coincide. Their equality therefore
induces a formal Poisson isomorphism $\poiss:\p^*\to P^*$ given by the composition
\begin{equation}\label{eq:EH}
\sfk[[P^*]]\cong(\UU_J)'/\hbar(\UU_J)' = \UU'/\hbar\UU'\cong\sfk[[\p^*]]
\end{equation}
where the first and last isomorphisms are given by \eqref{eq:iso formula}.
\end{prop}
The explicit form of the isomorphism $\poiss$ is given by the following
result of \EEM \cite[\S 3.3.2]{EEM}.
\begin{thm}\label{th:EEM}
Assume further that $\Phi=\Psi(\hbar\Omega_{12},\hbar\Omega_{23})$
where $\Psi$ is a Lie associator, and that $J$ lies in $\UU\otimes\UU'
\cap\UU'\otimes\UU$. Then,
\begin{enumerate}
\item The \sscl $\jmath=J\mod\hbar\,\UU\otimes\UU'$ lies in $P[[\p^*]]_+
\subset\Ug\wh{\otimes}\sfk[[\p^*]]$, that is, is a formal map $\p^*\to P$.
\item The composition of the Poisson isomorphism $\poiss:\p^*\to P^*$
with the big cell map $\beta:P^*\to P$ is the map $e_\jmath:\p^*\to P$ defined by
\[e_\jmath(\lambda)=\jmath(\lambda)^{-1}\cdot e^{\nnu(\lambda)}\cdot\jmath(\lambda)\]
\end{enumerate}
In particular, if $\sfr$ is non--degenerate, the map $\beta^{-1}\circ e
_\jmath:\p^*\to P^*$ is an isomorphism of formal Poisson manifolds.
\end{thm}
\begin{pf}
We outline the proof for the reader's convenience.
By assumption, $R=J_{21}^{-1}\cdot e^{\hbar\Omega/2}\cdot J$ lies in
$\UU\otimes\UU'=\UU_J\otimes(\UU_J)'$, and similarly $R_{21}\in\UU
_J\otimes(\UU_J)'$.
Consider now the identity
\begin{equation}\label{eq:basic id}
R_{21}\cdot R=J^{-1}\cdot e^{\hbar\Omega}\cdot J
\end{equation}
Let $b\in\UU_J\otimes(\UU_J)'/\hbar\UU_J\otimes(\UU_J)'\cong U\p[[P^*]]$ be
the \sscl of the left--hand side, and $a\in\UU\otimes\UU'/\hbar\UU\otimes
\UU'\cong U\p[[\p^*]]$ that of the right--hand side. Clearly, $b\circ\poiss=a$. It
therefore suffices to show that $b=\beta$ and $a=e_\jmath$.
The identity $\Delta_J\otimes\id(R)=R_{13}\cdot R_{23}$ implies that the
\sscl $R'$ of $R$ lies in $P[[P^*]]_+$, and $\id\otimes\Delta_J(R)=R_{13}
\cdot R_{12}$ that $R'$ is an antihomomorphism $P^*\to P$. Its differential
at the identity is readily seen to be the map $\p^*\to\p$ given by $\lambda
\to\id\otimes\lambda(\sfr)$, so that $R'(g^*)=\RR(g^*)^{-1}$. Similarly, the
\sscl of $R^{21}$ is the homomorphism $\LL:P^*\to P$, and it follows that
$b=\beta$.\comment{Notice an interesting peculiarity. The big cell map
$\beta:P^*\to P$ is not a Poisson map (afaik) and therefore one does
not expect to quantise it to an algebra homomorphism
$$\beta^*_\hbar:\sfk[[P]]_\hbar=(\UU_J)^*\longrightarrow(\UU_J)'=\sfk[[P^*]]_\hbar$$ or equivalently an element $B^*_\hbar\in
\UU_J\otimes(\UU_J)'$ satisfying $\Delta_J\otimes\id(B^*_\hbar)=(B^*_\hbar)
_{13}(B^*_\hbar)_{23}$. However, $\beta$ {\bf is} being quantised as the element
$RR_{21}$ of $\UU_J\otimes(\UU_J)'$. Clarify this.}
Since the \sscl of $e^{\hbar\Omega}$ is $e^\nu\in P[[\p^*]]_+$, we have
$a=e_\jmath$ and there remains to prove that $\jmath$ lies in $P[[\p^*]]
_+$, that is satisfies $\Delta_0\otimes\id(\jmath)=\jmath_{1,3}\cdot\jmath
_{2,3}$. This is a consequence of the reduction of the twist equation \eqref
{eq:twist eq} mod $\hbar\UU\otimes\UU\otimes\UU'$, as follows.
Note first that since $J\in1+\hbar\UU\otimes\UU$, $J_{1,2}\in 1+\hbar\UU
\otimes\UU\otimes\UU'$. Next, it is easy to see that for any $x\in\UU'$,
$\Delta_0(x)\in 1\otimes x+\hbar\UU\otimes\UU'$, hence $J_{1,23}\in
J_{1,3}+\hbar\UU\otimes\UU\otimes\UU'$. Finally, $\hbar\Omega_{12}\in
\hbar\UU\otimes\UU\otimes\UU'$, hence $\Phi=\Psi(\hbar\Omega_{12},
\hbar\Omega_{23})=\Psi(0,\hbar\Omega_{23})=1$ mod $\hbar\UU\otimes
\UU\otimes\UU'$.
\end{pf}
\section{Analytic linearisation via Stokes data} \label{se:alt boalch}%
Let $G$ be a complex reductive group, and $B_\pm\subset
G$ a pair of opposite Borel subgroups intersecting along the
maximal torus $H$. Let $\g,\b_\pm,\h$ be the Lie algebras of
$G,B_\pm$, and $H$ respectively, $\sfPhi\subset\h^*$ the corresponding
root system, and $\sfPhi_\pm\subset\sfPhi$ the set of positive
and negative roots, so that $\b_\pm=\h\bigoplus_{\alpha\in\sfPhi
_\pm}\g_\alpha$. Fix an invariant inner product $(\cdot,\cdot)$
on $\g$, and let $\sfr\in\b_+\otimes\b_-$ be the corresponding
canonical element (see \eqref{eq:standard cybe}). Then, $(\g,
\sfr)$ is a quasitriangular Lie bialgebra, and $G$ and $G^*=B
_-\times_H B_+$ are dual Poisson--Lie groups. Moreover, the
homomorphisms $L,R:G^*\to G$ defined in \ref{ss:EEM}
correspond to the first and second projection, respectively.
Let $A\in\h\reg$, and consider the connection
\[\nabla=d-\left(\frac{A}{z^2}+\frac{B}{z}\right)dz\]
Set $\h^\IR=\{t\in\h|\alpha(t)\in\IR,\alpha\in\sfPhi\}$, and
let $\C=\{t\in\h|\,\alpha(t)>0,\,\alpha\in\sfPhi_+\}\subset\h^\IR\reg$
be the fundamental chamber corresponding to $\sfPhi_+$.
Note that the rays $\pm\iota\IR_{>0}$ are admissible if $A\in\h
\reg^\IR+\iota\h^\IR\subset\h\reg$. Moreover, by \ref{ss:Stokes mat class},
the Stokes matrices $S_\pm$ corresponding to $r=-\iota\IR_{>0}$
lie in $B_\pm(A,r)=B_\mp$ if $A\in -\C+\iota\h^\IR$. Let
\[\calS:\g\to G^
\qquad\qquad
B\to\left(S_+^{-1} \cdot e^{-\iota\pi[B]}, S_-\cdot e^{\iota\pi[B]}\right)\]
be the Stokes map defined in \ref{ss:stokes map}.
Let $\nu:\g^*\to\g$ be the identification determined by $(\cdot,\cdot)$,
and set $\nnuc=-\nu/2\pi\iota$.
\begin{thm}
If $A\in -\C$, the map $\calS\circ\nnuc:\g^*\to G^*$ is a Poisson map.
\end{thm}
\begin{pf}
Since $\calS\circ\nnuc$ is complex analytic, it is sufficient to prove that
its formal Taylor series at $0$ is a Poisson map.
Set $\mu=-A\in\C$, and let $J_+=J_+(\mu)$ the differential twist defined
in \ref{ss:differential twist}. By Theorem \ref{th:J}, $J_+\in 1+\half{\hbar}
\sfj_++\hbar^2\UU^{\otimes 2}$, where $\sfj_+-\sfj^{21}_+=\sfr-\sfr^{21}$,
and $J_+$ kills the KZ associator $\Phi\KKZ$.
Write $\Omega=\Omega_0+\sum_{\alpha\in\sfPhi}\Omega_\alpha$,
where $\Omega_0=\sum_i t_i\otimes t^i$, with $\{t_i\},\{t^i\}$ dual
bases of $\h$ \wrt $(\cdot,\cdot)$, and $\Omega_\alpha=x_\alpha
\otimes x_{-\alpha}$, with $x_{\pm\alpha}\in\g_{\pm\alpha}$ such
that $(x_\alpha,x_{-\alpha})=1$. Then, one can show that $\log J_+$
is a Lie series in the variables $\hbar\Omega_0,\hbar\Omega_\alpha$.
Since the subspace $\adm_n=\{x\in\UU^{\otimes n}|\,\hbar x\in(\UU')
^{\otimes n}\}$ is a Lie algebra for any $n\geq 1$, and $\hbar\Omega
_0,\hbar\Omega_\alpha\in\adm_2$, it follows that $\log J_+\in\adm_2$.
Since $J_+$ lies in $\UU'\otimes\UU\cap\UU\otimes\UU'$ by \ref{ss:differential twist},
we may apply Theorem \ref{th:EEM} to the pair $(\Phi\KKZ,J_+)$. Let
$\jmath_+\in\UU\otimes\UU'/\UU\otimes\UU'=G[[\g^*]]_+$ be the \sscl
of $J_+$, and $e_{\jmath_+}\in G[[\g^*]]_+$ the map $\lambda\to\jmath
_+(\lambda)^{-1}\cdot e^{\nu(\lambda)}\cdot\jmath_+(\lambda)$. By
Theorem \ref{th:scl J}
\[\begin{split}
e_{\jmath_+}(\lambda)
&=
\wh{C}_-(-\nu(\lambda)/2\pi\iota;-\mu)\cdot e^{\nu(\lambda)}\cdot \wh{C}_-(-\nu(\lambda)/2\pi\iota;-\mu)^{-1}\\
&=
\left(
\wh{C}_-(\nu^\vee(\lambda);A)\cdot e^{2\pi\iota\nu^\vee(\lambda)}\cdot \wh{C}_-(\nnu^\vee(\lambda);A)^{-1}
\right)^{-1}\\
&=
\left(
\wh{S}_-(\nu^\vee(\lambda);A)\cdot e^{2\pi\iota[\nu^\vee(\lambda)]}\cdot\wh{S}_+(\nnu^\vee(\lambda);A)
\right)^{-1}\\
&=
\left(\wh{S}_+(\nnu^\vee(\lambda);A)^{-1}\cdot e^{-\pi\iota[\nu^\vee(\lambda)]}\right)
\cdot
\left(\wh{S}_-(\nnu^\vee(\lambda);A)\cdot e^{\pi\iota[\nu^\vee(\lambda)]}\right)^{-1}\\
&=
L(\wh{\calS}(\nu^\vee(\lambda);A))\cdot R(\wh{\calS}(\nu^\vee(\lambda);A))^{-1}
\end{split}\]
where the
the third equality follows from
the monodromy relation (Proposition \ref{pr:monodromy reln}), and the
fifth from the definition of the Stokes map, as well as the assumption that
$A\in-\C$, so that $S_\pm(B;A)\in N_\pm(A,r)=N_\mp$.
It follows that the composition $\beta^{-1}\circ e_{\jmath_+}$ is equal to
$\wh{\calS}\circ\nu^\vee$, and is therefore a Poisson map by Theorem
\ref{th:EEM}.
\end{pf}
\section{Isomonodromic deformations}\label{Section:isomono}
\newcommand {\cl}{^{\operatorname{scl}}}
Let $S_\pm\in\Ug^{\otimes 2}\fml^o$ be the Stokes matrices of the dynamical KZ
equations, and $S_\pm\cl\in G[[\g^*]]_+$ their semiclassical limit.
For any $\alpha\in\sfPhi$, let $Q_\alpha\in S^2\g\subset\IC[\g^*]$
be given by $Q_\alpha=x_\alpha\cdot x_{-\alpha}=Q_{-\alpha}$.\comment
{Work out what the corresponding vector field looks like}
\begin{prop}\hfill\break
\begin{enumerate}
\item As a function of $\mu\in\h\reg^\IR$, $S_\pm\cl$ satisfies the
following PDE
\[d_\h S_\pm\cl=
\frac{1}{2\pi\iota}
\sum_{\alpha\in\sfPhi_+}\frac{d\alpha}{\alpha}\{Q_\alpha,S_\pm\cl\}\]
\item Regard $B\in\g$ as a function of $\mu\in\hreg^\IR$. Then, the
Stokes matrices of (the classical ODE) are locally constant as
$\mu$ varies in $\hreg$ if, and only if $B$ satisfies the nonlinear
PDE
\[d_\h B=
-\frac{1}{2\pi\iota}\sum_{\alpha\in\sfPhi_+}\frac{d\alpha}{\alpha}H_\alpha\]
where $H_\alpha=\{Q_\alpha,-\}$ is the Hamiltonian vector field corresponding
to $Q_\alpha$.
\end{enumerate}
\end{prop}
\begin{pf}
(1)
By Proposition \ref{pr:Spm}, $S_\pm$ satisfy
\[d_\h S_\pm=
\frac{1}{4\pi\iota}\sum_{\alpha\in\sfPhi_+}\frac{d\alpha}{\alpha}
\left[\onetwo{\hbar\Kalpha},S_\pm\right]\]
Note that $\hbar^2\Kalpha\in\UU'$, and that its image in $\UU'/\hbar
\UU'$ is $2Q_\alpha$. As pointed out in \ref{se:filtered A}, $\hbar\ad
(\Kalpha)$ is a derivation of $\UU'$. Since $[\hbar\Kalpha,-]=\hbar^
{-1}[\hbar^2\Kalpha,-]$, $\hbar\ad(\Kalpha)$ induces the derivation
$\{Q_\alpha,-\}$ on $\IC[\g^*]$. The result now follows from the fact
that $\hbar\Kalpha^{(1)}\in\hbar\UU\otimes\UU'$, so that its image
in $\Ug\fmls{\g^*}$ is zero.
(2)
\[\begin{split}
d_\h S_\pm\cl
&=
\frac{1}{2\pi\iota}
\sum_{\alpha\in\sfPhi_+}\frac{d\alpha}{\alpha}\{Q_\alpha,S_\pm\cl\}+
d_{\g^*} S_\pm\cl(d_\h B)\\
&=
d_{\g^*} S_\pm\cl\left(
\frac{1}{2\pi\iota}\sum_{\alpha\in\sfPhi_+}\frac{d\alpha}{\alpha}H_\alpha+d_\h B
\right)\end{split}\]
\end{pf}
This is the time-dependent Hamiltonian description of the isomonodromic deformation given by \cite{JMU, Bo2}.
Here we give a quantum algebra proof, which enables us to interpret the symplectic nature of the isomonodromic deformation from the perspective of the gauge action of Casimir operators on quantum Stokes matrices.
\Omit{
\section{Gauge actions and isomonodromy equations}\label{Section:isomono}
\subsection{Gauge actions on quantum Stokes matrices}
Let $U'_0:={\rm Ker}(\varepsilon)\cap U(\g)\fml ^\circ$ and let
$V:=\{u_\hbar \in\hbar^{-1}U'_0\subset U(\g)\fml\}~|~u_\hbar =O(\hbar)\}$ be the Lie subalgebra for
the commutator. The reduction module $\hbar$ of the Lie algebra $V$ is $V/\hbar V=(\hat{S}(\g)_{>1},\{\cdot,\cdot\})$.
For any $u_\hbar\in V$, the gauge action of $e^{u_\hbar}$ on the set $\mathcal{H}$ of quantum $R$-matrices is given by $e^{u_\hbar}\ast R:=(e^{u_\hbar})^{\otimes 2}R{(e^{u_\hbar})^{\otimes 2}}^{-1}$, $R\in \mathcal{H}$,
and its infinitesimal action acts by vector fields on $\mathcal{H}$ by $\delta_{u_\hbar}(R)=[u_\hbar^{(1)}+u_\hbar^{(2)},R], \ R\in\mathcal{H}.$
In particular, this equation becomes
\begin{eqnarray*}\delta_{u_\hbar}(S_{\hbar\pm})=[u_\hbar^{(1)}+u_\hbar^{(2)},S_{\hbar\pm}]
\end{eqnarray*}
at $S_{\hbar\pm}\in\mathcal{H}$ the quantum Stokes matrices (associated with any $\mu\in\C\subset\h\reg^\IR$).
Denote by $u\in \hat{S}(\g)$ the semiclassical limit of $u_\hbar\in V$, then the reduction modulo $\hbar$ of the above equation becomes \[\delta_u(S_\pm)=\{1\otimes u,S_\pm\}.\] Here $S_\pm:U(\g)\otimes\hat{S}(\g)$ (the Stokes maps) are the semiclassical limit of $S_{\hbar\pm}$, and the bracket $\{1\otimes u,S_\pm\}$ takes the Lie bracket on the second component $\hat{S}(\g)$.
If we write $S_\pm$ as a map from $\g^*$ to $G$, the above (infinitesimal gauge action) equation takes the form
\begin{eqnarray}\label{infgauge}
\delta_u(S_\pm)(x)=(S_\pm)_*(H_u(x)),
\end{eqnarray}
where $u\in \hat{S}(\g)$ and $H_u$ is the Hamiltonian vector field on $\g^*$ generated by $u$, i.e., $H_u=\{u,\cdot\}$.
\subsection{Isomonodromic deformation equations.}
By Theorem \ref{pr:Spm}, as functions of $\mu\in\C\subset\h\reg^\IR$, the quantum Stokes matrices $S_{\hbar\pm}:\C\rightarrow\Ug^{\otimes 2}\fml^o$
satisfy
\begin{eqnarray}\label{Casimireq}
d_\h S_{\hbar\pm}=
\frac{\sfh}{2}\sum_{\alpha\in\sfPhi_+}\frac{d\alpha}{\alpha}
\left[\onetwo{\Kalpha},S_{\hbar\pm}\right].
\end{eqnarray}
Set $\omega_\hbar:=\frac{\hbar}{2}\sum_{\alpha\in \Phi_+} \Kalpha
\frac{d\alpha}{\alpha}\in U(\g)\fml\otimes\Omega^1(\C).$ One checks that $\omega_\hbar\in V\otimes \Omega^1(\C)$ (recall $V$ is a Lie subalgebra of $U(\g)\fml$ for the commutator), and the right hand side of \eqref{Casimireq} is the infinitesimal gauge action of $\omega_\hbar$ on $S_\pm\in\mathcal{H}$. ("time-dependent or $\C$-dependent" gauge action).
We denote by $\omega\in \Omega^1(\C)\otimes \hat{S}(\g)$ the semiclassical limit of the $\omega_\hbar$, which is a one-form on $\C$ whose coefficients are quadratic polynomials on the Poisson space $\g^*$ with the Kirillov-Kostant-Souriau bracket. (under the PBW isomorphism, $\omega$ coincides with $\frac{1}{2}\sum_{\alpha\in \Phi_+} \Kalpha
\frac{d\alpha}{\alpha}$). By taking the corresponding Hamiltonian vector field generated by the second component in $S(\g)$, $\omega$ corresponds to an element $H_\omega\in \Omega^{1}(\C)\otimes \mathfrak{X}(\g^*)$. Then it follows from the discussion above, especially equation \eqref{infgauge}, the reduction module $\hbar$ of equation \eqref{Casimireq} gives rise to
\begin{eqnarray}\label{classicalC}
d_{\h} S_\pm(x)+(S_\pm)_*(H_{\omega}(x))=0, \ \forall x\in \g^*.
\end{eqnarray}
Here $S_\pm$ is viewed as a map $\C\times \g^*\rightarrow G$, $H_\omega\in \Omega^{1}(\C)\otimes \mathfrak{X}(\g^*)$, and $d_\h S_\pm$, $(S_\pm)_*(H_\omega)$ are viewed as sections of $\Omega^{1}(\C)\otimes S_\pm^{-1}(TG)$.
This equation recovers the isomonodromic deformation equation \cite{JMU, Bo2} as follows.
Choose $\mu_t\in \C$ a one parameter family. Assume $x(\mu_t)\in\g^*$ is an isomonodromic flow, i.e., $x(\mu_t)$ is such that the Stokes matrices $S_\pm({\mu_t},x(\mu_t))$ of the connection $\nabla_t=d-(\frac{\mu_t}{z^2}+\frac{x(\mu_t)}{z})$ is constant with respect to $t$. Taking the derivative of the equation $S_{\pm}(\mu_t,x(u_t))={\rm const}$ (with respect to $t$) at $t_0$, we get that the isomonodromic flow $x(\mu_t)\in\g^*$ should satisfies
$$
\frac{d S_\pm(\mu_t,x_0)}{dt}|_{t=t_0}+\frac{dS_\pm(\mu_0,x(\mu_t))}{dt}|_{t=t_0}=0.
$$
From the arbitrarity of the one parameter family $\mu_t$ in $\C$, we deduce that the isomonodromic equation takes the form \[d_{\h}S_\pm(x(u))+(S_\pm)_*(d_{\h}x(u))=0.\]
Comparing this with equation \eqref{classicalC} leads to the identity
$$
(S_\pm)_*(d_{\h}x(\mu))=(S_\pm)_*(H_{\omega}(x(\mu))).
$$
Because for any $\mu\in \C$, the Stokes map $(S_-,S_+):\g^*\rightarrow G^*$ is a local isomorphism, we therefore obtain
\begin{thm}\label{isoequation}
The isomonodromic deformation equation takes the form $d_{\h}x(u)=H_{\omega}(x(u)).$
\end{thm}
This is the time-dependent Hamiltonian description of the isomonodromic deformation given by \cite{JMU, Bo2}.
Here we give a quantum algebra proof, which enables us to interpret the symplectic nature of the isomonodromic deformation from the perspective of the gauge action of Casimir operators on quantum Stokes matrices.
}
\Omit{
\section{The centraliser property and Gelfand-Zeitlin systems}\label{GZsystems} %
Let us assume $\g$ is a complex simple Lie algebra. A nested set on its Dynkin diagram $D$ is a collection of pairwise compatible, connected subdiagrams of $D$ containing $D$. If we denote by $\mathcal{N}_D$ the partially ordered set of nested sets on $D$, ordered by reverse inclusion, then $\mathcal{N}_D$ has a unique maximal element $\{D\}$, and its minimal elements are the maximal nested sets.
Fix a maximal nested set on $D$, with the subdiagram $D_n\subset \cdot\cdot\cdot\subset D_1\subset D_0=D$.
For any subdiagram $D_i\subset D$, let $\g_i\subset \g$ be the subalgebra generated by the root subspaces $\g_{\pm\alpha}$, $\alpha\in D_i$. Thus we get a chain $\g_n\subset\cdot\cdot\cdot \g_1\subset \g_0=\g$ of Lie sublagebras of $\g$. For each $i$, let $\sfPhi_i\subset \sfPhi$ be the root system, $\h_i\subset \h$ the Cartan subalgebra of $\g_i$, and $\frak l_i =\g_i +\h$ the corresponding Levi subalgebra of $\g$. We denote by $\sfPhi_{\rm KZ\g_i}$ the KZ associator for $\g_i$.
\subsection{The centraliser property and relative Drinfeld twists}
Let us denote by $\alpha_1$ the simple root in $D\setminus D_1$, thus $\sfPhi_1\subset\sfPhi$ is the root system generated by the simple
roots $\{\alpha_i\}_{i\ne 1}$.
The inclusion of root systems $\sfPhi_1 \subset \sfPhi$ gives rise to a projection $\pi:\h\rightarrow\h_1$ by requiring that $\alpha(\pi(\mu))=\alpha(\mu)$ for any $\alpha\in\sfPhi$. Therefore we have an isomorphism $\h\cong\mathbb{C}\times \h_1$, whose
inverse is given by $(\omega, \bar{\mu})\rightarrow \omega \lambda^\vee_1+\iota(\bar{\mu})$. Here $\lambda_i$ denotes the $i$-th fundamental coweight, and $\iota:\h_1\rightarrow \h$ is the embedding with image ${\rm Ker}(\alpha_1)$ given by $\iota(\bar{\mu})=\bar{\mu}-\alpha_n(\bar{\mu})\lambda^\vee_1.$
Let us denote by
$\K=\sum_{\alpha\in\sfPhi_+}\K_\alpha$
(resp. $\K_1=\sum_{\alpha\in{\sfPhi_1}_+}\K_\alpha$)
the (truncated) Casimir operators of $\g$ (resp. $\g_1$).
\begin{prop}[\cite{TL}]\label{pr:Fuchs infty}\hfill
\begin{enumerate}
\item For any $\olmu\in\h_1$, there is a unique holomorphic
function
\[H_\infty:\{w\in \IP^1|\,|w|>R_\olmu\}\to\Ug\fml^o\]
such that $H_\infty(\infty)=1$ and, for any determination of
$\log(\alpha_1)$, the function $\Upsilon_\infty=H_\infty(\alpha_1
\cdot \alpha_1^{\half{\hbar}(\K-\K_1)}$ satisfies
\[\left(d-\half{\hbar}\sum_{\alpha\in\sfPhi_+\setminus\sfPhi_1}
\frac{d\alpha_1}{\alpha_1-w_\alpha}\K_\alpha\right)\Upsilon_\infty
=\Upsilon_\infty\,d\]
\item The function $H_\infty(\alpha_1,\olmu)$ is holomorphic on
the simply--connected domain $\D_\infty\subset\IP^1\times\h_1$
given by
\begin{equation}\label{eq:D infty}
\D_\infty=\{(w,\ol\mu)|\,|w|>R_\olmu\}
\end{equation}
and, as a function on $\D_\infty$, $\Upsilon_\infty$ satisfies
\[\left(d-\half{\hbar}\sum_{\alpha\in\sfPhi_+}\frac{d\alpha}{\alpha}
\Kalpha\right)\Upsilon_\infty=
\Upsilon_\infty
\left(d-\half{\hbar}\sum_{\alpha\in{\sfPhi_1}_+}\frac{d\alpha}{\alpha}
\Kalpha\right)\]
\end{enumerate}
\end{prop}
let $\J_\g:={\J_\g}_\pm(\mu):\C\to\Ug
^{\otimes 2}\fml^o$ (resp. $\J_{\g_1}:={\J_{\g_1}}_\pm(\olmu):\C_1\to\Ug_1^{\otimes 2}\fml^o$) be the differential twist for $\g$ (resp. $\g_1$) defined as \ref{ss:differential twist}, where $\C\subset\h^\IR$ (resp. $\C_1\subset \h^\IR_1$) be the fundamental Weyl chamber. Let $r_{\g}$ (resp. $r_{\g_1}$) be the skewsymmetric part of the standard $r$-matrix for $\g$ (resp. $\g_1$) with respect to the choice of positive root system ${\sfPhi}_+$ (resp. ${\sfPhi}_{1+}$). The following proposition relates the differential twists of $\g$ and $\g_1$.
\begin{prop}[\cite{TL}]\label{th:centraliser}
Let $C_{\g_1}$ be defined by the following identity
\[\Delta(\Upsilon_\infty(\alpha_1,\olmu))^{-1}\cdot
\J_\g(\mu)\cdot
\Upsilon_\infty(\alpha_1,\olmu)^{\otimes 2}=
C_{\g_1}\cdot\J_{\g_1}(\olmu).\]
Then $C_{\g_1}\in\Ug^{\otimes 2}\fml^o$ is a constant element commuting with the diagonal action of $\frak l_1$, and satisfies the following properties
\begin{enumerate}
\item $\veps\otimes\id(C_{\g_1})=1=\id\otimes\veps(C_{\g_1})$.
\item $C_{\g_1}\equiv 1^{\otimes 2}\mod\hbar$.
\item $({\Phi\KKZ}_\g)_{C_{\g_1}}={\Phi\KKZ}_{\g_1}$.
\item $\Alt_{2}C_{{\g_1}}=\hbar\left(r_{\g}-r_{\g_1}\right)\mod\hbar^{2}$.
\end{enumerate}
Here the third identity is the relative twist equation
\[C_{\g_1}^{2,3}C_{\g_1}^{1,23}{\Phi\KKZ}_{\g}(C_{\g_1}^{1,2}C_{\g_1}^{12,3})^{-1}={\Phi\KKZ}_{\g_1}.\]
\end{prop}
Such a $C_{\g_1}$ is called a relative twist with respect to the pair $\g_1\subset\g$ and is studied in \cite{TL0}. Note that we have two relative twists $C_{\g_1\pm}$ depending on the choice of $\J^+$ and $\J^-$.
\subsection{Relative twists as quantum connection matrices}
Let us consider the KZ type equation
\begin{eqnarray}\label{eq:realtivesconn}
\frac{d\Upsilon}{dz}=
\left(\ad \lambda_1^{\vee (1)}+\sfh\frac{\Omega}{z}\right)\Upsilon.
\end{eqnarray}
Here recall that $\lambda_i$ denotes the $i$-th fundamental coweight.
Following \cite{TL}, analog to Proposition \ref{pr:Fuchs 0} and Theorem \ref{th:Stokes infty}, the equation has a $\E$--valued
canonical solution $\Psi_0$ at $z=0$ and canonical solutions $\Psi_{\infty\pm}$ on the Stokes sectors $\IH_\pm$ with prescribed asymptotics. Then the quantum connection matrices of \eqref{eq:realtivesconn} are defined as the ratio of $\Psi_0$ and $\Psi_{\infty\pm}$. They actually coincide with the relative twists $C_{\g_1\pm}$ given in Proposition \ref{th:centraliser}. That is
\begin{prop}\cite{TL}
The following holds,
\[C_{\g_1\pm} = \Psi^{-1}_0\cdot \Psi_{\infty\pm}\]
\end{prop}
In other words, the quantum connection matrices $C_{\g_1\pm}$ of the equation \ref{eq:realtivesconn} give rise to relative twists with respect to the pair $\g_1\subset\g$.
\subsection{Semiclassical limit}
Let $\P$ be the holomorphically trivial, principal $G$--bundle on
$\IP^1$, and consider the meromorphic connection on $\P$ given
by
\begin{equation*}
\nabla_{\lambda_1}=d-\left(\frac{\lambda_1^\vee}{z^2}+\frac{B}{2\pi i z}\right)dz.
\end{equation*}
where $B\in\g$. Then on each half plane (the Stokes sector) $\IH_\pm$, the equation $\nabla_{\lambda_1}\gamma=0$ has a canonical fundamental solution $\psi_\pm$ with prescribed asymptotics. See e.g. \cite{Bo2} Appendix. On the other hand, similar to Lemma \ref{le:nr dkz}, if $\nabla$ is non--resonant, there is a canonical fundamental solution $\psi_\infty$ at $z=\infty$. We can therefore define the connection matrix of $\nabla$ via the identity
\[C_{1\pm} = \psi^{-1}_0\cdot \psi_{\infty\pm}.\]
The connection map $C_{1\pm}:\g\nr\to G$ is given by
mapping $B$ to the connection matrices of $\nabla_{\lambda_1}=d-\left(\lambda_1/z^2+
B/2\pi\iota z\right)$.
The same argument as in Section \ref{sec:scl} will relate the canonical solutions $\Psi_\pm$ (resp. $\Psi_0$) of \eqref{eq:realtivesconn} with the solutions $\psi_\pm$ (resp. $\psi_0$) of $\nabla_{\lambda_1}\gamma=0$. In particular, we have
\begin{prop}
The semiclassical limit $\scl{C_{\g_1\pm}^{-1}}$ is the connection
matrix map $C_{1\pm}:\g^*\to G$.
\end{prop}
\subsection{The centraliser property and Ginzburg-Weinstein linearisations compatible with maximal nested sets}
Denote by $G_i$ the simply connected Poisson Lie group associated to the quasi-triangular Lie bialgebra $(\g_i,r_{\g_i})$. The Lie group morphism (inclusion) $\mathcal{T}_{i}:G_{i}\rightarrow G_{i-1}$ is a Poisson Lie group morphism. We denote by $\tau_{i}:\g_{i}\rightarrow \g_{i-1}$ the corresponding infinitesimal Lie algebra morphism, and by $\mathcal{T}_{i}^*:G^*_{i-1}\rightarrow G^*_{i}$ the dual Poisson Lie group morphism.
As in Theorem \ref{th:centraliser}, we define a relative twist $C_{\g_i}$ for each pair $\g_i\subset \g_{i-1}$, where $1\le i\le n$. Then $J_i:=C_{\g_n}\cdot\cdot\cdot C_{\g_{i+1}}$ is an admissible twist of $\g_i$ killing ${\Phi\KKZ}_{\g_i}$, and such that $\Alt_{2}J_i=\hbar r_{\g_i} \mod\hbar^{2}$. Denote by $C_i:=\scl{C_{\g_i}}$ the semiclassical limit of $C_{\g_i}$, and let $j_i:=C_n\cdot\cdot\cdot C_{i+1}$ (the pointwise multiplication) be the semiclassical limit of the twist $J_i$. Following \cite{EEM} Proposition 1.4, we have
\begin{prop}\label{compatibleGW1}
The map $\phi_i:\g^*_i\to G^*_i$ uniquely determined by
\[ j_i(\lambda)\cdot e^{\lambda^\vee}\cdot j_i(\lambda)^{-1}=
L(\phi_i(\lambda))\cdot R(\phi_i(\lambda))^{-1}\]
is a local Poisson isomorphism.
\end{prop}
On the other hand, because each $C_i=\scl{C_{\g_i}}$ is interpreted as the connection matrix map of an ordinary differential equation, it has some further properties. In particular, the restriction of $C_i$ on $\g^*_{i+1}\subset \g^*_i$ maps to the identity of $G$, i.e., $C_i(x)=id$ for any $x\in\g^*_{i+1}$. As an immediate consequence, we have
\begin{prop}\label{compatibleGW2}
The Poisson isomorphisms $\phi_i's$ in Proposition \ref{compatibleGW1} are compatible with the chosen maximal nested sets, in the sense that the resulting diagram commutes
\[\label{eq:diagram0}
\begin{CD}
\g^* @>{\tau_{1}^*}>>\g_{1}^* @>{\tau_{2}^*}>> \cdots @>{\tau_{n}^*}>> \g_{n}^*\\
@VV{\phi_0}V @VV{\phi_1}V @. @VV{\phi_n}V \\
G^* @>>{{\mathcal{T}}_{1}^*}>G_{1}^* @>> {{\mathcal{T}}^*_{2}}> \cdots @>>{{\mathcal{T}}_{n}^*}> G_{n}^*
\end{CD}
\]
\end{prop}
The above two propositions were first proved in \cite{Xu} Section 6.6 via the symplectic geometric property of irregular Riemann-Hilbert correspondence developed by Boalch \cite{Bo1, Bo3}. Here we give a quantum algebra proof.
Therefore from the Poisson geometry perspective, the (semiclassical limit of) centralizer property of the differential twist gives rise to Ginzburg-Weinstein linearisation compatible with maximal nested sets.
\subsection{Relation to Gelfand-Zeitlin systems}
Consider the Lie algebra ${\rm u}(n)$ of ${\rm U}(n)$, consisting of skew-Hermitian matrices, and identify ${\rm Herm}(n)\cong {\rm u}(n)^*$ via the pairing $\langle A,\xi\rangle=2{\rm Im}({\rm tr}A\xi)$. Note that ${\rm u}(n)^*$ carries a canonical linear Poisson structure. On the other hand, the
unitary group ${\rm U}(n)$ carries a standard structure as a Poisson Lie
group (see e.g. \cite{LW}). The dual Poisson Lie group ${\rm U}(n)^*$, the group of complex
upper triangular matrices with strictly positive diagonal entries,
is identified with ${\rm Herm}^+(n)$, by
taking the upper triangular matrix $X\in U(n)^*$ to the positive Hermitian matrix $(X^*X)^{1/2}\in
{\rm Herm}^+(n)$.
These two Poisson manifolds carry densely defined Hamiltonian torus actions which make them integrable systems, known as Gelfand-Zeitlin systems. See \cite{GS, FR} for more details.
In \cite{FR}, Flaschka and Ratiu conjectured the
existence of a distinguished Ginzburg-Weinstein diffeomorphism from ${\rm Herm}(n)$ to ${\rm Herm}^+(n)$, intertwining Gelfand-Zeitlin
systems. One natural candidate is the Stokes map $\calS:\g^*\rightarrow G^*$ for $\g={\rm gl}_n$, which restricts to a Ginzburg-Weinstein diffeomorphism between
the Poisson manifolds ${\rm Herm}(n)$ and ${\rm Herm}^+(n)$ (provided the irregular data $\mu$ is purely imaginary). However, the Stokes map $\calS$ in general is not compatible with the Gelfand-Zeitlin systems. As noted by Boalch \cite{Bo1}
\vspace{3mm}
{\em "Note that the hope of \cite{FR}, that the property of fixing a positive Weyl chamber
would pick out a distinguished Ginzburg-Weinstein isomorphism, does
not hold: the dependence of the monodromy map on the irregular type is
highly non-trivial."}
\vspace{3mm}
The centralizer property of the differential twist constructed by the first author in \cite{TL} enables us to pick out a distinguished Ginzburg-Weinstein isomorphism. Geometrically, {\bf mention its relation with the DCP/wonderful compactification.}
Let $\g={\rm gl}_n(\mathbb{C})$, and let $$0=\g_n\subset \g_{n-1}\cdot\cdot\cdot \g_1\subset \g_0=\g$$ be the Gelfand-Zeitlin chain, i.e., $\g_i$ is consisting of $(n-i)$-th principal submatrices of $\g$. We have seen that the centralizer property of the differential twists allows us to define relative twists $C_{\g_i}'s$, which are in turn connection matrices of certain meromorphic differential equations. Set $\Gamma:=C_n\cdot\cdot\cdot C_1$ as the pointwise multiplication of all their semiclassical limit $C_i:=\scl{C_{\g_i}}$. Following \cite{Xu} Theorem 4.1, we have
\begin{prop}
The composition $${\rm Ad}_\Gamma \circ {\rm exp} : {\rm Herm}(n)\cong u(n)^*\rightarrow {\rm Herm}^+(n)\cong U(n)^*$$
is a Poisson
diffeomorphism compatible with the Gelfand-Zeitlin systems.
\end{prop}
Flaschka-Ratiu conjecture \cite{FR} was first proved
by Alekseev-Meinrenken \cite{AM} using the Moser method in symplectic geometry. It is interesting to compare the diffeomorphism in the proposition with the Alekseev-Meinrenken diffeomorphism.
}
\newpage
|
1,116,691,500,742 | arxiv | \section{Introduction}
In \cite{Wit82}, Witten's study of instantons in the context of
supersymmetry of systems with deformed Hamiltonians gave rise to the
notion of a deformed cohomology ring. This ``quantum cohomology ring''
has since then been formulated precisely in terms of Gromov-Witten
invariants of symplectic manifolds (see \cite{McDSal} for
details). Necessarily, much of the attention paid to quantum
cohomology has been from the point of view of symplectic geometry, {\it e.g.\ }
\cite{RuaTia95, McDSal}. There has also been a great deal of natural
interest in the realm of algebraic geometry, {\it e.g.\ } \cite{KonMan94,
CraMir94}.
Nevertheless, there is strong motivation to pursue an approach which
emphasizes and investigates the parallels between classical and
quantum cohomology. The quantum cohomology ring of a manifold $M$ is
additively essentially the same as the classical cohomology ring of
$M$, but possesses a multiplication which is a deformation of the
classical cup product (see section \ref{secqcoh}). The strong analogy
between the algebraic structures of these two rings is responsible for
the fact that the Euler class has a quantum analogue which we refer to
as the ``quantum Euler class,'' defined in $\S 1$ below. We show here
that this element of the quantum cohomology ring carries with it
information about the semisimplicity, or lack therof, of the quantum
cohomology ring.
The issue of semisimplicity of quantum cohomology rings has already
been under investigation from other points of view, as in \cite{Dub96,
KonMan94}. In \cite{Dub96}), Dubrovin defines a {\bf Frobenius
manifold} $M$ to be a manifold such that each fiber of the tangent
bundle $TM$ has a Frobenius algebra (FA) structure, which varies
``nicely'' from fiber to fiber. This context allows for a close
investigation of the nature of the quantum deformations of classical
cohomology, which is generally realized as $T_0M$, the tangent plane
at ``the origin'' in $M$. Moreover, the fact that $M$ is a Frobenius
manifold is equivalent to the existence of a ``Gromov-Witten
potential'' on $M$ satisfying various differential equations,
including the ``WDVV'' equations [{\it ibid}, p. 133]. Special
manifolds, which Dubrovin calls {\bf massive Frobenius manifolds},
have the additional property that for a generic point $t \in M$, the
FA $T_tM$ is semisimple. In this case, a variety of additional results
relating to the classification of Frobenius manifolds hold [{\it
ibid}, Lecture 3].
Kontsevich and Manin discuss aspects of Frobenius manifolds in
\cite{KonMan94}, but deal with a different notion of
semisimplicity. Working with a manifold $M$ which is essentially the
cohomology ring of some space, they define a particular section $K
\colon M \rightarrow TM$ and, at each point $\gamma \in M$, the linear
operator $B(\gamma) \colon T_{\gamma}M \rightarrow T_{\gamma}M$ which
is ``multiplication by $K(\gamma)$.'' They also define a particular
extension $\tilde{T}M$ of $TM$ and show that if, over a subdomain of
$M$, the operator $B(\gamma)$ is semisimple ({\it i.e.\ } has distinct
eigenvalues), then $\tilde{T}M$ exhibits some special properties. The
notion of semisimplicity of $B(\gamma)$ is referred to as
``semisimplicity in the sense of Dubrovin'' in \cite{TiaXu96} and
other locations.
The quantum Euler class defined here also provides a section of the
tangent bundle of a Frobenius manifold, although its exact connection
with semisimplicity in the sense of Dubrovin is not yet clear.
The general structure and content of this article are as follows: The
expository presentation of classical cohomology in \S
\ref{secclasscoh} highlights the algebraic structures which are
generalized and deformed in \S\S \ref{secFA} - \ref{secqcohhyp}. In
particular, we offer a new canonical description of the Euler class
$e$. The approach of \S \ref{secclasscoh} is extended to the general
case of Frobenius algebras in \S \ref{secFA}, where the generalized
analogue of the Euler class -- ``the characteristic element'' -- is
shown to satisfy the following:
\vspace{.5\baselineskip}
\noindent
{\bf Theorem \ref{thomegaunit}} \ {\it The characteristic element of a
Frobenius algebra $A$ is a unit if and only if $A$ is semisimple.}
\vspace{.5\baselineskip}
Strictly speaking, quantum cohomology should be viewed as a ring
extension, and not an algebra. Section \ref{secFE} provides the
algebraic framework necessary to generalize the material of \S
\ref{secFA} to the case of a Frobenius extension (FE), {\it i.e.\ } when the
base ring is not a field. This having been done, \S \ref{secqcoh}
sketches the elements of the definition of quantum cohomology,
emphasizing its structure as a deformation of classical cohomology,
and in particular as a FE. The ``quantum Euler class'' $e_q$, which is
a deformation of $e$, is defined here to be the characteristic element
of the FE structure of the quantum cohomology ring. Utilizing the
material in \S \ref{secFE}, the semisimplicity test \ref{thomegaunit}
can be applied to quantum cohomology rings.
In the classical and quantum cohomology rings of the complex
Grassmannians, the Euler class and quantum Euler class take on
additional significance. Section \ref{secqcohgrass} outlines how these
rings can be described as Jacobian algebras, where the ideal of
relations is generated by the partial derivatives of the appropriate
Landau-Ginzburg potential $W$ ($W_q$ in the quantum case). In this
context we prove the following result:
\vspace{.5\baselineskip}
\noindent
{\bf Theorem \ref{thhesseuler}} \ {\it The classical and quantum Euler
classes are equal, up to sign, to the determinants of the Hessians of
$W$ and $W_q$, respectively.}
\vspace{.5\baselineskip}
\noindent
This connects the classical and quantum Euler classes to
Morse-theoretic considerations regarding the functions $W$ and
$W_q$. In a sense, it brings them back to the roots of quantum
cohomology in \cite{Wit82}, which utilizes a Morse-theoretic
approach. In addition, this result leads to a new proof of proposition
\ref{prqcohsemi} which, modulo technicalities, states that the quantum
cohomology of any finite complex Grassmannian manifold is semisimple.
Finally, \S \ref{secqcohhyp} applies the semisimplicity test
\ref{thomegaunit} to the quantum cohomology of hyperplanes, providing
good contrast to the situation for the Grassmannians.
\section{Classical Cohomology and the Euler Class}
\label{secclasscoh}
Let $X$ denote a connected $K$-oriented $n$-dimensional compact
manifold, where $n$ is even. Throughout this article, except where
noted otherwise, homology and cohomology groups will use coefficients
in a field $K$ of characteristic $0$. Denote by $[X] \in H_n(X)$ the
fundamental orientation class of $X$, and let $\langle-,-\rangle
\colon H^*(X) \otimes H_*(X) \rightarrow K$ denote the Kronecker
index. The kernel of the linear form $\mu^* \colon H^*(X) \rightarrow
K$ , where $\mu$ denotes the generator of $H^n(X)$ satisfying $\langle
\mu, [X] \rangle = 1$, contains no nontrivial ideals. This form can be
used to define the ``intersection form'' $H^*(X) \otimes H^*(X)
\rightarrow K$, by $a \otimes b \mapsto \mu*(a \cup b)$. The
intersection form is nondegenerate.
Notice that we may view $H_*(X)$ as a (left) $H^*(X)$-module via the
cap product $\cap \colon H^*(X) \otimes H_*(X) \rightarrow
H_*(X)$. Viewing $H^*(X)$ as the regular (left) module over itself, we
see that the Poincar\'{e} duality map
\[
D \colon H^*(X) \rightarrow H_*(X), \ \ \zeta \mapsto \langle-,\zeta
\cap [X]\rangle
\]
is an $H^*(X)$-module isomorphism.
Let $\Delta \colon X \rightarrow X \times X$ denote the diagonal
map. The transfer map $\Delta^! \colon H^*(X) \rightarrow H^*(X)
\otimes H^*(X)$ is defined to be the map which makes the following
diagram commutative:
\[
\begin{diagram}
\node{H^*(X)} \arrow{e,t}{\Delta^!} \arrow{s,l}{D} \node{H^*(X)
\otimes H^*(X)} \\ \node{H_*(X)} \arrow{e,t}{\Delta_*}
\node{H_*(X) \otimes H_*(X)} \arrow{n,l}{D^{-1} \otimes D^{-1}}
\end{diagram}
\]
Here, we implicitly use the isomorphism $H_*(X \times X) \cong H_*(X)
\otimes H_*(X)$, and the corresponding isomorphism for
cohomology. Modulo this latter isomorphism, the cup-product in
$H^*(X)$ is given by $\Delta^* \colon H^*(X) \otimes H^*(X)
\rightarrow H^*(X)$.
Let $j \colon (X \times X, \emptyset) \rightarrow (X \times X,\, X
\times X \setminus \Delta(X))$ denote inclusion of pairs. Consider the
element $\tau := \Delta_!(1) = (D^{-1} \otimes D^{-1}) \circ
\Delta_*([X])$. By the canonical isomorphism of the tangent bundle
$TX$ to the normal bundle of $\Delta(X)$ in $X \times X$
\cite{MilSta}, this is just the image under $j^*$ of the Thom class of
$TX$. It follows that $\Delta^* \circ \Delta^!(1) \in H^*(X)$ is in
fact the Euler class $e(X)$.
We recall the well known formula \cite{MilSta}
\[
e(X) \ = \ \sum_i e_ie_i^{\#},
\]
where $e_i$ ranges over a basis for $H^*(X)$, and $e^{\#}_j$ ranges
over the corresponding dual basis relative to the intersection form,
{\it i.e.\ } $\mu^*(e_i \cup e_j^{\#}) = \delta_{ij}$.
\section{Frobenius Algebras and the Characteristic Element}
\label{secFA}
Let $K$ be a field of characteristic $0$ and let $A$ be a
finite-dimensional (as a vector space) commutative algebra over $K$,
with unity $1_A$. Let $\beta \colon A \otimes A \rightarrow A$ denote
multiplication in $A$, and let $\bar{\beta} \colon A \rightarrow
\mathrm{End}(A)$ denote the regular representation of $A$, {\it i.e.\ }
$\bar{\beta}(a)$ is ``multiplication by $a$.'' View $A$ as the regular
module over itself, and view the vector space dual $A^*$ as an
$A$-module via the action $A \otimes A^* \rightarrow A^*$ given by $a
\otimes \zeta \, \mapsto \, a \cdot \zeta := \zeta \circ \bar{\beta}(a)$.
$A$ is referred to as a {\bf Frobenius algebra} (FA) if there exists
an $A$-module isomorphism $\lambda \colon A \rightarrow A^*$, {\it i.e.\ } a
nondegenerate pairing. In \cite[pages 414-418]{CurRei} this is shown
to be equivalent to the existence of a linear form $f \colon A
\rightarrow K$ whose kernel contains no nontrivial ideals, and to the
existence of a nondegenerate linear form $\eta \colon A \otimes A
\rightarrow K$ which is associative, {\it i.e.\ } $\eta(ab \otimes c) = \eta(a
\otimes bc)$. In fact, we may take $f := \lambda(1_A)$ and $\eta := f
\circ \beta$, and we will henceforth presume that $\lambda, f$ and
$\eta$ are related in this way. When it is useful to emphasize the FA
structure of $A$ endowed by particular $f, \eta,$ and $\lambda$, the
algebra $A$ will be denoted by $(A,f)$.
For the next result, view $A \otimes A$ as an $A$-module via the usual
module action $\beta \otimes I \colon A \otimes A \otimes A
\rightarrow A \otimes A$.
\begin{thm}
\label{thFAcoalg} A finite dimensional commutative algebra $A$ with
$1_A$ is a FA if and only if it has a cocommutative comultiplication
$\alpha \colon A \rightarrow A \otimes A$, with a counit, which is a
map of $A$-modules.
\end{thm}
\begin{prf}
A complete proof appears in \cite{Abr96}. Here, we simply note that if
$A$ is a Frobenius algebra with pairing $\lambda$, then the
comultiplication $\alpha$ is defined to be the map $(\lambda^{-1}
\otimes \lambda^{-1}) \circ \beta^* \circ \lambda$:
\[
\begin{diagram}
\node{A} \arrow{e,t}{\alpha} \arrow{s,l}{\lambda} \node{A \otimes
A}\\ \node{A^*} \arrow{e,t}{\beta ^*} \node{A^* \otimes A^*}
\arrow{n,r}{\lambda^{-1} \otimes \lambda^{-1}}
\end{diagram}
\]
\enlargethispage*{10\baselineskip}
\end{prf}
\pagebreak
Define the {\bf characteristic element of $(A,f)$} to be the element $
\omega_{A,f} := \beta \circ \alpha(1_A) \in A. $ This is a canonical
element which is shown in \cite{Abr96} to be of the form
\[
\omega_{A,f} \, = \, \sum_i e_ie^{\#}_i \, ,
\]
where $e_i$ ranges over a basis for $A$ and $e_j^{\#}$ ranges over the
corresponding dual basis relative to $\eta$.
It is easy to show that theorem \ref{thFAcoalg} still holds if
``commutative'' is replaced by ``skew-commutative,'' as would be the
case for $H^*(X)$. We see that in that case $f, \lambda, \alpha,
\omega$ correspond to $\mu^*, D, \Delta^!, e(X)$, respectively.
Given FA's $(A,f)$ and $(B,g)$, we can form the {\bf direct sum} $(A
\oplus B, f \oplus g)$, where $A \oplus B$ denotes the ``orthogonal
direct sum'' of algebras, and $f \oplus g$ acts by $f \oplus g(a
\oplus b) := f(a)+g(b) \in K$. $(A \oplus B, f \oplus g)$ is in fact
a FA \cite{Abr96}.
\begin{prop} \mbox{\rm \cite{Abr96}}
\label{promegarespect} The characteristic element respects direct
sum structure. Specifically,
\[
\omega_{A' \oplus A'' , f' \oplus f''} = \omega_{A',f'} \oplus
\omega_{A'',f''} \in A' \oplus A''
\]
\end{prop}
The minimal essential ideal $\mathcal{S} = \mathcal{S}(A)$ of a ring $A$ is called the
{\bf socle}. When $A$ is indecomposable, the socle is
$\mathrm{ann}({\mathcal{N}})$, where $\mathcal{N} = \mathcal{N}(A) \subset A$ is the ideal of
nilpotents. See \cite[\S 9]{AndFul} for details.
\begin{prop}
\label{promegasocle} In a FA $A$, the ideal $\omega A$ is the socle
of A.
\end{prop}
This result is independent of the choice of FA structure.
The construction in the following proof is essentially taken from
Sawin \cite{Saw95}, although this result does not explicitly appear
there.
\begin{prf}
Because the socle of a finite-dimensional commutative algebra is the
direct sum of the socles of its indecomposable constituents \cite[\S
9]{AndFul}, it suffices to prove this proposition for the
indecomposable cases. Furthermore, we showed in \cite{Abr96} that the
socle $\mathcal{S}$ of a FA is a principal ideal, any of whose elements is a
generator, so it suffices to show that $\omega$ lies in the
socle. Notice that $\omega$ is not $0$; we have $f(\omega) = (A : K)
\in K$, and this is not $0$ in $K$, since $K$ has characteristic $0$.
If $A$ is a field extension then $\mathcal{N}(A) = \{0\}$, so the socle $\mathcal{S} =
\mathrm{ann}(\mathcal{N}) = A$. But $\omega$ is not zero, so it is a unit, and
thus $\omega A = A = \mathcal{S}$.
If $A$ is not a field extension, define a chain of ideals $\mathcal{S}=S_1
\subset S_2 \subset \cdots \subset S_n=A$, where each $S_k$ is the
preimage in $A$ of the socle of $A/S_{k-1}$. Choose a basis for
$S_1$. Now, starting with $i=1$, iteratively take the basis for $S_i$
and extend it to a basis for $S_{i+1}$. Denote the elements of the
basis for $S_n=A$ by $e_1, \ldots , e_n$, and let $e_1^{\#}, \ldots ,
e_n^{\#}$ denote the corresponding dual basis elements. Suppose $e_i
\in S_k \setminus S_{k-1}$ and that $a \in A$ is any nilpotent
element. Then $ae_i \in S_{k-1}$, and therefore can be expressed as a
linear combination of basis elements other than $e_i$. It follows that
$f(ae_ie_i^{\#}) = 0$, so $e_ie_i^{\#}\mathcal{N}(A) \subset \mathrm{Ker\,} f$. But $\mathrm{Ker\,}
f$ can contain no nontrivial ideals, as mentioned above, so we must
have $e_ie_i^{\#}\mathcal{N}(A) = 0$, ie. $e_ie_i^{\#} \in \mathcal{S}$. This follows
for each $i$, so $\omega = \sum_i e_ie_i^{\#} \in \mathcal{S}$.
\end{prf}
\begin{thm}
\label{thomegaunit} The characteristic element $\omega$ of a FA $A$
is a unit if and only if $A$ is semisimple.
\end{thm}
\begin{prf}
First, recall from the proof of \ref{promegasocle} that $\omega$ is
not $0$. Because $A$ is commutative, it is semisimple if and only if
it is a direct sum of fields. In such a case, the component of
$\omega$ in each component of $A$ is nonzero (each component is a FA
\cite{Abr96}), and hence a unit. Since a direct sum of units is a
unit, $\omega$ is a unit.
If some component $A'$ of $A$ is not a field, then it contains
nontrivial nilpotents. In this case, $\mathcal{S}(A') = \mathrm{ann}(\mathcal{N}(A'))$
is nilpotent, so $\omega$ has a nilpotent component, and cannot be a
unit.
\end{prf}
In a skew-commutative context, such as $H^*(X)$, the characteristic
element is not necessarily nonzero. For instance, if $X$ is an
odd-dimensional compact oriented manifold then the characteristic
element, {\it i.e.\ } the Euler class, is $0$. However, if the characteristic
element is in fact nonzero, then \ref{thomegaunit} still holds.
\section{Frobenius Extensions}
\label{secFE}
Suppose $A/R$ is a finite-dimensional (as a module) commutative ring
extension with identity. By analogy with FA's, if there exists a
module isomorphism $\lambda \colon A \rightarrow A^*$, we call $A$ a
{\bf Frobenius extension} (FE). As in the case of FA's, this is
equivalent to the existence of maps $\eta$ and $\alpha$. There is also
a ``FE form'' $f := \lambda(1_A) \colon A \rightarrow R$, but in this
context it is not sufficient for the kernel of $f$ to contain no
nontrivial ideals. The characteristic element $\omega_{A,f}$ may be
defined as for FA's, but note that theorem \ref{thomegaunit} no longer
applies. This section provides an approach for dealing with this
circumstance.
Suppose $\theta \colon R \rightarrow S$ is a surjective homomorphism
of rings (sending $1_R \mapsto 1_S$). Let $(A,f)$ denote a FE, and
define $B = \theta_*(A)$ to be $A \otimes_R S$. In this ring, we have
$ra \otimes s \, = \, a \otimes \theta(r)s$ for all $r \in R,\, s \in
S$, and $a \in A$. Let $\tilde{\theta} \colon A \rightarrow B$ denote the
ring homomorphism $a \mapsto a \otimes 1_S$. Define the linear form
$\bar{f} \colon B \rightarrow S$ by
\[
\bar{f}(a \otimes s) := \theta \circ f(a)s.
\]
The form $\bar{f}$ is well-defined, since
\[
\bar{f}(ra \otimes s) \, = \, \theta \circ f(ra)s \, = \, \theta(rf(a))s
\, = \, \theta(r)(\theta \circ f(a))s \, = \,
\bar{f}(a \otimes \theta(r)s),
\]
and $\bar{f}$ satisfies the commutative diagram \[
\begin{diagram}
\node{A} \arrow{e,t}{\tilde{\theta}} \arrow{s,r}{f} \node{B}
\arrow{s,r}{\bar{f}} \\ \node{R} \arrow{e,t}{\theta} \node{S}
\end{diagram}
\]
Let $e_1, \ldots, e_n$ denote a basis for $A$, and let $e_i^{\#},
\ldots, e_n^{\#}$ denote the corresponding dual basis relative to
$\eta_A$.
\begin{prop}
\label{prinducedFE} The form $\bar{f}$ endows $B=\theta_*(A)$ with a
FE structure, and
\[
\omega_{B,\bar{f}} = \tilde{\theta}(\omega_{A,f}).
\]
\end{prop}
\begin{prf}
It suffices to show that the set $\{\tilde{\theta}(e_1), \ldots,
\tilde{\theta}(e_n)\}$ is a basis for $B$, and that its dual basis relative
to the form $B \otimes B \rightarrow S$, $a \otimes b \mapsto
\bar{f}(ab)$ is $\{\tilde{\theta}(e_1^{\#}), \ldots,
\tilde{\theta}(e_n^{\#})\}$. The existence of a dual basis will show $B$ is
a FE. The particular form of the basis and dual basis, together with
the fact that $\tilde{\theta}$ is a homomorphism, will prove the claim
about $\omega_{B,\bar{f}}$.
We first prove the orthogonality relations:
\[
\bar{f} \left( \tilde{\theta}(e_i)\tilde{\theta}(e_j^{\#}) \right) = \bar{f} \left(
\tilde{\theta}(e_ie_i^{\#}) \right) = \theta \circ f(e_ie_i^{\#}) =
\theta(\delta_{ij}) = \delta_{ij} \in S.
\]
To prove that we have a basis as claimed, note that the elements
$\tilde{\theta}(e_1), \ldots, \tilde{\theta}(e_n)$ clearly span $B$, since
$\tilde{\theta}$ is surjective. Suppose that for some $\{s_i\} \in S$ we
have $\sum_i s_i\tilde{\theta}(e_i) = 0$. Then, for all $j$,
\[
0 = \bar{f}(0) = \bar{f} \left( \sum_i
s_i\tilde{\theta}(e_i)\tilde{\theta}(e_j^{\#}) \right) = s_j.
\]
It follows that $\tilde{\theta}(e_1), \ldots, \tilde{\theta}(e_n)$ are
independent, and thus form a basis. The orthogonality relations show
that $\{\tilde{\theta}(e_1^{\#}), \ldots, \tilde{\theta}(e_n^{\#})\}$ is a basis
as well.
\end{prf}
In the next result, let $\theta \colon R \rightarrow K$ be any
surjective $K$-linear ring homomorphism, where $K$ is a field.
\begin{prop} \hspace{2pt}
\label{prinducedomega} The element $\omega_{A,f}$ is either a unit
in $A$ or a zero divisor.
\begin{description}
\item[(i)] If $\omega_{A,f}$ is a unit in $A$ then $B =
\theta_*(A)$ is semisimple. \item[(ii)] If $\omega_{A,f}$ is a
zero divisor and \ $\mathrm{ann}(\omega_{A,f}) \nsubseteq \mathrm{Ker\,}
\tilde{\theta}$, then $\theta_*(A)$ is not semisimple.
\end{description}
\end{prop}
\begin{prf}
If $\omega_{A,f}$ is a unit, then there exists a $u \in A$ such that
$\omega_{A,f} u = 1_A$. But then, by \ref{prinducedFE}
\[
\omega_{B,\bar{f}}\tilde{\theta}(u) = \tilde{\theta}(\omega_{A,f})\tilde{\theta}(u) =
\tilde{\theta}(1_A) = 1_B,
\]
so $\omega_{B,f}$ is a unit as well.
All FE structures on $A$ are given by $(A,f \circ \bar{\beta}(u))$, for some
unit $u \in A$ \cite[Proposition 2, {\it mutatis
mutandis}]{Abr96}. Thus, if $\omega$ is not a unit in $A$, then the
map $\omega_{A,f} \cdot f$ is not a FE form. This implies that there
exists an $a \in A$ such that $f(\omega_{A,f} aA) = \omega_{A,f} \cdot
f(aA) = \{0\}$. But $f$ is a FE form, so it must be that $\omega_{A,f}
a = 0$. If follows that $\tilde{\theta}(\omega_{A,f})\tilde{\theta}(a) =
0$. Since, by assumption, there exists some $a \in
\mathrm{ann}(\omega_{A,f})$ such that $a \notin \mathrm{Ker\,} \tilde{\theta}$, we
see that $\tilde{\theta}(\omega_{A,f}) = \omega_{B,\bar{f}}$ is a zero
divisor as well. Both statements (i) and (ii) now follow from theorem
\ref{thomegaunit}.
\end{prf}
\section{Quantum Cohomology and the Quantum Euler Class}
\label{secqcoh}
Let $X$ be a $2n$-dimensional compact oriented manifold which, in
addition, is symplectic, and let $H'_2(X)$ denote the free part of
$H_2(X,\mathbb{Z})$. Taking $B_1, \ldots, B_n$ to denote a basis of $H'_2(X)$,
the group algebra $\Lambda := K[H'_2(X)]$ may be expressed as
$K[q^{B_1}, \ldots, q^{B_n}]$, where $q$ is a formal variable and the
addition of exponents is the group operation of $H'_2(X)$. This is
essentially an algebraic version of the Novikov ring (see \cite[\S
9.2]{McDSal}). As an additive group, the {\bf quantum cohomology} ring
$\mathbf{QH^*(X)}$ has the same structure as $H^*(X) \otimes \Lambda$,
but has a ``deformed'' multiplication, which we describe briefly:
The classical cup product of two elements $a, b \in H^*(X)$ is given
by
\[
a \cup b = \sum_i(\alpha \cdot \beta \cdot \gamma_i)c_i \, ,
\]
where $c_i$ runs over a basis for $H^*(X)$ and $\alpha, \beta,
\gamma_i$ are the Poincar\'{e} duals of $a, b, c_i$, respectively, and
``$ \cdot$'' denotes the homology intersection index. The quantum
multiplication
\[
\ast \colon QH^*(X) \otimes QH^*(X) \rightarrow QH^*(X)
\]
is defined on elements $a,b \in H^*(X) \hookrightarrow QH^*(X)$ by
\[
a \ast b \, := \, \sum_{i,B} \Phi_B(\alpha, \beta, \gamma_i)q^Bc_i \,
,
\]
and extended by linearity to all of $QH^*(X)$. Here, $B$ ranges over
$H'_2(X)$, and $\Phi_B(\alpha, \beta, \gamma_i)$ denotes the Gromov
(Gromov-Witten) invariants. Intuitively, these count intersections
(subject to dimension requirements!) of the cells $\alpha, \beta,
\gamma_i$ not with themselves, but with the fourth cell $B$. When $B =
0$, the Gromov invariant is the classical intersection index. Thus,
\[
a \ast b \ = \ a \cup b \, + \mathrm{other\ terms}.
\]
For details regarding the definition of quantum cohomology, and in
particular proofs of the associativity of $\ast$, see
\cite{McDSal,RuaTia95}.
Extend $\mu^* \colon H^*(X) \rightarrow K$ (defined in section
\ref{secclasscoh}) by linearity over $\Lambda$ to a form $\mu^* \colon
QH^*(X) \rightarrow \Lambda$.
\begin{prop}
\label{prqcohFE} The form $\mu^*$ endows $QH^*(X)$ with a FE
structure.
\end{prop}
\begin{prf}
See \cite{Abrth} for a rigorous proof.
\end{prf}
Let $\iota \colon H^*(X) \hookrightarrow QH^*(X)$ denote the obvious
inclusion map. Note that although $H^*(X)$ and $QH^*(X)$ share
essentially the same basis $\{e_i\}$ and the same FA form, the
respective dual bases are not necessarily equal. In other words, the
fact that the element $e_i^{\#}$ is the dual in $H^*(X)$ to $e_i$ does
not necessarily imply that $\iota(e_i^{\#})$ is dual to $\iota(e_i)$
in $QH^*(X)$. However, it does hold that the $q^0$ term of
$\iota(e_i)^{\#}$ is in fact $\iota(e_i^{\#})$. It follows that the
$q^0$ term of the characteristic element $\omega_q$ of $(QH^*(X),
\mu^*)$ is $e(X)$. In other words, we have:
\begin{center}
The characteristic element $\omega_q$ is a deformation of the
classical Euler class.
\end{center}
Because of this, we refer to $\omega_q$ as {\bf the quantum Euler
class}, and denote it by $e_q(X)$. Unlike $e(X)$, the quantum Euler
class may very well be a unit. Strictly speaking, however, the
semisimplicity result \ref{thomegaunit} does not apply to $QH^*(X)$,
because it is infinite dimensional (as a vector space) over $K$ and
only a ring extension (not an algebra) over $\Lambda.$ We may,
however, utilize the approach of section \ref{secFE}. Define the
homomorphism $\theta \colon \Lambda \rightarrow K$ as follows: For
each generator $B_i$ of $H'_2(X)$ choose any nonzero $r_i \in K$ and
define $\theta(q^{B_i}) := r_i$. Extending $\theta$ by linearity over
$K$ gives a surjective ring homomorphism, often referred to as
``specialization.'' Theorem \ref{thomegaunit} now applies to
$\theta_*[QH^*(X)]$, which is a FA.
\section{The Quantum Cohomology of the Grassmannians}
\label{secqcohgrass}
Let \ensuremath{G_{k,n}}\ denote the Grassmannian manifold of complex $k$-dimensional
subspaces in $\mathbb{C}^n$. Define the Chern polynomial of $X = \ensuremath{G_{k,n}}$ to be
\[
c_t(\ensuremath{G_{k,n}}) := \sum_{i=1}^k x_it^i = \prod_{i=1}^k(1 + \lambda_it),
\]
where $t$ is a formal variable and the $x_i$'s are the Chern classes
of the canonical bundle \ensuremath{{\bf S}_{k,n}}. The $\lambda_i$ are referred to as the
{\bf Chern roots} of \ensuremath{G_{k,n}}\ (but they are {\it not} roots of
$c_t$!). Obviously, $x_i$ is the $i$'th elementary symmetric
polynomial $\sigma_i(\lambda_1, \ldots, \lambda_k)$ in the Chern
roots. Define
\[
W(\lambda_1, \ldots, \lambda_k) = \sum_{i=1}^k
\frac{1}{n+1}\lambda_i^{n+1}
\]
\begin{center} and \end{center}
\[
\begin{array}{rcl}
W_q(\lambda_1, \ldots, \lambda_k) & = & \displaystyle{ \sum_{i=1}^k
\frac{1}{n+1}\lambda_i^{n+1} + (-1)^kq\lambda_i} \\ \\ & = &
W(\lambda_1, \ldots, \lambda_k) + (-1)^kqx_1.
\end{array}
\]
The function $W_q$ is called the Landau-Ginzburg potential of
\ensuremath{G_{k,n}}. Because $W$ and $W_q$ are symmetric functions in the
$\lambda_i$, they may also be viewed as functions of $x_1, \ldots,
x_k$. Define $dW$ to be the ideal $( \frac{\partial W}{\partial x_1},
\ldots, \frac{\partial W}{\partial x_k} )$, and define $dW_q$
similarly. Then
\[
H^*(\ensuremath{G_{k,n}}) \cong K[x_1, \ldots, x_k] / dW
\]
\begin{center} and \end{center}
\[
QH^*(\ensuremath{G_{k,n}}) \cong K[q,q^{-1}][x_1, \ldots, x_k] / dW_q .
\]
Denote by $H$ and $H_q$ the determinants of the Hessians
\[
\H = \left( \frac{\partial^2 W}{\partial x_i \partial x_j} \right) \ ,
\ \ \H_q = \left( \frac{\partial^2 W_q}{\partial x_i \partial x_j}
\right)
\]
of $W$ and $W_q$, respectively.
In this section, we will prove the following:
\begin{thm}
\label{thhesseuler} $e(\ensuremath{G_{k,n}}) = (-1)^{n \choose 2}H$ \ and \
$e_q(\ensuremath{G_{k,n}}) = (-1)^{n \choose 2}H_q$.
\end{thm}
Suppose that an algebra $A$ (not necessarily a FA) is finite
dimensional as a vector space and is given by the presentation $A
\cong K[x_1, \ldots, x_n]/R$, where $R = (f_1, \ldots, f_p)$ is some
finitely-generated ideal in $K[x_1, \ldots, x_n]$. Note that we
continue to assume that $K$ has characteristic $0$. Because $A$ is
finite dimensional, we must have $p \geq n$. The {\bf Jacobian ideal}
$J = J(R)$ of $R$ is defined to be the ideal generated by the
determinants of the $n \times n$ minors of the matrix
\[
\left( \frac{\partial(f_1, \ldots, f_p)}{\partial(x_1, \ldots, x_n)}
\right) \mathrm{mod}\, R.
\]
The ideal $J$ is well-defined since it is a Fitting ideal of the
module $\Omega_{A/K}$ of K\"{a}hler differentials of $A$ (see \cite[\S
1.1, \S 10.3]{Vas}).
The following result of Scheja and Storch \cite{SchSto75} is reported
in more generality in \cite[{\it ibid}]{Vas} (although for the
definition of ``complete intersection'' we refer the reader to
\cite{Kun}):
\begin{prop}
\label{prjacobiansocle} $J \neq \{0\}$ if and only if $A$ is a
complete intersection and $J$ generates the socle of $A$.
\end{prop}
Now assume that $A \cong K[x_1, \ldots, x_n]/R$ is a FA with
characteristic element $\omega$ for some choice of FA structure.
\begin{prop}
\label{promegajacobian} $J \neq \{0\}$ if and only if $J = \omega
A$. If $p=n$, then $J \neq \{0\}$ if and only if
\[
\mathrm{det} \left( \frac{\partial f_i}{\partial x_j} \right)
\mathrm{mod}\, R = u \omega
\]
for some unit $u \in A$.
\end{prop}
\begin{prf}
This proposition follows immediately from \ref{promegasocle} and
\ref{prjacobiansocle}.
\end{prf}
\begin{prop}
\label{prHessomega} For each \ensuremath{G_{k,n}}\ there is a $\kappa \in K$ such
that $H = \kappa e(\ensuremath{G_{k,n}})$ and $H_q = \kappa e_q(\ensuremath{G_{k,n}})$.
\end{prop}
\begin{prf}
Because $H$ and $e(\ensuremath{G_{k,n}})$ are the $q^0$ terms of $H_q$ and
$e_q(\ensuremath{G_{k,n}})$, respectively, it suffices to prove the proposition for
the quantum case.
The polynomial $W_q$ is homogeneous of degree $2(n+1)$ \cite[\S
8.4]{McDSal}. In other words, each summand of $W_q$ has degree
$2(n+1)$ in $QH^*(\ensuremath{G_{k,n}})$, where $q$ is taken to have degree
$2n$. Also, $|x_i| = 2i$ for each $i$. Thus, for fixed $i, j$ we have
\[
\left| \left( \frac{\partial^2 W_q}{\partial x_i \partial x_j} \right)
\right| \ = \ |W_q| - |x_i| - |x_j| \ = \ |W_q| - 2i - 2j.
\]
We now show by induction that $H_q$ is homogeneous of degree
$2k(n-k)$. Each $s \times s$ minor $M$ of $\H$ is a matrix with
entries $m_{ij} := \H_{ij}$ where $i$ and $j$ run over elements of
some ordered subsets $I, J \subset \{1, \dots, k\}$ respectively, and
$\#I = \#J = s$. Define $M\widehat{(i,j)}$ to be the minor of $M$
which does not include the entry $m_{ij}$. We have already shown that
when $s=1$, the single entry of each $M$ is homogeneous of degree
$|W_q| - 2i - 2j$. Assume that for all minors $M$ of $\H$ of size less
than $(s+1) \times (s+1)$, the determinant of $M$ is homogeneous of
degree
\[
s |W_q| - \, 2\sum_{i \in I}i \ - \ 2\sum_{j \in J}j.
\]
Now consider any $(s+1) \times (s+1)$ minor $M$ of $\H$ with index
sets $I, J$. Take any $i' \in I$. Then
\[
\mathrm{det} M = \sum_{j' \in J}
\mathrm{sgn}(i',j')m_{i'j'}\mathrm{det}M\widehat{(i',j')},
\]
where $\mathrm{sgn}$ is the appropriate function $I \times J
\rightarrow \{+1, -1\}$. By the induction hypothesis,
\[
\begin{array}{rcl}
\left| m_{i'j'}\mathrm{det}M\widehat{(i',j')} \right| & = &
|m_{i'j'}| + |M\widehat{(i',j')}| \\ \\ & = & |W_q| - 2i' - 2j'
+ s |W_q| - {\displaystyle \left( 2\sum_{i \in
I\setminus\{i'\}}i \right) - \left( 2\sum_{j \in
J\setminus\{j'\}}j \right) }\\ \\ & = & (s+1) |W_q| - \
{\displaystyle 2 \sum_{i \in I}i \ - \ 2\sum_{j \in J}j }.
\end{array}
\]
But this is independent of the choice of $j'$, so det$M$ is
homogeneous of degree $(s+1) |W_q| - \, 2\sum_{i \in I}i \, - \,
2\sum_{j \in J}j$. In particular, we can take $M = \H$, and thus $H$
is homogeneous of degree
\[
k |W_q| \ - \ 2\sum_{i=1}^k i \ - \ 2s\sum_{j=1}^kj \ = \ k(2n+2) -
2k(k+1) \ = \ 2k(n-k).
\]
Of course, $e_q$ is also homogeneous of degree $2k(n-k)$ since $e_q =
\sum_i e_ie_i^{\#}$, where $e_i$ runs over a basis for $H^*(\ensuremath{G_{k,n}})$,
and since $|e_i^{\#}| = 2k(n-k) - |e_i|$.
Consider the algebra $A:=K[q][x_1,\ldots,x_k]/dW_q$. By the
definitions of $H_q$ and $e_q$, and the nature of the relations given
by $dW_q$, both $H_q$ and $e_q$ may be viewed as elements of $A$. Now,
the proof of \ref{prinducedFE} applies equally well to the algebra
$A$, so it is a FE, and $e_q$ is in fact the characteristic element of
$A$. Proposition \ref{promegajacobian}, shows that $H_q = ve_q(\ensuremath{G_{k,n}})$
for some unit $v \in A$. Of course, $v$ may also be viewed as an
element of $QH^*(\ensuremath{G_{k,n}})$ which simply has no $q^i$-terms with $i<0$.
Write $v = v' + v''$ where $v'$ is homogeneous of degree $0$, and
$v''$ contains no terms of degree $0$. Since $H_q = v'e_q + v''e_q$
and both $H_q$ and $v'e_q$ are homogeneous of degree $2k(n-k)$, we see
that $v''e_q$ must also be homogeneous of this degree. By degree
considerations, we must have $v''e_q = 0$, and thus $H_q = v'e_q$.
Write $v' = \sum_{j \geq 0} v_j q^j$. Since $v'$ is homogeneous of
degree $0$ and $|q^j| = 2jn$, we see that $|v_j| = -2jn$. But $v'$ is
an element of $A$, so we must have $v_j = 0$ for $j \neq 0$. Thus $v'$
may in fact be viewed as a degree $0$ element in $H^*(\ensuremath{G_{k,n}})$. In other
words, $v'$ is an element $\kappa \in K$.
\end{prf}
Take $K=\mathbb{R}$ or $\mathbb{C}$, and for any nonzero $r \in K$ let $\theta_r$
denote a specialization homomorphism $K[q,q^{-1}] \rightarrow K,\ q
\mapsto r$ as above. In the following paragraph, any reference to
$QH^*(\ensuremath{G_{k,n}})$ or any element $a$ therein should be interpreted as
referring to $(\theta_r)_*[QH^*(\ensuremath{G_{k,n}})]$ and $\tilde{\theta}_r(a)$,
respectively.
In this context, the relationship between the distinguished element
and the Hessian provides $e_q(\ensuremath{G_{k,n}})$ with a nontrivial geometric
interpretation: Denote the critical points of $W_q$ by $z_1, \ldots,
z_j$, and note that $H_q$ may be viewed as a function $K^k \rightarrow
K$, as may all the elements of $QH^*(\ensuremath{G_{k,n}})$. It is well known that,
for each $j$, $H_q(z_j)=0$ if and only if the critical point $z_j$ is
degenerate \cite{Mil}. Because the elements of $QH^*(\ensuremath{G_{k,n}})$, viewed as
functions, are completely determined by their values on the critical
points of $W_q$, we see that $H$ (and hence $e_q(\ensuremath{G_{k,n}})$) is a unit in
$QH^*(\ensuremath{G_{k,n}})$ if and only if the critical points of $W_q$ are all
nondegenerate.
This relationship between $e_q(\ensuremath{G_{k,n}})$ and $H$ also yields a new
approach to the following known result \cite{SieTia94}:
\begin{prop}
\label{prqcohsemi} For all \ensuremath{G_{k,n}}\ and all nonzero $r \in \mathbb{R}$, the
algebra
\begin{center}
$(\theta_r)_*[QH^*(\ensuremath{G_{k,n}})]$ is semisimple.
\end{center}
\end{prop}
The proof is based on calculations appearing in \cite{Ber94}.
\begin{prf}
The Jacobian matrix $V = (\partial x_i / \partial \lambda_j)$
associated to the elementary symmetric functions $x_i$ is a
Vandermonde matrix, and has determinant $\prod_{i<j}(\lambda_i -
\lambda_j) \neq 0$. Let $\nabla_x$ denote the gradient vector operator
with respect to $x_1, \ldots, x_k$, and let $\nabla_{\lambda}$ denote
the gradient operator with respect to $\lambda_1, \ldots,
\lambda_k$. Viewing the gradient operators as row vectors, we have
$\nabla_x(W_q)V = \nabla_{\lambda}(W_q)$. Let $\nabla_x(W_q)_i$ denote
the $i$'th entry of $\nabla_x(W_q)$, and let $V_i$ denote the $i$'th
row of $V$. Then the Hessian of $W_q$ with respect to the $\lambda$'s
is
\[
\begin{array}{rcl}
\nabla_{\lambda}^T \nabla_{\lambda}(W_q) & = & \nabla_{\lambda}^T
(\nabla_x(W_q)V) \\ \\ & = & V^T \nabla_x^T \nabla_x(W_q) V +
\sum_i \nabla_x(W_q)_i \, \nabla_{\lambda}^T V_i.
\end{array}
\]
Evaluating at the critical points of $W_q$ ({\it i.e.\ } assuming
$\nabla_x(W_q)=0$), and expressing everything in terms of the
$\lambda_i$'s, we see that
\[
H = \mathrm{det} \left( \nabla_{\lambda}^T \nabla_{\lambda}(W_q)
\right) \mathrm{det} (V^{-2}) = \frac{n^k \prod_{i=1}^k
\lambda_i^{n-1}} {(\prod_{i<j}(\lambda_i - \lambda_j))^2} .
\]
Now, because $V$ is invertible, the relation $\nabla_x(W_q)=0$ is
equivalent to $\nabla_{\lambda}(W_q)=0$. In other words, for each $i$
we have $\lambda_i^n = (-1)^{k+1}q$ at the critical points of
$W_q$. This implies that
\[
x_k \prod_{i=1}^k \lambda_i^{n-1} = \prod_{i=1}^k \lambda_i^n =
(-1)^{k(k+1)}q^k.
\]
Since $q \neq 0$, the numerator of $H$, and thus $H$ itself, is
nonzero at the critical point of $W_q$. It follows that $H$, as an
element of $QH^*(\ensuremath{G_{k,n}})$, has an inverse, and therefore, by
\ref{prHessomega}, so does $e_q(\ensuremath{G_{k,n}})$. By proposition
\ref{prinducedomega}, $\theta_r[QH^*(\ensuremath{G_{k,n}})]$ is semisimple.
\end{prf}
As discussed above, the Chern classes $x_1, \ldots, x_k$ arising from
the bundle \ensuremath{{\bf S}_{k,n}}\ are the elementary symmetric polynomials in the Chern
roots $\lambda_1, \ldots, \lambda_k$. An analogous situation holds for
the ``normal'' classes $y_1, \ldots, y_{n-k}$, which arise from the
quotient bundle \ensuremath{{\bf Q}_{k,n}}. Define $\mu_1, \ldots, \mu_{n-k}$ to be the
Chern roots corresponding to the formal polynomial
\[
\sum_{i=1}^{n-k} y_it^i.
\]
Then for all $i$, we have $y_i = \sigma_i(\mu_1, \ldots,
\mu_{n-k})$. In fact, the $\lambda_i$'s and $\mu_i$'s are the first
Chern classes of the line bundles in the splitting of \ensuremath{{\bf S}_{k,n}}\ and \ensuremath{{\bf Q}_{k,n}},
respectively \cite[\S 21]{BottTu}. Together with the well known
bundle-isomorphism of the tangent bundle $\ensuremath{{\bf T}_{k,n}} \cong \ensuremath{{\bf S}_{k,n}}^* \otimes
\ensuremath{{\bf Q}_{k,n}}$, this fact allows us to write the characteristic classes
$c_i(\ensuremath{{\bf T}_{k,n}})$ in terms of the $x_i$'s and $y_i$'s: The Chern polynomial
for \ensuremath{{\bf T}_{k,n}}\ is
\[
\sum_{i=1}^{k(n-k)} c_i(\ensuremath{{\bf T}_{k,n}})t^i \ = \ \prod_{i,j}(1 + (\mu_j -
\lambda_i)t ),
\]
where $i$ and $j$ in the product range over possible indices
\cite[{\it ibid}]{BottTu}. In other words, for each i we have
$c_i(\ensuremath{{\bf T}_{k,n}}) = \sigma_i(\{\mu_j-\lambda_i\}_{i,j})$. This shows that
each $c_i(\ensuremath{{\bf T}_{k,n}})$ is symmetric in the $\lambda_i$'s and the $\mu_j$'s,
and can therefore be written in terms of the $x_i$'s and $y_j$'s.
In particular, the Euler class $e(\ensuremath{G_{k,n}})$ can be lifted to a polynomial
\[
P \in K[x_1, \ldots, x_k, y_1, \ldots, y_{n-k}]
\]
or, using the relations between the $x_i$'s and $y_j$'s, to a
polynomial
\[
P' \in K[x_1, \ldots, x_k].
\]
$P'$ is referred to as the ``Euler polynomial.''
Bertram \cite{Ber94} has proven the following:
\begin{prop}
\label{prliftHess} For each $(k,n)$, the Euler polynomial $P'$ is a
lifting of $(-1)^{n \choose 2}H_q \in QH^*(\ensuremath{G_{k,n}})$.
\end{prop}
We can now prove theorem \ref{thhesseuler}.
\begin{prf} {\bf (of theorem \ref{thhesseuler})}
Proposition \ref{prHessomega} shows that $e_q:=e_q(\ensuremath{G_{k,n}}) = \kappa H_q$
for some $\kappa \in K$. Let $\pi \colon QH^*(\ensuremath{G_{k,n}}) \rightarrow
H^*(\ensuremath{G_{k,n}})$ denote the module homomorphism sending $q \mapsto 0$. By
definition, $P'$ is a lifting of $e:=e(\ensuremath{G_{k,n}})$, so (by
\ref{prliftHess}) we have
\[
\begin{diagram}
\node{K[x_1, \ldots, x_k]} \arrow{s} \arrow{se} \\ \node{H^*(\ensuremath{G_{k,n}})}
\node{QH^*(\ensuremath{G_{k,n}})} \arrow{w,t}{\pi}
\end{diagram} \quad
\begin{diagram}
\node{P'} \arrow{s} \arrow{se} \\ \node{e(\ensuremath{G_{k,n}})} \node{(-1)^{n
\choose 2}H_q } \arrow{w,t}{\pi}
\end{diagram}
\]
where the unlabeled arrows are the canonical projection maps. Now
$\pi(e_q) = e$, by definition of the quantum multiplication $\ast$, so
proposition \ref{prHessomega} shows that
\[
(-1)^{n \choose 2}e \, = \, \pi \left( (-1)^{n \choose 2}e_q \right)
\, = \, \pi \left( (-1)^{n \choose 2}\kappa H_q
\right) \, = \, \kappa e,
\]
and thus $\kappa = (-1)^{n \choose 2}$.
\end{prf}
\section{Quantum Cohomology of Hyperplanes}
\label{secqcohhyp}
For the sake of contrast with the Grassmannians, this section provides
another class of examples of a quantum cohomology ring, and determines
which of these are semisimple. In \cite{TiaXu96}, Tian and Xu discuss
a more general class of examples along these lines from the point of
view of semisimplicity in the sense of Dubrovin (as defined in the
introduction).
Let $X \subset \mathbb{C} P^{n+r}$ be a smooth complete intersection of
degree $(d_1, \ldots, d_r)$ and dimension $n \geq 2$ satisfying $n
\geq \sum(d_i-1) - 1$. Let $\Gamma$ denote the hyperplane class
generating $H^2(X,\mathbb{Z})$. By the ``primitive cohomology $H^n(X)_0$ of
$X$'' we mean $H^n(X)$ if $n$ is odd, and the subspace of $H^n(X)$
orthogonal to $\Gamma^{n/2}$ if $n$ is even. Beauville shows in
\cite{Bea95} (although he unnecessarily presumes $q=1$), that
$QH^*(X)$ is the algebra over $K[q,q^{-1}]$ generated by $\Gamma$ and
$H^n(X)_0$, subject to the relations
\[
\Gamma^{n+1}\, = \, d_1^{d_1} \cdots d_r^{d_r}\Gamma^{d-1}q
\]
and, for all $a,b \in H^n(X)_0$,
\[
\Gamma a = 0 \ \ \ \mathrm{and}\ \ \ ab = \langle a,b \rangle
\frac{1}{d} \left(\Gamma^n - d_1^{d_1} \cdots d_r^{d_r}\Gamma^{d-2}q
\right).
\]
Here, $\langle \cdot, \cdot \rangle$ denotes the classical
intersection form $a \otimes b \mapsto f(a \cup b)$, where
$f:=(\Gamma^n)^*$.
\begin{prop}
Let $X$ denote a hyperplane of degree $d$. For any nonzero $r \in K$,
if $d > 2$ then $(\theta_r)_*[QH^*(X)]$ is not semisimple. If $d=2$
then $(\theta_r)_*[QH^*(X)]$ is semisimple.
\end{prop}
\begin{prf}
Denote $(H^n(X)_0:K)$ by $R$, and choose a basis $e_1, \ldots, e_R$
for $H^n(X)_0$. Together with the elements $1, \Gamma, \Gamma^2,
\ldots, \Gamma^n$, this provides a full vector-space basis for
$QH^*(X)$. Thus the characteristic element of $(QH^*(X),f)$ is
\[
\begin{array}{rcl}
\omega & = & \sum_{i=0}^n \Gamma^i \Gamma^{n-i} + \sum_{i=1}^R
e_ie_i^{\#} \\ \\ & = & (n+1)\Gamma^n +
\frac{R}{d}\left(\Gamma^n - d^d \Gamma^{d-2}q \right).
\end{array}
\]
Notice that if $d > 2$ then $\omega$ is divisible by $\Gamma$, so
$\omega e_1 = 0$ (for example). Since $e_1 \notin \mathrm{Ker\,} \tilde{\theta}$ (it
is a basis element for any choice of coefficients!), proposition
\ref{prinducedomega} shows that $(\theta_r)_*[QH^*(X)]$ is not
semisimple for any choice of $r \in K$.
If $d=2$ then we have
\[
\omega \, = \, (n+1+\frac{R}{2})\Gamma^n - 2Rq,
\]
and thus $\Gamma\omega = 4(n+1)q\Gamma$. Order the basis for $QH^*(X)$
as follows: $1, \Gamma, \Gamma^2, \ldots, \Gamma^n,$ $e_1, \ldots,
e_R$. Then the matrix $[\bar{\beta}(\omega)]$ corresponding to $\omega$
under the regular representation $\bar{\beta}$ is
\[
\left( \begin{array}{ccccc|ccc} -2Rq & & & & n+1+\frac{R}{2} & \\ &
4(n+1)q & & $\LARGE{0}$ & \\ & & 4(n+1)q & & & & \\ & $\LARGE{0}$ &
& \ddots & & \\ & & & & 4(n+1)q & \\ \hline & & & & & -2nq & & 0 \\
& & & & & & \ddots & \\ & & & & & 0 & & -2nq \end{array} \right) .
\]
Since the determinant of this matrix is a unit in $K[q,q^{-1}]$, we
see that $\omega$ is a unit in $QH^*(X)$; by \ref{prinducedomega} this
shows that $(\theta_r)_*[QH^*(X)]$ is semisimple for any choice of
$r$.
\end{prf}
\section*{Acknowledgements}
The author thanks Jack Morava for his guidance of the research leading
to this article. Aaron Bertram, Steve Sawin and Geng Xu were also
helpful in correspondence and conversation, and Peter Landweber
graciously assisted with proof-reading.
|
1,116,691,500,743 | arxiv | \section{Introduction}
The HESS air Cerenkov telescope located in Namibia has recently made a ground-breaking detection
\citep{Aharonian2006} of
a diffuse flux (over a solid angle of approximately 4$^\circ \, \times 2^\circ $)
of $\gamma$-rays in the 0.2-20 TeV (1 TeV = $10^{12}$ eV) range emanating from the region
of the Galactic center
(GC) and distributed along the Galactic plane. This flux has been shown by the HESS collaboration
to be correlated
with the density of molecular gas (mostly H$_2$) pervading the region, as determined from CS
measurements \citep{Tsuboi1999}, and argued, therefore, to originate primarily from neutral
pion (and other meson) decay. These putative
mesons would themselves be generated in collisions between cosmic ray protons or heavier ions
(generically, cosmic ray hadrons; CRHs) and
the ambient gas, the latter of which is contained in a number of giant molecular clouds complexes (GMCs), including
the Sagittarius (Sgr) B complex.
We label this broad idea `the hadronic scenario'.
Highly significantly for our purposes, the HESS collaboration has separately
measured the $\gamma$-ray flux
emanating from a
$0.^\circ 5 \, \times 0.^\circ 5$ field centred on and covering
the Sgr B GMC itself (defined in Galactic co-ordinates by $0.^\circ 3<l<0.^\circ 8$ and $-0.^\circ 3<b<0.^\circ 2$)
and the spectrum of this emission.
Within the hadronic scenario,
the well-known particle physics of pion (and heavier meson) production and subsequent decay together with the HESS
$\gamma$-ray measurements allow us to make a prediction for the rate at which secondary electrons
and positrons (generically, secondary leptons)
would be injected into the Sgr B cloud through the same CR collision mechanism that, by hypothesis, supplies the
$\gamma$-rays (CRH collisions lead to production of charged and neutral pions -- and charged and neutral
kaons -- in roughly equal numbers and
the charged
pion and kaon decay chains ultimately terminate in the e$^\pm$ alluded to, together with neutrinos).
With the additional input of the local magnetic field strength, the gas density, and
the CRH spectral shape (viz., that the latter is a power-law
in momentum as generically predicted by shock acceleration theory with a {spectral index } and {normalization } that match it smoothly
on to the HESS observations)
we can predict the synchrotron radio spectrum from e$^\pm$ secondaries generated over the entire Sgr B GMC.
In fact, we can, on the basis of these inputs,
self-consistently calculate the entire broadband spectrum of the Sgr B complex
from radio to TeV energies, accounting for all relevant radiative processes (secondary electron synchrotron, bremsstrahlung
and inverse Compton radiation as well as neutral meson decay).
The hadronic interpretation of the HESS data has recently been
challenged by \citet{Yusef-Zadeh2006} who, remarking on an independent correlation between Fe K$\alpha$
X-ray line emission and TeV $\gamma$-ray emission across the GC region,
have suggested a model where a population (or populations) of {\it primary}
electrons is responsible for the non-thermal emission seen from a number of molecular clouds, including
Sgr B2.
These authors suggest, in particular, that the TeV emission originate in inverse Compton (IC)
scattering of sub-millimeter
radiation from dust by the high energy component of the putative electron population.
We label the broad idea that primary electrons are responsible for the bulk of the Sgr B phenomenology
`the leptonic scenario'.
We will examine the compatibility of both the hadronic and leptonic scenarios
with the broadband non-thermal spectrum of Sgr B and other Sgr B phenomenology
in detail in the current work.
\section{Inferred Environment and Morphology of Sgr B}
On the basis of CS measurements \citep{Tsuboi1999}, the HESS collaboration report that the
mass of molecular
material within the half degree by half degree field covering the Sgr B Complex is
between 6 and 15 million solar masses. In the following
we consider both these bounding values in modeling emission at various wavelengths from the region.
Sgr B is one of a number of GMCs bound
in relatively tight orbits (projected distance of $\sim$ 100 pc) around Sgr A*,
the radio source associated with the supermassive ($\sim 3.7 \times 10^6 {\,M_\odot}$)
black hole at the center of the Galaxy (at an assumed distance of 8.5 kpc).
Together these complexes constitute the central molecular zone, a structure that contains
fully $\sim$10\% of the total molecular material of the Galaxy ($3 - 8 \times 10^7 {\,M_\odot}$
according to the CS measurements: \cite{Aharonian2006}, \cite{Tsuboi1999}).
The GC GMCs are characterised by supersonic internal velocity dispersions
($\sim 15 - 50$ km s$^{-1}$: \cite{Morris1996}),
high molecular gas kinetic temperature and increased metallicity
(\cite{Mayer-Hasselwander1998} and references therein).
The Sgr B complex contains two bright
sub-regions, Sgr B1 and Sgr B2, that show up in radio continuum observations
\citep{Mehringer1995,Yusef-Zadeh2006}. The latter is
the largest molecular cloud in the Galaxy and contains one its largest
complexes of HII regions, being dominated by numerous ultra-compact and hyper-compact
H II regions which inhabit three dense cores (labeled North, Main, and South).
These are, themselves, located inside a structure labeled the Envelope \citep{Gordon1993}.
There have been $\sim$57 UCHII
sources identified in Sgr B2(M) alone (Gaume \& Claussen (1990), Gaume,
et al. (1995) and De Pree, Goss, \& Gaume (1998)). The cores are small
($\sim$0.5 pc), light ($10^{3-4}M_{\odot}$,
corresponding to $\sim$5\% of the Sgr B2 mass) and dense ($10^{6-
7}$ cm$^{-3}$). On the other hand, the envelope is massive ($7.6\times10^{5}M_{\odot}$), and less dense
(10$^{5}$ cm$^{-3}$). The average $n_{H_2}$ across
Sgr B2 is $\sim 10^6$ cm$^{-3}$.
Sgr B1, a region of diffuse
thermal emission, is comparatively less studied than Sgr B2. It is an HII region of
optical depth much less than one at $\sim$ cm wavelengths, suggesting that it is older, evolved structure
no longer containing the dense, hot
star forming UCHII regions found in Sgr B2
\citep{Mehringer1995}.
\subsection{Average Molecular Hydrogen Number Density of Sgr B}
In general, for molecular cloud material distributed through the CMZ at a distance $R_{\rm GC}$ from the
GC, the minimal number density required in order that the cloud's self-gravity overcome
tidal shearing by the central bar potential is given
by \citep{Bania1986,Gusten1989,Stark2004,Morris2007}:
\begin{equation}
n_{H_2}^{\rm min} \simeq 10^4 {\rm cm}^{-3} \left(\frac{75 {\rm pc}}{R_{\rm GC}}\right)^{1.8} \, .
\label{eqn_critDnsty}
\end{equation}
The average\footnote{Note that this is the average over mass at a given density,
not the volumetric average} molecular hydrogen number density over the entire
Sgr B complex $\left< n_{H_2} \right>$
is close to this critical value (given the $\sim$ 100 pc separation between Sgr B and the GC)
and lies in the
$10^{(3.5-4)}$ cm$^{-3}$ range \citep{Lis1989,Lis1990,Paglione1998}.
We adopt a working figure of $\left< n_{H_2} \right> = 10^4$ cm$^{-3}$ for definiteness.
As may be adjudged from the number densities quoted above, however,
one should keep in mind that the distribution of gas across Sgr B is highly clumpy.
Nevertheless, we have tested our models of synchrotron emission for injection into such a clumpy medium
(as parameterized on the basis of the results set out in Figure 6 of \cite{Paglione1998})
and find that integrated results are in agreement to within 20\% (small on the scale of
other uncertainties) with those obtained using a one-zone model
-- as henceforth employed -- with
average values for molecular density (and magnetic field).
\subsection{Average Magnetic Field of Sgr B}
\label{section_BSgrB}
The determination of magnetic field strengths through the GC region
has been of long-standing interest; a very recent -- though not entirely disinterested review --
is that by \citet{Morris2007}.
The magnetic phenomenology of the GC is dominated by the so-called
non-thermal filaments (NTFs; \cite{Yusef-Zadeh1987,Morris1996}).
The NTFs are remarkable structures: they are
magnetic flux tubes of up to 30 pc length but only fractions of pc wide,
running predominantly perpendicular to the Galactic plane,
and illuminated by synchrotron emission.
They are found at distances of up to 150 pc from the GC --
across a swathe of Galactic longitude, then, apparently conincident with the CMZ -- but not outside this region.
Given the almost invariant curvature of the filaments and their regularity -- despite clear interactions
with the turbulent GC ISM \citep{Morris2007} --
these structures must be quite rigid, consequently allowing the
inference of a very large field strength of $\sim$ mG scale \citep{Yusef-Zadeh1987,Morris1996}.
A long-standing and on-going controversy concerns the question of whether the $\sim$mG scale magnetic
field strength and structure determined for the NTFs is actually pervasive or not.
It is beyond the scope of this paper to consider this issue in any detail.
Instead, we shall adopt the approach of considering both the upper and lower scales for global GC magnetic field strength
argued for by various researchers in the following, viz., $\sim$ 1 mG and $\sim 10 \, \mu$G.
The assumed strength of this global field shall be of importance when we come to consider
whether a central object (located close to or at Sgr A*) is possible as the source for the populations of
relativistic particles we infer to be pervading Sgr B.
In any case, the poloidal
field structure traced by the NTFs
is observed (via far-IR polarization measurements: see \cite{Morris1996}, \cite{Novak2003})
to break down in GC molecular clouds.
In fact, the field direction inside GC molecular clouds tends to be drawn out along a direction following the ridge
of molecular and thermal radio emission,
predominantly {\it parallel} with the Galactic plane, therefore and at right angles to the
NTF fields.
Such a configuration might arise naturally from the shearing
action of the tidal field at the GC on the orbital motion on the molecular gas,
drawing an initially poloidal configuration of field lines fixed in the matter into a toroidal one
as an inadvertant consequence of differential gas rotation (in a situation where
gravitational forces overcome magnetic ones).
As regards Sgr B in particular,
the average magnetic field of the complex, especially
the average field magnitude over large scales, is
somewhat uncertain.
A determination of the line-of-sight magnetic
field strength $B_{\rm los}$ in the Sgr B2 region of the cloud
has also been made on the basis of
Zeeman mapping the H I line in absorption
\citep{Crutcher1996}. This is 0.48 mG with a spatial variation of around 50 \%.
Given the
resolution of those observations (5$\arcsec$ for Sgr B2), this
estimate may apply to the outer envelope of the cloud
complex \citep{Crutcher1996}.
Further, the figure of $\sim$ 0.5 mG is actually a {\it lower limit}
to the average local field strength as line-of-sight field reversals
can only reduce the field strength inferred from Zeeman splitting and not increase it \citep{Novak1997}.
Also note that, statistically for a large ensemble of molecular clouds, $B_{\rm los} = 1/2 \, | {\bf B}|$
\citep{Crutcher1996}.
Sub-millimetre polarimetry has also been used to trace the topology of the Sgr B2 field
\citep{Novak1997,Chuss2005} and, in concert
with the Zeeman measurement mentioned above, determines a lower limit to the large scale Sgr B2 field of 150 $\mu$G
(to be distinguished from the local, presumably somewhat disordered, field directly measured as being at $\gtrsim$ 0.5 mG
by the Zeeman technique alone).
The field structure of Sgr B is revealed by the most recent measurements to be quite complicated \citep{Chuss2005}.
The field of Sgr B2 in particular seems to follow a spiral, suggesting an ``extreme example of local forces
dominating the field structure" \citep{Chuss2005}.
Finally, using results obtained from an ensemble of 27 separate Zeeman
measurements of individual molecular clouds and assuming
uniform-density spherical clouds, \citet{Crutcher1999}
has derived the following scaling between magnetic field strength
and particle number density:
\begin{equation}
B = 0.1 \left(\frac{n_{H_2}}{{10^4\mbox{ cm}^{-3}}}\right)^{0.47} \mbox{~~~mG}.
\label{eqn_Crutcher}
\end{equation}
The Zeeman observation of Sgr B2 \citep{Crutcher1996} define
one of the data points from which this fit is obtained
(with an assumed $H_2$ number density of $10^{3.4}$ cm$^{-3}$),
but it should be noted that the Sgr B2 data point actually falls furthest
from (and considerably above) the fitted scaling relation (see \citet{Crutcher1999}, Figure 1).
Nevertheless,
the relation does capture the
approximate $\sqrt{n_{H_2}}$ scaling
for magnetic field strength
that
is predicted by many lines of theory (see \citet{Crutcher1999} and references therein),
including
ambipolar-diffusion-driven star formation.
The scaling relation predicts, in our assumed average
$\left< n_{H_2} \right> = 10^4$ cm$^{-3}$, a field of 0.1 mG.
\subsection{Hydrogen Column Density to and through Sgr B}
The hydrogen column density {\it to} Sgr B is estimated to be $\sim \, 1 \times 10^{23}$ H cm$^{-2}$
\citep{Murakami2001,Revnivtsev2004}. The column density {\it through} the densest part of the cloud,
the central 2' radius region of Sgr B2, is estimated to be $\gtrsim \, 8 \times 10^{23}$ H cm$^{-2}$
\citep{Lis1989,Murakami2001}. These two values bound the possible range for the average column density
to the emission in the one-zone models investigated in this paper (we actually fit to this parameter: see below).
\subsection{Temperature of Gas in Sgr B}
There is
evidence for a least two separate gas components across the CMZ:
a 150-200K component as traced by multi-transition ammonia lines (see \cite{Yusef-Zadeh2006} and references therein)
and a cooler and denser component with small volume filling factor. There is also
a 70 K dust component across the region.
Like the density structure, the temperature structure of Sgr B is complicated
and it is therefore difficult to nominate a `typical' temperature for the gas in this complex.
In general across Sgr B2 there is a
strong temperature gradient ranging from about 100K
at higher Galactic latititudes and longitudes down to about 40K at lower latititudes and longitudes.
The dense cores in Sgr B2
are at higher temperatures, $\sim$ 300 K, due presumably to heating by young stars
and the surrounding, lower density envelope may be hotter still (600-700K: \cite{Wilson2006}).
Furthermore, \citet{Wilson2006} have detected the (J,J) = (18,18) line of NH$_3$
in Sgr B2 which indicates a very hot component at 600-700K.
Sgr B1 hovers at around 40-50K but does have some hot
spots of gas at temperatures up to 90K \citep{Ott2007}.
Also, as mentioned above,
the centimeter radio continuum maps show that Sgr B2 has a significant number
($\sim$60) of compact, ultra compact (UC) and hyper compact (HC) HII regions,
indicating extremely hot gas, at temperatures of few times $10^{3}$ K on size
scale of 10's to 100's of AU \citet{dePree1998},
probably due to massive star formation in the dense cores of Sgr B2 Main and North.
Keeping in mind that, given the clumpiness of the density distribution, much of the total mass within the complex
is located in the sub-structures named above, we estimate that the
typical temperature for gas in the Sgr B complex lies in the range 150-200K.
\section{Radio Observations of Sgr B}
As described in detail below,
we wish to compare the radio flux predictions we arrive at with
radio data we have collected from various sources.
For fluxes at 330 and 74 MHz ($\sim$90 cm and $\sim$400 cm)
we make use of archival
Very Large Array\footnote{The Very Large Array (VLA), as part of the National Radio Astronomy Observatory, is operated
by Associated Universities, Inc., under cooperative agreement with the National Science Foundation.}
data that covers the region in question \citep{Brogan2003}.
At 843 MHz we make use of unpublished data obtained in the course of
the Sydney University Molonglo Sky Survey \citep{Bock1999}\footnote{The Molonglo Observatory Synthesis Telescope (MOST)
is operated by the University of Sydney}.
Finally,
we have recently observed the dense cores of Sgr B at 1.384 and 2.368 GHz (13 and 20 cm) with the Australia
Telescope Compact Array\footnote{The Australia Telescope Compact Array (ATCA)
is operated by the Australia Telescope National Facility, CSIRO, as a National Research Facility.} \citep{Jones2007}.
In the future we will perform observations with this array in a short baseline configuration that should allow the detection
of the non-thermal flux emanating from a relatively large solid angle we predict on the basis of the present work. For the
moment, however, the data we have collected were obtained in a baseline configuration relatively insensitive to such flux
and provide, therefore, only a rather weak lower limit to the radio flux from the whole Sgr B complex at these wavelengths.
On the other hand, we have used these 13 and 20 cm measurement to probe the cosmic ray populations in a number of
sub-structures within Sgr B2: see \citet{Jones2007}.
We have obtained 74 and 330 MHz VLA maps \citep{Brogan2003}
of the Sgr B region both obtained with a beam of $\sim$1$\arcmin\times$2$\arcmin$.
The
short baseline coverage with which these maps were obtained
implies a sensitivity to diffuse flux from scales of up to
6$^{\circ}$ at 74 MHz and 1-2$^{\circ}$ at 330 MHz.
Fluxes at both 74 and 330 MHz were obtained by integrating within the
entire $0.5^{\circ}\times0.5^{\circ}$ region defined in \citet{Aharonian2006}
and a smaller region of $0.35^{\circ}\times0.30^{\circ}$.
Unfortunately for our purposes,
74 MHz emission from the GC region is considerably
free-free absorbed by intervening HII regions \citep{Brogan2003,Kassim1990}
meaning that intrinsic flux is badly determined.
Moreover, the Sgr B complex is not obviously detected
above background
in the map at 74 MHz so the total flux at this frequency
over the solid angle of the Sgr B complex only defines
a rough upper limit to the flux attributable to the object in question.
The determination of the 330 MHz flux was made by
taking the statistics
over the region, using the Miriad task \emph{histo}, and multiplying the
mean flux within the region by the number of beams within it.
The flux derived is 45 Jy with an estimated (RMS) error for the primary beam corrected
image of 4 Jy.
Following a similar procedure -- but noting the provisos outlined above --
the 74 MHz flux from the $0.5^{\circ}\times0.5^{\circ}$ region is 570 $\pm 120$ Jy.
We convolved the 330 MHz map to the HESS resolution, using a Gaussian with a
standard deviation of 0.07$^{\circ} \, = 252''$.
This is shown as the gray scale of Figure \ref{fig_90cm-smeared-HESS-boxes-overlay}.
The $0.5^{\circ}\times0.5^{\circ}$ field defined above and from which total radio fluxes
were calculated is shown as the larger of the two rectangles
and the circle is the smoothed beam size.
The peak at 330 MHz region is spatially correlated
with the peak of the HESS emission which is given by the contours and sits on top of the Sgr B2 structure.
The western lobe of the peanut-shaped radio emission feature sits on top of Sgr B1.
Using the above techniques, we also calculated the total 330 MHz flux from the smaller rectangle displayed in
Figure \ref{fig_90cm-smeared-HESS-boxes-overlay} which turns out to be 24 Jansky.
The ratio of radio to $\gamma$-ray fluxes agrees
to better than 20\% between the two regions,
meaning that the correlation between radio and $\gamma$ emission is no better (actually slightly worse) for the
smaller region when averaged over this scale.
We will perform a more sophisticated analysis of the spatial correlation between
radio and $\gamma$-rays in a future work.
In the map one notes a clear radio signal from the SNR/pulsar wind nebula G0.9+0.1 which does not
seem to be correlated at all with $\gamma$-ray emission, but one should keep in mind here that this object {\it has} been detected
by HESS as a point source and its flux has been artificially removed from the $\gamma$-ray map as discussed below in
\S \ref{section_TeV}.
From unpublished data obtained in the course of the
SUMSS data at 843 MHz we have also obtained
the total flux from the HESS-defined $0^\circ.5 \times 0^\circ.5$ field covering Sgr B.
Because of the strong sidelobes caused by the emission from Sgr A, however, we
can only make a crude flux estimate of 20 $\pm 10$ Jy.
The spectral index between the central values of the 330 and 843 MHz fluxes is -0.9, with an allowable range
of -0.4 to -1.4 clearly indicating
a non-thermal spectrum even granted the large errors on the 843 MHz flux.
\section{Other Observations of Sgr B}
\subsection{X-ray Observations of Sgr B}
Galactic center molecular clouds, Sgr B2 in particular, have been intensively observed over the last decade in X-rays.
The X-ray data covering Sgr B2 reveal both strong continuum emission and a bright source of
fluorescent Fe K$\alpha$ line radiation within the cloud Sgr~B2
that displays an unusually large
equivalent width of $\approx 2$--$3$ keV.
In our data fitting we use the
sub-set of the collection of broad-band X-ray data points assembled by \citet{Revnivtsev2004}
(and presented in their Figure 2; we re-display these points
in our Figures \ref{fig_plotH2aBB}
and \ref{fig_plotL2eBB})
that approximately give only the {\it continuum} emission from Sgr B2.
Note that these data points -- which are due to ASCA/GIS\citep{Koyama1996,Murakami2000},
GRANAT/ART-P, and INTEGRAL/IBIS over
ascending photon energy -- are for observations on somewhat smaller angular scales than the
$0.^\circ 5 \, \times 0.^\circ 5$ field whose broad-band emission we are trying to model.
They do, however,
cover a good fraction of the total mass in the Sgr B Complex given the clumpiness of the gas distribution
(around $2 \times 10^6 \, {\,M_\odot}$).
In any case, because of the uncertainty introduced by this mismatch we do not try to reproduce
the continuum X-ray emission in minute detail
in our modeling below. Instead we seek only to broadly match the normalization and
spectral index of this emission.
Note also that {\it we do not seek to reproduce the Fe K$\alpha$ line emission at all in this paper}.
A cloud radiates via X-ray
fluorescence when it is illuminated, either internally or externally,
by a source of $\sim$ 8 keV X-rays or $\sim$ 30 keV CR ions or electrons.
Now, a steady X-ray source embedded within the
cloud produces an upper limit to the equivalent width of only $\sim 1$ keV
\cite[see, e.g.,][]{Fabian1977,Vainshtein1980,Fromerth2001}.
A number of authors have taken the large equivalent width of Sgr~B2, then, as evidence for illumination by
an external X-ray source located towards the actual GC, usually identified with Sgr A*
(see \cite{Revnivtsev2004} and references therein).
In this scenario, then,
we are witnessing the X-ray echo, delayed by
300--400 years relative to the direct signal from the black hole, due
to the light travel time from the Galactic center out to Sgr~B2's
position (in which time, by hypothesis,
the direct source has fallen back into relative quiescence).
\citet{Fryer2006} have recently suggested that the large dynamic range
(6 orders of magnitude: \citet{Revnivtsev2004,Fryer2006})
for Sgr A* required in this scenario is very difficult to account for theoretically and a more
plausible transient X-ray source of the appropriate magnitude and in the appropriate direction from Sgr B2
may have been provided by the collision of
the expanding shell of the Sgr~A East supernova remnant
and the so-called 50 km s$^{-1}$ molecular cloud.
An alternative explanation to the hypothesised external X-ray illuminator has been advanced by
\citet{Valinia2000,Yusef-Zadeh2002} and \cite{Yusef-Zadeh2006}, viz.
that non-thermal electrons collide with ambient matter to produce inner-shell ionizations
of iron atoms, leading to 6.4\,keV line emission, and
simultaneously generate bremsstrahlung radiation to supply the observed continuum X-ray emission.
Again the large equivalent width of the line emission is important here:
the low energy electron interpretation of the Fe K$\alpha$ emission
would require a metallicity significantly higher -- a factor of 3-4 -- than solar
and \citet{Revnivtsev2004} in fact raise this as a difficulty with the model.
But it is already known
that GC MCs have metallicities 1.5-2 solar, in general. Thus as
\citet{Yusef-Zadeh2006} remark, this scenario only requires that the metalicity of Sgr B2 be 1.5-2
that of surrounding clouds which, given the amount of star formation taking place in this object,
is perfectly credible.
In any case, because we do not actually attempt to reproduce the Fe K$\alpha$ line emission
from Sgr B2 in this paper, it is beyond the scope of this paper to go further into this controversy
(this issue will be addressed in a later work: \cite{Crocker2006}).
Rather, following the lead of \citet{Yusef-Zadeh2006}, in the modeling we present below,
we will attempt only to reproduce (broadly) the continuum X-ray spectrum of Sgr B2 with bremsstrahlung emission from
low energy electrons, either primary or secondary.
Whether the Fe K$\alpha$ emission seen is due to an external X-ray illuminator
or, as in the model of \citet{Yusef-Zadeh2006}, is actually due to low-energy
cosmic ray electron collisions, is immaterial for our purposes: the models for
non-thermal particle populations we eventually arrive at would seem
to work in either situation.
\subsection{TeV $\gamma$-ray Observations of Sgr B}
\label{section_TeV}
Observations of the Galactic centre region by the HESS instrument shortly after its inception
led to
the
detection of a point-like source of $\sim$TeV $\gamma$-rays at
the dynamical centre of the Galaxy
(HESS J1745-2290: \cite{Aharonian2004}) and
compatible with the position of the supermassive black hole
Sagittarius A* or the unusual supernova remnant (SNR) Sgr A East \citep{Crocker2005}. A deeper
observation of the region in 2004 revealed a second source: the SNR/pulsar wind nebula G0.9+0.1 \citep{Aharonian2005}.
Most recently, the HESS collaboration has demonstrated that, after
subtracting these two point sources from their map of the GC,
residual, fainter features are evident, in particular,
emission extending along the Galactic plane for roughly 2$^\circ$ \citep{Aharonian2006}.
This foreground-subtracted map is displayed as the contours in Figure \ref{fig_90cm-smeared-HESS-boxes-overlay}.
The morphology of this diffuse emission correlates well with the
distribution of molecular material in clouds as traced by CS emission \citep{Tsuboi1999}
and the spectrum of this emission, which is detected by HESS over more than two orders of magnitude in photon energy,
is constant, within systematic errors,
across the entire region with a spectral index of $\sim$2.3. This is appreciably harder
than emission detected at $\sim$ GeV energies across the Galactic plane -- the origin of the latter being
roughly compatible with creation in collisions of a CRH population of the same shape as that observed locally, viz.,
a spectral index of $\sim$2.7.
Moreover, the HESS-detected $\gamma$-ray emission above 1 TeV is a factor of 3-9 times higher than in the Galactic
disk and would seem to require, therefore, a
different or additional cosmic ray
population in this region.
\subsection{GeV $\gamma$-ray Observations of Sgr B}
In contrast to its clear detection in $\gamma$-rays at $\sim$ TeV energies, Sgr B has not been detected in the 30 MeV - 30 GeV
energy range, despite being in the field of view of the EGRET telescope's lengthy observations of the GC region
\citep{Mayer-Hasselwander1998}. Indeed, it was explicitly noted by \citet{Mayer-Hasselwander1998} that
no localized excess associated with the Sgr B complex was detected by EGRET excluding the possibility of a significantly
enhanced CR density in these clouds -- {\it in the appropriate energy range}, of course.
The closest
source EGRET did detect in these pointings, 3EG J1746-2851,
was at first thought to have a localization
marginally compatible
with Sgr A* or Sgr A East, but was later shown to be perceptibly off-set to Galactic east \citep{Hooper2005,Pohl2005}.
This imples that this $\sim$ GeV source can not be identified with the point source seen by HESS
at/near Sgr A* (HESS J1745-2290:
\cite{Crocker2005}). The extent of the 3EG J1746-2851 source was closely investigated by \citet{Mayer-Hasselwander1998}.
They found that, though a point source could not be ruled out, a best fit single source model marginally implied
emission (above 1 GeV) such that a flux enclosure angle of radius
0$^\circ$.6 was required to encompass 68 \% of the total flux.
This is a solid angle larger than the largest under consideration here
(given by the
entire Sgr B TeV emission region of $0^\circ.5 \times 0^\circ.5$). Furthermore,
the fact that the 3EG J1746-2851 source was only marginally determined to be extended
(its HWHM was $0^\circ.7$ to be contrasted with the HWHM of the pulsar (GeV) point sources Vela, Geminga, and Crab at
$0^\circ.55$) means that at GeV+ energies the Sgr B complex would have been indistinguishable from a point source
were it detected by EGRET.
In concert, the non-detection by EGRET of Sgr B and its detection in the same pointings of
the source 3EG J1746-2851
imply an upper limit to the $\gamma$-ray emission from Sgr B of the same
level as the detected flux from 3EG J1746-2851
(though it must be acknowledged here that putative Sgr B flux limits provided by
low energy EGRET data points ($<300$ MeV)
are more indicative than strict because of the growth in the EGRET psf towards lower energies).
Moreover, these two, in concert with total molecular mass determinations for Sgr B,
further imply that
the density of CRHs through Sgr B at $\sim$ GeV-100 GeV energies must not be significantly higher than the local CRH density
in the same energy range as noted by \citet{Mayer-Hasselwander1998}.
This is a significant constraint for our model fitting to obey as explained below.
\subsection{Cosmic Ray Ionization Rate in Sgr B2}
Recent determinations \citep{vanderTak2006}
of the cosmic ray ionization rate $\zeta_{CR}$ in the Sgr B2 Envelope
from observations
of H$_3$ O$^+$ emission lines at 364 and 307 GHz with the Atacama Pathfinder Experiment (APEX)
telescope\footnote{The
Atacama Pathfinder Experiment (APEX)
telescope is operated by Onsala Space Observatory, Max Planck Institut f{\" u}r
Radioastronomie (MPIfR), and European Southern Observatory (ESO).} show that
$\zeta_{CR}$ here is around 4 $\times 10^{-16}$ s$^{-1}$ (with a factor of four
uncertainty). The central value is thus an order of magnitude larger than
the value determined for local molecular clouds up to a few kpc from the Sun \citep{vanderTak2000}
and in the vicinity of the Solar System
on the basis of extrapolation of
measurements taken by the Pioneer and Voyager spacecraft, viz. 3 $\times 10^{-17}$ s$^{-1}$
\citep{Webber1998}. Van der Tak et~al. (2006) have also determined a
lower limit for $\zeta_{CR}$ in Sgr B2(Main) of around 4 $\times 10^{-17}$ s$^{-1}$ and estimate
that the actual value of this quantity is $\sim$ 1 $\times 10^{-16}$ s$^{-1}$
The authors conclude that {\it the ionization rates of dense molecular clouds are mainly determined
by their location in the Galaxy through variations in the ambient CR flux} and that, as a second order
effect, $\zeta_{CR}$ may be $\sim 3$ times lower in dense molecular clouds than in diffuse clouds.
In any case, these observations would seem to justify adopting an average $\zeta_{CR}$ for the entire
Sgr B Complex of 4 $\times 10^{-16}$ s$^{-1}$ -- and this may actually be an underestimate.
\subsubsection{Cosmic Ray Heating in Sgr B}
In passing we note that, employing the results obtained by \citet{Suchkov1993} for the temperature of
molecular gas heated by cosmic rays and cooled by molecular line emission (see their Figure 1),
the cosmic ray ionization rate given above, together with the assumed average
molecular hydrogen number density of $10^{4}$ cm$^{-3}$, translates into a temperature of $\sim$ 30 K.
It seems, then, that given the temperature range also stated above cosmic ray heating cannot
be the dominant heating mechanism in Sgr B.
\section{Secondary Particle Spectra from CRH Collisions}
We wish to model the injection of electrons, positrons and $\gamma$-rays into the Sgr B environment through the
decay of charged and neutral pions (and heavier mesons) which are themselves created in collisions between hadronic cosmic rays
(protons and
heavier ions all the way up to Fe) and ambient gas (which is roughly 93\% H nuclei -- mostly in H$_2$ molecules in the
molecular cloud environment --
and 7\% He nuclei). To this end, we have employed two Monte Carlo event generators, TARGET 2.1a \citep{Engel2003}
and SIBYLL 2.1 \citep{Engel2000}, that simulate
proton-proton collisions at a given energy, to create numerical yield data for secondaries.
We employed TARGET, in particular, to generate yields from single p-p collisions between 1.259 and 100 GeV
and SIBYLL to generate yields from 100 to $10^6$ GeV and these
two calculations were smoothed together at 100 GeV.
We then used a MATHEMATICA routine to interpolate the yield data for energies intermediate to those directly
simulated.
Secondary spectra due to collisions between an arbitrary distribution of beam CR particles
with gas in the molecular cloud can be written as
\begin{equation}
q_2 (E_2) = \int \, dE_p \, \frac{d N_p}{d E_p}(E_p) \, n_H \, \epsilon(E_p) \, \sigma_{pH}(E_p) \, \frac{d N_2}{d E_2}(E_p,E_2) \, ,
\end{equation}
where $q_2$ is in secondaries/eV/s/cm$^{-3}$ with secondary $\in \{e^-, e^+, \gamma \}$;
\begin{equation}
d N_p(E_p)/d E_p = \frac{4\pi \, J_p(E_p)}{\beta(E_p) \, c}
\end{equation}
is the number density of cosmic ray p's per
unit energy pervading the medium; $J_p$ is the differential flux of cosmic rays protons at total energy $E_p$ in
cm$^{-2}$ s$^{-1}$ eV$^{-1}$ sr$^{-1}$; $n_H$ is the number density
of H nucleus targets in the gas;
$\epsilon(E_p)$ is the (unitless) energy-dependent modification factor introduced
by \citet{Mori1997} to account for
the presence of heavier ions in both target and
beam\footnote{This factor is determined from the observed terrestrial CRH spectrum and abundances.
It is constant with a value of $\sim$ 1.5 from CR proton energies 1 GeV - 100 GeV, climbing
slowly thereafter to be $\sim$ 1.85 at $10^5$ GeV.
This factor may, of course, actually
be an underestimate in the case of the GC environment
where a higher supernova rate could imply more heavy ions in the CRH spectrum than detected locally
and a larger admixture of heavier nuclei in the target gas.};
$\sigma_{pH}(E_p)$ is the total, inelastic cross-section
for collisions between a beam proton at energy $E_2$ and a stationary
target H nucleus (in cm$^{2}$ and as parameterized by \citet{Block2000}; and
$d N_2(E_p,E_2)/d E_2$ in units eV$^{-1}$ is the (differential) yield function obtained from the interpolation of the
Monte Carlos, i.e., the
distribution of secondaries with respect to
secondary particle energy per interaction of a primary cosmic ray
p having energy $E_p$.
By way of reference, we have calculated the mean energy of the parent beam proton of a $\gamma$-ray
observed with energy $E_\gamma$ in a power-law distribution of given spectral index. We tabulate the results of this
calculation in an appendix. Similar tables are provided for secondary electrons and positrons.
\subsection{`Knock-on' Electron Production}
At low energies -- significantly below $10^8$ eV -- one must account for the fact that the primary source
of secondary leptons is no longer from meson decay but rather from thermal electrons directly `knocked-on'
by Coulombic collisions of primary protons and heavier CR ions.
In calculating spectra of such electrons we use
the results set out in
\citet{Dogiel1990}.
\subsection{Relating $\gamma$-ray and Secondary Electron/Positron Production}
With the above technology we can find, for a given input CRH spectrum, the resulting emissivity of secondary particles.
In fact, we can tie together the emissivity of $\gamma$-rays and secondary leptons
(due to both particle decay and the knock-on process) at a particular energy,
given that we know the shape of the initiating CRH spectrum. In other words, we may write:
\begin{equation}
q_e (E_e, {\rm spectrum}) = R_{e \gamma}(E_e,{\rm spectrum}) \, q_\gamma (E_e, {\rm spectrum})
\end{equation}
where $q_e$ denotes the emissivity of secondary electrons + positrons and we have implicitly defined
\begin{equation}
R_{e \gamma}(E_e,{\rm spectrum}) \equiv \frac{q_e (E_e, {\rm spectrum})}{q_\gamma (E_e, {\rm spectrum})}
\end{equation}
in which the notation $q_2 (E_e, {\rm spectrum})$ indicates that the emissivity of the secondary
is, in general, a function of the input spectrum of CRHs. For a given input spectrum shape, we can
numerically calculate $R_{e \gamma}(E_e,{\rm spectrum})$ using the results from our interpolation of the
pp collision Monte Carlos and, in addition, a contribution from the `knock-on' electrons identified above
(note that in the calculation of $R_{e \gamma}$ $n_{H_2}$ factorizes out).
So, for instance, this quantity can be calculated for a
power-law spectrum of CRHs parameterized by spectral index $\gamma$
(the overall normalization of the CRH spectrum factorizes out):
\begin{equation}
R_{e \gamma}(E_e,\gamma) = \frac{q_e (E_e, \gamma)}{q_\gamma (E_e, \gamma)} \, .
\end{equation}
Even more generally, we can relate the emissivity of electrons + positrons at one energy to the
$\gamma$-ray emissivity at another:
\begin{equation}
q_e (E_e, {\rm spectrum}) = R_{e \gamma}(E_e,{\rm spectrum}) \, R_\gamma(E_e,E_\gamma,{\rm spectrum}) \,
q_\gamma (E_\gamma, {\rm spectrum})
\end{equation}
where
\begin{equation}
R_\gamma(E_e,E_\gamma,{\rm spectrum}) \equiv \frac{q_\gamma (E_e, {\rm spectrum})}{q_\gamma (E_\gamma, {\rm spectrum})} \, ,
\end{equation}
which, again, we can of course particularize to the useful case of a power law spectrum of input CRHs.
\section{Steady-State Primary and Secondary Electron Distributions}
For the Sgr B environment, we have $n_H \sim 10^4$ cm$^{-3}$ and average
magnetic fields $B \sim 10^{-4}$ Gauss, implying relatively short energy loss
times for electrons and positrons:
e.g., the loss time $t_{\rm loss} (E_e)$ for leptons
in this environment reaches a {\it maximum} of
$\sim 3500$ years for energies around 10 GeV (see Figure \ref{fig_plotSgrBLossTimes}).
Note that in this work we do not consider non-steady-state lepton distributions at significant length.
We introduce this restriction basically to avoid opening up the parameter space
by too much: non-steady state models -- which will necessarily involve
fine-tuning given the short loss timescales mentioned above (see \S \ref{sectn_dblpwrlw}) --
will be investigated in detail in a later
work \citep{Crocker2006}.
Now, noting that (i) synchrotron radiation by electrons of $\nu$ GHz frequency
will be generated by
electrons of energy
$E_e^{\rm corres} \simeq {\rm GeV} \sqrt{(\nu/{\rm GHz}) \, (10^{-4} \, {\rm Gauss}/B)}$
in the expected magnetic field strength,
(ii) the loss time at GeV is $\sim 3000$ years,
and (iii) taking an absolute upper limit
on the diffusion co-efficient in the molecular cloud environment
to be given by the value appropriate to the Galactic plane, viz.
$D(E) \sim 5.2 \times 10^{28} \, (E/3 \, {\rm GeV})^{0.34}$ cm$^{-2}$ s$^{-1}$
\citep{Ptuskin2006}, an upper limit on
the diffusive transport scale is given by
$\sqrt{2 D({\rm GeV}) t_{\rm loss} ({\rm GeV})} \sim 20$ pc, approximately equal to
the radius of the Sgr B Complex.
This is likely to be a considerable overestimate of the diffusive transport scale
given that, in the turbulent magnetic environment of
GC molecular clouds, the diffusion coefficient is likely to be considerably
suppressed with respect to its value in the Galactic plane.
The above means that diffusive transport
can be neglected and, therefore, the ambient number density
of electrons + positrons, per unit energy,
$d n_e(E_e, \vec{r})/d E_e$
(in $e^\pm$ cm$^{-3}$ eV$^{-1}$), at various positions $\vec{r}$ within
the molecular cloud complex, can be obtained in steady-state
by numerical integration :
\begin{equation}
\frac{d n_e}{d E_e}(E_e, \vec{r}) =
{\int_{E_e}^\infty q_e(E_e', \vec{r}) dE_e' \over - dE_e(E_e)/dt}
\label{eqn_prcsdSpctrm}
\end{equation}
where $q_e(E_e', \vec{r})$ is the injected emissivity
of electrons + positrons that, in general, might
be either primary (i.e., directly accelerated {\it in situ}) or secondary and
$dE/dt(E_e)$ is the total rate of energy loss of electrons at energy $E_e$ due to ionization,
bremmstrahlung, synchrotron and, possibly, IC emission (because of the energies involved, we neglect positron annihilation,
and assume electrons and positrons suffer identical energy losses).
Electrons and positrons lose energy by ionization losses in neutral molecular hydrogen at a rate
\begin{equation}
{dE_e\over dt}_{\rm ioniz}\simeq - \frac{1.5 \times 10^{-8}}{\beta_e} \times \left({n_{H_2}\over {\rm cm^{-3}}}\right) \times
\ln\left[\frac{E_e^2 \, \beta_e^2 \, (\gamma_e - 1)}{2 \, E_{exctn}^2}\right]\mbox{ ~~ eV/s} \, ,
\end{equation}
where $E_{exctn}$ is the average excitation of the medium
(in the molecular cloud environment of interest we set $E_{exctn} = 15$ eV -- see \cite{Schlickeiser2002} p. 99).
Electrons and positrons lose energy by bremsstrahlung in molecular hydrogen at a rate
\begin{equation}
{dE_e \over dt}_{\rm bremss} = - 1.7 \times 10^{-15} \times \left({n_{H_2}\over {\rm cm^{-3}}}\right) \times \left(\frac{E_e}{\rm eV}\right) \mbox{ ~~ eV/s} \, .
\end{equation}
The synchrotron energy loss rate is
\begin{equation}
{dE_e \over dt}_{\rm synch}= - 1.0\times 10^{-3} \times \left({B_\perp \over {\rm 1 \; Gauss}}\right)^2 \times
\left(\gamma_e^2 \, -1 \right) \mbox{ ~~ eV/s},
\end{equation}
where $B_\perp$ is the component of magnetic field perpendicular to the electron's direction.
For an isotropic electron population $\langle B_\perp \rangle=0.78 B$.
\subsection{Cooled Primary Distribution}
In the case that $q_e$ is attributable to a steady-state injection of {\it primary} electrons, we have no independent, empirical
handle on this {\it injected} population -- we only see emission from the {\it cooled} distribution of electrons.
This is to be contrasted with the situation for secondary leptons where high-energy $\gamma$-ray emission
from neutral meson decay can always provide, in principle, such a handle as explained above.
The cooled electron distribution
is steepened at high energies
by synchrotron radiation and flattened at low energies by ionization losses.
However, for an intermediate range of energies
around $\sim$ GeV
(depending on the exact details of the magnetic field and ambient gas density) -- roughly the same energy
range as electrons that are synchrotron radiating at $\sim$ GHz wavelengths --
bremsstrahlung emission is the dominant cooling process
and, because the bremsstrahlung cooling rate
has a linear dependence on electron energy, it does not modify an injection spectrum.
In particular,
cooling of a power-law distribution of electrons injected with a spectral index $\gamma$
will not modify this spectral index around the $\sim$ GeV scale in a typical molecular cloud environment.
These two facts, together
with the relation between spectral index of the synchrotron-radiating
electron (power-law) distribution and the spectral index of the generated radio spectrum given below
in Eq.(\ref{eqn_SPIN}),
imply that
spectral index measurements between radio fluxes taken in $\sim$ GHz range
may give a handle on the {\it injected} electron distribution spectral index.
Furthermore,
once this fixed normalization point has been obtained, provided the magnetic and
gas environment of the region in which the synchrotron emitting electrons is located is known,
the spectral distortions introduced by ionization and synchrotron, at low and high energies respectively,
may be calculated
so that the overall shape of the cooled electron spectrum may be determined.
Finally, the overall normalization of the cooled
distribution
may also be obtained given that the distance to the emitting region is known.
Putting all these together -- as we do in a numerical routine -- and assuming {\it a priori}
that the injection spectrum for primary electrons should be a power-law in momentum (as described above)
allows one to obtain the injection spectrum: this is simply the power-law
distribution characterised by an overall normalization and spectral index that, once cooled by the ionization, bremsstrahlung,
and synchrotron processes in the known $n_{H_2}$ and magnetic field reproduces the observed radio spectral index and overall flux.
\section{Radiative Processes of Interest}
\subsection{Synchrotron Emission}
The synchrotron emission coefficient (for both primary and secondary electrons) can be calculated using standard
formulae in synchrotron
radiation theory \citep{Rybicki1979}
\begin{eqnarray*}
j_\nu(\nu,\vec{r})&=& 1.87 \times 10^{-30} \,\frac{\rm Watt}{{\rm Hz \, sr \, cm}^3} \, \times
\left({B_\perp\over {\rm \; 1 \; gauss}}\right)\int_{m_ec^2}^\infty F(\nu/\nu_c)\,
\frac{d n_e(E_e, \vec{r})/d E_e}{\rm cm^{-3}} \, dE \, \, , \\
\nu_c &=& 4.19\times 10^6 \times (E / m_ec^2)^2 \times (B_\perp/{\rm \; 1 \; gauss}) \, \, {\rm Hz},\\
F(x) &=& x\int_x^\infty K_{{5 \over 3}}(\xi)d\xi,
\end{eqnarray*}
where
$K_{{5 \over 3}}(x)$ is the modified Bessel function of order 5/3.
Note that, to a good approximation, for a synchrotron-radiating, power-law distribution of electrons with
spectral index $\alpha$, the observed radio spectrum will also be governed by a power law of index $\alpha$ where
\begin{equation}
\alpha = \frac{\gamma - 1}{2} \, .
\label{eqn_SPIN}
\end{equation}
\subsection{Bremsstrahlung and Inverse Compton Emission}
The differential power density originating from bremsstrahlung collisions of $e^\pm$ pervading a medium with atomic number $Z$ and
number density $n_Z$ can be determined to be
\begin{equation}
\frac{d P_\gamma^{\rm brems} }{d E_\gamma}(E_\gamma,\vec{r}) = 1.7 \times 10^{-15} {\rm \frac{eV}{s \, cm^3 \, eV}}
\times
\int_{m_e c^2 + E_\gamma}^\infty \, \frac{d n_e(E_e, \vec{r})/d E_e}{\rm cm^{-3}} \, d E_e \, .
\end{equation}
Similarly, IC emission by a population
of electrons $d N_e(E_e)/d E_e$ off a target, thermal photon field characterised by temperature $T$ is approximately
given by
\begin{equation}
\frac{d P_\gamma^{\rm IC} }{d E_\gamma}(E_\gamma) \simeq 1.36 \times 10^{-5} \, {\rm s}^{-1}
\times
\left(\frac{T}{\rm K}\right)^{5/2}
\times
\left(\frac{E_\gamma}{\rm eV}\right)^{-1/2}
\times
\frac{d N_e}{d E_e}\left(E_e = 3.35 \times 10^7 \, {\rm eV} \, \sqrt{ E_\gamma/{\rm eV} \times {\rm K}/T}\right) \, ,
\end{equation}
where the Thomson regime is assumed requiring
$\sqrt{E_\gamma/{\rm eV} \times T/{\rm K}} < m_e/\sqrt{2.7 \, k_B \, {\rm K \, eV}} \simeq 3.35 \times 10^7$.
For the Sgr B photon background we assume -- to generate an upper limit --
energy densities in ultra-violet and
dust-reprocessed infra-red light fields
the same as that determined for Sgr A East SNR at much smaller Galactic radii ($\sim$ 10 pc),
viz., both 5.7 eV cm$^{-3}$ \citep{Melia1998}.
The temperatures of these
distributions are taken to be at 30 000 and 20 K respectively.
Despite this over-generous attribution of energy into these light fields we find that
IC cooling and emission are always sub-dominant to other cooling and radiative processes
across the entire lepton spectrum (see below).
\subsection{Relating Synchrotron, Bremsstrahlung, and Inverse Compton Emission to $\gamma$-ray Emission}
With the technology outlined above we can predict on the basis of an observed flux of $\gamma$-rays
from a particular astrophysical object or region --
which must, by hypothesis, orgininate in
hadronic interactions of a primary CRH population (i.e, mostly from neutral pion decay) -- the
radio flux from the same object/region due to secondary leptons.
Apart from the $\gamma$-ray flux, the inputs to the synchrotron prediction are
(i) the
{\bf B} field strength in the object/region, (ii) the number density of target particles,
in our case,
molecular hydrogen, $n_{H_2}$, and (iii) the
shape of the cosmic ray spectrum between
(a) the energies of CRHs which are responsible for the observed $\gamma$-rays
and
(b) the energies of CRHs which are responsible for
the secondary e$^\pm$'s whose synchrotron emission we might detect in the radio in some given frequency range.
Significantly, because the synchrotron expectation is normalized directly to the $\gamma$-rays, a
determination of either the total mass of target particles
or the distance to the object/region containing the targets
is not necessary in this calculation.
We can use very similar considerations as above to determine predictions for bremsstrahlung emission by
secondary leptons. Likewise, with the additional input of the target photon field (radio, IR, optical, UV)
we can predict the IC emission by this population of secondaries.
Putting all this together, we can assemble the predicted broad-band spectrum of a particular astrophysical
object or region. The calculation is only self-consistent, of course,
if, at the $\gamma$-ray energy where we normalize our predictions,
the $\gamma$-rays do strictly originate only in CRH collisions, though a small level of `pollution' due to
IC or bremsstrahlung at the $\lesssim 20\%$ level is perfectly acceptable given other uncertainties.
\section{Predicted Sgr B Phenomenology in Hadronic Scenario}
\subsection{Predicted Sgr B Broad-band Spectrum in Hadronic Scenario}
As explained above, adopting the expected, average values {\bf B} $= 10^{-4}$ Gauss and $n_{H_2} = 10^4$ cm$^{-3}$ and assuming a
power law in momentum for the parent CRH population that, by hypothesis, is responsible for initiating the $\sim$ TeV $\gamma$-ray
emission (with spectral index $\simeq 2.3$ and a normalization consistent with this emission), we may determine the broadband
spectrum of Sgr B due to all pertinent electromagnetic emission mechanisms (initiated by primary CRHs and secondary leptons).
This spectrum is displayed in Figure \ref{fig_plotH2aBB}. Here one can see immediately that
the model fails to explain any data apart from
the gamma-ray flux -- though it is not, of course, excluded by the other data.
Of particular note, secondary synchrotron does not account for the radio flux detected at 330 MHz
(radio datum second from left) or 843 MHz.
One might seek to `dial-up' the magnetic field strength in the model in order to account for the radio data, but in this
endeavor two problems are encountered: (i) the spectral index of the predicted spectrum is rather too flat to go through
both 330 and 843 MHz data points (the spectral index between predicted central flux values is -0.31, cf.
the observational value of -(0.9 $\pm$ 0.5)) , even allowing for 1-$\sigma$ excursions of both experimental values and, more tellingly,
the magnetic
field strength demanded is unreasonably high at 10 mG (as supposed average over the entire Sgr B Complex), implying
a magnetic field energy density of $2.6 \times 10^6$ eV/cm$^{-3}$.
\subsection{Cosmic Ray Ionization Rate in Hadronic Scenario}
Even if we do not rule out such a field {\it a priori}, there is another reason to reject this model:
the contribution of the primary CRH population inferred from the $\gamma$-rays is far too small to account for the observed
cosmic ray ioniztion rate $\zeta_{CR}$ in Sgr B2, viz., 4 $\times 10^{-16}$ s$^{-1}$ as given above. And
as stated already, this value
should roughly apply over much of the Sgr B Complex.
We may calculate the predicted CRH ionization rate for Sgr B using the technology set out in
\citet{Webber1998}.
Employing the Bethe cross-section
\begin{equation}
\sigma_{\rm Bethe} (\beta, Z) = 1.23 \times 10^{-20} {\rm cm}^2 \frac{Z^2}{\beta^2}
\left[6.2 + \ln \left(\frac{\beta^2}{1 - \beta^2}\right) - 0.43 \, \beta^2 \right]
\end{equation}
the ionization rate is
\begin{equation}
\zeta_{CR}(J_p, m_{CR}) = \frac{5}{3} \,
\int_{T_p^{\rm min}} ^{\infty} \, 4 \pi \, J_p(T_p) \, \sigma_{\rm Bethe} \, d T_p \, ,
\label{eqn_InztnRt}
\end{equation}
where $J_p$ is the differential flux of cosmic rays protons at kinetic energy $T_p$ in
cm$^{-2}$ s$^{-1}$ eV$^{-1}$ sr$^{-1}$ and we take $T_p^{\rm min} \equiv $ 10 MeV.
To account for the effect of heavier ions in the CRH flux, one may introduce a constant multiplicative factor on the RHS of
equation (\ref{eqn_InztnRt}) which is 1.89 for the local CR flux \citep{Spitzer1968}
We note in passing that the above can be directly adopted for calculation of ionization by electrons, though in this case
the appropriate energy at which to cut-off the ionization rate integral is $T_e^{\rm min} \equiv $ 3 MeV \citep{Webber1998}.
Using these results, the predicted
$\zeta_{CR}$, is only 6 $\times 10^{-18}$ s$^{-1}$ and, if the entire non-thermal radio flux is accounted for by secondaries
generated by this CRH population, there is no room left for another low-energy particle population whose collisions
might account for the $\zeta_{CR}$ measurement.
Even allowing for a possible additional contribution from heavier ions in the GC
environment (relative to their contribution in the Solar System environment) could not make up the
$\sim$ two order of magnitude deficit
with respect to the $\zeta_{CR}$ measured by \citet{vanderTak2006} in Sgr~B2.
As a further confirmation that the TeV $\gamma$-ray and ionization data are irreconcilable within the current CRH model
one may turn around the above logic: for a power-law distribution of CRHs with a normalization
sufficient to account for the observed $\zeta_{CR}$ one may determine
what $\sim$TeV $\gamma$-ray spectrum is implied. Carrying out this exercise one finds that the
$\gamma$-ray flux from Sgr B is overpredicted by around two orders of magnitude.
Of course, it is to be admitted here that to posit a meaningful relation between the ultra-relativistic
cosmic rays that generate the TeV $\gamma$-rays and the sub-relativistic particle population primarily responsible
for ionizing the ambient gas might be considered speculative.
Nevertheless, the result stands that these two phenomenological inputs are irreconcilable with the simplest
expectation from shock acceleration theory, viz., that the ambient CRHs be governed by a single power law in momentum.
\subsection{Double Power-Law Hadronic Scenario}
\label{sectn_dblpwrlw}
One simple and reasonable modification of the above scenario is to allow for two power-law CRH populations: a steep
population, dominant at low energy and with a normalization given
by the requirement that it reproduce the observed ionization rate
and with a spectral index as flat as possible given the
EGRET constraint (implying a value of at least 2.7) and a flatter population of spectral index
of $\sim$ 2.2 that becomes dominant at high energy and explains the TeV $\gamma$-ray emission.
Note that in the GC region it is not at all unreasonable
that such a complicated overall spectrum might pertain:
the region posseses two and possibly
three sources of cosmic rays --
unlike other regions in the Galaxy there
is activity here associated with the super massive black hole, there is a
very high supernova rate and the GC
molecular clouds represent a huge reservoir of energy in their turbulent motions.
In any case,
we have investigated a double power law
scenario and indeed find that for a rather strong average magnetic field, viz., 2.2-3.7 mG (the range implied
by the range for the possible total mass of Sgr B Complex),
the broadband non-thermal phenomenology is well reproduced.
The required energy density in this assumed population is $\sim 2.3$ eV cm$^{-3}$, very similar to the local CR
energy density \citep{Webber1998} and, interestingly, the spectral index of this inferred population
is very similar to the local one (though one should note that the inferred spectrum, $\propto p^{-2.7}$,
is substantially in excess of the local CRH spectrum at 10-100 MeV, explaining how it is possible the former can
produce ten times the ionization rate of the local spectrum, despite possessing approximately the same total
energy). We have also checked the steady-state positron production rate in this scenario and find that
the scenario predicts a 511 keV $\gamma$-ray production rate from
electron-positron annihilation of $\sim 3 \times 10^{46}$ yr $^{-1}$, well
inside the limit by INTEGRAL observations ($\sim 10^{50}$ yr $^{-1}$ out to an angular radius of 8$^\circ$:
\cite{Knodlseder2003,Jean2003}).
The question of the naturalness of this scenario is addressed in the Discussion section but one should note
immediately that the energy density in the required magnetic field
is 400-1600 times larger than that in the ``expected" field given by the Crutcher scaling.
In a future publication \citep{Crocker2006} we will also investigate
an inflected primary CRH spectrum -- asymptoting to flat power laws at high and low energies but rather steeper at
energies in the GeV - 100 GeV range -- that holds out the promise of reproducing the broadband phenomenology
but for a rather smaller magnetic field.
\subsection{Non Steady State Scenario}
As stated, we have assumed a steady-state secondary lepton distribution in our calculations above requiring implicitly
that the age of the injected lepton distribution is greater than the cooling timescale.
If this condition is not satisfied,
then, in place of Eq.(\ref{eqn_prcsdSpctrm}) one has, to tolerable accuracy \citep{Fatuzzo2005}, that
the secondary distribution is given by (injection spectrum) $\times$ (age of lepton population).
Now, in consultation with Figure \ref{fig_plotSgrBLossTimes}, one also notes that the
loss time is a strong function of energy with different loss processes dominant at different
enegry scales.
In fact, one can see from this Figure that it would be possible to fine-tune the assumed age
of the lepton distribution in such a way that the leptons predominantly responsible for the
observed radio emission (i.e., those at $\sim$ GeV) be out-of-equilibrium, whereas those at lower and
higher energies would be in steady state.
Now, although the absolute emissivity of the lepton distribution reaches its greatest value when steady state is reached,
the greatest emissivity of GeV-scale leptons {\it relative} to the $\gtrsim$ TeV energy $\gamma$-ray emission
would be achieved in such a
fine-tuned situation (that would require a lepton population of age $<3000$ years).
Thus, a larger $\sim$ GHz radio emission could be achived for the same observed $\sim$ TeV $\gamma$-ray
emission in this somewhat fine-tuned scenario relative to a completely steady state model assumed above.
This scenario will be investigated at greater length in a forthcoming publication \citep{Crocker2006}.
\section{Predicted Sgr B Phenomenology in Leptonic Scenario}
The above considerations naturally lead one to consider the possibility that primary electrons,
presumably accelerated in-situ given the short loss times they experience in
this strong magnetic field and dense environment, can be invoked to explain the Sgr B phenomenology
under consideration.
\subsection{The Model of Yusef-Zadeh et al.}
Recently \citet{Yusef-Zadeh2006},
motivated
by their observation of a three-way correlation
between the distribution of molecular material across the GC region, Fe K$\alpha$ line
emission, and TeV emission, have considered a model in which a power-law population of primary
electrons
is invoked to explain the keV to TeV emission from a number of dense molecular regions in the vicinity
of the GC including Sgr B1 and B2 inside the Sgr B Complex.
Broadly, these authors' procedure is to
determine the local energy density in non-thermal electrons required in order that
these particles produce, via their Fe K-shell ionizing collisions, the observed
flux of Fe K$\alpha$ (6.4 keV) X-ray photons. The authors then set the local magnetic field strength to be
in equipartition with the energy density in this population and determine the synchrotron emission by the
electrons in this field. The magnetic fields arrived at in this way for Sgr B1 and B2 are
at the $\sim$ few $\times 10^{-5}$ G level. \citet{Yusef-Zadeh2006} then determine the inverse Compton (IC)
spectrum of photons up-scattered from mm wavelengths to TeV+ energies by the inferred high energy component
of the same electron population. This mechanism requires electrons with
energies in excess of 30 TeV as remarked by \citet{Yusef-Zadeh2006}
themselves -- a severe challenge given that
cooling (dominated by synchrotron emission) is so efficient at these energies
with
a loss timescale at 30 TeV of
only a few decades (for the expected magnetic field strengths given below).
We have repeated the analysis of \citet{Yusef-Zadeh2006} to confirm their findings.
Our detailed re-examination reveals, however,
that their models for Sgr~B1 and Sgr~B2 encounter a number of difficulties.
Not the least of these
is that, in the case of Sgr B2 at least, the model requires
an average magnetic field strength that is
0.045 mG, on the order of one order of magnitude less than actually
measured -- and set as a {\it lower limit} -- for this object by \citet{Crutcher1996}
(and also much less than the B field inferred on the basis of the Crutcher scaling, Eq.(\ref{eqn_Crutcher})).
Another problem for the Sgr~B2 model of \citet{Yusef-Zadeh2006} is that, in the average
molecular hydrogen density assumed by them, $10^4 $ cm$^{-3}$,
their assumed power-law, primary electron distribution seems to produce too much
bremsstrahlung emission at both $\sim$ 100 MeV and $\sim$ TeV energies, surpassing the EGRET upper limit
and also the HESS data points (Yusef-Zadeh et al. actually invoked IC to explain the $\sim$ TeV data,
but bremsstrahlung will make an unavoidable -- and actually dominant --
contribution in this same energy range given that $\sim$30 TeV+
electrons are necessary in order that the IC mechanism work).
This objection might be dealt with, however,
by invoking a somewhat reduced $n_{H_2} \simeq 5 \times 10^3$ cm$^{-3}$.
Yet another problem, though, is in what seems to be a lack of self-consistency in the way these authors
deal with the cooling of the injected, primary electrons. In their calculation of the energy density in
their fitted, equipartition electron population, Yusef-Zadeh et al. assume a pure power-law
in electron {\it kinetic} energy between 10 keV and 1 GeV. Leaving aside the potential issue that
the assumption of a power law in kinetic energy (rather than momentum) is a rather unnatural one, the problem here is that
ionization loses at low energy should significantly distort an injected spectrum away from such power-law
behavior
(unless the time from injection of the particles is significantly less than the ionization cooling timescale, $\ll$ 100 yr).
Indeed, cooling of the distribution assumed by \citet{Yusef-Zadeh2006} would seem to be necessary in order
that bremsstrahlung emission by lower emergy electrons not over-produce continuum X-rays -- but then the general procedure
of these authors would seem to lack self-consistency as, on the one hand, the X-ray emission is apparently
calculated with the cooled
distribution but the total energy density in the distribution is calculated with an uncooled (i.e., pure power law) distribution.
\subsection{A New Leptonic Model for Sgr B}
Because of these difficulties, and to further explore the parameter space,
we therefore relax the constraint imposed by \citet{Yusef-Zadeh2006}
that the energy density in the electron population be the same as in the
magnetic field.
We arrive at a satisfactory, alternative
lepton scenario to that proposed by these authors
in the following fashion: adopting the expected values $n_{H_2} = 10^4$ cm$^{-3}$,
$B = 10^{-4}$ Gauss and $N_{H_2} = 8 \times 10^{23}$ cm$^{-2}$, we find the {\it cooled} electron distribution (as parameterized
by injection {spectral index } and normalization) that reproduces the required $\zeta_{CR}$ and roughly reproduces the continuum
X-ray spectrum of Sgr B (via bremsstrahlung).
We find the spectral index at injection mustbe close to $\gamma = 2.85$.
Fixing $\gamma$ and also the overall electron population normalization to the values determined by the procedure above,
we then determine the B field required in order that the electron spectrum found above reproduce the 330 MHz VLA datum.
A very
reasonable field of 1.3 $10^{-4}$ G is necessary
here for the case of the minimum possible Sgr B mass (the maximum mass case is excluded because it produces significantly
too much
$\sim$ GeV bremsstrahlung emission contravening the EGRET limit).
We subsequently check that the new value of B does not substantially alter $\zeta_{CR}$ and the X-ray flux for the already-determined
{spectral index } and normalization.
We then check the $\sim$ GeV bremsstrahlung emission from this distribution (I.C. is sub-dominant at this energy)
in the expected $\langle n_{H_2} \rangle$.
Finding that the spectrum now exceeds the EGRET limit, we dial the $\langle n_{H_2} \rangle$ downwards until
Sgr B just escapes detection. The required $\langle n_{H_2} \rangle$ is $5 \times 10^3$ cm$^{-3}$.
This is very close to the critical number density established by Eq.(\ref{eqn_critDnsty})
for molecular gas at a distance of $r_{\rm Sgr \, B} \sim 100 pc$ from the actual GC, viz.,
$\sim 6 \times 10^3$ cm$^{-3}$.
Again, we determine whether the $\langle n_{H_2} \rangle$ substantially alters $\zeta_{CR}$ and the X-ray flux for
the previously determined {spectral index } and normalization. We find that $\zeta_{CR}$ is largely unaltered, but the X-ray flux has
now decreased. We then vary the final degree of freedom open to us to adjust, the average column density through the cloud
and the ISM to the emission region. A satisfactory fit can be determined by dialling this down to be $4 \times 10^{23}$ cm$^{-2}$
-- a value well within the phenomenologically-allowable range.
We have now fully constrained the system and can {\it predict}
the 843 MHz flux. This prediction is 22.3 Jansky,
fully consistent with the poorly-determined
observational value of 20 $\pm 10$ Jy.
Given that estimated fluxes might, {\it a priori},
vary by orders of magnitude, this represents good agreement.
Note that through this procedure we have arrived at an allowable parameter set that is probably not unique
and, given the complexities and non-negligible uncertainties surrounding the input data, we have not attempted a
$\chi^2$ analysis.
Non-trivially, however, we have found a one-zone model for the Sgr B complex that reproduces the broad features
of its low-energy, non-thermal phenomenology: a single electron population, injected as a power law in momentum
with reasonable spectral index and
then cooled by the processes of ionization, bremsstrahlung and synchrotron radiation into a steady state distribution
will reproduce the observed cosmic ray ionization rate, the 330 MHz datum and
the broad features of the continuum X-ray emission, all for very reasonable values of the ambient magnetic field,
molecular hydrogen number density and column density.
We note that the energy density in the cooled electron distribution is 2 eV cm$^{-3}$.
This is around 10 times the energy density in CR electrons through the Galactic disk
as determined by \citet{Webber1998}, but
considerably sub-equipartition
with respect to
the energy density in the fitted 1.3 $10^{-4}$ G magnetic field, viz. $\sim$ 400 eV cm$^{-3}$.
Note, however, that
the energy density represented by turbulent motions of the Sgr B gas
could easily be
in equipartition with such a field.
In fact, adopting
a line width of $10 $km s$^{-1}$, as detected for the envelope of Sgr B2 \citep{Lis1989}, e.g.,
one determines an equipartition field of $\sim$ 0.7 mG \citep{Novak1997}.
One also notes that the inferred energy density is considerably less than that found
for the Sgr B1 and B2 regions in the analysis of \citet{Yusef-Zadeh2006}, 21 and 51 eV cm$^{-3}$, respectively.
On the other hand it is considerably in excess of the energy density in the
broad scale, non-thermal electron population recently inferred by
\citet{LaRosa2005}. These authors have, on the basis of the application of an
argument
of equipartition (between magnetic field energy density and relativistic particles)
to observations of a large-scale ($6^\circ \times 2^\circ$),
diffuse flux of non-thermal radio emission (detected at 74 and 330 MHz) across the GC region, inferred
an average magnetic field strength through this region of $\sim$ 15 $\mu$G and
a non-thermal electron energy density of $\sim 0.06$ eV cm$^{-3}$ (assuming
an electron energy density 1/100 that in protons).
The cooled electron distribution also represents a total energy of $1.1 \times 10^{48}$ erg.
The energy required, however, to have been injected with the un-cooled initial spectrum --
and since lost mainly into ionization (thus heating) of the cloud matter -- is $1.4 \times 10^{50}$ erg.
The one respect in which our leptonic model can be said to fail is at TeV energies: it produces far too little
bremsstrahlung and IC emission in this energy range to be able to explain the HESS observations. On the other
hand it is certainly not in conflict with these observations.
\section{A Mixed Model for the Broadband Emission of Sgr B}
We are finally, then, led to consider a hybrid model in which a relatively steep population (at injection)
of low energy electrons
is responsible for the bulk of the measured ionization rate, the 330 and 843 MHz flux and the X-ray
continuum flux (via bremsstrahlung),
whereas a hard population of CRH's dominates the emission at
TeV energies producing the bulk of gamma rays at this energy through neutral pion production and decay.
For the broad-band spectrum of such a mixed model, please see Figure \ref{fig_plotMxdBB}.
\section{Discussion}
Both of our models -- the double power law CRH model and the mixed lepton/hadron model --
require a hard, high-energy (TeV+) spectrum of hadrons to explain the HESS $\gamma$-rays.
An IC/primary lepton model for this emission -- at least as suggested by \citet{Yusef-Zadeh2006} -- does
not seem to work.
As initially noted in \citet{Aharonian2006} and as subsequently explored in
\citet{Busching2006} and \citet{Ballantyne2007}, that the spectral index
of the diffuse $\gamma$-ray emission detected by HESS
does not seem to vary over the extent of the CMZ, and, moreover,
is so similar to that detected for the central point-like source coincident with Sgr A*,
supports the notion that the CRH population responsible for this emission
has its origins at relatively small distances from the central black hole and diffuses out
into the CMZ from this position.
The total energy in this population has been estimated
to be $10^{50}$ erg \citep{Aharonian2006} marginally consistent
with an origin in a single supernova.
Furthermore, adopting a diffusion coefficient
appropriate to cosmic ray diffusion through the Galactic disk, viz.
$D \sim 10^{30}$ cm$^{2}$ s$^{-1}$ at several TeV,
\citet{Aharonian2006} note that the break-down of the correlation between
$\gamma$-ray emission and molecular density at angular scales of $\sim$ 1.3$^\circ$
from the actual GC (a position to the East of Sgr B) implies a definite time
for the injection event of around 10 000 years, perhaps consistent with the unusual SNR
Sgr A East being the source of the CRH population \citep{Crocker2005}.
There are a number of caveats here, however.
Firstly,
one might expect in such a scenario that
-- at the periphery of the diffuse emission region (just before the breakdown of
the $\gamma$-ray/molecular density correlation) -- the spectral index of the emission should harden, but such an effect is
not seen.
Furthermore, the adoption of a diffusion coefficient appropriate to the Galactic disk carries with it
a large uncertainty (as noted by \citet{Aharonian2006}).
Certainly, if there is a coherent global GC magnetic field of $\sim$ mG strength, then, at the least,
the timing of the assumed injection event has to be blow out substantially, and in fact,
the whole diffusion-away-from-central-source picture may not be tenable \citep{Morris2007}.
A steady-state picture for the origin of these CRHs -- though one still assuming a central source -- may be
attractive, therefore. One such model that has recently been investigated in the context
of stochastic acceleration on the turbulent magnetic field close to the central black hole \citep{Liu2006}
is that of \citet{Ballantyne2007}.
\subsection{Origin of Putative Steep, Low-Energy Hadron Population}
As described at length above, one model that reproduces the broadband phenomenology
of the Sgr B Complex consists of a double power law CRH population, the steeper power-law
becoming dominant at low energy.
Despite the modest CRH energy density required in this scenario ($\sim 2$ eV cm$^{-3}$),
one should certainly question the naturalness of such an overall spectrum: if the
CRHs are assumed to originate outside the cloud and to be diffusing into it, then exactly
the opposite sort of spectral behavior as outlined would be expected, viz., a progressive
flattening towards low energies as lower-energy particles find it increasingly difficult
to diffuse into the cloud
before catastrophic energy loss via hadronic collision on ambient matter in the cloud:
see \citet{Gabici2006} and also the companion paper to the current work \citep{Jones2007}.
On the other hand, if the CRHs are accelerated {\it inside} the GMC,
it may be
that
lower energy CRHs are unable to leave the cloud and naturally accumulate, thereby steepening the
overall spectrum towards lower energies, precisely as we require in our model.
So, in this scenario, a hard, CRH population
-- perhaps originating outside the cloud but
with sufficient rigidity to penetrate the cloud at high energies
-- is dominant at HESS energies whereas a population accelerated in situ,
and perhaps trapped by the cloud's turbulent magnetic fields,
is dominant at lower energies and responsible for most of the Sgr B phenomenology we report.
The total energy in the double power law CRH spectrum is between 1.3-3.3 $\times 10^{48}$ erg, a small
fraction of the mechanical energy available from a SN explosion, e.g..
Another question hangs over the magnetic field required in
the double power law hadronic scenario. This is, at $\sim$2-4 mG,
possessed of an energy density 400-1600 times larger than that of
the ``expected" field given by the Crutcher scaling as noted above.
This scaling relation, however, does not seem to describe the situation in
Sgr B2 particularly well, as also already noted.
Furthermore, as stressed, the direct Zeeman splitting determination of the Sgr B2 field, at $\sim$ 0.5 mG,
actually defines a lower limit to the local average field sgtrength.
In fact, both \citet{Lis1989} and
\citet{Crutcher1996} actually countenance average magnetic field strengths as high as $\sim$2 mG
for the Sgr B2 cloud on the basis of the theoretical prejudice that the cloud be magnetically
supported against gravitational collapse --
so we cannot exclude that such field strengths may actually apply on large scales in the complex.
Finally, it must also be admitted that the
large magnetic field required
may be an artefact of our assumption of a 1-zone model:
if p's are accelerated inside the cloud then they will naturally accumulate more
in regions where B fields are higher and the synchrotron emissivity resulting
from secondaries created by collisions in these regions will also be relatively higher
in these locations
-- so the assumption of an ``average" value
for the B field may not adequately reflect these effects.
\subsection{Origin of Putative Electron Population}
In terms of fitting to the broad-band spectrum, the other successful model discovered above
is one that invokes a steep ($E^{-2.9}$) primary lepton spectrum at injection together with a hard
CRH population to explain the HESS $\gamma$-rays.
One notes that
the required (steep) electron injection index required in this scenario
is well beyond the range normally attributed to shock acceleration,
which is typically 2.0-2.4 and, indeed, even a combination of
shocks might not allow such an overall steep distribution easily.
There do seem, however, to be two plausible instances in which
such a steep spectrum might be arrived at:
(i) stochastic acceleration off a turbulent
magnetic field within the cloud (i.e., acceleration by plasma wave turbulence,
a second-order Fermi acceleration process; see, e.g. \cite{Petrosian2004}), and (ii)
shock acceleration with an energy-dependent loss or diffusion.
Again, concrete examples of these two general mechanisms will be
evaluated in a later work \citep{Crocker2006}.
The energy at injection
represented by the putative non-thermal electron population invoked above, $1.4 \times 10^{50}$ erg,
is too big, e.g., to have been supplied by a single, ordinary supernova event (the total energy
that goes into non-thermal particles populations and fields being around $5 \times 10^{49}$ erg; see,
e.g. \cite{Duric1995} -- but, of non-thermal particles, most energy goes into hadrons).
In this context, however, one notes the possibility that multiple supernovae have occurred
within the Sgr B Complex, it being a site of very active star formation and harboring many massive and hot stars.
In fact, \citet{Koyama2006a} and \citet{Koyama2006b} have very recently claimed the discovery of a new SNR
designated Suzaku J1747.0-2824.5 (G0.61+0.01) on the basis of a detection
of an excess of 6.7 keV FeXXV K$\alpha$ emission from inside the Sgr B region with
the X-ray Imaging Spectrometer on board the orbitting {\it Suzaku}
X-ray telescope.
Furthermore, on the basis of the CO mapping performed by \citet{Oka1998}, \citet{Koyama2006a} and \citet{Koyama2006b}
claim the existence of a radio shell in this same region
possessed of a kinetic energy of order $10^{52}$ erg, supporting the possibility of multiple
supernovae within the Sgr B Complex.
\section{Conclusion}
A major result of this work is that we get far
too little radio flux from secondary leptons (normalized to the
HESS $\gamma$-ray data)
to explain the VLA and SUMSS observations for reasonable magnetic field values
and assuming a power-law (in momentum) behavior of the initiating CRH primaries.
Furthermore --
making the (perhaps naive) assumption that the CRHs be governed by a single power law
in momentum from ultra-relativistic energies down to the sub-relativistic regime --
the cosmic ray ionization rate $\zeta_{CR}$ implied by the CRH population
inferred from the HESS $\gamma$-ray data is far too small to be reconciled with
recent determinations of this quantity for Sgr B2.
Conversely, a simple interpretation of the ionization rate in
this cloud being maintined solely by the CRH distribution would
then overproduce TeV photons via pp scattering events compared
to what is measured by HESS (again, assuming a pure power law).
Another major result is that it seems almost certain that one needs a hard, high-energy cosmic-ray {\it hadron}
population to explain the HESS TeV+ $\gamma$-ray data: a particular model introduced by \citet{Yusef-Zadeh2006}
that would seek to explain this emission by inverse Compton scattering by a population of
primary cosmic ray electrons seems to break down when considered in detail.
Certainly, as for the comsic ray hadron case,
a single power-law distribution of primary electrons cannot account for all Sgr B phenomenology.
We have investigated two scenarios that can be reconciled with all the data:
\begin{enumerate}
\item A scenario invoking two CRH
power law populations (with a steep spectrum dominant at low energy). This scenario requires an ambient
magnetic field in the Sgr B Complex in the range 2-4 mG that, while apparently not excluded by existing
Zeeman splitting and polarimetry data, may be uncomfortably high.
Future submillimetre polarimetry measurements
may soon rule in or rule out the necessary field strength.
Zeeman splitting measurements at mm wavelengths with the next generation of instruments
such as the Atacama Large Millimeter Array (ALMA: \cite{Wootten2006})
may also have sufficient sensitivity to detect these high field strengths, even in lower density regions.
\item Following a path blazed by
\citet{Yusef-Zadeh2006}, we have investigated the idea that
primary electrons play a significant role and, in fact, explain the bulk
of the non-thermal Sgr B phenomenology. Unfortunately we find that
the particular instantiation of a leptonic model arrived at by
\citet{Yusef-Zadeh2006} does not seem to be phenomenologically viable (requiring as it does
too small a magnetic field to be reconcilable with the data) or self-consistent (in its treatment of
spectral distortion due to ionization cooling at low energies).
We have found, however, a leptonic scenario which does satisfactorily account for much of the phenomenology of Sgr B
(viz., the CR ionization rate, the X-ray continuum emission, and the 330 and 843 MHz radio emission)
but a hadronic component, contrary to the opinions of \citet{Yusef-Zadeh2006},
is apparently necessary to explain the gamma-ray emission as mentioned above.
\end{enumerate}
\section{Acknowledgements}
The authors thank Jim Hinton for providing the HESS data in numerical form
and Crystal Brogan for supplying the 74 and 330 MHz data VLA data in numerical form.
RMC thanks Roger Clay
for discussions about cosmic ray diffusion, Gavin Rowell for advice about the analysis of the $\gamma$-ray
data, and Troy Porter for advice about the psf of EGRET.
\clearpage
\section{Appendix A: Average Parent Proton Energies of Secondaries}
\begin{deluxetable}{lcccccc}
\tablewidth{0pt}
\tablecaption{The average parent CRH energy of a $\gamma$-ray observed at $E_\gamma$ expressed as
a multiple of $E_\gamma$ for various spectral indices.\label{table_parentEnergyGammaRays}}
\tablehead{
\colhead{}
& \colhead{2.0}
& \colhead{2.2}
& \colhead{2.4}
& \colhead{2.6}
& \colhead{2.8}
& \colhead{3.0}
}
\startdata
$10^7$ eV &11212.1 &3307.03 &1492.45 &906.29 &642.835 &494.906 \\
$10^8$ eV &381.042 &93.5398 &39.2354 &24.2151 &18.2917 &15.2974 \\
$10^9$ eV &402.378 &124.055 &52.8439 &29.721 &20.0141 &15.0221 \\
$10^{10}$ eV &347.006 &136.529 &64.3183 &36.7628 &24.5029 &18.111 \\
$10^{11}$ eV &196.61 &95.9581 &49.9323 &28.7306 &18.4379 &13.0247 \\
$10^{12}$ eV &90.378 &59.6075 &39.3149 &26.5121 &18.5825 &13.6492 \\
$10^{13}$ eV &28.9777 &24.4022 &20.311 &16.792 &13.8674 &11.5044
\enddata
\end{deluxetable}
\begin{deluxetable}{lcccccc}
\tablewidth{0pt}
\tablecaption{The average parent CRH energy of a secondary electron observed at $E_e$ expressed as
a multiple of $E_e$ for various spectral indices.\label{table_parentEnergyElectrons}}
\tablehead{
\colhead{}
& \colhead{2.0}
& \colhead{2.2}
& \colhead{2.4}
& \colhead{2.6}
& \colhead{2.8}
& \colhead{3.0}
}
\startdata
$10^7$ eV &20509.2 & 4435.49 & 1315.68 & 548.547 & 305.384 & 209.977 \\
$10^8$ eV &839.363 & 226.115 & 95.561 & 56.3558 & 40.0325 & 31.4774 \\
$10^9$ eV &647.868 & 206.759 & 88.6688 & 49.7365 & 33.3554 & 24.8667 \\
$10^{10}$ eV &647.707 & 260.024 & 117.842 & 62.529 & 38.4941 & 26.6127 \\
$10^{11}$ eV &362.318 & 194.075 & 106.836 & 62.655 & 39.8574 & 27.4842 \\
$10^{12}$ eV &143.603 & 103.072 & 73.0004 & 51.8339 & 37.439 & 27.8002 \\
$10^{13}$ eV &39.3796 & 35.013 & 30.7712 & 26.7642 & 23.0849 & 19.7976
\enddata
\end{deluxetable}
\begin{deluxetable}{lcccccc}
\tablewidth{0pt}
\tablecaption{The average parent CRH energy of a secondary positron observed at $E_e$ expressed as
a multiple of $E_e$ for various spectral indices.\label{table_parentEnergyPositrons}}
\tablehead{
\colhead{}
& \colhead{2.0}
& \colhead{2.2}
& \colhead{2.4}
& \colhead{2.6}
& \colhead{2.8}
& \colhead{3.0}
}
\startdata
$10^7$ eV &4711.5 & 999.145 & 368.44 & 216.004 & 163.045 & 138.591 \\
$10^8$ eV &441.384 & 114.894 & 51.1813 & 32.9168 & 25.4181 & 21.4796 \\
$10^9$ eV &489.759 & 153.778 & 66.9793 & 38.7002 & 26.6889 & 20.347 \\
$10^{10}$ eV &525.323 & 205.124 & 92.4019 & 49.6369 & 31.1947 & 22.01 \\
$10^{11}$ eV &305.33 & 160.058 & 87.2869 & 51.3001 & 32.9662 & 23.0505 \\
$10^{12}$ eV &126.953 & 89.2382 & 62.0621 & 43.4609 & 31.1185 & 23.0179 \\
$10^{13}$ eV &36.8475 & 32.5944 & 28.563 & 24.8512 & 21.5276 & 18.6259
\enddata
\end{deluxetable}
\clearpage
|
1,116,691,500,744 | arxiv | \section{#1}}
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\begin{document}
\begin{titlepage}
\begin{flushright}
IPhT-T13/259
\end{flushright}
\medskip
\begin{center}
{\Large \bf Supersymmetric Janus Solutions in Four Dimensions}
\bigskip
\bigskip
{\bf Nikolay Bobev,${}^{(1)}$ Krzysztof Pilch,${}^{(2)}$ and Nicholas P. Warner${}^{(2,3,4)}$ \\ }
\bigskip
${}^{(1)}$
Perimeter Institute for Theoretical Physics \\
31 Caroline Street North, ON N2L 2Y5, Canada
\vskip 5mm
${}^{(2)}$ Department of Physics and Astronomy \\
University of Southern California \\
Los Angeles, CA 90089, USA \\
\bigskip
$^{(3)}$ Institut de Physique Th\'eorique, CEA Saclay \\
CNRS-URA 2306, 91191 Gif sur Yvette, France\\
\bigskip
$^{(4)}$ Institut des Hautes Etudes Scientifiques \\
Le Bois-Marie, 35 route de Chartres \\
Bures-sur-Yvette, 91440, France \\
\bigskip
[email protected],[email protected],[email protected] \\
\end{center}
\begin{abstract}
\noindent
\end{abstract}
\noindent We use maximal
gauged supergravity in four dimensions to construct the gravity dual of a class of supersymmetric conformal interfaces in the theory on the world-volume of multiple M2-branes. We study three classes of examples in which the $(1+1)$-dimensional defects preserve $(4,4)$, $(0,2)$ or $(0,1)$ supersymmetry. Many of the solutions have the maximally supersymmetric $AdS_4$ vacuum dual to the $\Neql8$ ABJM theory on both sides of the interface. We also find new special classes of solutions including one that interpolates between the maximally supersymmetric vacuum and a conformal fixed point with $\Neql1$ supersymmetry and $G_2$ global symmetry. We find another solution that interpolates between two distinct conformal fixed points with $\Neql1$ supersymmetry and $G_2$ global symmetry. In eleven dimensions, this $G_2$ to $G_2$ solution corresponds to a domain wall across which a magnetic flux reverses orientation.
\end{titlepage}
\setcounter{tocdepth}{2}
\tableofcontents
\section{Introduction}
Defects in conformal field theories have long been recognized as useful probes of interesting physics and have been broadly used in all areas where CFTs are ubiquitous, ranging from condensed matter and statistical physics to particle theory. In this paper we will study, holographically, superconformal interface defects in the maximally supersymmetric theory on the world-volume of multiple M2-branes. This theory was constructed in \cite{Aharony:2008ug} and is a Chern-Simons matter theory with two $SU(N)$ gauge groups of equal and opposite Chern-Simons levels $k$ with $N$ being the number of M2-branes.\footnote{Since we are interested in a limit where the number of M2-branes is large we will use the ABJM theory. See \cite{Bagger:2007jr,Gustavsson:2007vu} for earlier work on the problem of finding the world volume theory of multiple M2-branes.} For $k=1,2$ the theory preserves ${\cal N}=8$ superconformal symmetry and, at large $N$, has a holographic description in terms of eleven-dimensional supergravity on the background $AdS_4\times S^7$. For $k>2$, the supersymmetry is broken to ${\cal N}=6$ and the gravity dual background is $AdS_4\times S^7/\mathbb{Z}_k$, where the $\mathbb{Z}_k$ acts on the Hopf fiber of $S^7$ written as a $U(1)$ bundle over $\mathbb{CP}^3$.
There are two types of codimension-one defects in conformal field theory: the ones that only support degrees of freedom present in the bulk and ones that support new degrees of freedom confined to the defect. Here we study the first kind of defects and refer to them as interfaces or Janus configurations. Such Janus configurations have been constructed before for ${\cal N}=4$ SYM theory in four dimensions. Indeed, the holographic description of Janus configurations was initiated in \cite{Bak:2003jk} where a non-supersymmetric Janus solution was constructed directly in IIB supergravity. The field theory interpretation of this interface was clarified in detail in \cite{Clark:2004sb}, and it was shown in \cite{Papadimitriou:2004rz} how to calculate correlation functions in the presence of this interface holographically. This construction was later generalized and a number of supersymmetric and superconformal interfaces in ${\cal N}=4$ SYM were found in field theory \cite{Clark:2004sb,D'Hoker:2006uv,Gaiotto:2008sd}.
The supergravity duals of some of these defects were constructed in \cite{Clark:2005te,D'Hoker:2006uu,D'Hoker:2007xy,Suh:2011xc}. There also had been a number of studies on codimension-one defects in ${\cal N}=4$ SYM which support extra degrees of freedom, see, for example, \cite{DeWolfe:2001pq} for a holographic approach to such defects and \cite{Gaiotto:2008sa,Gaiotto:2008ak} for a detailed field theory analysis. It will be very interesting to study such generalizations of our Janus configurations both from the point of view of the field theory and in supergravity. In particular the low-energy theory for the well-known M2-M5 intersection will be described by such an interface with $(4,4)$ supersymmetry.\footnote{See the recent work \cite{Berman:2009kj,Berman:2009xd,Fujimori:2010ec,Faizal:2011cd,Okazaki:2013kaa} for a discussion on supersymmetric boundary conditions with various amounts of supersymmetry in supersymmetric Chern-Simons theories coupled to matter.} However we will not study these types of defects in the current work.
The dual gravitational description of the lowest dimension operators in the spectrum of the $\Neql8$ ABJM theory is given by $\Neql8$, $SO(8)$ gauged supergravity in four dimensions \cite{de Wit:1982ig}. Since we are interested in describing Janus configurations that support only such low-dimension ABJM operators (or degrees of freedom), this supergravity theory will be our main tool for constructing the gravity dual solutions to superconformal interfaces. We employ the usual Janus Ansatz of \cite{Bak:2003jk} with its domain-wall metric having an $AdS_3$ slicing.\footnote{See \cite{Karch:2000gx} for early work on holography for asymptotically $AdS_{D+1}$ solutions with $AdS_{D}$ slicing.} The metric of the Janus solutions is asymptotically $AdS_4$ and the only other non-trivial fields of the $\Neql8$ supergravity theory will be the scalars that vary as a function of the domain-wall radial variable.
Using this Ansatz and solving the BPS equations of the $\Neql8$ supergravity theory we find Janus solutions that preserve conformal invariance in $(1+1)$ dimensions and $1/2$, $1/8$ or $1/16$ of the maximal $(8,8)$ supersymmetry. In particular we find a Janus configuration with $(4,4)$ supersymmetry and an $SO(4)\times SO(4)$ ${\cal R}$-symmetry, a $(0,2)$ defect with $SU(3)\times U(1)$ global symmetry with $U(1)$ ${\cal R}$-symmetry as well as a $(0,1)$ defect with $G_2$ global symmetry.
Our $1/2$-BPS $SO(4)\times SO(4)$ Janus solutions can be uplifted, using existing technology, to a solution of eleven-dimensional supergravity and they represent a one-parameter generalization of the Janus solution found in \cite{D'Hoker:2009gg}. It is also interesting to note that our more general Janus solutions have not been captured by the classification given in \cite{D'Hoker:2008wc,D'Hoker:2009gg}. The detailed comparison and the eleven-dimensional uplift can be found in Appendix \ref{appendixB} of this paper.
The reason we restrict to the three classes of examples listed above is that all of them can be described in a unified fashion by considering consistent truncations of the maximal $\Neql8$ theory in four dimensions (which has 70 scalars) to a sector with a given global symmetry and only one scalar and one pseudoscalar that can be combined into a complex scalar parametrizing a $SU(1,1)/U(1)$ scalar manifold. One of the features of all our Janus solutions is that they come in continuous families in which one of the parameters is the asymptotic value of the phase of the complex supergravity scalar. This parameter is rather simple from the point of view of four-dimensional supergravity but in eleven dimensions and in the dual field theory it makes very significant changes to the physics. In eleven dimensions this phase controls the relative amount of metric deformation versus internal magnetic $3$-form flux and on the M2-brane the phase determines the combination of fermonic and a bosonic bilinear operators that are turned on and develop a non-trivial profile in the bulk ABJM theory.
In addition to the Janus solutions discussed above we also find a holographic realization of the phenomenon of RG flow domain walls, that is, a codimension-one defect that spatially separates two distinct superconformal fixed points related by an RG flow. See \cite{Gaiotto:2012np} and \cite{Dimofte:2013lba} for recent work on such configurations in two- and three-dimensional CFT's. The examples we present are interfaces between the maximally supersymmetric ABJM theory with $SO(8)$ global symmetry and one of two distinct $\mathcal{N}=1$ SCFT with $G_2$ global symmetry, which are related to the $SO(8)$ theory by an RG flow \cite{Bobev:2009ms}. The two distinct $\mathcal{N}=1$ SCFT are related by a reversal of the sign of the eleven-dimensional magnetic flux for their dual $AdS_4$ solutions and we also present an Janus solution which interpolates between them. On the interface all of these examples preserves $(0,1)$ superconformal symmetry and the $G_2$ global symmetry. To the best of our knowledge these are the first examples of a holographic description of RG flow domain walls. We plan to explore more general examples in the upcoming work \cite{BPWRG}.
Previous efforts to construct supersymmetric Janus solutions were generally made using IIB or eleven-dimensional supergravity \cite{D'Hoker:2006uu,D'Hoker:2007xy,Suh:2011xc,D'Hoker:2009gg}. The advantage of gauged supergravity is that it is extremely efficient in encoding some of the very complicated background fields of the higher-dimensional supergravity theories. As a result, it does not introduce a new level of difficulty if one wants to study superconformal defects that preserve less than half of the maximal supersymmetry since the system of BPS equations always reduces to a coupled system of ODEs for the four-dimensional metric coefficients and the supergravity scalars. If one were to study these solutions directly in eleven-dimensional supergravity one would typically have the daunting task of solving a system of coupled, non-linear PDE's. Another advantage of the four-dimensional description is that it should allow for more efficient calculations of correlation functions in the dual field theory in the presence of the Janus defect \cite{Papadimitriou:2004rz}.
In the next section we first review the holographic dictionary for M2-branes and how the four-dimensional scalars of interest are embedded in eleven-dimensional supergravity. In Section \ref{Sect:JanusSols} we summarize the basic structure of the class of supergravity truncations that we wish to study and in Section \ref{Sect:DetailSusy} we perform the detailed analysis of the supersymmetry and derive a universal set of BPS equations for all our Janus solutions. In Section \ref{sec:SO4} we present an analytic supergravity solution corresponding to a $(4,4)$-supersymmetric interface with $SO(4)\times SO(4)$ ${\cal R}$-symmetry. In Section \ref{sec:SU3} we find numerical solutions describing a $(0,2)$ Janus with $SU(3)\times U(1)\times U(1)$ global symmetry. We then find Janus solutions and RG flow domain walls with $(0,1)$ supersymmetry and $G_2$ global symmetry in Section \ref{sec:G2}. We conclude, in Section \ref{sec:Conclusions}, with a discussion of some problems for future study. In Appendix \ref{appendixA} we summarize various technical aspects of $\Neql8$ supegravity and in Appendix \ref{appendixC} we discuss alternative choices for the supergravity truncations in the $SU(3)\times U(1)^2$ and $G_2$ sector and show that they do not yield Janus solutions. Appendix~\ref{appendixB} contains details of the eleven-dimensional uplift of our $(4,4)$-supersymmetric Janus solutions and a detailed comparison with the results of \cite{D'Hoker:2009gg}. We also show that most of our new $(4,4)$-supersymmetric Janus solutions are not covered by the earlier classification in \cite{D'Hoker:2008wc}.
\section{The holographic dictionary and eleven-dimensional supergravity}
\label{Sect:HolDict}
Before diving into the details of the new Janus solutions it is valuable to review some of the subtleties in the holographic dictionary for the $\Neql8$ supergravity and to recall how the supergravity scalars encode different aspects of the eleven-dimensional theory.
First, the seventy-dimensional scalar manifold of the $\Neql8$ theory consists of $35$ scalars in the ${\bf 35}_s$ of $SO(8)$ and $35$ pseudoscalars in the $\mathbf{35}_c$ of $SO(8)$. To linear order in the $S^7$ truncation of eleven-dimensional supergravity, the former correspond to metric perturbations and the latter correspond to modes of the tensor gauge field, $A^{(3)}$. At higher orders these modes, of course, mix through the non-linear interactions.
The basic holographic dictionary\footnote{Here we will ignore subtle issues about monopole operators in the ABJM theory and treat them as bosonic/fermionic bilinear operators for simplicity. Alternatively one can view our discussion as applicable to the BLG theory \cite{Bagger:2007jr,Gustavsson:2007vu}.} implies that the scalars are dual to the dimension-one operators which may be thought of as bosonic bilinears of the form
\begin{equation}
O_{b}^{AB} ~=~ \text{Tr}(X^{A}X^{B}) - \frac{1}{8}\delta^{AB} \text{Tr}(X^C X^C)\;, \qquad A,B,C=1,\ldots,8\;,
\label{Obdef}
\end{equation}
and the pseudoscalars are dual to dimension-two operators, which can be thought of as fermionic bilinears of the form
\begin{equation}
O_{f}^{\dot A \dot B} ~=~ \text{Tr}(\lambda^{\dot A}\lambda^{\dot B}) - \frac{1}{8}\delta^{\dot A \dot B} \text{Tr}(\lambda^{ \dot C} \lambda^{ \dot C})\;, \qquad \dot A,\dot B, \dot C=1,\ldots,8\;.
\label{Ofdef}
\end{equation}
However, as discussed in \cite{Klebanov:1999tb}, there are subtleties in this dictionary coming from the choice of how one quantizes the modes.
The problem is how to distinguish between operator perturbations of the field theory Lagrangian and the development of vevs of the same operator. Usually non-normalizable supergravity modes correspond to coupling constants in perturbations of the Lagrangian of the dual theory, while normalizable supergravity modes correspond to states of the field theory, described by vevs. However, as discussed in \cite{Klebanov:1999tb}, this ``standard quantization'' does not necessarily apply in four dimensions if the scalars in the gravitational bulk theory have masses in the range $-9/4<m^2L^2<-5/4$, where $L$ is the scale of the $AdS_4$ fixed point. One can equally well choose ``alternative quantization,'' which reverses the standard dictionary with non-normalizable modes describing vevs and normalizable modes representing perturbations of the Lagrangian. For the 70 scalars of the ${\cal N}=8$ supergravity theory we have $m^2L^2=-2$ around the maximally supersymmetric vacuum dual to the ABJM theory and thus one can choose alternative quantization. On the other hand, it was shown in \cite{Breitenlohner:1982jf} that to preserve the supersymmetry in ${\cal N}=4$ supergravity (and therefore to preserve the supersymmetry in ${\cal N}=8$ supergravity) the supergravity pseudoscalars must be quantized in {\it exactly the opposite way} to the supergravity scalars. Thus, if the supergravity scalars follow the rules of standard quantization then the supergravity pseudoscalars must undergo alternative quantization, and vice versa.
As noted in \cite{Bobev:2011rv}, there are thus two possible choices of holographic dictionary for the seventy spin-$0$ particles of supergravity but there is only one choice in which the scaling dimensions of the supergravity modes match precisely with the scaling dimensions of the operators or couplings of the dual M2-brane theory. The correct holographic dictionary is thus:
\begin{itemize}
\item The non-normalizable ($\Delta =1$) modes of the $35$ pseudoscalars describe fermion masses on the M2-brane while for the $35$ scalars the $\Delta =1$ modes correspond to vevs of boson bilinears.
\item The normalizable ($\Delta =2$) modes of the $35$ pseudoscalars describe vevs of fermion bilinears on the M2-brane while for the $35$ scalars the $\Delta =2$ modes correspond to bosonic masses.
\end{itemize}
This is the {\it only} dictionary that is consistent with the following three features of the maximally supersymmetric $AdS_4$ vacuum (where all the supergravity scalars and pseudoscalars vanish) and the Hilbert space erected on it: a) ${\cal N}=8$ supersymmetry, b) the relationship between supergravity scalars and bosonic couplings/vevs on the M2-brane and supergravity pseudoscalars and fermionic couplings/vevs on the M2-brane, and c) the scaling dimensions of supergravity fields match the scaling dimensions of dual couplings or vevs.
To summarize, suppose that the $AdS_4$ has the Poincar\'e form:
\begin{equation}
ds_{AdS_4} ~=~ \frac{1}{\rho^2}\, (-dt^2~+~ dx^2~+~ dy^2 )~+~ \frac{d\rho^2}{\rho^2} \,.
\label{AdS4Met}
\end{equation}
Denote the 35 scalars by $\Phi_i$ and the 35 pseudoscalars by $\Psi_i$, then they will generically have the following asymptotic expansion close to the $AdS_4$ boundary at $\rho \to 0$:
\begin{equation}\begin{split}
\Phi_ i \approx \phi_i^{(v)} \rho + \phi_i^{(s)} \rho^2 + \mathcal{O}(\rho^3)~,\\
\Psi_ i \approx \psi_i^{(s)} \rho + \psi_i^{(v)} \rho^2 + \mathcal{O}(\rho^3)~.
\end{split}
\label{quantconds}
\end{equation}
The coefficients $\phi_i^{(v)}$ and $\phi_i^{(s)}$ are related to the vev and the source for the bosonic bilinear operator of dimension $\Delta=1$ and $\psi_i^{(v)}$ and $\psi_i^{(s)}$ are related to the vev and the source for the fermionic bilinear operator of dimension $\Delta=2$. It should, however, be remembered that if a supergravity mode involves a non-zero, non-normalizable part (${\cal O}(\rho )$) then it can source the normalizable part (${\cal O}(\rho^2)$) and so disentangling the independent physical meaning of the normalizable components can be subtle and one should use holographic renormalization. There is, of course, no such difficulty if the non-normalizable part vanishes.
The truncations of four-dimensional supergravity that we consider here consists of a complex scalar, $z$, in an $SU(1,1)/U(1) = SL(2,\mathbb{R})/SO(2)$ coset. The real part of $z$ is a supergravity scalar and the imaginary part of $z$ is a pseudoscalar. Thus the real part of $z$, at linear order, encodes metric perturbations in eleven dimensions and is dual to operators of the form \eqref{Obdef} and the imaginary part of $z$, at linear order, encodes flux perturbations and is dual to a linear combination of the operators in \eqref{Ofdef}. The precise holographic dictionary is then governed by (\ref{quantconds}). One of the interesting features of all our solutions is the phase of $z$ and the choice of its boundary values. From the perspective of both eleven-dimensional supergravity and for the field theory on the M2-branes, the families of such solutions represent very different physics.
\section{The BPS defects: The family of Janus solutions}
\label{Sect:JanusSols}
\subsection{The bosonic background}
We are seeking the gravity duals of $(1+1)$-dimensional conformal defects in $(2+1)$-dimensional conformal field theories.
This means that we are looking for solutions with four-dimensional metrics that are sectioned by $AdS_3$:
\begin{equation}
ds^2 = e^{2A}ds_{AdS_3(\ell)} ~+~ d\mu^2\,,
\label{JanMet}
\end{equation}
with boundary conditions that produce $AdS_4$ at $\mu= \pm \infty$. While the radius of the $AdS_3$ sections can be scaled away, we find it convenient to have this radius appear as an explicit parameter, $\ell$. In the Poincar\'e patch we therefore have:
\begin{equation}
ds_{AdS_3(\ell)} ~=~ e^{2r/\ell}(-dt^2~+~ dx^2)~+~ dr^2 \,.
\label{AdS3Met}
\end{equation}
Note that the metric, (\ref{JanMet}), is precisely that of an $AdS_4$ of radius $L$ if one has:
\begin{equation}
A ~=~ \log\Big(\frac{L}{\ell} \cosh \Big(\frac{\mu}{L} \Big)\Big) \,.
\label{Aasymp}
\end{equation}
This will therefore determine the boundary conditions at $\mu = \pm \infty$.
Since we are working in gauged supergravity, the only other non-trivial aspect to the background will be scalar fields in the four-dimensional theory. Furthermore, we restrict to sectors of gauged $\Neql8$ supergravity that are invariant under some group $G \subset SO(8) \subset E_{7(7)}$ and we choose this invariance group, $G$, so that it only commutes with an $SL(2, \mathbb{R})/SO(2)$ coset in $E_{7(7)}/SU(8)$. There are three intrinsically different possibilities for such an embedding and these are described in Section \ref{E7embed}. Here we simply use the $SL(2, \mathbb{R})$ structure and the fact that the embedding is characterized by a positive integer, $k$, known as the embedding index.
Our scalar sub-sector thus always reduces to $SL(2, \mathbb{R})/SO(2) =SU(1, 1)/U(1)$, which we can parameterize by
\begin{equation}
g ~=~ \exp\left(
\begin{array}{cc}
0 & \alpha\, e^{i\zeta} \\ \alpha\, e^{-i\zeta} & 0
\end{array} \right) ~=~ \left(
\begin{array}{cc}
\cosh \alpha &\sinh \alpha \, e^{i\zeta} \\ \sinh \alpha \, e^{-i\zeta} & \cosh \alpha
\end{array} \right) \,,
\label{gmat}
\end{equation}
for some real variables $\alpha$ and $\zeta$ with $\alpha \ge 0$, $-\pi \le \zeta < \pi$ . The kinetic term, ${\cals A}$, and the composite $U(1)$ connection, ${\cal B}$, are then given by
\begin{equation}
g^{-1} dg ~=~ \left(
\begin{array}{cc}
{\cal B} & {\cals A} \\ \bar {\cals A} & -{\cal B}
\end{array} \right) ~=~ \left(
\begin{array}{cc}
i \sinh^2 \alpha \, d \zeta &(d \alpha + \frac{i}{2} \,\sinh 2 \alpha \, d \zeta)\, e^{i\zeta} \\ (d \alpha - \frac{i}{2} \,\sinh 2 \alpha\, d \zeta)\, e^{-i\zeta} & -i \sinh^2 \alpha \, d \zeta
\end{array} \right) \,.
\label{kinmat}
\end{equation}
The standard normalized scalar kinetic term in the Lagrangian is then $\frac{1}{2} |{\cals A}|^2$.
In the foregoing discussion we used the $SL(2, \mathbb{R})$ group element, $g$, in one copy of the fundamental representation. However in $\Neql 8$ supergravity the kinetic term is normalized based upon the fundamental representation of $E_{7(7)}$ and this will generically decompose into larger representations of $SL(2,\mathbb{R}) \subset E_{7(7)}$. The index of the representation\footnote{See, for example, \cite{McKayPatera, Schellekens:1986mb}.} gives the embedding index or winding number, $k$, that multiplies both the canonically normalized $SL(2,\mathbb{R})$ kinetic term, ${\cals A}$, as well as the connection, ${\cal B}$, that arise from the corresponding canonically normalized $E_{7(7)}$ terms. Thus we will find this (positive) integer consistently arising throughout our discussions of various embeddings. The complete scalar Lagrangian also involves a scalar potential inherited from the potential of the ${\cal N}=8$ theory and this, of course, depends upon the details of the embedding of $SL(2,\mathbb{R})$ in $E_{7(7)}$.
The Lagrangian can be conveniently described by parametrizing everything in complex variables. Indeed, the coset $SL(2, \mathbb{R})/SO(2)$ is a K\"ahler manifold with canonical complex coordinate, $z$, defined by
\begin{equation}
z ~=~ \tanh \alpha\, e^{i\zeta} \,.
\label{zparam}
\end{equation}
The scalar Lagrangian is then parametrized by a K\"ahler potential, ${\cal K}(z, \bar z)$, and a holomorphic superpotential, ${\cal V}(z)$. Specifically, the Lagrangian of the theories of interest can be expressed in the form:\footnote{All models we consider arise as truncations of the $\mathcal{N}=8$ supergravity. However, it should be possible to rewrite them as four-dimensional, $\mathcal{N}=2$ gauged supergravity theories. This underpins the holomorphic structure that we are exploiting.}
\begin{equation}
e^{-1}{\cal L}\ ~=~ \coeff{1}{2}\, R ~-~ g^{\mu \nu} {\cal K}_{z\bar{z}} \partial_\mu z \, \partial_\nu \bar{z}~-~ g^2\mathcal{P} (z, \bar z )\,,
\label{genLag}
\end{equation}
where $g$ is the coupling constant of the gauged supergravity and
\begin{equation}
\mathcal{K}_{z\bar{z}} ~=~ \partial_{z}\partial_{\bar{z}}\mathcal{K}\;.
\label{derivK}
\end{equation}
We will also define $\mathcal{K}^{z\bar{z}}$ to be the inverse of $\mathcal{K}_{z\bar{z}}$. The potential, $\mathcal{P}(z,\bar{z})$, can be obtained from a holomorphic superpotential, ${\cal V}(z)$, via:
\begin{equation}
\mathcal{P} = e^{\mathcal{K}}(\mathcal{K}^{z\bar{z}}\nabla_{z}\mathcal{V}\nabla_{\bar{z}}\overline{\mathcal{V}} -3 \mathcal{V}\overline{\mathcal{V}})\;,
\label{Pform}
\end{equation}
where the covariant derivatives are defined in the usual way:
\begin{equation}
\nabla_{z}\mathcal{V} = \partial_{z}\mathcal{V} + (\partial_{z}\mathcal{K})\mathcal{V}\;, \qquad \nabla_{\bar{z}}\overline{\mathcal{V}} = \partial_{\bar{z}}\overline{\mathcal{V}} + (\partial_{\bar{z}}\mathcal{K})\overline{\mathcal{V}}\;.
\end{equation}
For the $SL(2, \mathbb{R})/SO(2)$ coset we have
\begin{equation}
\mathcal{K} = -k\, \log(1-z\bar{z})\;,
\label{Kahlerpot}
\end{equation}
where $k \in \mathbb{Z}_+$ is the embedding index of the $SL(2, \mathbb{R})$ in the $E_{7(7)}$ of $\Neql8$ supergravity. Thus the scalar kinetic term is given by the canonical sigma-model expression:
\begin{equation}\label{Kform}
\mathcal{K}_{z\bar{z}} ~=~ \dfrac{k}{(1-z\bar{z})^2}\;.
\end{equation}
As we will see, the holomorphic superpotential, ${\cal V}(z)$, is generically a polynomial of degree $k$, or less.
\subsection{The $SL(2, \mathbb{R})$ embeddings in $E_{7(7)}$ defined through invariance}
\label{E7embed}
Underlying our Janus solutions are consistent truncations of $\Neql8$ supergravity down to the scalar coset $SL(2, \mathbb{R})/SO(2)$. As we remarked earlier, we will find all such truncations that arise from $G$-invariant sectors of the ${\cal N}=8$ theory where $G \subset SO(8) \subset E_{7(7)}$ and so we require that $G$ only commute with $SL(2,\mathbb{R})$ in the $E_{7(7)}$. Once one has found the subgroup $G$ it will generically be contained in a larger, possibly non-compact group, $\widehat G$ so that $\widehat G \times SL(2,\mathbb{R})$ is a maximal embedding in $E_{7(7)}$. Such maximal embeddings are well-known and, for example, a list may be found in \cite{Schellekens:1986mb}. The complete list with $SL(2,\mathbb{R})$ factors is
\begin{itemize}
\item[(i)] $ (SO(4) \times SO(4)) \times SL(2,\mathbb{R}) ~\subset~ SO(6,6) \times SL(2,\mathbb{R})~\subset~ E_{7(7)}$\,, \ with $k =1$
\item[(ii)] $ (SU(3) \times U(1)\times U(1)) \times SL(2,\mathbb{R}) ~\subset~ F_{4(4)} \times SL(2,\mathbb{R})~\subset~ E_{7(7)}$\,, \ with $k =3$
\item[(iii)] $ (G_2) \times SL(2,\mathbb{R}) ~\subset~ G_2 \times SL(2,\mathbb{R}) ~\subset~ E_{7(7)}$\,, \ with $k =7$
\end{itemize}
where the first group in parenthesis defines $G \subset SO(8)$ and the second inclusion defines $\widehat G$. The integer, $k$, is the embedding index of the $SL(2, \mathbb{R})$ factor.
We thus have three distinct classes of models that we discuss systematically in the subsequent sections. These three consistent truncations have been considered before but not in the context of Janus solutions. Holographic flows of (i), and their eleven-dimensional uplifts, were extensively analyzed in \cite{Pope:2003jp}. The $SU(3)$-invariant sector has been studied in many papers \cite{Bobev:2009ms,Warner:1983vz,Ahn:2000mf,Ahn:2000aq,Ahn:2001by,Ahn:2001kw,Ahn:2002eh,Ahn:2002qga,Bobev:2010ib} and one can obtain (ii) and (iii) through further truncations of this sector. However, unlike some of the earlier analysis of such truncations, here we do not necessarily restrict ourselves to $G$-invariant supersymmetries and consider the more general possibility of supersymmetries that transform in a non-trivial representation, ${\cal R}_{\epsilon}$, of $G$.
All of these $SL(2, \mathbb{R})/SO(2)$ embeddings in $E_{7(7)}/SU(8)$ have a very important feature: The $SO(2)$ generator lies in the purely imaginary part of $SU(8)$ which means that it is {\it not} generically a symmetry of the gauged theory and that it rotates between the scalar and pseudoscalar sectors of the $\Neql{8}$ supergravity theory. Thus our complex scalar, $z$, has a real part that is a supergravity scalar and an imaginary part that is a pseudoscalar. In the UV limit of the holographic dual theory the real part of $z$ therefore encodes details of a boson bilinear and the imaginary part of $z$ encodes a fermion bilinear.
As described in Section \ref{Sect:HolDict}, the action of the $SO(2)$ is very interesting from the perspective of the holographic field theory in that in the UV it interpolates between bosonic and fermionic bilinears and thus changes the physics underlying the entire flow. Similarly, in eleven-dimensional supergravity, the $SO(2)$ action interpolates between metric fields and $3$-form fluxes and so, once again, changing the phase of $z$ makes dramatic changes in the boundary conditions and overall structure of the eleven-dimensional solution. Indeed, it was this observation that was a major motivation for the analysis in \cite{Pope:2003jp,Bena:2004jw}.
\section{Solving the BPS equations for $G$-invariant Janus solutions}
\label{Sect:DetailSusy}
We now take the general supersymmetry structure of the ${\cal N}=8$ theory and make the detailed reduction to the class of truncations described in Section \ref{E7embed}.
\subsection{Some supergravity preliminaries}
\label{gensusy}
Our metric has ``mostly plus'' signature and the gamma matrices are defined by $\{ \gamma^a \,, \gamma^b \} = 2\eta^{a b}$ where $\eta = {\rm diag} (- 1,+1,+1,+1)$. Thus $\gamma^a$, $a=1,2,3$ will be hermitian and $\gamma^0$ is anti-hermitian. We choose an explicit Majorana representation in which the $\gamma^a$, $a=0,1,2,3$ are real and in this representation, the helicity projector, $\gamma_5$, is purely imaginary and anti-symmetric.
Following the standard practice in four dimensions, spinors will be written in terms of the chiral projections of the corresponding Majorana spinors as described in \cite{deWit:1978sh}. For example:
\begin{equation}\label{chirdefs}
\begin{split}
\epsilon^i & ~\equiv~ \coeff 1 2\,(1+\gamma_5){ \epsilon}^i_M\,,\qquad
\epsilon_i ~\equiv~ \coeff 1 2\,(1-\gamma_5){ \epsilon}^i_M\,,\\
\bar \epsilon^{\,i} & ~\equiv~ \coeff 1 2\,{ \bar\epsilon}^i_M(1+\gamma_5)\,,\qquad
\bar \epsilon_{i} ~\equiv~ \coeff 1 2\, {\bar\epsilon}^{\,i}_M(1-\gamma_5)\,,\\
\end{split}
\end{equation}
where $\epsilon^i_M$, $i=1,\ldots,8$, are the underlying Majorana spinors. Since $\gamma_5$ is purely imaginary in this Majorana representation, complex conjugation raises and lowers the $SU(8)$ indices of the $\Neql8$ theory.
The supersymmetry variations of the 8 gravitinos and the 56 gauginos in the ${\cal N}=8$ theory are given by \cite{de Wit:1982ig}
\begin{equation}
\delta\psi_\mu{}^i = 2 D_\mu\epsilon^i ~+~ \sqrt {2} \, g\,A_1{}^{ij}\gamma_\mu\epsilon_j\,,
\label{deltagravitino}
\end{equation}
and
\begin{equation}
\delta\chi^{ijk} = -\cals A_\mu{}^{ijkl}\,\gamma^\mu\,\epsilon_l ~-~ 2\, g\, A_{2\,l}{}^{ijk}\epsilon^l\,,
\label{deltagaugino}
\end{equation}
respectively. The definitions of the various $E_{7(7)}$ tensors above are summarized in Appendix~\ref{appendixA}.
Since we are considering backgrounds that are invariant under some subgroup, $G \subset SO(8)$, the unbroken supersymmetries will lie in some representation, ${\cal R}_{\epsilon}$, of $G$. We will denote the helicity components, $\epsilon^l$ and $\epsilon_l$, in ${\cal R}_{\epsilon}$ generically by $\epsilon$ and $\epsilon^*$ respectively and since the $SO(8)$ has a real action on $\epsilon^l$ and $\epsilon_l$, both sets of helicity components must transform in the same $SO(8)$ representation.
Our task will ultimately be to solve the BPS conditions $\delta\psi_\mu{}^i=0$ and $\delta\chi^{ijk} =0$ within the truncations of interest. We will do this in detail below. As often happens with the BPS equations, we find that the solutions also automatically solve the equations of motion.
\subsection{The gaugino variation}
The fields are assumed to be invariant under the $SO(2,2)$ action on the $AdS_3$ and so the scalars can only depend upon the coordinate $\mu$ in (\ref{JanMet}). This means that the vanishing of the gaugino variation (\ref{deltagaugino}) only involves $\gamma^3$ and can be generically re-written as:
\begin{equation}
\gamma^3\,\epsilon= M\,\epsilon^*\,,\qquad \gamma^3\,\epsilon^*= M^*\,\epsilon\,,
\label{gauginovar}
\end{equation}
where we have used the reality of $\gamma^3$. In particular, this implies $MM^* = 1$ and hence we have
\begin{equation}
M = e^{i \Lambda}\,,
\label{MLambda}
\end{equation}
for some real phase, $\Lambda$. We can therefore define $\varepsilon$ by
\begin{equation}
\epsilon = e^{i \Lambda/2} \,\varepsilon \,,
\label{varespsdefn}
\end{equation}
and then we have
\begin{equation}
\gamma^3\varepsilon=\varepsilon^*\,,\qquad \gamma^3\varepsilon^*=\varepsilon\,.
\label{varepsproj}
\end{equation}
Explicitly, multiplying (\ref{deltagaugino}) by $\gamma^3$ we find that the quantity $M$ is given by:
\begin{equation}
M ~=~ \big(g \,{\cal K}^{z\bar{z}} \, e^{{\cal K}/2} \, \nabla_z {\cal V} \big)^{-1}\, \bar{z}' \,,
\label{Mexplicit}
\end{equation}
and so (\ref{MLambda}) implies:
\begin{equation}
\label{eqzdermag}
z'\bar{z}' ~=~ g^2 \, (\mathcal{K}^{z\bar{z}})^2\, e^{\mathcal{K}}\, \nabla_{z}\mathcal{V}\, \nabla_{\bar{z}}\overline{\mathcal{V}}\;.
\end{equation}
\subsection{The gravitino variation}
\label{Sect:Gravvar}
In looking for the Poincar\'e supersymmetries parallel to the $(1+1)$-dimensional flat sections of the $AdS_3$ metric (\ref{AdS3Met}), we assume that the supersymmetries are independent of $t$ and $x$. This means that the spin-$\frac{3}{2}$ variations along $t$ and $x$ reduce to
\begin{equation}
\Big(A'\,\gamma^3\,+{1\over\ell}\, e^{-A}\,\gamma^2 \Big)\,\epsilon+g\,\overline {\cals W }\,\epsilon^*= 0\,,
\label{gravvar}
\end{equation}
where ${\cals W}$ is the appropriate eigenvalue of $\frac{1}{\sqrt{2}} A_{1\,ij}$.
Indeed, ${\cals W }$ is related to the holomorphic superpotential via:
\begin{equation}
{\cals W } = e^{{{\cal K}}/2} \, {\cal V}\,.
\label{WVreln}
\end{equation}
Taking the complex conjugate of (\ref{gravvar}) and iterating, one obtains the quadratic constraint:
\begin{equation}
(A')^2= - {1\over\ell^2} \, e^{-2A}+g^2\,|\cals W|^2\,.
\label{quadconstr}
\end{equation}
However, the two projection conditions (\ref{gauginovar}) and (\ref{gravvar}) must be compatible with one another. In particular, one can use (\ref{gauginovar}) to eliminate $\gamma^3 \epsilon$ in favor of $\epsilon^*$ and obtain a projection condition solely involving $\gamma^2$, which must have the form:
\begin{equation}
\gamma^2 \epsilon= i\,\kappa\,e^{i \Lambda}\, \epsilon^* \quad \Leftrightarrow \quad \gamma^2 \varepsilon= i\,\kappa\,\varepsilon^*\,,\qquad |\kappa | = 1\,.
\label{3proj}
\end{equation}
After using (\ref{varepsproj}) in this projection condition one finds that compatibility ($\gamma^2 \gamma^3 = - \gamma^3 \gamma^2$) requires:
\begin{equation}
\kappa^2 = 1\,.
\label{projcomp}
\end{equation}
Explicitly, using (\ref{3proj}) and (\ref{gauginovar}) in (\ref{gravvar}) we find:
\begin{equation}
\Big(A' ~+~ {i \kappa \over\ell}\, e^{-A} \Big)\,e^{i \Lambda} = - g\,\overline {\cals W } = - g\, e^{{{\cal K}}/2} \, \overline {{\cal V}} \,,
\label{Lambdaexplicit}
\end{equation}
which provides a ``square root'' of (\ref{quadconstr}). In particular, note that we now know that $\kappa = \pm 1$ and is thus a constant.
The variation along the $AdS_3$ radial direction is
%
\begin{equation}\label{varr}
2\,\partial_r\epsilon+A'\,e^A\,\gamma^2\gamma^3\,\epsilon+g\, e^A \,\overline{\cals W}\,\gamma^2\epsilon^*= 0\,.
\end{equation}
Using \eqref{gravvar}, this reduces to
\begin{equation}\label{derreps}
2\,\partial_r\epsilon= {1\over \ell}\,\epsilon\,,
\end{equation}
and is solved by
\begin{equation}
\label{epsrdep}
\epsilon= e^{r/2\ell}\,\tilde \varepsilon\,,
\end{equation}
where $\tilde \varepsilon$ is independent of $r$.
Finally, in general one knows that $\bar \epsilon \gamma^\mu \epsilon$ is a timelike (or null) Killing vector and so consistency with (\ref{JanMet}), (\ref{AdS3Met}), (\ref{varespsdefn}) and (\ref{epsrdep}) means that we must have
\begin{equation}\label{varmu}
\epsilon= e^{(A(\mu)+ r/\ell + i \Lambda)/2}\,\varepsilon_0\,,
\end{equation}
where $\varepsilon_0$ could have a phase that depends upon $\mu$. Explicit calculations in each example show that the phase dependence of $\epsilon$ is determined precisely by $\Lambda$ in (\ref{MLambda}) and (\ref{varespsdefn}) and thus $\varepsilon_0$ is simply a constant spinor satisfying:
\begin{equation}
\gamma^3\varepsilon_0 =\varepsilon_0^*\,,\qquad \gamma^2\varepsilon_0 = i\, \kappa\, \varepsilon_0^* \,,
\label{vareps0proj}
\end{equation}
as a consequence of (\ref{varepsproj}) and (\ref{3proj}).
\subsection{The supersymmetries}
\label{Sect:susies}
As we remarked earlier, the unbroken supersymmetries will lie in some representation, ${\cal R}_{\epsilon}$, of $G$ and $\epsilon$ and $\epsilon^*$ respectively denote the helicity components, $\epsilon^l$ and $\epsilon_l$, of any spinor in ${\cal R}_{\epsilon}$. The elements of ${\cal R}_{\epsilon}$ can be distinguished by comparing (\ref{gravvar}) and (\ref{deltagravitino}): The supersymmetries are then simply determined by the space of $\epsilon_j$ upon which $A_1^{ij}$ has the eigenvalue $\frac{1}{\sqrt{2}} \overline{{\cal W}}$. This determines the number, $\widehat {\cal N}$, of supersymmetries, $\epsilon_j$, that go into the foregoing calculation. However, it is still possible for ${\cal R}_{\epsilon}$ to be a reducible representation of $G$ and the phases $e^{i \Lambda}$ and $\kappa$ can differ between irreducible components of ${\cal R}_{\epsilon}$. For the present we will assume that are dealing with $\widehat {\cal N}$ supersymmetries in one irreducible component of ${\cal R}_{\epsilon}$ and hence $e^{i \Lambda}$ and $\kappa$ are the same for all $\widehat {\cal N}$ supersymmetries. We will return to this issue in Section \ref{sec:SO4} where ${\cal R}_{\epsilon}$ will have two irreducible components.
The supersymmetric Janus solutions require that we impose the additional conditions (\ref{gauginovar}) and (\ref{3proj}). These each cut the four independent (real, Majorana) components down by half, leaving a single real component. In particular, these constraints imply
\begin{equation}
\gamma^2 \gamma^3 \varepsilon = -i\, \kappa\,\varepsilon \,.
\label{helproj}
\end{equation}
However, since $\epsilon$ represents some set of $\epsilon^l$, the helicity condition (\ref{chirdefs}) implies that
\begin{equation}
\gamma_5 \, \varepsilon = \varepsilon \qquad \Rightarrow \qquad \gamma_5 \, \varepsilon^* = - \varepsilon^* \,.
\label{helcond}
\end{equation}
and since $\gamma_5 = i\gamma^0 \gamma^2 \gamma^2 \gamma^3$, (\ref{helproj}) implies that the spinors are projected onto $(1+1)$-dimensional chiral components:
\begin{equation}
\gamma^0 \gamma^1 \varepsilon = \kappa\,\varepsilon \qquad \Rightarrow \qquad \gamma^0 \gamma^1 \varepsilon^* = \kappa\,\varepsilon^* \,,
\label{chirproj}
\end{equation}
where we have again used the reality of the $\gamma^a$. The conditions (\ref{gauginovar}) and (\ref{3proj}) thus serve to impose the Majorana condition in $(1+1)$ dimensions and so the four real components of $\epsilon$ are reduced to a single, real component of definite chirality (\ref{chirproj}), determined by $\kappa$, in $(1+1)$ dimensions. The theory on the interface thus has $(\widehat {\cal N},0)$ supersymmetry for $\kappa= +1$ and $(0,\widehat {\cal N})$ supersymmetry. for $\kappa= -1$.
As we will see in Section \ref{Sect:JanusEqns}, the choice of $\kappa$ enters directly into the BPS equations underlying the Janus solution and once a choice has been made and a solution has been constructed, the helicity of the supersymmetries of that solution is fixed. This observation becomes particularly important when ${\cal R}_{\epsilon}$ has more than one irreducible piece.
\subsection{The Janus BPS equations}
\label{Sect:JanusEqns}
Taking the real and imaginary parts of (\ref{Lambdaexplicit}) one obtains:
\begin{align}
A' =& - \coeff{1}{2}\,g\, e^{{{\cal K}}/2} \, \left( e^{i\Lambda}\, {\cal V} ~+~ e^{-i\Lambda}\, \overline {{\cal V}} \,\right) \,,
\label{dAsusy1} \\[6 pt]
e^{-A} =& - \coeff{1}{2}\,i \kappa \, g\,\ell\, e^{{{\cal K}}/2} \, \left( e^{i\Lambda}\,{\cal V} ~-~ e^{-i\Lambda}\, \overline {{\cal V}} \,\right) \,.
\label{eAres1}
\end{align}
We can now use (\ref{Lambdaexplicit}) to eliminate $M = e^{i \Lambda}$ in (\ref{Mexplicit}) to obtain the BPS equations for the scalars:
\begin{equation} \label{BPSz}
\begin{split}
z' & ~=~ -\mathcal{K}^{z\bar{z}}\, \big( \overline{{\cal V}}^{-1} \nabla_{\bar{z}} \overline{{\cal V}} \big) \, \left(A' + i \, \kappa \,\dfrac{e^{-A}}{\ell}\right)\,, \\
\bar{z}' & ~=~ -\mathcal{K}^{z\bar{z}}\, \big( {\cal V}^{-1} \nabla_{z} {\cal V} \big) \, \left(A' - i \, \kappa \, \dfrac{e^{-A}}{\ell}\right) \;.
\end{split}
\end{equation}
These four equations represent a first-order system for the four unknown quantities $z(\mu)$, $\bar z(\mu)$, $A(\mu)$ and $\Lambda(\mu)$.
Note that this shows that the supersymmetric $AdS_4$ critical points are determined by:
\begin{equation}
\nabla_{z} {\cal V} ~=~0\;.
\label{susycrit}
\end{equation}
Moreover, because ${\cal V}$ is holomorphic and ${\cal K}$ is real, if $z_0$ satisfies (\ref{susycrit}) then so does $\bar{z}_0$. Thus if $z_0$ has a non-trivial imaginary part, then the supersymmetric critical point comes in a pair related by $z_0 \to {\bar z}_0$. We will see an example of this in Section \ref{G2truncation}. In terms of eleven-dimensional supergravity, this complex conjugation corresponds to reversing the sign of the internal (magnetic) components of the tensor gauge field, $A^{(3)}$. This can be explicitly demonstrated within the $G_2$ truncation as well as for the $SO(4)\times SO(4)$ one, see \eqref{fourflux} and \eqref{intflux}.\footnote{More generally, it is clear at linear order in the consistent truncation and at non-linear order it holds because both the pseudoscalars and the internal components of $A^{(3)}$ are odd under the parity symmetry that flips all the internal coordinates.}
One can eliminate $\Lambda$ from (\ref{dAsusy1}) and (\ref{eAres1}) and rederive (\ref{quadconstr}). One can then view (\ref{BPSz}) and (\ref{quadconstr}) as three equations for the three physical quantities $z(\mu)$, $\bar z(\mu)$ and $A(\mu)$.
One can easily show that for any holomorphic superpotential, ${\cal V}$, these BPS equations imply the equations of motion derived from the action (\ref{genLag}), or (\ref{simpLag}).
\subsection{The general behavior of the Janus solutions}
\label{FOsystem}
The first order system, \eqref{BPSz}, can be given a more intuitive form if one writes it in terms of the real fields, $\alpha(\mu)$ and $\zeta(\mu)$, and the real superpotential, $W$, defined by
\begin{equation}
W^2 ~\equiv~ | {\cal W}|^2 ~=~ e^{\cal K} \, |{\cal V}|^2 \;.
\label{normW}
\end{equation}
One can then express the potential as
\begin{equation}
\label{Simppot}
\cals P= \frac{1}{k}\,\bigg[ \left({\partial W \over\partial\alpha}\right)^2+ \frac{4}{\sinh^{2}(2\alpha)}\left({\partial W\over\partial\zeta}\right)^2\bigg] ~-~ 3 \, W^2 \,,
\end{equation}
where $k$ is the embedding index that appears in the normalization of the K\"ahler form \eqref{Kform}. The scalar BPS equations may then be written:
\begin{align}
\alpha' & = -{1\over k}\,\bigg({A'\over W}\bigg) \,{\partial W\over\partial\alpha}~+~{2 \, \kappa \over k}\, \bigg({e^{-A}\over\ell}\bigg) \,{1\over\sinh(2\alpha) }\,\frac{1}{W}\,{\partial W\over\partial\zeta}\,,\label{simpalphaeqn} \\[10 pt]
\zeta' & =-{4\over k}\,\bigg({A'\over W}\bigg) \,{1 \over\sinh^2(2\alpha) }\,{\partial W\over\partial\zeta} ~-~ {2\, \kappa \over k}\, \bigg({e^{-A}\over\ell}\bigg) \,{1\over\sinh(2\alpha)}\,\frac{1}{W}\,{\partial W\over\partial\alpha} \label{simpzetaeqn} \,.
\end{align}
These scalar equations must be solved together with \eqref{quadconstr}, which in the real notation reads:\footnote{Similar BPS equations for holographic domain walls with curved slices were written down in \cite{Clark:2005te,LopesCardoso:2001rt,LopesCardoso:2002ec}.}
\begin{equation}
(A')^2= g^2\,W^2 ~-~{e^{-2A}\over\ell^2} \,.
\label{Aprimeeqn}
\end{equation}
The fact that \eqref{Aprimeeqn} is quadratic in $A'$ means that the solution may have a branch cut ambiguity when $A'(\mu) = 0$. We will see that the interesting Janus solutions do indeed move across these branches in the ``center'' of the solution: In particular, we will see that the interesting solutions have $A'(\mu) = \pm c_{\pm}$, where $c_{\pm} >0$, as $\mu \to \pm \infty$.
Observe that if one takes the limit $\ell \to \infty$, in which the $AdS_3$ sections (\ref{AdS3Met}) become flat, then the BPS equations become the familiar steepest descent equations of holographic RG flows (see, for example, \cite{Freedman:1999gp}). Note that in this limit (\ref{Aprimeeqn}) yields $A' = \pm W$ and this sign ambiguity is transmitted to (\ref{simpalphaeqn}) and (\ref{simpzetaeqn}). This sign choice is then determined in holographic RG flows by boundary conditions. Notice also that in the limit $\ell \to \infty$ there is a simplification in solving the system of equations \eqref{simpalphaeqn}, \eqref{simpzetaeqn}, and \eqref{Aprimeeqn}. The equations for the scalars \eqref{simpalphaeqn} and \eqref{simpzetaeqn} form a closed system which one can integrate and only after that solve \eqref{Aprimeeqn} for $A$.
In Janus solutions we typically want to start from asymptotically $AdS_4$ boundary conditions, which means we start and finish at some critical points of $W$ near which $e^{A(\mu)}$ is very large and positive. Since $W$ is manifestly positive this means that we must correlate $A' = \pm W$ as $\mu \to \pm \infty$ and then we have:
\begin{equation}
\alpha' = \mp {1\over k} \,{\partial W\over\partial\alpha} \,, \qquad
\zeta' = \mp{ 4 \over k \, \sinh^2(2\alpha) }\,{\partial W\over\partial\zeta} \,.
\end{equation}
This means that near $\mu =- \infty$ the solution starts as a steepest ascent from a critical point and then as $\mu \to +\infty$ the flow changes to steepest descent into another, or possibly the same, critical point. Indeed, we will typically start and finish at the same critical point and as the solution ascends out of that point the second terms in (\ref{simpalphaeqn}) and (\ref{simpzetaeqn}) start to play a role and the solution begins to loop around in the $(\alpha, \zeta)$ plane and at some point $A'$ passes through zero onto the other branch of the $A'$ equation and the evolution starts descending back to the critical point.
In the study of holographic RG flows, it was found that there were flows to ``Hades'' \cite{Freedman:1999gp} in which either the scalar fields ran off to infinite values of ${\cal P}$, or the metric function $A(\mu)$ diverged at some finite value of $\mu$. It was subsequently shown in \cite{Freedman:1999gk,Gubser:2000nd} that many of these flows to ``Hades'' had a simple physical interpretation in terms of a flow to the Coulomb branch in the dual field theory while others represented unphysical singularities \cite{Gubser:2000nd}. Here we also find that some of the Janus solutions involve flows to points at which $A(\mu)$ diverges and solutions with similar properties were found in \cite{Gutperle:2012hy}. It is possible that these might represent conformal interfaces between Coulomb branches and other phases of the theory on the M2-branes. This certainly deserves investigation but it will probably require the construction of the eleven-dimensional uplift. For the present we will confine our attention to regular Janus solutions that start and finish at conformal fixed points, for which the physical interpretation is much clearer.
\section{The $SO(4)\times SO(4)$-invariant Janus}
\label{sec:SO4}
\subsection{The truncation}
The $SO(4)\times SO(4)$ invariant truncation of $\Neql8$ supergravity was discussed extensively in \cite{Pope:2003jp}. The non-compact generators of the $SL(2,\mathbb{R}) \subset E_{7(7)}$ are defined by:
\begin{equation}
\Sigma_{IJKL} ~\sim~ \big(\, z \, \delta^{1234}_{[IJKL]} ~+~ \bar z \,\delta^{5678}_{[IJKL]} \,\big)\,,
\label{E7gens1}
\end{equation}
and the embedding index is equal to unity: $k=1$. The $SO(2)$ or $U(1)$ action is simply the $SU(8)$ transformation:
\begin{equation}
U ~=~ {\rm diag}\,(e^{i \beta}, e^{i \beta}, e^{i \beta}, e^{i \beta}, e^{-i \beta}, e^{-i \beta}, e^{-i \beta}, e^{-i \beta})\,,
\label{SU8a}
\end{equation}
which rotates $z$ by the phase $e^{4i \beta}$.
The scalar potential is given by
\begin{equation}
\mathcal{P} = -2\, (2 + \cosh 2 \alpha) ~=~ -\frac{2\,(3- |z|^2)}{1-|z|^2}\;.
\label{PSO4}
\end{equation}
{\it A priori} one does not expect (\ref{SU8a}) to generate a symmetry of the action but in this instance it does and given the consequences in eleven-dimensions this is a very surprising symmetry \cite{Pope:2003jp}.
The effective particle action that encodes all field equations is:
\begin{align}
\cals L ~=~ & e^{3A}\left[3 (A')^2-(\alpha')^2-{1\over 4}\sinh^2(2\alpha)(\zeta')^2+2g^2(2+\cosh(2\alpha))
\right]-{3\over\ell^2}e^A \\
~=~ & e^{3A}\left[3 (A')^2 ~-~ \dfrac{z'\bar{z}'}{1-z\bar{z}} ~+~ 2g^2\,\Big(\dfrac{3-|z|^2}{1-|z|^2}\Big)
\right]-{3\over\ell^2}e^A\,,
\label{EffLag1}
\end{align}
where we have used the K\"ahler potential (\ref{Kahlerpot}) with $k=1$.
The holomorphic superpotential is extremely simple:
\begin{equation}
\mathcal{V} ~=~ \sqrt{2} \qquad \Rightarrow \qquad {\cal W} = \sqrt{\frac{2}{1-|z|^2}} \,.
\label{Vres1}
\end{equation}
At the $SO(8)$ critical point one finds
\begin{equation}
\nabla_{z}\mathcal{V} |_{SO(8)} = 0\;.
\end{equation}
There are no other critical points of the potential or the superpotential in this truncation.
In the ${\cal N}=8$ theory, the eight gravitinos and the supersymmetry parameters, $\epsilon^i$, transform in the $\bfs 8$ of $SO(8)$,\footnote{We have already adopted a convention for the $SO(8)$ representation of the scalars to be $\bfs 35_s$ and pseudoscalars to be $\bfs 35_c$. This implicitly means that the $\epsilon^i$ transform in the $\bfs8_v$. One can, of course, permute all of this by triality.} which decomposes into $(\bfs 4,\bfs 1)+(\bfs 1,\bfs 4)$ under $SO(4)\times SO(4)$.
As noted in \cite{Pope:2003jp}, the $A_1^{ij}$ tensor is simply $\cosh \alpha \, \delta^{ij}$ so the spin-3/2 variations are diagonal:
\begin{equation}
\Big(A'\,\gamma^3\,+{1\over\ell}\, e^{-A}\,\gamma^2 \Big)\,\epsilon^j+g\,\overline {\cals W }\,\epsilon_j= 0\,, \qquad j=1,\ldots,8\,.
\label{so4gravvar}
\end{equation}
This means that ${\cal R}_\epsilon$ consists of all eight spinors but it is a reducible representation of $G=SO(4)\times SO(4)$.
The spin-1/2 variations, on the other hand, do distinguish between the irreducible components of ${\cal R}_\epsilon$:
\begin{equation}\label{s3proj}
\gamma^3\epsilon^j = M\,\epsilon_j\,,\qquad
\gamma^3\epsilon^{j+4} = M^*\,\epsilon_{j+4}\,,\qquad j=1,\ldots,4\,,
\end{equation}
where (using (\ref{zparam}) and (\ref{Vres1}) in (\ref{Mexplicit}) for $k=1$) we find
\begin{equation}\label{}
M= e^{i \Lambda}= {1\over\sqrt 2\,g}(\text{csch}\,\alpha\,\alpha'-i\cosh\alpha\,\zeta')\,.
\end{equation}
Following the analysis of Section \ref{Sect:Gravvar}, we can now use either one of the $\gamma^3$-projection conditions in equation (\ref{so4gravvar}) to obtain the $\gamma^2$-projection conditions:
\begin{equation}\label{s2proj}
\gamma^2\epsilon^j = i\,\kappa\,e^{i \Lambda}\, \epsilon_j\,,\qquad
\gamma^2\epsilon^{j+4} = - i\,\kappa\,e^{-i \Lambda}\,\epsilon_{j+4}\,,\qquad j=1,\ldots,4\,.
\end{equation}
Since ${\cal W}$ is real, the $\gamma^2$-projections on $\epsilon^{j+4}$ can be obtained from those of $\epsilon^{j}$ by complex conjugating (\ref{so4gravvar}). We therefore see that the effective sign of $\kappa$ changes between the two irreducible pieces of ${\cal R}_\epsilon$ and, in particular:
\begin{equation}
\gamma^0 \gamma^1 \, \epsilon^j = \kappa\,\epsilon^j \,, \qquad \gamma^0 \gamma^1 \, \epsilon^{j+4} = - \kappa\, \epsilon^{j+4}\,,\qquad j=1,\ldots,4 \,.
\label{so4chirproj}
\end{equation}
Thus the $(\bfs 4,\bfs 1)$ and $(\bfs 1,\bfs 4)$ correspond to supersymmetries with opposite $(1+1)$-dimensional helicity and hence we have an interface theory with $\widehat {\cal N}_L =\widehat {\cal N}_R= 4$, or $(4,4)$ supersymmetry. This is consistent with the unbroken supersymmetries of the corresponding eleven-dimensional lift discussed in Appendix~\ref{appendixB}.
Writing the symmetry action in terms of $SU(2)^4$, the action of the ${\cal R}$-symmetry on the supersymmetries, the bosons, $X^A$, and the fermions, $\lambda^{\dot{A}}$, decomposes as:
\begin{equation}
\begin{split}
\epsilon^i: \qquad & \bfs8_v ~=~ (\bfs2,\bfs2,\bfs1,\bfs1) \ \oplus \ (\bfs1,\bfs1,\bfs2,\bfs2)\,, \\
X^A: \qquad & \bfs8_s ~=~ (\bfs2,\bfs1,\bfs2,\bfs1) \ \oplus \ (\bfs1,\bfs2,\bfs1,\bfs2) \,, \\
\lambda^{\dot{A}}: \qquad & \bfs8_c ~=~ (\bfs2,\bfs1,\bfs1,\bfs2) \ \oplus \ (\bfs1,\bfs2,\bfs2,\bfs1) \,.
\end{split}
\end{equation}
The group theory implies that the $(\bfs2,\bfs2,\bfs1,\bfs1)$ supersymmetries must relate the $(\bfs2,\bfs1,\bfs2,\bfs1)$ bosons to the $(\bfs1,\bfs2,\bfs2,\bfs1)$ fermions and the $(\bfs1,\bfs2,\bfs1,\bfs2)$ bosons to the $(\bfs2,\bfs1,\bfs1,\bfs2)$ fermions. On the other hand, the $(\bfs1,\bfs1,\bfs2,\bfs2)$ supersymmetries must relate the $(\bfs2,\bfs1,\bfs2,\bfs1)$ bosons to the $(\bfs2,\bfs1,\bfs1,\bfs2)$ fermions and the $(\bfs1,\bfs2,\bfs1,\bfs2)$ bosons to the $(\bfs1,\bfs2,\bfs2,\bfs1)$ fermions. Thus each set of four symmetries naturally decomposes the bosons and fermions into two copies of a standard $\Neql 4$ representation, however the two different sets of four supersymmetries pair the boson and fermion decompositions differently.
\subsection{The BPS solutions}
\label{SO4sols}
As noted above, we have:
\begin{equation}
M= e^{i \Lambda} = {1\over\sqrt 2\,g}(\text{csch}\,\alpha\,\alpha'-i\cosh\alpha\,\zeta')\,.
\label{SO4M}
\end{equation}
One then finds that (\ref{dAsusy1}) simplifies to
\begin{equation}
\tanh\alpha \, A' ~+~ \alpha' ~=~ 0\;,
\label{simpcomp1}
\end{equation}
which can be integrated to yield
\begin{equation}
A = - \log (\sinh\alpha) + c_A\;,
\label{Asolnalpha}
\end{equation}
where $c_A$ is an integration constant.
Reality of this solution naturally requires that one has $\alpha >0$ and that $c_A$ is real. Alternatively, one could allow $\alpha < 0$ by making a purely imaginary shift in $c_A$. However, once $c_A$ is chosen, this option disappears and so we will require:
\begin{equation}
\alpha ~>~ 0 \,, \qquad c_A \in \mathbb{R} \;.
\label{constr1}
\end{equation}
The fact that (\ref{EffLag1}) is independent of $\zeta$ means that there is a conserved Noether charge:
\begin{equation}
e^{3A}\, \sinh^2 2 \alpha \, \, \zeta' ~=~ \text{const.}\;.
\label{Nother1}
\end{equation}
Using (\ref{Asolnalpha}) in (\ref{BPSz}) leads to a trivial identity in $\alpha'$ and it fixes the constant in (\ref{Nother1}):
\begin{equation}
\zeta' = - \dfrac{\kappa e^{-c_A}}{\ell} \dfrac{\sinh\alpha}{\cosh^2\alpha}\;.
\label{zetaeqn1}
\end{equation}
The last of the BPS equations, (\ref{quadconstr}), is simply
\begin{equation}\label{theAqss}
(A')^2=-\dfrac{e^{-2A}}{\ell^2} +2g^2\cosh^2\alpha\;,
\end{equation}
and using (\ref{Asolnalpha}) one obtains:
\begin{equation}
(\alpha')^2 ~=~ - \frac{e^{-2c_A}}{\ell^2}\,\frac{\sinh^4\alpha}{\cosh^2\alpha} ~+~ 2g^2\sinh^2\alpha\;.
\label{alphaeqn1}
\end{equation}
Define the parameter
\begin{equation}
a ~\equiv~ \sqrt{2}\,g\,\ell\, e^{c_A} \,, \label{adefn1}
\end{equation}
then (\ref{alphaeqn1}) is easily integrated to obtain, for $a<1$:
\begin{equation}
\sinh (\alpha(\mu)) ~=~ \kappa_{\alpha} \, \frac{a}{\sqrt{1- a^2}} \, \frac{1}{\cosh\big(\sqrt{2}\, g (\mu-\mu_0)\big)}\,,
\label{alphasol1}
\end{equation}
or, for $a>1$:
\begin{equation}
\sinh (\alpha(\mu)) = \kappa_{\alpha} \, \frac{a}{\sqrt{a^2 - 1}} \, \frac{1}{\sinh\big(\sqrt{2}\, g (\mu-\mu_0)\big)}\,,
\label{alphasol2}
\end{equation}
where $\kappa_{\alpha}^2 =1$.
The requirement (\ref{constr1}) that $\alpha > 0$ means that for the solutions (\ref{alphasol1}) we must take:
\begin{equation}
\kappa_\alpha ~=~ +1 \,,
\label{constr2}
\end{equation}
while for the solutions (\ref{alphasol2}) we must take either $\kappa_{\alpha} =+1$ and $\mu > \mu_0$ or $\kappa_{\alpha} =-1$ and $\mu < \mu_0$. Without loss of generality we will take the former choice and hence always choose (\ref{constr2}). The parameter $\mu_0$ is an integration constant and without loss of generality one can also take $\mu_0=0$. As we noted earlier, the parameter $\ell$ is spurious and, if it is finite, we can scale the metric so that $\ell=1$.
One can now solve (\ref{zetaeqn1}) and the result is:
\begin{align}
\tan (\zeta(\mu)-\zeta_0) &~=~ - \kappa \, \kappa_{\alpha} \sqrt{1- a^2}\, \sinh\big(\sqrt{2} \, g (\mu-\mu_0)\big) \,, \qquad a <1 \,; \label{zetasol1} \\
\tan (\zeta(\mu)-\zeta_0) &~=~ - \kappa \, \kappa_{\alpha} \sqrt{a^2-1}\, \cosh\big(\sqrt{2} \, g (\mu-\mu_0)\big) \,, \qquad a > 1 \,.\label{zetasol2}
\end{align}
Finally the solution for the metric function $A(\mu)$ is obtained from (\ref{Asolnalpha})
\begin{align}
e^{A(\mu)} &~=~ \kappa_{\alpha}\, \frac{ \sqrt{1- a^2}}{\sqrt{2}\, g \, \ell} \, \cosh \big(\sqrt{2}\, g (\mu-\mu_0)\big) \,, \qquad a <1 \,; \label{Asol1} \\
e^{A(\mu)} &~=~ \kappa_{\alpha}\, \frac{ \sqrt{a^2-1}}{\sqrt{2}\, g \, \ell} \, \sinh \big(\sqrt{2}\, g (\mu-\mu_0)\big) \,, \qquad a > 1 \,. \label{Asol2}
\end{align}
Scaling out $\ell$ by absorbing it in $c_A$, and then replacing this $c_A$ via (\ref{adefn1}) means that the free parameters in the solution are:
\begin{equation}\label{parameters}
a \;, \qquad \zeta_0\;, \qquad g\;,
\end{equation}
and there is also the sign choice, $\kappa$ ($\kappa_\alpha$ was fixed in (\ref{constr2})).
For $a < 1$ we get Janus solutions that are smooth for $-\infty < \mu < \infty$. The profiles of these solutions are all fairly similar in appearance. From (\ref{alphasol1}) it is evident that the scalar field, $\alpha$, is globally positive, vanishing at $\mu = \pm \infty$ and with a peak value of $ \frac{a}{\sqrt{1- a^2}}$ at $\mu = \mu_0$. From (\ref{zetasol1}) we see that the phase, $\zeta - \zeta_0$, goes between $ \frac{\kappa \pi}{2}$ and $- \frac{\kappa \pi}{2}$ as $\mu$ goes from $-\infty$ to $+\infty$. Similarly, (\ref{Asol1}) shows that $A(\mu) \sim \pm \sqrt{2} g \mu$ as $\mu \to \pm \infty$ and reaches a minimum value at $\mu = \mu_0$. Typical profiles are shown in Figure \ref{plotsAalphazeta}. The meaning of the parameters for this family of Janus solutions is as follows. The parameter $a<1$, controls the ``height of the bump" in the scalar $\alpha$. In field theory this parameter should map to the strength of the coupling between the $(1+1)$-dimensional defect and the $(1+2)$-dimensional bulk field theory. The parameter, $\zeta_0$, determines which linear combination of the fermionic bilinear and bosonic bilinear operators in field theory we turn on. Finally the parameter $g$ is the usual scale of $AdS_4$ which maps to the rank of the two CS gauge groups in the ABJM theory, that is, to the number of M2-branes.
For $a > 1$ and taking $\kappa_\alpha = +1$, $\mu > \mu_0$ in (\ref{alphasol2}) we get solutions in which $\alpha$ vanishes at $\mu = + \infty$ and runs off to $+\infty$ at $\mu = \mu_0$. From (\ref{Asol2}) we see that the metric function diverges: $A(\mu) \to -\infty$ at $\mu = \mu_0$ and the geometry becomes singular. It is also interesting to note that $A'(\mu)$ never vanishes. From (\ref{zetasol2}) we see that the phase, $\zeta - \zeta_0$, asymptotes to $- \frac{\kappa \pi}{2}$ as $\mu$ goes $+\infty$ and at $\mu = \mu_0$ this phase limits to some finite value whose sign is that of $-\kappa$. Thus the phase swings through a finite range of less than $\frac{\pi}{2}$. These singular ``flows to Hades" may have an interesting physical interpretation but we will refrain from discussing them further here.
\begin{figure}[t]
\centering
\includegraphics[width=5.25cm]{SO4Aplot.pdf}
\hfill
\includegraphics[width=5.25cm]{SO4alphaplot.pdf}
\hfill
\includegraphics[width=5.25cm]{SO4zetaplot.pdf}
\caption{Typical profiles for the $SO(4)^2$ Janus solutions. We have set $\mu_0=\zeta_0=0$, $\ell=1$, $g=1/\sqrt{2}$, and $\kappa = -1$. The curves are for $a=0.25$ (red), $a=0.85$ (blue), $a=0.95$ (purple), and $a=0.99$ (green).}
\label{plotsAalphazeta}
\end{figure}
\subsection{Holographic analysis and interpretation}
\label{subsec:HoloSO4}
While the singular solutions that run off to Hades (flows with $a>1$) might ultimately admit some interpretation involving domain walls between the $SO(8)$ invariant conformal phase and a Coulomb phase, we will focus here on the smooth flows with $a<1$ that evidently represent domain walls separating two $SO(8)$ invariant conformal fixed points.
We will therefore take $a<1$ and fix $\mu_0=0$ and $\kappa_\alpha=1$. To expand around $\mu \to \pm \infty$ it is convenient to define a new radial variable
\begin{equation}
\mu = \displaystyle\frac{\mp1}{\sqrt{2}g} \log\left(\displaystyle\frac{\sqrt{1-a^2}}{2a}\rho\right)~,
\end{equation}
and it is clear that for $\rho \to 0$ one has $\mu\to \pm \infty$. The scalars and the metric function have the following expansions for $\mu\to \pm \infty$ (the signs below are correlated)
\begin{equation}
\begin{split}
\alpha (\rho) &\approx \rho + \frac{1}{4}\left(\frac{1}{3} - \frac{1}{a^2}\right) \rho^3 + \mathcal{O}(\rho^5)~,\\
\zeta (\rho) &\approx \left(\zeta_0 \mp \kappa\displaystyle\frac{\pi}{2}\right) \pm \frac{\kappa}{a} \rho \mp \kappa\frac{(1+3a^2)}{12a^3} \rho^3 + \mathcal{O}(\rho^5)~,\\
A (\rho) &\approx -\log \rho + \log\frac{a}{\sqrt{2}g\ell} + \frac{1}{4}\left( \frac{1}{a^2}-1\right) \rho^2 + \mathcal{O}(\rho^4)~.\\
\end{split}
\end{equation}
For holographic purposes and for comparison with the eleven-dimensional solution of \cite{D'Hoker:2009gg} it is convenient to work with the scalars
\begin{equation}
x=\text{Re}(z)=\tanh\alpha \, \cos\zeta~, \qquad\qquad y = \text{Im}(z)=\tanh\alpha \, \sin\zeta~.
\end{equation}
One can expand the scalars $x(\mu)$ and $y(\mu)$ as
\begin{equation}\label{xydef}
\begin{split}
x (\rho) &\approx \cos\left(\zeta_0\mp \kappa \frac{\pi}{2}\right) \rho \mp \frac{\kappa}{a} \sin\left(\zeta_0\mp \kappa \frac{\pi}{2}\right)\rho^2 + \mathcal{O}(\rho^3)~,\\
y (\rho) &\approx \sin\left(\zeta_0\mp \kappa \frac{\pi}{2}\right) \rho \pm \frac{\kappa}{a} \cos\left(\zeta_0\mp \kappa \frac{\pi}{2}\right)\rho^2 + \mathcal{O}(\rho^3)~.
\end{split}
\end{equation}
Recalling the holographic dictionary from Section \ref{Sect:HolDict}, our general Janus solution is somewhat non-standard since the phase $\zeta_0$ ``rotates" scalars into pseudoscalars ({\it i.e.} bosonic bilinears into fermionic bilinears). For the solution at hand the scalar $x(\mu)$ in \eqref{xydef} is dual to a bosonic bilinear operator ${{\cal O}}_1$ of dimension 1 and the scalar $y(\mu)$ in \eqref{xydef} is dual to a fermionic bilinear operator ${{\cal O}}_2$ of dimension 2. These may be written as:
\begin{equation}
\label{O12}
\begin{split}
{{\cal O}}_1 & ~=~ \text{Tr}\big( (X^{1})^2+(X^{2})^2+(X^{3})^2+(X^{4})^2 -((X^{5})^2+(X^{6})^2+(X^{7})^2+(X^{8})^2)\big) \,,\\
{{\cal O}}_2 & ~=~ \text{Tr}\big((\lambda^{1})^2+(\lambda^{2})^2+(\lambda^{3})^2+(\lambda^{4})^2-((\lambda^{5})^2+(\lambda^{6})^2+(\lambda^{7})^2+(\lambda^{8})^2)\big) \,.
\end{split}
\end{equation}
By tuning the initial value of the phase $\zeta_0$ we obtain a famly of Janus solutions that are sourced in the boundary field theory by a linear combination of ${{\cal O}}_1$ and ${{\cal O}}_2$.
The four-dimensional reduction of the eleven-dimensional Janus solution discussed in \cite{D'Hoker:2009gg} was argued to have a normalizable mode for the pseudoscalar and the text suggests that the metric corrections were of the same, or lower, order. As we describe in detail in Appendix \ref{appendixB}, the solution of \cite{D'Hoker:2009gg} corresponds to our solution with $\zeta_0= \kappa \pi/{2}$. On the other hand, it is evident from our analysis in (\ref{xydef}) that if the pseudoscalar mode ($y (\rho)$) is normalizable then the scalar mode ($x (\rho)$) must be non-normalizable, or {\it vice versa}. Moreover, whatever the value of $\zeta_0$, both the scalar $x(\mu)$ and the pseduoscalar $y(\mu)$ always develop a non-trivial profile and therefore we have both operators ${{\cal O}}_1$ and ${{\cal O}}_2$ turned on in the dual field theory. To illustrate the importance of the parameter $\zeta_0$ we have presented plots of $x(\mu)$ and $y(\mu)$ for different values of $\zeta_0$ in Figure \ref{plotsxySO4}.
The apparent conflict with the asymptotic analysis of \cite{D'Hoker:2009gg} could stem from the difficulty of correctly identifying the internal metric perturbations from the eleven-dimensional perspective because of the warp factors. It is evident in \cite{D'Hoker:2009gg} that they have a non-trivial warp factors in front of the $AdS_3$ and $S^7$ metric in a manner that closely parallels ours. This shows that metric perturbations and hence the scalars are indeed playing a role in the Janus solution of \cite{D'Hoker:2009gg} and perhaps the expansion of these modes proved rather subtle.
Returning to our flows, note that, for generic choices of $\zeta_0$, we have both a source and a vev for the operators in the dual field theory. Naively one might think that inserting a codimension-one defect in the field theory should not induce a deformation of the Lagrangian of the parent theory far away from the defect and thus the only deformation of the parent theory should be by a vev. However it is clear that in our solutions the situation is more general and one has both a source and a vev deformation of the ABJM theory at asymptotic infinity. This implies that in the dual field theory one has relevant couplings turned on which are function of the distance to the interface. Such position dependent couplings may change the nature of relevant and marginal operators as discussed recently in \cite{Dong:2012ena,Dong:2012ua} (see also \cite{Gutperle:2012hy} for a discussion in the present context). It would be very interesting to understand the physics of such position dependent relevant deformations from the point of view of the dual strongly coupled field theory.
It is also curious to note that the ``oblique'' mixtures of scalars and dual operators defined by:
\begin{equation}\label{xyoblique}
\begin{split}
\tilde x (\rho) ~\equiv~ \cos\left(\zeta_0\mp\kappa\frac{\pi}{2}\right) \, x (\rho) + \sin\left(\zeta_0\mp\kappa\frac{\pi}{2}\right) y (\rho) &~\approx~ \rho + \mathcal{O}(\rho^3)\,, \\
\tilde y (\rho) ~\equiv~ \cos\left(\zeta_0\mp\kappa\frac{\pi}{2}\right) \, y (\rho) - \sin\left(\zeta_0\mp\kappa\frac{\pi}{2}\right) x (\rho) &~\approx~ \pm \frac{\kappa}{a} \rho^2 + \mathcal{O}(\rho^3)\,,
\end{split}
\end{equation}
suggests a simpler holographic interpretation in terms of a pure vev. However, the standard holographic dictionary discussed in Section \ref{Sect:HolDict} does not seem to admit a simple interpretation of the dual of such mixtures of scalars and pseudoscalars.
\begin{figure}[t]
\centering
\includegraphics[width=7.5cm]{plotxSO4.pdf}
\qquad
\includegraphics[width=7.5cm]{plotySO4.pdf}
\hfill
\caption{Plots for $x(\mu)$ and $y(\mu)$ for $\mu_0=0$, $g=\frac{1}{\sqrt{2}}$, $\kappa=-1$, and $a =\frac{1}{2}$. The parameter $\zeta_0$ takes the values $\zeta_0=0$ (blue), $\zeta_0=\pi/4$ (red), and $\zeta_0=\pi/2$ (green).}
\label{plotsxySO4}
\end{figure}
Note that in the holographic RG flows studied in \cite{Pope:2003jp} (see also \cite{Bena:2004jw}) the phase $\zeta$ was a constant throughout the flow. For the Janus interfaces we study here $\zeta$ is necessarily a non-trivial function of $\mu$. This will probably complicate the analysis if one tries to find the Janus-like generalization of the large family of solutions in \cite{Bena:2004jw}.
\section{The $SU(3) \times U(1) \times U(1)$-invariant Janus}
\label{sec:SU3}
\subsection{The truncation}
The $SU(3) \times U(1) \times U(1)$-invariant truncation is easily extracted from the $SU(3)$ invariant truncation that has been widely studied. In particular, it can be obtained from \cite{Bobev:2009ms,Warner:1983vz,Bobev:2010ib}. The non-compact generators, $\Sigma_{IJKL}$, of $E_{7(7)}$ can be associated with differential forms on $\mathbb{R}^8$:
\begin{equation}
\Sigma ~\equiv~\frac{1}{24}\, \Sigma_{IJKL} \, dx^I \wedge dx^J \wedge dx^K\wedge dx^L \,.
\label{E7forms}
\end{equation}
Define the complex variables $z_1=x_1+ix_2\,,\ldots,z_4= x_7+ix_8$ and introduce the $2$-forms
\begin{equation}
\label{complstr}
J^\pm= {i\over 2}\left(\sum_{j=1}^3 dz_j\wedge d \bar z_j\right)\pm {i\over 2}\,dz_4\wedge d\bar z_4\,,
\end{equation}
The non-compact generators of the $SU(1,1) \subset E_{7(7)}$ are then defined by:
\begin{equation}\label{Fone}
F^+= \frac{1}{4}\, (J^+ + J^-)\wedge (J^+ + J^-) \,,\qquad F^-= \frac{1}{4}\, (J^+ + J^-)\wedge (J^+ - J^-) \,,
\end{equation}
and the real-form generators of $SL(2,\mathbb{R})$ are obtained by taking real and imaginary parts. The embedding index, $k$, of this $SL(2,\mathbb{R})$ in $E_{7(7)}$
is $3$.
The $SO(2)$ or $U(1)$ action is simply the $SU(8)$ transformation acting on the real variables, $(x_1,\dots, x_8)$ by:
\begin{equation}
U ~=~ {\rm diag}\,(e^{i \beta}, e^{i \beta}, e^{i \beta}, e^{i \beta}, e^{i \beta}, e^{i \beta}, e^{-3i \beta}, e^{-3i \beta})\,,
\label{SU8b}
\end{equation}
which rotates $F^+$ by the phase $e^{4i \beta}$ and $F^-$ by the phase $e^{-4i \beta}$.
These forms are manifestly invariant under the $U(3)$ that acts on $(z_1,z_2, z_3)$ and the $U(1)$ acting on $z_4$. This $U(3) \times U(1)$ is also manifestly a subgroup of the $SO(8)$ symmetry acting on the $\mathbb{R}^8$ and hence is a subgroup of the gauge symmetry.
The scalar potential is given by
\begin{equation}
\mathcal{P} = -6\, \cosh 2 \alpha ~=~ -\frac{6\,(1 + |z|^2)}{1-|z|^2}\;.
\label{PSU3}
\end{equation}
Once again, one does not expect (\ref{SU8b}) to generate a symmetry of the action but here we find that it does. This means that there may well be new interesting classes of holographic RG flows along the lines of \cite{Pope:2003jp,Bena:2004jw} in which metric structure can be rotated into internal fluxes.
The effective particle action that encodes all field equations is:
\begin{equation}\label{su3lag}
\begin{split}
\cals L ~=~ & e^{3A}\left[3 (A')^2 ~-~ 3\, (\alpha')^2 ~-~ {3\over 4}\sinh^2(2\alpha)(\zeta')^2~+~ 6g^2 \, \cosh(2\alpha)
\right]-{3\over\ell^2}e^A \\
~=~ & 3\, e^{3A}\left[(A')^2 ~-~ \dfrac{z'\bar{z}'}{1-|z|^2} ~+~ 2 g^2\,\Big(\dfrac{1+|z|^2}{1-|z|^2}\Big)~-~ {1\over\ell^2}e^{-2A}
\right] \,,
\end{split}
\end{equation}
where we have used the K\"ahler potential (\ref{Kahlerpot}) with $k=3$. Once again the unexpected symmetry of the action makes it independent of $\zeta$ and so there is a conserved Noether charge:
\begin{equation}
e^{3A}\, \sinh^2 2 \alpha \, \, \zeta' ~=~ {\rm const. }
\label{Nother2}
\end{equation}
The tensor $A_1^{ij}$ of the $\Neql8$ theory is, once again, diagonal but there are only two equal eigenvalues, ${\cal W}$, that can be written in terms of a holomorphic superpotential, ${\cal V}$, as in (\ref{WVreln}). (We discuss the other six eigenvalues in Appendix \ref{appendixC}.) This means that the number of supersymmetries, as discussed in Section \ref{Sect:susies}, is $\widehat {\cal N} =2$ and the theory on the $(1+1)$-dimensional defect has $(0,2)$ supersymmetry. The residual ${\cal R}$-symmetry is the $U(1)$ symmetry that acts on $z_4 = x_7 + i x_8$ (as defined above) and lies outside the global $U(3)$ symmetry.
The holomorphic superpotential is a cubic:
\begin{equation}
\mathcal{V} ~=~ \sqrt{2} (z^3 +1) \qquad \Rightarrow \qquad {\cal W} = \frac{\sqrt{2}\, (z^3 +1)}{(1-|z|^2)^{3/2}}\,.
\label{Vres2}
\end{equation}
Apart from the $SO(8)$ critical point there are no other critical points of the potential or the superpotential within this truncation.
\subsection{Janus solutions}
\label{SU3numerics}
\begin{figure}[t]
\centering
\includegraphics[width=8cm]{SU3spag.pdf}
\caption{The $SU(3)\times U(1)\times U(1)$ space of solutions. The horizontal and vertical axes are $ \alpha\cos\zeta$ and $\alpha\sin\zeta $ and the contour lines are of the real superpotential, $W$. The maximally supersymmetric $AdS_4$ solution is the black dot in the middle. The green dots denote turning points of $A(\mu)$ in the solutions. If a turning point lies in the yellow region, the solution is a regular Janus solution. Other colored regions correspond to different types of singular solutions.}
\label{SU3WPplots}
\end{figure}
Unfortunately, unlike in the $SO(4) \times SO(4)$ sector, one cannot solve analytically the BPS equations \eqref{simpalphaeqn}--\eqref{Aprimeeqn} for flows based on the superpotential \eqref{Vres2}. In this section we use numerical methods to explore the space of solutions and identify those solutions that describe domain walls between conformal phases. Such solutions are asymptotic to $AdS_4$ as $\mu \to \pm \infty$ and have a turning point $A'(\mu_0) =0$ at some finite $\mu_0$. This means that in our analysis we may miss some flows to Hades, like those found in Section~\ref{sec:SO4}.
The representative numerical solutions presented in Figures~\ref{SU3WPplots} and \ref{SU3functPlots} are obtained as follows: We start by imposing the turning point\footnote{In our discussion, the ``turning point'' will mean the minimum of $A(\mu)$, which generically does not coincide with a turning point of $\alpha(\mu)$.} condition, $A'(0)=0$ at $\mu=\mu_0=0$, for some finite values $\alpha_*=\alpha(0)$ and $\zeta_*=\zeta(0)$. Next, for a fixed sign $\kappa=- 1$, we solve the BPS equations \eqref{simpalphaeqn}, \eqref{simpzetaeqn}, and \eqref{Aprimeeqn}, for $A(0)$, $\alpha'(0)$ and $\zeta'(0)$. This determines a complete set of initial conditions for the second order equations that follow from the Lagrangian \eqref{su3lag}. Setting $g=1/\sqrt 2$ and $\ell=1$, we then integrate numerically those equations to large positive and negative values of $\mu$. Finally, we check that the resulting numerical solutions solve the BPS equations. The advantage of numerically integrating the second order equations is that they do not contain any branch cuts. The choice of the branch cut in \eqref{Aprimeeqn} for a particular side of a solution is controlled at the outset by the initial conditions and the numerical integration can be carried out smoothly through the entire range of positive and negative values of the radial variable, $\mu$.
The space of numerical solutions to the BPS equations in the $(\alpha\cos\zeta,\alpha\sin\zeta)$-plane is illustrated in Figure~\ref{SU3WPplots}. The turning point is always denoted by a green dot and the blue and purple parts of curves correspond to negative and positive values of $\mu$, respectively. Since there is a clear symmetry of the BPS equations under $\mu\to -\mu$ and $\kappa\to -\kappa$, each ``blue-purple'' curve also has the same ``purple-blue'' counterpart obtained by switching the signs in the initial conditions.
Representative profiles for the scalars $\alpha$ and $\zeta$ and the metric function $A$ for some of the Janus solutions are shown in Figure~\ref{SU3functPlots}. They illustrate more precisely the dependence of the solutions on the radial variable $\mu$.
We find four classes of solutions. There are regular Janus solutions that asymptote to the maximally supersymmetric $AdS_4$ vacuum as $\mu \to \pm \infty$. These solutions exist when the turning point represented by the green dot lies in the yellow region in Figure \ref{SU3WPplots}. There are solutions which asymptote to $AdS_4$ as $\mu \to \infty$ or $\mu \to -\infty$, but are singular at a finite negative or positive value of $\mu$. The turning point for these solutions lies in the grey or orange regions, respectively. Finally, the solutions for which the turning point is in the pink region in Figure \ref{SU3WPplots} become singular on both sides of the defect at finite positive and negative values of $\mu$.
It is also clear from the plots that as in the $SO(4)\times SO(4)$ invariant regular Janus solutions we always find $\lim_{\mu\to \infty}(\zeta(\mu)-\zeta(-\mu))=\pi$. In the dual field theory this implies that on both sides of the Janus interface we turn on the same linear combination of a bosonic and a fermonic bilinear in the ABJM theory.
The asymptotic expansion of the Janus solutions here for $\mu \to \pm\infty$ is similar to the one discussed in Section \ref{subsec:HoloSO4}. Depending on the value of $\zeta_0$, we again have a different linear combination of the bosonic and fermonic bilinear operators $\mathcal{O}_1$ and $\mathcal{O}_2$:
\begin{equation}
\mathcal{O}_1 = \mathcal{O}_b^{77} + \mathcal{O}_b^{88}\;, \qquad\qquad \mathcal{O}_2 = \mathcal{O}_f^{77} + \mathcal{O}_f^{88}\;,
\end{equation}
where the bilinears on the right hand sides are defined in \eqref{Obdef} and \eqref{Ofdef}.
\begin{figure}[t]
\centering
\includegraphics[width=5.3cm]{su3A.pdf}
~
\includegraphics[width=5.3cm]{su3al.pdf}
~
\includegraphics[width=5.3cm]{su3zeta.pdf}
\caption{ Typical profiles of $A(\mu)$, $\alpha(\mu)$ and $\zeta(\mu)$ for $SU(3)\times U(1)^2$ Janus solutions. }\label{SU3functPlots}
\end{figure}
\section{The $G_2$-invariant Janus}
\label{sec:G2}
This sector of $\Neql8$ supergravity has a richer structure than the sectors considered above because there are two $G_2$-invariant, supersymmetric critical points, denoted by $G_2^\pm$, that differ by the sign of the pseudoscalar. In eleven dimensions, the sign of the four-dimensional pseudoscalar determines the sign of the internal, or magnetic, components of the three-form flux. Thus the $G_2^+$ and $G_2^-$ critical points represent supergravity phases with opposite magnetic fields.
The families of Janus solutions are also correspondingly much richer and include, in addition to solutions representing domain walls between two copies of the $SO(8)$-invariant phase, solutions that involve, or are dominantly controlled by, any combination of the three supersymmetric critical points. Indeed, we will find classes of solutions that start out in the $SO(8)$ phase at $\mu = \pm \infty$ but are perturbed by relevant operators that drive the solution, via the standard holographic RG flow \cite{Bobev:2009ms}, to either one or both $G_2$ phases. We will argue that in a certain limit they should give rise to new families of $SO(8)/G_2^\pm$ domain walls and a special $G_2^+/G_2^-$ domain wall.
\subsection{The truncation}
\label{G2truncation}
The $G_2$-invariant truncation can also be extracted from the $SU(3)$ invariant truncation. Indeed, the $SO(7)$-invariant self-dual tensor is given by \cite{Bobev:2009ms,Warner:1983vz,Godazgar:2013nma}:
\begin{equation}
\begin{split}
\Sigma^+_{IJKL} ~=~ & \Big( \delta^{1234}_{IJKL} +
\delta^{5678}_{IJKL} + \delta^{1256}_{IJKL} + \delta^{3478}_{IJKL}
+ \delta^{3456}_{IJKL} + \delta^{1278}_{IJKL} \\
& -(\delta^{1357}_{IJKL} + \delta^{2468}_{IJKL} )+
(\delta^{2457}_{IJKL} + \delta^{1368}_{IJKL}) +
(\delta^{1458}_{IJKL} +\delta^{2367}_{IJKL}) + (
\delta^{1467}_{IJKL} + \delta^{2358}_{IJKL}) \Big) \,.
\end{split}
\end{equation}
and the $SO(7)$-invariant anti-self-dual tensor, $\Sigma^-_{IJKL}$, can be obtained from this by making the reflection $x_8 \to -x_8$.
The $SO(2)$ or $U(1)$ action is simply the $SU(8)$ transformation acting on the real variables, $(x_1,\dots, x_8)$ by:
\begin{equation}
U ~=~ {\rm diag}\,(e^{i \beta}, e^{i \beta}, e^{i \beta}, e^{i \beta}, e^{i \beta}, e^{i \beta}, e^{i \beta}, e^{-7i \beta})\,,
\label{SU8c}
\end{equation}
which rotates $\Sigma^+_{IJKL} \pm \Sigma^-_{IJKL}$ by a phase $e^{\pm 4i \beta}$. These $E_{7(7)}$ Lie algebra elements generate the $SL(2,\mathbb{R})$, with embedding index, $k=7$.
The detailed structure of this supergravity sector can be read-off from \cite{Bobev:2009ms,Bobev:2010ib}. In particular, the tensor $A_1^{ij}$ of the $\Neql8$ theory is diagonal and has two distinct sets of eigenvalues according to the branching ${\bfs 8}_v\rightarrow \bfs 7 +\bfs 1$ of the gravitino representation under $G_2$. However, only one eigenvalue
\begin{equation}
\label{G2superpot}
{\cal W}= \sqrt{2} \left[\, \cosh ^7 \alpha + 7 \cosh ^3 \alpha \sinh ^4 \alpha \, e^{4 i \zeta }+ 7 \cosh^4 \alpha \sinh ^3 \alpha \, e^{3 i \zeta } + \sinh ^7 \alpha \, e^{7 i \zeta } \,\right] \,,
\end{equation}
corresponding to the singlet of $G_2$, can be written in terms of a holomorphic superpotential,\footnote{One can read-off this superpotential from Eqs.\ (2.34) and (2.35) in \cite{Bobev:2010ib} by setting $z=0$ and identifying $\zeta_{12}$ in \cite{Bobev:2010ib} with the $z$ below.}
\begin{equation}
{\cal V} ~=~ \sqrt{2}(z^7+7z^4+7z^3+1)\;,
\label{G2V}
\end{equation}
as in \eqref{WVreln}. This means that the number of supersymmetries, as discussed in Section~\ref{Sect:susies}, is $\widehat {\cal N} =1$ and the theory on the $(1+1)$-dimensional defect has $(0,1)$ supersymmetry.
The effective, one-dimensional Lagrangian is:
\begin{equation}\label{G2Lag}
{\cal L} ~=~ e^{3A}\Big[\, 3 (A')^2 -7\,\Big[(\alpha')^2 + \frac{1}{4}\sinh^2(2\alpha)(\zeta')^2\,\Big] ~-~ g^2\,{\cal P} \, \Big]~-~ {3\over\ell^2}e^A \,.
\end{equation}
where the supergravity potential, ${\cal P}$, can be obtained from \eqref{normW} and \eqref{Simppot}, or, equivalently from \eqref{Pform} with \eqref{Kahlerpot} and $k=7$.
\begin{figure}[t]
\centering
\includegraphics[width=7cm]{G2PotBT.pdf}
\qquad
\includegraphics[width=7cm]{G2WotBT.pdf}
\caption{Contour plots of the potential ${\cal P}$ (left) and the real superpotential, $W = |{\cal W}|,$ (right). The horizontal and vertical axes are $\alpha\cos\zeta$ and $\alpha\sin\zeta$. The $SO(8)$, $SO(7)^{+}$, $SO(7)^{-}$ and $G_2$ invariant critical points are denoted by black, blue, orange and red dots, respectively. The shading of various domains is described in Section \ref{G2Janusflows}.}
\label{G2WPplots}
\end{figure}
The scalar potential, ${\cal P}$, has a number of critical points \cite{Warner:1983vz} shown in Figure~\ref{G2WPplots}:
\begin{itemize}[noitemsep,topsep=0pt]
\item [(i)] the maximally supersymmetric point (black dot) at $z=0$;
\item [(ii)] the non-supersymmetric point with $SO(7)^+$ symmetry (blue dot) at $\alpha= {1\over 8}\log 5$ and $\zeta=0$;
\item [(iii)] two non-supersymmetric points with $SO(7)^-$ symmetry (orange dots) at $\alpha= {1\over 2}\mathop{\rm arccsch} 2$ and $ \zeta= \pm {\pi\over 2}$;
\item [(iv)] two supersymmetric $G_2$-invariant points, $G^\pm_2$, (red dots) at
\begin{equation}
\alpha= \frac{1}{2} \mathop{\rm arcsinh}\left(\sqrt{\frac{2\sqrt{3}-2}{5}}\right)\approx 0.2588 \,,\quad \zeta= \pm \mathop{\rm arccos}{1\over 2}\sqrt{3-\sqrt 3} \approx \pm 0.9727\,.
\label{G2pts}
\end{equation}
\end{itemize}
The $SO(8)$ and $G_2^\pm$ supersymmetric points are also critical points of the superpotential $\cals W$. For future reference we note that the slope of the function $A$ for the two supersymmetric critical points (where we fix $g=1/\sqrt{2}$) is given by:
\begin{equation}
\lim_{\mu\to\pm\infty}A'(\mu)|_{SO(8)} = \pm 1\;, \qquad\qquad \lim_{\mu\to\pm\infty}A'(\mu)|_{G_2} = \pm\left(\frac{3^92^{10}}{5^{10}}\right)^{\frac{1}{8}}\approx \pm1.0948 \;.
\end{equation}
This determines the $AdS$ radius of the corresponding vacua.
The non-supersymmetric points are perturbatively unstable \cite{Bobev:2010ib} and they do not give rise to any supersymmetric Janus sulutions.\footnote{By solving numerically the second order equations for \eqref{G2Lag}, we have, in fact, found some non-supersymmetric Janus solutions and RG flow domain walls in those sectors. However, it is very likely that these solutions are unstable and we refrain from discussing them here.} Similarly, there are no supersymmetric Janus solutions with $\cals R_\epsilon=\bfs 7$ of $G_2$. See Appendix~\ref{appendixC} for some additional details.
\subsection{Janus solutions}
\label{G2Janusflows}
\begin{figure}[t]
\centering
\includegraphics[width=8cm]{Unionplot.pdf}
\caption{ The ``phase diagram" of $AdS_3$ sliced domain wall solutions in the $G_2$ invariant sector of ${\cal N}=8$ gauged supergravity.}
\label{AllTogether}
\end{figure}
Not surprisingly, the BPS equations \eqref{simpalphaeqn}--\eqref{Aprimeeqn} in the $G_2$ sector can only be solved numerically. Using the same method as in Section~\ref{SU3numerics}, we have carried out an extensive search for different classes of solutions shown in Figure~\ref{AllTogether} and these will be discussed in more detail below.
Just as in the $SU(3)\times U(1)^2$ sector, we find that there is a ``basin of attraction'' around the maximally supersymmetric $SO(8)$ critical point where the solutions typically start and/or finish. We also find good numerical evidence for classes of solutions that start and/or finish at the $G_2^\pm$ points.
The details of the solutions are primarily controlled by the location, $(\alpha_*, \zeta_*)$, in the scalar manifold of the turning point of $A(\mu)$ in the Janus solution ({\it i.e.} by the values of $(\alpha, \zeta)$ at which $A'(\mu)$ momentarily vanishes). As usual, this point will be marked by a green dot in all the contour plots.
\subsubsection{Symmetric solutions}
\label{SymmJanus}
\begin{figure}[t]
\centering
\includegraphics[width=7cm]{G2zetapi.pdf}
\qquad
\includegraphics[width=7cm]{G2zetazero.pdf}
\caption{The symmetric solutions with (a) $\zeta_0=\pi$ and (b) $\zeta_0=0$. As the turning point approaches the point $\alpha_0=\alpha_{cr}$, $\zeta_0=0$, the Janus solution asymptotes to the $G_2^+/G_2^-$ Janus solution.}
\label{symmetricG2plots}
\end{figure}
The simplest class of solutions have the turning point of $A(\mu)$ on the real axis of the scalar manifold: $\zeta_*=0$ or $\zeta_*=\pi$, and thus are invariant under the $\mathbb{Z}_2$ symmetry generated by $\zeta\rightarrow-\zeta$.
Representative solutions with the turning point on the negative real axis, $\zeta_*=\pi$, are shown in the first plot in
Figure~\ref{symmetricG2plots}. We find only closed loops of $SO(8)$/$SO(8)$ Janus solutions that are similar to those in the previous two sections, but there is one significant difference. In the previous Janus solutions, the net change of the phase, $\Delta\zeta=\zeta(+\infty)-\zeta(-\infty)$, between the two sides was always equal to $\pi$, but here the net change of phase for solutions in Figure \ref{symmetricG2plots}, measured by the opening angle of the loops, is less than $\pi$ and depends on the initial data. We attribute this to a non-trivial dependence of the potential, $\cals P$, on the phase, $\zeta$, and hence the absence of a conserved quantity such as \eqref{Nother1}
or \eqref{Nother2}.
\begin{figure}[t]
\centering
\includegraphics[width=7cm]{g2alzeta0.pdf}
\qquad
\includegraphics[width=7cm]{g2dAlzeta0.pdf}
\caption{Plots of $\alpha(\mu)$ and $A'(\mu)$ for three solutions with $\zeta_0=0$ and $\alpha_0=0.15$ (blue), $\alpha_0=0.1756087990472$ (red) and $\alpha_0= 0.1756087990474$ (green).}
\label{aldAz0}
\end{figure}
The more interesting class of solutions arises when the turning point lies on the positive real axis, $\zeta_*=0$. This is evident from the second plot in Figure~\ref{symmetricG2plots}. Once again, for small values of $\alpha_*$, we find closed loops of $SO(8)$/$SO(8)$ Janus solutions with different values of $\Delta\zeta<\pi$. However, as the turning point approaches the point at the intersection of the four colored regions at $\alpha_*=\alpha_{cr}$, where
\begin{equation}\label{}
0.1756087990472\ldots \leq \alpha_{cr}\leq 0.1756087990473\ldots\,,
\end{equation}
the solution also begins to swing close to the $G_2$ critical points. In particular, for $\alpha_*$ very close, but smaller than $\alpha_{cr}$, one obtains what looks like a ``limiting loop:'' At each end it is almost exactly a steepest ascent from the $SO(8)$ to the $G_2^\pm$ points along the ridges of the real superpotential, $W$, and then it swings between the two $G_2$ points. If one examines the plot of $\alpha(\mu)$ and $A'(\mu)$ in Figure~\ref{aldAz0}, one sees that such a solution (plotted in red) involves a rapid evolution from the $SO(8)$ to $G_2^\pm$ critical points, where it spends a long period before it swings between the two $G_2$ points relatively rapidly. Numerical results suggest that by fine tuning $\alpha_*$ to $\alpha_{cr}$ the solution can be made to approach the $G_2^\pm$ points arbitrarily close and stay there arbitrarily long.
On the other side of the special point, where $\alpha_*>\alpha_{cr}$, we find solutions that become singular on both sides at finite values of $\mu$. Once more, as $\alpha_*$ approaches $\alpha_{cr}$, those solutions approach the $G_2^{\pm}$ points arbitrarily close and run off to infinity afterwards along the ridge of $W$, see Figure~\ref{symmetricG2plots} and the green plots in Figure~\ref{aldAz0}.
Since the two families, $\alpha_*<\alpha_{cr}$ and $\alpha_*>\alpha_{cr}$, of solutions depend continuously on $\alpha_*$, and given the behavior of those solutions close to the $G_2$ points, we expect that there exists a unique separating solution for $\alpha_*=\alpha_{cr}$ that describes a $G_2^-/G_2^+$ interface.
It appears that such a solution might be rather special in that it stays close to the $G_2^-$ and $G_2^+$ points infinitely long and then makes a quick transition between the two points close to $\mu=0$. Given the limited numerical accuracy and very slow convergence, we cannot predict whether that transition will be smooth, as for the approximating solution in Figure~\ref{aldAz0}, or whether it will become a discrete jump. In other words, looking at the plots in Figure~\ref{aldAz0}, the question is whether in the limit $\alpha_*\rightarrow\alpha_{cr}$, as the two sides of the plots asymptote the $G_2$ values over an increasing range of $\mu$, the transition around $\mu=0$ shrinks to zero width.
On the other side, there is a compelling physical argument for the existence of a $G_2^-/G_2^+$ interface solution. First, the loops to the left of the $SO(8)$ point and the smaller loops to the right represent Janus interfaces between $SO(8)$ phases. As $\alpha_*$ approaches $\alpha_{cr}$, the solution gets more and more controlled by the $G_2$ points. The limiting loops describe solutions in which the theory is initially perturbed so that it undergoes a rapid and standard holographic RG flow, as in \cite{Bobev:2009ms}, to settle in a $G_2$ phase on each side of the defect, where it remains for a significant interval in $\mu$. The limiting solution is thus a $G_2^-$ to $G_2^+$ Janus and the only role of the $SO(8)$ point is to provide a way to generate the $G_2^\pm$ phases on either side of the defect.
What makes this solution especially interesting is the fact that the two $G_2$ phases on either side of the defect are physically distinct: They have different signs for $\zeta$, which means that they have different signs for the pseudoscalar. In eleven dimensions this means that the two phases have opposite signs for the components of the $A^{(3)}$ gauge field on the $S^7$.\footnote{
Indeed, given that the complete set of uplift formulae for the $G_2$ invariant $AdS_4$ critical point is now known \cite{Godazgar:2013nma}, one can demonstrate this explicitly: Our phase parameter $\zeta$ is a called $\alpha$ in \cite{Godazgar:2013nma} and from formulae (73)--(76) and (96) of \cite{Godazgar:2013nma} one can see that $A^{(3)}$ changes sign if one changes the sign of the phase.} This is thus the M-theory analog of a conformal domain wall between two opposing magnetic fields.
\subsubsection{Asymmetric solutions}
\label{AsymmJanus}
\begin{figure}[t]
\centering
\includegraphics[width=7cm]{G2zetapi04.pdf}
\qquad
\includegraphics[width=7cm]{G2zetapio2.pdf}
\caption{A representative set of solutions for $\zeta_0=\pi/4$ and $\pi/2$. As the turning point approaches the orange or grey boundary the $SO(8)/SO(8)$ Janus solutions asymptote to a $G_2/SO(8)$ or $SO(8)/G_2$ Janus solution, respectively. }
\label{OrgBdryApproach}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=7cm]{g2alz1.pdf}
\qquad
\includegraphics[width=7cm]{g2dAz1.pdf}
\caption{Plots of $\alpha(\mu)$ and $A'(\mu)$ for three solutions with $\zeta_0=\pi/4$ and $\alpha_0=0.15$ (blue), $\alpha_0=0.21332461$ (red) and $\alpha_0= 0.21332464$ (green).}
\label{aldAzetapio4}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=7cm]{g2alz0.pdf}
\qquad
\includegraphics[width=7cm]{g2dAz0.pdf}
\caption{Plots of $\alpha(\mu)$ and $A'(\mu)$ for three solutions with $\zeta_0=\pi/2$ and $\alpha_0=0.15$ (blue), $\alpha_0=0.18337147$ (red) and $\alpha_0= 0.18337149$ (green).}
\label{aldAzetapio2}
\end{figure}
One can obviously move the turning point of $A(\mu)$ for the Janus solution into the upper or lower half-plane of the scalar manifold. These classes of solutions are related to each other by complex conjugation and so we focus on solutions with $\zeta_* >0$. Once again, if the turning point lies within the yellow region, see Figure~\ref{AllTogether}, the solutions are loops that start and finish at the $SO(8)$ point. As above, we interpret them as Janus solutions between two copies of the $SO(8)$ phase, where, depending on the asymptotic value of the angle, $\zeta$, at infinity, different mixtures of dual operators have been added to the field theory Lagrangian or are developing vevs within the phase on each side of the domain wall.
There are two interesting boundaries of the yellow region: the orange boundary and the grey boundary.
As the turning point approaches the grey boundary, see Figure~\ref{OrgBdryApproach} and Figure~\ref{aldAzetapio4}, the purple side of the solution, $\mu>0$, becomes more and more controlled by the $G_2^+$ point. At the grey boundary, the $SO(8)$ phase on the $\mu>0$ side rapidly undergoes an RG flow to establish a $G_2$ phase. The solution then loops back to the $SO(8)$ point via the $A(\mu)$-turning point. Thus the right-hand side of the interface ($\mu>0$) is in the $G_2$ phase while the left-hand side ($\mu<0$) is controlled by the $SO(8)$ point. This therefore represents a Janus interface with the $G_2$ phase on the right and the $SO(8)$ phase on the left. This description is also evident from the values of $A'(\mu)$ on either side of the interface in Figure \ref{aldAzetapio4}.
As the turning point approaches the orange boundary, see Figure~\ref{OrgBdryApproach} and Figure~\ref{aldAzetapio2},
the solution for $\mu<0$ becomes increasingly controlled by the $G_2^+$ point. At the boundary, the $SO(8)$ phase described by that side of the solution rapidly undergoes an RG flow to establish a $G_2^+$ phase for $\mu\rightarrow-\infty$ while the phase for $\mu\rightarrow+\infty$ is controlled by the $SO(8)$ point. This therefore represents an interface with the $SO(8)$ phase on the left and the $G_2^+$ phase on the right.
If the $A(\mu)$-turning point, $(\alpha_*,\zeta_{*})$, crosses into an orange or grey region then one end of the solution runs to Hades and the other end goes back to the the $SO(8)$ point, and if the turning point moves into a pink region then both ends of the solution run to Hades. Figure \ref{AllTogether} displays the features of the various domains we have described here.
For the $SO(8)$/$SO(8)$ Janus solutions the asymptotic analysis at $\mu \to \pm \infty$ is again similar to that of Section \ref{subsec:HoloSO4}. The operators ${\cal O}_{1,2}$ in \eqref{Obdef}--\eqref{Ofdef} are given by:
\begin{equation}\label{O1O2G2}
{\cal O}_1 = {\cal O}_b^{88}\;, \qquad\qquad {\cal O}_2 = {\cal O}_f^{88}\;.
\end{equation}
The value of $\zeta$ at $\mu \to \pm \infty$ controls the linear combination of the operators $\mathcal{O}_1$ and $\mathcal{O}_2$ that is being turned on. The new feature however is that for a generic Janus solution in the $G_2$ truncation we have $\lim_{\mu\to \infty}(\zeta(\mu)-\zeta(-\mu))\neq\pi$. This means that different linear combination of the bosonic and fermonic bilinear operators $\mathcal{O}_1$ and $\mathcal{O}_2$ are driving the flow on each side of the interface.
The three-dimensional field theory dual to the $G_2$ critical point is poorly understood since it is strongly coupled and has only ${\cal N}=1$ superconformal symmetry \cite{Bobev:2009ms}. The limited information we have about this theory comes from holography and therefore it is hard to identify the field theory deformations that trigger the $SO(8)$/$G_2^\pm$ and $G_2^+$/$G_2^-$ Janus solutions.
\section{Conclusions}
\label{sec:Conclusions}
We have seen, once again, that gauged supergravity can be an immensely powerful tool for constructing interesting holographic solutions. While the truncation to gauged $\Neql8$ supergravity limits one to the holographic duals of essentially bilinear operators and thereby limits the classes of flows that can be studied, the fact that the higher-dimensional fields are relatively simply and highly efficiently encoded in the four-dimensional theory means that one can find many solutions that would represent a formidable, if not impossible, task from the perspective of the higher-dimensional supergravities. Even with the four-dimensional solutions that we constructed at hand it is generally not a simple task to construct their eleven-dimensional uplift. Due to the large global symmetry and the previous results in the literature on consistent truncations it is possible to uplift the $SO(4)\times SO(4)$ Janus solutions to eleven dimensions with little effort (see Appendix \ref{appendixB}). The uplift of arbitrary solutions in the $SU(3)\times U(1)\times U(1)$ and $G_2$ truncations is generically not known. The uplift of the metric is relatively straightforward to perform using the uplift formula of \cite{de Wit:1986iy}. It is much more subtle to obtain the fluxes of the eleven-dimensional solution and the recent results on consistent truncations of eleven-dimensional supergravity \cite{Nicolai:2011cy,deWit:2013ija,Godazgar:2013nma} may provide useful methods for attempting such a construction.
It would be nice to have a better field-theory understanding of the interface defects we have constructed holographically. We provided evidence that, in addition to vevs, the presence of the defect introduces a deformation of the Lagrangian and it is important to clarify how this happens in the dual field theory. The analysis for the field theory duals to the $SO(8)$/$SO(8)$ Janus solutions should proceed along the lines of the calculations performed in \cite{D'Hoker:2006uv} for ${\cal N}=4$ SYM. It will be much more challenging to understand the $SO(8)$/$G_2^{\pm}$ and $G_2^{+}$/$G_2^{-}$ solutions in field theory due to the minimal amount of supersymmetry and the limited field theory information about the $G_2$ fixed points. More generally it will be nice to have a field theory classification of superconformal defects in the ABJM theory. There has been recent interesting work on boundary conditions in ${\cal N}=2$ theories in three dimensions \cite{Okazaki:2013kaa} and one should be able to use similar techniques to systematically classify at least the 1/2-BPS defects as was done in \cite{Gaiotto:2008sd,Gaiotto:2008sa,Gaiotto:2008ak} for ${\cal N}=4$ SYM.
Even within the extremely simple class of $SU(1,1)/U(1)$ coset models studied here we have found a plethora of new Janus solutions. Of particular interest are the interfaces between different superconformal fixed points and especially the $G_2^+$/$G_2^-$ interface between two domains of opposite magnetic fields. This leads to the obvious question of possible generalizations. We have done some calculations within the larger $SU(3)$-invariant sector that has been much studied in ordinary holographic RG flows \cite{Bobev:2009ms,Warner:1983vz,Ahn:2000mf,Ahn:2000aq,Ahn:2001by,Ahn:2001kw,Ahn:2002eh,Ahn:2002qga,Bobev:2010ib}. It is evident that there are indeed Janus solutions that involve not only the $SO(8)$ and $G_2^\pm$ phases but incorporate the $\Neql2$ supersymmetric $SU(3) \times U(1)^\pm$ critical point as well. We are continuing to investigate these flows \cite{BPWRG} and because of the $U(1)$ ${\cal R}$-symmetry at the $\Neql2$ points, the holographic field theory phase is better understood \cite{Benna:2008zy} and perhaps can lead to some non-trivial tests within the theory. Then there are the flows to Hades: From the field theory perspective it seems difficult for there to be a conformal interface between a superconformal phase and a Coulomb phase. However, it would certainly be interesting to see if such an interface is predicted by holography.
Although we have concentrated on examples of four-dimensional gravitational actions that arise as a consistent truncation of the $\Neql8$ gauged supergravity it should be emphasized that our construction works for any holomorphic superpotential, $\cal{V}$, and any real number, $k$. Therefore any four-dimensional supergravity theory with a $SU(1,1)/U(1)$ scalar manifold and a holomorphic superpotential will admit Janus solutions of the type discussed here. If $\cal{V}$ has any non-trivial critical points there will also be RG flow domain walls analogous to those that we found in the $G_2$ truncation.
Going beyond ABJM theory and $\Neql8$ supergravity in four dimensions there are obvious questions about the extent to which our results can be generalized to gauged supergravity theories in higher dimensions. Starting at the top, it is relatively easy to see that there are no supersymmetric Janus solutions in seven-dimensional maximal gauged supergravity. We have explicitly looked for such solutions and have shown that they do not exist. If there were Janus solutions they would be dual to codimension-one superconformal defects in the six-dimensional $(2,0)$ theory. The reason for this negative result is that the five-dimensional superconformal group $F(4)$, which should be the symmetry group of the defect, is not a subgroup (see \cite{D'Hoker:2008ix} for a proof) of the $OSp(8|4)$ superconformal symmetry group of the six-dimensional $(2,0)$ theory. This implies that there are no superconformal codimension one defects in the $(2,0)$ theory and its $(1,0)$ orbifold generalizations.
In five-dimensional, gauged $\Neql8$ supergravity the possibilities are much richer and Janus solutions are already known \cite{Clark:2005te,Suh:2011xc}. Here we are, of course, dealing with a consistent truncation of IIB supergravity and the holographic dual of $\Neql4$ Yang Mills theory. The interfaces are $(2+1)$-dimensional and the superconformal ones, for which the theory living on the two sides of the defect is ${\cal N}=4$ SYM, were classified in \cite{Clark:2004sb,D'Hoker:2006uv}. There are $1/2$, $1/4$ and $1/8$-BPS superconformal interfaces and some of their gravity duals are known. The $1/2$-BPS Janus was found in IIB supergravity in \cite{D'Hoker:2007xy} and the $1/8$-BPS Janus was found first in five-dimensional supergravity in \cite{Clark:2005te} and then uplifted to ten dimensions in \cite{D'Hoker:2006uu,Suh:2011xc}. The five-dimensional supergravity dual of the $1/4$-BPS Janus will be presented in \cite{BPW5d}.
It is therefore evident that there is still much to be learned about Janus solutions by using gauged supergravity theory in four and five dimensions and that this paper represents a fraction of the interesting results that are within reach.
\bigskip
\bigskip
\leftline{\bf Acknowledgements}
\smallskip
We would like to thank Costas Bachas, Chris Beem, Eric D'Hoker, John Estes, Davide Gaiotto, Jaume Gomis, Michael Gutperle, Murat G\"unaydin, Darya Krym, and Balt van Rees for helpful discussions. Most of this work was done while NB was a postdoc at the Simons Center for Geometry and Physics and he would like to thank this institution for the great working atmosphere. The work of NB is supported by Perimeter Institute for
Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. The work of KP and NPW is supported in part by DOE grant DE-FG03-84ER-40168. NPW is grateful to the IPhT, CEA-Saclay, the Institut des Hautes Etudes Scientifiques (IHES), Bures-sur-Yvette, and Perimeter Institute for Theoretical Physics for hospitality while this work was completed. NPW would also like to thank the Simons Foundation for their support through a Simons Fellowship in Theoretical Physics. We would all like to thank the KITP, Santa Barbara for warm hospitality during the initial stages of this project.
\begin{appendices}
|
1,116,691,500,745 | arxiv | \section{Parameter Settings}
\label{sec:setting}
\mbox{{\textsc{Ex-Ray}}}\@{} has three hyper-parameters, $\alpha$ to control the weight changes of cross-entropy loss in function (\ref{e:cnmloss}) (in Section~\ref{s:sfd}), $\beta$ to control the similarity comparison between masks in condition (\ref{e:fprule1}) in Section~\ref{s:similarity}
, and $\gamma$ the accuracy threshold in cross-validation checks of masks in Section~\ref{s:similarity}.
We use 0.8, 0.8, and 0.1, respectively, by default.
In our experiments, we use ABS and NC as the upstream scanners.
The numbers of optimization epochs are 60 for ABS and 1000 for NC.
The other settings are default unless stated otherwise.
\section{Effects of Hyperparameters}
\label{sec:hyperparameter}
We study \mbox{{\textsc{Ex-Ray}}}\@{} performance with various hyperparameter settings, including the different layer to which \mbox{{\textsc{Ex-Ray}}}\@{} is applied, different trigger size settings (in the upstream scanner) and different SSIM score bounds (in filter backdoor scanning to ensure the
generated kernel does not over-transform an input), and the
$\alpha$, $\beta$, and $\gamma$ settings of \mbox{{\textsc{Ex-Ray}}}\@{}.
Table~\ref{t:layer} shows the results for layer selection.
The row ``Middle'' means that we apply \mbox{{\textsc{Ex-Ray}}}\@{} at the layer in the middle of a model.
The rows ``Last'' and``2nd last'' show the results at the last and the second-last convolutional layers, respectively.
Observe that
layer selection may affect performance to some extent and the second to the last layer has the best performance. Tables~\ref{t:size} shows that a large trigger size degrades \mbox{{\textsc{Ex-Ray}}}\@{}'s performance but \mbox{{\textsc{Ex-Ray}}}\@{} is stable in 900 to 1200. Table~\ref{t:ssim} shows that the SSIM score bound has small effect on performance in 0.7-0.9. Note that an SSIM score smaller than 0.7 means the transformed image is quite different (in human eyes).
Figures~\ref{f:acc_alpha}, ~\ref{f:acc_beta}, and ~\ref{f:acc_gamma} show the performance variations with $\alpha$, $\beta$, and $\gamma$, respectively. The experiments are on the mixture of trojaned models with polygon triggers and the clean models from TrojAI round 2.
For $\beta$ and $\gamma$, we sample from 0.7 to 0.95 and for $\alpha$ we sample from 0.1 to 2.4. Observe that changing $\alpha$ and $\gamma$ does not have much impact on the overall accuracy.
When we change $\beta$ from 0.7 to 0.95, the overall accuracy is still consistently higher than 0.83. These results show the stability of \mbox{{\textsc{Ex-Ray}}}\@{}.
\begin{figure}[]
\centering
\footnotesize
\includegraphics[width=0.4\textwidth]{figs/alpha4_crop.pdf}
\caption{Accuracy changes with $\alpha$ on TrojAI R2
}
\label{f:acc_alpha}
\end{figure}
\begin{figure}[]
\centering
\footnotesize
\includegraphics[width=0.4\textwidth]{figs/beta4_crop.pdf}
\caption{Accuracy changes with $\beta$ on TrojAI R2}
\label{f:acc_beta}
\end{figure}
\begin{figure}[]
\centering
\footnotesize
\includegraphics[width=0.4\textwidth]{figs/gamma4_crop.pdf}
\caption{Accuracy changes with $\gamma$ on TrojAI R2}
\label{f:acc_gamma}
\end{figure}
\begin{table}[]
\caption{\mbox{{\textsc{Ex-Ray}}}\@{} with different trigger sizes; (T:276, C:552) means there are 276 trojaned models and 552 clean models}
\label{t:size}
\centering
\footnotesize
\setlength{\tabcolsep}{5pt}
\begin{tabular}{crrrrrrrrrrr}
\toprule
& \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}TrojAI R2\\ (T:276,C:552)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}TrojAI R3\\ (T:252,C:504)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}TrojAI R4\\ (T:143,C:504)\end{tabular}} \\
\cmidrule{2-4} \cmidrule{6-8} \cmidrule{10-12}
& \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} \\
900 & 198 & 19 & 0.883 & & 157 & 19 & 0.849 & & 95 & 58 & 0.836 \\
1200 & 203 & 30 & 0.876 & & 175 & 39 & 0.847 & & 105 & 53 & 0.859 \\
1600 & 210 & 46 & 0.864 & & 200 & 46 & 0.870 & & 108 & 77 & 0.827 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[]
\caption{\mbox{{\textsc{Ex-Ray}}}\@{} with different SSIM scores; (T:276, C:552) means there are 276 trojaned models and 552 clean models}
\label{t:ssim}
\footnotesize
\setlength{\tabcolsep}{5pt}
\centering
\begin{tabular}{crrrrrrrrrrr}
\toprule
\begin{tabular}[c]{@{}c@{}}SSIM\\ Score\end{tabular} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}TrojAI R2\\ (T:276,C:552)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}TrojAI R3\\ (T:252,C:504)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}TrojAI R4\\ (T:361,C:504)\end{tabular}} \\
\cmidrule{2-4} \cmidrule{6-8} \cmidrule{10-12}
& \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} \\
0.9 & 145 & 4 & 0.837 & & 115 & 9 & 0.807 & & 234 & 47 & 0.799 \\
0.8 & 160 & 13 & 0.844 & & 149 & 39 & 0.812 & & 242 & 46 & 0.809 \\
0.7 & 204 & 32 & 0.874 & & 178 & 90 & 0.783 & & 175 & 13 & 0.770
\\
\bottomrule
\end{tabular}
\end{table}
\begin{table*}[]
\caption{\mbox{{\textsc{Ex-Ray}}}\@{} with different layer; (T:276,C:552) means that there are 276 trojaned models and 552 clean models}
\label{t:layer}
\footnotesize
\centering
\setlength{\tabcolsep}{5pt}
\begin{tabular}{crrrrrrrrrrrrrrrrrrrrrrr}
\toprule
& \multicolumn{7}{c}{TrojAI R2} & \multicolumn{1}{c}{} & \multicolumn{7}{c}{TrojAI R3} & \multicolumn{1}{c}{} & \multicolumn{7}{c}{TrojAI R4} \\
\cmidrule{2-8} \cmidrule{10-16} \cmidrule{18-24}
& \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Polygon Trigger\\ (T:276,C:552)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Filter Trigger\\ (T:276,C:552)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Polygon Trigger\\ (T:252,C:504)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Filter Trigger\\ (T:252,C:504)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Polygon trigger\\ (T:143,C:504)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Filter trigger\\ (T:361,C:504)\end{tabular}} \\
\cmidrule{2-4} \cmidrule{6-8} \cmidrule{10-12} \cmidrule{14-16} \cmidrule{18-20} \cmidrule{22-24}
& TP & FP & Acc & & TP & FP & Acc & & TP & FP & Acc & & TP & FP & Acc & & TP & FP & Acc & & TP & FP & Acc \\
Middle & 215 & 54 & 0.861 & & 220 & 97 & 0.815 & & 205 & 100 & 0.806 & & 153 & 58 & 0.792 & & 102 & 75 & 0.821 & & 257 & 77 & 0.791 \\
Second Last & 198 & 19 & 0.883 & & 204 & 32 & 0.874 & & 182 & 68 & 0.818 & & 149 & 39 & 0.812 & & 105 & 53 & 0.859 & & 242 & 46 & 0.809 \\
Last & 141 & 6 & 0.83 & & 171 & 16 & 0.854 & & 159 & 65 & 0.791 & & 141 & 27 & 0.817 & & 84 & 37 & 0.852 & & 196 & 37 & 0.766 \\
\bottomrule
\end{tabular}
\end{table*}
\iffalse
\subsection{The stability of SFD}
We evaluate SFD's stability on three aspects, SFD's results on different layer, on patch triggers with different sizes and filter triggers with different ssim scores.
Table~\ref{t:layer} shows the,
the first column shows the which layer we apply SFD on. The row 'Layer-half' means that we apply SFD after half of all layers. The row 'Layer-2' shows the result when we run the SFD after the second to the last convolutional layer. The row 'Layer-1' shows the result when we run the SFD after the last convolutional layer.
\fi
\section{More Details of the Model Repair Experiment
}
\label{sec:repair}
We show the trigger size for each label pair for an trojaned model in Table~\ref{t:unlearnexample1} and for an benign model in Table~\ref{t:unlearnexample2}.
Table~\ref{t:unlearnexample1} (a) shows the trigger size between each pair of labels. The columns denote the victim label and the rows denote the the target label.
For example, the gray cell in Table~\ref{t:unlearnexample1} (a) shows the trigger size to flip class 1 to class 0. Table~\ref{t:unlearnexample1} (b) follows the same format and shows the result for trojaned model after unlearning. In the trojaned model the injected trigger flips class 1 to class 0. Before unlearning, class 1 and class 0 have the smallest trigger size 21. Unlearning increases the trigger size between the two to 106, which is above the average trigger size between any pairs. Intuitively, one can consider the backdoor is fixed.
In the benign model, the natural trigger flips class 3 to class 5. As shown in Table~\ref{t:unlearnexample2}, unlearning only increases the trigger size from 24 to 59 and 59 is still one of the smallest trigger size among all label pairs for the fixed model.
\begin{figure*}[]
\subfigure[Before unlearning]
{
\begin{minipage}[t]{.45\textwidth}
\footnotesize
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
0 & - & 48 & 34 & 75 & 42 & 52 & 62 & 46 & 48 & 52 \\ \hline
1 & \cellcolor[HTML]{C0C0C0}21 & - & 74 & 91 & 88 & 96 & 72 & 80 & 81 & 45 \\ \hline
2 & 32 & 54 & - & 66 & 39 & 57 & 54 & 61 & 78 & 60 \\ \hline
3 & 34 & 53 & 35 & - & 42 & 27 & 46 & 50 & 72 & 47 \\ \hline
4 & 29 & 45 & 29 & 49 & - & 36 & 46 & 48 & 63 & 48 \\ \hline
5 & 40 & 70 & 35 & 46 & 43 & - & 53 & 49 & 81 & 56 \\ \hline
6 & 29 & 48 & 23 & 41 & 44 & 61 & - & 66 & 70 & 59 \\ \hline
7 & 40 & 77 & 55 & 78 & 40 & 52 & 81 & - & 82 & 60 \\ \hline
8 & 21 & 44 & 42 & 75 & 50 & 59 & 60 & 65 & - & 47 \\ \hline
9 & 29 & 62 & 78 & 85 & 69 & 70 & 73 & 62 & 73 & - \\ \hline
\end{tabular}
\end{minipage}
}
~
\subfigure[After unlearning]
{
\begin{minipage}[t]{.45\textwidth}
\footnotesize
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
0 & - & 84 & 56 & 103 & 77 & 96 & 82 & 96 & 56 & 78 \\ \hline
1 & \cellcolor[HTML]{C0C0C0}106 & - & 132 & 162 & 150 & 140 & 113 & 134 & 124 & 70 \\ \hline
2 & 92 & 111 & - & 88 & 79 & 79 & 62 & 99 & 109 & 106 \\ \hline
3 & 105 & 92 & 66 & - & 86 & 60 & 58 & 82 & 124 & 86 \\ \hline
4 & 96 & 92 & 55 & 81 & - & 72 & 53 & 77 & 99 & 95 \\ \hline
5 & 119 & 100 & 64 & 70 & 91 & - & 58 & 101 & 136 & 90 \\ \hline
6 & 107 & 97 & 86 & 88 & 99 & 93 & - & 113 & 113 & 97 \\ \hline
7 & 94 & 101 & 92 & 124 & 87 & 87 & 81 & - & 126 & 100 \\ \hline
8 & 50 & 72 & 68 & 104 & 98 & 106 & 81 & 101 & - & 79 \\ \hline
9 & 104 & 87 & 129 & 119 & 117 & 115 & 110 & 108 & 123 & - \\ \hline
\end{tabular}
\end{minipage}
}
\caption{Injected trigger distance matrix before and after unlearning}
\label{t:unlearnexample1}
\end{figure*}
\begin{figure*}[]
\subfigure[Before unlearning]
{
\begin{minipage}[t]{.45\textwidth}
\footnotesize
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
0 & - & 43 & 44 & 47 & 38 & 42 & 67 & 68 & 37 & 48 \\ \hline
1 & 62 & - & 89 & 88 & 79 & 78 & 70 & 85 & 76 & 47 \\ \hline
2 & 53 & 58 & - & 37 & 35 & 42 & 51 & 71 & 72 & 62 \\ \hline
3 & 62 & 66 & 40 & - & 40 & \cellcolor[HTML]{C0C0C0}24 & 48 & 59 & 72 & 56 \\ \hline
4 & 61 & 57 & 38 & 48 & - & 31 & 52 & 52 & 85 & 64 \\ \hline
5 & 69 & 66 & 43 & 33 & 46 & - & 51 & 61 & 73 & 57 \\ \hline
6 & 66 & 55 & 32 & 35 & 44 & 38 & - & 80 & 87 & 62 \\ \hline
7 & 74 & 77 & 67 & 61 & 38 & 39 & 74 & - & 92 & 68 \\ \hline
8 & 29 & 44 & 55 & 61 & 48 & 51 & 62 & 65 & - & 46 \\ \hline
9 & 79 & 57 & 84 & 72 & 67 & 67 & 83 & 76 & 72 & - \\ \hline
\end{tabular}
\end{minipage}
}
~
\subfigure[After unlearning]
{
\begin{minipage}[t]{.45\textwidth}
\footnotesize
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline
0 & - & 79 & 56 & 89 & 63 & 90 & 65 & 99 & 59 & 76 \\ \hline
1 & 120 & - & 134 & 114 & 123 & 154 & 77 & 119 & 122 & 48 \\ \hline
2 & 95 & 101 & - & 75 & 58 & 74 & 59 & 82 & 121 & 82 \\ \hline
3 & 104 & 98 & 57 & - & 58 & \cellcolor[HTML]{C0C0C0}59 & 40 & 90 & 124 & 80 \\ \hline
4 & 104 & 100 & 60 & 88 & - & 79 & 50 & 73 & 117 & 79 \\ \hline
5 & 95 & 91 & 58 & 84 & 76 & - & 52 & 77 & 131 & 86 \\ \hline
6 & 114 & 115 & 77 & 129 & 102 & 89 & - & 131 & 137 & 93 \\ \hline
7 & 104 & 120 & 110 & 103 & 68 & 92 & 66 & - & 129 & 78 \\ \hline
8 & 36 & 61 & 74 & 80 & 66 & 112 & 66 & 86 & - & 70 \\ \hline
9 & 128 & 109 & 117 & 103 & 112 & 120 & 73 & 124 & 105 & - \\ \hline
\end{tabular}
\end{minipage}
}
\caption{Natural trigger distance matrix before and after unlearning}
\label{t:unlearnexample2}
\end{figure*}
\section{Additional Checks to Validate That Feature Difference Sets Share Substantial Commonality}
\label{sec:additionalcheck}
To further validate $M_1$ and $M_2$ in Section~\ref{s:similarity} both denote feature differences between the victim and target classes, we perform the following additional checks.
While $M_1$ is derived by comparing the victim class and the target class samples, we copy the feature maps indicated by $M_1$ between the victim samples and their compromised versions with trigger to see if the intended class flipping can be induced; similarly, while $M_2$ is derived by comparing the victim class samples and their compromised versions, we copy the feature maps indicated by $M_2$ between the victim samples and the target samples to see if the intended class flipping can be induced.
If so, the two are functionally similar and the trigger is natural.
The check is formulated as follows.
\begin{align}\label{e:fprule2}
\begin{split}
& Acc( h(g(X_V)\cdot M_2 + g(X_T)\cdot \neg M_2), V) > \gamma\ \land\\
& Acc( h(g(X_T)\cdot M_2 + g(X_V)\cdot \neg M_2), T) > \gamma\ \land\\
& Acc( h(g(X_V)\cdot M_1 + g(X_V+t)\cdot \neg M_1), V) > \gamma \ \land\\
& Acc(h(g(X_V+t)\cdot M_1 + g(X_V)\cdot \neg M_1), T) > \gamma \\
\end{split}
\end{align}
Here, $Acc()$ is a function to evaluate prediction accuracy on a set of samples and $\gamma$ a threshold (0.8 in the paper).
We use $g(X_V)$ to denote applying $g$ on each sample in $X_V$ for representation simplicity.
\section{Parameter Settings}
\label{sec:setting}
SFD has three hyper-parameters, $\alpha$ to control the weight changes of cross-entropy loss in function (\ref{e:cnmloss}) (in Section~\ref{s:sfd}), $\beta$ to control the similarity comparison between masks in condition (\ref{e:fprule1}) in Section~\ref{s:similarity}
, and $\gamma$ the accuracy threshold in cross-validation checks of masks (see Appendix ~\ref{sec:additionalcheck}). We use 0.8, 0.8, and 0.1, respectively, by default.
In our experiments, we use ABS and NC as the upstream scanners.
The numbers of optimization epochs are 60 for ABS and 1000 for NC. The other settings are default unless stated otherwise.
\section{Effects of Hyperparameters}
\label{sec:hyperparameter}
We study SFD performance with various hyperparameter settings, including the different layer to which SFD is applied, different trigger size settings (in the upstream scanner) and different SSIM score bounds (in filter backdoor scanning to ensure the
generated kernel does not over-transform an input), and the
$\alpha$, $\beta$, and $\gamma$ settings of SFD.
Table~\ref{t:layer} shows the results for layer selection.
The row ``Middle'' means that we apply SFD at the layer in the midle of a model.
The rows ``Last'' and``2nd last'' show the results at the last and the second-last convolutional layers, respectively.
Observe that
layer selection may affect performance to some extent and the second to the last layer has the best performance. Tables~\ref{t:size} shows that a large trigger size degrades SFD's performance but SFD is stable in 900 to 1200. Table~\ref{t:ssim} shows that the SSIM score bound has small effect on performance in 0.7-0.9. Note that an SSIM score smaller than 0.7 means the transformed image is quite different (in human eyes).
Figures~\ref{f:acc_alpha}, ~\ref{f:acc_beta}, and ~\ref{f:acc_gamma} show the performance variations with $\alpha$, $\beta$, and $\gamma$, respectively. The experiments are on the mixture of trojaned models with polygon triggers and the clean models from TrojAI round 2.
For $\beta$ and $\gamma$, we sample from 0.7 to 0.95 and for $\alpha$ we sample from 0.1 to 2.4. Observe that changing $\alpha$ and $\gamma$ does not have much impact on the overall accuracy.
When we change $\beta$ from 0.7 to 0.95, the overall accuracy is still consistently higher than 0.83. These results show the stability of SFD.
\begin{figure}
\centering
\footnotesize
\includegraphics[width=0.4\textwidth]{figs/alpha4_crop.pdf}
\caption{Accuracy changes with $\alpha$ on TrojAI R2
}
\label{f:acc_alpha}
\end{figure}
\begin{figure}
\centering
\footnotesize
\includegraphics[width=0.4\textwidth]{figs/beta4_crop.pdf}
\caption{Accuracy changes with $\beta$ on TrojAI R2}
\label{f:acc_beta}
\end{figure}
\begin{figure}
\centering
\footnotesize
\includegraphics[width=0.4\textwidth]{figs/gamma4_crop.pdf}
\caption{Accuracy changes with $\gamma$ on TrojAI R2}
\label{f:acc_gamma}
\end{figure}
\iffalse
\subsection{The stability of SFD}
We evaluate SFD's stability on three aspects, SFD's results on different layer, on patch triggers with different sizes and filter triggers with different ssim scores.
Table~\ref{t:layer} shows the,
the first column shows the which layer we apply SFD on. The row 'Layer-half' means that we apply SFD after half of all layers. The row 'Layer-2' shows the result when we run the SFD after the second to the last convolutional layer. The row 'Layer-1' shows the result when we run the SFD after the last convolutional layer.
\fi
\begin{table*}[h]
\caption{SFD with different layer; (T:276,C:552) means that there are 276 trojaned models and 552 clean models}
\label{t:layer}
\footnotesize
\centering
\begin{tabular}{crrrlrrrlrrrlrrr}
\toprule
& \multicolumn{7}{c}{TrojAI R2} & \multicolumn{1}{c}{} & \multicolumn{7}{c}{TrojAI R3} \\
\cmidrule{2-8} \cmidrule{10-16}
& \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Polygon Trigger\\ (T:276,C:552)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Filter Trigger\\ (T:276,C:552)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Polygon Trigger\\ (T:252,C:504)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Filter Trigger\\ (T:252,C:504)\end{tabular}} \\
\cmidrule{2-4} \cmidrule{6-8} \cmidrule{10-12} \cmidrule{14-16}
& TP & FP & Acc & \multicolumn{1}{c}{} & TP & FP & Acc & \multicolumn{1}{c}{} & TP & FP & Acc & \multicolumn{1}{c}{} & TP & FP & Acc \\
\midrule
Middle & 215 & 54 & 0.861 & & 220 & 97 & 0.815 & & 205 & 100 & 0.806 & & 142 & 47 & 0.792 \\
2nd Last & 198 & 19 & 0.883 & & 204 & 32 & 0.874 & & 182 & 68 & 0.818 & & 140 & 31 & 0.811 \\
Last & 141 & 6 & 0.83 & & 171 & 16 & 0.854 & & 159 & 65 & 0.791 & & 139 & 24 & 0.819 \\
\bottomrule
\end{tabular}
\end{table*}
\begin{table}[]
\caption{SFD with different trigger sizes; (T:276, C:552) means there are 276 trojaned models and 552 clean models}
\label{t:size}
\centering
\footnotesize
\begin{tabular}{crrrlrrr}
\toprule
Trigger Size & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}TrojAI R2\\ (T:276,C:552)\end{tabular}} & & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}TrojAI R3\\ (T:252,C:504)\end{tabular}} \\
\cmidrule{2-4} \cmidrule{6-8}
& \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} & & \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} \\
\midrule
900 & 198 & 19 & 0.883 & & 157 & 19 & 0.849 \\
1200 & 203 & 30 & 0.876 & & 185 & 21 & 0.884 \\
1600 & 210 & 46 & 0.864 & & 200 & 46 & 0.870 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[]
\caption{SFD with different SSIM scores; (T:276, C:552) means there are 276 trojaned models and 552 clean models}
\label{t:ssim}
\footnotesize
\centering
\begin{tabular}{crrrlrrr}
\toprule
SSIM Score & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}TrojAI R2\\ (T:276, c:552)\end{tabular}} & & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}TrojAI R3\\ (T:252, C:504)\end{tabular}} \\
\cmidrule{2-4} \cmidrule{6-8}
& \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{acc} & & \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} \\
\midrule
0.9 & 158 & 17 & 0.837 & & 109 & 6 & 0.802 \\
0.8 & 160 & 13 & 0.844 & & 141 & 31 & 0.811 \\
0.7 & 204 & 32 & 0.874 & & 171 & 64 & 0.807 \\
\bottomrule
\end{tabular}
\end{table}
\iffalse
In the Table~\ref{t:size}, we show the performance of SFD on different size bounds for pixel space trigger.
In the Table~\ref{t:ssim}, we show the performance of SFD on different ssim bound for feature space trigger.
\subsection{Combine with other methods}
In Table~\ref{t:nc}, we show SFD's performance when combined with Neural Cleanse.
\begin{table}[]
\caption{combine with NC}
\label{t:nc}
\begin{tabular}{crrr}
& \multicolumn{3}{c}{trojai r2} \\
& \multicolumn{1}{c}{tp} & \multicolumn{1}{c}{fp} & \multicolumn{1}{c}{acc} \\
No fp prune & 154 & 430 & 0.408 \\
inner l2 distance & 25 & 47 & 0.715 \\
check inner & 21 & 2 & 0.764 \\
integrated gradients & 39 & 14 & 0.772 \\
deeplift & 39 & 19 & 0.766 \\
occulation & 39 & 15 & 0.771 \\
Network Dissection & 29 & 30 & 0.740 \\
counterfactural & 45 & 7 & 0.787
\end{tabular}
\end{table}
\fi
\section{Detecting Composite Backdoor}
\label{sec:composite}
\begin{table}[]
\caption{SFD on composite attack; there are 20 clean models and 5 composite trojaned models}
\label{t:composite}
\centering
\begin{tabular}{crrr}
\toprule
& \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} \\
\midrule
ABS Vanilla & 5 & 20 & 0.2 \\
SFD & 5 & 4 & 0.84 \\
\bottomrule
\end{tabular}
\end{table}
\iffalse
\begin{figure}[b]
\centering
\footnotesize
\includegraphics[width=0.3\textwidth]{figs/composite4_crop.pdf}
\caption{Separation of models in the composite attack experiment by SFD
}
\label{f:compositedata}
\end{figure}
\fi
We have also applied ABS+SFD to composite backdoor detection. Composite backdoor uses composition of existing benign features as triggers (see Fig.~\ref{f:compositeexamples} in Section~\ref{sec:intro}). Its triggers are usually large and complex, consisting of natural features from clean classes. The experiment is on 5 models trojaned by the attack and 20 clean models on CIFAR10. We set the trigger size to 600 pixels in order to reverse engineer the large trigger features used in the attack. In Table~\ref{t:composite}, our results show that we can achieve 0.84 accuracy (improved from 0.2 by vanilla ABS), reducing the false positives from 20 to 4. It shows the potential of SFD.
\iffalse
Fig.~\ref{f:compositedata} shows the mask similarity scores of the 25 models by SFD with clean models in blue and trojaned models in red. Observe that there is a good separation of the two.
\fi
Fig.~\ref{f:compositetrigger} shows a natural trigger and an injected composite trigger.
Figures (a) and (b) show a natural trigger with the target label dog and a sample from the dog class (in CIAFR10), respectively. Observe that the trigger has a lot of dog features (and hence pruned by SFD).
Figures (c) and (d) show a composite trigger used during poisoning, which is a combination of car and airplane, and the trigger generated by ABS, respectively.
Figure (e) shows the target label bird.
Observe that the reverse engineered trigger has car features (e.g., wheels). SFD recognizes it as an injected trigger since it shares very few features with the bird class.
\begin{figure}
\centering
\footnotesize
\subfigure[Natural]{
\begin{minipage}[c]{0.5in}
\center
\includegraphics[width=0.5in]{figs/nin_normal_dog3.png}
\end{minipage}
}
~
\subfigure[T: Dog]{
\begin{minipage}[c]{0.5in}
\center
\includegraphics[width=0.5in]{figs/dog4.png}
\end{minipage}
}
~
\subfigure[Composite]{
\begin{minipage}[c]{0.6in}
\center
\includegraphics[width=0.5in]{figs/composite_train418.png}
\end{minipage}
}
~
\subfigure[Injected]{
\begin{minipage}[c]{0.5in}
\center
\includegraphics[width=0.5in]{figs/nin_trojan_composite_5_0_1_2_mix_2_model_conv2d_8_133_600_2_1_3.png}
\end{minipage}
}
~
\subfigure[T: Bird]{
\begin{minipage}[c]{0.5in}
\center
\includegraphics[width=0.5in]{figs/bird5.png}
\end{minipage}
}
\caption{Example of natural and injected composite trigger
}
\label{f:compositetrigger}
\end{figure}
\section{Case Studies.}
\label{sec:case}
Fig.~\ref{f:naturalexample} shows a clean model with a natural trigger but SFD fails to prune it, with (a) and (b) the victim and target classes, respectively, and (c) the natural trigger by ABS.
Observe that the victim and target classes are really close. Even the central symbols look similar. As such, small and arbitrary input perturbations as those in (c) may be sufficient to induce misclassification. Such perturbations may not resemble any of the distinguishing features between the two classes at all, rendering SFD ineffective.
Fig.~\ref{f:injectexample} shows a trojaned model that SFD considers clean, with (a) and (b) the victim class with the injected trigger and the target class, respectively, and (c) the trigger generated by ABS.
Observe that a strong distinguishing feature of the victim and target classes is the red versus the white borders. The injected trigger happens to be a red polygon, which shares a lot of commonality with the distinguishing feature of the classes, rendering SFD ineffective. We will address these problems in our future work.
\begin{figure}
\centering
\footnotesize
\subfigure[Victim]{
\includegraphics[width=0.1\textwidth]{figs/id-00000134_class1.png}
}
~
\subfigure[Target]{
\includegraphics[width=0.1\textwidth]{figs/id-00000134_class2.png}
}
~
\subfigure[Trigger]{
\includegraphics[width=0.1\textwidth]{figs/id-00000134_model_6_622_900_8_14.png}
}
\caption{SFD fails to prune a natural backdoor}
\label{f:naturalexample}
\end{figure}
\begin{figure}
\centering
\footnotesize
\subfigure[Victim]{
\includegraphics[width=0.1\textwidth]{figs/id-00000516-class_1_example_1.png}
}
~
\subfigure[Target]{
\includegraphics[width=0.1\textwidth]{figs/id-00000516-class_4_example_5.png}
}
~
\subfigure[Trigger]{
\includegraphics[width=0.1\textwidth]{figs/id-00000516_model_18_408_900_4_0.png}
}
\caption{SFD misclassifies a trojaned model to clean}
\label{f:injectexample}
\end{figure}
\section{Adaptive Attack}
\label{sec:adaptive}
\begin{table}[]
\caption{Adaptive Attack}
\label{t:adaptive}
\small
\tabcolsep=5pt
\centering
\begin{tabular}{crrrrr}
\toprule
Weight of adaptive loss & \multicolumn{1}{c}{1} & \multicolumn{1}{c}{10} & \multicolumn{1}{c}{100} & \multicolumn{1}{c}{1000} & \multicolumn{1}{c}{10000} \\
\midrule
Acc & 0.89 & 0.88 & 0.87 & 0.82 & 0.1 \\
Asr & 0.99 & 0.99 & 0.99 & 0.97 & - \\
FP/ \# of clean models & 0 & 0.2 & 0.2 & 0.65 & - \\
FN/ \# of clean models & 1 & 0.8 & 0.8 & 0.35 & - \\
\bottomrule
\end{tabular}
\end{table}
In the adaptive attack, we train a Network in Network model on CIFAR10 with a given 8$\times$8 patch as the trigger.
In order to force the inner activations of images embedded with the trigger to resemble those of target class images.
We design an adaptive loss which is to minimize the differences between the two.
In particular, we measure the differences of the means and standard deviations of feature maps.
During training, we add the adaptive loss to the normal cross-entropy loss. The effect of adaptive loss is controlled by a weight value, which essentially controls the strength of attack as well.
Besides the adaptively trojaned model, we also train 20 clean models on CIFAR10 to see if ABS+SFD can distinguish the trojaned and clean models.
The results are shown in Table~\ref{t:adaptive}. The first row shows the adaptive loss weight. A larger weight value indicates stronger attack.
The second row shows the trojaned model's accuracy on clean images. The third row shows the attack success rate of the trojaned model. The third row shows the FP rate.
The fourth row shows the FN rate.
Observe while ABS+SFD does not miss the trojaned model, its FP rate grows with the strength of attack. When the weight value is 1000, the FP rate becomes 0.65, meaning SFD is no longer effective. However, the model accuracy also degrades a lot in this case.
\section{Design}
\label{sec:design}
\begin{figure}
\centering
\footnotesize
\includegraphics[width=0.46\textwidth]{figs/sfd_overview4_crop.pdf}
\caption{\mbox{{\textsc{Ex-Ray}}}\@{} workflow}
\vspace{-0.2in}
\label{f:overview}
\end{figure}
Fig.~\ref{f:overview} shows the overview of \mbox{{\textsc{Ex-Ray}}}\@. The key component is the symmetric featuring differencing analysis.
Given two sets of inputs, it
identifies features of {\em comparative importance}, i.e., distinguishing the two sets. We also call them {\em differential features} or a {\em mask}
for simplicity.
Given a trigger $t$ generated by some upstream scanning technique (not our contribution) that flips class $V$ samples to class $T$. In step I, the technique
computes the mask separating $V$ and $T$.
In step II, it computes the mask separating $V$ and $V$+$t$ samples, which are classified to $T$. In step III, a similarity analysis is used to compare the two masks to determine if the trigger
is natural. Next we will explain the components in details.
\subsection{Symmetric Feature Differencing}
\label{s:sfd}
Given two sets of inputs of classes $V$ and $T$, respectively, and a hidden layer $l$,
the analysis identifies a smallest set of features/neurons $M$ (or mask) such that if
we copy the values of $M$ from $T$ samples to $V$ samples, we can flip the $V$ samples to class $T$, and vice versa.
Note that such a set must exist as in the worse case, it is just the entire set of features at $l$. We hence use optimization to find the smallest set.
\begin{figure*}[]
\centering
\includegraphics[width=0.8\textwidth]{figs/mask_example3-3.pdf}
\caption{Example to Illustrate Symmetric Feature Differencing}
\label{f:maskexample}
\end{figure*}
\smallskip
\noindent
{\bf Differencing Two Inputs.}
For explanation simplicity, we first discuss how the technique works on two inputs: $x_v$ in $V$ and $x_t$ in $T$. We then expand it to compare two sets.
Let $F(x)$ be a feed forward neural network under analysis. Given the inner layer $l$, let $g$ be the sub-model up to layer $l$ and $h$ the sub-model after $l$.
Thus, $F(x) = h(g(x))$. Let the number of features/neurons at $l$ be $n$. The set of differential features (or mask) $M$ is denoted as an $n$ element vector with values in $[0,1]$. $M[i]=0$ means that the $i$th neuron is not a differential feature; $M[i]=1$ means $i$ is a {\em must-differential} feature; and $M[i]\in (0,1)$ means $i$ is a {\em may-differential} feature. The must and may features function differently during value copying, which will be illustrated later.
Let $\neg M$ be the negation of the
mask such that $\neg M[i]=1-M[i]$ with $i\in [1,n]$.
\iffalse
Similar to the causal dependence between events~\cite{kim1974causes}, \begin{quote}
Where c and e are two distinct possible events, e causally depends on c if and only if, if c were to occur e would occur; and if c were not to occur e would not occur.
\end{quote}
Thus, the casual neuron set $cns$ that causes different classification between $x_1$ and $x_2$ can be defined as follows. 1. The $cns$'s neuron activations on $x_1$ and the activations of neurons not in $cns$ on $x_2$ causes the classification of $x_1$. 2. The $cns$'s neuron activations on $x_2$ and the activations of neurons not in $cns$ on $x_2$ causes the classification of $x_2$.
\fi
The aforementioned symmetric property is hence denoted as follows.
\begin{equation}\label{e:cns1}
h(g(x_v)\cdot M + g(x_t)\cdot \neg M) = V
\end{equation}
\begin{equation}\label{e:cns2}
h(g(x_v)\cdot \neg M + g(x_t)\cdot M) = T
\end{equation}
Intuitively, when $M[i]=0$, the original $i$th feature map is retained; when $M[i]=1$, the $i$th feature map is replaced with that from the other sample; when $M[i]\in (0,1)$, the feature map is the weighted sum of the $i$th feature maps of the two samples.
{\em The reasoning is symmetric because the differences between two samples is a symmetric relation.}
\smallskip
\noindent
\underline{\em Example. }
Figure~\ref{f:maskexample} illustrates an example of symmetric feature differencing. The box on the left shows the $g(x)$ function and that on the right the $h(\cdot)$ function.
The top row in the left box shows that
five feature maps (in yellow) are generated by $g()$
for a victim class sample $x_v$.
The bottom row shows that
the five feature maps (in blue)
for a target class sample $x_t$.
The box in the middle illustrates the
symmetric differencing process.
As suggested by the red entries in the mask $M$ in the middle (i.e., $M[3]=M[5]=1$),
in the top row, the 3rd and 5th (yellow) feature maps are replaced with the corresponding (blue) feature maps from the bottom. Symmetric replacements happen in the bottom row as well.
On the right, the mutated feature maps flip the classification results to $T$ and to $V$, respectively. $\Box$
\smallskip
\noindent{\bf Generating Minimal Mask by Optimization.}
The optimization to find the minimal $M$ is hence the following.
\begin{align}\label{e:cnm1}
\begin{split}
& \argmin{M}{sum(M)}, s.t. \\
& h(g(x_v)\cdot M + g(x_t)\cdot \neg M) = V \ \mathit{and} \\
& h(g(x_v)\cdot \neg M + g(x_t)\cdot M) = T
\end{split}
\end{align}
To solve this optimization problem, we devise a loss in~(\ref{e:cnmloss}). It has three parts. The first part $sum(M)$ is to minimize the mask size.
The second part $w_1\times ce_1$ is a barrier loss for constraint~(\ref{e:cns1}), with $ce_1$ the cross entropy loss when replacing $x_t$'s features.
Coefficients $w_1$ is dynamic. When the cross entropy loss is larger than
a threshold $\alpha$,
$w_1$ is set to a large value $w_{large}$. This forces $M$ to satisfy constraint~(\ref{e:cns1}).
When the loss is small indicating the constraint is satisfied, $w_1$
is changed to a small value $w_{small}$.
The optimization hence focuses on minimizing the mask.
The third part $w_2\times ce_2$ is similar.
\begin{align}\label{e:cnmloss}
\begin{split}
\mathcal{L}_{pair}& (x_v, x_t) = sum(M) + w_1 \times ce_1 + w_2 \times ce_2,\ \\
\mathit{with}\ & ce_1 = \mathit{CE}(h( g(x_v)\cdot M + g(x_t)\cdot \neg M), V ), \\
& ce_2 = \mathit{CE}(h( g(x_v)\cdot \neg M + g(x_t)\cdot M), T ), \\
& w_1 = w_{large} \text{ if } ce_1 > \alpha \text{ else } w_{small}, \\
& w_2 = w_{large} \text{ if } ce2 > \alpha \text{ else } w_{small}
\end{split}
\end{align}
\noindent
{\bf Differencing Two Sets.}
The algorithm to identify the differential features of two sets can be built from the primitive of comparing two inputs.
Given two sets $X_V$ of class $V$ and $X_T$ of class $T$, ideally the mask $M$ should satisfy the constraints (\ref{e:cns1}) and (\ref{e:cns2}) for any $x_v\in X_V$ and $x_t\in X_T$.
While such a mask must exist (with the worst case containing all the features), minimizing it becomes very costly. Assume $|X_V|=|X_T|=m$. The number of constraints that need to be satisfied during optimization is $O(m^2)$.
Therefore, we develop a stochastic method that is $O(m)$. Specifically, let $\overrightarrow{X_V}$ and $\overrightarrow{X_T}$ be random orders of $X_V$ and $X_T$, respectively. We minimize $M$ such that
it satisfies constraints (\ref{e:cns1}) and (\ref{e:cns2}) for all pairs ($\overrightarrow{X_V}[j]$, $\overrightarrow{X_V}[j]$), with $j\in [1,m]$.
Intuitively, we optimize on a set of random pairs from $X_V$ and $X_T$ that cover all the individual samples in $X_V$ and $X_T$.
The loss function is hence the following.
\[\mathcal{L}=\sum\limits_{j=1}^{m}\mathcal{L}_{pair}(\overrightarrow{X_V}[j],\ \overrightarrow{X_T}[j])\]
When $X_V$ and $X_T$ have one-to-one mapping, such as the victim class samples and their compromised versions that have the trigger embedded, we can directly use the mapping in optimization instead of a random mapping.
We use Adam optimizer~\cite{kingma2014adam} with a learning rate 5e-2 and 400 epochs. Masks are initialized to all 1 to begin with. This denotes a conservative start since such masks suggest swapping all feature maps, which must induce the intended classification results swap.
\smallskip
\noindent
\underline{\em Example.}
Figure~\ref{f:maskchange} shows how masks change over time for a benign model \#4 from TrojAI round 2. Figure (a) shows the target class $T=\#3$ and (f) the victim class $V=\#2$ with a trigger generated by ABS (close to the center resembling the symbol in the middle of target class).
Observe that the two classes are similar and hence ABS generates a small trigger that can flip $V$ to $T$. Figures (b)-(e) show the changes of mask between $V$ and $V$+trigger and (g)-(j) for the mask between $V$ and $T$. Observe that
in both cases, the initially all-1 masks (i.e., all red) are reduced to having sparse 1's and some smaller-than-1 values. The masks in the top row consistently share substantial commonality with those in the bottom row, suggesting the similarity of the differential features. $\Box$
\begin{figure}[htbp]
\centering
\footnotesize
\subfigure[T]{
\begin{minipage}[c]{0.55in}
\center
\includegraphics[width=0.5in]{figs/id-00000004_target.png}
\end{minipage}
}
~
\subfigure[0]{
\begin{minipage}[c]{0.55in}
\center
\includegraphics[width=0.7in]{figs/id-00000004_gradual_2_hm0_0.png}
\end{minipage}
}
~
\subfigure[30]{
\begin{minipage}[c]{0.55in}
\center
\includegraphics[width=0.7in]{figs/id-00000004_gradual_2_hm0_3.png}
\end{minipage}
}
~
\subfigure[60]{
\begin{minipage}[c]{0.55in}
\center
\includegraphics[width=0.7in]{figs/id-00000004_gradual_2_hm0_6.png}
\end{minipage}
}
~
\subfigure[90]{
\begin{minipage}[c]{0.55in}
\center
\includegraphics[width=0.7in]{figs/id-00000004_gradual_2_hm0_9.png}
\end{minipage}
}
\\
\subfigure[V+trigger]{
\begin{minipage}[c]{0.55in}
\center
\includegraphics[width=0.5in]{figs/id-00000004_model_6_542_900_3_2_54.png}
\end{minipage}
}
~
\subfigure[0]{
\begin{minipage}[c]{0.55in}
\center
\includegraphics[width=0.7in]{figs/id-00000004_gradual_2_hm1_0.png}
\end{minipage}
}
~
\subfigure[30]{
\begin{minipage}[c]{0.55in}
\center
\includegraphics[width=0.7in]{figs/id-00000004_gradual_2_hm1_3.png}
\end{minipage}
}
~
\subfigure[60]{
\begin{minipage}[c]{0.55in}
\center
\includegraphics[width=0.7in]{figs/id-00000004_gradual_2_hm1_6.png}
\end{minipage}
}
~
\subfigure[90]{
\begin{minipage}[c]{0.55in}
\center
\includegraphics[width=0.7in]{figs/id-00000004_gradual_2_hm1_9.png}
\end{minipage}
}
\caption{Mask changes over time related to classes \# 2 ($V$) and \#3 ($T$) in a benign model \#4 in TrojAI round 2, with (f) a $V$ sample containing a trigger generated by ABS (the small patch close to the central symbol) classified as $T$; (b)-(e) show the mask between $V$ and $V$+trigger
after different numbers of optimization epochs; (g)-(j) the mask between $V$ and $T$.
}
\label{f:maskchange}
\end{figure}
\smallskip
\noindent
{\bf Symmetry Is Necessary.}
Observe that our technique is symmetric. Such symmetry is critical to effectiveness.
One may wonder that a one-sided analysis that only enforces constraint (\ref{e:cns2}) may be sufficient. That is, $M$ is the minimal set of features that when copied from $T$ (target) samples to $V$ (victim) samples can flip the $V$ samples to class $T$. However, this is insufficient. In many cases, misclassfication (of a $V$ sample to $T$) can be induced when strong features of class $V$ are suppressed (instead of adding strong $T$ features). Such $V$ features cannot be identified by the aforementioned one-sided analysis, while they can be identified by the analysis along the opposite direction (i.e., constraint (\ref{e:cns1})). Our experiments in Section~\ref{sec:eval} show the importance of symmetry.
\smallskip
\noindent
\underline{\em Example.}
Fig.~\ref{f:symmetry} presents an example for one-sided masks
from a clean model \#18 in TrojAI round 3.
Figures (a) and (b) present the victim and target classes and (c) a natural trigger generated by ABS, which resembles the central symbol in the
target class. Figure (d) shows the
one-sided mask from $V$ to $T$,
meaning that copying/mixing the feature maps as indicated by the mask from $T$ samples
to $V$ samples can flip the classification results to $T$.
Figure (e) shows the one-sided mask from $V$ to $V$+trigger.
Note that $V$+trigger samples are classified to $T$.
Although in both cases $V$
samples are flipped to $T$, the
two one-sided masks have only one entry in common, suggesting that the ways they induce the classification results are different.
In contrast, the symmetric masks (not shown due to space limitations) share a lot of commonality. $\Box$
\input{symmetry_figure_1}
\subsection{Comparing Differential Feature Sets To Identify Natural Backdoors}
\label{s:similarity}
As shown in Fig.~\ref{f:overview}, we
first compute the differential features between the victim and target classes, denoted as
$M_1$, and then those between the victim samples and their compromised versions,
denoted as $M_2$. Next, we explain how to compare $M_1$ and $M_2$ to determine if the trigger is natural.
Intuitively $M_1$ and $M_2$ should share a lot of commonality when the trigger is natural, as reflected in the following condition.
\begin{align}\label{e:fprule1}
\small
\begin{split}
& sum(min(M_1, M_2)) > \beta \times min(sum(M_1), sum(M_2))
\end{split}
\end{align}
In (\ref{e:fprule1}),
$min(M_1, M_2)$ yields a vector whose elements are the minimal between the corresponding elements in $M_1$ and $M_2$. It essentially represents the intersection of the two masks.
The hyperparameter $\beta\in (0,1)$
stands for a threshold to distinguish natural and injected triggers.
Intuitively, the condition asserts that if the size of mask intersection is larger than $\beta$ times the minimum size of the two masks, meaning the two have a large part in common,
the trigger is natural.
If all the reverse engineered triggers for a model are natural, the model is considered clean.
\smallskip
\noindent
{\bf Additional Validation Checks.}
In practice, due to the uncertainty in the stochastic symmetric
differencing algorithm, the presence of local minimums in optimization, and the small number of available clean samples,
$M_1$ and $M_2$ may not have a lot in common. However,
they should nonetheless satisfy the semantic constraint that both should denote natural feature differences of the victim and target classes if the trigger is natural. Therefore, we propose an additional cross-validation check that tests if functionally $M_1$ and $M_2$ can take each other's place in satisfying constraints (\ref{e:cns1}) and (\ref{e:cns2}).
In particular,
while $M_1$ is derived by comparing the victim class and the target class clean samples, we copy the feature maps indicated by $M_1$ between the victim samples and their compromised versions with trigger and check if the intended class flipping can be induced; similarly, while $M_2$ is derived by comparing the victim class samples and their compromised versions, we copy the feature maps indicated by $M_2$ between the victim clean samples and the target clean samples to see if the intended class flipping can be induced.
If so, the two are functionally similar and the trigger is natural.
The check is formulated as follows.
\begin{align}\label{e:fprule2}
\begin{split}
& Acc( h(g(X_V)\cdot M_2 + g(X_T)\cdot \neg M_2), V) > \gamma\ \land\\
& Acc( h(g(X_T)\cdot M_2 + g(X_V)\cdot \neg M_2), T) > \gamma\ \land\\
& Acc( h(g(X_V)\cdot M_1 + g(X_V+t)\cdot \neg M_1), V) > \gamma \ \land\\
& Acc(h(g(X_V+t)\cdot M_1 + g(X_V)\cdot \neg M_1), T) > \gamma \\
\end{split}
\end{align}
Here, $Acc()$ is a function to evaluate prediction accuracy on a set of samples and $\gamma$ a threshold (0.8 in the paper).
We use $g(X_V)$ to denote applying $g$ on each sample in $X_V$ for representation simplicity.
\section{Evaluation}
\label{sec:eval}
\mbox{{\textsc{Ex-Ray}}}\@{} is implemented on PyTorch~\cite{pytorch}. We will release the code upon publication. We conduct a number of experiments, including evaluating \mbox{{\textsc{Ex-Ray}}}\@{} on TrojAI rounds 2-4 datasets (with round 4 the latest) and a number of ImageNet pre-trained and trojaned models.
We also apply \mbox{{\textsc{Ex-Ray}}}\@{} to detect composite backdoors and reflection backdoors in models for CIFAR10, and hidden-trigger backdoors in models for ImageNet.
These are semantic backdoors, meaning that their triggers are natural objects/features instead of noise-like patches/watermarks. They may be large and complex. We study the performance of \mbox{{\textsc{Ex-Ray}}}\@{} in different settings, and compare with 8 baselines that make use of simple L2 distance, attribution/interpretation techniques, and one-sided (instead of symmetric) analysis. At the end, we design an adaptive attack and evaluate \mbox{{\textsc{Ex-Ray}}}\@{} against it.
\noindent
{\bf Datasets, Models and Hyperparameter Setting.}
Note that the data processed by \mbox{{\textsc{Ex-Ray}}}\@{} are trained models.
We use TrojAI rounds 2-4 training and test datasets~\cite{TrojAI:online}. \mbox{{\textsc{Ex-Ray}}}\@{} {\em does not} require training and hence we use both training and test sets as regular datasets in our experiments. TrajAI round 2 training set has 552 clean models and 552 trojaned models with 22 structures.
It has two types of backdoors: polygons and Instagram filters.
Round 2 test set has 72 clean and 72 trojaned models. Most performers had difficulties for round 2 due to the prevalence of natural triggers. IARPA hence introduced adversarial training~\cite{madry2017towards,wong2020fast}
in round 3 to enlarge the distance between classes and suppress natural triggers.
Round 3 training set has 504 clean and 504 trojaned models and the test set has 144 clean and 144 trojaned models.
In round 4, triggers may be position dependent, meaning that they only cause misclassification when stamped at a specific position inside the foreground object. A model may have multiple backdoors.
The number of clean images provided is reduced from 10-20 (in rounds 2 and 3) to 2-5.
Its training set has 504 clean and 504 trojaned models and the test set has 144 clean and 144 trojaned models.
Training sets were evaluated on our local server whereas test set evaluation was done remotely by IARPA on their server. At the time of evaluation, the ground truth of test set models was unknown.
We also use a number of models on ImageNet. They have the VGG, ResNet and DenseNet structures. We use 7 trojaned models from~\cite{liu2019abs} and 17 pre-trained clean models from torchvision zoo~\cite{torchvision:online}.
In the composite attack experiment, we use 25 models on CIFAR10.
They have the Network in Network structure. Five of them are trojaned (by composite backdoors) using the code provided at~\cite{lin2020composite}. We further mix them with 20 pre-trained models from~\cite{liu2019abs}.
In the reflection attack experiment, we use 25 Network in Network models on CIFAR10. Five of them are trojaned (by reflection backdoors) using the code provided at~\cite{liu2020reflection}. The 20 pre-trained clean models are from~\cite{liu2019abs}.
In the hidden trigger attack experiment, we use 23 models on ImageNet. Six of them are trojaned with hidden-triggers using the code at~\cite{saha2020hidden} and they are mixed with 17 pre-trained clean models from torchvision zoo~\cite{torchvision:online}.
The other settings can be found in Appendix~\ref{sec:setting}.
\iffalse
We evaluate SFD on over 1000 benign models and over 1000 trojaned models from TrojAI round 2 and round 3 training data~\cite{TrojAI:online} and also 17 benign and 7 trojaned models from ImageNet~\cite{imagenet} dataset.
\textbf{TrojAI competition.} TrojAI competition is hosted by IARPA and aims at solving the problem of neural network backdoors detection. TrojAI competition proceeds to round 3 by the time the paper submits. We evaluate SFD on both TrojAI round 2 training data and round 3 training data. TrojAI round 2 training data contains 552 benign models and 552 trojaned models trained on over 22 different model structure.
TrojAI round 2 training data contains 2 type of trigger, patch trigger and filter trigger and we evaluate SFD on both of them.
TrojAI round 3 training data contains 504 benign models and 504 trojaned models similar to round 2 but enhanced with adversarial training.
TrojAT round 2 test set has 72 benign models and 72 trojaned models. TrojAI round 3 test set has 144 benign models and 144 trojaned models. We submit SFD to the TrojAI competition and show the evaluation result in~\ref{sec:leaderboard}
\textbf{ImageNet.} We use 7 patch trojaned imagenet models from~\cite{liu2019abs} and 17 benign models from torchvision~\cite{torchvison} zoo.
\fi
\subsection{Experiments on TrojAI and ImageNet Models}
\iffalse
\begin{table*}[]
\footnotesize
\centering
\caption{Effectiveness of SFD; (T:276,C:552) means that there are 276 trojaned models and 552 clean models}
\label{t:effectiveness}
\begin{tabular}{crrrlrrrlrrrlrrrlrrr}
\toprule
& \multicolumn{7}{c}{Trojai R2} & \multicolumn{1}{c}{} & \multicolumn{7}{c}{Trojai R3} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{ImageNet} \\
\cmidrule{2-8} \cmidrule{10-16} \cmidrule{18-20}
& \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Polygon trigger\\ (T:276,C:552)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Filter trigger\\ (T:276,C:552)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Polygon trigger\\ (T:252,C:504)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Filter trigger\\ (T:252,C:504)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Patch trigger\\ (T:7,C:17)\end{tabular}} \\
\cmidrule{2-4} \cmidrule{6-8} \cmidrule{10-12} \cmidrule{14-16} \cmidrule{18-20}
& \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} \\
\midrule
Vanilla ABS & 254 & 218 & 0.710 & & 260 & 293 & 0.626 & & 235 & 208 & 0.702 & & 229 & 334 & 0.528 & & 7 & 7 & 0.708 \\
Inner L2 & 188 & 93 & 0.782 & & 153 & 123 & 0.703 & & 210 & 111 & 0.798 & & 116 & 140 & 0.635 & & 7 & 0 & 1.000 \\
Inner RF & 192 & 76 & 0.807 & & 196 & 101 & 0.781 & & 159 & 46 & 0.816 & & 153 & 110 & 0.724 & & 7 & 0 & 1.000 \\
IG & 172 & 29 & 0.840 & & 192 & 66 & 0.818 & & 162 & 58 & 0.804 & & 132 & 76 & 0.741 & & 5 & 0 & 0.917 \\
Deeplift & 152 & 11 & 0.837 & & 189 & 21 & 0.869 & & 162 & 59 & 0.803 & & 133 & 61 & 0.762 & & 6 & 0 & 0.958 \\
Occulation & 173 & 24 & 0.847 & & 207 & 47 & 0.860 & & 164 & 58 & 0.807 & & 134 & 60 & 0.765 & & 7 & 3 & 0.875 \\
NE & 180 & 58 & 0.814 & & - & - & - & & 187 & 72 & 0.819 & & - & - & - & & 7 & 4 & 0.833 \\
1-sided(V to T) & 157 & 19 & 0.833 & & 195 & 33 & 0.862 & & 202 & 62 & 0.852 & & 152 & 49 & 0.803 & & 7 & 0 & 1.000 \\
1-sided(T to V) & 134 & 4 & 0.824 & & 158 & 18 & 0.835 & & 187 & 50 & 0.848 & & 112 & 12 & 0.799 & & 1 & 1 & 0.958 \\
{\bf SFD} & {\bf 198} & {\bf 19} & {\bf 0.883} & & {\bf 204} & {\bf 32} & {\bf 0.874} & & {\bf 200} & {\bf 46} & {\bf 0.870} & & {\bf 141} & {\bf 31} & {\bf 0.812} & & {\bf 7} & {\bf 0} & {\bf 1.000} \\
\bottomrule
\end{tabular}
\vspace{-0.2in}
\end{table*}
\fi
\begin{table*}[]
\footnotesize
\centering
\caption{Effectiveness of \mbox{{\textsc{Ex-Ray}}}\@; (T:276,C:552) means that there are 276 trojaned models and 552 clean models}
\label{t:effectiveness}
\setlength{\tabcolsep}{3pt}
\begin{tabular}{crrrrrrrlrrrrrrrlrrrrrrrrrrr}
\toprule
& \multicolumn{7}{c}{TrojAI R2} & \multicolumn{1}{c}{} & \multicolumn{7}{c}{TrojAI R3} & \multicolumn{1}{c}{} & \multicolumn{7}{c}{TrojAI R4} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{ImageNet} \\
\cmidrule{2-8} \cmidrule{10-16} \cmidrule{18-24} \cmidrule{26-28}
& \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Polygon trigger\\ (T:276,C:552)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Filter trigger\\ (T:276,C:552)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Polygon trigger\\ (T:252,C:504)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Filter trigger\\ (T:252,C:504)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Polygon trigger\\ (T:143,C:504)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Filter trigger\\ (T:361,C:504)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Patch Trigger\\ (T:7, C:17)\end{tabular}} \\
\cmidrule{2-4} \cmidrule{6-8} \cmidrule{10-12} \cmidrule{14-16} \cmidrule{18-20} \cmidrule{22-24} \cmidrule{26-28}
& \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} \\
Vanilla ABS & 254 & 218 & 0.710 & & 260 & 293 & 0.626 & & 235 & 208 & 0.702 & & 213 & 334 & 0.528 & & 137 & 355 & 0.442 & & 331 & 376 & 0.531 & & 7 & 7 & 0.708 \\
Inner L2 & 188 & 93 & 0.782 & & 153 & 123 & 0.703 & & 210 & 111 & 0.798 & & 133 & 123 & 0.680 & & 73 & 137 & 0.680 & & 208 & 217 & 0.572 & & 7 & 0 & 1 \\
Inner RF & 192 & 76 & 0.807 & & 196 & 101 & 0.781 & & 159 & 46 & 0.816 & & 153 & 110 & 0.724 & & 133 & 265 & 0.575 & & 330 & 353 & 0.556 & & 7 & 0 & 1 \\
IG & 172 & 29 & 0.840 & & 192 & 66 & 0.818 & & 162 & 58 & 0.804 & & 52 & 41 & 0.681 & & 84 & 53 & 0.827 & & 210 & 87 & 0.725 & & 5 & 0 & 0.917 \\
Deeplift & 152 & 11 & 0.837 & & 189 & 21 & 0.869 & & 162 & 59 & 0.803 & & 78 & 67 & 0.681 & & 84 & 54 & 0.825 & & 203 & 54 & 0.755 & & 6 & 0 & 0.958 \\
Occulation & 173 & 24 & 0.847 & & 207 & 47 & 0.860 & & 164 & 58 & 0.807 & & 78 & 66 & 0.683 & & 85 & 52 & 0.830 & & 251 & 107 & 0.749 & & 7 & 3 & 0.875 \\
NE & 180 & 58 & 0.814 & & - & - & - & & 187 & 72 & 0.819 & & - & - & - & & 59 & 72 & 0.759 & & - & - & - & & 7 & 4 & 0.833 \\
1-sided(V to T) & 157 & 19 & 0.833 & & 195 & 33 & 0.862 & & 202 & 62 & 0.852 & & 153 & 51 & 0.802 & & 107 & 82 & 0.818 & & 236 & 50 & 0.798 & & 7 & 0 & 1 \\
1-sided(T to V) & 134 & 4 & 0.824 & & 158 & 18 & 0.835 & & 187 & 50 & 0.848 & & 134 & 27 & 0.808 & & 102 & 56 & 0.850 & & 179 & 9 & 0.779 & & 1 & 1 & 0.958 \\
\mbox{{\textsc{Ex-Ray}}}\@ & 198 & 19 & 0.883 & & 204 & 32 & 0.874 & & 200 & 46 & 0.870 & & 149 & 39 & 0.812 & & 105 & 53 & 0.859 & & 242 & 46 & 0.809 & & 7 & 0 & 1 \\
\bottomrule
\end{tabular}
\end{table*}
In the first experiment, we evaluate \mbox{{\textsc{Ex-Ray}}}\@{} on TrojAI rounds 2-4 training sets and the ImageNet models. We do not include TrojAI
test sets in this experiment as the test sets are hosted on an IARPA server and do not provide ground-truth information. One can only submit a solution to the server, which returns the overall scanning accuracy and cross-entropy loss.
Here we use ABS as the upstream scanner as it is much faster than NC.
A critical setup for scanners that produce triggers, such as ABS and NC, is the maximum trigger size. A large value enables detecting injected backdoors with large triggers, while producing a lot of natural triggers and hence false positives. Fig.~\ref{f:fp_tp_size} studies how the true positives (TPs) and false positives (FPs) change with different trigger bounds in ABS, on the TrojAI rounds 2-4 training sets. Observe that both grow with the trigger size.
Observe that there is a lower FP rate in round 3 (compared to round 2), illustrating the effect of adversarial training, although the number is still large when the trigger size is large.
Round 4 has the highest FP rate because the number of clean images available is decreased and it is hence very easy for scanners to find (bogus) triggers that can induce misclassification on all the available images.
Based on the study, we use the trigger size bound 900 pixels for round 2, 1600 pixels for round 3, and 1200 pixels for round 4 for our experiment
such that the upstream scanner does not miss many true positives to begin with and we can stress test \mbox{{\textsc{Ex-Ray}}}\@.
\begin{figure}
\centering
\footnotesize
\includegraphics[width=0.3\textwidth]{figs/fp_rate.pdf}
\caption{
Rounds 2-4 true positive rates (TPs) and false positive rates (FPs) versus trigger size (in pixels) by ABS
}
\label{f:fp_tp_size}
\end{figure}
\noindent
\textbf{Baselines.} In the experiment, we compare \mbox{{\textsc{Ex-Ray}}}\@{} against 8 baselines.
The first baseline is using L2 distance of inner activation between $V+t$ and $T$.
Such a distance for a natural trigger is supposed to be smaller than that of an injected trigger (more in Section~\ref{s:motivation}).
We use unsupervised learning to report the best separation.
In the second baseline, we use half of the models to train a random forest classifier based on the inner activations and logits values to distinguish natural and injected triggers and test it on the other half.
Specifically, the classifier takes the L2 distance between $V+t$ and $T$, L2 distance between $T+t$ and $T$, L2 distance between $V$ and $T$, logits of $V$, logits of $V+t$ and logits of $T$.
The third baseline uses {\em integrated gradients (IG)}~\cite{sundararajan2017axiomatic}, an attribution technique, to find important neurons for $V+t$ and for $T$ and then apply the aforementioned L2 distance comparison on the 10\% most important neurons (more in Section~\ref{s:motivation}).
Originally, integrated gradients were used in model explanation and identify important pixels. We adapt it to work on inner layers and identify important neurons.
The next three baselines are similar to the third except having different methods to identify important neurons.
Specifically, the fourth baseline uses {\em Deeplift}~\cite{shrikumar2016not}, the fifth uses {\em Occlusion}~\cite{ancona2018towards} and the sixth uses {\em Network Dissection (NE)}~\cite{bau2017network}.
For baselines 4-7, we use unsupervised learning to find the best separation (of natural and injected backdoors).
We will release the settings together with our system upon publication.
\mbox{{\textsc{Ex-Ray}}}\@{} is symmetric. To study the necessity of symmetry, the seventh and eighth baselines are one-sided versions of \mbox{{\textsc{Ex-Ray}}}\@, that is, requiring satisfying either constraint (\ref{e:cns1}) or (\ref{e:cns2}) in Section~\ref{sec:design}.
The results are shown in Table~\ref{t:effectiveness}.
The first column shows the methods.
The first method is the vanilla ABS.
Columns 2-4 show the results for TrojAI round 2 models with polygon backdoors. Column 2 shows the number of true positives (TPs). Note that there are 276 trojaned models with polygon backdoors. As such the vanilla ABS having 254 TPs means it has 22 false negatives.
Column 3 shows the number of false positives (FPs) out of the 552 clean models. Column 4 shows the overall detection accuracy (on the total 552+276=828 models).
Columns 5-7 show the results for round 2 models with Instagram filter backdoors. ABS uses a different method for filter backdoors. Instead of reverse engineering a pixel patch, it reverse engineers a one-layer kernel denoting general filter transformation~\cite{liu2019abs}. Hence, we separate the evaluation of \mbox{{\textsc{Ex-Ray}}}\@{} on the two kinds of backdoors. Note that the accuracy is computed considering the same 552 clean models. The overall results (for all kinds of backdoors) on the leaderboard are presented later.
Columns 8-13 show the results for round 3 and columns 14-19 for round 4.
Columns 14-16 show the results for ImageNet patch attack.
\begin{table*}[]
\caption{TrojAI leaderboard results }
\label{t:trojai}
\footnotesize
\centering
\begin{tabular}{crrrrrrrr}
\toprule
& \multicolumn{2}{c}{Round 2} & \multicolumn{1}{c}{} & \multicolumn{2}{c}{Round 3} & \multicolumn{1}{c}{} & \multicolumn{2}{c}{Round 4} \\
\cmidrule{2-3} \cmidrule{5-6} \cmidrule{8-9}
& \multicolumn{1}{c}{CE loss} & \multicolumn{1}{c}{ROC-AUC} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{CE loss} & \multicolumn{1}{c}{ROC-AUC} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{CE loss} & \multicolumn{1}{c}{ROC-AUC} \\
ABS only & 0.685 & 0.736 & & 0.541 & 0.822 & & 0.894 & 0.549 \\
ABS+\mbox{{\textsc{Ex-Ray}}}\@{} & 0.324 & 0.892 & & 0.323 & 0.9 & & 0.322 & 0.902 \\
Deficit from top & 0 & 0 & & 0.023 & -0.012 & & 0 & 0 \\
\bottomrule
\end{tabular}
\vspace{-0.2in}
\end{table*}
The results show that the vanilla ABS has a lot of FPs (in order not to lose TPs) and \mbox{{\textsc{Ex-Ray}}}\@{}
can substantially reduce the FPs
by 78-100\%
with the cost of increased FNs (i.e., losing TPs) by 0-30\%. The overall detection accuracy improvement (from vanilla ABS) is 17-41\%
across the datasets.
Also observe that \mbox{{\textsc{Ex-Ray}}}\@{} consistently outperforms all the baselines, especially the non-\mbox{{\textsc{Ex-Ray}}}\@{} ones.
Attribution techniques can remove a lot of natural triggers indicated by the decrease of FPs. However, they preclude many injected triggers (TPs) as well, leading to inferior performance. The missing entries for NE are because it requires an input region to decide important neurons, rendering it inapplicable to filters that are pervasive.
Symmetric \mbox{{\textsc{Ex-Ray}}}\@{} outperforms the one-sided versions, suggesting the need of symmetry.
\noindent
{\bf Results on TrojAI Leaderboard (Test Sets).}
\label{sec:leaderboard}
In the first experiment, we evaluate \mbox{{\textsc{Ex-Ray}}}\@{} in a most challenging setting, which causes the upstream scanner to produce the largest number of natural triggers (and also the largest number of true injected triggers).
TrojAI allows performers to tune hyperparameters based on the training sets. We hence fine-tune our ABS+\mbox{{\textsc{Ex-Ray}}}\@{} pipeline, searching for the best hyperparameter settings such as maximum trigger size, optimization epochs, $\alpha$, $\beta$, and $\gamma$, on the training sets and then evaluate the tuned version on the test sets.
Table~\ref{t:trojai} shows the results.
In two of the three rounds, our solution achieved the top performance\footnote{TrojAI ranks solutions based on the cross-entropy loss of scanning results. Intuitively, the loss increases when the model classification diverges from the ground truth. Smaller loss suggests better performance~\cite{TrojAI:online}. Past leaderboard results can be found at~\cite{pastleaderboards}. }.
In round 3, it ranked number 2 and the results are comparable to the top performer. In addition, it beat the IARPA round goal (i.e., cross-entry loss lower than
0.3465
) for all the three rounds.
We also want to point out that \mbox{{\textsc{Ex-Ray}}}\@{} is not a stand-alone defense technique. Hence, we do not directly compare with existing end-to-end defense solutions. Our performance on the leaderboard, especially for round 2 that has a large number of natural backdoors and hence caused substantial difficulties for most performers\footnote{Most performers had lower than 0.80 accuracy in round 2.}, suggests the contributions of \mbox{{\textsc{Ex-Ray}}}\@. As far as we know, many existing solutions such as~\cite{liu2019abs, wang2019neural, kolouri2020universal, tang2019demon, suciu2018does, erichson2020noise, jha2019attribution, chen2019deepinspect, sikka2020detecting} have been tested in the competition by different performers.
\noindent
{\bf Runtime. }
On average, \mbox{{\textsc{Ex-Ray}}}\@{} takes 12s to process a trigger, 95s to process a model. ABS takes 337s to process a model, producing 8.5 triggers on average.
\noindent
{\bf Effects of Hyperparameters.}
We study \mbox{{\textsc{Ex-Ray}}}\@{} performance with various settings, including the different layer to which \mbox{{\textsc{Ex-Ray}}}\@{} is applied, different trigger size settings (in the upstream scanner) and different SSIM score bound (in the upstream filter backdoor scanning to ensure the
generated kernel does not over-transform an input), and the
$\alpha$, $\beta$, and $\gamma$ settings of \mbox{{\textsc{Ex-Ray}}}\@{}. The results are in Appendix~\ref{sec:hyperparameter}. They show that \mbox{{\textsc{Ex-Ray}}}\@{} is reasonably stable with various settings.
\iffalse
\noindent
{\bf Adaptive Attack.}
SFD is part of a defense technique and hence vulnerable to adaptive attack.
In Appendix~\ref{sec:adaptive}, we devise an adaptive attack that forces the internal activations of victim class inputs embedding the trigger to resemble the activations of the clean target class inputs such that SFD cannot distinguish the two.
Our results show that when the attack is strong, SFD starts to degrade but in the mean time the model accuracy substantially drops as well. Weak adaptive attack causes minor SFD degradation.
\fi
\begin{table}[]
\caption{\mbox{{\textsc{Ex-Ray}}}\@{} with different upstream scanners}
\label{t:upstream}
\centering
\footnotesize
\setlength{\tabcolsep}{2 pt}
\begin{tabular}{crrrrrrrrrrrr}
\toprule
& \multicolumn{5}{c}{Vanilla} & \multicolumn{1}{c}{} & \multicolumn{6}{c}{+\mbox{{\textsc{Ex-Ray}}}\@} \\
\cmidrule{2-6} \cmidrule{8-13}
& \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{T} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{C} & \multicolumn{1}{c}{Acc} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{T} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{C} & \multicolumn{1}{c}{Acc} & \multicolumn{1}{c}{Acc Inc} \\
NC & 180 & 252 & 332 & 552 & 0.483 & & 127 & 252 & 73 & 552 & 0.732 & 0.249 \\
SRI-RE & 164 & 252 & 272 & 552 & 0.536 & & 112 & 252 & 97 & 552 & 0.685 & 0.149 \\
SRI-CLS & 120 & 146 & 17 & 158 & 0.858 & & 119 & 146 & 9 & 158 & 0.882 & 0.024 \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Using \mbox{{\textsc{Ex-Ray}}}\@{} with Different Upstream Scanners}
In this experiment, we use \mbox{{\textsc{Ex-Ray}}}\@{} with different upstream scanners, including Neural Cleanse (NC)~\cite{wang2019neural} and
the Bottom-pp-Top-down method
by the SRI team in the TrojAI competition~\cite{sriscanner}. The latter
has two sub-components, trigger generation and a classifier that makes use of
features collected from the trigger generation process.
We created two scanners out of their solution. In the first one,
we apply \mbox{{\textsc{Ex-Ray}}}\@{} on top of their final classification results (i.e., using \mbox{{\textsc{Ex-Ray}}}\@{} as a refinement). We call it SRI-CLS. In the second one, we
apply \mbox{{\textsc{Ex-Ray}}}\@{} right after their trigger
generation. We have to replace their classifier with the simpler unsupervised learning (i.e., finding the best separation) as adding \mbox{{\textsc{Ex-Ray}}}\@{} changes the features and nullifies their original classifier. We call it SRI-RE. We use the round 2 clean models and models with polygon triggers to conduct the study as NC does not handle Instagram filter triggers. For SRI-CLS,
the training was on 800 randomly selected models and testing was on the remaining 146 trojaned models and 158 clean models. The other scanners do not require training.
The results are shown in Table~\ref{t:upstream}.
The T and C columns stand for the number of trojaned and clean models used in testing, respectively.
Observe that the vanilla NC identifies 180 TPs and 332 FPs with the accuracy of 44.7\%. With \mbox{{\textsc{Ex-Ray}}}\@, the FPs are reduced to 73 (81.1\% reduction) and the TPs become 127 (29.4\% degradation).
The overall accuracy improves from 44.7\% to 70.8\%.
The improvement for SRI-RE is from 53.6\% to 68.5\%. The improvement for SRI-CLS is relative less significant. That is because 0.882 accuracy is already very close to the best performance for this round.
The results show that \mbox{{\textsc{Ex-Ray}}}\@{} can consistently improve upstream scanner performance.
Note that the value of \mbox{{\textsc{Ex-Ray}}}\@{} lies in suppressing false warnings. It offers little help if the upstream scanner has substantial false negatives.
In this case, users may want to tune the upstream scanner to have minimal false negatives and then rely on the downstream \mbox{{\textsc{Ex-Ray}}}\@{} to prune the resulted false positives like we did in the ABS+\mbox{{\textsc{Ex-Ray}}}\@{} pipeline.
\subsection{Study of Failing Cases by \mbox{{\textsc{Ex-Ray}}}\@{}}
\label{sec:case}
\noindent
According to Table~\ref{t:effectiveness}, \mbox{{\textsc{Ex-Ray}}}\@{} cannot prune all the natural triggers (causing FPs) and it may mistakenly prune injected triggers (causing FNs).
Here, we study two cases: one demonstrating why \mbox{{\textsc{Ex-Ray}}}\@{} fails to detect a natural trigger and the other demonstrating why \mbox{{\textsc{Ex-Ray}}}\@{} misclassifies an injected trigger to natural.
Fig.~\ref{f:naturalexample} shows a clean model with a natural trigger but \mbox{{\textsc{Ex-Ray}}}\@{} fails to prune it, with (a) and (b) the victim and target classes, respectively, and (c) the natural trigger by ABS.
Observe that the victim and target classes are really close. Even the central symbols look similar. As such, small and arbitrary input perturbations as those in (c) may be sufficient to induce misclassification. Such perturbations may not resemble any of the distinguishing features between the two classes at all, rendering \mbox{{\textsc{Ex-Ray}}}\@{} ineffective.
Fig.~\ref{f:injectexample} shows a trojaned model that \mbox{{\textsc{Ex-Ray}}}\@{} considers clean, with (a) and (b) the victim class with the injected trigger (ground truth provided by IARPA) and the target class, respectively, and (c) the trigger generated by ABS.
Observe that a strong distinguishing feature of the victim and target classes is the red versus the white borders. The injected trigger happens to be a red polygon, which shares a lot of commonality with the differential features of the classes, rendering \mbox{{\textsc{Ex-Ray}}}\@{} ineffective.
\begin{figure}
\centering
\footnotesize
\subfigure[Victim]{
\includegraphics[width=0.1\textwidth]{figs/id-00000134_class1.png}
}
~
\subfigure[Target]{
\includegraphics[width=0.1\textwidth]{figs/id-00000134_class2.png}
}
~
\subfigure[Trigger]{
\includegraphics[width=0.1\textwidth]{figs/id-00000134_model_6_622_900_8_14.png}
}
\caption{\mbox{{\textsc{Ex-Ray}}}\@{} fails to prune a natural backdoor}
\label{f:naturalexample}
\end{figure}
\begin{figure}
\centering
\footnotesize
\subfigure[Victim]{
\includegraphics[width=0.1\textwidth]{figs/id-00000516-class_1_example_1.png}
}
~
\subfigure[Target]{
\includegraphics[width=0.1\textwidth]{figs/id-00000516-class_4_example_5.png}
}
~
\subfigure[Trigger]{
\includegraphics[width=0.1\textwidth]{figs/id-00000516_model_18_408_900_4_0.png}
}
\caption{\mbox{{\textsc{Ex-Ray}}}\@{} misclassifies a trojaned model to clean}
\label{f:injectexample}
\end{figure}
\subsection{Detecting Semantic Backdoor Attacks}
\label{sec:composite}
\iffalse
\begin{figure}[b]
\centering
\footnotesize
\includegraphics[width=0.3\textwidth]{figs/composite4_crop.pdf}
\caption{Separation of models in the composite attack experiment by SFD
}
\label{f:compositedata}
\end{figure}
\fi
While early backdoor attacks on image classifiers used noise-like pixel patches/watermarks that are small, recent backdoor attacks showed that models can be trojaned with natural objects and features that may be large and complex. As such, scanners that generate small triggers to determine if a model has backdoor become ineffective. A possible solution is to enlarge the size bound such that the injected large triggers can be generated. However, this entails a lot of false positives as it admits many natural triggers. In this experiment, we show that by enlarging the trigger bound of ABS and using \mbox{{\textsc{Ex-Ray}}}\@{} to prune false positives, we can detect composite backdoors~\cite{lin2020composite}, hidden-trigger backdoors~\cite{saha2020hidden}, and reflection backdoors~\cite{liu2020reflection}.
\smallskip
\noindent
{\bf Detecting Composite Backdoor.}
Composite backdoor uses composition of existing benign features as triggers (see Fig.~\ref{f:compositeexamples} in Section~\ref{sec:intro}).
The experiment is on 5 models trojaned by the attack and 20 clean models on CIFAR10. We set the trigger size to 600 pixels in order to reverse engineer the large trigger features used in the attack. In Table~\ref{t:composite}, our results show that we can achieve 0.84 accuracy (improved from 0.2 by vanilla ABS), reducing the false positives from 20 to 4. It shows the potential of \mbox{{\textsc{Ex-Ray}}}\@{}.
\iffalse
Fig.~\ref{f:compositedata} shows the mask similarity scores of the 25 models by SFD with clean models in blue and trojaned models in red. Observe that there is a good separation of the two.
\fi
Fig.~\ref{f:compositetrigger} shows a natural trigger and an injected composite trigger.
Figures (a) and (b) show a natural trigger (generated by ABS) with the target label dog and a sample from the dog class (in CIAFR10), respectively. Observe that the trigger has a lot of dog features (and hence pruned by \mbox{{\textsc{Ex-Ray}}}\@{}).
Figures (c) and (d) show a composite trigger used during poisoning, which is a combination of car and airplane, and the trigger generated by ABS, respectively.
Figure (e) shows the target label bird.
Observe that the reverse engineered trigger has car features (e.g., wheels). \mbox{{\textsc{Ex-Ray}}}\@{} recognizes it as an injected trigger since it shares very few features with the bird class.
\begin{figure}
\centering
\footnotesize
\subfigure[Natural]{
\begin{minipage}[c]{0.5in}
\center
\includegraphics[width=0.5in]{figs/nin_normal_dog3.png}
\end{minipage}
}
~
\subfigure[T: Dog]{
\begin{minipage}[c]{0.5in}
\center
\includegraphics[width=0.5in]{figs/dog4.png}
\end{minipage}
}
~
\subfigure[Composite]{
\begin{minipage}[c]{0.6in}
\center
\includegraphics[width=0.5in]{figs/composite_train418.png}
\end{minipage}
}
~
\subfigure[Injected]{
\begin{minipage}[c]{0.5in}
\center
\includegraphics[width=0.5in]{figs/nin_trojan_composite_5_0_1_2_mix_2_model_conv2d_8_133_600_2_1_3.png}
\end{minipage}
}
~
\subfigure[T: Bird]{
\begin{minipage}[c]{0.5in}
\center
\includegraphics[width=0.5in]{figs/bird5.png}
\end{minipage}
}
\caption{Example of natural and injected composite trigger
}
\label{f:compositetrigger}
\end{figure}
\begin{table}[]
\caption{\mbox{{\textsc{Ex-Ray}}}\@{} on composite attack, with 20 clean models and 5 trojaned models}
\label{t:composite}
\centering
\footnotesize
\begin{tabular}{crrrrr}
\toprule
& \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{FN} & \multicolumn{1}{c}{TN} & \multicolumn{1}{c}{Acc} \\
\midrule
ABS& 5 & 20 & 0 & 0 & 0.2 \\
ABS+\mbox{{\textsc{Ex-Ray}}}\@{} & 5 & 4 & 0 & 16 & 0.84 \\
\bottomrule
\end{tabular}
\end{table}
\begin{figure}
\centering
\footnotesize
\subfigure[Trigger]{
\begin{minipage}[c]{0.6in}
\center
\includegraphics[width=0.5in]{figs/reflect_trigger.png}
\end{minipage}
}
~
\subfigure[Reflection]{
\begin{minipage}[c]{0.6in}
\center
\includegraphics[width=0.5in]{figs/reflect_inject.png}
\end{minipage}
}
~
\\
\subfigure[Natural]{
\begin{minipage}[c]{0.5in}
\center
\includegraphics[width=0.5in]{figs/reflect_natrual_trigger.png}
\end{minipage}
}
~
\subfigure[T: Deer]{
\begin{minipage}[c]{0.5in}
\center
\includegraphics[width=0.5in]{figs/reflect_deer.png}
\end{minipage}
}
~
\subfigure[Injected]{
\begin{minipage}[c]{0.5in}
\center
\includegraphics[width=0.5in]{figs/reflect_re_trigger.png}
\end{minipage}
}
~
\subfigure[T: Plane]{
\begin{minipage}[c]{0.5in}
\center
\includegraphics[width=0.5in]{figs/reflect_airplane.jpg}
\end{minipage}
}
\caption{Example of natural and injected reflection trigger
}
\label{f:reflection}
\end{figure}
\begin{table}
\caption{\mbox{{\textsc{Ex-Ray}}}\@{} on reflection attack, with 20 clean models and 5 trojaned models}
\label{t:reflection}
\centering
\footnotesize
\begin{tabular}{crrrrr}
\toprule
& \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{FN} & \multicolumn{1}{c}{TN} & \multicolumn{1}{c}{Acc} \\
\midrule
ABS & 5 & 17 & 0 & 3 & 0.32 \\
ABS+\mbox{{\textsc{Ex-Ray}}}\@{} & 5 & 3 & 0 & 17 & 0.88 \\
\bottomrule
\end{tabular}
\vspace{-.2in}
\end{table}
\smallskip
\noindent
{\bf Detecting Reflection Backdoor.}
Reflection may occur when taking picture behind a glass window. Reflection backdoors uses the reflection of an image as the trojan trigger. Figure~\ref{f:reflection} (a) and (b) shows an image of triangle sign and its reflection on an image of airplane. Reflection attack uses the reflection of a whole image as the trigger, which is large and complex.
We evaluate ABS+\mbox{{\textsc{Ex-Ray}}}\@{} on 5 models trojaned with reflection backdoors and 20 clean models on CIFAR10. The trigger size bound is set to 256 (very large for CIFAR images). Table~\ref{t:reflection} shows that we can achieve 0.88 accuracy (improved from 0.32 by ABS), reducing the false positives from 17 to 3.
Figure~\ref{f:reflection} (c) and (d) show a natural trigger (generated by ABS) with the target label deer and a sample from the deer class. Observe that the trigger resembles deer antlers.
Figure~\ref{f:reflection} (e) and (f) show the trigger generated by ABS with the target class airplane. Observe that the generated trigger has (triangle) features of the real trigger shown in Figure~\ref{f:reflection} (a) and (b). \mbox{{\textsc{Ex-Ray}}}\@{} classifies it as injected as it shares few features with airplane (Figure~\ref{f:reflection} (f)).
\smallskip
\noindent
{\bf Detecting Hidden-trigger Backdoor.}
Hidden-trigger attack does not directly use trigger to poison training data. Instead, it introduces perturbation on the images of target label such that the perturbed images induce similar inner layer activations to the images of victim label stamped with the trigger. Since the inner layer activations represent features, the model picks up the correlations between trigger features and the target label. Thus images stamped with the trigger are misclassified to the target label at
test time. The attack is a clean label attack.
Since the trigger is not explicit, the attack is more stealthy compared to data poisoning.
On the other hand, the trojaning process is more difficult, demanding larger triggers, causing problems for existing scanners.
For example, it requires the trigger size to be 60$\times$60 for ImageNet to achieve a high attack success rate. Figure~\ref{f:hidden} (a) and (b) show a trigger and an ImageNet sample stamped with the trigger, with the target label terrier dog shown in (f).
We evaluate ABS+\mbox{{\textsc{Ex-Ray}}}\@{} on 6 models trojaned with hidden-triggers and 17 clean models on ImageNet. The trigger size bound for ABS is set to 4000.
Table~\ref{t:hidden} shows that we can achieve 0.82 accuracy (improved from 0.3 by ABS), reducing the number of false positives from 16 to 3.
Figure~\ref{f:hidden} (c) and (d) show a natural trigger generated by ABS and its target label Jeans. Observe that the center part of natural trigger resembles a pair of jeans pants. Since the natural trigger mainly contains features of the target label, it is pruned by \mbox{{\textsc{Ex-Ray}}}\@.
Figure~\ref{f:hidden} (e) shows the generated trigger by ABS for the target label (f). The trigger has few in common with the target label and thus is classified as injected by \mbox{{\textsc{Ex-Ray}}}\@{}.
\begin{table}[t]
\caption{\mbox{{\textsc{Ex-Ray}}}\@{} on hidden-trigger attack, with 17 clean models and 6 trojaned models}
\label{t:hidden}
\centering
\footnotesize
\begin{tabular}{crrrrr}
\toprule
& \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{FN} & \multicolumn{1}{c}{TN} & \multicolumn{1}{c}{Acc} \\
\midrule
ABS & 6 & 16 & 0 & 1 & 0.30 \\
ABS+\mbox{{\textsc{Ex-Ray}}}\@{} & 5 & 3 & 1 & 14 & 0.82 \\
\bottomrule
\end{tabular}
\vspace{-.1in}
\end{table}
\begin{figure}
\centering
\footnotesize
\subfigure[Trigger]{
\begin{minipage}[c]{0.7in}
\center
\includegraphics[width=0.7in]{figs/hidden_trigger.png}
\end{minipage}
}
~
\subfigure[Stamped]{
\begin{minipage}[c]{0.7in}
\center
\includegraphics[width=0.7in]{figs/hidden_stamp2.png}
\end{minipage}
}
\\
~
\subfigure[Natural]{
\begin{minipage}[c]{0.7in}
\center
\includegraphics[width=0.7in]{figs/hidden_natural_1.png}
\end{minipage}
}
~
\subfigure[T: Jeans]{
\begin{minipage}[c]{0.7in}
\center
\includegraphics[width=0.7in]{figs/hidden_jeans2.jpg}
\end{minipage}
}
~
\subfigure[Injected]{
\begin{minipage}[c]{0.7in}
\center
\includegraphics[width=0.7in]{figs/hidden_re.png}
\end{minipage}
}
~
\subfigure[T: terrier]{
\begin{minipage}[c]{0.7in}
\center
\includegraphics[width=0.7in]{figs/hidden_target.png}
\end{minipage}
}
\caption{Example of natural and injected hidden-triggers. The triggers have been enlarged for readability.
}
\label{f:hidden}
\vspace{-.1in}
\end{figure}
\begin{table}[]
\vspace{-.1in}
\caption{Adaptive Attack}
\label{t:adaptive}
\footnotesize
\tabcolsep=5pt
\centering
\begin{tabular}{crrrrr}
\toprule
Weight of adaptive loss & \multicolumn{1}{c}{1} & \multicolumn{1}{c}{10} & \multicolumn{1}{c}{100} & \multicolumn{1}{c}{1000} & \multicolumn{1}{c}{10000} \\
\midrule
Acc & 0.89 & 0.88 & 0.87 & 0.82 & 0.1 \\
Asr & 0.99 & 0.99 & 0.99 & 0.97 & - \\
FP/ \# of clean models & 0 & 0.2 & 0.2 & 0.65 & - \\
TP/ \# of trojaned models & 1 & 1 & 1 & 1 & - \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Adaptive Attack}
\label{sec:adaptive}
\mbox{{\textsc{Ex-Ray}}}\@{} is part of a defense technique and hence vulnerable to adaptive attack.
We devise an adaptive attack that forces the internal activations of victim class inputs embedding the trigger to resemble the activations of the clean target class inputs such that \mbox{{\textsc{Ex-Ray}}}\@{} cannot distinguish the two.
In particular, we train a Network in Network model on CIFAR10 with a given 8$\times$8 patch as the trigger.
In order to force the inner activations of images stamped with the trigger to resemble those of target class images,
we design an adaptive loss which is to minimize the differences between the two.
In particular, we measure the differences of the means and standard deviations of feature maps.
During training, we add the adaptive loss to the normal cross-entropy loss. The effect of adaptive loss is controlled by a weight value, which essentially controls the strength of attack as well.
Besides the adaptively trojaned model, we also train 20 clean models on CIFAR10 to see if ABS+\mbox{{\textsc{Ex-Ray}}}\@{} can distinguish the trojaned and clean models.
The results are shown in Table~\ref{t:adaptive}. The first row shows the adaptive loss weight. A larger weight value indicates stronger attack.
The second row shows the trojaned model's accuracy on clean images. The third row shows the attack success rate of the trojaned model. The fourth row shows the FP rate.
The fifth row shows the TP rate.
Observe while ABS+\mbox{{\textsc{Ex-Ray}}}\@{} does not miss trojaned models, its FP rate grows with the strength of attack. When the weight value is 1000, the FP rate becomes 0.65, meaning \mbox{{\textsc{Ex-Ray}}}\@{} is no longer effective. However, the model accuracy also degrades a lot in this case.
\iffalse
\subsection{Breakdown of submissions on TrojAI Test Datasets (Leaderboard).}
\xz{lets don't put these results}
\begin{table*}[]
\footnotesize
\centering
\caption{SFD submission breakdown}
\label{t:effectiveness}
\setlength{\tabcolsep}{4pt}
\begin{tabular}{ccrrrrrrccrrrrrrccrrrrrr}
\toprule
& \multicolumn{7}{c}{Trojai R2} & & \multicolumn{7}{c}{Trojai R3} & & \multicolumn{7}{c}{Trojai R4} \\
\cmidrule{2-8} \cmidrule{10-16} \cmidrule{18-24}
& \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Polygon trigger\\ (T:276,C:552)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Filter trigger\\ (T:276,C:552)\end{tabular}} & & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Polygon trigger\\ (T:252,C:504)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Filter trigger\\ (T:252,C:504)\end{tabular}} & & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Polygon trigger\\ (T:143,C:504)\end{tabular}} & \multicolumn{1}{c}{} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}Filter trigger\\ (T:361,C:504)\end{tabular}} \\
\cmidrule{2-4} \cmidrule{6-8} \cmidrule{10-12} \cmidrule{14-16} \cmidrule{18-20} \cmidrule{22-24}
& TP & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} & & TP & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} & & TP & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{TP} & \multicolumn{1}{c}{FP} & \multicolumn{1}{c}{Acc} \\
Vanilla ABS & \multicolumn{1}{r}{254} & 218 & 0.71 & & 260 & 293 & 0.626 & \multicolumn{1}{r}{} & \multicolumn{1}{r}{235} & 208 & 0.703 & & 229 & 334 & 0.528 & \multicolumn{1}{r}{} & \multicolumn{1}{r}{137} & 355 & 0.442 & & 331 & 376 & 0.531 \\
SFD & \multicolumn{1}{r}{198} & 19 & 0.883 & & 204 & 32 & 0.874 & \multicolumn{1}{r}{} & \multicolumn{1}{r}{182} & 68 & 0.818 & & 140 & 31 & 0.811 & \multicolumn{1}{r}{} & \multicolumn{1}{r}{105} & 53 & 0.859 & & 224 & 35 & 0.801 \\
SFD+heuristics & \multicolumn{1}{r}{198} & 19 & 0.883 & & 245 & 38 & 0.917 & \multicolumn{1}{r}{} & \multicolumn{1}{r}{205} & 42 & 0.917 & & 228 & 37 & 0.919 & \multicolumn{1}{r}{} & \multicolumn{1}{r}{95} & 0 & 0.926 & & 325 & 44 & 0.907 \\
\bottomrule
\end{tabular}
\end{table*}
\fi
\subsection{Fixing Models with Injected and Natural Backdoors}
\label{sec:repair_short}
As mentioned in Section~\ref{sec:intro},
an important difference between injected and natural backdoors is that the latter is inevitable and difficult to fix.
To make the comparison,
we try to fix 5 benign models and 5 trojaned models on CIFAR10. The trojaned models are trojaned by label-specific data poisoning.
Here we use unlearning~\cite{wang2019neural} which stamps triggers generated by scanning methods on images of victim label to finetune the model and forces the model to unlearn the correlations between the triggers and the target label. The process is iterative, bounded by the level of model accuracy degradation.
The level of repair achieved is measured
by the trigger sizes of the fixed model.
Larger triggers indicate the corresponding backdoors become more difficult to exploit. The trigger size increase rate suggests the difficulty level of repair.
Table~\ref{t:unlearning} shows the average accuracy and average reverse engineered trigger size before and after fixing the models. All models have
the same repair budget.
We can see that natural triggers have a larger accuracy decrease. Natural trigger size only increases by 34.4 whereas injected trigger size increases by 78, supporting our hypothesis.
More detailed results can be found in Appendix~\ref{sec:repair}. Note that model repair is not the focus of the paper and trigger size may not be a good metric to evaluate repair success for the more complex semantic backdoors. The experiment is to provide initial insights. A thorough model repair solution belongs to our future work.
\begin{table}[htbp]
\footnotesize
\centering
\caption{Average trigger size change before and after unlearning}
\label{t:unlearning}
\begin{tabular}{cccccc}
\toprule
& \multicolumn{2}{c}{Natural Trigger} & & \multicolumn{2}{c}{Injected Trigger} \\
\cmidrule{2-3} \cmidrule{5-6}
& Before & After & & Before & After \\
Avg Acc & \multicolumn{1}{r}{88.7\%} & \multicolumn{1}{r}{85.9\%} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{86.4\%} & \multicolumn{1}{r}{85.4\%} \\
Avg Trigger Size & \multicolumn{1}{r}{25.8} & \multicolumn{1}{r}{60.2} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{19} & \multicolumn{1}{r}{97} \\ \bottomrule
\end{tabular}
\end{table}
\section{Introduction}
\label{sec:intro}
Backdoor attack (or trojan attack) to Deep Learning (DL) models injects malicious behaviors (e.g., by data poisoning~\cite{GuLDG19, liu2020reflection, lin2020composite, bagdasaryan2020blind} and neuron hijacking~\cite{trojannn}) such that a compromised model behaves normally for benign inputs and misclassifies to a {\em target label} when a {\em trigger} is present in the input. Depending on the form of triggers, there are
patch backdoor~\cite{GuLDG19} where the trigger is a pixel space patch; watermark backdoor~\cite{trojannn} where the trigger is a watermark spreading over an input image, filter backdoor~\cite{liu2019abs} where the trigger is an Instagram filter, and reflection attack~\cite{liu2020reflection}
that injects semantic trigger through reflection (like through a piece of reflective glass).
More discussion of existing backdoor attacks can be found in a few comprehensive surveys~\cite{li2020backdoor, liu2020survey, gao2020backdoor}.
Backdoors are a prominent threat to DL applications due to the low complexity of launching such attacks, the devastating consequences especially in safety/security critical applications, and the difficulty of defense due to model uninterpretability.
There are a body of existing defense techniques.
Neural Cleanse (NC)~\cite{wang2019neural} and Artificial Brain Stimulation (ABS)~\cite{liu2019abs} make use of optimization to reverse engineer triggers and determine if a model is trojaned. Specifically for a potential target label, they use optimization to
find a small input pattern, i.e., a trigger, that can cause any input to be classified as the target label when stamped.
DeepInspect~\cite{chen2019deepinspect}
uses GAN to reverse engineer trigger.
These techniques leverage the observation that triggers are usually small in order to achieve stealthiness during attack such that a model is considered trojaned if small triggers can be found.
In~\cite{jha2019attribution, erichson2020noise}, it was observed that clean and trojaned models have different behaviors under input perturbations, e.g., trojaned models being more sensitive. Such differences can be leveraged to detect backdoors. More discussion of existing defense techniques can be found in Section~\ref{sec:related_work}.
\iffalse
\begin{figure}
\centering
\footnotesize \iffalse
\subfigure[benign]{
\includegraphics[width=0.1\textwidth]{figs/id-00000004_trigger2_b.png}
}
~
\subfigure[target]{
\includegraphics[width=0.1\textwidth]{figs/id-00000004_trigger2_t.png}
}
~
\subfigure[trigger]{
\includegraphics[width=0.1\textwidth]{figs/id-00000004_trigger2.png}
}
\subfigure[benign]{
\includegraphics[width=0.1\textwidth]{figs/imagenet_base.png}
}
\fi
\subfigure[Input + trigger]{
\begin{minipage}[c]{0.9in}
\center
\includegraphics[width=0.7in]{figs/imagenet_trigger.png}
\end{minipage}
}
\subfigure[Zoomed in trigger]{
\begin{minipage}[c]{1.1in}
\center
\includegraphics[width=0.7in]{figs/imagenet_trigger_enhance.png}
\end{minipage}
}
~
\subfigure[Target]{
\begin{minipage}[c]{0.7in}
\center
\includegraphics[width=0.7in]{figs/imagenet_target_crop.jpg}
\end{minipage}
}
\caption{ImageNet model with natural trigger}
\label{f:naturaltrigger}
\end{figure}
\fi
\begin{figure}
\centering
\footnotesize
\subfigure[Input + trigger]{
\begin{minipage}[c]{1in}
\center
\includegraphics[width=01in]{figs/imagenet_bird_stamped4.png}
\end{minipage}
}
~
\subfigure[Zoomed in trigger]{
\begin{minipage}[c]{1in}
\center
\includegraphics[width=1in]{figs/imagenet_bird_trigger.png}
\end{minipage}
}
~
\subfigure[Target]{
\begin{minipage}[c]{1in}
\center
\includegraphics[width=1in]{figs/n01534433_34162_croped.jpeg}
\end{minipage}
}
\caption{ImageNet model with natural trigger}
\label{f:naturaltrigger}
\end{figure}
Although existing techniques have demonstrated effectiveness in various scenarios, an open problem is to
distinguish natural and benign features that can act as backdoors, called {\em natural backdoors} in this paper, from {\em injected backdoors}. As demonstrated in~\cite{liu2019abs}, natural backdoors may exist in clean models. They are usually activated using strong natural features of the target label as triggers, called {\em natural triggers} in this paper.
Fig.~\ref{f:naturaltrigger} presents a clean ImageNet model downloaded from~\cite{Keras:online}
with a natural trigger. Figure (a) presents a clean sample of hummingbird stamped with a small 25$\times$25 natural trigger at the top-left corner; (b) shows a zoom-in view of the trigger; and (c) a sample in the target class (i.e., junco bird).
Observe that the trigger demonstrates natural features of the target (e.g., eyes and beak of junco).
The trigger can induce misclassification in 78\%
of the hummingbird samples.
{\em In other words, natural backdoors are due to natural differences between classes and hence inevitable}.
For instance, replacing 80\% area of any clean input with a sample of class $T$ very likely causes the model to predict $T$.
The 80\% replaced area
can be considered a natural trigger.
Natural backdoors share similar
characteristics as injected backdoors, rendering the problem of distinguishing them very challenging.
For instance, both NC and ABS rely on the assumption that triggers of injected backdoors are very small. However, natural triggers could be small (as shown by the above example) and injected triggers could be large.
For example in semantic data poisoning~\cite{bagdasaryan2020blind}, reflection attack~\cite{liu2020reflection}, hidden-trigger attack~\cite{saha2020hidden} and composite attack~\cite{lin2020composite}, injected triggers can be as large as any natural object.
In the round 2 of TrojAI backdoor scanning competition\footnote{TrojAI is a multi-year and multi-round backdoor scanning competition organized by IARPA~\cite{TrojAI:online}. In each round, a large number of models of different structures are trojaned with various kinds of triggers, and mixed with clean models. Performers are supposed to identify the trojaned models. Most aforementioned techniques are being used or have been used in the competition.}, many models (up to 40\% of the 552 clean models)
have natural backdoors that are hardly distinguishable from injected backdoors, causing a large number of false positives for performers.
Examples will be discussed in Section~\ref{s:motivation}.
\begin{figure*}[htpb]
\centering
\subfigure[Victim + \newline injected\_trigger]{
\begin{minipage}[t]{1in}
\center
\includegraphics[width=1in]{figs/id-00000007_trigger_gt.png}
\end{minipage}
}
~
\subfigure[Target]{
\begin{minipage}[t]{1in}
\center
\includegraphics[width=1in]{figs/id-00000007_target.png}
\end{minipage}
}
~
\subfigure[Victim+ \newline trigger\_by\_ABS]{
\begin{minipage}[t]{1in}
\center
\includegraphics[width=1in]{figs/id-00000007_trigger.png}
\end{minipage}
}
~
\subfigure[Victim]{
\begin{minipage}[t]{1in}
\center
\includegraphics[width=1in]{figs/id-123-b2.png}
\end{minipage}
}
~
\subfigure[Target]{
\begin{minipage}[t]{1in}
\center
\includegraphics[width=1in]{figs/id-123-t2.png}
\end{minipage}
}
~
\subfigure[Natural trigger by ABS]{
\begin{minipage}[b]{1in}
\center
\fbox{
\includegraphics[width=0.95in]{figs/id-00000123_model_6_539_900_4_6.png}
}
\end{minipage}
}
\caption{
Trojaned model \#7 from TrojAI round 2 in (a)-(c) and clean model \#123 in (d)-(f)
}
\label{f:triggerexamples}
\end{figure*}
Distinguishing natural backdoors from injected ones is critical due to the following reasons. (1) It avoids
false warnings, which may be in a large number due to the prevalence of natural triggers. For example, any models that separate the aforementioned two kinds of birds may have natural backdoors between the two classes due to their similarity. That is, stamping any hummingbird image with junco's beak somewhere may cause the image to be classified as junco, and vice versa. However, the models and the model creators should not be blamed for these inevitable natural backdoors. A low false warning rate is very important for traditional virus scanners. We argue that DL backdoor scanners should have the same goal.
(2) Being correctly informed
about the presence of backdoors and their nature (natural or injected), model end users can employ proper
counter measures. For example,
since natural backdoors are inevitable, the users can use these models with discretion and have tolerance mechanism in place. In contrast, models with injected backdoors are just malicious and should not be used. We argue that in the future, when pre-trained models are published, some quality metrics about similarity between classes and hence natural backdoors should be released as part of the model specification to properly inform end users.
(3) Since model training is very costly, users may want to fix problematic models instead of throwing them away. Correctly distinguishing
injected and natural backdoors provides appropriate guidance in fixing.
Note that the former denotes
out-of-distribution behaviors, which can be neutralized using in-distribution examples and/or adversarial examples. In contrast, the latter may denote in-distribution ambiguity (like cats and dogs) that can hardly be removed, or dataset biases.
For example, natural backdoors in between two classes that are not similar (in human eyes) denote that the dataset may have excessive presence of the corresponding natural features. The problem can be mitigated by improving datasets.
As we will show in Section~\ref{sec:repair_short}, injected backdoors are easier to mitigate than natural backdoors.
We develop a novel technique to
distinguish natural and injected backdoors. We use the following threat model.
\smallskip
\noindent
{\bf Threat Model.} Given a set of models, including both trojaned and clean models, and a small set of clean samples for each model (covering all labels), we aim to identify the models with injected backdoor(s) that can flip samples of a particular class, called the {\em victim class}, to the {\em target class}. The threat model is more general than the typical {\em universal attack} model in the literature~\cite{wang2019neural, liu2019abs, kolouri2020universal}, in which backdoors can induce misclassification for inputs of any class.
We assume the models can be trojaned via different methods such as pixel space, feature space data poisoning (e.g., using Instagram filters as in TrojAI), composite attack, reflection attack, and hidden-trigger attack.
According to the definition, the solution ought to prune out natural backdoor(s). $\Box$
We consider backdoors, regardless of natural or injected, denote differences between classes. Our overarching idea is hence to first derive a comprehensive set of natural feature differences between any pair of classes using the provided clean samples; then when a trigger is found between two classes (by an existing upstream scanner such as ABS), we compare if the feature differences denoted by the trigger is a subset of the differences between the two classes. If so, the trigger is natural. Specifically,
let $l$ be a layer where features are well abstracted.
Given samples of any class pair, say $V$ and $T$, we aim to identify a set of neurons at $l$ such that {\em (1) if we replace the activations of the $V$ samples at those neurons with the corresponding activations of the $T$ samples, the model will classify these $V$ samples to $T$; (2) if we replace the activations of the $T$ samples at the same set of neurons with the corresponding activations of the $V$ samples, the model will classify the samples to $V$}.
We call the conditions the {\em differential feature symmetry}.
We use optimization to identify the smallest set of neurons having the symmetry. We call it the {\em differential features} or {\em mask} in this paper. Intuitively, the features
in the mask define the differences between the two classes.
Assume some existing backdoor scanning technique is used to generate a set of
triggers. Further assume a trigger $t$ causes
all samples in $V$ to be misclassified to $T$. We then leverage the aforementioned
method to compute the mask between $V$ samples and $V+t$ samples, i.e., the $V$ samples stamped with $t$. Intuitively, this mask denotes the features in the trigger.
The trigger $t$ is considered natural if its mask shares substantial commonality with the mask between $V$ and $T$.
\iffalse
\noindent
{\em Example for Intuition.} In the following, we use an example to explain the intuition. Assume a model extracts three features $a$, $b$, and $c$.
Class $V$ benign samples have $\langle a=0, b=1, c=1\rangle$ and class $T$ benign samples have $\langle a=0, b=0, c=0\rangle$.
Assuming replacing $b$'s value in $V$ samples with that in $T$ samples, or vice versa flips the classification results accordingly. Our analysis decides that \{$b$\} is the feature separating $V$ and $T$.
Assume we inject a backdoor $c=2$ to flip $V$ to $T$. That is, we force all $V$ samples with $\langle a=0, b=1, c=2\rangle$ to be $T$.
Our analysis identifies \{$c$\} is the mask between $V$ and $V$+trigger, which does not overlap with the feature differences between $V$ and $T$. $\Box$
\fi
Our contributions are summarized as follows.
\begin{itemize}
\item We study the characteristics of natural and injected backdoors using TrojAI models and models with
various semantic
backdoors.
\item We propose a novel symmetric feature differencing technique to distinguish the two.
\item We implement a prototype
\mbox{{\textsc{Ex-Ray}}}\@, which stands for (``{\em \underline{\sc Ex}amining Diffe\underline{\sc r}ential Fe\underline{\sc a}ture Symmetr\underline{\sc y}}''). It can be used as an addon to serve multiple upstream backdoor scanners.
\item
Our experiments using ABS+\mbox{{\textsc{Ex-Ray}}}\@{} on TrojAI rounds 2-4 datasets\footnote{Round 1 dataset is excluded due to its simplicity.} (each containing thousands of models),
a few ImageNet models trojaned by data poisoning and hidden-trigger attack,
and CIFAR10 models trojaned by composite attack
and reflection attack,
show that our method is highly effective in reducing false warnings (78-100\% reduction) with the cost of a small increase in false negatives (0-30\%), i.e., injected triggers are undesirably considered as natural.
It can improve multiple upstream scanners' overall accuracy including ABS (by 17-41\%), NC (by 25\%), and
Bottom-up-Top-down backdoor scanner~\cite{sriscanner} (by 2-15\%).
Our method also outperforms other natural backdoor pruning methods that compare L2 distances and leverage attribution/interpretation techniques.
It allows effective detection of composite attack, hidden-trigger attack, and reflection attack that are semantics oriented (i.e., triggers are not noise-like pixel patterns but rather objects and natural features).
\item
On the TrojAI leaderboard, ABS+\mbox{{\textsc{Ex-Ray}}}\@{} achieves top performance in 2 out of the 4 rounds up to the submission day, including the most challenging round 4, with average cross-entropy (CE)
loss around 0.32\footnote{The smaller the better.} and average AUC-ROC\footnote{An accuracy metric used by TrojAI, the larger the better.} around 0.90.
It successfully beat all the round targets (for both the training sets and the test sets remotely evaluated by IARPA), which are a CE loss lower than 0.3465
More can be found in Section~\ref{sec:leaderboard}.
\end{itemize}
\section{Conclusion}
We develop a method to distinguish natural and injected backdoors. It is built on a novel symmetric feature differencing technique that identifies a smallest set of features separating two sets of samples. Our results show that the technique is highly effective and enabled us to achieve
top results on the rounds 2 and 4 leaderboard of the TrojAI competition, and rank the 2nd in round 3.
It also shows potential in handling complex and composite semantic-backdoors.
\bibliographystyle{plain}
\section{Motivation}
\label{s:motivation}
In this section, we use a few cases in TrojAI round 2 to study the characteristics of natural and injected backdoors and explain the challenges in distinguishing the two. We then demonstrate our method.
According to the round 2 leader-board~\cite{TrojAI:online, pastleaderboards}, most performers cannot achieve higher than 0.80 AUC-ROC, suffering from substantial false positives. In this round, TrojAI models make use of 22 different structures such as ResNet152, Wide-ResNet101 and DenseNet201. Each model is trained to classify images of 5-25 classes.
Clean inputs are created by compositing a foreground object, e.g.,
a synthetic traffic sign,
with a random background image from the KITTI dataset~\cite{Geiger2013IJRR}.
An object is usually a shape with some symbol at the center. Half of the models are poisoned by stamping a polygon to foreground objects or applying an Instagram filter.
Random transformations, such as shifting, titling, lighting, blurring, and weather effects, may be applied to improve diversity.
Fig.~\ref{f:triggerexamples} (a) to (c) show model \#7 in round 2, with (a) presenting a victim class sample
stamped with a polygon trigger in purple, (b) a target class sample, (c) the trigger generated by ABS\footnote{Our technique requires an upstream trigger generation technique, such as ABS and NC~\cite{wang2019neural}. ABS~\cite{liu2019abs} works by optimizing a small patch in the input space that consistently flips all the samples in the victim class to the target class. It samples
internal neuron behaviors to determine the possible target labels for search space reduction.
}.
Observe that the trigger is much smaller than the foreground objects and hence ABS can correctly determines there is a backdoor. However, there are foreground object classes similar to each other such that the features separating them could be as small as the trigger.
Figures (d) and (e) show two different foreground object classes in model \#123 (a clean model).
Observe that both objects are blue octagons. The differences lie in the small symbols at the center
of octagons. When scanning this pair of classes to determine if samples in (d) can be flipped to (e) by a trigger, ABS finds a (natural) trigger as shown
in figure (f). Observe
that the trigger has pixel patterns resembling the symbol at the center of (e).
Both ABS and NC report model \#123 as trojaned (and hence a false positive) since they cannot distinguish the natural trigger from injected ones due to their similar sizes.
\begin{figure}
\centering
\vspace{0.2in}
\includegraphics[width=0.30\textwidth]{figs/fp_size6_crop.pdf}
\caption{\# FPs w.r.t trigger size}
\label{f:fp_wrt_size}
\end{figure}
Besides object classes being
too similar to each other, another reason
for false positives in backdoor scanning is
that injected triggers could be large and complex. To detect such backdoors, optimization based techniques have to use a large size-bound in trigger reverse engineering, which unfortunately induces a lot of false positives, as large-sized natural triggers can be easily found in clean models.
According to our analysis, the size of injected triggers in the 276 TrojAI round 2 models trojaned with polygon triggers ranges from 85 to 3200 pixels.
Figure~\ref{f:fp_wrt_size} shows the number of false positives generated by ABS when the maximum trigger (to reverse engineer) varies from 400 to 3200 pixels. Observe that the number of false positives grows substantially with the increase of trigger size. When the size is 3200, ABS may produce over 400 false positives among the 552 clean models.
\begin{figure}
\centering
\footnotesize
\subfigure[Composite example used in poisoning]{
\begin{minipage}[c]{0.9in}
\center
\includegraphics[width=0.9in]{figs/composite_a.png}
\end{minipage}
}
~
\subfigure[Target]{
\begin{minipage}[c]{0.9in}
\center
\includegraphics[width=0.9in]{figs/composit_b.png}
\end{minipage}
}
~
\subfigure[Example triggering the backdoor]{
\begin{minipage}[c]{0.9in}
\center
\includegraphics[width=0.9in]{figs/composite_c.png}
\end{minipage}
}
\caption{Composite attack on Youtube Face
}
\label{f:compositeexamples}
\end{figure}
In addition, there are backdoor attacks that inject triggers as large as regular objects. For example, composite attack~\cite{lin2020composite} injects backdoor by mixing existing benign features from two or more classes. Fig.~\ref{f:compositeexamples} shows a composite
attack on a face recognition model trained on the Youtube Face dataset~\cite{wolf2011face}. Figure (a) shows an image used in the attack, mixing two persons and having the label
set to that of (b). Note that
the trigger is no longer a fixed
pixel pattern, but rather the co-presence of the two persons or their features. Figure (c) shows an input that triggers the backdoor. Observe that it contains the same two persons but with different looks from (a). As shown in~\cite{lin2020composite}, neither ABS nor NC can detect such attack, as using a large trigger setting in reverse-engineering, which is needed for this scenario, produces too many false positives.
Note that although perturbation based scanning techniques~\cite{erichson2020noise, jha2019attribution} do not require optimization, they suffer from the same problem, as indicated by the leaderboard results. This is because if benign classes are similar to each other, their classification results are as sensitive to input perturbations as classes with backdoor, causing false positives in scanning;
if triggers are large, the injected misclassification may not be so sensitive to perturbation, causing false negatives.
\begin{figure}
\centering
\footnotesize
\subfigure[Between $V$ and $T$
]{
\begin{minipage}[c]{0.70in}
\center
\includegraphics[width=0.73in]{figs/id-00000007_model_6_472_900_0_6_2_hm1.png}
\end{minipage}
}
~
\subfigure[Between $V$ and $V$+trigger]{
\begin{minipage}[c]{0.70in}
\center
\includegraphics[width=0.73in]{figs/id-00000007_model_6_472_900_0_6_2_hm2.png}
\end{minipage}
}
~
\iffalse
\subfigure[Intersection between (a) and (b)]{
\begin{minipage}[c]{0.8in}
\center
\includegraphics[width=0.7in]{figs/id-00000007_model_6_472_900_0_6_2_hm3.png}
\end{minipage}
}
\\
\fi
\subfigure[Between $V$ and $T$]{
\begin{minipage}[c]{0.70in}
\center
\includegraphics[width=0.73in]{figs/id-00000123_model_6_539_900_4_6_2_hm1.png}
\end{minipage}
}
~
\subfigure[Between $V$ and $V$+trigger]{
\begin{minipage}[c]{0.70in}
\center
\includegraphics[width=0.73in]{figs/id-00000123_model_6_539_900_4_6_2_hm2.png}
\end{minipage}
}
\iffalse
~
\subfigure[Intersection between (d) and (e)]{
\begin{minipage}[c]{0.75in}
\center
\includegraphics[width=0.75in]{figs/id-00000123_model_6_539_900_4_6_2_hm3.png}
\end{minipage}
}
~
\fi
\caption{Differential feature masks for the trojaned model \#7 in (a)-(b) and the clean model \#123 in (c)-(d); $V$ and $T$ stand for victim and target classes, respectively.}
\label{f:neuronmask}
\end{figure}
\iffalse
\begin{figure}
\centering
\footnotesize
\subfigure[Mask between Victim($V$) and Target($T$)
]{
\begin{minipage}[c]{1.1in}
\center
\includegraphics[width=0.7in]{figs/id-00000007_model_6_472_900_0_6_2_hm1.png}
\end{minipage}
}
~
\subfigure[Mask between $V$ and $V$+trigger]{
\begin{minipage}[c]{0.8in}
\center
\includegraphics[width=0.7in]{figs/id-00000007_model_6_472_900_0_6_2_hm2.png}
\end{minipage}
}
~
\subfigure[Intersection between (a) and (b)]{
\begin{minipage}[c]{0.8in}
\center
\includegraphics[width=0.7in]{figs/id-00000007_model_6_472_900_0_6_2_hm3.png}
\end{minipage}
}
\\
\subfigure[Mask between $V$ and $T$]{
\begin{minipage}[c]{1.1in}
\center
\includegraphics[width=0.7in]{figs/id-00000123_model_6_539_900_4_6_2_hm1.png}
\end{minipage}
}
~
\subfigure[Mask between $V$ and $V$+trigger]{
\begin{minipage}[c]{0.8in}
\center
\includegraphics[width=0.7in]{figs/id-00000123_model_6_539_900_4_6_2_hm2.png}
\end{minipage}
}
~
\subfigure[Intersection between (d) and (e)]{
\begin{minipage}[c]{0.8in}
\center
\includegraphics[width=0.7in]{figs/id-00000123_model_6_539_900_4_6_2_hm3.png}
\end{minipage}
}
~
\caption{Feature difference masks for the trojaned model \#7 in (a)-(c) and the clean model \#123 in (d)-(f)}
\label{f:neuronmask}
\end{figure}
\fi
\smallskip
\noindent
{\bf Our Method.}
Given a trigger generated by some upstream optimization technique that flips samples in victim class $V$ to target class $T$, our technique decides if it is natural by checking if
the trigger is composed of features that distinguish $V$ and $T$. This is achieved by a symmetric feature differencing method. The method identifies a set of features/neurons (called mask) such that copying their values from one class to another flips the classification results.
Fig.~\ref{f:neuronmask} (a) shows the mask in the second last convolutional layer of the TrojAI round 2 trojaned model \#7 (i.e., the model in Fig.~\ref{f:triggerexamples} (a)-(c)). It distinguishes the victim and target classes. Note that {\em a mask is not specific to some input sample, but rather to an pair of classes}.
Each block in the mask denotes a feature map (or a neuron) with
red meaning that the whole feature map needs to be copied (in order to flip classification results); gray meaning that the map is not necessary; and light red meaning that the copied values are mixed with the original values.
As such, copying/mixing the activation values of the red/light-red neurons from the target class samples to the victim class samples can flip all the victim samples to the target class, and vice versa.
Intuitively, it denotes the features distinguishing the victim and target classes.
Fig.~\ref{f:neuronmask} (b) shows the mask that distinguishes the victim class samples and the victim samples stamped with the trigger, denoting the constituent features of the trigger.
Observe that (a) and (b) do not have a lot in common.
Intuitively, the trigger consists of many features that are not part of the distinguishing features of the victim and target classes. In contrast, Fig.~\ref{f:neuronmask} (c) and (d) show the corresponding masks for the clean model \#123 in Fig.~\ref{f:triggerexamples} (d)-(f). Observe that (c) and (d) have a lot in common, indicating the trigger consists of most the features distinguishing the victim and target classes and is hence natural.
\begin{figure}
\centering
\footnotesize
\subfigure[Target]{
\begin{minipage}[c]{0.6in}
\center
\includegraphics[width=0.8in]{figs/id-00000007_5_inner_bo_hm.png}
\end{minipage}
}
~
\subfigure[Victim+trigger (L2: 0.59)]{
\begin{minipage}[c]{0.8in}
\center
\includegraphics[width=0.8in]{figs/id-00000007_5_inner_tb_hm.png}
\end{minipage}
}
~
\subfigure[Target]{
\begin{minipage}[c]{0.6in}
\center
\includegraphics[width=0.9in]{figs/id-00000123_5_inner_bo_hm.png}
\end{minipage}
}
~
\subfigure[Victim+trigger (L2: 0.74)]{
\begin{minipage}[c]{0.8in}
\center
\includegraphics[width=1in]{figs/id-00000123_6_inner_tb_hm.png}
\end{minipage}
}
\caption{ Feature maps of the trojaned model \#7 in (a)-(b) and the clean model \#123 in (c)-(d). Each color block denotes the average value of a feature map after normalization
}
\label{f:featuremap}
\vspace{-0.2in}
\end{figure}
Directly comparing activation values does not work. Fig.~\ref{f:featuremap} (a) and (b) show the average feature maps at the second last convolutional layer for the target class samples and the victim class samples stamped with the trigger, respectively, for the aforementioned trojaned model \#7.
Each block denotes the mean of the normalized activation values in a feature map.
The L2 distance of the two is 0.59 as shown in the caption of (b).
It measures the distance between the stamped samples and the clean target class samples. Ideally, a natural trigger would yield a small L2 distance as it possesses the target class features.
Figures (c) and (d) show the corresponding information for a clean model, with the L2 distance 0.74 (larger than 0.59). Observe there is not a straightforward separation of the two. This is because such a simple method does not consider what features are critical. Empirical results can
be found in Section~\ref{sec:eval}.
\begin{figure}
\centering
\footnotesize
\subfigure[Target]{
\begin{minipage}[c]{0.7in}
\center
\includegraphics[width=0.8in]{figs/attr_id-00000007_model_6_472_900_0_6_inner_2_bo_hm.png}
\end{minipage}
}
~
\subfigure[Victim+ \newline trigger (L2:0.22)]{
\begin{minipage}[c]{0.7in}
\center
\includegraphics[width=0.8in]{figs/attr_id-00000007_model_6_472_900_0_6_inner_2_tb_hm.png}
\end{minipage}
}
~
\subfigure[Target]{
\begin{minipage}[c]{0.6in}
\center
\includegraphics[width=0.8in]{figs/attr_id-00000123_model_6_539_900_4_6_inner_2_bo_hm.png}
\end{minipage}
}
~
\subfigure[Victim+\newline trigger (L2:0.21)]{
\begin{minipage}[c]{0.8in}
\center
\includegraphics[width=0.8in]{figs/attr_id-00000123_model_6_539_900_4_6_inner_2_tb_hm.png}
\end{minipage}
}
\caption{Important neurons by an attribution technique with (a)-(b) for model \#7 and (c)-(d) for model \#123}
\label{f:attributemap}
\end{figure}
A plausible improvement is to use attribution techniques to identify the important neuron/features and only compare their values.
Fig.~\ref{f:attributemap} (a) and (b) present the 10\% most important neurons in the trojaned model \#7 identified by an attribution technique DeepLift~\cite{shrikumar2016not}, for the target class samples and victim samples with the trigger, respectively.
The L2 distance for the features in the intersection of the two (i.e., the features important in both) is 0.22 as shown in the caption of (b).
Figures (c) and (d) show the information for the clean model \#123 (with L2 distance 0.21). Even with the attribution method, the two are not that separable.
This is because these techniques identify {\em features that are important for a class or a sample} whereas our technique identifies {\em comparative importance}, i.e., {\em features that are important to distinguish two classes}.
The two are quite different as shown by the differences between Fig.~\ref{f:neuronmask} and Fig.~\ref{f:attributemap}.
More results can be found in Section~\ref{sec:eval}.
Applying a symmetric differential analysis similar to ours in the input space does not work well either. This is because
semantic features/objects may appear in different positions of the inputs. Differencing pixels without aligning corresponding features/objects likely yields meaningless results.
\section{Related Work}
\label{sec:related_work}
\noindent
\textbf{Backdoor Attack.}
Data poisoning~\cite{GuLDG19, chen2017targeted} injects backdoor by changing the label of inputs with trigger.
Clean label attack~\cite{shafahi2018poison, zhu2019transferable, turner2019label, zhao2020clean, saha2020hidden} injects backdoor without changing the data label.
Bit flipping attack~\cite{rakin2020tbt,rakin2019bit} proposes to trojan models by flipping weight value bits.
Dynamic backdoor~\cite{salem2020dynamic, nguyen2020input} focuses on crafting different triggers for different inputs and breaks the defense's assumption that trigger is universal.
Ren et al.~\cite{pang2020tale} proposed to combine adversarial example generation and model poisoning to improve the effectiveness of both attacks.
There are also attacks on NLP tasks~\cite{zhang2020trojaning, chen2020badnl, kurita2020weight}, Graph Neural Network~\cite{zhang2020backdoor, xi2020graph}, transfer learning~\cite{rezaei2019target, wang2018great, yao2019latent}, and federated learning~\cite{xie2019dba, wang2020attack, tolpegin2020data,bagdasaryan2020backdoor,fang2020local}.
\mbox{{\textsc{Ex-Ray}}}\@{} is a general primitive that may be of use in defending these attacks.
\noindent
\textbf{Backdoor Attack Defense.}
ULP~\cite{kolouri2020universal} trains a few universal input patterns and a classifier from thousands of benign and trojaned models.
The classifier predicts if a model has backdoor based on activations of the patterns.
Xu et al.~\cite{xu2019detecting} proposed to detect backdoor using a meta classifier trained on a set of trojaned and benign models.
Qiao et al.~\cite{qiao2019defending} proposed to
reverse engineer the distribution of triggers.
Hunag et al.~\cite{huang2020one} found that trojaned and clean models react differently to input perturbations.
Cassandra~\cite{zhang2020cassandra} and TND~\cite{wang2020practical} found that universal adversarial examples behave differently on trojaned and clean models and used this observation to detect backdoor.
TABOR~\cite{guo2019tabor} and NeuronInspect~\cite{huang2019neuroninspect} used an AI explanation technique to detect backdoor.
NNoculation~\cite{veldanda2020nnoculation} used broad spectrum random perturbations and GAN based techniques to reverse engineer trigger.
Besides backdoor detection, there are techniques aiming at removing backdoor.
Fine-prune~\cite{liu2018fine} removed backdoors by pruning out compromised neurons.
Borgnia et al.~\cite{borgnia2020strong} and Zeng et al.~\cite{zeng2020deepsweep} proposed to use data augmentation technique to mitigate backdoor effect.
Wang et al.~\cite{wang2020certifying} proposed to use randomized smoothing to certify robustness against backdoor attacks.
There are techniques that defend backdoor attacks by data sanitization where they prune out poisoned training inputs~\cite{cao2018efficient, jagielski2018manipulating, mozaffari2014systematic, paudice2018label}.
There are also techniques that detect if a given input is stamped with trigger~\cite{ma2019nic, tang2019demon, gao2019strip, chen2018detecting, li2020rethinking, liu2017neural, chou2020sentinet,tran2018spectral, fu2020detecting, chan2019poison, du2019robust, veldanda2020nnoculation}. They target a different problem as they require inputs with embedded triggers.
\mbox{{\textsc{Ex-Ray}}}\@{} is orthogonal to most of these techniques and can serve as a performance booster.
Excellent surveys of backdoor attack and defense can be found at~\cite{li2020backdoor, liu2020survey, gao2020backdoor, li2020deep}.
\noindent{\bf Interpretation/Attribution.}
\mbox{{\textsc{Ex-Ray}}}\@{} is also related to model interpretation and attribution, e.g., those identifying important neurons and features~\cite{sundararajan2017axiomatic,shrikumar2016not,ancona2018towards,bau2017network,ancona2017towards,shrikumar2016not,bach2015pixel,das2020massif,dong2017towards,fong2018net2vec,zhou2018interpretable,bau2018identifying,hohman2019s,ghorbani2019towards,yeh2019fidelity}.
The differences lie in that \mbox{{\textsc{Ex-Ray}}}\@{} focuses on finding distinguishing features of two classes.
Exiting work~\cite{kim2018interpretability} utilizes random examples and training samples of a class (e.g., zebra) to measure the importance of a concept (e.g., `striped') for the class. It however does not find distinguishing internal features.
\section{Introduction}
A paragraph of text goes here. Lots of text. Plenty of interesting
text. Text text text text text text text text text text text text text
text text text text text text text text text text text text text text
text text text text text text text text text text text text text text
text text text text text text text.
More fascinating text. Features galore, plethora of promises.
\section{Footnotes, Verbatim, and Citations}
Footnotes should be places after punctuation characters, without any
spaces between said characters and footnotes, like so.%
\footnote{Remember that USENIX format stopped using endnotes and is
now using regular footnotes.} And some embedded literal code may
look as follows.
\begin{verbatim}
int main(int argc, char *argv[])
{
return 0;
}
\end{verbatim}
Now we're going to cite somebody. Watch for the cite tag. Here it
comes. Arpachi-Dusseau and Arpachi-Dusseau co-authored an excellent OS
book, which is also really funny~\cite{arpachiDusseau18:osbook}, and
Waldspurger got into the SIGOPS hall-of-fame due to his seminal paper
about resource management in the ESX hypervisor~\cite{waldspurger02}.
The tilde character (\~{}) in the tex source means a non-breaking
space. This way, your reference will always be attached to the word
that preceded it, instead of going to the next line.
And the 'cite' package sorts your citations by their numerical order
of the corresponding references at the end of the paper, ridding you
from the need to notice that, e.g, ``Waldspurger'' appears after
``Arpachi-Dusseau'' when sorting references
alphabetically~\cite{waldspurger02,arpachiDusseau18:osbook}.
It'd be nice and thoughtful of you to include a suitable link in each
and every bibtex entry that you use in your submission, to allow
reviewers (and other readers) to easily get to the cited work, as is
done in all entries found in the References section of this document.
Now we're going take a look at Section~\ref{sec:figs}, but not before
observing that refs to sections and citations and such are colored and
clickable in the PDF because of the packages we've included.
\section{Floating Figures and Lists}
\label{sec:figs}
\begin{figure}
\begin{center}
\begin{tikzpicture}
\draw[thin,gray!40] (-2,-2) grid (2,2);
\draw[<->] (-2,0)--(2,0) node[right]{$x$};
\draw[<->] (0,-2)--(0,2) node[above]{$y$};
\draw[line width=2pt,blue,-stealth](0,0)--(1,1)
node[anchor=south west]{$\boldsymbol{u}$};
\draw[line width=2pt,red,-stealth](0,0)--(-1,-1)
node[anchor=north east]{$\boldsymbol{-u}$};
\end{tikzpicture}
\end{center}
\caption{\label{fig:vectors} Text size inside figure should be as big as
caption's text. Text size inside figure should be as big as
caption's text. Text size inside figure should be as big as
caption's text. Text size inside figure should be as big as
caption's text. Text size inside figure should be as big as
caption's text. }
\end{figure}
Here's a typical reference to a floating figure:
Figure~\ref{fig:vectors}. Floats should usually be placed where latex
wants then. Figure\ref{fig:vectors} is centered, and has a caption
that instructs you to make sure that the size of the text within the
figures that you use is as big as (or bigger than) the size of the
text in the caption of the figures. Please do. Really.
In our case, we've explicitly drawn the figure inlined in latex, to
allow this tex file to cleanly compile. But usually, your figures will
reside in some file.pdf, and you'd include them in your document
with, say, \textbackslash{}includegraphics.
Lists are sometimes quite handy. If you want to itemize things, feel
free:
\begin{description}
\item[fread] a function that reads from a \texttt{stream} into the
array \texttt{ptr} at most \texttt{nobj} objects of size
\texttt{size}, returning returns the number of objects read.
\item[Fred] a person's name, e.g., there once was a dude named Fred
who separated usenix.sty from this file to allow for easy
inclusion.
\end{description}
\noindent
The noindent at the start of this paragraph in its tex version makes
it clear that it's a continuation of the preceding paragraph, as
opposed to a new paragraph in its own right.
\subsection{LaTeX-ing Your TeX File}
People often use \texttt{pdflatex} these days for creating pdf-s from
tex files via the shell. And \texttt{bibtex}, of course. Works for us.
\section*{Acknowledgments}
The USENIX latex style is old and very tired, which is why
there's no \textbackslash{}acks command for you to use when
acknowledging. Sorry.
\section*{Availability}
USENIX program committees give extra points to submissions that are
backed by artifacts that are publicly available. If you made your code
or data available, it's worth mentioning this fact in a dedicated
section.
\bibliographystyle{plain}
|
1,116,691,500,746 | arxiv | \section{Introduction}
Magnetophotonic crystals (MPC's) are periodic structures that
contain magnetic materials and have a period comparable to the
wavelength of electromagnetic radiation
\cite{inoue_jap,sakaguchi,lyubchanskii_rev,zvezdin,inoue_rev,chernovtsev}.
The simplest example of such a periodic structure is a multilayer
having one-dimensional (1D) periodicity. The main advantage of MPC's
in contrast to conventional nonmagnetic photonic crystals (PC's) is
their possibility to tune the band edge position in the spectrum of
the electromagnetic radiation by means of an external static
magnetic field. Moreover, the geometric structure of MPC's allows to
obtain strong enhancement in a number of magneto-optical effects.
Among the magneto-optical effects that can be significantly
enhanced in MPC's, two phenomena are of great interest: (i) the Faraday
effect, which denotes rotation of the polarization ellipse of
light as it propagates collinearly with an externally applied static
magnetic field, and (ii) the nonreciprocity effect, which involves a
difference in phase retardation, polarization rotation, and absorption of
forward- vs.~backward-directed waves propagating through the system.
The Faraday effect can be seen as the lifting of degeneracy for the
left (LCP) and right (RCP) circular polarization states, causing the
LCP and RCP components to propagate with different phase velocities
in the magnetic medium. This difference in velocity of propagation
causes the polarization ellipse of the light to rotate as the light
propagates. The effect is linear with respect to the static magnetic
field strength. The enhancement of this rotation in MPC's originates
from localization of light provided by the multiple interference.
\cite{lyubchanskii_rev,inoue_rev} In fact, the total rotation angle
becomes greater in MPC's with microcavity structure where a magnetic
defect is introduced into the periodic system\cite{inoue_jap}.
Optical nonreciprocity refers to different properties of a medium
for electromagnetic waves propagating in opposite directions. It is
well known that the nonreciprocity effects are inherent to magnetic
media and it can be explained from the symmetry viewpoint.
\cite{kotov} Magnetic field, which is an axial vector, has the
symmetry of circular currents set out in a plane perpendicular to
its vector. For a medium placed in magnetic field, the rotation
directions in this perpendicular plane are non-equivalent. Therefore
the optical properties of a magnetic medium are described with a
non-symmetric permittivity tensor, and the equations for propagation
of LCP and RCP waves in the direction of the field have to be
different. Thus, circularly polarized waves of opposite handedness
(or traveling in opposite directions) are characterized by different
phase velocities and/or attenuations in the course of traveling
along the same optical path. Since the transformation of a
forward-propagating wave into a backward-propagating one with the
same handedness is given by the time-reversal operation, the
sensitivity of medium properties to the reversal of wave propagation
direction is commonly viewed as the time-reversal nonreciprocity.
Aside from the asymmetry of the permittivity tensor, the material
nonlinearity can be another source of apparent reciprocity failure.
As an example, a nonreciprocal response appears in a layered medium
in which frequency changing or self-focusing is asymmetrically
located and in which there is also nonuniform
dichroism\cite{potton}. The order in which the nonlinear and
dichroic layers are encountered by incident light will significantly
influence the balance between nonlinear and absorptive effects.
Another possible way to obtain nonreciprocal response is to combine
the nonlinearity in a PC with an asymmetrically arranged defect
(i.e., microcavity) inside it, containing some intensity-dependent
material (e.g. a Kerr-type medium). In this system the strong field
localization inside the defect can be achieved, and the internal
field intensity becomes sufficient to change the optical
characteristics of the microcavity through the Kerr effect. Since
the spatial field distribution is different for the waves incident
on a spatially asymmetric structure from opposite sides,
nonreciprocal response appears\cite{lidorikis,diao}. It is
convenient to call such kind of spatially asymmetric response the
reversible nonreciprocity, since no time-reversal symmetry breaking
takes place here.
It is also important to note that such nonlinear reversible
nonreciprocity is accompanied by optical bistability. Thus, strong
field localization in a defect within a PC alters the
electromagnetic radiation spectrum including the position of the
band edges. This dynamical band edge shift produces optical
bistability which consists in the existence of two stable
transmission or reflection states for the same input intensity; the
typical input-output characteristic of the system contains a
hysteresis loop\cite{grigoriev,smirnov}. In this case the
nonreciprocity manifests itself in the different intensity level of
input light sufficient to achieve bistable switching for the waves
impinging on the system from the opposite sides.
One of the prominent applications of reversible nonreciprocity is
the design of a nonlinear electromagnetic
diode\cite{hou,grigoriev,miroshnichenko,smirnov}. On analogy with an
electronic diode that transmits electric current in only one
direction due to its nonlinear current--voltage characteristics, the
nonlinear optical diode features unidirectional transmission of the
incoming light. By introducing nonlinearity into the MPC, such
unidirectional transmission can be achieved for one circular
polarization while remaining transparent for the polarization of
opposite handedness.
Hence it is of special interest to study the simultaneous effects
of time-reversal nonreciprocity and nonlinear spatial asymmetry on
the optical properties of PC's. In this paper, we consider an MPC
where a nonlinear defect which is placed either symmetrically or
asymmetrically inside the periodic structure. An important feature
of the studied system is the fact that the asymmetric bistable
transmission is accompanied by the polarization
conversion\cite{flytzanis,jonson,lyubchanskii}. The main objective
of our study is focused on achieving the bistability-induced abrupt
switching between two distinct polarization states. This can be
important for thin-film polarization optics devices and
polarization-sensitive integrated optics.
The rest of the paper is organized as follows. In
Section~\ref{sec:theory}, we formulate the problem under study and
introduce its solution based on the transfer matrix method of
multilayer optics. Sections~\ref{sec:res1} and \ref{sec:res2} follow
with the results for a nonlinear defect placed symmetrically and
asymmetrically into an MPC, respectively. Finally,
Section~\ref{sec:conclus} summarizes the paper.
\section{Problem formulation and solution\label{sec:theory}}
We consider a planar multilayer stack of infinite transverse extent
(Fig.~\ref{fig:fig1}). Each unit cell is composed of a bilayer which
consists of magnetic (with constitutive parameters $\varepsilon_1,
\hat \mu_1$) and nonmagnetic (with parameters $\varepsilon_2,
\mu_2$) layers. The magnetic layers are magnetized up to saturation
by an external static magnetic field $\vec M$ directed along the
$z$-axis (Faraday configuration). A defect is created by introducing
into the structure a layer with constitutive parameters
$\varepsilon_d$, $\mu_d$. We assume that this layer is a Kerr
nonlinear dielectric, which permittivity $\varepsilon_d$ linearly
depends on the intensity $|E|^2$ of the electric field. The defect
can be settled either symmetrically or asymmetrically in the middle
of the structure. The parameters $m$ and $n$ describe the number of
bilayers placed before and after the defect layer. In any case
the bilayers are arranged symmetrically with respect to the
defect layer, i.e. the structure begins and ends with layers of
the same type. We suppose that all layers have the same thickness
$D$. The outer half-spaces $z\le 0$ and $z\ge [2(m+n)+1]D$ are
homogeneous, isotropic, and have parameters $\varepsilon_0, \mu_0$.
Assume that the normally incident field is a linearly polarized
plane monochromatic wave of a frequency $\omega$ and an amplitude
$A$. For the sake of definiteness, we also suppose that the vector
$\vec E$ of the incident wave is directed along the $x$-axis.
\begin{figure}[htb]
\centerline{\includegraphics[width=10.0cm]{Fig1.eps}}
\caption{(Color online) Magnetophotonic structure with nonlinear
defect.} \label{fig:fig1}
\end{figure}
As an convenient material for magnetic layers, the family of
impurity-doped yttrium-iron garnet (YIG) Y$_3$Fe$_5$O$_{12}$ films
can be proposed. These magnetic oxides are well studied and widely
used in integrated magneto-optics because they are transparent in
the near infrared region\cite{lyubchanskii_rev,inoue_rev}. As an
example, a few types of multilayered films composed of magnetic
Bi-substituted YIG (Bi:YIG) and dielectric SiO$_2$ or glass FR-5
layers were investigated. \cite{sakaguchi,kotov} The MPC's based on
the other materials are also known. Thus, a new class of
semiconductor-magnetic hybrid nanostructures consist of GaAs with
MnAs nanoclasters (GaAs:MnAs) which are paired with GaAs/AlAs
superlatices is recently investigated experimentally in the range
900-1100~nm \cite{shimizu}. Also in the nonlinear regime the
structure based on the semimagnetic semiconductors such as
Cd$_{1-x}$Mn$_x$Te with the defect being a quantum well with
prescribed spectral characteristics was reported\cite{buss,cubian}.
From these papers it may be deduced that the magnetic materials
manifest their nonlinear properties at the light intensity about
1~GW/cm$^2$. In our present paper we consider the nonlinear defect
which is made of nonmagnetic material due to its greater
availability. As an example, AsGa or InSb can be selected for this
purpose. We prefer such structure configuration because these
materials require much lower intensities of the incident light to
enable the nonlinear effects. From the literature \cite{palik} it
can be deduced that the nonlinear response in the semiconductor
materials can be achieved at the light intensity about 1~kW/cm$^2$.
Although a defect is made of nonmagnetic material, the studied
structure that consists of magnetic layers and such nonlinear defect
exhibits a number of very interesting and unique properties that we
consider.
Our solution is based on the transfer matrix
formalism\cite{berreman} which is used to calculate the field
distribution inside the structure and the reflection and
transmission coefficients of the MPC. In the Faraday configuration,
when external static magnetic field is biased parallel to the
direction of wave propagation ($\vec k\parallel \vec M$), the
magnetic permeability $\hat \mu_1$ is a tensor quantity with nonzero
off-diagonal components:
$$\hat \mu_1=
\begin{pmatrix}
\mu_1^T & i\alpha & 0 \\
-i\alpha & \mu_1^T & 0 \\
0 & 0 & \mu_1^L
\end{pmatrix}.
$$
For the description of electromagnetic waves in this case it is
necessary to use a $4 \times 4$ transfer matrix
formulation\cite{vidil}. Thus, at the first stage, in the linear
case, the equation which defines the coupling of the tangential
field components at the input and output of the structure is written
in the next form\cite{tuz_josab_09a,tuz_josab_11}
\begin{equation}\label{eq:one}
\vec{\Psi} (0)=\mathfrak{M} \vec{\Psi}(\Lambda)=\left\{(
\mathbf{M}_1 \mathbf{M}_2)^m \mathbf{M}_d(\mathbf{M}_2
\mathbf{M}_1)^n\right\}\vec{\Psi}(\Lambda),
\end{equation}
where $\vec{\Psi}=\{ E_x,E_y,H_x,H_y\}^T$ is the vector containing
the tangential field components at the structure input and output;
the upper index $T$ is the matrix transpose operator; $\Lambda$ is
the total length of the structure, $\Lambda=[2(m + n) + 1]D$; $m$
and $n$ are the numbers of periods placed before and after the
defect element; $\mathbf{M}_1$, $\mathbf{M}_2$ and $\mathbf{M}_d$
are the transfer matrices of the rank four of the first, second, and
defect layers, respectively. The elements of the transfer matrices
in (\ref{eq:one}) are determined from the solution of the Cauchy
problem and are given in\cite{vidil}.
As the solution of the linear problem (\ref{eq:one}) is obtained,
the intensity of the reflected and transmitted fields and the
distribution of the field $\vec {E}_{in}(z)$ inside the MPC can be
calculated. Generally, when the defect layer consists of a Kerr
nonlinear dielectric, the permittivity $\varepsilon_d$ is
inhomogeneous, and depends on the intensity of the electric field at
each point of this layer as follows
\begin{equation}
\varepsilon_d(z)=\varepsilon_d^l+\varepsilon_d^{nl}
|E_{in}(z)|^2,~(2mD \le z \le (2m+1)D). \label{eq:two}
\end{equation}
Knowing the field intensity in the defect layer, both the actual
value of permittivity $\varepsilon_d$ and, consequently, the actual
value of transfer-matrix $\mathfrak{M}$ can be calculated. Thus we
deal with an equation on the unknown function of field intensity
distribution inside the defect layer. A magnitude of the incident
field $A$ is an independent parameter of this equation. Since the
parameter $\varepsilon_d^{nl}$ is small and the nonlinear
contribution to $\varepsilon_d$ varies with the longitudinal
distance on the scale of one-half wavelength we provide an approach
which regards $\varepsilon_d$ as independent on $z$ and treats the
dependence of $\varepsilon_d$ on the average intensity of the
electric field $\overline{|E_{in}|^2}$ inside the defect layer.
Quantitative reasoning of this approach is presented
in\cite{tuz_josab_11}. On the basis of this approximation, we
suppose that the permittivity of the medium depends on the average
intensity of the electric field as $\varepsilon_d=\varepsilon_d^l+
\varepsilon_d^{nl} |\overline{E_{in}|^2}$.
As a result, at the second stage, the nonlinear equation related to
the average field intensity distribution in the defect is obtained.
The numerical solution of this equation yields us the final field
distribution in the MPC and the values of the reflection $R$ and
transmission $T$ coefficients, which expressions can be found
in\cite{vidil}.
\section{Symmetric multilayers: Polarization bistability\label{sec:res1}}
Our objective here is to study the main features of optical
response for an MPC with a nonlinear defect placed symmetrically inside
it. For this reason we consider an MPC consisting of two sections
with the same number of bilayers in them ($m=n$). The sections are
located symmetrically on each side of the defect layer. The main
idea of such an arrangement is to obtain a significant field
localization inside the defect layer, which is achieved by
an appropriate choice of the number of periods $m$ and the
material parameters of layers.
The basic optical properties of the studied MPC are inherited from
the characteristics of perfectly periodic structures with
nonmagnetic layers. Recall that all periodic structures with layer
thicknesses comparable to the wavelength possess forbidden frequency
gaps (stopbands or band gaps) as a direct consequence of
Floquet--Bloch theorem.\cite{sakoda} These gaps are determined by
the modulation period and the average refractive index. Propagation
of waves with frequencies in the stopbands of an idealized infinite
structure is completely inhibited, and the band gaps are in this
sense perfect. For finite structures these gaps appear as frequency
regions with low transmittance and high reflectance, located between
high-transmittance passbands. If any distortion (a ``defect'') is
introduced inside a periodic structure, transmission resonances can
appear in the stopbands, with the field strongly localized in the
defect. The existence of such ``localization resonances'' is
explained by the fact that the defect forms a resonant Fabry--Perot
cavity enclosed between two Bragg mirrors.
\begin{figure}[htb]
\centerline{\includegraphics[width=10.0cm]{Fig2.eps}}
\caption{(Color online) Frequency dependences
($\varkappa=D/\lambda$) of the transmission coefficient ($T$) of the
LCP ($-$) and RCP ($+$) waves in the (a) linear (b) nonlinear case
for $m=n=5$, $\varepsilon_1=10$, $\mu_1^T=\mu_1^L=1$, $\alpha=0.05$,
$\varepsilon_2=\mu_2=\mu_d=1$, $\varepsilon_d^l=4$. For the
nonlinear case,
$\tilde\varepsilon_d^{nl}=\varepsilon_d^{nl}I_0=1.5\times 10^{-4}$,
which corresponds to the incident light intensity
$I_0=15\,\text{kW}/\text{cm}^2$ for
$\varepsilon_d^{nl}\simeq1.0\times10^{-5}\,\text{cm}^2/\text{kW}$.
} \label{fig:fig2}
\end{figure}
\begin{figure}[htb]
\centerline{\includegraphics[width=10.0cm]{Fig3.eps}}
\caption{(Color online) Same as Fig.~\ref{fig:fig2} for the
reflection coefficient ($R$).} \label{fig:fig3}
\end{figure}
The main distinctive feature of an MPC in contrast to the
nonmagnetic one is the appearance of circular polarization
eigenstates. Such circular polarization eigenstates are also
inherent in PC's with chiral isotropic
layers\cite{tuz_josab_09a,tuz_josab_11} but in the case of MPC's
they are controlled with an external static magnetic field. Thus the
MPC reacts differently to circularly polarized waves with opposite
handedness, with distinct optical spectra for each of them (see
Figs.~\ref{fig:fig2}a and \ref{fig:fig3}a). This way, in the Faraday
configuration, both the edges of the forbidden bands and the
frequencies of the localized defect modes become different for LCP
vs.~RCP incident wave. As a result, the defect resonances split into
doublets (see Figs.~\ref{fig:fig2}a--\ref{fig:fig3}) known as the
longitudinal Zeeman-like doublets\cite{jonson}. These doublets
originate from lifting of the degeneracy between resonant conditions
for the LCP and RCP waves in the underlying MPC by the external
magnetic field. It can be seen that there are two closely spaced
resonant modes in the stopband, one of which is an RCP eigenmode and
the other is an LCP eigenmode.
In the insets of Figs.~\ref{fig:fig2}--\ref{fig:fig3} the frequency
band where the doublet exists is given on a larger scale. Throughout
the paper we suppose that the working frequency is far from the
frequency of the ferromagnetic resonance of magnetic layers and
their losses are negligibly small. Under this assumption, at the
resonant frequencies, the magnitude of the transmission coefficient
of the corresponding circularly polarized mode reaches unity, and
the structure becomes completely transparent for the LCP wave when
$\varkappa^-\approx0.098$ and for the RCP wave when
$\varkappa^+\approx0.0995$. Obviously, the magnitude of splitting
(the frequency difference between the peaks $\Delta \varkappa =
\varkappa^+ - \varkappa^-$) can be easily tuned by changing the
strength of the external static magnetic field.
Now we consider the case when the MPC contains a Kerr-type nonlinear
defect. It is known that the introduction of such a defect into an
otherwise linear structure can induce bistable behavior in the
system. The nature of this bistability is studied in the theory of
the nonlinear Fabry--Perot resonators quite well.\cite{gibbs} The
resonant frequencies $\varkappa^\pm$ are sensitive to the refractive
index of the material within the cavity. Thus, when the frequency of
the incident wave is tuned near a resonant frequency, the field
localization induces growth of the light intensity inside the
cavity, which, by means of the Kerr effect, eventually alters the
refractive index enough to shift the resonant frequency. When this
shift brings the resonant condition closer to match the frequency of
the incident field, even more energy gets localized in the cavity.
This further enhances the shift of the resonance, creating positive
feedback that leads to formation of a hysteresis loop in the spectra
with respect to the incident field intensity. As a result, for a
fixed input field intensity, the frequency dependences for any
resonant mode have a typical shape of ``bent resonances''. In the
spectra of a nonlinear MPC this bending can be seen for both
resonant modes in the split doublets
(Figs.~\ref{fig:fig2}b--\ref{fig:fig3}b).
\begin{figure}[htb]
\centerline{\includegraphics[width=10.0cm]{Fig4.eps}}
\caption{(Color online) Frequency dependences
($\varkappa=D/\lambda$) of the magnitudes of the co-polarized (co)
and cross-polarized (cr) components of the transmission (a) and
reflection (b) coefficients of linearly polarized waves. The input
intensity $I_0$ in the nonlinear regime is taken to be 5, 10, and 15
$\text{kW}/\text{cm}^2$. Other parameters are as in
Fig.~\ref{fig:fig2}.} \label{fig:fig4}
\end{figure}
Now consider a linearly polarized wave incident on an MPC with
defect. One can represent it as a superposition LCP and RCP waves.
As a result, the corresponding optical spectra will contain both
resonances. This is demonstrated in Fig.~\ref{fig:fig4} for
individual polarization components of reflected and transmitted
light, as measured in typical experiments. Since the whole system
possesses axial symmetry in the considered case of normal incidence
and Faraday configuration, we can only distinguish between
co-polarized (e.g., $ss$ or $pp$, denoted $co$) and cross-polarized
($sp$ or $ps$, denoted $cr$) components. Since LCP and RCP cmponents
are present in a linearly polarized wave in equal proportion, the
magnitudes of the co-polarized and cross-polarized components are
equal to each other at the resonant frequencies,
$|T^{co}|=|T^{cr}|=|R^{co}|=|R^{cr}|=0.5$. These conditions are
satisfied in the both linear and nonlinear regimes. In the nonlinear
case, both localization resonances are bent. The ``angle'' of
bending clearly depends on the intensity of the incident field and
is almost the same for both resonances in the doublet.
\begin{figure}[htb]
\centerline{\includegraphics[width=12.0cm]{Fig5.eps}}
\caption{(Color online) Frequency dependences
($\varkappa=D/\lambda$) of
(a) the elipticity angle $\eta$ and (b) the polarization azimuth $\theta$ of the
transmitted and reflected fields. The incident light is linearly
polarized, and structure parameters are as in Fig.~\ref{fig:fig2}.
The vertical line marks the bistable polarization switching at
$\varkappa_0$.} \label{fig:fig5}
\end{figure}
Due to the above mentioned polarization sensitivity of a
magnetophotonic system, a linearly polarized wave will very likely
undergo a change in its polarization state during reflection or
transmission. This is confirmed in Fig.~\ref{fig:fig5}, which shows
the corresponding frequency dependences of the ellipticity angle
($\eta$) and the polarization azimuth ($\theta$) for the
transmitted (black lines) and reflected (red lines) fields.
According to the definition of the Stokes parameters, we introduce
the ellipticity $\eta$ so that the field is linearly polarized when
$\eta=0$, and $\eta=-\pi/4$ for LCP and $+\pi/4$ for RCP (note that
in the latter cases the preferential azimuthal angle of the
polarization ellipse $\theta$ becomes undefined). In all other cases
($0<|\eta|<\pi/4$), the field is elliptically polarized. In the
considered frequency band and in the linear regime, the transmitted
field experiences the rotation of its polarization ellipse and
sequentially changes between LCP and RCP through elliptical and
linear polarization states (Fig.~\ref{fig:fig4}, solid black lines).
On the contrary, the reflected field is linearly polarized almost in
the whole selected band except the frequencies $\varkappa^-$ and
$\varkappa^+$ where it becomes circularly polarized
(Fig.~\ref{fig:fig5}, solid red lines). Note that at these resonant
frequencies the polarization azimuth
$\theta_{ref}=\theta_{ref}(\varkappa)$ is a discontinuous function.
Such a drastic difference in the polarization states of the
transmitted vs.~reflected fields can be understood from the fact
that the operating frequencies lie in the stopband of the MPC where
an impinging wave is almost completely reflected from the structure.
As the incident field is linearly polarized, so, too, is the
reflected field. Due to the finite size of the structure a small
fraction of the wave's energy still gets transmitted through the
MPC, undergoing a $90^\circ$ rotation of its polarization ellipse
(Fig.~\ref{fig:fig5}b) for $\varkappa^-<\varkappa<\varkappa^+$. At
the resonant frequencies, it is evident that the matching circularly
polarized eigenmode passes through the system while for the
orthogonally polarized eigenmode the transmission is still
forbidden. Therefore, both transmitted and reflected fields become
circularly polarized within the localized modes frequencies. Note
that the reflected field has the same polarization state as the
transmitted field because the reflected wave propagates in the
opposite direction (see Ref.~\onlinecite{kotov} for clarity).
In the nonlinear regime the ellipticity angle and the polarization
azimuth become multivalued functions. Therefore, it is possible to
use multistability to switch not only between different
transmittances and reflectances but also between two (or, generally,
more than two) distinct polarization states in the transmitted
and/or reflected light.
The most intriguing scenario for such switching is expected when a
bent resonance at $\varkappa^+$ spectrally overlaps with the
original location of the other resonance at $\varkappa^-$. This
overlap is possible as the resonances are spectrally close to each
other. For example, let us fix the operating frequency $\varkappa_0$
at $\varkappa^-$. At this frequency the reflected and transmitted
fields ought to be LCP. As the intensity of input field rises, the
other resonance corresponding to $\varkappa^+$ and associated with
RCP undergoes red shift and eventually reaches $\varkappa_0$.
It becomes possible to couple the incident wave with frequency
$\varkappa_0$ with either of the eigenmodes. Since these have
opposite circular polarizations (they are associated with converting
a linearly polarized incident light into LCP and RCP), it can be
expected that switching between these two polarization states can be
achieved.
Indeed, Fig.~\ref{fig:fig5} shows that at a frequency
$\varkappa_0\approx \varkappa^{-}$ the bistable switching occurs
between RCP and near-LCP for the transmitted light and between
linear polarization and RCP for the reflected light. This agrees
with the above explanation and is seen in the behaviour of resonance
bending in the Stokes parameter space (Fig.~\ref{fig:fig5}). For the
reflected light the bending in ellipticity resembles that in the
reflectance (Fig.~\ref{fig:fig4}a). For the transmitted light the
bent resonances occur in the immediate vicinity of $\eta=\pm \pi/4$,
because only circularly polarized waves can fully couple to the MPC
eigenmodes to become transmitted through it.
Finally, note that Fig.~\ref{fig:fig4}b illustrates another
peculiarity of the reflection spectra of the structure under study,
namely, the formation of closed loops, which appear in the
cross-polarized component of the reflected field. In particular, the
closed loop appears in the lower-frequency resonance at
$\varkappa^-$. The physical mechanism of loop formation is the
difference between the values of $T_{co}$ and $R_{co}$ to either
side of the resonance. In the linear regime,
$|T_{co}(\varkappa^{-}-\delta)|<|T_{co}(\varkappa^{-}+\delta)|$
since transmittance between the resonances should be higher that to
the either side of both defects because it is influenced by the
Lorentzian tails of both resonances. Consequently,
\begin{equation}|R_{co}(\varkappa^{-}-\delta)|>|R_{co}(\varkappa^{-}+\delta)|.\label{eq:inequal}\end{equation}
(This inequality can also be influenced by non-symmetric placement
of the resonances in the band gap due to the violation of the
quarter-wave condition in the structures under study.) In the
nonlinear regime, the relation in Eq.~(\ref{eq:inequal}) holds, and
the resonance bending to the direction of lower frequencies will
cause a loop to form.
\section{Asymmetric configuration: Polarization conversion\label{sec:res2}}
Nonlinear multilayer structures with spatial asymmetry, are commonly considered to obtain directional sensitivity or reversible nonreciprocity in nonmagnetic PC's.
As a few examples, random or deterministically
aperiodic media, as well as periodic structures with asymmetrically
positioned defects, were recently reported to have direction-dependent or unidirectional transmission.\cite
{grigoriev,miroshnichenko,smirnov,tuz_josab_11,zhukovsky,bliokh}
The general result is that interaction between nonlinearity and
asymmetry manifests itself in the simultaneous occurrence of bistability (or multistability) and nonreciprocity.
\begin{figure}[htb]
\centerline{\includegraphics[width=10.0cm]{Fig6.eps}}
\caption{(Color online) Frequency dependences
($\varkappa=D/\lambda$) of the transmission ($T$) and reflection
($R$) coefficients of the LCP ($-$) and RCP ($+$) waves of the MPC
with asymmetrically placed ($m\ne n$) nonlinear defect. Here,
$\tilde\varepsilon_d^{nl}=\varepsilon_d^{nl}I_0=1.0\times 10^{-4}$,
i.e., $I_0=10\,\text{kW}/\text{cm}^2$ for
$\varepsilon_d^{nl}\simeq1.0\times10^{-5}\,\text{cm}^2/\text{kW}$.
Other parameters are as in Fig.~\ref{fig:fig2}. Solid and dash lines
correspond to ($m=5$, $n=6$) and ($m=6$, $n=5$) configurations,
respectively.} \label{fig:fig6}
\end{figure}
From a mathematical point of view, this all-optical reversible nonreciprocity
is a result of non-commutativity of matrix multiplication
in Eq.~(\ref{eq:one}) when the transfer-matrix of
the structure is calculated. In partuicular, optical properties
of a 1D periodic structure with a defect strongly depend on the
position of that defect layer inside the sample. Nevertheless, in
the linear regime, specific properties of the transfer matrix that
stem from time-reversal reciprocity of the Maxwell equations ensure
that the transmission through the system remains the same regardless
of whether the field is incident from the left or right side of the
structure.
The situation changes drastically if an optically
sensitive (e.g., Kerr-type nonlinear) material is used for the defect
layer. In this case, due to different field localization patterns within
the defect layer for the waves impinging from the left and
right sides of the structure, the nonlinear response becomes different.
This difference manifests itself in the different angles of bending
of the localization resonances.\cite{smirnov}
\begin{figure}[htb]
\centerline{\includegraphics[width=12.0cm]{Fig7.eps}}
\caption{(Color online) Frequency dependences
($\varkappa=D/\lambda$) of the elipticity angle (a) and the
polarization azimuth (b) of the transmitted and reflected fields of
the MPC with asymmetrically placed ($m\ne n$) nonlinear defect.
Parameters are as in Fig.~\ref{fig:fig6}. The vertical line marks
the bistable polarization switching at $\varkappa_0$. }
\label{fig:fig7}
\end{figure}
Our goal here is to study the simultaneous effect of the spatial
asymmetry and the time-reversal nonreciprocity on the behavior of
the localization resonances in the MPC. We modify the structure from
Section~\ref{sec:res1} to make the number of bilayers in two
subsections before and after the defect element different ($m\ne
n$). We additionally assume that the static magnetic field direction
always coincides with the wave propagation direction. This can be
assumed
without loss of generality because changing the
direction of wave propagation without changing the direction of the
static magnetic field reverses the handedness of the circularly
polarized states (RCP$\rightleftharpoons$LCP). Hence by considering
the response of the original structure characterized by $(m,n)$ and
its mirror-symmetric counterpart $(n,m)$ to LCP and RCP incident
wave solves the problem completely.
\begin{figure}[htb]
\centerline{\includegraphics[width=10.0cm]{Fig8.eps}}
\caption{(Color online) Frequency dependences
($\varkappa=D/\lambda$) of the magnitudes of the co-polarized (co)
and cross-polarized (cr) components of the transmission (a) and
reflection (b) coefficients of the linearly polarized waves of the
MPC with asymmetrically placed ($m\ne n$) nonlinear defect.
Parameters are as in Fig.~\ref{fig:fig6}.} \label{fig:fig8}
\end{figure}
Comparison of the results presented in
Figs.~\ref{fig:fig2}b--\ref{fig:fig3}b and Fig.~\ref{fig:fig6} shows
that adding one bilayer at either side of the MPC drastically
changes the spectra of the structure. These changes are associated
with the already mentioned different field distribution inside the
structure. The stark difference in the angles of the localization
resonance bending results from the all-optical reversible
nonreciprocity.
The accompanying change of the magnitude for the reflection and
transmission coefficients at the bent resonances (so that
$|T^\pm_\text{max}|<1$ and $|R^\pm_\text{min}|>0$) results from a
certain conflict in the design principles for resonant multilayers.
Namely, to increase the structure's sensitivity to the direction of
incidence, one needs to increase its the spatial asymmetry; yet to
increase the maximum transmission at a resonant peak, the structure
should remain close to symmetric\cite{zhukovsky,smirnov}. As a
consequence, at the frequencies of the localization resonances the
transmission is always below unity and the reflected field is always
elliptically rather than circularly polarized. Indeed, as seen in
Fig.~\ref{fig:fig7}, the ellipticity angle $|\eta_{ref}|<\pi/4$ in
the whole selected frequency band. The transmitted field is still
circularly polarized at the localization resonances. The
polarization azimuth $\theta_{ref}=\theta_{ref}(\varkappa)$ is now a
continuous function.
Hence, while the symmetric structure features polarization switching between two
circularly polarized states, the asymmetric one only enables switching between two elliptically polarized states.
However, it can be seen that changing the position of the defect
layer within the structure significantly alters the ratio between
the reflected and transmitted field, and in particular the relations
between co-polarized and cross-polarized components in them
(Fig.~\ref{fig:fig8}). While the magnitudes of the co-polarized and
cross-polarized transmission components remain equal to each other
($|T^{co}|=|T^{cr}|\le0.5$), the the relation between the reflection
components ($|R^{co}|$ and $|R^{cr}|$) varies in a much wider range.
In one structure configuration ($m=5$, $n=6$), the peak magnitudes
of the co-polarized and cross-polarized reflection components are
$|R^{co}|\approx 0.8$ and $|R^{cr}|\approx 0.2$. In the other
configuration ($m=6$, $n=5$) they are opposite: $|R^{co}|\approx
0.2$ and $|R^{cr}|\approx 0.8$. In the latter case there is an
obvious significant polarization transformation in the reflected
field so that a $90^\circ$ polarization rotation of the incident
light can be achieved with good conversion efficiency. This can find
useful application as thin-film tunable polarization-rotating
mirrors. Also, an appropriate choice of the asymmetric structure
configuration, material parameters, layer thicknesses, and magnetic
field strength would achieve switching between two orthogonal linear
polarization states in the reflected field. This can be important in
the design of tunable thin-film polarization splitters and
switchers.
\section{Conclusions\label{sec:conclus}}
In the present paper, we have studied the effects of
bistability, nonreciprocity, and polarization transformation in a
magnetophotonic crystal with a nonlinear defect placed either
symmetrically or asymmetrically inside the structure. The problem is
considered in the Faraday configuration, i.e, the external static
magnetic field is applied in the direction of the structure
periodicity and is collinear with the wave vector of the incident
wave.
The reflection and transmission coefficients of the structures, along with the field distribution inside them, are calculated using the transfer matrix approach. The nonlinear
problem is solved under the assumption that the nonlinear permittivity of the medium inside the defect layer depends on the average intensity of the electric field inside the defect.
In the case of symmetric structure configuration, it is shown that a
bistable response of a nonlinear magnetophotonic system features switching
between two circular polarization states within the localization
resonances (defect modes) for reflected and transmitted fields.
In the case of asymmetric structure configuration, this switching
appears between elliptically polarized states in the reflected
field, and between circularly polarized states in the transmitted field. The asymmetric structure also features strong $90^\circ$ polarization rotation in the reflected field, with a potential for bistable switching between linear polarizations.
From the specific parameters used in our numerical calculations, it is reasonable to conclude
that bistable response and stepwise polarization switching can already be achieved at the incident power densities of 10--100 kW/cm$^2$ with available materials in the considered structure configuration.
|
1,116,691,500,747 | arxiv | \section{Introduction}
Game theory is a well-established branch of applied mathematics
first developed by von Neumann and Morgenstern \cite{Neumann1944}. It
offers a mathematical
model of a situation in which decision-makers interact and helps us to
understand what happens in such situations. Although game theory was
originally developed in the context of economics, it has also been
applied to many other disciplines in social sciences like political science
\cite{Ordeshook1986}, and even to biology \cite{Smith1982}.
Meyer \cite{Meyer1999} and Eisert et al. \cite{Eisert1999} brought the game theory into the physics community
and created a new field, quantum game theory. They both quantized a classical
game and found interesting new properties which the original classical game
does not possess. Nevertheless, their quantized games seem quite different.
$PQ$ penny flipover studied by Meyer is a quantum sequential game, in
which players take turns in performing some operations on a quantum system.
On the other hand, quantum Prisoners' Dilemma studied by Eisert et al.
is a quantum simultaneous game, in which there are $n$ players and a
quantum system which consists of $n$ subsystems, and player $i$
performs an operation only on the $i$-th subsystem.
Since the seminal works of Meyer and Eisert et al., many studies have been
made to quantize classical games and find interesting phenomena
\cite{Eisert2000, Marinatto2000, Benjamin2001, Du2002, Flitney2002, Flitney2004}. Most of the quantum games ever studied are classified
into either quantum simultaneous games or quantum sequential games,
although not much has been done on the latter.
Now that we see that game theory is combined with quantum theory and there
are two types
of quantum games, several questions naturally arise: (a) Are quantum
games truly different from classical games? (b) If so, in what sense are they
different? (c) What is the relationship between quantum simultaneous
games and quantum sequential games?
To answer these questions, it is necessary to examine the whole
structure of game theory including classical games and quantum games,
not a particular phenomenon of a particular game.
A work by Lee and Johnson \cite{Lee2003} is a study along this line.
They developed a formalism of games including classical games and quantum
games. With the formalism they addressed the questions (a) and (b),
concluding that
``playing games quantum mechanically can be more efficient''
and that ``finite classical games consist of a strict subset of finite quantum games''. However, they did not give a precise definition of the phrase
``consist of a strict subset''.
The purpose of this paper is twofold. The first is to present a
foundation and terminology
for discussing relationships between various types of games.
The second is to answer the question (c). Our conclusions are the following:
(1) For any quantum simultaneous game $G$, there exists a
quantum sequential game equivalent to $G$. (2) For any finite quantum
simultaneous game $G$, there exists a finite quantum sequential game
equivalent to $G$. (3) For any finite quantum
sequential game $G$, there exists a finite quantum simultaneous game
equivalent to $G$.
This paper is organized as follows. In Section \ref{framework}, a
framework for describing various types of games and discussing
relationships between them is presented. In Section \ref{varioustypes},
we define within the framework a number of classical and quantum games,
including quantum simultaneous games and quantum sequential games. In
Section \ref{secEquivalence}, we define a notion of `equivalence', which
is one of relationships between two games. Some properties of
equivalence and a sufficient condition for equivalence are also
examined. Section \ref{GameClass} gives the definition of game classes
and some binary relations between game classes. In Section
\ref{MainTheorems}, we prove the three theorems mentioned above.
Finally, in Section \ref{secDiscussion}, we discuss some consequences
of the theorems.
\section{\label{framework}Framework for the theory}
In order to discuss relationships between different types of games, we
need a common framework in which various types of games are
described. As the first step in our analysis, we will construct such a
framework for our theory.
For the construction, a good place to start is to consider what is game
theory. Game theory is the mathematical study of \textit{game
situations} which is characterized by the following three features:
\begin{enumerate}
\item There are two or more decision-makers, or players.
\item Each player develops his/her strategy for pursuing his/her
objectives. On the basis of the strategy, he/she chooses his/her action
from possible alternatives.
\item As a result of all players' actions, some situation is
realized. Whether the situation is preferable or not for one
player depends not only on his/her action, but also on the other
players' actions.
\end{enumerate}
How much the realized situation is preferable for a player is
quantified by a real number called a payoff. Using this term, we can
rephrase the second feature as ``each player develops his/her strategy
to maximize the expectation value of his/her payoff''. The reason why
the expectation value is used to evaluate strategies is that we can
determine the resulting situation only probabilistically in general,
even when all players' strategies are known.
As a mathematical representation of the three features of game
situations, we define a normal form of a game.
\begin{defn}
A normal form of a game is a triplet $(N, \Omega, f)$ whose components
satisfy the following conditions.
\begin{itemize}
\item $N=\{1,2,\dots,n\}$ is a finite set.
\item $\Omega = \Omega_1\times\dots\times\Omega_n$, where
$\Omega_i$ is a nonempty set.
\item $f$ is a function from $\Omega$ to $\mathbf{R}^n$.
\end{itemize}
\end{defn}
Here, $N$ denotes a set of players. $\Omega_i$ is a set of player $i$'s
strategies, which prescribes how he/she acts. The $i$-th element
of $f(\omega_1,\dots,\omega_n)$ is the expectation value of the payoff
for player $i$, when player $j$ adopts a strategy $\omega_j$.
Next, we propose a general definition of games, which works as a
framework for discussing relationship between various kinds of games. We
can regard a game as consisting of some `entities' (like players, cards,
coins, etc.) and a
set of rules under which a game situation occurs. We model the
`entities' in the form of a tuple $T$. Furthermore, we represent the game
situation caused by the `entities' $T$ under a rule $R$ as a normal form
of a game,
and write it as $R(T)$. Using these formulations, we define a game as follows.
\begin{defn}\label{DefGame}
We define a game as a pair $(T,R)$, where $T$ is a tuple, and $R$ is a
rule which determines uniquely a normal form from $T$. When $G=(T,R)$
is a game, we refer to $R(T)$ as the normal form of the game $G$. We
denote the set of all games by \textbf{G}.
\end{defn}
The conception of the above definition will be clearer if
we describe various kinds of games in the form of the pair defined above. This
will be done in the next section.
Thus far, we have implicitly regarded strategies and actions
of \textit{individual} players as elementary components of a game. In
classical game theory, such modeling of
games is referred to as a noncooperative game, in contrast to a
cooperative game in which strategies and actions of \textit{groups} of
players are elementary. However, we will call a pair in Definition
\ref{DefGame} simply a game, because
we will deal with only noncooperative games in this paper.
\section{\label{varioustypes}Various types of games}
In this section, various types of classical games and quantum games are
introduced. First, we confirm
that strategic games, which is a well-established representation of games in classical game theory
(see e.g. \cite{Osborne1994}), can be
described in the
framework of Definition \ref{DefGame}. Then, we define two quantum
games, namely, quantum simultaneous games and quantum sequential
games.
\subsection{Strategic Games}
We can redefine strategic games using the framework of Definition
\ref{DefGame} as follows.
\begin{defn}
A strategic game is a game $(T,R)$ which has the following form.
\begin{enumerate}
\item $T=(N,S,f)$, and each component satisfies the following
condition.
\begin{itemize}
\item $N=\{1,2,\dots,n\}$ is a finite set.
\item $S = S_1\times\dots\times S_n$, where
$S_i$ is a nonempty set.
\item $f: S\mapsto\mathbf{R}^n$ is a function from $S$ to $\mathbf{R}^n$.
\end{itemize}
\item $R(T)=T=(N,S,f)$.
\end{enumerate}
If the set $S_i$ is finite for all $i$, then we call the game $(T,R)$ a
finite strategic game. We denote the set of all strategic games by \textbf{SG},
and the set of all finite strategic games by \textbf{FSG}.
\end{defn}
\begin{defn}
Let $G=((N,S,f),R)$ be a finite strategic game. Then the mixed extension of
$G$ is a game $G^*=((N,S,f),R^*)$, where the rule $R^*$ is described
as follows.
\begin{itemize}
\item $R^*(N,S,f)=(N,Q,F)$, where $Q$ and $F$ are of the following forms.
\item $Q=Q_1\times\dots\times Q_n$, where $Q_i$ is the set of all
probability distribution over $S_i$.
\item $F: Q\mapsto \mathbf{R}^n$ assigns to each $(q_1,\dots,q_n)\in
Q$ the expected value of $f$. That is, the value of $F$ is given by
\begin{align}
&F(q_1,\dots,q_n)\nonumber\\
&\quad = \sum_{s_1\in S_1}\dots\sum_{s_n\in
S_n}\left\{\prod_{i=1}^n q_i(s_i)\right\}f(s_1,\dots,s_n),
\end{align}
where $q_i(s_i)$ is the probability attached to $s_i$.
\end{itemize}
We denote the set of all mixed extensions of finite strategic games by
\textbf{MEFSG}.
\end{defn}
\subsection{Quantum Simultaneous Games}
\begin{figure}
\includegraphics{fig1.eps}
\caption{\label{figsim}The setup of a quantum simultaneous game. }
\end{figure}
Quantum simultaneous games are quantum games in which a quantum system
is used according to a protocol depicted in Fig. \ref{figsim}. In
quantum simultaneous games, there
are $n$ players who can not communicate with each other, and a
referee. The referee prepares a quantum system
in the initial state $\hat{\rho}_{\mathrm{init}}$. The quantum system is
composed of $n$ subsystems, where the Hilbert space for the $i$-th subsystem
is $\mathcal{H}_i$. The referee provides player $i$ with the
$i$-th subsystem. Each player performs some quantum operation on the
provided subsystem. It is determined in advance which operations are
available for each player. After all players finish their operations, they
return the subsystems to the referee. Then the referee performs a POVM
measurement $\{\hat{M}_r\}$ on the total system. If the $r$-th measurement
outcome is obtained, player $i$ receives a payoff $a_r^i$.
Many studies on quantum simultaneous games have been carried out. Early
significant studies include Refs.
\cite{Eisert1999,Eisert2000,Marinatto2000,Benjamin2001,Du2002}.
The protocol of the quantum simultaneous games is formulated in the form
of Definition \ref{DefGame} as below.
\begin{defn}
A quantum simultaneous game is a game $(T,R)$ which has the following
form.
\begin{enumerate}
\item
$T=(N,\mathcal{H},\hat{\rho}_{\mathrm{init}},\Omega,\{\hat{M}_r\},\{\boldsymbol{a}_r\})$,
and each component satisfies the following condition.
\begin{itemize}
\item $N=\{1,2,\dots,n\}$ is a finite set.
\item
$\mathcal{H}=\mathcal{H}_1\otimes\mathcal{H}_2\otimes\dots\otimes\mathcal{H}_n$,
where $\mathcal{H}_i$ is a Hilbert space.
\item $\hat{\rho}_{\mathrm{init}}$ is a density operator on
$\mathcal{H}$.
\item $\Omega=\Omega_1\times\Omega_2\times\dots\times\Omega_n$,
where $\Omega_i$ is a subset of the set of all CPTP
(completely positive trace preserving) maps
on the set of density operators on $\mathcal{H}_i$. In
other words, $\Omega_i$ is a set of quantum operations
available for player $i$.
\item $\{\hat{M}_r\}$ is a POVM on $\mathcal{H}$.
\item $\boldsymbol{a}_r=(a_r^1,a_r^2,\dots,a_r^n)
\in\mathbf{R}^n$. The index $r$ of $\boldsymbol{a}_r$ runs
over the same domain as that of $\hat{M}_r$.
\end{itemize}
\item $R(T)=(N,\Omega,f)$. The value of $f$ is given by
\begin{equation}
f(\mathcal{E}_1,\dots,\mathcal{E}_n) = \sum_r\boldsymbol{a}_r\mathrm{Tr}\left[\hat{M}_r(\mathcal{E}_1\otimes\dots\otimes\mathcal{E}_n)(\hat{\rho}_{\mathrm{init}})\right]
\end{equation}
for all $(\mathcal{E}_1,\dots,\mathcal{E}_n)\in\Omega$.
\end{enumerate}
If $\mathcal{H}_i$ is finite dimensional for all $i$,
then we refer to the game $(T,R)$ as a finite quantum simultaneous
game. We denote the set of all quantum simultaneous games by \textbf{QSim}, and
the set of all finite quantum simultaneous games by \textbf{FQSim}.
\end{defn}
\subsection{Quantum Sequential Games}
\begin{figure}
\includegraphics{fig2.eps}
\caption{\label{figseq}The setup of a quantum sequential game.}
\end{figure}
Quantum sequential games are another type of quantum games, in which a
quantum system is used according to a protocol depicted in
Fig. \ref{figseq}. In quantum
sequential games, there are $n$ players who can not communicate each
other and a referee. The referee prepares a quantum system in the
initial state $\hat{\rho}_{\mathrm{init}}$. The players performs quantum
operations on the quantum system in turn. The order of the turn may be
regular like $1\to 2\to 3\to 1\to 2\to 3\to\cdots$, or may be
irregular like $1\to 3\to 2\to 3\to 1\to 2\to\cdots$, yet it is
determined in advance. After all the $m$ operations are finished, the
referee performs a POVM measurement $\{\hat{M}_r\}$. If the $r$-th measurement
outcome is obtained, then player $i$ receives a payoff $a_r^i$.
Games which belong to quantum sequential games include $PQ$ penny
flipover \cite{Meyer1999}, quantum Monty Hall problem \cite{Flitney2002},
and quantum truel \cite{Flitney2004}.
The protocol of the quantum sequential games is formulated as follows.
\begin{defn}\label{QSeqGame}
A quantum sequential game is a game $(T,R)$ which has the following
form.
\begin{enumerate}
\item
$T=(N,\mathcal{H},\hat{\rho}_{\mathrm{init}},Q,\mu,\{\hat{M}_r\},\{\boldsymbol{a}_r\})$,
and each component satisfies the following condition.
\begin{itemize}
\item $N=\{1,2,\dots,n\}$ is a finite set.
\item $\mathcal{H}$ is a Hilbert space.
\item $\hat{\rho}_{\mathrm{init}}$ is a density operator on
$\mathcal{H}$.
\item $Q=Q_1\times Q_2\times\dots\times Q_m$, where $Q_k$ is a
subset of the set of all CPTP maps on the set of density
operators on $\mathcal{H}$. The total number of operations
is denoted by $m$.
\item $\mu$ is a bijection from $\bigcup_{i=1}^n\{(i,j)|1\le
j\le m_i\}$ to $\{1,\dots,m\}$, where $m_i$'s are natural
numbers satisfying $m_1+\dots + m_n=m$. The meaning of
$\mu$ is that the $j$-th operation for player $i$ is the
$\mu(i,j)$-th operation in total.
\item $\{\hat{M}_r\}$ is a POVM on $\mathcal{H}$.
\item $\boldsymbol{a}_r=(a_r^1,a_r^2,\dots,a_r^n)
\in\mathbf{R}^n$. The index $r$ of $\boldsymbol{a}_r$ runs
over the same domain as that of $\hat{M}_r$.
\end{itemize}
\item $R(T)=(N,\Omega,f)$. The strategy space
$\Omega=\Omega_1\times\dots\times\Omega_n$ is constructed as
\begin{equation}
\Omega_i=Q_{\mu(i,1)}\times Q_{\mu(i,2)}\times\dots\times Q_{\mu(i,m_i)}.
\end{equation}
The value of $f$ is given by
\begin{multline}
f\left((\mathcal{E}_{\mu(1,1)},\dots,\mathcal{E}_{\mu(1,m_1)}),\dots,(\mathcal{E}_{\mu(n,1)},\dots,\mathcal{E}_{\mu(n,m_n)})\right)\\
= \sum_r \boldsymbol{a}_r\mathrm{Tr}\left[\hat{M}_r \mathcal{E}_m\circ\mathcal{E}_{m-1}\circ\dots\circ\mathcal{E}_{1}(\hat{\rho}_{\mathrm{init}})\right]
\end{multline}
for all
\begin{equation*}
\left((\mathcal{E}_{\mu(1,1)},\dots,\mathcal{E}_{\mu(1,m_1)}),\dots,(\mathcal{E}_{\mu(n,1)},\dots,\mathcal{E}_{\mu(n,m_n)})\right)
\end{equation*}
in $\Omega$.
\end{enumerate}
If $\mathcal{H}$ is finite dimensional,
then we refer to the game $(T,R)$ as a finite quantum sequential
game. We denote the set of all quantum sequential games by \textbf{QSeq}, and
the set of all finite quantum sequential games by \textbf{FQSeq}.
\end{defn}
\section{\label{secEquivalence}Equivalence of games}
In this section, we define equivalence between two games. The basic idea
is that two games are equivalent if their normal forms have the same
structure, for the essence of a game is a game situation which is
modeled by a normal form. The difficulty of this idea is that a
strategy set $\Omega_i$ may have some redundancy; that is, two or more
elements in $\Omega_i$ may represent essentially the same strategy. If
this is the case, it does not work well to compare the strategy sets
directly to judge whether two games are equivalent or not. Instead, we
should define a new normal form in which the redundancy in the strategy
set is excluded from the original normal form, and then compare the new
normal forms of the two games.
As the first step to define equivalence between games, we clarify what
it means by ``two elements in $\Omega_i$ represent essentially the same
strategy''.
\begin{defn}
Let $(N, \Omega, f)$ be a normal form of a game. Two strategies
$\omega_i,\omega_i'\in\Omega_i$ for player $i$ are said to be
redundant if
\begin{align}
&f(\omega_1\dots\omega_{i-1},\omega_i,\omega_{i+1}\dots\omega_n)\nonumber\\
&\quad =
f(\omega_1\dots\omega_{i-1},\omega_i',\omega_{i+1}\dots\omega_n)
\end{align}
for all $\omega_1\in\Omega_1, \dots,\omega_{i-1}\in\Omega_{i-1},\
\omega_{i+1}\in\Omega_{i+1},\dots,\omega_{n}\in\Omega_n$. If two
strategies $\omega_i,\omega'_i\in\Omega_i$ are redundant, we write
$\omega_i\sim\omega'_i$.
\end{defn}
We can show that the binary relation $\sim$ is an \textit{equivalence
relation}. Namely, for all elements $\omega$, $\omega'$, and $\omega''$
of $\Omega_i$, the following holds:
\begin{enumerate}
\item $\omega\sim\omega$.
\item If $\omega\sim\omega'$ then $\omega'\sim\omega$.
\item If $\omega\sim\omega'$ and $\omega'\sim\omega''$ then $\omega\sim\omega''$.
\end{enumerate}
Since $\sim$ is an equivalence relation, we can define the quotient set
$\tilde{\Omega}_i$ of a strategy set $\Omega_i$ by $\sim$. The quotient set
$\tilde{\Omega}_i$ is the set of all equivalence classes in
$\Omega_i$. An equivalence class in $\Omega_i$ is a subset of $\Omega_i$
which has the form of $\{\omega\,|\,\omega\in\Omega_i, a\sim\omega \}$,
where $a$ is an element of $\Omega_i$. We denote by $[\omega]$ an
equivalence class in which $\omega$ is included, and we define
$\tilde{\Omega}$ as
$\tilde{\Omega}\equiv\tilde{\Omega}_1\times\dots\times\tilde{\Omega}_n$.
This $\tilde{\Omega}$ is a new strategy set which has no redundancy.
Next, we define a new expected payoff function $\tilde{f}$ which maps
$\tilde{\Omega}$ to $\mathbf{R}^n$ by
\begin{equation}
\tilde{f}([\omega_1],\dots,[\omega_n]) = f(\omega_1,\dots,\omega_n).
\end{equation}
This definition says that for $(C_1,\dots,C_n)\in\tilde{\Omega}$, the
value of $\tilde{f}(C_1,\dots,C_n)$ is determined by taking one element
$\omega_i$ from each $C_i$ and evaluating $f(\omega_1,\dots,\omega_n)$.
$\tilde{f}$ is well-defined. That is to say, the value of
$\tilde{f}(C_1,\dots,C_n)$ is independent of which element in $C_i$ one
would choose. To show this, suppose $(C_1,\dots,C_n)\in\tilde{\Omega}$
and $\alpha_i,\beta_i\in C_i$. Then $\alpha_i\sim\beta_i$ for every $i$, so that
\begin{align}
f(\alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n) &= f(\beta_1,\alpha_2,\alpha_3,\dots,\alpha_n)\\
&= f(\beta_1,\beta_2,\alpha_3,\dots,\alpha_n)\\
&\quad\vdots \nonumber\\
&= f(\beta_1,\beta_2,\beta_3,\dots,\beta_n).
\end{align}
Thus the value of $\tilde{f}(C_1,\dots,C_n)$ is determined uniquely.
Using $\tilde{\Omega}$ and $\tilde{f}$ constructed from the
original normal form $(N,\Omega,f)$, we define the new normal form as follows.
\begin{defn}
Let $(N,\Omega,f)$ be the normal form of a game $G$. We refer to
$(N,\tilde{\Omega},\tilde{f})$ as the reduced normal form of $G$.
\end{defn}
Whether two games are equivalent or not is judged by comparing the
reduced normal forms of these games, as we mentioned earlier.
\begin{defn}\label{equivalent}
Let $(N^{(1)}, \tilde{\Omega}^{(1)}, \tilde{f}^{(1)})$ be the reduced
normal form of a game $G_1$, and let $(N^{(2)},\tilde{\Omega}^{(2)},
\tilde{f}^{(2)})$ be the reduced normal form of a game $G_2$. Then,
$G_1$ is said to be equivalent to $G_2$ if the following holds.
\begin{enumerate}
\item \label{equivcond1}$N^{(1)}=N^{(2)}=\{1,\dots,n\}$.
\item \label{equivcond2}There exists a sequence $(\phi_1,\dots,\phi_n)$ of bijection $\phi_k:
\tilde{\Omega}_k^{(1)}\mapsto\tilde{\Omega}_k^{(2)}$, such that
for all $(C_1,\dots,C_n) \in
\tilde{\Omega}^{(1)}$
\begin{equation}
\tilde{f}^{(1)}(C_1, \dots, C_n)
= \tilde{f}^{(2)}(\phi_1(C_1),\dots,
\phi_n(C_n)). \label{equivcond}
\end{equation}
\end{enumerate}
If $G_1$ is equivalent to $G_2$, we write $G_1\parallel G_2$.
\end{defn}
To give an example of equivalent games, let us consider classical $PQ$
penny flipover \cite{Meyer1999}, in which both player $P$ and player $Q$
are classical players.
In this game, a penny is placed initially heads up in a box. Players
take turns ($Q\to P\to Q$) flipping the penny over or not. Each player
can not know what the opponent did, nor see inside the box. Finally the
box is opened, and $Q$ wins if the penny is heads up. This game can be
formulated as a finite strategic game whose payoff matrix is given in
Table \ref{payoffmatrix}.
\begin{table}
\caption{\label{payoffmatrix}Payoff matrix for $PQ$ penny flipover. $F$
denotes a flipover and $N$ denotes no flipover. The
first entry in the parenthesis denotes $P$'s payoff and the second one
denotes $Q$'s payoff.}
\begin{ruledtabular}
\begin{tabular}{ccccc}
& $Q$: $NN$ & $Q$: $NF$ & $Q$: $FN$ & $Q$: $FF$\\
\hline
$P$: $N$ & $(-1,1)$ & $(1,-1)$ & $(1,-1)$ & $(-1,1)$\\
$P$: $F$ & $(1,-1)$ & $(-1,1)$ & $(-1,1)$ & $(1,-1)$
\end{tabular}
\end{ruledtabular}
\end{table}
Intuitively, $Q$ does not benefit from the second move, so that it does
not matter whether $Q$ can do the second move or not. The notion of
equivalence captures this intuition; the above penny flipover game is
equivalent to a finite strategic game whose payoff matrix is given in
Table \ref{reducedpenny}. It represents another penny flipover game in which
both players act only once. Proof of the equivalence is easy and we omit
it.
\begin{table}
\caption{\label{reducedpenny}Payoff matrix for another $PQ$ penny
flipover in which both players act only once.}
\begin{ruledtabular}
\begin{tabular}{ccc}
& $Q$: $N$ & $Q$: $F$\\
\hline
$P$: $N$ & $(-1,1)$ & $(1,-1)$\\
$P$: $F$ & $(1,-1)$ & $(-1,1)$
\end{tabular}
\end{ruledtabular}
\end{table}
We now return to the general discussion on the notion of equivalence. The following is a basic property of the equivalence between two games.
\begin{lemma}
The binary relation $\parallel$ is an equivalence relation; namely, for
any games $G_1$, $G_2$, and $G_3$, the following holds.
\begin{enumerate}
\item $G_1\parallel G_1$ (reflexivity).
\item If $G_1\parallel G_2$, then $G_2\parallel G_1$ (symmetry).
\item If $G_1\parallel G_2$ and $G_2\parallel G_3$, then $G_1\parallel
G_3$ (transitivity).
\end{enumerate}
\end{lemma}
\begin{proof}
For the proof, let $(N^{(i)},\tilde{\Omega}^{(i)},\tilde{f}^{(i)})$ be
the reduced normal form of $G_i$.
The reflexivity is evident.
Let us prove the symmetry. Assume $G_1\parallel G_2$. Then $N^{(1)}=N^{(2)}$, and there exists a
sequence $(\phi_1,\dots,\phi_n)$ of bijection $\phi_k:
\tilde{\Omega}_k^{(1)}\mapsto\tilde{\Omega}_k^{(2)}$, such that
for all $(C_1,\dots,C_n) \in \tilde{\Omega}^{(1)}$,
\begin{equation}
\tilde{f}^{(1)}(C_1, \dots, C_n)
= \tilde{f}^{(2)}(\phi_1(C_1),\dots,
\phi_n(C_n)). \label{sym1}
\end{equation}
Then, there exists a sequence $(\phi_1^{-1},\dots,\phi_n^{-1})$ of
bijection
$\phi_k^{-1}:\tilde{\Omega}_k^{(2)}\mapsto\tilde{\Omega}_k^{(1)}$, such
that for any $(D_1,\dots,D_n)\in\tilde{\Omega}^{(2)}$,
\begin{equation}
\tilde{f}^{(2)}(D_1,\dots,D_n) = \tilde{f}^{(1)}(\phi_1^{-1}(D_1),\dots,\phi_n^{-1}(D_n)).
\end{equation}
Thus, $G_2\parallel G_1$.
We proceed to the proof of the transitivity. Assume $G_1\parallel G_2$ and
$G_2\parallel G_3$. Then $N^{(1)}=N^{(2)}$ and $N^{(2)}=N^{(3)}$, which
leads to $N^{(1)}=N^{(3)}$. Furthermore, (i) there exists a sequence
$(\phi_1,\dots,\phi_n)$ of bijection $\phi_k:
\tilde{\Omega}_k^{(1)}\mapsto\tilde{\Omega}_k^{(2)}$ such that for all
$(C_1,\dots,C_n) \in \tilde{\Omega}^{(1)}$,
\begin{equation}
\tilde{f}^{(1)}(C_1, \dots, C_n)
= \tilde{f}^{(2)}(\phi_1(C_1),\dots,
\phi_n(C_n)),
\end{equation}
and (ii) there exists a sequence
$(\psi_1,\dots,\psi_n)$ of bijection $\psi_k:
\tilde{\Omega}_k^{(2)}\mapsto\tilde{\Omega}_k^{(3)}$ such that for all
$(D_1,\dots,D_n) \in \tilde{\Omega}^{(2)}$
\begin{equation}
\tilde{f}^{(2)}(D_1, \dots, D_n)
= \tilde{f}^{(3)}(\psi_1(D_1),\dots,\psi_n(D_n)).
\end{equation}
Combining the statements (i) and (ii), we obtain the following statement:
there exists a sequence $(\psi_1\circ\nolinebreak\phi_1,\dots,\psi_n\circ\phi_n)$
of bijection $\psi_k\circ\phi_k:
\tilde{\Omega}_k^{(1)}\mapsto\tilde{\Omega}_k^{(3)}$ such that for all
$(C_1,\dots, C_n)\in\tilde{\Omega}^{(1)}$,
\begin{align}
&\tilde{f}^{(1)}(C_1,\dots,C_n) \nonumber\\
&\quad =
\tilde{f}^{(3)}(\psi_1\circ\phi_1(C_1),\dots,\psi_n\circ\phi_n(C_n)).
\end{align}
Thus we conclude that $G_1\parallel G_3$.
\end{proof}
In some cases, we can find that two games are equivalent by
comparing the normal forms of the games, not the reduced normal
forms. In the
following lemma, sufficient conditions for such cases are presented.
\begin{lemma}\label{lemma}
Let $(N^{(1)}, \Omega^{(1)}, f^{(1)})$ be the
normal form of a game $G_1$, and let $(N^{(2)},\Omega^{(2)},
f^{(2)})$ be the normal form of a game $G_2$. If the following
conditions are satisfied, $G_1$ is equivalent to $G_2$:
\begin{enumerate}
\item \label{lemmacond1} $N^{(1)}=N^{(2)}=\{1,\dots,n\}$.
\item \label{lemmacond2} There exists a sequence $(\psi_1,\dots,\psi_n)$ of bijection $\psi_k:
\Omega_k^{(1)}\mapsto\Omega_k^{(2)}$, such that
for all $(\omega_1,\dots,\omega_n) \in\Omega^{(1)}$,
\begin{equation}
f^{(1)}(\omega_1, \dots, \omega_n)
= f^{(2)}(\psi_1(\omega_1),\dots,
\psi_n(\omega_n)). \label{lemmapsi}
\end{equation}
\end{enumerate}
\end{lemma}
\begin{proof}
We will show that if the above conditions are satisfied, the conditions
in the definition \ref{equivalent} are also satisfied.
From the condition \ref{lemmacond1} in the lemma, the condition
\ref{equivcond1} in the definition \ref{equivalent} is obviously
satisfied.
To show that the condition \ref{equivcond2} in the definition
\ref{equivalent} is also satisfied, we define a map $\phi_i$ from
$\tilde{\Omega}^{(1)}_i$ to the set of all subsets of $\Omega_i^{(2)}$ as
\begin{equation}
\phi_i(C_i) = \{\psi_i(\omega')\,|\,\omega'\in C_i\}.
\end{equation}
We will show that $(\phi_1,\dots,\phi_n)$ is a sequence which satisfies
the condition \ref{equivcond2} in the definition \ref{equivalent}.
First, we show that the range of $\phi_i$ is a subset of
$\tilde{\Omega}^{(2)}_i$; that is, for any
$[\omega_i]\in\tilde{\Omega}^{(1)}_i$ there exists
$\xi_i\in\Omega^{(2)}_i$ such that
$\phi_i([\omega_i]) = [\xi_i]$. In fact, $\psi_i(\omega_i)$
is such a $\xi_i$:
\begin{equation}
\phi_i([\omega_i]) = [\psi_i(\omega_i)]. \label{hoge}
\end{equation}
Below, we will prove $\phi_i([\omega_i])
\subset [\psi_i(\omega_i)]$ first, and then prove
$[\psi_i(\omega_i)]\subset\phi_i([\omega_i])$.
To prove $\phi_i([\omega_i]) \subset [\psi_i(\omega_i)]$, we will show
that an arbitrary element $\sigma_i\in\phi_i([\omega_i])$ satisfies
$\sigma_i\in [\psi_i(\omega_i)]$. For this purpose, it is sufficient to
show that
$\sigma_i\sim\psi_i(\omega_i)$; that is, for an arbitrary
$\sigma_k\in\Omega^{(2)}_k$ $(k\neq i)$
\begin{align}
&f^{(2)}(\sigma_1,\dots,\sigma_{i-1},\sigma_i,\sigma_{i+1},\dots,\sigma_n)\nonumber\\
&\quad =
f^{(2)}(\sigma_1,\dots,\sigma_{i-1},\psi_i(\omega_i),\sigma_{i+1},\dots,\sigma_{n}). \label{fff}
\end{align}
Since $\psi_k$ is a bijection,
there exists $\omega_k\in\Omega_k^{(1)}$ such that
$\psi_k(\omega_k)=\sigma_k$. In addition, because
$\sigma_i\in\phi_i([\omega_i])$, there exists
$\omega'_i\in [\omega_i]$ such that $\sigma_i =
\psi_i(\omega'_i)$. Thus,
\begin{align}
&
f^{(2)}(\sigma_1,\dots,\sigma_{i-1},\sigma_i,\sigma_{i+1},\dots,\sigma_n)
\nonumber\\
&
\begin{aligned}
=
f^{(2)}(\psi_1(\omega_1),\dots,&\psi_{i-1}(\omega_{i-1}),\psi_i(\omega'_i),\\
&\psi_{i+1}(\omega_{i+1}),\dots,\psi_n(\omega_n))
\end{aligned}
\nonumber\\
&=
f^{(1)}(\omega_1,\dots,\omega_{i-1},\omega'_i,\omega_{i+1},\dots,\omega_n). \label{lemmaeq2}
\end{align}
The last equation follows from \eqref{lemmapsi}. Because
$\omega'_i\in [\omega_i]$ and $\omega_i\in [\omega_i]$, it follows that
$\omega'_i \sim \omega_i$. Hence,
\begin{align}
\eqref{lemmaeq2} &=
f^{(1)}(\omega_1,\dots,\omega_{i-1},\omega_i,\omega_{i+1},\dots,\omega_n)\nonumber\\
&
\begin{aligned}
=
f^{(2)}(\psi_1(\omega_1),\dots,&\psi_{i-1}(\omega_{i-1}),\psi_i(\omega_i),\\
&\psi_{i+1}(\omega_{i+1}),\dots,\psi_n(\omega_n))
\end{aligned}
\nonumber\\
&=
f^{(2)}(\sigma_1,\dots,\sigma_{i-1},\psi_i(\omega_i),\sigma_{i+1},\dots,\sigma_{n}),\label{lemmaeq3}
\end{align}
which leads to the conclusion that the equation \eqref{fff} holds for any $\sigma_i\in\phi_i([\omega_i])$.
Conversely, we can show that
$[\psi_i(\omega_i)]\subset\phi_i([\omega_i])$. Let $\sigma_i$ be
an arbitrary element of $[\psi_i(\omega_i)]$. Since $\psi_i$ is a
bijection, there exists $\omega'_i\in\Omega_i^{(1)}$ such that
$\psi_i(\omega'_i)=\sigma_i$. For such $\omega'_i$, it holds that
$\psi_i(\omega'_i)\sim\psi_i(\omega_i)$, because $\psi_i(\omega'_i)\in
[\psi_i(\omega_i)]$. Hence,
\begin{align}
&
f^{(1)}(\omega_1,\dots,\omega_{i-1},\omega'_i,\omega_{i+1},\dots,\omega_n)\nonumber\\
&
\begin{aligned}
=
f^{(2)}(\psi_1(\omega_1),\dots,&\psi_{i-1}(\omega_{i-1}),\psi_i(\omega'_i),\\
&\psi_{i+1}(\omega_{i+1}),\dots,\psi_n(\omega_n))
\end{aligned}
\nonumber\\
&
\begin{aligned}
=
f^{(2)}(\psi_1(\omega_1),\dots,&\psi_{i-1}(\omega_{i-1}),\psi_i(\omega_i),\\
&\psi_{i+1}(\omega_{i+1}),\dots,\psi_n(\omega_n))
\end{aligned}
\nonumber\\
&= f^{(1)}(\omega_1,\dots,\omega_{i-1},\omega_i,\omega_{i+1},\dots,\omega_n),
\end{align}
which indicates that $\omega'_i\sim\omega_i$. Thus,
$\omega'_i\in [\omega_i]$. Therefore, we conclude that if
$\sigma_i\in [\psi_i(\omega_i)]$, then
$\sigma_i=\psi_i(\omega'_i)\in\phi_i([\omega_i])$; that is,
$[\psi_i(\omega_i)]\subset\phi_i([\omega_i])$.
We have shown above that $\phi_i$ is a map from
$\tilde{\Omega}_i^{(1)}$ to $\tilde{\Omega}_i^{(2)}$. The next thing we
have to show is that $\phi_i$ is a bijection from
$\tilde{\Omega}_i^{(1)}$ to $\tilde{\Omega}_i^{(2)}$. We will show the
bijectivity of $\phi_i$ by proving injectivity and surjectivity
separately.
First, we show that $\phi_i$ is injective. Suppose
$[\omega_i],[\omega'_i]\in\tilde{\Omega}_i^{(1)}$ and
$[\omega_i]\neq [\omega'_i]$. Because
$[\omega_i]\neq [\omega'_i]$, it follows that
$\omega_i\nsim\omega'_i$, so that there exists
$(\omega_1,\dots,\omega_{i-1},\omega_{i+1},\dots,\omega_n)\in\Omega_1^{(1)}\times\dots\times\Omega_{i-1}^{(1)}\times\Omega_{i+1}^{(1)}\times\dots\times\Omega_n^{(1)}$
such that
\begin{align}
&f^{(1)}(\omega_1,\dots,\omega_{i-1},\omega_i,\omega_{i+1},\dots,\omega_n)\nonumber\\
&\quad \neq
f^{(1)}(\omega_1,\dots,\omega_{i-1},\omega'_i,\omega_{i+1},\dots,\omega_n).
\end{align}
For such $(\omega_1,\dots,\omega_{i-1},\omega_{i+1},\dots,\omega_n)$,
\begin{align}
&
\begin{aligned}
f^{(2)}(\psi_1(\omega_1),\dots,&\psi_{i-1}(\omega_{i-1}),\psi_i(\omega_i),\\
&\psi_{i+1}(\omega_{i+1}),\dots,\psi_n(\omega_n))
\end{aligned}
\nonumber\\
&=
f^{(1)}(\omega_1,\dots,\omega_{i-1},\omega_i,\omega_{i+1},\dots,\omega_n)\nonumber\\
&\neq
f^{(1)}(\omega_1,\dots,\omega_{i-1},\omega'_i,\omega_{i+1},\dots,\omega_n)\nonumber\\
&
\begin{aligned}
=
f^{(2)}(\psi_1(\omega_1),\dots,&\psi_{i-1}(\omega_{i-1}),\psi_i(\omega'_i),\\
&\psi_{i+1}(\omega_{i+1}),\dots,\psi_n(\omega_n)).
\end{aligned}
\end{align}
This indicates that $\psi_i(\omega_i)\nsim\psi_i(\omega'_i)$. Hence,
$[\psi_i(\omega_i)]\neq [\psi_i(\omega'_i)]$. Thus,
using \eqref{hoge}, we conclude that $\phi_i([\omega_i])\neq\phi_i([\omega'_i])$.
Next, we show that $\phi_i$ is surjective. Let $[\sigma]$ be an
arbitrary element of $\tilde{\Omega}_i^{(2)}$. Define
$\omega\in\Omega_i^{(1)}$ as $\omega\equiv\psi_i^{-1}(\sigma)$. Then,
\begin{equation}
\phi_i([\omega]) = [\psi_i(\omega)] = [\sigma].
\end{equation}
The first equation follows from \eqref{hoge}. Thus, for an arbitrary
$[\sigma]\in \tilde{\Omega}_i^{(2)}$, there exists
$[\omega]\in\tilde{\Omega}_i^{(1)}$ such that
$\phi_i([\omega])=[\sigma]$.
Lastly, we show that $(\phi_1,\dots,\phi_n)$ satisfies
\eqref{equivcond}. For an arbitrary
$([\omega_1],\dots,[\omega_n])\in\tilde{\Omega}^{(1)}$,
\begin{align}
\tilde{f}^{(1)}([\omega_1],\dots,[\omega_n]) &=
f^{(1)}(\omega_1,\dots,\omega_n)\label{leq1}\\
&= f^{(2)}(\psi_1(\omega_1),\dots,\psi_n(\omega_n))\label{leq2}\\
&= \tilde{f}^{(2)}([\psi_1(\omega_1)],\dots,[\psi_n(\omega_n)])\label{leq3}\\
&=
\tilde{f}^{(2)}(\phi_1([\omega_1]),\dots,\phi_n([\omega_n])).\label{leq4}
\end{align}
Equations \eqref{leq1} and \eqref{leq3} follow from the definition of
$\tilde{f}^{(1)}$ and $\tilde{f}^{(2)}$. Equation \eqref{leq2} follows
from \eqref{lemmapsi}. The last equation follows from \eqref{hoge}.
\end{proof}
\section{\label{GameClass}Game Classes}
This short section is devoted to explaining game classes and some binary
relations between game classes. These notions simplify the statements of
our main theorems.
First, we define a game class as a subset of \textbf{G}.
We defined previously \textbf{G}, \textbf{SG}, \textbf{FSG},
\textbf{MEFSG}, \textbf{QSim}, \textbf{FQSim}, \textbf{QSeq}, and
\textbf{FQSeq}. All of these are game classes. Note that \textbf{G}
is itself a game class.
Next, we introduce some symbols. Let \textbf{A} and \textbf{B} be game
classes. If for any game $G\in\mathbf{A}$ there
exists a game $G'\in\mathbf{B}$ such that $G\parallel G'$, then
we write $\mathbf{A} \trianglelefteq \mathbf{B}$. If
$\mathbf{A}\trianglelefteq \mathbf{B}$ and
$\mathbf{B}\trianglelefteq \mathbf{A}$, we write $\mathbf{A}\bowtie\mathbf{B}$. If $\mathbf{A}\trianglelefteq \mathbf{B}$ but
$\mathbf{B}\ntrianglelefteq \mathbf{A}$, we write $\mathbf{A} \vartriangleleft \mathbf{B}$.
Lastly, we prove the following lemma.
\begin{lemma}\label{preorder}
The binary relation $\trianglelefteq$ is a preorder. Namely, for any
game classes $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$, the following
holds.
\begin{enumerate}
\item $\mathbf{A}\trianglelefteq\mathbf{A}$ (reflexivity).
\item If $\mathbf{A}\trianglelefteq\mathbf{B}$ and
$\mathbf{B}\trianglelefteq\mathbf{C}$, then
$\mathbf{A}\trianglelefteq\mathbf{C}$ (transitivity).
\end{enumerate}
\end{lemma}
\begin{proof}
The reflexivity is evident. So we concentrate on proving the
transitivity.
Assume $\mathbf{A}\trianglelefteq\mathbf{B}$ and
$\mathbf{B}\trianglelefteq\mathbf{C}$. Because
$\mathbf{A}\trianglelefteq\mathbf{B}$, for any $G_a\in\mathbf{A}$ there
exists $G_b\in\mathbf{B}$ such that $G_a\parallel G_b$. For such $G_b$,
there exists $G_c\in\mathbf{C}$
such that $G_b\parallel G_c$, since $\mathbf{B}\trianglelefteq\mathbf{C}$. Using the transitivity of the relation
$\parallel$, we conclude that for any $G_a\in\mathbf{A}$ there
exists $G_c\in\mathbf{C}$ such that $G_a\parallel G_c$; that is,
$\mathbf{A}\trianglelefteq\mathbf{C}$.
\end{proof}
\section{\label{MainTheorems}Main Theorems}
In this section, we examine relationships between game classes
\textbf{QSim}, \textbf{QSeq}, \textbf{FQSim}, and \textbf{FQSeq}.
\begin{thm}
$\mathbf{QSim}\trianglelefteq\mathbf{QSeq}$.
\end{thm}
\begin{figure}
\includegraphics{fig3.eps}
\caption{\label{figthm1}A quantum sequential game $G^{\mathrm{seq}}$
which is equivalent to a quantum simultaneous game $G$ depicted in Fig. \ref{figsim}.}
\end{figure}
\begin{proof}
We prove the theorem by constructing a quantum sequential game
$G^{\mathrm{seq}}$ equivalent to a
given quantum simultaneous game $G$. We show the construction procedure
of $G^{\mathrm{seq}}$
first, and then prove the equivalence using Lemma \ref{lemma}.
The idea for the construction of $G^{\mathrm{seq}}$ is that a quantum
simultaneous game depicted in Fig. \ref{figsim} can always be seen as a
quantum sequential game, as indicated in Fig. \ref{figthm1}.
Suppose $G$ is in the following form:
\begin{gather}
G=(T,R),\\
T=(N,\mathcal{H},\hat{\rho}_{\mathrm{init}},\Omega,\{\hat{M}_r\},\{\boldsymbol{a}_r\}),\\
R(T)=(N,\Omega,f).
\end{gather}
Furthermore, suppose $G^{\mathrm{seq}}$ to be constructed is in the
following form:
\begin{gather}
G^{\mathrm{seq}}=(T^{\mathrm{seq}},R^{\mathrm{seq}}),\\
T^{\mathrm{seq}}=(N^{\mathrm{seq}},\mathcal{H}^{\mathrm{seq}},\hat{\rho}^{\mathrm{seq}}_{\mathrm{init}},Q^{\mathrm{seq}},\mu^{\mathrm{seq}},\{\hat{M}_r^{\mathrm{seq}}\},\{\boldsymbol{a}_r^{\mathrm{seq}}\}),\\
R^{\mathrm{seq}}(T^{\mathrm{seq}})=(N^{\mathrm{seq}}, \Omega^{\mathrm{seq}}, f^{\mathrm{seq}}).
\end{gather}
We construct each component of $T^{\mathrm{seq}}$ from $G$ as follows.
\begin{itemize}
\item $N^{\mathrm{seq}}=N=\{1,2,\dots,n\}$.
\item $\mathcal{H}^{\mathrm{seq}} = \mathcal{H} =
\mathcal{H}_1\otimes\mathcal{H}_2\otimes\dots\otimes\mathcal{H}_n$.
\item $\hat{\rho}_{\mathrm{init}}^{\mathrm{seq}}=\hat{\rho}_{\mathrm{init}}$.
\item $Q^{\mathrm{seq}}=Q_1^{\mathrm{seq}}\times
Q_2^{\mathrm{seq}}\times\dots\times Q_n^{\mathrm{seq}}$, where
\begin{equation*}
Q_i^{\mathrm{seq}} =
\{\mathcal{I}\otimes\dots\otimes\mathcal{I}\otimes(\mathcal{E})_i\otimes\mathcal{I}\otimes\dots\otimes\mathcal{I}\,|\,\mathcal{E}\in\Omega_i\}.
\end{equation*}
Here, $\mathcal{I}$ is the identity superoperator.
\item $\mu^{\mathrm{seq}}$ is a map from $\{(i,1)\,|\,1\le i\le n\}$ to
$\{1,2,\dots,n\}$. The value of $\mu^{\mathrm{seq}}$ is defined by
$\mu^{\mathrm{seq}}(i,1)=i$.
\item $\{\hat{M}_r^{\mathrm{seq}}\}=\{\hat{M}_r\}$.
\item $\{\boldsymbol{a}_r^{\mathrm{seq}}\}=\{\boldsymbol{a}_r\}$.
\end{itemize}
Note that $\Omega_i^{\mathrm{seq}}=Q_i$ because of the construction of
$Q^{\mathrm{seq}}$ and $\mu^{\mathrm{seq}}$.
Next, we prove that $G^{\mathrm{seq}}$ constructed above is equivalent
to $G$, using Lemma \ref{lemma}. From $N^{\mathrm{seq}}=N$, condition
\ref{lemmacond1} of the Lemma \ref{lemma} is satisfied. To show that condition
\ref{lemmacond2} of the Lemma \ref{lemma} is also satisfied, we define a map
$\psi_i: \Omega_i\mapsto\Omega_i^{\mathrm{seq}}$ by
\begin{equation}
\psi_i(\mathcal{E})\equiv
\mathcal{I}\otimes\dots\otimes\mathcal{I}\otimes
(\mathcal{E})_i\otimes\mathcal{I}\otimes\dots\otimes\mathcal{I}.
\end{equation}
Then, $\psi_i$ is clearly a bijection. Furthermore, for any
$(\mathcal{E}_1,\dots,\mathcal{E}_n)\in\Omega$,
\begin{equation}
f(\mathcal{E}_1,\dots,\mathcal{E}_n) =
f^{\mathrm{seq}}(\psi_1(\mathcal{E}_1),\dots,\psi_n(\mathcal{E}_n)).
\end{equation}
Thus condition \ref{lemmacond2} of the Lemma \ref{lemma} is satisfied.
\end{proof}
From the above proof, we can easily see that the following theorem is
also true.
\begin{thm}\label{Theorem2}
$\mathbf{FQSim}\trianglelefteq\mathbf{FQSeq}$.
\end{thm}
The converse of Theorem \ref{Theorem2} is the following theorem.
\begin{thm}\label{Theorem3}
$\mathbf{FQSeq}\trianglelefteq\mathbf{FQSim}$.
\end{thm}
\begin{figure}
\includegraphics{fig4.eps}
\caption{\label{figthm3}A finite quantum simultaneous game $G^{\mathrm{sim}}$
which is equivalent to a given finite quantum sequential game $G$.}
\end{figure}
\begin{proof}
We prove the theorem by constructing a finite quantum simultaneous game
$G^{\mathrm{sim}}$ equivalent to a
given finite quantum simultaneous game $G$. We show the construction
procedure of $G^{\mathrm{sim}}$ first, and then prove the equivalence
between $G^{\mathrm{sim}}$ and $G$.
Suppose $G$ is in the following form:
\begin{gather}
G=(T,R),\\
T=(N,\mathcal{H},\hat{\rho}_{\mathrm{init}},Q,\mu,\{\hat{M}_r\}_{r\in \mathcal{R}},\{\boldsymbol{a}_r\}),\\
R(T)=(N,\Omega,f).
\end{gather}
From the above $G$, We construct a finite quantum simultaneous game
$G^{\mathrm{sim}}$ which is in the following form:
\begin{gather}
G^{\mathrm{sim}}=(T^{\mathrm{sim}},R^{\mathrm{sim}}),\\
T^{\mathrm{sim}}=(N^{\mathrm{sim}},\mathcal{H}^{\mathrm{sim}},\hat{\rho}^{\mathrm{sim}}_{\mathrm{init}},\Omega^{\mathrm{sim}},\{\hat{M}_{(i,r)}^{\mathrm{sim}}\},\{\boldsymbol{a}_{(i,r)}^{\mathrm{sim}}\}),\\
R^{\mathrm{sim}}(T^{\mathrm{sim}})=(N^{\mathrm{sim}}, \Omega^{\mathrm{sim}}, f^{\mathrm{sim}}).
\end{gather}
Figure \ref{figthm3} indicates the setting for $G^{\mathrm{sim}}$. The
precise instruction on how to construct each component of
$T^{\mathrm{sim}}$ is given below:
\begin{itemize}
\item $N^{\mathrm{sim}}=N=\{1,2,\dots,n\}$.
\item
$\mathcal{H}^{\mathrm{sim}}=\mathcal{H}^{\mathrm{sim}}_1\otimes\mathcal{H}^{\mathrm{sim}}_2\otimes\dots\otimes\mathcal{H}^{\mathrm{sim}}_n$, where
$\mathcal{H}^{\mathrm{sim}}_i = \mathcal{H}^{\otimes
2m_i}$ and $m_i$ is the number explained in Definition
\ref{QSeqGame}. This construction means that in game
$G^{\mathrm{sim}}$, the referee provides player $i$ with a subsystem
which is itself composed of the $2m_i$ subsystems, each of which
is the same system as the one used in the original quantum
sequential game $G$. We write
the Hilbert space of the $j$-th subsystem of $2m_i$ subsystems
for player $i$ as
$\mathcal{H}^{\mathrm{sim}}_{(i,j)}$. Likewise,
we write a state vector in $\mathcal{H}^{\mathrm{sim}}_{(i,j)}$
as $|\psi\rangle_{(i,j)}$ and a operator on
$\mathcal{H}^{\mathrm{sim}}_{(i,j)}$ as $\hat{A}_{(i,j)}$.
\item Define a map $\nu:
\{1,2,\dots,m\}\mapsto\bigcup_{i=1}^n\{(i,m_i+j)\,|\, 1\le j\le
m_i\}$ by
\begin{equation}
\nu(k)\equiv \mu^{-1}(k)+ (0,m_{i(k)}),
\end{equation}
where $i(k)$ is the first element of $\mu^{-1}(k)$. In addition,
let $\{|1\rangle_{(i,j)},\dots,|d\rangle_{(i,j)}\}$ be an
orthonormal basis of $\mathcal{H}^{\mathrm{sim}}_{(i,j)}$, where
$d$ is the dimension of $\mathcal{H}$. Then,
$\hat{\rho}^{\mathrm{sim}}_{\mathrm{init}}$ is constructed as
\begin{widetext}
\begin{align}
\hat{\rho}^{\mathrm{sim}}_{\mathrm{init}} =
\left(\hat{\rho}_{\mathrm{init}}\right)_{\mu^{-1}(1)}&\otimes
|1\rangle_{\nu(m)}\langle 1|_{\nu(m)} \nonumber\\
&\bigotimes_{a=1}^{m-1}
\left(\frac{1}{\sqrt{d}}\sum_{i_a=1}^d|i_a\rangle_{\nu(a)}|i_a\rangle_{\mu^{-1}(a+1)}\right)
\left(\frac{1}{\sqrt{d}}\sum_{j_a=1}^d\langle
j_a|_{\nu(a)}\langle j_a|_{\mu^{-1}(a+1)}\right).
\end{align}
\end{widetext}
\item
$\Omega^{\mathrm{sim}}=\Omega^{\mathrm{sim}}_1\times\Omega^{\mathrm{sim}}_2\times\dots\times\Omega^{\mathrm{sim}}_{n}$,
where
\begin{multline}
\Omega_i^{\mathrm{sim}}=\{\mathcal{E}_{\mu(i,1)}\otimes\mathcal{E}_{\mu(i,2)}\otimes\dots\otimes\mathcal{E}_{\mu(i,m_i)}\otimes\mathcal{I}\otimes\mathcal{I}\otimes\dots\otimes\mathcal{I}\\
|\,\mathcal{E}_{\mu(i,1)}\in Q_{\mu(i,1)},\dots,
\mathcal{E}_{\mu(i,m_i)}\in Q_{\mu(i,m_i)}\}.
\end{multline}
\item In the game $G^{\mathrm{sim}}$, measurement outcomes are
described by a pair of variables $(i,r)$, where $i$ takes the value of
1 or 2, and $r$ is an element of $\mathcal{R}$ (the index set of
the POVM in the original game $G$). Corresponding POVM elements
are defined by
\begin{align}
\hat{M}^{\mathrm{sim}}_{(1,r)} &= \hat{K}\otimes (\hat{M}_r)_{\mu^{-1}(m)}\otimes
\hat{I}_{\nu(m)},\\
\hat{M}_{(2,r)}^{\mathrm{sim}} &= (\hat{I}-\hat{K})\otimes
(\hat{M}_r)_{\mu^{-1}(m)}\otimes \hat{I}_{\nu(m)}.
\end{align}
Here, $\hat{K}$ is defined by
\begin{widetext}
\begin{equation}
\hat{K}\equiv\bigotimes_{b=1}^{m-1}\left(\frac{1}{\sqrt{d}}\sum_{k_b=1}^d|k_b\rangle_{\mu^{-1}(b)}|k_b\rangle_{\nu(b)}\right)\left(\frac{1}{\sqrt{d}}\sum_{l_b=1}^d\langle l_b|_{\mu^{-1}(b)}\langle l_b|_{\nu(b)}\right).
\end{equation}
\end{widetext}
$\hat{M}_{(1,r)}^{\mathrm{sim}}$ and $\hat{M}_{(2,r)}^{\mathrm{sim}}$ turn out to be positive operators, if we note
that $\hat{K}$ and $\hat{I}-\hat{K}$ are projection operators
and $\hat{M}_r$ is a
positive operator. Furthermore, the completeness condition is
satisfied:
\begin{equation}
\sum_{i=1}^2\sum_{r\in\mathcal{R}} \hat{M}^{\mathrm{sim}}_{(i,r)}=\hat{I}.
\end{equation}
Thus, it is confirmed that
$\{\hat{M}^{\mathrm{sim}}_{(i,r)}\}$ is
a POVM.
\item We set $\boldsymbol{a}^{\mathrm{sim}}_{(i,r)}$ as
\begin{equation}
\boldsymbol{a}_{(i,r)}^{\mathrm{sim}}=\begin{cases}
d^{2m-2}\boldsymbol{a}_r & \text{if } i=1,\\
0 & \text{if } i=2.
\end{cases}
\end{equation}
\end{itemize}
Next, we prove that $G^{\mathrm{sim}}$ constructed above is equivalent
to $G$, using Lemma \ref{lemma}. From $N^{\mathrm{sim}}=N$, condition
\ref{lemmacond1} of the lemma is satisfied. To show that condition
\ref{lemmacond2} of the lemma is also satisfied, we define a map
$\psi_i: \Omega_i^{\mathrm{sim}}\mapsto\Omega_i$ by
\begin{align}
&\psi_i(\mathcal{E}_{\mu(i,1)}\otimes\mathcal{E}_{\mu(i,2)}\otimes\dots\otimes\mathcal{E}_{\mu(i,m_i)}\otimes\mathcal{I}\otimes\dots\otimes\mathcal{I})\nonumber\\
& \quad= (\mathcal{E}_{\mu(i,1)},\mathcal{E}_{\mu(i,2)},\dots,\mathcal{E}_{\mu(i,m_i)}).
\end{align}
Then, $\psi_i$ is a bijection. Furthermore, for any element
\begin{align}
&\left((\mathcal{E}_{\mu(1,1)}\otimes\dots\otimes\mathcal{E}_{\mu(1,m_1)}\otimes\mathcal{I}\otimes\dots\otimes\mathcal{I}),\dots,\right.\nonumber\\
&\qquad\left. (\mathcal{E}_{\mu(n,1)}\otimes\dots\otimes\mathcal{E}_{\mu(n,m_n)}\otimes\mathcal{I}\otimes\dots\otimes\mathcal{I})\right),
\end{align}
of $\Omega^{\mathrm{sim}}$,
one can show after a bit of algebra that
\begin{align}
& f^{\mathrm{sim}}\left( (\mathcal{E}_{\mu(1,1)}\otimes\dots\otimes\mathcal{E}_{\mu(1,m_1)}\otimes\mathcal{
I}\otimes\dots\otimes\mathcal{I}),\dots,\right.\nonumber\\
&\qquad\left.(\mathcal{E}_{\mu(n,1)}\otimes\dots\otimes\mathcal{E}_{\mu(n,m_n)}\otimes\mathcal{I}\otimes\dots\otimes\mathcal{I})\right) \nonumber\\
&= f\left(\psi_1(\mathcal{E}_{\mu(1,1)}\otimes\dots\otimes\mathcal{E}_{\mu(1,m_1)}\otimes\mathcal{I}\otimes\dots\otimes\mathcal{I}),\dots,\right.\nonumber\\
&\qquad\left.\psi_n(\mathcal{E}_{\mu(n,1)}\otimes\dots\otimes\mathcal{E}_{\mu(n,m_n)}\otimes\mathcal{I}\otimes\dots\otimes\mathcal{I})\right).
\end{align}
Thus, condition \ref{lemmacond2} of Lemma \ref{lemma} is satisfied.
\end{proof}
\section{\label{secDiscussion}Discussion}
Using Theorem \ref{Theorem2} and Theorem \ref{Theorem3}, we can deduce a
statement about \textbf{FQSeq} (\textbf{FQSim}) from a statement about
\textbf{FQSim} (\textbf{FQSeq}). More precisely, when a statement ``if
$G \in \mathbf{FQSim}$ then $G$ has a property $P$'' is true, another
statement ``if $G\in\mathbf{FQSeq}$ then $G$ has a property $P$'' is
also true, and vice versa. Here, $P$ must be such a property that if a
game $G$ has the property $P$ and $G \parallel G'$, then $G'$ also has
the property $P$. We call such $P$ a property preserved under $\parallel$. For
example, ``a Nash equilibrium exists'' is a property preserved under
$\parallel$.
Unfortunately, no results are known which have the form ``if
$G\in\mathbf{FQSim}$ (\textbf{FQSeq}) then $G$ has a property $P$, but
otherwise $G$ does not necessarily have the property $P$''. Consequently,
we cannot reap the benefits of the above-mentioned deduction. However,
numerous results exist which have the form ``for a certain subset
$\mathbf{S}$ of \textbf{FQSim} (\textbf{FQSeq}), if $G\in\mathbf{S}$
then $G$ has a property $Q$ preserved under $\parallel$, but otherwise
$G$ does not necessarily have
the property $Q$''. For such \textbf{S} and $Q$, Theorem \ref{Theorem2}
and Theorems \ref{Theorem3} guarantee that there exists a subset $\mathbf{S'}$
of \textbf{FQSeq} (\textbf{FQSim}) which satisfies the following: ``If
$G\in\mathbf{S'}$ then $G$ has the property $Q$, but otherwise $G$ does
not necessarily have the property $Q$''. In this sense, many of the results
so far on \textbf{FQSim} (\textbf{FQSeq}) can be translated into
statements on \textbf{FQSeq} (\textbf{FQSim}).
It is worth noting that efficiency of a game \footnote{Amount of
information exchange between players and a referee, required to play a
game. See Ref. \cite{Lee2003} for more detail.} is not a property preserved
under $\parallel$. A good example is in the proof of Theorem \ref{Theorem3}. In
an original quantum sequential game $G$, it is necessary to transmit a
qudit $m+1$ times, while $4m$ times are needed in the
constructed game $G^{\mathrm{sim}}$. Thus, $G^{\mathrm{sim}}$ is far
more inefficient than $G$, despite $G^{\mathrm{sim}}$ and $G$ are
equivalent games.
Relevant to the present paper is the study by Lee and Johnson
\cite{Lee2003}. To describe their argument, we have to introduce a new
game class.
\begin{defn}
A finite quantum simultaneous game with all CPTP maps available
is a subclass of finite quantum simultaneous games, in which a
strategy set $\Omega_i$ is the set of all CPTP maps on the set of
density operators on $\mathcal{H}_i$ for every $i$. We denote the set
of all finite quantum simultaneous game with all CPTP maps available by
\textbf{FQSimAll}.
\end{defn}
We can easily prove that
$\mathbf{FQSimAll}\vartriangleleft\mathbf{FQSim}$, by showing that the
range of expected payoff functions for a game in
\textbf{FQSimAll} must be connected, while the one for a game in
\textbf{FQSim} can be disconnected.
Lee and Johnson claimed that (i) ``any game could be played
classically'' and (ii) ``finite classical games consist of a strict subset
of finite quantum games''.
Using the terms of this paper, we may interpret these claims as follows.
\begin{thm}\label{prop1}
$\mathbf{SG}\bowtie\mathbf{G}$.
\end{thm}
\begin{thm}\label{prop2}
$\mathbf{MEFSG}\trianglelefteq\mathbf{FQSimAll}$.
\end{thm}
\begin{proposition}\label{prop3}
$\mathbf{FQSimAll}\ntrianglelefteq\mathbf{MEFSG}$.
\end{proposition}
We can prove Theorem \ref{prop1} and Theorem \ref{prop2},
regardless of whether or not our interpretations of the claims (i) and
(ii) are
correct. In contrast, we have not yet proven Proposition \ref{prop3}, which is
the reason why we call it a proposition. Nonetheless
Lee and Johnson gave a proof of the statement (ii), so that if the
interpretation that the statement (ii) means
Theorem \ref{prop2} and Proposition \ref{prop3} is correct, Proposition
\ref{prop3} will be a theorem.
Using Lemma \ref{preorder} and assuming that Proposition \ref{prop3} is true,
relationships between various game classes can be summarized as follows:
\begin{align}
\mathbf{MEFSG}\vartriangleleft &\mathbf{FQSimAll}\vartriangleleft \mathbf{FQSim}\bowtie\mathbf{FQSeq}\nonumber\\
&\trianglelefteq\mathbf{QSim}\trianglelefteq\mathbf{QSeq}\trianglelefteq\mathbf{SG}\bowtie\mathbf{G}. \label{order}
\end{align}
Replacing $\trianglelefteq$ in \eqref{order} with either
$\vartriangleleft$ or $\bowtie$ will be a possible extension of this
paper.
Besides that, it remains to be investigated what the characterizing
features of each game class in \eqref{order} are. Especially, further
research on games which is in \textbf{FQSim} (or equivalently
\textbf{FQSeq}) but not equivalent to any games in \textbf{MEFSG} would
clarify the truly quantum mechanical nature of quantum games.
\begin{acknowledgments}
I would like to thank Prof. Akira Shimizu for his helpful advice.
\end{acknowledgments}
|
1,116,691,500,748 | arxiv | \section{Free boson field representation}
\label{sec:freeBosonEM}
Our Result~\ref{thm:n+1phys} associated to~$p$-form
electromagnetism in~$n+1$ dimensions a real Hilbert space~$\mathbf{P}$
consisting of pairs~$X=[A]\oplus E$, where~$[A]$ is an equivalence
class of~$p$-forms modulo~$D_{p-1}$-exact~$p$-forms and~$E$ is
a~$p$-form such that~$D_{p-1}^*E=0$ (what one might call
\emph{twisted-divergenceless}).
The symplectic structure
on~$\mathbf{P}$ was
$$
\omega(X,X')=(E,A')-(E',A).
$$
We need to make this~$\mathbf{P}$ into a
complex Hilbert space~$\mathbf{H}$, and put a time-independent complex
inner product on it whose imaginary part is the symplectic
structure~$\omega$.
This is equivalent to putting a real inner product~$h$ on~$\mathbf{P}$ which
is time-independent and satisfies
$$
h(X,X')=\omega(X,JX')
\qquad\hbox{for all}\quad
X,X'\in\mathbf{P}
$$
where~$J\colon\mathbf{P}\to\mathbf{P}$ is a densely-defined complex structure,
that is, a real-linear map such that~$J^2=-1$ on a dense domain
of~$\mathbf{P}$. Now, because of the appearance below of inverse powers of
the twisted Laplacian~$L_p$, we will be forced to restrict our
attention to the oscillating sector~$\mathbf{P}_o$, and ignore the `free'
sector~$\mathbf{P}_f$ which was the intersection of~$\mathbf{P}$ with the
kernel of~$L_p$. Recall that the time evolution in~$\mathbf{P}_o$ is given
by
$$
T_o(t)
\left( \! \begin{array}{c} A \\ E \end{array} \! \right) =
\left( \begin{array}{cc}
\cos(t\sqrt{L_p}) & \sin(t\sqrt{L_p})\,/\,\sqrt{L_p}\\
-\sqrt L_p\,\sin(t\sqrt{L_p}) & \cos(t\sqrt{L_p})
\end{array} \right)
\left( \! \begin{array}{c} A \\ E \end{array} \! \right)
$$
Defining multiplication by~$i$ by the action of~$J$, we can
make~$\mathbf{P}$ into a complex vector space. The completion of the dense
domain of~$K$ in~$\mathbf{P}$ with respect to the norm
$$
\|X\|^2=(E,L_p^{-1/2}E)+(A,L_p^{1/2}A')
$$
is the complex Hilbert space~$\mathbf{H}$. The key facts about~$\mathbf{H}$ are
summarized in the following theorem.
\begin{theorem}\label{thm:complex}
Let~$\mathbf{E}_o$ be a real Hilbert space with inner
product~$(~\mid~)$, let~$L$ be a nonnegative self-adjoint operator
on~$\mathbf{E}_o$ with vanishing kernel, and consider the real Hilbert space
$$
\mathbf{A}_o\colon=\{A\in\mathbf{E}_o\colon\|A\|^2+\|L^{1/2}A\|^2<\infty\}.
$$
Define time evolution on $\mathbf{P}_o=\mathbf{A}_o\oplus\mathbf{E}_o$ by
$$
\partial_t(A\oplus E)=E\oplus -LA,
$$
which preserves the canonical symplectic structure on~$\mathbf{A}_o\oplus\mathbf{E}_o$,
namely
$$
\omega(A\oplus E,A'\oplus E')=(A\mid E')-(A'\mid E).
$$
Then, there is a densely-defined complex structure~$J\colon\mathbf{Y}\to\mathbf{Y}$
given by~$J=-L^{-1/2}K$, or
$$
J(A\oplus E)\colon=-L^{-1/2}E\oplus L^{1/2}A,
$$
commuting with~$K$ and whose domain
$$
\mathbf{Y}\colon =\{A\oplus
E\in\mathbf{P}_o\colon\|A\|^2+\|L^{1/2}A\|^2+\|E\|^2+\|L^{-1/2}E\|^2<\infty\}
$$
is dense in~$\mathbf{P}_o$, preserved by time evolution and satisfying
$$
\|Jx\|_\mathbf{Y}=\|x\|_\mathbf{Y}
\qquad\hbox{and}\quad
\omega(Jx,Jy)=\omega(x,y)
\qquad\hbox{for all}\quad x,y\in\mathbf{Y}.
$$
Finally, the completion of~$\mathbf{Y}$ with respect to the norm
$$
\|x\|_{\mathbf{H}}^2\colon =\omega(x,Jy)
$$
is a complex Hilbert space~$\mathbf{H}$ with inner product
$$
\langle x,y\rangle\colon =\omega(x,Jy)+i\omega(x,y)
$$
Time evolution defined on~$\mathbf{Y}$ then extends to a strongly-continuous
one-parameter group of unitary operators on~$\mathbf{H}$, with nonnegative,
self-adjoint generator~$H=L^{1/2}$.
\end{theorem}
\begin{proof}
First we need to show that~$\mathbf{Y}$ is dense in~$\mathbf{P}_o$. Since $\|A\oplus
E\|_\mathbf{Y}^2=\|A\oplus E\|_\mathbf{P}^2+\|L^{-1/2}E\|^2$, $\mathbf{Y}$ is dense
in~$\mathbf{A}_o\oplus\mathop{\mathrm{ran}} L^{1/2}$. To show that~$\mathbf{Y}$ is dense in~$\mathbf{P}_o$ we
need to show that~$\mathop{\mathrm{ran}} L^{1/2}$ is dense in~$\mathbf{E}_o$. Now, $L^{1/2}$ is
self-adjoint on~$\mathbf{E}$ and has vanishing kernel so, by
lemma~\ref{lem:ran_ker}, $\{\mathop{\mathrm{ran}} L^{1/2}\}^\perp=\ker L^{1/2}=\{0\}$. But
this implies that~$\mathop{\mathrm{ran}} L^{1/2}$ is dense in~$\mathbf{E}_o$.
Next, we need to show that $\mathbf{Y}$ is preserved by the time evolution of
equation~(\ref{eq:time_evolution_twisted}) or, equivalently, that
$\|T(t)\|_{\mathbf{Y}}<\infty$ for all~$t$. It is not hard to check that, in
fact, $\|T(t)(A\oplus E)\|_\mathbf{Y}=\|A\oplus E\|_\mathbf{Y}$ for all~$t$. An even
easier calculation shows that $\|J\|_\mathbf{Y}=1$, so~$J$ maps~$\mathbf{Y}$ to
itself.
Then, we need to show that~$J$ is compatible with~$\omega$. With the
analytical subtleties out of the way, it requires only straightforward
algebraic calculations to check that
\begin{itemize}
\item{1)}$\omega(Jx,Jy)=\omega(x,y)$ for all $x,y\in\mathbf{Y}$
\item{2)}$\omega\bigl(A\oplus E,J(A\oplus
E)\bigr)=\|L^{1/4}A\|^2+\|L^{-1/4}E\|^2\ge 0$.
\end{itemize}
Also, $\|x\|_\mathbf{H}^2=\omega(x,Jx)$ is clearly a Hilbert-space norm.
Another simple calculation shows that $\|T(t)(A\oplus
E)\|_\mathbf{H}=\|A\oplus E\|_\mathbf{H}$, so $T(t)$ is a one-parameter unitary
group. Strong continuity is also easily checked. Finally, the
self-adjoint generator of time-evolution is determined by the
condition~$J\partial_t(A\oplus E)=H(A\oplus E)$, that is, $JK=H$. It
is also a straightforward algebraic calculation to check that~$\langle
A\oplus E,H(A\oplus E)\rangle=(A|LA)+(E|E)$, which is
nonnegative. Since time evolution is unitary with respect
to~$\langle~,~\rangle$, it follows that~$H$ is also self-adjoint.
\end{proof}
We can now apply Theorem~\ref{thm:FreeBosField} to the complex dual
of~$\mathbf{H}$ (denoted~$\mathbf{H}^\dagger$) to obtain the free boson field
over~$\mathbf{H}^\dagger$. Note that because of the mis-match
between~$\mathbf{H}$ in Theorem~\ref{thm:FreeBosField} and~$\mathbf{H}^\dagger$
now, there is a sign difference in the definition of the generator of
time evolution, which was~$U(t)=e^{-itA}$ then and is~$T_o(t)=e^{JtH}$
now.
Although this construction seems natural enough, one might worry that
there may be more than one complex structure with the given
properties, but in fact it is unique, as asserted in the following
theorem.
\begin{theorem}
Let~$T_o(t)$ be a one-parameter group of symplectic transformations on
the linear symplectic space~$(\mathbf{P},\omega)$. Then there is at most one
complex structure~$J$ on~$\mathbf{P}$ which is invariant, positive, symplectic
and such that the self-adjoint generator~$H$ of~$T_o(t)$ in the
completion of~$\mathbf{P}$ as a complex Hilbert space,~$\mathbf{H}$, is nonnegative and with
vanishing kernel.
\end{theorem}
\begin{proof}
The self-adjoint generator~$H$ commutes both with the complex
structure~$J$ and with each element~$T_o(t)$ of the unitary
group. Hence, the spectral projections (see~\cite[Section VIII.3]{RS})
associated to~$H$ also commute with them. We can use these spectral
projections to restrict the problem to the subspaces~$\mathbf{P}_n$
of~$\mathbf{P}$
where~$H\ge 1/n$. The hypothesis of the theorem hold, but now the
self-adjoint generator~$H$ is strictly positive (that is, bounded below by a
positive constant). That uniqueness holds in this case is proved
in~\cite[Scholium 3.3]{BSZ}.
\end{proof}
At this point, we redefine the meaning of~$\mathbf{P}$. It is clear that,
while mathematically convenient at the classical level, the real
Hilbert space structure of Theorems~\ref{thm:N+1}
and~\ref{thm:n+1phys} is really not the right one for Fock
quantization, which is the one given in Theorem~\ref{thm:complex}. We
now give concrete electromagnetic counterparts for all the objects
appearing in the development of the abstract free boson field
representation of Section~\ref{sec:freeBosonField}.
\begin{itemize}
\item The classical phase space~$\mathbf{P}$ consists of pairs of the
form~$X=[A]\oplus E$ such that~$h(X,X)<\infty$. It has a continuous
symplectic structure
$$
\omega(X,X')=(E,A')-(E',A)
$$
and a continuous complex structure $J\colon\mathbf{P}\to\mathbf{P}$ given by
$$
J([A]\oplus E)=L_p^{-1/2}E\oplus(-L_p^{1/2}A).
$$
We denote~$\mathbf{P}$ by~$\mathbf{H}$ when we want to view it as a complex
Hilbert space. Multiplication by~$i$ in~$\mathbf{H}$ corresponds to the
action of~$J$ on~$\mathbf{P}$.
\item Real observables $F\in\mathbf{P}^*$ are associated to phase space points~$F^*=[Q]\oplus
J\in\mathbf{P}$ where, if~$X=[A]\oplus E$,
$$
F(X)=(J,A)-(Q,E)=\omega(F^*,X)
$$
Each such real observable defines a complex-linear
observable~$iF+FJ\in\mathbf{H}^\dagger$. We have
$$
(iF+FJ)(X)=\langle F^*,X\rangle=h(F^*,X)+i\omega(F^*,X).
$$
This is consistent with the symplectic structure on~$\mathbf{P}^*$
$$
\omega(F,G)=-\omega(F^*,G^*)
\qquad\hbox{for all}\quad
F,G\in\mathbf{P}^*.
$$
It is customary to refer to observables primarily by~$F^*=[Q]\oplus
J$.
\item The free boson field representation of~$\mathbf{H}^\dagger$---the
complex dual of~$\mathbf{H}$---has
characteristic functional
$$
\mu(F)=e^{-{1\over 4}[(Q,L_p^{1/2}Q)+(J,L_p^{-1/2}J)]}.
$$
\item The coherent states of the electromagnetic field are of the form
$$
\ket{[Q]\oplus J}
\qquad\hbox{with}\quad
[Q]\oplus J\in\mathbf{P}.
$$
The inner product of two coherent states is
$$
\bracket{[Q]\oplus J}{[Q']\oplus J'}=e^{[(J,Q')-(J',Q)]/2i}e^{-{1\over 4}[(Q-Q',L_p^{1/2}(Q-Q')+(J-J',L_p^{-1/2}(J-J'))]}.
$$
\item The Weyl operator~$W(F)$, where~$F^*=[Q]\oplus J$, is defined by the following
action on the coherent states:
$$
W(F)\ket{[Q']\oplus J'}=e^{[(J',Q)-(J,Q')]/2i}\ket{[Q+Q']\oplus(J+J')}.
$$
The Heisenberg operator~$\Phi(F)$
satisfying~$W(F)=e^{-i\Phi(F)}$ has diagonal matrix elements on
coherent states given by
$$
\matElem{[Q]\oplus J}{\Phi(F)}{[Q]\oplus J}=-\omega([Q']\oplus
J',[Q]\oplus J)=(Q',J)-(Q,J')
$$
where~$F^*=[Q']\oplus J'$. In other words, the interpretation of~$F$
in~$\Phi(F)$ and in~$\ket{F}$ is very different: since the coherent
state~$\ket{F}$ is a semiclassical state of the quantum theory which
is peaked about the value~$F$ of the field configuration, it follows
that~$\Phi(F)$ does not represent the quantization of the
observable~$F$, but of~$JF$. We will see this in more detail in the
next section.
\item Time evolution is handled as follows. We have
$$
T_o(t)=e^{tJL^{1/2}}\colon\mathbf{P}\to\mathbf{P}
$$
on the phase space. The time evolution of the observables is
$$
U(t)=e^{-tJL^{1/2}}\colon\mathbf{P}^*\to\mathbf{P}^*.
$$
Then,~$\Gamma\bigl(U(t)\bigr)\colon\mathbf{K}\to\mathbf{K}$ is defined by extending the
following action on the coherent states:
$$
\Gamma\bigl(U(t)\bigr)\ket{[Q]\oplus J}=\ket{T_o(t)([Q]\oplus F)},
$$
and for all~$F\in\mathbf{P}^*$ the equation
$$
\Gamma\bigl(U(t)\bigr)W(F)\Gamma\bigl(U(-t)\bigr)=W\bigl(U(t)F\bigr).
$$
\end{itemize}
\section{Field quasioperators}
\label{sec:punchline}
First we try to define~$\hat A$ as an operator-valued $p$-form on~$S$
or, equivalently, an operator with matrix elements valued
in~$\Omega^p(S)$. It turns out~$\hat A$ exists as a quasioperator, and
we construct it as follows. First,~$\widehat{A(x)}$ can be defined for
all~$x\in S$ as a quasioperator by directly quantizing the classical
observable~$A(x)$. Then~$\hat A$ is defined so that~$\hat
A(x)=\widehat{A(x)}$ for all~$x\in S$. This technique is also used to
define~$\hat B(x)$ and~$\hat E(x)$, and the upshot is that, almost by
definition, the formulas
$$
\hat B(x)=\mathrm{d}\hat A(x)
\qquad\hbox{and}\quad
\oint_\gamma\hat A=\widehat{\textstyle\oint_\gamma A}
$$
hold as equations between quasioperators. Because in the physics
literature one does not distinguish between~$\mathbf{P}$ and~$\mathbf{P}^*$,
and it would be extremely awkward to use notations such
as~$\delta_x^*$, we identify them by means of using the duality
map~$*$ related to the symplectic structure~$\omega$.
When the gauge group is~$U(1)$, the proper holonomy is not
$\oint_\gamma A$ but the exponentiated version~$e^{i\oint_\gamma A}\in
U(1)$. However, because~$\gamma$ is a curve and~$A$ is
square-integrable, we know that the na\"\i{}ve candidate
for~$e^{i\oint_\gamma\hat A}$ has vanishing matrix elements between
any two coherent states, which is a problem. However, the
normal-ordered version of this exponentiated hlonomy exists as a
nonzero quasioperator on~$\mathbf{K}$ with domain containing the span~$\mathbf{K}_0$
of the smooth coherent states~$\ket{X}$ where~$X=[Q]\oplus J$ is not
only in~$\mathbf{P}$, but it is also infinitely differentiable. We will
denote the space of~$C^\infty$ elements of~$\mathbf{P}$---called
\emph{smooth field configurations}---by~$\mathbf{P}_0$.
\subsection{Quantizing the classical fields}
The classical observable~$A(x)$ is the densely-defined linear
functional on~$\mathbf{P}$ given by
$$
X=[A]\oplus E\mapsto A(x)
$$
In fact,~$A\in\mathbf{P}$ is in the domain of this observable as long
as~$A$ is continuous. Since a more convenient sufficient condition is
that~$A$ be infinitely differentiable, we give the following definition.
\begin{definition}[smooth coherent states]
Let~$\mathbf{P}$ be the oscillating phase space of $p$-form
electromagnetism, and let~$\mathbf{K}$ be the associated Fock space. We say
that~$X=[A]\oplus E\in\mathbf{P}$ is a \emph{smooth field configuration}, and
write~$X\in\mathbf{P}_0$, if~$[A]$ and~$E$ are
infinitely-differentiable. A coherent state~$\ket{X}$
with~$X\in\mathbf{P}_0$ is called a \emph{smooth coherent state}. We denote
by~$\mathbf{K}_0$ the span of the smooth coherent states.
\end{definition}
\begin{proof}[Note]
The space~$\mathbf{P}_0$ is a domain of essential self-adjointness of the
Laplacian~$L_p$ inside~$\mathbf{P}$, and is therefore dense. Hence,~$\mathbf{K}_0$
is also dense in~$\mathbf{K}$.
\end{proof}
Note that the observable~$A(x)$ takes values in~$\Lambda^pT^*_xS$. We
get a real-valued observable by contracting it with a multivector~$v_x\in \Lambda^pT_xS$. We denote this contraction by~$A_v(x)$. The
quantum observable~$\hat A_v(x)$ should be a Heisenberg
operator~$\Phi(F)$ such that
$$
\matElem{X}{\Phi(F)}{X}=(v\delta_x,A)=\omega(F^*,X)=F(X),
$$
where~$v\delta_x$ is the distributional~$p$-form defined by the
equation~$A_v(x)=(v\delta_x,A)$ for all smooth~$A$. In other words,
since~$A(x)=\omega(0\oplus v\delta_x,A\oplus E)$, one should
define
$$
\widehat{A_v(x)}\sim\Phi(0\oplus v\delta_x),
$$
as a quasioperator.
Now, it follows from Equation~\ref{eq:Heisenberg} that
$$
{\matElem{X'}{\widehat{A_v(x)}}{X}\over\bracket{X'}{X}}={A_v(x)+A_v'(x)\over 2}+iL_p^{-1/2}{E'_v(x)-E_v(x)\over 2}
$$
since~$h(0\oplus v\delta_x,A\oplus E)=(L_p^{-1/2}E)_v(x)$. Hence, defining
a quasioperator-valued $p$-form~$\hat A$ by
\begin{equation}\label{eq:AMatElem}
{\matElem{X'}{\hat A}{X}\over\bracket{X'}{X}}={A+A'\over 2}+iL_p^{-1/2}{E'-E\over 2},
\end{equation}
one has
$$
\hat A_v(x)=\widehat{A_v(x)}
\qquad\hbox{for all}\quad v_x\in T_xS
$$
as an equation between quasioperators.
In a entirely analogous manner, one can quantize the electric
field. Indeed,~$\hat E_v(x)$ is the quantum counterpart of
$$
A\oplus E\mapsto E(x)=(v\delta_x,E),
$$
with
$$
E(x)=-\omega(v\delta_x\oplus 0,A\oplus E)
\qquad\hbox{and}\quad
h(v\delta_x\oplus 0,A\oplus E)=(L_p^{1/2}A)_v(x).
$$
This means that
$$
\widehat{E_v(x)}\sim-\Phi(v\delta_x\oplus 0)
$$
and, demanding~$\hat E_v(x)=\widehat{E_v(x)}$,
$$
{\matElem{X'}{\hat E}{X}\over\bracket{X'}{X}}={E+E'\over 2}+iL_p^{1/2}{A-A'\over 2}.
$$
In the same way one can derive
$$
\widehat{(L_p A)_v(x)}=(L_p\hat A)_v(x)
$$
as quasioperators with domain~$\mathbf{K}_0$.
\subsection{Wilson surfaces as quasioperators}
Now that~$\hat A$ is defined as a~$p$-form (albeit
quasioperator-valued), we can define its integral on a
compact, oriented~$p$-dimensional submanifold~$\gamma$ of space in
such a way that
$$
\oint_\gamma\hat A=\widehat{\textstyle\oint_\gamma A}.
$$
as a quasioperator equation. In the~$p=1$ case, these observables are
called Wilson loops in physics. We call them \emph{Wilson surfaces} in
general.
We do this by
observing that the
classical observable~$\oint_\gamma\hat A$ can be written as
$$
A\oplus E\mapsto\oint_\gamma A=(\Gamma_\gamma,A)
$$
where~$\Gamma$ is a distributional~$p$-form analogous to Dirac's
delta, uniquely defined by this equation and satisfying
$$
\oint_\gamma A=\omega(0\oplus\Gamma_\gamma,A\oplus E)
\qquad\hbox{and}\quad
h(0\oplus\Gamma_\gamma,A\oplus E)=\oint_\gamma L_p^{-1/2}E.
$$
So, letting
$$
\widehat{\textstyle\oint_\gamma A}\sim\Phi(0\oplus\Gamma_\gamma)
$$
it follows that
showing that
$$
{\matElem{X'}{\textstyle\oint_\gamma\hat
A}{X}\over\bracket{X'}{X}}=\oint_\gamma{\matElem{X'}{\hat
A}{X}\over\bracket{X'}{X}}.
$$
so
$$
\widehat{\textstyle\oint_\gamma A}=\oint_\gamma\hat A,
$$
as a quasioperator equation on~$\mathbf{K}_0$.
In particular, we find that
$$
\matElem{X}{{\textstyle\oint_\gamma\hat
A}}{X}=\oint_\gamma A,
$$
when~$X=[A]\oplus E$ is a smooth field configuration. In fact, this
follows from the less
obvious expression
$$
{\matElem{X'}{{\textstyle\oint_\gamma\hat A}}{X}\over\langle X'\mid X\rangle}=\oint_\gamma\biggl({A+A'\over
2}\biggr)+i\oint_\gamma{1\over\sqrt{L_p}}\biggl({E-E'\over
2}\biggr)
$$
when~$X,X'$ are smooth field configurations, which is an easy
consequence of Equation~(\ref{eq:AMatElem}).
Finally, in order to extend our work to the case of a $U(1)$
connection which is much more common in the physics literature, we
would need to define the operator
$$
e^{i\oint_\gamma \hat A}
$$
which quantizes the holonomy
$$
e^{i\oint_\gamma A}\in U(1).
$$
As we know, there is a serious problem coming from the fact that~$\oint_\gamma\hat
A\sim\Phi(0\oplus\Gamma_\gamma)$,
and~$\|0\oplus\Gamma_\gamma\|=\infty$. However, we have shown that the
normal-ordered
$$
\Wick{e^{i\oint_\gamma\hat A}}=\Wick{W(0\oplus\Gamma_\gamma)}
$$
does exist as a quasioperator on~$\mathbf{K}$ with domain~$\mathbf{K}_0$. In fact,
$$
{\matElem{X'}{e^{i\oint_\gamma\hat
A}}{X}\over\bracket{X'}{X}}=\exp{i\matElem{X'}{\oint_\gamma\hat A}{X}\over\bracket{X'}{X}}
$$
whenever~$X,X'$ are smooth coherent states.
\subsection{The vacuum Maxwell equations}
We are now ready to show that the field quasioperators that we have
defined satisfy the vacuum Maxwell
equations in the following sense:
\begin{theorem}
Let~$\ket{X(t)}=\Gamma\bigl(U(t)\bigr)\ket{X}$ for all~$X\in\mathbf{P}$. Then,
\begin{eqnarray*}
{\partial\over\partial t}\matElem{X'(t)}{\hat
A}{X(t)}
&=&
\matElem{X'(t)}{\hat E}{X(t)}\\
{\partial\over\partial t}\matElem{X'(t)}{\hat
E(x)}{X(t)}
&=&
-\matElem{X'(t)}{L_p\hat A}{X(t)}\\
\end{eqnarray*}
\end{theorem}
\begin{proof}
First, recall that
$$
\Gamma(U(t))\ket X=\ket{T_o(t)X},
$$
so that
$$
\ket{X(t)}=\ket{T_o(t)X}.
$$
Therefore, if~$X=[A]\oplus E$, we have~$\ket{X(t)}=\ket{[A](t)\oplus
E(t)]}$, where~$[A](t)$ and~$E(t)$ are the solutions of the classical Maxwell
equations with initial data~$[A]\oplus E$.
Now, from the known
expression for the matrix elements of~$\hat A$
$$
{\matElem{X'(t)}{\hat A}{X(t)}\over\bracket{X'(t)}{X(t)}}={A(t)+A'(t)\over
2}+{i\over\sqrt{L_p}}{E(t)-E'(t)\over 2}.
$$
On the left-hand side,~$\hat A$ and~$\hat E$ are~$p$-form-valued
operators on~$\mathbf{K}$ (quantum observables), while on the right-hand side
we have the classical solutions of the Maxwell equations evaluated at
time~$t$. Since the quantities on the right-hand side satisfy the
Maxwell equations, and~$\bracket{X'(t)}{X(t)}$ is independent of~$t$
because~$\Gamma(U(t))$ is unitary, we have
$$
{1\over\bracket{X'(t)}{X(t)}}{\partial\over\partial t}\matElem{X'(t)}{\hat A(x)}{X(t)}={E+E'\over
2}-i\sqrt{L_p}\Bigl({A-A'\over 2}\Bigr),
$$
but the right-hand side is precisely~$\matElem{X'(t)}{\hat
E(x)}{X(t)}\over\bracket{X'(t)}{X(t)}$.
Similarly,
$$
{\matElem{X'(t)}{\hat E(x)}{X(t)}\over\bracket{X'(t)}{X(t)}}={E+E'\over
2}-i\sqrt{L_p}\Bigl({A-A'\over 2}\Bigr)
$$
implies that
\begin{eqnarray*}
{1\over\bracket{X'(t)}{X(t)}}{\partial\over\partial t}\matElem{X'(t)}{\hat E(x)}{X(t)}
&=&
-L_p\Bigl({E+E'\over
2}\Bigr)-i\sqrt{L_p}\Bigl({A-A'\over 2}\Bigr)\\
&=&
-L_p\Bigl({A+A'\over
2}+{i\over\sqrt{L_p}}{E-E'\over 2}\Bigr),
\end{eqnarray*}
and the result follows.
\end{proof}
The calculations involved in the proof of this fact are deceptively
simple. The point is that these would be purely formal had we not
developed a framework where objects such as~$\hat A(x)$ are
well-defined. All the hard work is hidden in
Chapter~\ref{chap:linear}.
Finally, here is the promised formula for the time evolution of
electromagnetism in terms of Wilson loop quasioperators:
\begin{corollary}
$$
{\partial\over\partial t}{\matElem{X'}{e^{i\oint_\gamma\hat
A}}{X}\over\bracket{X'}{X}}={i\matElem{X'}{{\textstyle \oint_\gamma\hat E}}{X}\over\langle X'\mid X\rangle}\exp{i\matElem{X'}{{\textstyle \oint_\gamma\hat A}}{X}\over\langle X'\mid X\rangle}.
$$
\end{corollary}
\begin{proof}
Differentiating
$$
{\matElem{X'}{e^{i\oint_\gamma\hat
A}}{X}\over\bracket{X'}{X}}=\exp{i\matElem{X'}{{\textstyle \oint_\gamma\hat A}}{X}\over\langle X'\mid X\rangle}
$$
we get
$$
{\partial\over\partial t}{\matElem{X'}{e^{i\oint_\gamma\hat
A}}{X}\over\bracket{X'}{X}}={i\matElem{X'}{{\textstyle \oint_\gamma\hat E}}{X}\over\langle X'\mid X\rangle}\exp{i\matElem{X'}{{\textstyle \oint_\gamma\hat A}}{X}\over\langle X'\mid X\rangle}.
$$
\end{proof}
\chapter{Introduction}
This work is motivated by the open problem of representing gravitons
in loop quantum gravity~\cite{rovelli98}, a proposed quantum theory of
geometry and candidate for a theory of quantum gravity. The great
virtue of loop quantum gravity is that it is manifestly
background-free and diffeomorphism-invariant. Unfortunately, because
the usual construction of the graviton Fock space depends explicitly
on a background metric, it is difficult to say precisely how the
notion of \emph{graviton} arises in this formalism. At least at the
kinematical level, in loop quantum gravity states of quantum geometry
are described not in terms of gravitons but in terms of spin
networks~\cite{baez96}, which had been invented independently by
Penrose~\cite{penrose71} and can be seen as a generalization of the
Wilson loops introduced in the 1970's for the study of non-abelian
gauge theories~\cite{wilson74}. However, describing the dynamics of
quantum gravity in terms of spin networks remains a difficult open
problem. So, we are not yet in a position to study how this dynamics
reduces to that of gravitons in some limit, as presumably it should.
As a warmup, it is natural therefore to investigate the dynamics of
Wilson loops in a gauge theory which is better understood: vacuum
electromagnetism. However, until recently we were in the embarrassing
situation of not even knowing the precise relation between the loop
representation of electromagnetism and the usual Fock
representation. Here, of course, the theory is linear and formulated
on a fixed background metric, which drastically simplifies the
situation. The technical problem is that the loop representation is
based on a diffeomorphism-invariant vacuum, while the traditional Fock
vacuum is tied to a particular background metric, which implies that
photon (Fock) states are not part of the loop state space and Wilson
loop states are not part of the Fock state space. In particular, with
respect to the Fock vacuum, the photon 2-point correlation function blows up at
short distances at such a rate that Wilson loops are not well-defined
operators on Fock space.
Varadarajan~\cite{varadarajan00,varadarajan01} tackled this problem by
``smearing'' the loop~$\gamma$ using Gaussian convolution in Minkowski
space. Varadarajan's procedure puts photons and Wilson loops in a
common framework. Our goal in the present work is to understand
electromagnetic Wilson loops without the need for smearing, and on
general static, globally hyperbolic spacetimes. A related and
important outstanding problem in loop quantum gravity is that spin
network dynamics is poorly understood, and here we tackle the
analogous problem of electromagnetic Wilson loop dynamics in the Fock
representation.
The modern view of electromagnetism is that the electromagnetic
potential~$A$ is a connection on a~$U(1)$ or~$\mathbb{R}$ bundle over spacetime,
and the electromagnetic field is the curvature of this connection. A
\emph{Wilson loop observable} is what mathematicians call
the \emph{holonomy} of the connection around a closed
loop. In quantum theory, observables of a physical system are
represented by operators on a Hilbert space of states of the
system. In the case of electromagnetism in Minkowski spacetime, the
state space of the electromagnetic field is the so-called Fock
space. The main problem with the Wilson loop approach to quantum gauge
field theories is that, even in the simple case of electromagnetism,
Wilson loop operators are not defined on Fock space. Because in
quantum field theory there is a correspondence between observables and
states, this means that there are also no Wilson loop states in the
Fock space of electromagnetism.
Quantum field theory on curved spacetimes is a famously problematic
subject, as it combines the difficulties of quantum field theory,
notably ultaviolet divergences, with a lack of a well-defined vacuum
state due to the lack of global symmetries in a curved spacetime. For
a free quantum field theory on a static spacetime, such as we are
studying, these problems go away as there are no divegent interactions
and there is a unique time-invariant vacuum state. Because of this,
most physicist would say that vacuum electromagnetism on a static
spacetime is well-understood. This is more or less true for scalar
fields~\cite{wald94}, but then despite it being known~\cite[\S
4.7]{wald94} that
\begin{quote}
the requirement that the classical field equations have a well-posed
initial value formulation in curved spacetime is a highly nontrivial
restriction: the straightforward generalization to curved spacetime of
the standard spin-$s$ field equations in flat spacetime do not admit a
well posed initial value formulation for~$s>1$
\end{quote}
even researchers concerned only with electromagnetism and not with
scalar fields work on the assumption that the mathematical theorems on
scalar fields apply without modification to other fields~\cite{dimock92}.
\begin{quote}For globally hyperbolic manifolds, the usual classical
linear field equations will have global solutions if they are
well-behaved locally. We quote the result for scalar fields.\end{quote}
Part of the point of this thesis is to show that things are not so
simple: there are subtleties involved due to gauge invariance and
noncompact spacetimes which interact in unexpected ways. Our first
goal is to clear this up and give a rigorous general treatment of
vacuum electromagnetism on a static, globally hyperbolic
spacetime. The subtleties arise mainly from the difference between the
usual de~Rham cohomology and a certain \emph{twisted} $L^2$ cohomology
arising from gravitational time-dilation. Indeed, in a careful
treatment the electromagnetic vector potential is not a
smooth~$1$-form modulo exact smooth~$1$-forms, but a
normalizable~$1$-form modulo exact normalizable~$1$-forms. Similarly,
the Aharonov--Bohm effect arises not from closed smooth modulo exact
smooth vector potentials, but from closed normalizable modulo exact
normalizable ones. This distinction would be inconsequential if space
were compact, but this is not believed to be the case in physically
realistic models of spacetime.
In Chapter~\ref{sec:harmonic} we present a rogues' gallery of
pathologies and counterexamples which illustrate how these subtleties
can manifest themselves as physical effects, including the photon
acquiring a mass due to the interaction of gravitational time dilation
and the asymptotic geometry at spatial infinity.
When we quantize electromagnetism in Chapter~\ref{chap:qed}, we will
actually exclude the Aharonov--Bohm modes from our analysis.
Chapter~\ref{chap:linear} describes our quantization
procedure---essentially just Fock quantization, but done in a way that
emphasizes the role of coherent states. The reason for this is that
Wilson loop ``operators''
$$
{\textstyle\oint_\gamma\hat A}
\qquad\hbox{or}\quad
\Wick{e^{i\oint_\gamma\hat A}}
$$
are \emph{not} densely-defined operators on Fock space, but their
matrix elements
$$
\matElem{\phi}{\textstyle\oint_\gamma\hat A}{\psi}
\qquad\hbox{or}\quad
\matElem{\phi}{\Wick{e^{i\oint_\gamma\hat A}}}{\psi}
$$
exist when~$\phi,\psi$ are linear combinations of regular coherent
states---that is, coherent states corresponding to sufficiently smooth
classical solutions of Maxwell's equations. Such regular
coherent states span a dense subspace of Fock space, so they are
sufficiently general to study Wilson loop dynamics. We are then able
to prove formulas such as
$$
{\mathrm{d}\over\mathrm{d} t}{\textstyle\oint_\gamma\hat
A}={\textstyle\oint_\gamma\hat E}
$$
and
$$
{\mathrm{d}\over\mathrm{d} t}{\matElem{X'}{\Wick{e^{i\oint_\gamma\hat
A}}}{X}\over\langle X'\mid
X\rangle}=i{\matElem{X'}{{\textstyle\oint_\gamma\hat
E}}{X}\over\langle X'\mid X\rangle}\exp i{\matElem{X'}{{\textstyle\oint_\gamma\hat
A}}{X}\over\langle X'\mid X\rangle},
$$
where~$\ket{X},\ket{X'}$ are regular coherent states.
The plan of this dissertation is as follows: in Part~$I$ we study
classical vacuum electromagnetism, and in Part~$II$ the quantization
of vacuum electromagnetism. Part~$I$ consists of three chapters. In
Chapter~$2$ we study ordinary vacuum electromagnetism in
a~$(3+1)$-dimensional static, globally hyperbolic spacetime. In
Chapter~$3$ we generalize our results to~$(n+1)$-dimensional
spacetimes and also consider theories where the electromagnetic
potential is not a~$1$-form but any~$p$-form, including the massless
scalar field ($p = 0$) and the Kalb-Ramond field ($p = 2$), which
plays a role in string theory. Finally, in Chapter~$4$ we survey the
theory of~$L^2$ cohomology and suggest physical interpretations of
some of its main results. Part~$II$ consists of two
chapters. Chapter~$5$ is where we describe our coherent-state
quantization of linear dynamical systems and develop the concept of a
quasioperator. Lastly, in Chapter~$6$ this quantization method is
applied to vacuum electromagnetism and used to make sense of
unregularized Wilson loop quasioperators.
\part{Classical electromagnetism}
In this part we lay the classical groundwork for a a rigorous
quantization of the vacuum Maxwell equations and the analogous
equations for $p$-form electromagnetism with gauge
group~$\mathbb{R}$\index{$\mathbb{R}$!gauge group} on an arbitrary static, globally
hyperbolic, $(n+1)$\index{$n$!dimension of space}-dimensional
spacetime. In other words, we assume that spacetime is invariant under
time evolution and time reversal, and that the time evolution of
fields in spacetime is completely determined by initial data. In fact,
any such spacetime is topologically $M = \mathbb{R} \times
S$\index{$M$!spacetime}\index{$S$!space}, and has a metric of the form
$$
g_M = e^{2\Phi} (-\mathrm{d} t^2 + g)
\index{$g_M$!spacetime metric}
\index{$\Phi$!Newtonian potential}
\index{$\mathrm{d}$!exterior derivative}
\index{$t$!time coordinate}
\index{$g$!optical metric on space}
$$
where~$g$\index{$g_M$!spacetime metric} is a complete metric
on~$S$\index{$S$!space}, so that no lightlike geodesics run off to
spatial infinity in a finite amount of their affine parameter.
\index{quantization}
\index{vacuum Maxwell equations}
\index{electromagnetism!$p$-form}
\index{gauge group}
\index{static spacetime}
\index{globally hyperbolic spacetime}
\index{spacetime}
\index{time evolution}
\index{time reversal}
\index{initial data}
\index{metric}
\index{complete metric}
\index{lightlike geodesic}
\index{spatial infinity}
\index{affine parameter}
Because the Lie algebras of~$\mathbb{R}$\index{$\mathbb{R}$!Lie
group} and~$U(1)$\index{$U(1)$!Lie group} are canonically
isomorphic, there is no difference between the versions of
electromagnetism with
either gauge group as far as the local formulation
of the Maxwell equations is
concerned. Globally there is a difference, though, because
all~$\mathbb{R}$\index{$\mathbb{R}$!Lie group}-bundles are
trivializable
whereas~$U(1)$\index{$U(1)$!Lie group}-bundles may not
be. In~$3+1$
dimensions, the second Chern class
of a
nontrivial gauge bundle
manifests itself as a topological magnetic
charge whose field can be gauged away
locally, but not globally. While topological
charges are interesting, our primary goal is
to study the effects of spatial non-compactness on
quantization, and so we choose the gauge
group~$\mathbb{R}$\index{$\mathbb{R}$!Lie group} to eliminate the
possibility of nontrivial bundles. When a principal~$\mathbb{R}$\index{$\mathbb{R}$!gauge group}-bundle is trivialized,
connections on it are
ordinary~$1$-forms.
\index{Lie algebra}
\index{canonical isomorphism}
\index{electromagnetism}
\index{gauge group}
\index{Maxwell equations}
\index{bundle}
\index{trivializable bundle}
\index{bundle}
\index{second Chern class}
\index{nontrivial bundle}
\index{gauge bundle}
\index{topological charge}
\index{topological charge}
\index{topological charge}
\index{characteristic class}
\index{topological charge}
\index{non-compact space}
\index{quantization}
\index{gauge group}
\index{nontrivial bundle}
\index{gauge bundle}
\index{trivialization}
\index{connection}
\index{differential $1$-form}
\index{gauge group}
\index{electromagnetic potential}
\index{connection}
Technically, the subtlest aspects of our work arise from the
function~$\Phi$\index{$\Phi$!Newtonian potential} appearing in the
spacetime metric. This function measures the time dilation due to the
gravitational field, and reduces to the Newtonian gravitational
potential in the limit $\Phi \to 0$\index{$\Phi$!Newtonian potential}.
When $\Phi = 0$\index{$\Phi$!Newtonian potential}, $p$-form
electromagnetism uses rather familiar mathematics, mainly this portion
of the~$L^2$ de~Rham cohomology complex:
$$
\begin{CD}
{L^2 \Omega^{p-1}_S}@>{\mathrm{d}_{p-1}}>>{L^2 \Omega^p_S}@>{\mathrm{d}_p}>>{L^2 \Omega^{p+1}_S}
\end{CD}
\index{$L^2\Omega^p_S$!square-integrable $p$-forms on~$S$}
\index{$\mathrm{d}_p$!exterior derivative on~$p$-forms}
$$
where~$L^2 \Omega^p_S$\index{$L^2\Omega^p_S$!square-integrable
$p$-forms on~$S$} stands for the Hilbert space of
square-integrable $p$-forms
on~$S$\index{$S$!space}. The case $\Phi \ne 0$\index{$\Phi$!Newtonian
potential} requires some less familiar mathematics---except when~$p+1$
is half the dimension of spacetime\index{spacetime}, in which case
$p$-form electromagnetism is conformally invariant, allowing us to
eliminate~$\Phi$\index{$\Phi$!Newtonian potential} by an appropriate
rescaling of the fields. Even in the absence of
conformal invariance, the most elegant
approach is still to hide all the factors
involving~$\Phi$\index{$\Phi$!Newtonian potential} by a field
redefinition, and replacing the exterior
derivative with the `twisted'
differential
$$
D_k = e^{{1\over 2}(n-2p-1)\Phi} \mathrm{d}_k e^{-{1\over 2}(n-2p-1)\Phi}
\index{$D_k$!twisted exterior derivative on~$k$-forms}
\index{$\Phi$!Newtonian potential}
$$
obtained by conjugating the ordinary differential by the rescaling
factor. This gives rise to a `twisted' version
of~$L^2$ cohomology which, on a
noncompact space, can differ from the usual~$L^2$
cohomology which, in turn, can
differ from the smooth de~Rham cohomology.
\index{spacetime}
\index{metric}
\index{time dilation}
\index{gravitational field}
\index{Newtonian gravitational potential}
\index{electromagnetism!$p$-form}
\index{de~Rham cohomology}
\index{cohomology complex}
\index{Hilbert space}
\index{square-integrable}
\index{differential $p$-form}
\index{conformal invariance}
\index{electromagnetism!$p$-form}
\index{field rescaling}
\index{conformal invariance}
\index{field redefinition}
\index{exterior derivative}
\index{twisted differential}
\index{rescaling factor}
\index{twisted~$L^2$ cohomology}
\index{square-integrable cohomology}
\index{de~Rham cohomology}
With this machinery in place we model the phase space\index{phase
space} of classical~$p$-form
electromagnetism\index{electromagnetism!$p$-form}
on~$(n+1)$-dimensional spacetime\index{spacetime} as a real Hilbert
space\index{real Hilbert space} with continuous
Hamiltonian\index{Hamiltonian} and symplectic
stucture\index{symplectic structure}. In the process, we address the
Aharonov--Bohm effect\index{Aharonov--Bohm effect} in situations where
the twisted~$L^2$ cohomology\index{twisted~$L^2$ cohomology} differs
from the usual de~Rham cohomology\index{de~Rham cohomology}, a subtle
issue that is largely neglected in the literature.
Among the most rigorous published treatments of Maxwell's
equations\index{Maxwell's equations} on a fairly generic manifold
stands that of Dimock~\cite{dimock92}, which however is restricted to
$(3+1)$-dimensional spacetimes\index{spacetime} with compact Cauchy
surfaces\index{Cauchy surface}. At the time of his writing, he said
``nothing that follows is particularly new, but it seems that the
various pieces have not been put together''. A later paper reviewing
the canonical\index{canonical mechanics} and covariant\index{covariant
mechanics} formulations of the classical Maxwell theory\index{Maxwell
theory} on a generic globally hyperbolic spacetime\index{globally
hyperbolic spacetime} is the one by Corichi~\cite{corichi}, again
``intended to fill an existing gap in the literature''.
Dimock constructs the classical phase space\index{phase space} from
gauge equivalence classes\index{gauge equivalence class} of Cauchy
data\index{Cauchy data} and the symplectic structure\index{symplectic
structure} obtained from the Noether current\index{Noether
current}. Gauge fixing\index{gauge fixing} appears as a technical step
used to show that Maxwell's equations\index{Maxwell's equations} are
strictly hyperbolic\index{hyperbolic equations}, so that solutions are
determined by their Cauchy data\index{Cauchy data}. Dimock uses
``fundamental solutions''\index{fundamental solutions} (essentially
Green's functions\index{Green's functions}) to parameterize the phase
space\index{phase space}, a technique that only works for linear field
equations\index{linear field equations}. Time evolution\index{time
evolution} enters the picture through symplectic
transformations\index{symplectic transformations} induced on phase
space\index{phase space} by changes in the choice of Cauchy
surface\index{Cauchy surface}. In fact, Dimock makes ``no choice of
Hamiltonian\index{Hamiltonian} or special time coordinate'', following
the covariant canonical formalism\index{covariant canonical mechanics}
of~\cite{crnkovic87}. Dimock points out how the field
strength\index{field strength} does not provide a complete set of
observables\index{complete set of observables} when the first homology
class\index{first homology class} of the Cauchy surfaces\index{Cauchy
surface} is nontrivial. In Chapter~\ref{sec:3+1} we relate this
phenomenon to the Aharonov--Bohm effect\index{Aharonov--Bohm effect}
and in Chapter~\ref{sec:harmonic} we present a thorough overview of
the situation in the non-compact case. Dimock assumes a
trivial~$U(1)$-bundle saying ``presumably our results can be extended
to non-trivial bundles for which~$A$ is only defined locally'', while
we take the more drastic step of assuming an~$\mathbb{R}$-bundle.
For the purposes of this Part, Dimock's presentation of Maxwell's
equations\index{Maxwell's equations} does have a couple of important
limitations. First, the restriction to compact Cauchy
surfaces\index{compact Cauchy surface} may be unphysical, and
certainly excludes many cases of theoretical interest. We address the
thorny analytic issues associated to allowing noncompact Cauchy
surfaces\index{noncompact Cauchy surface} in Chapter~\ref{sec:3+1},
albeit with the additional assumption that spacetime is
static\index{static spacetime}, which Dimock does not need. The
topological implications of noncompactness\index{noncompactness} are
discussed in Chapter~\ref{sec:harmonic}. Dimock's use of compact
Cauchy surfaces\index{compact Cauchy surface} allows his to bring
Hodge's theorem\index{Hodge's theorem} to bear on the Cauchy
data\index{Cauchy data} and, using the Kodaira
decomposition\index{Kodaira decomposition}, to show that the
symplectic structure\index{symplectic structure} is
non-degenerate. Although Hodge's theorem\index{Hodge's theorem} does
not hold on a noncompact space\index{noncompact space} (see
Chapter~\ref{sec:harmonic}), we are nevertheless able to prove a form
of Kodaira's decomposition\index{Kodaira decomposition} in
Chapter~\ref{sec:3+1}.
Dimock also states without proof or reference that ``for globally
hyperbolic manifolds, the usual classical linear field equations will
have global solutions if they are well-behaved locally. We quote the
result for scalar fields''. We repaired this defect by reference to
Chernoff's work in Chapter~\ref{sec:harmonic}. In the proof of existence of
solutions with given Cauchy data Dimock states ``The equation [above]
has principal part~$g^{\mu\nu}\partial_\mu\partial_\nu$ and thus is
strictly hyperbolic''; hyperbolicity easily follows from Chernoff's
work. Finally, the phase space constructed by Dimock does not have a topology
other than that induced by imposing the continuity of the symplectic
structure. Therefore, it is not a real inner-product space like ours
is.
While not assuming compact Cauchy surfaces\index{compact Cauchy
surface}, Corichi's paper is ``not very precise about
functional-analytic issues'' in the author's own words. The
covariant\index{covariant mechanics} formulation is, like Dimock's,
based on the formalism of~\cite{crnkovic87}, and differs mostly in the
notation. The canonical\index{canonical mechanics} formulation is
written in a manifestly covariant way, in terms of the
foliation\index{foliation} generated by an arbitrary time coordinate
function. Both formulations of classical
electromagnetism\index{electromagnetism} are more general than ours,
and the relationship between Corichi's covariant and canonical
descriptions of phase space\index{phase space} is equivalent to
Dimock's treatment of Cauchy data\index{Cauchy data} in the
covariant\index{covariant mechanics} formalism.
The plan of this Part is as follows. We begin in
Chapter~\ref{sec:3+1} by setting up classical
electromagnetism\index{classical electromagnetism} with gauge
group\index{gauge group}~$\mathbb{R}$\index{$\mathbb{R}$!Lie group}, leading up to Theorems~\ref{thm:3+1}
and~\ref{thm:3+1phys}, in which we make the phase space\index{phase
space} for this theory into a real Hilbert space\index{real Hilbert
space} on which the classical Hamiltonian\index{Hamiltonian} is a
continuous nonnegative quadratic form\index{quadratic form}. In
Chapter~\ref{sec:N+1} we generalize this work to~$p$-form
electromagnatism\index{electromagnetism!$p$-form} in~$n+1$ dimensions
using the twisted de~Rham complex\index{de~Rham cohomology complex},
leading up to the analogous Theorems~\ref{thm:N+1}
and~\ref{thm:n+1phys}. In Chapter~\ref{sec:harmonic} we survey what is
known about~$L^2$ cohomology\index{square-integrable cohomology} on
noncompact spaces, and study a number of examples illustrating some of
the associated subtleties.
\chapter{Classical vacuum electromagnetism}
\label{sec:3+1}
In this chapter we discuss the classical vacuum Maxwell equations on a
$(3+1)$-dimensional static globally hyperbolic spacetime. In
particular, we explain how the classical phase space of
electromagnetism splits into two parts, one containing the oscillatory
modes of the electromagnetic field and the other containing the
`topological' modes responsible for the `Aharonov--Bohm' effect.
The plan of this chapter is as follows: we begin in
Section~\ref{sec:3+1-geometry} by describing in detail our assumptions
and notation concerning spacetime geometry, decompose spacetime in the
form~$M\cong\mathbb{R}\times S$, and confront a number of analytical issues
arising from trying to define the exterior derivative on
square-integrable differential forms. In Section~\ref{sec:maxwell} we
give an overview of the stationary action formulation of classical
mechanics, and use it to derive the Maxwell equations, Noether
current, Hamiltonian and symplectic structure, as well as kinematical,
dynamical and physical phase spaces. Finally, in
Section~\ref{physical.interpretation} we describe the splitting on the
physical phase space of classical vacuum electromagnetism into an
sector consisting of oscillating modes, and a sector consisting of
topological modes responsible for the Aharonov--Bohm effect.
After seeing that the spacetimes we are interested split in the
form~$M\cong\mathbb{R}\times S$, where~$S$ is \emph{space}, we define the
exterior derivative~$\mathrm{d}$ and coderivative~$\mathrm{d}^*$ so that they act on
square-integrable differential forms on space and satisfy
\begin{equation}
\label{eqn:adjoint}
\int_Sg(\alpha,\mathrm{d}\beta)\mathrm{vol}=\int_Sg(\mathrm{d}^*\alpha,\beta)\mathrm{vol}
\index{$S$!space}
\index{$g$!optical metric on space}
\index{$\mathrm{d}$!exterior derivative}
\index{$\mathrm{vol}$!volume form of optical metric}
\index{$\mathrm{d}^*$!exterior coderivative}
\end{equation}
whenever~$\alpha$ and~$\beta$ are square-integrable differential forms
of appropriate degrees. The key is to show that no `boundary terms at
infinity'\index{boundary terms at infinity} appear in the integration
by parts\index{integration by parts} implicit in
Equation~(\ref{eqn:adjoint}). This can be used to show that the
Laplacian on square-integrable differential forms is
essentially self-adjoint\index{essentially self-adjoint} and
nonnegative, properties necessary for rigorous quantization.
\index{vacuum Maxwell equations}
\index{static spacetime}
\index{globally hyperbolic spacetime}
\index{spacetime}
\index{classical phase space}
\index{oscillatory mode}
\index{electromagnetic field}
\index{topological modes}
\index{Aharonov--Bohm effect}
In the \emph{temporal gauge}\index{temporal gauge} (vanishing
electrostatic potential\index{electrostatic potential}) the
configuration space\index{configuration space} of classical
electromagnetism\index{electromagnetism} on~$M$\index{$M$!spacetime}
consists of~$\mathbb{R}$-connections\index{connection} on~$S$ modulo gauge
transformations\index{gauge transformation}, and so is isomorphic to a
space of $1$-forms modulo square-integrable exact
$1$-forms\index{exact~$1$-forms} on~$S$. In physics, such a~$1$-form
is called a \emph{vector potential}\index{vector potential}. We make
the configuration space\index{configuration space} into a real Hilbert
space\index{real Hilbert space} by defining it as
$$
\mathbf{A}={\mathop{\mathrm{dom}}\{\mathrm{d}\colon L^2\Omega^1_S\to L^2\Omega^2_S\}\over\overline{\mathop{\mathrm{ran}}}\{\mathrm{d}\colon
L^2\Omega^0_S\to L^2\Omega^1_S\}}
\index{$\mathbf{A}$!space of vector potentials}
\index{$\mathrm{d}$!exterior derivative}
\index{$L^2\Omega^p_S$!square-integrable $p$-forms on~$S$}
$$
with its natural real inner product. That is,~$\mathbf{A}$\index{$\mathbf{A}$!space of
vector potentials} consists of equivalence classes of
square-integrable $1$-forms with square-integrable exterior
derivatives, modulo exact $1$-forms. This space is naturally a real
Hilbert space\index{real Hilbert space}.
The canonical conjugate\index{canonical conjugate} of the vector
potential\index{vector potential}~$[A]$\index{$[A]$!gauge equivalence
class of vector potentials} is a divergenceless\index{divergenceless}
$1$-form~$E$\index{$E$!electric field}, called the \emph{electric
field}\index{electric field}. The space of electric fields
$$
\mathbf{E}=\ker\{\mathrm{d}^*\colon L^2\Omega^1_S\to L^2\Omega^0_S\}
\index{$\mathbf{E}$!space of electric fields}
\index{$\mathrm{d}^*$!divergence}
\index{$L^2\Omega^p_S$!square-integrable $p$-forms on~$S$}
$$
is also naturally a real Hilbert space. The phase space of classical
electromagnetism is, then, the real Hilbert space
$$
\mathbf{P}=\mathbf{A}\oplus\mathbf{E}.
\index{$\mathbf{P}$!phase space}
\index{$\mathbf{A}$!space of vector potentials}
\index{$\mathbf{E}$!space of electric fields}
$$
The spaces~$\mathbf{A}$\index{$\mathbf{A}$!space of vector potentials}
and~$\mathbf{E}$\index{$\mathbf{E}$!space of electric fields} are dual
to each other by
$$
\bigl([A],E\bigr)=\int_S g\bigl(A,E)\mathrm{vol},
\index{$(~,~)$!$p$-form inner product on~$S$}
\index{$S$!space}
\index{$g$!optical metric}
\index{$[A]$!gauge equivalence class of vector potentials}
\index{$E$!electric field}
\index{$\mathrm{vol}$!volume form of~$g$}
$$
which is independent of the representative~$A$ chosen for~$[A]$
because~$E$ is divergenceless. The symplectic
structure\index{symplectic structure} on~$\mathbf{P}$\index{$\mathbf{P}$!phase
space} is constructed from this duality pairing\index{duality pairing}
by antisymmetrization:
$$
\omega\bigl([A]\oplus E,[A']\oplus E'\bigr)=\int_S\bigl[g(A,E')-g(A',E)\bigr]\mathrm{vol}.
$$
Because of global hyperbolicity\index{global hyperbolicity}, any point
$X=[A]\oplus E$\index{$X$!field configuration}\index{$[A]$!gauge
equivalence class of vector potentials}\index{$E$!electric field} of
the physical phase space determines a unique solution of Maxwell's
equations\index{Maxwell's equations} on all
of~$M$\index{$M$!spacetime}. Time evolution\index{time evolution} is
given by a continuous one-parameter group\index{one-parameter group}
of continuous symplectic
transformations~$T(t)\colon\mathbf{P}\to\mathbf{P}$\index{$T(t)$!time
evolution}\index{$\mathbf{P}$!phase space}. Unlike the symplectic
structure\index{symplectic structure} and the
Hamiltonian\index{Hamiltonian}, the natural Hilbert space norm
on~$\mathbf{P}$\index{$\mathbf{P}$!phase space} is not preserved by this time
evolution\index{time evolution}.
As a result of gauge-fixing\index{gauge fixing}, when
restricted to the phase space\index{phase space} the
Laplacian\index{Laplacian} on~$1$-forms
is~$\Delta=\mathrm{d}^*\mathrm{d}$\index{$\Delta$!Laplacian}\index{$\mathrm{d}$!exterior
derivative}. The assumption that spacetime is static\index{static
spacetime} then implies that time evolution\index{time evolution}
commutes with~$\Delta$\index{$\Delta$!Laplacian}, and so the phase
space\index{phase space} admits the decomposition
$$
\mathbf{P}=\mathbf{P}_o\oplus\mathbf{P}_f
\index{$\mathbf{P}$!phase space}
\index{$\mathbf{P}_o$!oscillating sector of phase space}
\index{$\mathbf{P}_f$!free sector of phase space}
$$
where~$\mathbf{P}_f$\index{$\mathbf{P}_f$!free sector of phase space} is
the kernel of~$\Delta$\index{$\Delta$!Laplacian}
in~$\mathbf{P}$\index{$\mathbf{P}$!phase space} and consists of generalized
Aharonov--Bohm modes\index{Aharonov--Bohm modes}. From the point of
view of dynamics, the direct
summand~$\mathbf{P}_o$\index{$\mathbf{P}_o$!oscillating sector of phase space}
consists of `oscillating modes'\index{oscillating modes}
and~$\mathbf{P}_f$\index{$\mathbf{P}_f$!free sector of phase space} of `free
modes'\index{free modes}. Specifically,
on~$\mathbf{P}_o$\index{$\mathbf{P}_o$!oscillating sector of phase space} the
Hamiltonian\index{Hamiltonian} is a positive-definite quadratic form,
and so that `sector' of the electromagnetic
field\index{electromagnetic field} has the dynamics of an
infinite-dimensional harmonic oscillator\index{harmonic
oscillator}. The free sector\index{free
sector}~$\mathbf{P}_f$\index{$\mathbf{P}_f$!free sector of phase space} has
dynamics analogous to those of a free particle\index{free particle}.
For the free sector\index{free sector} one can successfully apply the
algebraic approach to quantization\index{algebraic quantization} of
Chapter~\ref{chap:linear}, but the existence of a Hilbert-space
representation\index{Hilbert-space represetation} on which time
evolution is unitarily implementable\index{unitary time evolution} is
not guaranteed unless~$\mathbf{P}_f$\index{$\mathbf{P}_f$!free sector of phase
space} is finite-dimensional. As we shall see in
Chapter~\ref{sec:harmonic}, that may not be the case on a noncompact
space\index{noncompact space} even if it is topologically trivial.
\include{electromagnetism}
\chapter{$p$-form electromagnetism in~$n+1$ dimensions}
\label{sec:N+1}
In this chapter we generalize the results of the last chapter to electromagnetism on spacetimes of arbitrary dimension~$n+1$.
As before, we take our spacetime to be of the form~$\mathbb{R} \times S$,
equipped with the Lorentzian metric
$$
g_M = e^{2\Phi}(-dt^2 + g)
$$
where~$g$ is a complete Riemannian metric on~$S$. The only difference
is that now $S$ is $n$-dimensional instead of 3-dimensional. However,
this means that Maxwell's equations are no longer conformally
invariant, so the `gravitational potential'~$\Phi$ plays a more
significant role. To see why, recall that the Maxwell action is still
given by
$$
{\mathcal{S}}[A_M] = -{1\over 2}\int_M g_M(F_M , F_M )\, \mathrm{vol}_M
$$
where
$$
F_M = \mathrm{d} t\wedge(\partial_t A - \mathrm{d} A_0) + \mathrm{d} A.
$$
By Equation~(\ref{eq:stpart2}),
$$
g_M(F_M,F_M)=
e^{-4\Phi} \, \bigl[ -g(\partial_t A - dA_0,\partial_t A - dA_0)
+ g(dA,dA) \bigr]
$$
and, by Equation~(\ref{eq:vol}),
$$
\mathrm{vol}_M = e^{(n+1)\Phi} \mathrm{vol} \wedge dt
$$
where~$\mathrm{vol}$ is the volume form on space. Hence, we have
$$
{\mathcal{S}}[A,A_0] = {1\over 2}\int_M
\bigl [g(\partial_t A - \mathrm{d} A_0,\partial_t A - \mathrm{d} A_0) -
g(\mathrm{d} A,\mathrm{d} A) \bigr]\, e^{(n-3)\Phi} \mathrm{vol} \wedge dt .
$$
The factors involving~$\Phi$ cancel only if $n = 3$, indicating
conformal invariance. In other dimensions, the most elegant way to
deal with these factors involving~$\Phi$ is to \emph{redefine} the
fields~$A_0$ and~$A$ by setting
$$
A_M = e^{-{1\over 2}(n-3)\Phi} (\mathrm{d} t \wedge A_0+A),
$$
and then to `twist' the exterior derivative of differential forms on
space, defining a new operator
$$
D = e^{-{1\over 2}(n-3)\Phi} \, \mathrm{d} \, e^{{1\over 2}(n-3)\Phi} .
$$
The action is then
$$
{\mathcal{S}}[A,A_0] = {1\over 2}\int_M
\bigl [g(\partial_t A - DA_0,\partial_t A - DA_0) -
g(DA,DA) \bigr] \mathrm{vol} \wedge \mathrm{d} t
$$
which is formally just like equation~(\ref{eq:action}) was in the
$(3+1)$-dimensional case, but with rescaled fields~$A$ and~$A_0$, and
with the twisted exterior derivative~$D$ replacing the usual~$\mathrm{d}$.
With the help of this formal equivalence, the whole theory goes
through almost exactly as before. In particular, if we let $L^2
\Omega^p$ be the Hilbert space consisting of all square-integrable
$p$-forms on $S$, there are mutually adjoint operators
$$
\xymatrix{L^2\Omega^0\ar@<.5 ex>[r]^{D_0} &
L^2\Omega^1\ar@<.5 ex>[l]^{D_0^*}\ar@<.5 ex>[r]^{D_1} &
L^2\Omega^2\ar@<.5 ex>[l]^{D_1^*}}
$$
Using the Kodaira decomposition for this sequence we obtain
$$
L^2\Omega^1=
\overline{\mathop{\mathrm{ran}} D_0}
\oplus
\ker L_1
\oplus
\overline{\mathop{\mathrm{ran}} D_1^*}
$$
where now the Laplacian is replaced by the `twisted Laplacian' $L_1$,
a nonnegative self-adjoint operator on 1-forms given by
$$
L_1 = D_1^* D_1 + D_0 D_0^*.
$$
In fact, having come this far, it would be a pity not to treat
`$p$-form electromagnetism', a generalization of Maxwell's equations
in which the electromagnetic vector potential is replaced by a
$p$-form. The general case was treated by Henneaux and
Teitelboim~\cite{henneaux86}. For $p = 0$, this theory is just the
massless neutral scalar field. For $p = 2$, it is the Kalb--Ramond
field arising naturally in string theory~\cite[Section
3.4.5]{green87}\cite{kalb74}, while for $p = 3$ it plays a part in
11-dimensional supergravity \cite{duff99}. All our formulas generalize
painlessly to these theories in the absence of charges. Starting
with the $p$-form~$A_M$ on spacetime, we define a field strength
tensor $F_M = \mathrm{d}_M A_M$, and take the action of the theory to be
$$
{\mathcal{S}}[A_M] = -{1\over 2}\int_M g_M(F_M , F_M )\, \mathrm{vol}_M .
$$
This action gives equations of motion and gauge symmetries having
the same form as in Maxwell theory. Furthermore, if we set
$$
A_M = e^{-{1\over 2}(n-2p-1)\Phi} (\mathrm{d} t \wedge A_0+A)
$$
and define the twisted differential~$D$ as follows:
$$
D = e^{{1\over 2}(n-2p-1)\Phi} \mathrm{d} e^{-{1\over 2}(n-2p-1)\Phi} ,
$$
we obtain
$$
{\mathcal{S}}[A,A_0] = {1\over 2}\int_{\mathbb{R}}
\bigl [(\partial_t A - DA_0,\partial_t A - DA_0) -
(DA,DA) \bigr]\mathrm{d} t
$$
in complete analogy with ordinary Maxwell theory. This allows us to
generalize all our results on Maxwell's equations to the $p$-form
case.
\include{pform}
\chapter{Hodge--de~Rham theory on noncompact manifolds}
\label{sec:harmonic}
As we have seen, the space of harmonic differential forms, consisting
of closed and coclosed differential forms, plays a special role in
the analysis of the phase space of Maxwell's equations: it corresponds
to the space of physical vector potentials with vanishing magnetic
field (Aharonov--Bohm effect), and also to static electric fields with
no finite sources (charge without charge).
When space is compact, it is well known that the Hodge-de~Rham theorem
identifies the square-integrable, smooth and real cohomologies of a
space, and that the space of square-integrable harmonic forms
coincides with the kernel of the Hodge Laplacian~$\Delta=\mathrm{d}\delta
+\delta\mathrm{d}$.
When space is noncompact everything becomes more complicated. To begin
with, the definition of the codifferential~$\delta$ involves
integration by parts. As a result, unless space is complete in the
optical metric it may be
impossible to define the codifferential (and hence the Laplacian)
without specifying boundary conditions at infinity. When the optical
metric on space is
complete, not only is there an unambiguous definition of the
codifferential and Laplacian, but the space of $L^2$~harmonic forms is
identified with the kernel of the Hodge Laplacian, and it has a
square-integrable cohomology interpretation. However, the
square-integrable cohomology is not a topological invariant, as it
depends crucially on the geometry at infinity.
These are the main questions one can ask about the Laplacian~$\Delta$
on a complete Riemannian manifold~\cite{lott97,carron01,carron02}:
\begin{enumerate}
\item Is the dimension of~$\ker\Delta_p$ finite or infinite? In
physical terms, this is
the dimension of the space of $p$-form Aharonov--Bohm modes.
\item What are sufficient conditions for~$\ker\Delta_p$ to be trivial
or finite-dimensional?
\item If $\ker\Delta_p$ is finite-dimensional, does it have a
topological interpretation?
\item Is~$0$ in the essential spectrum of~$\Delta_p$? Physically, this
signals the presence of \emph{infrared divergences} for massless
$p$-form fields. Conversely, if the essential spectrum is
bounded away from~$0$, we have a mass gap
for a free massless field induced by the spatial geometry at infinity! Note that it is possible for~$0$ to
be in the essential spectrum of the Laplacian even
if~$\ker\Delta_p$ is trivial, and that the most familiar example
of this is Euclidean~$\mathbb{R}^n$.
\end{enumerate}
\noindent The answer to all of these questions depends on the
behaviour of the curvature of the optical metric at infinity, so even a massless field may
acquire an `effective mass'. In this chapter we collect some known facts and open
issues about the space~$\ker\Delta_p$ of harmonic~$p$-forms and the
spectrum of the Laplacian on a complete Riemannian manifold~$S$, and
give physical interpretations of them. Although this chapter is a
review, it points out how rich the subject is compared to the
amount of attention it has received from physicists.
This chapter is based in part on the excellent review of harmonic
forms on noncompact manifolds by Carron~\cite{carron01} (in French),
which includes his finite-dimensionality results~\cite{carron99}
obtained from Sobolev-type inequalities involving the
curvature. Another paper of his~\cite{carron02} (in English) contains a shorter overview, and a
geometrical interpretation of the~$L^2$ cohomology of manifolds with
flat ends (which are known to have finite cohomologies). The~$L^2$
cohomology of hyperbolic manifolds is described by Lott~\cite{lott97}. The case of geometrically finite hyperbolic
manifolds was obtained by Mazzeo and Phillips~\cite{mazzeo90},
including a calculation of the essential spectrum of the
Laplacian. Mazzeo also calculated the cohomology and essential
spectrum of the Laplacian for conformally compact metrics~\cite{mazzeo88}. The~$L^2$ cohomology for rotationally symmetric
manifolds was obtained by Dodziuk~\cite{dodziuk79}.
An additional complication is the `twisting' of the cohomology
complex:
$$
\xymatrix{L^2\Omega^{k-1}_S\ar[r]^{\mathrm{d}}\ar[d]^{e^{{1\over
2}(n-2p-1)\Phi}}&L^2\Omega^k_S\ar[r]^{\mathrm{d}}\ar[d]^{e^{{1\over
2}(n-2p-1)\Phi}}&L^2\Omega^{k+1}_S\ar[d]^{e^{{1\over 2}(n-2p-1)\Phi}}\\
L^2\Omega^{k-1}_S\ar[r]^{D}&L^2\Omega^k_S\ar[r]^{D}&L^2\Omega^{k+1}_S}
$$
For compact $S$ the (smooth) function $\Phi$ is bounded,
multiplication by $e^{{1\over 2}(n-1-2p)\Phi}$ is bi-continuous on each
$L^2\Omega^k$, the twisted $L^2$ cohomology coincides with the
ordinary $L^2$ cohomology, and the latter with the de~Rham cohomology
by Hodge's theorem. For non-compact $S$, however, $\Phi$ might be
unbounded, in which case the twisted $L^2$ cohomology complex need not
be isomorphic to the ordinary $L^2$ cohomology complex, which we know
already can be very much unlike the de~Rham cohomology complex for
which we have some intuition. Note that if~$n+1=2(p+1)$ (when $p$-form
electromagnetism is conformally invariant) there is no twisting of the
cohomology complex, so the only subtleties are the differences between
the $L^2$ and de~Rham cohomologies.
Since the twisted Laplacian~$DD^*+D^*D$ has not been studied in nearly
as much detail as the ordinary Hodge Laplacian, we know little about
its behaviour. Therefore, when~$\Phi$ is unbounded, most of what we will
say in this chapter is directly applicable only to the cases
where~$p$-form electromagnetism is conformally invariant---\textit{i.e.},
$p$-form electromagnetism in~$2(p+1)$-dimensional spacetime, which
includes the classical case of~$1$-forms in~$3+1$ dimensions.
\include{topology}
\part{Quantum electromagnetism}
The apparent truism that a quantum mechanical theory\index{quantum
mechanical theory} needs to be cast in classical
language\index{classical language} in order to correlate its
predictions with our experience, a point that Niels Bohr\index{Niels
Bohr} made into a cornerstone of his philosophy of quantum
mechanics\index{philosophy of quantum mechanics}, has practical
consequences for the development of quantum descriptions of physical
systems\index{physical system}. This is because a physical system will
be described operationally or geometrically in inevitably classical
terms, and this information needs to be fashioned into a quantum
theory\index{quantum theory} whose predictions need to be, again,
reexpressed in classical terms. In addition, the process of
constructing a classical theory\index{classical theory} from
operational or geometric data is so well-understood that it is
convenient to construct the quantum theory by first constructing a
classical theory from the data and then `quantizing' it.
Quantization\index{quantization} is a catch-all term for any process
taking as input a classical mechanical system\index{classical
mechanical system}, and producing as output a quantum mechanical
system\index{quantum mechanical system} reducing to the original
classical system\index{classical system} in an appropriate
limit. Quantization would ideally be algorithmic or functorial, but it
turns out to be neither, although formulating quantization in
algebraic language\index{algebraic language} seems to bring it closest
to the goal of functoriality.
In algebraic terms, a classical mechanical system\index{classical
mechanical system} is defined by specifying a Poisson
algebra\index{Poisson algebra} of observables, while any associative
algebra can play the role of algebra of observables\index{algebra of
observables} for a quantum system\index{quantum system}. The Dirac
quantization prescription~\cite[Chapter IV]{dirac57}\index{Dirac
quantization prescription} `promotes' the commuting classical
observables\index{classical observables} to operators satisfying the
Heisenberg commutation relations\index{Heisenberg commutation
relations}
$$
[\hat f,\hat g]=i\hbar\{\widehat{f,g}\},
\index{$\hat f$!quantized observable}
\index{[~,~]!commutator}
\index{$\hbar$!Planck's constant}
\index{$f$!observable}
\index{$\{~,~\}$!Poisson bracket}
$$
where~$\{f,g\}$ is the Poisson bracket\index{Poisson bracket} of
the classical observables~$f$ and~$g$, $[\hat f,\hat g]$ is the
commutator\index{commutator} of their quantum counterparts, and
Planck's constant~$\hbar$\index{Planck's constant} measures the
departure from classical behaviour\index{classical behaviour} (where
observables commute). It is not hard to convince oneself that, because
the algebra of quantum observables\index{quantum observables} is
nonabelian, the operation~$f\mapsto\hat f$ cannot be an algebra
homomorphism\index{algebra homomorphism}. That
is,~$\widehat{fg}\neq\hat f\hat g$ in general. Physicists call this
fact `operator ordering\index{operator ordering} ambiguities'.
An operator algebra of quantum observables\index{quantum observables}
realizing the canonical commutation relations achieves
quantization\index{quantization} in a kinematical sense, but the
physical and dynamical content of the theory comes about by means of a
specific representation of the quantum observables as an algebra of
(unbounded) linear operators on a Hilbert space\index{Hilbert space}
of quantum states\index{quantum states}. Each representation is
associated to a choice of `vacuum expectation'\index{vacuum
expectation} on the algebra of observables\index{algebra of
observables} and it is known that, for systems with infinitely many
degrees of freedom, different states may lead to unitarily
inequivalent representations. The choice of representation can be
narrowed down by the need to recover an appropriate classical limit,
and by requiring that physical symmetries be implemented unitarily.
The classical limit is encoded in the correspondence
principle\index{correspondence principle}, by which we mean the
following. The Poisson algebra\index{Poisson algebra} of classical
observables\index{classical observables} consists of smooth functions
on a symplectic manifold (phase space\index{phase space}) playing the
role of state space\index{state space} for the classical
theory\index{classical theory}. The correspondence
principle\index{correspondence principle} requires that, for any phase
space point~$x\in\mathbf{P}$\index{$x$!phase space
point}\index{$\mathbf{P}$!phase space} and any
observable~$f$\index{$f$!observable}, there should be a quantum
state~$\ket{x}$\index{$\ket{x}$!quantized phase space point} such that
the expected value of~$\hat f$\index{$\hat f$!quantized observable} in
the state~$\ket{x}$ equals the classical value~$f(x)$, if not exactly,
at least in the limit~$\hbar\to 0$. That is,
$$
\matElem{x}{\hat f}{x}=f(x)+O(\hbar).
\index{$\matElem{x}{\hat f}{y}$!matrix element of~$\hat f$}
$$
There is one last requirement that a sensible quantization must
satisfy, and that is that physical symmetries be represented by
unitary operators on the Hilbert space of quantum states of the
system.
In the case where the classical phase space\index{classical phase
space} is a vector space, the linear observables can be identified
with the points of the phase space\index{phase space} itself, and so
the Heisenberg commutation relations\index{Heisenberg commutation
relations} can be implemented on the phase space. In
Chapter~\ref{chap:linear} we develop the
quantization\index{quantization} of an abstract linear system and
develop the concept of a quasioperator on Fock space, and in
Chapter~\ref{chap:qed} we apply this to Maxwell's
equations\index{Maxwell's equations} for the electromagnetic
field\index{electromagnetic field} and express the dynamics of the
quantized electromagnetic field in terms of Wilson loops
quasioperators.
The work most closely akin to ours is that of
Dimock~\cite{dimock92}. Like us, Dimock constructs a $C^*$-algebra of
observables for the electromagnetic field, but he does not exhibit any
states or Hilbert-space representations. He notes in passing that ``in
any case such [Hilbert-space] representations exist, say by a Fock
space construction''. We discuss below some ways in which a Fock space
representations may fail to exist.
Because Dimock describes the classical theory in the covariant
canonical formalism, he is forced to focus on ``the algebraic
structure of the theory, not in the specification of particular
states''. In our terms, Dimock quantizes the electromagnetic field as
a `general boson field'. He also constructs a classical Poisson
bracket, and his quantization procedure is equivalent to our general
linear quantization. Dimock does show that different Hilbert-space
representations lead to~$*$-isomorphic~$C^*$-algebras of
observables. This form of equivalence, however, obviates the possible
physical consequences of unitary inequivalence of Hilbert-space
representations, and for this reason Dimock's paper suffers from what
Earman and coauthors critically term ``algebraic imperialism''
in~\cite{earman}.
Dimock does not show that the classical canonical transformations
associated to changes in the choice of Cauchy surface are implemented
unitarily on the $C^*$-algebras of quantum observables, because that
is simply not true. In fact, Torre and Varadarajan~\cite{torre} show
that, even in the case of free scalar fields on a flat spacetime of
dimension higher than two, there is no unitary transformation between
the Fock representations associated to arbitrary initial and final
Cauchy surfaces. They point out that unitary implementability is
easily obtained if the Cauchy surfaces are related by a spacetime
isometry, though. They also mention related results of Helfer (no
unitary implementation of the $S$-matrix if the `in' and `out'
states are Hadamard states)~\cite{helfer96}, and of van~Hove (only a
small subgroup of the classical canonical transformations is unitarily
implementable)~\cite{hove51}.
Another paper addressing specifically the quantization of the
electromagnetic field is the one by Corichi~\cite{corichi}. Corichi
stresses that Fock quantization depends crucially on the linear
structure of phase space, and characterizes the Fock quantization
procedure as ``completely elementary''.
Here we perform Fock quantization of Maxwell's equations on a static,
globally hyperbolic spacetime with a trivial~$\mathbb{R}$ bundle on
it. Presumably this can be extended to stationary spacetimes, but not
beyond that because of the need for a nontrivial group of
isometries. The treatment of nontrivial or~$\mathrm{U}(1)$ bundles should
require only straightforward modifications, but one of the lessons of
our work is that sometimes there are surprises in store even for
topics as well-understood as electromagnetism.
In chapter~\ref{chap:qed}, because of the appearance of negative
powers of the Laplacian~$\Delta$ (or the twisted Laplacian~$L_p$ in
the general case) in the process, we will be forced to restrict Fock
quantization to the space~$\mathbf{P}_o$ of oscillating modes of the
electromagnetic field. Also, for mathematical convenience one often
assumes that~$\Delta\ge\epsilon>0$ for some~$\epsilon$, which is true
when space is compact but not necessarily when it is
noncompact. However, we do not do this as one cannot exclude the
possibility that the spectrum of~$L_p$ or~$\Delta$ reach all the way
to~$0$ because that is the case in physically interesting situations
such as Minkowski space.
\chapter{Coherent-state quantization of linear
systems}\label{chap:linear}
In this chapter we present a rigorous framework for quantization of
linear dynamics based on the ideas of Irving Segal.
Segal pioneered the idea of of formalizing quantum mechanics in
terms of algebras of observables\index{algebras of observables},
making Hilbert spaces play the subordinate role of supporting linear
representations of them. These Hilbert spaces can, in fact, be
constructed from the abstract algebra of observables by means of the
Gel'fand--Na\u{\i}mark--Segal
construction\index{Gel'fand--Na\u{\i}mark--Segal construction} using a
single \emph{state}\index{GNS state} or, in physics parlance,
\emph{vacuum expectation}\index{vacuum expectation value}.
Implicit in the work of Segal is a concept of \emph{general boson
field}\index{general boson field} associated to any linear phase
space\index{linear phase space}, which formalizes the Heisenberg
commutation relations\index{Heisenberg commutation relations} among
field operators in terms of exponentiated field operators, using the
so-called \emph{Weyl relations}\index{Weyl relations}. This has the
advantage of avoiding the technicalities of unbounded
operators\index{unbounded operators}. In addition, physical symmetries
are readily implemented as automorphisms of the Weyl algebra.
Segal introduced the related concept of \emph{free boson
field}\index{free boson field}, which can be constructed from a phase
space equipped with a compatible complex structure\index{compatible
complex structure}. Segal's free boson field axiomatizes the
properties of the usual of Fock space\index{Fock space}, and the
axiomatic approach makes it transparent that the Fock\index{Fock
representation}, Schr\"odinger\index{Schr\"odinger representation} and
Bargmann--Segal\index{Bargmann--Segal representation} representations
of linear quantum fields are all unitarily equivalent. Within this
framework, Segal also studied the problem of representing time
evolution unitarily on Fock space\index{Fock space}, and the stability
of the generator of unitary time evolution\index{unitary time
evolution}, namely whether the quantum Hamiltonian\index{Hamiltonian}
is bounded below.
Here we put together both ideas, and the result is a new construction
of the free boson field\index{free boson field} based on
\emph{coherent states}\index{coherent states}. In this construction we
not only associate to each linear functional\index{linear functional}
on phase space\index{phase space} a field operator\index{field
operator} but, given a choice of vacuum state\index{vacuum state}, we
can associate to each point in phase space\index{phase space point} a
coherent state\index{coherent state}. The collection of all coherent
states indexed by points of phase space spans the Hilbert space of
quantum states\index{quantum state space} of the theory, and the
result is what Segal called the \emph{general boson field}. The free
boson field, which as we have mentioned is unitarily equivalent to the
Fock representation, is obtained by means of a GNS state with Gaussian
statistics.
We find that the mathematical process of quantization can be
understood with reference to three physical guiding principles: the
canonical commutation relations, the correspondence principle, and the
unitary implementation of physical symmetries.
We proceed as follows: we first construct the Weyl algebra\index{Weyl
algebra} of observables associated to a linear phase
space\index{linear phase space}, and then choose a compatible complex
structure\index{compatible complex structure} on the Phase space,
which amounts to
selecting a vacuum expectation\index{vacuum
expectation} on the Weyl algebra, with the help of the correspondence
principle and the requirement that time evolution\index{time
evolution} be unitarily and stably implemented.
Coherent states\index{coherent states} are most useful because many
classical equations hold exactly between expectation
values\index{expectation value} on coherent states. Thus, by using
coherent states, our quantization\index{quantization} procedure never
loses sight of the correspondence principle\index{correspondence
principle}. In addition, the vacuum expectation value\index{vacuum
expectation value} acts as a generating function\index{generating
function} of the matrix elements\index{matrix element} of field
operators\index{field operator} between coherent states\index{coherent
state}, not only for ordinary field operators\index{field operator}
but also for their Wick powers\index{Wick power} (called
normal-ordered operators\index{normal-ordered operator} in
physics). As an unexpected bonus, using matrix elements\index{matrix
elements} between coherent states\index{coherent states} one can
define normal-ordered Wilson loops\index{Wilson loops} as
quasioperators\index{quasioperators} without the need for
regularization\index{regularization}.
Segal's treatment of the free boson field\index{free boson field} is
presented in~\cite{BSZ}. A comprehensive physical treatment of the
coherent states\index{coherent states} of the electromagnetic
field\index{electromagnetic field} can be found in~\cite[Chapter
11]{mandel95}.
\include{quantization}
\chapter{$p$-form Electromagnetism as a Free Boson Field}
\label{chap:qed}
In this chapter we show how the oscillating modes of~$p$-form
electromagnetism in~$(p+1)$-dimensions have a free boson field
representation, define certain physically interesting observables as
quasioperators, and prove that suitable analogues of the classical
equations of motion hold as quasioperator equations.
According to Section~\ref{sec:freeBosonField}, in order to construct a
free boson field representation we need a complex Hilbert
space~$\mathbf{H}$ consisting of classical observables of~$p$-form
electomagnetism. The same vector space with its real structure will be
denoted~$\mathbf{P}^*$ since the space of observables is the dual of the
physical phase space~$\mathbf{P}$. The complex inner
product~$\langle~,~\rangle$ on~$\mathbf{H}$ must have as its imaginary part
the classical symplectic structure~$\omega$ on~$\mathbf{P}^*$. The free
boson field on~$\mathbf{H}$ is the representation of~$\mathcal{W}(\mathbf{P}^*,\omega)$
produced by the GNS construction applied to a state~$\langle~\rangle$
with characteristic functional~$\mu(f)=\exp(-\|f\|^2/4)$ for
all~$f\in\mathbf{H}$. Now, in order for time evolution to be unitary, it is
sufficient that~$\mu$ be invariant under time evolution; in other
words,~$\mu$ and hence~$\|~\|$ must be constants of the motion.
However, the analysis of the classical theory produces a real phase
space, without a complex structure and not having necessarily even a
real Hilbert space structure. That is, the starting point for
quantization is a classical phase space~$\mathbf{P}$ whose
dual~$(\mathbf{P}^*,\omega)$ is a a real topological vector space
with a continuous symplectic structure~$\omega$. Time evolution acts
on phase space as a strongly continuous one-parameter group of
\emph{bounded} operators~$T(t)$ preserving the symplectic structure
on~$\mathbf{P}^*$. To quantize these symplectic dynamics involves
constructing from~$(\mathbf{P}^*,\omega)$ and~$T$ a complex Hilbert
space~$\mathbf{H}$ on which~$T(t)$ is a strongly-continuous one-parameter
group of \emph{unitary} operators.
Ordinarily, for instance when quantizing a massive linear field such
as the Klein--Gordon field,~$\mathbf{H}$ would carry a weaker norm
than~$\mathbf{P}^*$, and so~$\mathbf{P}^*$ would be contained
in~$\mathbf{H}$. However, as we shall see, when there are infrared
divergences (as is the case for massless fields such as the Maxwell
field) neither~$\mathbf{H}$ nor~$\mathbf{P}^*$ contain each other. However,
there is a common subspace of both~$\mathbf{P}^*$ and~$\mathbf{H}$ on which all
the mathematical objects we are discussing are well-defined. This
space is constructed as a subspace of~$\mathbf{P}^*$ in a well-prescribed
way and then completed to obtain~$\mathbf{H}$.
An additional complication is the existence of a nontrivial
Aharonov--Bohm sector. We have seen that the dynamics in this sector
are analogous to those of a free particle. In the case of the
electromagnetic field, we will see that the definition of~$\mathbf{H}$
involves negative powers of the Laplacian, and so the Aharonov--Bohm
sector must be quantized in a different way, if at all. Accordingly,
although we set out to quantize~$\mathbf{P}\simeq\mathbf{P}_o\oplus \mathbf{P}_f$,
we really only achieve a Fock quantization of the oscillating
sector~$\mathbf{P}_o$. We do not attempt to determine whether a free boson
field representation of the free modes is possible; we expect this to
be the case only when~$\mathbf{P}_f$ is finite-dimensional. Moreover, only
on~$\mathbf{P}_o$ is it possible to find a (densely-defined) complex
structure preserved by time evolution. On the space~$\mathbf{P}_f$ of
Aharonov--Bohm modes, the time evolution operator~$T_f(t)$ is a shear,
and there is no way to make it unitary.
The plan of this short chapter is as follows. In
Section~\ref{sec:freeBosonEM} we construct the free boson field
representation of the oscillating sector of~$p$-form
electromagnetism. In Section~\ref{sec:punchline} we use our
quasioperator technology from Section~\ref{sec:quasioperators} to make
sense of Wilson loop operators and their higher-dimensional
generalizations, as well as electromagnetic field operators at a
point, which are then shown to satisfy the Maxwell equations as
quasioperator equations. Most importantly, we end with a description
of the dynamics of the electromagnetic field in terms of Wilson loops,
without any need for `regularizing' or `smearing' these loops as in
the work of Varadarajan~\cite{varadarajan00,varadarajan01}.
\include{QED}
\ssp
\bibliographystyle{alpha}
\section{Geometric setting}
\label{sec:3+1-geometry}
In this section we describe the mathematical framework for our study
of classical electromagnetism\index{electromagnetism}, and explain the
mathematical reasons why various physical restrictions are imposed on
the class of spacetimes\index{spacetime} under consideration.
\subsection{Static globally hyperbolic spacetimes}
Let us begin by recalling the precise definition of a
static\index{static spacetime}, globally hyperbolic
spacetimes\index{globally hyperbolic spacetime}. In physical terms, a
spacetime\index{spacetime} is stationary\index{stationary spacetime}
if it is invariant under time translations\index{time translations}
and static\index{static spacetime} if, in addition, it is invariant
under time reversal\index{time reversal}. Our first definition casts
these intuitive concepts in the language of (pseudo-)Riemannian
geometry.
\begin{definition}[stationary and static spacetimes]
A Lorentzian manifold\index{lorentzian manifold} without timelike
loops\index{timelike loop} (also called a
\emph{spacetime})\index{spacetime} is
\emph{stationary}\index{stationary spacetime} if, and only if, it
admits a one-parameter group of isometries\index{isometry} with
smooth, timelike orbits. A stationary spacetime\index{stationary
spacetime} is \emph{static}\index{static spacetime} if, in addition,
it is foliated\index{foliation} by a family of spacelike
hypersurfaces\index{spacelike hypersurface} everywhere orthogonal to
the orbits of the isometries.
\end{definition}
\begin{proof}[Note]
Spacetimes\index{spacetime} with closed timelike loops\index{closed
timelike loop} lead to a breakdown of the ordinary
initial-value\index{initial-value problem} formulation of
dynamics\index{dynamics}, and so must be excluded from our
analysis. Diffeomorphism\index{diffeomorphism} with smooth, timelike
orbits are generated by an everywhere timelike vector field. A vector
field\index{vector field} generating isometries is called a
\emph{Killing vector field}\index{Killing vector field}, and the
isometries\index{isometry} generated by a timelike Killing
field\index{timelike Killing field} are called \emph{time
translations}\index{time translation}. A stationary\index{stationary
spacetime} spacetime~$M$\index{$M$!spacetime} is
diffeomorphic\index{diffeomorphic} to~$\mathbb{R}\times
S$\index{$\mathbb{R}$!time}\index{$S$!space} for some smooth
manifold\index{smooth manifold}~$S$\index{$S$!space} representing
`space'\index{space}; if, in addition, $M$\index{$M$!spacetime} is
static\index{static spacetime}, it admits a metric\index{metric} of the
form
$
g_M=-e^{2\Phi}\mathrm{d} t^2+g_S,
\index{$g_M$!spacetime metric}
\index{$\Phi$!Newtonian potential}
\index{$t$!time coordinate}
\index{$g_S$!space metric}
$
where~$\Phi$\index{$\Phi$!Newtonian potential} is a time-independent
function on~$S$\index{$S$!space}, and~$g_S$\index{$g_S$!space metric}
is a time-independent Riemannian metric\index{Riemannian metric}
on~$S$\index{$S$!space}. A stationary spacetime\index{stationaty
spacetime} would require cross-terms of the form~$e^\Phi(\mathrm{d}
t\otimes\alpha+\alpha\otimes\mathrm{d} t)$ in the
metric\index{metric},~$\alpha$ being a nonzero
time-independent~$1$-form on~$S$. For proofs of these statements see,
for instance,~\cite{wald84}.
\end{proof}
The concept of global hyperbolicity\index{global byperbolicity} is
more subtle, but it is related to the simple idea of
causality\index{causality}: that points of spacetime\index{spacetime}
are partially ordered\index{partial order} by the
relation\index{relation} `being to the future of'. The name `global
hyperbolicity'\index{global hyperbolicity} originally referred to a
property of systems of partial differential equations\index{partial
differential equation} on Euclidean space\index{Euclidean space}. By
reinterpreting those equations as coordinate\index{coordinate}
representations of equations adapted to a curved Lorentzian
manifold\index{Lorentzian manifold}, the
hyperbolicity\index{hyperbolicity} of the system became a geometric
property of the spacetime\index{spacetime} itself (see~\cite{geroch70}
and references therein). As we shall see, global
hyperbolicity\index{global hyperbolicity} of the
spacetime\index{spacetime} implies that the evolution
equations\index{evolution equation} of massless fields\index{massless
field} are globally hyperbolic\index{globally hyperbolic} systems of
partial differential equations\index{partial differential equation}.
Hyperbolic systems of partial differential equations\index{hyperbolic
partial differential equations} have a finite propagation
velocity\index{finite propagation velocity}, meaning that
compactly-supported initial data\index{initial data} evolve into
compactly-supported solutions after a finite time. Under the
reinterpretation of hyperbolic systems\index{hyperbolic system} as
propagation equations on Lorentzian manifolds\index{Lorentzian
manifold}, the finite propagation velocity\index{finite propagation
velocity} means that solutions with compactly-supported initial
data\index{initial data} are completely contained in the light
cones\index{light cone} of the support of their initial data. This is
one of the manifestations of causality\index{causality}.
The following definition formalizes the geometric ideas of
causality\index{causality} and global hyperbolicity\index{global
hyperbolicity}.
\begin{definition}[globally hyperbolic spacetime]
A piecewise-smooth curve\index{smooth curve} in a
spacetime\index{spacetime}~$M$\index{$M$!spacetime} is \emph{causal}
if its tangent vector\index{tangent vector} is everywhere
timelike\index{timelike}. A set is \emph{achronal}\index{achronal set}
if there are no causal curves between any two of its points. The
\emph{domain of dependence}\index{domain of dependence} of a set
consists of all points~$p\in M$\index{$p$!point of
spacetime}\index{$M$!spacetime} such that every
inextensible\index{inextensible curve} causal curve\index{causal
curve} through~$p$\index{$p$!point of spacetime} intersects the set. A
\emph{Cauchy surface}\index{Cauchy surface} in a
spacetime\index{spacetime}~$M$\index{$M$!spacetime} is a
closed\index{closed set} achronal set\index{achronal set} whose domain
of dependence\index{domain of dependence} is all
of~$M$\index{$M$!spacetime}. A spacetime\index{spacetime} is
\emph{globally hyperbolic}\index{globally hyperbolic spacetime} if,
and only if, it admits a Cauchy surface\index{Cauchy surface}.
\end{definition}
\begin{proof}[Note]
The domain of dependence\index{domain of dependence} is also called
the \emph{Cauchy development}\index{Cauchy development}. Both names,
`domain of dependence'\index{domain of dependence} and `Cauchy
development'\index{Cauchy development}, betray their origin in the
theory of partial differential equations\index{partial differential
equations}, as does the term `Cauchy surface'\index{Cauchy surface}.
A Cauchy surface\index{Cauchy surface} in a
spacetime\index{spacetime}~$M$\index{$M$!spacetime} is an achronal
set\index{achonal} intersecting every inextensible\index{inextensible
curve} causal curve\index{causal} in~$M$. It is not hard to see that
closed timelike curves\index{closed timelike curves} cannot intersect
an achronal hypersurface\index{achronal hypersurface}, and so
spacetimes\index{spacetime} with closed timelike curves\index{closed
timelike curve} cannot be globally hyperbolic\index{globally
hyperbolic}. For a static spacetime\index{static spacetime} with
metric\index{metric}
\begin{equation}\label{eq:optical}
g_M=e^{2\Phi}(-\mathrm{d} t^2+g),
\index{$g_M$!metric on spacetime}
\index{$\Phi$!Newtonian potential}
\index{$t$!time coordinate}
\index{$g$!optical metric on space}
\end{equation}
global hyperbolicity\index{global hyperbolicity} is equivalent to
completeness of the metric\index{metric}~$g=e^{-2\Phi}g_S$\index{$g$!optical metric on
space}\index{$\Phi$!Newtonian potential}\index{$g_S$!metric on space}. This
metric\index{metric}~$g$\index{$g$!optical metric} is sometimes called
\emph{optical metric}\index{optical metric} (see, for
instance,~\cite{MR1694235,MR1605624,MR776077,MR0207364}) because light
rays follow geodesics of this metric. More precisely, the
geodesics\index{geodesic} of~$g$\index{$g$!optical metric on space}
parameterized by arc length\index{arc length} lift to affinely
parameterized\index{affine parameterization} lightlike
geodesics\index{lightlike geodesic} of~$-\mathrm{d} t^2+g$\index{$t$!time
coordinate}\index{$g$!optical metric}, with the
time~$t$\index{$t$!time coordinate} corresponding to the
arc-length\index{arc lenght} parameter on geodesics\index{geodesic}
of~$g$\index{$g$!optical metric}. Hence, the propagation of
light\index{light propagation} in the geometric optics\index{geometric
optics} approximation is determined by~$g$\index{$g$!optical metric}
alone. We will consistently use the optical
metric\index{optical metrix}~$g$\index{$g$!optical metric}
on~$S$\index{$S$!space} rather than~$g_S$\index{$g_S$!metric on
space}.
\end{proof}
\subsection{Spacetime geometry and topology}
We model spacetime\index{spacetime} as a static\index{static
manifold}, globally hyperbolic\index{globally hyperbolic manifold},
$(3+1)$-dimensional Lorentzian manifold\index{Lorentzian
manifold}. That is, spacetime\index{spacetime} will be represented by
a smooth $(3+1)$-dimensional manifold~$M$\index{$M$!spacetime}
diffeomorphic to~$\mathbb{R}\times S$\index{$\S$!space} and admitting a
Lorentzian metric\index{Lorentzian metric} of the form given in
Equation~(\ref{eq:optical}).
For convenience, we also assume~$S$\index{$S$!space} is oriented. In
that case, the metric~$g$\index{$g$!optical metric} determines a
volume form~$\mathrm{vol}$\index{$\mathrm{vol}$!volume form on space}
on~$S$. Similarly, the
spacetime\index{spacetime}~$M$\index{$M$!spacetime} acquires a volume
form\index{volume form}~$\mathrm{vol}_M$\index{volume form on spacetime} from
the metric~$g_M$\index{$g_M$!metric on spacetime}. The canonical
volume forms\index{canonical volume form} are related by
\begin{equation}
\mathrm{vol}_M = e^{4\Phi} \mathrm{vol} \wedge \mathrm{d} t .
\index{$\mathrm{vol}_M$!volume form on spacetime}
\index{$\Phi$!Newtonian potential}
\index{$\mathrm{vol}$!volume form on space}
\index{$t$!time coordinate}
\label{eq:vol}
\end{equation}
If~$S$\index{$S$!space} were nonorientable\index{nonorientable space},
we could still carry through our whole discussion with minor
modifications, the most important of which being
that~$\mathrm{vol}$\index{$\mathrm{vol}$!volume form on space}
and~$\mathrm{vol}_M$\index{$\mathrm{vol}_M$!volume form on spacetime} would have to be
treated as densities\index{density}.
We religiously follow the convention of writing all differential
forms\index{differential form} on spacetime\index{spacetime} with a
subscript~`$M$'\index{$M$!spacetime}. We also write the so-called
temporal part\index{temporal part} with a subscript~`$0$', and the
spatial part\index{spatial part} with no subscript. We decompose
$k$-forms\index{$k$-forms} on~$M$\index{$M$!spacetime} into
spatial\index{spatial part} and temporal parts\index{temporal part}
thus:
\begin{equation}
\alpha_M=\mathrm{d} t\wedge\alpha_0+\alpha,
\index{$\alpha_M$!differential form on spacetime}
\index{$t$!time coordinate}
\index{$\alpha_0$!temporal part of~$\alpha$}
\index{$\alpha_S$!spatial part of~$\alpha$}
\label{eq:stpart}
\end{equation}
where~$\alpha_0$ is a $(k-1)$-form and~$\alpha$ is a $k$-form
on~$S$, both $t$-dependent.
The exterior derivative\index{exterior derivative} operators on
spacetime\index{spacetime}~$\mathrm{d}_M\colon C_0^\infty\Omega^k_M\to
C_0^\infty\Omega^{k+1}_M$ and on space $\mathrm{d}\colon
C_0^\infty\Omega^k_S\to C_0^\infty\Omega^{k+1}_S$,
where~$C^\infty_0\Omega^k_S$ denotes smooth, compactly supported
$k$-forms on~$S$, are related by $\mathrm{d}_M=\mathrm{d} t\wedge\partial_t+\mathrm{d}$; in
other words,
\begin{equation}
\label{eq:stderivative}
\mathrm{d}_M\alpha_M=\mathrm{d} t\wedge(\partial_t\alpha-\mathrm{d}\alpha_0)+\mathrm{d}\alpha.
\end{equation}
for all compactly-supported smooth~$k$-forms~$\alpha_M\in
C_0^\infty\Omega^k_M$.
We use~$g$ and~$g_M$ to denote the respective induced metrics
on $k$-forms, satisfying
\begin{equation}
g_M(\alpha_M,\beta_M)=e^{-2k\Phi}\bigl[g(\alpha,\beta)-g(\alpha_0,\beta_0)\bigr],
\label{eq:stpart2}
\end{equation}
and define the positive-definite bilinear forms
\begin{equation}
(\alpha_M,\beta_M)_M=\int_M g_M(\alpha_M,\beta_M)\mathrm{vol}_M
\qquad\hbox{and}\quad
(\alpha,\beta)=\int_S g(\alpha,\beta)\mathrm{vol}
\label{eq:inner}
\end{equation}
on~$C^\infty_0\Omega^k_M$ and~$C^\infty_0\Omega^k_S$, which are related by
\begin{equation}
(\alpha_M,\beta_M)_M=\int_\mathbb{R} e^{(4-2k)\Phi}\bigl[(\alpha,\beta)-(\alpha_0,\beta_0)\bigr]\mathrm{d} t.
\label{eq:inner2}
\end{equation}
We denote by~$\delta$ the formal adjoint of~$\mathrm{d}$ with respect to the
bilinear form~$(~,~)$. This means that the operator~$\delta\colon
C_0^\infty\Omega^{k+1}_S\to C_0^\infty\Omega^k_S$ is defined by
\begin{equation}
(\alpha,\mathrm{d}\beta)=(\delta\alpha,\beta)
\qquad\hbox{for all}\quad
\alpha\in C_0^\infty\Omega_S^{k+1}
\qquad\hbox{and}\quad
\beta\in C^\infty_0\Omega_S^k,
\label{eq:formal_adjoint}
\end{equation}
The compact support in
Equations~(\ref{eq:stderivative}) and~(\ref{eq:formal_adjoint}) has
the function of avoiding boundary terms on the implicit integration by
parts involved in the definition of~$\delta$.
\subsection{Issues of analysis on noncompact spaces}
\label{sec:analysis}
A restatement of Equation~(\ref{eq:formal_adjoint}) is the existence
of operators
\begin{equation}
\xymatrix{C^\infty_0\Omega^k_S\ar@<.5ex>[r]^\mathrm{d} &
C^\infty_0\Omega^{k+1}_S\ar@<.5ex>[l]^\delta}
\label{eq:formal_adjoint2}
\end{equation}
which are formal adjoints of each other. Our goal is to extend these
to densely defined operators between~$L^2\Omega^k$
and~$L^2\Omega^{k+1}$ which are adjoint to each other in the strict
sense of operator theory, where~$L^2\Omega^k$ denotes the space of
square-integrable~$k$-forms on~$S$. It turns out that this can be done
precisely because~$g$ is a complete metric on~$S$, which we have seen
is equivalent to global hyperbolicity of spacetime.
There are both physical and mathematical reasons for wanting to do
this. Mathematically, a mutually adjoint pair of unbounded operators
between two Hilbert spaces are much better behaved than
formally-adjoint operators between spaces of smooth diferential forms,
although the latter have more intuitive geometric appeal. From a
physical point of view, we do not wish to restrict ourselves to
compactly-supported fields in a noncompact space, but on the other
hand we need the fields to be square integrable in order for the
Hamiltonian and symplectic structure on phase space to be finite at
all times. These sorts of physical considerations demand that we
treat~$\mathrm{d}$ and~$\delta$ as unbounded operators between Hilbert spaces
of square-integrable differential forms. To prove that time evolution
maps the classical phase space to itself, we will also need to
extend~$\delta\mathrm{d}$ to an unbounded self-adjoint operator on
square-integrable $1$-forms. Finally, once we insist on
interpreting~$\mathrm{d}$ as an operator between spaces of square-integrable
forms, the electromagnetic gauge transformations will need to have
square-integrable generators.
All this requires a short detour into functional analysis, which is
contained in this subsection. While the facts we need are well-known
to the experts, they may be unfamiliar to some readers, so we review
them in a fair amount of detail. We omit most of the proofs, many of
which can be found in Reed and Simon's textbook~\cite{RS}. The reader
who is more interested in the physical use of these operators can skip
to Section~\ref{sec:maxwell}, with the observation that from then on
the operator~$\delta$ is denoted~$\mathrm{d}^*$, as in
\begin{equation}
\xymatrix{L^2\Omega^k_S\ar@<.5ex>[r]^{\mathrm{d}} &
L^2\Omega^{k+1}_S\ar@<.5ex>[l]^{\mathrm{d}^*}},
\label{eq:formal_adjoint3}
\end{equation}
in order to free the
symbol~$\delta$ for use in variational calculus. Making sense of
Equation~(\ref{eq:formal_adjoint3}) is the main purpose of this
subsection.
In going from Equation~(\ref{eq:formal_adjoint2}) to
Equation~(\ref{eq:formal_adjoint3}), the first thing we need to do is
establish that the operators~$\mathrm{d}\colon C^\infty_0\Omega^k_S\to
C^\infty_0\Omega^{k+1}_S$ and~$\delta\colon C^\infty_0\Omega^{k+1}_S\to
C^\infty_0\Omega^k_S$ appearing in Equations~(\ref{eq:formal_adjoint})
and~(\ref{eq:formal_adjoint2}) can be interpreted as densely-defined
operators between~$L^2\Omega^k_S$ and~$L^2\Omega^{k+1}_S$. This
follows from Lemma~\ref{lem:dense}.
\begin{lemma}\label{lem:dense}
The completion of~$C^\infty_0\Omega^k_S$ with respect to the inner product
$(~,~)$ is~$L^2\Omega^k_S$.
\end{lemma}
\begin{proof}
We can reduce this to the well-known case where $S = \mathbb{R}^n$ using a
partition-of-unity argument.
\end{proof}
The domain of the operator~$\mathrm{d}$ is the dense subspace
$$
\mathop{\mathrm{dom}}\mathrm{d}=C^\infty_0\Omega^k_S \subseteq L^2 \Omega^k_S .
$$
We then define the adjoint~$\mathrm{d}^*$ in the usual way, as
follows. First, the domain of~$\mathrm{d}^*$ consists of all~$\alpha\in
L^2\Omega^{k+1}_S$ for which there exists a~$\gamma\in L^2\Omega^k_S$
such that
$$
(\alpha,\mathrm{d} \beta) =(\gamma,\beta)
$$
for all~$\beta\in C^\infty_0\Omega^k_S$. If such a~$\gamma$ exists it is
unique because~$C^\infty_0\Omega^k_S$ is dense in~$L^2\Omega^k_S$, and we
then define~$\mathrm{d}^*\alpha$ to equal this~$\gamma$, so that
\begin{equation}
(\alpha,\mathrm{d} \beta)=(\mathrm{d}^*\alpha,\beta) \qquad \forall\beta\in
C^\infty_0\Omega^k_S.
\label{eq:adjoint}
\end{equation}
as desired. Note that, because~$\beta$ is required to be of compact
support,~$\alpha$ is not required to have compact support.
Similarly, the dense domain of~$\delta$ is~$C^\infty_0\Omega^{k+1}_S$,
the domain of~$\delta^*$ is not restricted to compactly-supported
forms, and~$\delta^*\beta$ can be defined by
$
(\alpha,\delta^* \beta)=(\delta \alpha,\beta) \qquad \forall\alpha\in C^\infty_0\Omega^{k+1}_S.
$
We can also define operators~$\overline\mathrm{d}$ and~$\overline \delta$,
the respective closures of~$\mathrm{d}$ and~$\delta$. For~$\mathrm{d}$ this goes as
follows. We define the graph of~$\mathrm{d}$ to be the linear subspace
$$
\mathop{\mathrm{Gr}}(\mathrm{d})=\{\alpha\oplus\mathrm{d}\alpha \mid \alpha \in C_0^\infty \Omega^k_S \}
\subseteq L^2\Omega^k_S\oplus L^2\Omega^{k+1}_S
$$
where the latter space is a Hilbert space in an obvious way. This
subspace is typically not closed, and we say that~$\mathrm{d}$ is closable if
the closure of~$\mathop{\mathrm{Gr}}(\mathrm{d})$ is the graph of an operator, which we then
denote~$\overline\mathrm{d}$. In other words,
\begin{equation}
\alpha\in\mathop{\mathrm{dom}}\overline\mathrm{d} \Leftrightarrow
\alpha={\displaystyle\lim_{n\to\infty}}\alpha_n
\qquad\mathrm{and}\quad
\mathrm{d}\alpha_n\to\overline\mathrm{d}\alpha
\qquad\hbox{for some}\quad
\alpha_n\in
C^\infty_0\Omega^k_S.
\label{eq:closure}
\end{equation}
We define the closure~$\overline \delta$ in essentially the same way.
Because of Equation~(\ref{eq:formal_adjoint}) both~$\mathrm{d}^*$
and~$\delta^*$ are densely defined, so the following lemma applies.
\begin{lemma}\label{lem:doubledual}
A densely defined operator $T$ is closable if, and only if, $T^*$ is
densely defined. In that case, $\overline T=T^{**}$.
\end{lemma}
Observe that~$T^*$ is automatically closed and $\overline
T^*=T^*$. As a result, $\overline\mathrm{d}=\mathrm{d}^{**}$ and
$\overline\delta=\delta^{**}$. We have
$\mathrm{d}\subseteq\overline\mathrm{d}\subseteq\delta^*$ and
$\delta\subseteq\overline\delta\subseteq\mathrm{d}^*$.
\begin{proof} See Reed and Simon's textbook~\cite[Theorem
VIII.1]{RS}.
\end{proof}
We have argued that
$$
\mathop{\mathrm{dom}}\overline\mathrm{d} \subseteq \mathop{\mathrm{dom}}\delta^*
\qquad\hbox{and}\quad
\mathop{\mathrm{dom}}\overline\delta \subseteq \mathop{\mathrm{dom}}\mathrm{d}^*,
$$
but $\overline\mathrm{d}$ and $\overline \delta$ will be mutual adjoints only
if these are actually equalities. Having them be mutual adjoints is
highly desirable, as otherwise there are at least two possible
self-adjoint extensions of the operator~$\delta\mathrm{d}$,
namely~$\mathrm{d}^*\overline\mathrm{d}$ and~$\overline\delta\delta^*$. This means
we need to understand how the equations $\mathop{\mathrm{dom}}\overline\mathrm{d} =
\mathop{\mathrm{dom}}\delta^*$ and $\mathop{\mathrm{dom}}\overline\delta = \mathop{\mathrm{dom}}\mathrm{d}^*$ could fail to
hold.
The answer has to do with boundary values. Suppose that~$S$ is a
relatively compact open subset of some larger Riemannian manifold~$X$,
and its boundary~$\partial S$ is a smooth submanifold of~$X$. In this
case the desired equalities \emph{never} hold, and there is a
well-developed theory of boundary values which explains why
\cite{evans98}. In brief, if~$\alpha,\beta$ are compactly supported
smooth forms on~$S$, integration by parts gives
$$
(\mathrm{d}\alpha,\beta) = (\alpha,\delta\beta)
\qquad\hbox{for all}\quad\alpha\in C^\infty_0\Omega^k_S
\qquad\hbox{and}\quad\beta\in C^\infty_0\Omega^{k+1}_S.
$$
From this, an approximation argument gives
$$
(\overline\mathrm{d}\alpha,\beta) = (\alpha,\overline \delta\beta)
\qquad\hbox{if}\quad
\alpha \in \mathop{\mathrm{dom}}\overline\mathrm{d}
\quad\hbox{and}\quad
\beta\in\mathop{\mathrm{dom}}\overline\delta .
$$
On the other hand, if~$\alpha,\beta$ are merely smooth forms
on~$S$ that extend smoothly to~$X$, integration by parts gives
\begin{equation}
(\mathrm{d}\alpha,\beta) - (\alpha,\delta\beta) = (\alpha,\beta)_{\partial S},
\qquad\hbox{for all}\quad\alpha\in C^\infty\Omega^k_S
\qquad\hbox{and}\quad\beta\in C^\infty\Omega^{k+1}_S
\label{eq:boundary}
\end{equation}
and from this, again by an approximation argument, one can show
$$
(\delta^*\alpha,\beta)=(\alpha,\mathrm{d}^*\beta)+(\alpha,\beta)_{\partial S}
\qquad\hbox{if}\quad
\alpha \in \mathop{\mathrm{dom}} \delta^*
\quad\hbox{and}\quad
\beta\in\mathop{\mathrm{dom}}\mathrm{d}^* .
$$
Thus we cannot have $\mathop{\mathrm{dom}}\overline\mathrm{d} = \mathop{\mathrm{dom}}\delta^*$ and
$\mathop{\mathrm{dom}}\overline\delta = \mathop{\mathrm{dom}}\mathrm{d}^*$ in this case: the nonzero boundary
term~$(\alpha,\beta)_{\partial S}$ gets in the way.
The same sort of problem can occur even when~$S$ is not a relatively
compact open subset of some larger Riemannian manifold. However, in
this more general situation the concept of `boundary value' needs to
be reinterpreted as `value at spacelike infinity'. In fact,
Equation~(\ref{eq:boundary}) can be used to define the notion of
boundary at infinity of~$S$. The domain of~$\overline\mathrm{d}$ can be
understood as the space of square-integrable differential forms with
square-integrable exterior derivatives and vanishing `values at
infinity', while the domain of~$\delta^*$ consists of
square-integrable differential forms with square-integrable exterior
derivatives and no restriction on values at infinity. Thus, the
desired equation~$\overline\mathrm{d} = \delta^*$ fails to hold if an element
of~$\mathop{\mathrm{dom}}\delta^*$ can fail to `vanish at infinity'. Similar remarks
apply to the equation~$\overline \delta =\mathrm{d}^*$. Simply put, the
problems arise when there are boundary terms at infinity when we
integrate by parts.
Luckily, the folowing result of Gaffney implies that these problems
never happen when~$g$ is a \emph{complete} Riemannian metric on~$S$.
\begin{proposition}[Gaffney] \label{prop:gaffney}
If $S$ is a complete oriented Riemannian manifold, then
$$
(\delta^*\alpha,\beta)=(\alpha,\mathrm{d}^*\beta)
$$
whenever $\alpha\in\mathop{\mathrm{dom}}\delta^*$ and $\beta\in\mathop{\mathrm{dom}}\mathrm{d}^*$.
\end{proposition}
Gaffney calls manifolds where the conclusion of
Proposition~\ref{prop:gaffney} holds ``manifolds with negligible
boundary''.
\begin{proof}
This can be found in Gaffney's paper \cite{gaffney54}; we will also
give a proof of a more general result in
Corollary~\ref{cor:essential}, based on work of
Chernoff~\cite{chernoff73}.
\end{proof}
\begin{corollary}
If $S$ is a complete oriented Riemannian manifold, then
$$
\overline\mathrm{d}=\delta^*
\qquad\mathrm{and}\quad
\overline\delta=\mathrm{d}^*.
$$
\end{corollary}
This means that~$\mathrm{d}$ and~$\delta$ have mutually adjoint
closures
$$
\xymatrix{L^2\Omega^k\ar@<.5ex>[r]^{\overline\mathrm{d}} &
L^2\Omega^{k+1}\ar@<.5ex>[l]^{\overline\delta}}.
$$
As we pointed out above, this implies that the operators~$\delta\mathrm{d}$
and~$\mathrm{d}\delta$ have unique self-adjoint closures.
\begin{proof} We will prove that $\overline\mathrm{d}=\delta^*$, as the
other equality then follows by lemma~\ref{lem:doubledual}. We already
know that $\overline\mathrm{d}\subseteq\delta^*$, so we need only show that
$\delta^*\subseteq\overline\mathrm{d}$. To this end, let
$\alpha\in\mathop{\mathrm{dom}}\delta^*$ and $\beta\in\mathop{\mathrm{dom}}\mathrm{d}^*$. By
Lemma~\ref{lem:dense}, $\alpha\in\mathop{\mathrm{dom}}\delta^*$ is the~$L^2$ limit of a
sequence~$\alpha_n$ of compactly-supported differential
forms. Gaffney's Proposition~\ref{prop:gaffney} allows us to write
$$
(\delta^*\alpha\mid\beta)=(\alpha\mid\mathrm{d}^*\beta)=\lim_{n\to\infty}(\alpha_n\mid\mathrm{d}^*\beta)
$$
By the definition of~$\mathrm{d}^*$ in Equation~(\ref{eq:adjoint}),
$$
(\delta^*\alpha\mid\beta)=\lim_{n\to\infty}(\alpha_n\mid\mathrm{d}^*\beta)=\lim_{n\to\infty}(\mathrm{d}\alpha_n\mid\beta).
$$
Since this holds for arbitrary~$\beta$ in the dense domain of~$\mathrm{d}^*$,
not only $\alpha_n\to\alpha$ but also $\mathrm{d}\alpha_n\to\delta^*\alpha$,
and so $\alpha\in\mathop{\mathrm{dom}}\overline\mathrm{d}$ and
$\overline\mathrm{d}\alpha=\delta^*\alpha$ by the definition of
$\overline\mathrm{d}$ in Equation~(\ref{eq:closure}).
\end{proof}
As we shall see, the uniqueness of the self-adjoint closure
of~$\delta\mathrm{d}$ (in other words, the essential self-adjointness
of~$\delta\mathrm{d}$) is necessary to make sense of the Fock quantization of
the electromagnetic field. By Gaffney's result, the essential
self-adjointness of~$\delta\mathrm{d}$ follows from completeness of~$S$
which, as we have pointed out, is equivalent to the global
hyperbolicity of the original static spacetime~$M$. Intuitively, if a
spacetime is globally hyperbolic there is no information coming from
or lost to infinity, so no boundary conditions are necessary to
uniquely determine time evolution of square-integrable differential
forms and, in fact, space has `negligible boundary' in the sense of
Gaffney. This, in retrospect, is the justification for the assumption
that spacetime is globally hyperbolic although, strictly speaking,
this is a sufficient but not a necessary condition for~$S$ to have
negligible boundary.
Because our assumption of global hyperbolicity implies
that~$\overline\mathrm{d}=\delta^*$ and~$\overline\delta=\mathrm{d}^*$ there is no
ambiguity in the closing of the operators~$\mathrm{d}$ and~$\delta$, and from
this point on we shall assume that~$\mathrm{d}$ and~$\delta$ have been closed
unless otherwise stated. We will slightly abuse notation by
writing~$\mathrm{d}$ to denote the closed version of the exterior
derivative. As noted before, its adjoint will be denoted~$\mathrm{d}^*$ so as
to preserve~$\delta$ for use in variational calculus.
Sometimes, as shorthand or in order to avoid confusion between
exterior derivative operators acting on different spaces, an
additional bit of notation will be necessary; namely, we will denote
by~$\mathrm{d}_k$ the operator~$\mathrm{d}\colon L^2\Omega^k_S\to L^2\Omega^{k+1}$,
so that~$\mathrm{d}_k^*$ will stand for~$\mathrm{d}^*\colon L^2\Omega^{k+1}_S\to
L^2\Omega^k_S$.
\section{Maxwell's theory}
\label{sec:maxwell}
In the rest of this section we derive the Maxwell equations by
applying Hamilton's principle of stationary action, and define the
phase space of the theory as the collection of gauge equivalence
classes of solutions of the equations of motion. The phase space is
constructed in three steps (see, for instance, \cite{relativistic,
partial}): a \emph{kinematical phase space} on which the Hamilton
least action principle can be formulated, but not supporting a
Hamiltonian or symplectic structure; a \emph{dynamical phase space} of
solitions of the equations of motion on which a conserved Hamiltonian
and Noether current are defined, but without a symplectic structure;
and a \emph{physical phase space} with no remaining gauge freedom,
which is a symplectic space.
For simplicity, we only consider Maxwell's equations in the case where
the electromagnetic vector potential is a connection on a
\emph{trivial} bundle over spacetime. Luckily, this is a vacuous
restriction when the gauge group is~$\mathbb{R}$, as we are assuming.
For~$U(1)$ electromagnetism, nontrivial bundles can be used to model
magnetic monopoles. Having a trivial bundle means we can treat the
vector potential as a 1-form~$A_M$ on spacetime; that is, the
covariant exterior derivative on~$M$ is~$\mathrm{d}_M+A_M\wedge$. The field
strength is the curvature~$2$-form
$$
F_M = \mathrm{d}_M A_M.
$$
and the Maxwell action is
\begin{equation}
\label{eq:actionM}
\mathcal{S}[A_M] = -{1\over 2}(F_M , F_M ),
\end{equation}
which is invariant under gauge transformations of the form
$$
A_M\mapsto A_M+\mathrm{d}_M\phi.
$$
The equations of motion follow from applying the Hamilton principle of
stationary action to Equation~(\ref{eq:actionM}).
To obtain a Hamiltonian formulation of the equations of motion one
needs to use an explicit foliation of spacetime into a family of
Cauchy surfaces related by a time translation symmetry. We can do this
because we have assumed that spacetime is globally hyperbolic and
static. We use Equation~(\ref{eq:stpart}) to split~$A_M$ and~$F_M$
into spatial and temporal parts:
$$
A_M=\mathrm{d} t\wedge A_0+A
\qquad\hbox{and}\quad
F_M=\mathrm{d} t\wedge F_0+F,
$$
whose physical interpretation is that~$F_0$ is the electic field and~$F$
the magnetic field, as we shall see below. By
Equation~(\ref{eq:stderivative})
$$
F_0=\partial_t A-\mathrm{d} A_0
\qquad\hbox{and}\quad
F=\mathrm{d} A.
$$
Gauge transformations leave~$F_M$ unchanged, but their effect
on~$A_0$ and~$A$ is
\begin{equation}
\label{eq:stgauge}
A\mapsto A+\mathrm{d}\phi
\qquad\hbox{and}\quad
A_0\mapsto A_0+\partial_t\phi.
\end{equation}
Using Equation~(\ref{eq:inner2}), Equation~(\ref{eq:actionM}) can be
rewritten as
\begin{eqnarray}
\mathcal{S}[A,A_0]=
{1\over 2}\int_\mathbb{R}
\bigl[(\partial_t A -\mathrm{d} A_0,\partial_t A -\mathrm{d} A_0) -
(\mathrm{d} A,\mathrm{d} A)\bigr] \mathrm{d} t.
\label{eq:action}
\end{eqnarray}
Note that a factor of~$e^{-4\Phi}$ in the metric on 2-forms from
Equation~(\ref{eq:stpart2}) has cancelled the factor of~$e^{4\Phi}$ in
the volume form on spacetime from Equation~(\ref{eq:vol}). This makes
the $3+1$-dimensional case of Maxwell's theory special, and it is
intimately related to the fact that Maxwell's equations are
conformally invariant in this dimension. Conformal invariance is
another reason why the decomposition~$g_M=e^{2\Phi}(-\mathrm{d} t^2+g)$ is
preferable to~$g_M=-e^{2\Phi}\mathrm{d} t^2+g_S$, at least in this case. The
action of Equation~(\ref{eq:action}) is the time-integral of the
Lagrangian
\begin{equation}
\mathcal{L}[A,A_0]={1\over 2}\bigl[(\dot A-\mathrm{d} A_0,\dot A-\mathrm{d} A_0)-(\mathrm{d}
A,\mathrm{d} A)\bigr],
\label{eq:Lagr}
\end{equation}
where~$\dot A=\partial_t A$.
Because of energy conservation, the integral of
Equation~(\ref{eq:action}) is likely to diverge unless it is restricted
to a finite interval of~$t$. This restiction is, in any case,
necessary to use the action principle to study time evolution between
two given instants of time. In addition to evaluating the action
integral over a finite interval of time, sufficient conditions for
Equations~(\ref{eq:stgauge})--(\ref{eq:Lagr}) to make sense include
that
$$
\phi(t),A_0(t)\in\mathop{\mathrm{dom}}\{\mathrm{d}\colon L^2\Omega^0_S\to L^2\Omega^1_S\}
\qquad\hbox{and}\quad
A(t)\in\mathop{\mathrm{dom}}\{\mathrm{d}\colon L^2\Omega^1_S\to L^2\Omega^2_S\},
$$
for almost, with~$t$ with all the~$L^2$ norms being square-integrable
over any compact interval of~$t$; and that their respective time
derivatives are in the same spaces. This imposes nontrivial smoothness
and decay restrictions on the electromagnetic potentials~$A_0,A$, and
also on the allowed generators~$\phi$ of gauge transformations. In the
case when space is compact, any smooth gauge generator will
automatically be bounded and square-integrable, but in the noncompact
case we are forced to exclude some gauge transformations which are too
large at infinity but would otherwise na\"{\i}vely be allowed. This restriction on the gauge generators
cannot manifest itself in physical effects on any bounded region of
spacetime.
\subsection{Overview of covariant mechanics}
Hamilton's principle states that physically allowed field
configurations~$X$ in a region~$R$ of a spacetime~$M$ are critical
points (not necessarily minima) of an action
functional~$\mathcal{S}_R[X]$. We assume that the action is \emph{local},
that is, that~$\mathcal{S}_R[X]$ is the integral over the spacetime
region~$R$ of a Lagrangian density~$\mathcal{L}[X]$ which, at each point of
spacetime, depends only on~$X$ and a finite number of its derivatives
(usually just the first) at that point. The action functional is often
calculated by evaluating the integral in Equation~(\ref{eq:action})
over a bounded region of spacetime, and almost always over a finite
interval of time. In fact the action calculated over all of time may
be infinite, and the variation of the action might also be ill-defined
unless restricted to be compactly supported in time, which amounts to
evaluating the action integral over a finite interval of time in the
first place. Hamilton's principle is formulated on a \emph{kinematical
phase space}~$\mathcal{X}_R$ large enough to contain all plausible field
configurations and small enough that~$\mathcal{S}_R[X]=\int_R\mathcal{L}[X]$ is
well-defined.
\begin{figure}
$$
\begin{xy}
(30,0)*{};
(25,15)**\crv{(20,5)&(25,10)};
(-5,15)**\crv{(15,10)&(5,20)};
(0,0)**\crv{(-5,10)&(-10,5)};
(30,0)**\crv{(20,5)&(10,-5)}?(.5)+(-2,-1)*{\scriptstyle S};
(0,10)*{\scriptstyle M};
(12,8)*\xycircle<14pt,7pt>{};
(12,8)*{\scriptstyle R};
\end{xy}
$$
\caption{Schematic representation of spacetime, space and the domain
of integration for the action functional}
\end{figure}
The stationary action principle implies the vanishing of the first
variation of the action on any region~$R$:
$$
0=\delta\mathcal{S}_R[X]=\int_R\delta\mathcal{L}[X]=-\oint_{\partial R}\theta[X]+\int_R E[X].
$$ It has been shown~\cite{zuckerman87,crnkovic87} that it is possible
and advantageous to choose~$\mathcal{X}$ to be an infinite-dimensional manifold
(possibly even a vector space) and interpret the variational
derivative~$\delta$ as an exterior derivative on~$\mathcal{X}$. The Lagrangian
density~$\mathcal{L}$ is then an~$(n+1)$-form on~$\mathcal{X}\times M$ proportional
to~$\mathrm{vol}_M$. The condition that~$\mathcal{L}$ be a \emph{local Lagrangian}
means that, at any point~$p\in M$,~$\mathcal{L}$ depends on~$X$ only through
the values of~$X$ and finitely many ot its derivatives at~$p$. The
exterior derivative on~$\mathcal{X}\times M$ is~$\delta+\mathrm{d}_M$, which implies
the anticommutation relation~$\delta\mathrm{d}_M+\mathrm{d}_M\delta=0$. The
quantity~$E$ is an~$(n+2)$-form on~$X\times M$ which is a~$1$-form
with respect to~$\mathcal{X}$ and proportional to the~$(n+1)$-form~$\mathrm{vol}_M$;
similarly,~$\theta$ is an~$(n+1)$-form which is a~$1$-form with
respect to~$\mathcal{X}$ and an~$n$-form with respect to~$M$. Tangent vectors
to~$\mathcal{X}$ are variations of field configurations. We denote a typical
such tangent vector by~$\partial_X$.
If~$\delta\mathcal{S}_R[X]$ is evaluated at a stationary field
configuration~$X$, on variations~$\partial_X$ vanishing on the
boundary~$\partial R$, the stationary action condition implies the
\emph{Euler--Lagrange equations of motion}~$E[X](\partial_X)=0$. We define the
\emph{dynamical phase space} associated to the region~$R$ as the variety
$$
\mathbf{D}_R=\{X\in\mathcal{X}\colon
E[X](\partial_X)=0\quad\hbox{on}\quad
R\qquad\hbox{if}\quad\partial_X=0\quad\hbox{on}\quad\partial R\}.
$$
The so-called \emph{Noether current}~$\theta[X]$ is defined only up to
an exterior derivative, and can be interpreted as a generator of
conserved quantities associated to continuous symmetries of solutions
to the Euler--Lagrange equations of motion. To see this, consider a
tangent vector to~$\mathbf{D}_R$, which is a variation of solutions to the
Euler--Lagrange equations of motion. Because the Euler-Lagrange
equations are satisfied throughout, we have
$$
\oint_{\partial M}\theta[X](\partial_X)=0.
$$
Suppose now that~$R\cong [0,1]\times T$. Then,
$$
\int_{T_0}\theta[X](\partial_X)-\int_{T_1}\theta[X](\partial_X)=\int\limits_{\rlap{$\scriptstyle[0,1]\times\partial
T$}}\,\theta[X](\partial_X),
$$
where the right-hand side represents the time integral of the flux
of the conserved quantity through~$\partial T$. In the case where~$R$
is a globally hyperbolic region with Cauchy surface~$T$, the latter
has negligible boundary in the sense of Gaffney, and
$$
\int_{T_0}\theta[X](\partial_X)=\int_{T_1}\theta[X](\partial_X),
$$
so~$\int_T\theta[X](\partial_X)$ is a conserved quantity of the
motion.
\begin{figure}
$$
\begin{xy}
(30,0)*{};
(0,0)**\crv~lc{(15,15)};
(30,0)**\crv{(12,11)};
(0,0)**\crv{(12,6)};
(30,0)**\crv{(18,-4)};
(0,0)**\crv{(18,-9)};
(30,0)**\crv~lc{(15,-15)};
(0,0)**\dir{-}?(.5)+(0,1.5)*{\scriptstyle{T}};
(11,-9)*{\scriptstyle R};
\end{xy}
$$
\caption{Schematic representation of a globally hyperbolic region
foliated by a family of Cauchy surfaces.}
\end{figure}
For instance, in the case where~$M\cong\mathbb{R}\times S$ is static
and~$R=[t_0,t_1]\times S$, the variation~$\partial_X$ might represent
the generator of a one-parameter group of isometries of~$S$ (a
translation or rotation) on the field configuration~$X$, and the
associated conserved quantity would be the corresponding momentum (linear or angular) of~$X$. If~$\partial_X$ represented the action of an internal symmetry
of the field variables at each point (a gauge transformation), the
conserved quantity would be the conserved charge associated to the
gauge symmetry.
The variational derivative of the Noether current is a skew-symmetric
$2$-form on~$\mathbf{D}_R$,
$$
\omega_T[X]=\int_T\delta\theta[X].
$$
Given two variations of solutions,
$$
\omega_T[X](\partial_X,\partial_X')
$$
is a conserved quantity of the solution~$X$. It is possible
that~$\omega_T[X]$ is degenerate, admitting variations of
solutions~$\partial_X$ such that
$$
\omega_T[X](\partial_X,-)=0.
$$
Each such degenerate direction~$\partial_X$ generates a gauge
transformation of the dynamical phase space. The space of gauge orbits
of~$\mathbf{D}_R$ is the \emph{physical phase space}~$\mathbf{P}_R$. As we have
pointed out, it is in
general not a manifold, but an `infinite-dimensional variety
with singularities'. By
construction,~$\omega_S$ would project to a non-degenerate symplectic
structure on~$\mathbf{P}_R$.
A more cogent approach to the physical phase space~$\mathbf{P}_R$ would be
as follows. Let~$N$ denote the space of degenerate directions
of~$\omega_T$. The smooth functions~$f$ on~$\mathbf{D}_R$ such
that~$\partial_Xf=0$ whenever~$\partial_X\in N$ constitute a
subalgebra of~$C^\infty(\mathbf{D}_R)$, the so-called gauge-invariant
observables on~$\mathbf{D}_R$. The spectrum of homomorphisms of this algebra
would be~$\mathbf{P}_R$, and we can map the algebra of gauge-invariant
observables homeomorphically to~$C^\infty(\mathbf{P}_R)$. Whether or
not~$\mathbf{P}_R$ turns out to be a manifold that can support a
symplectic structure, the algebra of gauge-invariant supports the
canonical Poisson structure
$$
\{f,g\}=\omega_T(\partial_f+N,\partial_g+N)
\qquad\hbox{for all}\quad f\in C^\infty(\mathbf{P}_R),
$$
where~$\partial_f$ is a tangent vector to~$\mathbf{D}_R$ such
that~$\omega_T(\partial_f,\partial_Y)=\delta f(\partial_Y)$ for all
tangent vectors to~$\mathbf{D}_R$. Conveniently,~$\partial_f$ is
defined precisely up to addition of elements of~$N$, so one can associate a
unique equivalence class in~$T\mathbf{D}_R/N$ to it,
namely~$\partial_f+N$. Since~$\omega_T$ is, in fact,
non-degenerate on~$\mathbf{D}_R$, the algebra of gauge-invariant observables
is a Poisson algebra, whose spectrum is the physical phase space.
This construction simplifies considerably when the action functional
is quadratic, as in that case the equations of motion and the Noether
current are linear, and all the spaces involved are vector spaces. In
addition, in a stationary, globally hyperbolic spacetime~$M$ there is a
preferred foliation~$M\simeq\mathbb{R}\times S$ by Cauchy surfaces isometric to~$S$. When
there is a single timelike Killing field, there is a canonical
identification of the different Cauchy surfaces, and time evolution
can be represented as a transformation of the field configuration on a
single Cauchy surface. It is then possible to define a Hamiltonian
function.
In the next few sections we construct the phase space of
electromagnetism using this method. First, the \emph{kinematical phase space} is a space~$\mathcal{X}$ of
\emph{field configurations} on which the Maxwell action can be
defined, or on which the Maxwell equations can be written. The precise
definition of the kinematical phase space is somewhat arbitrary, as
long as it is large enough to contain all the actual solutions of the
equations of motion. In the next section we shall see three acceptable
formulations of the least action principle on different kinematical
phase spaces before settling on one of them.
Next, setting the first variation of the action to zero yields the
Maxwell equations of motion, whose space of solutions if the
\emph{dynamical phase space}~$\mathbf{D}$ and is a linear subspace of the
kinematical phase space (in more general cases,~$\mathbf{D}$ is just a
subvariety of~$\mathcal{X}$). The dynamical phase space supports the
Hamiltonian and Noether current of the system, which can be used to
obtain conserved quantities of the system and a pre-symplectic
structure on~$\mathbf{D}$. The null directions of the pre-symplectic
structure are seen to correspond to gauge transformations.
Finally, the set~$\mathbf{P}$ of gauge orbits on~$\mathbf{D}$ is the
\emph{physical phase space} or, simply, the \emph{phase space}. When
there is no gauge freedom, the dynamical phase space coincides with
the physical phase space. After this reduction from~$\mathbf{D}$
to~$\mathbf{P}$, the pre-symplectic structure on~$\mathbf{D}$ becomes a
non-degenerate symplectic structure on~$\mathbf{P}$.
\subsection{Kinematical phase space}
In this section we consider three possible action principles for
electromagnetism on slightly different kinematical phase spaces. The
first is the Lagrangian formulation of
Equations~(\ref{eq:action}--\ref{eq:Lagr}). The second formulation is
the associated Hamiltonian formulation, with the electrostatic
potential~$A_0$ acting as a Lagrange multiplier enforcing the Gauss
law as a constraint. Since the latter is linear, it is possible and
even convenient to impose the Gauss law at the kinematical level
without a Lagrange multiplier. This is the third formulation.
All three kinematical phase spaces are equivalent in that the action
principles defined on them lead to the same space of solutions of the
equations of motion. However, the three kinematical phase spaces are
not isomorphic to each other. The first requires that~$A_0$ be in the
domain of~$\mathrm{d}$, and that~$E=\partial_t A-\mathrm{d} A_0$ be
square-integrable. The second alternative allows~$A_0$ to be just
square integrable, but~$E$ must now be in the domain of~$\mathrm{d}^*$. The
third formulation does without~$A_0$ altogether, but~$E$ must be in the
kernel of~$\mathrm{d}^*$.
\subsubsection{Lagrangian formulation}
The action of Equation~(\ref{eq:action}) is defined on pairs~$(A,A_0)$
where~$A$ is a~$1$-form and~$A_0$ is a~$0$-form on~$S$, both
time-dependent and such that the quantities $\|A(t)\|$, $\|\partial_t
A(t)\|$, $\|\mathrm{d} A(t)\|$, $\|A_0(t)\|$ and $\|\mathrm{d} A_0(t)\|$ are all
square-integrable with respect to~$t$ on any bounded interval. This is
a suitable definition of the kinematical phase space.
The first variation of the Lagrangian of Equation~(\ref{eq:Lagr}) is
the variational~$1$-form on~$\mathcal{X}$
$$
\delta\mathcal{L}[A,A_0]=\partial_t(E,\delta A)-(\dot
E+\mathrm{d}^*\mathrm{d} A,\delta A)-(\mathrm{d}^*E,\delta A_0),
$$
where $E=\partial_A\mathcal{L}[A,A_0]=\dot A-\mathrm{d} A_0$ is the
\emph{electric field}. According to Hamilton's
principle, for physically allowed~$A$ and~$A_0$, the
variation~$\delta\mathcal{L}$ must vanish. The fact that~$\mathcal{L}$ is
independent of~$\dot A_0$ implies that~$A_0$ is a non-dynamical
Lagrange multiplier field enforcing the constraint
$$
\mathrm{d}^*E=0.
$$
The dynamical fields are~$A$ and its canonical conjugate,~$E$, which
we combine into a \emph{field configuration}~$X=A\oplus E$.
\subsubsection{Hamiltonian formulation with a Lagrange multiplier}
In terms of the field configuration $X=A\oplus E$ and the Lagrange
multiplier~$A_0$, the original Lagrangian from
Equation~(\ref{eq:Lagr}) can be rewritten as
\begin{equation}
\mathcal{L}[X;A_0]=(E,\dot A)-{1\over 2}\bigl[(E,E)+(\mathrm{d} A,\mathrm{d}
A)\bigr]-(\mathrm{d}^* E,A_0).
\label{eq:Lagr2}
\end{equation}
This leads to an alternative---and inequivalent---definition of the
kinematical phase space, namely the collection of pairs~$(X,A_0)$
where~$X=A\oplus E$, and such that $\|A(t)\|$, $\|E(t)\|$, $\|\partial_t
A(t)\|$, $\|\mathrm{d} A(t)\|$, $\|\mathrm{d}^* E(t)\|$ and~$\|A_0\|$ are all
square-integrable over finite intervals of~$t$.
The Euler--Lagrange equations obtained from the first variation of the
Lagrangian of Equation~(\ref{eq:Lagr2}),
\begin{equation}
\delta\mathcal{L}[A\oplus E;A_0]=\partial_t(E,\delta A)+(\dot A-E-\mathrm{d}
A_0,\delta E)-(\dot E+\mathrm{d}^*\mathrm{d} A,\delta A)-(\mathrm{d}^* E,\delta A_0)
\label{eq:firstVar}
\end{equation}
yield the Maxwell equations in Hamiltonian form
\begin{eqnarray*}
\mathrm{d}^* E&=&0\\
\dot A-E&=&\mathrm{d} A_0\\
\dot E+\mathrm{d}^*\mathrm{d} A&=&0
\end{eqnarray*}
Observe that, when the Gauss law is satisfied, the action is
independent of the non-dynamical (and hence arbitrary)~$A_0$, and so
because of the equation~$\dot A=E+\mathrm{d} A_0$ time evolution is not
uniquely determined by the initial conditions. This is all closely
related to the existence of time-dependent gauge transformations,
which by Equation~(\ref{eq:stgauge}) result in a change of the
Lagrange multiplier field~$A_0$. We can use this gauge freedom to
eliminate the Lagrange multiplier~$A_0$, that is, we perform a time-dependent
gauge transformation to make~$A_0=0$. This is the so-called `temporal
gauge'. Then, the Maxwell equations take the form
\begin{eqnarray}
\label{eq:Max}
\mathrm{d}^* E&=0&\quad\hbox{(Gauss law constraint)}\\
\dot A-E&=0&\quad\hbox{(Faraday--Lenz law)}\label{eq:FL}\\
\dot E+\mathrm{d}^*\mathrm{d} A&=0&\quad\hbox{(Amp\`ere--Maxwell law)}
\label{eq:well}
\end{eqnarray}
on the kinematical phase space.
\subsubsection{Hamiltonian formulation without Lagrange multipliers}
The partial gauge-fixing of the previous case can be carried out at
the level of the action, leading to a third possible definition of the
kinematical phase space~$\mathcal{X}$, consisting of pairs~$X=A\oplus E$ such
that $\|A(t)\|$, $\|\partial_t A(t)\|$, $\|\mathrm{d} A(t)\|$, and $\|E(t)\|$
are square-integrable on finite intervals of~$t$, and
that~$\mathrm{d}^*E(t)=0$ for almost all~$t$.
We choose this as our preferred kinematical phase space. This means
that, for us,~$\mathcal{X}$ consists of pairs~$X=A\oplus E$ such that
$$
A(t)\oplus E(t)\in\mathop{\mathrm{dom}}\{\mathrm{d}\colon L^2\Omega^1_S\to
L^2\Omega^2_S\}\oplus\ker\{\mathrm{d}^*\colon L^2\Omega^1_S\to L^2\Omega^0_S\}
\qquad\hbox{for almost all}\quad
t
$$
and $\|X(t)\|$ is square-integrable on bounded intervals of~$t$.
On this space,
Equation~(\ref{eq:Max}) is automatically satisfied and the Lagrangian
\begin{equation}
\label{eq:Lagr3}
\mathcal{L}[X]=(E,\dot A)-{1\over 2}\bigl[(E,E)+(\mathrm{d} A,\mathrm{d}
A)\bigr]
\end{equation}
leads to the additional Maxwell Equations~(\ref{eq:FL})
and~(\ref{eq:well}).
\subsection{Dynamical phase space}
The space of solution of the Maxwell equations in the temporal
gauge~(\ref{eq:Max})--(\ref{eq:well}) is the \emph{dynamical phase
space} of the theory. Because the Maxwell equations are linear, the
space of their solutions is a linear subspace of the kinematical phase
space~$\mathcal{X}$. The global hyperbolicity of~$M$ implies that, in the
temporal gauge, the Maxwell equations form a hyperbolic system of
partial differential equations. Then, each solution of the equations
of motion is uniquely determined by its restriction to a surface of
constant~$t$ (initial data at time~$t$), so each such surface provides
a coordinatization of the dynamical phase space in terms of a pair
of~$1$-forms on~$S$.
In other words, we adopt the point of view that the dynamical phase
space consists of time-dependent solutions~$A\oplus E$ of the
equations of motion, that data~$X(t)=A(t)\oplus E(t)$ at
time~$t$ are a coordinatization of the phase space, and that time
evolution is a change of coordinates in phase space. Under this
interpretation, it can be argued that it is a bad thing to concentrate
too much on the time evolution of initial data. We proceed to do just
this, however.
From any of the definitions of the kinematical phase space~$\mathcal{X}$ in the
previous section it follows that, for almost all~$t$, initial data
$X(t)=A(t)\oplus E(t)$ are such that~$A(t)\in\mathop{\mathrm{dom}}\{\mathrm{d}\colon
L^2\Omega^1_S\to L^2\Omega^2_S\}$ and~$E(t)\in\mathop{\mathrm{dom}}\{\mathrm{d}^*\colon
L^2\Omega^1_S\to L^2\Omega^0_S\}$. This means that the space of
solutions of Maxwell's equations is isomorphic to a (dense, at least)
subspace of
$$
\mathbf{D}=\mathop{\mathrm{dom}}\{\mathrm{d}\colon L^2\Omega^1_S\to L^2\Omega^2_S\}\oplus\ker\{\mathrm{d}^*\colon
L^2\Omega^1_S\to L^2\Omega^0_S\}.
$$
The Hamiltonian
\begin{equation}
\label{eq:Ham}
\mathcal{H}_t={1\over 2}\Bigl[\bigl(E(t),E(t)\bigr)+\bigl(\mathrm{d} A(t),\mathrm{d}
A(t)\bigr)\Bigr]
\end{equation}
can be directly read off from the form of the Lagrangian in
Equation~(\ref{eq:Lagr3}) and it is preserved by time evolution. What
this means is that, although the Hamiltonian is defined on a
particular surface of constant~$t$, it is independent of~$t$ as long
as~$A\oplus E$ satisfies the equations of motion. In other words, the
Hamiltonian is time-dependent---and thus ill-defined as a single
functional---on the kinematical phase space, but is
coordinate-independent on the dynamical phase space. Moreover,~$\mathbf{D}$
imposes just the right decay and smoothness conditions on~$A(t)$
and~$E(t)$ so that~$\mathbf{D}$ is exactly the space of initial data~$X(t)$
satisfying the Gauss law~$\mathrm{d}^*E(t)=0$ and for which~$\mathcal{H}$ is finite.
The so-called \emph{Noether current} can also be read off directly, in
this case from the total derivative term in the first variation of the
Lagrangian, Equation~(\ref{eq:firstVar}). The Noether current is a
variational~$1$-form on the dynamical phase space~$\mathbf{D}$ which, for
electromagnetism, takes the form
$$
\theta_t=\bigl(E(t),\delta A(t)\bigr).
$$
The Noether current can be used to obtain conserved quantities
associated to continuous transformations of the fields. Indeed,
If~$X=A\oplus E$ is a solution of the equations of motion,
$$
\theta_t-\theta_0=\delta\mathcal{S}[X].
$$
This means that, if~$X$ depends on a parameter~$\tau$ such
that~$\partial_\tau\mathcal{S}[X]=0$, then
$$
\theta_t(\partial_\tau)=\bigl(E(t),\partial_\tau A(t)\bigr)
$$
is independent of~$t$ and so is a conserved quantity of the
equations of motion. This means~$\theta$ is well-defined
on~$\mathbf{D}$. Conversely, if~$X=A\oplus E$ were not a solution of the equations of
motion the Noether current would depend on~$t$, and so~$\theta$ really
should not be interpreted as a~$1$-form on~$\mathcal{X}$.
For instance, the one-parameter gauge transformation given by
$\partial_\phi A=\mathrm{d}\phi$ leaves the action invariant, and in that
case~$\theta(\partial_\phi)=(\mathrm{d}^*E,\phi)$. The conserved quantity
associated to gauge transformations of this form is,
therefore,~$\mathrm{d}^*E$. Although Gauss' law makes this seem trivial, this
conservation law is nontrivial when the Maxwell equations are coupled
to matter, in which case~$\mathrm{d}^*E$ equals the electric charge, and
therefore the conservation law associated to gauge invariance is
electric charge conservation. When the surface~$S$ at constant~$t$ has
nontrivial continuous isometries, $\partial_\tau A=L_\xi A$
where~$\xi$ is the Killing field generating the isometries and~$L_\xi$
denotes the Lie derivative with respect to it. In that case, the
Noether current evaluates to~$\theta(\partial_\tau)=(E,L_\xi A)$,
which is the conserved quantity associated to the isometry. This is
one way to define the momentum and angular momentum of the
electromagnetic field on homogeneous, rotationally invariant or
isotropic spaces, such as Minkowski space. It also shows that, when
space has no continuous isometries, there is no global generalization
of the linear and angular momenta of the electromagnetic field.
The variational exterior derivative of the
Noether current is the pre-symplectic structure
\begin{equation}
\label{eq:symplectic}
\omega_t=\bigl(\delta E(t),\delta A(t)\bigr)-\bigl(\delta A(t),\delta E(t)\bigr),
\end{equation}
which is an variational~$2$-form. Like the Hamiltonian, though
ostensibly defined for data on a surface of constant~$t$ and thus
time-dependent, the pre-symplectic structure~$\omega$ is finite and
conserved by time evolution if it is evaluated at a solution~$X$ on
two variations compatible with the equations of motion (that is, two
tangent vectors to~$\mathbf{D}$ at the same~$X\in\mathbf{D}$).
Because the dynamical phase space~$\mathbf{D}$ is defined by
Equation~(\ref{eq:Max}),~$\omega$ has null directions consisting
precisely of all variations of the form
$$
\partial_\phi A=\mathrm{d}\phi,
$$
which are the gauge transformations remaining after choosing the
temporal gauge. This means that~$\omega$ is indeed degenerate, and
that the degeneracy is related to gauge freedom.
\subsection{Physical phase space}
\label{sec:analytical.setting}
We have seen that the Gauss law constraint generates the gauge transformations
$$
A \mapsto A + \mathrm{d}\phi ,
$$
and two sets of initial data~$A\oplus E$ and~$A'\oplus E'$ are
physically equivalent if they differ by a transformation of this
form. Thus, taking the quotient of~$\mathbf{D}$ by this equivalence relation
we should obtain the physical phase space of the Maxwell theory,
\begin{equation}
\mathbf{P}={\mathop{\mathrm{dom}}\{\mathrm{d}\colon L^2\Omega^1_S\to
L^2\Omega^2_S\}\over\overline{\mathop{\mathrm{ran}}}\{\mathrm{d}\colon L^2\Omega^0_S\to
L^2\Omega^1_S\}}\oplus\ker\{\mathrm{d}^*\colon L^2\Omega^1_S\to
L^2\Omega^0_S\}.
\label{eq:phase}
\end{equation}
In words, the physical phase space consists of pairs~$[A]\oplus E$
where:~$[A]$ is an equivalence class of
square-integrable~$1$-forms on~$S$ with square-integrable exterior
derivatives modulo~$L^2$ limits of the exterior derivatives of
square-integrable functions on~$S$; and~$E$ is a
square-integrable~$1$-form on~$S$ with vanishing divergence.
Note that the Hamiltonian of Equation~(\ref{eq:Ham}) is manifestly
independent of any choice of representative in the gauge equivalence
class of~$A$. On the other hand, the (now nondegenerate) symplectic
structure~(\ref{eq:symplectic}) is gauge-independent only
because of Gauss' law, as
$$
(A+\mathrm{d}\beta,E)=(A,E)+(\beta,\mathrm{d}^* E)=(A,E).
$$
The first direct summand in Equation~(\ref{eq:phase}),
$$
\mathbf{A}={\mathop{\mathrm{dom}}\{\mathrm{d}\colon L^2\Omega^1_S\to
L^2\Omega^2_S\}\over\overline{\mathop{\mathrm{ran}}}\{\mathrm{d}\colon L^2\Omega^0_S\to
L^2\Omega^1_S\}},
$$
has a natural Hilbert-space norm
\begin{equation}
\label{eqn:norm_A}
\bigl\|[A]\bigr\|_\mathbf{A}^2=\inf_{\phi
\in\Omega^0}(A+\mathrm{d}\phi,A+\mathrm{d}\phi)+(\mathrm{d} A,\mathrm{d} A),
\end{equation}
which combines the natural norm on a quotient space with the natural
Sobolev norm on~$\mathop{\mathrm{dom}}\{\mathrm{d}\colon L^2\Omega^1_S\to L^2\Omega^2_S\}$. The
second summand is simply
$$
\mathbf{E}=\ker\{\mathrm{d}^*\colon L^2\Omega^1\to L^2\Omega^0\}
\qquad\hbox{with}\quad
\|E\|_\mathbf{E}^2=(E,E),
$$
and the natural norm on~$\mathbf{P}=\mathbf{A}\oplus\mathbf{E}$ is the sum of the two.
The Hamiltonian and symplectic structure on~$\mathbf{P}$ are continous
with respect to these norms.
\section{Free and oscillating modes}
\label{physical.interpretation}
Since the definition of the physical phase space~$\mathbf{P}$ is rather technical,
let us expound on it a bit. A point in the classical phase space is a
pair~$[A]\oplus E$ where: the vector potential~$[A]$ is an equivalence
class of square-integrable $1$-forms modulo gauge transformations,
with square-integrable exterior derivatives; and the electric
field~$E$ is a square-integrable $1$-form satisfying the Gauss law.
Our definition of the physical phase space ensures that it contains
precisely such pairs for which the Hamiltonian (physically, the
energy)~$\mathcal{H}$ is finite and the symplectic structure~$\omega$ is
well-defined. It also makes the gauge equivalence relation precise,
and makes precise the sense in which the Gauss law holds.
Note that the physical phase space~$\mathbf{P}$ does not necessarily
contain \emph{all} finite-energy initial data for Maxwell's equations,
since we are imposing the additional condition that $([A],[A]) <
\infty$ to make the symplectic structure well-defined. If we omitted
this condition we could define a real Hilbert space consisting of {\it
all} finite-energy initial data for Maxwell's equations, but the
symplectic structure would only be densely defined on this space. This
is a gauge-independent condition because~$([A],[A])$ smallest $L^2$
norm among all the vector potentials in the same gauge-equivalence
class; one could fix the gauge by choosing the representative~$A$ such
that~$(A,A)=([A],[A])$, but that is not necessary.
Observe now that the phase space~$\mathbf{P}$ and the Hamiltonian~$\mathcal{H}$
are defined very simply in terms of~$\mathrm{d}$ and~$\mathrm{d}^*$, and recall the
Kodaira orthogonal-direct-sum decomposition
$$
L^2\Omega^1_S=
\overline{\mathop{\mathrm{ran}}\mathrm{d}_0}
\oplus
\ker \Delta_1
\oplus
\overline{\mathop{\mathrm{ran}}\mathrm{d}^*_1}
$$
where $\Delta_1=\mathrm{d}^*_1\mathrm{d}_1+\mathrm{d}_0\mathrm{d}^*_0\colon L^2\Omega^1_S\to
L^2\Omega^1_S$. We prove a general version of the Kodaira
decomposition in Section~\ref{sec:Kodaira}. In the present section we
use the decomposition to write~$\mathbf{P}$ as the direct sum of a
part~$\mathbf{P}_f$ containing the \emph{Aharonov--Bohm modes} or
\emph{free modes}, and a part~$\mathbf{P}_o$ containing the more familiar
\emph{oscillating modes} of the electromagnetic field. We will see
that it is convenient to treat the classical dynamics of Maxwell
theory separately on these two parts, but putting the results together
we shall see that time evolution acts as a strongly continuous
1-parameter group of symplectic transformations on~$\mathbf{P}$. Note
that, at least in the classical theory, the separation of the
oscillating and free modes is a matter of convenience.
Before embarking on the mathematical details of the Kodaira
decomposition, let us explore its physical significance for the
classical phase space of electromagnetism.
\subsection{The space of pure-gauge potentials}
Observe that, in our definition of the physical phase space,
Equation~(\ref{eq:phase}), we have taken the space of `pure gauge' vector
potentials to be~$\overline{\mathop{\mathrm{ran}}\mathrm{d}_0}$. This is subtly different from
the common assumption that pure gauge potentials are derivatives of
arbitrary smooth scalar functions. Instead, we are saying they lie in
the {\em closure\/} of the space of derivatives of {\em
square-integrable\/} functions. While these nuances may seem merely
pedantic, they have have dramatic consequences in certain situations
which we discuss in Section~\ref{sec:harmonic}. The simplest example,
in~$2+1$ dimensions, is when~$S$ is the hyperbolic plane, which has an
infinite-dimensional space of square-integrable $1$-forms that are
exterior derivatives of smooth functions which are not
square-integrable, so the $1$-forms are not pure gauge by our
definition.
Physically, as we are restricting the class of allowed gauge
transformations (essentially to be compactly supported), in general
there will be vector potentials that would na\"\i{}vely be considered
pure gauge but should not, because they involve a change of gauge on
an effectively infinite volume. However, these additional modes cannot
be detected by any experiment carried out on a finite volume, and so
one could argue that they should be discarded after all. However,
these vector potentials are canonically conjugate to static electric
fields with finite energy, and so are required in the canonical
formulation of electromagnetism. This is even more important if these
electric field modes are to be quantized.
Mathematically, our definition is natural thanks to the Kodaira
decomposition, and it leads to consistent classical and quantum
theories, except possibly (see chapters~\ref{sec:harmonic}
and~\ref{chap:qed}) in the case when the space of harmonic vector
potentials is infinite-dimensional.
As for the electric field~$E$, the space~$\mathop{\mathrm{ran}}\mathrm{d}_0$ is orthogonal
to~$\ker \mathrm{d}^*_0$, so square-integrable electric fields satisfying
the Gauss' law constraint~$\mathrm{d}^*_0 E=0$ belong
to~$\ker\Delta_1\oplus\mathop{\mathrm{ran}}\mathrm{d}^*_1$.
\subsection{Aharonov--Bohm modes}
The space~$\ker \Delta_1$ consists of square-integrable
harmonic~$1$-forms. For any vector potential~$A$ in this space, the
magnetic field~$\mathrm{d} A$ vanishes. If the manifold~$S$ is compact, Hodge's
theorem asserts that this space is isomorphic to the first de~Rham
cohomology of~$S$, a topological invariant, and vector potentials in
this space can be detected by their holonomies around noncontractible
loops, as in the Aharonov--Bohm effect. Thus, in the compact case, it
makes perfect sense to call~$\ker \Delta_1$ the configuration space of
`Aharonov--Bohm' or `topological' modes of the electromagnetic field.
The situation is subtler if~$S$ is noncompact. In this case~$\ker
\Delta_1$ is called the `first~$L^2$ cohomology group' of~$S$.
The~$L^2$ cohomology of a non-compact Riemannian manifold can differ
from the de~Rham cohomology, and it depends on the metric, so it is
not a topological invariant. By analogy with the compact case we still
call harmonic vector potentials `Aharonov--Bohm' modes. As we shall
see, sometimes there are Aharonov--Bohm modes even when~$S$ is
contractible. On the other hand, sometimes there are no Aharonov--Bohm
modes when they would be expected on elementary topological
considerations. Finally, the space of Aharonov--Bohm modes may be
infinite-dimensional. These facts make it a bit trickier to understand
vector potentials in~$\ker \Delta$ as topological Aharonov--Bohm
modes. However, at least for certain large classes of well-behaved
manifolds, it still seems to be possible. We review some of these
results in Section~\ref{sec:harmonic}.
\subsection{Decomposition into free and oscillating modes}
We now apply the Kodaira decomposition to the physical phase
space~$\mathbf{P}$, in order to understand the Aharonov--Bohm modes more
deeply, as well as the meaning of the third
summand~$\overline{\mathop{\mathrm{ran}}\delta_1}$ in the Kodaira decomposition.
The Kodaira decomposition allows us write $\mathbf{P}$ as a direct sum $\mathbf{P}_o
\oplus \mathbf{P}_f$ of `oscillating' and `free' modes of the electromagnetic
field. The oscillating modes are familiar from electromagnetism on
Minkowski spacetime. The free modes are those relevant to the
Aharonov--Bohm effect; we call them `free' because the equations of
motion for these modes are mathematically analogous to those of a free
particle, as we shall see.
To see this in detail, first recall that
$$
\mathbf{P} = \mathbf{A} \oplus \mathbf{E}
$$
where
$$
\begin{array}{ccl}
\mathbf{A} &=& \mathop{\mathrm{dom}} \mathrm{d}_1 / \, \overline{\mathop{\mathrm{ran}}\mathrm{d}_0} \\
\mathbf{E} &=& \ker \mathrm{d}^*_1 .
\end{array}
$$
The Kodaira decomposition lets us split~$\mathbf{A}$ and~$\mathbf{E}$ into
`oscillating' and `free' parts:
$$
\begin{array}{ccl}
\mathbf{A} &\cong& \mathbf{A}_o \oplus \mathbf{A}_f \\
\mathbf{E} &=& \mathbf{E}_o \oplus \mathbf{E}_f ,
\end{array}
$$
where
$$
\begin{array}{lclllcl}
\mathbf{A}_o &=& \mathop{\mathrm{dom}} \mathrm{d}_1 \cap \overline{\mathop{\mathrm{ran}} \mathrm{d}^*_1}
&\quad&
\mathbf{A}_f &=&\ker \Delta \\
&&&&&& \\
\mathbf{E}_o &=& \overline{\mathop{\mathrm{ran}} \mathrm{d}^*_1}
&\quad&
\mathbf{E}_f &=& \ker \Delta .
\end{array}
$$
Note that the difference between~$\mathbf{A}_o$ and~$\mathbf{E}_o$ is coming from the
different norms:~$\bigl\|[A]\bigr\|^2+\|\mathrm{d} A\|^2$ versus~$\|E\|^2$. This
decomposition lets us write the classical phase space as a direct
sum of real Hilbert spaces
$$
\mathbf{P} = \mathbf{P}_o \oplus \mathbf{P}_f ,
$$
where
$$
\begin{array}{ccl}
\mathbf{P}_o &=& \mathbf{A}_o \oplus \mathbf{E}_o \\
\mathbf{P}_f &=& \mathbf{A}_f \oplus \mathbf{E}_f .
\end{array}
$$
This splitting respects the symplectic structure and also the
Hamiltonian on~$\mathbf{P}$, so time evolution acts independently on the
oscillating and free part of any initial data $[A]\oplus E \in \mathbf{P}$.
\subsection{The oscillating sector}
For modes $A\oplus E \in \mathbf{P}_o$, Maxwell's equations say:
$$
\left\{
\begin{array}{l}
\partial_t A = E \cr
\partial_t E =- \Delta A, \cr
\end{array}
\right.
$$
a generalization of the equations of motion for a harmonic oscillator.
This is why we call~$\mathbf{P}_o$ the phase space of `oscillating' modes.
The Hamiltonian on~$\mathbf{P}_o$ is also of harmonic oscillator type:
$$
H[A\oplus E] = {1\over 2} \bigl[(\mathrm{d} A | \mathrm{d} A)+(E | E)\bigr].
$$
If we rewrite the above version of
Maxwell's equations as a single integral equation,
we find it has solutions of the form
\begin{equation}
\left( \! \begin{array}{c} A \\ E \end{array} \! \right)
\mapsto T_o(t)
\left( \! \begin{array}{c} A \\ E \end{array} \! \right) =
\left( \begin{array}{cc}
\cos(t\sqrt{\Delta}) & \sin(t\sqrt{\Delta})\,/\,\sqrt{\Delta}\\
-\sqrt\Delta\,\sin(t\sqrt{\Delta}) & \cos(t\sqrt{\Delta})
\end{array} \right)
\left( \! \begin{array}{c} A \\ E \end{array} \! \right)
\label{eq:time_evolution}
\end{equation}
where we define functions of~$\Delta$ using the functional calculus
\cite{RS}.
The time evolution operators $T_o(t)$ form
a strongly continuous group of bounded operators on
$\mathbf{P}_o$. This follows from three facts:
\begin{itemize}
\item $\|T_o(t)\|$ is finite for all~$t$.
\item $T_o(t)T_o(s)=T_o(t+s)$ for all real $s,t$. This involves simple
formal manipulations (as if $\Delta$ were a positive number)
allowed by the functional calculus.
\item $\lim_{t\to 0}T_o(t)\phi=\phi$ for
all~$\phi\in\mathcal{X}_o$. This is a straightforward calculation.
\end{itemize}
\subsection{The free sector}
On the other hand, the space~$\mathbf{P}_f$ consists of initial data where the
vector potential and electric field are harmonic; these are the states
relevant to the Aharonov--Bohm effect. For modes $A\oplus E \in \mathbf{P}_f$,
Maxwell's equations become
$$
\left\{
\begin{array}{l}
\partial_t A = E \cr
\partial_t E = 0 \cr
\end{array}
\right.
$$
These are analogous to the equations of motion for a free particle on
the line, with~$A$ playing the role of position and~$E$ playing the
role of momentum. This is why we call~$\mathbf{P}_f$ the phase space of
`free' modes. The Hamiltonian on this space is also analogous to the
kinetic energy of a free particle:
$$
H[A\oplus E]={1\over 2}(E\mid E).
$$
Solving the equations of motion, we see that time evolution
acts on~$\mathbf{P}_f$ as follows:
\begin{equation}
\left( \! \begin{array}{c} A \\ E \end{array} \! \right)
\mapsto
T_f(t) \left( \! \begin{array}{c} A \\ E \end{array} \! \right) =
\left( \begin{array}{cc}
1 & t \\
0 & 1
\end{array} \right)
\left( \! \begin{array}{c} A \\ E \end{array} \! \right)
\label{eq:time_evolution_free}
\end{equation}
The time evolution operators $T_f(t)$ form a norm-continuous group of
bounded operators on $\mathbf{P}_f$. Indeed:
\begin{itemize}
\item $1\le\|T_f(t)\|^2\le 2+t^2$, so $\|T_f(t)\|$ is finite for
all~$t$.
\item $T_f(t)T_f(s)=T_f(t+s)$ for all real $s,t$, trivially.
\item $\lim_{t\to 0}T_f(t)=V(0)$ in the norm topology, since
it is easily seen that~$\|T_f(t)-T_f(0)\|=|t|$.
\end{itemize}
A key ingredient in these calculations is that~$(\mathrm{d} A|\mathrm{d} A)=0$ identically
on~$\mathbf{P}_f$.
\section{Summary}
\label{sec:Kodaira}
In this section we summarize the mathematical and physical content of
the present chapter in two results. The first, Theorem~\ref{thm:3+1},
gathers all the important analysis results concerning the exterior
derivative operator on square-integrable differential forms on a
complete Riemannian manifold. The second, Result~\ref{thm:3+1hys} describes the phase space of
vacuum electromagnetism in~$3+1$ dimensions as a real Hilbert space
with a continuous quadratic and nonnegative Hamiltonian, and a
continuous symplectic structure.
We can combine into a single theorem Gaffney's
Proposition~\ref{prop:gaffney} about the operators~$d$ and~$\delta$ on
a complete Riemannian manifold and the version of the Kodaira
decomposition (Proposition~\ref{prop:kodaira}) which was essential to
the physical interpretation of the phase space of Maxwell's theory in
the preceding section:
\begin{theorem}\label{thm:3+1}
Let~$S$ be a smooth manifold equipped with a complete Riemannian
metric~$g$. Then the formally adjoint operators
$$
\xymatrix{C^\infty_0\Omega^k_S\ar@<.5ex>[r]^{\mathrm{d}_k} &
C^\infty_0\Omega^{k+1}_S\ar@<.5ex>[l]^{\mathrm{d}^*_k}}
$$
have mutually adjoint closures
$$
\xymatrix{L^2\Omega^k_S\ar@<.5ex>[r]^{\mathrm{d}_k} &
L^2\Omega^{k+1}_S\ar@<.5ex>[l]^{\mathrm{d}^*_k}}.
$$
These closed operators satisfy
$$
\mathop{\mathrm{ran}}\mathrm{d}_{k-1} \subseteq \ker\mathrm{d}_k, \qquad
\mathop{\mathrm{ran}} \mathrm{d}^*_k \subseteq \ker \mathrm{d}^*_{k-1}
$$
and there is a Hilbert-space direct-sum decomposition
$$
L^2\Omega^k =
\overline{\mathop{\mathrm{ran}} d_{k-1}}
\oplus
\ker \Delta_k
\oplus
\overline{\mathop{\mathrm{ran}} \delta_k} .
$$
where the Laplacian on $k$-forms,
$$
\Delta_k = \delta_k d_k + d_{k-1} \delta_{k-1} ,
$$
is a nonnegative densely defined self-adjoint operator on~$L^2
\Omega^k$.
\end{theorem}
\begin{proof}
The properties of the operators~$\mathrm{d}$ and~$\mathrm{d}^*$ are the subject of
Section~\ref{sec:analysis}. We postpone proving the self-adjointness
of the Laplacian to Corollary~\ref{cor:essential} in the next
chapter. To prove the desired direct sum decomposition, we apply the
general Kodaira decomposition (Proposition~\ref{prop:kodaira} below) to
$$
\xymatrix{L^2\Omega^{k-1}\ar@<.5 ex>[r]^{\mathrm{d}_{k-1}} &
L^2\Omega^k\ar@<.5 ex>[l]^{\mathrm{d}^*_{k-1}}\ar@<.5 ex>[r]^{\mathrm{d}_k} &
L^2\Omega^{k+1}\ar@<.5 ex>[l]^{\mathrm{d}^*_k}}
$$
and obtain
$$
L^2\Omega^k =
\overline{\mathop{\mathrm{ran}} \mathrm{d}_{k-1}}
\oplus
\ker \Delta_k
\oplus
\overline{\mathop{\mathrm{ran}}\mathrm{d}^*_k} .
$$
where
$$
\Delta_k = \mathrm{d}^*_k\mathrm{d}_k + \mathrm{d}_{k-1} \mathrm{d}^*_{k-1}
$$
is the Laplacian on $1$-forms.
\end{proof}
It remains only to prove the following general form of the Kodaira
decomposition, which is itself a generalization of the usual Hodge
decomposition for differential forms on a compact Riemannian manifold.
\begin{proposition}[Kodaira decomposition]\label{prop:kodaira}
If
$$
\begin{CD}
{H}@>{S}>>{H'}@>{T}>>{H''}
\end{CD}
$$
are densely defined closed operators and $\mathop{\mathrm{ran}} S\subseteq\ker T$, then
$$
H' = \overline{\mathop{\mathrm{ran}} T^*}\oplus\ker (T^*T+SS^*)\oplus\overline{\mathop{\mathrm{ran}} S}.
$$
\end{proposition}
\begin{proof}
We break the proof down into a series of lemmas. In the following
results and proofs, all the spaces we will consider will be Hilbert
spaces. The proofs work equally well for real or complex Hilbert
spaces, but in our application they will be real.
\begin{lemma}\label{lem:ran_ker}
If
$$
\begin{CD}
{H}@>{T}>>{H'}
\end{CD}
$$
is a densely defined operator, then
$$
\ker T^*=(\mathop{\mathrm{ran}} T)^\perp
\qquad\hbox{and}\quad
\ker T=(\mathop{\mathrm{ran}} T^*)^\perp\cap\mathop{\mathrm{dom}} T.
$$
\end{lemma}
\begin{proof}
Since $(\phi\mid T\psi)'=(T^*\phi\mid\psi)$ for all $\phi\in\mathop{\mathrm{dom}} T^*$
and $\psi\in\mathop{\mathrm{dom}} T$, it follows that $\ker T^*\perp\mathop{\mathrm{ran}} T$ and $\ker
T\perp\mathop{\mathrm{ran}} T^*$. Since~$T$ is densely defined, $(\mathop{\mathrm{ran}}
T)^\perp\subseteq\mathop{\mathrm{dom}} T^*$.
\end{proof}
The following lemma guarantees that the closed operators~$\mathrm{d}$
and~$\mathrm{d}^*$ satisfy
$$
\mathop{\mathrm{ran}} \mathrm{d}_{k-1} \subseteq \ker \mathrm{d}_k, \qquad
\mathop{\mathrm{ran}} \mathrm{d}^*_k \subseteq \ker \mathrm{d}_{k-1} .
$$
\begin{lemma}\label{lem:semiexact}
If
$$
\begin{CD}
{H}@>{S}>>{H'}@>{T}>>{H''}
\end{CD}
$$
are densely defined operators and $\mathop{\mathrm{ran}} S\subseteq\ker T$, then
$$
\mathop{\mathrm{ran}} T^*\subseteq\ker S^*.
$$
\end{lemma}
\begin{proof}
Since $\mathop{\mathrm{ran}} S\subseteq\ker T$, for all $\phi\in\mathop{\mathrm{dom}} T^*$ and
$\psi\in\mathop{\mathrm{dom}} S$ we have
$$
(T^*\phi\mid S\psi)'=
(\phi\mid TS\psi)''=
(\phi\mid 0)''=
0=(0\mid\psi),
$$
so $\mathop{\mathrm{ran}} T^*\subseteq\ker S^*$.
\end{proof}
\begin{corollary}
If
$$
\begin{CD}
{H}@>{S}>>{H'}@>{T}>>{H''}
\end{CD}
$$
are densely defined closable operators and $\mathop{\mathrm{ran}} S\subseteq\ker T$, then
$$
\mathop{\mathrm{ran}}\overline S\subseteq\ker\overline T.
$$
\end{corollary}
\begin{proof}
By Lemma~\ref{lem:doubledual}, since~$S,T$ are closable,~$S^*$
and~$T^*$ are densely defined and $\overline S=S^{**}$ and $\overline
T=T^{**}$. Then, $\mathop{\mathrm{ran}} S\subseteq\ker T$ implies $\mathop{\mathrm{ran}}
T^*\subseteq\ker S^*$, so $\mathop{\mathrm{ran}} S^{**}\subseteq\ker T^{**}$.
\end{proof}
We are now ready to finish the proof of the Kodaira decomposition. The
hypotheses of Proposition~\ref{prop:kodaira} guarantee that~$S^*$
and~$T^*$ are densely defined closed operators and $T=T^{**}$ and
$S=S^{**}$ (Lemma~\ref{lem:doubledual}), so
$$
\ker T=
\ker T^{**}=
(\mathop{\mathrm{ran}} T^*)^\perp
\qquad\hbox{and}\quad
\ker S^*=(\mathop{\mathrm{ran}} S)^\perp.
$$
Then,
$$
H' =\ker T\oplus\overline{\mathop{\mathrm{ran}} T^*}=\ker S^*\oplus\overline{\mathop{\mathrm{ran}} S}
$$
which, together with the inclusions $\mathop{\mathrm{ran}} S\subseteq\ker T$ and $\mathop{\mathrm{ran}}
T^*\subseteq\ker S^*$ (Lemma~\ref{lem:semiexact}), implies
$$
H' =\overline{\mathop{\mathrm{ran}} T^*}\oplus(\ker T\cap\ker S^*)\oplus\overline{\mathop{\mathrm{ran}} S}.
$$
Finally, we know
$$
(\ker T\cap\ker S^*)\subseteq
\bigl(\ker(T^* T)\cap\ker(SS^*)\bigr)\subseteq
\ker(T^*T+S^*S).
$$
The result then follows from $\ker(T^*T+S^*S)\subseteq(\ker T\cap\ker
S^*)$. Assume $\psi\in\ker(T^* T+SS^*)$; then
$$
(\psi\mid 0)'
= (\psi\mid(T^*T+SS^*)\psi)'
= (\psi\mid T^*T\psi)' + (\psi\mid S^{**}S^*\psi)'
$$
so that
$$
0 = (T\psi\mid T\psi)'' + (S^*\psi\mid S^*\psi),
$$
which implies $\psi\in\ker T\cap \ker S^*$.
\end{proof}
We end this chapter with a `physical theorem' gathering all the results of
physical interest about the phase space of electromagnetism
that we proved
in
this chapter.
\begin{result}
\label{thm:3+1phys}
Let~$M$ be a $(3+1)$-dimensional static, globally hyperbolic
spacetime, with metric
$$
g_M=e^{2\Phi}(-\mathrm{d} t^2+g).
$$
Then, electromagnetism on~$M$ with gauge group~$\mathbb{R}$ has as its phase
space the real Hilbert space
$$
\mathbf{P}={\mathop{\mathrm{dom}}\{\mathrm{d}\colon L^2\Omega^1_S\to
L^2\Omega^2_S\}\over\overline{\mathop{\mathrm{ran}}}\{\mathrm{d}\colon L^2\Omega^0_S\to
L^2\Omega^1_S\}}\oplus\ker\{\mathrm{d}^*\colon L^2\Omega^1_S\to
L^2\Omega^0_S\},
$$
with continuous symplectic structure
$$
\omega(X,X')=(E,A')-(E',A)
$$
where~$X=[A]\oplus E$ and~$X'=[A']\oplus E'$ lie in~$\mathbf{P}$, and
$$
(\alpha,\beta)=\int_S g(\alpha,\beta)\mathrm{vol}
$$
is the canonical inner product induced on~$\Omega^k_S$ by the
optical metric~$g$ on~$S$. The Hamiltonian is the continuous quadratic
form
$$
H[X]={1\over 2}\bigl[(E,E)+(\mathrm{d} A,\mathrm{d} A)\bigr].
$$
There phase space splits naturally into two sectors,
$$
\mathbf{P}=\mathbf{P}_o\oplus\mathbf{P}_f,
$$
and the direct summands
$$
\mathbf{P}_f=\mathbf{P}\cap\ker\Delta
\qquad\hbox{and}\quad
\mathbf{P}_o=\mathbf{P}\cap\mathop{\mathrm{ran}}\mathrm{d}^*_1
$$
are preserved by time evolution.
On~$\mathbf{P}_o$, time evolution takes the form
$$
\left( \! \begin{array}{c} A \\ E \end{array} \! \right)
\mapsto T_o(t)
\left( \! \begin{array}{c} A \\ E \end{array} \! \right) =
\left( \begin{array}{cc}
\cos(t\sqrt{\Delta}) & \sin(t\sqrt{\Delta})\,/\,\sqrt{\Delta}\\
-\sqrt\Delta\,\sin(t\sqrt{\Delta}) & \cos(t\sqrt{\Delta})
\end{array} \right)
\left( \! \begin{array}{c} A \\ E \end{array} \! \right)
$$
while on~$\mathbf{P}_f$ it takes the form
$$
\left( \! \begin{array}{c} A \\ E \end{array} \! \right)
\mapsto
T_f(t) \left( \! \begin{array}{c} A \\ E \end{array} \! \right) =
\left( \begin{array}{cc}
1 & t \\
0 & 1
\end{array} \right)
\left( \! \begin{array}{c} A \\ E \end{array} \! \right).
$$
\end{result}
\section{Spacetime Geometry}
\label{sec:N+1-geometry}
We model spacetime as an $(n+1)$-dimensional smooth manifold~$M$ with
a Lorentzian metric of signature $(-+\cdots +)$. We assume that $M =
\mathbb{R}\times S$ for some smooth manifold~$S$, and that the metric on~$M$
is of the form
$$
g_M=-e^{2\Phi} dt^2 +g_S
$$
where~$g_S$ is a Riemannian metric on~$S$ and~$\Phi$ is a smooth
real-valued function on~$S$. As in Equation~(\ref{eq:optical}) we
write
$$
g_M=e^{2\Phi}(-dt^2 + g)
$$
where the `optical metric'~$g$ is given by $g = e^{-2\Phi} g_S$. We
assume that~$g$ makes~$S$ into a complete Riemannian manifold, since
this is a necessary and sufficient condition for~$M$ to be a globally
hyperbolic spacetime with the surfaces $\{t = c\}$ as Cauchy surfaces.
For a more complete discussion, refer back to
Section~\ref{sec:3+1-geometry}. As before, all fields on spacetime
carry the subscript~`$M$'; fields on space are written without subscript
or with the subscript~$0$. To study $p$-form electromagnetism we need
to fix an integer~$p$ with $0 \le p \le n$. Then, any
$k$-form~$\alpha_M$ on~$M$ can be uniquely decomposed as
$
\alpha_M = e^{-{1\over 2}(n-2p-1)\Phi} (\mathrm{d} t \wedge \alpha_0+ \alpha)
$
where~$\alpha$ is a time-dependent $k$-form on~$S$ and~$\alpha_0$ a
time-dependent $(k-1)$-form on~$S$. As explained in the previous
section, the strange-looking factor involving~$\Phi$ is chosen to
simplify things later.
The metric~$g_M$ induces a metric on the $k$-forms on spacetime, which
we also call~$g_M$, and similarly for the metric~$g$ on space. In terms
of spatial and temporal parts, these are related by:
\begin{equation}
g_M(\alpha_M,\alpha_M')
=e^{-(n+2k-2p-1) \Phi} \left[ -g(\alpha_0,\beta_0)+ g(\alpha,\beta) \right].
\label{eq:newinner}
\end{equation}
Assuming that~$S$ is oriented, the metrics~$g_M$ and~$g$ determine
volume forms~$\mathrm{vol}_M$ on~$M$ and~$\mathrm{vol}$ on~$S$, which are related by
\begin{equation}
\mathrm{vol}_M = e^{(n+1) \Phi} \mathrm{vol} \wedge \mathrm{d} t .
\label{eq:newvol}
\end{equation}
Again, it would be possible to deal with the nonorientable case by
working with densities instead of forms. As before, we define an
inner product $(\cdot \mid \cdot)$ on $k$-forms on space by
Equation~(\ref{eq:inner}), namely
$$
(\alpha,\beta) = \int_S g(\alpha,\beta)\,\mathrm{vol},
$$
and define~$L^2 \Omega^k$ to be the space of measurable
$k$-forms~$\alpha$ on~$S$ such that $(\alpha \mid \alpha) < \infty$.
We define the twisted exterior derivative $D_k \colon C_0^\infty \Omega^k_S
\to C_0^\infty \Omega^{k+1}_S$ by
\begin{equation}
\label{Dk}
D_k = e^{{1\over 2}(n-2p-1)\Phi} \mathrm{d}_k e^{-{1\over 2}(n-2p-1)\Phi} .
\end{equation}
This operator has a formal adjoint
\begin{equation}
\label{Dkdagger}
D_k^\dagger = e^{-{1\over 2}(n-2p-1)\Phi} \delta_{k+1} e^{{1\over
2}(n-2p-1)\Phi}
\end{equation}
meaning that
\begin{equation}
(D_k^\dagger \alpha,\beta) = (\alpha, D_k \beta)
\label{eq:formal_adjoint_n+1}
\end{equation}
whenever $\alpha \in C_0^\infty \Omega^{k+1}$ and
$\beta \in C_0^\infty \Omega^k$. In what follows we shall
omit the subscript~`$k$' from the operators~$D_k$ and~$D_k^\dagger$
when it is clear from context.
In Section~(\ref{sec:N+1-theorem}) we shall show that these operators have
mutually adjoint closures~$\overline{D_k}\colon L^2\Omega^k_S\to
L^2\Omega^{k+1}_S$ and~$D_k^*\colon L^2\Omega^{k+1}_S\to\Omega^k_S$,
and that the operators~$D_kD_k^\dagger$ and~$D_k^\dagger D_k$ are both
essentially self-adjoint, meaning that their respective
closures,~$\overline{D_k}D_k^*$ and~$D_k^*\overline{D_k}$, are their
unique self-adjoint extensions~\cite[\S VIII.2]{RS}.
\section{$p$-Form electromagnetism}
\label{sec:N+1-maxwell}
In $p$-form electromagnetism we take the vector potential as a $p$-form
on spacetime,~$A_M$, and take the action to be
$$
{\mathcal{S}}[A_M] = -{1\over 2}\int_M g_M(F_M,F_M)\, \mathrm{vol}_M
$$
where the field strength tensor~$F_M$ is given by
$$
F_M=\mathrm{d}_MA_M
$$
In terms of the twisted exterior derivative defined in
Equation~(\ref{Dk}), the field strength tensor equals
$$
\begin{array}{ccl}
F_M &=& (dt \wedge \partial_t + d) A_M \\
&=& (dt \wedge \partial_t + d) e^{-{1\over 2}(n-2p-1)\Phi}
(dt \wedge A_0 + A) \\
&=& e^{-{1\over 2}(n-2p-1)\Phi}(dt \wedge \partial_t + D)
(dt \wedge A_0 + A) \\
&=& e^{-{1\over 2}(n-2p-1)\Phi}
\bigl[dt \wedge (\partial_t A - DA_0) + DA\bigr] .
\end{array}
$$
With the help of equations~(\ref{eq:newinner})--(\ref{eq:newvol}),
this means that the action can be written as
\begin{eqnarray}
\label{eq:action_n+1}
{\mathcal{S}} &=& {1\over 2}\int_\mathbb{R} \int_M
\bigl[ g(\partial_t A - DA_0, \partial_t A - DA_0) -
g(DA,DA) \bigr] \mathrm{vol}_M \nonumber\\
&& \nonumber\\
&=&\displaystyle
{1\over 2}\int_\mathbb{R}
\bigl[(\partial_t A - DA_0, \partial_t A - DA_0) -
(DA, DA)\bigr] \, dt .
\end{eqnarray}
Note the complete analogy with Equation~(\ref{eq:action}). This
action gives the following equations of motion:
$$
\left\{
\begin{array}{rcl}
\partial_t DA & = & D A_0 \\
\partial_t^2A &=& - D^\dagger D A + \partial_t DA_0. \\
\end{array}
\right.
$$
The equations of $p$-form electromagnetism admit gauge symmetries of
the form
$$
A_M \mapsto A_M + \mathrm{d}_M \beta_M
$$
where~$\beta_M$ is a $(p-1)$-form on spacetime. Thus, to obtain
evolution equations, we work in temporal gauge, which amounts to
setting $A_0 = 0$. The above equations can then be written as
$$
\left\{
\begin{array}{rcc}
D^\dagger E & = & 0 \\
\partial_t A & = & E \\
\partial_t E & = & - D^\dagger D A. \\
\end{array}
\right.
$$
The Gauss law constraint $D^\dagger E = 0$ generates gauge
transformations of the form
$$
A \mapsto A + D \beta
$$
where~$\beta$ is a $(p-1)$-form on spacetime. Two pairs~$A\oplus E$
are physically equivalent if they differ by such a transformation.
Thus, ignoring analytical subtleties, the phase space of $p$-form
electromagnetism consists of pairs~$[A]\oplus E$ where~$[A]$ is an
equivalence class of $p$-forms on~$S$ modulo those of the
form~$D\beta$ (twisted-exact), and~$E$ is a $p$-form on~$S$
satisfying~$D^\dagger E = 0$ (twisted-divergenceless). The
Hamiltonian on this phase space is easily seen to be
$$
H\bigl[[A]\oplus E\bigr]={1\over 2}\bigl[(DA, DA) +(E, E) \bigr] .
$$
and the symplectic structure is
$$
\omega\bigl[[A]\oplus E,[A']\oplus E'\bigr]= (A, E') -(E, A') .
$$
Again as in the case of~$3+1$ dimensions, $(D\beta, E)=(\beta,
D^\dagger E)=0$ implies that the symplectic structure is
gauge-invariant.
All these formulas have analogues in Section \ref{sec:3+1}, so to
generalize all the results of that section we only need to generalize
Theorem \ref{thm:3+1} to the present context. In other words, first
we must show that the operators
$$
\xymatrix{C^\infty_0\Omega^k\ar@<.5ex>[r]^{D} &
C^\infty_0\Omega^{k+1}\ar@<.5ex>[l]^{D^\dagger}}
$$
have mutually adjoint closures, which we write as
$$
\xymatrix{L^2 \Omega^k\ar@<.5ex>[r]^{D} &
L^2 \Omega^{k+1}\ar@<.5ex>[l]^{D^*}}.
$$
Then we must prove a version of the Kodaira decomposition
saying that
$$
L^2\Omega^p =
\overline{\mathop{\mathrm{ran}} D_{p-1}}
\oplus
\ker L_p
\oplus
\overline{\mathop{\mathrm{ran}} D_p^*}
$$
where the twisted Laplacian on $k$-forms,
$$
L_p = D_p^* D_p + D_{p-1} D_{p-1}^* ,
$$
is a nonnegative self-adjoint operator on~$L^2 \Omega^p$. We do all
this in Section~\ref{sec:N+1-theorem} below.
Using these facts, we define the classical phase space for
$p$-form electromagnetism to be
$$
\mathbf{P} = \mathbf{A} \oplus \mathbf{E}
$$
where
$$
\begin{array}{ccl}
\mathbf{A} &=& \mathop{\mathrm{dom}} D_p / \, \overline{\mathop{\mathrm{ran}} D_{p-1}} \\
\mathbf{E} &=& \ker D_p .
\end{array}
$$
As with the Maxwell theory in~$3+1$ dimensions,~$\mathbf{P}$ becomes a real
Hilbert space space if we define
$$
\|[A]\oplus E\|^2=
( [A], [A] ) + ( d A, d A) + (E \mid E),
$$
where~$([A],[A]')$ can be defined on gauge equivalence classes using
the fact that, by the Kodaira decomposition,~$L^2\Omega^p /
\overline{\mathop{\mathrm{ran}} D_{p-1}}$ is canonically isomorphic to~$\mathop{\mathrm{ran}}
D_p^\perp$, which inherits an inner product by virtue of being a
subspace of~$L^2\Omega^p$.
As before, we can split the spaces~$\mathbf{A}$ and~$\mathbf{E}$
into `oscillating' and `free' parts:
$$
\begin{array}{ccl}
\mathbf{A} &=& \mathbf{A}_o \oplus \mathbf{A}_f \\
\mathbf{E} &=& \mathbf{E}_o \oplus \mathbf{E}_f ,
\end{array}
$$
where
$$
\begin{array}{lclllcl}
\mathbf{A}_o &=& \mathop{\mathrm{dom}} D_p \cap \overline{\mathop{\mathrm{ran}} D_p^* }
&\quad&
\mathbf{A}_f &=&\ker L_p \\
&&&&&& \\
\mathbf{E}_o &=& \overline{\mathop{\mathrm{ran}} D_p^*}
&\quad&
\mathbf{E}_f &=& \ker L_p .
\end{array}
$$
These decompositions let us write the classical phase space
as a direct sum of real Hilbert spaces:
$$
\mathbf{P} = \mathbf{P}_o \oplus \mathbf{P}_f ,
$$
where
$$
\begin{array}{ccl}
\mathbf{P}_o &=& \mathbf{A}_o \oplus \mathbf{E}_o \\
\mathbf{P}_f &=& \mathbf{A}_f \oplus \mathbf{E}_f .
\end{array}
$$
This is also a direct sum of symplectic vector spaces, and the
Hamiltonian is a sum of separate Hamiltonians on~$\mathbf{P}_o$ and~$\mathbf{P}_f$.
As a result, time evolution acts by symplectic transformations,
independently on the oscillating and free parts of any initial
data~$[A]\oplus E \in \mathbf{P}$.
For modes~$[A]\oplus E \in \mathbf{P}_o$, the Hamiltonian resembles that
of a harmonic oscillator:
$$
H[A\oplus E] = {1\over 2} \bigl[(DA, DA)+(E, E)\bigr]
$$
and the equations of motion are
$$
\left\{
\begin{array}{l}
\partial_t A = E \cr
\partial_t E =- LA, \cr
\end{array}
\right.
$$
where we write the twisted Laplacian~$L_p$ simply as~$L$.
The solutions of the corresponding integral equation are given by
\begin{equation}
\left( \begin{array}{c} A \\ E \end{array} \right)
\mapsto
\left( \begin{array}{cc}
\cos(t \sqrt{L} ) &
\sin(t \sqrt{L}) \, / \,\sqrt{L} \\
-\sqrt{L} \,\sin(t\sqrt{L}) &
\cos(t \sqrt{L})
\end{array} \right)
\left( \begin{array}{c} A \\ E \end{array} \right)
\label{eq:time_evolution_twisted}
\end{equation}
where we use the functional calculus to define functions of~$L$. The
proof that this is a strongly continuous~$1$-parameter group of
bounded operators is essentially the same as the one sketched after
Equation~(\ref{eq:time_evolution}).
For modes~$[A]\oplus E \in \mathbf{P}_f$, the Hamiltonian resembles that of a
free particle:
$$
H\bigl[[A]\oplus E\bigr] = {1\over 2}(E,E)
$$
and the equations of motion are
$$
\left\{
\begin{array}{l}
\partial_t A = E \cr
\partial_t E = 0. \cr
\end{array}
\right.
$$
The solutions of the equations of motion are given by
$$
\left( \begin{array}{c} A \\ E \end{array} \right)
\mapsto
\left( \begin{array}{cc}
1 & t \\
0 & 1
\end{array} \right)
\left( \begin{array}{c} A \\ E \end{array} \right)
$$
Note that, in the case of free modes, nothing besides the
definition of the Laplacian has changed from the case of~$1$-forms in
$3+1$ dimensions. In particular, time evolution is given by the very same
Equation~(\ref{eq:time_evolution_free}).
\section{Mathematical details}
\label{sec:N+1-theorem}
The results we need to make our work in the previous section
rigorous are all contained in this theorem:
\begin{theorem}\label{thm:N+1}
Let $S$ be a smooth $n$-dimensional manifold equipped with a complete
Riemannian metric $g$, and let $\Phi$ be a smooth real-valued function
on $S$. Fix an integer $0 \le p \le n$. Then for any integer $k$, the
operators
$$
\xymatrix{C^\infty_0\Omega^k_S\ar@<.5ex>[r]^{D_k} &
C^\infty_0\Omega_S^{k+1}\ar@<.5ex>[l]^{D_k^\dagger}}
$$
defined in equations (\ref{Dk}) and (\ref{Dkdagger})
have mutually adjoint closures, which we write as
$$
\xymatrix{L^2\Omega^k_S\ar@<.5ex>[r]^{D_k} &
L^2\Omega^{k+1}_S\ar@<.5ex>[l]^{D_k^*}}
$$
These closures satisfy
$$
\mathop{\mathrm{ran}} D_{k-1} \subseteq \ker D_k, \qquad
\mathop{\mathrm{ran}} D_k^* \subseteq \ker D_{k-1}^* ,
$$
and we obtain a direct sum decomposition
$$
L^2\Omega^k =
\overline{\mathop{\mathrm{ran}} D_{k-1}}
\oplus
\ker L_k
\oplus
\overline{\mathop{\mathrm{ran}} D_k^*} .
$$
where the twisted Laplacian on $k$-forms,
$$
L_k = D_k^* D_k + D_{k-1} D_{k-1}^* ,
$$
is a nonnegative densely defined self-adjoint operator on~$L^2 \Omega^k$.
\end{theorem}
\begin{proof} Because of the twisting of the exterior derivative
operator in Equation~(\ref{Dk}), one cannot simply apply the proof of
Theorem~\ref{thm:3+1}. The reason is that Gaffney's
Proposition~\ref{prop:gaffney} depends on the specific properties of
the `untwisted'~$\mathrm{d}$ and~$\delta$. However, the generalization is in fact
true, essentially because~$e^f \mathrm{d} e^{-f}$ and~$\mathrm{d}$ have the
same first-order part whenever $f$ is a smooth function.
This is made precise by an argument
due to Chernoff, which uses the concept of the `symbol' of a differential
operator. This argument implies both a generalization of
Proposition~\ref{prop:gaffney} and the self-adjointness of the twisted
Laplacian.
We begin by recalling Chernoff's formalism~\cite{chernoff73}, which is
the key to proving this theorem. Let~$S$ be a Riemannian manifold with
metric~$g$, and let~$E$ be any vector bundle on~$S$ whose fiber at
each point $x \in S$ is equipped with an inner
product~$\langle\cdot,\cdot\rangle_x$ depending smoothly on~$x$. The
space of smooth compactly supported sections of this vector bundle,
denoted~$C^\infty_0 E$, is given an inner product
$$
(\alpha\mid\beta) = \int_S \langle\alpha(x),\beta(x)\rangle_x \; \mathrm{vol}_S,
$$
where~$\mathrm{vol}_S$ is the canonical volume form on~$S$. The Hilbert space
completion of~$C^\infty_0 E$ with respect to this inner product is
denoted~$L^2 E$.
Assume that $T \colon C^\infty_0 E \to C^\infty_0 E$ is a first-order
linear differential operator on~$E$. Its formal adjoint~$T^\dagger$ is
again a first-order linear differential operator, defined by requiring
that
$$
(\alpha\mid T\beta)=(T^\dagger\alpha\mid\beta)
\qquad\hbox{for all}\quad
\alpha,\beta \in C^\infty_0 E.
$$
The `symbol' of~$T$ is defined by
$$
\sigma(\mathrm{d} f,\alpha)= T(f \alpha)- fT\alpha
$$
for any~$C^\infty_0$ function~$f$ and any~$\alpha \in C^\infty_0 E$.
Note that~$\sigma(\mathrm{d} f,\alpha)$ is a function on~$S$ whose value at any
point depends only on the values of~$\mathrm{d} f$ and~$\alpha$ at that point.
If~$T+T^\dagger$ is equal to multiplication by a smooth function, we say
the differential equation $\partial_t \alpha = T\alpha$ is a `symmetric
hyperbolic system'. At any point~$x \in S$, solutions of this
equation propagate at the speed
$$
c(x) =
\sup\bigl\{\|\sigma(\mathrm{d} f,\alpha)\|_x \colon \; \|\mathrm{d} f\|_x=\|\alpha\|_x=1\bigr\}
$$
where~$\|\mathrm{d} f\|_x$ is the norm of~$\mathrm{d} f$ at the point~$x$, defined using
the Riemannian metric~$g$, and~$\|\alpha\|_x$ is the norm of~$\alpha$
at the point~$x$, defined using the inner product on the fiber of~$E$
at~$x$.
Chernoff then essentially proves the following theorem. Note that the
Hilbert spaces appearing in this theorem are complex, so to apply it
to our real Hilbert spaces we need to complexify them.
\begin{lemma}[Chernoff] \label{lem:chernoff}
If the metric~$c^{-2}g$ makes~$S$ into a complete Riemannian manifold,
the symmetric hyperbolic system $\partial_t \alpha =T \alpha$ with
initial data in~$C^\infty_0 E$ has a unique solution on~$\mathbb{R}\times S$
which is in~$C^\infty_0 E$ for all $t \in \mathbb{R}$. Moreover, if~$T$ is
formally skew-adjoint ($T+T^\dagger=0$), then~$-iT$ and all its powers
are essentially self-adjoint on~$C_0^\infty E$.
\end{lemma}
\begin{proof}[Sketch of proof] The basic idea is that when we solve the
differential equation $\partial_t \alpha = T\alpha$, perturbations
propagate at speed~$1$ with respect to the metric~$c^{-2}g$. If this
metric is complete, information can never reach spacelike infinity in
a finite amount of time. Thus, given compactly supported smooth
initial data, the equation $\partial_t \alpha = T\alpha$ has a
solution~$\alpha(t,x)$ such that~$\alpha(t,\cdot)$ is compactly
supported for all~$t$---and smooth, by general results on hyperbolic
systems.
If~$T$ is formally skew-adjoint, one can show that the inner product
of two solutions is constant as a function of time:
$$
\begin{array}{ccl}
{d\over dt} (\alpha(t,\cdot) \mid \beta(t,\cdot)) &=&
(T \alpha(t,\cdot) \mid \beta(t,\cdot)) +
(\alpha(t,\cdot) \mid T \beta(t,\cdot)) \\
&=& (\alpha(t,\cdot) \mid T^\dagger \beta(t,\cdot)) +
(\alpha(t,\cdot) \mid T \beta(t,\cdot)) \\
&=& 0 .
\end{array}
$$
The crucial point here is that~$\alpha(t,\cdot)$ and~$\beta(t,\cdot)$
are compactly supported for all~$t$, so there are no boundary terms:
we only need the fact that~$T$ and~$T^\dagger$ are formal adjoints.
It follows that time evolution defines a one-parameter group of
inner-product-preserving transformations of~$C_0^\infty E$, which by
density extends uniquely to a one-parameter unitary group~$U(t)$
on~$L^2 E$. One can show that~$C_0^\infty E$ forms a `dense invariant
subspace of $C^\infty$ vectors' for~$U(t)$; in other words,
that~$C_0^\infty E$ is a dense subspace of~$L^2 E$, and that given
initial data~$\alpha$ in this subspace, the solution~$U(t) \alpha$
remains in this subspace for all times, defining an infinitely
differentiable function from~$\mathbb{R}$ to~$L^2 E$. By a theorem of
Nelson~\cite[Lemma 10.1]{nelson59}, this implies that~$-iT$ and all
its powers are essentially self-adjoint on the domain~$C_0^\infty E$,
and that the closure of~$-iT$ generates the one-parameter
group~$U(t)$. The only new thing to check here is the existence of
the derivatives~${d^n \over dt^n} U(t) \alpha$, which one can show by
repeatedly using the differential equation ${d\over dt} U(t) \alpha =
-iT U(t)\alpha$.
\end{proof}
This result applies without modification to first-order differential
equations like the Dirac equation. To apply it to our problem, we
resort to a well-known trick, taking $-iT$ to be the operator
$$
\left( \begin{array}{cc}
0 & D_k^\dagger \\
D_k & 0
\end{array} \right) .
$$
The essential self-adjointness of this operator will imply that
$D_k$ and $D_k^\dagger$ have mutually adjoint closures:
\begin{lemma}\label{lem:direct_sum}
Let $H_1$ and $H_2$ be Hilbert spaces and let
$$
\xymatrix{H_1 \ar@<.5ex>[r]^{A} &
H_2 \ar@<.5ex>[l]^{B}}
$$
be densely defined operators that are formal adjoints of one another:
$$
\langle A \phi, \psi \rangle_1 =
\langle \phi , B \psi \rangle_2
\qquad\hbox{for all}\quad
\phi \in \mathop{\mathrm{dom}} A , \psi \in \mathop{\mathrm{dom}} B.
$$
Let $H = H_1 \oplus H_2$ and let $S$ be the densely
defined operator
$$
\left( \begin{array}{cc}
0 & B \\
A & 0
\end{array} \right)
$$
on $H$. If $S$ is essentially self-adjoint, then $A$ and
$B$ have mutually adjoint closures.
\end{lemma}
\begin{proof}
It is easy to verify that the closure of $S$ is
$$
\left( \begin{array}{cc}
0 & \overline{B} \\
\overline{A} & 0
\end{array} \right)
$$
while the adjoint of the closure of $S$ is
$$
\left( \begin{array}{cc}
0 & (\overline{A})^* \\
(\overline{B})^* & 0
\end{array} \right) .
$$
If $S$ is essentially self-adjoint, these two
operators are equal. This implies that
$$
(\overline{A})^* = \overline{B}
$$
and
$$
(\overline{B})^* = \overline{A}
$$
so the closures of~$A$ and~$B$ are mutually adjoint.
\end{proof}
\begin{lemma}\label{lem:essential}
Suppose $S$ is a complete Riemannian manifold and $\Phi$
a smooth real-valued function on $S$. Let
$$
T \colon L^2 \Omega^k_S \oplus L^2 \Omega^{k+1}_S \to
L^2 \Omega^k_S \oplus L^2 \Omega^{k+1}_S
$$
be the densely defined operator
$$
\left( \begin{array}{cc}
0 & iD_k^\dagger \\
iD_k & 0
\end{array} \right) .
$$
Then $-iT$ and all its powers are essentially self-adjoint on
$C^\infty_0\Omega^k\oplus C^\infty_0\Omega^{k+1}$.
\end{lemma}
\begin{proof}
We show that the hypotheses of Lemma \ref{lem:chernoff} apply to the
operator $T$. Clearly $T$ is formally skew-adjoint, so it suffices to
check that the equation $\partial_t \alpha = T\alpha$ has propagation
speed $c = 1$.
First we consider the case where $\Phi = 0$, so $D = \mathrm{d}$ and
$D^\dagger = \delta$. The symbol of the operator $\mathrm{d}$ is
$$
\sigma_\mathrm{d}(\mathrm{d} f,\alpha)=
\bigl(\mathrm{d}(f\alpha)- f \mathrm{d}\alpha\bigr)= \mathrm{d} f \wedge\alpha
$$
for any $\alpha \in C_0^\infty \Omega^p_S$. The symbol of $\delta$ is
$$
\sigma_\delta(\mathrm{d} f,\beta)= -i_{\mathrm{d} f} \beta
\qquad\hbox{for any}\quad
\beta \in C_0^\infty \Omega^{p+1}_S,
$$
since
$$
\bigl( \sigma_\delta(\mathrm{d} f,\beta), \gamma \bigr) =
\bigl( \delta(f\beta) - f \delta \beta, \gamma \bigr) =
-\bigl( \beta, \mathrm{d}(f\gamma) - f \mathrm{d}\gamma \bigr) =
-\bigl( \beta, \mathrm{d} f \wedge\gamma \bigr) =
-\bigl( i_{\mathrm{d} f} \beta, \gamma \bigr)
$$
for any $\gamma \in C_0^\infty \Omega^p_S$.
It follows that the symbol of $T$ is
$$
\sigma_T(\mathrm{d} f,\alpha \oplus\beta)= i(i_{\mathrm{d} h}\beta \oplus \mathrm{d} f\wedge\alpha).
$$
To compute the propagation speed, note first that
$$
\begin{array}{ccl}
\|\sigma(\mathrm{d} f,\alpha\oplus\beta)\|_x^2
&=& \|\mathrm{d} f\wedge\alpha\|_x^2 + \|i_{\mathrm{d} f}\beta \|_x^2 \\
&\le& \|\mathrm{d} f\|^2 \left(\|\alpha\|_x^2 + \|\beta \|_x^2 \right) \\
&=& \|\mathrm{d} f\|^2 \left(\|\alpha \oplus \beta \|_x^2 \right)
\end{array}
$$
so the propagation speed is $\le 1$. In fact the propagation speed is
exactly $1$, since equality is achieved by letting $\mathrm{d} f=\mathrm{d} x_1$,
$\alpha = \mathrm{d} x_2\wedge\cdots \wedge \mathrm{d} x_{k+1}$, and $\beta = 0$
near $x$, where $\mathrm{d} x_1,\ldots,\mathrm{d} x_n$ is a coordinate frame orthogonal
at $x$.
To deal with the general case where $\Phi$ is nonzero, note that for
any first-order linear differential operator $X$ and any smooth
real-valued function~$h$, the operator $e^hXe^{-h}$ has the same
symbol as~$X$. In particular, the operators~$\mathrm{d}$ and~$D$ have the same
symbol, as do~$\delta$ and~$D^\dagger$. It follows that~$T$ always
has the same symbol as it does in the special case where~$\Phi = 0$,
so the propagation speed is always~$1$.
\end{proof}
\begin{corollary} \label{cor:essential}
Under the same hypothesis as Lemma \ref{lem:essential},
the operators
$$
\xymatrix{C^\infty_0\Omega^k_S\ar@<.5ex>[r]^{D_k} &
C^\infty_0\Omega^{k+1}_S\ar@<.5ex>[l]^{D_k^\dagger}}
$$
have mutually adjoint closures, and the operators~$D_k^\dagger D_k$
and~$D_k D_{k-1}^\dagger$ are essentially self-adjoint
on~$C_0^\infty \Omega^k$.
\end{corollary}
\begin{proof} The first part follows immediately from Lemmas
\ref{lem:direct_sum} and \ref{lem:essential}. For the second part, note
by Lemma~\ref{lem:essential} that $T^2 = D_k^\dagger D_k \oplus D_k
D_k^\dagger$ is essentially self-adjoint on~$C^\infty_0\Omega^k \oplus
C^\infty_0\Omega^{k+1}$. This implies that~$D_k^\dagger D_k$ and~$D_k
D_k^\dagger$ are essentially self-adjoint.
\end{proof}
We can now complete the proof of Theorem~\ref{thm:N+1}.
If we now use~$D_k$ and~$D_k^*$ to stand for the mutually adjoint
closures of the operators~$D_k$ and~$D_k^\dagger$,
Lemma~(\ref{lem:semiexact}) implies that
$$
\mathop{\mathrm{ran}} D_{k-1} \subseteq \ker D_k, \qquad
\mathop{\mathrm{ran}} D_k^* \subseteq \ker D_{k-1}^* ,
$$
so we can apply the Kodaira decomposition (Proposition~\ref{prop:kodaira})
to see that
$$
L^2\Omega^k =
\overline{\mathop{\mathrm{ran}} D_{k-1}}
\oplus
\ker L_k
\oplus
\overline{\mathop{\mathrm{ran}} D_k^*} .
$$
where
$$
L_k = D_k^* D_k + D_{k-1} D_{k-1}^* .
$$
To conclude we only need to show that~$L_k$ is a non-negative
self-adjoint operator. With respect to the Kodaira decomposition this
operator takes the block diagonal form
$$
\left( \begin{array}{ccc}
D_{k-1} D_{k-1}^* & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & D_k^* D_k
\end{array} \right) .
$$
It thus suffices to show that that $D_k^* D_k$ and $D_{k-1}
D_{k-1}^*$ are nonnegative and self-adjoint. By Lemma
\ref{lem:essential} we know these operators are essentially
self-adjoint when restricted to $C_0^\infty \Omega^k$. So all that
remains is to show that they are nonnegative. But $(x\mid D_k^* D_k
x)=(D_k x\mid D_k x)\ge 0$ for all $x\in\mathop{\mathrm{dom}} D_k^* D_k$.
\end{proof}
We end this section with a `physical theorem' entirely analogous to
the Result~\ref{thm:3+1phys} stated at the end of last chapter.
\begin{result}
\label{thm:n+1phys}
Let~$M$ be a $(n+1)$-dimensional static globally hyperbolic
spacetime, with metric
$$
g_M=e^{2\Phi}(-\mathrm{d} t^2+g).
$$
Then, $p$-form electromagnetism on~$M$ with gauge group~$\mathbb{R}$ has as
its phase space the real Hilbert space
$$
\mathbf{P}={\mathop{\mathrm{dom}}\{D_p\colon L^2\Omega^p_S\to
L^2\Omega^{p+1}_S\}\over\overline{\mathop{\mathrm{ran}}}\{D_{p-1}\colon L^2\Omega^{p-1}_S\to
L^2\Omega^p_S\}}\oplus\ker\{D_{p-1}^*\colon L^2\Omega^{p}_S\to
L^2\Omega^{p-1}_S\},
$$
where
$$
D_p=e^{{1\over 2}(n-2p-1)\Phi}\mathrm{d}_p e^{-{1\over 2}(n-2p-1)\Phi}
$$
is the \emph{twisted exterior derivative}. The phase space
admits a continuous symplectic structure
$$
\omega(X,X')=(E,A')-(E',A)
$$
where~$X=[A]\oplus E$ and~$X'=[A']\oplus E'$ lie in~$\mathbf{P}$ and
$$
(\alpha,\beta)=\int_S g(\alpha,\beta)\mathrm{vol}
$$
is the canonical inner product induced on~$\Omega^k_S$ by the
optical metric~$g$ on~$S$. The Hamiltonian is the continuous quadratic
form
$$
H[X]={1\over 2}\bigl[(E,E)+(D_p A,D_p A)\bigr].
$$
The phase space splits naturally into two sectors,
$$
\mathbf{P}=\mathbf{P}_o\oplus\mathbf{P}_f,
$$
and the direct summands
$$
\mathbf{P}_f=\mathbf{P}\cap\ker L
\qquad\hbox{and}\quad
\mathbf{P}_o=\mathbf{P}\cap\mathop{\mathrm{ran}} D^*_{p}
$$
are preserved by time evolution.
On~$\mathbf{P}_o$, time evolution takes the form
$$
\left( \! \begin{array}{c} A \\ E \end{array} \! \right)
\mapsto T_o(t)
\left( \! \begin{array}{c} A \\ E \end{array} \! \right) =
\left( \begin{array}{cc}
\cos(t\sqrt{L_p}) & \sin(t\sqrt{L_p})\,/\,\sqrt{L_p}\\
-\sqrt L_p\,\sin(t\sqrt{L_p}) & \cos(t\sqrt{L_p})
\end{array} \right)
\left( \! \begin{array}{c} A \\ E \end{array} \! \right)
$$
while on~$\mathbf{P}_f$ it takes the form
$$
\left( \! \begin{array}{c} A \\ E \end{array} \! \right)
\mapsto
T_f(t) \left( \! \begin{array}{c} A \\ E \end{array} \! \right) =
\left( \begin{array}{cc}
1 & t \\
0 & 1
\end{array} \right)
\left( \! \begin{array}{c} A \\ E \end{array} \! \right).
$$
\end{result}
\section{The general boson field}
The development that follows may seem idiosyncratic to those familiar
with the traditional quantization methods and the notations used in
physics. In particular, we insist on distinguishing the phase
space~$\mathbf{P}$ from its dual~$\mathbf{P}^*$. There are some good reasons
for this. At the present stage of development of mathematical physics,
the most compelling reason for studying the quantization of a linear
systems is as a springboard for quantization of nonlinear systems, or
as a testing ground for ideas suggested by the study of nonlinear
systems. Our approach is motivated by the fact that the ordinary
quantization of linear systems makes use of several identifications
that can only be made for a linear system. Adopting the view that a
classical mechanical system is characterized by its Poisson algebra of
observables, the cotangent space at each point of phase space acquires
a symplectic structure. When the phase space~$\mathbf{P}$ is linear, the
following identifications can be made: the dual~$\mathbf{P}^*$ can be
identified with the linear observables, and the restriction of the
Poisson bracket to~$\mathbf{P}^*$ is a symplectic structure. Also, the
cotangent spaces to each point of phase space are canonically
isomorphic to each other and to~$\mathbf{P}^*$, and the globally-defined
symplectic structure on~$\mathbf{P}^*$ makes~$\mathbf{P}$ isomorphic
to~$\mathbf{P}^*$ and also endows it with a symplectic structure. All of
these identifications, and even the possibility of considering
itself~$\mathbf{P}$ to be a symplectic vector space, are accidents of
linearity. Accordingly, we will avoid making use of these features as
much as possible. Every time we are forced to make use of one of these
identifications, it will be a sign that the procedure cannot be
readily generalized to nonlinear situations.
\subsection{Linear phase spaces}
We start by formalizing the notion of \emph{linear phase space}, which
is the necessary classical input of our quantization procedure.
\begin{definition}[linear phase space]\label{def:linearPhaseSpace}
A \emph{linear phase space} is a reflexive real topological vector
space~$\mathbf{P}$ whose dual~$\mathbf{P}^*$ is a \emph{symplectic vector
space\/}. That is,~$\mathbf{P}^*$ is a topological vector space equipped
with a \emph{symplectic structure}: a continuous, skew-symmetric
bilinear form~$\omega$ which is \emph{weakly nondegenerate} in the
sense that the duality map~$*\colon\mathbf{P}^*\to\mathbf{P}$ given by
$$
\omega(f,g)=f(g^*)
\qquad\hbox{for all}\quad
f,g\in\mathbf{P}^*
$$
is injective.
\end{definition}
\begin{proof}[Note]
Without the assumption that~$\mathbf{P}$ is reflexive, the duality map
would be~$*\colon\mathbf{P}^*\to\mathbf{P}^{**}$. This would have a bearing on
the definition of the Hilbert space of quantum states below.
A finite-dimensional vector space has a unique Hausdorff topology, and
any infinite-dimensional vector space can be topologized
algebraically~\cite[\S 1.2]{BSZ}; in either case the continuity
of~$\omega$ is vacuously true. In general, the dual~$\mathbf{P}^*$ of a
topological vector space is itself naturally a topological vector
space, with the $\hbox{weak-}*$ topology making every element
of~$\mathbf{P}$ a continuous linear functional on~$\mathbf{P}^*$. If~$\mathbf{P}$
has a normed topology,~$\mathbf{P}^*$ can also be given the (normed)
strong operator topology. In either case,~$\mathbf{P}\subseteq\mathbf{P}^{**}$
is a continuous inclusion.
\end{proof}
The right notion of automorphism of a linear phase space is the
following. Recall that, if~$T\colon\mathbf{P}\to\mathbf{P}$ is linear, there is
a unique linear map ~$T^*\colon\mathbf{P}^*\to\mathbf{P}^*$ called its
\emph{dual} such that
$$
(T^*f)(x)=f(Tx)
\qquad\hbox{for all}\quad
x\in\mathbf{P},f\in\mathbf{P}^*.
$$
\begin{definition}[automorphism of a linear phase space]\label{def:automorph}
An automorphism of the linear phase space~$\mathbf{P}$ is a continuous
invertible linear map~$T\colon\mathbf{P}\to\mathbf{P}$ whose dual
map~$T^*\colon\mathbf{P}^*\to\mathbf{P}^*$ preserves the symplectic structure on~$\mathbf{P}^*$.
\end{definition}
The \emph{space of states} of a classical system is its physical phase
space~$P$, namely the space of gauge equivalence classes of
solutions of its equations of motion. Similarly, its \emph{algebra of
observables} consists of smooth gauge-invariant functions of solutions
to the equations of motion,~$C^\infty(P)$. The classical algebra
of observables is naturally a Poisson algebra, but the physical phase
space~$P$ need not be a Poisson manifold, let alone a symplectic
vector space. For instance, in Yang--Mills theory~$P$ is some
sort of `singular infinite-dimensional variety', a concept without a
precise definition. Continuous non-gauge symmetries of the physical
system are are represented by automorphisms of the Poisson algebra of
observables generated through Poisson brackets with appropriate
observables: the conserved quantities associated to the symmetries via
Noether's theorem.
Suppose, then, that not only~$C^\infty(P)$ is a Poisson algebra with
Poisson bracket~$\{~,~\}$ but that~$P$ is a manifold. The Poisson
bracket defines a
bivector~$\omega\colon\Omega^2(P)\to\mathbb{R}$ given by
$$
\omega(\mathrm{d} f,\mathrm{d} g)=\{f,g\}
\qquad\hbox{for all}\quad
f,g\in C^\infty(\mathbf{P}).
$$
If~$\omega$ is non-degenerate at~$x\in P$, the space~$T^*_xP$
becomes a symplectic vector space. In physical terms,~$x$ is a field
configuration and~$T^*_xP$ is the space of linear observables in the
vicinity of this field configuration. This is the only symplectic
vector space that can be constructed in a natural way from the phase
space~$P$, and Definition~\ref{def:linearPhaseSpace} applies
with~$\mathbf{P}=T_xP$ and~$\omega=\omega_x$. In these favourable cases,
symmetries of field configurations~$x\in P$ are Poisson maps
leaving~$x$ fixed, which induce linear symplectic transformations
of~$T_x^*P$.
Identifying all the~$T_x^*P$ amounts to choosing a trivialization
of~$T^*P$, and this is natural only if~$P$ is a linear space admitting
a canonical flat connection. In that case, each of the~$T_x^*P$ is
canonically isomorphic to~$P^*$ itself. When the equations of motion
are linear, one can take~$\mathbf{P}=P$ in
Definition~\ref{def:linearPhaseSpace}, and restrict one's attention to
linear observables and symmetry transformations.
\subsection{Quantizing a linear phase space}
Linear quantization\index{quantization} is a process ``promoting''
each~$x\in\mathbf{P}$\index{$x$!point of phase space} to a unit
vector~$\ket{x}$\index{$\ket{x}$!quantized phase space point} in a
suitable Hilbert space~$\mathbf{K}$\index{$\mathbf{K}$!quantum state space}, and
each~$f\in\mathbf{P}^*$ to a self-adjoint operator~$\hat f$\index{$\hat
f$!quantized observable} on~$\mathbf{K}$, in such a way that the Heisenberg
commutation relations\index{Heisenberg commutation relations}
\begin{equation}\label{HeisCommRel}
[\hat f,\hat g]=i\omega(f,g)\mathbf{1}_\mathbf{K}
\qquad\hbox{for all}\quad
f,g\in\mathbf{P}^*
\end{equation}
hold. Equation~\ref{HeisCommRel} is a restricted form of the Dirac
quantization prescription\index{Dirac quantization prescription},
since it is applied only to linear observables on~$\mathbf{P}$, and not to
arbitrary ones as it was originally formulated. In addition, the
correspondence principle\index{correspondence principle} is required
to hold in the form
\begin{equation}
\label{eq:corr}
\matElem{x}{\hat f}{x}=f(x)
\qquad\hbox{for all}\quad
x\in\mathbf{P},f\in\mathbf{P}^*,
\end{equation}
without allowing for corrections of order~$\hbar$.
Finally, one would hope to represent every physical
symmetry~$T\colon\mathbf{P}\to\mathbf{P}$ as a unitary
operator~$U_T\colon\mathbf{K}\to\mathbf{K}$ in such a way that~$U_S U_T=U_{ST}$ for all
symplectic maps~$S,T\colon\mathbf{P}\to\mathbf{P}$. As we shall see, in
general this is only possible for a subgroup of linear symplectic
transformations of~$\mathbf{P}$ and, in fact, choosing a small subgroup of
physical symmetries that must be unitarily implemented can be enough
to determine~$\mathbf{K}$, sometimes uniquely. Time evolution is always
required to be a physical symmetry and, in this sense, the dynamics
determine the quantization.
\subsubsection{Canonical commutation relations}
The Heisenberg relations cannot be implemented on an algebra of
bounded operators~\cite[\S 13.6]{rudin91}, and so
Equation~\ref{HeisCommRel} must be understood as holding on the
(hopefully) dense domain of~$[\hat f,\hat g]$ in~$\mathbf{K}$. This is only
the first of a long list of nuisances that arise from necessarily
dealing with unbounded operators, but all the same we encode it as a
definition.
\begin{definition}[Heisenberg system]
A \emph{Heisenberg system}\index{Heisenberg system} on a symplectic
vector space~$(\mathbf{P}^*,\omega)$ is a real-linear map~$\Phi\colon
f\mapsto\Phi(f)$ from~$\mathbf{P}^*$ to the self-adjoint operators on some
complex Hilbert space~$\mathbf{K}$, satisfying the \emph{Heisenberg
commutation relations}\index{Heisenberg commutation relations}
$$
\bigl[\Phi(f),\Phi(g)\bigr]=i\omega(f,g)\mathbf{1}_\mathbf{K}.
\qquad\hbox{for all}\quad
f,g\in\mathbf{P}^*
$$
as an operator equation holding on the common domain
of~$\Phi(f)\Phi(g)$ and~$\Phi(f)\Phi(g)$, which is assumed to be
dense. The operator~$\Phi(f)$ is called the \emph{Heisenberg
operator}\index{Heisenberg operator} associated to~$f\in\mathbf{P}^*$.
\end{definition}
In other words, linear quantization is partially achieved by
constructing a Heisenberg system on the space of linear
observables~$(\mathbf{P}^*,\omega)$. However, there are lots of Heisenberg
systems that have nothing to do with physics, examples of which can be
found in~\cite{R&S,BSZ}, so for honest quantum physics one needs to
impose some additional regularity on the Heisenberg systems. This is
achieved in an somewhat circuitous way by considering the unitary
groups supposedly generated by the Heisenberg
operators. Heuristically, if~$\Phi(f)$ is a Heisenberg operator
on~$\mathbf{K}$, the operator~$W(f)=e^{-i\Phi(f)}$ is unitary and,
since~$\bigl[\Phi(f),\Phi(g)\bigr]$ commutes with both~$\Phi(f)$
and~$\Phi(g)$, the Baker--Campbell--Hausdorff
formula\index{Baker--Campbell--Hausdorff formula} applies, giving
$$
e^{-i\Phi(f)}e^{-i\Phi(g)}=e^{-i\Phi(f+g)}e^{-{1\over 2}[\Phi(f),\Phi(g)]}.
$$
We take this heuristic calculation as the motivation of our next
definition.
\begin{definition}[Weyl algebra]
The \emph{Weyl algebra}\index{Weyl algebra} on a symplectic vector
space space~$(\mathbf{P}^*,\omega)$, is the
complex~$*$-algebra~$\mathcal{W}(\mathbf{P}^*,\omega)$\index{$\mathcal{W}(\mathbf{P}^*,\omega)$!Weyl
algebra} generated by the set
$\mathcal{W}(\mathbf{P}^*)=\bigl\{\mathcal{W}(f)\bigr\}_{f\in\mathbf{P}^*}$,\index{$\mathcal{W}(\mathbf{P}^*)$!Weyl
operators of~$\mathbf{P}^*$}\index{$\mathcal{W}(f)$!Weyl operator
of~$f\in\mathbf{P}^*$} of \emph{Weyl operators}, modulo the
\emph{unitarity relations}
$$
\mathcal{W}(f)^*=\mathcal{W}(-f)
\qquad\hbox{for all}\quad
f\in\mathbf{P}^*
$$
and the \emph{Weyl relations}\index{Weyl relations}
$$
\mathcal{W}(f)\mathcal{W}(g)=e^{\omega(f,g)/2i}\mathcal{W}(f+g)
\qquad\hbox{for all}\quad
f,g\in\mathbf{P}^*.
$$
\end{definition}
\begin{proof}[Note]
Because the Weyl relations\index{Weyl relations} reduce products of
Weyl operators\index{Weyl operator} to single Weyl operators, the Weyl
algebra~$\mathcal{W}(\mathbf{P},\omega)$ coincides with the linear span
of~$\mathcal{W}(\mathbf{P})$. In fact,~$\mathcal{W}(\mathbf{P})$ is a basis
of~$\mathcal{W}(\mathbf{P},\omega)$.
\end{proof}
Heuristically, because of the Baker--Campbell--Hausdorff
formula\index{Baker--Campbell--Hausdorff formula} above, one would expect
that a Heisenberg system can be constructed from a representation of
the Weyl algebra as an algebra of operators on a suitable Hilbert
space. Such a representation is called a Weyl system\index{Weyl
system}. We will consistently use the fonts~$\mathcal{W}$ and~$W$ to
distinguish the \emph{abstract} Weyl algebra~$\mathcal{W}(\mathbf{P},\omega)$, and
its generators~$\mathcal{W}(x)$, from Weyl systems~$W$ associated to
\emph{concrete} Hilbert-space representations of the Weyl algebra.
\begin{definition}[Weyl system]
A \emph{Weyl system}\index{Weyl system} on the symplectic vector
space~$(\mathbf{P}^*,\omega)$ is a continuous mapping $W\colon\mathbf{P}^*\to
U(\mathbf{K})$, where~$U(\mathbf{K})$ is the group of unitary operators on the complex
Hilbert space~$\mathbf{K}$ with the strong operator topology,
and~$W$ satisfies the Weyl relations\index{Weyl relations}
$$
W(f)W(g)=e^{\omega(f,g)/2i}W(f+g)
\qquad\hbox{for all}\quad
f,g\in\mathbf{P}^*.
$$
\end{definition}
\begin{proof}[Note]
Since a Weyl system is required to be continuous in the strong
operator topology on~$U(\mathbf{K})$, the map~$t\mapsto W(tf)$ is a
strongly-continuous one-parameter subgroup of~$U(\mathbf{K})$. By Stone's
theorem~\cite[\S VIII.4]{RS}, this one-parameter subgroup has a
self-adjoint generator~$\Phi(f)$ such that $W(f)=e^{-i\Phi(f)}$.
\end{proof}
\begin{lemma}
\label{lem:WeylHeis}
If~$W\colon\mathbf{P}^*\to U(\mathbf{K})$ is a Weyl system on the symplectic vector
space~$(\mathbf{P}^*,\omega)$ then~$\Phi\colon\mathbf{P}^*\to L(\mathbf{K})$ is a
Heisenberg system on~$(\Phi^*,\omega)$. In addition, for
all~$x,y\in\mathbf{P}$, the operator~$\Phi(f)+i\Phi(g)$ is closed
and~$\Phi(f+g)$ is the closure of~$\Phi(f)+\Phi(g)$.
\end{lemma}
\begin{proof}[Sketch of proof]
Differentiating the Weyl relation\index{Weyl relations}
$$
W(tf)W(tg)=e^{t^2\omega(f,g)/2i}W\bigl(t(f+g)\bigr)
$$
twice and setting~$t=0$, one obtains that~$f\mapsto\Phi(f)$ is additive
and satisfies the Heisenberg commutation relations
$$
\bigl[\Phi(f),\Phi(g)\bigr]=i\omega(f,g)\mathbf{1}_\mathbf{K}.
$$
The proof of the closure properties of the Heisenberg operators is
in~\cite[\S 1.2]{BSZ}.
\end{proof}
At this point, a theorem of von~Neumann~\cite[\S VIII.5]{R&S}
guarantees that all Weyl systems\index{Weyl system} on a
finite-dimensional phase space are unitarily equivalent. At any rate,
we see that Weyl systems are the right formalization of
Equation~(\ref{HeisCommRel}), the Heisenberg commutation
relations. The following lemma shows one reason why it is convenient
to insist that physical symmetries be represented by \emph{linear}
symplectic maps on~$\mathbf{P}$.
\begin{lemma}
Suppose that~$\gamma\colon\mathcal{W}(\mathbf{P}^*,\omega)\to\mathcal{W}(\mathbf{P}^*,\omega)$ is
a~$*$-algebra endomorphism such that
$$
\hbox{for every}\quad
f\in\mathbf{P}^*,
\qquad
\gamma\bigl(\mathcal{W}(f)\bigr)=\mathcal{W}(g)
\qquad\hbox{for some}\quad
g\in\mathbf{P}^*,
$$
and suppose furthermore that the
map~$T^*\colon(\mathbf{P}^*,\omega)\to(\mathbf{P}^*,\omega)$ given by~$T^*f=g$ is
continuous. Then,~$T^*$ is in fact linear and preserves the symplectic
structure~$\omega$. If, in addition,~$\gamma$ is an automorphism,
then~$T^*$ is invertible, that is,~$T$ is an automorphism of the
linear phase space~$\mathbf{P}$.
\end{lemma}
What this means is that the formalization of quantization using Weyl
systems is best suited to the case when physical symmetries---in
particular, time evolution---are linear.
\begin{proof}
Assuming~$\gamma$ is a $*$-algebra endomorphism,
$$
\gamma\bigl(\mathcal{W}(f)\bigr)\gamma\bigl(\mathcal{W}(h)\bigr)=\gamma\bigl(\mathcal{W}(f)\mathcal{W}(h)\bigr)
$$
so, applying the definition of~$T^*$ on the left-hand side and the Weyl
relations on the right-hand side,
$$
\mathcal{W}(T^*f)\mathcal{W}(T^*h)=\gamma\bigl(e^{\omega(f,h)/2i}\mathcal{W}(f+h)\bigr).
$$
Now, the Weyl relations on the left-hand side and the properties
of~$\gamma$ on the right-hand side imply
$$
e^{\omega(T^*f,T^*h)/2i}\mathcal{W}(T^*f+T^*h)=e^{\omega(f,h)/2i}\mathcal{W}\bigl(T^*(f+h)\bigr).
$$
Since all the~$\{\mathcal{W}(f)\}_{f\in\mathbf{P}^*}$ are linearly independent by
construction, it follows that~$T^*$ is additive and
preserves~$\omega$. Finally, continuous additive functions are linear.
\end{proof}
The converse of this result is also true.
\begin{lemma}\label{lem:symmetry}
If~$T\colon\mathbf{P}\to\mathbf{P}$ is an automorphism of the linear phase
space~$\mathbf{P}$, then there exists a unique~$*$-algebra
automorphism~$\gamma(T)\colon\mathcal{W}(\mathbf{P}^*,\omega)\to\mathcal{W}(\mathbf{P}^*,\omega)$
determined by
$$
\gamma(T)\colon\mathcal{W}(T^*f)\mapsto\mathcal{W}(f)
\qquad\hbox{for all}\quad
f\in\mathbf{P}^*
$$
and such that~$\gamma(ST)=\gamma(S)\gamma(T)$.
\end{lemma}
In other words,~$\gamma$ is the unique representation of the group of
symplectic automorphisms of~$(\mathbf{P}^*,\omega)$ as~$*$-algebra
automorphisms of~$\mathcal{W}(\mathbf{P}^*,\omega)$ mapping the set of
generators~$\{\mathcal{W}(f)\colon f\in\mathbf{P}^*\}$ to itself. This result is
related to~\cite[Corollary 5.1.1]{BSZ}.
\begin{proof}
Applying~$\gamma(T)$ to both sides of the Weyl relation
$$
W(T^*f)W(T^*g)=e^{\omega(T^*f,T^*g)/2i}W\bigl(T^*(f+g)\bigr)
\qquad\hbox{for all}\quad
f,g\in\mathbf{P}^*
$$
we obtain
$$
W(f)W(g)=e^{\omega(T^*f,T^*g)/2i}W\bigl(f+g\bigr)
\qquad\hbox{for all}\quad
f,g\in\mathbf{P}^*,
$$
so~$\gamma(T)$ is an automorphism because~$T^*$ is symplectic. Also,
if~$S,T\colon\mathbf{P}\to\mathbf{P}$ are two automorphisms of~$\mathbf{P}$,
\begin{eqnarray*}
\gamma(S)\gamma(T)\mathcal{W}\bigl((ST)^*f\bigr)
&=&
\gamma(S)\gamma(T)\mathcal{W}(T^*S^*f)\\
&=&
\gamma(S)\mathcal{W}(S^*f)\\
&=&
\mathcal{W}(f)\\
&=&
\gamma(ST)\mathcal{W}\bigl((ST)^*f\bigr)
\end{eqnarray*}
for all~$f\in\mathbf{P}^*$.
\end{proof}
\begin{definition}[general boson field]
If~$(\mathbf{P}^*,\omega)$ is a symplectic vector space, the \emph{general boson
field} over it is the pair~$(\mathcal{W},\gamma)$ where~$\mathcal{W}\colon
f\mapsto\mathcal{W}(f)$ is the map from~$\mathbf{P}^*$ to~$\mathcal{W}(\mathbf{P}^*,\omega)$,
and~$\gamma$ is the representation of automorphisms of~$\mathbf{P}$
by~$*$-automorphisms of~$\mathcal{W}(\mathbf{P}^*,\omega)$ mentioned in
Lemma~\ref{lem:symmetry}.
\end{definition}
\begin{proof}[Note]
This definition is implicit in \cite[\S 5.3]{BSZ}.
\end{proof}
In sum, given any linear phase space space~$\mathbf{P}$ with
dual~$(\mathbf{P}^*,\omega)$ one can construct the associated Weyl
algebra~$\mathcal{W}(\mathbf{P}^*,\omega)$, which supports a
representation~$\gamma$ of the automorphisms of~$\mathbf{P}$
as~$*$-algebra automorphisms of~$\mathcal{W}(\mathbf{P}^*,\omega)$. In addition,
any Weyl system on~$(\mathbf{P}^*,\omega)$, that is, any strongly
continuous representation of~$\mathcal{W}(\mathbf{P}^*,\omega)$ as unitary
operators on a complex Hilbert space~$\mathbf{K}$ provides a realization of
the Heisenberg commutation relations. This is the general boson field
on~$\mathbf{P}$.
\subsubsection{Correspondence principle}
The general boson field realizes the canonical commutation relations
and the physical symmetries of a linear system, but it does not
provide a complete quantization of a linear phase space, as there are
a few lingering issues. The first is how to actually construct Weyl
systems. The second is whether the correspondence principle is
satisfied. The third is whether physical symmetries are implemented
unitarily on the supporting Hilbert space of the Weyl system. It turns
out that all three are related. In this section we will first use the
Gel'fand--Na\u{\i}mark--Segal construction to produce Weyl systems,
and then use the correspondence principle and unitary implementability
of physical symmetries to select the Weyl systems that produce
physically sensible quantizations.
The following example constructs the so-called \emph{Schr\"odinger
representation}\index{Schr\"odinger representation} of the Heisenberg
commutation relations\index{Heisenberg commutation relations} in one
dimension.
\begin{proof}[Example]
We choose units such that~$\hbar=1$. Let~$\mathbf{K}=L^2(\mathbb{R})$ and, for
each~$f=(a,k)\in\mathbb{R}^2$, define
$$
\bigl[W(f)\psi\bigr](x)=e^{-ik(x-a/2)}\psi(x-a)
\qquad\hbox{for all}\quad
\psi\in\mathbf{K},
$$
which clearly makes~$W(f)$ a unitary operator on~$\mathbf{K}$. Also,
$$
W(f)W(f')=e^{(ka'-k'a)/2i}W(f+f')
$$
so~$W$ is a Weyl system\index{Weyl system} on the linear phase
space~$\mathbf{P}=\mathbb{R}^2$ with
$$
\mathbf{P}^*=\bigl\{f=(a,k)\in\mathbb{R}^2\bigr\}
\qquad\hbox{and}\quad
\omega(f,f')=ka'-k'a.
$$
The Heisenberg operators are given by
$$
\Phi(f)\psi(x)=(kx-ia\partial_x)\psi(x).
$$
This Heisenberg system is called the Schr\"odinger
representation\index{Schr\"odinger representation}.
Given that~$\Phi$ is linear, it might seem odd that the momentum
coordinate~$k$ appears as the coefficient of the operator of
multiplication by~$x$, which we would usually with the position
operator. In addition, the symplectic structure~$\omega(f,f')=ka'-k'a$
seems backwards. We now proceed to explain these features of the
representation.
The configuration space is~$\mathbb{R}$ with coordinate function~$q\colon\mathbb{R}
\to\mathbb{R}$ satisfying~$q(x)=x$, and the phase space is~$\mathbf{P}=\mathbb{R}^2$ with
coordinate functions~$q,p\colon\mathbb{R}^2\to\mathbb{R}$ ($p$ being the momentum
coordinate function). Then,~$\mathrm{d} p$ and~$\mathrm{d} q$ are a basis
of~$\mathbf{P}^*$, and~$(a,k)$ are coordinates on~$\mathbf{P}^*$ with respect
to that basis. That is, we identify~$f=(a,k)$ with~$f=a\mathrm{d} p+k\mathrm{d}
q$. This is the correct pairing despite what our intuition might
suggest, namely pairing~$a$ with~$q$ since they both refer to the same
quantity (position), because~$ps+qk$ has homogeneous units of action
while~$qs+pk$ is not a homogeneous quantity. We are, in fact, omitting
factors of Planck's constant~$\hbar$ as we have chosen `natural
units' in which~$\hbar=1$ according to custom.
The linear observables~$q$ (position) and~$p$ (momentum) on~$\mathbf{P}$
have Poisson bracket
$$
\{q,p\}=1.
$$
Accordingly, the dual~$\mathbf{P}^*$ is generated by~$\mathrm{d} q,\mathrm{d} p$ with
symplectic structure
$$
\omega(\mathrm{d} q,\mathrm{d} p)=\{q,p\}=1.
$$
In other words,
$$
\mathbf{P}^*=\{f=k\mathrm{d} q+a\mathrm{d} p\colon a,k\in\mathbb{R}\}
$$
and the symplectic structure on~$\mathbf{P}^*$ is
$$
\omega(f,f')=\omega(k\mathrm{d} q+a\mathrm{d} p,k'\mathrm{d} q+a'\mathrm{d} p)=ka'-k'a.
$$
So, the apparently contradictory
$$
\{q,p\}=1
\qquad\hbox{and}\quad
\omega\bigl((a,k),(a',k')\bigr)=ka'-k'a
$$
are entirely consistent. Then, we have
$$
\Phi(\mathrm{d} q)\phi(x)=x\phi(x)
\qquad\hbox{and}\quad
\Phi(\mathrm{d} p)\phi(x)=-i\partial_x\phi(x)
$$
as expected, and if~$f=(a,k)$,
$$
\Phi(f)=a\Phi(\mathrm{d} p)+k\Phi(\mathrm{d} q).
$$
\end{proof}
It is clear how this representation can be extended to any finite
number of dimensions, and by the theorem of von~Neumann alluded to
after Lemma~(\ref{lem:WeylHeis}), these representations are unique up
to unitary equivalence. For the infinite-dimensional case relevant to
field theories, though, one needs to use the
Gel'fand--Na\u\i{}mark--Segal
construction\index{Gel'fand--Na\u\i{}mark--Segal construction}, which
is based on the concept of a \emph{state}\index{state!GNS state} and
leads to possibly unitarily inequivalent representations.
\begin{definition}[GNS state]\label{def:GNS}
A \emph{state}\index{state!GNS state} on a~$*$-algebra~$A$ is a linear
functional
$$
\langle~\rangle\colon A\to\mathbb{C}
$$
which is \emph{nonnegative}
$$
\langle a^*a\rangle\ge 0
\qquad\hbox{for all}\quad
a\in A,
$$
and \emph{normalized}
$$
\langle 1\rangle=1.
$$
\end{definition}
\begin{proof}[Note]
The usage here is completely analogous to that for linear functionals
on vector spaces. A purely algebraic definition of linear functional
on a vector space requires that it be defined everywhere, but when a
topology is introduced one finds it useful to consider discontinuous,
densely-defined linear functionals. In the same vein, as long as the
algebra~$A$ is not assumed to have a topology, one must require that
states be defined on all of~$A$. However, if~$A$ has a topology making
addition and multiplication continuous, then one can talk about
continuous or bounded states, and also about discontinuous,
densely-defined states. At this point,~$\mathcal{W}(\mathbf{P}^*,\omega)$ does not
have a topology defined on it so states on it should be defined
everywhere. On the other hand, the Weyl system~$W(\mathbf{P}^*,\omega)$
on~$\mathbf{K}$ is given the strong operator topology, and so densely-defined
states make sense on it. In fact, we will use a state
on~$\mathcal{W}(\mathbf{P}^*,\omega)$ to construct~$\mathbf{K}$, and it is not guaranteed
that the state will be everywhere defined on it.
\end{proof}
A state\index{state!GNS state} on~$\mathcal{W}(\mathbf{P}^*,\omega)$ defines a
nonnegative-definite sesquilinear form~$\langle~\mid~\rangle$
on~$\mathcal{W}(\mathbf{P}^*,\omega)$ by means of
$$
\langle \mathcal{W}\mid\mathcal{W}'\rangle\colon=\langle \mathcal{W}^*\mathcal{W}'\rangle
\qquad\hbox{for all}\quad
\mathcal{W},\mathcal{W}'\in\mathcal{W}(\mathbf{P}^*,\omega).
$$
Note that,
since~$\langle\mathcal{W}(f)\mid\mathcal{W}(g)\rangle=\langle\mathcal{W}(-f)\mathcal{W}(g)\rangle$,
\begin{equation}
\label{eq:InnProd}
\langle\mathcal{W}(f)\mid\mathcal{W}(g)\rangle=e^{i\omega(f,g)/2}\langle\mathcal{W}(g-f)\rangle
\qquad\hbox{for all}\quad
f,g\in\mathbf{P}^*.
\end{equation}
The associated nonnegative quadratic form
$$
|\mathcal{W}|^2=\langle\mathcal{W}|\mathcal{W}\rangle
$$
is finite on all of~$\mathcal{W}(\mathbf{P},\omega)$, since
$$
\bigl|\mathcal{W}(f)\bigr|=1
\qquad\hbox{for all}\quad
f\in\mathbf{P}^*.
$$
However, it can only be guaranteed to be a seminorm, because it is
possible that~$\langle~\rangle$ has a kernel. However, this kernel is
necessarily invariant under multiplication by elements
of~$\mathcal{W}(\mathbf{P}^*,\omega)$. Indeed, that~$|\mathcal{W}|=0$ is equivalent
to~$\langle\mathcal{W}(f)\mid\mathcal{W}\rangle=0$ for all~$f\in\mathbf{P}^*$. But then
$$
\langle\mathcal{W}(f)\mid\mathcal{W}(g)\mathcal{W}\rangle=\langle e^{\omega(f,g)/2i}\mathcal{W}(f-g)\mid\mathcal{W}\rangle
\qquad\hbox{for all}\quad
f\in\mathbf{P}^*
$$
implies that~$|\mathcal{W}(g)\mathcal{W}|=0$ for all~$g\in\mathbf{P}^*$.
By the standard
procedure---namely, taking the quotient of~$\mathcal{W}(\mathbf{P}^*,\omega)$ by
the null subspace of~$|~|$ and completing the result with respect
to~$|~|$ (which is a norm after quotienting by the null
subspace)---one can construct a complex Hilbert space~$\mathbf{K}$ with inner
product~$\langle~\mid~\rangle$. The invariance of the null space
of~$|~|$ under the multiplicative action of~$\mathcal{W}(\mathbf{P}^*,\omega)$
implies that~$\mathcal{W}(\mathbf{P}^*,\omega)$ acts on~$\mathbf{K}$.
This is a version of the Gel'fand--Na\u\i{}mark--Segal
construction\index{Gel'fand--Na\u\i{}mark--Segal construction}. We now
show that we can give a description of~$\mathbf{K}$ in terms of the phase
space~$\mathbf{P}$. For this, we draw the following definition
from~\cite[\S 5.3]{BSZ}.
\begin{definition}[characteristic functional]\label{def:genFunct}
If~$\langle~\rangle$ is a state on the Weyl
algebra~$\mathcal{W}(\mathbf{P}^*,\omega)$, its \emph{characteristic
functional}~$\mu\colon\mathbf{P}^*\to\mathbb{C}$ is given by
\begin{equation}
\label{eq:genFunct}
\mu(f)\colon=\langle\mathcal{W}(f)\rangle
\qquad\hbox{for all}\quad
f\in\mathbf{P}^*.
\end{equation}
We say the state~$\langle~\rangle$ is~\emph{regular} if, for
every~$f\in\mathbf{P}^*$, the function
$$
t\mapsto\mu(tf)
\qquad(t\in\mathbb{R})
$$
is twice differentiable at~$t=0$.
\end{definition}
\begin{proof}[Note]
We will find it convenient to introduce the following notation:
$$
\partial_f\mu(g)=\left.{\partial\over\partial t}\right|_{t=0}\mu(g+tf).
$$
\end{proof}
\begin{theorem}
\label{thm:genBosField}
Let~$(\mathbf{P}^*,\omega)$ be a symplectic vector space. Then, given a
regular state~$\langle~\rangle$ on~$\mathcal{W}(\mathbf{P}^*,\omega)$ with
characteristic function~$\mu$, there is an~$x\in\mathbf{P}$ such that
$$
i\partial_f\mu(0)=f(x)
\qquad\hbox{for all}\quad
f\in\mathbf{P}^*.
$$
Then, the collection of formal
symbols~$\Psi=\bigl\{\ket{x+f^*}\colon f\in\mathbf{P}^*\bigr\}$
generates a complex vector space with the following properties:
\begin{enumerate}
\item the sesquilinear form
\begin{equation}\label{eqn:innProd}
\langle x+f^*\mid x+g^*\rangle=e^{\omega(g,f)/2i}\mu(g-f)
\end{equation}
makes the span of~$\Psi$ into a complex pre-Hilbert space whose
Hilbert space completion is denoted~$\mathbf{K}$
\item there is a Weyl system~$W\colon\mathbf{P}^*\to U(\mathbf{K})$
on~$(\mathbf{P}^*,\omega)$, given by
\begin{equation}\label{eq:WeylSystem}
W(f)\ket{x+g^*}=e^{\omega(f,g)/2i}\ket{x+f^*+g^*}
\qquad\hbox{for all}\quad
f,g\in\mathbf{P}^*
\end{equation}
\item the unit vector~$\ket{x}\in\mathbf{K}$ is a cyclic vector of the Weyl
system~$W(\mathbf{P}^*,\omega)$
\item the associated Heisenberg system~$\Phi\colon\mathbf{P}^*\to L(\mathbf{K})$
satisfies
$$
\matElem{x+g^*}{\Phi(f)}{x+g^*}=f(x+g^*)
\qquad\hbox{for all}\quad
f,g\in\mathbf{P}^*.
$$
\end{enumerate}
\end{theorem}
The last property states that the Heisenberg system obtained from the
regular state~$\langle~\rangle$ satisfies the correspondence
principle. Namely, the expected value of the quantum
observable~$\Phi(g)$ in the quantum state~$\ket{x+f^*}$ equals the
value of the classical observable~$g\in\mathbf{P}^*$ in the classical
state~$x+f^*\in\mathbf{P}$.
Because we have not assumed that the symplectic structure~$\omega$
makes the duality map~$*\colon\mathbf{P}^*\to\mathbf{P}$ onto, it is possible
that~$x\neq f^*$ for any~$f\in\mathbf{P}^*$, in which case the collection
of indices~$\{x+f^*\colon f\in\mathbf{P}^*\}$ is an affine subspace
of~$\mathbf{P}$.
\begin{figure}
$$
\begin{xy}
(-5,5)*{\scriptstyle\mathbf{P}};
(-15,0)*{};(15,0)**\dir{.};
(0,-15)*{};(0,15)**\dir{.};
(-10,-15)*{};(15,10)**\dir{-}?(.3)*\dir{*}+(-2,1)*{\scriptstyle
x}?(.6)*\dir{>}+(0,-2)*\rlap{$\scriptstyle x+f^*$};
\end{xy}
$$
\caption{schematic representation of the relative coherent states as
an affine subspace of phase space}
\end{figure}
In other words, if~$\omega$ is only
weakly and not strongly nondegenerate, not every classical state
in~$\mathbf{P}$ has a counterpart in~$\mathbf{K}$.
The physical interpretation of the vector~$x\in\mathbf{P}$ is that of a
classical `background' field configuration,
since~$\matElem{x}{\Phi(f)}{x}=f(x)$ for every classical linear
observable~$f\in\mathbf{P}^*$. Clearly any other density operator in~$\mathbf{K}$
can be used to define a state leading to a unitarily equivalent Weyl
system, possibly with a different background field
configuration. If~$x$ is not of the form~$g^*$ for any~$g\in\mathbf{P}^*$,
it will actually be impossible to eliminate the background altogether
by a unitary change of representation.
Finally, the fact that the span of~$\Psi$ is dense in the Hilbert
space~$\mathbf{K}$ will be used consistently in the sequel to
characterize densely defined linear operators and sesquilinear forms
on~$\mathbf{K}$.
\begin{proof}
This proof has a curious way of pulling itself up by its own
bootstraps: the main conceptual difficulty is that, in order to show
that~$f\mapsto i\partial_f\mu(0)$ is a continuous
linear functional on~$\mathbf{P}^*$ one needs to have the Weyl system~$W$
in place. We proceed by constructing~$\Psi$ and~$\mathbf{K}$ before
the names~$\ket{x+f^*}$ are available, and then renaming the vectors
after~$x$ is shown to have the advertised properties.
We will temporarily denote by~$\psi_f\in\mathbf{K}$ the image of~$\mathcal{W}(f)$ under
the GNS construction described immediately before
Definition~\ref{def:genFunct}. We denote~$\Psi=\{\psi_f\colon
f\in\mathbf{P}^*\}$. It follows immediately from Equation~(\ref{eq:InnProd})
that
\begin{equation}\label{InnProd}
\langle\psi_f\mid\psi_g\rangle=e^{i\omega(f,g)/2}\mu(g-f)
\qquad\hbox{for all}\quad
f,g\in\mathbf{P}^*.
\end{equation}
The span of~$\Psi$, which consists of unit vectors, is dense
in~$\mathbf{K}$ with respect to this inner product. Recall also that, in the
lead-up to Definition~\ref{def:genFunct}, we showed that the action
of~$\mathcal{W}(\mathbf{P}^*,\omega)$ on itself by left multiplication projects
to~$\mathbf{K}$ by virtue of the invariance of the null space
of~$\langle~\mid~\rangle$. This action, namely the Weyl relations,
passes to the quotient as
\begin{equation}\label{eq:preWeylSystem}
W(f)\psi_g=e^{\omega(f,g)/2i}\psi_{f+g}
\qquad\hbox{for all}\quad
f,g\in\mathbf{P}^*.
\end{equation}
We are now ready to construct a Weyl system~$W\colon\mathbf{P}^*\to
U(\mathbf{K})$. The following lemma shows that the Hilbert space~$\mathbf{K}$
automatically supports a Weyl sistem on~$\mathcal{W}(\mathbf{P}^*,\omega)$.
\begin{lemma}\label{lem:Weyl}
Suppose that a regular state~$\langle~\rangle$ is given on the Weyl
algebra~$\mathcal{W}(\mathbf{P},\omega)$ and the GNS construction is performed
resulting in the Hilbert space~$\mathbf{K}$, as just described. Then, Equation~(\ref{eq:preWeylSystem})
defines a map~$W\colon\mathbf{P}^*\to U(\mathbf{K})$ which is a Weyl
system\index{Weyl system} on~$(\mathbf{P}^*,\omega)$. In addition,
the unit vector~$\psi_0\in\mathbf{K}$ is a cyclic vector of the Weyl
system~$W\colon\mathbf{P}^*\to U(\mathbf{K})$.
\end{lemma}
\begin{proof}
First, we need to show that~$W(f)\in U(\mathbf{K})$ for
all~$f\in\mathbf{P}^*$. Indeed, observe that~$W(f)$ maps~$\Psi$ to itself
and that, for all~$f,g,h\in\mathbf{P}^*$,
\begin{eqnarray*}
\langle W(f)\psi_g\mid W(f)\psi_h\rangle
&=&
e^{\omega(f,h-g)/2i}\langle\psi_{f+g}\mid\psi_{f+h}\rangle\\
&=&
e^{\omega(f,h-g)/2i}e^{i\omega(f+g,f+h)/2}\mu(h-g)\\
&=&
e^{i\omega(g,h)/2}\mu(h-g)=\langle\psi_g\mid\psi_h\rangle
\end{eqnarray*}
This implies that $W(f)$ is an invertible isometry on the span
of~$\Psi$. Then, by density of the span of~$\Psi$ in~$\mathbf{K}$ and
linearity, it follows that~$W(f)$ is unitary on~$\mathbf{K}$.
Now, we need to show that, for all~$f,g,h\in\mathbf{P}^*$,
$$
W(f)W(g)\,\psi_h=e^{\omega(f,g)/2i\,}W(f+g)\,\psi_h.
$$
The left-hand side is equal to
$$
W(f)e^{\omega(g,h)/2i}\psi_{g+h}=
e^{\omega(g,h)/2i}\,e^{\omega(f,g+h)/2i}\,\psi_{f+g+h},
$$
and the right-hand side is equal to
$$
e^{\omega(f,g)/2i}W(f+g)\psi_h=e^{\omega(f,g)/2i}\,e^{\omega(f+g,h)/2i}\,\psi_{f+g+h}.
$$
To show strong continuity of the Weyl system~$W$ we need to show that,
if~$f_n\to f$ in~$\mathbf{P}^*$, then~$W(f_n)\to W(f)$ in the strong
operator topology on~$U(\mathbf{K})$. To this end, we consider
$$
\bigl[W(f)-W(g)\bigr]\psi_h=e^{\omega(f,h)/2i}\psi_{f+h}-e^{\omega(g,h)/2i}\psi_{g+h}.
$$
Then,
$$
\bigl\|\bigl[W(f)-W(g)\bigr]\psi_h\bigr\|^2=2\mathrm{Re}\bigl[1-
e^{i\omega(f-g,h)+i\omega(f,g)/2}\mu(g-f)\bigr],
$$
which indeed vanishes as~$f-g\to 0$ because of the continuity
of~$\omega$ and~$\mu$ and the antisymmetry of~$\omega$.
Finally, the unit vector~$\psi_0\in\mathbf{K}$ is a cyclic vector of the Weyl
system~$W\colon\mathbf{P}^*\to U(\mathbf{K})$ because~$W(f)\psi_0=\psi_f$ for
all~$f\in\mathbf{P}^*$, and the collection of all~$\psi_f$ is dense
in~$\mathbf{K}$.
\end{proof}
We now study the Heisenberg system associated to the Weyl system
defined in Lemma~\ref{lem:Weyl}.
\begin{lemma}\label{lem:Heis}
In the hypotheses of
Lemma~\ref{lem:Weyl},~$\langle\psi_g\mid\Phi(f)\psi_g\rangle$
and~$\|\Phi(f)\psi_g\|$ are both finite for
all~$f,g\in\mathbf{P}^*$. Moreover,
$$
\langle\psi_g\mid\Phi(f)\psi_g\rangle=\omega(f,g)+\langle\psi_0\mid\Phi(f)\psi_0\rangle
$$
and
$$
\|\Phi(f)\psi_g\|^2-\|\Phi(f)\psi_0\|^2=\langle\psi_g\mid\Phi(f)\psi_g\rangle^2-\langle\psi_0\mid\Phi(f)\psi_0\rangle^2.
$$
\end{lemma}
\begin{proof}
Observe that, if~$\psi_g$ is in the domain of~$\Phi(f)$, then
$$
\langle\psi_g\mid\Phi(f)\psi_g\rangle=i\left.{\partial\over\partial t}\right|_{t=0}
\langle\psi_g\mid W(tf)\psi_g\rangle
$$
and
$$
\|\Phi(f)\psi_g\|^2=\langle\psi_g\mid\Phi(f)^2\psi_g\rangle=-\left.{\partial^2\over\partial t^2}\right|_{t=0}
\langle\psi_g\mid W(tf)\psi_g\rangle.
$$
Conversely, since~$\Phi(f)$ is a closed operator, the finiteness
of~$-\left.(\partial^2/\partial t^2)\right|_{t=0} \langle\psi_g\mid
W(tf)\psi_g\rangle$ would imply that~$\psi_g$ is in the domain
of~$\Phi(f)$. We now show this.
First we use the definition of the Weyl system given in
Lemma~\ref{lem:Weyl} to compute the matrix elements of the unitary
operator~$W(f)$ between arbitrary elements of~$\Psi$:
$$
\langle\psi_g\mid
W(f)\psi_h\rangle=e^{\omega(f,g+h)/2i+i\omega(g,h)/2}\mu(f-g+h)
\qquad\hbox{for all}\quad f,g,h\in\mathbf{P}^*.
$$
When~$f=0$, this matrix element reduces to Equation~(\ref{InnProd})
for~$\langle\psi_g\mid\psi_h\rangle$
Differentiating the matrix element~$\langle\psi_g\mid
W(tf)\psi_g\rangle$ twice with respect to~$t$ and setting~$t=0$ one obtains
$$
\|\Phi(f)\psi_g\|^2=\bigl[\omega(f,g)\bigr]^2+2i\omega(f,g)\partial_f\mu(0)-\partial^2_f\mu(0),
$$
which is finite by the assumption that~$\mu(tf)$ is
twice-differentiable. Particularizing to~$g=0$ we obtain
$$
\partial^2_f\mu(0)=-\|\Phi(f)\psi_0\|^2.
$$
One obtains the matrix elements of the Heisenberg
operator~$\Phi(f)$ by differentiating the matrix
element~$\langle\psi_g\mid W(tf)\psi_h\rangle$ with respect to~$t$ and
setting~$t=0$, namely:
$$
\langle\psi_g\mid
\Phi(f)\psi_h\rangle=\Bigl[{1\over 2}\omega(f,g+h)\mu(h-g)+i\partial_f\mu(h-g)\Bigr]e^{i\omega(g,h)/2}.
$$
If, in particular, $h=g$,
$$
\langle\psi_g\mid
\Phi(f)\psi_g\rangle=\omega(f,g)+i\partial_f\mu(0).
$$
The case~$g=0$ shows that
$$
i\partial_f\mu(0)=\langle\psi_0\mid\Phi(f)\psi_0\rangle,
$$
and the result follows by elementary algebraic manipulations.
\end{proof}
At this point, we can assert that
$$
i\partial_f\mu(0)=\langle\psi_0\mid\Phi(f)\psi_0\rangle=f(x)
$$
for some~$x\in\mathbf{P}$ (had we not assumed that~$\mathbf{P}$ is
reflexive, we could only deduce that~$x\in\mathbf{P}^{**}$). This takes
care of the first conclusion of the theorem. If we now make the
identification~$\psi_f\sim\ket{x+f^*}$, it follows that
$$
\matElem{x+g^*}{\Phi(f)}{x+g^*}=f(x+g^*)
$$
because~$\omega(f,g)=f(g^*)$.
\end{proof}
\begin{definition}[relative coherent states]
\label{def:coh}
Given a regular state~$\langle~\rangle$ on~$\mathcal{W}(\mathbf{P},\omega)$, the
element~$x\in\mathbf{P}$ such
that
$$
i\partial_f\mu(0)=f(x)
\qquad\hbox{for all}\quad
f\in\mathbf{P}^*
$$
is called the \emph{background} for~$\langle~\rangle$.
The image of~$\mathcal{W}(f)$ inside~$\mathbf{K}$ by the GNS construction, denoted
by~$\ket{x+f^*}$, is called a \emph{coherent state
relative}\index{relative coherent state} to the
state~$\langle~\rangle$. We denote the set of
relative coherent states by~$\Psi=\{\ket{x+f^*}\colon f\in\mathbf{P}^*\}$.
\end{definition}
\begin{proof}[Note]
One of the conclusions of Lemma~\ref{lem:Heis} is that the variance
(mean-square deviation from the mean) of
the observable~$\Phi(g)$ in state~$\ket{x+f^*}$ is
$$
\mathrm{Var}_{x+f^*}(g)=\matElem{x+f^*}{\Phi(g)^2}{x+f^*}-\matElem{x+f^*}{\Phi(g)}{x+f^*}^2,
$$
which is independent of~$f\in\mathbf{P}^*$. In other words, the standard
deviation of each observable~$\Phi(g)$ is the same on all relative
coherent states. Note that we are not claiming that the relative
coherent states are minimal-uncertainty states in the sense that they
saturate the inequality in Heisenberg's
uncertainty principle, but it is true that if any one relative
coherent state is a minimal-uncertainty state, all of them will
be. Our definition of relative coherent state includes as
special cases the ordinary coherent states of the harmonic oscillator
and quantum optics, but also the so-called `squeezed states' and many
others, which may or may not be pure states.
\end{proof}
The problem of quantizing a linear phase
space~$(\mathbf{P},\omega)$\index{linear phase space} can thus be partly
solved by finding a state\index{GNS state}~$\langle~\rangle$ on the
Weyl algebra~$\mathcal{W}(\mathbf{P},\omega)$\index{Weyl algebra}. This leads to a
Weyl system on~$(\mathbf{P}^*,\tilde\omega)$ and so to Heisenberg
operators~$\Phi(f)$ satisfying the canonical commutation relations and
the correspondence principle, albeit possibly with a nontrivial
background.
Still, the canonical commutation relations and the correspondence
principle together are far from sufficient to uniquely determine the
quantization and, unless~$\mathbf{P}$ is finite-dimensional, different
states may lead to unitarily inequivalent Weyl systems\index{Weyl
system}. The problem remains how to construct or identify
representations suitable for particular physical applications. In the
next section we investigate the implications of requiring that
physical symmetries, in particular time evolution, be implemented
unitarily.
\subsubsection{Unitary representation of physical symmetries}
Having quantized the phase space itself, we now consider the
quantization of dynamics and, more generally, physical symmetries. The
ultimate goal is to represent physical symmetries as unitary operators
on the quantum state space~$\mathbf{K}$. The linear phase spaces we are
considering have been defined as topological vector spaces whose duals
are symplectic vector spaces, and are associated to a natural concept
of automorphism. Here we limit our attention to those physical symmetries
which can be represented by automorphisms of the physical phase space
in the sense of~Definition~\ref{def:automorph}.
Putting together Lemma~\ref{lem:symmetry}, Equation~(\ref{eq:WeylSystem}) and
Definition~\ref{def:coh}, we obtain the following result.
\begin{lemma}\label{lem:Gamma}
Assume that~$\langle~\rangle$ is a regular state
on~$\mathcal{W}(\mathbf{P}^*,\omega)$, with background~$x\in\mathbf{P}$. Given any automorphism~$T\colon\mathbf{P}\to\mathbf{P}$ of the linear phase
space~$\mathbf{P}$, there is a densely defined linear
map~$\Gamma(T)\colon\mathbf{K}\to\mathbf{K}$ such that
\begin{equation}
\label{eq:symAct}
\Gamma(T)\ket{x+Tf^*}=\ket{x+f^*}.
\end{equation}
This map intertwines the unitary
operators~$W(f)$, that is,
\begin{equation}
\label{eq:intertwine}
\Gamma(T)W(T^*f)=W(f)\Gamma(T)
\qquad\hbox{for all}\quad
f\in\mathbf{P}^*,
\end{equation}
and satisfies~$\Gamma(ST)=\Gamma(S)\Gamma(T)$.
\end{lemma}
It is worth remarking that, when the background~$x\in\mathbf{P}$ is
a fixed point of the automorphism~$T\colon\mathbf{P}\to\mathbf{P}$, we have the
nicer formula
$$
\Gamma(T)\ket{Ty}=\ket{y}
\qquad\hbox{for all}\quad
y=x+f^*
\qquad\hbox{with}\quad
f\in\mathbf{P}^*.
$$
\begin{proof}
By Lemma~\ref{lem:symmetry}, there is a unique
automorphism~$\gamma(T)$ of the $*$-algebra~$\mathcal{W}(\mathbf{P},\omega)$ such
that
$$
\gamma(T)\mathcal{W}(T^*f)=\mathcal{W}(f)
\qquad\hbox{for all}\quad
f\in\mathbf{P}^*
$$
and satisfying~$\gamma(ST)=\gamma(S)\gamma(T)$. The GNS construction
preceding Definition~\ref{def:coh} produces a unique densely-defined
linear operator~$\Gamma(T)\colon\mathbf{K}\to\mathbf{K}$ defined on the dense span
of~$\Psi$ by Equation~(\ref{eq:symAct}). In
addition, since
$$
\Gamma(S)\Gamma(T)\ket{x+STf^*}=\Gamma(S)\Gamma(T)\ket{x+{T^*S^*f}^*}=\Gamma(S)\ket{x+Sf^*},
$$
$\Gamma(ST)=\Gamma(S)\Gamma(T)$ holds on the span of the relative
coherent states, which is a common dense domain of all three operators
and is left invariant by them.
As for the intertwining of the Weyl operators,
Equation~(\ref{eq:WeylSystem}) implies
$$
\Gamma(T)W(T^*f)\ket{x+Tg^*}=e^{\omega(T^*f,T^*g)/2i}\Gamma(T)\ket{x+T(f^*+g^*)}
$$
which, because~$T^*$ preserves~$\omega$ and by definition
of~$\Gamma(T)$, equals
$$
e^{\omega(f,g)/2i}\ket{x+f^*+g^*}.
$$
Similarly,
$$
W(f)\Gamma(T)\ket{x+Tg^*}=W(f)\ket{x+g^*},
$$
and the result follows, again by Equation~(\ref{eq:WeylSystem}).
\end{proof}
Perhaps surprisingly,~$\Gamma(T)$ is not necessarily an isometry
of~$\mathbf{K}$ despite the fact that it preserves the norm of all the
relative coherent states~$\ket{x+f^*}$. However, it should not be
surprising that unitarity is obtained when the characteristic
functional~$\mu$ is preserved by~$T^*$.
\begin{lemma}\label{lem:unitary}
In the hypotheses of Lemma~\ref{lem:Gamma}, the operator~$\Gamma(T)$
extends uniquely to a unitary operator on~$\mathbf{K}$ if, and only if,~$T$
preserves~$\langle~\rangle$ in the sense that
$$
\mu(T^*h)=\mu(h)
\qquad\hbox{for all}\quad
h\in\mathbf{P}^*.
$$
\end{lemma}
In other words, the invertible operator~$\Gamma(T)$ is unitary on~$\mathbf{K}$
if, and only if, the characteristic functional~$\mu$ is constant on orbits
of~$T^*$.
\begin{proof}
From Equation~(\ref{eqn:innProd}) it follows that
$$
\langle x+Tf^*\mid
x+Tg^*\rangle=e^{\omega(T^*g,T^*f)/2i}\mu\bigl(T^*(g-f)\bigr)
$$
so~$\Gamma(T)$ is an isometry on the span of the relative coherent
states if, and
only if,~$\mu(T^*h)=\mu(h)$ for
all~$h\in\mathbf{P}^*$. An isometry is unitary if and only if it is
invertible.
\end{proof}
It follows that, for a whole subgroup~$G$ of automorphisms of~$\mathbf{P}$
to be unitarily implemented on~$\mathbf{K}$ by~$\Gamma$, it is necessary and
sufficient that~$\mu$ be constant on the orbits of the whole
subgroup. It is possible that unitary representations other
than~$\Gamma$ exist, and in fact that is guaranteed when~$\mathbf{P}^*$ is
finite-dimensional. Now, if~$G$ is a continuous group generated by a
Poisson algebra~$\mathfrak{g}$ of classical observables on~$\mathbf{P}$, this is
equivalent to the characteristic functional~$\mu$ having vanishing
Poisson brackets with all the elements of~$\mathfrak{g}$. In particular, if time
evolution is to be implemented unitarily, the characteristic
functional of the state must be a constant of the motion. It is in
this precise sense that the dynamics can be said to determine the
quantization.
\subsection{Summary}
Putting Theorem~\ref{thm:genBosField}
and~Lemmas~\ref{lem:Gamma}--\ref{lem:unitary} together we obtain the
following theorem listing the properties of representations of the
general boson field.
\begin{theorem}
\label{thm:GenBosField}
Let~$(\mathbf{P},\omega)$ be a linear phase space, let~$\langle~\rangle$
be a regular GNS state on the Weyl algebra~$\mathcal{W}(\mathbf{P},\omega)$ with
characteristic function~$\mu$. Let the background~$x\in\mathbf{P}$
associated to~$\mu$ be defined by
$$
i\partial_f\mu(0)=f(x)
\qquad\hbox{for all}\quad
f\in\mathbf{P}^*,
$$
and let~$\Psi=\{\ket{x+f^*}\mid
f\in\mathbf{P}^*\}$. Then,
\begin{enumerate}
\item the sesquilinear form
$$
\bracket{x+f^*}{x+g^*}=e^{i\omega(f,g)/2}\mu(g-f)
$$
makes the span of~$\Psi$ into a complex pre-Hilbert space whose
Hilbert-space completion is denoted~$\mathbf{K}$
\item there is a Weyl system~$W\colon\mathbf{P}^*\to U(\mathbf{K})$
on~$(\mathbf{P}^*,\omega)$, given by
$$
W(f)\ket{x+g^*}=e^{\omega(f,g)/2i}\ket{x+g^*+f^*}
\qquad\hbox{for all}\quad
f\in\mathbf{P},g\in\mathbf{P}^*
$$
\item the associated Heisenberg system~$\Phi\colon\mathbf{P}^*\to L(\mathbf{K})$
satisfies
$$
\matElem{x+g^*}{\Phi(f)}{x+g^*}=f(x+g^*)
\qquad\hbox{for all}\quad
f,g\in\mathbf{P}^*
$$
\item there is a group homomorphism~$\Gamma$
mapping automorphisms~$T\colon\mathbf{P}\to\mathbf{P}$ to invertible linear
operators on~$\mathbf{K}$, given by
$$
\Gamma(T)\ket{x+Tf^*}=\ket{x+f^*}
\qquad\hbox{for all}\quad
f\in\mathbf{P}^*
$$
and satisfying
$$
\Gamma(T)W(T^*f)=W(f)\Gamma(T)
\qquad\hbox{for all}\quad
f\in\mathbf{P}^*
$$
\item the unit vector~$\ket{x}\in\mathbf{K}$ is a
cyclic vector of the Weyl system~$W(\mathbf{P}^*,\tilde\omega)$
\item $\Gamma(T)$ is unitary if, and only if,~$\mu$ is constant on
orbits of~$T$.
\end{enumerate}
\end{theorem}
Since~$\Gamma$ is defined on symplectic and not unitary
transformations the generators are not self-adjoint and it is not
clear that there is a meaningful notion of positivity of~$\Gamma$, in
contrast with the free boson field below. In other words, there seems
to be no way to define what a stable representation of the general
boson field is. Also, we have not specified a topology on the
automorphisms of~$\mathbf{P}$, so we cannot prove continuity of~$\Gamma$.
\begin{proof}
\begin{enumerate}
\item This is Equation~(\ref{eqn:innProd}) from
Theorem~\ref{thm:genBosField}.
\item This is the content of Lemma~\ref{lem:Weyl}, which was part of
the proof of Theorem~\ref{thm:genBosField}.
\item This is by Lemma~\ref{lem:Heis}, also part of the proof of Theorem~\ref{thm:genBosField}.
\item This is Lemma~\ref{lem:Gamma}.
\item This is part of the conclusions of Lemmas~\ref{lem:Weyl}
and~\ref{lem:Gamma}.
\item This is the content of Lemma~\ref{lem:unitary}.
\end{enumerate}
\end{proof}
\section{The free boson field}
\label{sec:freeBosonField}
In certain cases, the space~$\mathbf{P}^*$ of linear observables on the
physical phase-space a classical theory is not only a real symplectic
space, but also admits a complex Hilbert space~$\mathbf{H}$ such that and
the symplectic structure~$\omega$ is the imaginary part of the complex
inner product. This is the algebraic setting in which Segal~\cite{BSZ}
defined his concept of a free boson field, which is an axiomatic
definition of the usual Fock representation of free quantum fields. In
the present section we develop tools and techniques specific to Fock
quantization and that will be needed later on.
\begin{definition}[free boson field]
The \emph{free boson field} over a complex Hilbert space~$\mathbf{H}$
consists of
\begin{enumerate}
\item a complex Hilbert space~$\mathbf{K}$
\item a Weyl system~$W\colon\mathbf{H}\to U(\mathbf{K})$
\item a continuous representation~$\Gamma\colon U(\mathbf{H}^\dagger)\to U(\mathbf{K})$
satisfying
$$
\Gamma(U)W(z)\Gamma(U)^{-1}=W(Uz)
\qquad\hbox{for all}\quad
z\in\mathbf{H}
$$
\item a unit vector~$\nu\in\mathbf{K}$ which is invariant under~$\Gamma(U)$
for all~$U\in U(\mathbf{H}^\dagger)$ and a cyclic vector of~$W(\mathbf{H})$
\end{enumerate}
such that $\Gamma$ is positive in the sense that, if the one-parameter
group~$U(t)\subset U(\mathbf{H}^\dagger)$ has a nonnegative self-adjoint
generator~$A$, then~$\partial\Gamma(A)$, which denotes the
self-adjoint generator of the
group~$\Gamma\bigl(U(t)\bigr)\colon\mathbf{K}\to\mathbf{K}$, is a nonnegative
self-adjoint operator on~$\mathbf{K}$.
\end{definition}
\begin{proof}[Note]
The positivity condition can be weakened to apply only to a single
operator~$A$, and the free boson field is unique up to unitary
equivalence~\cite[\S 1.10]{BSZ}.
\end{proof}
Now, if~$\mathbf{H}$ is a complex Hilbert space with inner
product~$\langle~,~\rangle$ and norm~$\|~\|$, and one
defines~$h(f,g)=\mathrm{Re}{\langle f,g\rangle}$
and~$\omega(f,g)=\mathrm{Im}{\langle f,g\rangle}$, then~$\mathbf{H}$ becomes a
real Hilbert space with inner product~$h$ and norm~$\|~\|$,
and~$\omega$ is a continuous symplectic structure on~$\mathbf{H}$. If we
denote by~$\mathbf{P}$ the real dual of~$\mathbf{H}$, then~$\mathbf{P}$ is a linear
phase space in the sense of Definition~\ref{def:linearPhaseSpace},
with~$(\mathbf{P}^*,\omega)=(\mathbf{H},\omega)$. In addition, we have a
map~$*\colon\mathbf{H}\to\mathbf{P}$ defined by
$$
g^*(f)=\omega(f,g)
\qquad\hbox{for all}\quad
f,g\in\mathbf{H}.
$$
With this notation, the following is a consequence of
Theorem~\ref{thm:GenBosField}.
\begin{theorem}\label{thm:FreeBosField}
Let~$\mathbf{H}$ be a complex Hilbert space with inner product~$\langle
~,~\rangle$ and norm~$\|~\|$. Define~$h$, and~$\omega$
on~$\mathbf{H}\cong\mathbf{P}^*$ and~$*\colon\mathbf{H}\to\mathbf{P}$ as above. Then, the
representation of the general boson field on~$\mathcal{W}(\mathbf{P}^*,\omega)$
given by the regular state with characteristic functional
$$
\mu(f)=e^{-\|f\|^2/4}
\qquad\hbox{for all}\quad
f\in\mathbf{H}
$$
is the free boson field on~$\mathbf{H}$, with
\begin{enumerate}
\item $\mathbf{K}$ being the completion of the span
of~$\Psi=\{\ket{f^*}\colon f\in\mathbf{H}\}$ with respect to the complex
inner product
$$
\bracket{f^*}{g^*}=e^{\omega(g,f)/2i}e^{-\|g-f\|^2/4}
$$
\item $W$ being the Weyl system on~$\mathcal{W}(\mathbf{H},\omega)$ given by
$$
W(f)\ket{g^*}=e^{ig^*(f)/2}\ket{g^*+f^*}
\qquad\hbox{for all}\quad
f,g\in\mathbf{H}
$$
\item $\Gamma$ being defined by
$$
\Gamma(U)\ket{f^*}=\ket{(Uf)^*}
\qquad\hbox{for all}\quad
f\in\mathbf{H}
$$
\item $\nu=\ket{0}$
\end{enumerate}
In addition, the mean and variance of~$\Phi(g)$ in the
state~$\ket{x}$ are
$$
\matElem{f^*}{\Phi(g)}{f^*}=\omega(g,f)
\qquad\hbox{and}\quad
\mathrm{Var}_{f^*}(g)={1\over 2}\|g\|^2
\qquad\hbox{for all}\quad
x,f\in\mathbf{H}.
$$
\end{theorem}
\begin{proof}
All the numbered properties of the free boson field are immediate
consequences of Theorem~\ref{thm:GenBosField}. It only remains to show
positivity of the representation~$\Gamma$.
Assume that~$U=e^{-itA}\in U(\mathbf{H})$ with~$\langle Af,f\rangle=\langle
f,Af\rangle\ge 0$ for all~$f\in\mathbf{H}$. Then,
\begin{eqnarray*}
\matElem{f^*}{\partial\Gamma(A)}{f^*}
&=&
\left.i{\partial\over\partial t}\right|_{t=0}\matElem{f^*}{\Gamma(e^{-itA})}{f^*}\\
&=&
\left.i{\partial\over\partial t}\right|_{t=0}\bracket{f^*}{(e^{-itA}f)^*}\\
&=&
\left.i{\partial\over\partial t}\right|_{t=0}e^{\omega(e^{-itA}f,f)/2i}e^{-\|(e^{-itA}-1)f\|^2/4}\\
&=&
{1\over 2}\omega(-iAf,f)={1\over 2}\langle f,Af\rangle\ge 0\\
\end{eqnarray*}
for all~$f\in\mathbf{H}$.
As for the mean and variance of the Heisenberg observables, note that
\begin{equation}\label{eq:matElem}
\matElem{f^*}{W(g)}{h^*}=e^{\omega(g,h+f)/2i}e^{\omega(h,f)/2i}e^{-\|h+g-f\|^2/4}.
\end{equation}
In particular, if~$f=h$,
$$
\matElem{f^*}{W(g)}{f^*}=e^{-i\omega(g,f)}e^{-\|g\|^2/4}.
$$
But this is precisely the characteristic functional of a Gaussian
random variable with mean~$\omega(g,f)$ and variance~${1\over 2}\|g\|^2$.
\end{proof}
\subsection{Normal-ordered functions}
The Weyl system~$W\colon\mathbf{H}\to U(\mathbf{K})$ has an associated real-linear
Heisenberg system~$\Phi\colon\mathbf{H}\to L(\mathbf{K})$. From this real-linear map
we can construct complex-linear and complex-antilinear
maps~$a,a^\dagger\colon\mathbf{H}\to L(\mathbf{K})$ with the help of the complex
structure of~$\mathbf{H}$. The \emph{creation operator}
$$
a^\dagger(f)={\Phi(f)-i\Phi(if)\over\sqrt 2}
\qquad\hbox{for all}\quad
f\in\mathbf{H}
$$
is complex-linear, and its adjoint the \emph{annihilation operator}
$$
a(f)={\Phi(f)+i\Phi(if)\over\sqrt 2}
\qquad\hbox{for all}\quad
f\in\mathbf{H}
$$
is complex-antilinear. The creation and annihilation operators satisfy
the commutation relations
$$
\bigl[a(f),a(g)\bigr]=0,
\qquad
\bigl[a(f),a^\dagger(g)\bigr]=\langle f,g\rangle,
\qquad\hbox{and}\quad
\bigl[a^\dagger(f),a^\dagger(g)\bigr]=0
$$
for all $f,g\in\mathbf{H}$.
It is now easy to prove that coherent states are joint eigenstates of every~$a(f)$.
\begin{lemma}\label{lem:annihilation}
If~$f,g\in\mathbf{H}$ then
$$
a(g)\ket{h^*}={\langle g,h\rangle\over i\sqrt 2}\ket{h^*}.
$$
\end{lemma}
\begin{proof}
Equation~(\ref{eq:matElem}) implies that
$$
{\matElem{f^*}{W(g)}{h^*}\over\bracket{f^*}{h^*}}=e^{(\langle
f,g\rangle-\langle g,h\rangle)/2}e^{-\|g\|^2/4}
$$
so the matrix elements of the Heisenberg operators satisfy
\begin{equation}\label{eq:Heisenberg}
{\matElem{f^*}{\Phi(g)}{h^*}\over\bracket{f^*}{h^*}}={i\over 2}\bigl[\langle
f,g\rangle-\langle g,h\rangle\bigr]
\end{equation}
which implies
$$
{\matElem{f^*}{\Phi(ig)}{h^*}\over\bracket{f^*}{h^*}}=-{1\over 2}\bigl[\langle
f,g\rangle+\langle g,h\rangle\bigr]
$$
and so
$$
\matElem{f^*}{a(g)}{h^*}={\langle g,h\rangle\over i\sqrt 2}\bracket{f^*}{h^*}
$$
By the density of the span of the coherent states in~$\mathbf{K}$, the result
follows.
\end{proof}
We now use this property to prove a remarkable formula for the matrix
elements of `normal-ordered' functions of Heisenberg operators. We
first introduce the definition of normal-ordered powers of Heisenberg
operators, or Wick powers. The Wick powers are obtained by expressing
the Heisenberg operator in terms of creation and annihilation
operators, expanding the product and rearranging each monomial to have
all creation operators to the left of all the annihilation operators,
discarding all commutators.
\begin{definition}[Wick power]
If~$f\in\mathbf{H}$, the~$n$th \emph{Wick power} or \emph{normal-ordered power} of the Heisenberg operator~$\Phi(f)$ is
the operator on~$\mathbf{K}$ given by
$$
\Wick{\Phi(f)^n}={1\over
2^{n/2}}\sum_{m=0}^n{n\choose m}a^\dagger(f)^m a(f)^{n-m}.
$$
\end{definition}
We first show that Wick powers are densely defined on~$\mathbf{K}$; what is
more, their domain always contains the coherent states.
\begin{lemma}
For all~$n\in\mathbb{N}$ and all~$f\in\mathbf{H}$, the Wick
power~$\Wick{\Phi(f)^n}$ is densely defined on~$\mathbf{K}$.
\end{lemma}
In other words, for all~$f\in\mathbf{H}$, the coherent states
are~\emph{$C^\infty$ vectors for~$\Phi(f)$} \cite[\S X.6]{RS}.
\begin{proof}
That the domain of~$\Wick{\Phi(f)^n}$ is dense in~$\mathbf{K}$ will follow from
the fact that it contains the coherent states, whose span is dense
in~$\mathbf{K}$. Since the coherent states are eigenstates of the annihilation
operators, it is clear that any power of annihilation operators is
densely defined on~$\mathbf{K}$. Also, the creation operators are defined
on the coherent states because they are linear combinations of the
Heisenberg operators, to which Lemma~\ref{lem:Heis} applies. However,
the question is whether higher powers of the creation operators
are defined on coherent states. Since powers of annihilation operators
are polynomials in the Heisenberg operators, the result will follow if
we can show that arbitrary powers of Heisenberg operators are defined
on coherent states. The techniques used to prove Lemma~\ref{lem:Heis}
generalize to this situation.
Indeed, observe that~$W(f)=e^{-i\Phi(f)}$ implies that
$$
\|\Phi(f_1)\cdots\Phi(f_n)\ket{g^*}\|^2=\matElem{g^*}{\Phi(f_n)\cdots\Phi_{f_1}^2\cdots\Phi(f_n)}{g^*}
$$
equals
$$
\Bigl.i^{2n}{\partial^{2n}\over\partial
t_1\cdots\partial t_{2n}}\Bigr|_{t_i=0}\matElem{g^*}{W(t_1f_n)\cdots
W(t_nf_1)W(t_{n+1}f_1)\cdots W(t_{2n}f_n)}{g^*}.
$$
Since the matrix element is proportional
to~$\mu(t_1f_n+\cdots+t_{2n}f_n)$, it follows
that the squared norm $\|\Phi(f_1)\cdots\Phi(f_n)\ket{g^*}\|^2$ is a linear combination
of derivatives of~$\mu(0)$ of order up
to~$2n$.
It is easily checked
that the characteristic functional of the free boson
field,
$$
\mu(f)=e^{-\|f\|^2/4}
$$
is infinitely differentiable, and the
result follows.
\end{proof}
Just how well coherent states and Wick powers get along is made evident by
the following result.
\begin{lemma}\label{lem:Wick}
The matrix elements of Wick powers on coherent states satisfy
$$
{\matElem{f^*}{\Wick{\Phi(g)^n}}{h^*}\over\bracket{f^*}{h^*}}=\Bigl({\matElem{f^*}{\Phi(g)}{h^*}\over\bracket{f^*}{h^*}}\Bigr)^n
$$
whenever~$f,g,h\in\mathbf{H}$.
\end{lemma}
\begin{proof}
By repeated application of Lemma~\ref{lem:annihilation},
\begin{eqnarray*}
\matElem{f^*}{\Wick{\Phi(g)^n}}{h^*}
&=&
{1\over
2^{n/2}}\sum_{m=0}^n{n\choose m}\matElem{f^*}{a^\dagger(g)^m a(g)^{n-m}}{h^*}\\
&=&
{1\over
2^{n/2}}\sum_{m=0}^n{n\choose m}\Bigl({\langle f,g\rangle\over -i\sqrt 2}\Bigr)^m\Bigl({\langle
g,h\rangle\over i\sqrt 2}\Bigr)^{n-m}\bracket{f^*}{h^*}\\
&=&\bracket{f^*}{h^*}\Bigl({i\over 2}\bigl[\langle f,g\rangle-\langle g,h\rangle\bigr]\Bigr)^n\\
\end{eqnarray*}
and the result follows by Equation~(\ref{eq:Heisenberg}).
\end{proof}
\subsection{Quasioperators}
\label{sec:quasioperators}
Let us look again at Equation~(\ref{eq:Heisenberg}):
$$
{\matElem{f^*}{\Phi(g)}{h^*}\over\bracket{f^*}{h^*}}={i\over 2}\bigl[\langle
f,g\rangle-\langle g,h\rangle\bigr]
\qquad\hbox{for all}\quad f,g,h\in\mathbf{H}.
$$
In this equation the right-hand side, being multilinear, is much
better behaved as a function of~$f,g,h\in\mathbf{H}$ than one would expect
from the object on the left-hand side: recall that~$f\mapsto\ket{f^*}$
is not a linear map from~$\mathbf{H}$ to~$\mathbf{K}$, and also that~$\Phi(g)$ is
an unbounded operator on~$\mathbf{K}$. This is extremely useful, as
it allows one to make sense of the expression on the left-hand side in
cases where~$f$ is so singular that~$\Phi(f)$ does not exist as an
operator on~$\mathbf{K}$.
Specifically, suppose that we are given a classical linear
observable~$f$ which is too singular to be an element of~$\mathbf{P}^*=\mathbf{H}$. Often-used examples of this come readily to mind, since
typically~$\mathbf{P}$ is a space of square-integrable tensor-valued
differential forms on a manifold and these have no pointwise values
nor can they be integrated on submanifolds. Thus, classical
observables such as~$A\mapsto A(x)$ or~$A\mapsto\oint_\gamma A$ do
not, in general, admit quantum analogues defined by the techniques
introduced so far. In the case of the free boson field we can see
explicitly that, if~$\|g\|=\infty$, then any attempt at constructing
the unitary operator~$W(g)$ will fail,
as~$\lim_{\|g\|\to\infty}\matElem{f^*}{W(g)}{h^*}=0$ because it
contains a leading factor of~$e^{-\|g\|^2/4}$. Accordingly, there is
no coherent state~$\ket{g}$ nor is a nonzero Heisenberg
operator~$\Phi(g)$ obtainable by taking derivatives of~$W(g)$.
However, if there is a scale of
spaces~$\mathbf{H}_0\subseteq\mathbf{H}\subseteq\mathbf{H}_0^\dagger$,
Equation~(\ref{eq:Heisenberg}) makes sense
for~$g\in\mathbf{H}_0^\dagger$ as long as~$f,h\in\mathbf{H}_0$. If the
span of the coherent states~$\{\ket{f}\colon f\in\mathbf{H}_0\}$ is
dense in~$\mathbf{K}$, then~$\Phi(g)$ is well-behaved enough for most
practical purposes. We now make this idea precise by means of the
concept of \emph{quasioperator}, and prove that things are in fact as
we suggest.
\begin{definition}[quasioperator]
Let~$\mathbf{K}_0$ be a topological vector space with a dense continuous
inclusion into the Hilbert space~$\mathbf{K}$. A \emph{quasioperator} on~$\mathbf{K}$ with
domain~$\mathbf{K}_0$
is a continuous
sesquilinear form~$Q\colon\mathbf{K}_0\times\mathbf{K}_0\to\mathbb{C}$, antilinear in the first argument and
linear in the second.
\end{definition}
\begin{proof}[Note]
Whenever there is a scale of
spaces~$\mathbf{K}_0\subseteq\mathbf{K}\cong\mathbf{K}^\dagger\subseteq\mathbf{K}_0^\dagger$, we will refer to
elements of~$\mathbf{K}_0$ as the space of \emph{regular} elements of~$\mathbf{K}$,
and~$\mathbf{K}_0^\dagger$ as the space of \emph{singular} ones. In other
words, a quasioperator on~$\mathbf{K}$ maps regular elements of~$\mathbf{K}$ to
`singular elements of~$\mathbf{K}$'. While possibly hair-raising to the
mathematician, this manner of speaking is actually very useful in
physical reasoning. For instance, we call the Dirac delta a `singular
function' even though it is not, strictly speaking, a function.
\end{proof}
We now assume that the $\mathbf{H}_0\subseteq\mathbf{H}$ is a complex
topological vector space and that the inclusion map is continuous,
with dense range. We call the elements of~$\mathbf{H}_0$ \emph{regular
observables}. The map~$*\colon\mathbf{H}\to\mathbf{P}$ restricts to a
map~$*\colon\mathbf{H}_0\to\mathbf{P}$ whose image~$\mathbf{P}_0$ is the space of
\emph{regular field configurations}. The dual~$\mathbf{H}_0^\dagger$ is the
space of~\emph{singular observables}. Our goal is to extend the
Heisenberg system~$\Phi$ from~$\mathbf{H}$ to~$\mathbf{H}_0^\dagger$. If~$g$ is
a singular observable~$\Phi(g)$ will be defined as a quasioperator.
Recall now that the collection of coherent
states~$\Psi=\{\ket{f^*}\colon f\in\mathbf{H}\}$ spans a dense subspace
of the Fock space~$\mathbf{K}$. We will call the coherent states
in~$\Psi_0=\{\ket{f^*}\colon f\in\mathbf{H}_0\}$ \emph{regular coherent
states}. We now show that the span of the regular coherent states is
also dense in~$\mathbf{K}$.
\begin{lemma}
Let~$\mathbf{H}_0\subseteq\mathbf{H}$ be a topological vector space with a dense
continuous inclusion into~$\mathbf{H}$. Then, if~$f_n\in\mathbf{H}_0$ for all~$n$
and~$\lim_{n\to\infty}f_n=f$ in the topology of~$\mathbf{H}$, then
$$
\lim_{n\to\infty}\ket{f_n^*}=\ket{f^*}
$$
in the topology of~$\mathbf{K}$.
\end{lemma}
\begin{proof}
For all~$g\in\mathbf{H}$,
$$
\bracket{g^*}{f_n^*}-\bracket{g^*}{f^*}=e^{\omega(f_n,g)/2i}\mu(g-f_n)-e^{\omega(f,g)/2i}\mu(g-f).
$$
By the continuity of~$\omega$ and~$\mu$ on~$\mathbf{H}$ and the density
of the~$\ket{g^*}$ in~$\mathbf{K}$, the result follows.
\end{proof}
We now let~$\mathbf{K}_0$ be the span of~$\Psi_0$, consisting of finite linear
combinations of regular coherent states, topologized algebraically.
We are then ready to define~$\Phi(g)$ as a quasioperator on~$\mathbf{K}_0$.
\begin{lemma}\label{lem:quasioperator1}
For every~$g\in\mathbf{H}_0^\dagger$ there is a unique quasioperator~$\Phi(g)$
on~$\mathbf{K}$ with domain~$\mathbf{K}_0$ such that
$$
{\matElem{f^*}{\Phi(g)}{h^*}\over\bracket{f^*}{h^*}}
={i\over 2}\bigl[\langle
f,g\rangle-\langle g,h\rangle\bigr]
\qquad\hbox{for all}\quad f,h\in\mathbf{H}_0.
$$
\end{lemma}
Note that, when~$g\in\mathbf{H}$, the matrix elements of the ordinary
Heisenberg operator~$\Phi(g)$ provide a quasioperator of this form. In
this sense, this construction extends the definition of the Heisenberg
operator~$\Phi(g)$ from regular~$g$ to singular~$g$.
\begin{proof}
Consider the function from~$\Psi_0\times\Psi_0$
$$
\ket{f^*}\times\ket{h^*}\mapsto{i\over 2}\bigl[\langle
f,g\rangle-\langle g,h\rangle\bigr]\bracket{f^*}{h^*}
\qquad\hbox{for all}\quad
f,h\in\mathbf{H}_0,
$$
which is clearly jointly continuous in the topology of~$\mathbf{K}_0$. This function
extends by linearity to a continous sesquilinear form
on~$\mathbf{K}_0$, and therefore is associated to a quasioperator on~$\mathbf{K}$ with
domain~$\mathbf{K}_0$.
\end{proof}
An entirely analogous construction generalizes Wick powers of
Heisenberg operators,~$\Wick{\Phi(g)^n}$, from regular~$g\in\mathbf{H}$ to
singular~$g\in\mathbf{H}_0^\dagger$.
\begin{lemma}\label{lem:quasioperator2}
For every~$g\in\mathbf{H}_0^\dagger$ there is a unique quasioperator~$\Wick{\Phi(g)^n}$
on~$\mathbf{K}$ with domain~$\mathbf{K}_0$ such that
$$
{\matElem{f^*}{\Wick{\Phi(g)^n}}{h^*}\over\bracket{f^*}{h^*}}
=\biggl({\matElem{f^*}{\Phi(g)}{h^*}\over\bracket{f^*}{h^*}}\biggr)^n
\qquad\hbox{for all}\quad f,h\in\mathbf{H}_0.
$$
\end{lemma}
\begin{proof}
As before, the function
$$
\ket{f^*}\times\ket{h^*}\mapsto\biggl({i\over 2}\bigl[\langle
f,g\rangle-\langle g,h\rangle\bigr]\biggr)^n\bracket{f^*}{h^*}
\qquad\hbox{for all}\quad
f,h\in\mathbf{H}_0
$$
on~$\Psi_0\times\Psi_0$
is jointly continuous in the topology of~$\mathbf{K}_0$. Extending it to all of~$\mathbf{K}_0$
by linearity, it defines a quasioperator on~$\mathbf{K}$ with
domain~$\mathbf{K}_0$.
\end{proof}
We can now extend the normal-ordering operation by linearity to the
algebra of polynomials on a Heisenberg operator~$\Phi(g)$, that is,
if~$P(x)=\sum_{k=0}^n p_kx^k$ we define
$$
\Wick{P\bigl(\Phi(g)\bigr)}=\sum_{k=0}^n p_k\Wick{\Phi(g)^k}.
$$
Then, it is easily checked that
$$
\Wick{(P+Q)\bigl(\Phi(g)\bigr)}=\Wick{P\bigl(\Phi(g)\bigr)}+\Wick{Q\bigl(\Phi(g)\bigr)}
$$
for all polynomials~$P,Q\in\mathbb{C}[x]$. This holds both at the level of
operators on~$\mathbf{K}$, if~$g\in\mathbf{H}$, and as an equation between
quasioperators on~$\mathbf{K}_0$.
\begin{corollary}\label{cor:remarkable}
Let~$F\colon\mathbb{C}^n\to\mathbb{C}$ be an entire function. Then, for
all~$g\in\mathbf{H}_0^\dagger$, there is a unique
quasioperator~$\Wick{F(\Phi(g))}$ on~$\mathbf{K}$ with domain~$\mathbf{K}_0$ satisfying
$$
{\matElem{f^*}{\Wick{F(\Phi(g))}}{h^*}\over\bracket{f^*}{h^*}}=F\biggl({\matElem{f^*}{\Phi(g)}{h^*}\over\bracket{f^*}{h^*}}\biggr)
\qquad\hbox{for all}\quad
f,h\in\mathbf{H}_0.
$$
\end{corollary}
We have proved this formula for single Heisenberg operators in
Equation~(\ref{eq:Heisenberg}), and for monomials of the Heisenberg
operators in Lemma~\ref{lem:Wick}; it also holds for Heisenberg
quasioperators (Lemma~\ref{lem:quasioperator1}) and their Wick powers (Lemma~\ref{lem:quasioperator2}). We have defined the normal-ordering
operator on the entire algebra of polynomials on the Heisenberg
(quasi)operator~$\Phi(g)$ by linearity from the normal-ordered monomials and, since
the operation
$$
X\mapsto{\matElem{f^*}{X}{h^*}\over\bracket{f^*}{h^*}}
$$
is complex linear, our desired formula holds for all polynomials of
Heisenberg operators.
\begin{proof}
For the proof, we do as before and define a complex function
on~$\Psi_0\times\Psi_0$ by
$$
\ket{f^*}\times\ket{h^*}\mapsto F\biggl({\matElem{f^*}{\Phi(g)}{h^*}\over\bracket{f^*}{h^*}}\biggr)\bracket{f^*}{h^*}
$$
which is jointly continuous in the topology of~$\mathbf{K}_0$, and extends by
linearity to a sesquilinear form on~$\mathbf{K}_0$ defining a
quasioperator with the required properties.
\end{proof}
By analogy with~$W(f)=e^{-i\Phi(f)}$, we can now define
$$
\Wick{W(g)}=\sum_{n\ge 0}{(-i)^n\over n!}\,\Wick{\Phi^n(g)}.
$$
and then
$$
{\matElem{f^*}{\Wick{W(g)}}{h^*}\over\bracket{f^*}{h^*}}=\exp{\matElem{f^*}{-i\Phi(g)}{h^*}\over\bracket{f^*}{g^*}}.
$$
This means that~$\Wick{W(g)}$ is defined as a quasioperator on the
span of the regular coherent states.
We can now deduce the following useful formula.
\begin{lemma}
$$
\Wick{W(g)}={W(g)\over\matElem{0}{W(g)}{0}}
\qquad\hbox{for all}\quad
g\in\mathbf{H}
$$
as an equation between quasioperators on~$\mathbf{K}$ with domain~$\mathbf{K}_0$.
\end{lemma}
This shows that the definition of the normal-ordered Weyl
quasioperator~$W(g)$ for~$g\in\mathbf{H}_0^\dagger$ is analogous to
resolving a singularity of the form~$0/0$ by taking a limit.
\begin{proof}
We particularize Equation~(\ref{eq:matElem})
$$
\matElem{f^*}{W(g)}{h^*}=e^{\omega(g,h+f)/2i}e^{\omega(h,f)/2i}e^{-\|h+g-f\|^2/4}
$$
to~$g=0$
$$
\bracket{f^*}{h^*}=e^{\omega(h,f)/2i}e^{-\|h-f\|^2/4}
$$
and, to~$f=h=0$
$$
\matElem{0}{W(g)}{0}=e^{-\|g\|^2/4}.
$$
Then,
$$
{\matElem{f^*}{W(g)}{h^*}\over\bracket{f^*}{h^*}\matElem{0}{W(g)}{0}}=e^{(\langle f,g\rangle-\langle g,h\rangle)/2}.
$$
By Equation~(\ref{eq:Heisenberg}), the right-hand side is
$$
\exp{\matElem{f^*}{-i\Phi(g)}{h^*}\over\bracket{f^*}{h^*}}={\matElem{f^*}{\Wick{W(g)}}{h^*}\over\bracket{f^*}{h^*}}
$$
by Corollary~\ref{cor:remarkable} applied to~$W(g)=e^{-i\Phi(g)}$.
\end{proof}
\section{Cohomologies galore}
Let~$C^\infty\Omega^k_S$ denote the space of smooth~$k$-forms on
space. The exterior derivative
$$
\mathrm{d}\colon C^\infty\Omega^k_S \to C^\infty\Omega^{k+1}_S
$$
gives rise to the smooth (de~Rham) complex
$$
\xymatrix{C^\infty\Omega^{k-1}_S\ar[r]^{\mathrm{d}}&C^\infty\Omega^k_S\ar[r]^{\mathrm{d}}&C^\infty\Omega^{k+1}_S}
$$
and the smooth de~Rham cohomology is
$$
H^k(S) \colon = {Z^k(S)\over B^k(S)} \colon = {\ker\{\mathrm{d}\colon C^\infty\Omega^k_S\to
C^\infty\Omega^{k+1}_S\}\over \mathrm{d} C^\infty\Omega^{k-1}_S}.
$$
It is the content of de~Rham's theorem that~$H^k(S)$ is isomorphic to
the real cohomology of the manifold,~$H^k(S;\mathbb{R})$.
Recall that we used compactly-supported smooth differential forms to
derive the Maxwell equations. Denoting the space of smooth,
compactly-supported $k$-forms by~$C^\infty_0\Omega^k_S$, we have the
complex
$$
\xymatrix{C^\infty_0\Omega^{k-1}_S\ar[r]^{\mathrm{d}}&C^\infty_0\Omega^k_S\ar[r]^{\mathrm{d}}&C^\infty_0\Omega^{k+1}_S}
$$
and the compactly-supported smooth cohomology is defined by
$$
H^k_0(S) \colon = {Z^k_0(S)\over B^k_0(S)} \colon = {\ker\{{\mathrm{d}}\colon C^\infty_0\Omega^k_S\to
C^\infty_0\Omega^{k+1}_S\}\over \mathrm{d} C^\infty_0\Omega^{k-1}_S}.
$$
If~$S$ is the interior of a compact manifold~$M$ with
boundary~$\partial M$, then~$H^k_0(S)$ is isomorphic to the real
relative cohomology of~$M$, denoted~$H^k(M,\partial M;\mathbb{R})$.
In fact, the derivation of the Maxwell equations and the definition of
the codifferential~$\delta$ required an inner product on the space of
differential forms. If~$L^2\Omega^k_S$ denotes the space of
square-integrable $k$-forms on~$S$, then we have the complex
$$
\xymatrix{L^2\Omega^{k-1}_S\ar[r]^{\mathrm{d}}&L^2\Omega^k_S\ar[r]^{\mathrm{d}}&L^2\Omega^{k+1}_S}
$$
where~$\mathrm{d}$ is the densely-defined operator obtained by closing the
exterior differential defined on compactly-supported, smooth
differential forms. The reduced~$L^2$ cohomology is
$$
H^k_2(S) \colon = {Z^k_2(S)\over B^k_2(S)} \colon = {\ker\{{\mathrm{d}}\colon L^2\Omega^k_S\to
L^2\Omega^{k+1}_S\}\over \overline{\mathrm{d} L^2\Omega^{k-1}_S}}.
$$
and has a natural Hilbert-space topology. Note also that
$\overline{\mathrm{d} L^2\Omega^{k-1}_S}=\overline{\mathrm{d}
C^\infty_0\Omega^{k-1}_S}$. This cohomology space is not a topological
invariant, but it is quasi-isometrically invariant, even bi-Lipschitz
homotopy invariant~\cite{lott97}. We also know that, when the metric
on~$S$ is complete, $H^k_2(S)\simeq\ker\Delta_k$.
Finally, the absolute~$L^2$ cohomology is
$$
H^k_{2,a}(S) \colon = {Z^k_{2,a}(S)\over B^k_{2,a}(S)} \colon = {\ker\{\mathrm{d}\colon L^2\Omega^k(S)\to
L^2\Omega^{k+1}(S)\}\over \mathrm{d} L^2\Omega^{k-1}(S)}.
$$
This coincides with the reduced cohomology when~$0$ is not in the
essential spectrum of~$\Delta$ (in particular, when~$M$ is compact),
but otherwise it is infinite-dimensional. The absolute cohomology has
nicer algebraic properties than the relative cohomology, such as the
Mayer--Vietoris sequence, but it is not a Hilbert space because it
involves a quotient by a non-closed subspace~\cite{mazzeo90}. It is common
usage to refer to the reduced~$L^2$ cohomology as simply the~$L^2$
cohomology.
In short, the problem is that our intuition about cohomology is based
on compact spaces, and that there the (compactly-supported) smooth and
(absolute/reduced) square-integrable cohomologies all coincide, and
moreover are isomorphic to the cohomologies obtained by combinatorial
methods. Since in the non-compact case all of these cohomologies may
be different, the question arises of which cohomology to use. This
choice has physical implications for electromagnetism: both
classically, through Wheeler's concept of ``charge without charge''
arising through ``field lines trapped by the topology of spacetime'';
and quantumly, through the Aharonov--Bohm effect and mass gaps induced
by the metric when the spectrum of the Laplacian is bounded away
from zero. If we were using~$U(1)$ as the gauge group instead
of~$\mathbb{R}$, topology would manifest itself also through topological terms
in the action (``topological mass'') and topologically stable
solutions (solitons and monopoles).
As we have pointed out,~$p$-form electromagnetism on
an~$(n+1)$-dimensional spacetime is conformally invariant if the
relation~$n+1=2(p+1)$ is satisfied. In all other cases we have seen
that the phase space of classical electromagnetism can be described
most conveniently in terms of the twisted differential operator
$$
D = e^{{1\over 2}(n-2p-1)\Phi} \mathrm{d} e^{-{1\over 2}(n-2p-1)\Phi}.
$$
The following commutative diagram
$$
\xymatrix{C^\infty_0\Omega^{k-1}_S\ar[r]^{\mathrm{d}}\ar[d]^{e^{{1\over
2}(n-2p-1)\Phi}}&C^\infty_0\Omega^k_S\ar[r]^{\mathrm{d}}\ar[d]^{e^{{1\over
2}(n-2p-1)\Phi}}&C^\infty_0\Omega^{k+1}_S\ar[d]^{e^{{1\over 2}(n-2p-1)\Phi}}\\
C^\infty_0\Omega^{k-1}_S\ar[r]^{D}&C^\infty_0\Omega^k_S\ar[r]^{D}&C^\infty_0\Omega^{k+1}_S}
$$
where the downward arrows represent multiplication operators, is an
isomorphism of cohomology complexes as long as~$\Phi$ is smooth. If we
complete all the spaces in the~$L^2$ norm and close all operators we
still obtain two cohomology complexes, but the diagram
$$
\xymatrix{L^2\Omega^{k-1}_S\ar[r]^{\mathrm{d}}\ar[d]^{e^{{1\over
2}(n-2p-1)\Phi}}&L^2\Omega^k_S\ar[r]^{\mathrm{d}}\ar[d]^{e^{{1\over
2}(n-2p-1)\Phi}}&L^2\Omega^{k+1}_S\ar[d]^{e^{{1\over 2}(n-2p-1)\Phi}}\\
L^2\Omega^{k-1}_S\ar[r]^{D}&L^2\Omega^k_S\ar[r]^{D}&L^2\Omega^{k+1}_S}
$$
now has vertical arrows which, depending on the behaviour of~$\Phi$
and~$\mathrm{d}\Phi$ at spatial infinity, may be only densely defined, and so
definitely not isomorphisms. Therefore, the ordinary (top) and twisted
(bottom) chain complexes may not be isomorphic, and so the
`twisted' $L^2$ cohomology based on~$D$ may not be isomorphic to the
ordinary one based on~$\mathrm{d}$, even though the smooth cohomologies are
in fact isomorphic.
This may come about in several ways. On the one hand, the closure of
the multiplication operator depends on the behaviour of the
function~$\Phi$ at infinity. Indeed, for a~$k$-form~$\alpha$ to be in
the domain of~$\overline{e^{{1\over 2}(n-1-2p)\Phi}}$ it is necessary
that both~$\alpha$ and~$e^{{1\over 2}(n-1-2p)\Phi}\alpha$ be
square-integrable. Also, although multiplication by~$e^{{1\over
2}(n-1-2p)\Phi}$ is an isomorphism between spaces of smooth forms, its
closure need not be invertible if~$\Phi$ is unbounded. On the other
hand, the closure of~$D=\mathrm{d}+\bigl[p+1-{n+1\over
2}\bigr](\mathrm{d}\Phi)\wedge$ depends on the behaviour of~$\mathrm{d}\Phi$ at
infinity, which can be wild even if~$\Phi$ is bounded.
\section{Known results}
In this section we present a summary of known results on the
(reduced)~$L^2$ cohomology of a Riemannian manifold. Since Poincar\'e duality still holds in the form
$H^k_2(S)\simeq H^{n-k}_2(S)$, so only the cases~$0\le 2k\le n$
need be considered.
The zeroth cohomology~$H^0_2(S)$ is $1$-dimensional if~$S$ has finite
volume, and trivial otherwise. This is easy to understand since,
essentially, the question is whether constants are square-integrable
or not. This is the first difference with the compact case.
Other than in the extreme dimensions~$0$ and~$n$, very little can be
said in general. For instance, Anderson~\cite{anderson85} proves that,
if~$n>1$,~$a>|n-2p|$ and~$a\ge 1$, there are complete Riemannian
manifolds diffeomorphic to~$\mathbb{R}^n$, with curvature bounded by~$-a^2\le
K\le 1$ and such that their $p$th square-integrable cohomology~$H^p_2$
is infinite-dimensional.
For rotationally symmetric $n$-dimensional manifolds with metric
$$
ds^2=dr^2+f(r)^2d\theta^2,
$$
where~$d\theta$ is the standard metric on~$S^{n-1}$, the
square-integrable cohomology is~$H^k_2=\{0\}$ if~$k\neq 0,n/2,n$. As
we know, when $k=0,n$ the cohomology depends on the volume of
spacetime. Finally, when~$k=n/2$, $H^k_2=\{0\}$ if
$\int^\infty{ds\over f(s)}=\infty$, and infinite-dimensional
otherwise. This is because of conformal invariance of the cohomology
in the middle dimension, and the fact that convergence of the integral
correlates with conformal compactness~\cite{dodziuk79}. A remarkable
consequence of this is that, when space a two-dimensional cylinder,
the square-integrable cohomology~$H^1_2$ never matches the smooth
cohomology, which is one-dimensional and is generated by~$\mathrm{d}\theta$.
A complete Riemannian manifold is conformally compact if it is
diffeomorphic to the interior of a compact manifold~$M$ with boundary,
and the metrics of the two manifolds at corresponding points are
proportional by a function called the conformal factor:
$$
g_M=\rho^2 g.
$$
The conformal factor~$\rho$ has the effect of ``pushing the
boundary of~$M$ to infinity''. For this it is necessary that
$\int_\gamma \rho^{-1}\mathrm{d} s_M$ diverge whenever~$\gamma$ is a
(finite-length) curve in~$M$ with at least one endpoint on~$\partial
M$, so the conformal factor must vanish precisely on~$\partial
M$. Conformal compactification makes precise the idea of ``ideal
boundary at infinity'' of a noncompact manifold, and it was introduced
into general relativity as an important tool by Penrose.
Mazzeo~\cite{mazzeo88} studies the case where the conformal factor
satisfies the additional regularity condition that~$\mathrm{d}\rho$ does not
vanish on~$\partial M$, in which case the manifold is asymptotically
hyperbolic. He then proves that a complete conformally
compact~$n$-dimensional Riemannian manifold has finite-dimensional
cohomology groups except possibly for the middle dimensions, and gives a
topological interpretation of them:
$$
H^k_2\simeq\left\{
\begin{array}{ll}
H^k(M,\partial M,\mathbb{R}) & k<(n-1)/2 \\
H^k(M,\mathbb{R}) & k>(n+1)/2
\end{array}
\right.
$$
Moreover, if~$-a^2$ is the most negative limiting curvature at
infinity, then the essential spectrum of the Laplacian~$\Delta_k$ is
$$
\sigma_{\mathrm{ess}}(\Delta_k)=\left\{
\begin{array}{ll}
[a^2(n-2k-1)^2,\infty) & k<n/2 \\
\{0\}\cup[a^2/4,\infty) & k=n/2 \\
{[a^2(n-2k+1)^2,\infty)} & k>n/2
\end{array}
\right.
$$
In particular, if~$n=2k$, the $k$th cohomology group is
infinite-dimensional and, if $|n-2k|\le 1$, the essential spectrum
extends all the way to~$0$. For hyperbolic manifolds which are
geometrically finite (\textit{i.e.}, having no tubular ends), Mazzeo and
Phillips~\cite{mazzeo90} prove that the cohomology of the middle
dimensions~$k=(n\pm 1)/2$ is finite-dimensional and has a topological
interpretation. These results are extended by Lott~\cite{lott97} to
the case of hyperbolic $3$-manifolds which are diffeomorphic to the
interior of a compact manifold with boundary and geometrically
infinite. In particular, Lott proves that, if such a space is `nice'
(has incompressible ends and its injectivity radius does not go to
zero at infinity), the kernel of the Laplacian on~$1$-forms is
finite-dimensional. He also provides a variety of results on the
spectrum of the Laplacian on~$1$-forms.
These results have a direct physical interpretation when~$p$-form
electromagnetism is conformally invariant, as otherwise one has to
consider an appropriately twisted~$L^2$ cohomology complex for which
there are no known general results. We have pointed out that,
when~$\Phi$ and~$\mathrm{d}\Phi$ are both bounded, the twisted $L^2$
cohomology complex is isomorphic to the untwisted one, and so the
above-mentioned results can be applied directly. In the general case,
it is reasonable to assume that the behaviour of the twisted
cohomology will be at least as rich as that of the ordinary~$L^2$
cohomology. In the conformally invariant cases, we have the following
possible physical interpretations:
\begin{itemize}
\item the massless scalar field ($0$-form electromagnetism)
in~$1+1$ dimensions. In this case, since the space manifold~$S$ is
assumed to be noncompact, it is
diffeomorphic to~$\mathbb{R}$. Global hyperbolicity then requires that the optical metric give~$S$ infinite
length, and so~$H^0_2=\{0\}$ because the constant field is not square
integrable. In other words, square-integrable fields must go to zero
at infinity.
\item ordinary ($1$-form) electromagnetism in~$3+1$ dimensions. If
space is spherically symmetric there are no harmonic,
square-integrable~$1$-forms according to~\cite{dodziuk79}. This is not
a surprise since the first de~Rham cohomology is also trivial. In more
general cases, if the space manifold~$S$ is conformally compact the
spectrum of the Laplacian reaches all the way to~$0$ (physically, the
photon does not acquire a mass), but the
dimension of the kernel of the Laplacian is not known in
general. Anderson's example~\cite{anderson85} shows that it is
possible for this space of non-standard Aharonov-Bohm modes to be
infinite-dimensional.
\item when~$p$-form electromagnetism is conformally invariant the
dimension of space is~$p=(n-1)/2$, and we are always in one of the ``middle
dimension'' cases where the dimension of the space of harmonic vector
potentials remains unresolved, although for a large class of manifolds
it is known that the essential spectrum of the Laplacian is all
of~$[0,\infty)$ and so there is no mass gap.
\end{itemize}
\noindent
In case~$\Phi$ and~$\mathrm{d}\Phi$ are bounded, the dimension of the space
of ``twisted'' harmonic~$p$-forms is independent of~$\Phi$, and so we
can draw valid physical conclusions about non-standard Aharonov--Bohm
modes even in the absence of conformal invariance. The lower bounds to
the spectrum of the Laplacian may be critically dependent on~$\Phi$,
so any inferences we make from the~$\Phi=0$ case are probably
unwarranted, but still enticingly point to situations where the
phenomenon of mass gaps might occur. The physical interpretation of
the~$L^2$ cohomology results in the cases when electromagnetism is not
conformaly invariant follows.
\begin{itemize}
\item the massless scalar field in~$n+1$ dimensions has at most a
one-dimensional space of harmonic solutions. This depends on whether
the function
$$
f=e^{{1\over 2}(n-1)\Phi}
$$
is square-integrable with respect to the optical metric. Also, if
space is conformally compact and the curvature at infinity is bounded
below by~$-a^2$, then the essential spectrum of the Laplacian
is~$[a^2(n-1),\infty)$. This means that, if the dimension of space is
$n>1$, the free massless scalar field can have a
mass gap in the~$\Phi=0$ case.
\item ordinary electromagnetism in~$2+1$ dimensions can have an
infinite-dimensional space of harmonic vector potentials even in the
rotationally symmetric case, including when the optical metric on
space is that of the hyperbolic plane. In addition, if space is
conformally compact the~$\Phi=0$ mass gap is~$a^2/4$, where~$-a^2$ is the
lower bound to the curvature at infinity. When the optical metric on
space is conformally compact and of dimension~$4+1$ or higher, the
space of harmonic vector potentials is isomorphic to the first
cohomology of~$M$ relative to its boundary, and so there are no
non-standard Aharonov--Bohm modes. When the curvature at infinity is
bounded below by~$-a^2$, the essential spectrum of the ordinary
Laplacian is~$[a^2(n-3),\infty)$ if~$n\ge 3$, signaling the
possibility of topological mass gaps in~$4+1$ dimensions or higher, at
least when~$\Phi=0$.
\item for~$p$-form electromagnetism, there is an infinite-dimensional
space of harmonic vector potentials if space is a~$2p$-dimensional and
rotationally symmetric or conformally compact. In the latter case,
there is a~$\Phi=0$ mass gap of~$a^2/4$. If~$|n-2p|>1$ there are no
non-standard Aharonov-Bohm modes, but the~$\Phi=0$ mass gap is zero only
if~$n=2p\pm 1$.
\end{itemize}
|
1,116,691,500,749 | arxiv | \section{Introduction}
Deep learning has facilitated technological advances in a variety of domains, e.g.\ computer vision, natural language processing, and robotics. However, the notable success in achieving high test performances should not obscure the fact that we are still lacking a sufficient understanding of the functioning of deep neural networks. \newline
In the past years, it has become apparent that deep neural networks are quite brittle, in particular they are vulnerable to small - even imperceptible - perturbations of the input \cite{Akhtar, Dodge}. This observation at least partially contradicts the common belief that deep networks generalize well to unseen, similar examples. These harmful inputs can be the result of distributional shifts or general noise in the environment of the classifier \cite{Volpi}, e.g.\ unusual lighting or weather conditions for an autonomous car. Alternatively, they might be adversarial examples, i.e.\ perturbed natural images that were intentionally crafted by some adversary to cause misclassifications. In the past, these two types of harmful perturbations have been analyzed by mostly separate research communities, namely the corruption robustness and the adversarial robustness researchers. Recently it has become more and more evident that the vulnerability to these different perturbations is closely connected and should therefore not be analyzed separately \cite{Hendrycks}. For example, in \cite{Ford} the authors derived a promising estimate for the size of small worst-case adversarial perturbations with the help of the test error on additive Gaussian noise. \newline
With the seminal work of Biggio et al.\ \cite{Biggio4} and Szegedy et al.\ \cite{Szegedy} as a starting point, there has been tremendous research effort directed towards exploring methods that generate adversarial examples \cite{Assion, Eykholt} as well as defenses that aim at increasing robustness \cite{Guo, Metzen2}. The consistent success of strong targeted adversarial attacks in computer vision, like PGD \cite{Madry} and C\&W \cite{Carlini2}, suggests that an adversary can basically create any desired classification output by adding a suitable adversarial perturbation to the natural image. This has also been impressively shown by Metzen et al.\ \cite{Metzen} in the context of semantic segmentation. Here, the authors were able to create universal, i.e.\ input-agnostic, adversarial perturbations that lead to any desired target segmentation. These adversarial attack results bring us to the realization that the decision boundary of a conventional deep neural network seems to be close to almost every input image. In other words, the closeness of the decision boundary to natural data points explains the vulnerability of state-of-the-art ML models to certain input perturbations. \newline
A vast number of adversarial defenses therefore try to increase the minimal distance, also called margin in the input space, of natural images to the decision boundary. Among other things, researchers have proposed regularization penalties \cite{Gu, Cisse}, data augmentation techniques \cite{Tramer, Papernot3} and specific architectures \cite{Papernot4, Schott} to obtain a more desirable course of the decision boundary, and in consequence to get to more robust classifiers. Unfortunately, the majority of defenses have not succeeded and were "broken" shortly after their publication due to stronger adversaries \cite{Athalye2}.
In general, PGD adversarial training is still viewed as the most reliable and the most successful adversarial defense \cite{Madry}. But, it should be noted that adversarial training shows promising results under very limited threat models and tends to overfit on specific attacks instead of improving general robustness \cite{Kang}. \newline
This limited progress in increasing robustness has led more and more researchers to the question whether robustness can be obtained at all by deep neural networks \cite{Fawzi, Shafahi, Schmidt}. The well-known, but rather theoretical, statement that neural networks with a single hidden layer are universal function approximators has partly given us a false sense of security. For example, recent work indicates that commonly used neural network topologies with a large number of relatively low-dimensional hidden layers may not lead to universal function approximators \cite{Johnson}.
Additionally, publications have shown empirically, as well as theoretically, that the decision regions of modern ML classifiers, i.e.\ the regions of the input space that lead to a certain output class, tend to be connected sets \cite{Johnson, Fawzi2}.
This strong topological restriction on the decision regions might already limit the expressive power of these models and thus, their maximal achievable robustness. \newline
In addition, there have been efforts to derive general robustness bounds for classes of classification problems, independent of the used classification function. Fawzi et al.\ \cite{Fawzi} provide fundamental upper bounds on the achievable robustness assuming that the natural data comes from a smooth generative model. Under this assumption on the origin of the data, they show that any type of classifier is prone to adversarial perturbations as long as the latent space of the generative model is high-dimensional. \newline
Overall, we are still at an early stage of understanding the decision making and limitations of deep neural networks. Especially, the course of the decision boundary and factors that influence its course have to be explored in more detail. Findings in this area can then guide us towards promising adversarial defenses as well as general robustness bounds. In this paper, we want to continue along this path by providing an empirical study focusing on the distance of data points to the decision boundary and how this margin evolves over the training of the classifier. To the best of our knowledge, the change of the margin during the training process has not yet been analyzed in the existing literature. Our experiments with neural network classifiers on \textsc{Mnist}, \textsc{Fashion-Mnist}, and \textsc{Cifar-10} lead us to three central observations:
\begin{itemize}
\item The decision boundary moves closer to training as well as test images over training. This convergence of the decision boundary even continues in the late phases of training where the neural network already obtains low training and test error rates.
\item Adversarial training results in a significantly different development of the decision boundary. Here, the average distance to the decision boundary of the images stays at a relatively high level over training. The clear downward trend observable for standard training is damped considerably, which underlines the success of adversarial training in improving robustness for simple classification tasks.
\item Wrongly classified images from the natural data distribution are on average significantly closer to the decision boundary than correctly classified data points. During training the decision boundary is pushed towards these points, which implies that their already small distance to the decision boundary decreases even further with increasing epoch number. This observation holds for adversarial as well as standard training.
\end{itemize}
Due to the success of adversarial attacks on ML models, it is not surprising that the decision boundary is close to the majority of natural data points after training. But, our findings still challenge common beliefs about the training of neural network classifiers. It does not seem to be true that training moves the decision boundary away from the training data in order to facilitate generalization, or at least this is not true for all directions in the input space.
\section{Background} \label{sec: related_work}
In this section, we formalize the notion of decision boundary, and summarize related work on the decision boundary of deep neural networks. For our experiments it will be crucial to calculate the minimal distance of an image to the decision boundary of the classifier. Unfortunately, calculating the exact margin in the input space is in general intractable, thus one has to make use of a reasonable upper bound. We will use DeepFool \cite{Dezfooli} for this margin approximation. Due to its significant role within the following empirical study, we will recall the central intuition underlying the DeepFool adversarial attack in this section.
\subsection{The Decision Boundary}
We define a classifier as a function $f: \mathbb{R}^n \rightarrow \mathbb{R}^c$, where $n$ denotes the number of dimensions in the input space (e.g.\ number of pixels in an image) and $c$ is the number of classification classes. For an input point $x \in \mathbb{R}^n$ the output $f(x)$ can be interpreted as the vector of softmax values of the classifier. The classification decision is then given by
$$\hat{k}(x) = \underset{k=1,...,c} {\operatorname{argmax}} \; f_k(x).$$
With this notation, we can now define the decision boundary $\mathcal{D} \subseteq \mathbb{R}^n$ of $f$ as the set
\begin{align*}
\begin{split}
\mathcal{D} := \{ x \in \mathbb{R}^n \mid \exists & k_1, k_2 = 1,...,c, \; k_1 \neq k_2, \\
& f_{k_1}(x) = f_{k_2}(x) = \operatorname{max}_k f_k(x)\}.
\end{split}
\end{align*}
In other words, these are the points where the decision of the classifier is tied.
The margin $d_2(x) \in \mathbb{R}_{\geq 0}$ of a data point $x$ is then given by
\begin{align} \label{eq: 1}
\begin{split}
d_2(x) = \underset{\delta \in \mathbb{R}^n}{\min} &\; \| \delta \|_2 \\
&s.t. \; x + \delta \in \mathcal{D}
\end{split}
\end{align}
In the above margin definition, we make use of the $\ell_2$-norm, but one can define the margin with respect to any reasonable distance measure. In our empirical study, we will also consider the margin with respect to the $\ell_{\infty}$-norm, which we will denote $d_{\infty}$. \newline
The existence of adversarial examples for state-of-the-art neural networks indicates that natural images lie near the decision boundary $\mathcal{D}$ with high probability, hence $d_2(x)$ is small for most data points $x$. At the same time, deep neural networks also tend to be rather robust to random noise \cite{Yu}. It may thus be concluded that the decision boundary is close in a "few" directions, but further away with respect to the majority of perturbation directions.
In \cite{He2} the authors confirm this observation by adding randomly sampled orthogonal directions to benign as well as adversarial images. For the benign images, these sampled perturbations rarely change the classification decision. On the other hand, they observe that most of the adversarial examples generated by the FGSM attack \cite{Good1} are not at all robust to these random distortions. \newline
For our experiments, we do not want to rely on sampled perturbations for the calculation of $d_2(x)$, but rather utilize an optimization method. Since the margin optimization problem (\ref{eq: 1}) is intractable for deep neural networks, one has to search for an approximate solution. To be more precise, one searches for a small perturbation $\delta$ such that $\hat{k}(x+\delta) \neq \hat{k}(x)$. Ideally, we then have $d_2(x) \approx \| \delta \|_2$, although $x+ \delta$ is not necessarily an element of $\mathcal{D}$. \newline
We will obtain the margin estimate with the help of DeepFool, but one can also make use of other strong adversarial attacks for the generation of $\delta$. For example, in \cite{Nar} the authors use the PGD attack to approximate the distance of training data to the decision boundary. They find that the usual cross-entropy loss is one contributing factor to small margins and that a differential training procedure leads to more robust models. \newline
However, DeepFool has proven to generate particularly small adversarial perturbations which makes it suitable for margin approximation. Jiang et al.\ \cite{Jiang} used a simplified targeted version of DeepFool with a single iteration step to estimate $d_2(x)$ for images of the training data set. These distances then formed the basis for a measure which correlates with the generalization gap of a trained neural network. \newline
Apart from determining the distance to the decision boundary, it is also desirable to understand more general geometric properties of the decision boundary. Already in the early adversarial robustness literature, it has been hypothesized that the decision boundary of a deep neural network locally resembles the decision boundary of a linear classifier. In \cite{Good1} the authors claim that this "too" linear behavior explains the success of the FGSM adversarial attack which utilizes the linearization of the loss function of the network. More recently, Fawzi et al.\ \cite{Fawzi2} empirically showed that the decision boundary near natural images is flat in most directions, and curved only in very few directions. Furthermore, their results suggest that the decision boundary is biased towards negative curvatures. \newline
\subsection{DeepFool}
DeepFool is an untargeted, iterative adversarial attack which stops as soon as a perturbation $\delta$ has been found with $\hat{k}(x_0+\delta) \neq \hat{k}(x_0)$ for some given data point $x_0$. Moosavi-Dezfooli et al.\ \cite{Dezfooli} introduced DeepFool with the goal to provide a method that can calculate adversarial perturbations with similar efficiency as FGSM (or comparable attacks like BIM \cite{Kurakin} and PGD \cite{Madry}), but which at the same time leads to a more accurate approximation of the robustness of $f$ at $x_0$. The authors achieve this by making use of the well-understood orthogonal projection mapping of a point onto the decision boundary of an affine classifier. \newline
To be more specific, in every iteration step of DeepFool the class probability functions $f_k, \; k = 1,...,c,$ of the classifier $f$ are linearized around the current position $x_i$. Then, one calculates the smallest perturbation $\delta_i$ with respect to the $\ell_2$-norm which moves $x_i$ onto the decision boundary of the linearized model of $f$. This perturbation $\delta_i$ can be written down in closed form:
$$ \delta_i := \frac{ |f_{\hat{l}}(x_i) - f_{\hat{k}(x_0)}(x_i) |}{\| \nabla f_{\hat{l}}(x_i) - \nabla f_{\hat{k}(x_0)}(x_i) \|_2^2 } (\nabla f_{\hat{l}}(x_i) - \nabla f_{\hat{k}(x_0)}(x_i))$$
with index
$$\hat{l}=\hat{l}(x_i):= \operatorname{argmin}_{k \neq \hat{k}(x_0) } \frac{|f_{\hat{l}(x_i)}(x_i) - f_{\hat{k}(x_0)}(x_i)|}{\| \nabla f_{\hat{l}(x_i)}(x_i) - \nabla f_{\hat{k}(x_0)}(x_i) \|_2 }. $$
Now, let $N$ denote the stopping index of this iterative scheme for image $x_0$, i.e.\ $\hat{k}(x_N) \neq \hat{k}(x_0)$. Then, the desired adversarial perturbation is given by
$$ \delta = \sum_{i=0}^{N-1} \delta_i.$$
Overall, the DeepFool attack can be viewed as a function $\operatorname{DeepFool}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ which takes an image $x_0$ as input and returns a corresponding adversarial perturbation $\delta$ for the given classifier $f$.
In the original paper, the authors also formulate adaptations of the DeepFool algorithm to any $\ell_p$-distance measure for $p \in [1, \infty]$. Since we also want to consider the margin with respect to the $\ell_{\infty}$-norm in our empirical study, we will use the $\ell_{\infty}$ adaptation for these approximations. For the closed form formula of $\ell_{\infty}$-DeepFool we refer to the original paper \cite{Dezfooli}. \newline
\section{Experimental Results}
The objective of the experiments is to track the $\ell_2$-norm margin values $d_2$ as well as the $\ell_{\infty}$-norm margin values $d_{\infty}$ for training data as well as test data over the training process of a deep neural network. We obtain these approximate distances of images to the decision boundary with the $\ell_2$-DeepFool algorithm and the $\ell_{\infty}$-DeepFool algorithm, respectively. Hence, we assume that $$d_2(x) = \| \operatorname{DeepFool}_2(x) \|_2$$ and $$d_{\infty}(x) = \| \operatorname{DeepFool}_{\infty}(x) \|_{\infty}$$ for any image $x$.
In order to derive general observations, we will analyze the average margin and the corresponding standard error over large parts of the training as well as the test data set in every training epoch. To be more precise, for an image data set $\mathcal{D}$ we report
$$d_2^{avg}:= \frac{1}{|\mathcal{D}|}\sum_{x_i \in \mathcal{D}} d_2(x_i), \; \; d_{\infty}^{avg}:= \frac{1}{|\mathcal{D}|}\sum_{x_i \in \mathcal{D}} d_{\infty}(x_i)$$
and
\begin{align*}
\begin{split}
d_2^{se}:= \frac{1}{|\mathcal{D}|}\sqrt{\sum_{x_i \in \mathcal{D}}(d_2(x_i)-d_2^{avg})^2}, \\
d_{\infty}^{se}:=\frac{1}{|\mathcal{D}|}\sqrt{\sum_{x_i \in \mathcal{D}} (d_{\infty}(x_i)-d_{\infty}^{avg})^2},
\end{split}
\end{align*}
where $| \mathcal{D}|$ denotes the cardinality of the image set $\mathcal{D}$. For one of the following observations we will also plot the distributions $d_2(x)$ and $d_{\infty}(x)$ for all images $x \in \mathcal{D}$ in several training epochs. These distribution visualizations help us to better understand how the different images contributed to the calculated average margin and standard error. \newline
It should be noted that we only use successful adversarial perturbations for the margin approximation. Thus, if the DeepFool attack is not able to find a small adversarial perturbation for a given image, we iteratively perturb the image with Gaussian noise and retry DeepFool on the perturbed image until we find a successful adversarial example. \newline
To ensure consistency of our observations, we train several classifiers with different architectures and computer vision tasks. In particular, we conduct experiments on \textsc{Mnist}, \textsc{Fashion-Mnist}, and \textsc{Cifar-10}. In the following figures we will present the experimental results for a convolutional neural network trained on the \textsc{Fashion-Mnist} data set. The given graphs summarize our results for the first $40$ epochs of training. After this limited training period, the \textsc{Fashion-Mnist} model obtains good, but not yet competetive, error rates. However, all our central observations can already be made during the first $40$ epochs, and they continue to hold when we extend the training time.
Comparable graphs for different architectures and for the other tasks - \textsc{Mnist} and \textsc{Cifar-10} - can be found in Appendix \ref{sec: A}. Overall, our experiments suggest three major findings which we will discuss separately in the following sections.
\subsection*{Observation 1: Standard Training}
\begin{figure*}[t]
\begin{center}
\centerline{%
\includegraphics[width=0.35\textwidth]{01_fmnist_cnn_vanilla_error.png}%
\includegraphics[width=0.35\textwidth]{02_fmnist_cnn_vanilla_l2_dists_no_split.png}%
\includegraphics[width=0.35\textwidth]{03_fmnist_cnn_vanilla_linf_dists_no_split.png}%
}%
\end{center}
\caption{Experimental results of a convolutional neural network (CNN) on the \textsc{Fashion-Mnist} dataset: (1) Left: Development of training and test error; (2) Middle: Development of average $\ell_2$-margin $d_2^{avg}$ and standard error $d_2^{se}$ over $1000$ randomly picked images from the training and the test data set; (3) Right: Development of average $\ell_{\infty}$-margin $d_{\infty}^{avg}$ and standard error $d_{\infty}^{se}$ over $1000$ randomly picked images from the training and the test data set.}
\label{fig: results_vanilla}
\end{figure*}
Due to the vulnerability of state-of-the-art neural networks to adversarial examples, we know that at least a large portion of the training images as well as test images lie close to the decision boundary after training. As a consequence, we expect low values for the average $\ell_2$-norm margin $d^{avg}_2$ and the average $\ell_{\infty}$-norm margin $d^{avg}_{\infty}$ for a trained classifier. But it is still unclear how the margin metrics $d^{avg}_2$ and $d^{avg}_{\infty}$ evolve over the training process. To analyze this, we train a convolutional and a dense architecture with cross-entropy loss for the three given computer vision tasks. Sample results for the convolutional neural network (CNN) on \textsc{Fashion-Mnist} are shown in Figure \ref{fig: results_vanilla}.
In general, we observe a similar development of the average margins and their standard errors during the training process for all models and tasks. \newline
After the weight initialization - i.e.\ before the first training weight update - test and training images are very close to the decision boundary. After the first epoch, $d^{avg}_2$ as well as $d^{avg}_{\infty}$ jump to a higher level. Thus, already the first training epoch changes the course of the decision boundary decisively, although it does not yet lead to a classifier with optimal train and test accuracy.
In the subsequent epochs, the average margins have a clear downward trend and they never get back to the peak of the first epoch. Especially in the early training epochs, the level of $d^{avg}_2$ and $d^{avg}_{\infty}$ drops significantly. The margins decrease less strictly in later phases of the training, but they still decrease noticeably.
Overall, we see a strong negative correlation between the average distance to the decision boundary and the training (or test) accuracy. At the same time, the standard errors $d_2^{se}$ and $d_{\infty}^{se}$ remain relatively small and stable throughout the whole training. It should also be noted that there is no significant difference between the margin values of the training and the test set. Hence, the decision boundary does not appear to be closer to test images compared to training images. \newline
From these results, we can derive the general finding that the decision boundary moves closer to training as well as test images during training.
This observed convergence of the decision boundary is an undesirable side effect of training, since it shows that trained models with state-of-the-art test accuracy will end up with relatively low average margins. In particular, increasing the training time of a model might lead to a decrease in robustness. \newline
Unfortunately, we can not yet offer a clear and provable explanation for this behavior of the decision boundary during training. At first glance, one might try to justify this observation by assuming a lack of model capacity. In other words, the chosen model architectures might just not be able to simultaneously achieve good accuracies as well as sufficient decision boundary margins. The existence of this trade-off between performance and robustness would then automatically imply a decrease of the margins during training, since the cross-entropy function forces the network to achieve high accuracy in every training epoch. \newline
Alternatively, one might be tempted to view this phenomenon as a sign of overfitting on the training data, because overfitting also leads to a decision boundary which is getting unnecessarily close to natural data points. But, both hypotheses are hard to validate, and we even notice aspects of our experimental results contradicting these ideas. For example, the distances to the decision boundary decline throughout phases of the training where the test error is also decreasing significantly. This contradicts the overfitting explanation, because overfitting would manifest itself in a downturn of the test accuracy. On the other hand, the following section and its related experiments will show that the exact same model architectures can also lead to a totally different development of the decision boundary, which confutes the lack of model capacity explanation. \newline
Thus, the question remains why this phenomenon of decreasing average margins occurs and whether this movement of the decision boundary can be prevented.
\subsection*{Observation 2: Adversarial Training}
\begin{figure*}[t]
\begin{center}
\centerline{%
\includegraphics[width=0.35\textwidth]{04_fmnist_cnn_adversarial_error.png}%
\includegraphics[width=0.35\textwidth]{05_fmnist_cnn_adversarial_l2_dists_no_split.png}%
\includegraphics[width=0.35\textwidth]{06_fmnist_cnn_adversarial_linf_dists_no_split.png}%
}%
\end{center}
\caption{Experimental results of a convolutional neural network (CNN) with PGD adversarial training on the \textsc{Fashion-Mnist} dataset: (1) Left: Development of training and test error; (2) Middle: Development of average $\ell_2$-margin $d_2^{avg}$ and standard error $d_2^{se}$ over $1000$ randomly picked images from the training and the test data set; (3) Right: Development of average $\ell_{\infty}$-margin $d_{\infty}^{avg}$ and standard error $d_{\infty}^{se}$ over $1000$ randomly picked images from the training and the test data set.}
\label{fig: results_adversarial}
\end{figure*}
We again train convolutional and dense neural networks on \textsc{Mnist} and \textsc{Fashion-Mnist}, but this time perform adversarial training. We keep the cross-entropy loss function and network architectures as in the previous experiments. In every training batch, one now replaces 50\% of the images by adversarial images generated by the PGD attack. Then, we again track the average distances of natural train as well as test images to the decision boundary with respect to the $\ell_2$- and the $\ell_{\infty}$-norm. In Figure \ref{fig: results_adversarial} we summarize the results for the CNN architecture on \textsc{Fashion-Mnist}. \newline
As before, one notices a jump of the average margins after the first epoch of training compared to the level at network initialization. However, in the following epochs, we observe significant deviations from the results of standard training.
After the first epoch, $d^{avg}_2$ as well as $d^{avg}_{\infty}$ remain at a relatively high level compared to standard training. Even though a slight downward trend is observable, it is not as severe as in the setting considered before. It can also be noticed that this downward trend starts in later epochs compared to standard training. Our experimental results for \textsc{Mnist} even show a slight upward trend of the average margins with training time (see: Appendix \ref{sec: A}).
At the same time, the standard errors $d_2^{se}$ and $d_{\infty}^{se}$ are comparable to the past experiments, i.e.\ remain at a rather stable level throughout training for all computer vision tasks. \newline
We come to the conclusion that the previously observed steady decrease of the average margins is not an inevitable phenomenon. The injection of PGD adversarial examples into the training set leads to a decision boundary which is further away from natural data points in comparison to a decision boundary of a classically trained model. Furthermore, the experiments suggest that the robustness of the adversarially trained classifier does not significantly degrade throughout training time. At least for these simple computer vision tasks, we can therefore confirm the general belief that PGD adversarial training creates comparably robust models. \newline
A further interesting aspect is the high level of average margin size measured in the $\ell_2$-norm, although the PGD attack is concerned with finding small $\ell_{\infty}$-norm adversarial perturbations. Hence, the $\ell_{\infty}$-based adversarial training procedure was also able to increase the minimal $\ell_2$-distance to the decision boundary for a large number of training and test images. It is still an open research question to which extent adversarial training can be used to robustify classifiers against broad classes of adversarial attacks. Past studies suggested that adversarial training does not transfer well between imperceptibility metrics, in particular that $\ell_{\infty}$-based adversarial training does not necessarily lead to robustness with respect to other $\ell_p$-norms, e.g.\ \cite{Kang}. Our results indicate that there actually exist settings, where the positive impact of adversarial training transfers to broader classes of perturbations.
\subsection*{Observation 3: Wrongly Classified Images}
\begin{figure*}[t]
\begin{center}
\centerline{%
\includegraphics[width=0.35 \textwidth]{07_fmnist_cnn_vanilla_l2_hist_10_epochs.png}%
\includegraphics[width=0.35 \textwidth]{08_fmnist_cnn_vanilla_l2_hist_20_epochs.png}%
\includegraphics[width=0.35 \textwidth]{09_fmnist_cnn_vanilla_l2_hist_40_epochs.png}%
}%
\centerline{%
\includegraphics[width=0.35 \textwidth]{10_fmnist_cnn_adversarial_l2_hist_10_epochs.png}%
\includegraphics[width=0.35 \textwidth]{11_fmnist_cnn_adversarial_l2_hist_20_epochs.png}%
\includegraphics[width=0.35 \textwidth]{12_fmnist_cnn_adversarial_l2_hist_40_epochs.png}%
}%
\end{center}
\caption{Sample $L^2$-margin distributions for $1000$ training images and the training epochs $10, 20$ and $40$: (1) First row: $d_2(x)$ distribution for the classically trained CNN on \textsc{Fashion-Mnist}; (2) Second row: $d_2(x)$ distribution for the adversarially trained CNN on \textsc{Fashion-Mnist}.}
\label{fig: distributions}
\end{figure*}
\begin{figure*}[t]
\begin{center}
\centerline{%
\includegraphics[width=0.5 \textwidth]{13_fmnist_cnn_vanilla_l2_dists_split_correct_incorrect.png}%
\includegraphics[width=0.5 \textwidth]{14_fmnist_cnn_vanilla_linf_dists_split_correct_incorrect.png}%
}%
\end{center}
\caption{Development of average margins and standard errors of the classically trained CNN on \textsc{Fashion-Mnist}. For the calculation of these values we used $1000$ randomly picked images of the test set: (1) Left plot: development of $d_2^{avg}$; (2) Right plot: development of $d_{\infty}^{avg}$.}
\label{fig: average_margin_split}
\end{figure*}
Up until now, we have primarily focused on average decision boundary margins and standard errors during the training process. These aggregated statistical values only provide a limited understanding of the distances of training and test images to the decision boundary. We, therefore, analyze the distribution of $d_2(x)$ and $d_{\infty}(x)$ for images $x$ from the training as well as the test set at different training epochs. We again consider the classically trained models and the models trained with PGD adversarial training. Figure \ref{fig: distributions} shows sample distributions of $d_2(x)$ for the classically and the adversarially trained CNN on \textsc{Fashion-Mnist}. \newline
Additionally, we separate correctly and incorrectly classified images in these distribution graphs, which directly brings us to one apparent realization. The images that were assigned to the wrong class by the deep neural network tend to lie significantly closer to the decision boundary than correctly classified ones. The calculated margins also have a smaller empirical variance (or standard error), hence they are rather concentrated around the mean of the margin approximates. \newline
The distribution of correctly classified images and the distribution of incorrectly classified images wander more and more in the direction of the origin during standard training of a model. This underlines again that the decision boundary moves closer to natural images over the training process (see also: Figure \ref{fig: average_margin_split}). As a consequence, images which were wrongly classified by a fully trained model will be comparably easy to push over the decision boundary via a small perturbation. In the future, this observation might be useful for the detection of misclassified, natural images. \newline
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.95\linewidth]{15_fmnist_cnn_vanilla_train_correct_incorrect_sankey.png}
\end{center}
\caption{This Sankey plot visualizes the movement of natural images between correct and incorrect classification for the classically trained CNN over the different epochs of training. The plot is based on the whole \textsc{Fashion-Mnist} training set.}
\label{fig: sankey}
\end{figure}
A possible explanation for this large margin difference between correctly and incorrectly classified images is the fact that the cross-entropy loss punishes wrong decisions during training. Thus, the decision boundary is pushed towards these mistakes, in order to turn these wrong decisions into correctly classified images. This directly implies that the distance to the decision boundary of these wrongly classified images decreases from one epoch to the next, i.e.\ the general movement of the decision boundary towards natural data points is reinforced. This hypothesis is also supported by the given Sankey diagrams in Figure \ref{fig: sankey}. Here, we observe that the majority of incorrectly classified images in one training epoch has also been incorrect in the prior epoch. Hence, the average margin values and the margin distributions of the wrongly classified images of two adjacent epochs rely largely on the same images and thus, allow the statement that the decision boundary is pushed towards these images. \newline
The Sankey plots also show that the decision boundary eventually reaches large parts of the initially misclassified images. These images become correctly classified images, which also results in a smaller training and test error rate. This automatically leads to new correctly classified images with a low distance to the decision boundary in every training epoch. Since the decision boundary has just reached these images and is thus still very close, they have a negative impact on the average margin of the correctly classified images. However, it should be noted that the different distribution graphs also indicate a general trend of the correctly classified images towards smaller margins, which can not all be attributed to the switch of incorrectly classified images to correctly classified images. \newline
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.95\linewidth]{16_fmnist_cnn_adversarial_linf_dists_split_correct_incorrect.png}
\end{center}
\caption{Development of $\ell_{\infty}$ average margin $d_{\infty}^{avg}$ and standard error $d_{\infty}^{se}$ of adversarially trained CNN on \textsc{Fashion-Mnist}. For the calculation of these values we used $1000$ randomly picked images of the test set.}
\label{fig: average_margin_split_adversarial}
\end{figure}
It is not surprising that adversarially trained models still show the same phenomenon of decreasing margins for incorrectly classified images, while at the same time being able to stabilize the average margins of the training and test set. In Figure \ref{fig: average_margin_split_adversarial} we see that the average $\ell_{\infty}$-margins for the incorrectly classified images have a clear downward trend after the first training epochs. It becomes obvious that the decision boundary of an adversarially trained classifier tries to balance two goals encoded in the loss function and training procedure. It tries to achieve a high accuracy on the training set and simultaneously, to be robust in the neighborhood of training images. The decreasing margins of wrongly classified images resemble the attempt to increase accuracy, while the increasing, or at least stable, margins of correctly classified images show the desired increase in robustness.
\section{Future Work \& Conclusion}
In this empirical study, we have seen that the decision boundary of a state-of-the-art deep neural network moves closer to training and test images during training. The movement of the decision boundary even continues in late phases of training, although the test and training accuracy barely changes at this point. This leads to the conclusion that fully trained models are susceptible to adversarial perturbations as well as general corruption noise. \newline
On the other hand, adversarial training has the potential to prevent this undesired downward trend of the distances to the decision boundary. In general, the average margin of training and test data is at a higher level in this adapted training procedure. Besides, we even noticed a transfer of robustness between the $\ell_2$-norm and the $\ell_{\infty}$-norm for the \textsc{Mnist} and \textsc{Fashion-Mnist} task. \newline
Furthermore, for trained classifiers there exists a significant difference between correctly and incorrectly classified images concerning their distances to the decision boundary. Incorrectly classified images lie a lot closer to the decision boundary than correctly classified images, and here the decision boundary comes closer to these images over training, too. This observation remains present for both standard and adversarial training. \newline
This empirical study contributes to a better understanding of the decision boundary, but there are still a lot of open research questions - even related to the above results. Our observations put the widely studied problem of deep neural networks being susceptible to adversarial examples into a different light. The vulnerability itself indicates that the decision boundary of neural networks is close to most natural images after training. However, we have found that this property of neural networks is not predetermined by their initialization or architecture. Rather, this insufficiency is created during training. Therefore, from our perspective, it seems most promising to further study the influence of loss functions and training procedures on the margins. \newline
Nevertheless, in future experiments one has to analyze whether the given observations hold for more varying network architectures and complex computer vision tasks. It is also crucial to double check the exactness of the DeepFool margin approximates. At this point, one could make use of other strong adversarial attacks, or even apply formal verification techniques to derive lower bounds for the margins of data points. These further investigations will hopefully help us to identify provable explanations for the observed phenomena. \newline
Recently, poor calibration has also been proposed as a potential contributing factor to the robustness issues of ML models. It has been argued that, due to bad uncertainty estimates, the network is not able to identify shifts in the data distribution, that is, out-of-sample instances \cite{bradshaw2017adversarial, smith2018understanding}. This claim is supported by \cite{li2017dropout} that find better calibrated models are able to detect certain classes of adversarial examples. Therefore, it is interesting to also explore the connection between calibration and the development of the distance to the decision boundary over training in more detail. In our third observation we have already seen that the distance to the decision boundary can be a helpful uncertainty metric. There is a significant difference between correctly and incorrectly classified images concerning their margins, although the network usually assigns high confidence scores to both classes of images - which is a clear sign of poor calibration. \newline
In general, we hope that more researchers pick up on the idea to track changes of the decision boundary throughout training, instead of solely concentrating on fully trained networks. This adds a new dimension to the robustness evaluation of a classifier and gives us a better chance to detect different causes of insufficient adversarial and corruption robustness.
{\small
\bibliographystyle{ieee}
|
1,116,691,500,750 | arxiv | \section{Introduction}
Based on simple energetic considerations regarding the energy density of cosmic rays and the energy release per supernova explosion, SNRs have long been thought to be sources of galactic cosmic rays. This presumption is supported by numerous detections of non-thermal emission in the radio and X-ray band \citep[e.g.,][]{1995Natur.378..255K,1999ApJ...525..357S,2001ApJ...548..814S,2000PASJ...52.1157B} observed from the direction of known SNRs and interpreted to be synchrotron radiation of relativistic electrons with energies up to $100\,$TeV.
High-resolution observations of young SNRs performed with the \emph{Chandra} satellite show that this emission of non-thermal radiation is concentrated in narrow regions on the limbs \citep{2003ApJ...584..758V,2005ApJ...621..793B}. These regions of increased synchrotron emissivity close to the forward shock are called filaments and demonstrate the presence of high-energy electrons around their acceleration sites.
The most plausible process for the acceleration of electrons is diffusive shock acceleration (DSA), which leads to a power-law distribution of particles \citep[e.g.,][]{1978MNRAS.182..147B,1978ApJ...221L..29B,1983RPPh...46..973D}. Although no clear evidence for relativistic-ion acceleration exists at shocks, the DSA-mechanism is also considered to be responsible for the acceleration of cosmic-ray nuclei, as indicated by observations of non-relativistic ion acceleration at solar-wind shocks driven by coronal mass ejections \citep{2011ApJ...735....7R}. However, many details of the DSA are still vague such as the maximum energy of particles, the role of the magnetic field, and how the particles are injected into the acceleration process (also referred to as the injection problem). Apparently, the investigation of the properties of the non-thermal filaments may provide key information for a better understanding of the DSA-mechanism. In particular, knowing the magnetic-field strength gives constraints on the maximum particle energy achievable in the acceleration process, which can help answering the question whether SNRs can accelerate particles to energies above the knee in the cosmic-ray spectrum.
Accurate analysis of several SNRs shows that the filamentary structures are very thin compared with the radii of the remnants. This limitation of the filament widths is associated with a rapid decrease of the synchrotron emissivity that can be explained by energy losses of the electrons in a locally enhanced magnetic field. A number of authors have used that model to constrain the magnetic-field strength, the degree of turbulence and the obliquity \citep[e.g.,][]{2003ApJ...589..827B,2006A&A...453..387P,2010ApJ...714..396A}. \citet{2010ApJ...714..396A}, for instance, have investigated several filaments of the remnant of Cas A and found that the magnetic fields of the filaments are highly turbulent and nearly perpendicular to the shock normal. Another important result of this and other studies is an estimate of the downstream magnetic-field strength that is higher than simple shock compression would suggest. Such observations indicate an additional amplification of the magnetic field in the shock region of the SNRs. A possible amplification process could be a streaming instability in the upstream region as proposed by \citet{2000MNRAS.314...65L} and \citet{2004MNRAS.353..550B}, or the effects of preexisting turbulent density fluctuations on the propagating shock front \citep{2007ApJ...663L..41G}.
Besides energy losses, also the magnetic field itself can limit the filament widths. Based on the turbulence relaxation downstream of the forward shock and neglecting any amplification process, \citet{2005ApJ...626L.101P} have calculated that the turbulent magnetic field downstream can decay exponentially on a damping-length scale $l_{\mbox{\tiny d}}=10^{16}-10^{17}\,$cm that is small enough to produce the narrow observable filaments. Furthermore, from observations of the post-shock steepening of the synchrotron spectrum in Tycho's SNR it can be seen that also the damping of the magnetic field fairly well describes the corresponding X-ray data \citep{2007ApJ...665..315C}, and thus, may appear within the filaments.
Since the magnetic field controls the radiative cooling of electrons, high magnetic fields lead to strong cooling, and too few high-energy electrons remain capable of producing the gamma-ray emission, that is observed from regions near the edges of numerous SNRs \citep[e.g.,][]{2007ApJ...661..236A,2010A&A...516A..62A,2011ApJ...734...28A}. Any gamma rays observed in such a case are likely to be hadronic in origin. Weak cooling leads to a large number of high-energy electrons and the possibility of gamma-ray emission through inverse Compton or bremsstrahlung processes. All these implications of the magnetic-field structure on the particle acceleration, gamma-ray emission and magnetic-field amplification make it necessary to understand the non-thermal filaments in detail.
In this paper we investigate the properties of the filaments for both cases, filaments limited by electron energy losses or by damping of the magnetic field. For that purpose, using observational values of some characteristical SNR parameter, we calculate the X-ray emission of the filaments. The resulting filament profiles then allow us to make specific predictions regarding the magnetic-field strength. We additionally calculate the total non-thermal emission, which shall be referred to as "plateau", and compare their spectra with those of the filaments. It should be noted that in our models we only consider non-thermal synchrotron emission and restrict ourselves to the evolution of relativistic electrons in the downstream region. Furthermore, we assume the electrons to be already accelerated at the shock front and treat our problem to be independent of the acceleration process. We also do not consider any electron propagation into the upstream region and simplify the SNRs to be spherical objects of constant downstream-velocity profile. Recent hydrodynamical simulations suggest that this oversimplification of a constant velocity is an acceptable approximation only for SNRs of an age of less than several hundred years \citep{2012APh....35..300T}, implying that our models are restricted to SNRs being in the adiabatic phase and just entering the Sedov phase, respectively.
\section{Modelling the filaments}
We calculate the X-ray intensity as a function of the projected radius, $r_{\mbox{\tiny p}}$. It is an integral over the synchrotron emission coefficient, $j_\nu$, along the line of sight, $I_\nu=\int_{-\infty}^{\infty} j_\nu \, \mbox{d}y$. Using $r^2=y^2+r_{\mbox{\tiny p}}^2$ and taking into account that only emission originating inside the SNR contributes, the X-ray intensity can be written as
\begin{equation} \label{eqn: intensity distribution}
I_{\nu}(r_{\mbox{\tiny p}})=2\int_{r_{\mbox{\tiny p}}}^{r_{\mbox{\tiny s}}} \, \frac{j_{\nu}(r)}{\sqrt{1-\frac{r_{\mbox{\tiny p}}^2}{r^2}}} \, \mbox{d}r \; ,
\end{equation}
where $r$ and $r_{\textrm{\tiny s}}$ denote the positions inside the SNR and the radius of the SNR, respectively. Then, obtaining the corresponding spectrum of the filaments just involves an integration over the X-ray emission along the projected radius, whereas the spectrum of the plateau emission can be calculated as a volume integral over the synchrotron emission coefficient. In both cases, the isotropic synchrotron emission coefficient is needed, given by
\begin{equation}
j_{\nu}(r)=\frac{1}{4\pi}\int_{}^{\infty} N(r,E)P_{\nu}(r,E)\, \mbox{d}E \; ,
\end{equation}
where $N(r,E)$ and $P_{\nu}(r,E)$ are the isotropic differential electron number density and the spectral emissivity per electron, respectively. Thus, we need to derive the appropriate electron distribution within the filaments.
\subsection{The electron distribution}
In the following, we derive the electron distribution that is necessary to calculate the synchrotron emissivity. To do this, a transport equation describing the dynamics of a distribution of relativistic electrons affected by advection, diffusion and energy losses needs to be solved. According to simulations described in \citet{2012APh....35..300T}, we can approximate the advection velocity of young SNRs to be constant downstream of the shock, implying that energy losses due to adiabatic deceleration can be neglected. Note that the non-thermal emission come from a thin spherical shell near the edges of the SNRs. If we restrict our treatment to a region near the shock that is crossed by the advection flow on a timescale short compared with the age of the SNR, then we approximate the electron distribution with a one-dimensional steady-state solution.
It is convenient to introduce a comoving spatial coordinate, $z=r_{\mbox{\tiny s}}-r$, where $z=0$ marks the position of the shock front at all times. Hence, restricting ourselves on the downstream region, the one-dimensional transport equation for the isotropic differential number density, $N=N(z,E)$, reads as follows:
\begin{equation} \label{eqn:transport equation}
\frac{\partial}{\partial z}\left[D(z,E)\frac{\partial N}{\partial z}\right]-v\frac{\partial N}{\partial z}-\frac{\partial}{\partial E}\left[\beta(z,E)N\right]+Q(z,E)=0 \,.
\end{equation}
In this equation $v$ denotes the constant advection velocity of the electrons relative to the forward shock, $D(z,E)$ is the diffusion coefficient, $\beta(z,E)=\mbox{d}E/\mbox{d}t$ represents the electron energy loss due to the emission of radiation, and $Q(z,E)$ is the source term describing the injection of accelerated electrons into the propagation process.
Since the electrons are likely accelerated by the DSA-mechanism at the forward shock ($z=0$), we assume the electrons to be injected with a power-law dependence $E^{-s}$, where $s$ is the injection index. Hence, the source term reads
\begin{equation} \label{eqn:source term}
Q(z,E)=q_0E^{-s}\exp\left(-\frac{E}{E_{\mbox{\tiny cut}}}\right)\delta(z) \;,
\end{equation}
and includes a cut-off at energy $E_{\mbox{\tiny cut}}$, because the maximum possible energy can be limited by either the finite acceleration time of the SNR \citep{1991MNRAS.251..340D} or energy losses \citep{1984A&A...137..185W}.
Eq. (\ref{eqn:transport equation}) can be solved using Green's method, implying that the solution can be written in terms of Green's function,
\begin{equation} \label{eqn:solution}
N(z,E)=\int_0^{\infty}\mbox{d}z'\int_{}^{\infty}\mbox{d}E' \, g(z,z';E,E')Q(z',E') \;,
\end{equation}
where Green's function $g=g(z,z';E,E')$ satisfies
\begin{equation} \label{eqn:transport equation with Green's function}
\frac{\partial}{\partial z}\left[D(z,E)\frac{\partial g}{\partial z}\right]-v\frac{\partial g}{\partial z}-\frac{\partial}{\partial E}\left[\beta(z,E)g\right]=-\delta(z-z')\delta(E-E') \;.
\end{equation}
Assuming the diffusion coefficient and the energy losses to be separable in a spatial and in an energetic part,
\begin{equation}
D(z,E)=d(z)D(E) \; ,
\end{equation}
\begin{equation}
\beta(z,E)=-b(z)B(E) \; ,
\end{equation}
and that the spatial dependent terms are inversely proportional to each other,
\begin{equation} \label{eqn:alpha}
d(z)b(z)=\alpha=\mbox{const.} \; ,
\end{equation}
as well as introducing the substitutions
\begin{equation} \label{eqn: substitution Green's function}
g(z,z';E,E')=\frac{G(z,z';E,E')}{B(E)} \; ,
\end{equation}
\begin{equation} \label{eqn: substitution x}
x(z)=\int_0^z \frac{\mbox{d}y}{d(y)} \; ,
\end{equation}
\begin{equation} \label{eqn:substitution lamba}
\lambda(E)=\frac{1}{\alpha}\int_{E}^{\infty} \frac{\mbox{d}u}{B(u)} \; ,
\end{equation}
an analytical solution to Eq. (\ref{eqn:transport equation with Green's function}) can be found in the literature \citep[][see their Eq. (A20)]{1980ApJ...239.1089L} and reads
\begin{eqnarray} \label{eqn:Green's function}
G(x,x';\lambda,\lambda')&=&\frac{\Theta(\lambda-\lambda')}{2\alpha\sqrt{\pi}}\sqrt{\frac{1}{\int_{\lambda'}^{\lambda}\, D(t)\,\mbox{d}t}} \nonumber \\
&& \times \exp\left\{-\frac{\left[v(\lambda-\lambda')+x'-x\right]^2}{4\int_{\lambda'}^{\lambda}\, D(t)\, \mbox{d}t}\right\} \; ,
\end{eqnarray}
where $\Theta(\lambda-\lambda')$ is the step function and $\lambda'=\lambda(E')$. Note, this analytical solution is valid only if Eq. (\ref{eqn:alpha}) applies.
In general, the diffusion coefficient is an unknown parameter. But it is often assumed that the diffusion proceeds in the so-called Bohm-regime. Hence, the diffusion coefficient can be written in the extreme-relativistic limit $E\gg mc^2$ as $D=\eta r_{\mbox{\tiny L}}c/3$, where $r_{\mbox{\tiny L}}=E/(eB)$ and $\eta\geq 1$ are the gyroradius and gyrofactor, respectively. Here, $m$ is the mass of the electron, $e$ is its charge, $c$ is the speed of light, and $B$ is the magnetic-field strength. Therefore, we may write $D(E)=D_0E$, where $D_0=\eta c/(3eB)$. Additionally, we assume the emission of synchrotron radiation to be the main energy-loss process that is proportional to the square of the electron energy, $B(E)=b_0E^2$, where $b_0=4e^4B^2/(9m^4c^7)$. Using these assumptions, Eq. (\ref{eqn:substitution lamba}) rewrites as $$\lambda(E)=\frac{1}{\alpha}\int_E^{\infty}\frac{\mbox{d}u}{b_0u^2}=\frac{1}{\alpha b_0 E}\, ,$$ so that $$D(E)=\frac{D_0}{\alpha b_0 \lambda}$$ and hence, $$\int_{\lambda'}^{\lambda}\, D(t) \, \mbox{d}t=\frac{D_0}{\alpha b_0}\ln\left(\frac{\lambda}{\lambda'}\right)=\frac{D_0}{\alpha b_0}\ln\left(\frac{E'}{E}\right) \; .$$ According to Eq. (\ref{eqn: substitution Green's function}) and Eq. (\ref{eqn:Green's function}), Green's function then reduces to
\begin{eqnarray}
g(x,x';E,E')&=&\frac{\theta(E'-E)}{2\sqrt{\pi\alpha b_0 D_{0}}E^{2}}\sqrt{\frac{1}{\ln\left(\frac{E'}{E}\right)}} \nonumber \\
&& \times \exp\left\{ -\frac{\left[\frac{v}{\alpha b_0}\left(\frac{1}{E}-\frac{1}{E'}\right)+x'-x\right]^{2}}{\frac{4D_{0}}{\alpha b_0}\ln\left(\frac{E'}{E}\right)}\right\} \, .
\end{eqnarray}
Now, the electron distribution can be easily determined using Eq. (\ref{eqn:solution}) and the appropriate source distribution (\ref{eqn:source term}), so that
\begin{eqnarray}
N(x(z),E) &=& \int_{x(0)}^{^{x(\infty)}}\mbox{d}x'\int_{}^{\infty} \mbox{d}E' \, \frac{q_{0}\theta(E'-E)}{2\sqrt{\pi\alpha b_0 D_{0}}}\,E^{-2}E'^{-s}\sqrt{\frac{1}{\ln\left(\frac{E'}{E}\right)}} \nonumber \\
&& \times \exp\left\{ -\frac{E'}{E_{\mbox{\tiny cut}}}-\frac{\left[\frac{v}{\alpha b_0}\left(\frac{1}{E}-\frac{1}{E'}\right)+x'-x\right]^{2}}{\frac{4D_{0}}{\alpha b_0}\ln\left(\frac{E'}{E}\right)}\right\} \delta(x') \nonumber \\
&=& \frac{q_{0}}{2\sqrt{\pi\alpha b_0 D_{0}}}\,E^{-2}\int_{E}^{\infty}\frac{E'^{-s}}{\sqrt{\ln\left(\frac{E'}{E}\right)}} \nonumber \\
& & \times \exp\left\{-\frac{E'}{E_{\mbox{\tiny cut}}}-\frac{\left[\frac{v}{\alpha b_0 E}\left(1-\frac{E}{E'}\right)-x(z)\right]^{2}}{\frac{4D_{0}}{\alpha b_0}\ln\left(\frac{E'}{E}\right)}\right\} \mbox{d}E' \; .
\end{eqnarray}
Substituting $n=E'/E$, we can rewrite the desired result as
\begin{eqnarray}
N(x(z),E) &=& \frac{q_{0}}{2\sqrt{\pi\alpha b_0 D_{0}}}\,E^{-(s+1)}\int_{1}^{\infty}\frac{n^{-s}}{\sqrt{\ln(n)}} \nonumber \\
& & \times \exp\left\{-\frac{nE}{E_{\mbox{\tiny cut}}}-\frac{\left[\frac{v}{\alpha b_0 E}\left(1-\frac{1}{n}\right)-x(z)\right]^{2}}{\frac{4D_{0}}{\alpha b_0}\ln(n)}\right\} \mbox{d}n \; . \label{eqn:electron distribution}
\end{eqnarray}
\subsection{Parameter used in the models}
The models we use are based on different assumptions on the magnetic field. In the first model, which shall be referred to as "energy-loss model", we assume the magnetic field to be spatially constant, whereas the second one, which is referred to as "magnetic-field damping model", includes the damping of magnetic turbulence and assumes a spatially-dependent magnetic-field strength described by a profile following the relation
\begin{equation} \label{eqn:magnetic field}
B(z)=B_{\mbox{\tiny min}}+(B_{\mbox{\tiny max}}-B_{\mbox{\tiny min}}){\mbox{}}\exp\left(-\frac{z}{l_{\mbox{\tiny d}}}\right) \; ,
\end{equation}
where $l_{\mbox{\tiny d}}$ is the damping length and $z\geq 0$. Here, we choose the minimum value of the magnetic field to be similar to that of the interstellar medium, $B_{\mbox{\tiny min}}=10~\mu$G, whereas the maximum value, $B_{\mbox{\tiny max}}$, corresponds to the field at the shock. It should be noted that Eq. (\ref{eqn:magnetic field}) describes the averaged amplitude of the magnetic field in a given volume. Hence, we do not make any assumption on the magnetic-field direction and do not distinguish between parallel and transverse diffusion, as is done in detailed calculations of diffusion coefficients \citep[e.g.,][]{2006A&A...453..193M}.
According to the Rankine-Hugoniot conditions, the downstream advection velocity can be expressed by the shock velocity, $v_{\mbox{\tiny s}}$, through $v=v_{\mbox{\tiny s}}/4$, if we consider strong shocks with a high Mach number and a monatomic gas with adiabatic index of 5/3, leading to a shock compression ratio of 4. Thus, we neglect any non-linear effects expected to occur with efficient particle acceleration, which would modify the shock \citep{2004A&A...413..189E}. Furthermore, the filament width, $w$, is defined as the length, at which the intensity described by Eq. (\ref{eqn: intensity distribution}) is reduced by a factor $1/e$ of its maximum.
To have a realistic model, the values chosen for the shock velocity, filament width, and radius are based on reference values of real SNRs. In our case, we consider the young remnants of the historical supernovae SN 1006, Cas A, Tycho and Kepler. It should be noted that the real filament widths found in the literature have been measured in a certain X-ray band and not for an individual X-ray energy. However, as is shown in the calculation done in Sect. \ref{sect:results}, the shape of the filament profiles depends on the X-ray energy. Nevertheless, we relate the observational value of the width to a X-ray energy of 5 keV, since it is an energy, at which no significant contribution from thermal emission is expected.
Additionally, we treat the cut-off energy to be the maximum electron energy that can be achieved in the acceleration process. Unlike cosmic-ray nuclei whose energy is probably age-limited due to the finite acceleration time available, we assume the maximum electron energy to be loss-limited, since the electrons experience synchrotron losses during their acceleration. By equating the acceleration time scale to the synchrotron loss time, it is possible to derive an expression for the maximum electron energy in terms of the downstream magnetic-field strength and shock velocity \citep{2006A&A...453..387P}:
\begin{equation}\label{eqn:cutoff energy}
E_{\mbox{\tiny cut}}\equiv E_{\mbox{\tiny max}}\simeq (8.3~\mbox{TeV})\,\eta^{-1/2}\left(\frac{B(z=0)}{100~\mu\mbox{G}}\right)^{-1/2}\left(\frac{v_{\mbox{\tiny s}}}{1000~\mbox{km/s}}\right)\;.
\end{equation}
Since different mechanism can account for magnetic-field amplification \citep{2000MNRAS.314...65L, 2007ApJ...663L..41G}, the structure of the magnetic field is generally unknown within the shock region. We therefore simplify the problem by making the assumption that only the magnetic-field strength at the shock determines the maximum electron energy given by Eq. (\ref{eqn:cutoff energy}). In addition, we have taken a shock compression ratio of 4.
At last, we assume the injection index to be $s=2$, which results from the Rankine-Hugoniot conditions for strong shocks \citep{1978MNRAS.182..147B}, as well as Bohm diffusion ($\eta=1$), which implies the smallest possible value of the diffusion coefficient, as the mean free path of the particle is equal to the gyroradius.
All parameter used are summarized in Table \ref{tab:parameters}. Note, that it is possible that all four SNRs could exhibit similar shock velocities. To take the uncertainties of this quantity into account, we perform the calculation of SN 1006 and Kepler for two different shock velocities.
\begin{table*}
\caption{Parameters used to derive the filament profiles and spectra. Furthermore, $s=2$ and Bohm diffusion are assumed in all SNRs.} \label{tab:parameters}
\centering
\begin{tabular}{c c c c c c c c c c c c}
\hline \hline
SNR & Age & \multicolumn{2}{c}{Distance} & \multicolumn{2}{c}{$r_{\mbox{\tiny s}}$} & \multicolumn{2}{c}{$v_{\mbox{\tiny s}}$} & \multicolumn{3}{c}{$w$} & $B_{\mbox{\tiny min}}$ \\ \cmidrule(lr){3-4} \cmidrule(lr){5-6} \cmidrule(lr){7-8} \cmidrule(lr){9-11}
& [yr] & [kpc] & Reference & [arcmin]\tablefootmark{a} & [pc]\tablefootmark{b} & [km\,$\mbox{s}^{-1}$] & Reference & [arcsec] & [pc]\tablefootmark{b} & Reference & [$\mu$G]\tablefootmark{c} \\
\hline
SN 1006\tablefootmark{d} & 1000 & 2.2 & 1 & 15 & 10 & 4900 & 2 & 20 & 0.2 & 3 & 10 \\
& & & & & & (2900) & & & & & \\
Cas A & 330\tablefootmark{e} & 3.4 & 4 & 2.5 & 2.5 & 5200 & 5 & 1.5 & 0.03 & 6, 7 & 10 \\
Tycho & 440 & 2.5 & 8 & 4 & 3 & 5000 & 9 & 5 & 0.06 & 6 & 10 \\
Kepler\tablefootmark{f} & 410 & 4.8 & 10 & 1.5 & 2 & 5040 & 11 & 3.5 & 0.08 & 6 & 10 \\
& & (6.4) & & & (3) & (6720) & & & (0.11) & & \\
\hline
\end{tabular}
\tablefoot{
\tablefoottext{a}{Taken from \citet{2009BASI...37...45G}.}
\tablefoottext{b}{Directly inferred using the distance and the appropriate quantity given in angular units.}
\tablefoottext{c}{Only used in the magnetic-field damping model.}
\tablefoottext{d}{The shock velocity in the northwestern limb (value in brackets) is used, as well as in the northeastern limb, where strong electron acceleration appears to occur.}
\tablefoottext{e}{The supernova explosion may have been observed in 1680 \citep{1980JHA....11....1A}. Otherwise, the detection of radioactive $^{44}$Ti \citep{1997NuPhA.621...83H} and the analysis of the dynamics of this remnant \citep{2006ApJ...645..283F} also suggest an explosion date in the late 17th century.}
\tablefoottext{f}{Distance to this remnant is uncertain. Calculation is performed for the lower and upper limit (values in brackets) to the distance.}
}
\tablebib{
(1)~\citet{2003ApJ...585..324W}; (2)~\citet{2002ApJ...572..888G} + \citet{2009ApJ...692L.105K}; (3)~\citet{2003ApJ...589..827B}; (4)~\citet{1995ApJ...440..706R}; (5)~\citet{1998A&A...339..201V}; (6)~\citet{2005ApJ...621..793B}; (7)~\citet{2010ApJ...714..396A}; (8)~\citet{2011ApJ...729L..15T}; (9)~\citet{2000ApJ...545L..53H} + distance; (10)~\citet{1999AJ....118..926R}; (11)~\citet{2008ApJ...689..231V} + distance
}
\end{table*}
\section{Results} \label{sect:results}
\subsection{Energy-loss model}
In this model we treat the magnetic field to be constant, $B(z)=B=\mbox{const.}$, implying no spatial dependence of the energy-loss term, $b(z)=1$. Using $d(z)=\alpha/b(z)$, as well as Eq. (\ref{eqn: substitution x}), the spatial coordinate $x(z)$ then scales as $$x(z(r))=\frac{z}{\alpha}=\frac{r_{\mbox{\tiny s}}-r}{\alpha} \, .$$
Using the parameters given in Table \ref{tab:parameters}, we calculate the X-ray intensity as a function of the projected radius according to Eq. (\ref{eqn: intensity distribution}). Here, the magnetic-field strength is a free parameter whose value can be chosen so that the filament widths match those found in the observations. In addition, we also determine the cut-off energy of the electron spectrum, which is connected to the magnetic field through Eq. (\ref{eqn:cutoff energy}).
Reproducing the filament width for each SNR at a photon energy of 5~keV determines the magnetic-field strengths as given in Table \ref{tab:Constraints energy-loss model}. For our examples the downstream magnetic-field strength ranges from about $100~\mu$G up to about $500~\mu$G. Remnants with narrower filaments exhibit a higher downstream magnetic field. These magnetic fields then imply cut-off energies of the electron spectra in the energy range between 19~TeV and 37~TeV. The order of magnitude of the cut-off energies is in agreement with the results obtained from spectral modeling of the radio-to-X-ray spectra of young SNRs whose cut-off energies of their electron distribution must be in the TeV-band, since the cut-off frequencies are generally found in the X-ray band \citep{1999ApJ...525..368R}.
In Fig. (\ref{fig: filament profile loss}) we illustrate the profiles of the filaments for four different photon energies calculated with the parameters of Tycho. To be noted from the figure is a frequency dependence of the filament widths, which can be explained by the energy loss of the electrons in a constant magnetic field and by the advection process. The advection length represents the distance covered by the electrons within the synchrotron loss time, $\tau_{\mbox{\tiny syn}}=E/|\frac{\mbox{\tiny d}E}{\mbox{\tiny d}t}|$, and is given by
\begin{eqnarray}
l_{\mbox{\tiny ad}}&=&v\tau_{\mbox{\tiny syn}}=\frac{v_{\mbox{\tiny s}}}{4}\frac{9m^4c^7}{4e^4B^2E} \nonumber \\
&\simeq& (2\times 10^{-2}~\mbox{pc})\left(\frac{B}{200~\mu\mbox{G}}\right)^{-2} \left(\frac{v_{\mbox{\tiny s}}}{5000~\mbox{km}\,\mbox{s}^{-1}}\right) \left(\frac{E}{20~\mbox{TeV}}\right)^{-1}\, . \label{eqn:advection length}
\end{eqnarray}
Synchrotron radiation usually provides a continuum around a characteristic synchrotron frequency,
\begin{eqnarray}
\nu_{\mbox{\tiny c}}&=&\frac{3\nu_{\mbox{\tiny L}}\gamma^2}{2}=\frac{3eBE^2}{4\pi m^3c^5} \nonumber \\
&\simeq & (1.3\times 10^{18}~\mbox{Hz}) \left(\frac{B}{200~\mu\mbox{G}}\right)\left(\frac{E}{20~\mbox{TeV}}\right)^2\;, \label{eqn: synchrotron frequency}
\end{eqnarray}
where $\nu_{\mbox{\tiny L}}$ and $\gamma $ are the Larmor frequency and the Lorentz factor of the accelerated electrons, respectively. Hence, the relation $E\propto \nu^{1/2}$. Therefore, a dependence of the width on the frequency of the radiation of the form $l_{\mbox{\tiny ad}}\propto \nu^{-1/2}$ would result. But the electrons are also affected by diffusion. With the Bohm diffusion coefficient, $D=r_{\mbox{\tiny L}}c/3$, one thus obtains for the corresponding diffusion length
\begin{equation} \label{eqn:diffusion length}
l_{\mbox{\tiny diff}}=\sqrt{D\tau_{\mbox{\tiny syn}}}=\sqrt{\frac{3m^4c^8}{4e^5B^3}}\simeq (1.3\times 10^{-2}~\mbox{pc}) \left(\frac{B}{200~\mu\mbox{G}}\right)^{-3/2}\, ,
\end{equation}
which does not depend on the electron energy. Equating the advection and diffusion length, the relation
\begin{equation} \label{eqn: critical energy}
E_{\mbox{\tiny c}}\simeq (31~\mbox{TeV}) \left(\frac{v_{\mbox{\tiny s}}}{5000~\mbox{km}\,\mbox{s}^{-1}}\right)\left(\frac{B}{200~\mu\mbox{G}}\right)^{-1/2}\,
\end{equation}
can be derived. Electrons with energies $E>E_{\mbox{\tiny c}}$ can stream farther from the shock than advection alone would allow. According to Eq. (\ref{eqn: synchrotron frequency}) and Eq.(\ref{eqn: critical energy}), the characteristic synchrotron frequency for the electrons with $E>E_{\mbox{\tiny c}}$ is then higher than
\begin{equation}
\nu_{\mbox{\tiny c}}(E_{\mbox{\tiny c}})\simeq (3.1\times 10^{18}~\mbox{Hz})\left(\frac{v_{\mbox{\tiny s}}}{5000~\mbox{km}\,\mbox{s}^{-1}}\right)^2\;,
\end{equation}
corresponding to a X-ray energy of about 13~keV, if $v_{\mbox{\tiny s}}=5000~$km/s. Hence, the filament profiles at higher photon energies, which need the most energetic electrons, show approximately the same behaviour as can be seen in Fig. (\ref{fig: filament profile loss}) for the 5-keV and 10-keV profile.
In addition, the advection and diffusion length also explain the relation between the filament widths and the corresponding magnetic fields given in Table \ref{tab:Constraints energy-loss model}. Narrower filaments require a faster decrease in the synchrotron emissivity, implying a shorter advection and diffusion length. And according to Eq. (\ref{eqn:advection length}) and Eq. (\ref{eqn:diffusion length}), this is given for higher magnetic-field strengths.
Now, equipped with the X-ray intensity distribution establishing the filament profiles, and the volume emissivity, we calculate the spectra of the filament and plateau for each of our examples. To obtain the filament spectrum, we integrate the intensity along the projected radius from $r_{\mbox{\tiny p}}=r_{\mbox{\tiny s}}$ up to $r_{\mbox{\tiny p}}=r_{\mbox{\tiny s}}-w$. Because we do not know the electron source strength, $q_0$, we are not interested in absolute fluxes. However, we can show the qualitative behaviour described by the appropriate photon spectral indices, which should be sufficient for the comparison. Assuming the photon spectra to show a power-law characteristic, $N_\nu=F_\nu/h\nu\propto \nu^{-\Gamma}$, where $F_\nu$ and $\Gamma$ are the flux and photon spectral index, respectively, we can then describe the spectra between the photon energies $h\nu$ and $h\nu'$ through
\begin{equation} \label{eqn:spectral index}
\Gamma=\frac{\ln\left(\frac{\nu F_{\nu'}}{\nu' F_\nu}\right)}{\ln\left(\frac{\nu}{\nu'}\right)} \; .
\end{equation}
Here, we also want to calculate the differences between the photon spectral index of the filament spectrum, $\Gamma_{\mbox{\tiny f}}$, and that of the plateau emission, $\Gamma_{\mbox{\tiny p}}$, which are given at three different photon energies, $E_\nu$, in Table \ref{tab:Constraints energy-loss model}. Additionally, in Fig. (\ref{fig: indices loss}) we show the photon spectral indices calculated with the parameters of Tycho.
As can be seen from Table \ref{tab:Constraints energy-loss model}, as well as from Fig. (\ref{fig: indices loss}), the spectra at higher photon energies rarely differ significantly. The plateau shows a steeper spectrum at lower photon energies. Up to a photon energy of 1~keV the difference between the indices of filament and plateau is in the range $0.05-0.34$, whereas at energies higher than 1~keV the difference is always smaller than 0.1. This property can be explained by the effective radiation of the energetic electrons in the enhanced magnetic field. Accordingly, the most energetic electrons lose all of their energy inside the filaments, implying that the regions farther from the shock provide almost no contributions to the total emission of hard X-rays, so that both filament and plateau show nearly the same behaviour.
In Table \ref{tab:Constraints energy-loss model} we also show the indices of the filaments at three different photon energies. One can see that the parameters of Cas A, Tycho, Kepler and SN 1006 lead to the same spectral behaviour, if their shock velocities are similar. However, using the shock velocities measured in the northwestern limb of SN 1006 and resulting from the upper limit to the distance to Kepler, it turns out that the filament spectrum is softer and harder, respectively, than in the former case.
\begin{table*}
\caption{Constraints on the downstream magnetic-field strength, cut-off energy of the electron spectrum, photon spectral index of the filament, and the difference between the photon indices of the spectra of filament and plateau at three different photon energies, $E_\nu$, for four young SNRs. These values are calculated using the energy-loss model and the parameters given in Table \ref{tab:parameters}.} \label{tab:Constraints energy-loss model}
\centering
\begin{tabular}{c c c c c c c c c}
\hline \hline
SNR & $B$ & $E_{\mbox{\tiny cut}}$ & \multicolumn{3}{c}{$\Gamma_{\mbox{\tiny f}}$} & \multicolumn{3}{c}{$\Gamma_{\mbox{\tiny p}}-\Gamma_{\mbox{\tiny f}}$} \\ \cmidrule(lr){4-6} \cmidrule(lr){7-9}
& [$\mu$G] & [TeV] & $E_\nu=0.1~$keV & $E_\nu=1~$keV & $E_\nu=10~$keV & $E_\nu=0.1~$keV & $E_\nu=1~$keV & $E_\nu=10~$keV \\
\hline
SN 1006\tablefootmark{a} & 130 & 36 & 1.81 & 2.33 & 2.71 & 0.31 & 0.07 & 0.02 \\
& (110) & (23) & (1.99) & (2.49) & (2.94) & (0.25) & (0.05) & (0.01) \\
Cas A & 520 & 19 & 1.83 & 2.33 & 2.70 & 0.31 & 0.06 & 0.01 \\
Tycho & 310 & 24 & 1.82 & 2.33 & 2.71 & 0.32 & 0.06 & 0.01 \\
Kepler\tablefootmark{b} & 250 & 26 & 1.81 & 2.32 & 2.70 & 0.31 & 0.07 & 0.01 \\
& (230) & (37) & (1.75) & (2.23) & (2.61) & (0.34) & (0.08) & (0.02) \\
\hline
\end{tabular}
\tablefoot{
\tablefoottext{a}{The values in brackets were calculated using the shock velocity $v_{\mbox{\tiny s}}=2900~$km/s as measured in the northwestern limb.}
\tablefoottext{b}{The values in brackets were calculated using the upper limit of 6.4~kpc to the distance.}
}
\end{table*}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[angle=0]{Filament_Profile_Loss.eps}}
\caption{Non-thermal X-ray intensity as a function of the projected radius calculated for four different X-ray energies with the parameters of Tycho given in Table \ref{tab:parameters}, as well as $B=310~\mu$G. The forward shock is located at $r_{\mbox{\tiny s}}=3~$pc.}
\label{fig: filament profile loss}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[angle=0]{Indices_Loss.eps}}
\caption{Photon spectral indices of the spectra of filament and plateau using the parameters of Tycho.}
\label{fig: indices loss}
\end{figure}
\subsection{Magnetic-field damping model}
As already mentioned above, in the model of magnetic-field damping we assume the magnetic-field strength to follow a profile described in Eq. (\ref{eqn:magnetic field}), which can also be written as $$B(z)=B_{\mbox{\tiny min}}\left[1+\frac{B_{\mbox{\tiny max}}-B_{\mbox{\tiny min}}}{B_{\mbox{\tiny min}}}\exp\left(-\frac{z}{l_{\mbox{\tiny d}}}\right)\right] \,.$$ The spatial dependence of the energy-loss term then obeys the relation $$b(z)=\left[1+\frac{B_{\mbox{\tiny max}}-B_{\mbox{\tiny min}}}{B_{\mbox{\tiny min}}}\exp\left(-\frac{z}{l_{\mbox{\tiny d}}}\right)\right]^2 \, ,$$ corresponding to a spatial variation of the diffusion coefficient $$d(z)=\alpha \left[1+\frac{B_{\mbox{\tiny max}}-B_{\mbox{\tiny min}}}{B_{\mbox{\tiny min}}}\exp\left(-\frac{z}{l_{\mbox{\tiny d}}}\right)\right]^{-2} \,,$$ because the product $b(z)d(z)$ must be constant, as required by Eq. (\ref{eqn:alpha}). According to Eq. (\ref{eqn: substitution x}), the spatial coordinate $x(z)$ then scales as
\begin{eqnarray}
x(z)&=&\frac{1}{\alpha}\left\{z+\frac{l_{\mbox{\tiny d}}}{2}\frac{(B_{\mbox{\tiny max}}-B_{\mbox{\tiny min}})^2}{B_{\mbox{\tiny min}}^2}\left[1-\exp\left(-\frac{2z}{l_{\mbox{\tiny d}}}\right)\right]\right. \nonumber \\
&& \hspace{2cm}\left. +2l_{\mbox{\tiny d}}\frac{B_{\mbox{\tiny max}}-B_{\mbox{\tiny min}}}{B_{\mbox{\tiny min}}}\left[1-\exp\left(-\frac{z}{l_{\mbox{\tiny d}}}\right)\right] \right\}\,.
\end{eqnarray}
Again, using the parameters given in Table \ref{tab:parameters}, we calculate the X-ray intensity as a function of the projected radius. In this case, the damping length and the maximum field strength, $B_{\mbox{\tiny max}}$, are free parameters that can be chosen so that the filament widths match those observed. According to the calculation of \citet{2005ApJ...626L.101P}, the damping length should be in the range $l_{\mbox{\tiny d}}=10^{16}-10^{17}~$cm ($l_{\mbox{\tiny d}}=0.003-0.03~$pc). To also investigate the influence of the damping length on the results, we perform the calculation using two different values of $l_{\mbox{\tiny d}}$ in each of our examples. Here, the larger value used for $l_{\mbox{\tiny d}}$ may describe weak magnetic-field damping, whereas the smaller one may cause a strong damping of the field. But note that energy losses are still included.
Reproducing the filament width of each SNR at a photon energy of 5~keV requires the maximum field strengths given in Table \ref{tab:Constraints magnetic damping model}. Depending on the damping length, the maximum field strength can be found in the range between 50~$\mu$G and 260~$\mu$G, implying, according to Eq. (\ref{eqn:cutoff energy}), cut-off energies between 27~TeV and 62~TeV. Furthermore, it turns out that an increased damping length requires an increased field strength, because the electrons radiate efficiently in a larger volume, resulting in wider filaments. To retain the observed filament widths, it is then necessary to have a higher magnetic field that, on the other hand, also leads to a smaller cut-off energy of the electron distribution.
If the damping length is too small, the observed filament widths cannot be realized for any maximum field strength. In these cases the intensity first decreases but then increases again even for the 5-keV profile, so that the typical shape of the filament profiles is not given anymore. Therefore, we use damping lengths in the case of strong damping, for which the profiles just still exhibit the typical shape. For instance, using the given parameters of SN 1006, the damping length used in the calculation should not be smaller than 0.02~pc.
The filament profiles calculated with the parameters of Tycho for four different photon energies are illustrated in Fig. (\ref{fig: filament profile damping}). As can be seen from the figure, there is also a dependence of the filaments on the frequency of the X-rays. This dependence is based on the spatial variation of the magnetic-field strength. Only in regions very close to the shock front the electrons radiate in fields of high magnitude, so that even the most energetic of them can emit photons of several keV in energy only in a small volume. The electrons remain energetic when they propagate into the downstream region, where they radiate at lower frequencies. Hence, we expect increased emission of low-energy X-rays in regions farther from the shock. For instance, using the cut-off energy of the electron spectrum in Tycho obtained for the case of strong damping, $E_{\mbox{\tiny cut}}=45~$TeV, one finds that according to Eq. (\ref{eqn: synchrotron frequency}), even the most energetic electrons located in a magnetic field of $B=10~\mu$G have their synchrotron continuum around the characteristic frequency $\nu_{\mbox{\tiny c}}(E_{\mbox{\tiny cut}})=3.3\times 10^{17}~$Hz, corresponding to about 1.4~keV in X-ray energy. This issue can be seen for the 0.1-keV and 1-keV profile in Fig. (\ref{fig: filament profile damping}). The X-ray intensity does not decrease with decreasing projected radius as happens in the energy-loss model, but remains nearly constant and even increases, respectively.
Using the X-ray intensity distribution, as well as the volume emissivity, we now calculate the spectra of filament and plateau for each example. Again, we integrate the intensity along the projected radius between $r_{\mbox{\tiny p}}=r_{\mbox{\tiny s}}$ and $r_{\mbox{\tiny p}}=r_{\mbox{\tiny s}}-w$ in order to obtain the filament spectrum. The difference between the photon spectral indices of filament and plateau, $\Gamma_{\mbox{\tiny f}}-\Gamma_{\mbox{\tiny p}}$, at three photon energies is given in Table \ref{tab:Constraints magnetic damping model}, whereas in Fig. (\ref{fig: indices damping}) we illustrate the photon spectral indices calculated with the parameters of Tycho.
As can be seen again, the spectrum of the plateau is steeper than that of the filament. However, the difference between the spectra of filament and plateau depends on the chosen damping length. At relatively large damping lengths (weak damping) the difference between the indices over the hole energy range is smaller than 0.1, whereas at smaller damping lengths (strong damping) it also takes values in the range 0.1-0.2. The small differences at larger damping lengths are due to the higher magnetic fields that need to be chosen in order to retain the observed filament widths. Hence, considerable energy losses have to be taken into account. Similarly to the energy-loss model, this results in an almost equal behaviour of the spectra of filament and plateau at high photon energies. In contrast, the magnetic fields used at smaller damping lengths are low enough to result in spectra that show significant differences among each other.
Finally, one can also see from Table \ref{tab:Constraints magnetic damping model} that the filament spectrum becomes steeper with decreasing damping length, in particular at small X-ray energies. This is due to the lower magnetic-field strengths used in that case. Although the weaker magnetic fields imply higher cut-off energies, which harden the spectra, their influence is not sufficient enough to result in spectra similar to those found at larger damping lengths. Besides, as in the energy-loss model and independently of the damping length, Cas A, Tycho, Kepler and SN 1006 show roughly the same spectral behaviour, if the shock velocities are similar, whereas the filament spectrum obtained from the shock velocity of the northwestern limb of SN 1006 has a steeper profile. In contrast, the upper limit to the distance to Kepler implies a harder filament spectrum.
\begin{table*}
\caption{Constraints on the maximum magnetic-field strength, cut-off energy of the electron spectrum, photon spectral index of the filament, and the difference between the photon indices of the spectra of filament and plateau at three different photon energies, $E_\nu$, for four young SNRs. These values are calculated using the magnetic-field damping model and the parameters given in Table \ref{tab:parameters}.} \label{tab:Constraints magnetic damping model}
\centering
\begin{tabular}{c c c c c c c c c c}
\hline \hline
\multicolumn{10}{c}{Weak Damping} \\
\hline
SNR & $l_{\mbox{\tiny d}}$ & $B_{\mbox{\tiny max}}$ & $E_{\mbox{\tiny cut}}$ & \multicolumn{3}{c}{$\Gamma_{\mbox{\tiny f}}$} & \multicolumn{3}{c}{$\Gamma_{\mbox{\tiny f}}-\Gamma_{\mbox{\tiny p}}$} \\ \cmidrule(lr){5-7} \cmidrule(lr){8-10}
& [pc] & [$\mu$G] & [TeV] & $E_\nu=0.1~$keV & $E_\nu=1~$keV & $E_\nu=10~$keV & $E_\nu=0.1~$keV & $E_\nu=1~$keV & $E_\nu=10~$keV \\
\hline
SN 1006 & 0.03 & 65 & 50 & 1.78 & 2.12 & 2.64 & 0.04 & 0.08 & 0.07 \\
& (0.03) & (57) & (32) & (1.93) & (2.35) & (2.95) & (0.05) & (0.09) & (0.08) \\
Cas A & 0.015 & 260 & 27 & 1.71 & 1.98 & 2.51 & 0.04 & 0.02 & 0.01 \\
Tycho & 0.02 & 150 & 34 & 1.71 & 2.00 & 2.57 & 0.06 & 0.04 & 0.02 \\
Kepler & 0.03 & 135 & 36 & 1.71 & 2.01 & 2.59 & 0.05 & 0.04 & 0.02 \\
& (0.03) & (115) & (52) & (1.67) & (1.93) & (2.43) & (0.04) & (0.05) & (0.02) \\
\hline
\multicolumn{10}{c}{Strong Damping} \\
\hline
SNR & $l_{\mbox{\tiny d}}$ & $B_{\mbox{\tiny max}}$ & $E_{\mbox{\tiny cut}}$ & \multicolumn{3}{c}{$\Gamma_{\mbox{\tiny f}}$} & \multicolumn{3}{c}{$\Gamma_{\mbox{\tiny f}}-\Gamma_{\mbox{\tiny p}}$} \\ \cmidrule(lr){5-7} \cmidrule(lr){8-10}
& [pc] & [$\mu$G] & [TeV] & $E_\nu=0.1~$keV & $E_\nu=1~$keV & $E_\nu=10~$keV & $E_\nu=0.1~$keV & $E_\nu=1~$keV & $E_\nu=10~$keV \\
\hline
SN 1006\tablefootmark{a} & 0.02 & 64 & 51 & 1.81 & 2.16 & 2.66 & 0.04 & 0.10 & 0.13 \\
& (0.02) & (56) & (32) & (1.97) & (2.39) & (2.96) & (0.04) & (0.11) & (0.13) \\
Cas A & 0.004 & 115 & 40 & 1.81 & 2.12 & 2.58 & 0.07 & 0.16 & 0.19 \\
Tycho & 0.008 & 85 & 45 & 1.82 & 2.14 & 2.61 & 0.05 & 0.13 & 0.17 \\
Kepler\tablefootmark{b} & 0.01 & 80 & 47 & 1.81 & 2.14 & 2.62 & 0.04 & 0.11 & 0.14 \\
& (0.012) & (80) & (62) & (1.74) & (2.03) & (2.48) & (0.03) & (0.09) & (0.12) \\
\hline
\end{tabular}
\tablefoot{
\tablefoottext{a}{The values in brackets were calculated using the shock velocity $v_{\mbox{\tiny s}}=2900~$km/s as measured in the northwestern limb.}
\tablefoottext{b}{The values in brackets were calculated using the upper limit of 6.4~kpc to the distance.}
}
\end{table*}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[angle=0]{Filament_Profile_Damping.eps}}
\caption{Non-thermal X-ray intensity as a function of the projected radius calculated for four different X-ray energies with the parameters of Tycho given in Table \ref{tab:parameters}, as well as $l_{\mbox{\tiny d}}=0.008~$pc and $B_{\mbox{\tiny max}}=85~\mu$G. The forward shock is located at $r_{\mbox{\tiny s}}=3~$pc.}
\label{fig: filament profile damping}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics[angle=0]{Indices_Damping.eps}}
\caption{Photon spectral indices of the spectra of filament and plateau using the parameters of Tycho.}
\label{fig: indices damping}
\end{figure}
\section{Conclusions}
Compared to the magnetic-field damping model, the spectra of filament and plateau obtained in the energy-loss model exhibit larger spectral indices. This can be explained by the considerable energy losses leading to the evolution towards a softer electron distribution in the energy-loss model, and hence, resulting in X-ray spectra that are softer than those obtained in the damping model.
In case of a weak magnetic-field damping the difference between the spectral indices of filament and plateau over the full X-ray spectrum is smaller than 0.1, which is probably to small to be detectable. Only if there is a strong damping, our calculation suggests a measurable difference between the spectra in some SNRs, since the difference between the spectral indices of filament and plateau can take values of almost 0.2 at X-ray energies higher than 1~keV.
In the energy-loss model the difference between the indices of filament and plateau above the X-ray energy of 1~keV is even smaller than 0.1, so that a possible detection can be excluded here, too. On the other hand, the difference between the indices below 1~keV is larger than 0.1, and at a photon energy of 0.1~keV it is even approximately 0.3. This might suggest that there is a measurable difference in the spectra of filament and plateau at small X-ray energies, if the filaments are limited by energy losses. However, on account of the interstellar photoelectric absorption of the soft X-rays, these different spectral characteristics are probably not detectable, too. Furthermore, the plasma downstream of the forward shock is at high temperature, implying also thermal emission contributing to the soft X-ray band, and thus, complicating a clear identification of the non-thermal emission.
Hence, if there is no measurable difference between the spectra of filament and plateau, it is not possible to make definite predictions from the comparison of the spectra whether the filaments are limited by energy losses of the radiating electrons or by damping of the magnetic field. But if a significant difference appears, our calculations then suggest that the filaments are limited by the magnetic field itself.
It should be noted that our results presented here have been derived using Bohm diffusion. According to Eq. (\ref{eqn:diffusion length}), a larger diffusion coefficient with gyrofactor $\eta>1$ would imply a larger diffusion length, resulting in significant widening of the filaments, because now, the regions farther from the shock contain a sufficient number of high-energy electrons contributing to the intensity. Widening must then be compensated by a higher magnetic field to retain the observed filament widths. Hence, the magnetic-field strengths derived in our models represent lower limits for the chosen parameters. The calculation then shows that a larger diffusion coefficient results in softer spectra due to a lower cut-off energy, which decreases with increasing gyrofactor. However, the final results regarding the differences in spectral indices do not change fundamentally.
In a last step we want to compare the predictions derived here with observations. At first, we notice that, independently of the model, the parameters from the remnants of Cas A, Tycho, Kepler and SN 1006 lead to nearly the same spectral behaviour in case of similar shock velocities, as can be seen from the spectral indices in Table \ref{tab:Constraints energy-loss model} and Table \ref{tab:Constraints magnetic damping model}. However, the analysis of the filament spectra of these remnants reported by \citet{2003ApJ...589..827B,2005ApJ...621..793B} reveals significant differences among the spectral indices obtained from the fit of an absorbed power-law model. Compared to our spectra whose calculation has been done using an injection index resulting from an unmodified shock ($s=2$), the observation may be an indication for different electron injection indices in these remnants, implying shocks that are differently affected by non-linear effects due to differences in efficiency in the particle acceleration.
Regarding the magnetic-field strengths, we take, as an example for comparison with our results, the non-thermal filaments of Cas A analysed by \citet{2010ApJ...714..396A}. From the best-fit parameters used to fit the observed filament spectra, the magnetic field has been derived to be in the range $(30-70)~\mu$G. These values are consistent with those derived from the magnetic-field damping model, in which the magnetic field varies, according to Eq. (\ref{eqn:magnetic field}) and the values from Table \ref{tab:Constraints magnetic damping model}, between the field strengths $(10-260)~\mu$G for weak damping and $(10-115)~\mu$G for strong damping, respectively. For comparison, the constant magnetic field derived from the energy-loss model is several times higher, $B=520~\mu$G. This might suggest that the non-thermal filaments of Cas A are limited by the damping of the magnetic field.
Another comparison concerns the magnetic fields in SN 1006 and Tycho. Using the data from radio up to TeV-observations, \citet{2010A&A...516A..62A} have analysed the multi-wavelength spectrum of SN 1006 in the framework of a leptonic and hadronic origin for the gamma-ray emission, giving a magnetic-field of $\sim 30~\mu$G in the leptonic model and a magnetic field of $\sim 120~\mu$G in the hadronic model, respectively. Moreover, combining radio and X-ray data with recent TeV-observations performed with the VERITAS instrument, the magnetic field of Tycho has been estimated to be $\sim 80~\mu$G in a leptonic-dominated model, whereas a hadronic dominated model yields a magnetic field of $\sim 230~\mu$G \citep{2011ApJ...730L..20A}. Compared to our model predictions given in Table \ref{tab:Constraints energy-loss model} and Table \ref{tab:Constraints magnetic damping model}, we notice that the magnetic fields derived from the energy-loss model are in good agreement with those estimated from the hadronic model used to describe the observed spectra of SN 1006 and Tycho. In contrast, the predictions from the magnetic-field damping model suggest the leptonic model for the origin of the gamma-ray emission from these remnants. It should be noted that current gamma-ray observations do not reach the spatial resolution of those done in X-rays, so that the magnetic fields estimated using gamma-ray observations of SN 1006 and Tycho are averages over a region much larger than the filaments, implying that the observed values do not necessarily match those found for the filaments.
To discriminate between the energy-loss model and magnetic-field damping model, and hence between a leptonic and a hadronic origin of TeV-band gamma-ray emission, one may either search for differences between X-ray spectra of filaments and plateau, as calculated in this paper, or perform gamma-ray observations with higher spatial resolution.
\subsection*{Acknowledgement}
We acknowledge support by the "Helmholtz Alliance for Astroparticle Phyics HAP" funded by the Initiative and Networking Fund of the Helmholtz Association.
\bibliographystyle{aa}
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1,116,691,500,751 | arxiv | \section*{Appendix}
|
1,116,691,500,752 | arxiv | \section{Introduction}
More than 90 binary black holes (BBHs) have been detected in the data of the ground-based gravitational-wave detectors LIGO~\citep{TheLIGOScientific:2014jea} and Virgo~\citep{TheVirgo:2014hva} by the LIGO-Virgo-Kagra (LVK) collaboration and other groups~\citep{LIGOScientific:2021djp, Nitz:2021zwj, Olsen:2022pin}. This dataset has been used to infer the properties of the underlying population---or populations---of BBHs. Among the parameters of interest, the masses and spins of the black holes play a prominent role since they can shed light on the binary formation channels\footnote{Eccentricity is also a powerful indicator of a binary formation channel~\citep{PhysRev.136.B1224, PhysRevD.77.081502, Morscher:2014doa, Samsing:2017xmd, Rodriguez:2017pec, Rodriguez:2018pss, Gondan:2018khr, Zevin:2017evb}, but it is currently harder to measure due to the limited sensitivity of ground-based detectors at frequencies below 20~Hz.}~\citep[e.g.,][]{Vitale:2015tea,Farr:2017uvj,Zevin:2017evb,Farr:2017gtv,Wong:2020ise, Zevin:2020gbd, Bouffanais:2021wcr}.
There currently exist a few approaches toward measuring the population properties. The first is to use a functional form for a reasonable population distribution, parameterized by some phenomenological parameters. For example, the LVK parameterized the primary mass distribution of the black holes as a mixture of a power law distribution and a Gaussian component~\citep{LIGOScientific:2021psn, Talbot:2018cva, Fishbach:2017zga}. This model includes several hyperparameters (since they pertain to the population as a whole, not to the individual events), which are measured from the data: the slope of the power law, the minimum and maximum black hole mass (including a smoothing parameter), the mean and standard deviation of the Gaussian component, and the branching ratio between the power law and Gaussian components. It is worth noting that not all parametric models are equally strong: an example of a more flexible parametrization is the Beta distribution that the LVK has used to describe the population distribution of black hole spin magnitudes~\citep{Wysocki:2018}.
Those more elastic models might be more appropriate if one doesn't have strong observational or theoretical expectations about what the astrophysical distribution of a parameter should look like, or simply if they want to be more conservative. Just as Bayesian priors can significantly affect the posterior for parameters for which the likelihood is not very peaky when analyzing individual compact binary coalescences, a hypermodel that is too strong could leave imprints on the inferred hyperparameters.
A second approach is to use non-parametric models, based on e.g. Gaussian processes~\citep{Tiwari:2020vym, Edelman:2021zkw, Rinaldi:2021bhm, Mandel:2016prl,Vitale:2018yhm}. Those usually have many more free parameters, which allows them to fit features in the data that parmeterized models might miss. However, their larger number of parameters implies they might need more sources to reach a level of precision comparable to parametric models.
Ideally, when strong parametric models are used, one would like to check that the results are not impacted by the model itself, and instead reveal features that are genuinely present in the data. A possible approach is to run multiple models (parametric and non parametric) and verify that they agree to within statistical uncertainties.
Finally, recent work has focused on using as a model the predictions of numerical simulations. This typically involves applying machine learning~\citep{Wong:2020ise} or density estimation techniques~\citep{Zevin:2017evb, Bouffanais:2021wcr} to the binary parameter distributions output by rapid population synthesis codes or N-body simulations. While this approach is more astrophysically motivated, the population synthesis simulations have their own uncertainties and assumptions and often include so many free parameters that a complete exploration of the model space is not possible with current computational techniques~\citep[e.g.][]{Broekgaarden:2021efa}.
The impact of modeling on astrophysical inference of gravitational-wave sources has already become apparent in recent months. Several groups have investigated whether there is evidence for a fraction of black hole spin magnitude to be vanishingly small, finding results that depend on the model to a large extent~\citep{Callister:2022qwb,Galaudage:2021rkt,Tong:2022iws,Roulet:2021hcu,Mould:2022xeu} (Note that \cite{Callister:2022qwb} also provides a comprehensive summary of the status of that measurement).
In this paper we focus on the inference of the population distribution of tilt angles for the black hole binaries in the latest LVK catalog. This is the angle that each of the black hole spin vectors forms with the orbital angular momentum at some reference frequency (following the LVK data release, we will use $20$~Hz for the reference frequency\footnote{For the BBHs detected in the second half of the third observing run (O3b), the LVK has also released tilt posteriors evaluated at minus infinity, i.e. at very large orbital separations~\citep{Mould:2021xst}. We have run the \texttt{Isotropic + Beta}\xspace model of Sec.~\ref{sec.IsoPlusBeta} on O3b sources only, and found that the analyses with tilts calculated at $20$~Hz and minus infinity yield the same astrophysical \ensuremath{\cos{\tau}}\xspace distribution. We also find that using O3b only sources the \ensuremath{\cos{\tau}}\xspace distribution moves toward the left, compared to what is shown in Fig.~\ref{fig.IsoBeta_None_costau}, and peaks closer to 0.}).
In their latest catalog of BBHs, the LVK collaboration has characterized this distribution by using a mixture model composed of an isotropic distribution plus a Gaussian distribution that peaks at $\ensuremath{\cos{\tau}}\xspace=1$ (i.e. when the spin vector and the angular momentum are aligned) with unknown width~\citep{Talbot:2017yur, LIGOScientific:2021psn}. This model reflects expectations from astrophysical binary modeling. Indeed, numerical simulations suggest that binaries formed in galactic fields should have spins preferentially aligned with the angular momentum~\citep[e.g.,][]{1993MNRAS.260..675T, Kalogera:1999tq, Belczynski:2017gds, Zaldarriaga:2017qkw, Stevenson:2017tfq, Gerosa:2018wbw}, whereas binaries formed dynamically (i.e. in globular or star clusters) should have randomly oriented tilts~\citep[e.g.,][]{PortegiesZwart:2002iks, Rodriguez:2015oxa, Antonini:2016gqe, Rodriguez:2019huv, Gerosa:2021mno}.
While this is a reasonable model, we are interested in verifying whether it is actually supported by the data in hand, or whether we are instead getting posteriors that are strongly dependent on that model.
\cite{Callister:2022qwb} and \cite{Tong:2022iws} recently considered alternative models for the tilt angles, but they focused on whether there is a cutoff at negative cosine tilts (i.e. for anti-aligned spins), and if that answer depends on the model for the spin magnitude. On the other hand, we don't limit our investigation to the existence of negative tilts, but instead are interested in what---if anything---can be said about the tilt inference that is not strongly dependent on the model being used.
We consider different alternative models and verify that all of them yield Bayesian evidences (and maximum log likelihood values) which are comparable with the default model used by the LVK. We also include models that allow for a correlation between \ensuremath{\cos{\tau}}\xspace and other parameters (binary mass, mass ratio, or spin magnitude, in turn) and find that those too yield similar evidences. Critically, all of these alternative---and equally supported by the data---models yield noticeably different posterior distributions for the tilt population relative to the default LVK model. In particular, different models give different support to the existence and position of a feature at positive $\ensuremath{\cos{\tau}}\xspace$. On the other hand, they all agree on the fact that there is no excess of systems with $\ensuremath{\cos{\tau}}\xspace \simeq -1$.
We conclude that the current constraints on the distribution of tilt angles are significantly affected by the model used, and that more sources (or weaker models) are needed before any conclusions can be drawn about the astrophysical distribution of binary black hole tilts.
\section{Method}
We aim to measure the hyper parameters {\ensuremath{\vec{\lambda}} \xspace} that control the distribution of single-event parameters \ensuremath{\vec{\theta}} \xspace (the black hole masses, spins, redshifts, etc.) given the dataset \ensuremath{D}\xspace consisting of the 69 GWTC-3\xspace BBHs with false alarm ratio smaller than 1 per year---$\ensuremath{D}\xspace \equiv \{d_i, i=1\ldots 69\}$---reported by the LVK collaboration~\citep{LIGOScientific:2021psn}.
The posterior for {\ensuremath{\vec{\lambda}} \xspace} can be written as~\citep{Mandel:2018mve,Fishbach:2018edt,Vitale:2020aaz}:
\begin{equation}
p({\ensuremath{\vec{\lambda}} \xspace} | \ensuremath{D}\xspace) \propto {\pi({\ensuremath{\vec{\lambda}} \xspace})} \prod_{i=1}^{69} \frac{p(d_i |{\ensuremath{\vec{\lambda}} \xspace}) }{\alpha({\ensuremath{\vec{\lambda}} \xspace})}\,. \label{Eq.HyperPostOfLikeFirst} \nonumber
\end{equation}
where we have analytically marginalized over the overall merger rate, which is not relevant for our inference. The function $\alpha({\ensuremath{\vec{\lambda}} \xspace})$ represents the detection efficiency, i.e. the \textit{fraction} of BBHs that are detectable, given the population parameters {\ensuremath{\vec{\lambda}} \xspace}; $\pi({\ensuremath{\vec{\lambda}} \xspace})$ is the prior for the population hyper parameters, and $p(d_i |{\ensuremath{\vec{\lambda}} \xspace}) $ is the likelihood of the stretch of data containing the i-th BBH. This allows us to account for selection effects and infer the properties of the underlying, rather than the observed, population.
Using Bayes' theorem and marginalizing over the single-event parameters, the single-event likelihood can be written as
\begin{equation}
p(d_i |{\ensuremath{\vec{\lambda}} \xspace}) =\int \ensuremath{\mathrm{d}} \ensuremath{\vec{\theta}} \xspace p(d_i |\ensuremath{\vec{\theta}} \xspace ) \pi(\ensuremath{\vec{\theta}} \xspace|{\ensuremath{\vec{\lambda}} \xspace}) \propto \int \ensuremath{\mathrm{d}} \ensuremath{\vec{\theta}} \xspace\;\; \frac{ p(\ensuremath{\vec{\theta}} \xspace | d_i, \ensuremath{\mathcal{H}_{\mathrm{PE}}}\xspace)\pi(\ensuremath{\vec{\theta}} \xspace|{\ensuremath{\vec{\lambda}} \xspace}) }{\pi(\ensuremath{\vec{\theta}} \xspace|\ensuremath{\mathcal{H}_{\mathrm{PE}}}\xspace)}, \label{eq.Sampling}
\end{equation}
where $p(\ensuremath{\vec{\theta}} \xspace | d_i, \ensuremath{\mathcal{H}_{\mathrm{PE}}}\xspace)$ is the posterior distribution for the binary parameters \ensuremath{\vec{\theta}} \xspace of the i-th source. The population hyperparameters {\ensuremath{\vec{\lambda}} \xspace} are typically inferred using a hierarchical process that first involves obtaining posteriors for \ensuremath{\vec{\theta}} \xspace for each individual event under a non-informative prior, $\pi(\ensuremath{\vec{\theta}} \xspace|\ensuremath{\mathcal{H}_{\mathrm{PE}}}\xspace)$. The hypothesis \ensuremath{\mathcal{H}_{\mathrm{PE}}}\xspace represents the settings that were used during this individual-event parameter estimation step. The last term, $\pi(\ensuremath{\vec{\theta}} \xspace|{\ensuremath{\vec{\lambda}} \xspace})$ is the population prior, i.e., our model for how the parameters \ensuremath{\vec{\theta}} \xspace are distributed in the population, given the hyper parameters.
The integral in Eq.~\ref{eq.Sampling} can be approximated as a discrete sum
\begin{equation}
\int \ensuremath{\mathrm{d}} \ensuremath{\vec{\theta}} \xspace p(d_i |\ensuremath{\vec{\theta}} \xspace ) \pi(\ensuremath{\vec{\theta}} \xspace|{\ensuremath{\vec{\lambda}} \xspace}) \simeq {N_{\rm{samples}}}^{-1} \sum_{k}^{N_{\rm{samples}}}\frac{\pi(\ensuremath{\vec{\theta}} \xspace^k_i|{\ensuremath{\vec{\lambda}} \xspace}) }{\pi(\ensuremath{\vec{\theta}} \xspace^k_i|\ensuremath{\mathcal{H}_{\mathrm{PE}}}\xspace)} \nonumber
\end{equation}
where the ${N_{\rm{samples}}}$ samples are drawn from the posterior distribution of the i-th event. We use the posterior samples of the 69 BBHs reported in GWTC-3\xspace, as released in \cite{GWTC1Release,GWTC2Release,ligo_scientific_collaboration_and_virgo_2021_5117703,ligo_scientific_collaboration_and_virgo_2021_5546663}. For the sources reported in GWTC-1, we use the samples labelled \texttt{IMRPhenomPv2\_posterior} in the data release; for GWTC-2 we use \texttt{PublicationSamples}; for GWTC-2.1 we use \texttt{PrecessingSpinIMRHM}, and for GWTC-3 we use \texttt{C01:Mixed}. To sample the hyper posterior we use the \textsc{dynesty}~\citep{Speagle_2020} sampler available with the \texttt{GWPopulation} package~\citep{Talbot:2019okv}.
The detection efficiency $\alpha({\ensuremath{\vec{\lambda}} \xspace})$ can also be calculated through an approximated sum starting from a large collection of simulated BBHs for which the SNR (or another detection statistic) is recorded, as described in \cite{2019RNAAS...3...66F,LIGOScientific:2021psn}. We use the \texttt{endo3\_bbhpop-LIGO-T2100113-v12-\\1238166018-15843600.hdf5} sensitivity file released by the LVK~\citep{ligo_scientific_collaboration_and_virgo_2021_5546676} to calculate $\alpha(\lambda)$, using a false alarm threshold of 1 per year to identify detectable sources, consistently with~\cite{LIGOScientific:2021psn}.
\section{Reference model}
To represent the astrophysical distribution of primary mass, mass ratio, redshift and spin magnitudes we use the flagship models used by the LVK in \cite{LIGOScientific:2021psn}. Namely, the primary mass distribution is their ``power law $+$ peak''~\citep{Talbot:2018cva}; the mass ratio is a power law~\citep{Fishbach:2019bbm}; the two spin magnitudes are independently and identically distributed according to a Beta distribution~\citep{Wysocki:2018}, and the redshift is evolving with a power law~\citep{Fishbach:2018edt}.
For the (cosine\footnote{Even though we will only report results for the cosine of the tilt angle---\ensuremath{\cos{\tau}}\xspace---we might occasionally refer to tilts only, to lighten the text.}) tilt distributions, we consider several models of increasing complexity. The simplest model assumes that the tilt distribution is isotropic, i.e.
\begin{equation}
p(\cos\tau_1,\cos\tau_2) = \frac{1}{4}. \label{Eq.Iso}
\end{equation}
We consider this rather extreme model only to obtain a useful comparison point for the Bayesian evidence of the more sophisticated models described in the next sections.
Next, we consider the LVK's model (\texttt{LVK default}\xspace): a mixture between an isotropic component and Gaussian distribution with $\mu=1$ and an unknown standard deviation:
\begin{equation}
p(\cos\tau_1,\cos\tau_2 | \sigma, \ensuremath{\mathfrak{g}}\xspace) = \frac{1-\ensuremath{\mathfrak{g}}\xspace}{4} +\ensuremath{\mathfrak{g}}\xspace \prod_j^2{\mathcal{N}(\cos{\tau_j},\mu=1,\sigma)} \label{Eq.LVKModel}
\end{equation}
The Gaussian component is truncated and normalized in the range $[-1,1]$. The two hyperparameters of \texttt{LVK default}\xspace are thus the branching ratio \ensuremath{\mathfrak{g}}\xspace of the Gaussian component and its standard deviation $\sigma$, the same for both black holes.
We notice that in \cite{Talbot:2017yur} the two normal distributions can assume different values of $\sigma$. However, since in general the spins of the least massive objects are measured with extremely large uncertainty, there are no reasons to expect that imposing the same distribution to both tilts will introduce biases.
In Fig.~\ref{fig.LVK_costau} we show the resulting inference on the \ensuremath{\cos{\tau}}\xspace, which---modulo differences in sampling settings---is directly comparable to what is presented by the LVK in \citet{LIGOScientific:2021psn}. The colored area shows the 90\% credible interval (CI), the thick dashed line is the median, and the dim lines represent individual draws from the posterior. The two dashed lines represent the edges of the 90\% credible interval obtained by sampling the hyperparameters from their priors. It is worth noticing that the \texttt{LVK default}\xspace model excludes a priori the possibility of an excess of tilts relative to isotropy (i.e. a posterior larger than 0.5) at negative values, as well as a dearth of tilts relative to isotropy for $\ensuremath{\cos{\tau}}\xspace \gtrsim 0.45$.
Just as \citet{LIGOScientific:2021psn}, we find that the posterior is not inconsistent with a fully isotropic tilt distribution, while preferring an excess of positive alignment.
\begin{figure}
\includegraphics[width=0.5\textwidth]{lvk_costau.pdf}
\caption{Posterior for \ensuremath{\cos{\tau}}\xspace obtained using the reference \texttt{LVK default}\xspace model. The thin black lines represent individual posterior draws, whereas the colored band shows the 90\% credible interval. The thick dashed line is the median. The two thin dashed lines show 90\% credible interval obtained by drawing the model's hyperparameters from their priors.}
\label{fig.LVK_costau}
\end{figure}
We can better visualize what happens at the edges of the \ensuremath{\cos{\tau}}\xspace domain---i.e. for values of tilts close to aligned ($\ensuremath{\cos{\tau}}\xspace \simeq 1$) or anti-aligned ($\ensuremath{\cos{\tau}}\xspace \simeq -1$)---by plotting the ratio of the posterior for aligned spin vs anti-aligned spins. This is equivalent to making a histogram of the ratio of the value that the thin black curves in Fig.~\ref{fig.LVK_costau} take on the far right and far left side. Specifically, for each of the posterior draws, we calculate the ratio
\begin{equation}
Y\equiv \frac{p(\ensuremath{\cos{\tau}}\xspace \in [0.9,1])}{p(\ensuremath{\cos{\tau}}\xspace \in [-1,-0.9])}\label{eq:y}
\end{equation}
and histogram it. When $Y=1$, the \ensuremath{\cos{\tau}}\xspace distribution takes the same value at both of the edges; when $Y>1$ ($Y<1$) the data support populations with more aligned (anti-aligned) systems than anti-aligned (aligned) ones. This is shown in Fig.~\ref{fig.LVK_pedges}, where the solid grey line is obtained using samples from the hyperparameters' posterior whereas the dashed line is obtained by sampling their priors. The fact that there is a hard cutoff at $Y=1$ (the finite bin size causes the curves to extend to values slightly smaller than 1) is just a symptom of the fact that the \texttt{LVK default}\xspace model excludes a priori an excess of negative tilts and a dearth at positive tilts, as mentioned above.
The curve is consistent with $Y=1$, i.e. isotropic posteriors are perfectly consistent with the data, even though it should be appreciated that the model prefers that region a priori. The level of consistency can also be assessed with Figure~\ref{fig.iso_gaussian_models_pzeta}, which reports with dashed blue lines the marginalized posterior on the branching ratio of the isotropic component (as opposed to the Gaussian component, to allow direct comparisons with other models). While broad, it favors small values for the fraction of sources in the isotropic component, though fully isotropic distributions ($\ensuremath{\mathfrak{i}}\xspace=1$) are not excluded. The other curves in the figure will be discussed below. For all of our models, Tab.~\ref{tab:Evidences} reports the Bayesian evidence, maximum log-likelihood and the number of parameters for the \ensuremath{\cos{\tau}}\xspace model, as a differential relative to the default LVK model. The \texttt{Isotropic}\xspace model performs the worst, though not at the point that it can be ruled out with high confidence.
\begin{figure}
\includegraphics[width=0.5\textwidth]{lvk_pedges.pdf}
\caption{A normalized histogram of $Y$, the ratio between the probability of \ensuremath{\cos{\tau}}\xspace for $\ensuremath{\cos{\tau}}\xspace \in [0.9,1]$ and $\ensuremath{\cos{\tau}}\xspace \in [-1,-0.9]$, obtained using the reference \texttt{LVK default}\xspace model. The solid curve is obtained by sampling the hyperposteriors. The dashed line reports the same quantity, but drawing the model's hyperparameters from their priors. Values of $Y>1$ imply more support for aligned than anti-aligned spins.}
\label{fig.LVK_pedges}
\end{figure}
\begin{figure}
\includegraphics[width=0.5\textwidth]{iso_2comp_models_piso.pdf}
\caption{Marginalized posterior for the branching ratio of the \textit{isotropic component} -- \ensuremath{\mathfrak{i}}\xspace -- for all of the uncorrelated two-component models, Sec.~\ref{sec.IsoPlusGaussian},\ref{sec.IsoPlusBeta},~\ref{sec.IsoPlusTukey}. The figure is split into two panels to enhance clarity. In both panels, we also report the posterior obtained with the \texttt{LVK default}\xspace model (blue dashed line) for comparison.}
\label{fig.iso_gaussian_models_pzeta}
\end{figure}
\section{Alternative models}\label{sec.TwoCompUncor}
In this section we re-analyze the GWTC-3 BBH with different parameterized two-component mixture models for \ensuremath{\cos{\tau}}\xspace. We will report results for three-component models in Appendix~\ref{Sec.ThreeCompUncor}. It is assumed that the \ensuremath{\cos{\tau}}\xspace distribution is not correlated to any other of the astrophysical parameters. This assumption will be revisited in Appendix~\ref{Sec.Correlated}.
\subsection{\texttt{Isotropic + Gaussian}\xspace model}~\label{sec.IsoPlusGaussian}
To check if the data \textit{requires} the that the Gaussian component of the \texttt{LVK default}\xspace mixture model peaks at $\ensuremath{\cos{\tau}}\xspace=1$, we relax the assumption that the normal distribution must be centered at $+1$, That is, we treat the mean of the Gaussian component $\mu$ as another model parameter:
\begin{equation}
p(\cos\tau_1,\cos\tau_2 | \mu, \sigma,\ensuremath{\mathfrak{g}}\xspace) = \frac{1-\ensuremath{\mathfrak{g}}\xspace}{4} +\ensuremath{\mathfrak{g}}\xspace \prod_j^2{\mathcal{N}(\cos{\tau_j},\mu,\sigma)}
\label{Eq.IsoPlusGaussian}
\end{equation}
The prior for $\mu$ is uniform in the range $[-1,1]$ (Tab.~\ref{tab:priors} reports the priors for the hyperparameters of all models used in the paper).
\begin{figure}
\includegraphics[width=0.5\textwidth]{IsoMovingGaussian_None_ConstrainedMu_costau.pdf}
\caption{Same as Fig.~\ref{fig.LVK_costau}, but for the \texttt{Isotropic + Gaussian}\xspace model, when the mean of the Gaussian component is allowed to vary in the range $\mu \in [-1,1]$.}
\label{fig.IsoMovingGaussian_None_ConstrainedMu_costau}
\end{figure}
Figure ~\ref{fig.IsoMovingGaussian_None_ConstrainedMu_costau} shows the resulting posterior for the tilt angle. For the mean of the Gaussian component we measure $\mu=0.47^{+0.47}_{-1.04}$ (unless otherwise stated, we quote median and 90\% symmetric credible interval). Some of the uncertainty in this measurement is due to our choice to allow for large $\sigma$, since Gaussians with large $\sigma$ are rather flat, and can be centered anywhere without significantly affecting the likelihood.
If we restrict the prior space to only allow for narrower Gaussians, then $\mu$ is much more constrained. E.g. if we limit to samples with $\sigma<0.5$ ($\sigma<1$) then $\mu=0.29^{+0.36}_{-0.33}$ ($\mu=0.41^{+0.47}_{-0.39}$).
This can be also seen in a corner plot of the mean and standard deviation of the Gaussian component, Fig.~\ref{fig.IsoMovingGaussian_None_ConstrainedMu_sns_mu_sigma}. The data prefers positive means with standard deviations in the approximate range $\sigma \in [0.25,1.5]$. Smaller values of $\sigma$ are excluded, as are Gaussians centered at negative values of $\mu$, i.e. such that \ensuremath{{\chi_{\rm{eff}}}}\xspace---the mass-weighted projection of the total spin along the angular momentum~\citep{Damour:2001}---would be negative.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{IsoMovingGaussian_None_ConstrainedMu_sns_mu_sigma.pdf}
\caption{Joint and marginal posteriors for the mean and standard deviation of the Gaussian component for the \texttt{Isotropic + Gaussian}\xspace model, as well as for the branching ratio \ensuremath{\mathfrak{g}}\xspace, when the mean of the Gaussian component is allowed to vary in the range $\mu \in [-1,1]$.}
\label{fig.IsoMovingGaussian_None_ConstrainedMu_sns_mu_sigma}
\end{figure}
It is worth noting that the astrophysical \ensuremath{\cos{\tau}}\xspace posterior changes entirely if the prior for $\mu$ is extended to allow for values outside of the range $[-1,1]$ (while the resulting population model is, of course, still truncated and properly normalized in that domain).
For example, Fig.~\ref{fig.IsoMovingGaussian_None_UnconstrainedMu_costau} shows the posterior for the tilt distribution obtained with a wider $\mu$, uniform in the range $[-5,5]$. Here again it is the case that the distribution is consistent with having a peak for aligned spins. A look at the joint distribution of $\mu$ and $\sigma$, Fig.~\ref{fig.IsoMovingGaussian_None_UnconstrainedMu_sns_mu_sigma}, reveals that the peak at $\mu=1$ is not obtained because the Gaussian component peaks there, but rather by truncating Gaussians that peak at $\mu>1$ and have large standard deviations. This results in a much steeper shape for the posterior near the $\cos\tau=+1$ edge than what could possibly be obtained by forcing $-1\leq\mu\leq1$.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{IsoMovingGaussian_None_UnconstrainedMu_costau.pdf}
\caption{Same as Fig.~\ref{fig.LVK_costau}, but for the \texttt{Isotropic + Gaussian}\xspace model, when the mean of the Gaussian component is allowed to vary in the range $\mu \in [-5,5]$.}
\label{fig.IsoMovingGaussian_None_UnconstrainedMu_costau}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{IsoMovingGaussian_None_UnconstrainedMu_sns_mu_sigma.pdf}
\caption{Same as Fig.~\ref{fig.IsoMovingGaussian_None_ConstrainedMu_sns_mu_sigma}, but allowing the mean of the Gaussian component to vary in the range $\mu \in [-5,5]$.}
\label{fig.IsoMovingGaussian_None_UnconstrainedMu_sns_mu_sigma}
\end{figure}
The only feature that seems solid against model variations is the fact that there is no excess of systems at negative $\cos\tau$. We notice that for this model, unlike for the \texttt{LVK default}\xspace model, it \textit{would} have been possible for the \ensuremath{\cos{\tau}}\xspace posterior to show an excess at negative values. This is shown in Fig.~\ref{fig.IsoGaussian_unconstrained_pedges}, where we see that the prior (dashed curve) of $Y$ (Eq.~\ref{eq:y}) can extend to values smaller than $1$, i.e. can produce more anti-aligned spins than aligned ones. In fact, we see that the prior for this asymmetry probe is much less strong than in the default \texttt{LVK default}\xspace model, as it does not exclude $Y<1$. Here too the posterior of $Y$ is not inconsistent with $1$. Values of $Y$ smaller than 1, i.e. an excess of anti-aligned spins are severely suppressed relative to the prior, and so are large excess of positive tilts. The posterior for $Y$ has a rather broad peak in the range $\sim [1,2]$ corresponding to distributions that are consistent with being either isotropic or having a mild excess of positive alignment.
\begin{figure}
\includegraphics[width=0.5\textwidth]{IsoGaussian_unconstrained_pedges.pdf}
\caption{Same as Fig.~\ref{fig.LVK_pedges}, but for the \texttt{Isotropic + Gaussian}\xspace model with the Gaussian component's mean allowed to vary in the range $\mu \in [-5,5]$.}
\label{fig.IsoGaussian_unconstrained_pedges}
\end{figure}
In the top panel of Figure~\ref{fig.iso_gaussian_models_pzeta} we show the marginalized posteriors of the branching ratio for the isotropic component, \ensuremath{\mathfrak{i}}\xspace, of the models described in this section, together with the reference \texttt{LVK default}\xspace model. While small variations exist, they all have support across the whole prior range, with a preference for small values of \ensuremath{\mathfrak{i}}\xspace.
\subsection{\texttt{Isotropic + Beta}\xspace model}~\label{sec.IsoPlusBeta}
The results from the previous subsection suggest that a Gaussian distribution might not be the best way of describing the non-isotropic part of the tilt population. Therefore, we now replace the Gaussian component of the previous section with a Beta distribution:
\begin{equation}
p(\cos\tau_1,\cos\tau_2 | \alpha, \beta,\ensuremath{\mathfrak{b}}\xspace) = \frac{1-\ensuremath{\mathfrak{b}}\xspace}{4} +\ensuremath{\mathfrak{b}}\xspace \prod_j^2 \mathcal{B}(\cos \tau_j,\alpha,\beta)\label{Eq.IsoPlusBeta}
\end{equation}
We offset the input of the Beta distribution and scale its maximum value such that it spans the domain $[-1,1]$. We stress that we do \textit{not} limit the range of $\alpha$ and $\beta$ to non-singular values, i.e. we do allow them to be smaller than 1---Tab.~\ref{tab:priors}. In turn, this implies that we can get posteriors for \ensuremath{\cos{\tau}}\xspace that peak at the edges of the range. The resulting posterior for $\cos\tau$ is shown in Fig.~\ref{fig.IsoBeta_None_costau}, which also shows for comparison the 90\% CI when drawing hyperparameters from their priors, thin dashed lines. With this model, we recover a broad peak at small positive values of \ensuremath{\cos{\tau}}\xspace. One can convert the $\alpha$ and $\beta$ parameters of our rescaled Beta distribution to the corresponding mean as
\begin{equation} \mu_\beta= -1 + 2 \frac{\alpha}{\alpha+\beta}. \label{Eq.BetaMu}\end{equation} We find $\mu_\beta= 0.20^{+0.21}_{-0.18}$. While some of the posterior draws do peak at $+1$, overall the upper edge of the 90\% CI band does not show a peak in that region.
Compared to what seen in the previous section, this model finds more support for small values of $p(\ensuremath{\cos{\tau}}\xspace)$ at anti-aligned tilts, with a median value that is at the lower edge of the 90\% CI for the \texttt{Isotropic + Gaussian}\xspace models.
In Fig.~\ref{fig.IsoBeta_pedges} we show the posterior and prior of the asymmetry coefficient $Y$ defined in Eq.~\ref{eq:y}. For this model too we observe that the posterior disfavours configurations with an excess of anti-aligned black hole tilts, relative to the prior. We notice that in this case values of $Y>1$ do not necessarily represent a \textit{peak} at positive tilts, see Fig.~\ref{fig.IsoBeta_None_costau}, but only that negative tilts are even more suppressed.
\begin{figure}
\includegraphics[width=0.5\textwidth]{IsoBeta_None_costau.pdf}
\caption{Same as Fig.~\ref{fig.LVK_costau}, but for the \texttt{Isotropic + Beta}\xspace model.}
\label{fig.IsoBeta_None_costau}
\end{figure}
\begin{figure}
\includegraphics[width=0.5\textwidth]{IsoBeta_pedges.pdf}
\caption{Same as Fig.~\ref{fig.LVK_pedges}, but for the \texttt{Isotropic + Beta}\xspace model.}
\label{fig.IsoBeta_pedges}
\end{figure}
The marginalized posterior for the branching ratio of the isotropic component is shown in the bottom panel of Fig.~\ref{fig.iso_gaussian_models_pzeta} (histogram with dotted hatches). We find that, unlike the Gaussian-based models of Sec.~\ref{sec.IsoPlusGaussian} or the \texttt{LVK default}\xspace model, it does feature a very broad peak in the middle of the range.
\subsection{\texttt{Isotropic + Tukey}\xspace model}~\label{sec.IsoPlusTukey}
We end our exploration of two-component models with a mixture of an isotropic distribution and a distribution based on the Tukey window function. Mathematically:
\begin{eqnarray}
p(\cos\tau_1,\cos\tau_2 | \ensuremath{\mathfrak{t}}\xspace,T_{x0},T_k,T_r) = \frac{1-\ensuremath{\mathfrak{t}}\xspace}{4} && \nonumber \\
+\ensuremath{\mathfrak{t}}\xspace \prod_j^2 {\mathcal{T}(\cos{\tau_j},T_{x0},T_k,T_r)}
\label{Eq.IsoPlusTukey}
\end{eqnarray}
\begin{figure*}
\begin{align}
\mathcal{T}(x,T_{x0},T_k,T_r) &\propto
\begin{cases}
0,\; & x< \rm{max}(-1,T_{x0}-T_k) \\
\frac{1}{2} \left\{1+ \cos\left[\frac{\pi}{T_k T_r}\left(x-T_{x0}+T_k -T_k T_r \right)\right] \right\},&\; \mathrm{max}(-1,T_{x0}-T_k) \leq x< T_{x0} -T_k (1-T_r)\\
1,&\; T_{x0} -T_k (1-T_r) \leq x < T_{x0} +T_k (1-T_r)\\
\frac{1}{2} \left\{1+ \cos\left[\frac{\pi}{T_k T_r}\left(x-T_{x0}-T_k -T_k T_r \right)\right] \right\},&\; T_{x0} +T_k (1-T_r) \leq x\leq \rm{min}(+1,T_{x0}+T_k) \\
0,&\; x> \mathrm{min}(+1,T_{x0}+T_k)
\end{cases}\,.
\label{Eq.TukeyDef}
\end{align}
\end{figure*}
where we have defined the distribution $\mathcal{T}(x,T_{x0},T_k,T_r)$ in Eq.~\ref{Eq.TukeyDef}. While vaguely intimidating, this is simply a Tukey window symmetric around $T_{x0}$ and whose domain is $2 T_k$ wide. The parameter $T_r$ controls the shape of the window ($T_r=0$ gives a rectangular window while $T_r=1$ gives a cosine). The distribution is then truncated and normalized in the range $[-1,1]$. Figure~\ref{fig.Tukey_examples} shows four examples. Since the width, the shape, and the position can all be varied, this model is quite elastic and can latch onto both broad and narrow features. We highlight that in the default setting, we allow the uniform prior of $T_k$ to go up to 4, Tab~\ref{tab:priors}. This implies, that just as for the \texttt{Isotropic + Gaussian}\xspace model, there are parts of the parameter space where the non-isotropic component can be made very similar to, or indistinguishable from, the isotropic component. In this case, that happens when $T_k$ is large and $T_r$ is small. This distribution can also produce curves that ramp up from zero to a plateau, with various degrees of smoothness: the green thick line in Fig.~\ref{fig.Tukey_examples} is an example and---if $\ensuremath{\mathfrak{t}}\xspace$ were zero---would produce a \ensuremath{\cos{\tau}}\xspace distribution similar to the second row in Fig.~5 of~\citet{Callister:2022qwb}.
\begin{figure}
\includegraphics[width=0.5\textwidth]{Tukey_examples.pdf}
\caption{
Four examples of the distribution in Eq.~\ref{Eq.TukeyDef}.
}
\label{fig.Tukey_examples}
\end{figure}
Figure~\ref{fig.IsoPlusTukeyExt_None_costau} reports the resulting posterior distribution for \ensuremath{\cos{\tau}}\xspace, together with the prior (dotted black lines). The 90\% CI band features a plateau that extends from $\ensuremath{\cos{\tau}}\xspace \simeq -0.5$ to $+1$.
\begin{figure}
\includegraphics[width=0.5\textwidth]{IsoPlusTukeyExt_None_costau.pdf}
\caption{
Posterior for \ensuremath{\cos{\tau}}\xspace obtained using the \texttt{Isotropic + Tukey}\xspace model. The thin colored lines represent individual posterior draws, colored according to the corresponding branching ratio for the Tukey component, \ensuremath{\mathfrak{t}}\xspace. The light-colored band shows the 90\% credible interval, while the thick dashed line is the median. The two thin dashed lines show 90\% credible interval obtained by drawing the model's hyperparameters from their priors.}
\label{fig.IsoPlusTukeyExt_None_costau}
\end{figure}
In Fig.~\ref{fig.IsoPlusTukeyExt_None_TukeyPars}, we show the posterior distribution for the parameters controlling the Tukey channel, together with the corresponding branching ratio. The Tukey component is centered at $T_{x0}$, whose $5$th and $95$th percentile are $-0.47$ and $0.94$ respectively.
The marginal posterior for $T_k$ prefers values close to $1.8$, implying wider Tukey distributions. Smaller values of $T_k$ are possible only for small \ensuremath{\mathfrak{t}}\xspace as expected given that for small \ensuremath{\mathfrak{t}}\xspace the data cannot constrain the Tukey component, and the posterior must then resemble the uniform prior, which includes small $T_k$. The posteriors of these two parameters are correlated such that when $\ensuremath{\mathfrak{t}}\xspace$ is large, $T_k$ is also large, meaning the resulting \ensuremath{\cos{\tau}}\xspace distribution more closely resembles an isotropic distribution. However, when $T_k$ gets larger than $\sim 2$ then \ensuremath{\mathfrak{t}}\xspace is not constrained at all. This happens because when $T_k$ is that large the Tukey component is extremely close to an isotropic distribution, at which point the whole model is isotropic, and the branching ratio stops being a meaningful parameter. Finally, the posterior for $T_r$ is wide, with a preference for larger values, implying a Tukey distribution that ramps up and down smoothly rather than producing sharp features.
\begin{figure}
\includegraphics[width=0.5\textwidth]{IsoPlusTukeyExt_None_TukeyPars.pdf}
\caption{Joint and marginal posteriors for the hyperparameters and branching ratio of the Tukey component in the \texttt{Isotropic + Tukey}\xspace model.}
\label{fig.IsoPlusTukeyExt_None_TukeyPars}
\end{figure}
Figure ~\ref{fig.IsoPlusTukeyExt_None_TukeyPars} also reveals that large values of $T_k$ are responsible for the near entirety of the support at $T_{x0}<0$, since a Tukey that is basically a uniform distribution can be centered anywhere without affecting the likelihood. If we restrict the analysis to $T_k\leq 2$ ($T_k\leq 1$), we get that $T_{x0}$ is much better constrained to $T_{x0}=0.45_{-0.42}^{+0.46}$ ($T_{x0}=\ensuremath{0.27_{-0.26}^{+0.42}}\xspace$), which excludes negative values at $\sim$90\% credibility.
The fact that our generous hyperparameter priors allow for Tukey distributions that resemble isotropic ones also explain the peak at $Y=1$ in Fig.~\ref{fig.IsoTukey_pedges}. However, we also see that as for the previous models, the $Y$ posterior is heavily suppressed at $Y\lesssim 1$, indicating a dearth of black hole tilts there. We, once again the data does not exclude that the \ensuremath{\cos{\tau}}\xspace distribution is in fact isotropic, and the only solid conclusion one can make seems to be that there is no excess of systems at $\ensuremath{\cos{\tau}}\xspace \simeq -1$.
\begin{figure}
\includegraphics[width=0.5\textwidth]{IsoTukey_pedges.pdf}
\caption{Same as Fig.~\ref{fig.LVK_pedges}, but for the \texttt{Isotropic + Tukey}\xspace model.}
\label{fig.IsoTukey_pedges}
\end{figure}
\section{Conclusions}
In this paper we have re-analyzed the LVK's 69 BBHs of GWTC-3 using different models for the astrophysical distribution of the black hole spin tilt angle, i.e. the angle the spin vector forms with the orbital angular momentum at a reference frequency ($20$~Hz).
Black hole spin tilts can yield precious information about their astrophysical formation channels. It is usually expected that dynamical formation of binaries results in an isotropic distribution of the spin vectors~\citep[e.g.,][]{PortegiesZwart:2002iks, Rodriguez:2015oxa, Antonini:2016gqe, Rodriguez:2019huv, Gerosa:2021mno}. On the other hand, for black holes binaries formed in the field via isolated binary evolution, it is expected that the spins are nearly aligned with the angular momentum~\citep[e.g.,][]{1993MNRAS.260..675T, Kalogera:1999tq, Belczynski:2017gds, Zaldarriaga:2017qkw, Stevenson:2017tfq, Gerosa:2018wbw}, i.e. that tilts are small, as the angular momenta of the progenitor stars are aligned by star-star and star-disk interactions~\citep{1981A&A....99..126H, 1981A&A...102...17P}.
Indeed, if the binary forms in the field, the only mechanism that could yield significant black hole tilts are asymmetries in the supernovae explosions that create the black holes. These asymmetries can impart a natal kick large enough to tilt the orbital plane~\citep{1975Natur.253..698K, Kalogera:1999tq, Hurley:2002rf}. However, the black hole natal kick distribution is poorly understood both theoretically~\citep[e.g.,][]{Dominik:2012kk, Zevin:2017evb, Mapelli:2018wys, Repetto:2012bs, Giacobbo:2019fmo, Fragos:2010tm} and observationally~\citep{1995MNRAS.277L..35B, 1999A&A...352L..87N, 2001Natur.413..139M, 2002A&A...395..595M, Wong:2013vya}.
These expectations explain why in their most recent catalog, the LVK has modeled the astrophysical tilt distribution as a mixture of two components: an isotropic part and a Gaussian distribution centered at $\ensuremath{\cos{\tau}}\xspace=1$ and with a width that is measured from the data. This is a rather strong model, as it forces onto the data a Gaussian that \textit{must} be centered at $+1$. This might not be advisable, because individual tilt measurements are usually broad, which implies that the functional form of the astrophysical model can leave a discernible imprint on the posterior.
Given the rather large uncertainties about how much spin misalignment can be produced in each channel, and the fact that the BBH population currently on hand might contain back holes formed in different channels~\citep{Zevin:2020gbd, Wong:2020ise, Bouffanais:2021wcr, Franciolini:2022iaa}, it is legitimate to question whether the data \textit{requires} that (as opposed to: is consistent with) the \ensuremath{\cos{\tau}}\xspace distribution peaks at $+1$. We find that it does not.
We consider {three} 2-component mixture models: \texttt{Isotropic + Gaussian}\xspace, made of an isotropic component and a Gaussian component whose mean is \textit{not} fixed at $+1$, but rather measured from the data; \texttt{Isotropic + Beta}\xspace, made of an isotropic component and a (singular) Beta component; \texttt{Isotropic + Tukey}\xspace, made of an isotropic component and a distribution based on the Tukey window.
We find that the only model that yields a peak at $\ensuremath{\cos{\tau}}\xspace=1$ is the \texttt{Isotropic + Gaussian}\xspace, and only if we allow the mean of the Gaussian to take values outside of the \ensuremath{\cos{\tau}}\xspace domain $[-1,1]$: Fig.~\ref{fig.IsoMovingGaussian_None_UnconstrainedMu_sns_mu_sigma} shows that in this case the Gaussian component peaks at $\mu>1$ and is very broad, yielding a peak at $+1$, Fig.~\ref{fig.IsoMovingGaussian_None_UnconstrainedMu_costau}.
The other two models yield either a posterior that peaks at $\ensuremath{\cos{\tau}}\xspace \simeq 0.2$ and no peak at $+1$, or a plateau from $\ensuremath{\cos{\tau}}\xspace \simeq -0.5$ to $+1$.
For all of the above models the data is not decisively ruling out a fully isotropic tilt distribution, but is \textit{inconsistent} with an excess of systems with large and negative tilts.
This agrees with previous results for population models fitting the distribution of \ensuremath{{\chi_{\rm{eff}}}}\xspace, which find only a small fraction of sources with negative \ensuremath{{\chi_{\rm{eff}}}}\xspace, implying negative tilts~\citep{Roulet:2018jbe, Miller:2020zox, Roulet:2021hcu, Callister:2022qwb}. Indeed, if we recast our inferred distributions for \ensuremath{\cos{\tau}}\xspace and spin magnitude to the resulting \ensuremath{{\chi_{\rm{eff}}}}\xspace distribution, we obtain results consistent with \cite{LIGOScientific:2021psn}.
In the appendices below we report on other models. In Appendix ~\ref{Sec.ThreeCompUncor} we consider {three} 3-component mixture models: \texttt{Isotropic + 2 Gaussians}\xspace, made of an isotropic component and 2 Gaussian components; \texttt{Isotropic + Gaussian + Beta }\xspace, made of an isotropic component a (non-singular) Beta component and a Gaussian component; \texttt{Isotropic + Gaussian + Tukey }\xspace, made of an isotropic component, a distribution based on the Tukey window and a Gaussian component. These more elastic models are consistent a broad plateau in the \ensuremath{\cos{\tau}}\xspace posterior that extends from $\sim-0.5$ to $+1$. Whether there also are peaks or features at small positive \ensuremath{\cos{\tau}}\xspace and/or at $+1$ depends on the exact model.
In Appendix ~\ref{Sec.Correlated} we augment some of the 2-component mixture models to allow for correlations between the tilt angles and another of the binary parameters: component masses, component spins, mass ratio and {total mass} in turn. One might expect some correlation as the mechanisms that can misalign the binary orbital plane or the black hole spins are affected by the binary parameters~\citep[e.g.][]{1994A&A...290..496J, Burrows:1995bb, Fryer:2005sz, Gerosa:2021mno}. To keep the number of model parameters limited---consistently with the relatively limited number of sources---we only consider linear correlations, and only correlate the tilt distribution with one other parameter at a time. However, even with these limitations we find that the current dataset cannot significantly constrain eventual correlations. For all of the models considered in this work, we report Bayesian evidences, Tab.~\ref{tab:Evidences}, which might be used to calculate odds ratios. We find that---within sampling and numerical uncertainties---all of the models are equally supported by the data.
We conclude that the current dataset is not yet large and informative enough to prove that the astrophysical tilt distribution has features, nor that {if} features exist they manifest as an excess of systems with nearly aligned spin vectors. On the contrary, most of the models we considered yield a broad peak in the astrophysical \ensuremath{\cos{\tau}}\xspace distribution at small and positive values. The only conclusion that is consistently found across all models is that there is no excess of systems with negative tilts, relative to what is expected in an isotropic distribution. Our results agree with the literature~\citep[e.g.][]{Callister:2022qwb,Tong:2022iws, LIGOScientific:2021psn,Mould:2022xeu} on the lack of an excess of $\ensuremath{\cos{\tau}}\xspace \simeq -1$ but disagree on other details (e.g. whether there is an hard cutoff in the \ensuremath{\cos{\tau}}\xspace distribution at $\ensuremath{\cos{\tau}}\xspace<0$ \citep[cfr][]{Callister:2022qwb}). The point of this work is to show that those disagreements are to be expected, given the information in the current dataset. The next observing run of LIGO, Virgo and KAGRA is scheduled to start in early 2023~\citep{KAGRA:2013rdx} and should yield hundreds of BBH sources. Those may yield a first firm measurement of the astrophysical distribution of the tilt angle, and possibly allow us to begin probing correlations with other astrophysical parameters.
\section*{Acknowledgments}
The authors would like to thank C. Adamcewicz, V. Baibhav, T. Dent, S. Galaudage, C. Rodriguez and M. Zevin for useful comments and discussion.
We would in particular like to thank T. Callister and D. Gerosa for many insightful comments and suggestions.
S.V. is supported by NSF through the award PHY-2045740.
S.B. is supported by the NSF Graduate Research Fellowship under Grant No. DGE-1122374.
CT is supported by the MKI Kavli Fellowship.
This material is based upon work supported by NSF's LIGO Laboratory which is a major facility fully funded by the National Science Foundation.
This paper carries LIGO document number LIGO-P2200275.
\section*{Data Availability}
A public repository with the hyperposteriors used in this work will be available on \href{toappear}{Zenodo}. We used publicly-available programs \textsc{Bilby}~\citep{Ashton:2018jfp, Romero-Shaw:2020owr}, \textsc{dynesty}~\citep{Speagle_2020} and \textsc{GWPopulation}~\citep{Talbot:2019okv}
\bibliographystyle{mnras}
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1,116,691,500,753 | arxiv | \section{Introduction}
\subsection{Log Calabi--Yau pairs}
One topic of much contemporary interest is the geometry of log Calabi--Yau pairs.\footnote{See Definition~\ref{def!lCY} for the precise notion of what we consider to be a log Calabi--Yau pair, which is a somewhat more restrictive definition than that considered by other authors.} In part, this is because the interior of a maximal log Calabi--Yau pair is expected to have remarkable properties predicted from mirror symmetry (see e.g.\ \cite[\S1]{hk}). It is therefore important to understand the classification of log Calabi--Yau pairs up to volume preserving equivalence.
\paragraph{The coregularity.}
The most important volume preserving invariant of a log Calabi--Yau pair $(X,\Delta_X)$ is an integer $0\leq \coreg(X,\Delta_X)\leq \dim X$, called the \emph{coregularity} of $(X,\Delta_X)$, which is the dimension of the smallest log canonical centre on a dlt modification of $(X,\Delta_X)$ (see Definition~\ref{def!coreg}). At one end of the spectrum are the pairs with $\coreg(X,\Delta_X)=\dim X$. These are necessarily of the form $(X,0)$, where $X$ is a variety with trivial canonical class $K_X\sim 0$, and hence this case reduces to the study of (strict) Calabi--Yau varieties. At the opposite end are the pairs satisfying $\coreg(X,\Delta_X)=0$, which are also known as \emph{maximal pairs}. These form the next most important case to understand, particularly given the role that maximal pairs play in mirror symmetry via the Gross--Siebert program. They are characterised by the property that the dual complex $\mathcal{D}(X,\Delta_X)$ has the largest possible dimension.
\paragraph{Toric models.}
The simplest examples of maximal log Calabi--Yau pairs are toric pairs, and these lie in a single volume preserving equivalence class. We say that $(X,\Delta_X)$ \emph{has a toric model} if it also belongs to the same volume preserving equivalence class as a toric pair. A \emph{toric model} for $(X,\Delta_X)$ is a particular choice of volume preserving map $\varphi \colon (X,\Delta_X) \dashrightarrow (T,\Delta_T)$ onto a toric pair $(T,\Delta_T)$.
\begin{rmk}\label{rmk!toric-models}
We note three immediate consequences for a $d$-dimensional maximal log Calabi--Yau pair $(X,\Delta_X)$ with a toric model.
\begin{enumerate}
\item $X$ is rational, since it is birational to a toric variety.
\item Every irreducible component $D\subset\Delta_X$ is rational. This is because, after choosing a suitable toric model $\varphi\colon (X,\Delta_X)\dashrightarrow (T,\Delta_T)$, $D$ maps birationally onto a component of $\Delta_T$.
\item $\mathcal{D}(X,\Delta_X)$ is pl-homeomorphic to a sphere $\mathbb{S}^{d-1}$, by \cite[Theorem~13]{kx}.
\end{enumerate}
\end{rmk}
\subsection{Main result} \label{sec!main-result}
In this paper we consider log Calabi--Yau pairs of the form $(\mathbb{P}^3,\Delta)$ where $\Delta$ is a quartic surface. The behaviour of $(\mathbb{P}^3,\Delta)$ depends upon the trichotomy $\coreg(\mathbb{P}^3,\Delta)=2$, $1$ or~$0$, which is equivalent to the condition that a general pencil of quartic surfaces passing though $\Delta$ defines a type I, type II or type III degeneration of K3 surfaces respectively. For the cases with $\coreg(\mathbb{P}^3,\Delta)\leq1$, we prove the following result.
\begin{thm}\label{thm!main-result} Suppose that $(\mathbb{P}^3,\Delta)$ is a log Calabi--Yau pair with $\coreg(\mathbb{P}^3,\Delta)\leq 1$. Then there is a volume preserving map $\varphi\colon(\mathbb{P}^3,\Delta)\dashrightarrow (\mathbb{P}^1\times\mathbb{P}^2,\Delta')$, where
\[ \Delta'= (\{0\}\times\mathbb{P}^2) + (\mathbb{P}^1\times E) + (\{\infty\}\times\mathbb{P}^2) \in |{-K_{\mathbb{P}^1\times\mathbb{P}^2}}| \]
for a plane cubic curve $E\subset \mathbb{P}^2_{x,y,z}$ such that
\begin{enumerate}
\item $\coreg(\mathbb{P}^3,\Delta)=1$ if and only if $E$ is smooth,
\item if $\coreg(\mathbb{P}^3,\Delta)=0$ (i.e.\ $(\mathbb{P}^3,\Delta)$ is maximal) then $E=\mathbb{V}(xyz)$. In particular, $\Delta'$ is the toric boundary of $\mathbb{P}^1\times \mathbb{P}^2$ and thus $(\mathbb{P}^3,\Delta)$ has a toric model.
\end{enumerate}
\end{thm}
\paragraph{Outline of the proof.}
A log Calabi--Yau pair $(\mathbb{P}^3,\Delta)$ has $\coreg(\mathbb{P}^3,\Delta)\leq1$ if and only if $\Delta$ has a singularity which is strictly (semi-)log canonical. Thus to prove Theorem~\ref{thm!main-result}, we start by consulting the extensive literature on the classification of singular quartic surfaces (\cite{shah,umezu,umezu2,urabe,wall} etc.) and organise all such pairs into eleven different deformation families of pairs $(\mathbb{P}^3,\Delta)$ depending on the singularities of $\Delta$. These are
\begin{description}
\item[(A.1-4)] the first four families described in Proposition~\ref{prop!type-A-quartics}, corresponding to irreducible quartic surfaces with a simple elliptic (or cusp) singularity,
\item[(B.1-3)] the first three families described in Proposition~\ref{prop!type-B-quartics}, corresponding to irreducible non-normal quartic surfaces,
\item[(C.1-4)] the four families described in \S\ref{sec!type-C-quartics}, corresponding to reducible quartic surfaces.
\end{description}
We then construct ten explicit volume preserving maps (i)-(x), as shown in Figure~\ref{fig!flowchart}, which link the different families together.
\begin{figure}[htbp]
\begin{center}
\begin{tikzpicture}[scale=1.4]
\node (a) at (0,1) {(A.3)};
\node (b) at (2,1) {(A.2)};
\node (h) at (4,1) {(B.2)};
\node (i) at (6,1) {(B.3)};
\node (j) at (10,1) {(A.1)};
\node (f) at (0,0) {(A.4)};
\node (g) at (2,0) {(B.1)};
\node (c) at (4,0) {(C.1)};
\node (d) at (6,0) {(C.2)};
\node (e) at (8,0) {(C.3)};
\node (k) at (10,0) {(C.4)};
\draw[->] (a) to node[above] {\small (i)} (b);
\draw[->] (b) to node[above] {\small (ii)} (c);
\draw[->] (f) to node[above] {\small (iii)} (g);
\draw[->] (g) to node[above] {\small (iv)} (c);
\draw[->] (h) to node[right] {\small (v)} (c);
\draw[->] (c) to node[above] {\small (vi)} (d);
\draw[->] (i) to node[right] {\small (vii)} (d);
\draw[->] (d) to node[above] {\small (viii)} (e);
\draw[->] (e) to node[above] {\small (ix)} (k);
\draw[->] (j) to node[right] {\small (x)} (k);
\end{tikzpicture}
\caption{The volume preserving maps that link the eleven different families.}
\label{fig!flowchart}
\end{center}
\end{figure}
Ultimately this shows that every pair admits a volume preserving map onto a pair from the family (C.4) which, by definition (cf.\ \S\ref{sec!type-C-quartics}), consists of all pairs $(\mathbb{P}^3,\Delta)$ whose boundary divisor $\Delta=D_1+D_2$ is the union of a plane $D_1$ and the cone over a plane cubic curve $D_2$. At this point the proof of Theorem~\ref{thm!main-result} follows easily (see \S\ref{sec!the-proof}).
\subsection{The two-dimensional cases} \label{sec!2d}
As a toy example, and because it also illustrates the basic process of our proof, we describe the 2-dimensional analogue of Theorem~\ref{thm!main-result}.
\paragraph{Classification of two-dimensional log Calabi--Yau pairs.}
If $(X,\Delta_X)$ is a two-dimensional log Calabi--Yau pair then, after replacing $(X,\Delta_X)$ by a minimal resolution of singularities and consulting the classification of surfaces, it follows that $(X,\Delta_X)$ is given by one of the following.
\begin{enumerate}
\item If $\coreg(X,\Delta_X)=2$ then $X$ is either an abelian surface or a K3 surface and $\Delta_X=0$.
\item If $\coreg(X,\Delta_X)=1$ then either
\begin{enumerate}
\item $X$ is a rational surface and $\Delta_X\in|{-K_X}|$ is a smooth elliptic curve, or
\item $\pi\colon X \to E$ is a (not necessarily minimal) ruled surface over a smooth elliptic curve $E$, and $\Delta_X=D_1+D_2\in|{-K_X}|$ is the sum of two disjoint sections of $\pi$.
\end{enumerate}
\item If $\coreg(X,\Delta_X)=0$ then $X$ is a rational surface and $\Delta_X\in|{-K_X}|$ is a (possibly reducible) reduced nodal curve of arithmetic genus 1.
\end{enumerate}
The maximal pairs are also known in the literature as \emph{Looijenga pairs} and they always have a toric model \cite[Proposition~1.3]{ghk}. Thus there is a single volume preserving equivalence class of two-dimensional maximal log Calabi--Yau pairs.
\begin{eg}\label{eg!P2}
The classification above shows that there are precisely four possibilities for a log Calabi--Yau pair of the form $(\mathbb{P}^2,\Delta)$. We either have
\begin{enumerate}
\item $\coreg(\mathbb{P}^2,\Delta)=1$, which holds if $\Delta$ is a smooth cubic curve, or
\item $\coreg(\mathbb{P}^2,\Delta)=0$, which holds if either
\begin{enumerate}
\item $\Delta_a :=\Delta$ is an irreducible nodal cubic curve,
\item $\Delta_b :=\Delta$ is the sum of a conic and (non-tangent) line, or
\item $\Delta_c :=\Delta$ is a triangle of lines.
\end{enumerate}
\end{enumerate}
It follows from the existence of toric models that the three maximal cases are all volume preserving equivalent and, indeed, it is simple to construct explicit volume preserving maps that relate them.
Recall that a quadratic transformation $\varphi\colon \mathbb{P}^2\dashrightarrow \mathbb{P}^2$ is determined by a linear system $|\mathcal{O}_{\mathbb{P}^2}(2)-p_1-p_2-p_3|$ of conics that pass through three non-collinear (but possibly infinitely near) basepoints $p_1,p_2,p_3\in\mathbb{P}^2$. We can define a volume preserving quadratic transformations $(\mathbb{P}^2,\Delta_a)\stackrel{\varphi_1}{\dashrightarrow}(\mathbb{P}^2,\Delta_b)\stackrel{\varphi_2}{\dashrightarrow}(\mathbb{P}^2,\Delta_c)$ by picking basepoints as illustrated in Figure~\ref{fig!P2} (which also shows the basepoints of $\varphi_1^{-1}$ and $\varphi_2^{-1}$).
\begin{figure}[htbp]
\begin{center}
\begin{tikzpicture} [scale=1.5]
\draw [thick, domain=0:1.5, samples=100] plot ({\x}, {sqrt(\x*(\x-1)^2)});
\draw [thick, domain=0:1.5, samples=100] plot ({\x}, {-sqrt(\x*(\x-1)^2)});
\node at (1,0) {$\bullet$};
\node at (1/2,{ sqrt(1/8)}) {$\bullet$};
\node at (1/2,{-sqrt(1/8)}) {$\bullet$};
\draw[thick, dashed,->] (1.75,0) to node[above]{$\varphi_1$} (2.25,0);
\draw [thick, domain=0:360, samples=360] plot ({(cos(\x)+1)/2+2.5}, {sin(\x)/2});
\draw [thick] (1/2+2.5,-0.7) -- (1/2+2.5,0.7);
\node at (3, 0.25) {$\bullet$};
\node at (3,-0.25) {$\bullet$};
\node at (3.5,0) {$\bullet$};
\begin{scope}[xshift = 3cm]
\draw [thick, domain=0:360, samples=360] plot ({(cos(\x)+1)/2+2.5}, {sin(\x)/2});
\draw [thick] (1/2+2.5,-0.7) -- (1/2+2.5,0.7);
\node at (3, 0.5) {$\bullet$};
\node at (2.5,0) {$\bullet$};
\node at (3.5,0) {$\bullet$};
\draw[thick, dashed,->] (4,0) to node[above]{$\varphi_2$} (4.5,0);
\draw [thick] (-0.2+5,-0.6) -- (1.2+5,-0.6);
\draw [thick] (0+5,-0.7) -- (0.6+5,0.7);
\draw [thick] (1+5,-0.7) -- (0.4+5,0.7);
\node at (5 + 0.2,-0.7+1.4/3) {$\bullet$};
\node at (5 + 0.4,-0.7+2.8/3) {$\bullet$};
\node at (5.5,-0.6) {$\bullet$};
\end{scope}
\end{tikzpicture}
\caption{Volume preserving maps between the three maximal pairs of the form $(\mathbb{P}^2,\Delta)$.}
\label{fig!P2}
\end{center}
\end{figure}
In other words, the basepoints of $\varphi_1$ are $p_1,p_2,p_3\in \Delta_a$, where $p_1$ is the node of $\Delta_a$ and $p_2,p_3$ are general points. Similarly, the basepoints of $\varphi_2$ are $p_1,p_2,p_3\in \Delta_b$, where $p_1$ is one of the nodes of $\Delta_b$ and $p_2,p_3$ are general points on the conic component of $\Delta_b$. In each case, for $\varphi_i$ to be volume preserving the basepoints are required to belong to $\Delta$, and in order to pull out a new irreducible component we let one of the basepoints coincide with a node of $\Delta$ (i.e.\ a minimal log canonical centre of $(\mathbb{P}^2,\Delta)$).
\end{eg}
Our proof of Theorem~\ref{thm!main-result} proceeds in a similar (but more involved) manner. For a given pair $(\mathbb{P}^3,\Delta)$, we find a collection of points and curves contained in the log canonical centres of $(\mathbb{P}^3,\Delta)$ which form the baselocus for volume preserving map $\varphi\colon(\mathbb{P}^3,\Delta)\dashrightarrow(\mathbb{P}^3,\Delta')$ such that $\Delta'$ is `simpler' than $\Delta$ (where `simpler' is to be interpreted in accordance with the structure of the graph in Figure~\ref{fig!flowchart}).
\subsection{Relationship to other work}
\subsubsection{Characterising maximal pairs with a toric model}
Finding criteria which characterise maximal log Calabi--Yau pairs with a toric model is a difficult problem which originated in work of Shokurov. Theorem~\ref{thm!main-result}(2) is a special case of the following conjecture.
\begin{conj}\label{conj!toric-model}
Suppose that $(X,\Delta_X)$ is a maximal log Calabi--Yau pair and $X$ is a rational $3$-fold. Then $(X,\Delta_X)$ has a toric model.
\end{conj}
Unfortunately the simple and appealing statement of Conjecture~\ref{conj!toric-model} fails miserably as soon as one tries to relax any of the given assumptions.
\begin{rmk}\label{rmk!restrictions}
In light of the three consequences of Remark~\ref{rmk!toric-models}, we note that the following conditions in the statement of Conjecture~\ref{conj!toric-model} are essential.
\begin{enumerate}
\item It is necessary to assume that $X$ is rational, since there exist examples of non-rational maximal log Calabi--Yau 3-fold pairs constructed by Kaloghiros \cite{kal} and Svaldi \cite[Example~5]{kal}. (This is in contrast to the 2-dimensional setting, in which maximal pairs are always rational.)
\item It is necessary to assume that $\dim X=3$, since the examples of Kaloghiros can easily be used to produce a maximal log Calabi--Yau pair of the form $(\mathbb{P}^4,\Delta)$ where $\Delta$ contains an irreducible component which is a non-rational quartic 3-fold.
\item We cannot relax the condition $K_X+\Delta_X\sim 0$ in Definition~\ref{def!lCY} to include the case that $K_X+\Delta_X\sim_\mathbb{Q} 0$, since it is easy to construct a 3-fold pair $(X,\Delta_X)$ with $2(K_X+\Delta_X)\sim 0$ but for which $\mathcal{D}(X,\Delta_X)\simeq \mathbb{R}\mathbb{P}^2$. (For example, take the quotient of the toric variety $(\mathbb{P}^1)^3$ by the involution acting by $(x,y,z)\mapsto(x^{-1},y^{-1},z^{-1})$ on the dense open torus $(\mathbb{C}^\times)^3\subset (\mathbb{P}^1)^3$.)
\end{enumerate}
\end{rmk}
Concerning Remark~\ref{rmk!restrictions}(3), we note that Filipazzi, Mauri \& Moraga \cite{coreg} have shown that a maximal pair $(X,\Delta_X)$ (in our context of Definition~\ref{def!lCY}) satisfies either $K_X+\Delta_X\sim 0$ or $2(K_X+\Delta_X)\sim 0$, and that these two possibilities are distinguished by the orientability of $\mathcal{D}(X,\Delta_X)$. Moreover, Remark~\ref{rmk!toric-models}(3) will never provide an obstacle to Conjecture~\ref{conj!toric-model} since $\mathcal{D}(X,\Delta_X)\simeq \mathbb S^2$ for maximal 3-fold pairs $(X,\Delta_X)$ with $K_X+\Delta_X\sim 0$ by \cite[\S33]{kx}.
\subsubsection{Cremona equivalence of rational quartic surfaces with a plane}
Mella \cite{mella} has shown that every rational quartic surface $\Delta\subset \mathbb{P}^3$ is Cremona equivalent to a hyperplane $H\subset\mathbb{P}^3$. That is to say that there exists a birational map $\varphi \colon \mathbb{P}^3\dashrightarrow \mathbb{P}^3$ which maps $\Delta$ birationally onto $H$. Our theorem strengthens this result of Mella (at least in the case that $(\mathbb{P}^3,\Delta)$ is log canonical) by showing that a rational quartic $\Delta\subset \mathbb{P}^3$ can be mapped onto a hyperplane by a volume preserving map for the pair $(\mathbb{P}^3,\Delta)$. Most of the maps that Mella constructs do not extend to volume preserving maps of $(\mathbb{P}^3,\Delta)$, and thus we need to proceed rather more carefully. Roughly speaking, in order for $\varphi$ to be volume preserving we need to ensure that the $k$-dimensional components of the baselocus of $\varphi$ are contained in $(k+1)$-dimensional log canonical centres of $(\mathbb{P}^3,\Delta)$.
\subsubsection{Volume preserving subgroups of $\Bir(\mathbb{P}^n)$}
A log Calabi--Yau pair $(\mathbb{P}^n,\Delta)$ (up to volume preserving equivalence) determines a subgroup $\Bir^{\text{vp}}(\mathbb{P}^n,\Delta)\subseteq \Bir(\mathbb{P}^n)$ (up to conjugation), where $\Bir^{\text{vp}}(\mathbb{P}^n,\Delta)$ is the subgroup consisting of volume preserving birational self-maps of $(\mathbb{P}^n,\Delta)$. It is an interesting question to know how big (or small) this subgroup can be, depending on the geometry of $(\mathbb{P}^n,\Delta)$, and whether one can describe a set of maps that generate it.
A complete picture is known in the case of $\mathbb{P}^2$. For the pairs $(\mathbb{P}^2,\Delta)$ of coregularity one, any map $\varphi\colon (\mathbb{P}^2,\Delta)\dashrightarrow (\mathbb{P}^2,\Delta)$ induces a birational map $\varphi|_\Delta\colon \Delta\dashrightarrow \Delta$ which is necessarily an isomorphism. Thus $\Bir^{\text{vp}}(\mathbb{P}^2,\Delta)$ coincides with the \emph{decomposition group} of the smooth plane cubic curve $\Delta$, which has been studied by Pan \cite{pan}. For the pairs of coregularity zero, Blanc \cite{blanc} has given a very explicit description of the group $\Bir^{\text{vp}}(\mathbb{P}^2,\Delta)$ when $\Delta=\mathbb{V}(xyz)$ is the triangle of coordinate lines.
In dimension 3, Araujo, Corti \& Massarenti \cite{acm} consider the case of a very general quartic surface $\Delta\subset\mathbb{P}^3$ (in particular, $\Delta$ is smooth and has Picard rank 1), and show that $\Bir^{\text{vp}}(\mathbb{P}^3,\Delta)$ consists only of those automorphisms of $\mathbb{P}^3$ that preserve $\Delta$. Moreover, they also give an explicit description of $\Bir^{\text{vp}}(\mathbb{P}^3,\Delta)$ in the case that $\Delta$ is a general quartic surface with a single ordinary double point.
\subsubsection{Pairs $(\mathbb{P}^3,\Delta)$ of coregularity two}
Theorem~\ref{thm!main-result} only treats the case of pairs $(\mathbb{P}^3,\Delta)$ of coregularity at most one. The remaining case $\coreg(\mathbb{P}^3,\Delta)=2$ occurs if and only if $\Delta$ is an irreducible quartic surface with at worst Du Val singularities.
Aside from the results of \cite{acm} mentioned above, giving an explicit classification of all such pairs up to volume preserving equivalence will be difficult, and significantly more involved than simply classifying quartic surfaces up to birational equivalence. For example, Oguiso \cite{oguiso} has given an example of two smooth isomorphic quartic surfaces $\Delta_1,\Delta_2\subset \mathbb{P}^3$ for which there is no map $\varphi\in \Bir(\mathbb{P}^3)$ (let alone a volume preserving one) that maps $\Delta_1$ birationally onto $\Delta_2$.
\subsection{Notation}
We use $\dP_d$ to denote a del Pezzo surface of degree $d$, possibly with Du Val singularities. We often need to consider curves which are either smooth elliptic curves, or reduced nodal curves of arithmetic genus 1. Since repeating this each time we want to use it is a bit of a mouthful we call such a curve an \emph{ordinary curve}.
\subsection{Acknowledgements}
I would like to thank Anne-Sophie Kaloghiros for some very helpful correspondence and comments on the topic of this paper.
\section{Log Calabi--Yau pairs}
We begin with some useful results concerning the geometry of log Calabi--Yau pairs.
\begin{defn}\label{def!lCY}
A log Calabi--Yau pair $(X,\Delta_X)$ is a log canonical pair consisting of a proper variety $X$ over $\mathbb{C}$ and a reduced effective integral Weil divisor\footnote{More generally, it is sometimes assumed that $\Delta_X$ has $\mathbb{Q}$-coefficients and that $K_X+\Delta_X\sim_\mathbb{Q} 0$ is \emph{only $\mathbb{Q}$-linearly trivial}, but we will always assume that $\Delta_X$ is integral (cf. Remark~\ref{rmk!restrictions}(3)).}
$\Delta_X$ such that $K_X+\Delta_X\sim 0$.
\end{defn}
A global section of $H^0(X,K_X+\Delta_X)\cong \mathbb{C}$ defines a meromorphic volume form $\omega_{\Delta_X}$ on $X$ with $\operatorname{div}(\omega_{\Delta_X})=\Delta_X$, and which is uniquely determined up to scalar multiplication.
\subsection{Volume preserving maps}
The natural notion of birational equivalence between log Calabi--Yau pairs is that of volume preserving equivalence (cf.\ \cite[Definition~2.23]{kollar}).
\begin{defn} \label{def!vp}
A proper birational morphism of pairs $f\colon (Z,\Delta_Z) \to (X,\Delta_X)$ is called \emph{crepant} if
$f_*(\Delta_Z)=\Delta_X$ and $f^*(K_X+\Delta_X) \sim K_Z+\Delta_Z$. A birational map of pairs $\varphi\colon (Y,\Delta_Y) \dashrightarrow (X,\Delta_X)$ is called \emph{crepant} if it admits a resolution of the form
\begin{equation}
\begin{tikzcd}
& (Z,\Delta_Z) \arrow[rd, "g"] & \\
(Y,\Delta_Y) \arrow[leftarrow, ru, "f"] \arrow[rr, dashed, "\varphi"] & & (X,\Delta_X)
\end{tikzcd}
\end{equation}
where $f$ and $g$ are crepant birational morphisms.
\end{defn}
In the context of log Calabi--Yau pairs $(X,\Delta_X)$ and $(Y,\Delta_Y)$, crepant birational maps are also known as \emph{volume preserving maps},\footnote{We use this terminology despite the following potential for confusion: the condition for $\varphi\colon(\mathbb{P}^3,\Delta_1)\dashrightarrow (\mathbb{P}^3,\Delta_2)$ to be volume preserving depends on the choice of $\Delta_1,\Delta_2\subset\mathbb{P}^3$. It is \emph{not} necessarily the case that $\varphi^*\omega_{\Delta_2}= \omega_{\Delta_2}$.} since $\varphi^*\omega_{\Delta_X}=\omega_{\Delta_Y}$ for an appropriate rescaling of the naturally defined volume form on each side \cite[Remark 5]{ck}.
\begin{rmk}
An easy consequence of the definition is that a volume preserving map preserves discrepancies, i.e.\ that $a_E(X,\Delta_X)=a_E(Y,\Delta_Y)$ for any exceptional divisor $E$ over both $X$ and $Y$, where $a_E(X,\Delta_X)\in\mathbb{Q}$ denotes the discrepancy of $E$ over $(X,\Delta_X)$. Moreover a composition of volume preserving maps is volume preserving.
\end{rmk}
\subsection{Dlt modifications}
The main problem with considering log canonical pairs $(X,\Delta_X)$ in general is that they can exhibit rather complicated singularities. Life becomes easier if we focus on pairs with \emph{divisorial log terminal} (dlt) singularities. This is always possible by passing to a dlt modification.
\begin{prop}[\cite{ck} Theorem 7]
Given a log Calabi--Yau pair $(X,\Delta_X)$, there exists a volume preserving map $\varphi\colon (\widetilde X,\Delta_{\widetilde X}) \to (X,\Delta_X)$ where $(\widetilde X,\Delta_{\widetilde X})$ is a $\mathbb{Q}$-factorial dlt pair and $\widetilde X$ has at worst terminal singularities.
\end{prop}
One of the most pleasing consequences of working with a dlt pair $(X,\Delta_X)$ is that it is easy to understand the log canonical centres of $(X,\Delta_X)$ and they satisfy some very pleasing forms of adjunction.
\begin{thm}[cf.\ \cite{kollar} Theorems 4.6, 4.16 \& 4.19]
If $(X,\Delta_X)$ is a dlt log Calabi--Yau pair and $\Delta_X=\sum_{i=1}^k D_i$ then
\begin{enumerate}
\item the log canonical centres of $(X,\Delta_X)$ are precisely the irreducible components of $D_J := \bigcap_{j\in J}D_j$ for any subset $J\subseteq \{1,\ldots,k\}$ (where $D_\emptyset=X$),
\item every such log canonical centre is normal and has pure codimension $\#J$,
\item for any log canonical centre $Z\subset X$ there is a naturally defined\footnote{In favourable situations, for example if $Z=D_1\cap\cdots\cap D_k$ is an intersection of Cartier divisors in $X$, then $\Delta_{Z} := (\Delta_X - D_1-\cdots-D_k)|_{Z}$ is obtained by repeated application of the adjunction formula. In general it is given by a variant of the \emph{different} $\Delta_{Z} := \Diff^*_{Z}(\Delta_X)$.} divisor class $\Delta_{Z}$ on $Z$ such that $(Z,\Delta_{Z})$ is a dlt log Calabi--Yau pair,
\item if $\varphi\colon (X,\Delta_X)\dashrightarrow(Y,\Delta_Y)$ is a volume preserving map which restricts to a birational map of log canonical centres $\varphi|_{Z_X}\colon (Z_X,\Delta_{Z_X})\dashrightarrow(Z_Y,\Delta_{Z_Y})$, then $\varphi|_{Z_X}$ is also volume preserving.
\end{enumerate}
\end{thm}
Thus the boundary divisor $\Delta_X$ of a dlt log Calabi--Yau pair can be thought of as a collection of log Calabi--Yau pairs of dimension $d-1$, glued together along their boundary components. One can make a similar study of log canonical log Calabi--Yau pairs, but in general the picture is significantly more complicated (see \cite[\S4]{kollar} for details).
\subsection{The coregularity}
Since volume preserving maps preserve discrepancies, they map log canonical centres onto log canonical centres. In particular, one can use this to show that the dimension of a minimal log canonical centre on a dlt modification is a volume preserving invariant\footnote{This is rather crude invariant in general, since more is true: any two minimal log canonical centres $Z,Z'\subset X$ of a log Calabi--Yau pair $(X,\Delta_X)$ are birational to one another \cite[Theorem~4.40]{kollar} and, in fact, the volume preserving equivalence class of $(Z,\Delta_Z)$ is an even finer invariant.} of $(X,\Delta_X)$.
\begin{defn}\label{def!coreg}
The \emph{coregularity} $\coreg(X,\Delta_X)$ is defined to be the dimension of a minimal log canonical centre in a dlt modification $\varphi\colon (\widetilde X,\Delta_{\widetilde X})\to (X,\Delta_X)$. A log Calabi--Yau pair $(X,\Delta_X)$ is called \emph{maximal} if $\coreg(X,\Delta_X)=0$.
\end{defn}
Given a log canonical centre $Z\subset X$ of a dlt pair $(X,\Delta_X)$ then $\coreg(X,\Delta_X)=\coreg(Z,\Delta_Z)$, since any smaller log canonical centre $Z'\subset Z\subset X$ restricts to a log canonical centre of $(Z,\Delta_Z)$ by \cite[Theorem~4.19(3)]{kollar}.
\subsection{The dual complex $\mathcal{D}(X,\Delta_X)$}
Although we will not use it, we briefly recall the \emph{dual complex} $\mathcal{D}(X,\Delta_X)$ of a log Calabi--Yau pair $(X,\Delta_X)$ since it was mentioned in the introduction. This is a simplicial complex which encodes the geometry of the log canonical centres of $(X,\Delta_X)$ obtained by associating a $(k-1)$-dimensional simplex $\sigma_Z$ to each $k$-codimensional log canonical centre $Z\subsetneq X$, which are then glued together according to inclusion. Thus $\mathcal{D}(X,\Delta_X)$ has dimension $\dim \mathcal{D}(X,\Delta_X)=\dim X - \coreg(X,\Delta_X) - 1$, and this is of maximal possible dimension if $\coreg(X,\Delta_X)=0$ (which is one explanation for the terminology `maximal pair'). A key theorem of Koll\'ar \& Xu relates volume preserving maps of pairs to homeomorphisms of their dual complexes.
\begin{thm}[\cite{kx} Theorem 13] \label{thm!vp-dual-cx}
A volume preserving map $\varphi\colon (X,\Delta_X)\to (Y,\Delta_Y)$ induces a piecewise linear homeomorphism of dual complexes $\varphi_*\colon \mathcal{D}(X,\Delta_X)\to \mathcal{D}(Y,\Delta_Y)$ .
\end{thm}
\section{A rough classification of quartic surfaces} \label{sect!quartics}
We now recall some results on the classification of quartic surfaces. Quartic surfaces can have one of many thousands of different singularity types \cite{degtyarev}, but they have been well-studied and the study of the classification of singular quartic surfaces goes back to Jessop \cite{jess}. Moreover, since then other authors have also given very precise descriptions of the type of singularities that a quartic surface can have, e.g.\ \cite[Corollary 2.3 \& Theorem 2.4]{shah}.
We divide log Calabi--Yau pairs $(\mathbb{P}^3,\Delta)$ of coregularity $\leq1$ into eleven different families according to the singularities of $\Delta$, as described in \S\ref{sec!main-result}. Each family is taken to be closed under degeneration and they are not supposed to be mutually exclusive. Moreover, every such pair belongs to one of these eleven families. Clearly every reducible quartic surface $\Delta$ is either the union of a plane and cubic surface (C.1) or two quadrics (C.2), or a degeneration of one of these two cases. The fact that every log canonical pair with irreducible boundary divisor belongs to one of the other seven families follows from Proposition~\ref{prop!type-A-quartics} and Proposition~\ref{prop!type-B-quartics}.
\subsection{Two-dimensional semi-log canonical singularities}
In Table~\ref{table!log-canonical} we present the classification of two-dimensional strictly (semi-)log canonical hypersurface singularities $\mathbb{V}(f(x,y,z))\subset\mathbb{A}^3_{x,y,z}$, up to local analytic isomorphism, cf.\ \cite{slc}.
\begin{table}[htp]
\caption{Two-dimensional strictly semi-log canonical hypersurface singularities}
\begin{center}
\def1.5{1.3}
\begin{tabular}{|c|cc|cc|} \hline
Isolated & Type & Name & Normal form for $f$ & Condition \\ \hline
\multirow{4}{*}{Yes} & \multirow{3}{*}{Simple elliptic} & $\widetilde E_6$ (or $T_{333}$) & $\lambda xyz = x^3 + y^3 + z^3$ & $\lambda^3\neq 27$ \\
& & $\widetilde E_7$ (or $T_{244}$) & $\lambda xyz = x^2 + y^4 + z^4$ & $\lambda^4\neq 64$ \\
& & $\widetilde E_8$ (or $T_{236}$) & $\lambda xyz = x^2 + y^3 + z^6$ & $\lambda^6\neq 432$ \\ \cline{2-5}
& Cusp & $T_{pqr}$ & $xyz = x^p + y^q + z^r$ & $\tfrac{1}{p}+\tfrac{1}{q}+\tfrac{1}{r}<1$ \\ \hline
\multirow{5}{*}{No} & Normal crossing & $A_\infty$ & $xy = 0$ & \\
& Pinch point & $D_\infty$ & $x^2 + y^2z = 0$ & \\ \cline{2-5}
& \multirow{3}{*}{Degenerate cusp} & $T_{\infty\infty\infty}$ & $xyz = 0$ & \\
& & $T_{p\infty\infty}$ & $xyz = x^p$ & $p\geq2$ \\
& & $T_{pq\infty}$ & $xyz = x^p + y^q$ & $\tfrac{1}{p}+\tfrac{1}{q}<1$ \\ \hline
\end{tabular}
\end{center}
\label{table!log-canonical}
\end{table}%
When we refer to $p\in \Delta$ as an `\emph{$\widetilde E_k$ singularity}' we implicitly take that to include the possibility that $p\in \Delta$ is a degeneration of an $\widetilde E_k$ singularity. For example, the cusp singularities $T_{pqr}$ with $3\leq p,q,r\leq \infty$ are $\widetilde E_6$ singularities, the cusp singularities $T_{2qr}$ with $4\leq q,r\leq \infty$ are $\widetilde E_7$ singularities, and the cusp singularities $T_{23r}$ with $6\leq r\leq \infty$ are $\widetilde E_8$ singularities.
\paragraph{Coregularity of $(\mathbb{P}^3,\Delta)$.}
Now let $(X,\Delta_X)$ be a 3-fold pair and $p\in \Delta_X\subset X$ is a point at which $X$ is smooth, but $\Delta_X$ has an $\widetilde E_6$, $\widetilde E_7$ or $\widetilde E_8$ singularity. We consider the weighted blowup
\[ \pi\colon \left(E\cong\mathbb{P}(a,b,c)\subset \widetilde X\right) \to (p\in X), \]
with weights $\{a,b,c\}=\{1,1,1\}$, $\{2,1,1\}$ or $\{3,2,1\}$ given to the local coordinates $x,y,z$ for the normal form presented in Table~\ref{table!log-canonical}, so that $\deg f(x,y,z)\geq3$, $4$ or $6$ in each case respectively. In all cases $E$ is a log canonical centre of $(X,\Delta_X)$ and, setting $\Delta_{\widetilde X} := f^{-1}_*\Delta_X + E$, the map $\pi\colon (\widetilde X,\Delta_{\widetilde X})\to (X,\Delta_X)$ is volume preserving. Moreover the curve $\Gamma := E\cap f^{-1}_*\Delta_X$ is also a log canonical centre of $(X,\Delta_X)$, and is a smooth elliptic curve if $p\in \Delta_X$ is a simple elliptic singularity, and a reduced nodal curve otherwise.
In particular if $\coreg(\mathbb{P}^3,\Delta)=1$ then $\Delta$ can only have either simple elliptic singularities or a double curve with a finite number of pinch points. Similarly, $\coreg(\mathbb{P}^3,\Delta)=0$ if and only if $\Delta$ has a cusp singularity or a degenerate cusp singularity.
\subsection{Irreducible quartic surfaces with isolated singularities}
There are four distinct ways in which an irreducible quartic surface can have an isolated strictly log canonical singularity. Singularities of type $\widetilde E_6$ and $\widetilde E_7$ each appear in an essentially unique way, but singularities of type $\widetilde E_8$ can appear in one of two different ways (cf.\ \cite{jess, noether, shah, wall}).
\begin{prop}\label{prop!type-A-quartics}
Suppose that $\Delta=\mathbb{V}(F(x,y,z,t))\subset \mathbb{P}^3$ is a reduced irreducible quartic surface such that $\Delta$ has at least one isolated simple elliptic (or cusp) singularity and possibly some additional Du Val singularities. Then, up to projective equivalence, one of the following occurs.
\begin{enumerate}
\item (\cite[Theorem~8.1(iii)]{wall}) $\Delta$ is a rational surface with exactly one such singularity $p\in \Delta$. The type of singularity $p\in \Delta$ and the form of the equation $F(x,y,z,t)$ are given by one of the following four cases.
\begin{description}
\item[(A.1)] type $\widetilde E_6$: $\deg F\geq3$ with respect to the weights $(0,1,1,1)$ for $(t,x,y,z)$,
\item[(A.2)] type $\widetilde E_7$: $\deg F\geq4$ with respect to the weights $(0,1,1,2)$ for $(t,x,y,z)$,
\item[(A.3)] type $\widetilde E_8$: $\deg F\geq6$ with respect to the weights $(0,1,2,3)$ for $(t,x,y,z)$, or
\item[(A.4)] type $\widetilde E_8$: $\deg F\geq6$ with respect to the weights $(0,1,2,2,3)$ for $(t,x,y,z,tz+x^2)$.
\end{description}
\item (\cite[Theorems 1 \& 2]{umezu2}) $\Delta$ is an elliptic ruled surface with exactly two such singularities $p_1,p_2\in \Delta$ which are necessarily simple elliptic singularities of the same type. The type of singularities $p_1,p_2\in \Delta$ and the form of the equation $F(x,y,z,t)$ are given by one of the following two cases.
\begin{description}
\item[(A.2*)] $2\times\widetilde E_7$: $\deg F\geq4$ with respect to weights $(0,1,1,2)$ and $(2,1,1,0)$ for $(t,x,y,z)$, or
\item[(A.3*)] $2\times\widetilde E_8$: $\deg F\geq6$ with respect to weights $(0,1,2,3)$ and $(3,2,1,0)$ for $(t,x,y,z)$.
\end{description}
\end{enumerate}
Moreover, in order for $p\in\Delta$ to be log canonical, in each case the appropriate weighted tangent cone must define the cone over an ordinary curve.
\end{prop}
We treat cases (A.2*) and (A.3*) as degenerate cases of (A.2) and (A.3).
\begin{rmk}
The two different types of $\widetilde E_8$ singularity can be distinguished by the fact that the quartics in family (A.3) contain a line $\mathbb{V}(y,z)$ which through $P\in \Delta$, whereas the generic member of family (A.4) does not contain any line. From the point of view of GIT stability, that quartics in family (A.3) are unstable but those in (A.4) are stable \cite{shah,wall}.
\end{rmk}
\paragraph{Parameterisation of the rational cases.}
The study of rational quartics with these four types of simple elliptic singularities goes back to Noether \cite{noether}. The rational parameterisation of quartics with a triple point (A.1) is straightforward. In each of the other cases (A.2-4) Noether produced a rational parameterisation which we now describe.
First note that $(\Delta,0)$ is a log Calabi--Yau pair by adjunction, and we consider the volume preserving minimal resolution of log Calabi--Yau pairs $\mu\colon (\widetilde \Delta,D)\to (\Delta,0)$ where $D\subset \widetilde \Delta$ the reduced exceptional curve (or reduced exceptional cycle) over the simple elliptic (or cusp) singularity $p\in \Delta$. By the classification of two-dimensional log Calabi--Yau pairs \S\ref{sec!2d}, $\widetilde \Delta$ is a rational surface. Let $(\widetilde \Delta_0, D_0) := (\widetilde \Delta, D)$ and, for $i=1,\ldots,k$, let $f_i\colon(\widetilde \Delta_{i-1},D_{i-1})\to(\widetilde \Delta_{i},D_{i})$ be a sequence of volume preserving blowdowns (i.e.\ obtained by setting $D_i:=f_i(D_{i-1})$) ending with a minimal surface $\widetilde \Delta_k$.
\begin{equation} \label{eq!delta-res}
\begin{tikzcd}
(\Delta,0) \arrow[leftarrow, r, "\mu"] & (\widetilde \Delta,D) =: (\widetilde \Delta_0,D_0) \arrow[r, "f_1"] & (\widetilde \Delta_1,D_1) \arrow[r, "f_2"] & \cdots \arrow[r, "f_k"] & (\widetilde \Delta_k,D_k)
\end{tikzcd}
\end{equation}
By choosing a sequence of contractions carefully it is possible to show that we can always find a sequence ending with $\widetilde \Delta_k=\mathbb{P}^2$ and therefore $D_k\subset\mathbb{P}^2$ is either a smooth or nodal cubic curve. Let $f\colon \widetilde \Delta\to \mathbb{P}^2$ be the composition of the $f_i$ and consider $\NS(\widetilde \Delta) = \mathbb{Z}\langle h, e_1,\ldots e_k\rangle$ given with its standard basis, where $h=f^*\mathcal{O}_{\mathbb{P}^2}(1)$ and $e_i$ is the total transform of class of the exceptional divisor of $f_i$. Then we have $D\sim 3h-e_1-\ldots-e_k$ and the map $\mu\colon\widetilde \Delta\to \Delta\subset \mathbb{P}^3$ is induced by a nef divisor class $A=\mu^*\mathcal{O}_{\Delta}(1)$ satisfying $h^0(\widetilde\Delta,A)=4$, $A^2=4$ and $A\cdot D=0$. The possibilities for $A$ are given in Table~\ref{table!type-A-quartics}.
\begin{table}[htp]
\begin{center}
\caption{Irreducible rational quartic surfaces with an isolated log canonical singularity.}
\label{table!type-A-quartics}
\resizebox{!}{!}{
\def1.5{1.5}
\begin{tabular}{|c|ccc|c|} \hline
Case & Singularity & $\widetilde \Delta$ & $A$ & Noether \cite{noether} \\ \hline
(A.1) & $\widetilde E_6$ & $\Bl_{12}\mathbb{P}^2$ & $4h-\sum\limits_{i=1}^{12} e_i$ & \\
(A.2) & $\widetilde E_7$ & $\Bl_{11}\mathbb{P}^2$ & $6h-\sum\limits_{i=1}^7 2e_i-\sum\limits_{i=8}^{11}e_i$ & $F_4^{(1)}$ \\
(A.3) & $\widetilde E_8$ & $\Bl_{10}\mathbb{P}^2$ & $9h-\sum\limits_{i=1}^{8}3e_i-2e_9-e_{10}$ & $F_4^{(3)}$ \\
(A.4) & $\widetilde E_8$ & $\Bl_{10}\mathbb{P}^2$ & $7h-3e_1-\sum\limits_{i=2}^{10}2e_i$ & $F_4^{(2)}$ \\ \hline
\end{tabular}}
\end{center}
\end{table}%
\paragraph{The ruled elliptic cases.}
The remaining ruled elliptic cases (A.2*) and (A.3*) were described by Umezu~\cite[Theorems 1\& 2]{umezu2}. She gives a similar construction of them, by taking a minimal volume preserving resolution of singularities and blowing down to a minimal ruled elliptic surface.
\subsection{Irreducible quartic surfaces with non-isolated singularities} \label{sec!type-B-quartics}
The classification of non-normal quartic surfaces is contained in Jessop's book \cite{jess}, but it has also been considered in more modern times by Urabe \cite{urabe}. We follow Urabe's treatment and his subdivision into eight classes. Only two families in Urabe's classification have a general member which has worse than semi-log canonical singularities: (I) corresponding to the cone over a plane quartic curve, and (II-2) corresponding to a ruled elliptic surface with a line of cuspidal singularities. We recall the remaining cases.
\begin{prop}[\cite{urabe}]\label{prop!type-B-quartics}
Suppose that $\Delta\subset \mathbb{P}^3$ is a reduced irreducible non-normal quartic surface with semi-log canonical singularities. Then $\Delta$ has double points along a curve $\Sigma\subset \Delta$ and possibly some Du Val singularities outside of $\Sigma$. Moreover $\Sigma$ is (possibly a degeneration of) one of the following cases.
\begin{enumerate}
\item $\Delta$ is a rational surface and the curve $\Sigma$ is
\begin{description}
\item[(B.1)] a line \cite[(III-C)]{urabe},
\item[(B.2)] a plane conic \cite[(III-B)]{urabe},
\item[(B.3)] a twisted cubic \cite[(III-A-2) \& (III-A-3)]{urabe},
\item[(B.4)] the union of three concurrent lines \cite[(III-A-1)]{urabe}, or
\end{description}
\item $\Delta$ is a elliptic ruled surface and the curve $\Sigma$ is
\begin{description}
\item[(B.1*)] a pair of skew lines \cite[(II-1)]{urabe}.
\end{description}
\end{enumerate}
\end{prop}
\begin{rmk}
As one may see, cases (B.4) and (B.1*) do not appear in the list of families considered in \S\ref{sec!main-result}. The generic member of family (B.4) is isomorphic to Steiner's Roman surface. Given that this surface necessarily has a triple point at the intersection of the three lines, we treat it as a special case of (A.1). We treat case (B.1*) as a special case of (B.1).
\end{rmk}
\paragraph{Parameterisation of the rational cases.}
Urabe also provides an explicit construction for all cases of his classification, which is analogous to the results of Noether and Umezu discussed in isolated singularity case above. We recall the description for the rational cases.
The normalisation $\nu\colon (\overline\Delta,\overline D) \to (\Delta,0)$ is a volume preserving map of log Calabi--Yau pairs, where $\overline D=\nu^{-1}(\Sigma)$ is the preimage of the double curve. Let $\mu\colon(\widetilde\Delta, D)\to(\overline \Delta,\overline D)\to(\Delta,0)$ be the volume preserving minimal resolution of singularities which factors through $\nu$. As in \eqref{eq!delta-res} above, we consider a sequence of volume preserving blowdowns $f=f_k\circ\cdots\circ f_1$ from $(\widetilde \Delta_0,D_0):=(\widetilde \Delta,D)$ to a minimal pair $(\widetilde \Delta_k,D_k)$ and we keep the same notation for $\NS(\widetilde\Delta)=\mathbb{Z}\langle h,e_1,\ldots, e_k\rangle$ and the divisor class $A=\mu^*\mathcal{O}_\Delta(1)$.
For the case (B.1), Urabe shows that it is always possible to find a sequence of nine contractions ending in $\widetilde D_9=\mathbb{P}^2$, and for which $A=4h-2e_1-\sum_{i=2}^9e_i$. In the three remaining cases, we have that $\mu^*|\mathcal{O}_\Delta(1)|\subsetneq|A|$ is a strict linear subsystem of $|A|$. In these cases the normalisation $\overline\Delta$ can be realised as $\nu\colon (\overline\Delta\subset\mathbb{P}^{h^0(A)-1})\dashrightarrow (\Delta\subset \mathbb{P}^3)$, where $\nu$ is the projection from a general linear subspace of $\mathbb{P}^{h^0(A)-1}$ which is disjoint from $\overline\Delta$. In particular, one of the following cases occurs.
\begin{description}
\item[(B.2)] $\overline\Delta \cong \Bl_5\mathbb{P}^2\subset\mathbb{P}^4$ is an anticanonically embedded $\dP_4$ and $\widetilde \Delta\to\overline\Delta$ is at worst the crepant resolution of some Du Val singularities.
\item[(B.3)] $\widetilde\Delta=\overline\Delta\subset\mathbb{P}^5$ is either $\mathbb{P}^1\times\mathbb{P}^1$ embedded by $\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(2,1)$, or $\mathbb{F}_2$ embedded by $|s+f|$ where $s$ is a (positive) section of $\mathbb{F}_2$ and $f$ is the class of a fibre. By blowing up one more point on $\widetilde \Delta$ we can treat these both as one case, where $\widetilde \Delta=\Bl_2\mathbb{P}^2$ is a $\dP_7$. (The difference between the two cases is then whether these two points are infinitely near or not.)
\item[(B.4)] $\widetilde\Delta=\overline\Delta\cong\mathbb{P}^2\subset\mathbb{P}^5$ is embedded by $\mathcal{O}_{\mathbb{P}^2}(2)$ (i.e.\ the second Veronese embedding of $\mathbb{P}^2$).
\end{description}
We summarise these results in Table~\ref{table!type-B-quartics}.
\begin{table}[htp]
\begin{center}
\caption{Irreducible rational quartic surfaces with a double curve $\Sigma\subset \Delta$.}
\label{table!type-B-quartics}
\resizebox{\textwidth}{!}{
\def1.5{1.5}
\begin{tabular}{|c|cccc|c|} \hline
Case & Type of $\Sigma$ & $\widetilde \Delta$ & $A$ & $h^0(A)$ & Urabe \cite{urabe} \\ \hline
(B.1) & Line & $\Bl_9\mathbb{P}^2$ & $4h - 2e_1-\sum\limits_{i=2}^{9}e_i$ & 4 & (III-C) \\
(B.2) & Conic & $\Bl_5\mathbb{P}^2$ & $3h - \sum\limits_{i=1}^{5}e_i$ & 5 & (III-B) \\
(B.3) & Twisted cubic & $\Bl_2\mathbb{P}^2$ & $3h-2e_1-e_2$ & 6 & (III-A-2/3) \\
(B.4) & Three concurrent lines & $\mathbb{P}^2$ & $2h$ & 6 & (III-A-1) \\ \hline
\end{tabular}}
\end{center}
\end{table}%
\subsection{Reducible quartic surfaces}
\label{sec!type-C-quartics}
The remaining families correspond to pairs with a reducible boundary divisor and we divide them up into the following four families. The subdivision into these four particular cases may look somewhat artificial or arbitrary. Our only reason for considering it is that it corresponds to the logical structure of our proof of Theorem~\ref{thm!main-result}.
\begin{description}
\item[(C.1)] $\Delta$ is the union of a plane and a cubic surface,
\item[(C.2)] $\Delta$ is the union of two quadrics,
\item[(C.3)] $\Delta$ is the union of a plane and a singular cubic surface,
\item[(C.4)] $\Delta$ is the union of a plane and the cone over a cubic curve.
\end{description}
\section{Low degree maps in $\Bir(\mathbb{P}^3)$}
The \emph{bidegree} of a 3-dimensional birational map $\varphi\in\Bir(\mathbb{P}^3)$ is given by $(\deg\varphi,\,\deg\varphi^{-1})\in\mathbb{Z}_{\geq1}^2$. Maps with low bidegree are well-understood and there are some very detailed classification results. For example, Pan, Ronga \& Vust \cite{prv} show that quadratic maps can have bidegree $(2,d)$ for $d=2,3,4$, and these three types of map comprise three irreducible families $\mathcal F_{(2,d)}$ of dimensions $29,28,26$ respectively. Moreover they give a complete description of the strata of $\mathcal F_{(2,d)}$, by exhibiting all possible ways in which the baselocus $\Bs(\varphi)$ can degenerate. Deserti \& Han \cite{dh} provide a similar analysis for a large part of the landscape of maps of degree 3.
\subsection{Strategy} \label{sec!strategy}
In this section we exhibit a few examples of maps $\varphi\in\Bir(\mathbb{P}^3)$ of low bidegree that we will use to construct some of the links between in our families in \S\ref{sec!proof}. For each map $\varphi$ we construct a resolution of the following form.
\begin{equation*}
\begin{tikzcd}
& X \arrow[rd, "\psi'"] & \\
\mathbb{P}^3 \arrow[leftarrow, ru, "\psi"] \arrow[rr, dashed, "\varphi"] & & \mathbb{P}^3
\end{tikzcd}
\end{equation*}
Then, to simplify a given pair $(\mathbb{P}^3,\Delta)$, we will find boundary divisors $\Delta_X\subset X$ and $\Delta'\subset\mathbb{P}^3$ such that $(\mathbb{P}^3,\Delta')$ belongs to a simpler family (according to Figure~\ref{fig!flowchart}), and
\begin{equation}\label{eq!vp}
\begin{tikzcd}
& (X,\Delta_X) \arrow[rd, "\psi'"] & \\
(\mathbb{P}^3,\Delta) \arrow[leftarrow, ru, "\psi"] \arrow[rr, dashed, "\varphi"] & & (\mathbb{P}^3,\Delta')
\end{tikzcd}
\end{equation}
is a diagram of volume preserving maps of Calabi--Yau pairs, as in Definition~\ref{def!vp}.
Throughout the following calculations we let $H$ denote the hyperplane class on the lefthand copy of $\mathbb{P}^3$ (i.e.\ the domain of $\varphi$), and $H'$ the hyperplane class on the righthand copy (i.e.\ the range of $\varphi$). By abuse of notation we refer to the strict transform of a subvariety (whenever it makes sense) by the same name as for the original.
\subsection{The generic map of bidegree $(2,2)$} \label{sec!bidegree22}
The generic map $\varphi \colon \mathbb{P}^3\dashrightarrow \mathbb{P}^3$ of bidegree $(2,2)$ is defined by $|2H-C-p|$, the linear system of quadrics passing through a plane conic $C\subset\mathbb{P}^3$ and a general point $p\in \mathbb{P}^3$. Let $E'$ be the plane containing $C$ and let $F'$ be the quadric cone through $C$ with vertex at $p$. The map $\varphi$ is resolved by a symmetric diagram of the form
\[ \mathbb{P}^3 \stackrel{\psi}{\longleftarrow} X \stackrel{\psi'}{\longrightarrow} \mathbb{P}^3 \]
where
\begin{enumerate}
\item $\psi$ blows up $p$ with exceptional divisor $E\cong\mathbb{P}^2$ and $C$ with exceptional divisor $F\cong\mathbb{F}_2$,
\item $\psi'$ contracts $E'\subset X$ onto a point $p'\in \mathbb{P}^3$ and $F'\subset X$ onto a conic $C'\subset \mathbb{P}^3$.
\end{enumerate}
It follows that $-K_X \sim 4H-2E-F \sim 4H'-2E'-F'$ and we have the following relations between divisor classes.
\[ \begin{pmatrix}
2 & -1 & -1 \\
1 & 0 & -1 \\
2 & -2 & -1
\end{pmatrix}\begin{pmatrix}
H \\ E \\ F
\end{pmatrix} \sim \begin{pmatrix}
H' \\ E' \\ F'
\end{pmatrix} \]
Moreover, $\varphi^{-1}$ is defined by the linear system $|2H'-C'-p'|$.
\subsection{The generic map of bidegree $(3,2)$} \label{sec!bidegree32}
Consider three pairwise skew lines $\ell_1,\ell_2,\ell_3\subset\mathbb{P}^3$ and a fourth line $\ell_0$ which meets each of the first three. The generic map $\varphi \colon \mathbb{P}^3\dashrightarrow \mathbb{P}^3$ of bidegree $(3,2)$ is defined by the linear system $|3H-2\ell_0-\ell_1-\ell_2-\ell_3|$. Let $F'\subset\mathbb{P}^3$ be the unique quadric surface containing all four lines $\ell_0,\ldots,\ell_3\subset F'$ and let $E'_i\subset \mathbb{P}^3$ be the
plane containing $\ell_0$ and $\ell_i$ for $i=1,2,3$. Then $\varphi$ can resolved by a diagram of the form
\[ \mathbb{P}^3 \stackrel{\sigma}{\longleftarrow} Y \stackrel{\tau}{\longleftarrow} X \stackrel{\tau'}{\longrightarrow} Y' \stackrel{\sigma'}{\longrightarrow} \mathbb{P}^3 \]
where
\begin{enumerate}
\item $\sigma$ is the blowup of $\ell_0\subset \mathbb{P}^3$ with exceptional divisor $F_0\subset Y$,
\item $\tau$ is the blowup of $\ell_1,\ell_2,\ell_3\subset Y$ with exceptional divisors $F_1,F_2,F_3\subset X$,
\item $\tau'$ contracts $E'_1,E'_2,E'_3\subset X$ onto points $p_1',p_2',p_3'\in Y'$,
\item $\sigma'$ contracts $F'\subset Y'$ onto a line $\ell'\subset \mathbb{P}^3$.
\end{enumerate}
Let $\psi = \tau\circ\sigma$ and $\psi' = \tau'\circ\sigma'$. Then $-K_X \sim 4H-F_0-F_1-F_2-F_3 \sim 4H'-2E'_1-2E'_2-2E'_3-F'$ and we have the following relations between divisor classes.
\[ \begin{pmatrix}
3 & -2 & -1 & -1 & -1 \\
1 & -1 & -1 & 0 & 0 \\
1 & -1 & 0 & -1 & 0 \\
1 & -1 & 0 & 0 & -1 \\
2 & -1 & -1 & -1 & -1
\end{pmatrix}\begin{pmatrix}
H \\ F_0 \\ F_1 \\ F_2 \\ F_3
\end{pmatrix} \sim \begin{pmatrix}
H' \\ E'_1 \\ E'_2 \\ E'_3 \\ F'
\end{pmatrix} \qquad \begin{pmatrix}
2 & -1 & -1 & -1 & -1 \\
1 & -1 & -1 & -1 & 0 \\
1 & -1 & 0 & 0 & -1 \\
1 & 0 & -1 & 0 & -1 \\
1 & 0 & 0 & -1 & -1
\end{pmatrix}\begin{pmatrix}
H' \\ E'_1 \\ E'_2 \\ E'_3 \\ F'
\end{pmatrix} \sim \begin{pmatrix}
H \\ F_0 \\ F_1 \\ F_2 \\ F_3
\end{pmatrix} \]
Moreover, $\varphi^{-1}$ is defined by the linear system $|2H'-\ell'-p_1'-p_2'-p_3'|$ and is the generic map of bidegree $(2,3)$.
\subsection{Maps of bidegree $(3,3)$}
\subsubsection{The generic map of bidegree $(3,3)$} \label{sec!bidegree33}
The generic map of bidegree (3,3) is the classical \emph{cubo-cubic Cremona transformation} and has been studied by many authors, e.g\ \cite{katz}. Let $C\subset \mathbb{P}^3$ be a smooth curve of degree 6 and genus 3 defined by the $3\times3$-minors of a $3\times 4$-matrix with linear entries. The trisecant lines to $C$ span a ruled surface $E'\subset \mathbb{P}^3$ of degree 8 with multiplicity 3 along $C$. Then the birational map $\varphi\colon \mathbb{P}^3\dashrightarrow \mathbb{P}^3$ defined by the linear system $|3H-C|$ can be resolved by a symmetric diagram of the form
\[ \mathbb{P}^3\stackrel{\psi}{\longleftarrow} X \stackrel{\psi'}{\longrightarrow} \mathbb{P}^3 \]
where $\psi$ is the blowup of $C$ with exceptional divisor $E$ and $\psi'$ contracts $E'\subset X$ onto a curve $C'\subset \mathbb{P}^3$ which is isomorphic to $C$. We have $-K_X \sim 4H-E\sim 4H'-E'$ the following relations between divisor classes.
\[ \begin{pmatrix}
3 & -1 \\
8 & -3
\end{pmatrix}\begin{pmatrix}
H \\ E
\end{pmatrix} \sim \begin{pmatrix}
H' \\ E'
\end{pmatrix} \]
Moreover, $\varphi^{-1}$ is defined by the linear system $|3H'-C'|$.
\subsubsection{A special map of bidegree $(3,3)$} \label{sec!sp-bidegree33}
We consider a degenerate case of the previous example in which $C=\Gamma\cup\ell_1\cup\ell_2\cup\ell_3$ is the union of a twisted cubic curve $\Gamma$ and three lines $\ell_1,\ell_2,\ell_3$ which are secant lines to $\Gamma$. This kind of cubo-cubic Cremona transformation was considered by Mella \cite[Proof of Proposition 2.2]{mella} and we briefly recall the description.
Let $E',F'_1,F'_2,F'_3\subset \mathbb{P}^3$ be the uniquely determined quadric surfaces such that $\ell_1,\ell_2,\ell_3\subset E'$ and $\Gamma,\ell_i,\ell_j\subset F'_k$ for $\{i,j,k\}=\{1,2,3\}$. The birational map $\varphi\colon \mathbb{P}^3\dashrightarrow \mathbb{P}^3$ defined by the linear system $|3H-\Gamma-\ell_1-\ell_2-\ell_3|$ can be resolved by a symmetric diagram of the form
\[ \mathbb{P}^3 \stackrel{\sigma}{\longleftarrow} Y \stackrel{\tau}{\longleftarrow} X \stackrel{\tau'}{\longrightarrow} Y' \stackrel{\sigma'}{\longrightarrow} \mathbb{P}^3 \]
where
\begin{enumerate}
\item $\sigma$ is the blowup of $\Gamma\subset\mathbb{P}^3$ with exceptional divisor $E\subset Y$,
\item $\tau$ is the blowup of $\ell_1,\ell_2,\ell_3\subset Y$ with exceptional divisors $F_1,F_2,F_3\subset X$,
\item $\tau'$ contracts $F'_1,F'_2,F'_3\subset X$ onto disjoint lines $\ell_1',\ell_2',\ell_3'\subset Y'$,
\item $\sigma'$ contracts $E'\subset Y'$ onto a twisted cubic curve $\Gamma'\subset \mathbb{P}^3$.
\end{enumerate}
Let $\psi = \tau\circ\sigma$ and $\psi' = \tau'\circ\sigma'$. We have $-K_X \sim 4H-E-F_1-F_2-F_3 \sim 4H'-E'-F_1'-F_2'-F_3'$ and the following relations between divisor classes.
\[ \begin{pmatrix}
3 & -1 & -1 & -1 & -1 \\
2 & 0 & -1 & -1 & -1 \\
2 & -1 & 0 & -1 & -1 \\
2 & -1 & -1 & 0 & -1 \\
2 & -1 & -1 & -1 & 0
\end{pmatrix}\begin{pmatrix}
H \\ E \\ F_1 \\ F_2 \\ F_3
\end{pmatrix} \sim \begin{pmatrix}
H' \\ E' \\ F'_1 \\ F'_2 \\ F'_3
\end{pmatrix} \]
Moreover, $\varphi^{-1}$ is defined by the linear system $|3H'-\Gamma'-\ell_1'-\ell_2'-\ell_3'|$.
\section{Connecting the eleven families} \label{sec!proof}
We now constructing the ten types of volume preserving map (i)-(x), appearing in Figure~\ref{fig!flowchart}, that connect our eleven families.
\subsection{The map (x) between (A.1) and (C.4)}
\label{sec!triple}
Although it appears last in Figure~\ref{fig!flowchart}, we begin by explaining map (x) since it is by far the easiest case to deal with. Suppose that $\Delta$ has an $\widetilde E_6$ singularity, or in other words a triple point. Thus we may write $\Delta = \mathbb{V}(a_3t + b_4)$ for some polynomials $a_3,b_4\in\mathbb{C}[x,y,z]$. We let $\Delta'=\mathbb{V}(a_3t)$ and consider the birational map
\[ \varphi\colon(\mathbb{P}^3,\Delta) \dashrightarrow (\mathbb{P}^3,\Delta'), \qquad \varphi(t,x,y,z) = \left(t+a_3^{-1}b_4,x,y,z\right). \]
which is volume preserving. To see this we can restrict to the affine patch of $\mathbb{P}^3$ where $z=1$ (on both sides of $\varphi$) and then we compute that
\[ \varphi^*(\omega_{\Delta'}) = \varphi^*\left(\frac{dt\wedge dx\wedge dy}{a_3t}\right) = \frac{d(t+a_3^{-1}b_4)\wedge dx\wedge dy}{a_3(t+a_3^{-1}b_4)} = \frac{dt\wedge dx\wedge dy}{a_3t+b_4} = \omega_{\Delta}. \]
Lastly, note that $\Delta'=\mathbb{V}(t) +\mathbb{V}(a_3)$ is the union of a plane and the cone over a plane cubic curve, and thus $(\mathbb{P}^3,\Delta')$ is a member of family (C.3).
\subsection{The maps (i) and (ii) between (A.2), (A.3) and (C.1)}
Next we produce the maps (i) and (ii) which connect the families (A.2), (A.3) and (C.1). To this end, it is convenient to introduce two new deformation families of log Calabi--Yau pairs:
\begin{description}
\item[(D.1)] $\left(\mathbb{P}(1,1,2,3), \, \Delta\right)$ where $\Delta$ is the union of a plane and a $\dP_1$,
\item[(D.2)] $\left(\mathbb{P}(1,1,1,2), \, \Delta\right)$ where $\Delta$ is the union of a plane and a $\dP_2$.
\end{description}
\begin{prop}
There exist volume preserving maps $\varphi_1,\ldots,\varphi_4$ linking these five families, as in the following diagram. Thus map (i) in Figure~\ref{fig!flowchart} is given by $\varphi_2^{-1}\circ\varphi_3\circ \varphi_1$ and map (ii) is given by $\varphi_4\circ\varphi_2$.
\begin{center}\begin{tikzpicture}[scale=1.2]
\node (a) at (0,0) {(A.3)};
\node (b) at (0,1.5) {(D.1)};
\node (c) at (2,0) {(A.2)};
\node (d) at (2,1.5) {(D.2)};
\node (e) at (4,1.5) {(C.1)};
\draw[->] (a) to node[left] {$\varphi_1$} (b);
\draw[->] (1.8,1.2) to node[left] {$\varphi_2^{-1}$} (1.8,0.3);
\draw[->] (2.2,0.3) to node[right] {$\varphi_2$} (2.2,1.2);
\draw[->] (b) to node[above] {$\varphi_3$} (d);
\draw[->] (d) to node[above] {$\varphi_4$} (e);
\end{tikzpicture}\end{center}
\end{prop}
\begin{proof}
We construct each of the maps $\varphi_1,\ldots,\varphi_4$ in turn.
\paragraph{The map $\varphi_1$.}
We start with a quartic $\Delta\subset\mathbb{P}^3$ in the family (A.3). By Proposition~\ref{prop!type-A-quartics} it is defined by an equation $F_4(t,x,y,z)$ of the form
\[ F_4(t,x,y,z) = a_0t^2z^2 + b_0ty^3 + c_1txz + d_2tz + e_0x^3z + f_2x^2 + g_3x + h_4 \]
for some homogeneous polynomials $a_0,\ldots,h_4\in\mathbb{C}[y,z]$.
The two-ray game that begins with the $(1,2,3)$-weighted blowup at the singularity $p\in \mathbb{P}^3$ initiates a Sarkisov link to $\mathbb{P}(1,1,2,3)$, resulting in the birational map
\[ \varphi_1\colon \mathbb{P}^3 \dashrightarrow \mathbb{P}(1,1,2,3)
\qquad \varphi_1(t,x,y,z) = (y,z,xz,tz^2). \]
Now let $u=xz$ and $v=tz^2$ be coordinates on $\mathbb{P}(1,1,2,3)$ and consider the sextic equation
\[ G_6(y,z,u,v) = a_0v^2 + b_0vy^3 + c_1uv + d_2vz + e_0u^3 + f_2u^2 + g_3uz + h_4z^2. \]
We see that $\varphi_1$ maps $\Delta$ birationally onto $\mathbb{V}(G_6)$. Moreover, in the affine patches of $\mathbb{P}^3$ and $\mathbb{P}(1,1,2,3)$ where $y=1$ we compute that
\[ \varphi_1^*\left(\frac{dz\wedge du\wedge dv}{zG_6(1,z,u,v)}\right) = \frac{dz\wedge d(xz)\wedge d(tz^2)}{zG_6(1,z,xz,tz^2)} = \frac{dz\wedge dx\wedge dt}{F_4(t,x,1,z)} \]
and thus it follows that $\varphi_1$ produces a volume preserving map of pairs if we consider the boundary divisor $\Delta' = \mathbb{V}(zG_6)\subset \mathbb{P}(1,1,2,3)$. In particular, $\Delta'$ is the union of a plane $D_1=\mathbb{V}(z)$ and a sextic $D_2=\mathbb{V}(G_6)$, which is a (possibly degenerate) $\dP_1$.
\paragraph{The map $\varphi_2$.}
The construction of $\varphi_2$ is very similar to that of $\varphi_1$. By Proposition~\ref{prop!type-A-quartics}, we are considering a quartic $\Delta$ defined by an equation of the form $F_4(t,x,y,z) = a_0t^2z^2 + b_2tz + c_4$ for some homogeneous polynomials $a_0,b_2,c_4\in\mathbb{C}[x,y,z]$.
This time the $(1,1,2)$-weighted blowup at the singularity $p\in \mathbb{P}^3$ initiates the map
\[ \varphi_2\colon \mathbb{P}^3\dashrightarrow \mathbb{P}(1,1,1,2), \qquad \varphi_2(t,x,y,z)=(x,y,z,tz). \]
Letting $u=tz$ be the coordinate on $\mathbb{P}(1,1,1,2)$ then, by a similar calculation to the one above, we see that $\varphi_2$ produces a volume preserving map, where the boundary divisor $\Delta' = \mathbb{V}(zG_4)\subset \mathbb{P}(1,1,1,2)$ is the union of a plane and a (possibly degenerate) $\dP_2$ defined by the quartic equation $G_4 = a_0u^2 + b_2u + c_4$.
\paragraph{The map $\varphi_3$.}
Suppose that $(\mathbb{P}(1,1,2,3), \, \Delta)$ is a log Calabi--Yau pair with coordinates $t,x,y,z$ and boundary divisor $\Delta=\mathbb{V}(tF_6)$ consisting of a plane $D_1=\mathbb{V}(t)$ and $D_2=\mathbb{V}(F_6)$, a $\dP_1$. The component $D_2$ must contain a line $\ell\subset D_2$, since it arises as a degeneration of a $\dP_1$ which in general contains 240 lines. Without loss of generality we can take $\ell = \mathbb{V}(y,z)$, and thus we may write $F_6 = a_4y + b_3z + c_0z^2$ for some homogeneous polynomials $a_4,b_3,c_0\in\mathbb{C}[t,x,y]$. We can contract $\ell$ by applying the birational map
\[ \varphi_3\colon \mathbb{P}(1,1,2,3) \dashrightarrow \mathbb{P}(1,1,1,2) \qquad \varphi(t,x,y,z) = (t,x,y^{-1}z,y) \]
and, if we let $u=y^{-1}z$, then it is easy to check that this produces a volume preserving map of pairs, where the boundary divisor $\Delta'=\mathbb{V}(tG_4)\subset\mathbb{P}(1,1,1,2)$ is given by the union of a plane and a (possibly degenerate) $\dP_2$ defined by the equation $G_4 = a_4 + b_3u + c_0u^2y$.
\paragraph{The map $\varphi_4$.}
The construction of $\varphi_4$ is very similar to that of $\varphi_3$. Indeed if $t,x,y,z$ are coordinates on $\mathbb{P}(1,1,1,2)$ and $\Delta=\mathbb{V}(tF_4)$, then, arguing as before, we may assume that the component $D_2=\mathbb{V}(F_4)$ contains a line $\ell=\mathbb{V}(y,z)$. Then the map
\[ \varphi_4 \colon \mathbb{P}(1,1,1,2) \dashrightarrow \mathbb{P}^3 \qquad \varphi(t,x,y,z) = (t,x,y,y^{-1}z) \]
is easily seen to produce a volume preserving map, for a boundary divisor $\Delta'\subset \mathbb{P}^3$ given by the union of the plane $\mathbb{V}(t)$ and the cubic surface obtained by contracting $\ell\subset D_2$.
\end{proof}
\subsection{The map (iii) between (A.4) and (B.1)}
\label{sec!stE8}
It follows from the description given in Proposition~\ref{prop!type-A-quartics}, that in this case the equation defining $\Delta\subset \mathbb{P}^3$ can be written in the form
\[ \Delta = \mathbb{V}\big(\mu_1 (tz+x^2)^2 + \mu_2 ty^3 + (2a_1x + b_2)(tz+x^2) + c_2x^2 + d_3x + e_4\big) \]
for some $\mu_1,\mu_2\in\mathbb{C}$ and homogeneous polynomials $a_1,b_2,c_2,d_3,e_4\in\mathbb{C}[y,z]$. Moreover we may assume that both $\mu_1,\mu_2\neq0$, or else $\Delta$ has either a triple point (A.1) or a degenerate $\widetilde E_7$ singularity (A.2). Thus by a change of coordinates both $\mu_1$ and $\mu_2$ can be rescaled to 1.
\paragraph{Finding a conic in $\Delta$.}
By introducing the coordinates $u/w = (tz+x^2)/z^2$ and $v/w = x/z$, we see that $\Delta$ is birational to a surface $W\subset \mathbb{P}^2_{u,v,w}\times \mathbb{P}^1_{y,z}$ defined by a bihomogeneous equation of degree $(2,4)$
\[ W = \mathbb{V}\left( z^4u^2 + 2a_1z^3uv + (y^3 + b_2z)zuw + (c_2z - y^3)zv^2 + d_3zvw + e_4w^2 \right). \]
The fibres of the map $\xi\colon W\to \mathbb{P}^1_{y,z}$ are conics, which are reducible over the discriminant locus
\[ \Xi =\mathbb{V}\left(
z^3\left((y^3+b_2z)^2-4e_4z^2\right)\left(y^3+a_1^2z-c_2z\right) - z^4\left(a_1(y^3+b_2z)-d_3z\right)^2\right), \]
and, since the top term of this polynomial in powers of $y$ is given by $y^9z^3 + \cdots$, there must be a root of the form $y=\lambda z$. This implies that the equation defining $W$ factorises as
\[ z^4(u + \alpha v + \beta w)(u + \gamma v + \delta w) \mod y \]
for $\alpha,\beta,\gamma,\delta\in\mathbb{C}$ where $\alpha + \gamma = 2a_1(\lambda,1)$, $\beta + \delta = b_2(\lambda,1)+\lambda^3$, $\alpha\gamma = c_2(\lambda,1)-\lambda^3$, $\alpha\delta + \beta\gamma = d_3(\lambda,1)$ and $\beta\delta = e_4(\lambda,1)$.
In terms of the quartic equation defining $\Delta$, we find that it can be rewritten
\[ (tz + x^2 + \alpha xz + \beta z^2)(tz + x^2 + \gamma xz + \delta z^2) + \cdots \]
\[\cdots + (y-\lambda z)\left( t(y^2+\lambda yz + \lambda^2z^2) + (2a_0x + b_1)(tz+x^2) + c_1x^2 + d_2x + e_3\right) \]
where $a_0,b_2,c_1,d_2,e_3$ are defined by setting $a_1 = a_0(y-\lambda z) + a_1(\lambda,1)z$, $b_3 = b_2(y-\lambda z) + b_3(\lambda,1)z^3$, etc. In particular we see that $\Delta$ contains a conic curve $C= \mathbb{V}(y-\lambda z,\, tz + x^2 + \alpha xz + \beta z^2)$.
\paragraph{Reduction to (B.1).}
To reduce to a simpler case, we apply the quadratic map
\[ \varphi\colon \mathbb{P}^3\dashrightarrow \mathbb{P}^3, \qquad \varphi(t,x,y,z) = \left((y-\lambda z)^{-1}(tz+x^2+\alpha xz+\beta z^2),x,y,z\right) \]
given by the linear system of all quadrics passing through $C$ and the infinitely near point $p\in C$. This induces a volume preserving map $\varphi\colon (\mathbb{P}^3,\Delta)\dashrightarrow (\mathbb{P}^3,\Delta')$ where $\Delta'$ is the quartic defined by the equation
\[ \left(tz+y^2+\lambda yz+\lambda^2z^2+2a_0xz+b_1z\right)\left(ty-\lambda tz-\alpha xz-\beta z^2\right) + \cdots \]
\[\cdots + (\gamma x + \delta z)tz^2 + c_1x^2z + d_2xz + e_3z - (y^2+\lambda yz + \lambda^2z^2)x^2 \]
which is singular along the line $\ell=\mathbb{V}(y,z)$.
\subsection{The map (iv) between (B.1) and (C.1)}
\label{sec!line}
Throughout this subsection we assume that $\Delta\subset\mathbb{P}^3$ has a line of double points along $\ell_0=\mathbb{V}(y,z)$, so that the equation of $\Delta$ can be written in the form
\begin{equation}
a_2t^2 + b_2tx + c_2x^2 + d_3t + e_3x + f_4 \label{eq!double-line}
\end{equation}
for homogeneous polynomials $a_2,\ldots,f_4\in\mathbb{C}[y,z]$. We deal with these surfaces in three steps.
\paragraph{Singularities away from $\ell_0$.}
First we consider the cases in which $\Delta$ is singular at a point outside of the line $\ell_0$.
\begin{lem}
If $\Delta$ is singular at some point $p\in\Delta\setminus \ell_0$ then $(\mathbb{P}^3,\Delta)$ is volume preserving equivalent to a pair from case (A.2).
\end{lem}
\begin{proof}
Without loss of generality we may assume that $p=(0,0,0,1)$ and the equation \eqref{eq!double-line} of $\Delta$ can be rewritten in the form
\[ g_2(t,x,y)y^2 + h_2(t,x,y)yz + i_2(t,x,y)z^2 \]
for some homogeneous polynomials $g_2,h_2,i_2\in\mathbb{C}[t,x,y]$ which don't depend on $z$. By considering the linear system of quadrics passing through the point $p$ and the conic $\mathbb{V}(g_2,z)\subset\Delta$ we get a volume preserving map of bidegree (2,2)
\[ \varphi\colon (\mathbb{P}^3,\Delta)\dashrightarrow (\mathbb{P}^3,\Delta'), \qquad \varphi(t,x,y,z) = (tz,xz,yz,g_2) \]
where $\Delta' = \mathbb{V}(y^2z^2 + h_2yz + g_2i_2)$ is a quartic with an $\widetilde E_7$ singularity at $(0,0,0,1)$.
\end{proof}
Thus we may assume that $\Delta\setminus \ell_0$ is smooth, or else proceed via the (A.2) case.
\paragraph{Degenerate cusp singularities along $\ell_0$.}
Now consider the normalisation $\nu\colon \overline \Delta\to \Delta$ with ramification curve $\overline D= \nu^{-1}(\ell_0)$. Since $(\overline \Delta,\overline D)$ is a log Calabi--Yau pair we either have that
\begin{enumerate}
\item $\overline D\subset \overline\Delta$ is a smooth elliptic curve and $\Delta$ has four pinch points along $\ell_0$ corresponding to the four ramification points of the double cover $\nu|_{\overline D}\colon \overline D\to \ell_0$, or
\item $\overline D\subset \overline\Delta$ is a nodal curve of arithmetic genus 1 and $\Delta$ has a degenerate cusp singularity at the point $\nu(p)\in \ell_0$ corresponding to the image of a node of $p\in \overline D$.
\end{enumerate}
We now dispense with the cases in which $\Delta$ has a degenerate cusp by showing that $(\mathbb{P}^3,\Delta)$ can be treated as a special case of a previously considered family.
\begin{lem}
If $\Delta$ has a degenerate cusp singularity along $\ell_0$ then the pair $(\mathbb{P}^3,\Delta)$ is a special case of either (A.1) or (A.3).
\end{lem}
\begin{proof}
Suppose that $\Delta$ has a degenerate cusp singularity of type $T_{pq\infty}$ at the point $(1,0,0,0)\in\Delta$. If both $p,q\geq3$ then $\mult_p\Delta\geq 3$, so $\Delta$ has a triple point and we can consider it as a degeneration of the case (A.1). Otherwise we have a $T_{2q\infty}$ singularity. Looking at the equation \eqref{eq!double-line} of $\Delta$ in a neighbourhood of this point, we see that $a_2\in\mathbb{C}[y,z]$ is a nonzero square, so without loss of generality we can take $a_2=z^2$. Then we also have that $b_2=b_1z$ for some $b_1\in\mathbb{C}[y,z]$. Thus the equation of $\Delta$ has degree $\geq6$ with respect to the weights $1,2,3$ for $x,y,z$ and we can treat it as a degeneration of case (A.3).
\end{proof}
\paragraph{The generic case.}
Thus we may reduce ourselves to considering the generic case in which $\Delta$ is smooth outside of $\ell_0$ and $\overline D\subset \overline \Delta$ is a smooth elliptic curve. In particular, the normalisation $\nu\colon \overline \Delta\to \Delta$ is equal to the minimal resolution $\overline\Delta = \widetilde \Delta$. Recall that we have the following construction of Urabe, described in \S\ref{sec!type-B-quartics}, which consists of volume preserving maps of log Calabi--Yau pairs
\[ (\Delta,0)\stackrel{\mu}{\longleftarrow}(\widetilde \Delta, D)=: (\widetilde \Delta_0,D_0)\stackrel{f_1}{\longrightarrow}\cdots \stackrel{f_9}{\longrightarrow}(\widetilde\Delta_9,D_9) = (\mathbb{P}^2,D_9) \]
and the map $\widetilde \Delta\to\Delta$ is realised by the nef divisor class $A=\mu^*\mathcal{O}_\Delta(1)=4h-2e_1-e_2-\cdots-e_9$. In particular, under our additional assumptions we must have that $D_9\subset\mathbb{P}^2$ is a smooth cubic curve and, if $p_i\in\mathbb{P}^2$ is the point that corresponds to the centre of the blowup $f_i$, then none of the $p_i$ can be infinitely near to another or else $\Delta$ would have a Du Val singularity away from $\ell_0$.
Since $D\sim 3h-e_1\cdots-e_9$ we have $A\sim D + (h - e_1)$ and $A\cdot(h-e_1)=2$, and thus the pencil of hyperplanes in $\mathbb{P}^3$ that pass through $\ell_0=\mu(D)$ cuts out a pencil of residual conics $|h-e_1|$ in $\widetilde\Delta$. This pencil has eight reducible fibres, corresponding to eight decompositions of the form $(h-e_1) = (h-e_1-e_i) + (e_i)$ for $i=2,\ldots,9$. Pick three lines $\ell_1,\ell_2,\ell_3\subset \Delta$ from three of these reducible fibres, e.g.\ $\ell_i=\mu(e_{i+1})$ for $i=1,2,3$, which are pairwise skew and all of which intersect $\ell_0$ transversely.
Now we consider the map $\varphi\colon \mathbb{P}^3\dashrightarrow \mathbb{P}^3$ of bidegree (3,2) which is given by the linear system $|{\mathcal{O}_{\mathbb{P}^3}(3)-2\ell_0-\ell_1-\ell_2-\ell_3}|$. Considering the resolution $\mathbb{P}^3\stackrel{\psi}{\leftarrow} X \stackrel{\psi'}{\rightarrow} \mathbb{P}^3$ of $\varphi$, as in the notation of \S\ref{sec!bidegree32}, we see that $K_X + \Delta_X \sim 0$ where $\Delta_X := \psi^{-1}_*(\Delta) + F_0$. Thus, setting $\Delta':=\psi'_*(\Delta_X)$, we obtain a volume preserving map of pairs $\varphi\colon(\mathbb{P}^3,\Delta)\dashrightarrow(\mathbb{P}^3,\Delta')$. Since we have
\[ \psi^{-1}_*(\Delta) \sim 3H'-E_1'-E_2'-E_3'-F' \quad \text{and} \quad F_0 \sim H' - E_1' - E_2' - E_3' \]
it follows that $\Delta'=D_1'+D_2'$ is the union of a cubic surface $D_1'=\psi'_*(\psi^{-1}_*(\Delta))$ and a plane $D_2'=\psi'_*(F_0)$, where $\ell'\subset D_1'$ and $p_1',p_2',p_3'\in D_1'\cap D_2'$.
\subsection{The map (v) between (B.2) and (C.1)}
\label{sec!conic}
We now consider the case of an irreducible quartic $\Delta\subset \mathbb{P}^3$ with double points along a plane conic $C = \mathbb{V}(t,q_2)$. In particular, $\Delta = \mathbb{V}( q_2^2 + a_1tq_2 + b_2t^2 )$ for some $a_1,b_2\in \mathbb{C}[x,y,z,t]$. Moreover, we may assume that $C$ is irreducible, or else $\Delta$ is singular along a line and we can treat $(\mathbb{P}^3,\Delta)$ as a special case of family (B.1). We now construct a volume preserving map $\varphi\colon (\mathbb{P}^3,\Delta)\dashrightarrow (\mathbb{P}^3,\Delta')$ of bidegree (2,2) whose baselocus consists of $C$ and a well-chosen point infinitely near to $C$.
To do this, we let $X=\mathbb{V}(tu-q_2)\subset \mathbb{P}^4_{t,x,y,z,u}$ and consider a factorisation $\varphi = g\circ f^{-1}$
\[ (\mathbb{P}^3,\Delta)\stackrel{f}{\dashleftarrow} (X,\Delta_X) \stackrel{g}{\dashrightarrow} (\mathbb{P}^3,\Delta'), \]
where $f,g$ are volume preserving maps with the following properties.
\begin{enumerate}
\item The boundary divisor $\Delta_X=\mathbb{V}(t(u^2+a_1u+b_2))\in |{-K_X}|$ consists of two irreducible components: a quadric cone $D_1 = \mathbb{V}(t,q_2)$ which is singular at $p=(0,0,0,0,1)\in \mathbb{P}^4$, and a $\dP_4$ given by $D_2=\mathbb{V}(ut-q_2, u^2+a_1u+b_2)$. These two components meet in an ordinary curve $\Gamma=D_1\cap D_2$.
\item The map $f$ is the projection from $p\in X$, which contracts $D_1$ onto $C$ and maps $D_2$ birationally onto $\Delta$.
\item The map $g$ is the projection from a general point $q\in\Gamma$. Since $g$ maps $D_1$ birationally onto a plane and $D_2$ birationally onto a cubic surface, we set $\Delta' := g(\Delta_X)$.
\end{enumerate}
Thus $(\mathbb{P}^3,\Delta')$ is a pair in the family (C.1).
\subsection{The map (vi) between (C.1) and (C.2)}
We suppose that $\Delta=D_1+D_2$ is the union of a plane $D_1$ and a cubic surface $D_2$. We may assume that $D_2$ is smooth, or else we can proceed directly to case (C.3).
Since $D_2$ is smooth, we can choose three skew lines $\ell_1,\ell_2,\ell_3$ amongst the 27 lines of $D_2$ and a fourth line $\ell_0$ which meets each of the first three. Now consider the map $\varphi\colon \mathbb{P}^3\dashrightarrow \mathbb{P}^3$ of bidegree (3,2) defined by the linear system $|{\mathcal{O}_{\mathbb{P}^3}(3)-2\ell_0-\ell_1-\ell_2-\ell_3}|$, as described in \S\ref{sec!bidegree32}. Setting $\Delta_X := \psi^{-1}_*(\Delta)$ and $\Delta':=\psi'_*(\Delta_X)$, we get a resolution as in \eqref{eq!vp} giving a volume preserving map $\varphi\colon (\mathbb{P}^3,\Delta)\dashrightarrow(\mathbb{P}^3,\Delta')$. Since
\[ \psi^{-1}_*(D_1)= 2H'-E_1'-E_2'-E_3'-F' \quad \text{and} \quad \psi^{-1}_*(D_2) = 2H'-E'_1-E'_2-E'_3. \]
it follows that $\Delta' = D_1'+D_2'$ is given by the union of two quadric surfaces, where $D_i'=\psi'_*(\psi^{-1}_*(D_i))$ for $i=1,2$, such that $\ell'\subset D_1'$ and $p_1',p_2',p_3'\in D_1'\cap D_2'$.
\subsection{The map (vii) between (B.3) and (C.2)}
\label{sec!twisted-cubic}
We suppose that $\Delta\subset \mathbb{P}^3$ has double points along a twisted cubic curve $\Sigma$. If $\Sigma$ is a degenerate twisted cubic curve then it splits into the union of a line and a (possibly reducible) conic. In particular we can treat $(\mathbb{P}^3,\Delta)$ as a special case of family (B.1). We now follow the argument of Mella \cite{mella} very closely.
Since we can assume $\Sigma$ is smooth, without loss of generality it is given by the equations
\[ \Sigma = \mathbb{V}(xz-y^2,\; xt-yz, \; yt-z^2) \subset \mathbb{P}^3. \]
Let $\xi_1,\xi_2,\xi_3$ denote the three quadrics generating $I(\Sigma)$. The equation of $\Delta$ must be of the form
\begin{equation} \Delta = \mathbb{V}(a_1\xi_1^2 + a_2\xi_1\xi_2 + a_3\xi_1\xi_3 + a_4\xi_2^2 + a_5\xi_2\xi_3 + a_6\xi_3^2) \label{eqGamma} \end{equation}
for some constants $a_1,\ldots,a_6\in \mathbb{C}$. Moreover the quadrics satisfy the syzygies
\begin{equation} z\xi_1 - y\xi_2 + x\xi_3 = 0, \qquad
t\xi_1 - z\xi_2 + y\xi_3 = 0 \label{eqX} \end{equation}
and thus we can write the blowup of $\Sigma$ as $\sigma\colon X\subset \mathbb{P}^3\times\mathbb{P}^2_{\xi_1,\xi_2,\xi_3}\to \mathbb{P}^3$ where $X$ is the complete intersection of codimension two cut out by the two equations \eqref{eqX}.
We note that the fibres of the projection $\pi\colon X\to \mathbb{P}^2_{\xi_1,\xi_2,\xi_3}$ are precisely the secant lines to $C$, and that $\sigma^{-1}_*\Delta = \pi^{-1}(\Gamma)$ is the preimage of the conic $\Gamma\subset \mathbb{P}^2$ with the equation \eqref{eqGamma}. Pick three non-collinear points on $\Gamma$ corresponding to three lines $\ell_1,\ell_2,\ell_3\subset \Delta$ which are secant lines to $\Sigma$.
We consider the birational map $\varphi\colon \mathbb{P}^3\dashrightarrow \mathbb{P}^3$ which is the degenerate cubo-cubic Cremona transformation with baselocus $\Sigma\cup\ell_1\cup\ell_2\cup\ell_3$, as described in \S\ref{sec!sp-bidegree33}. Setting $\Delta_X := \psi^{-1}_*(\Delta) + E$ and $\Delta':=\psi'_*(\Delta_X)$ gives the resolution of a volume preserving map $\varphi\colon (\mathbb{P}^3,\Delta)\dashrightarrow(\mathbb{P}^3,\Delta')$, as in \eqref{eq!vp}. Since $\psi^{-1}_*(\Delta) = 2H' - E'$ and $E = 2H' - F_1'-F_2'-F_3'$ we see that $\Delta'=D_1'+D_2'$ is given by the union of two quadric surfaces: $D_1'=\psi'_*(\psi^{-1}_*(\Delta))$ contains the twisted cubic $\Sigma'\subset \mathbb{P}^3$ and $D_2'=\psi'_*(E)$ contains the three secant lines $\ell_1',\ell_2',\ell_3'$ to $\Sigma'$.
\subsection{The map (viii) between (C.2) and (C.3)}
Suppose that $\Delta=D_1+D_2$ is the union of two quadric surfaces intersecting in an ordinary curve $\Sigma = D_1\cap D_2$. Let $C\subset D_1$ be a conic, let $p\in \Sigma$ be a general point and consider the map $\varphi\colon \mathbb{P}^3\dashrightarrow \mathbb{P}^3$ of bidegree (2,2) with $\Bs(\varphi)=C\cup p$, as described in \S\ref{sec!bidegree22}.
Setting $\Delta_X := \psi^{-1}_*(\Delta)$ and $\Delta':=\psi'_*(\Delta_X)$ we get a volume preserving map $\varphi\colon (\mathbb{P}^3,\Delta)\dashrightarrow(\mathbb{P}^3,\Delta')$. Since $\psi^{-1}_*(D_1)= H'$ and $\psi^{-1}_*(D_2) = 3H'-2E'-F'$ we see that $\Delta'$ is given by the union of a plane and a cubic surface, where the cubic surface passes through the conic $\Gamma'$ and has a double point at $p'\in\mathbb{P}^3$.
\subsection{The map (ix) between (C.3) and (C.4)}
Consider the case in which $\Delta=D_1+D_2$ consists of a plane $D_1$ and singular cubic surface $D_2$ intersecting in an ordinary curve $\Sigma=D_1\cap D_2$. If $D_2$ is either a cone, non-normal, or singular at a point of $\Sigma$, then $\Delta$ has a point of multiplicity 3 and we can treat it as a special case of family (A.1). Therefore we may assume that $D_2$ has a double point $p\in D_2\setminus \Sigma$. Without loss of generality $D_1=\mathbb{V}(t)$ and $D_2$ is singular at $p=(1,0,0,0)$. Thus $D_2=\mathbb{V}(a_2t+b_3)$ for some $a_2,b_3\in\mathbb{C}[x,y,z]$.
We let $\Delta'=\mathbb{V}(b_3t)$ and consider the birational map
\[ \varphi\colon(\mathbb{P}^3,\Delta) \dashrightarrow (\mathbb{P}^3,\Delta'), \qquad \varphi(t,x,y,z) = \left(\frac{a_2tz}{a_2t+b_3},\, x,\, y,\, z\right). \]
This is volume preserving, since in the affine patch of $\mathbb{P}^3$ where $z=1$ we see that
\[ \varphi^*(\omega_{\Delta'}) = \varphi^*\left(\frac{dx\wedge dy \wedge dt}{b_3t}\right) = \frac{dx\wedge dy \wedge dt}{t(a_2t+b_3)} = \omega_\Delta. \]
\section{Conclusion} \label{sec!the-proof}
We have constructed all of the volume preserving maps that appear in Figure~\ref{fig!flowchart}. This now easily implies our main result.
\begin{proof}[Proof of Theorem~\ref{thm!main-result}]
We can immediately reduce to the case of $(\mathbb{P}^3,\Delta)$ in which $\Delta=\mathbb{V}(ta_3(x,y,z))$ is the union of a plane $D_1=\mathbb{V}(t)$ and $D_2=\mathbb{V}(a_3)$ the cone over an ordinary cubic curve. Without loss of generality we may assume that $a_3$ is not divisible by $z$ and then
\[ \varphi\colon (\mathbb{P}^3,\Delta) \dashrightarrow (\mathbb{P}^1\times\mathbb{P}^2,\Delta') \qquad \varphi(t,x,y,z) = (t,z)\times(x,y,z) \]
is a volume preserving map, where $\Delta'= (\{0\}\times\mathbb{P}^2) + (\mathbb{P}^1\times E) + (\{\infty\}\times\mathbb{P}^2)$ is the boundary divisor appearing in the statement of Theorem~\ref{thm!main-result} and $E$ is the cubic curve $E=\mathbb{V}(a_3) \subset \mathbb{P}^2_{x,y,z}$. If $E$ is smooth then $\coreg(\mathbb{P}^1\times\mathbb{P}^2,\Delta')=1$. Otherwise $E$ is nodal, $\coreg(\mathbb{P}^1\times\mathbb{P}^2,\Delta')=0$ and by applying birational maps of the form $\id\times\varphi\colon\mathbb{P}^1\times\mathbb{P}^2\dashrightarrow \mathbb{P}^1\times\mathbb{P}^2$ we can find a volume preserving map which sends $\mathbb{P}^1\times E$ onto $\mathbb{P}^1\times\mathbb{V}(xyz)$, as in Example~\ref{eg!P2}.
\end{proof}
|
1,116,691,500,754 | arxiv | \section{Introduction}
The gluon being a colored object, Quantum Chromodynamics
(QCD) allows the existence of exotic resonances, such as glueballs or hybrid mesons.
Glueballs are composed of gluons only, while hybrid mesons contain a quark-antiquark pair as well. In particular, the study of heavy hybrid mesons is an active domain
in theoretical and in experimental particle physics. BELLE and BABAR have already reported the discovery of several intriguing $c\bar c$- or $b\bar b$-like resonances: One can quote the X(3872)~\cite{choi}, but also the Y(4260)~\cite{Aub05} and $\Upsilon(10890)$ resonances~\cite{Che08}, that could be interpreted either as $Q\bar Q$ hybrid mesons or as $Q\bar Q q\bar q$ tetraquarks -- see Refs.~\cite{revexp} for complete reviews. Notice that $Q$ ($q$) denotes a heavy (light) quark. Moreover, one expects that future
experiments like COMPASS, BESIII, GLUEX and PANDA should be very efficient in the detection of heavy hybrid mesons, especially of $c\bar c$-type.
There are two possible descriptions of hybrid mesons: First, a genuine three-body object made of a quark, an antiquark and a constituent gluon; Second, a two-body object made of a quark and an antiquark in the potential due to the gluon field in an excited state. In the framework of constituent models, it has been shown that these two pictures of the same object are, to a large extent, equivalent \cite{Bui06a,Bui06b,Bui07}. In this paper, we model the heavy hybrid meson as a $Q\bar Q$ pair within an excited gluon field.
In general, the string energy, and therefore the potential energy between the static
quark and antiquark in the excited gluon field is given by \cite{Arv83,All98}
\begin{equation}
\label{eq:potex}
V(r)=\sqrt{a^2 r^2+b},
\end{equation}
where $a$ is the usual string tension while $b=2 \pi a K + C$ is a term exhibiting the string excitation number $K$ and a constant $C$. These values depend on the model adopted: a pure string theory \cite{Arv83} or a more phenomenological approach \cite{All98,Bui07}. In the present work, we choose the form predicted in Ref.~\cite{Bui07}, $K=2 n_g+l_g$ and $C=3 \pi a$, which is in very good agreement with lattice QCD for the standard value $a=0.2$~GeV$^2$ \cite{Jug98,Jug03}. Finally,
\begin{equation}
\label{eq:bform}
b=2 \pi a (2 n_g+l_g+3/2),
\end{equation}
where $n_g$ and $l_g$ are respectively the radial and orbital quantum numbers of the constituent gluon simulating the excitation of the string. For the study of heavy hybrid mesons, it is therefore very interesting to calculate the eigenenergies of the Schr\"{o}dinger equation governed by the potential (\ref{eq:potex}), or equivalently by the Hamiltonian
\begin{equation}
\label{eq:H}
H=\frac{\bm{p}^2}{2\mu} + \sqrt{a^2 r^2+b},
\end{equation}
where $\mu=m_Q m_{\bar Q}/(m_Q+m_{\bar Q})$ is the reduced mass and where the parameter $b$ is given by Eq.~(\ref{eq:bform}).
The aim of this report is to give an analytical expression for the mass of a heavy hybrid meson in the picture of an excited color field and to derive interesting physical consequences. Our analytical method relies on the auxiliary (or einbein) field method (AFM) which has proved to be very powerful for such kinds of calculations \cite{Sil08a,Sil08b}. The application of the AFM to Hamiltonian~(\ref{eq:H}) is presented in Sec.~\ref{sec:Eigen}, leading to an analytical mass formula. Using this formula, we show in Sec.~\ref{sec:Phenom} that it is possible to predict the general behavior of the heavy hybrid meson masses as a function of the quantum numbers of the system and search for possible towers of states. Our results are summarized in Sec.~\ref{sec:Conc}. The excellent accuracy of the AFM mass formula is discussed in Appendix~\ref{sec:comp} through a comparison with exact numerical results.
\section{Eigenenergies}
\label{sec:Eigen}
\subsection{Analytical expression}
\label{subsec:analexp}
Assuming that the potential~(\ref{eq:potex}) is the dominant interaction in a heavy hybrid meson, the mass of this system is $M_{\textrm{hm}}=m_Q+m_{\bar Q}+E$, where $E$ is an eigenvalue of the Hamiltonian~(\ref{eq:H}). Using scaling laws (see Ref.~\cite{Sil08b}), dimensionless variables $\epsilon$ and $\beta$ can be defined as
\begin{equation}
\label{eq:eps}
E=\left( \frac{2 a^2}{\mu} \right)^{1/3} \epsilon (\beta), \quad \textrm{with} \quad
\beta=b \left( \frac{\mu}{2 a^2} \right)^{2/3}.
\end{equation}
$\epsilon (\beta)$ is an eigenvalue of the dimensionless Hamiltonian
\begin{equation}
\label{eq:ph}
h=\frac{\bm{q}^2}{4} + \sqrt{x^2+\beta}.
\end{equation}
Let us follow the general procedure of the AFM \cite{Sil08a} in order to find approximate expressions
for the eigenvalues of the Hamiltonian~(\ref{eq:ph}). We first choose an auxiliary function $P(x)=x^2$; the auxiliary field $\nu$ is then defined by
\begin{equation}
\label{eq:funcK}
\nu = K(x) = \frac{V'(x)}{P'(x)} = \frac{1}{2 \sqrt{x^2+\beta}}.
\end{equation}
For the moment $\nu$ is an operator, and Eq.~(\ref{eq:funcK}) can be inverted to give
$x$ as a function of $\nu$: $x = I(\nu)$. Explicitly
\begin{equation}
\label{eq:funcI}
I(\nu)=\sqrt{\frac{1}{4 \nu^2}-\beta}.
\end{equation}
The AFM needs the definition of a Hamiltonian $\tilde{h}(\nu) = \bm{q}^2/4 + \nu P(x) +
V(I(\nu))-\nu P(I(\nu))$. In our particular case,
\begin{equation}
\label{eq:tildeH}
\tilde{h}(\nu) = \frac{\bm{q}^2}{4} + \nu x^2 + \frac{1}{4 \nu} +\beta \nu.
\end{equation}
If we choose the auxiliary field in order to extremize $\tilde{h}(\nu)$, $\delta \tilde{h}/
\delta \nu |_{\nu = \hat{\nu}} = 0$, then the value of this Hamiltonian for such an
extremum is precisely the original Hamiltonian: $\tilde{h}(\hat{\nu}) = h$. Instead of
considering the auxiliary field as an operator, let us consider it as a real number.
In this case, the eigenenergies of $\tilde{h}$ are exactly known for all $(n,l)$ quantum
numbers:
\begin{equation}
\label{eq:eigenener}
\epsilon(\nu)=\sqrt{N^2 \nu}+ \frac{1}{4 \nu} + \beta \nu,
\end{equation}
where, as usual, $N=2n+l+3/2$ is the principal quantum number of the state.
The philosophy of the AFM is very similar to a mean field procedure. We first seek the value
$\nu_0$ of the auxiliary field which minimizes the energy, $\partial \epsilon(\nu)/\partial \nu
|_{\nu=\nu_0}$, and consider that the value $\epsilon(\nu_0)$ is a good approximation
of the exact eigenvalue \cite{Sil08a,Sil08b}.
At this stage it is useful to define the new variable
\begin{equation}
\label{eq:defx0}
x_0 = N^{-1/3} \nu_0^{-1/2}
\end{equation}
and the parameter
\begin{equation}
\label{eq:defY}
Y=\frac{16\beta}{3N^{4/3}}.
\end{equation}
The minimization condition is now concerned with $x_0$ and
results from the fourth order reduced equation
\begin{equation}
\label{eq:redeq}
4 x_0^4 - 8 x_0 - 3Y = 0.
\end{equation}
The solution of this equation can be obtained by standard algebraic techniques.
It looks like
\begin{equation}
\label{eq:valx0}
x_0 = G(Y) = \frac{1}{2} \sqrt{V(Y)} + \frac{1}{2} \sqrt{ 4 (V(Y))^{-1/2}
- V(Y)},
\end{equation}
with
\begin{equation}
\label{eq:defVY}
V(Y)=\left(2 + \sqrt{4 + Y^3} \right)^{1/3} - Y \left(2 + \sqrt{4 + Y^3}
\right)^{-1/3}.
\end{equation}
Substituting this value into the expression of $E(\nu_0)$ leads to the analytical
form of the searched eigenenergies, namely
\begin{eqnarray}
\label{eq:Enu0}
\epsilon_{{\rm AF}}=\epsilon(\nu_0)&=&2\sqrt{\frac{\beta}{3 Y}} \left[G^2(Y)+\frac{1}{G(Y)} \right].
\end{eqnarray}
The problem is entirely solved.
As it is shown in Ref.~\cite{Sil08b}, the same formula would be obtained for the choice
$P(r)=\textrm{sgn} (\lambda)r^\lambda$ with $\lambda > -2$, but with different
forms for the quantity $N$. With the choice $\lambda = 2$ made above, $N=2 n+l+3/2$.
In this case, using results from Ref.~\cite{Bui08}, it can be shown that formula~(\ref{eq:Enu0})
gives an upper bound of the exact result. For $\lambda = -1$, $N=n+l+1$ and
the formula gives a lower bound. The qualities of these bounds are examined in Appendix~\ref{sec:comp}. It is also shown in this appendix that the expression
\begin{equation}
\label{eq:exprN}
N=A(\beta) n + l + C(\beta),
\end{equation}
with $A(\beta)$ and $C(\beta)$ defined by Eq.~(\ref{eq:coefacd}), leads to an analytical formula which reproduces very accurately (up to 1\%) the exact results.
\subsection{Asymptotic expansions}
\label{subsec:asymp}
Equation~(\ref{eq:Enu0}) is complicated but quite accurate. In order to get a better insight into this formula, it is interesting to compute several limits:
\begin{align}
\label{eq:lYg}
\lim_{Y\gg 1} \epsilon_{\textrm{AF}} &= \sqrt{\beta} +\frac{N}{\sqrt{2}\beta^{1/4}}-\frac{N^2}{16\beta}, \\
\label{eq:lYp}
\lim_{Y\ll 1} \epsilon_{\textrm{AF}} &= \frac{3 N^{2/3}}{2^{4/3}} + \frac{\beta}{2^{2/3}N^{2/3}}-\frac{\beta^2}{3N^2}.
\end{align}
The first two terms in the r.h.s. of Eq.~(\ref{eq:lYg}) are the solution of a harmonic potential, while Eq.~(\ref{eq:lYp})
with $\beta=0$ corresponds to the solution of a linear interaction obtained with the AFM~\cite{Sil08a}.
For the lowest excited state of the gluon field ($K=0$) and for physical values of the parameters ($a\approx 0.15$-$0.20$~GeV$^2$, $m_c\approx 1.1$~GeV, $m_b\approx 4.5$~GeV), it comes that $\beta\approx 7$-$20$. In this case, $\beta$ is large enough for the choice $N=2 n+l+3/2$ to be relevant (see Eq.~(\ref{eq:coefacd})). A harmonic oscillator band number $B=2 n+l$ can thus be introduced to label the states. We can further assume that hybrid mesons with $K=0$ and $B \gtrsim 4$ cannot be easily produced and discriminated from the low-lying $K=1$ hybrids which would lie in the same mass range. So in the following, we will only consider that $\beta \in [7,20]$, $B \le 4$, and $K=0$. It is indeed more probable that hybrids with the lowest possible excitation of the gluon field will be first observed.
As we are interested in the study of towers of states, it could be expected that the power expansion~(\ref{eq:lYp}), valid for large $N$, should be used. But, $\beta$ is so large for heavy quarks that it is actually not the case. For example, when $\beta=20$, one has $Y\approx 5\gg 1$ even with the large value $N=10$. By comparing with accurate numerical solutions obtained for the Hamiltonian $h$ with the Lagrange mesh method \cite{sem01}, it has been checked that the power expansion~(\ref{eq:lYg}) is far better. Moreover, the ratio of the third term over the second one in this expansion is at most 10\%. So with very good approximation the eigenvalues of $h$ are given by
\begin{equation}
\label{eq:epsapp}
\epsilon_{\textrm{app}} = \sqrt{\beta} +\frac{B+3/2}{\sqrt{2}\beta^{1/4}}.
\end{equation}
The approximate formula~(\ref{eq:epsapp}) only depends on $B$. This exact degeneracy is actually broken by the non-harmonic character of the Hamiltonian $h$, but the breaking is small:
It can be numerically checked that the maximal relative error of Eq.~(\ref{eq:epsapp}) with respect to the exact values of $\epsilon-\sqrt{\beta}$ is around 10\%.
So, $B$ is a relevant classification number.
\subsection{Final mass formula}
The Coulomb interaction must also play a role in heavy hybrids, since heavy quarks are expected to orbit close to each other and must feel strongly this short range interaction. The Hamiltonian~(\ref{eq:ph}) might then be completed with the potential $\kappa/x$ where
\begin{equation}
\label{eq:kappa}
\kappa = \frac{\alpha_S}{6}\left( \frac{\mu^2}{4 a} \right)^{1/3},
\end{equation}
$\alpha_S$ being the strong coupling constant and the $1/6$ factor coming from color Casimir operator. Since $\alpha_S \lesssim 0.2$ in the heavy meson sector \cite{Bui06b}, it appears that $\kappa\lesssim 0.02$-$0.07$. For the large values of $\beta$ considered here, the contribution of the potential $\kappa/x$ can be computed in perturbation. Using the AFM results, one obtains
\begin{equation}
\label{eq:meankap}
\left\langle \frac{\kappa}{x} \right\rangle \approx \frac{2^{1/4}\kappa}{\beta^{1/8} \sqrt{B+3/2}}.
\end{equation}
This contribution is at most around 4\% of $\epsilon-\sqrt{\beta}$ for $c\bar c$ hybrids and 10\% for $b\bar b$ hybrids. So, in first approximation, it is relevant to neglect the Coulomb interaction in the search for towers of hybrid mesons.
Under these conditions, the mass of a hybrid meson is given by
\begin{eqnarray}
\label{eq:mqqg}
M_{\textrm{hm}} &\approx& m_Q+m_{\bar Q}+\sqrt{2\pi a (K+3/2)} \nonumber \\
&&+\sqrt{\frac{a^{3/2}}{\mu\sqrt{2\pi (K+3/2)}}}\, (B+3/2).
\end{eqnarray}
For high excitation of the gluon field ($K\gg 0$), $\beta$ increases and the accuracy of the approximation~(\ref{eq:mqqg}) improves. Using the AFM, an accurate mass formula for a ordinary meson with just the linear confinement is given by \cite{Sil08a}
\begin{equation}
\label{eq:mqq}
M_{\textrm{om}}\approx m_Q+m_{\bar Q}+ \frac{3}{2}\left( \frac{a^2}{\mu} \right)^{1/3}
\left( \frac{\pi}{\sqrt{3}}n +l+ \frac{\sqrt{3}\pi}{4} \right)^{2/3}.
\end{equation}
Notice that $K=0$ is not the ground state of the flux tube, but rather its first excited level. The ground state simply corresponds to $b=0$ in Eq.~(\ref{eq:potex}), that is a linear confining potential, and leads to formula (\ref{eq:mqq}).
\section{Phenomenology}
\label{sec:Phenom}
Some comments have to be done at this stage. Hamiltonian~(\ref{eq:H}) describes a genuine heavy hybrid meson (no mixing with other hadronic states) within a spin-independent formalism. It will thus lead to qualitative global predictions rather than to a detailed mass spectrum. All spin effects are neglected but they should be weak for heavy hybrids since they are proportional to $1/(m_Q m_{\bar Q})$ -- see for example Ref.~\cite{kalash} for a numerical check of that point. We think however that such global predictions are quite robust precisely because they do not depend on a fine-tuned model. Nevertheless, it is important to wonder whether our method will preferentially apply to some $J^{PC}$ quantum numbers or not. Non-exotic quantum numbers, like $1^{--}$ for example, must be examined very carefully because such quantum numbers allow for a possibly strong mixing with ordinary mesons.
Such a mixing is by definition absent for exotic quantum numbers like $1^{-+}$, $2^{+-}$, $3^{-+}$, \dots, although mixing with tetraquarks (or even glueballs) cannot be excluded. To our knowledge, the mixing between hybrid mesons and tetraquarks is far from being well-known theoretically, and we will not discuss it in the present work.
\begin{figure}[t]
\begin{center}
\includegraphics*[width=8.5cm]{fig0.eps}
\end{center}
\caption{Comparison between experimental $c\bar c$ and $b\bar b$ radial trajectories (full circles) and the predictions of Eq.~(\ref{eq:mqq}) for $a=0.2$~GeV$^2$, $m_c=1.152$~GeV, and $m_b=4.620$~GeV. Experimental data come from the PDG~\cite{PDG} and the quantum numbers assignment of Ref.~\cite{lak} for the $c\bar c$ mesons is followed. Dashed lines are plotted to guide the eyes.}
\label{fig1}
\end{figure}
Let us first estimate the value of the our model's parameters. The spin effects in mesons are minimal in the $1^{--}$ channel; it is thus relevant to fit the parameters by requiring Eq.~(\ref{eq:mqq}) to reproduce the radial trajectories of the $1^{--}$ $c\bar c$ and $b\bar b$ mesons. We first take $a=0.2$~GeV$^2$ so that potential~(\ref{eq:potex}) optimally fits the corresponding lattice QCD data~\cite{Bui06b}. Then, it is readily seen in Fig.~\ref{fig1} that the experimental data are well described by setting $m_c=1.152$~GeV and $m_b=4.620$~GeV -- the $\Upsilon(1S)$ is poorly reproduced because of the neglect of the Coulomb term. Such values are not unrealistic when compared to the PDG values of $1.27^{+ 0.07}_{- 0.11}$~GeV and $4.20^{+ 0.17}_{- 0.07}$~GeV~\cite{PDG}, and quite common in effective approaches.
As already said in the introduction, the status of the $1^{--}$ resonance $Y(4260)$ is not clear yet. If it is an ordinary meson, it should have $S=1$, $n=3$ and $l=0$ \cite{Lla05}, and Eq.~(\ref{eq:mqq}) then leads to a mass of 4.512~GeV, quite far from the experimental value. If it is a hybrid meson, the quantum numbers must be $S=1$, $n=0$ and $l=1$ \cite{Gen07}, and Eq.~(\ref{eq:mqqg}) leads to a more similar mass of 4.239~GeV. But, the repulsive Coulomb term would slightly decrease this agreement. The ordinary meson interpretation for the $1^{--}$ resonance $\Upsilon(10890)$ has been discarded in Ref.~\cite{Che08}; Eq.~(\ref{eq:mqqg}) is compatible with the hybrid meson picture since it leads to a mass of 10.894~GeV for the corresponding state. Our model does not disagree with a hybrid meson assignment for both states, but no definitive conclusion can be drawn at this stage, mainly because of the neglect of the mixing with ordinary mesons.
We now propose a \textit{modus operandi} to separate the hybrid mesons from the ordinary ones. Let us first assume that we have a sufficient amount of experimentally found $Q\bar Q$-like resonances -- this should be possible in a near future thanks to the forthcoming experiments we mentioned in the introduction. Then, for each $J^{PC}$ state, one has to find the lowest possible value of $B$ that is compatible with these quantum numbers in a hybrid meson picture. We recall that $P=(-1)^{l+l_g}$ and $C=(-1)^{l+S_{q\bar q}+1}$ in this case, and that $l_g=0$ when $K=0$. Once this step is achieved, one can plot the experimental masses versus $B$. Our prediction is that the hybrid mesons, or at least states that are dominated by a hybrid meson component, will be located along a straight line while other states like ordinary mesons or tetraquarks will not (typically for example, one has $M_{\textrm{om}}-m_Q-m_{\bar Q}\propto n^{2/3}$ and $l^{2/3}$).
Formula~(\ref{eq:mqqg}) can be rewritten as follows for $K=0$
\begin{equation}\label{bmass}
M_{{\rm hm}}(B) = \sqrt{\frac{a^{3/2}}{\mu\sqrt{3\pi }}}\ (B-B_0)+M_{{\rm hm}}(B_0).
\end{equation}
It is plausible that an exotic $1^{-+}$ hybrid meson, thus with $B=1$, will be first discovered. Effective approaches as well as lattice QCD indeed reach the conclusion that the $1^{-+}$ state is the lightest exotic $Q\bar Q$ hybrid meson (see \textit{e.g.} Refs.~\cite{revlat,kalash}). Then, by setting $B_0=1$ and $M_{{\rm hm}}(B_0)=M_{1^{-+}}$, Eq.~(\ref{bmass}) states that the first exotic states will be located on a straight line, as schematically represented in Fig.~\ref{fig2}. Note that in our approach, the $1^{-+}$ $c\bar c$ state lies below the experimental estimation of the $DD^*$ threshold, which is of about 4.3 GeV~\cite{PDG} ($D^*$ denotes the P-wave excitation of the $D$ meson).
The results of formula~(\ref{bmass}) can also be compared with lattice QCD. Most of the efforts in lattice QCD were devoted to the computation of the lowest-lying $1^{-+}$ hybrid meson masses. We can quote: 4.420$\pm$0.013~GeV~\cite{clat0}, $4.369 \pm0.136$~GeV~\cite{clat1}, and $4.405\pm0.038$~GeV~\cite{clat2} for the $1^{-+}$ $c\bar c$ states, and $10.82\pm0.08$~GeV~\cite{blat1} and $10.977\pm0.123$~GeV~\cite{blat2} for the $1^{-+}$ $b\bar b$ states. It is worth saying that the $1^{-+}$ and $1^{--}$ $b\bar b$ hybrid mesons are degenerate in Ref.~\cite{blat1}, showing the weakness of spin effects for systems of bottom quarks. Formula~(\ref{bmass}) is in agreement with the current estimations as shown in Fig.~\ref{fig2}; it would be very interesting that other masses become available in lattice QCD in order to check whether they are located on the straight lines we predict or not.
\begin{figure}[t]
\begin{center}
\includegraphics*[width=8.5cm]{fig0b.eps}
\end{center}
\caption{Masses of exotic $Q\bar Q$ hybrids versus the band number $B$ as predicted by formula~(\ref{bmass}) (circles and dashed lines). Parameters of Fig.~\ref{fig1} are used. An explicit example of exotic $J^{PC}$ trajectory is given; the lattice data of Refs.~\cite{clat0,clat1,clat2,blat1,blat2} are also plotted for comparison (crosses).}
\label{fig2}
\end{figure}
\section{Conclusion}
\label{sec:Conc}
Starting from a potential model of hybrid mesons which is in agreement with lattice QCD, we predict that the mass of the lowest-lying genuine heavy hybrid mesons is a linear function of the band number $B=2 n+l$, $n$ and $l$ being the quantum numbers of the quark-antiquark pair. The slope and the intercept at the origin depend on the excitation quantum number $K$ of the gluon field. These heavy hybrid mesons form towers of states organized as lines in plots where the masses are presented versus the harmonic oscillator band number. We believe that this property can be an interesting tool to disentangle resonances which are dominated by a heavy hybrid meson component from other hadronic states in future experiments, just as the existence of Regge trajectories is an important guide to identify light mesons. Apart from experiment, an important check of the present results would be the computation of higher-spin exotic hybrid mesons on the lattice. We hope that such results will be available in the future.
\acknowledgments
CS and FB thank the F.R.S.-FNRS for financial support.
\begin{appendix}
\section{Comparison to exact results}
\label{sec:comp}
The AFM cannot give strong constraints on the dependence
of $N$ in terms of $(n,l)$. In particular, had we chosen $P(r)=\textrm{sgn}(\lambda)\,
r^\lambda$, the better choice for $N$ would have been $N = A(\lambda)n + l +
C(\lambda)$, with the quantities $A(\lambda)$ and $C(\lambda)$ given in
Ref.~\cite{Sil08a}. The square root potential $\sqrt{x^2+\beta}$ ensures a smooth transition from a
linear form ($\lambda=1$ but in this case we have only approximate expressions) to
a quadratic form ($\lambda=2$ and in this case the values are exact) as $\beta$
increases from $0$ to $\infty$.
The procedure we adopt is based on the following points:
\begin{itemize}
\item We calculate the exact values $\epsilon_{nl}^*(\beta)$ for $0 \leq n \leq
n_{\max}$, $0 \leq l \leq l_{\max}$ and for a given set of $\beta$ values. This
program is achieved using a very powerful method known as the Lagrange mesh
method (described in detail in Ref.~\cite{sem01}). For our purpose, we consider
that $n_{\max}=l_{\max}=4$ is a good choice. For any calculated value, we have
an accuracy better than $10^{-5}$.
\item We calculate the approximate values $\epsilon_{nl}(\beta)$ using Eqs.~(\ref{eq:Enu0})
with $Y$ given by Eq.~(\ref{eq:defY}) in which $N$ is deduced from Eq.~(\ref{eq:exprN}),
for the same set of $\beta$ values. Building the $\chi$-square
\begin{equation}
\label{eq:chis}
\chi(\beta) = \frac{1}{(n_{\max}+1)(l_{\max}+1)}\sum_{n=0}^{n_{\max}} \sum_{l=0}^{l_{\max}}
\left(\epsilon_{nl}^*(\beta)-\epsilon_{nl}(\beta)\right)^2,
\end{equation}
we request the coefficients $A(\beta)$ and $C(\beta)$ of $N$ to minimize this function.
The obtained values are represented by black dots in Figs.~\ref{fig:exp}.
\item In order to obtain functions which are as simple as possible, continuous in $\beta$,
and which reproduce at best the above calculated values, we choose hyperbolic forms
and require a best fit on the set of the sample. Explicitly, we find
\begin{equation}
\label{eq:coefacd}
A(\beta) = \frac{8 \beta + 102}{4 \beta + 57}, \quad
C(\beta) = \frac{30 \beta + 53}{20 \beta + 39}.
\end{equation}
These integers are rounded numbers whose magnitude is chosen in order to not exceed too much 100.
The corresponding values are plotted as continuous curves in Figs.~\ref{fig:exp}.
They have been constrained to exhibit the right behavior $A \to 2$ and
$C \to 3/2$ for very large values of $\beta$. Formulas~(\ref{eq:coefacd}) give $A(0)=102/57\approx 1.789$ and
$C(0)=53/39\approx 1.359$. These values are such that
$A(0)\approx \pi/\sqrt{3}\approx 1.814$ and $C(0)\approx \sqrt{3}\pi/4\approx 1.360$, as expected from the results of Ref.~\cite{Sil08a} in the case of a nonrelativistic linear potential.
\end{itemize}
\begin{table}[ht]
\caption{\label{tab:comp} Comparison between the exact values $\epsilon_{nl}^*(\beta)$ (2nd line)
and analytical approximate expressions $\epsilon_{nl}(\beta)$ for the
eigenvalues of Hamiltonian~(\ref{eq:ph}) with $\beta=1$. For each set $(n,l)$, the exact result
is obtained by numerical integration. 3rd line: approximate results
are given by Eqs.~(\ref{eq:Enu0}) with Eqs.~(\ref{eq:defY}), (\ref{eq:exprN}) and
(\ref{eq:coefacd}); 1st line: upper bounds obtained with $N=2 n+l+3/2$; 4th line:
lower bounds obtained with $N=n+l+1$.}
\begin{ruledtabular}
\begin{tabular}{cccccc}
& $l=0$ & $l=1$ & $l=2$ & $l=3$ & $l=4$\\
\hline
$n=0$ & 1.94926 & 2.49495 & 2.99541 & 3.46197 & 3.90193 \\
& 1.91247 & 2.45074 & 2.94841 & 3.41419 & 3.85430 \\
& 1.89549 & 2.44621 & 2.95032 & 3.41969 & 3.86189 \\
& 1.65395 & 2.22870 & 2.75000 & 3.23240 & 3.68492 \\
$n=1$ & 2.99541 & 3.46197 & 3.90193 & 4.32027 & 4.72059 \\
& 2.89556 & 3.34652 & 3.77899 & 4.19405 & 4.59335 \\
& 2.85420 & 3.32970 & 3.77678 & 4.20097 & 4.60620 \\
& 2.22870 & 2.75000 & 3.23240 & 3.68492 & 4.11355 \\
$n=2$ & 3.90193 & 4.32027 & 4.72059 & 5.10556 & 5.47723 \\
& 3.74112 & 4.14232 & 4.53310 & 4.91307 & 5.28251 \\
& 3.69078 & 4.11913 & 4.52783 & 4.91998 & 5.29790 \\
& 2.75000 & 3.23240 & 3.68492 & 4.11355 & 4.52250 \\
$n=3$ & 4.72059 & 5.10556 & 5.47723 & 5.83725 & 6.18692 \\
& 4.50374 & 4.87138 & 5.23246 & 5.58628 & 5.93264 \\
& 4.44883 & 4.84403 & 5.22459 & 5.59242 & 5.94903 \\
& 3.23240 & 3.68492 & 4.11355 & 4.52250 & 4.91485 \\
$n=4$ & 5.47723 & 5.83725 & 6.18692 & 6.52732 & 6.85935 \\
& 5.20859 & 5.55148 & 5.88996 & 6.22329 & 6.55111 \\
& 5.15078 & 5.52098 & 5.87970 & 6.22821 & 6.56756 \\
& 3.68492 & 4.11355 & 4.52250 & 4.91485 & 5.29295 \\
\end{tabular}
\end{ruledtabular}
\end{table}
Since our results are exact for $\beta \to \infty$, one has obviously $\chi=0$
in this limit. The error is maximal for small values of $\beta$ but, over the
whole range of $\beta$ values, the results given by
our analytical expression can be considered as excellent.
Just to exhibit a quantitative comparison, we report in Table~\ref{tab:comp} the exact $\epsilon_{nl}^*(\beta)$ and approximate $\epsilon_{nl}(\beta)$ values
obtained for $\beta=1$, a value for which the corresponding potential is neither
well approximated by a linear one nor a harmonic one.
As can be seen, our approximate expressions are better than $1\%$ for any
value of $n$ and $l$ quantum numbers. Such a good description is
general and valid whatever the parameter $\beta$ chosen.
The upper bounds obtained with $P(r)=r^2$ are far better than the lower bounds computed with $P(r)=-1/r$.
This is expected since the potential $\sqrt{a^2 r^2+b}$ is closer to a harmonic interaction than to a Coulomb one.
Better lower bounds could be obtained with $P(r)=r$. But,
the exact form of $N$ is not known for this potential, except for $l=0$ for which $N$ can be
expressed in term of zeros of the Airy function. With the approximate form
$N=(\pi/\sqrt{3}) n+l+\sqrt{3}\pi/4$ \cite{Sil08a,Sil08b}, we have checked that results obtained are good but
the variational character cannot be guaranteed.
\begin{figure}[ht]
\begin{center}
\includegraphics*[width=6.4cm]{fig1a.eps}
\includegraphics*[width=6.4cm]{fig1b.eps}
\caption{\label{fig:exp} Best values of the coefficients $A(\beta)$ and $C(\beta)$
to parameterize the eigenvalues of Hamiltonian~(\ref{eq:ph}):
numerical fit with Eq.~(\ref{eq:chis}) (dots); functions~(\ref{eq:coefacd}) (solid line).}
\end{center}
\end{figure}
\end{appendix}
|
1,116,691,500,755 | arxiv | \section{Introduction and Related Work}
Licensed shared access (LSA)~\cite{ECC2014} has been adopted in Europe as a promising paradigm to dynamically share licensed spectrum between different networks and technologies.
LSA proposes a two-tier approach where the initial target use case considered mobile network operators (MNOs) leasing spectrum in the 2.3--2.4~GHz band from incumbent technologies like Programme Making and Special Events (PMSE)~\cite{ECC2015}.
However, recent initiatives from industry and spectrum regulators have proposed a symmetric use case, where PMSE users could lease spectrum from MNOs, targeting reliable short-term use of spectrum for concerts, conferences, etc.~\cite{PMSExG2017}.
Though the adoption of LSA brings significant benefits from a technical perspective, a number of business challenges arise for the key stakeholders of the market (\emph{i.e.}, regulator, incumbent spectrum user, and LSA licensee). These include the MNO's costs of additional infrastructure and the required modifications of the existing systems to support and manage the sharing procedure, as well as the license fees~\cite{tehrani2016}. Thus, the stakeholders must perform a \mbox{techno-economic} analysis in order to assess whether LSA is worth the investment. However, business research on LSA is scarce~\cite{ahokangas2014,ahokangas2014b,matinmikko2018} and focuses on the qualitative domain, without offering quantitative results on whether LSA schemes are \mbox{techno-economically attractive.}
The work closest to ours is~\cite{voicu2018}, where an MNO that operates under the LSA framework leases spectrum to a number of PMSE users that belong to two distinct quality-of-service (QoS) classes, admitting either low or high QoS requirements. As in~\cite{voicu2018}, we study scenarios where all users have either high or low QoS requirements, as well as mixed QoS requirements (\emph{i.e.}, some users have low and some users have high QoS requirements). We extend the approach of~\cite{voicu2018}, aiming at unlocking the potential of QoS-aware pricing in this LSA market, where we adopt price differentiation based on the QoS class. Our key contributions are the following. From the perspective of the PMSE users, we model the behaviour of the users regarding how they choose between the two QoS classes, as well as their available budgets for the two QoS classes. Through this process, we are able to predict the distribution of the users between the two QoS classes for each possible combination of \mbox{considered prices.}
From the perspective of the MNO, we identify the prices that correspond to the maximum revenue that can be achieved for each QoS scenario. A consistent result arises independently of i)~the distribution of the budgets, ii)~the way that the users choose between the QoS classes, and iii)~the values of the technical parameters. The MNO can always tune the prices so that the maximum revenue for the high QoS scenario is the highest, followed by the mixed QoS scenario and finally by the low QoS scenario. This result highlights the potential of \mbox{QoS-aware} pricing for the MNO, since the MNO has motivation to sacrifice some of the users with low QoS in order to support more users with high QoS and charge them more. This is also interesting from a regulatory point of view, since we identify a constant tussle in the LSA market, where the goal of the MNO (\emph{i.e.}, revenue maximisation) is not aligned with the goal of the market regulator (\emph{i.e.}, social welfare maximisation). Finally, we quantify the impact of the budget parameters on the revenue of the QoS scenarios, providing insights for which markets have the potential to be more profitable for the MNO.
\section{The Techno-Economic Problem}
We first summarise the techno-economic input from~\cite{voicu2018} that we are going to use for our analysis. Then, we introduce our extensions. We assume a monopolistic market with one MNO and $N$ PMSE users that are interested in leasing spectrum from the unique MNO. Consistent with one of the business models in~\cite{PMSExG2017}, the PMSE users also utilise the network infrastructure of the MNOs. Furthermore, the PMSE users are classified into two distinct QoS classes: there are at most \mbox{$N_L$ PMSE users} with low QoS requirements (e.g., audio speech applications) and at most $N_H$ PMSE users with high QoS requirements (e.g., high definition audio productions). We are interested in analysing from a techno-economic point of view the following three \mbox{QoS scenarios:}
\begin{itemize}
\item \emph{Low QoS Scenario}: The MNO can support at most $N_L$ users, where all of them have the same low QoS requirements $Q_L$.
\item \emph{High QoS Scenario}: The MNO can support at most $N_H$ users, where all of them have the same high QoS requirements $Q_H$.
\item \emph{Mixed QoS Scenario}: The MNO supports users with mixed QoS requirements, \emph{i.e.}, at most $N_\mathit{L,M}$ users with $Q_L$ and at most $N_\mathit{H,M}$ users \mbox{with $Q_H$.}
\end{itemize}
Given the maximum number of supported PMSE users for the three QoS scenarios, the goal of the MNO is to define a pricing policy and choose the scenario that will maximise its revenue. Among the four pricing policies that have been considered in~\cite{voicu2018}, we apply QoS-aware pricing, where the differentiation in the price is based on the QoS class that each user belongs to \cite{huang2013}. Depending on the assumptions and the model, QoS-aware pricing may maximise e.g. the revenue of the MNO or the social welfare \cite{shetty2010, wang2017}.
We adopt a type of QoS-aware pricing which corresponds to an application of the \emph{second degree of price discrimination}~\cite{maille2014}. In this form of discrimination, there are at least two distinct prices, which correspond to at least two different types of services. Any customer who wants the same type of service will pay the same price. In our case, we propose that the discrimination is based on the QoS class that each PMSE user belongs to; each user that targets $Q_L$ pays $P_L \in [P_{L,\min},P_{L,\max}]$, whereas each user that targets $Q_H$ pays $P_H$. We also define parameter $K=\frac{P_H}{P_L}$ which is always above 1. Then, the revenue of the MNO for each of the three QoS scenarios is:
\begin{align}
\text{Low QoS Scenario: }& N_LP_L, \label{eq:lowQoS} \\
\text{High QoS Scenario: }& N_HP_H=N_HKP_L,\label{eq:highQoS} \\
\text{Mixed QoS Scenario: }& N_\mathit{L,M}P_L+N_\mathit{H,M}P_H=N_\mathit{L,M}P_L+N_\mathit{H,M}KP_L. \label{eq:mixedQoS}
\end{align}
Clearly, the scenario that maximises the MNO's revenue can be computed by the following formula:
\begin{equation*} \max\{N_L, N_HK, N_\mathit{L,M}+N_\mathit{H,M}K\}.\end{equation*}
In~\cite{voicu2018}, there has been an extensive study of the revenue for the three QoS scenarios. For different values of the technical parameters including carrier frequency $f$, propagation environment, base station (BS) transmit power level, and bandwidth, the maximum number of supported PMSE users for the three QoS scenarios has been computed. Then, the revenue after the application of QoS-aware pricing has been estimated for a fixed value of $P_L$ and a range of values of $P_H$. A key assumption during the whole analysis was that the MNO always serves the maximum number of users that can be \mbox{technically supported.
We generalise this study towards the following two directions. First, we introduce an additional degree of freedom studying markets with different values of $P_L$. Second, we relax the assumption that the market always performs at its maximum capacity by proposing a methodology to compute the exact number of PMSE users that will be admitted in each QoS scenario. In order to do so, we need to model the behaviour of the users. Initially, we need to model how a user chooses between the two QoS classes. Therefore, we introduce a metric $w$ that quantifies the preference of each user $i$ for each QoS class by weighing the importance that the user gives to the price and the QoS.
For the high QoS class, $w$ is defined as follows:
\begin{equation*}
w_{H,i}=a_i\frac{P_L}{P_L+P_H}+(1-a_{i})\frac{Q_H}{Q_L+Q_H},
\end{equation*}
where the user-specific parameter $a_i $ follows a uniform distribution in (0,1). When $a_i$ is above 0.5, user $i$ considers as the most important factor the price that it has to pay, otherwise the most decisive factor is the QoS that it gets. We note that we use fractions for a relative comparison of the two factors that influence the decision of the user, which is why $w$ also ranges between 0 \mbox{and 1.}
Similarly, for the low QoS class, $w$ is defined as:
\begin{equation*}
w_{L,i}=a_i\frac{P_H}{P_L+P_H}+(1-a_i)\frac{Q_L}{Q_L+Q_H}.
\end{equation*}
Note that $w_{H,i}+w_{L,i}=1$, meaning that each user $i$ needs to compute just one of them. If $w_{H,i}$ is higher than 0.5, then user $i$ prefers the high QoS class. Otherwise, it prefers the low QoS class.
Another aspect that was not modelled in~\cite{voicu2018} is the user's available budget for each QoS class. Though we are not aware of specific studies for the distribution of the budgets of the PMSE users, we expect that it follows a (variation of the) normal distribution. This is in accordance with adjacent telecommunication markets~\cite{maille2014}. More specifically, we model the distribution of the budget for the low QoS $B_L$ as a truncated normal distribution with minimum value \mbox{$P_{L,\min}=\$10$ ~\cite{Ofcom2018}.} We need a minimum value, otherwise a user can never get access to this QoS class, so it is not of interest for this market. We study 6 cases for $B_L$, where the mean $\mu_L=\{0.5,0.7,0.9\}P_{L,\max}$ and the standard deviation $\sigma_L=\{0.2,0.4\}P_{L,\max}$, with $P_{L,\max}=\$120$~\cite{Ofcom2018}.
Then, we model the distribution of the budget for the high QoS $B_H$ as a truncated normal distribution with minimum value $B_{L}$. The motivation for this minimum threshold is that the user's budget for the high QoS class should be at least equal to its budget for the low QoS class. For $B_H$, we also consider 6 cases, where the mean $\mu_H=\{0.2,0.4,0.6\}\frac{Q_H}{Q_L}B_{L}$ and the standard deviation $\sigma_H=\{0.2,0.4\}\frac{Q_H}{Q_L}B_{L}$. The quantity $\frac{Q_H}{Q_L}B_{L}$ is used as a benchmark, since, as we know from adjacent markets \cite{maille2014}, a typical user is expected to be willing to spend at most $\frac{Q_H}{Q_L}$ times more to get the class $Q_H$ instead of the class $Q_L$. Moreover, since the budget of the users for more expensive services is expected to be tighter, the coefficients of $\mu_H$ are typically lower than the ones of $\mu_L$.
\subsection*{Maximum Number of PMSE Users}
Table~\ref{table_qos} summarises the values of the technical parameters from \cite{voicu2018} used to estimate the maximum number of PMSE users that can be \mbox{technically supported.}
Each PMSE user has either high or low QoS requirements. We define the QoS requirements in terms of the target Application-layer throughput $R$, where high QoS and low QoS correspond to 4.61~Mbps and 150~kbps, respectively. These values are consistent with the highest and lowest PMSE audio throughput requirements in~\cite{3GPP2018, PMSExG2017}, where low throughput values correspond to audio speech applications, while high throughput values are required for high definition audio productions~\cite{Pilz2018}. Based on these values of the technical parameters, Table~\ref{table_max_users} summarises from \cite{voicu2018} the maximum number of users that can be supported for the three QoS scenarios. Since the number of users for the carrier frequencies of 2600~MHz and 3800~MHz are quite similar, we analyse only three cases: i)~800~MHz for the indoor propagation environment, ii)~800~MHz for the outdoor propagation environment, and iii)~3800~MHz for the indoor \mbox{propagation environment.}
\begin{table}[t]
\caption{PMSE user QoS requirements and technical parameters.}
\label{table_qos}
\centering
\begin{tabular}{|p{4cm}|p{2.2cm}|p{2.3cm}|p{3.5cm}|}
\hline
\multirow{2}{*}{\parbox{4cm}{\centering\textbf{Parameter}}} & \multicolumn{3}{c|}{\textbf{Value}} \\
\cline{2-4}
& \parbox[c][0.9cm]{2.2cm}{\centering\textbf{Low QoS Scenario}} & \parbox{2.2cm}{\centering\textbf{High QoS Scenario}} & \parbox{3.8cm}{\centering\textbf{Mixed QoS \\Scenario}} \\
\hline
PMSE user QoS requirements as Application-layer throughput $R$ & 150~kbps \cite{3GPP2018, PMSExG2017} & 4.61~Mbps \cite{3GPP2018, PMSExG2017} & 4.61 Mbps for 50\% of the users in the high QoS scenario and 150~kbps for other users\\
\hline
bandwidth $C$ & \multicolumn{3}{l|}{20 MHz~\cite{3GPP2017}}\\
\hline
carrier frequency $f$ & \multicolumn{3}{l|}{800, 2600, 3800 MHz~\cite{3GPP2017}}\\
\hline
BS transmit power $T$ & \multicolumn{3}{l|}{30 dBm~\cite{3GPP2017, 3GPP2017a} (same for all BSs)}\\
\hline
propagation environment & \multicolumn{3}{l|}{indoor, outdoor} \\
\hline
\end{tabular}
\end{table}
\begin{table} [t]
\caption{Max. number of users that can be supported for the three QoS scenarios for the different values of the technical parameters.}
\label{table_max_users}
\begin{center}
\footnotesize
\begin{tabular}{ |p{4.3cm} || c|| c || c | c |}
\hline
\multirow{3}{*}{\diagbox[width=4.74cm, height=1cm]{\textbf{Frequency,}\\ \textbf{Environment}}{\textbf{Scenario}}} & \textbf{Low QoS} & \textbf{High QoS} & \multicolumn{2}{c|}{\textbf{Mixed QoS}}\\
& & & \multicolumn{2}{c|}{ } \\
\cline{4-5}
& \textbf{Users} $N_L$ & \textbf{Users} $N_H$ & \textbf{Users} $N_\mathit{L,M}$ & \textbf{Users} $N_\mathit{H,M}$ \\
\hline
{\cellcolor{green!25}} $f$=800~MHz, indoor & {\cellcolor{green!25}}65 & {\cellcolor{green!25}}6 & {\cellcolor{green!25}}21 & {\cellcolor{green!25}}3 \\
\hline
{\cellcolor{green!25}} $f$=800~MHz, outdoor & {\cellcolor{green!25}}7 & {\cellcolor{green!25}}2 & {\cellcolor{green!25}}4 & {\cellcolor{green!25}}1 \\
\hline
$f$=2600~MHz, indoor & 36 & 4 & 13 & 2 \\
\hline
$f$=2600~MHz, outdoor & 31 & 4 & 12 & 2 \\
\hline
{\cellcolor{green!25}} $f$=3800~MHz, indoor & {\cellcolor{green!25}}37 & {\cellcolor{green!25}}4 & {\cellcolor{green!25}}13 & {\cellcolor{green!25}}2 \\
\hline
$f$=3800~MHz, outdoor & 33 & 4 & 12 & 2 \\
\hline
\end{tabular}
\end{center}
\vspace{-0.8cm}
\end{table}
\section{Revenue Analysis: A Case Study}
In this section, we illustrate the evolution of the revenue for the three QoS scenarios for the example of the carrier frequency $f$=3800~MHz and the indoor propagation environment. We assume that the market consists of 41 PMSE users so that, provided that all of them have the necessary budget to pay for the prices $P_L$ and $P_H$, the maximum number of supported users can be admitted (\emph{i.e.}, either $N_L$=37, or $N_H$=4). For a given set of prices $P_L$ and $P_H$, we assume that the users follow a so-called \emph{non-strict} version for the choice of the QoS class. In this non-strict version, a user initially applies for getting access to the QoS class that it prefers more based on the value of the weighted metric $w$. It gets access to this QoS class provided that the following two conditions hold: i)~it can afford to pay the price that the MNO has announced and ii)~the MNO has not reached the maximum number of PMSE users that it can support for this QoS class. If the user does not get access to the QoS class of its first choice, then it applies for the other QoS class and it gets admitted provided that the same conditions hold. In the following section, we also consider a \emph{strict} version for the choice of the QoS class, where each user applies for only one QoS class, \emph{i.e.}, the one that corresponds to the highest value of the weighted metric $w$.
\begin{figure}[t!]
\centering
\subfloat[$P_L$=\$30]{\includegraphics[width=0.4\textwidth]{fig_1_Pl_30} \label{fig_example_PL30}}
~
\subfloat[$P_L$=\$60]{\includegraphics[width=0.4\textwidth]{fig_1_Pl_60} \label{fig_example_PL60}}\\
\subfloat[$P_L$=\$90]{\includegraphics[width=0.4\textwidth]{fig_1_Pl_90} \label{fig_example_PL90}}
~\subfloat[$P_L$=\$120]{\includegraphics[width=0.4\textwidth]{fig_1_Pl_120} \label{fig_example_PL120}}
\\\caption{Evolution of the revenue for the three QoS scenarios. Technical parameters: carrier frequency $f$=3800~MHz and indoor propagation environment. Parameters of the distribution of the budgets $B_L$ and $B_H$: $\mu_L=0.7P_{L,\max}$, $\sigma_L=0.4P_{L,\max}$, $\mu_H=0.4\frac{Q_H}{Q_L}B_{L}$, $\sigma_H=0.2\frac{Q_H}{Q_L}B_{L}$. The choice of the QoS class is non-strict.}
\label{fig_revenue_example}
\end{figure}
After deciding whether a user will be admitted and, if so, in which QoS class, the MNO computes the revenue for the three QoS scenarios. We consider four values of $P_L$, corresponding to 30, 60, 90, and 120 \$ for 48-hour access \cite{Ofcom2018}. For a given $P_L$, we apply QoS-aware pricing where $P_H=KP_L$, with parameter $K \in \{2, 3, \dots, \floor[\Big]{\frac{Q_H}{Q_L}}=30\}$.
Fig.~\ref{fig_revenue_example} shows the evolution of the revenue for the three QoS scenarios for the four values of $P_L$. Each subfigure corresponds to the revenue as a function of parameter $K$, for a given $P_L$. The results are averaged based on the simulation of 1000 markets, each consisting of 41 users. As we notice from Fig.~\ref{fig_example_PL30}, when parameter $K$ is below~7, the low QoS scenario generates the highest revenue. This is justified since the price differentiation between $Q_H$ and $Q_L$ is small enough to not overcome the difference between the actual number of users that are supported for $Q_H$ and $Q_L$. For higher values of $K$, the high QoS scenario generates the highest revenue, followed by the mixed QoS scenario. Also, the revenue for both the high QoS and the mixed QoS scenario increases linearly with $K$. This is expected from the corresponding equations \eqref{eq:highQoS} and \eqref{eq:mixedQoS} provided that the number of users $N_H$ and $N_{H,M}$ does not change with $K$. Finally, for the low QoS scenario, the revenue does not change with $K$, so any fluctuation is due to changes in the number of users.
Fig.~\ref{fig_example_PL60} shows the revenue for $P_L=\$60$, where we notice some differences in the trends. First, though $P_L$ was doubled compared to Fig.~\ref{fig_example_PL30}, the revenue for the low QoS scenario was not doubled. This means that the budget $B_L$ of some of the users is below~\$60 and, therefore, they cannot afford to pay for this QoS class. Due to this, the high QoS scenario generates the highest revenue starting with a smaller value of $K$ (it is for $K>6$, whereas for $P_L=\$30$ it was for $K>7$). Moreover, for high values of $K$, the revenue for the high QoS scenario starts increasing sub-linearly and then it decreases. This is again due to budget constraints, this time for the budget $B_H$. The trend of a sub-linear increase is also noticed for the mixed QoS scenario, though it starts for higher values of $K$ compared to the high QoS scenario. This is expected since, for the mixed QoS scenario, the maximum number of users with high QoS that can be admitted is 2 instead of~4 for the high QoS scenario (see Table~\ref{table_max_users}). Therefore, for higher values of $K$, it is easier to find 2 instead of 4 users with $Q_H$.
\begin{figure}
\centering
\subfloat[Budget low]{\includegraphics[width=0.4\textwidth]{fig_2_Bl} \label{fig_budget_low}}
~
\subfloat[Budget high]{\includegraphics[width=0.415\textwidth]{fig_2_Bh} \label{fig_budget_high}}
\\
\caption{Distribution of the budgets $B_L$ and $B_H$.}
\label{fig_budget}
\end{figure}
Figs.~\ref{fig_example_PL90} and~\ref{fig_example_PL120} verify the above mentioned trends. The revenue for the low QoS scenario starts decreasing as $P_L$ increases further to $\$90$ and $\$120$, since many users cannot afford to pay these prices. The message learnt for the MNO is that, for the low QoS scenario, a high price does not lead to high revenues. Due to this, the high QoS scenario generates the highest revenue, even with very low values of $K$. Also, the maximum revenue for the high QoS scenario is admitted for a value of $K$ that decreases as $P_L$ increases. The same trends hold for the mixed QoS scenario, but with a higher value of $K$ due to fewer users with high QoS. Due to this and a steep decrease for the revenue of the high QoS scenario, the mixed QoS scenario is the most profitable when both $P_L$ and $K$ are high.
\section{Revenue Analysis: General Results}
Through the detailed analysis of the previous section, we are able to compute the expected revenue of the three QoS scenarios for every possible combination of the techno-economic parameters. Though this methodology provides a \mbox{fine-grained} view for each case, we need to extract general conclusions. Indeed, for a given set of techno-economic parameters, the ultimate challenge for the MNO is to choose the prices $P_L$ and $P_H$ so that its revenue will be maximised. Therefore, we can consider this fine-grained analysis as an internal process for the MNO to compute: i)~the value of $P_L$ that maximises its revenue for the low QoS scenario, ii)~the value of $P_H$, \emph{i.e.}, parameter $K$ and $P_L$, that maximises its revenue for the high QoS scenario, and iii)~the values of $P_L$ and $P_H$ that maximise its revenue for the mixed QoS scenario. Then, the MNO can choose which QoS scenario maximises globally its revenue.
Though the MNO controls the technical parameters and the price, the distribution of the users' budgets as well as the users' preferences for the two QoS classes are private information. The complementary problem of how to estimate this piece of information is not addressed in this paper. However, we present a broad number of scenarios for the parameters that each user controls, so as to estimate the revenue for the three QoS scenarios under different users' behaviours.
\begin{figure}[t!]
\centering
\subfloat[Revenue]{\includegraphics[width=0.4\textwidth]{fig_3_MaxRev_37} \label{fig_revenue_non_strict}}
~
\subfloat[$P_L$]{\includegraphics[width=0.38\textwidth]{fig_3_Pl_37} \label{fig_PL_non_strict}}
\\
\subfloat[$P_H$]{\includegraphics[width=0.4\textwidth]{fig_3_Ph_37} \label{fig_PH_non_strict}}
~
\subfloat[$K=\frac{P_H}{P_L}$]{\includegraphics[width=0.38\textwidth]{fig_3_K_37} \label{fig_k_non_strict}}
\\
\caption{Max. revenue and the corresponding values for $P_L$ and $P_H$ for the three QoS scenarios. Technical parameters: $f$=3800~MHz, indoor. The choice of the QoS class is non-strict.}
\label{fig_non_strict}
\vspace{-0.4cm}
\end{figure}
Initially, we generalise the results of the previous section where we consider 36 budget scenarios for the distribution of the users' budgets $B_L$ and $B_H$. The number of budget scenarios arises since the 4-tuple \{$\mu_L$, $\sigma_L$, $\mu_H$, $\sigma_H$\} can get $3\cdot2\cdot3\cdot2=36$ possible values. Fig.~\ref{fig_budget} represents the evolution of the budget distribution. We progressively update the elements of the 4-tuple in four loops, with the following order from the outermost loop to the innermost loop: i)~$\mu_L$, ii)~$\sigma_L$, iii)~$\mu_H$, and iv)~$\sigma_H$. Due to this, as we can see from Fig.~\ref{fig_budget_low}, $\mu_L$, depicted as a red line, increases every 12 budget scenarios, remaining the same for scenarios 1-12, 13-24, and 25-36. Let us consider scenarios 1-12: due to a higher value of $\sigma_L$, scenarios 7-12 have higher upper quartiles and whiskers than scenarios 1-6. For the case of $B_H$ (Fig.~\ref{fig_budget_high}), we notice that every 6 scenarios where $\mu_L$ and $\sigma_L$ are fixed (\emph{i.e.}, scenarios 1-6, 7-12, etc.), the upper quartile increases. Moreover, the maximum upper whiskers correspond to scenarios 6, 12, etc., where $B_H$ has the highest coefficients for $\mu_H$ and $\sigma_H$.
Fig.~\ref{fig_non_strict} presents the maximum revenue and the corresponding values for $P_L$ and $P_H$ for the three QoS scenarios. As in Fig.~\ref{fig_revenue_example}, we consider the \emph{non-strict} version for the choice of the QoS class and the results are obtained for the carrier frequency $f$=~3800 MHz and the indoor propagation environment. For all combinations of budgets $B_L$ and $B_H$ in Fig.~\ref{fig_revenue_non_strict}, the maximum revenue of the MNO is achieved for the high QoS scenario, followed by the mixed QoS scenario and then by the low QoS scenario. This result highlights the existence of a tussle for this market between the social welfare (\emph{i.e.}, supporting the maximum number of PMSE users) and the revenue maximisation. Focusing on the revenue from the high QoS scenario, we notice that, for budget scenarios 1-6, the maximum is for the last scenario (scenario 6) and this trend is repeated every six scenarios. The explanation is based on the previous analysis for the distribution of the budget $B_H$. The same trend holds for the mixed QoS scenario, implying that the dominant component for the mixed QoS revenue is the revenue that arises from the users with $Q_H$. Finally, for the low QoS scenario, there is a repeating trend for budget scenarios 1-12, 13-24, and 25-36. We recall from Fig.~\ref{fig_budget_low} that all budget scenarios of each of these cycles correspond to the same $\mu_L$ of the budget distribution $B_L$. Moreover, the revenue during each cycle slightly decreases, admitting three local maxima for budget scenarios 1, 13, 25, where $\mu_H$ and $\sigma_H$ have the lowest values (see Fig.~\ref{fig_budget_high}).
\begin{figure}
\centering
\subfloat{\includegraphics[width=0.35\textwidth]{fig_legend_4_1}}
~
\subfloat{\includegraphics[width=0.4\textwidth]{fig_legend_4_2}}
\\
\addtocounter{subfigure}{-2}
\subfloat[Revenue, $f$=3800~MHz, indoor]{\includegraphics[width=0.4\textwidth]{fig_4_MaxRev_37_strict} \label{fig_revenue_comparison}}
~
\subfloat[Users, $f$=3800~MHz, indoor]{\includegraphics[width=0.37\textwidth]{fig_4_NoUsers_37_strict} \label{fig_users_comparison}}
\\
\subfloat[Revenue, $f$=800~MHz, indoor]{\includegraphics[width=0.4\textwidth]{fig_4_MaxRev_65_strict} \label{fig_revenue_comparison_65}}
~
\subfloat[Users, $f$=800~MHz, indoor]{\includegraphics[width=0.37\textwidth]{fig_4_NoUsers_65_strict} \label{fig_users_comparison_65}}
\\
\subfloat[Revenue, $f$=800~MHz, outdoor]{\includegraphics[width=0.4\textwidth]{fig_4_MaxRev_7_strict} \label{fig_revenue_comparison_7}}
~
\subfloat[Users, $f$=800~MHz, outdoor]{\includegraphics[width=0.365\textwidth]{fig_4_NoUsers_7_strict} \label{fig_users_comparison_7}}\\
\caption{Comparison of the max. revenue and the corresponding number of users for the non-strict and the strict choice of the QoS class.}
\label{fig_comparison_65_7}
\end{figure}
Fig.~\ref{fig_PL_non_strict} presents the corresponding value of $P_L$ for which the maximum revenue for each QoS scenario is achieved. It is interesting that for the high QoS scenario, $P_L$ is always equal to $\$120$, \emph{i.e.}, the maximum that the MNO can set throughout the study. For the mixed QoS scenario, $P_L$ is higher than the corresponding price for the low QoS scenario. This is expected, since in the mixed QoS scenario, the MNO can admit at most 13 users with $Q_L$, instead of 37 users for the low QoS scenario (see Table~\ref{table_max_users}). We also notice that the evolution of $P_L$ is similar for both low and mixed QoS scenarios, with the highest values being for budget scenarios 31-36, where $\mu_L$ and $\sigma_L$ get the highest \mbox{values (see Fig.~\ref{fig_budget_low}).}
Then, we show in Fig.~\ref{fig_PH_non_strict} the corresponding value of $P_H$. As expected, it is higher for the mixed QoS scenario where at most 2 users with $Q_H$ can be supported than for the high QoS scenario where $N_H=4$. Moreover, the curves follow the same trend with the revenue. Finally, Fig.~\ref{fig_k_non_strict} depicts the evolution of parameter $K=\frac{P_H}{P_L}$, where the trends are similar with the trends for $P_H$. Clearly, there is room for the MNO to apply higher price differentiation for the case of the mixed QoS scenario compared to the high QoS scenario. Our analysis suggests that in budget scenarios where $\mu_H$ and $\sigma_H$ get the highest values, the MNO has motivation to charge the mixed QoS users with $Q_H$ at the maximum level of price differentiation, \emph{i.e.}, 30 times more than the users with~$Q_L$.
We repeat the same analysis for the \emph{strict} preference of the QoS class, where each user has a single choice for the QoS class. Fig.~\ref{fig_revenue_comparison} compares the maximum revenue for the non-strict and the strict version. The conclusion that arises is that, for all QoS scenarios and all budget scenarios, the revenue is higher for the non-strict version. This is justified due to the fact that the set of revenues for the MNO for the non-strict version is a superset of the strict version: it additionally includes the revenue that each user can bring for its second QoS preference in case it has not been admitted for its first QoS preference. We identify the factors that can justify the difference in the revenue between the non-strict and the strict version, as follows.
The first one is that the number of PMSE users for the non-strict version can be higher than for the strict version. This is clearly the case for the low QoS scenario where, as we can see from Fig.~\ref{fig_users_comparison}, there is a significant drop in the number of users with $Q_L$ for the strict version. However, it is worth mentioning that even in the case of the non-strict version, the maximum revenue for the low QoS scenario does not coincide with the theoretical maximum of PMSE users that can be supported, which is 37. This means that either some users do not have the necessary budget $B_L$ to pay for a particular price $P_L$, or it is more profitable for the MNO to support fewer users with $Q_L$ but at a higher price. Furthermore, it is interesting to notice that, e.g., budget scenarios 1-6 correspond to a higher number of users with $Q_L$ than scenarios 7-12. Given that these scenarios have the same mean $\mu_L$, we conclude that the standard deviation $\sigma_L$ for scenarios 1-6, which is smaller than for scenarios 7-12, is the reason for the difference in the number of users. Indeed, for the users with $Q_L$, it is more profitable for the MNO if the standard deviation $\sigma_L$ is smaller, since, for prices $P_L$ that are close to $\mu_L$, more users can afford to pay for it.
The second factor is that, in the non-strict version, the MNO may have motivation to support fewer users provided that it can charge them more. This is the case with the mixed QoS scenario, where, for some budget parameters (budget scenarios 26-28), the MNO in the non-strict version prefers to support fewer users with $Q_L$ (dark blue solid line) than in the strict version (dark blue dashed line).
We finally proceed with the results for the other two technical cases, \emph{i.e.}, carrier frequency $f$=800~MHz and indoor/outdoor propagation environment. We present the maximum revenue and the corresponding number of users for the three QoS scenarios in Figs.~\ref{fig_revenue_comparison_65}\textendash\ref{fig_users_comparison_7}, omitting the corresponding values of $P_L$ and $P_H$ due to space constraints. As in Fig.~\ref{fig_revenue_comparison}, the high QoS scenario generates always the highest revenue. This is a strong result independent of the technical parameters and the distribution of the budgets. Regarding the corresponding number of users, the two key conclusions that we extracted from Fig.~\ref{fig_users_comparison} still hold. First, the number of users that maximises the revenue for the low QoS scenario does not coincide with the maximum number of users (\emph{i.e.}, 65 users for indoor and 7 users for outdoor). Second, the number of users with $Q_L$ for the mixed QoS scenario is in general lower for the non-strict version compared to the strict version, since the MNO has motivation to support fewer users with $Q_L$ in order to admit more users with $Q_H$ and charge them with high values of $K$. This trend becomes clearer in Fig. \ref{fig_users_comparison_7}, where the non-strict version of the mixed QoS scenario (dark blue solid line) is almost always below the strict version of the mixed \mbox{QoS scenario} (dark blue dashed line).
\section{Conclusions and Outlook}
The goal of this work was to unlock the potential of QoS-aware pricing for an MNO that operates under the LSA regime. The business model for the MNO was to lease spectrum to PMSE users, differentiating their prices based on whether they belong to the high or the low QoS class. We analysed three QoS scenarios: i)~all users have the same low QoS requirements, ii)~all users have the same high QoS requirements, and iii)~a mixed QoS scenario.
From the perspective of the PMSE users, we made two contributions. First, we modelled the behaviour of the users regarding how they choose between the two QoS classes, quantifying the importance that each user gives to the QoS class versus the price that it has to pay. Second, we modelled the distribution of the budget of the users for the two QoS classes. The added value of these models is that we were able to perform a fine-grained analysis, predicting the distribution of the users between the two QoS classes for each possible combination of considered prices.
From the perspective of the MNO, the challenge was to choose the prices $P_L$ and $P_H$ so as to compute the maximum revenue that can be achieved for each QoS scenario. Our analysis revealed a consistent result that holds independent of i) the distribution of the budgets, ii) the way that the users choose between the QoS classes, and iii) the values of the technical parameters. The MNO can always tune the prices so that the maximum revenue for the high QoS scenario is the highest, followed by the mixed QoS scenario and finally by the low QoS scenario. This result highlights the potential of QoS-aware pricing for the MNO. For the high and mixed QoS scenarios where QoS price differentiation can be applied, the MNO can consistently generate higher revenue than for the low QoS scenario. This is also interesting from a regulatory point of view, since the MNO has motivation to support few users charging them at a higher price instead of supporting more users at a lower price. Therefore, we identified a constant tussle in the LSA market, where the goal of the MNO (\emph{i.e.}, revenue maximisation) is not aligned with the goal of the market regulator (\emph{i.e.}, social welfare maximisation).
Through the analysis of the revenues for the different budget scenarios, we identified the impact of the budget parameters on the revenue of the QoS scenarios. The revenue for the high and mixed QoS scenarios admits local maxima when both the mean and the standard deviation of the budget distribution $B_H$ are high (budget scenarios 6, 12, etc.). On the other hand, the revenue for the low QoS scenario admits local maxima when the mean of the budget distribution $B_L$ and both parameters of the budget distribution $B_H$ are small (budget scenarios 1, 13, 25). These trends hold for any values of the technical parameters. We argue that they are useful in particular for an MNO who evaluates the business opportunities in different markets before entering into them since they provide insights for which markets have the potential to be more profitable.
Finally, we conclude with two key messages extracted from our study for the mixed QoS scenario. First, there is higher room for price differentiation for the mixed QoS scenario, since fewer users with $Q_H$ can be admitted compared to the high QoS scenario. Second, for the non-strict version of the choice of the QoS class, the MNO usually prefers to sacrifice some of the users with $Q_L$ in order to support more users with $Q_H$ and charge them more. Both conclusions reinforce the message learnt, \emph{i.e.}, that the application of QoS-aware pricing unlocks significant revenue opportunities.
As future work, it is interesting to extend this study by introducing an additional (intermediate) QoS class and evaluate the robustness of the results. This also requires a modification for the way that the users choose among the three QoS classes. Another interesting direction is to consider an oligopoly market with two or three MNOs, analysing the churn of the users and the evolution of the revenue as the MNOs update their pricing policies.
\section*{Acknowledgment}
The authors would like to thank Shaham Shabani who conducted the simulations in ns-3 for estimating the maximum number of PMSE users for all QoS scenarios.
\bibliographystyle{IEEEtran}
|
1,116,691,500,756 | arxiv | \section{Introduction}
Let $G$ be a graph with vertex set $V = V(G)$. Throughout this paper, we only consider graphs without multiple edges and loops. Two vertices are \emph{neighbors} if they are adjacent. A \emph{dominating set} of a graph $G$ is a set $S$ of vertices of $G$ such that every vertex in $G$ is dominated by a vertex in $S$, where a vertex \emph{dominates} itself and its neighbors. Equivalently, $S \subseteq V$ is a dominating set of $G$ if every vertex of $V \setminus S$ is adjacent to a vertex of $S$. The minimum cardinality of a dominating set of $G$ is the \emph{domination number} of $G$, denoted by $\gamma(G)$. If $X,Y \subseteq V(G)$, then the set $X$ \emph{dominates} the set $Y$ if every vertex in $Y$ is dominated by at least one vertex in $X$. We refer the reader to the books~\cite{HaHeHe-20,HaHeHe-21} to study an overview of dominating sets in graphs.
In~2020, Haynes, Hedetniemi, Hedetniemi, McRae, and Mohan~\cite{coal0} presented a graph theoretic model of a coalition, and introduced the concept of a \emph{coalition} in graphs. They defined a pair of sets $V_1, V_2 \subseteq V$ to be a \emph{coalition} in $G$ if none of them is a dominating set of $G$ but $V_1 \cup V_2$ is. Such a pair $V_1$ and $V_2$ is said to \emph{form a coalition}, and are called \emph{coalition partners}. A vertex partition $\Psi = \{V_1,\ldots, V_k\}$ of $V$ is a \emph{coalition partition} of $G$, abbreviated a $c$-\emph{partition} in~\cite{coal0}, if every set~$V_i\in \Psi$ is either a dominating set of $G$ with cardinality $|V_i|=1$, or is not a dominating set but for some $V_j\in \Psi$, $V_i$ and $V_j$ form a coalition. The maximum cardinality of a coalition partition of $G$ is called the \emph{coalition number} of $G$, denoted by $\mathcal{C}(G)$. A coalition partition of $G$ of cardinality $\mathcal{C}(G)$ is called a \emph{$\mathcal{C}$-partition of $G$}. A motivation of this graph theory model of a coalition is given by Haynes et al. in their series of papers on coalitions in~\cite{coal0,coal1,coal2,coal3}.
Given a coalition partition $\Psi = \{V_1, \ldots, V_k\}$ of $G$, a coalition graph ${\rm CG}(G, \Psi)$ is associated on $\Psi$ such that there is a one-to-one correspondence between its vertices and the members of $\Psi$, where two vertices of ${\rm CG}(G, \Psi)$ are adjacent if and only if the corresponding sets form a coalition in $G$.
For notation and graph theory terminology not defined herein, we in general follow~\cite{HeYe-book}. Specifically, let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and of order~$n(G) = |V(G)|$ and size $m(G) = |E(G)|$. For a set of vertices $S\subseteq V(G)$, the subgraph induced by $S$ is denoted by $G[S]$. Two vertices in $G$ are \emph{neighbors} if they are adjacent. The \emph{open neighborhood} $N_G(v)$ of a vertex $v$ in $G$ is the set of neighbors of $v$, while the \emph{closed neighborhood} of $v$ is the set $N_G[v] = \{v\} \cup N(v)$. We denote the \emph{degree} of $v$ in $G$ by $\deg_G(v) = |N_G(v)|$. The minimum and maximum degree in $G$ is denoted by $\delta(G)$ and $\Delta(G)$, respectively. An \emph{isolated vertex} is a vertex of degree~$0$, and an \emph{isolate}-\emph{free graph} is a graph that contains no isolated vertex. A vertex of degree~$1$ is called a \emph{leaf}, and its unique neighbor a \emph{support vertex}. A graph is \emph{isolate}-\emph{free} if it contains no isolated vertex. A vertex of degree~$n(G)-1$ in $G$ is a \emph{universal vertex}, also called a \emph{full vertex} in the literature, of $G$. For a set $S \subseteq V(G)$, its \emph{open neighborhood} is the set $N_G(S) = \cup_{v \in S} N_G(v)$, and its \emph{closed neighborhood} is the set $N_G[S] = N_G(S) \cup S$. If the graph $G$ is clear from the context, we omit writing it in the above expressions. For example, we simply write $V$, $E$, $n$, $m$, $N(v)$ and $N(S)$ rather than $V(G)$, $E(G)$, $n(G)$, $m(G)$, $N_G(v)$ and $N_G(S)$, respectively.
A \emph{vertex cover} of a graph $G$ is a set $S$ of vertices such that every edge in $E(G)$ is incident with at least one vertex in~$S$. The \emph{vertex covering number} $\beta(G)$, also denoted $\tau(G)$ in the literature, is the minimum cardinality of a vertex cover of $G$. For a positive integer $k$, we let $[k] = \{1, \ldots, k\}$.
\subsection{Motivation and known results}
In this paper, we continue the study of coalitions in graphs. Our immediate aim is to answer, in part or fully, the following intriguing problems posed by the Haynes et al. in their recent series of paper given in~\cite{coal0,coal1,coal2}. Haynes et al.~\cite{coal0} posed the following open problem.
\begin{problem}{\rm (\cite{coal0})}
\label{prob1}
{\rm Characterize the graphs $G$ satisfying $\mathcal{C}(G)=n(G)$.}
\end{problem}
For $r, s \ge 1$, a \emph{double star} $S(r,s)$ is a tree with exactly two (adjacent) vertices that are not leaves, with one of the vertices having $r$ leaf neighbors and the other $s$ leaf neighbors. The double star $S(2,2)$, for example, is illustrated in Figure~\ref{f:fig1}(a). The bull graph $B$, illustrated in Figure~\ref{f:fig1}(b), is a graph obtained from a triangle by adding two disjoint pendant edges. Let $F_1$ be obtained from a bull graph by deleting one of the vertices of degree~$1$, and let $F_2$ be obtained from a $4$-cycle by adding a pendant edges. The graphs $F_1$ and $F_2$ are illustrated in Figures~\ref{f:fig1}(c) and~\ref{f:fig1}(d), respectively.
\begin{figure}[htb]
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\path (13.5,2) coordinate (x3);
\path (15.5,1) coordinate (x4);
\path (15.5,0) coordinate (x5);
\draw (x1)--(x2)--(x4)--(x5)--(x1);
\draw (x2)--(x3);
\draw (x1) [fill=white] circle (\vr);
\draw (x2) [fill=white] circle (\vr);
\draw (x3) [fill=white] circle (\vr);
\draw (x4) [fill=white] circle (\vr);
\draw (x5) [fill=white] circle (\vr);
\draw (14.5,-0.75) node {{\small (d) $F_2$}};
\end{tikzpicture}
\end{center}
\begin{center}
\vskip -0.5 cm
\caption{Four graphs of small orders}
\label{f:fig1}
\end{center}
\end{figure}
Let $\cal F$ be a family of graphs consisting of 18 graphs: $K_1$, $K_2$, $\overline{K_2}$, $K_1\cup K_2$, $P_3$, $K_3$, $K_{1,3}$, $2K_2$, $P_4$, $C_4$, $F_1$, $K_4-e$, $P_2\cup P_3$, $F_2$, $B_1$, $P_5$, $S(1,2)$, and $S(2,2)$. Haynes et al.~\cite{coal1} proved the following result.
\begin{theorem}{\rm (\cite{coal1})}
\label{t:known1}
If $\Psi$ is a coalition partition of a path $P_k$, then ${\rm CG}(P_k,\Psi) \in {\cal F}$.
\end{theorem}
A path is called a \emph{universal coalition path} if all 18 graphs of $\cal F$ can be defined by the coalition partitions of the path. Haynes et al.~\cite{coal1} posed the following two open problems.
\begin{problem}{\rm (\cite{coal1})}
\label{prob2}
{\rm Given a positive integer $k$, how many coalition graphs can be defined by the coalition partitions of a path $P_k$?}
\end{problem}
\begin{problem}{\rm (\cite{coal1})}
\label{prob3}
{\rm Does there exist a positive integer $k$ such that all 18 graphs of $\cal F$ can be defined by the coalition partitions of $P_k$? If so, what is the smallest universal coalition path?}
\end{problem}
In this paper, we characterize all graphs $G$ with $\delta(G)=1$ and $\mathcal{C}(G)=n$. Moreover, we characterize all trees $T$ with $\mathcal{C}(T)=n$ and all trees $T$ with $\mathcal{C}(T)=n-1$. This solves part of Problem~\ref{prob1}. On the other hand, we solve Problem~\ref{prob2} and Problem~\ref{prob3}. In particular, we theoretically and empirically determine the number of coalition graphs that can be defined by the coalition partitions of path $P_k$. Consequently, we show that there is no universal coalition path.
The coalition number of a path and a cycle is determined in~\cite{coal0}.
\begin{theorem}{\rm (\cite{coal0})}
\label{thm:path}
The following hold for a path $P_n$ and a cycle $C_n$. \\ [-24pt]
\begin{enumerate}
\item[{\rm (a)}] $C(P_n) = n$ if $n \le 4$, $C(P_n) = 4$ if $n = 5$, $C(P_n) = 5$ if $6 \le n \le 9$, and $C(P_n) = 6$ if $n \ge 10$.
\item[{\rm (b)}] $C(C_n) = n$ if $3 \le n \le 6$, $C(C_n) = 5$ if $n = 7$, and $C(C_n) = 6$ if $n \ge 8$.
\end{enumerate}
\end{theorem}
As consequence of Theorem~\ref{thm:path}, we have the following result.
\begin{corollary}{\rm (\cite{coal0})}
\label{cor:path}
If $G$ is a path or a cycle, then $C(G) \le 6$.
\end{corollary}
We shall need the following properties of a $c$-partition in a graph.
\begin{proposition}{\rm (\cite{coal0})}
\label{prop:Delta}
If $\Psi$ is a $c$-partition of a graph $G$, then $\Delta({\rm CG}(G,\Psi)) \le \Delta(G) + 1$.
\end{proposition}
By Proposition~\ref{prop:Delta}, if $\Psi$ is a $c$-partition of a graph $G$ and $S\in \Psi$, then $S$ is in at most $\Delta(G)+1$ coalitions in $\Psi$. The following result is stated without proof in~\cite{coal2}. For completeness we present a proof of this elementary property of a $c$-partition in a graph.
\begin{proposition}{\rm (\cite{coal2})}
\label{p:path}
If $\Psi$ is a $c$-partition of a graph $G$ and $v$ is an arbitrary vertex of $G$, then $\beta({\rm CG}(G,\Psi)) \le \deg_G(v) + 1$.
\end{proposition}
\begin{proof} Let $G$ with a graph with $c$-partition $\Psi = \{V_1, V_2, \ldots, V_k\}$, and let $v \in V(G)$. Let $\Psi'$ be a subset of $\Psi$, such that $V_i \in \Psi'$ if and only if $V_i$ contains a vertex of $N[v]$. Hence, $|\Psi'| \le |N[v]| = \deg_G(v) + 1$. Now consider an arbitrary edge $V_iV_j$ in ${\rm CG}(G,\Psi)$. Since the set $V_i \cup V_j$ is a dominating set of $G$, at least one of $V_i$ and $V_j$ contains a vertex from $N[v]$ in $G$. Thus, at least one of $V_i$ and $V_j$ is in $\Psi'$. Hence, $\Psi'$ is a vertex cover of ${\rm CG}(G,\Psi)$, and so $\beta({\rm CG}(G,\Psi)) \le |\Psi'| \le \deg_G(v) + 1$.~$\Box$
\end{proof}
\medskip
The following upper bounds on the coalition number of a graph are established in~\cite{coal2}.
\begin{theorem}{\rm (\cite{coal2})}
\label{thm:bound1}
If $G$ is a graph with $\delta = \delta(G)$ and $\Delta = \Delta(G)$, then the following hold. \\ [-24pt]
\begin{enumerate}
\item[{\rm (a)}] $C(G) \le \frac{1}{4}( \Delta + 3)^2$. \vspace{0.1cm}
\item[{\rm (b)}] If $\delta < \frac{1}{2}\Delta$, then $C(G) \le (\delta + 1)(\Delta - \delta + 2)$.
\end{enumerate}
\end{theorem}
As a consequence of Corollary~\ref{cor:path} and Theorem~\ref{thm:bound1}(b), we have the following result.
\begin{corollary}{\rm (\cite{coal0,coal2})}
\label{cor:bound1}
If $G$ is a graph with $\delta(G) = 1$, then $C(G) \le 2(\Delta(G) + 1)$.
\end{corollary}
\section{Trees with $\mathcal{C}(T)=n$}
In this section, we characterize trees $T$ of order~$n$ satisfying $C(T) = n$. First we characterize graphs $G$ of order $n$ and $\delta(G)=0$ such that $\mathcal{C}(G)=n$.
\begin{theorem}
\label{thmf0}
If $G$ is a graph of order~$n$ with $\delta(G) = 0$, then $\mathcal{C}(G) = n$ if and only if $G \cong K_1 \cup K_{n-1}$.
\end{theorem}
\begin{proof}
Clearly, $\mathcal{C}(K_1\cup K_{n-1}) = n$. Conversely, suppose that $G$ is a graph of order $n$ with $\delta(G) = 0$ and $\mathcal{C}(G) = n$. Thus, $\{\{v\} \colon v\in V(G)\}$ is a $\mathcal{C}$-partition of $G$. Let $x$ be an isolated vertex of $G$. If $v$ and $v'$ are two distinct vertices in $V(G) \backslash \{x\}$, then since the set $\{v,v'\}$ does not dominate the graph $G$, the pair $\{v\}$ and $\{v'\}$ do not form a coalition. Therefore, $\{v\}$ and $\{x\}$ form a coalition for each $v \in V (G)\backslash\{x\}$, implying that $V(G) \backslash\{x\}$ is a clique in $G$ of cardinality $n - 1$. Thus, $G \cong K_1 \cup K_{n-1}$.~$\Box$
\end{proof}
We next define a family $\mathcal{F}_1$ of graphs.
\begin{definition}{\rm (The family $\mathcal{F}_1$)}
\label{defn1}
{\rm Let $G$ be an isolate-free graph constructed as follows. Let $V(G) = \{x,y,w\} \cup P \cup Q$, where $P \cap Q \cap \{x,y,w\} = \emptyset$ and $|P \cup Q| \ge 1$. Further, if $Q \ne \emptyset$, then $|Q| \ge 2$. Let the edge set $E(G)$ be defined as follows. Let $N_G(x) = \{y\}$ and let $N_G(w) = P \cup Q$. Join each vertex $p \in P$ to every vertex in $(P \cup Q) \setminus \{p\}$. If $Q \ne \emptyset$, then add all edges from the vertex $y$ to every vertex in $Q$, and so $Q \cup \{x\} \subseteq N_G(y)$. Further if $Q \ne \emptyset$, then add edges between vertices in $Q$, including the possibility of adding no edge (in which case $Q$ is an independent set), in such a way that $G[Q]$ does not contain a full vertex. Thus, every vertex $q \in Q$ is not adjacent in $G$ to at least one vertex in $Q \setminus \{q\}$. Finally, add any number of edges between the vertex~$y$ and vertices in $P$, including the possibility of adding no edge between~$y$ and vertices in $P$. }
\end{definition}
We note that if $G \in \mathcal{F}_1$ is a disconnected graph of order~$n$, then $G \cong K_2 \cup K_{n-2}$. We are now in a position to characterize graphs $G$ of order~$n$ with $\delta(G)=1$ and with no full vertex that satisfy $\mathcal{C}(G)=n$.
\begin{theorem}
\label{thmd1}
If $G$ is a graph of order~$n$ with $\delta(G)=1$ and with no full vertex, then $\mathcal{C}(G)=n$ if and only if $G \in {\cal F}_1$.
\end{theorem}
\begin{proof}
Suppose firstly that $G \in {\cal F}_1$. Adopting our notation in Definition~\ref{defn1}, we have that if $p \in P \cup \{w\}$, then $\{x,p\}$ is a dominating set of $G$, and if $q \in Q \cup \{w\}$, then $\{y,q\}$ is a dominating set of $G$, implying that every subset $\{v\}$, where $v \in N[w]$, forms a coalition with $\{x\}$ or $\{y\}$. Therefore, $\mathcal{C}(G)=n$.
Conversely, let $G$ be a graph of order~$n$ with $\delta(G)=1$ and with no full vertex, and suppose that $\mathcal{C}(G)=n$. Let $x$ be a leaf of $G$ and let $y$ be the unique neighbor of $x$ in $G$. Let $\Psi$ be a $\mathcal{C}$-partition of $G$, and so $|\Psi| = n$ and $\Psi$ is a $c$-partition of $G$. Thus, each member of $\Psi$ is a singleton set (of cardinality~$1$), that is, if $v \in V(G)$, then $\{v\} \in \Psi$. We show that $G\in {\cal F}_1$. We know that $\{x\} \in \Psi$ and $\{y\}\in \Psi$. Suppose that $\{x\}$ and $\{y\}$ form a coalition in $G$. Since $x$ is a leaf and $N(x)=\{y\}$, all members of $V \setminus \{x,y\}$ must be adjacent to $y$. Thus, the degree of~$y$ is $n-1$, and so $y$ is a full vertex of $G$, a contradiction. Hence, $\{x\}$ and $\{y\}$ do not form a coalition. Let $w$ be a vertex not dominated by~$y$, and so $w \ne y$ and $w$ is not adjacent to~$y$. Let $N(w)=\{w_1,\dots,w_k\}$.
We show that $V = \{x,y\} \cup N[w]$. Suppose, to the contrary, that there exists a vertex $v \in V \backslash (\{x,y\} \cup N[w])$. If $z$ is an arbitrary vertex distinct from $x$ and $y$, then the pair $\{v\}$ and $\{z\}$ do not form a coalition since the vertex $x$ is not dominated by the set $\{v,z\}$. Moreover, $\{v\}$ does not form a coalition with $\{x\}$ or $\{y\}$ since the vertex $w$ is not dominated by the set $\{v,x,y\}$. Hence, $\{v\}$ has no coalition partner, a contradiction. Therefore, $V = \{x,y\} \cup N[w]$. Recall that $w \ne y$ and $w$ is not adjacent to~$y$. Since~$G$ has no full vertices, every set of $\Psi$ must be a coalition partner of some other set of $\Psi$. Let
\[
{\cal W} = \{S\in \Psi \colon N[w] \cap S\ne \emptyset\}.
\]
Since~$\mathcal{C}(G)=n$, we note that ${\cal W} = \left\{\{w\},\{w_1\},\ldots,\{w_k\}\right\}$. Necessarily, each member of $\cal W$ forms a coalition with~$\{x\}$ or~$\{y\}$, and there is no coalition between the members of $\cal W$. In particular, $\{w\}$ is a coalition partner of each of~$\{x\}$ and~$\{y\}$. Let ${\cal W}_P \subseteq {\cal W} \setminus \left\{\{w\}\right\}$ be the collection of all sets of $\Psi$ that form a coalition with $\{x\}$, and let $P = \{ p \in N(w) \colon \{p\} \in {\cal W}_P \}$. Moreover, let ${\cal W}_Q = \left({\cal W} \setminus \left\{\{w\}\right\}\right) \setminus {\cal W}_P$ and let $Q = \{ q \in N(w) \colon \{q\} \in {\cal W}_Q \}$. We note that $\{P,Q\}$ is a weak partition of $N(w)$, that is, $N(w) = P \cup Q$, where $P \cap Q = \emptyset$ and where in a weak partition we allow some of the sets in the partition to be empty (in our case, possibly $P = \emptyset$ or $Q = \emptyset$). No set in ${\cal W}_Q$ forms a coalition with $\{x\}$, and therefore every member of ${\cal W}_Q$ forms a coalition with $\{y\}$.
Let $p \in P$. Thus, $\{p\} \in {\cal W}_P$ forms a coalition with $\{x\}$, and so the set $\{x,p\}$ is a dominating set of $G$, implying that the vertex $p$ dominates the set $N[w]$, and so $p$ is adjacent to every vertex in $N(w) \backslash \{p\}$. Let $q \in Q$. Thus, $\{q\} \in {\cal W}_Q$ and the set $\{x,q\}$ is not a dominating set of $G$, implying that the vertex~$q$ is not adjacent to at least one vertex in $N(w) \setminus \{q\}$. As observed earlier, every vertex in $P$ is adjacent to every vertex in $N(w) = P \cup Q$, implying that all non-neighbors of $q$ in $N(w)$ belong to the set $Q$. Further, since $\{q,y\}$ is a dominating set of $G$, the vertex~$y$ is adjacent to every vertex in $Q$ that is not adjacent to~$q$. In particular, if $q'$ is a vertex in $Q \setminus \{q\}$ that is not adjacent to $q$, then since $\{y,q'\}$ is a dominating set of $G$, the vertex $y$ is adjacent to the vertex~$q$. Therefore, the vertex $y$ dominates the set $Q$. According to the definition of family~${\cal F}_1$, we infer that $G\in {\cal F}_1$.~$\Box$
\end{proof}
\medskip
Next we characterize graphs $G$ of order~$n$ with $\delta(G)=1$ and exactly one full vertex that satisfy $\mathcal{C}(G)=n$.
\begin{theorem}
\label{col1}
If $G$ is a graph of order~$n \ge 3$ with $\delta(G)=1$ and with exactly one full vertex, then $\mathcal{C}(G)=n$ if and only if $G$ is obtained from the graph $K_1 \cup K_{n-1}$ by adding an edge joining the isolated vertex to an arbitrary vertex of the complete graph $K_{n-1}$.
\end{theorem}
\begin{proof}
For $n \ge 3$, let $G$ be obtained from $K_1 \cup K_{n-1}$ by adding an edge $xy$ where $x$ is the vertex in the copy of $K_1$ and $y$ is a vertex in the complete graph $K_{n-1}$. In the resulting graph $G$, the vertex $x$ has degree~$1$ and the vertex $y$ has degree~$n-1$. Thus, $y$ is a full vertex. If $w \in V(G) \setminus \{x,y\}$, then $\{w,x\}$ is a dominating set of $G$, and so $\{w\}$ forms a coalition with the set $\{x\}$, implying that $\mathcal{C}(G)=n$. This proves the sufficiency.
To prove the necessity, let $G$ be a graph of order~$n \ge 3$ with $\delta(G)=1$ and with exactly one full vertex and suppose that $\mathcal{C}(G)=n$. Let $\Psi$ be a $\mathcal{C}$-partition of $G$. Let $u$ and $v$ be vertices of $G$ where $\deg_G(u)=1$ and $\deg_G(v)=n-1$, and let $w$ be an arbitrary vertex in $V(G) \setminus \{u,v\}$. Since $v$ is a full vertex, the set $\{v\}$ does not form a coalition with any vertex of $G$. Hence since the set $\{w\}$ is not a dominating set of $G$, the only coalition partner of the set $\{w\}$ in $\Psi$ is the set $\{u\}$. This implies that the vertex $w$ is adjacent to every vertex in $G$ except for the vertex~$u$. This is true for all vertices $w \in V(G) \setminus \{u,v\}$, implying that the graph $G' = G - v \cong K_1 \cup K_{n-2}$. Rebuilding the graph $G$ from $G'$ by adding back the vertex~$y$ and all edges joining $y$ to the $n-1$ vertices in $V(G')$, the desired result follows.~$\Box$
\end{proof}
We are now in a position to characterize all trees $T$ of order~$n$ with $\mathcal{C}(T)=n$.
\begin{theorem}
\label{thmtree}
If $T$ is a tree of order~$n$, then $\mathcal{C}(T) = n$ if and only if $T$ is a path of order at most~$4$.
\end{theorem}
\begin{proof}
By Theorem~\ref{thm:path}(a), if $T = P_n$ where $n \in [4]$, then $C(P_n) = n$. Conversely, let $T$ be a tree of order~$n$ satisfying $\mathcal{C}(T) = n$. The result is immediate if $n = 1$ or $n = 2$, and so we may assume that $n \ge 3$. If there is a full vertex in $T$, then $T \cong K_{1,n-1}$. In this case, by Theorem~\ref{col1} we must have that $n = 3$, yielding $T = P_3$, as desired. Hence we may assume that $T$ does not have a full vertex, implying that $n \ge 4$. In this case by Theorem~\ref{thmd1}, the tree $T$ belongs to the family~${\cal F}_1$. Adopting our earlier notation in Definition~\ref{defn1}, if $Q \ne \emptyset$, then $|Q| \ge 2$, and the graph $T[\{y,q_1,q_2,w\}]$ contains a $4$-cycle where $\{q_1,q_2\} \subseteq Q$, contradicting the fact that $T$ is a tree. Hence, $Q = \emptyset$. If $|P| \ge 2$, then the graph $T[P \cup \{w\}]$ contains a copy of $K_3$, once again contradicting the fact that $T$ is a tree. Hence, $|P| = 1$, implying that $T$ is the path $P_4$. Therefore, if $\mathcal{C}(T) = n$, then $T = P_n$ where $n \in [4]$.~$\Box$
\end{proof}
\section{Trees with $\mathcal{C}(T)=n-1$}
In this section, we characterize all trees $T$ with $\mathcal{C}(T) = n-1$. Recall that for $r, s \ge 1$, a double star $S(r,s)$ is a tree with exactly two (adjacent) vertices that are not leaves, with one of the vertices having $r$ leaf neighbors and the other $s$ leaf neighbors. We shall prove the following result.
\begin{theorem}
\label{thm:tree2}
If $T$ is a tree of order~$n$, then $\mathcal{C}(T) = n-1$ if and only if $T \in \{K_{1,3},P_5,P_6,S(2,1)\}$.
\end{theorem}
\begin{proof}
Let $T$ be a tree of order~$n$. If $T \in \{K_{1,3},P_5,P_6,S(2,1)\}$, then it is straightforward to check that $\mathcal{C}(T) = n-1$. To prove the necessity, suppose that $\mathcal{C}(T) = n-1$. By Theorem~\ref{thmtree}, $\mathcal{C}(T) \le n-1$ if and only if $T = K_{1,3}$ or the order of $T$ is at least~$5$. If $T$ contains a full vertex, then $T$ is a star $K_{1,n-1}$ and $\mathcal{C}(T) = 3$. By supposition, $\mathcal{C}(T) = n-1$, implying that in this case, $T = K_{1,3}$ as desired. Hence, we may assume that $T$ does not contain a full vertex, implying that $n \ge 5$.
Let $\Psi$ be a $\mathcal{C}$-partition of $G$, and so $|\Psi| = n-1$ and $\Psi$ is a $c$-partition of $G$. Thus, each member of $\Psi$ is a singleton set of cardinality~$1$, except for one member in $\Psi$ of cardinality~$2$. Let $\Psi = \{V_1, \ldots, V_{n-1},U\}$ where $|V_i| = 1$ for $i \in [n-1]$ and $|U| = 2$. Let $x$ be a leaf of $T$ and let $y$ be the unique neighbor of $x$. We proceed further with the following series of structural properties of the tree $T$.
\begin{claim}
\label{c:claim-1}
$U \ne \{x,y\}$.
\end{claim}
\noindent\textbf{Proof. } Suppose, to the contrary, that $U = \{x,y\}$. Let $v \in V(T) \setminus U$, and so $\{v\} \in \Psi$. If $\{v'\}$ is a coalition partner of $\{v\}$ for some $v' \in V(T) \setminus U$, then the vertex $x$ is not dominated by the set $\{v,v'\}$, a contradiction. Hence, the coalition partner of $\{v\}$ is the set $U$. In particular, $\{v,y\}$ is a dominating set for all $v \in V(T) \setminus U$. Let $A$ be the set of vertices in $V(T) \setminus U$ that are not adjacent to the vertex~$y$, and let $B$ be the set of vertices in $V(T) \setminus U$ that are dominated by~$y$. If $A = \emptyset$, then $y$ would be a full vertex, a contradiction. Hence, $A \ne \emptyset$. As observed earlier, if $a \in A$, then $\{a\}$ and $\{x,y\}$ form a coalition, implying that the vertex~$a$ is adjacent to every other vertex of $A$. Since this is true for all vertices $a \in A$, we infer that $A$ is a clique. If $B = \emptyset$, then $T$ would be disconnected, a contradiction. Hence, $B \ne \emptyset$. If $b \in B$, then $\{b\}$ and $\{x,y\}$ form a coalition, implying that the vertex~$b$ is adjacent to every vertex of $A$. Since this is true for all vertices $b \in B$, every vertex in $A$ is adjacent to every vertex in $B$. Hence, $\{a,x\}$ is a dominating set for every vertex $a \in A$, and $\{b,y\}$ is a dominating set for every vertex $b \in B$, implying that $\mathcal{C}(T) = n$, a contradiction.~{\tiny ($\Box$)}
\medskip
By Claim~\ref{c:claim-1}, $U \ne \{x,y\}$, and so $|U \cap \{x,y\}| \le 1$. Since $T$ has no full vertex, no vertex dominates $T$. Let $w$ be a vertex that is not dominated by $y$. Let $N(w) =\{w_1,\ldots,w_k\}$, and so $\deg_G(w) = k$ and $N(w) \cap \{x,y\} = \emptyset$. If two neighbors of $w$ are adjacent, then these two vertices together with $w$ induce a triangle in $T$, a contradiction. Hence since $T$ is a tree, the set $N(w)$ is an independent set. If the vertex $y$ is adjacent to two vertices in $N(w)$, then these two vertices together with $y$ and $w$ induce a $4$-cycle in $T$, a contradiction. Hence the vertex $y$ is adjacent to at most one vertex in $N(w)$.
\begin{claim}
\label{c:claim-2}
If $U \subseteq N(w)$, then $T = P_5$.
\end{claim}
\noindent\textbf{Proof. } Suppose that $U \subseteq N(w)$, and so $U \cap \{x,y\} = \emptyset$. Renaming vertices if necessary, we may assume that $U = \{w_1,w_2\}$. Suppose that $k \ge 3$. If $U$ forms a coalition with the set $\{x\}$, then $U$ dominates the set $N(w)$, contradicting our earlier observation that the set $N(w)$ is an independent set. Hence, $U$ does not form a coalition with $\{x\}$, implying that $U$ forms a coalition with the set $\{y\}$. Thus, $\{y,w_1,w_2\}$ is a dominating set in $G$. Since $N(w)$ is an independent set, the vertex $y$ therefore dominates the set $N(w) \setminus \{w_1,w_2\}$. Thus, $y$ is adjacent to at least~$k-2$ vertices in $N(w)$. As observed earlier, $y$ is adjacent to at most one vertex in $N(w)$. Therefore, $k = 3$ and $y$ is adjacent to exactly one vertex in~$N(w)$, namely to the vertex~$w_3$. Since $\{x,w_3\}$ does not dominate the set $U$, the pair $\{x\}$ and $\{w_3\}$ do not form a coalition. Since $\{y,w_3\}$ does not dominate the set $U$, the pair $\{y\}$ and $\{w_3\}$ do not form a coalition. However, then the set $\{w_3\} \in \Psi$ does not form a coalition with any other set in $\Psi$, a contradiction. Hence, $k = 2$, that is, $U = N(w) = \{w_1,w_2\}$.
Suppose that $n \ge 6$. Let $z \in V(T) \setminus \{x,y,w,w_1,w_2\}$. By our earlier observations, $\{z\}$ forms a coalition with $\{x\}$ or with $\{y\}$. Thus $\{x,z\}$ or $\{y,z\}$ is a dominating set of $G$. However neither $y$ nor $z$ belong to $N(w)$, and so $\{x,y,z\}$ is not a dominating set of $G$, a contradiction. Hence, $n = 5$, that is, $V(T) = \{x,y,w,w_1,w_2\}$. Since $T$ is connected, the vertex $y$ is adjacent to at least one of $w_1$ and $w_2$. However as observed earlier, the vertex $y$ is adjacent to at most one of $w_1$ and $w_2$. Consequently, the vertex $y$ is adjacent to exactly one of $w_1$ and $w_2$, implying that $T$ is a path $P_5$.~{\tiny ($\Box$)}
\begin{claim}
\label{c:claim-3}
If $U \subseteq N[w]$, then $T = P_5$.
\end{claim}
\noindent\textbf{Proof. } Suppose that $U \subseteq N[w]$, and so $U \cap \{x,y\} = \emptyset$. By Claim~\ref{c:claim-2}, if $U \subseteq N(w)$, the $T = P_5$ as desired. Hence renaming vertices if necessary, we may assume that $U = \{w,w_1\}$. Recall that $n \ge 5$. Let $z \in V(T) \setminus \{x,y,w,w_1\}$. Thus, $\{z\}$ forms a coalition with $\{x\}$ or with $\{y\}$. Thus $\{x,z\}$ or $\{y,z\}$ is a dominating set of $G$. Since $y$ is not adjacent with $w$, the vertex $y \notin N[w]$, and therefore the vertex $z$ is necessary adjacent to $w$. Since $z$ is an arbitrary vertex in $V(T) \setminus \{x,y,w,w_1\}$, we have that $V(T) = \{x,y\} \cup N[w]$. We now consider the vertex $w_2$. If $\{w_2\}$ forms a coalition with $\{x\}$, then $\{x,w_2\}$ is a dominating set, implying that $w_1$ and $w_2$ are adjacent, contradicting our earlier observation that $N(w)$ is an independent set. Hence, $\{w_2\}$ forms a coalition with $\{y\}$, then $\{y,w_2\}$ is a dominating set, implying that $y$ and $w_1$ are adjacent. If $k \ge 3$, then the vertex $w_3$ is not dominated by $\{y,w_2\}$, a contradiction. Hence, $k = 2$, and so $T$ is a path $P_5$.~{\tiny ($\Box$)}
\medskip
By Claim~\ref{c:claim-3}, we may assume that $|U \cap N[w]| \le 1$, for otherwise $T = P_5$ and the desired result follow. By our earlier assumptions, $|U \cap \{x,y\}| \le 1$.
\begin{claim}
\label{c:claim-4}
If $y \in U$, then $T \in \{P_5,S(2,1)\}$.
\end{claim}
\noindent\textbf{Proof. } Suppose that $y \in U$, and so $x \notin U$. Let $U = \{y,z\}$. By definition of a coalition partition, the set $U$ is not a dominating set of $G$ since the only sets $S \in \Psi$ that dominate $G$ have cardinality $|S| = 1$. Recall that $w$ was chosen earlier as an arbitrary vertex that is not adjacent to~$y$. Renaming vertices if necessary, we can choose $w$ to be a vertex not dominated by the set $U$. Thus, $\{x,y,z\} \cap N[w] = \emptyset$. Recall that $N(w) = \{w_1,\ldots,w_k\}$. By our earlier observations, the set $N(w)$ is an independent set. Further since there is no $4$-cycle in $T$, every vertex in $V(T) \setminus N[w]$ is adjacent to at most one vertex in $N(w)$.
Since the set $\{x,y,z\}$ is not a dominating set of $T$, the sets $\{x\} \in \Psi$ and $U \in \Psi$ do not form a coalition. Hence, the set $\{x\} \in \Psi$ forms a coalition with a set $\{x'\} \in \Psi$, where $x' \notin \{x,y,z\}$. Since $\{x,x'\}$ is a dominating set of $T$, necessarily $x' \in N[w]$. Since the vertex $z$ is not dominated by the set $\{x,w\}$, we note that $x' \ne w$, implying that $x' \in N(w)$. Renaming vertices if necessary, we may assume that $x' = w_1$. Thus, $\{x,w_1\}$ is a dominating set of $G$. Since $N(w)$ is an independent set, this in turn implies that $k = |N(w)| = 1$ and that $w_1$ is adjacent to the vertex~$z$. Suppose that $n \ge 6$, and so there exists a vertex $q \in V(T) \setminus \{x,y,w,w_1,z\}$. Since $\{x,q\}$ is not a dominating set of $T$, the set $\{q\} \in \Psi$ does not form a coalition with $\{x\}$, implying that the set $\{q\} \in \Psi$ forms a coalition with the set $U \in \Psi$. However, the vertex $w$ is not dominated by the set $\{q,y,z\}$, a contradiction. Hence, $n = 5$, and so $V(T) = \{x,y,w,w_1,z\}$. As observed earlier, $\{xy,zw_1,ww_1\} \subset E(T)$. Since $T$ is connected, the only remaining edge of $T$ is either $yw_1$ or $yz$. If $yw_1 \in E(T)$, then $T = S(2,1)$, while if $yz \in E(T)$, then $T = P_5$.~{\tiny ($\Box$)}
\medskip
By Claim~\ref{c:claim-4}, we may assume that $y \notin U$, for otherwise the desired result follows.
\begin{claim}
\label{c:claim-5}
If $x \notin U$, then $T \in \{P_5,S(2,1)\}$.
\end{claim}
\noindent\textbf{Proof. } Suppose that $x \notin U$. By assumption, $y \notin U$. Hence, $\{x\} \in \Psi$ and $\{y\} \in \Psi$. Since $T$ has no full vertices, the pair $\{x\}$ and $\{y\}$ do not form a coalition. By our earlier assumptions, $|U \cap N[w]| \le 1$. Let $U = \{z_1,z_2\}$ and let $z_1 \in U \setminus N[w]$. By supposition, $U \cap \{x,y\} = \emptyset$, and so $z_1 \notin \{x,y\}$. Since $U$ forms a coalition with $\{x\}$ or $\{y\}$, the set $\{x,y,z_1,z_2\}$ is a dominating set of $T$, implying that $z_2 \in N[w]$ in order to dominate the vertex~$w$.
Suppose that there is a vertex $z \in V(T) \setminus \left(\{x,y,z_1\} \cup N[w]\right)$. Since $z \notin U$, $\{z\} \in \Psi$. By our earlier observations, the pair $\{z\}$ and $\{x\}$ form a coalition or the pair $\{z\}$ and $\{y\}$ form a coalition. However, the set $\{x,y,z\}$ does not dominate the vertex~$w$, and so $\{z\}$ does not form a coalition with $\{x\}$ or with $\{y\}$, a contradiction. Hence, $V(T) = \{x,y,z_1\} \cup N[w]$.
Suppose that $k \ge 2$. Renaming vertices in $N(w)$ if necessary, we may assume that $w_1 \ne z_2$, that is, $w_1 \notin U$. Thus, $\{w_1\} \in \Psi$. Since $\{x,w_1\}$ is not a dominating set of $T$ noting that $N(w)$ is an independent set of cardinality~$k \ge 2$, the pair $\{w_1\}$ and $\{x\}$ do not form a coalition, implying that the pair $\{w_1\}$ and $\{y\}$ form a coalition. Thus, the vertex~$y$ dominates the set $N(w) \setminus \{w_1\}$. Since $y$ is adjacent to at most one vertex in $N(w)$, we infer that $k = 2$, $yw_2 \in E(T)$, and $yw_1 \notin E(T)$. Analogously, if $w_2 \ne z_2$, then $yw_1 \in E(T)$ and $yw_2 \notin E(T)$, a contradiction. Hence, $w_2 = z_2$. Thus, $\{w\} \in \Psi$. Since $\{x,w\}$ does not dominate the vertex~$z_1$, the pair $\{w\}$ and $\{x\}$ do not form a coalition, implying that the pair $\{w\}$ and $\{y\}$ form a coalition. This in turn implies that $yz_1 \in E(T)$. The tree is now determined and $E(T) = \{xy,yz_1,yw_2,w_2w,ww_1\}$. Further, $\Psi = \{ \{x\}, \{y\}, \{w\}, \{w_1\}, \{z_1,w_2\} \}$. However, $\{x\}$ does not form a coalition with any other set in $\Psi$, a contradiction.
Hence, $k = 1$, and so $V(T) = \{x,y,w,w_1,z_1\}$ and $\{xy,ww_1\} \subset E(T)$. Since $T$ is a tree, exactly two of the edges in the set $\{yw_1,yz_1,w_1z_1\}$ are present in $T$. If $\{yw_1,yz_1\} \subset E(T)$ or $\{yw_1,w_1z_1\} \subset E(T)$, then $T = S(2,1)$. If $\{yz_1,w_1z_1\} \subset E(T)$, then $T = P_5$.~{\tiny ($\Box$)}
\medskip
By Claim~\ref{c:claim-5}, we may assume that $x \in U$, for otherwise the desired result follows. Since $x$ is an arbitrary vertex of degree~$1$ in $T$, we may therefore assume that the set $U$ contains all vertices of degree~$1$ in $T$. Since $|U| = 2$, this implies that $T$ is a path $P_n$ where $n \ge 5$. By Theorem~\ref{thm:path}(a), $C(P_n) = n-1$ if $n \in \{5,6\}$ and $C(P_n) \le n-2$ if $n \ge 7$. Hence since $T = P_n$ and $\mathcal{C}(T) = n-1$, we infer that $T \in \{P_5,P_6\}$. This completes the proof of Theorem~\ref{thm:tree2}.~$\Box$
\end{proof}
\medskip
As a consequence of Theorem~\ref{thm:tree2}, we have the following result.
\begin{corollary}
If $T$ is a tree of order $n$ with $n\ge 7$, then $\mathcal{C}(T) \le n-2$.
\end{corollary}
\section{Coalition graphs}
In this section, we solve Problem~\ref{prob2} and Problem~\ref{prob3}. In particular, we determine the number of the coalition graphs that can be defined by all coalition partitions of the path $P_k$ for a given positive integer $k$ (see Theorem~\ref{thma}) and then as a consequence of Theorem~\ref{thma}, we see that there is no universal coalition path.
A coalition graph corresponding to a coalition partition of a path is called a {\rm CP}-\emph{graph} in \cite{coal1}. We say a path $P_k$ defines a ${\rm CP}$-graph $G$ if there is a coalition partition $\Psi$ of $P_k$ such that ${\rm CG}(P_k,\Psi)$ is isomorphic to $G$. In \cite{coal1}, Haynes et al. proved the following theorem.
\begin{theorem}{\rm (\cite{coal1})}
\label{lemCP}
A graph $G$ is a ${\rm CP}$-graph if and only if $G\in {\cal F}$.
\end{theorem}
According to Theorem~\ref{lemCP}, all 18 graphs of ${\cal F}$ are ${\rm CP}$-graph. For $k \ge 2$, let the ordered list $s_1, s_2,\ldots, s_k$ be the vertices of the path $P_k$ whose edges are $\{s_i,s_{i+1}\}$ where $i \in [k-1]$. Now, we prove the following proposition.
\begin{proposition}
\label{lem33}
The following properties hold. \\ [-24pt]
\begin{enumerate}
\item[{\rm (a)}] The path $P_1$ defines the ${\rm CP}$-graph $K_1$.
\item[{\rm (b)}] The path $P_2$ defines the ${\rm CP}$-graph $\overline{K_2}$.
\item[{\rm (c)}] The path $P_3$ defines the ${\rm CP}$-graph $K_1\cup K_2$.
\item[{\rm (d)}] For any integer $k\ge 4$, the path $P_k$ does not define the ${\rm CP}$-graphs $\overline{K_2}$ and $K_1\cup K_2$.
\end{enumerate}
\end{proposition}
\begin{proof} It is immediate that the only path that defines the ${\rm CP}$-graph $K_1$ is $P_1$. This establishes~(a). For the path $P_2$, the set $\left\{\{s_1\},\{s_2\}\right\}$ is a coalition partition and the corresponding coalition graph is $\overline{K_2}$. This establishes~(b). For the path $P_3$, the set $\left\{\{s_1\}, \{s_2\}, \{s_3\}\right\}$ is a coalition partition such that none of the sets $\{s_1\}$ and $\{s_3\}$ is a dominating set but they form a coalition. Moreover, the set $\{s_2\}$ is a singleton dominating set. Hence, the coalition graph corresponding to $\left\{\{s_1\}, \{s_2\}, \{s_3\}\right\}$ is $K_1\cup K_2$. This establishes~(c). Since the graphs $\overline{K_2}$ and $K_1\cup K_2$ have at least one isolated vertex, their corresponding coalition partition must contain at least one singleton dominating set. We know that for any $k \ge 4$, the domination number of $P_k$ is at least two. Therefore, every coalition partition of $P_k$ with $k\ge 4$ does not contain any singleton dominating set. This proves part~(d).~$\Box$
\end{proof}
\medskip
By Proposition~\ref{prop:Delta} and Theorem~\ref{thm:path}(a), we can readily deduce the following result.
\begin{corollary}
\label{cor1}
For any positive integer $k\le 3$, the path $P_k$ does not define the ${\rm CP}$-graphs $P_3, K_3, K_{1,3}$, $2K_2$, $P_4, C_4, F_1, K_4-e, P_2\cup P_3, F_2, B_1, P_5, S(2,1), S(2,2)$.
\end{corollary}
Next we prove several propositions about paths and their corresponding {\rm CP}-graphs.
\begin{proposition}\label{prop1}
For $k \ge 4$, the path $P_k$ defines the ${\rm CP}$-graph $K_2$.
\end{proposition}
\begin{proof}
For $k \ge 4$, let $\Psi = \{A, B\}$, where $A=\{s_1,s_2,\ldots, s_{\lfloor\frac{k}{2}\rfloor}\}$ and $B=\{s_{\lfloor\frac{k}{2}\rfloor+1},\ldots, s_k\}$. The sets $A$ and $B$ form a coalition, and so ${\rm CG}(P_k,\Psi)\cong K_2$.~$\Box$
\end{proof}
\begin{proposition}\label{prop2}
For $k \ge 4$, the path $P_k$ defines the ${\rm CP}$-graphs $P_3$ and $C_4$.
\end{proposition}
\begin{proof}
To prove the proposition, we provide two coalition partitions $\Psi_1$ and $\Psi_2$ for $P_k$ with $k\ge 4$ whose corresponding coalition graphs are $P_3$ and $C_4$, respectively. Let $\Psi_1=\{A, B, C\}$, where
\[
A=\{s_1,s_2\}, \hspace*{0.5cm} B=\bigcup_{i=2}^{\lceil\frac{k}{2}\rceil}\{s_{2i-1}\} \hspace*{0.5cm} \mbox{and} \hspace*{0.5cm} C=\bigcup_{i=2}^{\lfloor\frac{k}{2}\rfloor}\{s_{2i}\}.
\]
The sets $A$ and $B$ form a coalition, as do the sets $A$ and $C$. Moreover, the sets $B$ and $C$ do not form a coalition. Hence, ${\rm CG}(P_k,\Psi_1)\cong P_3$. Let $\Psi_2=\{A, B, C, D\}$, where
\[
A=\{s_1\}, \hspace*{0.5cm} B=\{s_2\}, \hspace*{0.5cm} C= \bigcup_{i=0}^{\lfloor\frac{k}{2}\rfloor-2}\{s_{k-(2i+1)}\} \hspace*{0.5cm} \mbox{and} \hspace*{0.5cm} D=\bigcup_{i=0}^{\lceil\frac{k}{2}\rceil-2}\{s_{k-2i}\}.
\]
The set $A$ forms a coalition with both sets $C$ and $D$, while the set $B$ forms a coalition with both sets $C$ and $D$. Moreover, the sets $A$ and $B$ do not form a coalition, and the sets $C$ and $D$ do not form a coalition. Hence, ${\rm CG}(P_k,\Psi_2)\cong C_4$.~$\Box$
\end{proof}
\begin{proposition}\label{prop3}
For $k \ge 5$, the path $P_k$ defines the ${\rm CP}$-graph $F_1$.
\end{proposition}
\begin{proof}
Let $\Psi=\{A, B, C, D\}$ be the coalition partition of $P_k$ defined as follows. For $k=6$, let $A=\{s_1,s_5\}$, $B=\{s_3,s_6\}$, $C=\{s_4\}$ and $D=\{s_2\}$. For $k \ge 8$ even, let
\[
A=\left(\bigcup_{i=2}^{\frac{k}{2}-2}\{s_{k-2i}\}\right)\cup\{s_1, s_k\}, \hspace*{0.5cm} B=\left(\bigcup_{i=2}^{\frac{k}{2}-2}\{s_{k-(2i-1)}\}\right)\cup \{s_2\}, \hspace*{0.5cm}
C=\{s_{k-2}\} \hspace*{0.5cm} \mbox{and} \hspace*{0.5cm}
D=\{s_3, s_{k-1}\}.
\]
For $k \ge 5$ odd, let
\[
A=\left(\bigcup_{i=2}^{\frac{k-3}{2}}\{s_{k-2i}\}\right)\cup\{s_1, s_k\}, \hspace*{0.5cm} B=\left(\bigcup_{i=2}^{\frac{k-3}{2}}\{s_{k-(2i-1)}\}\right)\cup \{s_2\}, \hspace*{0.5cm}
C=\{s_{k-2}\} \hspace*{0.5cm} \mbox{and} \hspace*{0.5cm}
D=\{s_{k-1}\}.
\]
The set $A$ forms a coalition with each of the sets $B$, $C$ and $D$, and the sets $B$ and $D$ form a coalition. Moreover, the sets $B$ and $C$ do not form a coalition, and the sets $C$ and $D$ do not form a coalition. Hence, ${\rm CG}(P_k,\Psi)\cong F_1$.~$\Box$
\end{proof}
\begin{proposition}
\label{prop4}
For $k \ge 6$, the path $P_k$ defines the ${\rm CP}$-graphs $K_{1,3}$, $K_3$, $K_4-e$, and $P_2 \cup P_3$.
\end{proposition}
\begin{proof}
To prove the proposition, we provide four coalition partitions $\Psi_1, \Psi_2, \Psi_3$ and $\Psi_4$ for $P_k$ with $k\ge 6$ whose corresponding coalition graphs are $K_{1,3}, K_3, K_4-e$ and $P_2\cup P_3$, respectively.
Let $\Psi_1=\{A, B, C, D\}$, where $A=\{s_1,s_2, \ldots, s_{k-4}\}\cup\{s_k\}$, $B=\{s_{k-3}\}$, $C=\{s_{k-2}\}$ and $D=\{s_{k-1}\}$. The set $A$ forms a coalition with each of the sets $B$, $C$ and $D$. Moreover, there are no coalitions among the sets $B$, $C$ and $D$. Hence, ${\rm CG}(P_k,\Psi) \cong K_{1,3}$. Let $\Psi_2=\{A, B, C\}$, where
\[
A=\left(\bigcup_{i=2}^{{\lfloor\frac{k}{3}\rfloor-1}}\{s_{k-3i}\}\right)\cup \{s_1, s_{k-2}\}, \hspace*{0.5cm}
B=\left(\bigcup_{i=1}^{\lfloor\frac{k}{3}\rfloor-1}\{s_{k-(3i+1)}\}\right)\cup \{s_2, s_{k}\} \hspace*{0.5cm}
\]
and
\[
C=\left(\bigcup_{i=1}^{{\lfloor\frac{k+1}{3}\rfloor-2}}\{s_{k-(3i+2)}\}\right)\cup \{s_{k-1}, s_{k-3}\}.
\]
The set $A$ forms a coalition with each of the sets $B$ and $C$. Moreover, the sets $B$ and $C$ form a coalition. Hence, ${\rm CG}(P_k,\Psi_2)\cong K_3$. Let $\Psi_3=\{A, B, C, D\}$, where the coalition partition is defined as follows. For $k \ge 6$ even, let $C=\{s_{k-3}\}$ and $D=\{ s_{k-2}\}$, and let
\[
A=\left(\bigcup_{i=0}^{\frac{k}{2}-3}\{s_{2i+1}\}\right)\cup\{s_{k-1}\}
\hspace*{0.25cm} \mbox{and} \hspace*{0.25cm} B=\left(\bigcup_{i=1}^{\frac{k}{2}-2}\{s_{2i}\}\right)\cup \{s_k\}.
\]
For $k \ge 7$ odd, let $C=\{s_{k-3}\}$ and $D=\{ s_{k-2}\}$, and let
\[
A=\left(\bigcup_{i=0}^{\frac{k-5}{2}}\{s_{2i+1}\}\right)\cup\{s_{k}\}
\hspace*{0.25cm} \mbox{and} \hspace*{0.25cm}
B=\left(\bigcup_{i=1}^{\frac{k-5}{2}}\{s_{2i}\}\right)\cup \{s_{k-1}\}.
\]
The set $A$ forms a coalition with each of the sets $B$, $C$ and $D$. Moreover, the set $B$ forms a coalition with each of the sets $C$ and $D$. However, the sets $C$ and $D$ do not form a coalition. Hence, ${\rm CG}(P_k,\Psi_3)\cong K_4-e$. Let $\Psi_{4}=\{A, B, C, D, F\}$, where the coalition partition is defined as follows. For $k \ge 6$ even, let $B=\{s_2, s_{k-4}\}$, $C=\{ s_{k-3}\}$, and $D = \{ s_{k-2}\}$ and let
\[
A=\left(\bigcup_{i=0}^{\frac{k-6}{2}}\{s_{2i+1}\}\right)\cup\{s_{k}\}
\hspace*{0.25cm} \mbox{and} \hspace*{0.25cm}
F=\left(\bigcup_{i=2}^{\frac{k-6}{2}}\{s_{2i}\}\right)\cup \{s_{k-1}\}.
\]
For $k \ge 7$ odd, let $B=\{s_2, s_{k-5}\}$, $C=\{ s_{k-4}\}$, and $D=\{ s_{k-3}\}$, and let
\[
A=\left(\bigcup_{i=0}^{\frac{k-7}{2}}\{s_{2i+1}\}\right)\cup\{s_{k-1}\}
\hspace*{0.25cm} \mbox{and} \hspace*{0.25cm}
F=\left(\bigcup_{i=2}^{\frac{k-7}{2}}\{s_{2i}\}\right)\cup \{s_{k-2}, s_{k}\}.
\]
The sets $A$ and $C$, the sets $A$ and $D$, and the sets $B$ and $F$ are the only pairs among the sets $A$, $B$, $C$, $D$, and $F$ in $\Psi_{4}$ that form a coalition. Hence, ${\rm CG}(P_k,\Psi_4)\cong P_2\cup P_3$.~$\Box$
\end{proof}
\begin{proposition}
\label{prop5}
For $k \ge 7$, the path $P_k$ defines the ${\rm CP}$-graphs $P_4$ and $P_5$.
\end{proposition}
\begin{proof}
To prove the proposition, we provide two coalition partitions $\Psi_1$ and $\Psi_2$ for $P_k$ with $k \ge 7$ whose corresponding coalition graphs are $P_4$ and $P_5$, respectively. Let $\Psi_1=\{A, B, C, D\}$, where $C=\{s_{k-4}\}$ and $D=\{s_{k-1}\}$, and where
\[
A=\left(\bigcup_{i=3}^{\lceil\frac{k}{2}\rceil-2}\{s_{k-2i}\}\right)\cup\{s_1, s_{k-2}, s_k\}
\hspace*{0.25cm} \mbox{and} \hspace*{0.25cm}
B=\left(\bigcup_{i=3}^{\lfloor\frac{k}{2}\rfloor-1}\{s_{k-(2i-1)}\}\right)\cup \{s_2, s_{k-3}\}.
\]
The sets $C$ and $A$, the sets $A$ and $B$, and the sets $B$ and $D$ are the only pairs among the sets $A$, $B$, $C$, and $D$ in $\Psi_{1}$ that form a coalition. Hence, ${\rm CG}(P_k,\Psi_1) \cong P_4$. Let $\Psi_{2}=\{A, B, C, D, F\}$, where $C=\{ s_3\}$, $D=\{ s_4\}$, and $F = \{s_{5}\}$, and where
\[
A=\left(\bigcup_{i=3}^{\lfloor \frac{k}{2} \rfloor}\{s_{2i}\}\right)\cup\{s_{1}\}
\hspace*{0.25cm} \mbox{and} \hspace*{0.25cm}
B=\left(\bigcup_{i=3}^{\lceil \frac{k}{2} \rceil-1}\{s_{2i+1}\}\right)\cup\{s_{2}\}.
\]
The sets $C$ and $A$, the sets $A$ and $D$, the sets $D$ and $B$, and the set $B$ and $F$ are the only pairs among the sets $A$, $B$, $C$, $D$, and $F$ in $\Psi_{2}$ that form a coalition. Hence, ${\rm CG}(P_k,\Psi_2) \cong P_5$.~$\Box$
\end{proof}
\begin{proposition}
\label{prop6}
For $k \ge 8$, the path $P_k$ defines the ${\rm CP}$-graph $2K_2$.
\end{proposition}
\begin{proof}
Let $\Psi=\{A, B, C, D\}$ be the coalition partition of $P_k$ defined as follows.
For $k = 8$, let $A = \{s_1,s_7\}$, $B = \{s_2,s_8\}$, $C = \{s_3,s_4\}$ and $D = \{s_5,s_6\}$.
For $k=9$, let $A = \{s_1,s_8\}$, $B = \{s_2,s_4,s_9\}$, $C = \{s_3,s_5\}$ and $D = \{s_6,s_7\}$.
For $k=10$, let $A = \{s_1,s_9\}$, $B = \{s_2,s_4,s_{10}\}$, $C = \{s_3,s_5,s_6\}$ and $D = \{s_7,s_8\}$.
For $k=11$, let $A = \{s_1,s_{10}\}$, $B = \{s_2,s_4,s_{11}\}$, $C = \{s_3,s_5,s_7\}$ and $D = \{s_6, s_8,s_9\}$.
For $k \ge 12$, let $A=\{s_1, s_{k-3}, s_{k-1}\}$ and $B=\{s_2, s_4, s_k\}$, and let
\[
C=\left(\bigcup_{i=2}^{\lceil\frac{k}{2}\rceil-3}\{s_{2i+1}\}\right)\cup\{s_3\}
\hspace*{0.25cm} \mbox{and} \hspace*{0.25cm}
D=\left(\bigcup^{\lceil\frac{k}{2}\rceil-2}_{i=3}\{s_{2i}\}\right)\cup\{s_{k-2}\}.
\]
The sets $A$ and $C$, and the sets $B$ and $D$, are the only pairs among the sets $A$, $B$, $C$, and $D$ in $\Psi$ that form a coalition. Hence, ${\rm CG}(P_k,\Psi) \cong 2K_2$.~$\Box$
\end{proof}
\begin{proposition}
\label{prop7}
For $k \ge 9$, the path $P_k$ defines the ${\rm CP}$-graphs $F_2$, $B_1$, and $S(2,1)$.
\end{proposition}
\begin{proof}
To prove the proposition, we provide three coalition partitions $\Psi_1, \Psi_2$ and $\Psi_3$ for $P_k$ with $k\ge 9$ whose corresponding coalition graphs are $F_2$, $B_1$, and $S(2,1)$, respectively. Let $\Psi_{1}=\{A, B, C, D, F\}$, where $C=\{ s_3,s_6\}$, $D=\{ s_4, s_7\}$, and $F= \{s_{5}\}$, and where
\[
A=\left(\bigcup_{i=4}^{\lceil \frac{k}{2} \rceil-1}\{s_{2i+1}\}\right)\cup\{s_{1}\}
\hspace*{0.25cm} \mbox{and} \hspace*{0.25cm}
B=\left(\bigcup_{i=4}^{\lfloor \frac{k}{2} \rfloor}\{s_{2i}\}\right)\cup\{s_{2}\}.
\]
The set $A$ forms a coalition with each of the sets $C$ and $D$, and the set $B$ forms a coalition with each of the sets $C$, $D$ and $F$. However, these are the only pairs of sets in $\Psi_{1}$ that form a coalition, implying that ${\rm CG}(P_k,\Psi_1) \cong F_2$. Let $\Psi_{2}=\{A, B, C, D, F\}$, where $C=\{ s_3\}$, $D=\{ s_5\}$, and $F= \{s_{7}\}$, and where
\[
A=\left(\bigcup_{i=4}^{\lfloor \frac{k}{2} \rfloor}\{s_{2i}\}\right)\cup\{s_{1}, s_4\}
\hspace*{0.25cm} \mbox{and} \hspace*{0.25cm}
B=\left(\bigcup_{i=4}^{\lceil \frac{k}{2} \rceil-1}\{s_{2i+1}\}\right)\cup\{s_{2}, s_6\}.
\]
The set $A$ forms a coalition with each of the sets $B$, $D$ and $F$, and the set $B$ forms a coalition with each of the sets $A$, $C$ and $D$. However, these are the only pairs of sets in $\Psi_{2}$ that form a coalition, implying that ${\rm CG}(P_k,\Psi_2) \cong B_1$. Let $\Psi_{3}=\{A, B, C, D, F\}$, where $C=\{ s_3\}$, $D=\{ s_4\}$, and $F = \{s_{7}\}$, and where
\[
A=\left(\bigcup_{i=3}^{\lfloor \frac{k}{2} \rfloor}\{s_{2i}\}\right)\cup\{s_{1}\}
\hspace*{0.25cm} \mbox{and} \hspace*{0.25cm}
B=\left(\bigcup_{i=4}^{\lceil \frac{k}{2} \rceil-1}\{s_{2i+1}\}\right)\cup\{s_{2}, s_5\}.
\]
The set $A$ forms a coalition with each of the sets $B$, $C$ and $D$, and the set $B$ forms a coalition with each of the sets $A$ and $F$. However, these are the only pairs of sets in $\Psi_{2}$ that form a coalition, implying that ${\rm CG}(P_k,\Psi_2) \cong S(2,1)$.~$\Box$
\end{proof}
\begin{proposition}
\label{prop8}
For $k \ge 10$, the path $P_k$ defines the ${\rm CP}$-graph $S_{2,2}$.
\end{proposition}
\begin{proof}
By Theorem~\ref{thm:path}(a), we have $C(P_n) = 6$ for all $k \ge 10$. Therefore, there is a coalition partition of $P_k$ such that the corresponding ${\rm CP}$-graph has six vertices. On the other hand, the only ${\rm CP}$-graph with six vertices is $S_{2,2}$. Hence, the path $P_k$ with $k \ge 10$ defines the ${\rm CP}$-graph~$S_{2,2}$.~$\Box$
\end{proof}
\medskip
In Propositions~\ref{prop1}--\ref{prop8}, we presented results that determine which ${\rm CP}$-graphs can be defined by the path $P_k$. But there are still cases that have not yet been identified. For example, does $P_8$ define the ${\rm CP}$-graphs $F_2$ and $B$ or does $P_6$ define the ${\rm CP}$-graphs $2K_2$ and $P_4$? To this end, we empirically checked the remaining cases. Table~\ref{T11} summarizes the results of Corollary~\ref{cor1} and Propositions~\ref{lem33}-\ref{prop8}, and the empirical results. In the table, we used the letters $Y, N, y$ and $n$. Let $T(i,j)$ be the cell of the table in row $i$ and column $j$. Let ${\rm CP}_i$ be the ${\rm CP}$-graph in row $i$, and let $P_j$ be the path in column $j$. When $T(i,j)$ is equal to a uppercase letter, it means that using Corollary~\ref{cor1} and Propositions~\ref{lem33}-\ref{prop8}, we have obtained the value of $T(i,j)$, and when $T(i,j)$ is equal to a lowercase letter, it means that we empirically have obtained it. If $T(i,j)\in \{Y, y\}$, it means that the ${\rm CP}$-graph ${\rm CP}_i$ can be defined by the path $P_j$, and if $T(i,j)\in \{N, n\}$, it means that the ${\rm CP}$-graph ${\rm CP}_i$ cannot be defined by the path $P_j$.
\begin{table}[hbt]
\centering
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
& $P_1$ & $P_2$ & $P_3$ & $P_4$ & $P_5$ & $P_6$ & $P_7$ & $P_8$ & $P_9$ & $P_{k\ge 10}$ \\ \hline
$K_1$&$Y$ & $N$ & $N$ & $N$ & $N$ & $N$ & $N$ & $N$ & $N$ & $N$ \\ \hline
$K_2$ & $N$ & $n$ & $Y$ & $Y$ & $Y$ & $Y$ &$Y$ &$Y$ &$Y$ &$Y$ \\ \hline
$\overline{K_2}$ &$N$ & {$Y$} & $n$ & $N$ &$N$ & $N$ &$N$ & $N$ & $N$ & $N$ \\ \hline
$K_1\cup K_2$ & $N$ & $N$ & {$Y$} & $N$ &$N$ & $N$ &$N$ & $N$ & $N$ & $N$ \\ \hline
$P_3$ & $N$ & $N$ & {$N$} & $Y$ & $Y$ & $Y$ & $Y$ & $Y$ & $Y$ & $Y$ \\ \hline
$K_3$ & $N$ & $N$ & $N$& $n$ & $y$& $Y$ & $Y$ & $Y$ & $Y$ & $Y$ \\ \hline
$K_{1,3}$ & $N$ & $N$ & $N$ & $n$ & $n$ & $Y$ & $Y$ & $Y$ & $Y$ & $Y$ \\ \hline
$2K_2$ & $N$ & $N$ & $N$ & $n$ & $n$ & $y$ & $y$ & $Y$ & $Y$ & $Y$ \\ \hline
$P_4$ & $N$& $N$& $N$& $n$& $y$ & $y$ & $Y$ & $Y$ & $Y$ & $Y$ \\ \hline
$C_4$ & $N$& $N$ & $N$& $Y$ & $Y$ & $Y$ & $Y$ & $Y$ & $Y$ & $Y$ \\ \hline
$F_1$ & $N$& $N$& $N$& $n$ & $Y$ & $Y$ & $Y$ & $Y$ & $Y$ & $Y$ \\ \hline
$K_4-e$ & $N$& $N$& $N$ & $n$ & $n$ & $Y$ & $Y$ & $Y$ & $Y$ & $Y$ \\ \hline
$P_2\cup P_3$ & $N$ & $N$ & $N$ & $n$ & $n$ & $Y$ & $Y$ & $Y$ & $Y$ & $Y$ \\ \hline
$F_2$ & $N$& $N$ & $N$& $n$ & $n$ & $n$ & $n$ & $n$ & $Y$ & $Y$ \\ \hline
$B$ & $N$& $N$ & $N$& $n$ & $n$ & $n$ & $n$ & $n$& $Y$ & $Y$ \\ \hline
$P_5$ & $N$ & $N$ & $N$ & $n$ & $n$ & $n$ & $Y$ &$Y$ & $Y$ & $Y$ \\ \hline
$S(2,1)$ & $N$ & $N$ & $N$ & $n$ & $n$ & $n$ & $y$& $y$ & $Y$ & $Y$ \\ \hline
$S_{2,2}$ & $N$ & $N$ & $N$ & $n$ & $n$ & $n$ & $n$ & $n$ & $n$ & $Y$ \\ \hline
\end{tabular} \caption{The ${\rm CP}$-graphs that can be defined by any path $P_k$ with $k\ge 1$.}
\label{T11}
\end{table}
In the following, for the cases that $T(i,j)=y$, we present a coalition partition of the path $P_j$ corresponding to the ${\rm CP}$-graph ${\rm CP}_i$. \\ [-22pt]
\begin{enumerate}
\item[$\bullet$] The path $P_8$ and the ${\rm CP}$-graph $S(2,1)$: $\{\{1,4\}, \{2,6,8\},\{3\}, \{5\}, \{7\}\}$.
\item[$\bullet$] The path $P_7$ and the ${\rm CP}$-graph $S(2,1)$: $\{\{1,4\}, \{2,6\},\{3\}, \{5\}, \{7\}\}$.
\item[$\bullet$] The path $P_7$ and the ${\rm CP}$-graph $2K_2$: $\{\{1,7\}, \{2\},\{3,4\}, \{5,6\}\}$.
\item[$\bullet$] The path $P_6$ and the ${\rm CP}$-graph $P_4$: $\{\{1,4\}, \{2\},\{3,5\}, \{6\}\}$.
\item[$\bullet$] The path $P_6$ and the ${\rm CP}$-graph $2K_2$: $\{\{1,6\}, \{2\},\{3,4\}, \{5\}\}$.
\item[$\bullet$] The path $P_5$ and the ${\rm CP}$-graph $P_4$: $\{\{1\}, \{2\},\{3,4\}, \{5\}\}$.
\item[$\bullet$] The path $P_5$ and the ${\rm CP}$-graph $K_3$: $\{\{1,5\}, \{2\},\{3,4\}\}$.
\end{enumerate}
Let ${\rm NC}(P_i)$ be the number of ${\rm CP}$-graphs that can be defined by $P_i$. From Table \ref{T11}, we can readily obtain the following result.
\begin{theorem}
\label{thma}
It holds that ${\rm NC}(P_1)={\rm NC}(P_2)=1$, ${\rm NC}(P_3)=2$, ${\rm NC}(P_4)=3$, ${\rm NC}(P_5)=6$, ${\rm NC}(P_6)=10$, ${\rm NC}(P_7)={\rm NC}(P_8)=12$, ${\rm NC}(P_9)=14$, and ${\rm NC}(P_k)=15$ for any integer $k\ge 10$.
\end{theorem}
By Theorem \ref{thma}, there is no path $P_k$ that defines all 18 ${\rm CP}$-graphs, yielding the following result.
\begin{theorem}
There is no universal coalition path.
\end{theorem}
\section{Conclusion}
In this paper, we characterized all graphs $G$ of order~$n$ with $\delta(G)\le 1$ and $\mathcal{C}(G)=n$. Furthermore, we characterized all trees $T$ of order~$n$ with $\mathcal{C}(T)=n$ and all trees $T$ of order~$n$ with $\mathcal{C}(T)=n-1$. On the other hand, we theoretically and empirically determined the number of coalition graphs that can be defined by all coalition partitions of a given path $P_k$. Furthermore, we showed that there is no universal coalition path. It remains an open problem to characterize all trees $T$ of order~$n$ with $\mathcal{C}(T)=n-k$ for all $k$ where $2 \le k \le n-2$. It would also be interesting to determine whether there is a linear-time algorithm to compute the coalition number of a given tree.
\medskip
|
1,116,691,500,757 | arxiv | \section{Introduction}
Inverse problems are encountered in a wide range of scientific and engineering applications, especially in the context of medical imaging. The aim is to recover an unknown model parameter $\boldsymbol{x}^*\in\mathbb{X}$ containing critical information about the structural details of an underlying subject from data $\boldsymbol{y} = \Op{A}(\boldsymbol{x}^*)+ \boldsymbol{e} \in \mathbb{Y}$, representing noisy, indirect, and potentially incomplete set of measurements. The forward operator $\Op{A}:\mathbb{X}\rightarrow \mathbb{Y}$ and the distribution of the measurement noise $\boldsymbol{e}\in \mathbb{Y}$ are typically known and they jointly form a simulator for the data acquisition process. Generally, $\mathbb{X}$ is the Hilbert space of functions defined on some $\Omega\subseteq \mathbb{R}^d$, while $\mathbb{Y}$ is another Hilbert space of functions defined on a suitable data manifold $\mathbb{M}$. Inverse problems are typically ill-posed in the absence of any further information apart from the measurements alone, meaning that different model parameters can give rise to the same measurement. Variational reconstruction \cite[Part II]{scherzer2009variational} is a generic, yet adaptable framework for solving inverse problems, wherein ill-posedness is tackled by solving
\begin{equation}
\underset{\boldsymbol{x} \in \mathbb{X}}{\min}\text{\,\,}\left\|\boldsymbol{y}-\Op{A}(\boldsymbol{x})\right\|_2^2+\lambda\, \mathcal{R}(\boldsymbol{x}).
\label{var_recon}
\end{equation}
The goal here is to alleviate the aforementioned inherent indeterminacy by incorporating some prior knowledge about the model parameter using a hand-crafted regularizer $\mathcal{R}$ in addition to seeking data-consistency. The role of the regularizer is to penalize unlikely or undesirable solutions. The variational framework is said to be well-posed if it admits a unique solution which varies continuously in the measurement.
While variational methods enjoy rigorous theoretical guarantees for stability and convergence, and have remained the state-of-the-art for several decades, they are limited in their ability to adapt to a particular application at hand. With the emergence of deep learning, modern approaches for solving inverse problems have increasingly shifted towards data-driven reconstruction \cite{data_driven_inv_prob}, which generally offers significantly superior reconstruction quality as compared to the traditional variational methods. Data-adaptive reconstruction methods can broadly be classified into two categories: (i) end-to-end trained over-parametrized models that either attempt to map the measured data to the true model parameter (such as AUTOMAP proposed in \cite{automap}), or remove artifacts from the output of an analytical reconstruction method \cite{postprocessing_cnn}, and (ii) learning the image prior using a neural network based on training data of images and then using such a learned regularizer in a variational model for reconstruction \cite{ar_nips,nett_paper,kobler2020total,meinhardt2017learning}. The first approach relies on learning the reconstruction method from a large training dataset that consists of many ordered pairs of model-parameter and corresponding noisy data. Since obtaining a vast amount of paired examples is difficult in medical imaging applications, over-parametrized models trained end-to-end in a supervised manner might run into the danger of overfitting and generalize poorly on unseen data. The second category still requires one to solve a high-dimensional variational problem where the objective involves a trained neural network, a task that is typically computationally demanding.
\indent One promising way to circumvent the limited data problem is to build network architectures by incorporating the physics of the acquisition process \cite{lpd_tmi,jonas_learned_iterative}. The learned primal-dual (LPD) approach proposed in \cite{lpd_tmi} is data-efficient as compared to fully data-driven approaches and can generalize well when trained on a moderate amount of examples. However, an unrolled LPD network trained by minimizing the squared-$\ell_2$ error between the network output and the target essentially returns an approximation to the conditional-mean estimator, which is the statistical expectation of the target image conditioned on the measurement. Owing to this implicit averaging, the resulting estimate tends to suffer from blurring artifacts with the loss of important details in the reconstruction. The proposed adversarially trained LPD method, referred to as ALPD, circumvents this problem by seeking proximity in the space of distribution instead of aiming to minimize the squared-$\ell_2$ distortion in the image space.
\section{Main Contributions: training objective and protocol}
For supervised learning, one needs paired training examples of the form $\left\{\boldsymbol{x}_i,\boldsymbol{y}_i\right\}_{i=1}^{n}$ sampled i.i.d. from the joint probability distribution $\pi_X(\boldsymbol{x})\pi_{\text{data}}(\boldsymbol{y}|\boldsymbol{x})$ of the image and the measurement. In contrast, the proposed training protocol is unsupervised, i.e., it assumes availability of i.i.d. samples $\left\{\boldsymbol{x}_i\right\}_{i=1}^{n_1}$ and $\left\{\boldsymbol{y}_i\right\}_{i=1}^{n_2}$ from the marginal distributions $\pi_X$ and $\pi_Y$ of the ground-truth image and measurement data, respectively. The image and the data samples are unpaired, i.e., $\boldsymbol{y}_i$ does not necessarily correspond to the noisy measurement of $\boldsymbol{x}_i$. In the context of CT, $\boldsymbol{x}_i$'s could be the high-/normal-dose reconstructions obtained using the classical filtered back-projection algorithm, whereas $\boldsymbol{y}_i$'s correspond to low-dose projection.
We begin with a description of the proposed unsupervised approach and how it differs from and relates to supervised and classical variational methods. Subsequently, we motivate the training loss using the maximum-likelihood (ML) principle and explain the reconstruction network parametrization, which follows the same philosophy proposed in \cite{lpd_tmi}.
\subsection{Proposed training protocol for ALPD}
\label{sec:adv_learning_proposed}
Similar to supervised training, one key component of the proposed unsupervised approach is to first build a parametric reconstruction network $\mathcal{G}_{\theta}:\mathbb{Y}\rightarrow \mathbb{X}$ (see Sec. \ref{g_phi_pdhg_sec} for details) that takes the measurement as input and produces a reconstructed image as the output. However, unlike supervised training, it is not possible to train $\mathcal{G}_{\theta}$ by minimizing a chosen distortion measure between $\mathcal{G}_{\theta}(\boldsymbol{y}_i)$ and $\boldsymbol{x}_i$, since $\boldsymbol{x}_i$ is not the ground-truth image corresponding to $\boldsymbol{y}_i$. Our training framework essentially seeks to achieve the following three objectives:
\begin{enumerate}[leftmargin=*]
\item The reconstructions produced by $\mathcal{G}_{\theta}$ should be close to the ground-truth images in the training dataset in terms of distribution (measured with respect to the Wasserstein distance);
\item $\mathcal{G}_{\theta}$ should be encouraged to be the right-inverse of $\Op{A}$, so that the forward operator applied on the output of $\mathcal{G}_{\theta}$ is close to the measured data; and
\item $\mathcal{G}_{\theta}$ should approximately be a left-inverse of $\Op{A}$, i.e., $\mathcal{G}_{\theta}$ must recover the ground-truth from noise-free measurement.
\end{enumerate}
More concretely, we propose to learn $\mathcal{G}_{\theta}:\mathbb{Y}\rightarrow \mathbb{X}$ by minimizing the training loss
\begin{multline}\label{alpd_train_loss}
J(\theta) = \mathcal{W}\left((\mathcal{G}_{\theta})_{\#}\pi_{Y},\pi_X\right)+\lambda_{\mathbb{Y}}\,\,\mathbb{E}_{\pi_{Y}}\left[\left\|\Op{A}(\mathcal{G}_{\theta}(Y))-Y \right\|_2^2\right]
\\ + \lambda_{\mathbb{X}}\,\,\mathbb{E}_{\pi_{X}}\left[\left\|\mathcal{G}_{\theta}\left(\Op{A}(X)\right)-X \right\|_2^2\right].
\end{multline}
The penalty parameters $\lambda_{\mathbb{X}}$ and $\lambda_{\mathbb{Y}}$ control the relative weighting of the three objectives. Notably, in the absence of noise in the measurement, the first objective becomes superfluous, i.e., any $\mathcal{G}_{\theta}$ that satisfies the third objective automatically satisfies the first one too. For noisy measurements, the combination of the first and the third objectives helps compute a stable estimate which does not overfit to noise, whereas the second objective ensures that the reconstruction explains the data well. Similar to \cite{wgan_main}, we make use of the Kantorovich-Rubinstein (KR) duality for approximating the Wasserstein distance term in \eqref{alpd_train_loss}. This requires training a critic network $\mathcal{D}_{\alpha}:\mathbb{X} \rightarrow{\mathbb{R}}$ that scores an image on the real line based on how closely it resembles the ground-truth images in the dataset. More precisely, the KR duality helps estimate the Wasserstein distance by solving
\begin{equation}
\mathcal{W}\left((\mathcal{G}_{\theta})_{\#}\pi_{Y},\pi_X\right)
=\sup_{\alpha}
\mathbb{E}_{\pi_X}\left[\mathcal{D}_{\alpha}\left(X\right)\right]-\mathbb{E}_{(\mathcal{G}_{\theta})_{\#}\pi_{\mathbb{Y}}}\left[\mathcal{D}_{\alpha}\left(X\right)\right]
\text{ where $\mathcal{D}_{\alpha}\in \mathbb{L}_1$.}
\label{wasserstein_dist}
\end{equation}
Here, $\mathbb{L}_1$ denotes the space of 1-Lipschitz functions. In practice, both $\mathcal{G}_{\theta}$ and $\mathcal{D}_{\alpha}$ are updated in an alternating manner instead of fully solving \eqref{wasserstein_dist} for each $\mathcal{G}_{\theta}$ update. The 1-Lipschitz condition is enforced by penalizing the gradient of the critic with respect to the input \cite{wgan_gp}. Estimating the Wasserstein distance in \eqref{wasserstein_dist} and the training loss for $\mathcal{G}_{\theta}$ in \eqref{alpd_train_loss} requires samples from the marginals, thereby rendering the training framework unsupervised. The detailed steps involved in training the networks are listed in Algorithm \ref{algo_acr_train}.
\indent At this point, it is instructive to interpret the training objective \eqref{alpd_train_loss} through the lens of the variational framework, by recasting the variational problem as a minimization over the parameter $\theta$ of $\mathcal{G}_{\theta}$ instead of $\boldsymbol{x}$. Given the data distribution $\pi_Y$, it is natural to estimate $\theta$ that minimizes the expected variational loss:
\begin{equation}
\underset{\theta}{\min}\,\,J_1(\theta) := \mathbb{E}_{\pi_Y}\left[\left\|\Op{A}(\mathcal{G}_{\theta}(Y))-Y \right\|_2^2+\lambda\,\mathcal{R}\left(\mathcal{G}_{\theta}(Y)\right)\right].
\label{var_opt_interpret}
\end{equation}
Now, suppose the existence of an \textit{ideal} regularizer in \eqref{var_opt_interpret}, which returns a small score when the input is drawn from $\pi_X$ and a large score when the distribution of the input differs from $\pi_X$. For such a regularizer, the difference
\begin{equation}
\mathcal{L}(\mathcal{R})=\mathbb{E}_{\pi_X}\left[\mathcal{R}(X)\right] - \mathbb{E}_{\pi_Y}\left[\mathcal{R}(\mathcal{G}_{\theta}(Y))\right],
\label{var_opt_interpret1}
\end{equation}
should be small. As a matter of fact, a consequence of the KR duality is that
\[
\inf_{\mathcal{R}} \mathcal{L}(\mathcal{R})
= -\mathcal{W}\left((\mathcal{G}_{\theta})_{\#}\pi_{Y},\pi_X\right),
\]
provided that $\mathcal{R}$ is constrained to be 1-Lipschitz. Substituting this in \eqref{var_opt_interpret1} and ignoring terms independent of $\theta$ reduces \eqref{var_opt_interpret} to minimizing $\theta \mapsto \hat{J}_1(\theta)$ where
\begin{equation}
\hat{J}_1(\theta) := \mathbb{E}_{\pi_Y}\left[\left\|\Op{A}(\mathcal{G}_{\theta}(Y))-Y \right\|_2^2\right]+\lambda\,\mathcal{W}\left((\mathcal{G}_{\theta})_{\#}\pi_{Y},\pi_X\right).
\label{var_opt_interpret_red}
\end{equation}
If $\lambda:=1/\lambda_{\mathbb{Y}}$, then we obtain the inequality $J(\theta)\geq \lambda_{\mathbb{Y}}\, \hat{J}_1(\theta)$ for the training loss $J$. This indicates that our training loss majorizes (up to a scaling) the objective $\hat{J}_1$ in \eqref{var_opt_interpret_red}, which emerges naturally from the variational loss under the assumption of a 1-Lipschitz ideal regularizer.
The penalty terms in the $\mathbb{X}$- and $\mathbb{Y}$-domains in \eqref{alpd_train_loss} are unmistakably reminiscent of the cycle-consistency losses in cycle-GANs \cite{cycle_gan_main} that are widely used learning paradigms for unpaired image-to-image translation problems. Similar unsupervised approaches involving GANs were also proposed in \cite{conditional_im2im_cvpr} for conditional image-to-image synthesis tasks. The proposed approach can indeed be thought of as a simpler variant of cycle-GAN with the generator learned in only one direction ($\mathbb{Y}\rightarrow \mathbb{X}$) instead of two. Notably, it was recently shown in \cite{sim_cyclegan} that the cycle-GAN training loss can be derived as the optimal transport loss corresponding to the case where the transport cost is equal to the variational loss with $\mathcal{R}(\boldsymbol{x})=\left\|\boldsymbol{x}-\mathcal{G}_{\theta}(\boldsymbol{y})\right\|_2$.
\subsection{A maximum-likelihood (ML) perspective}
The ML principle seeks to solve
\begin{equation}
\underset{\theta}{\max}\,\, \left[\frac{1}{n_1}\sum_{i=1}^{n_1}\log \pi^{(\theta)}_X(\boldsymbol{x}_i)+\frac{1}{n_2}\sum_{i=1}^{n_2}\log \pi^{(\theta)}_Y(\boldsymbol{y}_i)\right],
\label{overall_ml}
\end{equation}
where $\pi^{(\theta)}_X$ and $\pi^{(\theta)}_Y$ are the distributions induced by appropriately postulated probabilistic models on $X$ and $Y$, respectively. In the following, we explain the statistical models and use them to derive a tractable lower-bound (generally referred to as the evidence lower-bound (ELBO)) on the ML objective in \eqref{overall_ml}. Our analysis reveals that the resulting ELBO is equivalent to the ALPD training loss in \eqref{alpd_train_loss} in spirit, except for the measure of distance for comparing the distributions of the reconstruction and the ground-truth. The KL-divergence-based distance measure is replaced with the Wasserstein-1 distance since it lends itself to continuous differentiability with respect to the parameters of the reconstruction network, thereby facilitating a stable gradient-based parameter update.
\subsubsection{Bound on the data likelihood}
The ELBO for $Y$ is derived by treating $Y$ as the observed variable and $X$ as the unobserved/latent variable. We then derive two expressions for the conditional distribution of $X$ given $Y = \boldsymbol{y}$, one from the measurement process and the other from the reconstruction process. The lower-bound is tight when these are close to each other.
\begin{description}[leftmargin=*]
\item[Measurement process:]
Model parameters are generated by $X \sim \pi_X$ whereas the measured data are generated by the $\mathbb{Y}$-valued random variable $\left(Y|X=\boldsymbol{x}\right)\sim \mathcal{N}\left(\Op{A}(\boldsymbol{x}),\sigma_e^2\,\boldsymbol{I}\right)$ for given $\boldsymbol{x} \in \mathbb{X}$.
Let $\pi^{(m)}_{X,Y}$ denote the induced joint distribution of $(X,Y)$ with $\pi^{(m)}_{X}(\boldsymbol{x}):=\pi_X(\boldsymbol{x})$ and $\pi^{(m)}_{Y}(\boldsymbol{y})$ denoting its marginals.
Also, let $\pi^{(m)}_{X|Y}(\boldsymbol{x}|\boldsymbol{y})$ and $\pi^{(m)}_{Y|X}(\boldsymbol{y}|\boldsymbol{x})$ denote the corresponding conditional distributions.
\item[Reconstruction process:]
Data are generated by $Y \sim \pi_Y$ and reconstructed model parameters are generated by the $\mathbb{X}$-valued random variable $\left(X|Y=\boldsymbol{y}\right)\sim \mathcal{N}\left(\mathcal{G}_{\theta}(\boldsymbol{y}),\sigma_1^2\,\boldsymbol{I}\right)$ for given $\boldsymbol{y} \in \mathbb{Y}$. Denote the associated joint distribution by $\pi^{(r)}_{X,Y}(\boldsymbol{x},\boldsymbol{y})$ and its marginals and corresponding conditionals are denoted similarly as for the measurement process.
\end{description}
The log-likelihood of $Y$ is given by $\mathcal{L}^{(y)}_{\text{ML}}(\theta)=\frac{1}{n_2}\sum_{i=1}^{n_2}\log \pi^{(m)}_Y(\boldsymbol{y}_i)$, which is the empirical average of the natural logarithm of the model-induced probability density computed over samples of the true distribution of $Y$.
Using the statistical model above, $\mathcal{L}^{(y)}_{\text{ML}}(\theta)$ can be expressed as
\begin{align*}
\log \pi^{(m)}_Y(\boldsymbol{y})
&= \log\left(\int_{\mathbb{X}}\pi^{(m)}_{X,Y}(\boldsymbol{x},\boldsymbol{y})\,\mathrm{d}\boldsymbol{x}\right)=
\log\left(\mathbb{E}_{\pi^{(r)}_{X|Y}}\left[\frac{\pi^{(m)}_{X,Y}(X,\boldsymbol{y})}{\pi^{(r)}_{X|Y}(X|\boldsymbol{y})}\right]\right).
\end{align*}
Since $\log$ is a concave function, applying Jensen's inequality leads to
\begin{align*}
\log \pi^{(m)}_Y(\boldsymbol{y})
&\geq \mathbb{E}_{\pi^{(r)}_{X|Y}}\left[\log\left(\frac{\pi_X(X)\pi^{(m)}_{Y|X}(\boldsymbol{y}|X)}{\pi^{(r)}_{X|Y}(X|\boldsymbol{y})}\right)\right].
\end{align*}
The above can further be simplified as
\begin{align}
\log \pi^{(m)}_Y(\boldsymbol{y})
&\geq \mathbb{E}_{\pi^{(r)}_{X|Y}}\left[\log\left(\pi^{(m)}_{Y|X}(\boldsymbol{y}|X)\right)\right]+\mathbb{E}_{\pi^{(r)}_{X|Y}}\left[\log\left(\frac{\pi_X(X)}{\pi^{(r)}_{X|Y}(X|\boldsymbol{y})}\right)\right]
\nonumber\\
&= \mathbb{E}_{\pi^{(r)}_{X|Y}}\left[\log\left(\pi^{(m)}_{Y|X}(\boldsymbol{y}|X)\right)\right]-\operatorname{KL}\left(\pi^{(r)}_{X|Y=\boldsymbol{y}},\pi_X\right).
\label{log_likeli_y_lb_simple}
\end{align}
Under the postulated statistical model, we have that
\begin{equation}
\pi^{(m)}_{Y|X}(\boldsymbol{y}|\boldsymbol{x}) := \mathcal{N}\left(\Op{A}(\boldsymbol{x}),\sigma_e^2\,\boldsymbol{I}\right)
\text{ and }
\pi^{(r)}_{X|Y}(\boldsymbol{x}|\boldsymbol{y}):=\mathcal{N}\left(\mathcal{G}_{\theta}(\boldsymbol{y}),\sigma_1^2\,\boldsymbol{I}\right).
\label{cons_of_model}
\end{equation}
If the forward operator $\Op{A}$ is linear (which reduces to a matrix in the finite-dimensional case), then by \eqref{cons_of_model} one can simplify the bound in \eqref{log_likeli_y_lb_simple}:
\begin{equation}
-\log \pi^{(m)}_Y(\boldsymbol{y})\leq
\operatorname{KL}\left(\pi^{(r)}_{X|Y=\boldsymbol{y}},\pi_X\right) + \frac{1}{2\,\sigma_e^2}\Bigl\|\boldsymbol{y}-\Op{A}\bigl(\mathcal{G}_{\theta}(\boldsymbol{y})\bigr)\Bigr\|_2^2+ c_1.
\label{log_likeli_y_lb_final1}
\end{equation}
Here, $c_1$ is a constant independent of $\theta$ (see Proposition \ref{prop_gauss_expect} for a proof).
\subsubsection{Bound on the image likelihood}
The ELBO corresponding to $X$ can be derived by treating $X$ as the observed variable and the clean (synthetic) data $U=\Op{A}(X)$ as the latent variable.
\begin{description}[leftmargin=*]
\item[Backward process:]
Here $U\sim \pi_{U}(\boldsymbol{u})$ and $\left(X|U=\boldsymbol{u}\right)\sim \mathcal{N}\left(\mathcal{G}_{\theta}(\boldsymbol{u}),\sigma_2^2\,\boldsymbol{I}\right)$ for given $\boldsymbol{u}$, with possibly $\sigma_2\ll\sigma_1$.
\item[Forward process:]
$X\sim \pi_{X}$ and $\left(U|X=\boldsymbol{x}\right)\sim \delta\left(\boldsymbol{u}-\Op{A}(\boldsymbol{x})\right)$ (Dirac measure concentrated at $\boldsymbol{u}=\Op{A}(\boldsymbol{x})$).
\end{description}
Proceeding similarly to the analysis used for deriving a bound on the data likelihood, we can show that (with the superscripts $(f)$ and $(b)$ indicating the forward and backward processes, respectively)
\begin{equation}
\log \pi^{(b)}_X(\boldsymbol{x})
\geq \mathbb{E}_{U\sim \pi^{(f)}_{U|X=\boldsymbol{x}}}\left[\log\left(\pi^{(b)}_{X|U}(\boldsymbol{x}|U)\right)\right]-\underbrace{\operatorname{KL}\left(\pi^{(f)}_{U|X=\boldsymbol{x}},\pi_U\right)}_{\text{does not depend $\theta$}}.
\label{log_likeli_x_lb_simple}
\end{equation}
Using the postulated distributions to simplify the first term in \eqref{log_likeli_x_lb_simple} leads to
\begin{equation}
-\log \pi^{(b)}_X(\boldsymbol{x})\leq \frac{1}{2\sigma_2^2}\Bigl\|\boldsymbol{x}-\mathcal{G}_{\theta}\bigl(\Op{A}(\boldsymbol{x})\bigr)\Bigr\|_2^2+\operatorname{KL}\left(\pi^{(f)}_{U|X=\boldsymbol{x}},\pi_U\right).
\label{log_likeli_x_lb_final1}
\end{equation}
\subsubsection{Evidence bound}
The idea is now to combine \eqref{log_likeli_y_lb_final1} and \eqref{log_likeli_x_lb_final1}.
Then, observe that minimizing the so-called (negative) evidence bound on the overall negative log-likelihood in \eqref{overall_ml} can be phrased as follows:
\begin{equation}
\min_{\theta}\operatorname{KL}\left(\pi^{(r)}_{X|Y=\boldsymbol{y}},\pi_X\right) + \frac{1}{2\,\sigma_e^2}\Bigl\|\boldsymbol{y}-\Op{A}\bigl(\mathcal{G}_{\theta}(\boldsymbol{y})\bigr)\Bigr\|_2^2+ \frac{1}{2\sigma_2^2}\Bigl\|\boldsymbol{x}-\mathcal{G}_{\theta}\bigl(\Op{A}(\boldsymbol{x})\bigr)\Bigr\|_2^2.
\label{overall_ml_final}
\end{equation}
This is identical to minimizing the ALPD training loss in \eqref{alpd_train_loss} but using the KL divergence instead of the Wasserstein-1 to quantify similarity in distribution.
\begin{prop}
\label{prop_gauss_expect}
Let $X\in\mathbb{R}^{d_x}$, $Y\in\mathbb{R}^{d_y}$, and let $\Op{A}\in \mathbb{R}^{d_y \times d_x}$ be a $d_y \times d_x$ matrix. Then, $\mathbb{E}_{X\sim \mathcal{N(\boldsymbol{\mu}, \boldsymbol{K})}}\left[\left\|Y-\Op{A}X\right\|_2^2\right]=\left\|Y-\Op{A}\boldsymbol{\mu}\right\|_2^2+\text{trace}\,(\Op{A}^\top \Op{A}\boldsymbol{K})$.
\end{prop}
\noindent\textbf{Proof}: Expanding the squared $\ell_2$-norm, we have that
\begin{eqnarray}
\mathbb{E}\left[\left\|Y-\Op{A}X\right\|_2^2\right]&=&\mathbb{E}\left[Y^\top Y - 2\,Y^\top \Op{A}X+X^\top \Op{A}^\top \Op{A} X\right].
\label{exp_quad_form_1}
\end{eqnarray}
Since the expectation is a linear operation, the expected value of the second term in \eqref{exp_quad_form_1} is $\left(-2\,Y^\top \Op{A}\boldsymbol{\mu}\right)$. The expected value of the third term can be evaluated as
\begin{eqnarray}
\mathbb{E}\left[X^\top \Op{A}^\top \Op{A} X\right]&=& \mathbb{E}\left[\text{trace}\,(\Op{A}^\top \Op{A} XX^\top)\right]
=\text{trace}\,(\Op{A}^\top \Op{A}(\boldsymbol{K}+\boldsymbol{\mu}\boldsymbol{\mu}^\top))\nonumber\\
&=&\text{trace}\,(\Op{A}^\top \Op{A}\boldsymbol{K})+\boldsymbol{\mu}^\top \Op{A}^\top \Op{A} \boldsymbol{\mu}.
\label{exp_quad_form_2}
\end{eqnarray}
Substituting \eqref{exp_quad_form_2} in \eqref{exp_quad_form_1} leads to the desired result.\hfill $\blacksquare$
\begin{algorithm}[t]
\caption{Adversarial training of an iterative reconstruction network.}
\begin{algorithmic}
\STATE {\bf 1.} {\bf Input:} Gradient penalty $\lambda_{\text{gp}}$, initial reconstruction network parameter $\theta$ and critic parameter $\alpha$, batch-size $n_{b}$, Adam optimizer parameters $\left(\eta,\beta_1,\beta_2\right)$, the number of $\mathcal{D}$ updates per $\mathcal{G}$ update (denoted as $K$), penalty parameters $\lambda_{\mathbb{X}}$ and $\lambda_{\mathbb{Y}}$.
\STATE {\bf 2.} {\bf for mini-batches $m=1,2,\cdots$, do (until convergence)}:
\begin{itemize}[leftmargin=*]
\item Sample $\boldsymbol{x}_j\sim\pi_X$, $\boldsymbol{y}_j\sim \pi_Y$, and $\epsilon_j\sim \text{uniform}\,[0,1]$; for $1 \leq j \leq n_b$. Compute $\boldsymbol{x}^{(\epsilon)}_j=\epsilon_j \boldsymbol{x}_j+\left(1-\epsilon_j\right)\mathcal{G}_{\theta}(\boldsymbol{y}_j)$.
\item Critic loss: $\mathcal{L}_{\mathcal{D}}=\frac{1}{n_b}\sum_{j=1}^{n_b}\left[\mathcal{D}_{\alpha}(\mathcal{G}_{\theta}(\boldsymbol{y}_j))-\mathcal{D}_{\alpha}(\boldsymbol{x}_j) + \lambda_{\text{gp}}\left(\left\|\nabla \mathcal{D}_{\alpha}\left(\boldsymbol{x}^{(\epsilon)}_j\right)\right\|_2-1\right)_{+}^2\right]$.
\item \textbf{for $k=1,\cdots,K$, do}: update critic as $\alpha\leftarrow\text{Adam}_{\eta,\beta_1,\beta_2}\left(\alpha,\nabla_{\alpha}\mathcal{L}_{\mathcal{D}}\right)$.
\item Compute the loss for the reconstruction network for the current mini-batch:
\begin{flalign*}
\mathcal{L}_{\mathcal{G}}=\frac{1}{n_b}\sum_{j=1}^{n_b}\left[-\mathcal{D}_{\alpha}(\mathcal{G}_{\theta}(\boldsymbol{x}_j))+\lambda_{\mathbb{X}}\left\|\mathcal{G}_{\theta}\left(\Op{A}(\boldsymbol{x}_j)\right)-\boldsymbol{x}_j\right\|_2^2+\lambda_{\mathbb{Y}}\left\|\Op{A}\left(\mathcal{G}_{\theta}(\boldsymbol{y}_j)\right)-\boldsymbol{y}_j\right\|_2^2\right].
\end{flalign*}
\item Update reconstruction network parameters: $\theta\leftarrow\text{Adam}_{\eta,\beta_1,\beta_2}\left(\theta,\nabla_{\theta}\mathcal{L}_{\mathcal{G}}\right)$.
\end{itemize}
\STATE {\bf 3.} {\bf Output:} The trained iterative reconstruction network $\mathcal{G}_{\theta}$.
\end{algorithmic}
\label{algo_acr_train}
\end{algorithm}
\subsection{Parametrizing the reconstruction and the critic networks}
\label{g_phi_pdhg_sec}
For parametrizing the reconstruction network $\mathcal{G}_{\theta}$, we adopt the same strategy as in \cite{lpd_tmi}, which is briefly explained here to make the exposition self-contained. The architecture of $\mathcal{G}_{\theta}$ is built upon the idea of \textit{iterative unrolling}, the origin of which can be traced back to the seminal work by Gregor and LeCun \cite{lecun_ista} on learned sparse approximation. Specifically, our reconstruction network $\mathcal{G}_{\theta}$ is parametrized by unrolling the Chambolle-Pock (CP) algorithm \cite{cham_pock} for non-smooth convex optimization. The CP algorithm is an iterative primal-dual scheme aimed at minimizing objectives of the form $f(\mathcal{K}\boldsymbol{x})+g(\boldsymbol{x})$, where $\mathcal{K}$ is a bounded linear operator, and $g$ and $f^*$ (the convex conjugate of $f$) are proper, convex, and lower semi-continuous. For convex $\mathcal{R}(\boldsymbol{x})$ in \eqref{var_recon}, a wide range of problems are solvable by the CP algorithm, the update rules of which are given by
\begin{eqnarray}
\boldsymbol{h}^{(\ell+1)} &=& \text{prox}_{\sigma\,f^*}(\boldsymbol{h}^{(\ell)}+\sigma\,\mathcal{K}(\bar{\boldsymbol{x}}^{(\ell)})),
\boldsymbol{x}^{(\ell+1)} =\text{prox}_{\tau\,g}(\boldsymbol{x}^{(\ell)}-\tau\,\mathcal{K}^*(\boldsymbol{h}^{(\ell+1)})),\nonumber\\
\bar{\boldsymbol{x}}^{(\ell+1)} &=& \boldsymbol{x}^{(\ell+1)}+\gamma (\boldsymbol{x}^{(\ell+1)}-\boldsymbol{x}^{(\ell)}), \text{\,\,for\,\,}0\leq\ell \leq L-1,
\label{cp_update}
\end{eqnarray}
starting from a suitable initial point $\boldsymbol{x}^{(0)}=\bar{\boldsymbol{x}}^{(0)}$. In order to construct an architecture for $\mathcal{G}_{\theta}$, we essentially replace the proximal operators in \eqref{cp_update} by trainable convolutional neural networks (CNNs). More specifically, the output of $\mathcal{G}_{\theta}$ is computed by applying the following two steps repeatedly $L$ times:
\begin{equation*}
\boldsymbol{h}^{(\ell+1)}=\Gamma_{\theta_{\text{d}}^{(\ell)}}(\boldsymbol{h}^{(\ell)},\sigma^{(\ell)}\Op{A}(\boldsymbol{x}^{\ell}),\boldsymbol{y}), \text{\,and\,} \boldsymbol{x}^{(\ell+1)}=\Lambda_{\theta_{\text{p}}^{(\ell)}}(\boldsymbol{x}^{(\ell)},\tau^{(\ell)}\Op{A}^*(\boldsymbol{h}^{\ell+1})),
\end{equation*}
The learnable parameters $\left\{\theta_{\text{p}}^{(\ell)}, \theta_{\text{d}}^{(\ell)},\sigma^{(\ell)},\tau^{(\ell)}\right\}_{\ell=0}^{L-1}$ are denoted using the shorthand notation $\theta$. The CNNs $\Gamma$ and $\Lambda$ are composed of a cascade of convolutional layers followed by a parametric ReLU activation. For the CT reconstruction experiment conducted in Sec. \ref{sec:numerical_expr}, we set $\boldsymbol{h}^{(0)}=\boldsymbol{0}$, and take the initial estimate $\boldsymbol{x}^{(0)}$ as the filtered back-projection (FBP) reconstruction. The number of layers is selected as $L=15$ and the filters in $\Gamma$ and $\Lambda$ are taken to be of size $5\times 5$ to increase the overall receptive field of the model to make it suitable for sparse-view CT. The critic $\mathcal{D}_{\alpha}$ (consisting of $\sim$ 2.76 million parameters) is a simple feed-forward CNN with four cascaded modules; each consisting of a convolutional layer, an instance-normalization layer, and a leaky-ReLU activation with negative-slope 0.2; followed by a global average-pooling layer in the end.
\begin{figure*}[h!]
\centering
\begin{minipage}[t]{\textwidth}
\centering
\vspace{0pt}
\includegraphics[width=0.35\textwidth]{expr_phantom/example_ellipse_phantom}
\qquad
\includegraphics[width=0.35\textwidth,trim=30 27 0 0,clip,angle=90]{expr_phantom/example_ellipse_phantom_sinogram}
\vskip-0.1\baselineskip
{\small Example of training data, image and its corresponding projection data}
\end{minipage}
\\[1em]
\begin{minipage}[t]{0.35\textwidth}
\centering
\vspace{0pt}
\includegraphics[width=\linewidth]{expr_phantom/fbp_shepplogan_001}
\vskip-0.5\baselineskip
{\small FBP: 19.51 dB, 0.13}
\end{minipage}
\begin{minipage}[t]{0.35\textwidth}
\centering
\vspace{0pt}
\includegraphics[width=\linewidth]{expr_phantom/adv_lpd_5x5_filters_15layers_shepplogan_001}
\vskip-0.5\baselineskip
{\small TV: 29.18 dB, 0.84}
\end{minipage} \\
\begin{minipage}[t]{0.35\textwidth}
\centering
\vspace{0pt}
\includegraphics[width=\linewidth]{expr_phantom/lpd_5x5_filters_15layers_shepplogan_001}
\vskip-0.5\baselineskip
{\small LPD: 27.89 dB, 0.96}
\end{minipage}
\begin{minipage}[t]{0.35\textwidth}
\centering
\vspace{0pt}
\includegraphics[width=\linewidth]{expr_phantom/adv_lpd_5x5_filters_15layers_shepplogan_001}
\vskip-0.5\baselineskip
{\small ALPD: 28.27 dB, 0.90}
\end{minipage}
\caption{\small{Comparison of supervised and unsupervised training on the Shepp-Logan phantom. The PSNR (dB) and SSIM are indicated below the images. ALPD does a better job of alleviating over-smoothing, unlike its supervised variant (LPD).}}
\label{ct_image_figure_shepplogan}
\end{figure*}
\section{Numerical results}
\label{sec:numerical_expr}
For numerical evaluation of the proposed approach, we consider the classical inverse problem of sparse-view CT reconstruction. First, we demonstrate a proof-of-concept using phantoms containing random ellipses of different intensities for training the networks. Subsequently, we present a comparative study of the proposed ALPD approach with state-of-the-art model- and data-driven reconstruction methods. Parallel-beam projection data along 200 uniformly spaced angular directions, with 400 lines/angle, are simulated using the ODL library \cite{odl} with a GPU-accelerated \textit{astra} back-end. Subsequently, white Gaussian noise with a standard-deviation of $\sigma=2.0$ is added to the projection data to simulate noisy measurements. For supervised training, the phantoms and their corresponding noisy parallel-beam projections are aligned, whereas they are shuffled for unsupervised learning to eliminate the pairing information.
The penalty parameters in \eqref{alpd_train_loss} are selected as $\lambda_{\mathbb{X}}=\lambda_{\mathbb{Y}}=10.0$, and the gradient penalty in Algorithm \ref{algo_acr_train} is also taken as $\lambda_{\text{gp}}=10.0$. The parameters in the Adam optimizer for updating both $\mathcal{G}_{\theta}$ and $\mathcal{D}_{\alpha}$ are chosen as $\left(\eta,\beta_1,\beta_2\right)=\left(5\times 10^{-5}, 0.50,0.99\right)$. The same set of hyper-parameters are used for training on both phantoms and real CT images. The critic $\mathcal{D}_{\alpha}$ is updated once per $\mathcal{G}_{\theta}$ update and the batch-size is taken as one (i.e., $K=1$ and $n_b=1$ in Algorithm \ref{algo_acr_train}).
\subsection{Training on ellipse phantoms}
\label{phantom_train_sec}
In this experiment, we generate a set of 2000 2D phantoms, each of size $512\times 512$ and containing 5 ellipses of random eccentricities at random locations and orientations, for training the networks. Each ellipse has an intensity value chosen uniformly at random in the range [0.1, 1]. The intensity of the background is taken as 0.0 and the intensities of the ellipses add up in the regions where they overlap. A representative phantom and its corresponding noisy sparse-view parallel-beam projection are shown in the first row of Figure \ref{ct_image_figure_shepplogan}.
\indent The main objective of this experiment is to study the differences between supervised and unsupervised learning in terms of their ability to reproduce images containing homogeneous regions separated by sharp edges. For performance evaluation, we consider reconstruction of the Shepp-Logan phantom which essentially consists of elliptical homogeneous regions delineated by sharp boundaries. Since the total-variation (TV) regularizer, which seeks sparsity in the gradient image, is tailor-made for such phantoms, we consider the reconstructed image produced by TV as the `gold-standard' in this case and compare the proposed ALPD approach with its supervised counterpart vis-\`a-vis the TV reconstruction. To compute the TV solution, we use the ADMM-based solver in the ODL library with the penalty parameter $\lambda=10.0$, which leads to the best reconstruction in our setting.
\noindent A visual comparison of the reconstructed images using LPD and ALPD (in Figure \ref{ct_image_figure_shepplogan}) indicates that ALPD does a better job of recovering the three small tumors on the top region of the Shepp-Logan phantom. The ALPD reconstruction, although slightly inferior to TV in terms of PSNR, looks almost identical, while the supervised LPD reconstruction looks significantly blurry, making it difficult to discern the small tumors.
\begin{figure*}[t]
\begin{minipage}[t]{0.32\textwidth}
\centering
\vspace{0pt}
\begin{tikzpicture}[spy using outlines={circle,red,magnification=4.0,size=1.50cm, connect spies}]
\node {\includegraphics[width=\linewidth]{mayo_ct_200x400_sigma_2.0/ground_truth_001}};
\spy on (0.02,-0.68) in node [left] at (1.9,1.25);
\spy on (-0.54,-0.76) in node [left] at (-0.4,1.25);
\end{tikzpicture}
\vskip-0.5\baselineskip
{\small Ground-truth}
\end{minipage}
\begin{minipage}[t]{0.32\textwidth}
\centering
\vspace{0pt}
\begin{tikzpicture}[spy using outlines={circle,red,magnification=4.0,size=1.50cm, connect spies}]
\node {\includegraphics[width=\linewidth]{mayo_ct_200x400_sigma_2.0/fbp_001}};
\spy on (0.02,-0.68) in node [left] at (1.9,1.25);
\spy on (-0.54,-0.76) in node [left] at (-0.4,1.25);
\end{tikzpicture}
\vskip-0.5\baselineskip
{\small FBP: 21.63 dB, 0.24}
\end{minipage}
\begin{minipage}[t]{0.32\textwidth}
\centering
\vspace{0pt}
\begin{tikzpicture}[spy using outlines={circle,red,magnification=4.0,size=1.50cm, connect spies}]
\node {\includegraphics[width=\linewidth]{mayo_ct_200x400_sigma_2.0/tv_001}};
\spy on (0.02,-0.68) in node [left] at (1.9,1.25);
\spy on (-0.54,-0.76) in node [left] at (-0.4,1.25);
\end{tikzpicture}
\vskip-0.5\baselineskip
{\small TV: 29.25 dB, 0.79}
\end{minipage}
\\[1em]
\begin{minipage}[t]{0.32\textwidth}
\centering
\vspace{0pt}
\begin{tikzpicture}[spy using outlines={circle,red,magnification=4.0,size=1.50cm, connect spies}]
\node {\includegraphics[width=\linewidth]{mayo_ct_200x400_sigma_2.0/ar_001}};
\spy on (0.02,-0.68) in node [left] at (1.9,1.25);
\spy on (-0.54,-0.76) in node [left] at (-0.4,1.25);
\end{tikzpicture}
\vskip-0.5\baselineskip
{\small AR: 31.83 dB, 0.84}
\end{minipage}
\begin{minipage}[t]{0.32\textwidth}
\centering
\vspace{0pt}
\begin{tikzpicture}[spy using outlines={circle,red,magnification=4.0,size=1.50cm, connect spies}]
\node {\includegraphics[width=\linewidth]{mayo_ct_200x400_sigma_2.0/lpd_5x5_filters_15layers_001}};
\spy on (0.02,-0.68) in node [left] at (1.9,1.25);
\spy on (-0.54,-0.76) in node [left] at (-0.4,1.25);
\end{tikzpicture}
\vskip-0.5\baselineskip
{\small LPD: 33.39 dB, 0.88}
\end{minipage}
\begin{minipage}[t]{0.32\textwidth}
\centering
\vspace{0pt}
\begin{tikzpicture}[spy using outlines={circle,red,magnification=4.0,size=1.50cm, connect spies}]
\node {\includegraphics[width=\linewidth]{mayo_ct_200x400_sigma_2.0/adv_lpd_5x5_filters_15layers_001}};
\spy on (0.02,-0.68) in node [left] at (1.9,1.25);
\spy on (-0.54,-0.76) in node [left] at (-0.4,1.25);
\end{tikzpicture}
\vskip-0.5\baselineskip
{\small ALPD: 32.48 dB, 0.84}
\end{minipage}
\caption{\small{Comparison of ALPD with state-of-the-art model- and data-driven reconstruction methods on Mayo clinic data. The corresponding PSNR (dB) and SSIM are indicated below the images and the key differences in the reconstructed images are highlighted. The ALPD reconstruction is visibly sharper as compared to LPD, enabling easier identification of clinically important features.}}
\label{ct_image_figure_mayo2}
\end{figure*}
\begin{table}[t]
\centering
\begin{tabular}{l c l r r r}
\multicolumn{1}{l}{\textbf{Method}}
& \multicolumn{1}{c}{\quad\textbf{PSNR (dB)}\quad}
& \multicolumn{1}{c}{\quad\textbf{SSIM}\quad}
& \multicolumn{1}{c}{\quad\textbf{\# param.}\quad}
& \multicolumn{1}{c}{\quad\textbf{Time (ms)}\quad} \\
\toprule
FBP & 21.2866 & 0.2043 & 1 & 14.0\\
TV & 30.3476 & 0.8110 & 1 & 21\,315.0 \\%[1.2ex]
\midrule
\multicolumn{5}{l}{\emph{Trained against supervised data}} \\
FBP + U-Net & 31.8008 & 0.7585 & 7\,215\,233 & 18.6 \\
LPD & 35.1561 & 0.9048 & 854\,040 & 184.4 \\%[1.2ex]
\midrule
\multicolumn{5}{l}{\emph{Trained against unsupervised data}} \\
AR & 33.6207 & 0.8750 & 19\,347\,890 & 41\,058.3\\
ALPD & 33.7386 & 0.8559 & 854\,040 & 183.9\\%[1.2ex]
\bottomrule
\end{tabular}
\\[2ex]
\caption{\small{Average performance of reconstruction methods.
LPD has the best overall performance (in terms of PSNR and SSIM), but this reconstruction method needs to be trained against
supervised data, i.e., pairs of high quality images (ground-truth) and corresponding noisy data. ALPD has slightly worse PSNR and SSIM values, but it can be trained against unsupervised data, which is vastly easier to get hold of as compared to supervised data. Note also that ALPD has significantly fewer parameters than AR, indicating that it can be trained against smaller datasets.}}
\vspace{-0.2in}
\label{sparse_ct_table}
\end{table}
\subsection{Sparse-view CT on Mayo-Clinic data}
We perform a comparison of the proposed ALPD method with competing model- and data-driven reconstruction techniques on human abdominal CT scans released by the Mayo Clinic for the low-dose CT grand challenge \cite{mayo_ct_challenge}. The dataset consists of CT scans corresponding to 10 patients, from which we extract 2D slices of size $512\times 512$ for our experiment. A total of 2250 slices extracted from 3D scans for 9 patients are used to train the networks in the data-driven methods, while 128 slices extracted from the scan for the remaining one patient are used for performance validation and comparison. The acquisition geometry and measurement noise distribution are kept the same as stated in Sec. \ref{phantom_train_sec}. For the sake of bench-marking the performance, we consider two model-based techniques, namely the classical FBP and TV reconstruction. As two representative state-of-the-art data-driven methods, we consider adversarial regularization (AR) \cite{ar_nips}, and the LPD method \cite{lpd_tmi} trained on paired data. The performance of a U-Net-based learned post-processing applied on FBP is reported in Table \ref{sparse_ct_table} along with the aforementioned techniques as a baseline for fully data-driven methods.
\indent Similar to what we noted for the Shepp-Logan phantom, the ALPD reconstruction outperforms LPD in terms of recovering sharp boundaries in the images, thus facilitating better delineation of clinically important features (see Fig. \ref{ct_image_figure_mayo2}). In terms of PSNR and SSIM, ALPD performs slightly worse than LPD, but it outperforms other competing techniques both qualitatively and quantitatively, as seen from the average PSNR and SSIM values reported in Table \ref{sparse_ct_table}. Notably, ALPD has the same reconstruction time as LPD, which is a couple of orders of magnitude lower than variational methods such as TV and AR that require computing iterative solutions to a high-dimensional optimization problem.
\section{Conclusions}
We proposed an unsupervised training protocol that learns a parametric reconstruction operator for solving imaging inverse problems from samples of the marginal distributions of the image and the measurement. The reconstruction operator is parametrized by an unrolled iterative scheme, namely the Chambolle-Pock method, originally developed for solving non-smooth convex optimization in \cite{cham_pock} and subsequently adopted for network parametrization in the supervised learning framework in \cite{lpd_tmi}. The proposed learning strategy, nevertheless, is not limited to the specific parametrization of the reconstruction operator chosen in this work and extends, in principle, to other iterative reconstruction schemes. Experimental evidence suggests that the proposed method does not suffer from the curse of over-smoothing as it minimizes a distortion measure in the distribution space instead of seeking pixel-wise proximity. Minimizing the Wasserstein-1 distance requires the introduction of a critic network, leading to a more resource-intensive training, which pays off in terms of superior performance and a more flexible training framework that it offers.
\bibliographystyle{splncs04}
|
1,116,691,500,758 | arxiv | \section{Introduction}
\label{sec:intro}
\par
Atomic nuclei are composed of protons and neutrons that interact with one another through the nuclear and electromagnetic forces.
The former, which is much stronger than the latter, dominates the properties of atomic nuclei.
Nevertheless, in specific studies, it is crucial to evaluate contribution of the latter to the properties of atomic nuclei.
The mass difference of the mirror nuclei and energy of the isobaric analog state are such examples.
In this report, we focus on the recent studies of the Coulomb energy density functionals (EDFs)
of electron systems in the context of atomic nuclei
\cite{Naito2018Phys.Rev.C97_044319,Naito2019Phys.Rev.C99_024309}
in the density functional theory (DFT) \cite{Hohenberg1964Phys.Rev.136_B864,Kohn1965Phys.Rev.140_A1133}.
\par
In the DFT for electron systems, the correlation energy is considered as well as the Hartree and exchange energies,
whereas in the nuclear DFT it is not considered explicitly.
The correlation EDF is tested for nuclear systems, where the local density approximation (LDA) is used,
and we have used the experimentally observed charge-density distribution for quantitative calculations of selected nuclei to avoid an error coming from the density \cite{Naito2018Phys.Rev.C97_044319}.
\par
To calculate Coulomb exchange energy, we carry out self-consistent Skyrme Hartree-Fock calculations by using the Perdew-Burke-Ernzerhof generalized gradient approximation (PBE-GGA) Coulomb exchange functional \cite{Perdew1996Phys.Rev.Lett.77_3865} instead of using the exact-Fock term,
and the optimal value of the free parameter $ \mu $ that appears in the PBE-GGA functional is also discussed \cite{Naito2019Phys.Rev.C99_024309}.
\section{Correlation Functional}
\label{sec:corr}
\par
The Coulomb correlation energies calculated by the charge density distribution \cite{DeVries1987At.DataNucl.DataTables36_495} in the LDA, $ E_{\urm{Cc}} $, for selected nuclei from light to heavy region are shown in Table \ref{tab:corr}.
For comparison, the Coulomb exchange energies calculated in the LDA, $ E_{\urm{Cx}} $, and the ratio $ E_{\urm{Cc}} / E_{\urm{Cx}} $ are also shown.
\par
It is seen that in these calculations, $ E_{\urm{Cc}} $ is all around $ 2 \, \% $ of $ E_{\urm{Cx}} $.
However, Bulgac and Shaginyan \cite{Bulgac1996Nucl.Phys.A601_103,Bulgac1999Phys.Lett.B469_1} evaluated that in atomic nuclei,
$ E_{\urm{Cc}}/E_{\urm{Cx}} $ would be around $ -40 \, \% $ to $ -20 \, \% $, instead of $ 2 \, \% $.
Hypothetically, if there is only Coulomb interaction
since correlation always further decreases the energy of the whole system,
we have the signs of the Hartree, exchange, and correlation energies as
$ E_{\urm{Cd}} > 0 $, $ E_{\urm{Cx}} < 0 $, and $ E_{\urm{Cc}} < 0 $, respectively.
In reality, the correlation EDF is not separable at all.
In Refs.~\cite{Bulgac1996Nucl.Phys.A601_103,Bulgac1999Phys.Lett.B469_1}
the correlation EDFs are written in terms of the response functions,
and such response functions are determined by the total interaction, i.e.,~mainly by the attractive nuclear part,
instead of the repulsive Coulomb part.
The total correlation energy is still negative,
mainly due to the contribution of the nuclear interaction.
As a result,
the contribution of Coulomb interaction becomes positive,
i.e.,~for the Coulomb energies, $ E_{\urm{Cc}} $ has the different sign as $ E_{\urm{Cx}} $.
In short, the correlation energy density functionals of electron systems cannot be applied directly to atomic nuclei.
It is also noted that the Coulomb correlation functional in the GGA gives around $ 30 $--$ 80 \, \% $ of $ E_{\urm{Cc}} $ in the LDA.
\begin{table}[!htb]
\centering
\caption{
Coulomb correlation energies $ E_{\urm{Cc}} $ for selected nuclei.
Energies are shown in $ \mathrm{MeV} $.
Data are taken from Ref.~\cite{Naito2018Phys.Rev.C97_044319}.}
\label{tab:corr}
\begin{tabular}{rD{.}{.}{4}D{.}{.}{5}D{.}{.}{6}}
\hline \hline
\multicolumn{1}{c}{Nuclei} & \multicolumn{1}{c}{LDA $ E_{\urm{Cx}} $} & \multicolumn{1}{c}{LDA $ E_{\urm{Cc}} $} & \multicolumn{1}{c}{$ E_{\urm{Cc}}^{\urm{LDA}} / E_{\urm{Cx}}^{\urm{LDA}} $} \\ \hline
$ \nuc{O}{16}{} $ & -2.638 & -0.05218 & 1.978 \, \% \\
$ \nuc{Ca}{40}{} $ & -7.087 & -0.1329 & 1.875 \, \% \\
$ \nuc{Ca}{48}{} $ & -7.113 & -0.1332 & 1.873 \, \% \\
$ \nuc{Ni}{58}{} $ & -10.28 & -0.1879 & 1.828 \, \% \\
$ \nuc{Sn}{116}{} $ & -18.41 & -0.3361 & 1.826 \, \% \\
$ \nuc{Sn}{124}{} $ & -18.24 & -0.3356 & 1.840 \, \% \\
$ \nuc{Pb}{208}{} $ & -30.31 & -0.5524 & 1.823 \, \% \\
\hline \hline
\end{tabular}
\end{table}
\section{Exchange Functional}
\label{sec:exch}
\par
The GGA Coulomb exchange functionals have been proposed as
\begin{equation}
\label{eq:x-PBE}
E_{\urm{Cx}}
\left[
\rho_{\urm{ch}}
\right]
=
- \frac{3}{4}
\frac{e^2}{4 \pi \epsilon_0}
\left( \frac{3}{\pi} \right)^{1/3}
\int
\left[
\rho_{\urm{ch}} \left( \ve{r} \right)
\right]^{4/3}
F \left(s \left( \ve{r} \right) \right)
\, d \ve{r},
\end{equation}
where $ \rho_{\urm{ch}} $ is the charge density distribution,
$ F $ is the enhancement factor depending on the density gradient,
and $ s $ denotes the dimensionless density gradient
\begin{equation}
\label{eq:s}
s = \frac{\left| \bm{\nabla} \rho_{\urm{ch}} \right|}{2 k_{\urm{F}} \rho_{\urm{ch}}},
\qquad
k_{\urm{F}}
=
\left(
3 \pi^2 \rho_{\urm{ch}}
\right)^{1/3}.
\end{equation}
In particular,
\begin{equation}
\label{eq:F-PBE}
F \left( s \right)
=
1 + \kappa
-
\frac{\kappa}{1 + \mu s^2 / \kappa},
\end{equation}
is used in the PBE-GGA Coulomb exchange functional
to satisfy some physical conditions,
and $ F \equiv 1 $ corresponds to the LDA one,
i.e.,~the Hartree-Fock-Slater approximation \cite{Dirac1930Proc.Camb.Phil.Soc.26_376,Slater1951Phys.Rev.81_385}.
The parameter $ \kappa = 0.804 $ is determined for any value of $ \mu $ by the H\"{o}lder inequality.
In contrast, two different values of $ \mu $ have been widely used in the studies of atoms \cite{Perdew1996Phys.Rev.Lett.77_3865} and solids \cite{Perdew2008Phys.Rev.Lett.100_136406}, respectively.
For the PBE-GGA functional, $ \mu = 0.21951 $ is determined by the random phase approximation of the homogeneous electron gas.
Since this $ \mu $ can be a different value for nuclear systems, the free parameter of the PBE-GGA Coulomb exchange functional, $ \mu $, is multiplied by a factor $ \lambda $.
For the nuclear part, the SAMi functional \cite{Roca-Maza2012Phys.Rev.C86_031306} is used in the self-consistent calculation.
For comparison, the exact-Fock energies are also calculated \cite{Roca-Maza2016Phys.Rev.C94_044313}.
\par
The deviation of the Coulomb exchange energy $ E_{\urm{Cx}} $ of PBE-GGA from that of LDA, $ \Delta E_{\urm{Cx}}^{\urm{LDA}} $,
\begin{equation}
\label{eq:ecx_diff}
\Delta E_{\urm{Cx}}^{\urm{LDA}}
=
\frac{E_{\urm{Cx}} - E_{\urm{Cx}}^{\urm{LDA}}}{E_{\urm{Cx}}}
\end{equation}
are shown as a function of mass number $ A $ in
Fig.~\ref{fig:systematic_dmagic}.
\par
It is found that in the light-mass region, to reproduce the exact-Fock results, $ \lambda = 1.50 $ or more is required,
while in the medium-heavy- and heavy-mass regions $ \lambda = 1.25 $ reproduces well the exact-Fock results.
The PBE-GGA result with $ \lambda = 1.00 $ reproduces the exact-Fock result in the case of the super-heavy nucleus $ \nuc{126}{310}{} $
since the ratio of the surface region to the volume region in the super-heavy nuclei is smaller than that in the medium-heavy or heavy nuclei.
\begin{figure}[!htb]
\centering
\includegraphics[width=1.0\linewidth]{./figure_SAMi_systematic_dmagic_final.eps}
\caption{
Deviation between the PBE-GGA and the LDA, $ \Delta E_{\urm{Cx}}^{\urm{LDA}} $
defined as Eq.~\eqref{eq:ecx_diff}.
Taken from Ref.~\cite{Naito2019Phys.Rev.C99_024309}.}
\label{fig:systematic_dmagic}
\end{figure}
\section{Conclusion}
\label{sec:conc}
\par
In these works \cite{Naito2018Phys.Rev.C97_044319,Naito2019Phys.Rev.C99_024309}, the Coulomb exchange and correlation EDFs in electron systems are applied to the nuclear systems.
On the one hand, the Coulomb correlation energy density functionals of electron systems are not applicable for atomic nuclei,
because these functionals are not separable and the nuclear interaction determines properties of atomic nuclei mainly.
On the other hand, the PBE-GGA Coulomb exchange functional with $ \lambda = 1.25 $ reproduces the exact-Fock energy in the self-consistent Skyrme Hartree-Fock calculations for atomic nuclei.
It should be emphasized that the numerical cost of the self-consistent calculations with the PBE-GGA exchange functional is $ O \left( N^3 \right) $, whereas that with the exact-Fock term is $ O \left( N^4 \right) $.
\par
\begin{acknowledgement}
TN and HL would like to thank the RIKEN iTHEMS program
and the JSPS-NSFC Bilateral Program for Joint Research Project on Nuclear mass and life for unravelling mysteries of the $ r $-process.
TN acknowledges the financial support from Computational Science Alliance, The University of Tokyo,
Universit\`{a} degli Studi di Milano,
and the JSPS Grant-in-Aid for JSPS Fellows under Grant No.~19J20543.
HL acknowledges the JSPS Grant-in-Aid for Early-Career Scientists under Grant No.~18K13549.
GC and XRM acknowledge funding from the European Union's Horizon 2020 research and innovation program under Grant No.~654002.
\end{acknowledgement}
|
1,116,691,500,759 | arxiv | \section{Introduction}
\begin{comment}
outline
cooperation in real time without verbal comm
- social forces \cite{Helbing_1995};
- modeling with temporal network \cite{rnn}; with obstacles \cite{ss-lstm} ;with interatciont \cite{social_lstm}
- capture multimoalidty in interactions \cite{unfreezing} ; modeling motion in difference homotopy classes as different behaviors \cite{schulz} ; generative models to capture multimoality\cite{social_vrnn}
- intent estimation (hihg level task): bayesian \cite{hu2018framework}, black box \cite{mlp} gametheretic \cite{gonzalez}
- motion prediction (microinteractions)
but none of those on a physically cooperative task involving both agents acting together
\end{comment}
\par Humans internally develop and rely on models of the world around them to make goal-oriented decisions. They have internalized strategies that drive their decision-making based on their goals and observations. In collaborative tasks, human strategies must accommodate their teammates' behaviors in order to successfully meet their mutual objectives.
Robots that collaborate with humans must emulate this ability to be effective teammates \cite{Dafoe2020,Kragic2018}. Achieving this capability is especially difficult due to the inherent multimodal nature of human interactions. However, even when robots are equipped with an underlying prior on human actions, human actions are extremely noisy and difficult to predict. This situation is especially true in complex, continuous action and state space tasks.
Learning the sampling distributions over future motion sequences from successful human demonstrations not only helps the robot determine its next action and potential outcomes, but also allows the robot to effectively come up with viable actions that result in more fluent interactions (e.g. not pushing the table towards each other, or pulling in opposite directions).
\begin{figure}[h!]
\centering
\includegraphics[width=0.3\textwidth]{fig/splash.png}
\caption{A Variational Recurrent Neural Network (VRNN) is trained with successful human-human demonstrations (A) for the cooperative table-carrying task. The robot uses the model as an online path planner to plan waypoints (B) when executing the task with a human-in-the-loop (C).}
\label{network}
\end{figure}
\par An example of a human-robot task that involves both agents exerting joint effort is the human-robot cooperative table-carrying task, which is the primary focus of this paper. As a physically challenging and decentralized collaborative task, both agents must align their goals without explicit verbal communication, and each agent should anticipate and adapt to their collaborator's actions to allow for fluent collaboration. We are motivated by this task to improve online, cooperative planning by predicting realistic future trajectories of human-robot teams navigating the task by leveraging demonstration data. The difficulty of the task lies in how each agent can exert an action independently, but the joint action influences the state of the task due to physical constraints. While it is possible to gauge human intent in this scenario, the joint state (of both agents) do not explicitly imply how each human individually wants to move. Furthermore, a multimodality of motions can be generated from this interactive task.
\par We explore how we can leverage offline human demonstration data to effectively improve motion prediction generation for this task of human-robot collaborative table-carrying, and propose leveraging state-of-the-art tools in generative modeling to generate realistic, human-like motion predictions to aid a robot planner in successfully cooperating with a human in the task.
\par \textit{Contributions}: In this work, we frame the human-robot cooperative table-carrying problem and develop a data-driven framework to capture the following system requirements: (1) variance over time in generated plans, (2) multimodality in interactions, and (3) motion realism to reflect what a human would actually do while cooperating with a partner. Due to the physical and simultaneous nature of the task, our model makes no explicit assumptions about the other human's high-level behaviors, but rather learns from sequences of demonstrated behaviors performed on the task. We leverage this behavior to develop a learned sampling-based planner for a robot agent that learns realistic, human-like motion, as evaluated with several metrics that have not yet, to the best of our knowledge, been deployed to evaluate tasks involving human-robot interaction. Furthermore, we demonstrate that our planner works with a human to successfully complete the table-carrying task in human-in-the-loop scenarios, both in simulation and on real robots.
\begin{figure*}[ht]
\centering
\vspace{2mm}
\includegraphics[width=0.8\textwidth]{fig/network.png}
\caption{Model architecture of the VRNN used for cooperative carrying.}
\label{network}
\end{figure*}
\section{Related Work}
\par The problem of modeling and predicting human behavior for interactive tasks is widely studied and has been approached from various viewpoints. For cooperative tasks wherein spatio-temporal relationships with other agents or environments are important considerations, predicting motion is a challenging aspect of the cooperation.
Early works in human motion prediction use state transition models to encode assumptions of the physical state of the agents. These models have been physics-based, such as the ``social force" model \cite{Helbing_1995} to account for uncertainty in human interaction.
\par More recently, temporal networks have been used to capture time dependencies in motion prediction. Recurrent Neural Networks (RNNs), which allow state history to be retained in a neural network, have been widely used as tools to model sequence problems. In the human-robot interaction space, RNNs have been deployed in various trajectory prediction problems \cite{rnn}. To improve the use of these models for obstacle awareness, a hierarchical architecture was proposed to leverage scene information \cite{ss-lstm}. Recurrent networks have also been vastly used in the social scene prediction problem, and can be greatly improved by modeling uncertainty in motion prediction \cite{social_lstm}. Furthermore, by modeling interaction as a hidden state, such temporal models can more explicitly capture the time dependencies of interaction.
\par In working with humans, many techniques have been applied for human intent estimation. Particularly, some works have used Bayesian frameworks \cite{hu2018framework} along with black box modeling techniques \cite{mlp} to capture human intent in human-robot interaction scenarios. Others explicitly break down the task and employ game-theoretic modeling to consider turn-by-turn interactions \cite{fisac}. However, many of these interaction-aware works notably do not capture cooperative behaviors, particularly for tasks such as cooperative table-carrying, which require some degree of modeling the interaction, which is mostly physical and non-verbal, on the order of $\sim$1 second or less.
\par Inspired by recent breakthroughs in generative modeling that have shown promising success in modeling the motion prediction problem in interactive scenarios \cite{social_vrnn}, \cite{schmerling2017}, \cite{Zhan2018}, \cite{boris}, we address the task of cooperative table carrying, wherein two agents navigating a scene cluttered with obstacles must reach their goal destination without collision. Cooperative table-carrying is a challenging sequential task that requires joint effort; therefore, it is important that our model can capture (1) variance over time, (2) multimodality in interactions, and (3) motion realism to reflect what a human would actually do so that the robot can be a better partner.
\section{Problem Formulation: Human-Robot Cooperative Carrying}
\par We consider a single human agent and a single robot agent both maintaining contact at opposite ends of a mechanical load being carried. Unlike in scenarios where it is desirable for each agent to move around without contact with one another~\cite{social_vrnn}, the poses of each agent in the cooperative carrying task are coupled by parameters of the load being carried. The dynamics of the joint state $x_t \in \mathcal{X} \subset \mathbb{R}^n$, which we model as the state of the mechanical load, are
$$
x_{t+1} = f(x_t, u_{t}^R, u_{t}^H) \eqno{(1)}
$$
where $u_{t}^i \in \mathcal{U}_i \subset \mathbb{R}^{m_i}$ is the action from each agent $i \in \{R, H\}$ at time step $t$.
\par We aim to learn a probabilistic model for the robot that captures the multimodality of future trajectories of the carried load up to time step $T$, $x_{t+1:T}$, conditioned on previous $H$ states of the table, $x_{t-H:t}$, as well as information about the human-robot team's surroundings. We represent knowledge of obstacle and goal locations relative to the table as $d^{ego}_{goal}~\in~\mathbb{R}^2$, which is the heading vector to the goal in the table's ego frame; and $d^{ego}_{obs}\in \mathbb{R}^2$, the heading vector to the nearest (L2 distance) obstacle in the table's ego frame. The task demands accounting for high variability in motion conditioned on environment, as well as multimodality in human actions; for this reason, we choose to include latent hidden variables to model the sequential task, $p(x_{t+1} | h_{t-1}, x_t, z_t)$, where $z_t$ is a latent variable introduced to help structure the learning.
\subsection{Motion Prediction Model}
\par Considering the temporal nature of the task, we seek to leverage temporal networks for model predictions. Vanilla temporal models such as Recurrent Neural Networks do not possess the capability to model large deviations in observations with a deterministic hidden state; therefore, to capture the variations in the sequence of observations, we adopt the Variational Recurrent Neural Network architecture developed in prior work (Figure \ref{network}). The model input $s_{t-H:t}$ is an 8-dimensional vector sequence of previous states over the observational period, $H$:
$$ s_{t-H:t} =
\begin{bmatrix}
\Delta p_x & \Delta p_y & \Delta cos\theta & \Delta sin\theta & d^{ego}_{goal} & d^{ego}_{obs}
\end{bmatrix}_{t-H:t}^T
$$
where $\Delta p_x$, $\Delta p_y$, $\Delta cos\theta$, $\Delta sin\theta$ is the discrete time difference in the world frame of the table's $ x$, $y$, and orientation $\theta$ (smoothed via cosine and sine transforms), respectively. The model predicts the next change in state,
$$ x_{t+1:t+T} =
\begin{bmatrix}
\Delta p_x & \Delta p_y & \Delta cos\theta & \Delta sin\theta
\end{bmatrix}_{t+1:t+T}^T
$$
\par We train our network using two steps: first, we obtain context in the form of the last hidden state, $h_{t-1}$ by providing the history $s_{t-H:t}$; then, we obtain motion plan rollouts by sampling from a prior on $h_{t-1}$. During the first step, we feed the first $H$ steps of history, $s_{t-H:t}$ into an encoder network. The network begins with a hidden state initialization of zeros. As seen in Fig. \ref{network}: the inputs are encoded per time step using a two-layer Multi-Layer Perceptron (MLP) with a 64-unit embedding layer. This is fed into a two-layer MLP encoder network, $q(z_t \mid s_{\leq T}, z_{\leq T})$, with hidden dimension of size 64, which is shaped into the mean and log-variance parameters of the parameterized Gaussian latent variable $z_t$ with dimension size 6. This variable~$z_t$ is obtained via reparameterizing the encoder output and sampling. The $z_t$ is embedded with an MLP and fed into a two-layer MLP decoder network, \mbox{$p(x_t \mid z_{<t}, x_{<t})$} of size 64, which then outputs the reconstruction of the table state $x_t$. Using a Gated Recurrent Unit (GRU) with 2 layers and 64 hidden units, we are able to update the hidden state conditioned on each time step of the embedded model input and latent sampled.
\par Starting from the current time step $t$, the hidden state from the previous step is fed through a prior network, $\phi^{prior}_{z_t}(h_{t-1})$ which notably conditions the sample on all previous states via the last hidden state, $h_{t-1}$; after sampling from the prior for $z_t$, the next table state is produced via the decoder network, and the recurrence step can be performed again to update the hidden state. That is, using this structure, we can roll out future motion plans up to time $T$ by conditioning on state and observation history. Note that we can vary $H$ and $T$ since we are leveraging a recurrent network structure. The loss function is given by minimizing the variational lower-bound:
\begin{equation}
\begin{split}
\mathbb{E}_{q(z_{\leq T} \mid s_{\leq T})} = & \sum_{t=1}^{T} \Bigl( - KL( q(z_{t} \mid s_{\leq T}, z_{\leq T}) \big\| p(z_t \mid s_{<t}, z_{<t})) \\
& + \log p(x_t \mid z_{<t}, s_{<t}) \Bigr)
\end{split}
\end{equation}
where the log term becomes the L2 loss on the predicted states.
\begin{figure}[thp]
\centering
\vspace{2mm}
\includegraphics[ width=0.2\textwidth]{fig/dynamics-table.png}
\caption{Table state in the simulator. One agent is denoted by an orange circle, the other by a blue triangle.}
\label{tabledynamics}
\end{figure}
\begin{figure*}[thp]
\centering
\subfloat[Predictions on a map unseen by the model during training. \label{1a}]{ \adjincludegraphics[width=0.33\textwidth, trim={{0.1\width} {0.01\height} {0.1\width} {0.01\height}}, clip]{51_unseen.png}
\hfill
\adjincludegraphics[width=0.33\textwidth, trim={{0.1\width} {0.01\height} {0.1\width} {0.01\height}}, clip]{112_unseen.png}
\adjincludegraphics[width=0.33\textwidth, trim={{0.1\width} {0.01\height} {0.1\width} {0.01\height}}, clip]{160_unseen.png}
}
\centering
\centering
\subfloat[Examples of predictions on randomly initialized map configurations.
\label{1b}]{ \adjincludegraphics[width=0.33\textwidth, trim={{0.1\width} {0.01\height} {0.1\width} {0.1\height}}, clip]{391_oneobs.png}
\adjincludegraphics[width=0.33\textwidth, trim={{0.1\width} {0.01\height} {0.1\width} {0.1\height}}, clip]{177_twoobs.png}
\adjincludegraphics[width=0.33\textwidth, trim={{0.1\width} {0.1\height} {0.1\width} {0.01\height}}, clip]{373.png}}
\\
\centering
\caption{Table state predictions (orange lines) plotted at various time steps on a map with an obstacle configuration unseen during model training (top) and on three maps with randomized configurations (bottom), plotted against ground truth trajectories (green lines). The red squares denote the obstacles; grey circles denote the initial table location; the yellow rectangles denote the goal regions.}
\label{predtraj}
\end{figure*}
\subsection{Training and Validation of Prediction Model}
\par We implemented a cooperative table-carrying simulator in Pygame for the table-carrying task. Given input forces from two agents, $u^{H}$ and $u^{R}$, we model the table (displayed in Figure \ref{tabledynamics}) to follow double-integrator dynamics, where $u^i, i \in{R, H}$ are inertial forces that contribute to table acceleration. We add damping parameters on table velocity to mimic frictional forces. The simulator records force inputs and the table state at a maximum frequency of 30Hz. Actions are recorded from both humans during dataset collection using joystick controllers. The action space is continuous, with $u \in [-1, 1]$ normalized with the joystick maximum and minimum displacement in both the $x$ and $y$ axes.
\par To train our model, we collected a human-human demonstration dataset from the simulator on a variety of initial and goal conditions, and maps. At the start of each demonstration, we initialize the simulator randomly, parameterized by 3 different initial table poses, 7 different obstacle configurations, and 3 different goal poses -- thus, a total of 63 different possible configurations. The maps are configured such that the agents always generally start at the left side of the map, and navigate through an obstacle-filled space to a goal location on the right side of the map. A total of 369 trajectories of human demonstrations on 5 different pairs of people were collected; 15 of them were separated into a holdout test set, and another 15 trajectories were demonstrations on a new, unseen map to which the model did not have exposure. For each map, we told the demonstrators to non-verbally and cooperatively move the table together from the start to end location while avoiding obstacles. The models were trained on a 12th Gen. Intel i9 12900K processor with 2 NVIDIA GeForce 2080 Ti GPUs.
\section{Experiments}
\subsection{Experimental Setup}
\par We evaluated the model from two aspects: (1) its ability to accurately predict future trajectories and (2) to navigate simulated and real human-in-the-loop environments. To evaluate the model's trajectory prediction, we compare our sampled rollouts against 15 held-out trajectories and 15 unseen maps. During evaluation, we provide the model with the first $H = 1$ second (30 steps in the simulator) of the ground truth trajectory, and then sample rollouts for up to $T = 3$ seconds. We also evaluate the model with human-in-the-loop simulations, wherein one agent is running a planner while a human provides action inputs (see Fig. \ref{framework}). The agent re-plans at each time step after sampling from predicted trajectories and selects the optimal trajectory leading to the minimum cost. The agent is provided with the cost function from the environment, which repels from obstacles and attracts to the goal. After selecting a plan, the robot uses a simple proportional controller to navigate to the planned waypoint.
\par We compare the motion realism and diveristy of generated rollouts from our method with those generated by an ``ideal" team that leverages centralized planning. As a baseline, we use a sampling-based algorithm, Rapidly-exploring Random Tree (RRT), to sample paths for both agents to follow. During the human-in-the-loop evaluations, we use RRT in a decentralized fashion as a baseline (referred to as Dec-RRT): one agent plans with RRT while the other agent is controlled with human inputs through a joystick.
\par Finally, we demonstrate that our planner is able to successfully move cooperatively with a human in a zero-shot transfer to the real environment. Using two LoCoBot WX250s rigidly joined together via a rod pin connection, we deploy our planner on one robot while a human provides control inputs via joystick to the other robot.
\begin{figure}[h!]
\centering
\includegraphics[width=0.4\textwidth]{hil_frameworkV2.png}
\caption{Our framework leveraging learned motion prediction models for planning and control.}
\label{framework}
\end{figure}
\begin{table*}[h!]
\centering
\vspace{3mm}
\begin{tabular}{|| l | c c c c c | c c c c c ||}
\hline
& \multicolumn{5}{c|}{Holdout set} & \multicolumn{5}{c||}{Unseen map} \\
& L2 $(10^3)$ & FD $(10^3)$ & Var $x$ & Var $y$ & Var $\theta$ & L2 $(10^3)$ & FD $(10^3)$ & Var $x$ & Var $y$ & Var $\theta$ \\
\hline \hline
Centralized RRT & 1001.08 & 30.53 & 352.33 & 31.78 & 0.64 & 1137.02 & 45.66 & 386.99 & 30.88 & 0.70 \\
VRNN & 0.14 & 0.55 & 0.13 & 3.03 & 1.72 & 0.14 & 0.57 & 0.13 & 3.03 & 1.72 \\
\hline
\end{tabular}
\caption{Comparison of plans generated from the VRNN vs. the centralized baseline RRT. Lower FD indicates that the trajectories resemble human demonstration data more closely, while higher variance indicates more diversity in the corresponding dimension.}
\label{table:1}
\end{table*}
\begin{table*}[h!]
\centering
\begin{tabular}{|| c | c | c c c | c c c | c c c ||}
\hline
& & \multicolumn{3}{c|}{$x$} & \multicolumn{3}{c|}{$y$} & \multicolumn{3}{c||}{$\theta$} \\
& Planner & L2 $(10^3)$ & FD $(10^3)$ & Var & L2 $(10^3)$ & FD $(10^3)$ & Var & L2 & FD & Var \\
\hline\hline
\multirow{3}{*}{Holdout set} & GT & -- & -- & 28132.02 & -- & -- & 2026.07 & -- & -- & 4.39\\
& Dec-RRT & 1.27 & 330.17 & 20832.18 & 0.95 & 253.37 & 1552.11 & 87.73 & 230.56 & 2.01\\
& VRNN & 0.99 & 143.97 & 24837.02 & 0.75 & 549.50 & 1761.54 & 79.8 & 986.75 & 4.43\\ [1ex]
\hline
\multirow{3}{*}{Unseen map} & GT & -- & -- & 2594.42 & -- & -- & 183.91 & -- & -- & 2.47\\
& Dec-RRT & 0.25 & 39.04 & 2901.47 & 0.27 & 30.65 & 215.14 & 35.63 & 169.1 & 2.59\\
& VRNN & 1.75 & 2937.48 & 16172.11 & 0.76 & 149.98 & 866.03 & 73.27 & 380.08 & 6.16 \\ [1ex]
\hline
\end{tabular}
\caption{Comparison of human-in-the-loop interactions generated from our method with the Decentralized RRT (Dec-RRT) planner.}
\label{table:2}
\end{table*}
\subsection{Evaluation Metrics}
\par Quantifying rollouts and interactions of a human-robot interactive task is a difficult problem that requires evaluation on multiple fronts. Our metrics are based on metrics from prior work on motion realism in computer vision literature, as well as task-based metrics such as success rate and time to completion. We consider ideal cooperative behavior to be (1) realistic - as close as possible to what two humans cooperating on the task would do in a demonstration, (2) diverse - such that a variety of behaviors can emerge, and (3) able to successfully complete a task together.
\par We evaluate the plans generated by our model on two ground truth (GT) test datasets and on real-time human interactions using the following metrics:
\begin{itemize}
\item \textit{L2}: Distance to ground truth motion in SE(2) space ($p_x, p_y, \theta$).
\item \textit{Fretchet distance (FD)}: FD can be used to capture motion realism by measuring the distribution distance between generated and ground-truth motion sequences as used in \cite{learntodance}, \cite{ng}. We calculate FD between the distributions along the time dimension of each planner's rollouts for all trajectories in the test set. For human-in-the-loop trials, we further capture the distribution difference on each dimension ($p_x, p_y, \theta$) of the trajectories.
\item \textit{Variance}: Each planner's ability to generate diversity in motion is characterized by computing the batch-averaged temporal variance on each dimension ($p_x, p_y, \theta$).
\end{itemize}
For the human-in-the-loop trials, we also use task-related metrics to analyze each planner's ability to work with a human:
\begin{itemize}
\item \textit{Success rate (\%)}: The percentage of runs in which the human and planner successfully reached the goal location.
\item \textit{Time to completion (s)}: The time in seconds it took to successfully complete the task.
\end{itemize}
\subsection{Comparison of Sampled Plans}
\par Table \ref{table:1} show a comparison between plans generated by the VRNN and plans generated by the baseline Centralized RRT. The high FD score on the centralized baseline RRT suggests that while the RRT planner can plan for task success, it may not necessarily have the capability to do so in a human-like manner; the distribution of trajectories generated from the RRT planner is further from those of the learned model of human interactions, signaling that centralized behavior from high-performing motion planners is not representative of ground truth cooperative behavior in human-human demonstrations.
\subsection{Human-in-the-loop Trials}
\par Evaluation of the VRNN planner in simulation with a human-in-the-loop suggests that our model is capable of generating more human-like motion plans than the baseline while performing better in task-related metrics. As seen in \ref{table:2}, lower FD and L2 scores indicate that the motion plans generated by the VRNN are more human-like, and exhibit more diversity in rotation and pose. Additionally, it outperforms the Dec-RRT planner on both task success rate and time taken to complete the task \ref{table:3}. Compared to the performance of Dec-RRT on the unseen map, the VRNN's higher FD and L2 scores suggest that it generates plans that are less similar to the GT trajectories; however, the VRNN planner exhibits more variance across $x$, $y$, and rotation, all while performing better in task success rate \ref{table:3}.
\begin{table}[htbp]
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{|| l | c c | c c ||}
\hline
& \multicolumn{2}{c|}{Holdout set} & \multicolumn{2}{c||}{Unseen map} \\
& Success ($\%$) & Time (s) & Success ($\%$) & Time (s) \\
\hline
Dec-RRT & 36.7 & 18.16 & 60 & 16.50 \\
VRNN & 60 & 17.38 & 86.67 & 18.37 \\
\hline
\end{tabular}
}
\caption{Task success rate ($\%$) and average time (s) to completion for successful trajectories (i.e. wherein the human-robot team reached the goal without collision with obstacles).}
\label{table:3}
\end{table}
\begin{figure*}[!h]
\centering
\vspace{1mm}
\adjincludegraphics[width=0.75\textwidth, trim={{0.0\width} {0.0\height} {0.0\width} {0.0\height}}, clip]{fig/real_demo.png}
\caption{Sim-to-real transfer of our model trained in simulation to control a LoCoBot in the real world. Our model (blue rectangle) successfully cooperates with the human subject (orange circle) in carrying the table from the initial position (blue circle) to the goal region (yellow rectangle), while avoiding the obstacle (red rectangle).}
\label{fig:real_demo}
\end{figure*}
\subsection{Real robot demonstration}
\par We demonstrate the utility of our method on a real robot setup. Using two Interbotix LoCoBots, we show that our framework can be used to predict motion and adapt to real-time human inputs shown in Fig. \ref{fig:real_demo}. The human provides joystick inputs while the robot leverages the planner to anticipate the team's motion. The robots successfully navigate the unseen map scenario. For video of real robot results, please see the \href{https://youtu.be/CqWh-yWOgeA}{supplemental video}.
\section{Conclusion}
\par In this work, we developed a planner for human-robot cooperative load-carrying which leverages human demonstration data to predict realistic motion plans. We demonstrated in two different evaluations (i.e., generated rollouts and trajectories from human-in-the-loop tests) that plans generated by the VRNN model match closely with the human demonstration data, have large variance in motion, and can perform well on an unseen map. We also showed that the model can transfer successfully to the real robot case in zero-shot transfer.
\par An important next step in our work should evaluate the model on more challenging unseen maps to show that our model can generalize to multiple environments for the cooperative case. Future work will also improve the observation model used for this task. Furthermore, despite successfully modeling the table dynamics, there is potentially much more information that can be gathered, despite being noisy, from observing actions of the human-robot team. A promising avenue of future research could be learning to predict future action trajectories, and leveraging those to improve our model of interaction for cooperative human-robot tasks.
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|
1,116,691,500,760 | arxiv | \section{The key buried signatures from the central accretion disk}
The primary radiative output of active galactic nuclei (AGNs) is
observed at the ultraviolet/optical wavelengths. This is attributed to
be from an accretion disk around a supermassive black hole. While
this putative accretion disk has been modeled extensively, it is well
known that there are disagreements between observations and model
predictions in a few major respects
(e.g. refs. \cite{Antonucci99,Koratkar99}). A crucial observational
difficulty here has been that we still do not have enough spatial
resolution to isolate the accretion disk from the surrounding
regions. Important spectral features of the disk are thus often buried
under the strong emission from these regions --- in particular, from
the broad-line-emitting region (BLR) and from a slightly larger-scale
torus-like region with hot dust grains.
One such key spectral region is the near-infrared. In the fundamental
hypothesis of the standard, most extensively studied model
\cite{Shakura73}, the disk is optically thick and heated by local
energy dissipation, and this sets the effective disk temperature $T$
as a function of radius $r$ as $T \propto r^{-3/4}$ over a broad range
of radii. This leads to the well-known blue spectral shape at long
wavelengths, $F_{\nu} \propto \nu^{+1/3}$, in the simple case of local
blackbody emission. In more sophisticated, bare-disk atmosphere
models (e.g. ref. \cite{Hubeny00}), the same blue limit is reached
longward of $\sim$1 $\mu$m essentially independent of parameters
suitable for quasars. Then we should be able to robustly test disk
models here. Furthermore, the standard disk is also well known to be
gravitationally unstable in the outer radii \cite{Shlosman87}, which
may correspond to those emitting in the near-infrared
\cite{Goodman03}. The spectrum may show a break due to a possible
disk truncation and become even bluer toward longer wavelengths. Thus
the near-infrared disk spectrum is quite important for the tests of
disk models. However this is almost exactly where the dust thermal
emission from the torus starts to dominate the spectrum (set by dust
sublimation temperature). Thus this important near-infrared disk
spectrum usually cannot be observed.
Another key spectral signature is the Balmer edge. Among the opacity
edge features generally predicted by disk atmosphere models, the
Balmer edge has the advantage (over the Lyman edge) of being much less
prone to foreground absorptions. However, high-order Balmer emission
lines and Balmer continuum (and also FeII blends) from the BLR bury
the Balmer edge feature in the disk spectrum making it essentially
unobservable.
However, we argue here that we can separate the disk emission from the
surrounding emissions by using optical and near-infrared polarization
of quasars. With this measurement, effectively gaining very high
spatial resolution, we can study these key spectral signatures from
the accretion disk.
\begin{figure}[b]
\begin{center}
\includegraphics[width=12cm]{type2vs1rev.eps}
\caption{\label{fig_type2vs1}Schematic diagrams for the geometry of
dominant scattering regions for (a) Type 2s (b) Type 1s with some line
polarization (c) Type 1s with no line polarization. In each panel, the
double arrow shows the position angle of continuum polarization.}
\end{center}
\end{figure}
\section{The optical polarization of quasars or Type 1 AGNs}
Perhaps the most well-known optical polarization in AGNs is the one
seen in Type 2 objects, where the torus obscures our line of sight to
the accretion disk and the BLR. The broad emission lines are seen in
the polarized flux in many of these, with polarization position angle
(PA) perpendicular to the radio jet axis \cite{Antonucci93}. The
interpretation is that the gas which resides {\it outside} the BLR,
along the jet axis above and below the torus, scatters the light from
the accretion disk {\it and} the BLR into our line of sight
(Fig.1a). Thus they both show up in the polarized flux.
Here we are interested {\it not} in these Type 2 objects, but rather
in Type 1 objects, namely Seyfert 1 galaxies and quasars. In those
objects, our line of sight is much less inclined, giving a direct view
of the bright nuclear region interior to the torus. This gives rise to
a different nuclear polarization component to dominate. The optical
continuum is often polarized at polarization degrees $P$ of
$\lesssim$1\% level, with PAs {\it parallel} to the jet axis
(e.g. ref.\cite{Berriman90}). In many Seyfert 1s, the broad lines are
also polarized but often at lower $P$ and at different PA than
continuum (it rotates across the line wavelengths;
e.g. ref.\cite{Smith04may}). These imply that the scattering region is
more or less similar in size to the BLR. The parallel polarization
quite possibly indicates that the scattering region is in a
flattened/equatorial optically-thin geometry having its symmetry axis
along the jet direction (Fig.1b).
At least in some quasars, however, similar continuum polarization is
seen but with no or very little line emission in the polarized
flux. In this case, scattering is considered to be caused {\it
interior} to the BLR (Fig.1c), by electrons (since the site is inside
the dust sublimation radius). Then the polarized flux would in fact be
an electron-scattered copy of the intrinsic spectrum of the central
engine, with all the emissions from the BLR and torus eliminated. This
polarized flux enables us to isolate the accretion disk spectra from
the contaminating emissions.
\begin{figure}[b]
\begin{center}
\includegraphics[width=10cm]{balmedge.eps}
\caption{\label{fig_balm}Optical spectropolarimetry of five quasars
from ref.\cite{Kishimoto04}. The solid line is the polarized flux in
units of 10$^{-18}$ ergs/cm$^2$/sec/\AA. The dotted line is the total
flux scaled to match the polarized flux at the red side. The
wavelength of the Balmer discontinuity, 3646\AA, is indicated as a
folded line in each panel. }
\end{center}
\end{figure}
\begin{figure}[b]
\begin{center}
\includegraphics[width=10cm]{mobj_nfnnrm.eps}
\caption{\label{fig_nearIR}
Overlay of the polarized and total flux spectra observed in six
different quasars, from ref.\cite{Kishimoto08}. We plot scaled $\nu
F_{\nu}$ data: Q0144-3938 (redshift $z$=0.244), green; 3C95
($z=$0.616), blue; CTS A09.36 ($z$=0.310), light blue; 4C 09.72
($z$=0.433), red; PKS 2310-322 ($z$=0.337), light green; Ton 202
($z$=0.366), purple. Total flux spectra, shown as bold traces in the
optical and as squares in the near-infrared, are normalized at 1$\mu$m
in the rest frame, by interpolation. Polarized flux spectra, shown as
light points in the optical and as bold points in the near-infrared
(vertical error bars, 1-$\sigma$), are separately normalized, also at
1$\mu$m, by fitting a power-law to the near-infrared polarized flux spectra.
The normalized polarized flux spectra are arbitrarily shifted
downwards by a factor of three relative to the normalized total flux
spectra, for clarity.}
\end{center}
\end{figure}
\begin{figure}[t]
\includegraphics[width=10cm]{slope_refnamed.eps}\hspace{0.5cm}%
\begin{minipage}[b]{5.5cm}\caption{\label{fig_slope}
Spectral index of polarized flux spectra, from
ref.\cite{Kishimoto08}. We plot $\alpha$ (in $F_{\nu} \propto
\nu^{\alpha}$) against $\nu L_{\nu}$ for total light at 0.51
$\mu$m. The index was measured by a power-law fit for each
near-infrared polarized flux spectrum (note the different wavelength
range covered depending on the redshift) and is shown with 1-$\sigma$
error bars. A weighted mean of the spectral index measurements is
shown dashed; the shaded area represents its deduced 1-$\sigma$
uncertainty. The mean or median slopes of the ultraviolet/optical
total flux spectra derived in various studies
\cite{Neugebauer87,Francis91,VandenBerk01,Cristiani90,Zheng97} are
also shown.}
\end{minipage}
\end{figure}
\section{Intrinsic disk spectra as revealed by polarimetry}
Figure~\ref{fig_balm} shows the polarized flux spectra of such
quasars. In contrast to the total flux spectra, we see essentially no
emission lines in the polarized flux spectra. Thus the polarized flux
is very likely to show the intrinsic spectral behavior of the disk
without the BLR emission contamination. The objects were chosen to be
at redshift $\gtrsim$0.3 to make sure that the Balmer edge region is
covered with good sensitivity. The feature is indeed seen, all in
absorption: there is a downturn at around 4000\AA\ and an upturn at
around 3600\AA\ in all the objects shown. Thus the fundamental
implication here is that the emission is thermal and optically thick
in nature.
Then we have extended the work to longer wavelengths: since the
polarization originates interior to the dusty torus, we should also be
able to eliminate the dust emission and uncover the underlying
near-infrared disk spectrum. Figure~\ref{fig_nearIR} shows the
results for six quasars \cite{Kishimoto05,Kishimoto08}. Some of the
quasars are those shown in Figure~\ref{fig_balm}, while the others
have newly been found to be suitable for this work (i.e. no or very
little line flux in polarized light) from our optical polarimetric
survey with the ESO3.6m telescope and follow-up spectropolarimetry
with the VLT \cite{Kishimoto08}. In all six objects, the total flux
spectra in $\nu F_{\nu}$ show an up-turn longward of $\sim$1 $\mu$m
due to the onset of dust emission. However, the polarized flux spectra
all show systematically a rapid decrease toward long wavelengths with
a shape of approximately power-law form. The spectral indices
$\alpha$ measured in $F_{\nu} \propto \nu^{\alpha}$ for the
near-infrared polarized flux are shown in Figure~\ref{fig_slope}, and
compared with those observed at the optical/ultraviolet wavelengths.
The uncovered near-infrared colors are clearly much bluer than those
at the shorter wavelengths. Surprisingly they are all consistent with
the shape $\nu^{+1/3}$, with an weighted mean of $\alpha =
+0.44\pm0.11$.
The systematic behavior of the polarized flux, and the fact that PAs
are observed to be essentially constant over the whole wavelengths
(from the ultraviolet to near-infrared) in each object
\cite{Kishimoto08}, strongly argue against there being any secondary
polarization component arising in the near-infrared. Therefore, the
near-infrared polarized flux spectra are very likely to reveal the
intrinsic spectra of accretion disks. The measured slopes, which are
as blue as the predicted spectral shape of $\nu^{+1/3}$, strongly
suggest that, at least in the outer near-infrared emitting radii, the
standard but unproven picture of the disk being optically thick and
locally heated is approximately correct. In this case, other model
problems at shorter wavelengths should be directed to the lack of our
understanding of the inner regions of the same disk.
\section{Outlook}
The optical and near-infrared polarization measurements of quasars
have turned out to be quite revealing, and have been delineating the
fundamental aspects of the accretion disk in their central engine. A
further question is: does the revealed near-infrared spectrum show an
indication of disk truncation in this outer region? Although
statistically insignificant, the data do suggest that the slope is
slightly bluer than the shape $\nu^{+1/3}$. Theoretical modeling is
underway with the current data. Further measurements may provide
totally new insight on the outer edge of the disk and how material is
being supplied to the nucleus.
\section*{References}
\providecommand{\newblock}{}
|
1,116,691,500,761 | arxiv | \section{Introduction}
Particle-based numerical methods are routinely used in plasma physics
calculations \cite{Birdsall1985,Hockney1988}. In many cases these methods are
more efficient and simpler to implement than the corresponding continuum
Eulerian methods. However, particle methods face the well known
statistical sampling limitation of attempting to simulate a physical system containing $N$
particles using $N_p \ll N$ computational particles.
Particle methods do not seek to reproduce the exact
individual behavior of the particles, but rather to approximate
statistical macroscopic quantities like density, current,
and temperature. These quantities are determined from the particle
distribution function. Therefore, a problem of relevance for
the success of particle-based simulations is the reconstruction of the
particle distribution function from discrete particle data.
The difference between the distribution function reconstructed from a
simulation using $N_p$ particles and the exact distribution function
gives rise to a discretization error generically known as ``particle noise'' due to its random-like
character. Understanding and reducing this error is a complex problem
of importance in the validation and verification of particle
codes, see for example Refs.~\cite{Nevins2005,Krommes2007,McMillan2008} and
references therein for a discussion in the context of
gyrokinetic calculations. One obvious way to reduce particle noise is by
increasing the number of computational particles. However, the
unfavorable scaling of the error with the number of particles, $\sim
1/\sqrt{N_p}$ \cite{Krommes1993,Aydemir1994}, puts a severe limitation on this
straightforward approach.
This has motivated the development of various noise reduction techniques
including finite size particles (FSP) \cite{Hockney1966,ABL70}, Monte-Carlo methods \cite{Aydemir1994}, Fourier-filtering
\cite{Jolliet2007}, coarse-graining \cite{Chen2007}, Krook operators \cite{McMillan2008},
smooth interpolation \cite{Shadwick2008}, low noise
collision operators \cite{Lewandowski2005}, and Proper Orthogonal
Decomposition (POD) methods \cite{delCastillo2008} among others.
In the present paper we propose a wavelet-based method for noise
reduction in the reconstruction of particle distribution
functions from particle simulation data.
The method, known as Wavelet Based Density Estimation (WBDE), was
originally introduced in Ref.~\cite{Donoho1996} in the context of statistics
to estimate probability densities given a finite number of independent
measurements. However, to our knowledge, this method has not been
applied before to particle-base computations. WBDE, as used here, is based on truncations of the wavelet representation of the Dirac
delta function associated with each particle. The method
yields almost optimal results for functions with unknown local smoothness without compromising computational efficiency, assuming that the particles' coordinates are statistically independent.
As a first step in the application of the WBDE method to plasma
particle simulations, we limit attention to ``passive denoising''. That
is the WBDE method is treated as a post-processing technique
applied to independently generated particle data. The problem of ``active denoising'', e.g. the application
of WBDE methods in the evaluation of self-consistent fields in
particle in cell simulations, will not be addressed. This
simplification will allow us to assess the efficiency of the proposed noise
reduction method in a simple setting. Another simplification
pertains the dimensionality.
Here, for the sake of simplicity, we limit attention to the reconstruction and denoising problem
in two dimensions. However, the extension of the WBDE method to
higher dimensions is in principle straightforward.
Collisions, or the absence of them, play an important role in plasma
transport problems. Particle methods handle the collisional and
non-collisional parts of the dynamics differently. Fokker-Planck-type collision operators
are typically introduced in particle methods using Langevin-type
stochastic differential
equations. On the other hand, the non-collisional part of the dynamics is described using
deterministic ordinary differential equations. Collisional dominated
problems tend to washout small scale structures whereas
collisionless problems typically develop fine scale filamentary
structures in phase space. Therefore, it is important to test the dependence
of the efficiency of denoising reconstruction methods on the level of
collisionality.
Here we test the WBDE method in strongly collisional, weakly collisional and
collisionless regimes. For the strongly collisional regime we consider
particle data of force-free collisional relaxation
involving energy and pinch-angle scattering. The weakly collisional
regime is illustrated using guiding-center particle data of a
magnetically confined plasma in toroidal geometry.
The collisionless regime is studied using particle in cell (PIC)
data corresponding to bump-on-tail and two streams
instabilities in the Vlasov-Poisson system.
Beyond the role of collisions, the data sets that we are considering open the possibility of exploring the role of external and self-consistent fields in the reconstruction of the particle density.
In the collisional relaxation problem no forces act on the particles, in the guiding-center problem particles interact with an external magnetic field, and in the Vlasov-Poisson problem particle interactions are incorporated
through a self-consistent electrostatic mean field. One of the goals of this paper is to compare the WBDE method with the Proper Orthogonal Decomposition (POD) density reconstruction method proposed in Ref.~\cite{delCastillo2008}.
The rest of the paper is organized as follows.
In Sect.~II we review the main properties of kernel density estimation (KDE) and show its relationship with finite size particles (FSP). We then review basic notions on orthogonal wavelet and multiresolution analysis and outline a step by step algorithm for WBDE. Also, for completeness, in this section we include a brief description of the POD
reconstruction method proposed in Ref.~\cite{delCastillo2008}.
Section~III discusses applications of the WBDE method and the
comparison with the POD method.
We start by post-processing a simulation of plasma relaxation by random collisions against a background thermostat.
We then turn to a $\delta f$ Monte-Carlo simulation in toroidal geometry, whose phase space has been reduced to two dimensions.
Finally, we analyze the results of particle-in-cell (PIC) simulations of a 1D Vlasov-Poisson plasma.
The conclusions are presented in Sec.~IV.
\section{Methods}
This section presents the wavelet-based density estimation (WBDE)
algorithm. We start by reviewing basic ideas on kernel density
estimation (KDE) which is closely related to the use of finite size
particles (FSP) in PIC simulations. Following this, we
we give a brief introduction to wavelet analysis and discuss the WBDE algorithm.
For completeness, we also include a brief summary of the POD
approach.
\subsection{Kernel density estimation}
Given a sequence of independent and identically distributed measurements, the nonparametric density estimation problem consists in finding the underlying probability density function (PDF), with no
a priori assumptions on its functional form. Here we discuss general ideas on
this difficult problem for which a variety of statistical methods have been developed.
Further details can be found in the statistics literature, e.g. Ref.~\cite{Silverman1986}.
Consider a number $N_p$ of statistically independent particles with phase space coordinates $(\mathbf{X}_n)_{1\leq n \leq N_p}$ distributed in $\mathbb{R}^d$ according to a PDF $f$. This data
can come from a PIC or a Monte-Carlo, full $f$ or $\delta f$ simulation.
Formally, the sample PDF can be written as
\begin{equation}\label{dirac_estimate}
f^\delta(\mathbf{x}) = \frac{1}{N_p}\sum_{n=1}^{N_p} \delta(\mathbf{x}-\mathbf{X}_n)
\end{equation}
where $\delta$ is the Dirac distribution.
Because of its lack of smoothness, Eq.~(\ref{dirac_estimate}) is far from the actual distribution $f$ according to most reasonable definitions of the error. Moreover, the dependence of $f^\delta$ on the statistical fluctuations in $(\mathbf{X}_n)$ can lead to an artificial increase of the collisionality of the plasma.
The simplest method to introduce some smoothness in $f^\delta$ is to use a histogram.
Consider a tiling of the phase space by a Cartesian grid with $N_g^d$ cells. Let $\left\{B_\lambda\right\}_{\lambda\in\Lambda}$ denote
the set of all cells with characteristic function
$\chi_\lambda$ defined as $\chi_\lambda=1$ if $x \in B_\lambda$ and $\chi_\lambda=0$ otherwise.
Then the histogram corresponding to the tiling is
\begin{equation}\label{histogram_estimate}
f^H(\mathbf{x}) = \sum_{\lambda \in \Lambda} \left(\frac{1}{N_p}\sum_{n=1}^{N_p} \chi_\lambda(\mathbf{X}_n) \right) \chi_\lambda(\mathbf{x})
\end{equation}
which can also be viewed as the orthogonal projection of $f^\delta$ on the space spanned by the $\chi_\lambda$. The main difference between $f^{\delta}$ and $f^H$ is that the latter cannot vary at scales finer than the grid scale which is of order $N_g^{-1}$.
By choosing $N_g$ small enough, it is therefore possible to reduce the variance of $f^H$ to very low levels,
but the estimate then becomes more and more biased towards a piecewise continuous function, which is not smooth enough to be the true density. Histograms correspond to the nearest grid point (NGP) charge assignment scheme used in the early days of plasma physics computations \cite{Hockney1966}.
One of the most popular methods
to achieve higher level of smoothness is kernel density estimation (KDE) \cite{Parzen1962}.
Given $(\mathbf{X}_n)_{1\leq n \leq N_p}$, the kernel estimate of
$f$ is defined as
\begin{equation}\label{kernel_estimate}
f^K(\mathbf{x}) = \frac{1}{N_p}\sum_{n=1}^{N_p} K(\mathbf{x}-\mathbf{X}_n)\, ,
\end{equation}
where the smoothing kernel $K$ is a positive definite, normalized, $\int K =1$, function.
Equation~(\ref{kernel_estimate}) corresponds to the convolution of $K$ with the Dirac delta measure corresponding to each particle.
A typical example is the Gaussian kernel
\begin{equation}\label{gaussian_kernel}
K_h(\mathbf{x}) = \frac{1}{(\sqrt{2\pi}h)^d} e^{-\frac{\Vert \mathbf{x} \Vert^2}{2h^2}}
\end{equation}
where the so-called ``bandwidth'', or smoothing scale, $h$, is a free parameter.
The optimal smoothing scale depends on how the error is measured.
For example, in the one dimensional case, to minimize the mean $L^2$-error between the estimate and the true density, the smoothing volume $h^d$ should scale like ${N_p}^{-\frac{1}{5}}$, and the resulting error scales like $N_p^{-\frac{2}{5}}$ \cite{Silverman1986}.
As in the case of histograms, the choice of $h$ relies on a trade-off between variance and bias.
In the context of plasma physics simulations the kernel $K$ corresponds to the charge assignment function \cite{Hockney1988}.
A significant effort has been devoted in the choice of the function $K$ since it has a strong impact on computational efficiency and on the conservation of global quantities. Concerning $h$, it has been shown that it should not be much larger than the Debye length $\lambda_D$ of the plasma to obtain a realistic and stable simulation \cite{Birdsall1985}.
Given a certain amount of computational resources, the general tendency has thus been to reduce $h$ as far as possible in order to fit more Debye lengths inside the simulation domain,
which means that the effort has been concentrated on reducing the bias term in the error.
Since the force fields depend on $f$ through integral equations,
like the Poisson equation, that tend to reduce the high wavenumber noise,
we do not expect the disastrous scaling $h \propto {N_p}^{-\frac{1}{5}}$,
which would mean $N_p \propto \lambda_D^{5d}$ in $d$ dimensions, to hold.
Nevertheless, the problem remains that if we want to preserve high resolution features of $f$ or of the electromagnetic fields, we need
to reduce $h$, and therefore greatly increase the number of particles to prevent the simulation from drowning into noise.
Bandwidth selection has long been recognized as the central issue in kernel density estimation \cite{Chiu1991}.
We are not aware of a theoretical or numerical prediction of the optimal value of $h$ taking into account the noise term.
To bypass this difficulty, it is possible to use new statistical methods which do not force us to choose a global smoothing parameter.
Instead, they adapt locally to the behavior of the density $f$ based on the available data.
Wavelet based-density estimation, which we will introduce in the next two sections, is one of these methods.
\subsection{Bases of orthogonal wavelets}
Wavelets are a standard mathematical tool to analyze and compute non stationary signals. Here we recall basic concepts and definitions. Further details can be found in Ref.~\cite{MF92} and references therein. The construction takes place in the Hilbert space
$L^2(\mathbb{R})$
of square integrable functions.
An orthonormal family $(\psi_{j,i}(x))_{j\in\mathbb{N},i\in\mathbb{Z}}$ is called a wavelet family when its members are dilations and translations of a fixed function $\psi$ called the mother wavelet:
\begin{equation}\label{wavelet_1}
\psi_{j,i}(x) = 2^{j/2}\psi(2^j x-i)
\end{equation}
where $j$ indexes the scale of the wavelets and $i$ their positions, and $\psi$ satisfies $\int \psi = 0$.
In the following we shall always assume that $\psi$ has compact support of length $S$.
The coefficients $\langle f \mid \psi_{j,i} \rangle = \int f \psi_{j,i} $ of a function $f$ for this family are denoted by $(\tilde{f}_{j,i})$.
These coefficients describe the fluctuations of $f$ at scale $2^{-j}$ around position $\frac{i}{2^j}$.
Large values of $j$ correspond to fine scales, and small values to coarse scales.
Some members of the commonly used Daubechies 6 wavelet family
are shown in the left panel of Fig.~1.
It can be shown that the orthogonal complement in $L^2(\mathbb{R})$ of the linear space spanned by the wavelets
is itself orthogonally spanned by the translates of a function $\varphi$, called the scaling function.
Defining
\begin{equation}\label{phi_shrink}
\varphi_{L,i} = 2^{\frac{L}{2}} \varphi(2^L x - i)
\end{equation}
and the scaling coefficients $\bar{f}_{L,i} = \langle f \mid \varphi_{L,i} \rangle$,
one thus has the reconstruction formula:
\begin{equation}\label{wavelet_reconstruction}
f = \sum_{i=-\infty} ^{\infty} \bar{f}_{L,i} \varphi_{L,i} + \sum_{j=L}^{\infty}\sum_{i=-\infty}^{\infty} \tilde{f}_{j,i} \psi_{j,i}
\end{equation}
The first sum on the right hand side of Eq.~(\ref{wavelet_reconstruction}) is a smooth approximation of $f$ at the coarse scale, $2^{-L}$, and the second sum corresponds to the addition of details at successively finer scales.
If the wavelet $\psi$ has $M$ vanishing moments:
\begin{equation}\label{vanishing_moments}
\int x^m \psi(x) dx = 0
\end{equation}
for $0 \leq m < M$, and if $f$ is locally $m$ times continuously differentiable around some point $x_0$,
then a key property of the wavelet expansion is that the coefficients located near $x_0$ decay when $j\to\infty$ like $2^{-j(m+\frac{1}{2})}$ \cite{Jaffard1991}.
Hence, localized singularities or sharp features in $f$ affect only a finite number of wavelet coefficients within each scale.
Another important consequence of (\ref{vanishing_moments}) of special relevance to particle methods
is that for $0 \leq m < M$, the moments $\int x^m f(x) dx$ of the particle distribution function depend only on its scaling coefficients, and not on its wavelet coefficients.
If the scaling coefficients $\overline{f}_{J,i}$ at a certain scale $J$ are known,
all the wavelet coefficients at coarser scales ($j \leq J$) can be computed using the fast wavelet transform (FWT) algorithm \cite{SM00}.
We shall address the issue of computing the scaling coefficients themselves in section \ref{critical_discussion}.
The generalization to $d$ dimensions involves tensor products of wavelets and scaling functions at the same scale.
For example, given a wavelet basis on $\mathbb{R}$, a wavelet basis on $\mathbb{R}^2$ can be constructed in the following way:
\begin{eqnarray}
\psi^1_{j,i_1,i_2}(x_1,x_2) &=& 2^j\psi(2^j x_1-i_1) \varphi(2^j x_2-i_2) \\
\psi^2_{j,i_1,i_2}(x_1,x_2) &=& 2^j\varphi(2^j x_1-i_1) \psi(2^j x_2-i_2) \\
\psi^3_{j,i_1,i_2}(x_1,x_2) &=& 2^j\psi(2^j x_1-i_1) \psi(2^j x_2-i_2 ) \, ,
\end{eqnarray}
where we refer to the exponent $\mu = 1, 2, 3$ as the direction of the wavelets.
This name is easily understood by looking at different wavelets shown in Fig.~\ref{daubechies_wavelets_1D} (right).
The corresponding scaling functions are simply given by $2^j \varphi(2^j x_1 - i_1)\varphi(2^j x_2 - i_2)$.
Wavelets on $\mathbb{R}^d$ are constructed exactly in the same way, but this time using $2^d-1$ directions.
To lighten the notation we write the $d$-dimensional analog of Eq.~(\ref{wavelet_reconstruction}) as
\begin{eqnarray}\label{wavelet_reconstruction_2D}
f = \sum_{\lambda\in\Lambda_{\phi,L}} \overline{f}_\lambda \phi_\lambda + \sum_{\lambda\in\Lambda_{\psi,L}} \tilde{f}_\lambda \psi_\lambda
\end{eqnarray}
where $\lambda = (j,\mathbf{i},\mu)$ is a multi-index,
with the integer $j$ denoting the scale and the integer vector $\mathbf{i} = (i_1,i_2,\ldots)$ denoting the
position of the wavelet.
The wavelet multiresolution reconstruction formula in Eq.~(\ref{wavelet_reconstruction}) involves an infinite sum over the
position index $i$. One way of dealing with this sum is to determine a priori the non-zero coefficients in Eq.~(\ref{wavelet_reconstruction}),
and work only with these coefficients, but still retaining the full wavelet basis on $\mathbb{R}^d$ as presented above.
Another alternative, which we have chosen because it is easier to implement, is to periodize the wavelet transform on a bounded domain \cite{SM00}.
Assuming that the coordinates have been rescaled so that all the particles lie in $[0,1]^d$,
we replace the wavelets and scaling functions by their periodized counterparts:
\begin{eqnarray}\label{periodized_wavelets}
\psi_{j,i}(x) & \to & \sum_{l=-\infty}^{\infty} \psi_{j,i}(x+l) \\
\varphi_{j,i}(x) & \to & \sum_{l=-\infty}^{\infty} \varphi_{j,i}(x+l) \, .
\end{eqnarray}
Throughout this paper we will consider only periodic wavelets.
For the sake of completeness we mention a third alternative which is technically more complicated.
It consists in constructing a wavelet basis on a bounded interval \cite{Cohen1993}.
The advantage of this approach is that it does not introduce artificially large wavelet coefficients at the boundaries for functions $f$ that
are not periodic.
\subsection{Wavelet based density estimation}
The multiscale nature of wavelets allows them to adapt locally to the smoothness of the analyzed function \cite{SM00}.
This fundamental property has triggered their use in a variety of problems.
One of their most fruitful applications has been the denoising of intermittent signals \cite{DJ94}.
The practical success of wavelet thresholding to reduce noise relies on the observation that the expansion of signals in a wavelet basis is typically sparse.
Sparsity means that the interesting features of the signal are well summarized by a small fraction of large wavelet coefficients.
On the contrary, the variance of the noise is spread over all the coefficients appearing in Eq.~(\ref{wavelet_reconstruction_2D}).
Although the few large coefficients are of course also affected by noise, curing the noise in the small coefficients is already a very good improvement.
The original setting of this technique, hereafter referred to as global wavelet shrinkage, requires the noise to be additive, stationary, Gaussian and white.
It found a first application in plasma physics in Ref.~\cite{Farge2006}, where coherent bursts were extracted out of plasma density signals.
Since Ref.~\cite{DJ94}, wavelet denoising has been extended to a number of more general situations, like non-Gaussian or correlated additive noise, or to denoise the spectra of locally stationary time series \cite{Sachs1996}.
In particular, the same ideas were developed in Ref.~\cite{Vannucci1995,Donoho1996} to propose a wavelet-based density estimation (WBDE) method based on independent observations.
At this point we would like to stress that WBDE assumes nothing about the Gaussianity of the noise or whether or not it is stationary.
In fact, under the independence hypothesis -- which is admittedly quite strong -- the statistical properties of the noise are entirely determined by standard probability theory.
We refer to Ref.~\cite{Vidakovic1999} for a review on the applications of wavelets in statistics.
In Ref.~\cite{Gassama2007}, global wavelet shrinkage was applied directly to the charge density of a 2D PIC code, in a case were the statistical fluctuations were quasi Gaussian and stationary.
In particular, an iterative algorithm \cite{AAMF04}, which crucially relies on the stationnarity hypothesis, was used to determine the level of fluctuations. However,in the next section we will show an example where the noise is clearly non-stationary, and this procedure fails.
Let us now describe the WBDE method as we have generalized it to several dimensions.
The first step is to expand the sample particle distribution function, $f^\delta$, in Eq.~(\ref{dirac_estimate}) in a wavelet basis
according to Eq.~(\ref{wavelet_reconstruction_2D}) with the wavelet coefficients
\begin{eqnarray}\label{empirical_wavelet_coefficients}
\overline{f}_{\lambda} & = & \langle f^\delta \mid \varphi_\lambda \rangle = \frac{1}{N_p} \sum_{n=1}^{N_p} \varphi_\lambda(X_n) \\
\tilde{f}_{\lambda} & = & \langle f^\delta \mid \psi_\lambda \rangle = \frac{1}{N_p} \sum_{n=1}^{N_p} \psi_\lambda(X_n) \, .
\end{eqnarray}
Since this reconstruction is exact, keeping all the wavelet coefficients does not improve the smoothness of $f^\delta$.
The simple and yet efficient remedy consists in keeping only a subset of the wavelet coefficients in Eq.~(\ref{wavelet_reconstruction_2D}).
A straightforward prescription would be to discard all the wavelet coefficients at scales finer than a cut-off scale $L$.
This approach corresponds to a generalization of the histogram method in Eq.~(\ref{histogram_estimate}) with $N_g = 2^L$. Because the characteristic functions $\chi_\lambda$ of the cells in a dyadic grid are the scaling functions associated with the Haar wavelet family, Eqs.~(\ref{wavelet_reconstruction_2D}) and (\ref{histogram_estimate}) are in fact equivalent for this wavelet family. Accordingly, like in the histogram case, we would have to choose $L$ quite low to obtain a stable estimate, at the risk of losing some sharp features of $f$.
Better results can be obtained by keeping some wavelet coefficients down to a much finer scale $J > L$.
However, to prevent that statistical fluctuations contaminate the estimate, only those coefficients whose modulus are above a certain threshold should be kept.
We are thus naturally led to a nonlinear thresholding procedure.
In the one dimensional case, values of $J$, $L$, and of the threshold within each scale that yield theoretically optimal results have been given in Ref.~\cite{Donoho1996}.
This reference discusses the precise smoothness requirements on $f$, which can accommodate well localized singularities, like shocks and filamentary structures known to arise in collisionless plasma simulations.
There remains the question of how to compute the $\tilde{f}_{j,i}$ based on the positions of the particles.
Although more accurate methods based on (\ref{empirical_wavelet_coefficients}) may be developed in the future, our present approximation relies on the computation of a histogram, which creates errors of order $N_g^{-1}$.
The complete procedure is described in the following {\bf
Wavelet-based density estimation} algorithm:
\begin{enumerate}
\item \label{histogram_approx} construct a histogram $f^H$ of the particle data with $N_g = 2^{J_g}$ cells in each direction,
\item \label{scaling_function_approx} approximate the scaling coefficients at the finest scale $J_g$ by :
\begin{equation}\label{scaling_function_approximation}
\overline{f}_{J_g,\mathbf{i}} \simeq 2^{-{J_g}/{2}} f^H(2^{-J_g} \mathbf{i})
\end{equation}
\item compute all the needed wavelet coefficients using the FWT algorithm,
\item keep all the coefficients for scales coarser than $L$, defined by $2^{dL}\sim N_{p}^{\frac{1}{1+2r_{0}}}$ where $r_{0}$ is the order of regularity of the wavelet (1 in our case),
\item discard all the coefficients for scales strictly finer than $J$ defined by $2^{dJ}\sim\frac{N_{p}}{\log_{2}N_{p}}$,
\item \label{thresholding_function} for scales $j$ in between $L$ and $J$, keep only the wavelet coefficients $\tilde{f}_{\lambda}$ such that $\vert\tilde{f}_{\lambda}\vert\geq T_j = C\sqrt{\frac{j}{N_{p}}}$ where $C$ is a constant that must in principle depend on the smoothness of $f$ and on the wavelet family \cite{Donoho1996}.
\end{enumerate}
In the following, except otherwise indicated, $C=\frac{1}{2}$. For the wavelet bases we
used orthonormal Daubechies wavelets with 6 vanishing moments and thus support of size $S=12$ \cite{Daubechies1992}.
In our case, $r_0 = 1$, which means that the wavelets have a first derivative but no second derivative, and the size of the wavelets at scale $L$ for $d=1$ is roughly $N_p^{-\frac{1}{3}}$.
Since $N_p \gg 1$, it follows from the definition at stage 5 of the algorithm that the size of the wavelets at scale $J$ is orders of magnitude smaller than that.
Using the adaptive properties of wavelets, we are thus able to detect small scale structures of $f$ without compromising the stability of the estimate.
Note that the error at stage \ref{scaling_function_approx} could be reduced by using Coiflets \cite{Daubechies1993} instead of Daubechies wavelets,
but the gain would be negligible compared to the error made at stage \ref{histogram_approx}.
We will denote the WBDE estimate of $f$ as $f^W$.
In the one-dimensional case,
\begin{equation}\label{explicit_wavelet_estimate}
{f^{W}}=\sum_{i=1}^{2^{L}}\overline{f}_{L,i}\varphi_{L,i}+\sum_{j=L}^{J}\sum_{i=1}^{2^{j}}\tilde{f}_{j,i} \rho_{j}(\tilde{f}_{j,i})\psi_{j,i}
\end{equation}
where $\rho_{j}$ is the thresholding function as defined by stage \ref{thresholding_function} of the algorithm : $\rho_{j}(y) = 0$ if $\vert y \vert \leq T_j$ and $\rho_{j}(y) = 1$ otherwise.
Finally, let us propose two methods for applying WBDE to postprocess $\delta f$ simulations.
Recall that the Lagrangian equations involved in the $\delta f$ schemes are identical to their full $f$ counterparts.
The only difficulty introduced by the $\delta f$ method lies in the evaluation of phase space integrals of the form $\delta I = \int A \cdot (f-f_0)$,
where $A$ is a function on phase space and $f_0$ is a known reference distribution function.
In these integrals, $f-f_0$ should be replaced by $\delta f$, which is in turn written as a product $wf$, where $w$ is a ``weighting'' function.
Numerically, $w$ is known via its values at particles positions, $w(X_n)$, and the usual expression for $\delta I$ is thus $\delta I = \sum_{n=1}^{N_p} A(X_n) w(X_n)$.
We cannot apply WBDE directly to $\delta f$, since this function is not a density function
An elegant approach would be to first apply WBDE to the unweighted distribution $f^\delta$ to determine the set of statistically significant wavelet coefficients, and to include the weights only in the final reconstruction (\ref{explicit_wavelet_estimate}) of $f^W$.
A simpler approach, which we will illustrate in section \ref{delta_5d_example}, consists in renormalizing $\delta f$, so that
\int\vert\delta f\vert = 1$,
and treat it like a density.
\subsection{Further issues related to practical implementation}
\label{critical_discussion}
In this section we discuss how the WBDE method handles two issues of direct relevance to plasma simulations:
conservation of moments and computational efficiency.
As mentioned before, due to the vanishing moments of the wavelets in Eq.~(\ref{vanishing_moments}),
the moments up to order $M$ of the particle distribution distribution are solely determined by its scaling function coefficients.
As a consequence, we expect the thresholding procedure to conserve these moments, in the sense that
\begin{equation}\label{moments_def}
\mathcal{M}_{m,k}^W = \int x_k^m f^W(\mathbf{x})\mathrm{d}\mathbf{x} \simeq \int x_k^m f^\delta(\mathbf{x})\mathrm{d}\mathbf{x} = \mathcal{M}_{m,k}^\delta
\end{equation}
for $0 \leq m \leq M-1$ and for all $i\in \{1,\ldots,d\}$.
This conservation holds up to round-off error if the wavelet coefficients can be computed exactly.
Due to the type of wavelets that we have used, we were not able to achieve this in the results presented here.
There remains a small error related to stages 1 and 2 of the algorithm, namely the construction of $f^H$ and the approximation of the scaling function coefficients by Eq.~(\ref{scaling_function_approximation}).
They are both of order $N_g^{-1}$.
We will present numerical examples of the moments of $f^W$ in the next section.
Conservation of moments is closely related to a peculiarity of the denoised distribution function resulting from the WBDE algorithm:
it is not necessarily everywhere positive.
Indeed, wavelets are oscillating functions by definition, and removing wavelet coefficients therefore cannot preserve positivity in general.
Further studies are needed to assess if this creates numerical instabilities when $f^W$ is used in the computation of self-consistent fields.
The same issue was discussed in Ref.~\cite{Denavit1972} where a kernel with two vanishing moments was used to linearly smooth the distribution function.
The fact that this kernel is not everywhere positive was not considered harmful in this reference.
We acknowledge that it may render the resampling of new particles from $f^W$, if it is needed in the future, more difficult.
There are ways of forcing $f^W$ to be positive, for example by applying the method to $\sqrt{f}$ and then taking the square of the resulting estimate, but this implies the loss of the moment conservation, and we have not pursued in this direction.
\begin{figure}
\includegraphics[width=0.5\columnwidth,height=0.4\columnwidth]{wavelets3}
\hspace{0.05cm}
\includegraphics[width=0.44\columnwidth]{wavelets4}
\caption{
\label{daubechies_wavelets_1D}Daubechies 6 wavelet family.
Left, bold red: scaling function $\varphi$ at scale $j = 5$.
Left, bold blue: wavelet $\psi$ at scale $j = 5$.
Left, thin black, from left to right: wavelets at scales 6, 7, 8 and 9.
Right : (a) 2D scaling function $\varphi(x_1)\varphi(x_2)$.
(b) first 2D wavelet $\psi(x_1)\varphi(x_2)$.
(c) second 2D wavelet $\varphi(x_1)\psi(x_2)$.
(d) third 2D wavelet $\psi(x_1)\psi(x_2)$.
}
\end{figure}
The number of arithmetic operations to perform a fast wavelet transform from scale $2^{-J}$ to scale $2^{-L}$ with the FWT in $d$ dimensions is $2S 2^{d(J-L)}$, where $S$ is the length of the wavelet filter (12 for the Daubechies filter that we are using).
The definitions of $J$ and $L$ imply that $2^{d(J-L)}$ scales like $\frac{N_p^{\frac{2}{3}}}{\log{N_p}}$.
The cost of the binning stage of order $N_{p}$, so that the total cost for computing $f^W$ is $O(N_{p})$, not larger than the cost of one time step during the simulation that produced the data.
The amount of memory needed to store the wavelet coefficients during the denoising procedure is proportional to $N_g^d$, which should at least scale like $2^{dJ}$, and therefore also like $N_p$.
If one wishes to use a finer grid to ensure high accuracy conservation of moments, the storage requirements grow like $N_g^d$.
Thanks to optimized in-place algorithms, the amount of additional memory needed during the computation does not exceed $3S$.
Another consequence of using the FWT algorithm is that $N_g$ must be an integer multiple of $2^{J-L}$.
For comparison purposes, let us recall that most algorithms to compute the POD have a complexity proportional to $N_g^3$ when $d=2$
To conclude this subsection, Fig.~\ref{example_1D} presents an example of the reconstruction of a 1D discontinuous density that illustrates the difference between the KDE and WBDE methods.
The probability density function is uniform on the interval $\left[\frac{1}{3},\frac{2}{3}\right]$ and the estimates were computed on $\left[0,1\right]$ to include the discontinuities.
The sample size was $2^{14}$, and the binning used $N_g = 2^{16}$ cells to compute the scaling function coefficients.
For this 1D case the value $C=2$ was used to determine the thresholds (step \ref{thresholding_function} of the algorithm).
The KDE estimate is computed using a Gaussian kernel with smoothing scale $h=0.0138$ \cite{KDE2003}.
The relative mean squared errors associated with the KDE and WBDE estimates are respectively $19.6\times 10^{-3}$ and $6.97\times 10^{-3}$.
The error in the KDE estimate comes mostly from the smoothing of the discontinuities.
The better performance of WBDE stems from the much sharper representation of these discontinuities.
It is also observed that the WBDE estimate is not everywhere positive.
The approximate conservation of moments is demonstrated on Table \ref{example_1D_moments}.
Note that the error on all these moments for $f^W$ could be made arbitrary low by increasing $N_g$.
The overshoots could also be mitigated by using nearly shift invariant wavelets \cite{NK01}.
\begin{figure}
\includegraphics[width=0.47\columnwidth]{example_1D_kernel}
\hspace{0.05cm}
\includegraphics[width=0.47\columnwidth]{example_1D_wavelets}
\caption{
\label{example_1D}
Estimation of the density of a sample of size $2^{14}$ drawn uniformly in $[1/3,2/3]$,
using Gaussian kernels (left) or wavelets (right).
The discontinuous analytical density is plotted with a dashed line in the two cases.
}
\end{figure}
\subsection{Proper Orthogonal Decomposition Method}
For completeness, in this subsection we present a brief review of the POD density reconstruction method. For the sake of comparison with the WBDE method, we limit attention to the time independent case. Further details, including the reconstruction of time dependent densities using POD methods can be found in Ref.~\cite{delCastillo2008}.
The first step in the POD method is to construct the histogram $f^H$ from the particle data.
This density is represented by an $N_x \times N_y$ matrix $\hat{f}_{ij}$ containing the fraction of particles
with coordinates $(x,y)$ such that $X_i \leq x< X_{i+1}$ and $Y_i \leq y< Y_{i+1}$.
In two dimensions, the POD method is based on the singular value decomposition of the histogram.
According to the SVD theorem \cite{golub_van_loav_1996}, the matrix $\hat{f}$ can always be factorized as
$\hat{f}= U W V^t$ where $U$ and $V$ are $N_x \times N_x$ and $N_y \times N_y$ orthogonal matrices,
$U U^t =V V^t = I$, and $W$ is a diagonal matrix,
$W = {\rm diag} \left( w_1, w_2, \ldots w_N \right )$, such that $w_1
\geq w_2\geq \ldots \geq w_N \geq 0$.
with $N= {\rm min} (N_x,N_y)$.
In vector form, the decomposition can be expressed as
\begin{equation}
\label{svd_vector}
\hat{f}_{ij}= \sum_{k=1}^N\, w_k \, u^{(k)}_i\, v^{(k)}_j \, ,
\end{equation}
where the $N_x$-dimensional vectors, $u_i^{(k)}$, and the $N_y$-dimensional vectors, $v_j^{(k)}$, are the orthonormal
POD modes and correspond to the columns of the matrices $U$ and $V$ respectively. Given the decomposition in Eq.~(\ref{svd_vector}), we define the rank-$r$ approximation of
$\hat{f}$ as
\begin{equation}
\label{lr_svd}
\hat{f}^{(r)}_{ij}= \sum_{k=1}^r\, w_k \, u^{(k)}_i\, v^{(k)}_j \, ,
\end{equation}
where $1 \leq r < N$,
and define the corresponding rank-$r$ reconstruction error as
\begin{equation}
\label{nre}
e(r) = || \hat{f}-\hat{f}^{(r)} ||^2 = \sum_{i=r+1}^N w_i^2 \, ,
\end{equation}
where $|| A|| = \sqrt{\sum_{i j} A_{ij}^2}$ is the Frobenius norm.
Since $\hat{f}^{(r=N)}=\hat{f}$, we define $e(N)=0$.
The key property of the POD is that the approximation in Eq.~(\ref{lr_svd}) is
optimal in the sense that
\begin{equation}
e(r) = {\rm min} \left \{
||\hat{f}-g||^2 \, \left | {\rm rank} (g) = r \right. \right \} \, .
\end{equation}
That is, of all the possible rank-$r$ Cartesian product approximations of $\hat{f}$, $\hat{f}^{(r)}$ is the closest to $\hat{f}$
in the Frobenius norm.
The SVD spectrum, $\{ w_k\}$, of noise free coherent signals decays very rapidly after a few modes, but the spectrum of noise dominated signals is relatively flat and decays very slowly.
When a coherent signal is contaminated with low level noise, the SVD spectrum
exhibits an initial rapid decay followed by a weakly decaying spectrum known as the noisy plateau.
In the POD method the denoised density is defined as the truncation $f^P=\hat{f}^{(r_c)}$, where
$r_c$ corresponds to the rank where the noisy plateau starts. In general it is difficult to provide a precise a priori estimate of $r_c$, and this is one of the potential limitations of the POD method. One possible quantitative criterion used in Ref.~\cite{delCastillo2008} is to consider the relative decay of the spectrum, $\Delta(k)=(w_{k+1}-w_k)/(w_{2}-w_1)$, for $k>1$, and define $r_c$ by the condition $\Delta(r_c)\leq \Delta_c$ where $\Delta_c$ is a predetermined threshold.
\begin{table}
\begin{tabular}{l*{6}{c}r}
& $m=0$ & $m=1$ & $m=2$ & $m=4$ \\
\hline
$f^K$ & $1.81\cdot 10^{-5}$ & $1.70\cdot 10^{-5}$ & $7.52\cdot 10^{-4}$ & $3.90\cdot 10^{-3}$ \\
$f^W$ & $1.08\cdot 10^{-11}$ & $1.52\cdot 10^{-5}$ & $2.93\cdot 10^{-5}$ & $5.52\cdot 10^{-5}$ \\
\end{tabular}
\caption{
\label{example_1D_moments}
Relative absolute difference between the moments of $f^\delta$ and those of $f^K$ and $f^W$, for the distribution function corresponding to Fig.~\ref{example_1D}.
}
\end{table}
\section{Applications}
In this section, we apply the WBDE method to reconstruct and denoise the particle distribution function starting from discrete particle data. The data corresponds to three different groups of simulations: collisional thermalization with
a background plasma, guiding center transport in toroidal geometry, and Vlasov-Poisson electrostatic instabilities. The first two groups of simulations were analyzed using POD methods in Ref.~\cite{delCastillo2008}. One of the goals of this section is to compare the POD method with the WBDE method in these cases and in a new Vlasov-Poisson data set. This data set allows the testing of the reconstruction algorithms in a collisionless system that incorporates the self-consistent evaluation of the forces acting on the particles, as opposed to the collisional, test particle problems analyzed before.
When comparing the two methods it is important to keep in mind that POD
has one free parameter, namely
the number $r$ of singular vectors that are retained to reconstruct the denoised distribution function.
In the cases studied here we used a best guess for $r$ based on the properties of the reconstruction. In Ref.~\cite{delCastillo2008} the POD method was developed and applied to time independent and time dependent data sets. However, in the comparison with the WBDE method, we limit attention to $2$-dimensional time independent data sets.
The accuracy of the reconstruction of the density at a fixed time $t$ will be monitored using the absolute mean square error
\begin{equation}\label{error_e}
e = \sum_{i,j} \vert f^{est}(x_i,y_j;t) - f^{ref}(x_i,y_j;t) \vert^2 \, ,
\end{equation}
where $(x_i,y_j)$ are the coordinates of the nodes of a prescribed $N_g \times N_g$ grid in the space, and $f^{est}$ denotes the estimated density computed from a sample with $N_p$ particles.
For the WBDE method
$f^{est}=f^W$, and for the POD method $f^{est}=f^P$.
In principle, the reference density, $f^{ref}$, in Eq.~(\ref{error_e}) should be the density function obtained from the exact solution of the corresponding continuum model, e.g. the Fokker-Planck or the Vlasov-Poisson system. However, when no explicit solution is available, we will set $f^{ref}=f^{H}$ where $f^{H}$ is the histogram corresponding to a simulation with a maximum number of particles available which in the cases reported here correspond to $N_p=10^6$.
We will also use the normalized error
\begin{equation}
\label{error_e0}
e_0=\frac{e}{\sum_{i,j} \vert f^{ref}(x_i,y_j;t) \vert^2 \,} \, .
\end{equation}
\subsection{Collisional thermalization with
a background plasma}
This first example models the relaxation of a non equilibrium plasma by collisional damping and pitch angle scattering on a thermal background. The plasma is spatially homogeneous and is represented by an ensemble of $N_p$ particles in a three-dimensional velocity space. Assuming a strong magnetic field, the dynamics can be reduced to two degrees of freedom: the magnitude of the particle velocity, $v$, and the particle pitch, $\lambda=\cos \theta$, where $\theta$ is the angle between the particle velocity and the magnetic field. In the continuum limit the particle distribution function is governed by the Fokker-Planck equation which in the particle description corresponds to the stochastic differential equations \begin{equation}
\label{mc_1}
d \lambda = - \lambda \nu_D\, dt - \sqrt{\nu_D \left ( 1 - \lambda^2
\right) } \, d \eta_\lambda \, ,
\end{equation}
\begin{equation}
\label{mc_2}
d v = -\left[ \alpha \, \nu_s \, v - \frac{1}{2 v^2}\, \frac{d}{dv}\left( \nu_{||} v^4 \right) \right] \, dt + \sqrt{v^2\, \nu_{||}} \,
d\eta_{v}\, ,
\end{equation}
describing the evolution of $v \in (0, \infty)$ and $\lambda \in [-1,1]$ for each particle,
where $d \eta_\lambda$ and $d \eta_v$ are independent Wiener
stochastic processes and $\nu_D$, $\nu_s$ and $\nu_{\parallel}$ are functions of $v$. For further details on the model see Ref.~\cite{delCastillo2008} and references therein.
We considered simulations with $N_p=10^3$, $10^4$, $10^5$ and $10^6$ particles. The initial conditions of the ensemble of particles were obtained by sampling a distribution of the form
\begin{equation}
\label{f_ic}
f(v,\lambda ,t=0)= C v^2 \exp \left \{ -\frac{1}{2}\left[ \frac{(\lambda-\lambda_0)^2}{\sigma_\lambda^2}
+ \frac{(v-v_0)^2}{\sigma_v^2}\right] \right \} \, ,
\end{equation}
where a $v^2$ factor has been included in the definition of the initial condition so that the volume element is simply $\mathrm{d}v \mathrm{d}\mu$, $C$ is a normalization constant, $\lambda_0=0.25$, $v_0=5$, $\sigma_\lambda=0.25$ and $\sigma_v=0.75$.
This relatively simple problem is particularly well suited for the WBDE method because
the simulated particles do not interact and therefore
statistical correlations can not build-up between them.
Before applying the WBDE method, we analyze the sparsity of the wavelet expansion of $f^\delta$, and compare the number of modes kept and the reconstruction error for different thresholding rules.
The plot in the upper left panel of Fig.~\ref{compression_curve_time_dependent} shows the absolute values of the wavelet coefficients in decreasing order at different fixed times.
The wavelet coefficients exhibit a clear rapid decay beyond the few significant modes corresponding to the gross shape of the Maxwellian distribution. A similar trend is observed in the coefficients of the POD expansion shown in
the upper right panel of Fig.~\ref{compression_curve_time_dependent}.
However, in the wavelet case the exponential decay starts after more than $100$ modes, whereas in the POD case the exponential decay starts after only one mode.
\begin{figure
\includegraphics[width=0.49\columnwidth]{compression}
\includegraphics[clip,width=0.49\columnwidth]{compression_svd}
\includegraphics[width=0.49\columnwidth]{error}
\includegraphics[width=0.49\columnwidth]{error_svd}
\caption{
\label{compression_curve_time_dependent}
Wavelet and POD analyses of collisional relaxation particle data at different fixed times, with $N_p = 10^5$.
Top left: absolute values of the wavelet coefficients sorted by decreasing order (full lines), and thresholds given by the Waveshrink algorithm (dashed lines).
Top right: singular values of the histogram used to construct $f^P$.
Bottom left:
error estimate $\frac{e^{1/2}}{N_g^2} $ with respect to the run for $N_p = 10^6$ as a function of the number of retained wavelet coefficients (full lines), error obtained when using the Waveshrink threshold (dashed lines),
and error obtained using the WBDE method (dash-dotted lines).
Bottom right:
error estimate $\frac{e^{1/2}}{N_g^2}$ for $f^P$ as a function of the number $l$
of retained singular values.
}
\end{figure}
The two panels at the bottom of Fig.~\ref{compression_curve_time_dependent} show the square root of the reconstruction error normalized by $N_g$, $\sqrt{e}/N_g^2$, in the WBDE and POD methods. Because in this case we do not have access to the exact solution of the corresponding Fokker-Planck equation at the prescribed time, we used $f^H$ computed using $N_p=10^6$ particles as the reference density $f^{ref}$ in Eq.~(\ref{error_e}).
The error observed when applying a global threshold to the wavelet coefficients (bottom left panel in Fig.~\ref{compression_curve_time_dependent}) is minimal when around $100$ modes are kept whereas in the POD case
(bottom right panel in Fig.~\ref{compression_curve_time_dependent}) the minimal error is reached with about two or three modes.
Fig.~\ref{compression_curve_time_dependent} also shows the wavelet threshold obtained by applying the iterative algorithm
based on the stationary Gaussian white noise hypothesis \cite{AAMF04,Farge2006}.
The error corresponding to this threshold is larger than the optimal error because the noise in this problem is very non-stationary due to the lack of statistical fluctuations in the regions were particles are absent.
In contrast, the error corresponding to the WBDE procedure (dash-dotted line) is typically smaller than the optimal error obtained by global thresholding.This is not a contradiction, because the WBDE procedure is not a global threshold, but a level dependent threshold.
\begin{figure
\includegraphics[width=1.0\columnwidth]{paper}
\caption{\label{time_dependent_contour_plots}
Contour-plots of estimates of $f$ for the collisional relaxation particle data.
First row: Histogram method estimated using $N_p=10^{5}$ particles.
Second row: Histogram method estimated using $N_p=10^{6}$ particles.
Third row: POD method estimated using $N_p=10^5$ particles.
Fourth row: WBDE method estimated using $N_p=10^5$ particles.
The three columns correspond to $t=28$, $t=44$ and $t=72$ respectively.
The plots show twenty isolines, equally spaced in the interval $[0,0.4]$.
}
\end{figure}
Figure~\ref{time_dependent_contour_plots} compares at different times the densities estimated with the WBDE and the POD (retaining only three modes) methods using $N_p=10^5$ particles with the histograms computed using $N_p=10^5$ and $10^6$ particles. The key future to observe is that
the level of smoothness of $f^W$ and $f^P$ corresponding to $N_p=10^5$ is similar, if not greater, than the level of smoothness in $f^H$ computed using ten times more particles, i.e. $N_p=10^6$ particles.
Table~\ref{time_dependent_error} summarizes the normalized reconstruction errors for $N_p=10^5$ according Eq.~(\ref{error_e}) using $f^H$ with $N_p=10^6$ as $f^{ref}$.
The WBDE and POD denoising methods offer a significant improvement, approximately by a factor $2$, over the
raw histogram method.
\begin{table}
\vspace{3 cm}
\begin{tabular}{l*{6}{c}r}
& $t = 28$ & $t = 44$ & $t = 72$ \\
\hline
$f^H$ & $0.14$ & $0.17$ & $0.12$ \\
$f^P$ & $0.068$ & $0.090$ & $0.094$ \\
$f^W$ & $0.064$ & $0.094$ & $0.088$ \\
\end{tabular}
\caption{
\label{time_dependent_error}
Normalized root mean squared error $e_0$ (\ref{error_e0}) for the histogram, POD and WBDE estimates
of the particle distribution function for $N_p=10^5$ at three different times of
the Maxwellian relaxation problem.
}
\end{table}
A more detailed comparison of the estimates can be achieved by focusing on the Maxwellian final equilibrium state
\begin{equation}
\label{f_max}
f_M(v)= \frac{2}{\sqrt{\pi} } v^2 e^{-v^2} \, ,
\end{equation}
where, as in Eq.~(\ref{f_ic}), the $v^2$ metric factor has been included in the definition of the distribution.
For this calculations we considered sets of particles sampled from Eq.~(\ref{f_max}) in the compact domain
$[-1,1]\times[0,4]$. Since $f_M$ is an exact equilibrium solution of the Fokker-Plack equation, the ensemble of particles will be in statistical equilibrium but it will exhibit fluctuations due to the finite number of particles.
Figure~\ref{wavelets_e} shows the dependence of the square root of the reconstruction error, $e$ (normalized by $N_g^2$) on
the number of particles $N_p$ and the grid resolution $N_g$ for the WBDE and POD methods. The main advantage of this example is that the exact density $f^M$ can be used as the reference density $f^{ref}$ in the evaluation of the error.
\begin{figure
\includegraphics[width=0.8\columnwidth]{err}
\caption{\label{wavelets_e}
Reconstruction error, $\frac{e^{1/2}}{N_g^2} $, as a function of $N_{p}$ for the collisional relaxation particle data
corresponding to the Maxwellian equilibrium state.
Bold solid lines correspond to the WBDE method, bold dashed lines correspond to the POD method, and thin dashed lines
correspond to the histogram method.}
\end{figure}
\subsection{Collisional guiding center transport in toroidal geometry}
\label{delta_5d_example}
The previous example focused on collisional dynamics. However, in addition to collisions, plasma transport involves external and self-consistent electromagnetic fields and it is of interest to test the particle density reconstruction algorithms in these more complicated settings. As a first step on this challenging problem we consider a plasma subject to collisions and an externally applied fixed magnetic field in toroidal geometry.
The choice of the field geometry and structure was motivated by problems of interest to magnetically confined fusion plasmas. The data was presented and analyzed using POD method in
Ref.~\cite{delCastillo2008}.
The phase space of the simulation is five dimensional. However, as in Ref.~\cite{delCastillo2008}, we limit attention to the denoising of the particles distribution function along two coordinates corresponding to the poloidal angle $\theta\in[0,2\pi]$ and the cosine of the pitch angle $\mu\in[-1,1]$.
The remaining three coordinates have been averaged out for the purpose of this study.
The $\theta$ coordinate is periodic, but the pitch coordinate $\mu$ is not.
An important issue to consider is that the data was generated using a $\delta f$ code (DELTA5D). Based on an expansion on $\rho/L \ll1$ (where $\rho$ is the characteristic Larmor radius and $L$ a typical equilibrium length scale) the distribution function is decomposed into a Maxwellian part $f_M$ and a first-order perturbation $\delta f$ represented as a collection of particles (markers)
\begin{equation}
\delta f({\bf x}) = \sum_n W_n \delta({\bf x}-{\bf X}_n) \, ,
\end{equation}
like in Eq.~(\ref{dirac_estimate}) except that each marker is assigned a time dependent weight $W_n$ whose time evolution depends on the Maxwellian background \cite{parker}.
The direct use of $\delta f({\bf x})$ is problematic in the WBDE method because $\delta f$ is not a probability density. To circumvent this problem the WBDE method was applied after normalizing the $\delta_f$ distribution so that
$\int \vert\delta f\vert^H =1$, on a $128\times128$ grid.
Figure~\ref{fig:d5d_3d_comp} shows contour plots of the histogram $f^H$ corresponding to $N_p=32 \times 10^3$, $64 \times 10^3$, and $1024 \times 10^3$ along with the WBDE and POD reconstructed densities. The POD reconstructions were done using $r=3$ modes, as in Ref.~\cite{delCastillo2008}.
It is observed that comparatively high levels of smoothness can be achieved with considerably less particles by using either the WBDE or POD reconstruction methods.
The WBDE method provides better results for the $\delta f \sim 0$ contours. This is because
POD modes are tensor product functions, that have difficulties in approximating the triangular shape of these contour lines.
Note that the boundary artifacts due to periodization of the Daubechies wavelets do not seem to be very critical. The large wavelet coefficients associated with the discontinuity between the values of $\delta f$ at $\mu=\pm1$ are not thresholded,
so that the discontinuity is preserved in the denoised function.
Figure~\ref{fig:d5d_RMS} compares the reconstruction errors in the WBDE, POD, and histogram methods as functions of the number of particles. To evaluate the error we used $f^H$ computed using $N_p=1024 \times 10^3$ as the reference density $f^{ref}$.
As in the collisional transport problem, the error is reduced roughly by a factor $2$ for both methods compared to the raw histogram. Note that the scaling with $N_p$ is slightly better for WBDE than for POD.
\begin{figure
\includegraphics[width=1.0\columnwidth]{paper_delta5}
\caption
{
\label{fig:d5d_3d_comp}
Contour plots of estimates of $f$ for the collisional guiding center transport particle data:
Histogram method (first row), POD method (second row), and WBDE method (third row).
The left, center and right
columns correspond to $N_p = 32\cdot 10^3$ (left), $N_p = 128\cdot 10^3$ (middle) and $N_p = 1024\cdot 10^3$ (right)
respectively.
The plots show seventeen isolines equally spaced within the interval $[-0.5,0.5]$.
}
\end{figure}
\begin{figure
{\includegraphics[width=0.8\columnwidth]
{error_delta5d}
}
\caption
{
\label{fig:d5d_RMS}
Error estimate, $\frac{e^{1/2}}{N_g^2} $, for collisional guiding center transport particle data
according to the histogram, the POD, and the wavelet methods.
}
\end{figure}
\subsection{Collisionless electrostatic instabilities}\label{PIC_example}
In this section we apply the WBDE and POD methods to reconstruct the single particle distribution function from discrete particle data obtained from PIC simulations of a Vlasov-Poisson plasma.
We consider a one-dimensional, electrostatic, collisionless electron plasma with an ion neutralizing background in a finite size domain with periodic boundary conditions.
In the continuum limit the dynamics of the distribution function is governed by the system of equations
\begin{eqnarray}
\partial_t f + v \partial_x f + \partial_x \phi \partial_v f=0
\\
\partial_x^2 \phi = \zeta \int f(x,v,t) dv -1 \, ,
\end{eqnarray}
where the variables have been non-dimensionalized using the Debye length as length scale and the plasma frequency as time scale, and $L$ is the length of the system normalized with the Debye length.
Following the standard PIC methodology \cite{Birdsall1985}, we solve the Poisson equation on a grid and solve the particle equations using a leap-frog method.
The reconstruction of the charge density uses a triangular shape function.
We consider two initial conditions:
the first one leads to a bump on tail instability, and the second one to a two streams instability.
\subsubsection {Bump on tail instability}\label{bump_tail}
For the bump on tail instability we initialized ensembles of particles by sampling the distribution function
\begin{equation}
\label{bump_on_tail_init}
f_0(x,v) = \frac{2}{3 \pi \zeta}\frac{1-2 q v+ 2 v^2}{\left( 1 + v^2 \right)^2}\, .
\end{equation}
using a pseudo-random number generator.
This equilibrium is stable for $q \leq 1$ and unstable for $q>1$.
The dispersion relation and linear stability analysis for this equilibrium studied in Ref.~\cite{delCastillo1998} was used to benchmark the PIC code as shown in Fig.~\ref{pic_validation}.
In all the computations presented here $q=1.25$ and $N_p=10^4$, $10^5$ and $10^6$.
The spatial domain size was set to $\zeta=16.52$ to fit the wavelength of the most unstable mode.
\begin{figure
\includegraphics[width=0.9\columnwidth]{gamma}
\caption
{
\label{pic_validation}
Electrostatic energy as a function of time in the Vlasov-Poisson PIC simulations of the bump on tail instability for different numbers of particles.
The straight lines denote the growth rate predicted by linear stability theory \cite{delCastillo1998}.
}
\end{figure}
Since the value of $q$ is relatively close to the marginal value, the instability grows weakly
and is concentrated in a narrow band in phase space centered around the point where the bump is located,
$v \approx 1$ in this case.
In order to unveil the nontrivial dynamics we focus the analysis in the band $v\in(-3,3)$,
and plot the departure of the particle distribution function from the initial background equilibrium.
The POD method is applied directly to $\delta f^H=f^H(x,v,t)-f_0(x,v)$,
but the WBDE method is applied to the full $f^H(x,v,t)$, and $f_0(x,v)$ is subtracted only for visualization.
Note that because we are considering only a subset of phase space,
the effective numbers of particles, $N_p=7318$, $N_p=73143$ and $N_p=731472$,
are smaller than the nominal numbers of particles, $N_p=10^4$, $N_p=10^5$ and $N_p=10^6$ respectively.
Figure~\ref{bump_tail_snapshots} shows contour plots of $\delta f$, for different number of particles.
Since the instability is seeded only by the random fluctuations in the initial condition, increasing $N_p$ delays the onset of the linear stability and this leads to a phase shift of the nonlinear saturated regime.
To aid the comparison of the saturated regime for different numbers of particles we have eliminated this phase shift by centering the peak of the particle distributions in the middle of the computational domain.
A $256 \times 256$ grid was used in the WBDE method, and a $50\times 50$ grid was used for the histogram and the POD methods.
The thresholds for the POD method where $r=1$, $r=2$, and $r=3$ for $N_p=10^4$, $N_p=10^5$ and $N_p=10^6$, respectively.
Except for the case where $N_p=10^4$, both the POD and WBDE estimates are very smooth, in agreement with the expected behavior of $f$ for this instability.
It is observed that the level of smoothness of the histogram estimated using $10^6$ particles is comparable to the level of smoothness achieved after denoising using only $10^5$ particles.
One should mention that for scales between $L$ and $J$ occurring in
the WBDE algorithm we find that none of the wavelet coefficients are above the thresholds at each scale.
In fact, a simple KDE estimate with a large enough smoothing scale would probably do the job pretty well for this kind of instabilities which do not induce abrupt variations in $f$.
Table~\ref{bump_tail_error} shows the POD and WBDE reconstruction errors for $N_p=10^4$ and $N_p=10^5$.
The error is computed using formula (\ref{error_e0}), taking for $f_{ref}$ the histogram obtained from the simulation with $N_p=10^6$.
\begin{figure
\includegraphics[width=1.0\columnwidth]{snapshots2_bt}
\caption
{
\label{bump_tail_snapshots}
Contour plots of estimates of $\delta f$ for the bump-on-tail instability PIC data at $t=149$:
Histogram method (first row), POD method (second row), and WBDE method (third row),
The left, center and right
columns correspond to $N_p=10^4$, $N_p=10^5$ and $N_p=10^6$ particles respectively.
The plots show thirteen contour lines equally spaced within the interval $[-0.012 0.012]$.
}
\end{figure}
Figure~\ref{pic_moments} shows the relative error on the second order moment :
$$\frac{\vert \mathcal{M}_{v,2}^W - \mathcal{M}_{v,2}^\delta \vert}{ \mathcal{M}_{v,2}^\delta }$$
where $\mathcal{M}^W_{v,2}$ is defined by (\ref{moments_def}).
A similar quantity is also represented for $f^H$ and $f^P$.
The time and number of particles are kept fixed at $t=149$ and $N_p=10^6$, and the grid resolution is varied.
As expected, $f^H$ and $f^W$ conserve the second order moment with accuracy $O(N_g^{-1})$.
The errors corresponding to $f^P$ is of the same order of magnitude but seems to reach a plateau for $N_g \simeq 1024$.
This may be due to the fact that for $N_g \geq 1024$, there is less than one particle per cell of the histogram used to compute $f_P$.
\begin{figure
\includegraphics[width=0.8\columnwidth]{moments}
\caption
{
\label{pic_moments}
Relative error on the second order moment as a function of the grid resolution, $N_g$, in the POD, WBDE, and histogram methods for the bump on tail instability particle data at $t=149$, with $N_p=10^6$ particles.
}
\end{figure}
\subsubsection {Two-streams instability}\label{two_streams}
As a second example we consider the standard two-streams instability with an initial condition consisting of two counter-propagating cold electron beams initially located at $v=-1$ and $v=1$.
This case is conceptually different to the previous one
because the initial condition depends trivially on the velocity. Therefore, there is no statistical error in the sampling of the distribution and the noise builds up only due to the self-consistent interactions between particles.
In other words, there is initially a strong correlation between particles' coordinates, which will eventually
almost vanish. This situation offers a way to test robustness of the WBDE method with respect to the underlying decorrelation hypothesis.
\begin{table}
\begin{tabular}{l*{6}{c}r}
& $N_p=10^4$ & $N_p=10^5$ \\
\hline
$f^H$ & $ 0.443 $ & $ 0.140 $ \\
$f^P$ & $ 0.163 $ & $ 0.090 $ \\
$f^W$ & $ 0.173 $ & $ 0.086 $ \\
\end{tabular}
\caption{
\label{bump_tail_error}
Comparison of normalized root mean squared errors $e_0$ (\ref{error_e0}) for the raw histogram and for the WBDE and POD methods, for the bump-on-tail instability at $t=149$,
depending on the number of particles.
The simulation with $N_p=10^6$ is used as a reference to compute the error.
}
\end{table}
The analysis is focused on four stages of the instability corresponding to $t=40$, $60$, $100$, and $400$.
Fig.~\ref{two_streams_snapshot} shows a comparison of the raw histogram, the POD and the WBDE reconstructed particle distribution functions at these four instants.
Grid sizes were $N_g=1024$ for the WBDE estimate, and $N_g=128$ for the two others.
For $t=40$, no noise seems to have affected the particle distribution yet,
therefore a perfect denoising procedure should conserve the full information about the particle positions.
Although WBDE introduces some artifacts in regions of phase space that should contain no particles at all,
it remarkably preserves the global structure of the two streams.
This is possible thanks to the numerous wavelet coefficients close to the sharp features in $f$ that are above the thresholds, in contrast to the bump-on-tail case.
On the next snapshot at $t=60$, the filaments have overlapped and the system is beginning to loose
its memory due to numerical round-off errors.
The fastest filaments still visible on the histogram are not preserved by WBDE, but the most active regions are well reproduced.
At $t=100$, the closeness between the histogram and the WBDE estimate is striking.
To put it somewhat subjectively, one may say that WBDE did not consider most of the rough features present at this stage as 'noise', since they are not removed.
Only with the last snapshot at $t=400$ does the WBDE estimate begin to be smoother than the histogram, suggesting that the nonlinear interaction between particles has introduced randomization in the system.
\begin{figure*
{
\includegraphics[width=1.75\columnwidth]{snapshots2.png}
}
\caption
{
\label{two_streams_snapshot}
Contour plots of estimates of $f$ for the two streams instability PIC data at
times $t=40$, $t=60$, $t=100$ and $t=400$ (left to right).
Histogram method (first row), WBDE method (second row), and POD method (third row),
The gray level tone varies uniformly in the interval $[0,A]$, where $A=0.15$, $A=0.08$, $A=0.05$ and $A=0.025$ in the first, second, third and fourth columns respectively.
}
\end{figure*}
The POD method is able to track very well the small and large scale structures of the particle density using a significantly smaller number of modes. In particular, for $t=40$, $60$, $100$, and $400$ only
$r=28$, $r=27$, $r=18$, and $r=5$ modes were kept. The decrease of the number of modes with time is a result of the lost of fine scale features in the distribution function. Despite this, a limitation of the POD method is the lack of a thresholding algorithm to determine the optimal number of modes a priori.
\section{Summary and Conclusion}
Wavelet based density estimation was investigated as a post-processing tool to reduce the noise in the reconstruction of particle distribution functions starting from discrete particle data. This is a problem of direct relevance to particle-based transport calculations in plasma physics and related fields. In particular, particle methods present many advantages over continuum methods, but have the potential drawback of introducing noise
due to statistical sampling.
In the context of particle in cell methods this problem is typically approached using finite size particles. However, this approach, which is closely related to the kernel density estimation method in statistics, requires the choice of a smoothing scale, $h$, (e.g., the standard deviation for Gaussian shape functions) whose optimal value is not known a priori. A small $h$ is desirable to fit as many Debye wavelengths as possible, whereas a large $h$ would lead to smoother distributions. This situation results from the compromise between bias and variance in statistical estimation.
To address this problem we proposed a wavelet based
density estimation (WBDE) method that does not require an a priori selection of a global smoothing scale and that its able to adapt locally to the smoothness of the density based on the given discrete data.
The WBDE was introduced in statistics \cite{Donoho1996}. In this paper we extended the method to higher dimension and applied it for the first time to particle-based calculations. The resulting method exploits the multiresolution properties of wavelets, has very weak dependence on adjustable parameters, and relies mostly on the raw data to separate the relevant information from the noise.
As a first example, we analyzed a plasma collisional relaxation problem modeled by stochastic differential equations.
Thanks to the sparsity of the wavelet expansion of the distribution function, we have been able to extract the information out of the statistical fluctuations by nonlinear thresholding of the wavelet coefficients.
At late times, when the particle distribution approaches a Maxwellian state, we have been
able to quantify the difference between the denoised particle distribution function
and its analytical counterpart, thus demonstrating the improvement with respect to the raw histogram.
The POD-smoothed and wavelet-smoothed particle distribution functions were shown to be roughly equivalent in this respect.
These results were then extended to a more complex situation simulated with a $\delta f$ code.
Finally, we have turned to the Vlasov-Poisson problem, which includes interactions between particles
via the self-consistent electric field.
The POD and WBDE methods were shown to yield quantitatively close results in terms of mean squared error
for a particle distribution function resulting from nonlinear saturation after occurrence of a bump-on-tail instability.
We have then studied the denoising algorithm during nonlinear evolution after the two-streams instability starting from two counter-streaming cold electron beams.
This initial condition violates the decorrelation hypothesis underlying the WBDE algorithm, and thus offers a good way to test its robustness regarding this aspect.
The WBDE method was shown to yield qualitatively good results without changing the threshold values.
One limitation of the present work comes from the way denoising quality is measured.
We have considered the quadratic error on the distribution function $f$ as a first indicator of the quality of our denoising methods.
However, it may be more relevant to compute the error on the force fields, which determine the evolution of the simulated plasma.
These forces depend on $f$ through integrals, and statistical analysis of the estimation of $f$ using weak norms,
like was done in \cite{Victory1991} in the deterministic case, could therefore be of great help to obtain
threshold parameters more efficient than those considered in this study.
The computational cost of our method scales linearly with the number of particles and with the grid resolution.
Therefore, WBDE is an excellent candidate to be performed at each time step during the course of a simulation.
Once the wavelet expansion of the denoised particle distribution function is known,
it is possible to continue using the wavelet representation to solve the Poisson equation \cite{Jaffard1992} and to compute the forces.
The moment conservation properties that we have demonstrated in this paper should mitigate the unavoidable dissipative effects implied by the smoothing stage.
In Ref.~\cite{McMillan2008}, a dissipative term was introduced in a global PIC code to avoid unlimited growth of particle weights in $\delta f$ codes,
and this was shown to improve long time convergence of the simulations.
It would be of interest to assess if the nonlinear dissipation operator corresponding to WBDE has the same effect.
\subsection*{Acknowledgements}
We thank D. Spong for providing the DELTA5D Monte-Carlo guiding center simulation data in Fig.6, originally published in Ref.~\cite{delCastillo2008}.
We also thank Xavier Garbet for his comments on the paper and for pointing out
several key references.
MF and KS acknowledge financial support by ANR under contract M2TFP, M\'ethodes multi\'echelles pour la turbulence dans les fluides et les plasmas.
DCN and GCH acknowledge support from the Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the U.S.
Department of Energy under contract DE-AC05-00OR22725. DCN also gratefully acknowledges the support and hospitality of the \'Ecole Centrale de Marseille for the three, one month visiting positions during the elaboration of this work.
This work, supported by the European Communities under the contract of Association between EURATOM, CEA and the French Research Federation for fusion studies, was carried out within the framework of the European Fusion Development Agreement. The views and opinions expressed herein do not necessarily reflect those of the European Commission.
|
1,116,691,500,762 | arxiv | \section{Introduction}
The dwarf spheroidal (dSph) satellite companions of the Milky Way (MW) contain a wealth of information regarding the formation and ongoing evolution of the Local Group. The $\sim 15$ known MW dSphs are the nearest, smallest, and faintest known galaxies ($0.5 \leq$ R/kpc $\leq 1$; $10^4 \leq$ L/L${\sun}\leq 10^7$; $20 \leq$ D/kpc $\leq 250$; \citealt{mateo98} and references therein; \citealt{willman05a,belokurov06a}). From their resolved stellar populations one can study in detail dSph morphologies, star formation histories, chemical abundances, and kinematics. Given their surface brightness profiles, the stellar velocity dispersions ($\sigma_v \sim 10$ km s$^{-1}$) of these galaxies indicate a larger mass than could be contributed by any reasonable equilibrium stellar population. Applying the virial theorem, one derives for the MW dSphs masses of $\sim 10^7-10^8$ M$_{\sun}$ and mass-to-light (M/L) ratios ranging from five to several hundred (\citealt{mateo98} and references therein). To the extent that the virial theorem is applicable despite the tidal field exerted by the Milky Way (\citealt{pp95,oh95}), the derived M/L ratios place at least some dSphs among the most dark-matter dominated galaxies known, generally with negligible baryonic contribution to their total mass. In this sense the MW dwarfs offer an ideal testbed with which to constrain the behavior of dark matter at perhaps the most difficult scale to model---the lower extremum of the halo mass function.
Efforts to characterize the internal kinematics of dSphs have expanded in scope over the past quarter-century. In a pioneering study, \citet{aaronson83} considered the possible dark matter (DM) content of dSphs based on the radial velocity (RV) dispersion of just three Draco stars. The need to invoke DM to describe dSph kinematics was placed on firmer footing after samples of tens of stars \citep{aaronson87,mateo91,mateo93,suntzeff93,edo95,queloz95,hargreaves94,hargreaves96b,vogt95,mateo98b} confirmed Aaronson's result. RV samples for nearly 100 stars accumulated over several epochs \citep{armandroff95,edo96,hargreaves96a,hargreaves96b} showed that the magnitudes of dSph velocity dispersions were not unduly influenced by the presence of binary stars or motion within the stellar atmospheres. Recent RV samples for more than 100 stars per dSph \citep{kleyna01,kleyna02,kleyna03,kleyna04,tolstoy04,munoz06,battaglia06,walker06a,walker06b,koch06b}, yield the general result that dSph velocity profiles appear isothermal over most of the visible faces of dSphs, though there is some dispute over the velocity behavior near the nominal tidal radius \citep{kleyna04,munoz05}. Moreover, the growing number of measurements for stars at and even well beyond the nominal tidal radius (e.g., \citealt{munoz06}; \citealt{sohn06}; Mateo et al. in preparation) has re-energized debate about the degree to which external tidal forces influence dSph stellar kinematics.
Studies by \citet{merritt97,wilkinson02,wang05} have shown theoretically that velocity samples for $\sim 1000$ stars are capable of distinguishing among spherical mass models while relaxing some of the assumptions (e.g., constant mass-to-light ratio and velocity isotropy) characteristic of early analyses. The present observational challenge thus is to acquire data sets consisting of $\ge 1000$ stellar RVs with spatial sampling that extends over the entire face of the dSph. In order to resolve narrow dSph velocity distributions one must work at high resolution. Stellar targets sufficiently bright (V$\leq 21$) for measuring radial velocity and chemical abundances are generally confined to the dSph red giant branch. The development of multi-object fiber and/or slit-mask spectrographs with echelle resolution has fueled a recent boom in this field. Recent and ongoing work using VLT/FLAMES/GIRAFFE \citep{tolstoy04,battaglia06,munoz06}, WHT/WYFFOS \citep{kleyna04}, Keck/DEIMOS \citep{sohn06,koch06b}, MMT/Hectochelle (Mateo et al.\ in preparation) has yielded RV samples for hundreds of stars in several MW dSphs. These studies have produced several results that challenge previous assumptions. \citet{tolstoy04} find evidence for multiple stellar populations in Sculptor that follow distinct distributions in position, metallicity, \textit{and} RV. \citet{battaglia06} produce a similar result in a study of Fornax. \citet{munoz06} discover several widely separated Carina members, which they interpret as evidence of tidal streaming; they also attribute a secondary clump in Carina's velocity distribution to contamination by stars once associated with the Large Magellanic Cloud. Mateo et al.\ (in preparation) and Sohn et al.\ (2006) detect a kinematic signature of tidal streaming from the outer regions of Leo I. It is becoming clear that dSphs are complex systems that can be characterized adequately only with data sets of the sort coming from large surveys.
Here we introduce an independent dSph kinematic survey undertaken with a new instrument, the Michigan-MIKE Fiber System (MMFS). MMFS provides multi-object observing capability using the dual-channel Magellan Inamori Kyocera Echelle (MIKE) spectrograph at the Magellan 6.5m Clay Telescope. Ours is the only large-scale dSph survey to measure RV using primarily the Mg-triplet ($\lambda \sim 5170$ \AA) absorption feature; others use the infrared calcium triplet near 8500 \AA. Thus far we have used MMFS to measure 6415 RVs to a median precision of $\pm 2.0$ km s$^{-1}$ for 5180 stars in four MW dSphs. Of the measured stars, which span the magnitude range $17 \le V \le 20.5$, approximately 470 are members of the Carina dSph, 1900 are members of Fornax, 990 are members of Sculptor, and 400 are members of Sextans. The remaining 1400 measured stars are likely dwarfs contributed by the MW foreground. In addition, the MMFS spectra are of sufficient quality that we can measure magnesium line strengths via a set of spectral indices that correlate with stellar parameters such as effective temperature and surface gravity, and which trace chemical abundances of iron-peak and alpha elements. The spectral indices provide independent constraints on dSph membership and allow us to probe the chemical abundance distribution within a given dSph.
Our focus in this paper is to provide a thorough description of MMFS, the observations, data reduction procedure, derivation of measurement uncertainties, and comparison with previous velocity measurements. This material will be referenced in companion papers (Walker et al.\ in preparation) in which we present all data and analyses.
\section{The Michigan-MIKE Fiber System}
\label{sec:mmfs}
In its standard slit observing mode the MIKE spectrograph delivers high-resolution spectra spanning visible and near-infrared wavelengths. MIKE's first optical element is a coated glass dichroic that reflects/transmits incoming light into one of two independent channels. The ``blue'' channel has wavelength coverage 3200-5000 \AA, while the ``red'' channel covers 4900-10000 \AA. The dual-beam design allows for independent optimization of throughput and dispersion characteristics over each range of spectrum (see \citet{bernstein03} for a complete description of MIKE).
For multi-object observations using Magellan+MMFS, MIKE is modified from its standard slit configuration to receive light from up to 256 fibers. In fiber mode MIKE is backed $2$ m from the Nasmyth port, where it remains fixed with respect to gravity. Inserted at the port is a conical structure onto which an (interchangeable) aluminum plug plate is mounted. The plug plate holds fibers against a telecentrator lens that ensures the 1.4-arcsec fiber apertures accept light parallel to the optical axis and along the focal surface, regardless of position in the target field. A drum lens at the entrance to each fiber forms an image of the pupil on the the $175$-$\mu$m fiber core. The drum lens and the natural focal ratio degradation of the fiber convert the incoming beam from F/11 to F/3.5 (more similar to MIKE's camera optics). Fibers run from the plug plate through a junction box, where they are bundled and sorted into one of two 128-fiber assemblies that enter the two channels of the spectrograph. Light losses are minimized by the short fiber length; from end to end, blue (red) fibers extend just 2.43 m (2.29 m).
At the spectrograph end, the two 128-fiber assemblies replace MIKE's standard injection optics. Due to the spectrograph's fixed optical elements, blue and red channels remain optimized for smaller and larger wavelengths, respectively. In order to fit spectra from all 128 fibers onto a single detector in each channel, narrow-band filters are used to isolate a single order of spectrum for each target object. We use filters in both channels that isolate the spectral region $5130-5185$ \AA. In late-type stellar spectra this region includes the prominent MgI triplet (MgT) absorption feature and an assortment of FeI and NiI absorption lines. The MgT filter yields useful spectra through either MIKE channel, though the blue channel offers higher spectral resolution ($\sim 0.059$ \AA\ pix$^{-1}$; $R \sim 25000$) and greater throughput than does the red channel ($\sim 0.073$ \AA\ pix$^{-1}$; $R\sim 20000$). MMFS thus has the ability to acquire high-resolution spectra from up to 256 objects simultaneously. The usable field of view subtends an angular diameter of $20$ arcmin.
\section{Observations}
\label{sec:observations}
\subsection{Target Selection}
For our survey, MMFS target star candidates are selected from V,I imaging data taken prior to spectroscopic observing runs. We select MMFS target candidates from the dSph red giant branch (RGB) of the resulting color-magnitude diagram (CMD). We expect a fraction of the selected candidates to be foreground Milky Way dwarfs with magnitudes and colors placing them on or near the dSph RGB. The fraction of interlopers naturally increases for lines of sight along which the dSph surface density is relatively small. The efficiency with which our CMD-based selection identifies bona fide dSph red giants thus varies from galaxy to galaxy and with position within each galaxy, and foreground stars must subsequently be identified and removed from kinematic samples.
Details of the photometry used to select red giant candidates in the Fornax and Sculptor dSphs are published elsewhere. Fornax candidates were selected using the V,I photometric data described in \citet{walker06a}. Sculptor candidates were selected using the V,I photometry of \citet{coleman05a}, who generously provided their data set.
\subsubsection{Sextans}
For the Sextans dSph, red giant candidates were chosen based on V,I photometry taken with the MDM 2.4 m Hiltner Telescope and 8k Mosaic CCD detector. Observations took place the nights of 2004 February 11-17. Three of the seven nights were photometric; the remaining nights had mixed observing conditions. We observed 25 Sextans fields, each with dimensions $24\arcmin \times 24\arcmin$, forming a square array with adjacent field centers separated by $20\arcmin$. We exposed for 360s in I and 540s in V. Standard reduction steps included overscan, bias, and flat-field corrections using twilight sky flats. We used the two-dimensional stellar photometry program DoPHOT \citep{schechter93} to identify stars and measure instrumental magnitudes. We placed the photometry in the Kron-Cousins system \citep{bessell76} using 50 photometric standard stars \citep{landolt92} observed during the run. We used the $4\arcmin$ overlap between adjacent fields to remove photometric zero-point offsets. Formal error values returned by DoPHOT and multiple measurements from overlapping fields indicate the typical photometric accuracy is $\pm 0.04$ mag.
\subsubsection{Carina}
Over several MMFS observing runs, red giant candidates in the Carina dSph were chosen using three independent photometric data sets. Carina targets observed with MMFS in 2004 March were selected from B,R photometry obtained with the CTIO 4m telescope with Mosaic CCD detector during 1999 September. Carina targets observed with MMFS in 2005 February were chosen in part from the same B,R mosaic photometry, but targets in outer fields were selected from V,I photometry taken by Kaspar von Braun at the 2.4 m Swope Telescope (Las Campanas Observatory) in 2004 December. Carina targets observed with MMFS in 2006 March were selected from V,I photometry taken by Patrick Seitzer with the CTIO 0.9 m Schmidt Telescope in 2005 December. We used DoPHOT to reduce the photometry in each case.
In order to place the three photometric data sets for Carina on a common V,I system, first we used 122 photometric standard stars in the Carina field of \citet{stetson00} to apply zero-point corrections to the Schmidt instrumental V,I magnitudes. Residuals after applying these shifts have rms values 0.033 mag in V and 0.035 mag in I. We followed the same procedure in calibrating the 2004 V,I photometry (residuals have rms 0.046 mag in V and 0.048 mag in I). Next we used a piecewise linear transformation to place the B,R mosaic photometry on the Stetson V,I system. Coefficients were determined using 286 Stetson standard stars detected in the mosaic photometry. Over the color range $0.6 \leq$ V-I $\leq 1.8$, residuals have rms $\sim 0.03$ mag for the V and V-I transformations. We monitored for consistency by comparing the calibration results for stars detected in both the mosaic and Schmidt data. For 637 stars present in both data sets, flux deviations have mean values $\langle V_{schmidt}-V_{mosaic} \rangle = 0.011$ mag and $\langle I_{schmidt}-I_{mosaic} \rangle = 0.003$ mag, with rms values of 0.072 mag and 0.088 mag, respectively. These values are reasonable considering the undersampling and blending in the Schmidt data.
\subsubsection{Globular Clusters}
For the purpose of calibrating spectral-line index/abundance relations, we have also observed stellar targets in seven Galactic globular clusters. For clusters NGC 104 (47 Tucana), NGC 288, and NGC 7099 (Messier 30) we chose targets on the basis of V,I photometry obtained in 2005 July with the 40-inch telescope at Las Campanas Observatory, Chile. For clusters NGC 5927, NGC 6121 (Messier 4), NGC 6171 (Messier 107), and NGC 6397 we selected targets using V,I photometry obtained in 2005 June with the Magellan/Baade 6.5m Telescope at Las Campanas. For all clusters, instrumental magnitudes were measured with DoPHOT and then placed on the V,I system using stars in common with Stetson standard fields. For NGC 6171, which is absent from the Stetson catalog, we applied the same transformations determined for NGC 5927, a cluster observed on the same night and at similar airmass. Direct comparison to Stetson stars indicates that the globular cluster photometry is accurate to within $\sim 0.04-0.08$ magnitudes.
\subsection{CMDs and maps for dSph targets}
Figures \ref{fig:cmd} plots CMDs for Carina, Fornax, Sculptor and Sextans. Polygons enclose stars we consider to be red-giant candidates, and therefore eligible MMFS targets. For Carina, Sculptor and Sextans the selection region extends from near the tip of the red giant branch to the horizontal branch at $V \sim 20.5$. For Fornax, the greater stellar density allows us to increase efficiency by selecting from only those red-giant candidates having $V < 20$. Figure \ref{fig:selection_map} maps the locations of red giant candidates in each dSph. Circles in Figure \ref{fig:selection_map} indicate MMFS fields that have been observed as of 2006 November. Initial MMFS runs gave priority to target-rich central fields, with goals of building sample size and evaluating instrument performance. Later runs gave preference to fields along the morphological major axis and/or at large radius. The partial overlap of many adjacent fields allows us to quantify velocity variabilty using repeat measurements.
The $1.4\arcsec$ fiber apertures require that target coordinates be accurate to within $\sim 0.2\arcsec$. DoPHOT's centroid algorithm returns the (x,y) CCD position of each identified star. We convert these to equatorial coordinates using the IRAF routines TFINDER and CCTRANS. We then tie the astrometry to the USNO-B1 system by identifying up to several hundred USNO stars per CCD frame. From the derived coordinates of the same stars in overlapping fields we find a typical rms scatter of 0.24\arcsec.
\begin{figure*}
\epsscale{1}
\plotone{smallf1.eps}
\caption{ V,I color-magnitude diagrams for the four dSphs observed with MMFS. Stars enclosed by polygons are considered red giant candidates and eligible MMFS targets}
\label{fig:cmd}
\end{figure*}
\begin{figure*}
\epsscale{1}
\plotone{smallf2.eps}
\caption{ Maps of red giant candidates identified in Figure \ref{fig:cmd}. Circles indicate fields observed with MMFS (field number corresponds to values in Column 2 of Table \ref{tab:observations}. Dotted ellipses correspond to (nominal) tidal radii, as identified by \citet{king62} model fits to the surface brightness profiles by \citet{ih95}. For Sextans the tidal radius ($r_t \sim 160\arcmin$) lies beyond the plotted region. All sky positions are given in standard coordinates with origin at the dSph center. We observed three distinct sets of target stars in each of the densely populated Fornax fields 4 and 9.}
\label{fig:selection_map}
\end{figure*}
\subsection{Spectroscopic Observing Procedure}
In a typical night we take $30$-$50$ zero-second exposures just prior to evening twilight. During twilight we take $3$-$5$ exposures of the scattered solar spectrum, to allow for wavelength calibration and provide a velocity zero point, followed by $3 \times 60$s quartz lamp exposures and a $1 \times 300$s Th-Ar arc-lamp exposure.
Under optimal observing conditions we take $3 \times 2400$s exposures of target fields, immediately followed by $3 \times 60$s flat field (quartz) exposures and $1 \times 300$s Th-Ar. Other observations taken during the night include single-fiber, $1 \times 30$s exposures of radial velocity standard stars, immediately followed by quartz and Th-Ar calibration exposures taken in the usual manner. The procedure allows for observation of 2-4 dSph fields per night.
Table \ref{tab:observations} logs all MMFS observations taken for dSph and globular cluster fields as of 2006 November. The first two columns identify the field by galaxy name and field number (observed fields are mapped by field number in Figure \ref{fig:selection_map}). Columns 3-7 in Table \ref{tab:observations} list, respectively, the heliocentric Julian date at the midpoint of the first exposure, UT date at the midpoint of the first exposure, total exposure time, number of red giant candidates to which we assigned fibers, and the number of these for which the radial velocity was measured succesfully. The final column identifies repeat observations of fields for which we judged during the course of the run that the initial observation was insufficient.
\begin{deluxetable*}{lccccccl}
\tabletypesize{\scriptsize}
\tablewidth{0pc}
\tablecaption{ MMFS observations of dSph and globular cluster fields as of 2006 November}
\tablehead{\colhead{Galaxy}&\colhead{Field}&\colhead{HJD $-2.45 \times 10^6$}&\colhead{UT Date}&\colhead{Exposure Time }&\colhead{Targets}&\colhead{Velocities}&\colhead{Notes}\\
\colhead{}&\colhead{}&\colhead{(days)}&\colhead{}&\colhead{(s)}&\colhead{}&\colhead{}&\colhead{}
}
\startdata
Sextans&8&3086.781&2004 Mar. 22&$2 \times 1800$&120&32\\
Carina&18&3087.533&2004 Mar. 23&$4 \times 2400$&173&83\\
Sextans&8&3087.716&2004 Mar. 23&$4 \times 2400$&120&&repeat from 2004 Mar. 22\\
Carina&1&3088.540&2004 Mar. 24&$3 \times 2400$&219&92\\
Sextans&14&3088.704&2004 Mar. 24&$4 \times 2400$&139&49\\
Carina&1&3089.540&2004 Mar. 25&$4 \times 2400$&220&&repeat from 2004 Mar. 24\\
Sextans&1&3089.724&2004 Mar. 25&$2 \times 2400$&196&66\\
&&&&$1 \times 3000$\\
Carina&12&3090.551&2004 Mar. 26&$4 \times 2400$&223&77\\
Sextans&7&3090.704&2004 Mar. 26&$2 \times 2700$&105&48\\
&&&&$1\times 3600$\\
Carina&11&3091.520&2004 Mar. 27&$2 \times 3000$&190&115\\
&&&&$1 \times 3600$\\
Sextans&5&3091.698&2004 Mar. 27&$1 \times 3000$&140&50\\
&&&&$1 \times 3600$\\
Carina&1&3092.519&2004 Mar. 28&$2 \times 3000$&220&&repeat from 2004 Mar. 24, Mar. 25\\
&&&&$1 \times 3600$\\
Sextans&1&3092.704&2004 Mar. 28&$1 \times 2400$&200&&repeat from 2004 Mar. 25\\
&&&&$1 \times 3000$\\
&&&&$1 \times 3600$\\
Carina&26&3093.531&2004 Mar. 29&$1 \times 2700$&85&21\\
&&&&$1 \times 3000$\\
\\
\hline
\\
Sculptor&20&3286.629&2004 Oct. 8&$4 \times 1800$&135&110\\
Fornax&15&3286.755&2004 Oct. 8&$1 \times 1800$&220&191\\
&&&&$1 \times 2400$\\
&&&&$1 \times 2700$\\
&&&&$1 \times 3600$\\
Sculptor&3&3287.552&2004 Oct. 9&$2 \times 1800$&224&168\\
&&&&$1 \times 2400$\\
&&&&$1 \times 2600$\\
Sculptor&31&3287.694&2004 Oct. 9&$2 \times 1800$&87&69\\
&&&&$2 \times 2400$\\
Fornax&15&3287.826&2004 Oct. 9&$1 \times 2100$&222&&repeat from 2004 Oct. 8\\
&&&&$1 \times 2400$\\
&&&&$1 \times 2200$\\
Sculptor&4&3288.571&2004 Oct. 10&$2 \times 2400$&224&196\\
&&&&$1 \times 2500$\\
Sculptor&72&3288.679&2004 Oct. 10&$2 \times 2400$&31&19\\
&&&&$1 \times 2100$\\
Fornax&15&3288.801&2004 Oct. 10&$1 \times 2900$&222&&repeat from 2004 Oct. 8,9\\
&&&&$1 \times 3000$\\
Sculptor&3&3289.525&2004 Oct. 11&$1 \times 3600$&224&&repeat from 2004 Oct. 9\\
Sculptor&5&3289.593&2004 Oct. 11&$3 \times 2400$&224&176\\
Sculptor&15&3289.712&2004 Oct. 11&$3 \times 2400$&138&128\\
&&&&$1 \times 2700$\\
Sculptor&20&3289.868&2004 Oct. 11&$1 \times 3000$&135&&repeat from 2004 Oct. 8\\
\\
\hline
\\
Sextans&12&3408.709&2005 Feb. 7&$4 \times 2400$&75&34\\
Carina&47&3409.650&2005 Feb. 8&$3 \times 2400$&129&109\\
Sextans&34&3409.774&2005 Feb. 8&$3 \times 2400$&83&75\\
Carina&13&3410.547&2005 Feb. 9&$4 \times 1800$&223&139\\
Carina&108&3410.668&2005 Feb. 9&$2 \times 1800$&95&73\\
Sextans&7&3410.758&2005 Feb. 9&$5 \times 2400$&185&127\\
Carina&25&3411.553&2005 Feb. 10&$6 \times 2400$&223&148\\
Sextans&1&3411.776&2005 Feb. 10&$4 \times 2700$&223&113\\
Sextans&366&3412.725&2005 Feb. 11&$3 \times 2700$&44&27\\
Sextans&38&3412.846&2005 Feb. 11&$2 \times 2500$&52&38\\
Carina&25&3415.550&2005 Feb. 14&$4 \times 1800$&224&&repeat from 2005 Feb. 10\\
Carina&108&3415.678&2005 Feb. 14&$3 \times 1800$&95&&repeat from 2005 Feb. 9\\
Sextans&1&3415.787&1005 Feb. 14&$4 \times 2400$&224&&repeat from 2005 Feb. 10\\
Carina&118&3416.555&2005 Feb. 15&$3 \times 2400$&119&90\\
Sextans&5&3416.683&2005 Feb. 15&$4 \times 2400$&177&117\\
Sextans&26&3416.837&2005 Feb. 15&$1 \times 1800$&118&61\\
Carina&13&3417.544&2005 Feb. 16&$2 \times 2400$&224&&repeat from 2005 Feb. 9\\
Carina&222&3417.630&2005 Feb. 16&$3 \times 2400$&81&72\\
Sextans&15&3417.761&2005 Feb. 16&$4 \times 2400$&119&91\\
Carina&26&3418.552&2005 Feb. 17&$4 \times 2400$&80&48\\
Sextans&23&3418.704&2005 Feb. 17&$3 \times 2400$&104&72\\
\\
\hline
\\
NGC 7099&1&3663.533&2005 Oct. 20&$3 \times 900$&59&58&\\
47 Tuc&1&3663.584&2005 Oct. 20&$3 \times 600$&92&71&\\
&&&&$1 \times 300$\\
Sculptor&49&3663.657&2005 Oct. 20&$3 \times 2400$&57&27\\
Sculptor&45&3663.763&2005 Oct. 20&$3 \times 2400$&52&32\\
NGC 7099&1&3664.492&2005 Oct. 20&$4 \times 300$&59&38&\\
47 Tuc&1&3664.510&2005 Oct. 21&$3 \times 300$&92&78&\\
Sculptor&239&3664.558&2005 Oct. 21&$3 \times 2400$&26&13\\
Sculptor&95&3664.656&2005 Oct. 21&$4 \times 1800$&33&24\\
Fornax&10&3664.771&2005 Oct. 21&$3 \times 1800$&223&201\\
&&&&$2 \times 2100$\\
NGC 288&1&3665.513&2005 Oct. 22&$4 \times 300$&63&40&\\
Sculptor&58&3665.559&2005 Oct. 22&$3 \times 2400$&41&20\\
Sculptor&27&3665.662&2005 Oct. 22&$3 \times 2400$&110&87\\
Fornax&29&3665.772&2005 Oct. 22&$4 \times 2400$&216&177\\
NGC 288&1&3666.504&2005 Oct. 22&$4 \times 300$&64&63&\\
Sculptor&54&3666.546&2005 Oct. 23&$3 \times 2400$&65&48\\
Fornax&34&3666.655&2005 Oct. 23&$4 \times 2400$&163&147\\
Fornax&592&3666.796&2005 Oct. 23&$2 \times 2700$&127&109\\
&&&&$1 \times 2400$\\
\\
\hline
\\
Carina&135&3800.581&2006 Mar. 6&$4 \times 2400$&202&47\\
Sextans&110&3800.762&2006 Mar. 6&$3 \times 2400$&77&43\\
Carina&175&3801.535&2006 Mar. 7&$4 \times 2400$&195&103\\
Sextans&99&3801.696&2006 Mar. 7&$3 \times 2400$&74&51\\
Sextans&79&3801.820&2006 Mar. 7&$1 \times 2400$&69&42\\
NGC 5927&1&3801.8611&2006 Mar. 7&$3 \times 300$&111&111&\\
Carina&175&3802.542&2006 Mar. 8&$3 \times 2400$&195&&repeat from 2006 Mar. 7\\
Sextans&99&3802.656&2006 Mar. 8&$3 \times 2400$&74&&repeat from 2006 Mar. 7\\
Sextans&79&3802.764&2006 Mar. 8&$3 \times 2400$&69&&repeat from 2006 Mar. 7\\
M4&1&3802.858&2006 Mar. 8&$3 \times 300$&64&64&\\
NGC 6397&1&3802.894&2006 Mar. 8&$2 \times 150$&31&31&\\
&&&&$1 \times 300$\\
Carina&98&3803.601&2006 Mar. 9&$3 \times 2400$&210&75\\
Sextans&54&3803.716&2006 Mar. 9&$3 \times 2400$&59&32\\
NGC 6171&1&3803.861&2006 Mar. 9&$3 \times 300$&61&55&\\
Carina&105&3804.524&2006 Mar. 10&$4 \times 2400$&166&45\\
Sextans&6&3804.673&2006 Mar. 10&$6 \times 2400$&197&61\\
Carina&135&3805.515&2006 Mar. 10&$2 \times 2400$&202&&repeat from 2006 Mar. 6\\
Carina&93&3805.597&2006 Mar. 11&$4 \times 2400$&166&90\\
Sextans&100&3805.736&2006 Mar. 11&$4 \times 2400$&59&22\\
\\
\hline
\\
Sculptor&1&4017.5547&2006 Oct. 9&$4 \times 2700$&223&186\\
Sculptor&160&4018.5527&2006 Oct. 10&$3 \times 2700$&28&21\\
Sculptor&186&4018.6665&2006 Oct. 10&$3 \times 2700$&25&10\\
Sculptor&337&4019.5623&2006 Oct. 11&$3 \times 2400$&18&11\\
Fornax&4.1&4019.6829&2006 Oct. 11&$3 \times 2400$&224&208\\
Fornax&9.1&4020.6892&2006 Oct. 12&$3 \times 2700$&197&185\\
Fornax&4.3&4021.6101&2006 Oct. 13&$3 \times 2000$&224&218\\
&&&&$1 \times 1700$\\
Fornax&9.2&4022.7825&2006 Oct. 14&$3 \times 2700$&192&109\\
&&&&$1 \times 1800$\\
Fornax&54&4023.6057&2006 Oct. 15&$2 \times 2700$&80&74\\
&&&&$1 \times 2000$\\
Fornax&54&4024.5654&2006 Oct. 16&$1 \times 2700$&80&&repeat from 2006 Oct. 15\\
Fornax&81&4024.6206&2006 Oct. 16&$3 \times 2700$&101&95\\
NGC 7099&1&4025.5010&2006 Oct. 16&$3 \times 300$&59&50&\\
NGC 288&1&4025.5530&2006 Oct. 17&$3 \times 300$&64&60&\\
Fornax&9.3&4025.6353&2006 Oct. 17&$3 \times 2700$&224&205\\
Sculptor&743&4026.5581&2006 Oct. 18&$3 \times 2700$&25&9\\
Fornax&33&4026.6887&2006 Oct. 18&$3 \times 2700$&105&88\\
Fornax&31&4027.5469&2006 Oct. 19&$3 \times 2700$&129&119\\
Fornax&4.4&4027.6787&2006 Oct. 19&$3 \times 2700$&223&216\\
Fornax&70&4027.8198&2006 Oct. 19&$3 \times 1900$&72&60\\
\enddata
\label{tab:observations}
\end{deluxetable*}
\section{Reduction of MMFS Spectra}
\label{sec:reduction}
\subsection{Initial Processing}
We extract MMFS spectra and measure radial velocities using standard packages and subroutines available in the IRAF\footnote{IRAF is distributed by the National Optical Astronomy Observatories, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.} astronomical software package. After $3 \times 3$ binning of the $2$k $\times 4$k detectors at readout, raw data frames from either detector are $683 \times 1365$ pix$^{2}$, and the circular resolution element has FWHM $\sim 3.5$ binned pixels in the spectral direction. We reduce blue and red spectra independently and following identical procedures.
After applying overscan and bias corrections, we use the IMCOMBINE task to average the series of target frames for each field. Individual frames are scaled and weighted to account for differences in exposure time and airmass. Bad pixels, including most of those affected by cosmic rays, are rejected using a $4 \sigma$ clip around the median of the scaled frames. We then use the APSCATTER task to subtract scattered light from target frames. We identify spectral apertures in the well-exposed quartz frame (averaged in the same manner as the target frames) associated with each target field, and then use the APALL task to extract one-dimensional spectra assuming the target, quartz, and Th-Ar frames associated with a given field have identical spectral apertures.
\subsection{Wavelength Calibration}
\label{subsec:wavelength}
We determine the wavelength/pixel relation using absoprtion line features in the solar spectrum observed at evening twilight. In the observed region the solar spectrum contains twice as many useful lines as the Th-Ar spectrum. The MMFS wavelength/pixel relation is sufficiently stable over an observing run that the benefit of using more lines outweighs disadvantages that arise because twilight exposures are available only at dusk and dawn. To quantify any drift in the wavelength/pixel relation during the night we measure the relative redshifts of the twilight-calibrated Th-Ar spectra taken with each target field, and we compensate for these redshifts by applying low-order corrections to individual wavelength solutions. Because this method is a departure from the conventional Th-Ar calibration, we describe it in detail.
In our twilight spectra we identify solar absorption lines using the NOAO FTS solar atlas \citep{kurucz84}. Figure \ref{fig:calibration} displays an example twilight-solar spectrum and labels lines used for wavelength calibration. We exclude several prominent lines (including the MgI line at 5167.327 \AA) because they are blended with nearby lines. The wavelength solution is determined from a 4$^{th}$-order polynomial fit; residuals have mean rms $\sim 0.01$\AA\ ($0.6$ km s$^{-1}$) for blue spectra and $\sim 0.02$\AA\ ($1.2$ km s$^{-1}$) for red. For reference, typical residuals from fits to 9 emission lines in our Th-Ar spectra are larger by a factor of two.
\begin{figure}
\epsscale{1.2}
\plotone{smallf3.eps}
\caption{ Example twilight solar spectrum used in wavelength calibration. Absorption lines used in the wavelength solution are identified.}
\label{fig:calibration}
\end{figure}
Next we apply initial wavelength solutions to all frames using the DISPCOR task. We assign all target, quartz, and Th-Ar frames the (aperture-dependent) wavelength solutions determined from the (temporally) nearest twilight spectrum. We then use FXCOR to measure zero-point drift in the wavelength/pixel relation (during the night and/or over the course of the run) from shifts in the wavelengths of emission lines in the Th-Ar spectra. The measured zero-point shifts for Th-Ar frames taken during the observing run in 2005 February are shown as a function of aperture number in Figure \ref{fig:tharshift}. We find that drift in the wavelength solution results in small ($\leq 2$ km s$^{-1}$) zero-point offsets that depend linearly on aperture number. We note that the zero-point offsets, and hence the wavelength/pixel relation, remain stable over the span of several nights. We correct for zero-point drift using the DOPCOR task to shift the wavelength solution in each aperture of the associated target and quartz frames by a value determined from the best linear fit to the zero-point offset vs.\ aperture data.
\begin{figure*}
\plotone{smallf4.eps}
\caption{ Stability of the MMFS wavelength/pixel relation. Plotted for each multispectrum Th-Ar frame in the 2005 February MMFS run is the velocity shift measured via cross-correlation against the Th-Ar frame associated with a single twilight frame. Prior to cross-correlation, wavelengths in all Th-Ar frames were given identical wavelength/pixel solutions based on solar absorption lines identified in the twilight spectra. Blue/red points indicate blue/red MIKE detectors. Each panel indicates the heliocentric Julian date ($-2.45 \times 10^6$ days) at the midpoint of the Th-Ar exposure.}
\label{fig:tharshift}
\end{figure*}
Figure \ref{fig:twilights} displays solar radial velocity results measured from the several twilight exposures obtained during each MMFS observing run. Panels on the right side give velocity results for twilight-calibrated twilight frames. For comparison, panels on the left side give velocities measured using conventional Th-Ar-calibration. We note the consistent velocity precision obtained with twilight calibration. We suspect the degradation in the precision from Th-Ar calibration is related to the fact that the Th-Ar lamp system used exclusively by MMFS has operated at different voltages and with different power supplies over the six observing runs. The sample standard deviation of velocities measured using twilight calibration has remained steady at $\sim 0.8$ km s$^{-1}$ (blue) and $\sim 1.6$ km s$^{-1}$ (red). Figure \ref{fig:twilights} also provides evidence of small channel- and in some cases aperture-dependent zero-point offsets. The red channel tends to yield slightly more positive velocities than the blue channel. However, it is straightforward to measure and correct for these offsets (see Section \ref{subsec:zeropoint}), which are smaller than the typical velocity measurement error.
\begin{figure}
\epsscale{1.2}
\plotone{smallf5.eps}
\caption{ Comparison of the Th-Ar wavlength calibration method (left panels) with the twilight calibration method (right panels). For each of the MMFS observing runs, blue/red markers indicate (solar rest frame) radial velocities measured from twilight spectra in each aperture in blue/red channels. In 2005 October, Th-Ar exposures were not taken immediately after twilight exposures, precluding Th-Ar calibration of those twilight spectra. Due to the previous comparisons, we did not perform Th-Ar calibration of the 2006 October spectra.}
\label{fig:twilights}
\end{figure}
\subsection{Throughput Corrections and Sky Subtraction}
\label{subsec:corrections}
Because fibers differ in throughput by up to a factor of two, we create a response frame by dividing each aperture of the quartz frame by the average (over all apertures) quartz continuum. Dividing the target frame by the response frame corrects for variations in fiber throughput and pixel sensitivity.
At this point we use the CONTINUUM task to remove from the target frames any remaining effects due to cosmic rays and/or bad pixels and columns. We fit a $10^{th}$-order legendre polynomial to the corrected spectrum in each aperture and replace with the function value any pixel values that deviate from the fit by more than $6 \sigma$.
Sky noise near the Mg-triplet is contributed mainly by scattered solar light. Left uncorrected, sky contamination produces a secondary velocity signal at $0$ km s$^{-1}$. During observation of every dSph field we assign $\sim 32$ science fibers ($\sim 16$ per MIKE channel) to regions of blank sky. We use the SCOMBINE task to average the apertures corresponding to these sky spectra in the target frames. Prior to averaging we discard the three highest and two lowest values among the sky spectra at each pixel. Removal of the high pixel values reduces the contribution from resolved background galaxies \citep{wyse92}. We subtract the average sky spectrum from each aperture in the target frame.
Black, solid lines in the left panels of Figures \ref{fig:bscl5spectra} and \ref{fig:bsex6spectra} display sky-subtracted spectra representing the range of S/N achieved during typical observations of dSph fields. Red, dotted lines in the left panels of either figure indicate the spectra prior to sky subtraction. The spectra in Figure \ref{fig:bscl5spectra} were obtained at new moon, while the spectra in Figure \ref{fig:bsex6spectra} were obtained at first-quarter moon. Prior to sky subtraction, spectra from the latter observation clearly are contaminated by solar absorption lines. After sky subtraction, residual solar lines typically deviate from the continuum level by an amount similar to the intrinsic continuum noise level.
\begin{figure*}
\epsscale{1}
\plotone{smallf6.eps}
\caption{ Processed spectra and corresponding radial velocity cross-correlation functions from observations during a dark night. Solid black lines in the left-hand panels display examples of fully processed spectra from a Sculptor target field. Red lines represent each spectrum prior to sky subtraction. Right-hand panels show each spectrum's cross-correlation against our radial velocity template and list the Tonry-Davis R value measured from the CCF.}
\label{fig:bscl5spectra}
\end{figure*}
\begin{figure*}
\epsscale{1}
\plotone{smallf7.eps}
\caption{ Panels plot the same relations as in Figure \ref{fig:bscl5spectra}, but for example spectra from a Sextans field observed in the presence of significant moonlight.}
\label{fig:bsex6spectra}
\end{figure*}
\subsection{Final Processing}
\label{subsec:finalprocessing}
In most MMFS observing runs there have been one or more fields for which we deemed the original observation useful but insufficient (see Table 1). Reasons range from poor observing conditions to inadequate exposure time available prior to morning twilight. In such cases we re-observed the field using fiber assignments identical to those of the original observation. During the reduction steps already described, repeat observations---often separated by several days from the original observation---are treated independently to allow for drift in the wavelength solution and different sky levels. At this point we combine all observations of the same field taken during the same run. First we use the DOPCOR task to correct the wavelength solutions of repeat observations for the changing component of Earth's motion (typical shifts are less than 1 km s$^{-1}$) toward the target field. The SCOMBINE task then averages spectra in corresponding apertures of the original and repeat target frames. Spectra in individual apertures are weighted by the mean pixel value prior to computing the aperture average.
In the final processing step prior to velocity measurement, we remove the continuum shape of the spectra. The CONTINUUM task fits a $10^{th}$-order legendre polynomial to the spectrum in each aperture of the target frames, then subtracts the function values from pixel values to produce the final spectra.
\subsection{Signal-to-Noise Ratio}
\label{snratio}
The variance in our spectra has independent components contributed by source, sky background, and read noise. For the $i^{th}$ pixel, let $N_{S,i}$ represent the number of detected photons from the target star, let $N_{B,i}$ represent the number of detected photons from the sky background, and let $\sigma_R^2$ be the noise associated with detector readout (typically between 2-4 electrons per pixel for MIKE). In practice we estimate $N_{B,i}$ from $s$ individual sky spectra. Assuming the estimate of $N_{B,i}$ contributes a variance component $N_{B,i}/s$, the signal-to-noise ratio (S/N) at the $i^{th}$ pixel is given by
\begin{equation}
\biggl [\frac{S}{N} \biggr]_i=\frac{N_{S,i}}{\sqrt{N_{S,i}+N_{B,i}+N_{B,i}/s+\sigma_R^2}}.
\label{eq:snratio}
\end{equation}
We estimate source and background photon fluxes from smooth fits to the the continua of our sky-subtracted and averaged sky spectra, respectively. At the $i^{th}$ pixel the continuum fit to the sky-subtracted spectrum has digital unit (DU) value\footnote{We follow the convention whereby $\hat{q}$ denotes the estimate of $q$.} $n_{S,i}=\hat{N}_{S,i}/(R_iG)$, where $R_i$ is the value of the response frame (Section \ref{subsec:corrections}) and $G$ is the detector gain (between 0.5 and 1.5 electrons/DU over the several MMFS observing runs). The continuum fit to the averaged sky spectrum has DU value $n_{B,i}=\hat{N}_{B,i}/(R_iG)$. From the estimates $\hat{N}_{S,i}$ and $\hat{N}_{B,i}$ we calculate S/N at each pixel using Equation \ref{eq:snratio}. Figure \ref{fig:snratio} plots, for all spectra that eventually yield an acceptable velocity measurement, the distribution of the mean S/N calculated from pixels spanning the range $5141 - 5177$ \AA. The median S/N per pixel among dSph spectra acquired with MIKE's blue (red) channel is 3.7 (2.0) per pixel. Multiplying these values by $\sqrt{3.5}$ gives the mean S/N per resolution element.
\begin{figure}
\epsscale{1.2}
\plotone{smallf8.eps}
\caption{ Distributions of the mean signal-to-noise ratio (S/N) per pixel, calculated over the wavelength range $5141-5177$ \AA. Distributions are plotted separately for dSph, globular cluster, RV standard and solar twilight spectra. In each panel the thin solid (dotted) lines give distributions for spectra obtained with MIKE's blue (red) channels, while the thick solid line is the combined distribution. The resolution element is 3.5 pixels and the dispersion is 0.08 \AA /pix (0.12 \AA /pix) for blue (red) channels.}
\label{fig:snratio}
\end{figure}
\subsection{Measurement of Radial Velocities}
\label{subsec:ccf}
We measure radial velocity using the FXCOR task, which cross-correlates every continuum-subtracted spectrum against a stellar template spectrum of known redshift. For the template we use the sum of 428 spectra from bright, late-type stars, including 155 spectra of radial velocity standard stars and 273 spectra of stars from 7 Milky Way globular clusters. All spectra contributing to the template were obtained with echelle spectrographs, either during a previous radial velocity study \citep{walker06a} or with MMFS as part of the present study, and all have spectral resolution equal to or higher than those of the target spectra. We co-add these individual spectra after shifting each to a common redshift.
For cross-correlation, all spectra are re-binned linearly with $\log$($\lambda/$\AA) at a common dispersion of $3.11 \times 10^{-6}$ pix$^{-1}$. Fourier transforms of the target and template spectra are filtered such that low-frequency continuum residuals and undersampled features at high frequencies do not contribute to the resulting cross-correlation function (CCF). The right panels of Figures \ref{fig:bscl5spectra} and \ref{fig:bsex6spectra} display CCFs calculated from the spectra shown in the left panels. The relative velocity between object and template is the velocity value at the center of a Gaussian profile fit to the tallest CCF peak (within a window of width 700 km s$^{-1}$, centered on the velocity of the target dSph). From the CCF, the known redshift of the template, the time of observation, and coordinates of the object, FXCOR returns the radial velocity in the solar rest frame.
\subsection{Velocity Zero Point}
\label{subsec:zeropoint}
The solar velocity measurements in Figure \ref{fig:twilights} show evidence of small residual offsets that depend on channel and aperture. We correct for these offsets by demanding that the many repeat velocity measurements obtained from solar twilight spectra yield mean velocities of 0 km s$^{-1}$ in each aperture in each run. We obtained sets of $2-7$ independent velocity measurements per aperture, per observing run, from solar twilight spectra (for a total of 6816 twilight spectra). We subtract the mean velocity of each such set from all velocity measurements (including those from twilight exposures, science targets and radial velocity standard stars) obtained using the same aperture in the same observing run; this places velocities measured in a given run on a common zero point. The mean value subtracted from blue (red) apertures is $0.48$ km s$^{-1}$ ($1.07$ km s$^{-1}$ ), and the corrections have a standard deviation of $0.48$ km s$^{-1}$ ($0.78$ km s$^{-1}$). The standard deviation from zero among corrected twilight RVs is 0.26 km s$^{-1}$ (0.60 km s$^{-1}$) for blue (red) apertures; we adopt this value as the baseline velocity error in the error analysis (Section \ref{subsec:errors}).
Because the template spectrum is composed of late-type stellar spectra, velocities measured from solar spectra are susceptible to object/template mismatch. While we consider the the zero-point corrections described above to be sufficient for removing channel-and aperture-dependent offsets within a given run, we next use dSph targets observed with the same channel in two or more runs to correct for variation in the velocity zero point between observing runs. For such stars measured on blue (red) channels, Figure \ref{fig:runshiftb} (Figure \ref{fig:runshiftr}) plots the difference between measured velocities as a function of the earlier measured velocity. Histograms indicate the distribution of measurement differences. In order to bring the center of each distribution to zero we apply constant offsets to all velocities measured in a given channel during a given run. Arbitrarily we choose the run/channel having the smallest mean velocities to serve as the absolute zero point. Table \ref{tab:runshift} summarizes the applied corrections. For each MMFS run listed in column 1, columns 2 and 3 list the number of stars on blue and red channels that were observed in the same channel during at least one other run. Columns 3 and 4 list the blue and red offsets added to every velocity from the run. The maximum correction is $1.7$ km s$^{-1}$, slightly smaller than the median measurement error (Section \ref{subsec:errors}).
\begin{figure}
\epsscale{1.2}
\plotone{smallf9.eps}
\caption{ Distribution of measurement deviations for stars measured with the blue channel in multiple observing runs. Left-hand sub-panels plot velocity deviation as a function of the (chronologically) first measurement; histograms in the right-hand subpanels give the distribution of $\Delta$V. We observed a given star in up to three different runs. Filled circles and solid histograms result from comparison with the earlier of the two other runs; open circles and dotted histograms result from comparison with the latter of the two other runs.}
\label{fig:runshiftb}
\end{figure}
\begin{figure}
\epsscale{1.2}
\plotone{smallf10.eps}
\caption{ Same as Figure \ref{fig:runshiftb} but for stars observed with the red channel in multiple runs.}
\label{fig:runshiftr}
\end{figure}
\renewcommand{\arraystretch}{0.6}
\begin{deluxetable}{cccccc}
\tabletypesize{\scriptsize}
\tablewidth{0pc}
\tablecaption{ Run-dependent velocity zero-point corrections}
\tablehead{\colhead{MMFS run}&\colhead{N$_{blue}$}&\colhead{N$_{red}$}&\colhead{Blue correction}&\colhead{Red correction}\\
\colhead{}&\colhead{}&\colhead{}&\colhead{(km s$^{-1}$)}&\colhead{(km s$^{-1}$)}
}
\startdata
Mar. 2004&178&45&-1.65&-1.45\\
Oct. 2004&125&37&0.0&-0.57\\
Feb. 2005&194&44&-1.59&-1.31\\
Oct. 2005&185&73&-1.70&0.0\\
Mar. 2006&68&2&0.0&0.0\\
Oct. 2006&198&110&-1.08&0.0\\
\enddata
\label{tab:runshift}
\end{deluxetable}
As a check on the absolute RV zero point we consider 58 velocity measurements of 22 RV standard stars (after applying the zero point shifts described above). Table \ref{tab:std_veldata1} lists the individual MMFS velocity measurements for each standard, and for comparison lists previously published velocity measurements, from \citet{edo91,udry99,beers00,walker06a}. The mean deviation between the measured RV and the published value is 0.07 km s$^{-1}$, which indicates the zero point of the MMFS velocities is consistent with RV standards. In the case of HD 45282, \citet{beers00} estimate a measurement error of $\pm 10$ km s$^{-1}$; thus despite a deviation of 6 km s$^{-1}$, the MMFS measurement is consistent with the published value. Repeat MMFS measurements of RV standards show excellent agreement; the largest deviation among repeat measurements is 3.4 km s$^{-1}$, and 93\% of repeat measurements differ by less than 3 km s$^{-1}$.
\renewcommand{\arraystretch}{0.6}
\begin{deluxetable*}{lcrrrrrrrr}
\tabletypesize{\scriptsize}
\tablewidth{0pc}
\tablecaption{ MMFS Velocities of Radial Velocity Standards}
\label{tab:std_veldata1}
\tablehead{\colhead{ID}&\colhead{HJD}&\colhead{$V_{published}$}&\colhead{$V$}&\colhead{$V-V_{published}$}\\
\colhead{}&\colhead{(-2.45 $\times 10^6$ days)}&\colhead{(km s$^{-1}$)}&\colhead{(km s$^{-1}$)}&\colhead{(km s$^{-1}$)}
}
\startdata
HD 83212 & 3088.6241&$ 110.0\pm 1.0$\tablenotemark{3}&$ 107.4\pm 0.4$&$ -2.6\pm 1.1$\\
& 3088.6276&$ 110.0\pm 1.0$&$ 107.5\pm 0.4$&$ -2.5\pm 1.1$\\
& 3088.6321&$ 110.0\pm 1.0$&$ 109.2\pm 0.7$&$ -0.8\pm 1.2$\\
& 3088.6363&$ 110.0\pm 1.0$&$ 109.3\pm 0.7$&$ -0.7\pm 1.2$\\
HD 103545 & 3088.6441&$ 177.0\pm 1.5$\tablenotemark{4}&$ 178.9\pm 3.3$&$ 1.9\pm 3.6$\\
& 3088.6491&$ 177.0\pm 1.5$&$ 176.9\pm 1.3$&$ -0.1\pm 2.0$\\
& 3090.6633&$ 177.0\pm 1.5$&$ 177.6\pm 1.2$&$ 0.6\pm 1.9$\\
& 3090.6666&$ 177.0\pm 1.5$&$ 179.5\pm 3.4$&$ 2.5\pm 3.7$\\
& 3409.8842&$ 177.0\pm 1.5$&$ 177.2\pm 1.5$&$ 0.2\pm 2.1$\\
& 3417.8769&$ 177.0\pm 1.5$&$ 176.9\pm 1.2$&$ -0.1\pm 1.9$\\
& 3418.8698&$ 177.0\pm 1.5$&$ 177.8\pm 1.4$&$ 0.8\pm 2.0$\\
HD 45282 & 3089.4900&$ 301.0\pm 10.0$\tablenotemark{3}&$ 307.1\pm 2.4$&$ 6.1\pm 10.3$\\
HD 80170 & 3089.6553&$ 0.5\pm 0.2$\tablenotemark{2}&$ 0.5\pm 0.7$&$ 0.0\pm 0.8$\\
& 3089.6584&$ 0.5\pm 0.2$&$ -1.6\pm 0.3$&$ -2.1\pm 0.4$\\
& 3092.6314&$ 0.5\pm 0.2$&$ -2.8\pm 0.3$&$ -3.3\pm 0.4$\\
& 3092.6335&$ 0.5\pm 0.2$&$ -0.1\pm 0.7$&$ -0.6\pm 0.8$\\
HD 92588 & 3089.6641&$ 42.5\pm 0.3$\tablenotemark{2}&$ 40.8\pm 0.4$&$ -1.8\pm 0.5$\\
& 3089.6683&$ 42.5\pm 0.3$&$ 42.1\pm 0.7$&$ -0.4\pm 0.8$\\
& 3091.6315&$ 42.5\pm 0.3$&$ 41.9\pm 0.7$&$ -0.6\pm 0.8$\\
& 3091.6343&$ 42.5\pm 0.3$&$ 41.1\pm 0.3$&$ -1.4\pm 0.4$\\
HD 124358 & 3090.8064&$ 325.0\pm 1.0$\tablenotemark{3}&$ 323.6\pm 2.7$&$ -1.4\pm 2.9$\\
& 3090.8111&$ 325.0\pm 1.0$&$ 323.0\pm 0.7$&$ -2.0\pm 1.2$\\
HD 157457 & 3090.8115&$ 17.8\pm 0.3$\tablenotemark{2}&$ 16.1\pm 0.3$&$ -1.7\pm 0.4$\\
& 3090.8134&$ 17.8\pm 0.3$&$ 16.6\pm 0.8$&$ -1.2\pm 0.8$\\
HD 118055 & 3091.6431&$ -101.0\pm 1.0$\tablenotemark{3}&$ -101.8\pm 0.4$&$ -0.8\pm 1.1$\\
& 3091.6471&$ -101.0\pm 1.0$&$ -102.2\pm 0.9$&$ -1.2\pm 1.3$\\
& 3415.9002&$ -101.0\pm 1.0$&$ -103.2\pm 0.4$&$ -2.2\pm 1.1$\\
& 3417.8831&$ -101.0\pm 1.0$&$ -102.7\pm 0.4$&$ -1.7\pm 1.1$\\
& 3801.8875&$ -101.0\pm 1.0$&$ -100.2\pm 0.4$&$ 0.8\pm 1.1$\\
& 3801.8891&$ -101.0\pm 1.0$&$ -100.1\pm 0.4$&$ 0.9\pm 1.1$\\
HD 176047 & 3287.5020&$ -42.5\pm 0.2$\tablenotemark{2}&$ -41.8\pm 0.3$&$ 0.7\pm 0.4$\\
HD 196983 & 3287.5157&$ -9.1\pm 0.3$\tablenotemark{2}&$ -8.3\pm 0.3$&$ 0.8\pm 0.4$\\
& 4018.5032&$ -9.1\pm 0.3$&$ -9.3\pm 0.3$&$ -0.2\pm 0.4$\\
& 4027.4955&$ -9.1\pm 0.3$&$ -9.8\pm 0.3$&$ -0.7\pm 0.4$\\
& 4027.5121&$ -9.1\pm 0.3$&$ -7.0\pm 0.7$&$ 2.1\pm 0.7$\\
HD 219509 & 3287.5246&$ 67.5\pm 0.5$\tablenotemark{2}&$ 66.3\pm 1.0$&$ -1.2\pm 1.1$\\
CPD-432527 & 3287.8984&$ 19.7\pm 0.9$\tablenotemark{2}&$ 20.4\pm 0.3$&$ 0.7\pm 0.9$\\
& 3410.5182&$ 19.7\pm 0.9$&$ 17.9\pm 0.3$&$ -1.8\pm 0.9$\\
& 3665.8825&$ 19.7\pm 0.9$&$ 19.3\pm 0.3$&$ -0.4\pm 0.9$\\
HD 48381 & 3288.8948&$ 40.5\pm 0.2$\tablenotemark{2}&$ 41.6\pm 0.3$&$ 1.1\pm 0.3$\\
& 3289.8915&$ 40.5\pm 0.2$&$ 43.2\pm 0.3$&$ 2.7\pm 0.3$\\
& 3289.8939&$ 40.5\pm 0.2$&$ 43.4\pm 0.3$&$ 2.9\pm 0.3$\\
& 3666.8791&$ 40.5\pm 0.2$&$ 39.9\pm 0.3$&$ -0.6\pm 0.3$\\
SAO 201636 & 3408.8782&$ 264.3\pm 1.4$\tablenotemark{4}&$ 264.8\pm 1.0$&$ 0.5\pm 1.7$\\
& 3409.8670&$ 264.3\pm 1.4$&$ 264.2\pm 0.7$&$ -0.1\pm 1.6$\\
HD 83516 & 3409.8590&$ 43.5\pm 0.2$\tablenotemark{2}&$ 41.8\pm 0.3$&$ -1.7\pm 0.3$\\
& 3412.9035&$ 43.5\pm 0.2$&$ 42.0\pm 0.3$&$ -1.5\pm 0.3$\\
HD 111417 & 3410.8967&$ -19.1\pm 0.2$\tablenotemark{2}&$ -19.1\pm 0.6$&$ 0.0\pm 0.7$\\
& 3803.8877&$ -19.1\pm 0.2$&$ -18.7\pm 0.6$&$ 0.4\pm 0.7$\\
HD 43880 & 3411.5240&$ 43.6\pm 2.4$\tablenotemark{1}&$ 40.6\pm 0.3$&$ -2.9\pm 2.4$\\
& 3412.5257&$ 43.6\pm 2.4$&$ 40.3\pm 0.3$&$ -3.3\pm 2.4$\\
& 3801.5077&$ 43.6\pm 2.4$&$ 43.3\pm 0.3$&$ -0.3\pm 2.4$\\
& 3801.5090&$ 43.6\pm 2.4$&$ 43.2\pm 0.3$&$ -0.4\pm 2.4$\\
HD 93529 & 3411.9006&$ 143.0\pm 1.0$\tablenotemark{3}&$ 144.3\pm 0.3$&$ 1.3\pm 1.1$\\
HD 23214 & 3415.5220&$ -5.1\pm 0.9$\tablenotemark{4}&$ -6.8\pm 0.3$&$ -1.7\pm 1.0$\\
HD 21581 & 3416.5260&$ 151.3\pm 1.3$\tablenotemark{4}&$ 151.7\pm 0.5$&$ 0.4\pm 1.4$\\
SAO 217998 & 3417.5155&$ 19.0\pm 1.6$\tablenotemark{4}&$ 17.9\pm 0.3$&$ -1.1\pm 1.6$\\
HD 223311 & 4018.5258&$ -20.2\pm 0.3$\tablenotemark{2}&$ -20.0\pm 0.4$&$ 0.2\pm 0.5$\\
\enddata
\tablenotetext{1}{\citet{edo91}}
\tablenotetext{2}{\citet{udry99}}
\tablenotetext{3}{\citet{beers00}}
\tablenotetext{4}{\citet{walker06a}}
\end{deluxetable*}
\subsection{Quality Control}
\label{subsec:qc}
The right panels of Figures \ref{fig:bscl5spectra} and \ref{fig:bsex6spectra} demonstrate the range in quality of the CCF peaks we obtain from the MMFS spectra. In order to help eliminate poorly measured velocities from the sample, we inspect all CCFs by eye and assign to each a pass/fail grade according to the following criteria. First, the tallest peak (within the search window centered on the dSph systemic velocity) in a satisfactory CCF must be unambiguous. CCFs in which the height of one or more secondary peaks reaches 0.8 times that of the tallest peak are flagged as unsatisfactory. Second, the tallest peak in a satisfactory CCF must be symmetric about the measured velocity. Gaussian fits to asymmetric peaks can result in substantial velocity errors; we therefore flag as unsatisfactory those CCFs having significant residuals with respect to the Gaussian fit. According to these criteria, the CCFs in the top two panels of both Figure \ref{fig:bscl5spectra} and Figure \ref{fig:bsex6spectra} are considered unsatisfactory.
A second, quantitative quality control filter is provided by the Tonry-Davis value, $R_{TD}$ (calculated by FXCOR), which indicates the height of the tallest CCF peak relative to the average CCF peak \citep{tonry79}. Figure \ref{fig:tdr} plots, separately for all the dSph stellar and blank-sky spectra measured in blue and red channels, the distributions of $R_{TD}$ associated with CCFs judged by eye to be satisfactory (solid histograms) and unsatisfactory (dotted histograms). We expect blank-sky spectra to yield poor CCFs; this is generally the case, as indicated by the small number of sky spectra judged by eye to give satisfactory CCFs and the relatively low $R_{TD}$ values associated with these measurements. Among stellar spectra, qualitatively satisfactory CCFs have $2.8 \leq R_{TD} \leq 40$, with median $R_{TD}=9.12$. We define a critical value, $R_{TD,c}$, to be the value of $R_{TD}$ at which, absent other information, the probability that a stellar CCF is satisfactory equals the probability that it is unsatisfactory. For spectra obtained with MIKE's blue channel this occurs at $R_{TD,c}=4.0$; for the red channel $R_{TD,c}=4.5$. On the basis of Figure \ref{fig:tdr}, we allow a given stellar velocity measurement into the final sample only if it is derived from a CCF that 1) is judged by eye to be satisfactory, and 2) has $R_{TD}\geq R_{TD,c}$.
\begin{figure}
\epsscale{1.2}
\plotone{smallf11.eps}
\caption{ Distributions of $R_{TD}$ associated with measured cross-correlation functions. Panels indicate whether the distribution corresponds to stellar or blank-sky spectra obtained in MIKE's blue or red channel. In each panel, the solid/dotted histogram is the distribution of $R_{TD}$ associated with CCFs judged by eye inspection to be satisfactory/unsatisfactory.}
\label{fig:tdr}
\end{figure}
\section{Velocity Measurement Errors}
\label{subsec:errors}
\subsection{Gaussian Errors}
\label{subsec:gaussianerrors}
For the majority of stars in our sample we have obtained only a single velocity measurement and direct estimation of the measurement error is impossible. Instead we use the many existing cases of independent repeat measurements to relate the errors to quantifiable features of individual spectra. Let $V_{ij}$ be the $i^{th}$ of $n_j$ independent radial velocity measurements obtained for the $j^{th}$ of $N$ stars in the data set, and suppose the $j^{th}$ star has true radial velocity $V_{*j}$. We adopt a model in which the velocity measurement error, $V_{ij}-V_{*j}$, is the sum of two independent Gaussian components. First, noise in the stellar spectrum gives rise to noise in the cross-correlation function. We assume the error component due to CCF noise follows a Gaussian distribution with variance $\sigma_{CCF}^2$. The finite spectral resolution of the instrument contributes a second error component that is independent of signal-to-noise ratio and defines the maximum precision of the instrument. We assume this ``baseline'' error follows a Gaussian distribution with constant variance, $\sigma_0^2$. The error in a given velocity measurement is then
\begin{equation}
V_{ij}-V_{*j}=(\sigma_{CCF,ij}^2+\sigma_0^2)^{1/2}\epsilon_{ij}.
\label{eq:errormodel}
\end{equation}
The values $\epsilon_{ij}$, which account for the inherent randomness of measurement, follow the standard normal (Gaussian with mean zero and unit variance) distribution. The resulting error bars are symmetric about $V_{ij}$, and given by $\pm \sigma_{V,ij}$, where $\sigma_{V,ij}=(\sigma_{CCF,ij}^2+\sigma_0^2)^{1/2}$.
We model $\sigma_{CCF}$ as in \citet{walker06a}:
\begin{equation}
\sigma_{CCF,ij}=\frac{\alpha}{(1+R_{TD,ij})^x}.
\label{eq:tdr}
\end{equation}
Except for the introduction of the parameter $x$, this is essentially the model of \citet{tonry79}, in which $\sigma_{CCF} \propto (1+R_{TD})^{-1}$.
After substituting for $\sigma_{CCF}$ in Equation \ref{eq:errormodel}, taking the base-ten logarithm gives
\begin{equation}
\log[(V_{ij}-V_{*j})^2]=\log \biggl[\frac{\alpha^2}{(1+R_{TD,ij})^{2x}}+\sigma_0^2 \biggr ] + \log[\epsilon_{ij}^2].
\label{eq:logerrormodel1}
\end{equation}
For normally distributed $\epsilon_{ij}$, the values $\epsilon_{ij}^2$ follow the $\chi^2$ distribution (one degree of freedom). From Monte Carlo simulations, $\log [\epsilon_{ij}^2]$ has mean value $\langle \log[\epsilon_{ij}^2] \rangle = -0.55$. If we define $\delta_{ij}\equiv \log [\epsilon_{ij}^2] +0.55$, then
\begin{equation}
\log[(V_{ij}-V_{*j})^2]=\log \biggl[\frac{\alpha^2}{(1+R_{TD,ij})^{2x}}+\sigma_0^2 \biggr ] + \delta_{ij}-0.55,
\label{eq:logerrormodel2}
\end{equation}
and $\langle \delta_{ij}\rangle = 0$. For the baseline variance we adopt the standard deviation of velocities measured from twilight spectra. The CCFs from these spectra generally have $R_{TD} \geq 30$, a regime in which $\sigma_{CCF}$ is negligible. From Section \ref{subsec:zeropoint}, $\sigma_0=0.26$ km s$^{-1}$ ($\sigma_0=0.60$ km s$^{-1}$) for blue (red) channels. The parameter pair $\{x,\alpha\}$ can then be estimated, via nonlinear regression, as that which minimizes the sum
\begin{equation}
\displaystyle\sum_{j=1}^N\displaystyle\sum_{i=1}^{n_j} \biggl (\log[(V_{ij}-V_{*j})^2]- \biggl [ \log \biggl (\frac{\alpha^2}{(1+R_{TD,ij})^{2x}}+\sigma_0^2 \biggr )-0.55 \biggr ] \biggr )^2
\label{eq:nonlinear}
\end{equation}
We estimate $\{x,\alpha\}$ separately for blue and red channels. We consider only dSph candidate stars with multiple, independent velocity measurements that pass quality control filters (Section \ref{subsec:qc}). For the blue (red) channel we use 1249 (561) measurements of 583 (257) stars. We replace the unknown $V_{*j}$ with an estimate computed from the weighted mean, $\hat{V}_{*j}=\sum_{i=1}^{n_j}w_{ij}V_{ij}/\sum_{i=1}^{n_j}w_{ij}$. For weights we use $w_{ij}=(\sigma_{FX,ij}^2+\sigma_0^2)^{-1}$, where $\sigma_{FX,ij}$ is the initial velocity error returned by FXCOR (to first order, $\sigma_{FX,ij}$ is proportional to $\sigma_{CCF,ij}$). This substitution introduces a bias, as it ignores the variance in the estimate $\hat{V}_{*j}$. However, analysis of repeat measurements (Section \ref{subsec:repeats}) demonstrates that resulting error estimates are reasonable.
The top panels in Figure \ref{fig:xalphafit} plot (for blue and red channels) the nonlinear regression data in the $\log[(V_{ij}-\hat{V}_{*j})^2]$ versus $(1+R_{TD,ij})$ plane. Overplotted is the best-fit curve. We note the presence of outliers that have $|V_{ij}-\hat{V}_{*j}|$ up to $\sim 250$ km s$^{-1}$ (middle panels of Figure \ref{fig:xalphafit}). The broad velocity distribution of these outliers suggests an error component not considered in the model specified by Equation \ref{eq:errormodel}. We address non-Gaussian error in Section \ref{subsec:nongaussianerrors}.
Table \ref{tab:errors} summarizes the estimated parameters of the Gaussian error model. For the blue (red) channel, the best-fit curve to the regression data corresponds to the parameter pair $\{x,\alpha\}=\{1.69,81.0$ km s$^{-1}\}$ ($\{1.65,107.9$ km s$^{-1}\}$). Using these values, we calculate error bars $\pm \sigma_{V,ij}$ as a function of $R_{TD}$ (bottom panels of Figure \ref{fig:xalphafit}). Velocities measured from minimally acceptable CCFs (those with $R_{TD}=R_{TD,c}$), have $\sigma_{V,ij}=5.3$ km s$^{-1}$ (blue) and $\sigma_{V,ij}=6.5$ km s$^{-1}$ (red). Among all acceptable measurements, median errors are $\sigma_{V,ij}=1.6$ km s$^{-1}$ (blue) and $\sigma_{V,ij}=2.4$ km s$^{-1}$ (red).
\begin{figure}
\epsscale{1.2}
\plotone{smallf12.eps}
\caption{ \textit{Top:} Data from repeat velocity measurements, used for estimating error model parameter pair $\{x,\alpha\}$, with best-fit curve overplotted. Left/right panels show data for blue/red channels. \textit{Middle:} Deviation of measured velocity from estimated true velocity. \textit{Bottom}: Size of error bar resulting from parameter pair $\{x,\alpha\}$ that yields the best-fit curve. In all panels, dotted, vertical lines indicate the minimally acceptable value of $R_{TD}$ (section \ref{subsec:qc}).}
\label{fig:xalphafit}
\end{figure}
\renewcommand{\arraystretch}{0.6}
\begin{deluxetable*}{lrrrrrrr}
\tabletypesize{\scriptsize}
\tablewidth{0pc}
\tablecaption{ Summary of Parameters for Gaussian Error Model}
\tablehead{\colhead{Channel}&R$_{TD,c}$&\colhead{$N_{repeats}$}&\colhead{$\sigma_0$}&\colhead{$\hat{x}$}&\colhead{$\hat{\alpha}$}&\colhead{$\sigma_{V,median}$}&\colhead{$\sigma_{V,max}$}\\
\colhead{}&\colhead{}&\colhead{}&\colhead{(km s$^{-1}$)}&\colhead{}&\colhead{(km s$^{-1}$)}&\colhead{(km s$^{-1}$)}&\colhead{(km s$^{-1}$)}
}
\startdata
Blue&4.0&1249&0.26&1.69&81.0&1.6&5.3\\
Red&4.5&561&0.60&1.65&107.9&2.4&6.5
\enddata
\label{tab:errors}
\end{deluxetable*}
\subsection{Non-Gaussian Errors}
\label{subsec:nongaussianerrors}
The adopted error model assumes the measurement error is the sum of Gaussian components. Figure \ref{fig:xalphafit} shows that the several outliers from the main distribution of regression data (top panels) correspond to outliers in the $V_{ij}-\hat{V}_{*j}$ distribution (middle panels). The distribution of $V_{ij}-\hat{V}_{*j}$ among these outliers indicates an additional, non-Gaussian error component not considered in Equation 2. For stars with repeat observations it is straightforward to identify and discard affected measurements; this is not possible for stars with only a single measurement. However, we show that only a negligible number of measurements affected by large non-Gaussian errors are likely to be included in dSph samples.
Potential sources of non-Gaussian errors include astrophysical velocity variability (e.g., binary stars), but the most plausible origin of large $V_{ij}-\hat{V}_{*j}$ outliers is the selection of a false peak from a noisy CCF. The outlying data points occur preferentially at small $R_{TD}$, a regime in which CCFs are relatively noisy and odds of selecting a false peak increase. Adjacent peaks in CCFs measured from our spectra are separated typically by $50-100$ km s$^{-1}$. Assuming false peaks occur at no preferred velocity within the CCF, velocities derived from false peaks will result in large deviations that follow a uniform velocity distribution. This is consistent with the observed distribution of $V_{ij}-\hat{V}_{*j}$ among outliers (middle panels in Figure \ref{fig:xalphafit}).
To what extent does false peak selection affect our data set? Among the dSph candidate stars with multiple measurements, we consider cases of false peak selection to be associated with measurements having $|V_{ij}-\hat{V}_{*j}| > 30$ km s$^{-1}$ and $R_{TD} < 6$; we do not consider $V_{ij}-\hat{V}_{*j}$ outliers with $R_{TD} \geq 6$ to be false-peak measurements because all such cases can be attributed to a poor estimate of $V_{*j}$ due to an associated repeat measurement having $R_{TD} < 6$. For the blue (red) channel, 22 of 1249 (6 of 561) repeat measurements that pass both of the initial quality control filters (Section \ref{subsec:qc}) meet the false-peak criterion. Combining results from both channels yields 486 additional repeat measurements that pass the quality control filters; ten of these meet the false peak criterion. Extrapolating to the entire sample, we expect fewer than $2\%$ of measurements passing quality control filters to be derived from false CCF peaks. Cases that are identifiable via comparison to repeat measurements are removed from the sample, but we expect $\sim 70$ cases of false peak selection to remain among the 4175 stars for which we obtained only a single measurement. However, if the distribution of unidentified false-peak velocities is uniform over a range spanning at least 400 km s$^{-1}$, as Figure \ref{fig:xalphafit} suggests, we expect only $10-15\%$ to fall within the range of a typical dSph velocity distribution. This results in only a handful of false-peak velocities spread among the four dSph member samples. This level of contamination is below that expected to be introduced by cases of ambiguous dSph membership.
\subsection{Repeat Measurements}
\label{subsec:repeats}
After removing identified cases of false $CCF$ peaks, the final MMFS data set contains repeat velocity measurements for 1011 dSph target stars. There are 2246 independent measurements of these stars, including up to five measurements for some stars. If we calculate the weighted mean velocity for the $j^{th}$ star as $\bar{V}_{j}=\frac{\sum_{i=1}^{n_j}(w_{ij}V_{ij})}{\sum_{i=1}^{n_j}w_{ij}}$, using weights $w_{ij}=\sigma_{V,ij}^{-2}$, then the Values $V_{ij}-\bar{V}_j$ should follow a Gaussian distribution with variance
\begin{equation}
\sigma_{\Delta V,ij}^2 \equiv Var\bigl (V_{ij}-\bar{V}_j) =\frac{1}{w_{ij}} \bigl (1-\frac{w_{ij}}{K} \bigr )^2+\frac{1}{K^2}\sum_{k\neq i}^{n_j}w_{kj},
\label{eq:repeatvariance}
\end{equation}
where $K \equiv \Sigma_{i=1}^{n_j} w_{ij}$. Figure \ref{fig:dsph_velrepeats} plots the distribution of $(V_{ij}- \bar{V}_{j})/\sigma_{\Delta V,ij}$ for all dSph targets with repeat velocity measurements. A Gaussian fit to the empirical distribution has standard deviation $\sigma=1.28$. The excess of this value with respect to the nominal value of one is due in large part to the presence of outliers and is not surprising given the likely presence of binaries in the sample.
\begin{figure}
\epsscale{1.2}
\plotone{smallf13.eps}
\caption{ For all stars with repeat MMFS measurements, the distribution of $V-\bar{V}$, normalized by the estimated error. Overplotted is the best-fitting Gaussian distribution (solid line), which is nearly identical to the (vertically scaled) normal distribution with unit variance (dotted line).}
\label{fig:dsph_velrepeats}
\end{figure}
\section{Comparison with Previous Work}
The MMFS sample has stars in common with several previously published RV data sets---\textbf{Carina}: \citet{mateo93,majewski05,munoz06}. \textbf{Fornax}: \citet{mateo91,walker06a}. \textbf{Sculptor}: \citet{armandroff86,queloz95}. \textbf{Sextans}: \citet{dacosta91,suntzeff93,hargreaves94,kleyna04}. Among previous studies, only \citet{mateo91}, \citet{queloz95}, and \citet{walker06a} measure RV using the magnesium triplet. All others use the infrared calcium triplet (CaT) near 8500 \AA. The MMFS sample offers the first opportunity to compare directly large numbers of velocity measurements obtained using both techniques.
To identify MMFS stars in common with other samples, we searched published data sets for coordinate matches, tolerating offsets of up to $0.2\arcsec - 1.5\arcsec$ depending on the sample. For each detected matching pair, Figure \ref{fig:previouswork1} plots the velocity deviation $\Delta V \equiv V_{\rm{MMFS}}-V_{\rm{other}}$ as a function of $V_{\rm{MMFS}}$. Figure \ref{fig:previouswork2} plots the same quantities over a window of width 50 km s$^{-1}$, centered on the systemic velocity of the dSph. Also shown in both figures are internal comparisons using stars with repeat MMFS measurements. In each panel of Figures \ref{fig:previouswork1} and \ref{fig:previouswork2} the marker type indicates the relative quality of the RV measurement. Filled squares denote stars for which both the MMFS and comparison measurements have quoted uncertainties less than the median uncertainty in their respective sample. Open squares/triangles denote stars for which the MMFS measurement has uncertainty less/greater than the median MMFS uncertainty and the comparison measurement has uncertainty greater/less than the median from its sample. Crosses denote stars for which both the MMFS and comparison measurements have uncertainties greater than the respective sample median. We find that the best measurements (solid squares) indeed provide the closest agreement between samples. Figure \ref{fig:previouswork3} plots for each comparison sample the distribution of $\Delta V/\sqrt{\sigma_{\rm{MMFS}}^2+\sigma_{\rm{other}}^2}$.
Table \ref{tab:comparison} summarizes the comparisons. For each comparison sample, columns 3-6 list the primary absorption features used, the spectral resolution (if available), the number of stars in common with the MMFS sample, and the number of MMFS measurements for the common stars. Columns 7-8 list the mean velocity offset and standard deviation calculated from all matching pairs. Because these values are susceptible to the effects of even a single outlier (which may be due to a mis-identified match, a binary star, or a false CCF peak), columns 9-10 list biweight estimates \citep{beers90} of the mean offset and scatter. While the biweight is resistant to outliers, it is known to perform poorly when the number of matches is small ($N < 10$).
\renewcommand{\arraystretch}{0.6}
\begin{deluxetable*}{llllrrrrrrrr}
\tabletypesize{\scriptsize}
\tablewidth{0pc}
\tablecaption{ Comparison of RV data with previous results}
\tablehead{\colhead{Galaxy}&\colhead{Author}&\colhead{$\lambda$}&\colhead{Resolution}&\colhead{$N_*$}&\colhead{$N_{V}$}&\colhead{$\langle \Delta V \rangle$}&\colhead{$\sigma_{\rm{rms}}$}&\colhead{$\langle \Delta V \rangle_{\rm{BW}}$}&\colhead{$\sigma_{\rm{BW}}$}&\colhead{slope}&\colhead{p}\\
\colhead{}&\colhead{}&\colhead{}&\colhead{}&\colhead{}&\colhead{}&\colhead{(km s$^{-1}$)}&\colhead{(km s$^{-1}$)}&\colhead{(km s$^{-1}$)}&\colhead{(km s$^{-1}$)}&\colhead{}&\colhead{}
}
\startdata
Carina&\citet{mateo93}&MgT&21000&5&6& -2.34& 3.54& -0.45& 3.75& 0.27& 0.31200\\
Carina&\citet{majewski05}&CaT&2600,7600\tablenotemark{a}&17& 20& 40.54& 123.32& -2.04& 12.51& -0.23& 0.39328\\
Carina&\citet{munoz06}\tablenotemark{b}&CaT&19000&11&17& 2.28& 7.94& 0.62& 5.06& 0.49& 0.01460\\
Carina&\citet{munoz06}\tablenotemark{c}&CaT&6500&218&252& -0.98& 26.88& 0.51& 6.05& 0.13& 0.15372\\
Sextans&\citet{dacosta91}&CaT&2600&6&14& -5.58& 14.88& -2.41& 20.71& 1.15& 0.08198\\
Sextans&\citet{suntzeff93}&CaT&2800&4&14& -5.53& 6.35& -1.86& 7.50& 0.018& 0.88844\\
Sextans&\citet{hargreaves94}&CaT&---&11&32& -0.42& 5.26& -0.61& 4.64& 0.050& 0.68364\\
Sextans&\citet{kleyna04}&CaT&---&45&92& -6.84& 24.80& -0.80& 5.46& 0.34& 0.00178\\
Fornax&\citet{mateo91}&MgT&21000&3&3& 2.41& 2.82& 0.40& 1.40& 0.37& 0.14845\\
Fornax&\citet{walker06a}&MgT&21000&58&72& 3.33& 7.76& 1.62& 5.23& 0.094& 0.07913 \\
Sculptor&\citet{armandroff86}&CaT&---&1&1&13.32&\nodata&\nodata&\nodata&\nodata&\nodata\\
Sculptor&\citet{queloz95}&MgT&16000&5&6& 2.94& 6.56 & 0.42 & 2.62 & 0.90& 0.02979\\
\enddata
\tablenotetext{a}{Instrumental setup differed over two observing runs; see \citet{majewski05} for details.}
\tablenotetext{b}{using Magellan+MIKE}
\tablenotetext{c}{using VLT+GIRAFFE}
\label{tab:comparison}
\end{deluxetable*}
\begin{figure*}
\epsscale{1}
\plotone{smallf14.eps}
\caption{ Comparison of MMFS velocity measurements with previous results. For stars in common with the specified RV sample, each panel plots the difference in measured velocity ($\Delta V \equiv V_{\rm{MMFS}}-V_{{other}}$) as a function of the MMFS velocity. The bottom subpanel within each panel plots internal comparisons from stars with multiple MMFS measurements. Filled squares denote stars for which both the MMFS and comparison measurements have quoted uncertainties less than the median uncertainty in their respective sample. Open squares/triangles denote stars for which the MMFS measurement has uncertainty less/greater than the median MMFS uncertainty and the comparison measurement has uncertainty larger/less than the median from its sample. Crosses denote stars for which both the MMFS and published measurements have uncertainties greater than the respective sample median. Histograms in right-hand sub-panels, individually normalized for legibility, indicate distributions of $\Delta V$.}
\label{fig:previouswork1}
\end{figure*}
\begin{figure*}
\epsscale{1}
\plotone{smallf15.eps}
\caption{ Comparison of MMFS velocity measurements with previous results, within a window spanning the approximate velocity range of dSph member stars. Symbols have the same meanings as in Figure \ref{fig:previouswork1}. The dotted line in each panel provides the best fit to the plotted points. The steepness of the slopes in the Kleyna et al.\ (2004) and Mu\~{n}oz et al.\ (2006 w/ MIKE) comparisons are significant at the 99.8\% and 98.5\% levels, respectively.}
\label{fig:previouswork2}
\end{figure*}
\begin{figure*}
\epsscale{1}
\plotone{smallf16.eps}
\caption{ For MMFS stars in common with previously published samples, distribution of velocity deviations $\Delta V \equiv V_{\rm{MMFS}}-V_{{other}}$ normalized by combined errors $\sigma=\sqrt{\sigma_{MMFS}^2+\sigma_{other}^2}$.}
\label{fig:previouswork3}
\end{figure*}
We find generally acceptable agreement with the results of \citet{dacosta91,mateo91,mateo93,suntzeff93,hargreaves94,queloz95,majewski05}. Despite large apparent deviations, we include the \citet{dacosta91} and \citet{majewski05} samples in this list because both studies report large measurement uncertainties (10-20 km s$^{-1}$ and 4-10 km s$^{-1}$, respectively). The somewhat large offset and standard deviation with respect to the \citet{queloz95} Sculptor sample are due to a single star; the remaining five matching stars show excellent velocity agreement. The single MMFS star in common with the \citet{armandroff86} Sculptor sample happens to be the same star giving the most deviant result in the Queloz et al.\ comparison; notice that for this star the Armandroff et al.\ and Queloz et al.\ measurements agree. Unless the MMFS measurement is in error, it is likely that this star is a binary, consistent with the conclusion of Queloz et al.\ that Sculptor has a binary fraction of $\sim 20\%$. Other than this, we can conclude little from the single star in common with the \citet{armandroff86} sample. Similarly, the three stars in common with the \citet{mateo91} Fornax sample offer little basis for comparison.
The comparison to the Fornax sample of \citet{walker06a} shows evidence for a zero-point offset of $\sim 2$ km s$^{-1}$, and the scatter is somewhat larger than expected from quoted measurement errors. Here we can rule out mis-identified coordinate matches, as MMFS targets are chosen from the same target list used by \citet{walker06a}. While it remains possible that some disagreement is due to binary stars, we note that the most deviant measurements correspond to stars for which the \citet{walker06a} measurement has higher uncertainty than the median for that sample. If the measurement errors for the \citet{walker06a} stars are 1.3 times larger than quoted, the distribution of $\Delta V$ becomes that which we expect for Gaussian errors. We note that the error model given by Equation \ref{eq:errormodel} improves upon that used in \citet{walker06a}, which does not take the baseline variance into account.
The \citet{kleyna04} and \citet{munoz06} samples offer relatively large numbers of common stars (45 and 229, respectively) with which to compare MMFS velocities. The Mu\~{n}oz et al.\ sample consists of two data sets acquired independently using Magellan+MIKE (in slit mode; 11 MMFS stars) and VLT+GIRAFFE (fibers; 218 MMFS stars); we consider each sample separately. The most striking feature we find in these comparisons is the apparent correlation of $\Delta V$ with $V_{\rm{MMFS}}$. With respect to the VLT+GIRAFFE data, this manifests as a tendency toward ($V_{\rm{MMFS}}-V_{\rm{VLT}})<0$ km s$^{-1}$ when $V_{\rm{MMFS}}< 75$ km s$^{-1}$ (Figure \ref{fig:previouswork1}). We note that the stars with largest VLT uncertainties and smallest MMFS uncertainties (open squares) are most responsible for this trend. Moreover, for the several stars that have incompatible MMFS and VLT velocities \textit{and} have multiple MMFS measurements, we find that the MMFS measurements show good internal agreement.
In comparison to the Mu\~{n}oz et al.\ MIKE data for Carina and the \citet{kleyna04} data for Sextans, we find apparent correlations between $\Delta V$ and $V_{\rm{MMFS}}$ over the velocity distribution characteristic of the dSph member stars. Dotted lines in each panel of Figure \ref{fig:previouswork2} give least-squares linear fits to the plotted points. Columns 11-12 in Table \ref{tab:comparison} give the slope of each line and the probability that such a slope would be observed in the absence of a correlation (determined via 100000 Monte Carlo realizations that assume the same $V_{\rm{MMFS}}$ values and draw $\Delta V$ randomly from a Gaussian distribution with variance equal to the standard deviation of $\Delta V$ for points plotted in Figure \ref{fig:previouswork2}). The apparent trends observed with respect to the Mu\~{n}oz et al.\ (MIKE) and Kleyna et al.\ comparisons have slopes $\rm{d}(\Delta V)/\rm{d}V_{\rm{MMFS}}=0.49$ and $0.34$, respectively, and these are significant at the 98.5\% 99.8\% levels. Over the range of velocities consistent with dSph membership, these slopes amount to systematic deviations of up to 10 km s$^{-1}$.
That we observe such behavior in the largest available comparison samples suggests there may be systematic differences between survey data. At this point we can only speculate regarding the source of such a systematic trend and whether it might originate in the MMFS or the two previously published samples. Perhaps the most significant observational difference between ours and the other large dSph surveys is that we observe at the MgT rather than the CaT. Whereas late-type stars are brighter in the infrared, the primary benefit we enjoy by using the MgT is the relatively clean sky. In contrast, the CaT region is prone to contamination by atmospheric -OH emission lines (see \citealt{osterbrock96}). At redshifts characteristic of Carina and Sextans ($\sim 225$ km s$^{-1}$), strong sky lines fall directly on top of lines 1 and 2 of the CaT. \citet{kleyna04} note that, when using CaT lines individually to measure velocities, lines 1 and 2 tend to give spurious Sextans-like velocities of 233 and 223 km s$^{-1}$, respectively, due to contribution from sky residuals to the cross-correlation function (see their Figure 1). In the end, \citet{kleyna04} use lines 2 and 3 jointly for their velocity measurements. If the inclusion of line 2 tends to ``pull'' their velocities toward 223 km s$^{-1}$, this could account for the observed gradient in Figure \ref{fig:previouswork2}. Mu\~{n}oz et al.\ (w/ MIKE) measure Carina velocities using $\sim 15$ absorption features, including all three CaT lines. Because Carina's radial velocity in the solar rest frame is nearly identical to that of Sextans, one might expect a similar effect from sky residuals. Indeed we observe in comparison to the Mu\~{n}oz et al.\ (w/MIKE) sample a slope similar in magnitude and significance to those obtained in the comparison to the Kleyna et al.\ sample.
\section{Spectral Line Indices}
Our MMFS observing program was designed chiefly to measure stellar velocities at low S/N. While our spectra are not suitable for measuring equivalent widths, nearly all are of sufficient quality for estimating line strengths via spectral indices. Let $S(\lambda)$ represent the observed spectrum and suppose $\lambda_{f_1}$ and $\lambda_{f_2}$ are lower and upper boundaries, respectively, of a bandpass containing an absorption feature of interest. A ``pseudo-'' equivalent width is given by the spectral line index defined, for example, by Equation A2 of Cenarro et al.\ (2001; see also \citealt{gonzalez93,cardiel98}):
\begin{equation}
W(\mathrm{\AA})\equiv \int_{\lambda_{f_{1}}}^{\lambda_{f_{2}}}[1-S(\lambda)/C(\lambda)]d\lambda,
\label{eq:index}
\end{equation}
where $C(\lambda)$ is is an estimate of the continuum flux. Letting $\lambda_{r_1}$,$\lambda_{r_2}$ and $\lambda_{b_1}$,$\lambda_{b_2}$ denote boundaries of nearby continuum regions redward and blueward, respectively, of the feature bandpass, $C(\lambda)$ is given by \citet{cenarro01}:
\begin{equation}
C(\lambda) \equiv S_b \frac{\lambda_r-\lambda}{\lambda_r-\lambda_b}+S_r \frac{\lambda-\lambda_b}{\lambda_r-\lambda_b}.
\end{equation}
Here $\lambda_r$ and $\lambda_b$ are the central wavelengths of the red and blue continuum bandpasses, respectively, and $S_b$ and $S_r$ are estimates of the mean level in the two continuum bands:
\begin{eqnarray*}
S_b \equiv (\lambda_{b_{2}}-\lambda_{b_{1}})^{-1}\int_{\lambda_{b_{1}}}^{\lambda_{b_{2}}}S(\lambda)\rm{d}\lambda;\\
S_r \equiv (\lambda_{r_{2}}-\lambda_{r_{1}})^{-1}\int_{\lambda_{r_{1}}}^{\lambda_{r_{2}}}S(\lambda)\rm{d}\lambda.
\label{eq:sbsr}
\end{eqnarray*}
The Lick system of indices \citep{burstein84,faber85,worthey94}, while in wide use, is not applicable to our spectra, as the relevant Lick magnesium passband alone is larger than our entire spectral range! Instead we take advantage of the high MMFS spectral resolution and define a set of sixteen indices, each of which measures the flux in a single, resolved spectral line.
To aid in the determination of feature and continuum bandpasses appropriate for the MMFS spectra we produced a high-S/N composite spectrum for each galaxy by averaging its individual stellar spectra (with relative redshifts and probable nonmembers removed). Figures \ref{fig:specbandsblue} and \ref{fig:specbandsred} display composite dSph spectra for blue and red channels. We define feature bandpasses to cover the sixteen atomic absorption lines most prominent in these spectra. In practice the width of each feature bandpass is approximately proportional to the mean line strength in the composite spectra. In most cases, blue and red continuum bandpasses are adjacent to the feature bandpass and span at least 0.25 \AA\ ($\sim 5$ pixels). Where possible (in the absence of a nearby feature) we use larger continuum passbands. Feature and continuum passbands for the chosen indices appear as shaded regions over the blue (red) composite galaxy spectra in Figure \ref{fig:specbandsblue} (Figure \ref{fig:specbandsred}). Table \ref{tab:indexbands} lists bandpass boundaries for the sixteen MMFS indices.
\begin{figure*}
\plotone{smallf17.eps}
\caption{ Composite spectra produced by averaging (with redshifts and spectra from probable nonmembers removed) individual dSph stellar and twilight spectra acquired with MIKE's blue channel. Black shaded regions correspond to feature bandpasses for sixteen MMFS line indices. Blue and red shaded regions correspond to the associated continuum bandpasses. To avoid overlap, each panel shows a unique subset of the sixteen indices.
}
\label{fig:specbandsblue}
\end{figure*}
\begin{figure*}
\plotone{smallf18.eps}
\caption{ Same as Figure \ref{fig:specbandsblue} but for spectra acquired with MIKE's red channel.}
\label{fig:specbandsred}
\end{figure*}
\begin{deluxetable*}{lccc}
\tabletypesize{\scriptsize}
\tablewidth{0pc}
\tablecaption{ MMFS Spectral Line Indices}
\tablehead{\colhead{Index}&\colhead{Central Bandpass}&\colhead{Red Continuum}&\colhead{Blue Continuum}\\
&\colhead{(\AA)}&\colhead{(\AA)}&\colhead{(\AA)}
}
\startdata
FeI$_{5142}$&$5141.50-5142.00$&$ 5141.10-5141.40$&$ 5142.05-5142.35$\\
FeI/NiI$_{5143}$&$5142.10-5143.40$&$5141.10-5141.50$&$5143.80-5144.60$\\
NiI$_{5146}$&$5146.00-5146.80$&$ 5145.65-5145.90$&$ 5146.90-5147.30$\\
FeI$_{5148}$&$5147.85-5148.45$&$ 5147.60-5147.80$&$ 5148.60-5148.90$\\
FeI$_{5151}$&$5150.40-5151.20$&$ 5148.90-5150.00$&$ 5151.25-5151.65$\\
FeI$_{5152}$&$5151.65-5152.45$&$ 5151.20-5151.60$&$ 5152.50-5152.95$\\
CoI/TiII$_{5154}$&$5153.85-5154.40$&$5153.55-5153.80$&$5154.55-5154.90$\\
NiI$_{5156}$&$5155.55-5156.00$&$ 5155.25-5155.50$&$ 5156.15-5156.40$\\
FeI$_{5159}$&$5158.65-5159.35$&$ 5158.00-5158.45$&$ 5159.45-5159.95$\\
FeI$_{5162}$&$5161.90-5162.70$&$ 5161.00-5161.60$&$ 5162.85-5163.50$\\
FeI$_{5166}$&$5165.90-5166.60$&$ 5165.60-5165.85$&$ 5166.65-5166.85$\\
MgI$_{5167}$&$5166.80-5168.00$&$ 5164.25-5165.25$&$ 5169.55-5170.70$\\
FeI/FeII$_{5169}$&$5168.50-5169.45$&$5168.00-5168.40$&$5169.50-5170.00$\\
FeI$_{5172}$&$5171.30-5171.90$&$ 5170.70-5171.20$&$ 5171.95-5172.15$\\
MgI$_{5173}$&$5171.90-5173.35$&$ 5169.55-5170.70$&$ 5174.35-5175.50$\\
TiI$_{5174}$&$5173.50-5174.00$&$ 5173.15-5173.40$&$ 5174.10-5174.50$\\
\enddata
\label{tab:indexbands}
\end{deluxetable*}
\subsection{Measurement of MMFS Indices}
Further processing of individual stellar spectra is required prior to measuring line indices. For dSph and globular cluster stellar spectra, we begin with the sky-subtracted frames obtained during the data reduction steps described in Section \ref{sec:reduction}. We use the DOPCOR task and the measured radial velocities to shift wavelength solutions such that all spectra have zero redshift in the solar rest frame. We then truncate spectra such that all cover the range $5141 - 5177$ \AA\ with linear dispersion 0.071 \AA\ pix$^{-1}$ (blue) and 0.107 \AA\ pix$^{-1}$ (red). These choices of dispersion values minimize pixel re-binning. From the resulting spectra, which share a common redshift, range, and dispersion, we measure the sixteen MMFS indices using Equations \ref{eq:index}-\ref{eq:sbsr}. In order to estimate index measurement errors, we assume the amounts by which the measured values $W$ differ from ``true'' index values $W_{*}$ follow Gaussian distributions with variances $\sigma_{W}^2$. For the $i^{th}$ measurement of the $j^{th}$ star,
\begin{equation}
W_{ij}-W_{*j}=\sigma_{W,ij}\epsilon_{ij}.
\label{eq:inderror}
\end{equation}
As in Equation \ref{eq:errormodel}, the values $\epsilon_{ij}$ follow the standard normal distribution, and we adopt the error approximation given by \citet{cardiel98},
\begin{equation}
\sigma_{W} \approx \beta \frac{c_1-c_2W}{\langle SN \rangle},
\label{eq:indexerror}
\end{equation}
modified by our introduction of scaling parameter $\beta$. Constants $c_1$ and $c_2$ depend on the index definition and are given by Equations 43-44 of \citet{cardiel98}, and we use our Equation \ref{eq:snratio} to estimate $\langle SN \rangle$, the mean S/N per pixel. After taking logarithms and replacing $\log [\epsilon_{ij}^2]$ with $\delta_{ij}-0.55$ (Section \ref{subsec:gaussianerrors}), we obtain $\log[(W_{ij}-W_{*j})^2]-\log (S_{ij}^2)=\log (\hat{\beta}^2)-0.55$. We estimate $\beta$ uniquely for each of the sixteen indices on both channels via linear regression, using the approximation $W_{*j} \sim \sum_{i=1}^{n_j}W_{ij}w_{ij}/\sum_{i=1}^{n_j}w_{ij}$ (where $w_{ij}$ is the inverse of the variance obtained from Equation \ref{eq:indexerror} with $\beta=1$). For the sixteen blue indices, estimates all fall within the range $3.45 \leq \hat{\beta} \leq 4.89$. For the eight red indices, $2.72 \leq \hat{\beta} \leq 3.95$. Among all indices, the resulting errors have median values in the range $0.05 \leq \sigma_{W}/\rm{\AA} \leq 0.10$.
\subsection{Individual Index Results}
The upper-left panels in Figures \ref{fig:indicesblue} and \ref{fig:indicesred} display, for each index, the measured value as a function of the mean S/N per pixel (calculated over the range $5141-5177$ \AA). Because the red channel has poor sensitivity toward the blue end of our spectral coverage (see the composite red spectra in Figure \ref{fig:specbandsred}), we consider only the eight red indices with feature bandpass redward of 5158 \AA. For a given index, the width of the distribution of $W$ values correlates with both (S/N) and the width of the feature bandpass
The remaining panels in Figures \ref{fig:indicesblue} and \ref{fig:indicesred} display empirical relationships between individual MMFS indices and other stellar parameters: V-I color, radial velocity, and MgI$_{5173}$ index. For several indices there is a clear trend toward higher index values at redder colors (upper-right panel); this behavior indicates the expected dependence of line strength on effective temperature. The lower-left panels plot $W$ against the measured radial velocity; for clarity in these panels the plotted points correspond exclusively to spectra from Carina targets. Along the radial velocity axis, most points are clumped in relatively narrow distributions about Carina's systemic velocity ($\sim 225$ km s$^{-1}$). That the velocity outliers with respect to the narrow Carina distribution tend also to be MgI index outliers indicates the ability of the MgI indices to distinguish dSph members from interlopers. We expect interlopers with large MgI line strengths to be due to Milky Way dwarf stars with high surface gravity. Thus the spectral indices provide a basis for dwarf/giant discrimination similar to that derived from photometric techniques that use narrow-band filters (e.g., DDO51) sensitive to surface gravity \citep{geisler84,majewski00}.
Finally, when individual indices are plotted against one-another, indices corresponding to the same element(s) display monotonic, linear relationships with slopes approximately equal to the ratio of feature bandpass widths (if indices are stated as a fractions of their bandpass widths, these relations tend toward 1:1). For example, the lower-right panels of Figures \ref{fig:indicesblue} and \ref{fig:indicesred} plot each index against the MgI$_{5173}$ index. In this case the MgI$_{5167}$ index exhibits the expected monotonic, linear relation. Noting the behavior of the FeI indices, it is immediately striking that many exhibit a bifurcation. With respect to the trend followed by stars with small and intermediate MgI$_{5173}$ index values, stars with the largest MgI$_{5173}$ values have systematically small FeI indices. Such stars tend to be the outliers in the radial velocity and MgI index joint distributions (lower-left panels); thus the bifurcation is due primarily to the presence of interloping stars, and dSph members generally follow a linear relationship in the iron-magnesium index plane.
\begin{figure*}
\plotone{smallf19.eps}
\caption{ Measured index value vs.\ spectral and stellar parameters for blue-channel spectra. Each panel corresponds to one of the sixteen MMFS indices. Points in the lower-left panel correspond to spectra from only Carina target stars (interlopers included), and show a clear separation in MgI indices between dSph and interloper populations.}
\label{fig:indicesblue}
\end{figure*}
\begin{figure*}
\plotone{smallf20.eps}
\caption{ Same as Figure \ref{fig:indicesblue} but for red-channel spectra. }
\label{fig:indicesred}
\end{figure*}
\subsection{Composite Indices}
We have noted that indices for different absorption lines corresponding to the same element(s) correlate linearly. We can improve precision by computing their sum. For blue-channel spectra we define a composite Fe index:
\begin{eqnarray}
\Sigma \rm{Fe}_{blue} \equiv 0.160(\rm{Fe}I_{5142}+\rm{Fe}I/NiI_{5143}+\rm{Fe}I_{5148}\\
+\rm{Fe}I_{5151}+\rm{Fe}I_{5152}+\rm{Fe}I_{5159}+\rm{Fe}I_{5162}\nonumber\\
+\rm{Fe}I_{5166}+\rm{Fe}I/\rm{Fe}II_{5169}+\rm{Fe}I_{5172})\nonumber,
\label{eq:masterfeblue}
\end{eqnarray}
where the sum is scaled by the ratio of the FeI$_{5143}$ bandpass width to the sum of all FeI bandpass widths (this scaling effectively weights each index by its bandpass width). Similarly, for red-channel spectra the composite Fe index is given by
\begin{eqnarray}
\Sigma \rm{Fe}_{red} \equiv 0.253(\rm{Fe}I_{5159}+\rm{Fe}I_{5162}+\rm{Fe}I_{5166}\\
+\rm{Fe}I/\rm{Fe}II_{5169}+\rm{Fe}I_{5172})\nonumber,
\label{masterfered}
\end{eqnarray}
where in this case the straight sum is scaled by the ratio of the FeI$_{5169}$ bandpass width to the sum of the included FeI bandpass widths.
For both blue- and red-channel spectra we compute the composite Mg index as
\begin{equation}
\Sigma \rm{Mg} \equiv 0.547(\rm{Mg}I_{5167}+\rm{Mg}I_{5173}),
\label{eq:mastermg}
\end{equation}
where the scale factor is the ratio of the MgI$_{5173}$ bandpass width to the sum of the two MgI bandpass widths. We compute errors $\sigma_{\Sigma W}$ associated with composite indices by adding in quadrature the estimated errors $\sigma_{W}$ of the contributing individual indices, and then scaling by the constant multipliers in Equations \ref{eq:masterfeblue}-\ref{eq:mastermg}. Figures \ref{fig:masterindicesblue} and \ref{fig:masterindicesred} plot for composite indices the same relations shown for individual indices in Figures \ref{fig:indicesblue} and \ref{fig:indicesred}. The composite indices show the same features and behaviors as the relevant individual indices, only with less scatter. Figure \ref{fig:masterindices_errors} plots estimated errors as a function of mean S/N per pixel.
We note that $\Sigma \rm{Fe}_{blue}$ includes a small contribution from a NiI absorption line at 5143 \AA. This line is blended with two FeI lines in the MMFS spectra. Because the FeI/NiI$_{5143}$ index behaves similarly to the other FeI indices (see Figures \ref{fig:indicesblue} and \ref{fig:indicesred}) and because it has among the highest S/N of any individual Fe index, we choose to include it in $\Sigma \rm{Fe}_{blue}$. For similar reasons we choose to include the FeI/FeII$_{5169}$ in both $\Sigma \rm{Fe}_{blue}$ and $\Sigma \rm{Fe}_{red}$ despite a small contribution from a blended FeII feature. Exclusion of either index from the composite Fe index does not significantly alter any of the correlations or features seen in Figures \ref{fig:masterindicesblue} and \ref{fig:masterindicesred}, nor the distribution of errors seen in Figure \ref{fig:masterindices_errors}.
For stars with multiple index measurements, Figure \ref{fig:indrepeats} plots the distribution of $\Delta \Sigma W \equiv \Sigma W_{ij} - \bar{\Sigma W_{j}}$, normalized by the associated errors, where $\bar{\Sigma W_{j}}$ is the weighted mean. The best-fitting Gaussian distributions (solid curves) are nearly identical to the standard normal distribution (dotted curves), indicating that the estimated errors are valid.
\begin{figure}
\epsscale{1.2}
\plotone{smallf21.eps}
\caption{ Dependence of Fe and Mg composite indices on spectral and stellar parameters for blue-channel spectra. Points correspond to the same spectra plotted in Figure \ref{fig:indicesblue} (again, points in the lower-left panel correspond only to Carina targets). Errorbars in the lower-right panel indicate median error bars.}
\label{fig:masterindicesblue}
\end{figure}
\begin{figure}
\epsscale{1.2}
\plotone{smallf22.eps}
\caption{ Same as Figure \ref{fig:masterindicesred} but for red-channel spectra.}
\label{fig:masterindicesred}
\end{figure}
\begin{figure}
\epsscale{1.2}
\plotone{smallf23.eps}
\caption{ For composite Fe (top panels) and Mg (bottom panels) indices measured on blue (left) and red (right) channels, sizes of derived error bars are plotted against mean spectral S/N per pixel. Due to the dependence of $\sigma$ on line strength (Equation \ref{eq:indexerror}), interloping dwarf stars with systematically stronger absorption lines have systematically smaller derived errors, producing the apparent bifurcations in the bottom two panels. }
\label{fig:masterindices_errors}
\end{figure}
\begin{figure}
\epsscale{1.2}
\plotone{smallf24.eps}
\caption{ For stars with repeat index measurements within a given MIKE channel, the distribution of measurements about the weighted mean, normalized by measurement errors. Solid curves represent the best-fitting Gaussian distributions. Dotted curves represent a standard normal distribution.}
\label{fig:indrepeats}
\end{figure}
\begin{acknowledgments}
\section{Conclusion}
We have described the instrumentation, data acquisition and data reduction procedures for a new spectroscopic survey that targets large numbers of stars in nearby dSphs. In forthcoming papers (Walker et al.\ in preparation) we present the entire data set described herein and provide kinematic analyses that consider both equilibrium and tidal interaction models. We use the spectral indices measured from stars in seven globular clusters to calibrate relations between the MMFS spectral indices and iron abundance, and to identify dSph interlopers independently of velocity. In the end we have velocity and line-strength measurements for nearly 3800 probable members in the four observed dSphs. Alone and/or in combination with data published from other contemporary spectroscopic surveys, the MMFS data set enables us to map the two-dimensional behavior of dSph stellar kinematics and chemistry with unprecedented precision.
We thank Matthew Coleman and Gary Da Costa for providing photometric data that was used to select spectroscopic targets in Sculptor. We thank Kaspar von Braun and Patrick Seitzer for providing photometric observations that we used to select spectroscopic targets in Carina. We thank the following people for invaluable assistance in developing MMFS: Stephen Shectman, Steve Gunnels, Alex Athey and Vince Kowal, as well as the excellent staff at Las Campanas Observatory, including Emilio Cerda, Oscar Duhalde, Patricio Jones, Marc Leroy, Gabriel Martin and Mauricio Navarette, and Magellan telescope operators Victor Meri\~{n}o, Hernan Nu\~{n}ez, Hugo Rivera, and Geraldo Valladares. This work is supported by NSF Grants AST 05-07453, AST 02-06081, and AST 94-13847.
\end{acknowledgments}
|
1,116,691,500,763 | arxiv | \section{Introduction \label{SecIntroduction}}
Standard surface muon beams have a relatively poor phase space quality. An improvement of the muon beam quality would open the way for new experiments in low energy particle physics, atomic physics and material research. Examples of such experiments include muonium 1S-2S spectroscopy, muon electric dipole moment (EDM) measurement and muon spin rotation ($\mu$SR) applications. Due to the limited lifetime of the muons ($\tau=2198$~\si{\nano \second}), a fast cooling scheme is required, and thus conventional beam cooling methods such as stochastic~\cite{VanderMeer1985} or electron cooling~\cite{Budker1978} cannot be applied.
A new phase space compression scheme (muCool device) has been proposed \cite{Taqqu2006}, which transforms a standard surface muon beam into a beam of high brightness and low energy. Within about 10~\si{\micro \second}, the incoming $\mu^+$ are stopped in He gas, manipulated by electric and magnetic fields in order to decrease their spatial extent, and extracted into vacuum again. The 6D phase space is reduced by a factor of $10^{10}$ with an efficiency of $10^{-3}$, mainly limited by the muon lifetime. Thus, the brightness of the incoming beam is enhanced by a factor of $10^{7}$.
The decrease of the muon swarm extension is obtained by making the drift velocity of the $\mu^+$ in the gas position-dependent. The drift velocity vector of $\mu^+$ in gas, in the presence of electric and magnetic fields, can be written as \cite{Blum}:
\begin{equation}
\vec{v}_D = \frac{\mu |\vec{E}|}{1 + \omega^2 / \nu^2} \left[ \vec{\hat{E}} + \frac{\omega}{\nu} \vec{\hat{E}} \times \vec{\hat{B}} + \frac{\omega^2}{\nu^2} \left(\vec{\hat{E}} \cdot \vec{\hat{B}} \right) \vec{\hat{B}} \right],
\label{EqDriftVelocity}
\end{equation}
where $\mu=e/\nu m$ is the mobility of the muon, $\vec{\hat{E}}$ and $\vec{\hat{B}}$ are the unit vectors of the electric and magnetic fields, $\omega = eB/m$ is the cyclotron frequency of the muon and $\nu$ the average collision frequency of the muon with the He gas atoms.
The drift velocity $\vec{v}_D$ can be made position-dependent by applying a position-dependent electric field $\vec{E}$ and/or by making the collision frequency $\nu$ position-dependent, which changes the weights of the three components in Eq.~\ref{EqDriftVelocity}.
A sketch of the He gas target where such a compression occurs is shown in Fig.~\ref{FigWorkingPrinciple}, together with the coordinate system used throughout this paper. The target consists of several stages and is placed in a homogeneous magnetic field, oriented along the $+z$-direction, i.e. $\vec{B}=(0,0,B)$, with $B=5$~\si{Tesla}.
The incoming $\mu^+$ beam is stopped in the first stage that is at cryogenic temperatures, with a vertical ($y$-direction) gas density gradient induced by a temperature gradient between $T=4$~\si{\kelvin} at the bottom and $T=12$~\si{\kelvin} at the top of the roughly 3~\si{\centi \meter} high target \cite{Wichmann}. The gas density gradient gives rise to a position-dependent collision frequency $\nu=\nu(x,y,z)$.
\begin{figure}
\includegraphics[trim=0 0 0 0cm,clip=true,width=1.0\columnwidth]{Fig1.png}%
\caption{\label{FigWorkingPrinciple}Overview of the proposed phase space compression scheme. A standard surface muon beam is stopped in a cryogenic helium gas with a vertical temperature gradient inside a 5~T magnetic field. The stopped muon swarm is then compressed by means of $\vec{E}$- and $\vec{B}$-fields in the $y$-direction (``transverse compression''), and at the same time it is steered in $x$-direction towards the longitudinal compression stage. There, the He gas is at room temperature, and there is a component of the electric field parallel to the magnetic field axis. Thus, the muon swarm is compressed towards the center of this stage. Finally, the muons with a swarm size of $\mathcal{O}$(mm) are extracted through a small orifice into vacuum, where they are re-accelerated and sent to another experiment.}
\end{figure}
An electric field $\vec{E}=(E_x,E_y,0)$ with $E_x=E_y \approx 1~\si{\kilo \volt \per \centi \meter}$ is applied perpendicular to the magnetic field in this stage, thus only the first two terms in Eq.~\ref{EqDriftVelocity} remain. The muons in a region of higher gas density (colder temperatures) follow the $\vec{\hat{E}}$-direction because $\nu$ is large, whereas muons in a region of lower gas density (warmer temperatures) primarily follow the $\vec{\hat{E}}\times\vec{\hat{B}}$-direction. With our choice of $\vec{E}$, $\vec{B}$ and $\nu(x,y,z)$ we achieve compression of the stopped muon swarm in vertical ($y$) direction (called ``transverse compression"), with superimposed drift in $+x$-direction towards the second compression stage, where longitudinal (in $z$-direction) compression occurs.
The second stage is at room temperature (much lower gas density) and the electric field is parallel to the magnetic field: $\vec{E}=(0,0,\mp E_z)$ pointing towards the center of the target, as indicated in Fig.~\ref{FigWorkingPrinciple}, with $E_z\approx50$~\si{\volt \per \centi \meter}. Therefore, in the second stage Eq.~\ref{EqDriftVelocity} simplifies to $\vec{v}_D \approx \mu \vec{E}$, and the muon swarm is compressed along the $\pm z$-direction (``longitudinal compression") towards $z=0$. Adding an electric field in the $y$-direction gives rise to a superimposed drift in $+x$-direction, which guides the compressed muon swarm towards an orifice of about 1~mm diameter, from where the muons are extracted into vacuum. After the extraction into the vacuum, the $\mu^+$ are accelerated to keV energy, while keeping eV energy spread and are successively extracted from the magnetic field.
Initially, the various stages of the compression target can be tested separately, simplifying the experimental approach considerably. Transverse compression has been successfully demonstrated and will be presented in a separate publication. Longitudinal compression has been demonstrated to be feasible~\cite{Bao2014}.
However, several problems prevented quantification of the muon swarm compression efficiency. The major issue was impurities in the helium gas target, which captured low-energy muons and thus lowered the compression efficiency. Moreover, a large background allowed to demonstrate only qualitatively the feasibility of the longitudinal compression.
In this paper we present an improved demonstration of the longitudinal compression with a quantification of the compression efficiency.
Furthermore, we demonstrate that the muon swarm can be steered in $+x$-direction by adding a vertical ($y$-direction) electric field. This is crucial to guide the $\mu^+$ through the different compression stages and finally towards the point of extraction. The experiments were performed at the $\pi$E1 beam line at the Paul Scherrer Institute (PSI), which delivers about $10^4$ $\mu^+$/s at $11$~MeV/c.
\begin{figure} []
\includegraphics[width=\columnwidth]{Fig2.png}%
\caption{\label{FigSetup2014}Sketch of the He gas target used to measure the longitudinal compression (not to scale). The muon beam passes through an entrance detector D1, target window (region I) and enters the He gas. Some of the muons are stopped in the gas and subjected to the electric field, pointing towards the center of the target. A large fraction of the muons passes through the target and stops in region II. Several scintillators (S1, $\ldots$, S21, T1, T2) detect the positrons from muons decaying in their vicinity.}
\end{figure}
\section{Test of longitudinal compression \label{SecSetup}}
In order to test longitudinal compression, an 11~MeV/c muon beam was injected longitudinally into a target containing He gas of a few mbar pressure at room temperature. The setup is sketched in Fig.~\ref{FigSetup2014}. The muons first have to pass through a 55 \si{\micro \meter} thick entrance detector (D1), giving the initial time $t_0 = 0$, then through a 2~\si{\micro \meter} Mylar target window enclosing the He gas. Only a small fraction $\mathcal{O}(1\%)$ of the muons producing the signal in D1 stop in the gas, while the remaining muons stop in D1 or target window (region I in Fig.~\ref{FigSetup2014}) or downstream part of the target (region II in Fig.~\ref{FigSetup2014}).
The longitudinal compression target had a transverse cross section of $12 \times 12$~\si{\square \milli \meter}, and an ``active" length, where the electric field was defined, of about 200~\si{\milli \meter}. Side walls of the target were lined with gold electrodes that created a V-shaped electric potential with a minimum at the center of the target cell at $z=0$. This choice of the electric potential produced an electric field pointing towards $z=0$, which caused the muons to move along the $\pm z$-direction towards the potential minimum \footnote{Because of $\nabla \cdot E=0$, at $z=0$ the electric field will also have a radial component. This component is included automatically in the COMSOL Multiphysics\textregistered~simulation but it is practically so small that it does not affect significantly the muon motion. }.
The exact electric field was simulated using a finite elements method within COMSOL Multiphysics\textregistered~\cite{Comsol} and imported in the GEANT4 simulation package~\cite{Geant4}, allowing us to study muon motion in the realistic electric field.
The GEANT4 simulation included the most relevant low energy processes down to eV energies, namely low energy elastic collisions ($\mu^+-$He and muonium$-$He) and charge exchange processes (muonium formation and ionization)~\cite{Bao2014}.
Cross sections for the elastic collisions and charge exchange were implemented by energy and velocity scaling ~\cite{Senba1989}, respectively, of the proton cross sections from ~\cite{Krstic2006,Nakai1987}.
A simulation of the muon distribution inside the target at two different times $t$ is shown in Fig.~\ref{FigSpaceDistSim}.
The distribution at $t=0.15$~\si{\micro \second} represents the initial muon stop distribution, while the distribution at $t=2$~\si{\micro \second} shows the muon distribution when the longitudinal compression is almost completed. To highlight the muon swarm compression, the number of counts has been scaled with $\exp(t/\tau)$, where $\tau=2198$~\si{\nano \second} is the muon lifetime, in order to compensate for the muon decay.
\begin{figure}[]
\includegraphics[width=1.0\columnwidth]{Fig3.png}%
\caption{\label{FigSpaceDistSim}
Simulated muon distribution along the $z$-axis for various times. The data has been corrected for the finite muon lifetime by multiplying the counts with $e^{t/2198\si{\nano \second}}$.
As can be seen, for $t=2$~\si{\micro \second} most of the muons are close to the center of the active region.
The shaded areas show the fraction of the $\mu^{+}$ stopped in the walls of the target. At $t=2$~\si{\micro \second}, the distribution of the $\mu^+$ adhering to the target walls is not flat anymore. The reason for the peak around $z=0$ is that some of the $\mu^+$ are scattered into the wall while drifting towards the center of the target.
}
\end{figure}
Under such conditions, at $t=2$~\si{\micro \second}, 63\% of the muons that were in the active region at $t=0.15$~\si{\micro \second} are still within the active region, with 90\% of them already within $z=\pm5$~mm. The 50\% of all the muons in the active region at $t=2$~\si{\micro \second} are already compressed in the center within an even smaller region of $z=\pm1$~mm. Therefore, the number of muons within the region of $z=\pm1$~mm increases by about a factor of 20 between 0.15~\si{\micro \second} and 2~\si{\micro \second} (neglecting muon decay).
The other 37\% of the muons that were initially in the active region are lost through two main mechanisms: 26\% through muonium formation and 11\% through scattering out of the active region due to low-energy elastic collisions.
The muon bound in the neutral muonium atom is not contained by the 5 Tesla magnetic field, causing it to fly into the walls of the target. The losses due to scattering of $\mu^+$ are only relevant when the muon beam is injected into the longitudinal target at keV energies because the $\mu^+$ mean free path at these energies can be up to few cm. These losses will therefore be absent in the final setup, where muons enter the longitudinal compression stage from the transverse compression stage at eV energies. Indeed, at such energies, the $\mu^+$ mean free path is sub-mm. Similarly, the losses due to the muonium formation will be strongly reduced in the final setup.
The muon swarm movement was also measured experimentally by placing 21 identical scintillators (S1 to S21) along the target $z$-axis, as shown in Fig.~\ref{FigSetup2014}. The scintillators detected positrons from muon decays and were read out by Silicon Photomultipliers (SiPMs). The acceptance of some of these scintillators as a function of muon decay $z$-position is plotted in the Fig.~\ref{FigGeomAcc}. Each detector has an average geometrical acceptance in $z$-direction of $16.5$~mm (FWHM).
\begin{figure}[]
\includegraphics[width=1.0\columnwidth]{Fig4.png}%
\caption{\label{FigGeomAcc}
Simulated geometrical acceptance of the scintillators S1-S21 (only every second detector is shown) versus the $z$-position. Shown is the normalized probability that a positron from a $\mu^+$ decay will be detected in the corresponding scintillator. For comparison, the geometrical acceptance of the T1 \& T2 coincidence is also plotted (red line).
The maximum of the efficiency for each detector was normalized to 1 to highlight the different geometrical resolutions.
}
\end{figure}
\begin{figure}[]
\includegraphics[width=1.0\columnwidth]{Fig5.png}%
\caption{\label{FigLongSpatialDistribution}Measured (dots) and simulated (lines) positron hits in the 21 scintillators shown in Fig.~\ref{FigSetup2014}, for $t=150~\si{\nano \second}$ and $t=2000~\si{\nano \second}$. The center-to-center distance between two adjacent scintillators is $10$~mm. The simulated data is the same as in Fig.~\ref{FigSpaceDistSim}, but convoluted with the geometrical acceptance of the detectors. The width of the peak at $t=2$~\si{\micro \second} is due to the large geometrical acceptance of the 21 scintillators and does not reflect the width of the muon swarm directly (compare with Fig.~\ref{FigSpaceDistSim}).
The shaded areas show the simulated contribution to the positron hits from muons stopped in regions I and II.
The data and the simulations have been corrected for the finite muon lifetime by multiplying the counts with $e^{t/2198\si{\nano \second}}$.}
\end{figure}
The number of measured positrons in each scintillator is presented for $t=150$~\si{\nano \second} (black dots) and $t=2000$~\si{\nano \second} (green dots) in Fig.~\ref{FigLongSpatialDistribution}. Note that the number of counts has been scaled with $e^{t/2198\si{\nano \second}}$ to compensate for the muon decay.
In Fig.~\ref{FigLongSpatialDistribution} the measured $\mu^+$ distributions (dots) are compared to the GEANT4 simulations (lines). The simulated number of positron hits in each of the detectors S1-S21 is obtained by convoluting the simulated spatial distribution of the muon swarm at the corresponding time, given in Fig.~\ref{FigSpaceDistSim}, with the detection efficiency of the appropriate detector, given in Fig.~\ref{FigGeomAcc}. These positron hits originate not only from the muons in the active region, but also from muons from the regions where the electric field is not well defined, giving rise to a substantial background.
This background is dominated by positron hits from the $\mu^+$ that stop in the regions I and II (as defined in Fig.~\ref{FigSetup2014}). The background is larger for the detectors placed at the periphery of the active region (S1, S2 and S20, S21), as shown in the Fig.~\ref{FigLongSpatialDistribution} (shaded areas). The shape of the background caused by these muons can be simulated. However, the exact number of $\mu^+$ which stop in the regions I and II depends strongly on the momentum distribution of the initial muon beam, which is not sufficiently well known.
Therefore, two measured distributions along the $z$-axis (for $t=150$~\si{\nano \second} and $t=2000$~\si{\nano \second}) are fitted simultaneously with the sum of 4 contributions:
\begin{enumerate}
\item Background arising from the region I
\item Background arising from the region II
\item Linear background
\item Simulation of all the $\mu^+$ that stop in the gas (which includes $\mu^+$ in the active region).
\end{enumerate}
The shape of these 4 contributions is known, under assumption that all the detectors (S1 to S21) have the same detection efficiency. Each of the contributions has to be scaled with a different scaling factor to account for the different stopping probability in region I, region II, gas, and prompt stop at the target lateral walls.
The additional linear background allows us to account for possible misalignment between the target and the magnetic field axis, which would lead to different numbers of muon wall stops at the position of the various scintillators.
Even though we observe fair agreement between the measurement and the simulation (reduced chi-square $\chi^2_{\rm red}=2.3$, for 37 degrees of freedom), it is difficult to extract precise values of the compression efficiency and of the width of the muon swarm from these measurements, given the limited geometrical resolution of the detectors S1 to S21 and the large background from regions I and II. The relatively large $\chi^2_{\rm red}$ could be attributed to the small variations of the detector efficiencies.
In order to better quantify the compression efficiency we turn our attention to the two telescope detectors T1 \& T2 in coincidence that were placed in the center of the target, at $z=0$, as shown in Fig.~\ref{FigSetup2014}. Massive brass shielding all around the target ensured that coincidence hits in T1 \& T2 originated only from muons decaying within the small region between about $z=\pm3$~\si{\milli \meter} in the center of the target, as shown in Fig.~\ref{FigGeomAcc}.
From the time difference $t=t_1-t_0$ between the positron hit in T1 and T2 in coincidence (at time $t_1$) and the entrance detector, at time $t_0$, a time spectrum can be obtained as shown in Fig.~\ref{FigLongTimeSpectraSimulationData}. The time spectra were recorded for different applied electric potentials.
\begin{figure}
\includegraphics[width=1.0\columnwidth]{Fig6.png}%
\caption{\label{FigLongTimeSpectraSimulationData}Measured (dots) and simulated (line) time spectra for 5~mbar He gas pressure and potentials of $-500$~V (red), 0~V (black) and $+500$~V (green).
The data has been normalized to the number of incoming muons in the entrance detector D1 and fitted simultaneously with only 2 free parameters, a common normalization and a common background. In total $5\cdot10^8$ muons have been simulated.
The data and the simulations have been corrected for the finite muon lifetime by multiplying the counts with $e^{t/2198\si{\nano \second}}$.
}
\end{figure}
It can be seen in Fig.~\ref{FigLongTimeSpectraSimulationData} that if no electric field is applied (black points) the number of muons decaying in the center of the target (in T1 \& T2 acceptance region) stays constant (after compensation for the $\mu^+$ decay).
When a negative potential (red points) is applied in the center of the target cell, the measured number of counts increases with time. In that case, the stopped muons are attracted towards the potential minimum, so that more muons decay within the acceptance region of T1 \& T2. This means that the muon swarm extent has been decreased in $z$-direction, representing a complementary way to demonstrate longitudinal compression. On the contrary, if the ``wrong" polarity is applied (green points), the muons drift away from the center of the target, out of the acceptance region of T1 \& T2. The very few counts at late times in that case are due to some small background (mostly muons stopping in the wall of the target). The reduction to a nearly background-free measurement represents a major improvement compared with the earlier measurements from 2011~\cite{Bao2014} and compared to the measurement of the $\mu^+$ $z$-distribution of Fig.~\ref{FigLongSpatialDistribution}.
The measurements of Fig.~\ref{FigLongTimeSpectraSimulationData} are compared to GEANT4 simulations. The three simulations ($+$, $-$ and $0$ voltage) are fitted simultaneously to the corresponding measured time spectra. Only two free parameters were used: a common scaling factor and a common flat background. The common scaling factor is needed to remove the uncertainties related to the positron detection and muon stopping efficiencies.
The flat background was included in the fit to account for potential misalignment of the target with respect to the magnetic field axis, which would lead to increased muon stops in the walls.
The simultaneous fit of the 3 curves has a reduced chi-square $\chi^2_{\rm red} = 1.41$ (for $46$ degrees of freedom). Introduction of additional losses during the compression process, as detailed in the next section, improves the $\chi^2_{\rm red}$ value to $0.95$. Alternatively, a smaller $\chi^2_{\rm red}$ can also be obtained by a minor tuning of the detector acceptance, related to uncertainties in the position, tilt and energy threshold of the T1 \& T2 scintillators.
\section{Additional muon losses?}
As mentioned in the introduction, in the previous 2011 experiment~\cite{Bao2014}, the data quality also suffered from low-energy muon losses due to impurities present in the helium gas, which were caused by the use of Araldite glue and PCB boards in the target construction. This manifested itself by a fast termination of the compression (around $t=0.5$~\si{\micro \second}), because the low-energy muons were captured by the contaminant molecules forming muonic ions or replacing a proton from such a contaminant molecule.
By implementing a ``chemical capture" rate for muons with energies $\leq10$~\si{\electronvolt} in the simulation, it was possible to roughly mimic this effect.
In the experiments presented here, care was taken to ensure high gas purity. This was achieved by realizing the target from glass plates, using a low-outgassing glue and purifying the He gas in a cold-trap before feeding it into the target.
In order to prove the hypothesis that indeed the compression efficiency is lowered if contaminants are present in the gas, we introduced in a controlled way different amounts of contaminant gases into the pure helium gas. The result of this test is presented in the time spectra of Fig.~\ref{FigImpurities}. It is observed that the compression stops at earlier times when the contaminants are introduced, and consequently the compression efficiency decreases (i.e. the number of muons that are brought to the center of target).
\begin{figure}[!h]
\includegraphics[trim=0cm 0cm 0cm 0cm, clip=true,width=1.0\columnwidth]{Fig7.pdf}%
\caption{\label{FigImpurities}Measured time spectra for 5~mbar pure He gas, and various admixtures of contaminants: (black) no additional contaminants, (dashed blue) 0.01~mbar H$_2$, (dashed orange) 0.02~mbar H$_2$, (dashed green) 0.01~mbar of O$_2$.
All data were normalized to the number of incoming muons and corrected for the finite muon lifetime by multiplying the counts with $e^{t/2198\si{\nano \second}}$.}
\end{figure}
The sensitivity of the measurements of Fig.~\ref{FigLongTimeSpectraSimulationData} to the muon loss mechanisms has been investigated assuming constant (in time) loss rates $R$ during compression, that causes the ``free" muon population to decrease according to $e^{-Rt}$. Various simulations have been performed with $R$ ranging from $R=0$ up to $R=0.5$~\si{\per \micro \second}. The loss rate $R=0.5$~\si{\per \micro \second} would correspond roughly to the measurement with $0.01$~mbar H$_2$ contamination (blue dashed line in the Fig.~\ref{FigImpurities}).
\begin{figure}[]
\includegraphics[trim=0cm 0cm 0cm 0cm, clip=true,width=1.0\columnwidth]{Fig8a.png}
\vfill
\includegraphics[trim=0cm 0cm 0cm 0cm, clip=true,width=1.0\columnwidth]{Fig8b.png}
\caption{\label{FigVerificationSimulation}(Top) Simulated time spectra for loss rates of $R=(0,0.125,0.25,0.333)$~\si{\per \micro \second} (lines). The $-500$~V measurement is represented by square points. The data and the simulations have been corrected for the finite muon lifetime by multiplying the counts with $e^{t/2198\si{\nano \second}}$.
(Middle) Residuals (normalized to the data uncertainty) for the time spectra from the top figure. For simplicity, error bars are shown only for the curve with the reduced $\chi^2=0.95$. For other curves error bars are of the similar size.
(Bottom) Reduced $\chi^2$ as a function of the loss rate $R$. A second order polynomial was fitted to these points (green line).}
\end{figure}
For each loss rate $R$ the simultaneous fit of the simulations for $+$, $-$ and $0$ voltages to the corresponding measurement has been performed (as in Fig.~\ref{FigLongTimeSpectraSimulationData}, but for $R \neq 0$). Figure~\ref{FigVerificationSimulation} (top) shows the fitted time spectra (only for negative voltage) for several loss rates $R$. For each of the loss rates $R$, the $\chi^2_{\rm red}$ between the measurement (only $-500$~V data) and simulations has been calculated and plotted in Fig.~\ref{FigVerificationSimulation} (bottom). A parabola was then fitted to these points (green line). The best agreement (minimum $\chi^2_{\rm red}$) between simulation and measurement is obtained for $R=0.14$~\si{\per \micro \second}, corresponding to an additional loss at $t=2$~\si{\micro \second} of $1-e^{-0.14\si{\per \micro \second} \cdot 2 \si{\micro \second}}=24$\%. The $\chi^2_{\rm red, min}+1$ is obtained for $R=0.35$~\si{\per \micro \second} corresponding to an additional loss at $t=2$~\si{\micro \second} of $51$\%. Therefore, we conclude that the total additional loss after 2~\si{\micro \second} is $= 24^{+27}_{-24}\%$.
If these losses would be caused by capture of the muons by gas impurities, the partial pressure of the impurities would be $1\cdot 10^{-3}-3\cdot 10^{-3}$~mbar. The relation between fitted $R$ and impurity concentration is obtained by fitting the simulation for various loss rates $R$ to the measured time spectra of Fig.~\ref{FigImpurities} with the additional $0.01$~mbar H$_2$ contamination. Because such an impurity level has not been observed in the experiment, we may conclude that the origin of these "effective losses" $R$ cannot be attributed solely to the impurities.
Other explanations, such as uncertainties of the detector acceptance and of the cross sections implemented in the simulation are more favored.
\section{$\vec{E} \times \vec{B}$-Drift}
The next important step towards the realization of the complete muCool device is the demonstration of the so-called $\vec{E}\times\vec{B}$-drift. This drift guides the muons from one compression stage to the next, and is used to extract them finally into vacuum through a small orifice.
For this purpose we modified the target cell to generate an electric field with an additional vertical component $E_\text{y}=120$~\si{\volt / \centi \meter}. The non-vanishing central term in Eq.~\ref{EqDriftVelocity} leads to a drift of the muons in $+x$-direction. In total seven scintillators were mounted around the target at $z=0$ along the $x$-direction to monitor this drift. For simplicity, in the sketch of Fig.~\ref{FigDriftSetup} and in the plots of Fig.~\ref{FigDrift} we only show three of them, namely ``Left", ``Middle", ``Right". The target cell was enlarged to a cross section of $24\times12$~\si{\square \mm} and, additionally, the muon beam was injected off-center, as indicated in Fig.~\ref{FigDriftSetup} with the yellow circle. These two modifications allowed $\mu^+$ to drift for longer times before hitting the right wall of the target, thus enhancing the sensitivity of the measured time spectra to the muon drift.
\begin{figure}[]
\includegraphics[width=0.4\textwidth]{Fig9.png}
\caption{\label{FigDriftSetup} Sketch of the setup to measure the $\vec{E}\times\vec{B}$-drift. The muon beam was injected off-axis to allow the muons to drift a longer distance before hitting the lateral wall. The electric field has a $z$-component for longitudinal compression and a $y$-component to drift the muons in $x$-direction. The three scintillators (``Left", ``Middle", ``Right") are positioned in the center of the target at $z=0$ and monitor the muon swarm movement in $x$-direction.}
\end{figure}
The simulated spatial distribution of the muon swarm as a function of time is given in Fig.~\ref{FigMuonDrift}. This distribution, convoluted with the corresponding detector acceptance, gives rise to the simulated time spectra shown in Fig.~\ref{FigDrift}, together with the corresponding measurements.
At early times, the time spectra are dominated by the muon swarm compression in $z$-direction, thus the number of detected positrons increases in all scintillators. However, after about 2~\si{\micro \second}, the number of detected positrons in the scintillator ``Left" decreases, indicating that the muons are moving out of the acceptance region of this scintillator. On the other hand, scintillator ``Right" detects increasingly more positrons. This finding indicates that the muon swarm slowly drifts in $x$-direction towards the prospective point of extraction.
\begin{figure}[]
\includegraphics[width=0.5\textwidth]{Fig10.png}
\caption{\label{FigMuonDrift}Muon positions in the $xz$-plane for various times. The time is given by the color scale. Muon beam centered at $x=-6$~mm with 3~mm radius is stopped uniformly along the $z$-axis. The muons drift in the $+x$-direction while compression occurs in $z$-direction.}
\end{figure}
\begin{figure}[]
\includegraphics[trim=0 0 2cm 2cm,clip=true,width=0.49\textwidth]{Fig11.png}
\caption{\label{FigDrift}Measured (dots) time spectra for the three scintillators ``Left", ``Middle" and ``Right". The increase and decrease of positron counts in the detectors ``Right" and ``Left", respectively, indicates that the muons are moving towards the right.
Simulated time spectra for additional muon loss rate $R=0$ (continuous lines) and $R=0.125$~\si{\per \micro \second} (dashed lines) are fitted to the measured time spectra. The $\chi^2_{\rm red}$ for $R=0$ is $3.44$ for $433$ degrees of freedom. Introducing the additional loss rate $R=0.125$ ~\si{\per \micro \second} improves the $\chi^2_{\rm red}$ to $2.23$.
Note that the data and the simulations have been corrected for the finite muon lifetime by multiplying the counts with $e^{t/2198\si{\nano \second}}$.
}
\end{figure}
Also in this case, the measured time spectra of all seven ``drift" detectors were fitted simultaneously with the simulation allowing for one common scaling factor, and a different flat background for each detector. A fair agreement between data and simulation has been observed ($\chi^2_{\rm red}=3.44$ for $433$ degrees of freedom).
To study the effect of the additional muon losses on the measured time spectra, the loss rate $R$ has been introduced in the simulation, analogously to the procedure described in the previous section. The obtained time spectra for the various $R$ were then fitted to the data. The best agreement between simulation and the data is obtained for $R=0.125$~\si{\per \micro \second}, consistent with the loss rate $R$ reported in the previous section.
The best fit, which gives a $\chi^2_{\rm red}=2.23$, is shown in Fig.~\ref{FigDrift} (dashed lines).
Even with the additional muon losses introduced, some systematic discrepancies between the data and the simulation still remain.
Yet the main goal, namely to demonstrate the feasibility of the $\vec{E}\times\vec{B}$-drift, has been achieved.
The difference between the simulation and the measurement can be attributed either to the simplified modelling of the additional losses (without any energy dependence) or a misalignment of the beam with respect to the magnetic field axis.
According to the simulation, the drift velocity is about 2~\si{\mm / \micro \second}. This value can be increased in the final setup by increasing the strength of the electric field in $y$-direction.
\section{Conclusions}
The longitudinal compression stage of the muCool device under development at PSI has been demonstrated. An elongated muon swarm of $200$~\si{\milli \meter} length has been compressed to below $2$~\si{\milli \meter} length within 2~\si{\micro \second}. Good agreement between the simulation and the measurement has been observed. Some additional losses which have been parametrized by only one constant $R$ have been introduced to improve the agreement.
Furthermore, the ability to drift the $\mu^+$ beam in $\vec{E}\times\vec{B}$-direction towards the prospective position of the extraction hole has been demonstrated by performing a measurement with the electric field having also a component perpendicular to the magnetic field. The simulation of the compression and drift towards the extraction hole is in fair agreement with measurements when the small additional loss rate $R$ is introduced in the simulation.
In summary, better agreement between simulations and measurements is achieved by either including small additional losses in the simulation or, more likely, minor tuning of the detector acceptance or minor variation of the cross section of muonium formation, ionization and muonium-He scattering. In any case, even with this additional loss rate $R$, the proposed $\mu^+$ compression efficiency of $10^{-3}$ is attainable.
\begin{acknowledgements}
The experimental work was performed at the proton accelerator at PSI. We thank the machine and beamline groups for providing excellent conditions.
We gratefully acknowledge the outstanding support received from the workshops and support groups at ETH Zurich and PSI. Furthermore, we thank F. Kottmann, M. Horisberger, U. Greuter, R. Scheuermann, T. Prokscha, D. Reggiani, K. Deiters, T. Rauber, and F. Barchetti for their help. This work was supported by the SNF grants No. 200020\_159754 and 200020\_172639.
\end{acknowledgements}
|
1,116,691,500,764 | arxiv | \section*{Executive Summary}
\end{center}
Five contemporary technologies with potential
application to particle tracking in future high energy physics experiments are discussed.
\bigskip
Silicon sensors of the 3D technology have electrodes oriented perpendicular to their wafer
surfaces. These show promise for compensation of lost signal in high radiation
environments and for separation of pileup events by precision timing. New 3D geometries involving p-type trench electrodes spanning the entire length of the
detector, separated by lines of segmented n-type electrodes for readout, promise improved uniformity, timing resolution, and radiation resistance relative to
established devices operating effectively at the LHC. Present research
aims for operation with adequate
signal-to-noise ratio at fluences approaching $10^{18}~n_{\rm eq}/{\rm cm}^2$, with timing resolution
on the order of 10 ps.
\bigskip
The 3D technology is also
being realized in diamond substrates, where column-like electrodes may be placed
inside the detector material by use of a 130 fs laser with wavelength 800~nm. When
focussed to a 2 micron spot, the laser has energy density sufficient to convert diamond
into an electrically resistive mixture of different carbon phases. The drift
distance an electron-hole pair must travel to reach an electrode can be reduced
below the mean free path without reducing the number of pairs created. Initial
tests have shown that after $3.5 \times 10^{15}~n_{\rm eq}/{\rm cm}^2$, a 3D diamond sensor with $50~\mu {\rm m} \times 50~\mu {\rm m}$ cells collects more charge than would be collected by a
planar device and shows less damage due to the shorter drift distance.
The goal
of this project is to create a detector that is essentially immune to radiation
doses at the level of $10^{17}$ hadrons/cm$^2$.
\bigskip
A pixel architecture, named DoTPiX, has been proposed on the principle of a
single n-channel MOS transistor, in which a buried quantum well gate performs
two functions: as a hole-collecting electrode and as a channel current modulation
gate. The quantum well gate is made with a germanium layer deposited on a
silicon substrate. The active layers are of the order of 5 microns below the surface, permitting detection of minimum ionizing particles. This technology is intended to achieve extremely small
pitch size to enable trigger-free operation without multiple hits
in a future linear collider, as well as simplified reconstruction of tracks with low
transverse momentum near the interaction point.
\bigskip
Thin film detectors have the potential to be fully integrated,
while achieving large area coverage and low power consumption with low dead
material and low cost. Thin flim transistor technology uses crystalline growth
techniques to layer materials, such that monolithic detectors may be fabricated
by combining layers of thin film detection material with layers of amplification
electronics using vertical integration.
\bigskip
Lastly, a technology is under development in which a novel ultra-fast scintillating material employs a semiconductor stopping medium with embedded quantum dots. The candidate material, demonstrating very high light yield and fast emission, is a GaAs matrix with InAs quantum dots. The first prototype detectors have been produced, and pending research goals include demonstration of detection
performance with minimum ionizing particles, corresponding to signals of about
4000 electron-hole pairs in a detector of 20 micron thickness. A compatible electronics
solution must also be developed. While the radiation tolerance of the device is
not yet known, generally quantum dot media are among the most radiation hard
semiconductor materials.
\clearpage
\section{Introduction}
Research in particle tracking detectors for high energy physics application is underway with a goal of improving radiation hardness, achieving improved position, vertex, and timing resolution, simplifying integration, and optimizing power, cost, and material. The five technologies described here approach these goals in complementary ways.
\section{Silicon Sensors in 3D Technology}
{\sl Boscardin, Dalla Betta, Hoeferkamp, Seidel, Sultan}
\bigskip
Silicon sensors of the 3D technology~\cite{DaVia} are employed in LHC experiments~\cite{Abbott,Ravera} to provide
radiation tolerant particle tracking at integrated fluences in the regime of $10^{16} n_{\rm eq}/{\rm cm}^2$. The
decoupling of the depletion depth from the sensor thickness allows operation at bias voltages
below breakdown despite very high integrated fluence, with significant savings on power
dissipation, and the small inter-electrode distance suppresses the effect of radiation-induced charge
trapping. The ATLAS IBL sensors, for example, are implemented in p-type with $230 \mu$m thickness
and column electrodes of diameter approximately $10 \mu$m, separated by approximately 62 microns. A slim edge
of 200 microns is employed. Designs for application to the HL-LHC, where innermost tracking
will be exposed over the course of 10 years to fluence $2.3 \times 10^{16} n_{\rm eq}/{\rm cm}^2$~\cite{Contardo}, are more aggressive still,
in anticipation of conditions in which the carrier lifetime will be reduced to 0.3 ns, corresponding
to a mean free path of 30 microns. Up to 200 interactions per 25 ns bunch crossing are expected
at the HL-LHC. Small-pitch 3D pixels ($25 \times 100~\mu {\rm m}^2$ or $50 \times 50~\mu{\rm m}^2$) have been developed to this
purpose, with inter-electrode distances of approximately 30 microns~\cite{DallaBetta} and a slim edge of 150 microns, and are
currently in the pre-production phase for the ATLAS ITk.
\bigskip
Plans~\cite{Riegler} for future facilities such as the FCC-hh anticipate a lifetime integrated luminosity
of 30 ab$^{-1}$, predicting integrated fluence at the innermost tracking volume approaching $10^{18}n_{\rm eq}/{\rm cm}^2$.
Estimates~\cite{Drasal} of the pileup conditions are on the order of 1000 events per crossing.
Continued development of silicon sensors of the 3D technology presents prospects both for
restoration of signal loss in high radiation environments, and for separation of pileup signals by precision
timing. Measurements~\cite{Kramberger} carried out on $50 \times 50~\mu{\rm m}^2$ cell 3D sensors have shown signals with a
full width of 5 ns, and a rise time of 1.5 ns, with a timing resolution of 30 - 180 ps (depending on
the signal amplitude); this is a mode of operation comparable to that achieved by low gain
avalanche detectors --- but lacking gain --- with the advantage of higher radiation tolerance and
better fill factor. The standard column configuration of 3D has the disadvantages, however, that
the electric and weighting fields are non-uniform, leading to position dependence of the pulse
rise time; this is the limiting factor on the timing resolution. New geometries~\cite{Forcolin,Anderlini} involving p-type
trench electrodes spanning the entire length of the detector, separated by lines of segmented
n-type electrodes for readout, promise improved uniformity and better timing resolution combined
with further increased radiation tolerance. Nevertheless, at this time, trenched electrodes cause
higher capacitance and introduce larger dead volumes within the substrate. Device optimizations,
especially in terms of geometrical efficiency, remain to be carried out. In addition, this problem
can be tackled at the system level by tilting the sensor plane with respect to the particle direction,
so that a larger fraction of the charge is generated within the depleted volume, and using multiple
planes of sensors with an offset between the electrodes, so all tracks would traverse several planes
without crossing the electrodes~\cite{Mendicino}.
\bigskip
3D columnar pixels with internal gain~\cite{Feasibility}-\cite{Indication} offer an alternative approach to signal
restoration at high fluence. When implemented with very small inter-electrode separation, approximately 15
microns or less, these devices can achieve controlled charge multiplication at voltages on the order
of 100 V, both before and after irradiation. Moderate gain values can be achieved, sufficient to
compensate the loss of charge signal due to irradiation of these thin (approximately $100~\mu$m) devices. Design
optimization continues with a goal of achieving uniform gain throughout the cell active volume,
also benefiting from the wider operating range that is possible due to increasing the breakdown
voltage.
\bigskip
The goal of this research is to advance one or two 3D technologies in silicon for tracking particles,
able to operate with adequate signal-to-noise ratio at fluences approaching $10^{18}$ n$_{\rm eq}/{\rm cm}^2$, and timing
resolution on the order of 10 ps. Planned research activities include TCAD simulations, process
optimization and fabrication of several generations of prototypes, and thorough characterization
of the prototypes before and after irradiation to extreme fluences.
\section{3D Diamond Detectors}
{\sl Kagan, Trischuk}
\bigskip
By 2028, experiments operating at the HL-LHC must be prepared for an instantaneous luminosity of
$7.5 \times 10^{34}/{\rm cm}^2/{\rm s}$ and charge particle fluxes of GHz/cm$^2$. After these doses, all detector materials will
be trap-limited, with the average drift distance a free charge carrier travels before it gets trapped being
below $50~\mu$m~\cite{Tsung}. 3D sensors reduce the drift distance the charge carriers must travel to reach an electrode to much less
than the sensor thickness. This is
particularly beneficial in detectors with a limited distance free charge carriers travel, such as
trap-dominated sensor materials like heavily irradiated silicon and pCVD diamond, where the observed
signal size is related to the mean free path divided by the drift distance. Under these circumstances
one gains radiation tolerance (larger signals) by keeping the drift distance less than the mean free
path. With the 3D geometrical structure, charge carriers drift inside the bulk parallel to the surface
over a typical drift distance of $25-100~\mu$m instead of perpendicular to the surface over a distance of
$250-500~\mu$m.
\bigskip
The RD42 collaboration has studied novel 3D detector designs in diamond, to extend the radiation
tolerance of diamond to fluences greater than $10^{17}$ hadrons/cm$^2$, exceeding the HL-LHC doses. The
detector design places column-like electrodes inside the detector material using a 130 fs laser with a
wavelength of 800 nm. After focusing to a $2~\mu$m spot, the laser has the energy density to convert
diamond into an electrically resistive mixture of different carbon phases~\cite{Pimenov}. A Spatial Light
Modulator (SLM)~\cite{Sun} is used to correct spherical aberrations during fabrication. This helps to
achieve in $50~\mu{\rm m} \times 50~\mu{\rm m}$ cells a high column yield of $\ge 99.8$\%, a small
column diameter of $2.6~\mu$m, and a resistivity of the columns of the order of $0.1-1~\Omega$cm.
In this detector geometry, the drift
distance an electron-hole pair must travel to reach an electrode can be reduced below the mean free
path of an irradiated sensor without reducing the number of electron-hole pairs created. In a detector
with $25~\mu{\rm m} \times 25~\mu{\rm m}$ cells, the maximum drift distance for charge carriers that go into the saddle
point region is $25~\mu$m, and $17.5~\mu$m for charge carriers that avoid the saddle point.
The goal of this research project is to create a detector that is essentially immune to radiation
doses at the level of $10^{17}$ hadrons/cm$^2$. Initial tests have shown that after $3.5 \times 10^{15}$ n/cm$^2$, the 3D
geometry with $50~\mu{\rm m} \times 50~\mu{\rm m}$ cells has better than three times less charge loss than a planar diamond detector after
normalizing both unirradiated devices to a relative charge of 1. Furthermore the charge in the
unirradiated 3D device is twice as large as that in the planar device. Thus, in addition to having twice
the charge, the 3D device also has better than three times less damage, due to the shorter drift distance. In order to
achieve the $10^{17}$ hadrons/cm$^2$ goal, completion of the design of 3D diamond devices with
$25~\mu{\rm m} \times 25~\mu{\rm m}$ cells and testing of these devices after irradiation with $10^{17}$ hadrons/cm$^2$ is proposed.
\section{Submicron Pixels with a Quantum Well for Vertexing}
{\sl Fourches, Renard, Barbier}
\bigskip
Development of a submicron position sensitive vertex detector for the future linear collider experiments is proposed.
Although improved relative to
their predecessors, the present vertexing pixel detectors at the Large Hadron Collider suffer from low position resolution. The objective of vertex detectors is to enable the accurate secondary vertex determination that is crucial for b-tagging~\cite{Nagai,Pasquali}
in the case of high transverse momentum ($p_{\rm T}$) events. The heavy quark events are characterized by a relatively high lifetime that leads to
a secondary vertex distinct from the interaction point~\cite{Tanabe}. For accurate track reconstruction, it is necessary to improve point-to-point resolution
well below the 5 micrometer limit. In the framework of ILD, development of a pixel detector based on the DoTPiX structure is proposed.
\bigskip
Accurate track reconstruction with a vertex detector is possible using a small pitch detector, which in the case of the ILC can reduce the
multiplicity (in which a pixel is hit several times). This is crucial for the ILD where the readout of the detector is made only after several bunches.
With a track fit, displaced secondary vertices can be evaluated, using an impact parameter technique to select the right track, and the analysis of the full decay
of the particle can be done, using all necessary jets. In addition, isolated tracks can be tagged in order to reduce fake events.
The vertex detectors implemented in the LHC experiment are based on a hybrid design.
The high particle rate at the LHC induces a large dose in the detector where non-ionizing energy loss damages
the detector material and the electronic readout. Special techniques have been used to circumvent these effects with the use of hardened processes~\cite{Dentan}
and adequately doped silicon pixel structures~\cite{Terzo}.
To accommodate the LHC beam crossing time, detectors use a triggered readout involving a fast readout chip (ROC).
The on-pixel electronics has to be elaborate to collect the information of all the pixel hits' output when triggered.
For the technologies available from the late 1990's to the early 2000's, this requirement excluded small pitch pixel detectors.
Even with pitch of tens of microns, the number of channels (pixels) is of the order of tens of millions in the inner vertex detectors.
\bigskip
The constraints are different for the ILC where more precise reconstruction is the objective. The advantage of vertex detectors with much improved
resolution will be good secondary vertex reconstruction with an accuracy of 0.5 micrometers (or in time at the speed of light, of 1.6 fs).
This cannot be matched by a timing procedure, which can only estimate the position of the interaction point in the beam-crossing zone.
This zone will be reduced at the ILC compared with the LHC. Additionally, short-lived particles can be tagged at this stage.
Significant features of this proposal include the following.
\begin{itemize}
\item The detectors close to the primary interaction point can detect low-mass charged particles that can escape the tracker due to
the effect of the magnetic field on low-mass particles~\cite{Griso}.
Tagging of such particles can be established with a good vertex detector~\cite{Suehara}.
These particles can produce disappearing tracks. The energy of these long-lived particles cannot be determined easily as they escape calorimetry.
The only possibility, besides using time-of-flight, is to add extra layers to the vertex detector to match the trajectory.
\item The operation of a vertex detector in a trigger-free mode means that many bunch crossings will be combined (this is pile-up) before being output and reset.
This makes the use of very small pixels necessary to avoid multiple hits in a single pixel. The pitch has to be reduced to match these requirements,
and only a fully monolithic pixel can be used for this purpose.
\item The reconstruction of tracks with relatively low $p_{\rm T}$ near the interaction point will be easier with a pixel detector with large enough
aspect (height/width) ratio. A small pitch (less than 1 micrometer) with a height up to of 10 micrometers (the sensitive zone) opens up such a possibility.
\end{itemize}
\bigskip
A pixel design has been proposed~\cite{Fourches1,Fourches2}. The necessary simulations have been made to assess the functionality of the proposed device. The next step is to find
out what is the best process to obtain the functionality and to reach some specifications. The principle of this pixel architecture (named DoTPiX) is the single n-channel MOS transistor, in which a buried quantum well gate performs two functions (Fig.~\ref{Fourches-figure1}): as a hole-collecting electrode and as a channel current modulation gate. Extensive simulations were made with this pixel architecture~\cite{Fourches1}; to assess its functionality, the buried quantum well gate is made with a Ge layer deposited on a silicon substrate. We are currently developing the basic structure with UHV/CVD techniques. The proposed structure is in its fabrication phase to obtain a test vehicle. The active layers where electron-hole pairs are created is of the order of $5~\mu$m below the surface, enabling detection of minimum ionizing particles. The processed wafers (Fig.~\ref{Fourches-figure2}) should be compatible with advanced CMOS nodes including SOI (FDSOI) and nanowire devices. The surface roughness on the processed wafers (as measured by AFM) is low enough but should be slightly improved. We have established a working group with different institutes for this project.
\bigskip
\begin{figure}[htbp]
\centering
\includegraphics[width=1.0\linewidth]{Fourches-fig1.png}
\caption{The operational principle of the DoTPiX structure within a pixel array (row and column); the array readout is similar to those of CMOS sensors, with detection, readout, and reset modes. The end of column is connected to a preamplifier, for digital or hit/no hit readout mode. Power dissipation occurs only during readout, due to the biasing scheme. In detection mode, $V_{\rm gate} < V_{\rm drain}$ and $V_{\rm source}$, to collect holes in the buried gate.}
\label{Fourches-figure1}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=1.0\linewidth]{Fourches-fig2.png}
\caption{For the DoTPiX project: (left) the TCAD simulation structure; (center) Ge hole quantum well, and (right) results of the processing (on a full wafer), the deposition of a thin Ge layer.
This results from electron microscopy, STEM Energy Dispersive X Spectrometry (STEM-EDX). The Ge concentration reaches 95\% in the 21 nm buried layer. The wafer prepared this way should be CMOS compatible with attention to the thermal budget of the process.}
\label{Fourches-figure2}
\end{figure}
\clearpage
\section{Thin Film Detectors }
{\sl Kim, Metcalfe, Sumant}
\bigskip
Nanoscience technologies are developing new cutting edge materials and devices for a wide range of applications.
HEP can take advantage of the many advances by looking toward thin film fabrication
techniques to implement a new type of particle detector. Thin Film (TF) Detectors have the potential to be fully
integrated, large area, low power, with low dead material, and low cost. The present goal is to investigate potential
research paths using thin film technologies and to identify and characterize the performance benefits for
future particle experiments.
\bigskip
A new detector technology is proposed based on thin films that is aimed at dramatically
improving the precision of particle detectors by greatly reducing the mass of the detector~\cite{Metcalfe6}. Cleaner signatures
of the particles from the primary collision will be obtained by reducing those particles' interactions
with dead material, which will improve reconstruction efficiencies and resolutions.
Thin Film technologies could potentially replace the entire detector including all the services. If a thin
film detector could be printed in large areas (square meters), it is estimated that the cost would be reduced to less than 1\%
of the current cost. If the nuclear interaction length can be decreased by a factor of 10, then the track
reconstruction efficiencies would reach 99\% and enable a host of new measurements and searches.
\bigskip
Thin Film technology presents one possible solution to achieve
these performance milestones. TF transistors (TFT’s) were first conceived in the 1960’s by Paul Weimer~\cite{Weimer}.
By the 21st century, fabrication technologies had improved enough to make it competitive
with existing technologies. TFT’s are the basis of technologies such as Liquid Crystal Display (LCD)
screens, solar cells, and light emitting diodes. It is a rapidly growing technology area with a large market
base and has corresponding investment in large scale fabrication and industrialization. Ultimately, the
broader interest of these technologies enables HEP to leverage the investments in commercialization as well
as the R \& D into materials, tools, and techniques.
\bigskip
Some of the advantages of TF’s are optical transparency, mechanical flexibility, high spatial resolution,
large area coverage, and low cost relative to traditional silicon-based semiconductor technology. TF technology
uses crystalline growth techniques to layer materials. Monolithic sensors can be fabricated using layers
of thin film materials for particle detection with layers for amplification electronics. The advantages of a
detector made with this type of technology include single piece large area devices (on the order of a few
square meters), high resolution ($< 10~\mu$m), low cost (100 times less than that of Si-CMOS), low mass, and high curvature for a
cylindrical, edgeless design~\cite{Gnade}-\cite{Street}.
\bigskip
Fabrication processes such as chemical bath deposition and close-space sublimation on a substrate material
can produce thin films with a high degree of precision. Here, the crystalline structure is grown in
layers, avoiding drilling and etching techniques standard in traditional silicon fabrication; consequently, TF
processing is much less expensive.
\bigskip
Thin film electronics can be vertically integrated with a thin film sensor if the fabrication
techniques are compatible. This would allow vertical integration of sensor and pixel electronics. Further
vertical integration using through-vias would enable signals to pass from one layer to the next, thus enabling
several levels of electronic processing. Typical front-end ASIC functions could be integrated into the
monolithic structure as well as higher-end processing to perform functions such as data aggregation and
region-of-interest processing. Such processing would reduce the number of transmission lines integrated
into a top layer and further reduce the material inside the detector volume.
Figure~\ref{Metcalfe-figure} shows a potential vertical stack-up.
\bigskip
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{Metcalfe-figure.png}
\caption{The conceptual stack-up of a monolithic thin film detector that
incorporates suppot structure, sensor, pixel electronics, regional
processing electronics, and power/data transmission. Potentially, no
other structures would be needed in the detector volume.}
\label{Metcalfe-figure}
\end{figure}
The transistor is the most basic unit that determines the power consumption in electronics. Complementary Metal Oxide Semiconductor
(CMOS) is a low power technology and is the current mainstay for most of commercial
electronics. There are, however, many types of transistor technologies that can outperform CMOS.
Silicon Germanium (SiGe) Heterojunction Bipolar Transistors (HBT’s) are another class of transistor that
typically boast faster speeds and lower power consumption~\cite{Cressler}-\cite{Metcalfe5}. Fin-Field Effect Transistors (FinFET’s) are
being pursued as the next ultra-low power technology and are manufactured by companies such as IBM and
Motorola. However, the most transformational transistor is the Thin Film Transistor (TFT), which is breaking
records in terms of size and power. All of these technologies have the potential for reducing the power (and
the copper in the transmission lines) over the current technologies.
\bigskip
Thin Film Detectors have the potential to replace a wide range of detector types
from tracking to calorimetry. The present goals are to identify key areas of research within
Thin Film technologies, quantify the key requirements from different types of experiments, and evaluate the
potential physics impact.
\clearpage
\section{Scintillating Quantum Dots in GaAs for Charged Particle Detection}
{\sl Mahajan, Minns, Tokranov, Yakimov, Oktyabrsky, Gingu, Murat, Hedges}
\bigskip
Future collider experiments will require particle detectors with timing resolution better than 10 ps; this is beyond the limits of existing technologies \cite{BRN2019}. One possible avenue for innovation of charged-particle tracking relies on novel ultra-fast scintillating material utilizing semiconductor stopping media with embedded quantum dots \cite{Oktyabrsky2016, Dropiewski2020, DROPIEWSKI2020161472}.
\bigskip
Fabrication of such a detector requires a scintillator with unique properties: very high light yield and a fast emission time. We have identified a candidate sensor material based on self-assembled InAs quantum dots (QDs) embedded into a GaAs matrix \cite{Oktyabrsky2016}. QDs are known to be excellent light emitters with close to 100\% efficiency and emission times on the order of hundreds of picoseconds. To make a scintillator, however, one needs to embed QDs into a dense medium that is transparent to the QD photon emission. GaAs fulfills this requirement. An ideal sensor consists of two physically integrated systems:
\bigskip
\begin{enumerate}
\item{The scintillator: A charged particle travels through the InAs QD/GaAs scintillator and produces electron-hole pairs in the GaAs matrix ($2.4 \times{} 10^5$ pairs/MeV). The carriers are rapidly captured (within 2-5 ps) in the positively charged QDs due to high electron mobility of up to 8500 $\rm{cm}^2 / \rm{Vs}$ at room temperature. The infrared emission (1.1 eV photons) is red-shifted more than 300 mV from the bandgap of the GaAs matrix, resulting in low self-absorption ($\sim{}1 \rm{cm}^{-1}$) \cite{Dropiewski2020}.}
\item{The photodetectors: As the refraction index of GaAs is high (approximately 3.4), only $\sim{}2\%$ of the emitted light exits the
scintillator through one planar interface with air, and the rest gets reflected and travels inside the scintillator. Thus for efficient detection, the photodetector (PD) must be physically integrated with the scintillator. The proposed design has a matrix of InGaAs photodiodes fabricated directly on the surface of the scintillation matrix. The photodiode thickness is of the order of 1--2 microns, leading to efficient absorption of the QD emission. Photodiodes fully cover the scintillator area, resulting in uniform and efficient collection of the emitted light with close to unity fill factor.}
\end{enumerate}
A schematic drawing showing this system is shown in Fig.~\ref{fig:qds-schem}.
The first prototype detectors have been produced at SUNY Polytechnic Institute at Albany as thin wafers of $\sim20$ micron thickness, each with a single integrated small area PD. The measurements of single-channel performance with $\alpha$ particles have been published~\cite{Dropiewski2020, DROPIEWSKI2020161472, Minns2021, Mahajan2021}. Using $\mathrm{Am}^{241} ~\alpha$-sources (5.5 MeV) and fast preamplifiers, we have measured a fast-decay constant of 270 ps and a 38 ps time resolution at room temperature without bias voltage applied to the PD. This performance, shown in Fig.~\ref{fig:qds-timingres}, is currently limited by circuit noise and bandwidth. We measure $1.7 \times 10^4$ detected electrons per 1 MeV of deposited energy with this system. Alternatively, a slower low-noise preamplifier demonstrated $5.1 \times 10^4$ electrons/MeV of incident energy with longer ($\sim6$ ns) pulses. We expect performance to be near that of the theoretical optimum of $2\times10^5$ electrons/MeV. A comparison of the performance of this detector with that of other scintillators is shown in Fig.~\ref{fig:qds-performance}. The demonstrated performance is, at the time of writing, the fastest and highest light yield of all known scintillators.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\linewidth]{qds-schematic.png}
\caption{A schematic drawing of the proposed tracking sensor. A charged particle enters the GaAs scintillator, producing electron-hole pairs. The electrons are then quickly trapped by the positively charged InAs quantum dots (QDs). The QDs undergo photoluminescence (PL) and emit photons that travel through the medium. The emitted photons are collected by a photodiode (PD) array.}
\label{fig:qds-schem}
\end{figure}
\bigskip
\begin{figure}[htbp]
\centering
\includegraphics[width=0.7\linewidth]{qds-timingres-tushar2021.png}
\caption{Scope traces of events from $\rm{Am}^{241}$ events. Recorded pulses showing 100 ps rise time, 270 ps decay time, and 38 ps time resolution with average collected charge of $1.5\times10^5$ electrons. Adapted from Ref. \cite{Mahajan2021}.}
\label{fig:qds-timingres}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{qds-vs-theworld.png}
\caption{Light yield in photons/MeV of deposited energy versus decay time for various known scintillators. Faster performance (decreasing decay-time) extends rightward along the horizontal axis. Adapted from Ref. \cite{Mahajan2021}.}
\label{fig:qds-performance}
\end{figure}
Significant exploratory research and development is required to accurately assess expected performance of these detectors in future high-energy physics applications. First, we must demonstrate detection performance with minimum ionizing particles, corresponding to expected signals of about 4000 electron-hole pairs in a single detector of $20~\mu$m thickness. Given that the measurements with $\alpha$-particles are noise-limited, we also expect to encounter significant challenges developing a suitable electronics solution for optimal energy and timing performance for MIP detection. Furthermore, the radiation tolerance of this type of custom epitaxially-grown detector is not known, although the QD medium itself is among the most radiation hard semiconductor materials \cite{Oktybrsky2005}. We will ultimately need to assess the performance of these detectors in the high-radiation environments expected in future high-energy physics experiments.
\section{Conclusion}
Five contemporary technologies are under development for applications at future
high energy physics experiments. Collaborators interested in joining any of these efforts are welcome.
|
1,116,691,500,765 | arxiv | \section{Introduction}\label{sect:intro}
We investigate a generalized Weber problem in a dynamic and stochastic environment. The model combines a classical planar location problem with optimizing the size of the fleet of trucks in a logistic and services network
and describes interacting production and distribution facilities with logistic components.\\
This integrated
problem is related to location-allocation problems,
location-routing problems (LRPs), location-inventory problems (LIPs), and transporta\-tion-location-allocation problems
under random influences.
A key property of our model is to integrate strategic (facility location) and tactical/operational (allocation, scheduling) aspects of decision making. Such integrated logistics-location models occur e.g. in supply chain planning and operation, for surveys see
\cite{melo;nickel;saldanha-da-gama:09} and \cite{heckmann;nickel:19}.
Difficulties arising with similar integration procedures in combined location-routing problems are described in \cite{min;jayaraman;srivastava:98}[p.10].\\
Our model comprises a set of warehouses with known locations, a single production center which produces commodities which are demanded by the warehouses, and a set of trucks which transport the commodities from the production center to the warehouses.
The model captures additionally congestion which emerges at service stations with limited capacity for loading of trucks at the center and for unloading at the warehouses.
Optimality of the system is defined
with respect to \textbf{maximization} of the warehouses' throughput (which generates revenue) and \textbf{minimization} of the number of trucks (which generate costs) \textbf{over time } and \textbf{in a stationary system}. {Our main contributions are}:\\
For given demands from the warehouses and incorporating
the consequences of congestion at the center and the warehouses
\\
\textbf{(i)} we find an optimal location for the central production facility, and \\
\textbf{(ii)} determine the needed transportation capacity,
and\\
\textbf{(iii)}
solve jointly both optimization problems in a unified model, to combine strategic decision making (for location of the center) with decisions for tactical and operational issues (transportation capacity and scheduling rules for routing trucks), and\\
\textbf{(iv)} we demonstrate that the solution of the optimization problem reveals important robustness of that solution against changes or perturbations of several parameters on the tactical/operational level (insensitivity).\\
\underline{Summarizing}: We introduce a model which is tailored to integrate several hierarchical levels of decision making in complex systems. As indicated in the literature, neglecting these inter-dependencies often generates sub-optimal solutions, see the comments on literature in Section \ref{sect:Literature}, especially concerning
integration of strategic and tactical/operational aspects of planning on p. \pageref{page:Integration}.
\textbf{Structure of the paper.}
In Section \ref{sect:Literature} we review related literature.
Section \ref{sect:Problem} presents an overview of the problem setting and connections to
investigations of related problem settings.
Details are provided in Section \ref{sect:DetailsProblem}.
Our main findings are described in Section \ref{sect:mainresults}. In Section \ref{sect:reliable stations} we analyze the system in full detail.
In Section \ref{sect:NumberOfTrucks} we complement our structural results by an algorithm to determine the minimal number of trucks needed to satisfy the overall demand at the warehouses.
In Section \ref{sect:DiscussionModel} we discuss the assumptions used in Section \ref{sect:reliable stations} and we indicate that in many cases the results
hold in more general settings.
Section \ref{sect:MiniPassageTimes} is devoted to determine the optimal location of the center for minimizing the round-trip times of trucks.
Section \ref{sect:NumericManagerSensitive} contains numerical experiments to underpin the structural results obtained sofar and a discussion of robustnes properties of our results.
In Section \ref{sect:ManagerialDecision} we discuss the impact of our results on the consequences of intertwining, respectively separating strategic (location) and
tactical/operational (routing and scheduling) decisions.
We conclude by discussing generalizations and extensions of the location problem.
Necessary prerequisits from queueing network theory are provided in Appendix \ref{sect:fundamentals}.
Proofs are postponed to Appendix \ref{sect:proofs}.
Details of numerical experiments are presented in Section \ref{sect:AddNumeric}.\\
\textbf{Conventions:}
$\mathbb{N}=\{0,1,2,\dots\}$, $\mathbb{N}_+=\{1,2,\dots\}$.
Empty sums are $0$, empty products are $1$. We set
$0/0=0$. For any set $A$, $\mathcal{P}(A)$ is the set of subsets of $A$.
Increasing means non-decreasing and decreasing means non-increasing.
\section{Literature review}\label{sect:Literature}
\underline{Stochastic location models with congestion},
i.e.
location problems in the context of service processes under stochastic influences have been investigated since the 1970's by many authors. Research on location problems within the scope of queuing systems was initiated with Larson's paper \cite{larson:74} followed by work of Larson, Berman and coauthors (e.g. \cite{berman;larson;chiu:85} and \cite{berman;larson;parkan:87}) on discrete location problems. For surveys we refer to the relevant chapters of the collections
\cite{drezner;hamacher:04}[Chapter 11],
\cite{mirchandani;francis:90}[Chapter 13], (vehicle routing problems under stochastic side constraints),
and the recent very detailed \cite{berman;krass:19}.
We sketch here only two main research directions:\\
(1) In \cite{drezner;schaible;simchi-levi:90} and \cite{scott;jefferson;drezner:99} the authors consider
location problems where mobile servers move in a plane and demands of the clients occur as Poisson processes.
The mobile servers are described as queueing systems, e.g. of type $M|M|1|\infty$ or $M|G|s|\infty$.
The guiding principle is to incorporate travel times to and from the clients into the service times of the mobile server. A survey is provided in \cite{berman;krass:04}.
Locational analysis in a randomly changing environment with occurrence of queueing phenomena is investigated in \cite{dan;marcotte:19}.\\
(2) A class of models which is somehow dual to that described in (1):
The servers are fixed and clients move to the nearest service stations. Work in this direction is by \cite{berman;drezner:07} and investigated further by \cite{aboolian;berman;drezner:08} and \cite{aboolian;berman;drezner:09}, where additional references are provided.
In these papers the demand is generated at specified points according to a Poisson or a general renewal process and the servicing nodes usually are modeled as M/M/k/$\infty$ systems. For different cost functions the authors determine the optimal location of a set of servers on a set of nodes.
The field is reviewed in detail in \cite{berman;krass:19} and classified there as SLCIS ( $=$ Stochastic Location models with Congestion and Immobile Servers).
\underline{Location-inventory problems} (LIPs) aim ``to integrate strategic supply chain decisions with tactical and operational inventory management decisions \cite{farahani;bajgan;fahimnia;kaviani:15}''.
The basic LIP, as described in \cite{farahani;bajgan;fahimnia;kaviani:15}[Section2], encompasses a single supplier (production center), several warehouses (distribution centers), and retailers. The location of the supplier and the retailers are given and the decision problem is to determine the location of the distribution centers. \cite{farahani;bajgan;fahimnia;kaviani:15} provides a survey of research on basic LIPs and more evolved variants.
\underline{Location of local repair facilities} with a central service station and inventories at the repair stations is investigated in \cite{ommeren;bump;sleptchenko:06}.
Given the location of the center, the objective is to determine optimal locations of repair facilities, ensuring a certain inventory level at each station. The optimization problem is solved by approximations and local search.
In this article queueing models are used to describe the interrelation between local repair stations and the center.
\underline{Location routing problems} aim to combine location analysis and planning of vehicle routing.
Fundamentals of vehicle routing problems are described in \cite{laporte:88}.
A recent survey with emphasis on location decisions on networks
(discrete location problems) is
\cite{albareda-sambola;rodriguez-pereira:19}. Closer to our problem of location in the plane are \cite{salhi;nagy:09} and in a more versatile setting \cite{manzour-al-ajdad;torabi-salhi:12}.
\underline{Integration of strategic and tactical/operational aspects of planning} is a common topic of almost all the mentioned work.
\label{page:Integration}
In \cite{salhi;rand:89} for LRPs it is shown that separating decision on location and routing can lead to sub-optimal decisions.
Under the heading ``Why logistics matters in location modeling'' this problem is discussed indepth in \cite{heckmann;nickel:19}[Section 6.2], stating as main conclusion ``that making location decisions ignoring primary logistics activities \dots may result in excessive costs.''
\section{Problem description and classification}
\label{sect:Problem}
\underline{Problem setting.} We are given a set of warehouses (stations), indexed by $j\in \{2,3,\dots,J\}$, whose positions
$a_j=(a_{j1},a_{j2})\in\mathbb{R}^2$ in the plane are known. We are also given the aggregate demand (requirement) of $D_j$ truck loads per day generated by warehouse $j$ for a single commodity, $D_j>0$ .\\
We are to find a location $x=(x_1,x_2)\in\mathbb{R}^2$ for a production center (source of commodities), indexed henceforth by $1$, where the commodities are produced and dispatched.\\
To deliver the commodities from the production center to the warehouses, $N\geq 1$ identical trucks circulate in the system.
The dispatching rules are roughly as follows:
A truck loaded at the center is directed to one of the warehouses according to a given schedule or plan, is unloaded there, and returns to the central station to start another delivery cycle. The next destination,
determined by the scheduling regime, might be different.
The center has limitations on its capacity to ship the product, especially when loading several trucks simultaneously, and the warehouses have limited capacity to unload trucks in parallel.\\
To assess the system's performance we take into consideration the distances between the center and the warehouses, the number of trucks, the travel times,
the scheduling rules for sending out the trucks,
the loading and unloading times, and additionally the delay resulting from limitations of loading and unloading capacities, i.e., congestion at loading and unloading facilities.
\underline{The optimization problem} to be solved is to determine the location of the center such that the delivered amount of commodities is maximal and all requests of the warehouses are satisfied with a minimal number of trucks.
\underline{Related problems and classification.}
Following \cite{min;jayaraman;srivastava:98} and \cite{nagy;salhi:07} this problem is related to location-allocation problems because it is assumed that there are only radial trips from the center to the warehouses.
The problem is related to location-routing problems because it deals with ``location planning with tour planning aspects taken into account \cite{nagy;salhi:07}[p.1]'',
because we have to decide about subsequent tours for the trucks to visit different warehouses (the scheduling regimes/plans).
The relation to transportation-location problems in the sense of Cooper \cite{cooper:72}, \cite{cooper:76} is clear because we consider, among others, both problems. Moreover, the center has limitations on its capacity to ship the product \cite{cooper:72}[p. 94].
Additionally, our optimization criterion encompasses a time dimension as it is considered in \cite{tapiero:71} for the classical setting of Cooper.
Aspects of our problem setting which are not included in the above standard problem classes are:\\
$\bullet$ Limited unloading and loading capacities at the warehouses and the central production facility, i.e. congestion in the transportation network and queueing problems.\\
$\bullet$ Time dimension of transportation-location-allocation problems, as discussed e.g. in \cite{tapiero:71}[p. 383], in the optimization criterion.
In \cite{tapiero:71}[Section 5] questions concerning delivery time lags are sketched.
In our setting sequentially varied scheduling of trucks is allowed and for feasible scheduling of trucks emerges a necessary condition which optimal plans must satisfy.
We sketch a toy example which highlights the problems.
\begin{beisp}\label{ex:TimeIncluded}
Consider the case $J=3$, i.e. a center indexed $1$ and two warehouses, numbered $2, 3$, with demands
$D_2 = 10$ and $D_3 = 20$ truck loads per day for a commodity and $N=3$ trucks with equal capacity. Two reasonable schedules are\\
\textbf{(i)}
one truck serves station $2$ and two trucks serve station $3$,\\
\textbf{(ii)} all trucks are scheduled
to serve in a cycle first station $3$, thereafter station $2$, and finally station $3$ again; these
cycles are iterated.\\
Both schedules generate fair service for the warehouses because they guarantee that over time
warehouse $2$ will obtain $\rho_2=1/3$ and
warehouse $3$ will obtain $\rho_3=2/3$ of the overall delivered goods.\\
Because we investigate location decisions we have to consider long time horizons and we will therefore assume that the system has approached its stationary state.
Clearly, then a necessary condition for optimality of a scheduling regime for trucks is to partition in the long run sent-out goods according to:
$\rho_2=D_2/(D_2+D_3) \wedge \rho_3=D_3/(D_2+D_3)$.
\end{beisp}
Our problem fits into neither of the mentioned streams
on queueing-location problems because we incorporate local congestion at the warehouses explicitly.
Determining a center's location in \cite{berman;drezner:07}, \cite{aboolian;berman;drezner:08}, and \cite{aboolian;berman;drezner:09} is somehow similar to our problem but in these papers no two-way interactions (here generated by trucks) occur between sources of demands and the production center.
\section{Model and main results}\label{sect:ModelResults}
\subsection{Details of problem statement}\label{sect:DetailsProblem}
\underline{Modeling the warehouses-production network.}
Warehouse $j$ is equipped with $s_j\geq 1$ service facilities for unloading trucks and there is ample waiting space for trucks that arrive while all unloading facilities are busy,
$j\in\{2,\ldots,J\}$. The time for unloading a truck is exponential with rate $\mu_j$ per hour.\\
Center $1$ has $s_1\geq 1$ service facilities for loading trucks and ample waiting space for trucks that arrive while all loading facilities are occupied. Loading times are exponential with mean $\mu_1^{-1}$.
The queueing regime for trucks is First-Come-First-Served (FCFS).
We abbreviate the service rate functions by
\begin{equation}\label{eq:Muj}
\mu_j(n) :=
\mu_j\cdot min(n, s_j),~~ n=0,1,\dots, \qquad j=1,\dots,J.
\end{equation}
We consider general schemes for scheduling the radial trips of trucks which must meet only the following restrictions.
If $D:=\sum_{j=2}^J D_j$ denotes the total demand per day, the portion of demand that has to be delivered to warehouse $j$ is
\begin{equation}\label{eq:routingRho}
\rho_j=D_j/D\in (0,1],~~j\in\{2,\ldots,J\},\quad \sum_{j=2}^J \rho_j = 1.
\end{equation}
Then a truck loaded at the center is directed to warehouse $j$ with (average) frequency $\rho_j=D_j/D$.
In the mathematical model we realize this property of the dispatching rules by a randomized schedule which selects the next warehouse with probability $\rho_j$ for destination $j$
in a Markovian way.
We emphasize that this does not mean that scheduling of trucks should be randomized, but reflects that we are interested in gross characteristics of the system layout. We comment
on this later in the discussion of ``Random routing\dots''
on p. \pageref{page:RandRouting}.\\
Demand that is not satisfied immediately will be backordered at the respective warehouses.
The distance between warehouse $j$ and center at $x$
is denoted by $d_j(x):=d(a_j,x)$ where $d:\mathbb{R}^2\times\mathbb{R}^2\longrightarrow\mathbb{R}_+$ is a general convex function.\\
\underline{Details of the optimization problem.}
The aim is to fulfill all demands occurring at the warehouses
with a minimal number of trucks
and to maximize the utilization of the given resources. The latter means that we are to maximize the overall mean number of delivered goods per hour (= time unit for servicing)
which is the sum of the throughputs of the warehouses measured in truck loads.
The throughput of a warehouses depends on the coordinates $x=(x_1,x_2)$ of the center and the number of trucks $N$.
Formally:\\
Denote for $T\in\mathbb{R}_+$ by $A_j(N;x)(T)$ the amount of commodities which arrived at warehouse $j$ and is unloaded
within time horizon $[0,T]$ when the center is located at $x$
and $N$ trucks are cycling.
It will be shown that for $j=2,\dots,J$ the throughput
\begin{equation}
TH_j(N;x) := \lim\limits_{T\to \infty}\frac{1}{T}A_j(N;x)(T)
\end{equation}
for warehouse $j$ exists
and determines by standard ergodicity arguments
for Markov processes the overall mean number of truck departures per hour from $j$
in the stationary system. By stationarity this equals the mean number of truckloads delivered to $j$ per hour.
The total throughput of interest is
\[TH_w(N;x)=\sum_{j=2}^{J} TH_j(N;x)\}.
\]
``$ _w$'' indicates that we evaluate only throughputs of warehouses.
This leads to
\begin{opt}\label{opt:throughputoptimization} Determine
\begin{eqnarray*
\min_{N\in\mathbb{N}_+}\left(\max_{x\in\mathbb{R}^2} \Big\{TH_w(N;x)\Big\}\right)
&&~\text{and}~~~~
\arg\Big\langle\min_{N\in\mathbb{N}_+}\left(\max_{x\in\mathbb{R}^2} \Big\{TH_w(N;x)\Big\}\right)\Big\rangle\\
\text{subject to}
&&TH_j(N;x)\geq D_j , j=2,\dots,J.
\end{eqnarray*}
\end{opt}
It will turn out that the main effort in solving Optimization Problem
\ref{opt:throughputoptimization} is to solve a sequence of
maximization sub-problems. These are
\begin{opt}\label{opt:ThroughputMaximization} Determine
for each $N\geq 1$
\begin{eqnarray*
\max_{x\in\mathbb{R}^2} \Big\{TH_w(N;x)\Big\}
~~~~\text{and}~~~~
\arg\Big\langle\max_{x\in\mathbb{R}^2} \Big\{TH_w(N;x)\Big\}\Big\rangle.
\end{eqnarray*}
\end{opt}
\begin{rem}\label{rem:SmallCapacities}
If loading and unloading capacities are small compared to the demand that has to be delivered, there might be no solution of the Optimization Problem \ref{opt:throughputoptimization}.
This is due to bottlenecks in the network.
Nevertheless, all Optimization Sub-Problems
\ref{opt:ThroughputMaximization} have solutions.
If necessary, we assume that the available capacities guarantee that a feasible solution of the problem exists.
\end{rem}
\subsection{Main results and detailed analysis} \label{sect:mainresults}
Facility location in connection with queueing
problems usually leads to complex algorithms, see
\cite{berman;larson;chiu:85}.
In view of this, our first theorem is counter-intuitive.
\begin{theorem}\label{thm:loesung}
Consider locations $a_j=(a_{j1},a_{j2})\in\mathbb{R}^2, j=2,\dots,J,$ in the plane and associated weights $\rho_j$ from \eqref{eq:routingRho}.
Let $x^{\ast}\in\mathbb{R}^2$ be a solution of the standard Weber problem with weighted distances:
\begin{equation}\label{weberproblem}
\text{Find}~~ \min_{x\in\mathbb{R}^2}\ \Big\{\sum_{j=2}^J\rho_jd_j(x)\Big\}
~~\text{and}~~ x^{\ast}=\arg\Big\langle\min_{x\in\mathbb{R}^2} \Big\{\sum_{j=2}^J\rho_jd_j(x)\Big\}\Big\rangle.~~~
\end{equation}
Then $x^{\ast}$
is a solution of the Optimization Problem \ref{opt:ThroughputMaximization}
for any $N\geq 1$ as well.
\end{theorem}
The proof is postponed to Appendix \ref{sect:proofs}.
It relies on the observation that the model for the logistic and services network from Section \ref{sect:Problem} can be described in terms of a closed queueing network of Gordon-Newell type.
The proof of the next theorem will be given implicitly by proving correctness of Algorithm \ref{buzen2} below.
\begin{theorem}\label{thm:MiniMax}
If a solution of the Optimization Problem \ref{opt:throughputoptimization} exists for the capacities $\mu_{j}(\cdot), j=1,\dots,J$, it is uniquely determined if $x^*$ is given.
\end{theorem}
\begin{rem}\label{rem:StrikingResults}
The results of Theorem \ref{thm:loesung} and Theorem \ref{thm:MiniMax} are striking, so comments are necessary. \textbf{(i)}
If the Optimization Problem \ref{opt:throughputoptimization}
has a solution, i.e. the side constraints are satisfied with
capacities $\mu_{j}(\cdot), j=1,\dots,J$,
these service capacities $\mu_{j}(\cdot)$ (respectively the number of service channels $s_j$) at warehouses and center do not matter for optimizing the overall warehouse throughput with respect to the location of the center. Similarly, the absolute demands
$D_j$ and the number $N$ of trucks are not relevant for the
optimal location $x^*$.
The relevant information for the location decision only comprises \\
$\bullet$ the distances $d_j(x):=d(a_j,x)$, which determine travel times, and\\
$\bullet$ the proportions $\rho_j=D_j/D$ of goods to be dispatched to warehouse $j$.\\
\textbf{(ii)}
It is intuitive that increasing the loading capacity at the center increases throughput at any warehouse.
However, less intuitive is: If we fix the capacities at the center and at all but one dedicated warehouse and increase the unloading capacity at the dedicated warehouse then the throughput at \textbf{all} warehouses increases. Both facts are consequences of Theorem 14.B.13 of
\cite{shaked;shanthikumar:94}.\\
When capacities of loading/unloading facilities change
Theorem \ref{thm:loesung} guarantees that the decision for the optimal location remains optimal as long as the solution of the Optimization Problem \ref{opt:throughputoptimization} exists.\\
\textbf{(iii)}
Theorem \ref{thm:loesung} does not propose that warehouse throughput is independent of local properties of the warehouses. Details about functional dependencies will be provided below. Moreover, it is not clear in advance whether a prescribed overall throughput can be met with a given set of parameter values.
If the throughput can be met with the given capacities, Theorem \ref{thm:MiniMax} in connection with the main result of \cite{vanderwal:89} guarantees that by successively adding trucks we can increase the throughput until the total requirements can be dispatched.
Otherwise, if loading/unloading capacities do not suffice,
bottlenecks occur.
Our proofs will show that
we can increase the throughput by increasing the loading and unloading capacities at the nodes, see Section \ref{sect:NumberOfTrucks}. Theorem \ref{thm:loesung} states that in any case \underline{the selected location remains optimal}.
\end{rem}
\subsection{Analysis of the model as a Gordon-Newell network}\label{sect:reliable stations}
The locations in the problem setting of Section \ref{sect:Problem} can be arranged as a star-like graph with warehouses as exterior vertices $2,\dots,J$ and the production unit as central vertex $1$. Routes from the center to the exterior nodes, and vice versa back, correspond to edges (links, lanes).
The vertices contain the loading and unloading facilities
modeled as queueing systems.
The circulating trucks are modeled as customers requesting for service at these queueing systems.\\
Because the number of trucks circulating in the network is fixed these features establish a closed queueing network structure (Gordon-Newell network). Additionally,
we apply a standard feature to incorporate travel times into the model for the logistic network:
For each warehouse, roads from the center to that warehouse and back are modeled as two additional infinite server nodes with random or deterministic service times ( $=$ travel times).
Necessary definitions, facts, and formulas from network theory are summarized in Appendix \ref{sect:fundamentals}.
\subsubsection{The detailed model of the logistic and services network}\label{sect:DetailedModel}
We start with node $1$ (the center)
and nodes $2,\ldots,J$ (the warehouses) which are multi-server nodes with $s_j$ service channels and exponentially distributed service times with mean $\mu_j^{-1}, j=1, 2,\ldots,J$,
and with distances $d_j(x):=d(a_j,x), j=2,\ldots,J$.
$N\in\mathbb{N}_+$ customers (trucks) cycle in the network. For simplicity of presentation we assume that trucks are traveling with unit speed, i.e., $1$ km/hour.
(In examples we shall introduce realistic speeds.)
Whenever a truck is served (loaded) at center $1$ and is routed to warehouse $j$ it has to travel distance $d_j(x)$ to and from.
Traveling these distances is modeled as trucks being served by an infinite server station
for a deterministic time $d_j(x)$ or a random time with mean $d_j(x)$. The first infinite server, from $1$ to $j$, is denoted $ja$. After passing the road to $j$ the truck will be unloaded at warehouse $j$.
When unloaded at warehouse $j$ the truck
travels back to center $1$. This is modeled as being served by another infinite server station, denoted $jb$, with mean service time $d_j(x)$.\\
Summarizing: Any radial tour from the center to node $j$ consist of three nodes, representing: (i) traveling to $j$ (via $ja$), (ii) unloading at $j$,
and (iii) return from $j$ (via $jb$). Thus we have a network with $3 (J-1) + 1$ nodes and any round trip (radial tour to $j$) is of the form: ''$\mathbf{1 \to ja \to j \to jb \to 1}$''.\\
We refer henceforth to center, warehouses, and lanes jointly as ``stations''.
We assume that all service and travel times are independent and for all lanes and loading/unloading stations identically distributed. The dispatching rules and the radial structure of the trucks are modeled by routing probabilities $r(\cdot,\cdot)$ as follows:
\begin{description}\label{routing}
\item[(R1)]from center $1$ to lane $ja$: $r(1,ja):= \rho_j$, $j=2,\ldots,J$,~~ $\sum_{j=2}^J r(1,ja)=1$,
\item[(R2)]all other routing is deterministic: $r(ja,j)=r(j,jb)=r(jb,1)= 1, j=2,\ldots,J$, and
$r(k,m)=0$ otherwise\,.
\end{description}
Routing decisions at node 1 are independent of the network's previous history.
For simplicity of presentation we assume that the travel times of the trucks are exponentially distributed with mean $d_j(x)$ when $x$ is the location of the center. We discuss this in detail in Section \ref{sect:DiscussionModel}.\\
With these stations, routing, and customers we have constructed a star-like Gordon-Newell network.
Local states of nodes are: For station $j$ is $n_j$ the number of customers present (in service + waiting), for station $ja$ is $n_{ja}$ the number of trucks on the way from station $1$ to station $j$ and
for station $jb$ is $n_{jb}$ the number of trucks on the way from station $j$ to station $1$.
We abbreviate state-dependent service intensities as $\mu_j(\cdot),j=1, 2,\ldots,J,$ given in \eqref{eq:Muj} and for lanes by
\begin{equation}\label{eq:intensities}
\mu_{ja}(x)(n_{ja}) = d_j(x)^{-1}\cdot n_{ja}~~\text{and} ~~
\mu_{jb}(x)(n_{jb}) =d_j(x)^{-1}\cdot n_{jb},\quad j= 2,\ldots,J.
\end{equation}
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\put(3.35,3){$r(1,3a)$}
\put(3,1.4){$r(1,Ja)$}
\put(4.7,4.5){\framebox(2.4,0.6){$\exp(\mu_{2a}(x))$}}
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Summarizing, we have a Gordon-Newell network
with node set\\
$\bar{J}=\{1, 2,\ldots, J, 2a,3a,\ldots,Ja,2b,3b,\ldots,Jb\},$
individual service rates $\mu_{ja}(x) = \mu_{jb}(x) =
d_j(x)^{-1}$, and $\mu_{j}(\cdot),$
routing matrix via {\bf (R1), (R2)} (p.\pageref{routing})
and state space
\begin{equation}\label{statespace1}
S(N,{\bar J}) = \{(n_i:i\in {\bar J})\in \mathbb N^{\bar J}:
\sum_{i\in{\bar J}}n_i = N\}.
\end{equation}
\subsubsection{Utilization of resources: Computing throughputs}\label{sect:Throughputs}
Utilizing facts collected in Appendix \ref{sect:fundamentals}, we are in a position to determine the overall throughputs at the warehouse stations. These are measures for efficient utilization of the given resources. Proofs are postponed to Appendix \ref{sect:proofs}.
\begin{theorem}\label{thm:normingconstant}
Denote by $G(N,\bar{J};x)$ the normalization constant of the Gordon-Newell network with node set $\bar J$, if $N$ trucks are cycling and the center is located at
$x\in \mathbb R^2$. Then with $\eta_j$ from the routing {\bf (R1), (R2)} via traffic equation \eqref{trafficeqn} it holds with
$\eta_1= \frac{1}{4}, \eta_j= \frac{1}{4}\cdot \rho_j, j=2,\dots, J,$
\begin{equation*}
G(N,\bar{J};x)=\sum_{n=0}^N\left[\left(\sum_{n_1+\ldots
+n_J=N-n}\prod_{j=1}^J\left(\prod_{k=1}^{n_j}\frac{\eta_j}{\mu_j(k)}\right)\right)
\frac{2^n}{n!}\left(\sum_{j=2}^J
\eta_jd_j(x)\right)^n \right].
\end{equation*}
\begin{equation}\label{othruput1}
\text{The overall throughput of the network is}~~ TH(N;x)=\frac{G(N-1,\bar{J};x)}{G(N,\bar{J};x)}.\quad
\end{equation}
The total throughput at the warehouse stations is
\begin{equation}\label{othruput}
TH_w(N;x) = TH(N;x)\sum_{j=2}^J \eta_j = \frac{1}{4}\cdot TH(N;x).
\end{equation}
\end{theorem}
\begin{rem}\label{rem:Interpretation}
The representation of $G(N,\bar{J};x)$ has a remarkable interpretation. It is the same normalization constant as that for a Gordon-Newell network with $N$ customers, $J$ multi-server stations with the same service rates as given in \eqref{eq:Muj} and an attached
\underline{single infinite server}, which will be indexed by $J+1$, with visit ratio $\eta_{J+1} = 1/2$ and exponentially distributed service time with mean $(\sum_{j=2}^J \rho_jd_j(x))$.
\end{rem}
\begin{kor}\label{normingconstant-det}
Consider the system of Theorem \ref{thm:normingconstant} with deterministic or general random travel times
with means $d_j(x)$.
Then the normalization constant is the same $G(N,\bar{J};x)$ and the relevant throughputs are
\eqref{othruput1} and \eqref{othruput} as well.
\end{kor}
\subsection{Determining the number of trucks} \label{sect:NumberOfTrucks}
We demonstrate the power of Theorem \ref{thm:normingconstant} by showing how to determine efficiently the minimal number of trucks to fulfill the total demand. We assume that
the center's location $x$ and capacities $\mu_j(\cdot)$ are fixed and sufficiently high to satisfy demands $D_i$ eventually, i.e. with sufficiently many trucks.\\
Recall that in our development we assumed up to now that trucks travel with unit speed (1 km/hour). This implies that $d_j(x)$ is exactly the time to travel distance $d_j(x)$.
For the present demonstration we allow general speed $S>0$ for trucks. The mean time for traveling distance $d_j(x)$ is then
the mean service time $d_j(x)/S$ at the infinite servers $ja$ and
$jb$, $j=2,\dots,J$.\\
With notation from Definition \ref{defn:GN} and
Theorem \ref{thm:gordonnewell}, we apply Buzen's Algorithm
\ref{alg:Buzen} in a first step to a Gordon-Newell network consisting of stations $1, 2, \dots, J$. Then we apply the representation of $G(N,\bar{J};x)$ from
Remark \ref{rem:Interpretation}.
\begin{alg}
[Determine minimal number of trucks.]\label{buzen2}
Let $C\in(0,\infty)$ denote the capacity of trucks, i.e., the amount of the commodity, that each truck can carry, and
$S\in(0,\infty)$ the speed of the trucks.\\
\underline{Initialization}: {\sc Store}\\
$ G(0,j) := 1, j=1,\dots,J,~~~ \kappa := 2\cdot \sum_{j=2}^J \eta_jd_j(x)/S,~~~
D=\sum_{j=2}^J D_j$,\\ $H(1,\bar J;x):= 1$.
\noindent
{\sc Set} $N\leftarrow 1$.
\underline{Iterate (*)}
{\sc For} $N$ {\sc do}
\begin{eqnarray*}
&&\text{{\sc Store}}~~ G(N,1):= g_1(N)~\text{from}~ \eqref{eq:Buzen2}.\\
&&\text{{\sc Compute with}}~ \eta_1={1}/{4}, \eta_j={1}/{4} \rho_j, ~{\sc and}~
\mu_j(k) ~{\sc from}~ \eqref{eq:Muj}, j=2,\dots,J,\\
&&{\sc from}~ \eqref{buzen1}:\quad G(0,J), G(1,J),\dots G(N,J).\\
&&\text{{\sc Compute}}~ G(N,\bar{J};x)=\sum_{n=0}^N\left[G(N-n,J) \cdot \frac{\kappa^n}{n!}\right].\\
&& \text{{\sc If} }~~
C\cdot\frac{1}{4} \frac{H(N,\bar J;x)}{G(N,\bar{J};x)} \geq D\,:\quad \text{{\sc Then output}~~$N$,}\quad \text{{\sc Stop}. }\\
&& \text{\sc{else Store}}\\
&&{H(N+1,\bar J;x)}\leftarrow{G(N,\bar{J};x)}~
\text{{\sc and set}}~ N\leftarrow N+1.~~~
\text{{\sc Go to~~\underline{(*)}}}\\
&&\text{{\sc Output} is the minimal number of trucks needed to
guarantee the}\\
&&\qquad\qquad \text{required transport capacity to satisfy demand} ~D =D_2+\dots+D_J.
\end{eqnarray*}
\end{alg}
\begin{beweis}
Following Remark \ref{rem:Interpretation}, ${G(N,\bar{J};x)}$ from Theorem \ref{thm:normingconstant}
can be interpreted as normalization constant
in a Gordon-Newell network with nodes $1,\dots,J$ having visit ratios $\eta_j$ and
service rates $\mu_j(n_j)=\mu_j\cdot \min(n_j,s_j)$ at station $j$, and an additional infinite server node $J+1$ with visiting ratio $1/2$ and exponential-$(\sum_{j=2}^J \rho_jd_j(x))$ service time distribution.
Because the $\mu_j(n_j)$ and $n\cdot (\sum_{j=2}^J \rho_jd_j(x))$ are non-decreasing in $n_j$ and $n$, from van der Wal's theorem \cite{vanderwal:89} it follows that
the throughput of this artificial Gordon-Newell network is non-decreasing in $N$.
As can be seen from the proof in \cite{vanderwal:89} the throughput is strictly increasing in $N$. This guarantees that the algorithm stops after a finite number of iterations,
because we assumed that the capacities are high enough to satisfy all the demands eventually.
\end{beweis}
Recall that from Theorem \ref{thm:loesung} the optimal location for the center is independent of loading and unloading capacities and the number of trucks cycling. So, if for $N$ trucks and optimal location $x$ the demand $D$ exceeds the achievable maximal
throughput $TH_w(N;x)$ for the given $\mu_{j}$, one may increase the loading
and/or unloading capacities (for a proof see Theorem 14.B.13 of \cite{shaked;shanthikumar:94}).
\begin{rem}
The fact that for a given set of parameters
$\mu_j(\cdot), j=1,\dots,J,$ it may be impossible to realize the requested demands is a consequence of the observation that in closed queueing networks with nodes which are, roughly stated, not all of infinite server type, bottlenecks exist.
The throughput of bottlenecks converges under unbounded increasing population size to a finite value. This bounds the network's overall throughput which can be attained by increasing number of customers (trucks).
A short survey with more relevant details of classical bottleneck analysis is \cite{schweitzer;serazzi;broglia:93}.
\end{rem}
Because the maximal throughput at a bottleneck node can be increased by increasing the local service rate,
we have the following simple recipe.
\begin{prop}\label{mu1opt}
Consider the system of Theorem \ref{thm:normingconstant} with all parameters other than $\mu_j, j=1,\dots,J$ being fixed.
Then loading and unloading capacities $\mu_j$ for center and warehouses exist which guarantee that
$TH_w(N;x) \geq D$ holds for sufficiently many trucks available.
\end{prop}
\subsection{Discussion of the modeling assumptions}\label{sect:DiscussionModel}
\underline{Infinite server queue for modeling traffic on a lane} \cite{newell:82}[Chapter 6] is a standard device. For fixed mean travel time $d_j(x) <\infty$ of a vehicle moving with unit speed, we can allow any distribution to incorporate or forbid overtaking.
Extreme travel time distributions are exponential-$d_j(x)^{-1}$ (maximal entropy) and deterministic-$d_j(x)$ (minimal entropy). In any case:
The joint stationary queue length distribution of number of trucks on lanes, normalization constants, and throughputs remain the same.
A realistic model for travel times is obtained using random travel times with mean $d_j(x)$ and small variance which generates moderate overtaking.
We obtain throughput \eqref{othruput1} and \eqref{othruput} in any case.
We discuss consequences of this observation in Section \ref{sect:Sensitivity}\\
\underline{Exponential multi-server stations under FCFS} for loading and unloading the trucks are standard models. Other service disciplines may be more realistic in specific situations.
These often yield the same performance characteristics with respect to the optimization
criterion, see \cite{daduna:01a}[Theorem 9.9].
Moreover, for many service disciplines it is possible that loading and unloading times may have general distributions.
Two typical settings where throughput will be the same for any shape of the service time distributions as long as the mean service time is fixed (robustness of throughput), are (see \cite{daduna:01a}[Theorem 9.7 (3),Theorem 10.2, Remark 9.6]):\\\label{page:ServiceDisz}
(i) If there is ample capacity, i.e. all trucks present are served in parallel (= infinite server), general distributions for loading and unloading times are admitted.\\
(ii) If loading of all trucks present at the central station is is performed concurrently, the adequate model of servicing is Processor Sharing. This means: If there are $n_1$ trucks at the center each of them obtains a fraction of $1/{n_1}$ of the station's total capacity.\\
An interesting observation is that the mentioned robustness property (insensitivity) is not valid in the related
{\em combined location-routing problem} (LRP) where strategic and tactical decisions are intertwined. For more details
see the review paper \cite{nagy;salhi:07}[Section 1.2] and the early survey \cite{laporte:88}.\\
\underline{Location of the center coincides with one of the warehouse locations} is a possible scenario
because we have reduced the queueing-location problem to a pure location problem. If the center's position is $x=a_j$, then the travel times to and from this node are $d_j(x)=0$. Consequently, the service times at the infinite servers $ja$ and $jb$ are zero. This situation is covered by our framework with mean zero travel times from $1$ to $j$ and back. In this case $n_{ja} = n_{jb} =0$.\\
\underline{Random routing when departing from the center}
\label{page:RandRouting} is a modeling assumption which
adjusts the distribution of the available transportation capacity in the long run and in the stationary system according to the demands of the warehouses. As discussed in Example \ref{ex:TimeIncluded}, transitions into realizable schedules can be found easily.
A realistic schedule determines a sequence of visits for each truck to warehouses. These sequences are cyclically iterated.
A necessary condition for optimal scheduling
is that (in a stationary system and in the long run) the schedule \underline{must} reproduce the
frequencies $\rho_j=D_j/D$:\\
Recall Example \ref{ex:TimeIncluded}: Two warehouses with demands $D_2 = 10$ and $D_3 = 20$ truck loads per hour and $N=3$ trucks with equal capacity.
Both reasonable schedules described there can be modeled with deterministic routing schemes described in
\cite{kelly:79}[Section 3.4] to obtain a Markovian network model using a more elaborated state space.
The route of the trucks (in the sense of Kelly) for the second schedule would be:\\
$[1\to 3a\to 3\to 3b\to 1\to 2a\to 2\to 2b\to 1\to 3a\to 3\to 3b] \to [1\to 3a\to3b\to 1\ldots\to 3b]\to[\ldots]\to \dots$,
iterated indefinitely over time.\\
If we evaluate the joint queue length distribution of stations $1, 2, 3,$ in these models we obtain
exactly the stationary distribution of our present model with $\rho_2 = 1/3, \rho_3 = 2/3$.\\
The main conclusion from Theorem \ref{thm:loesung} is:
The optimal location of the center is the same \textbf{for all schedules which generate in the stationary state the values $\rho_j=D_j/D$ by cyclical sequencing}. This is in line with intuition.\\
\underline{Incorporating time into the optimality conditions,}
we follow the arguments of Tapiero \cite{tapiero:71}:
``The time dimension in problems of transportation-location-allocation is particularly important since decisions to construct production facilities are based on long range plans.
Also, although environmental conditions, demand, etc. may change over time, the decision to locate a plant in a particular place is made once and is not subject to frequent change. (page 383)''\\
``Long range plans'' in the sense of Tapiero \cite{tapiero:71} justify to assume that we consider a
\textbf{stationary} system. Note that this does not
mean to consider static systems. Only the random fluctuations of the system are invariant over time.\\
\subsection{Minimizing round-trip times} \label{sect:MiniPassageTimes}
Throughput maximization is probably the most important objective
in the logistic and services network. Another important objective is minimization of expected round-trip times, i.e.
the mean travel time for a truck between two successive departures from the center.
The \it expected passage-time $Z_j(N;x)$ from the center located at $x$ to station $j$ and back \rm is the sum of all expected waiting and service times, which a truck spends
at station $j$ plus traveling to and from $j$
and thereafter at station $1$. If $W_i(N;x)$ is the expected sojourn time at station $i$, then
$Z_j(N;x)=W_{ja}(N;x)+W_{j}(N;x)+W_{jb}(N;x)+W_{1}(N;x),\ \ j=2,\ldots,J.$
The overall \it expected passage-time \rm is
$ Z(N;x):=\sum_{j=2}^J r(1,ja)Z_j(N;x).$\\
With $\eta_j=(1/4)r(1,ja)$, $j=1,\ldots,J$ and
$\eta_1=\sum_{j=2}^J\eta_j$ (see \eqref{trafficsolution}) we get
\begin{equation}\label{ZNX1}
Z(N;x)=\sum_{j=2}^Jr(1,ja)Z_j(N;x)
=4\sum_{j=2}^J\eta_jZ_j(N;x)
=4\sum_{j\in\bar{J}}\eta_jW_j(N;x).
\end{equation}
From Little's Theorem \cite{chen;yao:01}[Formula (2.18)], we obtain
$\sum_{j\in\bar{J}}\eta_jW_j(N;x)=\frac{N}{TH(N;x)}.$
This yields the optimization problem:
\begin{displaymath}
\text{Find}\quad \arg\min_{x\in\mathbb{R}^2}\ \left\{Z(N;x)=4\frac{N}{TH(N;x)}\right\}.
\end{displaymath}
So $Z(N;x)$ attains its minimum when $TH(N;x)$ is
maximal and $Z(N;x)$ attains
its minimum at $x^\ast$ given in Theorem \ref{thm:loesung}. Hence, travel time minimization is reduced to a standard Weber problem.
This result holds for the generalizations from Section \ref{sect:DiscussionModel} as well because we consider mean passage times.
\section{Numerical examples and discussion} \label{sect:NumericManagerSensitive}
Our main result states: For the Weber problem in the logistics and services network under congestion the strategic decision for the center's location and the
tactical/operational decision for the fleet size decouple
as long as the relative demands $\rho_j:=D_j/D$ remain stable.
Nevertheless, it is of value to demonstrate the consequences of this invariance property by examples and to discuss consequences of the result for managerial decision making.
Distance measure is in any case Euclidean distance with specific weights which will vary.
\subsection{Numerical example}\label{sect:Numeric}
The first two examples mimic the location of $12$ midsize up to large towns in Northern Germany in a rectangle of size 400 km $\times$ 260 km (approximately).
We embed this rectangle into the positive lattice
$\mathbb{N}_0\times\mathbb{N}_0$ in the plane, and shift the town in the south-west corner to the point $(10,10)$.
The demands $D^{(\cdot)}_j$ (measured in truck loads per day) of the warehouses $j=2,\dots,J$, are
approximately chosen
(i) proportional to the number of inhabitants of the respective cities, $(D_j^{pro})$,
resulting in total demand of $81$ truck loads/day, and
(ii) according to the logarithm of the number of inhabitants (divided by 1000) of the respective cities, $(D_j^{log})$, resulting in total demand of $66$ truck loads/day.\\
We assume for simplicity of computations that all loading and unloading facilities are single servers.
Unload capacities at all warehouses are
$2$ truck loads/hour ($\mu_j=2$), loading capacity at the center is $4$ truck loads/hour ($\mu_1=4$).
The locations of the warehouses $(a_{j1},a_{j2}),j=2,\dots,J,$ are indicated in the next table. Demands of the respective locations are listed below the locations. (All numbers are rounded to integers).\\
\noindent
\small{
\begin{tabular}{|r|*{12}{r|}} \hline
$j =$& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12& 13\\ \hline
$a_{j1}$ & 10& 100& 170& 290& 410& 220& 260& 180& 320& 160& 40& 80\\ \hline
$a_{j2}$ & 10& 130& 190& 30& 70& 230& 190& 270& 250& 50& 40& 180\\ \hline
$D_j^{pro}$& 3& 6& 19& 2& 36& 2& 1& 2& 2& 5& 2& 1\\ \hline
$D_j^{log}$& 6& 6& 7& 5& 8& 5& 4& 5& 5& 6& 5& 4\\ \hline
\end{tabular}
}
\\
We applied the Weiszfeld algorithm
and obtained
(i) for weights $D_j^{pro}$ the location for the center
at $x=(x_1,x_2) = (288.156, 112.283)$,
and (ii) for weights $D_j^{log}$ the location for the center
at $x=(x_1,x_2) = (179.756, 155.904)$.
Without weights the center's location is $x=(179.210, 162.372)$.
We observed the following results of the optimization procedure.\\
(i) For demands $D_j^{pro}$ the distance between the center with weights and the center without weights is
$119.909$. The necessary number of trucks needed to fulfill the total demand of $81$ truck loads is $28$ for center with weights and $29$ for center without weights.
Nevertheless, the thoughput at the warehouses is greater with weights and 28 trucks than without weights and 29 trucks. The details are summarized in the following table. For completeness we added the probability that the loading server at the center is busy as a measure of congestion.\\
\begin{tabular}{|r|r|r|r|r|r|}\hline
demand $D_j^{pro}$&location,~ $\mu_1=4$&trucks&throughput/day& $P(X_1>0)$ \\ \hline
with weights& 288.156, 112.283 & 28 & 82.261& 0.857\\ \hline
no weights& 179.210, 162.372 & 29&81.342& 0.847\\ \hline
\end{tabular}
\vspace{0.3cm}\\
(ii) For demands $D_j^{log}$ the distance between the center with weights and the center without weigts is
$6.491$ and the necessary number of trucks needed to fulfill the total demand of $66$ truck loads is the same when the center's location is determined with or without weights. Nevertheless, the throughput at the warehouses is slightly greater with weights. Details are given in the next table.\\
\begin{tabular}{|r|r|r|r|r|r|}\hline
demand $D_j^{log}$&location,~$\mu_1=4$&trucks&throughput/day& $P(X_1>0)$ \\ \hline
with weights& 179.756, 155.904 & 19 & 67.871& 0.706990 \\ \hline
no weights& 179.210, 162.372 & 19&67.841& 0.706676 \\ \hline
\end{tabular}
\vspace{0.3cm}
\\
Interpretation:
The demand structure $D_j^{log}$ is rather homogeneous and therefore the difference between the locations of the center is insignificant.
The case of demand structure $D_j^{pro}$ is more interesting because of the great distance between the centers' location
which is a consequence of the more variable demand structure.
In any case the congestion, measured as $P(X_1>0)$, increased with increasing throughput which follows from easy computations. Our experiments show that the increase of congestion seems to be tolerable, especially if we can reduce the number of trucks in parallel.\\
It is easy to see that in both scenarios the loading server at the center is the bottleneck of the network.
We therefore investigated the influence of the loading capacity at the center in the above examples
and reduced the loading capacity from $\mu_1=4$ to $\mu_1=3$ truck loads/hour. The results for demands $D_j^{log}$ in the
next table are not surprising. The reduced server capacity is compensated by more trucks to deliver total demand of 66.\\
\vspace{0.02cm}
\begin{tabular}{|r|r|r|r|r|r|}\hline
demand $D_j^{log}$&location,~$\mu_1=3$&trucks&throughput/day& $P(X_1>0)$ \\ \hline
with weights& 179.756, 155.904 & 22 & 67.054& 0.931308 \\ \hline
no weights& 179.210, 162.372 & 22 &67.040& 0.931110 \\ \hline
\end{tabular}
\vspace{0.3cm}\\
For the more variable demand $D_j^{pro}$ (total 81) the results are given in the next table. Because the center is the bottleneck the results for throughput (should be less than $72$) and non-idling probabilities (should be less than $1$) are due to rounding errors.\\
\vspace{0.02cm}
\begin{tabular}{|r|r|r|r|r|r|}\hline
demand $D_j^{pro}$&location,~$\mu_1=3$&trucks&throughput/day& $P(X_1>0)$ \\ \hline
with weights& 288.156, 112.283 & -- & 72.000& 1.000\\ \hline
no weights& 179.210, 162.372 & --&72.000& 1.000\\ \hline
\end{tabular}
\vspace{0.3cm}\\
The algorithm stopped when it detected that $100$ trucks are not sufficient to satisfy the requested total demand of $81$ truckloads.
This is a consequence of the fact that the departure stream from the bottleneck server approaches asymptotically a Poisson process with intensity $3/hour$ which determines asymptotically (Number of trucks $\to \infty$) the maximal total throughput of the system. The resulting upper bound for the total throughput ($72$ truckloads) does not meet the requested total demand of $81$.
Increasing the service rate at the center to $3.38/hour$ yields
the following results.\\
\begin{tabular}{|r|r|r|r|r|r|}\hline
demand $D_j^{pro}$&location,~$\mu_1=3.38$&trucks&throughput/day& $P(X_1>0)$ \\ \hline
with weights& 288.156, 112.283 & 43 & 81.013& 0.998676\\ \hline
no weights& 179.210, 162.372 & 45& 81.021& 0.998780\\ \hline
\end{tabular}
\vspace{0.3cm}\\
Note, that overshot of throughput is slightly higher without weights and $45$ trucks than in case of weights with $43$ trucks needed to satisfy demand.
The moderate deviation of the number of trucks needed in the setting with and without weights is surprising.
But from observations in a series of experiments we concluded that this is not unusual.
Additionally, we performed experiments with $12$ warehouses located on the lattice
$\{10,\dots,410\}\times\{10, \dots,270\}$ and sampled independently according to
uniform distribution. The respective demands are selected according to uniform distribution $U(A)$ on different sets $A$ of feasible demands. In any case the unloading capacities are $\mu_j=2~\text{truck loads/hour}, j= 2,\dots,13$.
The loading capacities $\mu_1$ varied and are given in
Table \ref{tab:Summary} below. Details are presented in the tables of Section \ref{sect:AddNumeric}.\\
For comparison with the ratio (sample mean/sample variance) of
$D_j^{log}, j=2,\dots,13: (5.5/1.37)$ and $D_j^{pro}, j=2,\dots,13 : (6.75/109.30)$ in the previous experiments
we indicate for the respective demand distribution the mean (Exp) and variance (Var) in the first two rows of Table \ref{tab:Summary}.\\
In any of 4 blocks ((I),\dots,(IV)) we performed 10 experiments.
For extreme total demands it turned out that the demand can not be satisfied with the given capacities due to occurrence of bottlenecks (the number of such samples are indicated as ``Num$\infty$Tru''). We included these cases in the tables in Section \ref{sect:AddNumeric} and performed additional experiments to obtain in any block 10 complete data sets.
Within these we observed throughout that in approximately half of the samples for the center's location with weights less trucks (usually 1 truck less) are needed than without weights (precise numbers indicated as ``Num$<$Tru'').
We observed in both blocks (III) and (IV) a single experiment where 3 trucks less are sufficient (underlined). These high differences coincided with the maximal distance
(MaxDist) between the two centers' location (bold).\\
In all experiments where an equal number of trucks is needed for center with weights and for center without weights, this coincides with a higher throughput at the warehouses for the center's location selected with weights. This seems to be in line with intuition.
On the other side, when less trucks are needed for center with weights, in almost all cases the throughput with more trucks and location selected without weights produces more throughput, i.e. more overshot.\\
This demonstrates that decisions on the basis of integrated models leads to better utilization of the given resources. This is substantiated by the following observations:\\
Exceptions of the moderate decrease of needed resources (trucks) are the extreme cases in (III) and (IV). With 3 trucks less the throughput with weights exceeds the throughput without weights. Moreover, in both blocks
(III) and (IV) the second largest distance (bold) between the centers' location generated with 1 truck less (with weights) a higher throughput than without weights.
In Table \ref{tab:Summary}
we report for any block the minimal and maximal distance between the centers and the minimal and maximal demands.
Detailed results are presented in Section \ref{sect:AddNumeric} of the Appendix.\\
\begin{table}
\begin{tabular}{|r|r|r|r|r|}\hline
&(I)&(II)&(III)&IV\\ \hline
&$U(\{1,..,8\})$
&$U(\{1,..,16\})$ & $U(\{1,..,21\})$ & $U(\{1,11, 21\})$\\
\hline
Exp& 4.75 & 8.5& 11& 11 \\ \hline
Var& 5.25& 21.25& 36.67& 66.67 \\ \hline
$\mu_1$& 4& 5& 7& 7 \\ \hline
MinDist& 7.5896& 16.2628 & 11.0563 & 5.5115
\\ \hline
MaxDist&60.7717 & 88.8410 & 108.5533 & 85.6094
\\ \hline
MinDem& 41 & 93 & 94 & 42
\\ \hline
MaxDem& 63 & 123 & 163& 162
\\ \hline
Num$<$Tru& 4 & 6 & 5 & 6
\\ \hline
Num$\infty$Tru& 0 & 2 & 1 & 2
\\ \hline
\end{tabular}
\caption[Summary]{Summary of the experiments. Details in Section \ref{sect:AddNumeric}}. \label{tab:Summary}
\end{table}
\subsection{Sensitivity analysis and robustness}\label{sect:Sensitivity}
Our main results indicated that some of the
system's parameters are not relevant for the decision problems which are in the focus of our investigations.
\textbf{(a)} Discussing modeling assumptions in Section
\ref{sect:DiscussionModel} we indicated that
for fixed mean travel time $d_j(x) (<\infty)$ of vehicles we can allow any shape of travel time distribution.
The joint stationary queue length distribution, the normalization constants, and the throughputs of the
network are the same.\\
Consequently, our model is robust against changes of these data, e.g. against variability of travel times.
This flexibility is due to so-called {\it Insensitivity theory for queueing networks}.
This theory dates back to insensitivity in
{\it Verallgemeinerte Bedienungsschemata} \cite{koenig;matthes;nawrotzki:74} and in BCMP and Kelly networks \cite{schassberger:78a}, for details
see \cite{daduna:01a}[Section 9 and 10].
The relevant fact from insensitivity theory for our problem is:
{\it At an {\em infinite server} the stationary queue length distribution is {\em invariant} under variation of the
shape of the service time distribution as long \underline{as the mean is fixed}.}
This implies that we can compute stationary queue length distributions on the lanes using exponential service time - the result is the same for any other distribution with the same mean, see \cite{daduna:01a}[Theorem 9.7 (3) and Theorem 10.2]
and the remark thereafter.\\
A similar robustness property is observed when varying service
disciplines (i.e. reorganizing loading/unloading) at the stations to a certain extend (see p. \pageref{page:ServiceDisz}).
\textbf{(b)} Separability of Optimization Problem
\ref{opt:throughputoptimization} implies that decision for the location of the central production facility is robust against variations of all parameters of the integrated production-transportation-inventory system as long as the proportions of the demands $\rho_j=D_j/D, j=2,\dots, J,$ are not changed and the capacities $\mu_j(\cdot)$ are sufficiently high to meat the demand.
\textbf{(c)} On the other side, separability of Optimization Problem \ref{opt:throughputoptimization} implies that, when
the optimal location is fixed,
we can optimize for the number of trucks needed to satisfy demands $D_2, \dots, D_J$ by reallocation of capacities at the unloading service stations or by adding capacity at the loading station at the center.
\subsection{Managerial insights}\label{sect:ManagerialDecision}
Theorems \ref{thm:loesung} and \ref{thm:MiniMax} substantiate conclusions for managerial decision making.
\textbf{(1)} Strategic (location of central facility) and tactical and operational (routing, scheduling) decisions are usually thought to be independent and are consequently
separated. Then decision about the location of a central production facility neglects the actual and future capacities, the resulting congestion, and delays at downstream warehouses. Several authors have shown that for location-routing problems (LRPs) such a structural separation produces sub-optimal solutions for allocation problems, see e.g. \cite{nagy;salhi:07}.
For general supply chain analysis this is discussed in \cite{heckmann;nickel:19}.\\
Contrary to this, our Theorems \ref{thm:loesung} and \ref{thm:MiniMax} justify in a stylized but rather general model the separation of strategic decisions for locations from several tactical and operational decisions for scheduling and placing service capacities. In our setting, in a first step the optimal location can be determined by way of a standard Weber problem where future allocation/routing decisions are incorporated only via gross information (or gross assumptions) about expected demand.
Later on, in a second step the fine-tuning of the radial trips can be carried out according to the service resources at hand without the central location becoming sub-optimal.
In Section \ref{sect:DiscussionModel} it is demonstrated that results obtained in the stylized model are valid in more realistic settings as well.
\textbf{(2)} Once the system is built, the quality of the delivering process measured in the standard metrics throughput or round-trip time can be increased by local enhancement of service without making the central location sub-optimal.
\textbf{(3)} Shifting capacities between the nodes is possible without perturbing optimality of the center's location as long as the fractions of demand at the stations remain the same. More precisely: If we can fine-tune
scheduling at the warehouses by
placing a prescribed number $M>J$ of service facilities at the exterior stations, with at least one facility per station, according to some
further optimization criterion, our theorems state that the location of the central server remains optimal as long as $M$ provides a feasible distribution of capacities for loading and unloading.
This second step of fine-tuning is related to
distribution of servers in the Multiple Server Location problem introduced in \cite{berman;drezner:07} and investigated further
in \cite{aboolian;berman;drezner:08}, \cite{aboolian;berman;drezner:09}.
\textbf{(4)} Concurrent optimization for location of central facility and the number of trucks in the system for a target throughput can be carried out in a step-by-step procedure according to Theorem \ref{thm:MiniMax}:
With the center's location fixed we can algorithmically solve for the needed number of trucks,
see Section \ref{sect:NumberOfTrucks}.\\
\section{Conclusion and directions of further research}\label{sect:Conclusion}
We have developed a methodology for determining jointly
optimal solutions for location-allocation-routing problems which are usually considered to be problems on different levels of decision making: strategic versus tactical/operational level.
We translated the problem into a stochastic network problem and showed that (i) determining the center of a star-like network under constraints on the demands generated by the exterior nodes, and (ii) determining the optimal number of customers in the network, can be separated.\\
Starting from an exponential version of the problem, which allows for simple proofs, we have shown that more realistic models are covered by the result.
Probably, the most important result is that the decision upon the location of the production center can be decoupled from building the exterior stations and their equipment.\\
Many research problems are not yet tackled in the more general area of locating additional nodes in networks of queues. These can be easily identified by investigating the more involved location problems, like p-median or p-center problems, in the continuous as well as in the discrete setting of queueing networks.\\
Parts of our ongoing research related to the present paper are
(i) location theory in the discrete queueing network setting with prescribed network graphs,
and (ii) p-center location problems in the setting of this paper.
\begin{appendix}
\section{Appendix}
\subsection{Prerequisites from queueing
network theory}\label{sect:fundamentals}
\begin{defi}\label{defn:GN}
A \textbf{Gordon-Newell network} consists of stations $\{1,2,\ldots,I\}$. Station j has $s_j\geq 1$ service channels and ample waiting room under FCFS.
$N>0$ indistinguishable customers
cycle according to an irreducible Markov matrix
$R=(r(i,j),i,j=1,\ldots,I)$ in the network and request for service the nodes. The service time at node $j$ is
exponentially distributed with mean $\mu_j^{-1}$.
Whenever $n_j$ customers are present at node $j$ (in service or waiting), service is provided with rate
$ \mu_j(n_j) = \mu_j\cdot \min(n_j,s_j)\,.$\\
Let $X_j(t)$ denote the number of customers at station j at time $t\ge 0$ and
$X(t):=(X_j(t),j=1,\ldots,I)$ the joint queue
length vector at time $t$. $X=(X(t),t\ge 0)$ is the joint queue length process on state space
$S(N,I)=\{(n_1,\ldots,n_I)\in\mathbb{N}^I, n_1+\ldots +n_I=N\}.$
\end{defi}
\begin{theorem}\label{thm:gordonnewell}
\bf (\cite{jackson:63}, \cite{gordon;newell:67}) \it
The joint queue length process $X=(X(t):t\geq 0)$ of the
Gordon-Newell network is an ergodic Markov process.
Denote by
$\eta=(\eta_1,\ldots,\eta_I)$ the unique probability
solution of the traffic equation
\begin{align}\label{trafficeqn}
\eta_j=\sum_{i=1}^{I}\eta_i r(i,j),\ j\in \{1,\dots,I\}.
\end{align}
$\eta_j$ is the customers' visit ratio at node $j$.
With normalization constant $G(N,I)$, the unique stationary and limiting distribution $\pi=\pi(N,I)$
of X on S(N,I) is
\begin{align*}
\pi(n_1,\ldots,n_I)=G^{-1}(N,I)\prod_{j=1}^I\prod_{i=1}^{n_j}\frac{\eta_j}{\mu_j(i)},\quad (n_1,\ldots,n_I)\in S(N,I).
\end{align*}
\end{theorem}
\textbf{Remark.} Any non-zero solution $\eta$ of \eqref{trafficeqn} is admissible to compute $\pi$. Consequently, the next algorithm can be used with any such $\eta$.
\begin{alg} \label{alg:Buzen}\textbf{Buzen's Algorithm.}
\cite{bruell;balbo:80}[Section 2.2.1]
\begin{eqnarray}
&&\text{For}~j=1,\dots,J~\text{set}~ g_j(0) := 1, ~~ g_j(n_j) := \prod_{k=1}^{n_j}\frac{\eta_j}{\mu_j(k)}, n_j\geq 1.\label{eq:Buzen2}\\
&&\text{Set boundary values}~~\nonumber\\
&& G(0,j) := 1, ~~j= 1,\dots,J,\quad G(m,1) := g_1(m),~~ m=1,2,\dots.\nonumber\\
&&\text{Denote for}~~j\geq 1, m\geq 1,\nonumber\\
&& G(m,j) := \sum_{n_1+\dots +n_j=m}\prod_{j=1}^j\left(\prod_{k=1}^{n_j}\frac{\eta_j}{\mu_j(k)}\right)
=\sum_{n_1+\dots +n_j=m}\prod_{j=1}^j g_j(n_j)\,.\nonumber
\end{eqnarray}
{\em Buzen's algorithm to compute norming constants}
for $j\geq 1$ and $m\geq 1$ is
\begin{equation}\label{buzen1}
G(m,j) = \sum_{\ell=0}^{m} G(\ell,j-1) \cdot g_j(m-\ell) \,.
\end{equation}
\end{alg}
\begin{lem}
\bf\cite{chen;yao:01} \it In a Gordon-Newell network with $N\ge 1$ customers the mean number of departures per time unit from node $j$ (node-$j$ throughput) is
\begin{align}\label{localthru}
TH_j(N):=\sum_{(n_1,\ldots,n_I)\in S(N,I)}\pi(n_1,\ldots,n_I)
\mu_j(n_j)=\eta_j&\frac{G(N-1,I)}{G(N,I)}.\\
\label{globalthru}
\text{The (overall) throughput is}~~ TH(N)=\sum_{j=1}^I TH_j(N)=&\frac{G(N-1,I)}{G(N,I)}.
\end{align}
\end{lem}
\subsection{Proofs}\label{sect:proofs}
Recalling node set $\bar{J}=\{1,2a,3a,\ldots,Ja,2,3,\ldots,J,2b,3b,\ldots,Jb\}$,
the probability solution of the traffic equation \eqref{trafficeqn} for the network in Section \ref{sect:reliable stations} is
\begin{align}\label{trafficsolution}
\eta_1= \frac{1}{4}\ \ \mbox{and \ } \eta_j=\eta_{ja}=\eta_{jb}=\frac{1}{4}r(1,ja)
=\frac{1}{4}\rho_j, \quad j=2,\ldots,J.
\end{align}
The stationary distribution of the network is
$ \pi(n_j,j\in\bar{J};x) =$
\begin{align*}
&G^{-1}(N,\bar{J};x)\prod_{j=1}^J
\left(\prod_{k=1}^{n_j}\frac{\eta_j}{\mu_j(k)}\right)
\prod_{i=2}^J\left(\prod_{l=1}^{n_{ia}}\left(\frac{\eta_{ia}}{l\mu_{ia}(x)}\right)
\prod_{m=1}^{n_{ib}}\left(\frac{\eta_{ib}}{m\mu_{ib}(x)}\right)\right)\\
&=G^{-1}(N,\bar{J};x)\prod_{j=1}^J\left(\prod_{k=1}^{n_j}\frac{\eta_j}{\mu_j(k)}\right)
\prod_{i=2}^{J}\frac{1}{n_{ia}!n_{ib}!}(\eta_id_i(x))^{n_{ia}+n_{ib}}.
\end{align*}
where we utilized $\eta_{ja}=\eta_{jb}=\eta_j$ and
$\mu_{ja}(x)=\mu_{jb}(x)=d_j^{-1}(x)$ for $j=2,\ldots,J$.
The following representation of normalization constants will be of value.
\begin{lem}\label{lem:normingconstant}
The normalization constant of the system is
\begin{align}
G(N,\bar{J};x)&=\sum_{n=0}^N\left[\left(\sum_{n_1+\ldots
+n_J=N-n}\prod_{j=1}^J\left(\prod_{k=1}^{n_j}\frac{\eta_j}{\mu_j(k)}\right)\right)
\frac{2^n}{n!}\left(\sum_{j=2}^J
\eta_jd_j(x)\right)^n \right].\nonumber\\
&\text{With}~~~ C_n(N,\bar{J}):=\left(\sum_{n_1+\ldots +n_J=N-n}
\prod_{j=1}^J\left(\prod_{k=1}^{n_j}\frac{\eta_j}{\mu_j(k)}\right)\right)
\frac{2^n}{n!}\nonumber\\
&\text{ and}~~~ h_j(x) :=\eta_jd_j(x),\qquad
h(x) := \sum_{j=2}^J h_j(x)=\sum_{j=2}^J\eta_jd_j(x)\nonumber\\
&\text{we can write}~~~ G(N,\bar{J};x)=\sum_{n=0}^N C_n(N,\bar{J})h(x)^n. \label{ofvalue}
\end{align}
\end{lem}
\begin{beweis}
From the definition, we have $G(N,\bar{J};x)=$
\begin{align*}
&=\sum_{n_1+\ldots +n_{Jb}=N}\prod_{j=1}^J\left(\prod_{k=1}^{n_j}\frac{\eta_j}{\mu_j(k)}\right)\prod_{i=2}^{J}\frac{(\eta_id_i(x))^{n_{ia}+n_{ib}}}{n_{ia}!n_{ib}!}\\
&=\sum_{n=0}^N\left[\left(\sum_{n_1+\ldots+n_J=N-n}\prod_{j=1}^J \left(\prod_{k=1}^{n_j}\frac{\eta_j}{\mu_j(k)}\right)\right)\left(\sum_{n_{2a}+\ldots +n_{Ja}+\atop n_{2b}+\ldots +n_{Jb}=n}\prod_{j=2}^J\frac{h_j(x)^{n_{ja}+n_{jb}}}{n_{ja}!n_{jb}!}\right)\right].
\end{align*}
The statement follows from
\begin{eqnarray*}
&& \sum_{n_{2a}+\ldots +n_{Ja}+\atop n_{2b}+\ldots +n_{Jb}=n}\prod_{j=2}^J
\frac{h_j(x)^{n_{ja}+n_{jb}}}{n_{ja}!n_{jb}!}\\
&=& \sum_{n_{2a}+\ldots +n_{Ja}+\atop n_{2b}+\ldots +n_{Jb}=n}\prod_{j=2}^Jh_j(x)^{n_{ja}}\prod_{i=2}^Jh_i(x)^{n_{ib}} \frac{1}{\prod_{i=2}^Jn_{ia}\prod_{i=2}^Jn_{ib}}\\
&=&\sum_{n_{2a}+\ldots +n_{Ja}+\atop n_{2b}+\ldots +n_{Jb}=n}\underbrace{\prod_{j=2}^J\left(\frac{h_j(x)}{2h(x)}\right)^{n_{ja}}\prod_{i=2}^J\left(\frac{h_i(x)}{2h(x)}\right)^{n_{ib}} \frac{n!}{\prod_{i=2}^Jn_{ia}\prod_{i=2}^Jn_{ib}}}_{\mbox{density function of a multinomial distribution}}\cdot\frac{(2h(x))^n}{n!}\\
&=&\frac{2^n}{n!}h(x)^n.
\end{eqnarray*}
\end{beweis}
\begin{beweis} (\underline{of Theorem \ref{thm:normingconstant}}) The representation of the normalization constant is the first statement of Lemma \ref{lem:normingconstant},
the throughput \eqref{othruput1} is the standard result \eqref{globalthru},
and \eqref{othruput} follows from \eqref{localthru} and \eqref{trafficsolution}.
\end{beweis}
\begin{beweis} (\underline{of Corollary \ref{normingconstant-det}})
The simplest way is to approximate the general travel time distribution by a finite mixture of
Erlangian distributions. These mixtures constitute a class which is dense in the set
of all service times on
$[0,\infty)$, see \cite{daduna:01a}[Definition 9.2] and the references given there.
A Markovian state description of the network is obtained using supplementary variables.
Writing down the steady state throughput, we see that after some computations this
boils down to the same explicit expression as it occurs when writing down the throughput
expression for the companion exponential network.\\
The last step is a continuity argument: When a sequence of finite mixtures of Erlangian service time distributions approaches (in the sense of weak convergence)
the given service time distribution at some node, then that node's queue length distributions converge weakly as well.
This is equivalent to (multi-dimensional) point-wise convergence in a suitable multi-dimensional
real space. Finally, the normalization constants and the throughput continuously depend on the
densities of the joint queue length distributions, as visible from Lemma
\ref{lem:normingconstant}.
\end{beweis}
\begin{lem}\label{lem:ungleichung}
For all $k=0,1,\ldots,N-2$, $n=0,1,\ldots,N-k-2$, it holds that
\begin{displaymath}
C_{n+k+1}(N,\bar{J})C_k(N-1,\bar{J})\ge C_{n+k+1}(N-1,\bar{J})C_k(N,\bar{J}).
\end{displaymath}
\end{lem}
\begin{beweis}
By definition, we have
\begin{displaymath}
C_k(N,\bar{J})=\left(\sum_{n_1+\ldots+n_J=N-k}\prod_{j=1}^J
\left(\prod_{k=1}^{n_j}\frac{\eta_j}{\mu_j(k)}\right)^{n_j}\right)\frac{2^k}{k!},
\end{displaymath}
and hence with Theorem \ref{thm:gordonnewell}, $C_k(N,\bar{J})$ is the normalization constant $G(N-k,J)$ of a standard Gordon-Newell
network with $N-k$ customers, $J$ service stations and not normalized solution $(\eta_1,\ldots,\eta_J)$
of the associated traffic equation, multiplied with $2^k/k!$.
With these expressions and $\sum_{j=1}^J\eta_j=1/2$, it is easily verified that
the throughput of this Gordon-Newell network with $N$
customers is $TH(N)=(1/2)G(N-1,J)/G(N,J)$.
We then obtain
\begin{align*}
&\ C_{n+k+1}(N,\bar{J})C_k(N-1,\bar{J})\ge C_{n+k+1}(N-1,\bar{J})C_k(N,\bar{J})\\
\Leftrightarrow &\ G(N-n-k-1,J)G(N-k-1,J)\ge G(N-n-k-2,J)G(N-k,J)\\
\Leftrightarrow &\ \frac{G(N-k-1,J)}{G(N-k,J)}\ge\frac{G(N-n-k-2,J)}{G(N-n-k-1,J)}\\
\Leftrightarrow &\ TH(N-k)\ge TH(N-n-k-1).
\end{align*}
From \cite{vanderwal:89}, the throughput of a Gordon-Newell-network with service rates non-decreasing in the number of customers is a non-decreasing function in the network's population size. Thus, the lemma is proved.
\end{beweis}
\begin{beweis} \underline{(of Theorem \ref{thm:loesung})}
For all $x\in\mathbb{R}^2$ with $h(x)>h(x^\ast)$ and $N\in\mathbb{N}$ we will show
\begin{displaymath}
TH(N;x^\ast)=\frac{G(N-1,\bar{J};x^\ast)}{G(N,\bar{J};x^\ast)}
>\frac{G(N-1,\bar{J};x)}{G(N,\bar{J};x)}=TH(N;x)
\end{displaymath}
By Lemma \ref{lem:normingconstant} ,this is equivalent to
\begin{eqnarray*}
& & G(N-1,\bar{J};x^\ast)G(N,\bar{J};x)-G(N-1,\bar{J};x) G(N,\bar{J};x^\ast)>0\\
\Leftrightarrow & & \left(\sum_{n=0}^{N-1} C_n(\mathbb{N}-1,\bar{J})h(x^\ast)^n)\right)
\left(\sum_{k=0}^N C_k(N,\bar{J})h(x)^k\right)\\
& - & \left(\sum_{n=0}^{N-1} C_{n}(N-1,\bar{J})h(x)^n\right)
\left(\sum_{k=0}^N C_k(N,\bar{J})h(x^\ast)^k)\right)>0\\
\Leftrightarrow & &
\sum_{k=0}^N\sum_{n=0}^{N-1} C_k(N,\bar{J})
C_n(N-1,\bar{J})\left(h(x)^kh(x^\ast)^n-h(x)^nh(x^\ast)^k\right)>0.
\end{eqnarray*}
We consider the summand for $k=N$
\begin{align*}
& \sum_{n=0}^{N-1}C_N(N,\bar{J})C_n(N-1,\bar{J})
\left(h(x)^Nh(x^\ast)^n-h(x)^nh(x^\ast)^N\right)\\
& = \sum_{n=0}^{N-1}C_N(N,\bar{J})C_n(N-1,\bar{J})
h(x)^nh(x^\ast)^n\left(h(x)^{N-n}-h(x^\ast)^{N-n}\right).
\end{align*}
Because of $h(x)>h(x^\ast)$ we have $h(x)^{N-n}-h(x^\ast)^{N-n}>0$. So the whole summand is strictly positive
and the problem is reduced to prove
\begin{displaymath}
\sum_{k=0}^{N-1}\sum_{n=0}^{N-1}C_{k}(N,\bar{J})
C_{n}(N-1,\bar{J})(h(x)^kh(x^\ast)^n-h(x)^nh(x^\ast)^k)>0.
\end{displaymath}
For $k=n$ we get $h(x)^kh(x^\ast)^n-h(x)^nh(x^\ast)^k=0$. So the problem is reduced to
\begin{align*}
& \sum_{k=0}^{N-1}\sum_{n=0 \atop n\ne k}^{N-1}
C_k(N,\bar{J})C_n(N-1,\bar{J})(h(x)^k
h(x^\ast)^n-h(x)^nh(x^\ast)^k)>0\\
\Leftrightarrow &
\sum_{k=0}^{N-1}\sum_{n=k+1}^{N-1}
C_k(N,\bar{J})C_n(N-1,\bar{J})(h(x)^k
h(x^\ast)^n-h(x)^nh(x^\ast)^k)+\\
& \sum_{k=0}^{N-1}\sum_{n=0}^{k-1}
C_k(N,\bar{J})C_n(N-1,\bar{J})(h(x)^k
h(x^\ast)^n-h(x)^nh(x^\ast)^k)>0.
\end{align*}
Now consider the case $k=N-1$ in the first summand. The second sum is empty ($=0$). The same holds in the second
summand for $k=0$. So we have reduced the problem to
\begin{eqnarray*}
& & \sum_{k=0}^{N-2}\sum_{n=k+1}^{N-1}
C_k(N,\bar{J})C_n(N-1,\bar{J})(h(x)^k
h(x^\ast)^n-h(x)^nh(x^\ast)^k)\\
& + & \sum_{k=1}^{N-1}\sum_{n=0}^{k-1}
C_k(N,\bar{J})C_n(N-1,\bar{J})(h(x)^k
h(x^\ast)^n-h(x)^nh(x^\ast)^k)>0.
\end{eqnarray*}
by index-shift in both summands we get
\begin{eqnarray*}
& & \sum_{k=0}^{N-2}\sum_{n=0}^{N-k-2}
C_k(N,\bar{J})C_{n+k+1}(N-1,\bar{J})(h(x)^k
h(x^\ast)^{n+k+1}-h(x)^{n+k+1}h(x^\ast)^k)\\
& + & \sum_{k=0}^{N-2}\sum_{n=0}^{k}
C_{k+1}(N,\bar{J})C_n(N-1,\bar{J})(h(x)^{k+1}
h(x^\ast)^n-h(x)^nh(x^\ast)^{k+1})>0.
\end{eqnarray*}
In the second summand we apply the following summation formula twice:
\begin{displaymath}
\text{for}~~~a_k,b_k\in\mathbb{R},~~k=1,\ldots,N,~~~~
\sum_{k=0}^N\sum_{n=0}^k a_{k+1}b_n
=\sum_{k=0}^N\sum_{n=0}^{N-k} a_{n+k+1}b_k,
\end{displaymath}
and obtain
\begin{eqnarray*}
& & \sum_{k=0}^{N-2}\sum_{n=0}^{N-k-2}
C_k(N,\bar{J})C_{n+k+1}(N-1,\bar{J})(h(x)^k
h(x^\ast)^{n+k+1}-h(x)^{n+k+1}h(x^\ast)^k)\\
& + & \sum_{k=0}^{N-2}\sum_{n=0}^{N-k-2}
C_{n+k+1}(N,\bar{J})C_k(N-1,\bar{J})(h(x)^{n+k+1}
h(x^\ast)^k-h(x)^kh(x^\ast)^{n+k+1})>0.
\end{eqnarray*}
Because of $h(x)>h(x^\ast)$, we have $h(x)^{n+k+1}h(x^\ast)^k-h(x)^kh(x^\ast)^{n+k+1}>0$
and from Lemma \ref{lem:ungleichung} we have $C_{n+k+1}(N,\bar{J})C_k
(N-1,\bar{J})\ge C_{n+k+1}(N-1,\bar{J})C_k(N,\bar{J})$. So
\begin{eqnarray*}
& & \sum_{k=0}^{N-2}\sum_{n=0}^{N-k-2}
C_k(N,\bar{J})C_{n+k+1}(N-1,\bar{J})(h(x)^k
h(x^\ast)^{n+k+1}-h(x)^{n+k+1}h(x^\ast)^k)\\
& + & \sum_{k=0}^{N-2}\sum_{n=0}^{N-k-2}
C_{n+k+1}(N,\bar{J})C_k(N-1,\bar{J})(h(x)^{n+k+1}
h(x^\ast)^k-h(x)^kh(x^\ast)^{n+k+1})\\
&\ge &\sum_{k=0}^{N-2}\sum_{n=0}^{N-k-2}
C_k(N,\bar{J})C_{n+k+1}(N-1,\bar{J})(h(x)^k
h(x^\ast)^{n+k+1}-h(x)^{n+k+1}h(x^\ast)^k)\\
& + & \sum_{k=0}^{N-2}\sum_{n=0}^{N-k-2}
C_k(N,\bar{J})C_{n+k+1}(N-1,\bar{J})(h(x)^{n+k+1}
h(x^\ast)^k-h(x)^k h(x^\ast)^{n+k+1})=0,
\end{eqnarray*}
and the theorem is proved.
\end{beweis}
\subsection{Additional numerical experiments}\label{sect:AddNumeric}
We report in this section details of the experiments which have been summarized in Table \ref{tab:Summary} in Section \ref{sect:Numeric}. We performed four blocks
of experiments, distinguished by demand distributions which
are uniform $U(A)$ on finite demand sets $A$.\\
In any case $12$ locations are sampled uniformly from
$\{10,\dots,410\}\times \{10,\dots,270\}$.
Service is provided by single servers with intensities
$\mu_j=2, j=2,\dots,13$, for unloading servers at warehouses.
Service intensity $\mu_1$ at the center varies with the blocks.\\
\underline{Abbreviations:}\\
DistLoc $\equiv$ distance between center with weights and center without weights\\
Demand: to/$\wedge$ /$\vee$ $\equiv$ total demand/minimal demand/maximal demand \\
+/- $\equiv$ quantities for: center with weights(+)/ center without weights (-)
The extreme cases of large demand are highlighted by boldface numbers, the extreme differences for needed trucks are colored red.
The cases where the requested demand could not be delivered due to bounds determined by bottlenecks are indicated under ``Trucks''
as ``--/--''. Bottleneck was in any case the loading server at the center.
{\small
\begin{table}
\begin{tabular}{|r|r|r|r|r|}\hline
DistLoc&Demand&Trucks&Throughput/day& $P(X_1>0)$ \\ \hline
& to/$\wedge$/$\vee$ & +/-~~ & +/-~~~~~~~~ & +/- ~~~~ \\ \hline
19.0874& 52/1/8& 12/12& 53.4447/53.1933& .5567/.5541 \\ \hline
11.2037& 54/2/8& 13/13& 56.5579/56.2689& .5891/.5861
\\ \hline
14.4548& 50/1/8& 11/11& 52.1731/51.9762& .5435/.5414
\\ \hline
\textbf{60.7717}& 41/1/8& 9/10& 42.5902/43.5175& .4436/.4533
\\ \hline
7.5896& 44/1/7& 10/10& 46.1790/46.1360& .4810/.4806
\\ \hline
36.9812& 59/1/8& 16/17& 59.5479/61.9236& .6203/.6450
\\ \hline
52.2486& 49/1/8& 11/12& 50.6667/52.9925& .5278/.5520
\\ \hline
23.0776& 59/2/8& 14/14& 61.5728/61.0184& .6414/.6356
\\ \hline
53.3149& 45/1/7& 13/14& 45.8186/47.9310& .4773/.4993
\\ \hline
\textbf{57.3881}& 63/1/8& 19/19& 64.6827/64.0564& .6738/.6673
\\ \hline
\end{tabular}
\caption{\textbf{(I)}: Demand distribution $U(\{1,\dots,8\})$, loading rate $\mu_1=4$.}
\end{table}
\begin{table}
\begin{tabular}{|r|r|r|r|r|}\hline
DistLoc&Demand&Trucks&Throughput/day& $P(X_1>0)$ \\ \hline
& to/$\wedge$/$\vee$ & +/-~~ & +/- ~~~~~~~~~ & +/-~~~ ~~ \\ \hline
\textbf{88.8410}& 107/2/16& 29/31& 107.1006/108.5612& .8925/.9047
\\ \hline
29.8252& 111/2/16& 32/32& 112.6342/111.6704& .9386/.9306
\\ \hline
42.3096& 112/2/16& 34/34& 113.1859/112.4563& .9432/.9371
\\ \hline
54.8994& 98/2/15& 25/26& 98.0313/98.4849& .8169/.8207
\\ \hline
16.5874& 121/3/16& --/--& 120.0000/120.0000& 1.0000/1.0000
\\ \hline
24.7907& 93/2/15& 24/24& 95.3214/94.5225& .7943/.7877
\\ \hline
\textbf{76.2196} & 96/2/16& 27/28& 96.6996/96.7557& .8058/.8063
\\ \hline
4.9971& 123/2/16& --/--& 120.0000/119.9999& 1.0000/.9999
\\ \hline
41.8285& 97/3/16& 27/28& 97.8266/98.2422& .8152/.8187
\\ \hline
16.2628& 105/3/16& 29/29& 106.3469/105.8383& .8862/.8820
\\ \hline
51.2422& 99/1/16& 28/29& 99.2529/100.0747& .8271/.8340
\\ \hline
30.2271& 107/1/16& 31/32& 107.1240/107.9508& .8927/.8996
\\ \hline
\end{tabular}
\caption{\textbf{(II)}: Demand distribution $U(\{1,\dots,16\})$, loading rate $\mu_1=5$.}
\end{table}
\begin{table}
\begin{tabular}{|r|r|r|r|r|}\hline
DistLoc&Dem&Tru&Throughput/day& $P(X_1>0)$ \\ \hline
& to/$\wedge$/$\vee$ & +/-~~ & +/-~~~~~~~~~~ & +/-~~~~~ \\ \hline
16.8426& 163/3/20& 43/44& 163.2859/163.9906& .9719/.9761
\\ \hline
28.9316& 138/2/19& 37/38& 138.9829/140.1825& .8273/.8344
\\ \hline
\textbf{108.5533}& 101/2/21&
\underline{27/30}&
102.7856/101.7569& .6118/.6057
\\ \hline
19.0861& 113/1/18& 28/28& 113.9980/113.5362& .6786/.6758
\\ \hline
6.6153& 188/6/21& --/--& 167.9999/168.0000& .9999/1.0000
\\ \hline
11.0563& 127/4/18& 29/29& 129.1835/128.7957& .7689/.7666
\\ \hline
24.5071& 102/1/21& 24/24& 105.0053/103.6437& .6250/.6169
\\ \hline
26.8209& 94/1/20& 27/27& 96.2469/94.8691& .5729/.5647
\\ \hline
18.4939& 156/1/21&45/46& 156.3308/157.6311& .9305/.9383
\\ \hline
16.6062& 156/3/21& 32/32& 157.5397/156.5776& .9377/.9320
\\ \hline
\textbf{39.9891}& 119/2/21& 31/32& 120.2597/120.0498& .7158/.7146\\ \hline
\end{tabular}
\caption{\textbf{(III)}: Demand distribution $U(\{1,\dots,21\})$, loading rate $\mu_1=7$.}
\end{table}
\begin{table}
\begin{tabular}{|r|r|r|r|r|}\hline
DistLoc&Dem&Tru&Throughput/day& $P(X_1>0)$ \\ \hline
& to/$\wedge$/$\vee$ & +/-~~ & +/-~~~~~~~~~~ & +/-~~~~~ \\ \hline
35.0573& 142/1/21& 37/37& 144.2880/142.0597& .8589/.8456
\\ \hline
\textbf{85.6094}& 122/1/21&
\underline{33/36}&
124.7843/123.4753& .7428/.7350
\\ \hline
29.3856& 132/1/21& 32/33& 132.0441/134.0093& .7860/.7977
\\ \hline
.1719& 202/11/21& --/--& 167.9999/168.0000& .9999/1.0000
\\ \hline
21.9623& 162/11/21& 54/55& 162.0168/162.8296& .9644/.9692
\\ \hline
5.5901& 142/1/21& 31/31& 144.8758/144.8089& .8624/.8620
\\ \hline
37.6852& 42/1/11& 6/7& 42.6926/44.3910& .2541/.2642
\\ \hline
18.3488& 72/1/21& 17/17& 75.1295/74.8134& .4472/.4453
\\ \hline
5.5115& 152/1/21& 38/38& 152.1765/152.1388& .9058/.9056
\\ \hline
6.0370& 192/11/21&--/--& 168.0000/ 167.9999& 1.0000/.9999
\\ \hline
\textbf{48.1097}& 132/1/21& 33/34& 134.0373/133.2210& .7978/.7930
\\ \hline
21.2497& 122/1/21& 30/31& 122.7658/124.9653& .7307/.7438
\\ \hline
\end{tabular}
\caption{\textbf{(IV)}: Demand distribution $U(\{1,11,21\})$, loading rate $\mu_1=7$.}
\end{table}
}
\end{appendix}
\textbf{Acknowledgment:} We thank Peter Sieb for helpful discussions on the subject of this paper.
|
1,116,691,500,766 | arxiv | \section{Introduction}
The divisibility properties of the class numbers of number fields are very important
for understanding the structure of the ideal class groups of number fields.
For a given integer $n>1$, the Cohen-Lenstra heuristic \cite{CL84} predicts that a
positive proportion of imaginary quadratic number fields have class number divisible by $n$.
Proving this heuristic seems out of reach with the current state of knowledge.
On the other hand, many families of (infinitely many) imaginary quadratic fields
with class number divisible by $n$ are known.
Most of such families are of the type $\mathbb{Q}(\sqrt{x^2-t^n})$ or of the type
$\mathbb{Q}(\sqrt{x^2-4t^n})$, where $x$ and $t$ are positive integers with some
restrictions (for the former see
\cite{AC55, IT11p, IT11, KI09, RM97, RM99, NA22, NA55, SO00, MI12},
and for the later see \cite{Cohn, GR01, IS11, IT15, LO09, YA70}).
Our focus in this article will be on the family $K_{t,x}=\mathbb{Q}(\sqrt{x^2-t^n})$.
In 1922, T. Nagell~\cite{NA22} proved that for an odd integer $n$,
the class number of imaginary quadratic field $K_{t,x}$
is divisible by $n$ if $t$ is odd, $(t,x)=1$,
and $q\mid x$, $q^2\nmid x$ for all prime divisors $q$ of $n$.
Let $b$ denote the square factor of $x^2-t^n$, that is,
$x^2-t^n=b^2d$, where $d<0$ is the square-free part of $x^2-t^n$.
Under the condition $b=1$, N. C. Ankeny and S. Chowla \cite{AC55}
(resp.\ M. R. Murty \cite[Theorem~1]{RM97})
considered the family $K_{3,x}$ (resp.\ $K_{t,1}$).
M. R. Murty also treated the family $K_{t,1}$ with $b<t^{n/4}/2^{3/2}$ (\cite[Theorem~2]{RM97}).
Moreover, K. Soundararajan \cite{SO00} (resp.\ A. Ito \cite{IT11p})
treated the family $K_{t,x}$ under the condition that
$b<\sqrt{(t^n-x^2)/(t^{n/2}-1)}$ holds (resp.\ all of divisors of $b$ divide $d$).
On the other hand, T. Nagell~\cite{NA55} (resp.\ Y. Kishi~\cite{KI09}, A. Ito~\cite{IT11}
and M. Zhu and T. Wang~\cite{AC55}) studied the family $K_{t,1}$
(resp.\ $K_{3,2^k}$, $K_{p,2^k}$ and $K_{t,2^k}$)
unconditionally for $b$, where $p$ is an odd prime.
In the present paper, we consider the case when both
$t$ and $x$ are odd primes and $b$ is unconditional and prove the following:
\begin{thm}\label{T1}
Let $n\geq 3$ be an odd integer and $p,q$ be distinct odd primes with $q^2<p^n$.
Let $d$ be the square-free part of $q^2-p^n$. Assume that $q \not \equiv \pm 1 \pmod {|d|}$. Moreover, we assume
$p^{n/3}\not= (2q+1)/3, (q^2+2)/3$ whenever both $d \equiv 1 \pmod 4$ and $3\mid n$.
Then the class number of $K_{p,q}=\mathbb{Q}(\sqrt{d})$ is divisible by $n$.
\end{thm}
In Table 1 (respectively Table 2), we list $K_{p,q}$ for small values of $p,q$ for $n=3$ (respectively for $n=5$).
It is readily seen from these tables that the assumptions in Theorem \ref{T1} hold very often. We can easily prove, by reading modulo $4$, that the condition \enquote{$p^{n/3}\not= (2q+1)/3, (q^2+2)/3$} in Theorem \ref{T1} holds whenever $p \equiv 3 \pmod 4$.
Further, if we fix an odd prime $q$, then the condition \enquote{$q \not\equiv \pm 1 \pmod{|d|}$} in Theorem \ref{T1} holds almost always, and,
this can be proved using the celebrated Siegel's theorem on integral points on affine curves. More precisely, we prove the following theorem in this direction.
\begin{thm}\label{T2}
Let $n\geq 3$ be an odd integer not divisible by $3$. For each odd prime $q$ the class number of $K_{p,q}$ is divisible by $n$ for all but finitely many $p$'s. Furthermore, for each $q$ there are infinitely many fields $K_{p,q}$.
\end{thm}
\section{Preliminaries}
In this section we mention some results which are needed for the proof of the Theorem \ref{T1}. First we state a basic result from algebraic number theory.
\begin{prop}\label{P1}
Let $d \equiv 5 \pmod 8$ be an integer and $\ell$ be a prime. For odd integers $a,b$ we have
$$\left(\frac{a+b\sqrt{d}}{2}\right)^{\ell} \in \mathbb{Z}[\sqrt{d}] \mbox{ if and only if } \ell=3.$$
\end{prop}
\begin{proof}
This can be easily proved by taking modulo some power of two.
\end{proof}
We now recall a result of Y. Bugeaud and T. N. Shorey \cite{BS01} on Diophantine equations which is one of the main ingredient in the proof of Theorem \ref{T1}.
Before stating the result of Y. Bugeaud and T. N. Shorey, we need to introduce some definitions and notations.
Let $F_k$ denote the $k$th term in the Fibonacci sequence defined by $F_0=0, \ F_1= 1$
and $F_{k+2}=F_k+F_{k+1}$ for $k\geq 0$. Similarly $L_k$ denotes the $k$th term in the Lucas
sequence defined by $L_0=2, \ L_1=1$ and $L_{k+2}=L_k+L_{k+1}$ for $k\geq 0$.
For $\lambda\in \{1, \sqrt{2}, 2\}$, we define the subsets $\mathcal{F}, \ \mathcal{G_\lambda},
\ \mathcal{H_\lambda}\subset \mathbb{N}\times\mathbb{N}\times\mathbb{N}$ by
\begin{align*}
\mathcal{F}&:=\{(F_{k-2\varepsilon},L_{k+\varepsilon},F_k)\,|\,
k\geq 2,\varepsilon\in\{\pm 1\}\},\\
\mathcal{G_\lambda}&:=\{(1,4p^r-1,p)\,|\,\text{$p$ is an odd prime},r\geq 1\},\\
\mathcal{H_\lambda}&:=\left\{(D_1,D_2,p)\,\left|\,
\begin{aligned}
&\text{$D_1$, $D_2$ and $p$ are mutually coprime positive integers with $p$}\\
&\text{an odd prime and there exist positive integers $r$, $s$ such that}\\
&\text{$D_1s^2+D_2=\lambda^2p^r$ and $3D_1s^2-D_2=\pm\lambda^2$}
\end{aligned}\right.\right\},
\end{align*}
except when $\lambda =2$, in which case the condition ``odd''
on the prime $p$ should be removed in the definitions of $\mathcal{G_\lambda}$
and $\mathcal{H_\lambda}$.
\begin{thma}\label{A1}
Given $\lambda\in \{1, \sqrt{2}, 2\}$, a prime $p$ and positive co-prime integers $D_1$ and $D_2$, the number of positive integer solutions $(x, y)$ of the Diophantine equation
\begin{equation}\label{E1}
D_1x^2+D_2=\lambda^2p^y
\end{equation}
is at most one except for $$
(\lambda,D_1,D_2,p)\in\mathcal{E}:=\left\{\begin{aligned}
&(2,13,3,2),(\sqrt 2,7,11,3),(1,2,1,3),(2,7,1,2),\\
&(\sqrt 2,1,1,5),(\sqrt 2,1,1,13),(2,1,3,7)
\end{aligned}\right\}
$$
and $(D_1, D_2, p)\in
\mathcal{F}\cup \mathcal{G_\lambda}\cup \mathcal{H_\lambda}$.
\end{thma}
We recall the following result of J. H. E. Cohn \cite{Cohn1} about appearance of squares in the Lucas sequence.
\begin{thma}\label{A2}
The only perfect squares appearing in the Lucas sequence are $L_1=1$ and $L_3=4$.
\end{thma}
\section{Proofs}
We begin with the following crucial proposition.
\begin{prop}\label{P2}
Let $n,q,p,d$ be as in Theorem \ref{T1} and let $m$ be the positive integer with
$q^2-p^n=m^2d$. Then the element $\alpha =q+m\sqrt{d}$ is not an $\ell^{th}$ power of an element in the ring of integers of $K_{p,q}$ for any prime divisor $\ell $ of $n$.
\end{prop}
\begin{proof}
Let $\ell$ be a prime divisor of $n$. Since $n$ is odd, so is $\ell$.
We first consider the case when $d \equiv 2 \mbox{ or }3 \pmod 4$.
If $\alpha$ is an $\ell^{th}$ power, then there are integers $a,b$ such that
$$q+m \sqrt{d}=\alpha =(a+b\sqrt{d})^{\ell}.$$
Comparing the real parts, we have
$$q=a^{\ell}+\sum_{i=0}^{(\ell-1)/2} \binom{\ell}{2i} a^{\ell-2i}b^{2i}d^i.$$
This gives $a\mid q$ and hence $a=\pm q$ or $a=\pm 1$.\\
Case (1A): $a= \pm q$.\\
We have $q+m\sqrt{d}=(\pm q+b\sqrt{d})^{\ell}$. Taking norm on both sides we obtain
$$p^n=(q^2-b^2d)^{\ell}.$$
Writing $D_1=-d>0$, we obtain
\begin{equation*
D_1b^2+q^2=p^{n/ \ell}.
\end{equation*}
Also, we have
\begin{equation*
D_1m^2+q^2=p^n.
\end{equation*}
As $\ell$ is a prime divisor of $n$ so $(x,y)=(|b|,n/ \ell)$ and $(x,y)=(m,n)$ are
distinct solutions of (\ref{E1}) in positive integers for $D_1=-d>0,D_2=q^2, \lambda=1$.
Now we verify that $(1, D_1,D_2,p) \not \in \mathcal{E}$ and $(D_1,D_2,p) \not \in \mathcal{F}\cup \mathcal{G_\lambda}\cup \mathcal{H_\lambda}$. This will give a contradiction.
Clearly $(1,D_1,D_2,p) \not \in \mathcal{E}$. Further, as $D_1>3$, we see that $(D_1,D_2,p) \not \in \mathcal{G}_1$. From Theorem \ref{A2}, we see that $(D_1,D_2,p) \not \in \mathcal{F}$. Finally, if $(D_1,D_2,p) \in \mathcal{H}_1$ then there are positive integers $r,s$ such that
\begin{equation}\label{eq:1}
3D_1s^2-q^2=\pm 1
\end{equation}
and
\begin{equation}\label{eq:2}
D_1s^2+q^2=p^r.
\end{equation}
By (\ref{eq:1}), we have $q\not= 3$, and hence we have $3D_1s^2-q^2=- 1$.
From this together with (\ref{eq:2}), we obtain
$$4q^2=3p^r+1,$$
that is $$(2q-1)(2q+1)=3p^r.$$
This leads to $2q-1=1 \mbox{ or } 2q-1=3$, but this is not possible as $q$ is an odd prime. Thus $(D_1,D_2,p) \not \in \mathcal{H}_1$.\\
Case (1B): $a=\pm 1$.\\
In this case we have $q+m\sqrt{d}=(\pm 1+b \sqrt{d})^{\ell}$.
Comparing the real parts on both sides, we get $q \equiv \pm 1 \pmod{|d|}$ which contradicts to the assumption \enquote{$q \not\equiv \pm 1 \pmod{|d|}$}.
Next we consider the case when $d \equiv 1 \pmod 4$.
If $\alpha$ is an $\ell^{th}$ power of some integer in $K_{p,q}$,
then there are rational integers $a,b$ such that
$$q+m\sqrt{d}=\left( \frac{a+b\sqrt{d}}{2} \right)^{\ell},\ \ a\equiv b\pmod 2.$$
In case both $a \mbox{ and }b$ are even, then we can proceed as in the case
$d \equiv 2 \mbox{ or }3 \pmod 4$ and obtain a contradiction under the assumption
$q \not\equiv \pm 1 \pmod{|d|}$. Thus we can assume that both $a$ and $b$ are odd.
Again, taking norm on both sides we obtain
\begin{equation}\label{E4}
4p^{n/\ell}=a^2-b^2d.
\end{equation}
Since $a,b$ are odd and $p \neq 2$, reading modulo 8 in (\ref{E4}) we get
$d \equiv 5 \pmod 8$. As $\left( \frac{a+b\sqrt{d}}{2} \right)^{\ell}=q+m\sqrt{d} \in \mathbb{Z}[\sqrt{d}]$, by Proposition \ref{P1} we obtain $\ell=3$.
Thus we have
$$q+m\sqrt{d}=\left( \frac{a+b\sqrt{d}}{2} \right)^3.$$
Comparing the real parts, we have
\begin{align}\label{E44}
8q&=a(a^2+3b^2d).
\end{align}
Since $a$ is odd, therefore, we have $a\in\{\pm 1,\pm q\}$.\\
Case (2A): $a=q$.\\
By (\ref{E44}), we have $8=q^2+3b^2d$, and hence, $2\equiv q^2\pmod{3}$.
This is not possible.\\
Case (2B): $a=-q$.\\
By (\ref{E4}) and (\ref{E44}), we have
$$4p^{n/3}=q^2-b^2d\ \text{and}\ 8=-(q^2+3b^2d).$$
From these, we have $3p^{n/3}=q^2+2$, which violates our assumption.\\
Case (2C): $a=1$.\\
By (\ref{E44}) and $d<0$, we have $8q=1+3b^2d<0$.
This is not possible.\\
Case (2D): $a=-1$.\\
By (\ref{E4}) and (\ref{E44}), we have
$$4p^{n/3}=1-b^2d\ \text{and}\ 8q=-(1+3b^2d).$$
From these, we have $3p^{n/3}=2q+1$, which violates our assumption.
This completes the proof.
\end{proof}
We are now in a position to prove Theorem \ref{T1}.
\begin{proof}[\bf Proof of Theorem~$\ref{T1}$]
Let $m$ be the positive integer with $q^2-p^n=m^2d$ and put $\alpha =q+m\sqrt{d}$.
We note that $\alpha$ and $\bar{\alpha}$ are co-prime and $N(\alpha)=\alpha \bar{\alpha}=p^n$.
Thus we get $(\alpha)= \mathfrak{a}^n$ for some integral ideal $\mathfrak{a}$ of $K_{p,q}$.
We claim that the order of $[\mathfrak{a}]$ in the ideal class group of $K_{p, q} $ is $n$.
If this is not the case, then we obtain an odd prime divisor $\ell$ of $n$ and an integer
$\beta $ in $K_{p,q}$ such that $(\alpha)=(\beta)^{\ell}$.
As $q$ and $p$ are distinct odd primes, the condition \enquote{$q \not \equiv \pm 1 \pmod{|d|}$}
ensures that $d<-3$. Also $d$ is square-free, hence the only units in the ring of integers of
$K_{p,q}=\mathbb{Q}(\sqrt{d})$ are $\pm1$. Thus we have $\alpha =\pm \beta^{\ell}$.
Since $\ell$ is odd, therefore, we obtain $\alpha= \gamma^{\ell}$ for some integer
$\gamma$ in $K_{p,q}$ which contradicts to Proposition \ref{P2}.
\end{proof}
We now give a proof of Theorem \ref{T2}. This is obtained as a consequence of a well known theorem of Siegel (see \cite{ES, LS}).
\begin{proof}[\bf Proof of Theorem $\ref{T2}$]
Let $n>1$ be as in Theorem \ref{T2} and $q$ be an arbitrary odd prime. For each odd prime $p \neq q$, from Theorem \ref{T1}, the class number of $K_{p,q}$ is divisible by $n$ unless $q\equiv \pm 1 \pmod {|d|}$.
If $q\equiv \pm 1 \pmod {|d|}$, then $|d|\leq q+1$.
For any positive integer $D$, the curve
\begin{equation}\label{E5}
DX^2+q^2=Y^n
\end{equation}
is an irreducible algebraic curve (see \cite{WS}) of genus bigger than $0$. From Siegel's theorem (see \cite{LS}) it follows that there are only finitely many integral points $(X,Y)$ on the curve (\ref{E5}). Thus, for each $d<0$ there are at most finitely many primes $p$ such that
$$q^2-p^n=m^2d.$$
Since $K_{p,q}=\mathbb{Q}(\sqrt{d})$, it follows that there are infinitely many fields $K_{p,q}$ for each odd prime $q$. Further if $p$ is large enough, then for $q^2-p^n=m^2d$, we have $|d|>q+1$. Hence, by Theorem \ref{T1}, the class number of $K_{p,q}$ is divsible by $n$ for $p$ sufficiently large.
\end{proof}
\section{Concluding remarks}
We remark that the strategy of the proof of Theorem \ref{T1} can be adopted, together with the following result of W. Ljunggren \cite{LJ43}, to prove Theorem \ref{T4}.
\begin{thma}\label{TE}
For an odd integer $n$, the only solutions to the Diophantine equation
\begin{equation}
\frac{x^n-1}{x-1}=y^2
\end{equation}
in positive integers $x,y, n $ with $x>1$ is $n=5, x=3, y=11$.
\end{thma}
\begin{thm}\label{T4}
For any positive odd integer $n$ and any odd prime $p$, the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{1-p^n})$ is divisible by $n$ except for the case $(p, n)=(3, 5)$.
\end{thm}
Theorem \ref{T4} alternatively follows from the work of T. Nagell (Theorem 25 in \cite{NA55}) which was elucidated by J. H. E. Cohn (Corollary 1 in \cite{CO03}). M. R. Murty gave a proof of Theorem \ref{T4} under condition either \enquote{$1-p^n$ is square-free with $n>5$} or \enquote{$m<p^{n/4}/2^{3/2}$ whenever $m^2\mid 1-p^n$ for some integer $m$ with odd $n>5$} (Theorem 1 and Theorem 2 in \cite{RM97}, also see
\cite{RM99}).
Now we give some demonstration for Theorem \ref{T1}. All the computations in this paper were done using PARI/GP (version 2.7.6). Table 1 gives the list of imaginary quadratic fields $K_{p,q}$ corresponding to $n=3$, $p \leq 19$ (and hence discriminant not exceeding $19^3$). Note that the list does not exhaust all the imaginary quadratic fields $K_{p,q}$ of discriminant not exceeding $19^3$. Table 2 is the list of $K_{p,q}$ for $n=5$ and $p \leq 7$.\\
\begin{center}
\begin{longtable}{|l|l|l|l|l|l|l|l|l|l|}
\caption{Numerical examples of Theorem 1 for $n=3$.} \label{tab:long1} \\
\hline \multicolumn{1}{|c|}{$p$} & \multicolumn{1}{c|}{$q$} & \multicolumn{1}{c|}{$q^2-p^n$}& \multicolumn{1}{c|}{$d$} & \multicolumn{1}{c|}{$h(d)$}& \multicolumn{1}{|c|}{$p$} & \multicolumn{1}{c|}{$q$} & \multicolumn{1}{c|}{$q^2-p^n$}& \multicolumn{1}{c|}{$d$} & \multicolumn{1}{c|}{$h(d)$}\\ \hline
\endfirsthead
\multicolumn{10}{c}%
{{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\
\hline \multicolumn{1}{|c|}{$p$} & \multicolumn{1}{c|}{$q$} & \multicolumn{1}{c|}{$q^2-p^n$}& \multicolumn{1}{c|}{$d$} & \multicolumn{1}{c|}{$h(d)$}& \multicolumn{1}{|c|}{$p$} & \multicolumn{1}{c|}{$q$} & \multicolumn{1}{c|}{$q^2-p^n$}& \multicolumn{1}{c|}{$d$} & \multicolumn{1}{c|}{$h(d)$}\\ \hline
\endhead
\hline \multicolumn{10}{|r|}{{Continued on next page}} \\ \hline
\endfoot
\hline
\endlastfoot
3&5&-2&-2&1*& 5&3&-116&-29&6\\
5&7&-76&-19&1**& 7&3&-334&-334&12\\
7&5&-318&-318&12&7&11&-222&-222&12\\
7&13&-174&-174&12& 7&17&-54&-6&2*\\
11&3&-1322&-1322&42&11&5&-1306&-1306&18\\
11&7&-1282&-1282&12&11&13&-1162&-1162&12\\
11&17&-1042&-1042&12&11&19&-970&-970&12\\
11&23&-802&-802&12&11&29&-490&-10&2*\\
11&31&-370&-370&12&11&37&-38&-38&6*\\
13&3&-2188&-547&3&13&5&-2172&-543&12\\
13&7&-2148&-537&12&13&11&-2076&-519&18\\
13&17&-1908&-53&6&13&19&-1836&-51&2**\\
13&23&-1668&-417&12&13&29&-1356&-339&6\\
13&31&-1236&-309&12&13&37&-828&-23&3\\
13&41&-516&-129&12&13&43&-348&-87&6\\
13&47&-12&-3&1*&17&3&-4904&-1226&42\\
17&5&-4888&-1222&12&17&7&-4864&-19&1**\\
17&11&-4792&-1198&12&17&13&-4744&-1186&24\\
17&19&-4552&-1138&12&17&23&-4384&-274&12\\
17&29&-4072&-1018&18&17&31&-3952&-247&6\\
17&37&-3544&-886&18&17&41&-3232&-202&6\\
17&43&-3064&-766&24&17&47&2704&-1&1*\\
17&53&-2104&-526&12&17&59&-1432&-358&6\\
17&61&-1192&-298&6&17&67&-424&-106&6\\
19&3&-6850&-274&12&19&5&-6834&-6834&48\\
19&7&-6810&-6810&48&19&11&-6738&-6738&48\\
19&13&-6690&-6690&72&19&17&-6570&-730&12\\
19&23&-6330&-6330&48&19&29&-6018&-6018&48\\
19&31&-5898&-5898&48&19&37&-5490&-610&12\\
19&41&-5178&-5178&48&19&43&-5010&-5010&48\\
19&47&-4650&-186&12&19&53&-4050&-2&1*\\
19&59&-3378&-3378&24&19&61&-3138&-3138&24\\
19&67&-2370&-2370&24&19&71&-1818&-202&6\\
19&73&-1530&-170&12&19&79&-618&-618&12\\
\end{longtable}
\end{center}
\begin{center}
\begin{longtable}{|l|l|l|l|l|l|l|l|l|l|}
\caption{Numerical examples of Theorem 1 for $n=5$.} \label{tab:long2} \\
\hline \multicolumn{1}{|c|}{$p$} & \multicolumn{1}{c|}{$q$} & \multicolumn{1}{c|}{$q^2-p^n$}& \multicolumn{1}{c|}{$d$} & \multicolumn{1}{c|}{$h(d)$}& \multicolumn{1}{|c|}{$p$} & \multicolumn{1}{c|}{$q$} & \multicolumn{1}{c|}{$q^2-p^n$}& \multicolumn{1}{c|}{$d$} & \multicolumn{1}{c|}{$h(d)$}\\ \hline
\endfirsthead
\multicolumn{10}{c}%
{{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\
\hline \multicolumn{1}{|c|}{$p$} & \multicolumn{1}{c|}{$q$} & \multicolumn{1}{c|}{$q^2-p^n$}& \multicolumn{1}{c|}{$d$} & \multicolumn{1}{c|}{$h(d)$}& \multicolumn{1}{|c|}{$p$} & \multicolumn{1}{c|}{$q$} & \multicolumn{1}{c|}{$q^2-p^n$}& \multicolumn{1}{c|}{$d$} & \multicolumn{1}{c|}{$h(d)$}\\ \hline
\endhead
\hline \multicolumn{10}{|r|}{{Continued on next page}} \\ \hline
\endfoot
\hline
\endlastfoot
3&5&-218&-218&10&3&7&-194&-194&20\\
3&11&-122&-122&10&3&13&-74&-74&10\\
5&3&-3116&-779&10&5&7&-3076&-769&20\\
5&11&-3004&-751&15&5&13&-2956&-739&5\\
5&17&-2836&-709&10&5&19&-2764&-691&5\\
5&23&-2596&-649&20&5&29&-2284&-571&5\\
5&31&-2164&-541&5&5&37&-1756&-439&15\\
5&41&-1444&-1&1*&5&43&-1276&-319&10\\
5&47&-916&-229&10&5&53&-316&-79&5\\
7&3&-16798&-16798&60&7&5&-16782&-16782&100\\
7&11&-16686&-206&20&7&13&-16638&-16638&80\\
7&17&-16518&-16518&60&7&19&-16446&-16446&100\\
7&23&-16278&-16278&80&7&29&-15966&-1774&20\\
7&31&-15846&-15846&160&7&37&-15438&-15438&80\\
7&41&-15126&-15126&120&7&43&-14958&-1662&20\\
7&47&-14598&-1622&30&7&53&-13998&-13998&100\\
7&59&-13326&-13326&100&7&61&-13086&-1454&60\\
7&67&-12318&-12318&60&7&71&-11766&-11766&120\\
7&73&-11478&-11478&60&7&79&-10566&-1174&30\\
7&83&-9918&-1102&20&7&89&-8886&-8886&60\\
7&97&-7398&-822&20&7&101&-6606&-734&40\\
7&103&-6198&-6198&40&7&107&-5358&-5358&40\\
7&109&-4926&-4926&40&7&113&-4038&-4038&60\\
7&127&-678&-678&20&&&&& \\
\end{longtable}
\end{center}
In both the tables we use $*$ in the column for class number to indicate the failure of condition \enquote{$q\not\equiv \pm 1 \pmod{|d|}$} of Theorem \ref{T1}. Appearance of $**$ in the column for class number indicates that both the conditions \enquote{$q\not\equiv \pm 1 \pmod{|d|}$ and $p^{n/3}\ne (2q+1)/3, (q^2+3)/3$} fail to hold.
For $n=3$, the number of imaginary quadratic number fields obtained from the family provided by T. Nagell (namely $K_{t,1}$ with $t$ any odd integer) with class number divisible by $3$ and discriminant not exceeding $19^3$ are at most $9$, whereas, in Table 1 there are 59 imaginary quadratic fields $K_{p,q}$ with class number divisible by $3$ and discriminant not exceeding $19^3$ (Table 1 does not exhaust all such $K_{p,q}$). Out of these 59 fields in Table 1, the conditions of Theorem~\ref{T1} hold for 58. This phenomenon holds for all values of $n$.
\noindent\textbf{Acknowledgements.}
The third and fourth authors would like to appreciate the hospitality provided by Harish-Chandra Research Institute, Allahabad, where the main part of the work was done. The authors would like to thank the anonymous referee for valuable comments to improve the presentation of this paper.
|
1,116,691,500,767 | arxiv | \section{Introduction}
The idea behind phase averaging is to find a time integration method
where we can robustly take large timesteps for highly oscillatory PDEs
such as the rotating shallow water equations in the low Rossby number
regime. In phase averaging the nonlinearity is averaged over all
phases of the fast waves to obtain an approximation of the slow
dynamics with no fast oscillations present. \citet{majda1998averaging}
showed that for the rotating shallow water equations, taking the low
Rossby number limit in the phase averaged equations leads to the
quasigeostrophic equations. This gives us an equation set where we can
take much larger timesteps because the fast dynamics are no longer
present, and hence the fast wave CFL condition is not
present. However, the quasigeostrophic equations are not uniformly
valid, which is one reason why their 3D counterpart is not used for
operational weather forecasting. \citet{haut2014asymptotic} proposed to
use a phase average of the nonlinearity over a finite width averaging
window $T$. For small $T$, the original equations are recovered and for
large $T$ the full phase averaging is recovered which filters all fast
dynamics. They proposed to perform the phase averaging numerically,
replacing the phase integral with a numerical quadrature rule. The
important part is that each term in the quadrature rule can be
evaluated independently and hence in parallel: the idea is to use parallel
computation to perform averaging allowing larger timesteps to be
taken. \citet{Peddle2019PararealCF}
showed that, given a chosen timestepping
scheme and timestep size, there is an optimum averaging window
$T$. Below the optimum, the timestepping scheme becomes unstable, and
above the optimum, the averaging error dominates. For larger
timesteps, the optimal $T$ is larger, requiring more quadrature points
in the average and consequently requiring more computational cores.
The finite window phase averaged model trades computational cores for
larger timesteps at the expense of accuracy. For standard rotating
shallow water on the sphere test cases, we have found that the
averaged model is already very accurate. However, if the level of
accuracy is insufficient, a time-parallel predictor-corrector approach
can be used to increase accuracy. \citet{haut2014asymptotic} proposed
to use the averaged model as the coarse propagator in a highly
efficient Parareal iteration, demonstrating parallel speedups of a
factor of 100 in a rotating shallow water test case.
\citet{Peddle2019PararealCF} proved convergence of the iterative
Parareal procedure for highly oscillatory PDEs with quadratic
nonlinearity, making use of the optimal averaging window $T$ at finite
Rossby number. \citet{bauer2021higher} created a hierarchy of higher
order averaged models that increase accuracy through increasing the
number of variables; this type of hierarchy is ideal for
predictor-corrector approaches such as RIDC methods
\citep{ong2016algorithm} and PFASST \citep{minion2011hybrid} that
compute more accurate correction steps in parallel as new predictor
steps are being taken. These are motivations for the work in this
paper but here we focus on the impact of the averaging on the rotating
shallow water solution.
The rest of this paper is organised as follows. In
Section \ref{sec:model}, we describe the phase averaging procedure,
and how it can be applied to the rotating shallow water equations. We
also describe our approach to timestepping these equations. In
Section \ref{sec:results}, we present our numerical results,
examining the impact of averaging window $T$ and timestep $\Delta t$
on the errors associated with time integration. Finally,
in \ref{sec:conclusion} we provide a summary.
\section{Model description}
\label{sec:model}
\subsection{Shallow water equations}
In this section we describe the shallow water equations and link them
to the general notation framework for phase averaging that will be
used in subsequent sections.
We begin with the nonlinear shallow water equations on a two
dimensional surface that is embedded in three dimensions,
\begin{eqnarray}
\MM{u}_t + f\MM{u}^{\perp} + g\nabla\eta + (\MM{u}\cdot\nabla)\MM{u}
&=& 0, \label{ueq}\\
\eta_t + H\nabla\cdot\MM{u} + \nabla[\MM{u}(\eta - b)]
&=& 0, \label{etaeq}
\end{eqnarray}
where $\MM{u}$ is the horizontal velocity, $f$ is the Coriolis parameter, $\MM{u}^{\perp} = \MM{k} \times \MM{u}$ where $\MM{k}$ is the normal to the surface, and $g$ is the gravitational acceleration;
$\eta$ is the free surface elevation, $H$ is the mean layer thickness and $b$ is the height of the lower boundary, where the layer depth $h = H+\eta -b$; $\nabla$ and $\nabla\cdot$ are appropriate invariant gradient and divergence operators defined on the surface. Here we will concentrate on the case of the equations being solved on the surface of the sphere, so there are no boundary conditions to
consider.
Then we rewrite the equations as
\begin{eqnarray}
\MM{U}_t &=& \mathcal{L}\MM{U}+\mathcal{N}(\MM{U}), \label{split_eqs}
\end{eqnarray}
where the vector of unknowns $\MM{U}(t) = (\MM{u}, \eta)$. The matrix $\mathcal{L}$ represents a linear operator and $\mathcal{N}(\cdot)$ is a nonlinear operator which satisfy
\begin{eqnarray}
\mathcal{L}\MM{U}
&=& \left(\begin{array}{cc} -f(\cdot)^{\perp} & -g\nabla\\ -H\nabla\cdot & 0\\ \end{array}\right)
\left(\begin{array}{c} \MM{u}\\ \eta\\ \end{array}\right)
= \left(\begin{array}{cc} -f(\MM{u})^{\perp} & -g\nabla\eta\\ -H\nabla\cdot\MM{u} & 0\\ \end{array}\right),
\label{linear_operator}\\
\mathcal{N}(\MM{U})
&=& \left(\begin{array}{c} -(\MM{u}\cdot\nabla)\MM{u}\\ -\nabla[\MM{u}(\eta - b)]\\ \end{array}\right).
\label{nonlinear_operator}
\end{eqnarray}
\subsection{Phase averaging}
Now we consider an approximation to the equation \eqref{split_eqs} by averaging the nonlinearity over the fast oscillations.
First we introduce a coordinate transformation,
\begin{eqnarray}
\MM{V}(t)&=&e^{-\mathcal{L}t}\MM{U}(t), \label{v_eq}\\
\frac{\partial\MM{V}}{\partial t}(t) &=& e^{-\mathcal{L}t}\mathcal{N}(e^{\mathcal{L}t}\MM{V}(t)), \label{vt_eq}
\end{eqnarray}
where $e^{\mathcal{L}t}\MM{W}$ is solution at time $t$ to the linear part of the equation $\MM{U}_t = \mathcal{L}\MM{U}$ with an initial condition of $\MM{U}(0) = \MM{W}$.
To allow averaging the model over a finite time interval $s$ around time $t$, we modify the equations \eqref{v_eq} and \eqref{vt_eq} as
\begin{eqnarray}
\MM{V}(t,s)&=&e^{-\mathcal{L}(t+s)}\MM{U}(t, s), \label{v_eq_s}\\
\frac{\partial\MM{V}}{\partial t}(t, s) &=& e^{-\mathcal{L}(t+s)}\mathcal{N}(e^{\mathcal{L}(t+s)}\MM{V}(t, s)). \label{vt_eq_s}
\end{eqnarray}
An averaging approximation to \eqref{vt_eq_s} over the averaging window $T$ with respect to a weight function $\rho$ can be written as
\begin{eqnarray}
\frac{\partial\bar{\MM{V}}}{\partial t}(t, s) &=& \lim_{t \to \infty}\frac{1}{2T}\int^{T}_{-T}\rho\left(\frac{s}{T}\right)e^{-\mathcal{L}(t+s)}\mathcal{N}(e^{\mathcal{L}(t+s)}\bar{\MM{V}}(t, s)) \diff s, \label{vt_averaged}
\end{eqnarray}
where $\bar{\MM{V}}$ denotes the averaged $\MM{V}$. Then we discretise the right hand side as
\begin{eqnarray}
\frac{\partial\bar{\MM{V}}}{\partial t} \!\!\!\!
&\,& \,\simeq \quad {\sum_{k=-N}^{N}} w_{k}e^{-\mathcal{L}(t+s_{k})}\mathcal{N}(e^{\mathcal{L}(t+s_{k})}\bar{\MM{V}}) \\
&\,& := \quad <e^{-\mathcal{L}(t+s)}\mathcal{N}(e^{\mathcal{L}(t+s)}\bar{\MM{V}})>_s, \label{vt_discretised}
\end{eqnarray}
where $w_k$ are weight coefficients and $s_k = kT/N$. We use the notation \eqref{vt_discretised} in the following section.
\subsection{Time discretisation}
In this section we discretise the model in time using a 3rd order strong-stability-preserving Runge-Kutta (SSPRK) scheme which takes the form:
\begin{eqnarray}
x_1 &=& x^{n}+\Delta t f(x^{n}, t^{n}), \\
x_2 &=& \frac{3}{4} x^{n} + \frac{1}{4}[x_1 + \Delta t f(x_1, t^{n+1})], \\
x^{n+1} &=& \frac{1}{3} x^{n} + \frac{2}{3}[x_2 + \Delta t f(x_2, t^{n+1/2})],
\end{eqnarray}
where $x_t = f(x)$ and $\Delta t$ is the time step. Using the notation \eqref{vt_discretised}, the first stage of the Runge-Kutta scheme can be written in the equation of $\bar{\MM{V}}$ as
\begin{eqnarray}
\bar{\MM{V}}_{1}
&=& \bar{\MM{V}}^{n} + \Delta t <e^{-\mathcal{L}(t^n+s)}\mathcal{N}(e^{\mathcal{L}(t^n+s)}\bar{\MM{V}}^{n})>_s. \label{stage1_a}
\end{eqnarray}
Using the relationship $\bar{\MM{U}}^n = e^{\mathcal{L}t^n}\bar{\MM{V}}^n$, obtained from the equation \eqref{vt_eq}, the equation \eqref{stage1_a} becomes
\begin{eqnarray}
\bar{\MM{V}}_{1}
&=& \bar{\MM{V}}^{n} + \Delta t \, e^{-\mathcal{L}t^n} <e^{-\mathcal{L}s}\mathcal{N}(e^{\mathcal{L}s}\bar{\MM{U}}^{n})>_s. \label{stage1_b}
\end{eqnarray}
Applying $e^{\mathcal{L}t^{n+1}}$ to the both sides of the equation \eqref{stage1_b}, and defining $\bar{\MM{U}}_1 = e^{\mathcal{L}t^{n+1}}\bar{\MM{V}}_1$, we have the first stage of the scheme in the equation of $\bar{\MM{U}}$ as
\begin{eqnarray}
\bar{\MM{U}}_{1} &=& e^{\mathcal{L}\Delta t} \bar{\MM{U}}^{n} + \Delta t \, e^{\mathcal{L}\Delta t} <e^{-\mathcal{L}s}\mathcal{N}(e^{\mathcal{L}s}\bar{\MM{U}}^{n})>_s.
\label{stage1_c}
\end{eqnarray}
Similary, the second stage can be written as,
\begin{eqnarray}
\bar{\MM{V}}_{2}
&=& \frac{3}{4}\bar{\MM{V}}^{n} + \frac{1}{4}[\bar{\MM{V}}_{1} + \Delta t <e^{-\mathcal{L}(t^{n+1}+s)}\mathcal{N}(e^{\mathcal{L}(t^{n+1}+s)}\bar{\MM{V}}_1)>_s] \\
&=& \frac{3}{4}\bar{\MM{V}}^{n} + \frac{1}{4}[\bar{\MM{V}}_{1} + \Delta t \, e^{-\mathcal{L}t^{n+1}} <e^{-\mathcal{L}s}\mathcal{N}(e^{\mathcal{L}s}\bar{\MM{U}}_1)>_s]. \label{stage2}
\end{eqnarray}
Applying $e^{\mathcal{L}t^{n+\frac{1}{2}}}$ to the both sides of the equation \eqref{stage2}, and defining $\bar{\MM{U}}_2 = e^{\mathcal{L}t^{n+\frac{1}{2}}}\bar{\MM{V}}_2$, we have
\begin{eqnarray}
\bar{\MM{U}}_{2}
&=& \frac{3}{4} e^{\mathcal{L}\frac{\Delta t}{2}}\bar{\MM{U}}^{n} + \frac{1}{4} e^{-\mathcal{L}\frac{\Delta t}{2}} [\bar{\MM{U}}_{1} + \Delta t <e^{-\mathcal{L}s}\mathcal{N}(e^{\mathcal{L}s}\bar{\MM{U}}_1)>_s].
\label{stage2s}
\end{eqnarray}
Finally, the third stage can be written as
\begin{eqnarray}
\bar{\MM{V}}^{n+1}
&=& \frac{1}{3}\bar{\MM{V}}^{n} + \frac{2}{3}[\bar{\MM{V}}_{2} + \Delta t <e^{-\mathcal{L}(t^{n+\frac{1}{2}}+s)}\mathcal{N}(e^{\mathcal{L}(t^{n+\frac{1}{2}}+s)}\bar{\MM{V}}_2)>_s] \\
&=& \frac{1}{3}\bar{\MM{V}}^{n} + \frac{2}{3}[\bar{\MM{V}}_{2} + \Delta t \, e^{-\mathcal{L}t^{n+\frac{1}{2}}} <e^{-\mathcal{L}s}\mathcal{N}(e^{\mathcal{L}s}\bar{\MM{U}}_2)>_s].
\end{eqnarray}
Applying $e^{\mathcal{L}t^{n+1}}$ to the both sides, we obtain
\begin{eqnarray}
\bar{\MM{U}}^{n+1}
&=& \frac{1}{3} e^{\mathcal{L} \Delta t}\bar{\MM{U}}^{n} + \frac{2}{3} e^{\mathcal{L}\frac{\Delta t}{2}} [\bar{\MM{U}}_{2} + \Delta t <e^{-\mathcal{L}s}\mathcal{N}(e^{\mathcal{L}s}\bar{\MM{U}}_2)>_s],
\label{stage3}
\end{eqnarray}
where we used $\bar{\MM{U}}^{n+1} = e^{\mathcal{L}t^{n+1}}\bar{\MM{V}}^{n+1}$.
\subsection{Chebyshev exponentiation}
To implement the exponential operator $e^{\mathcal{L}t}$, we use a Chebyshev approximation of
\begin{eqnarray}
e^{\mathcal{L}t}\bar{\MM{U}} \approx \sum^{N}_{k=0}a_{k}P_{k}(\mathcal{L}t)\bar{\MM{U}}
\label{cheby}
\end{eqnarray}
where $N$ is the number of polynomials, $a_k$ are Chebyshev coefficients, and $P_k$ are Chebyshev polynomials. Here $P_k$ are transformed to create approximation on the interval $-iL \leq l \leq iL$ using the following recurrence relation:
\begin{eqnarray}
P_0(l) &=& 1, \\
P_1(l) &=& \frac{-il}{L}, \\
P_{k+1}(l) &=& \frac{2l P_k(l)}{iL} - P_{k-1}(l).
\end{eqnarray}
Therefore $P_k(\mathcal{L}t)\bar{\MM{U}}$ in the equation \eqref{cheby} are obtained recursively as
\begin{eqnarray}
P_0(\mathcal{L}t)\bar{\MM{U}} &=& \bar{\MM{U}}, \\
P_1(\mathcal{L}t)\bar{\MM{U}} &=& \frac{-it\mathcal{L}\bar{\MM{U}}}{L}, \\
P_{k+1}(\mathcal{L}t)\bar{\MM{U}} &=& \frac{2t P_k(\mathcal{L}t)\mathcal{L}\bar{\MM{U}}}{iL} - P_{k-1}(\mathcal{L}t)\bar{\MM{U}}.
\end{eqnarray}
The coefficients $a_k$ are computed by using the FFT.
In section \ref{sec:results}, $L$ is set to $|\lambda_{\mathrm{max}}|T$ where $\lambda_{\mathrm{max}}$ is the maximum frequency.
\section{Results}\label{sec:results}
In this section we show numerical results from a standard test case on the sphere described by \citet{williamson1992standard}.
Here we use their test case number 5 (flow over a mountain), where the model is initialised with the layer depth and velocity fields that are in geostrophic balance:
\begin{eqnarray}
h &=& H - \left(R\Omega u_0 + \frac{u_0^{2}}{2}\right)\frac{z^2}{gR^2},\\
\MM{u} &=& \frac{u_0}{R}(-y, x, 0),
\end{eqnarray}
where $R = 6.37122 \times 10^6$ m is the radius of the Earth, $\Omega = 7.292 \times 10^5$ s$^{-1}$ is the rotation rate of the Earth, $(x, y ,z)$ are the 3D Cartesian cooordinates, the maximum zonal wind speed $u_0$ = 20 m, $g$ = 9.8 m s$^{-2}$ and $H$ = 5960 m.
An isolated mountain is placed with its centre at latitude $\phi = \pi/6$ and longitude $\lambda = -\pi/2$. The height of the mountain is described as
\begin{eqnarray}
b &=& b_0\left(1-\frac{(\mathrm{min}[R_0^2, (\phi-\phi_c)^2+(\lambda-\lambda_c)^2])^{1/2}}{R_0}\right)
\end{eqnarray}
where $b_0$ = 2000 m and $R_0 = \pi / 9$.
Icosahedral grids with a piecewise cubic approximation to the sphere are used in the model.
The number of cells $N = 1280$ where the maximum cell center to cell center distance is 1054 km and the minimum distance is 720 km.
As there is no analytical solution for this problem, the model output at 15 days is compared to a reference solution generated from a semi-implicit nonlinear shallow water code provided by \citet[][]{gibson2019compatible}, using the same spatial resolution of $N = 1280$.
A timestep of 1 hour is used in the averaged model whereas a timestep of 180 s is used to generate the reference solution.
\begin{figure}[h]
\centering
\includegraphics[height=50mm]{figures/eta}
\caption{The free surface elevation $\eta$ at day 15 from the flow over mountain test case. The circle indicates the position of the mountain.}
\label{fig:eta}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[height=50mm]{figures/etadiff}
\caption{The errors in the elevation $\eta$ compared to the reference solution at day 15. The circle indicates the position of the mountain.}
\label{fig:etadiff}
\end{figure}
Figure \ref{fig:eta} shows the field of the free surface elevation $\eta$ at day 15 in the averaged model plotted in a latitude longitude space.
The averaging window $T$ is set to 1 hour in this plot.
It shows waves that travel around the globe as a result of the zonal flow interacting with the mountain.
Figure \ref{fig:etadiff} shows the errors in $\eta$ at day 15 compared to the reference solution.
Errors are prominent on the lee side of the mountain with some evidence of grid imprinting.
\begin{figure}[h]
\centering
\begin{subfigure}{0.47\hsize}
\centering
\includegraphics[height=65mm, clip]{figures/dt1_eta}
\end{subfigure}
\begin{subfigure}{0.47\hsize}
\centering
\includegraphics[height=65mm, clip]{figures/dt1_u}
\end{subfigure}
\caption{Normalised $L_2$ norm of the error in (left) $\eta$ and (right) $\MM{u}$ versus the averaging window $T$. The time step $\Delta t = 1$ hour for the averaged model and $\Delta t = 180$ s for the reference solution are fixed in all the simulations. Note the clear existence of optimal averaging windows at around $T = 2$ for $\eta$ and $T = 1.5$ for $\MM{u}$.}
\label{fig:peddle_plots}
\end{figure}
Figure \ref{fig:peddle_plots}a shows the $L_2$ norm of the error in $\eta$ normalised by the norm of the reference solution, plotted over various averaging windows.
The same spatial resolution of $N = 1280$, and the same time steps of $1$ hour for the averaged model and $180$ s for the reference solution, are used as in Figure \ref{fig:eta} and \ref{fig:etadiff}.
The result reveals the clear existence of an optimal averaging window at around $T = 2$.
This demonstrates that the behavior in the averaged model is consistent with the error bounds shown in \citet[][]{Peddle2019PararealCF}.
Figure \ref{fig:peddle_plots}b shows the same plot but for the error in $\MM{u}$.
We note that the optimal averaging window for $\MM{u}$ is at around $T = 1.5$ which is smaller than that in $\eta$; this will vary in the choice of norm.
\section{Summary and outlook}
\label{sec:conclusion}In this paper we presented a phase averaging framework
for the rotating shallow water equations, and a time integration
methodology for it. We presented results from the rotating shallow
water equations and analysed their errors, which confirm that there is
an optimal averaging window value for a given time stepsize $\Delta
t$. Naturally, the optimal averaging window for both height and
velocity fields combined depends on the choice of norm. In future work
we will explore the combination of phase averaging methods with
implicit or split timestep methods that allow us to take even larger
timesteps, will incorporate parallel rational approximation techniques
to speed up the exponential evaluations \citep{haut2016high}, and will
undertake parallel performance benchmarks.
|
1,116,691,500,768 | arxiv | \section{Introduction}
Disformal scalar fields were introduced by Bekenstein~\cite{Bekenstein:1992pj}. By assuming the weak equivalence principle and causality,
he showed that a scalar field $\pi$ can modify the geometry by defining the following metric $g_{\mu\nu}$:
\begin{equation}
g_{\mu\nu}=A(\pi,X)\tilde{g}_{\mu\nu} + B(\pi,X) \partial_\mu \pi \partial_\nu \pi\; ,
\label{eq:bekmetric}
\end{equation}
where $\tilde{g}_{\mu\nu}$ is a background metric independent of the scalar field, and
$X=(1/2)\tilde{g}^{\mu\nu}\partial_{\mu}\pi\partial_{\nu}\pi$ depends on the first derivatives of such a field.
This is the most general metric that respects the previous two basic constraints.
If $B(\pi,X)=0$, Eq. (\ref{eq:bekmetric}) defines a conformal relation between $g_{\mu\nu}$ and $\tilde{g}_{\mu\nu}$.
In such a case, the scalar field is named conformal, and the conformal factor $A(\pi,X)$ defines the conformal
coupling of $\pi$ to the rest of fields. Conformal scalar fields have been studied from a long time and its phenomenology
can be found in a large number of works from different approaches.
On the other hand, the term proportional to $B(\pi,X)$ is the genuine disformal coupling. The presence of a non-zero
disformal factor $B(\pi,X)$ is able to change dramatically the phenomenology of the scalar field. We will focus on such a case
by assuming that the disformal coupling dominates over the conformal one. The simplest possibility for
the disformal coupling is just a constant term $B(\pi,X)\propto 1/f^4$, where $f$ has dimensions of energy. In such a case the main interaction of $\pi$
with the rest of fields is through their energy momentum tensor $T^{\mu\nu}$:
\begin{equation}
\mathcal{L}_D \propto \frac{1}{f^4}\partial_\mu \pi\partial_\nu \pi T^{\mu\nu}\;.
\label{eq:coupling}
\end{equation}
Eq. (\ref{eq:coupling}) shows explicitly the basic property of the disformal model. It is an effective field theory governed by an energy dimension 8 operator. The high power of the leading interaction may seem artificial and unstable against radiative corrections. Indeed, this is the case except if additional symmetries are present.
First of all, the derivative coupling shown in
(\ref{eq:coupling}) preserves the shift symmetry of the field: $\pi(x^\mu)\rightarrow\pi'(x^\mu)\equiv \pi(x^\mu)+\Lambda$,
where $\Lambda$ is a constant. In this sense, the disformal scalar can be associated with the Nambu-Goldstone Boson (NGB) arising from the spontaneous breaking of a global symmetry. If the symmetry is exact, the $A$ and $B$ functions can only depend on the $X$ term:
$A(\pi,X)=A(X)$ and $B(\pi,X)=B(X)$. On the other hand, the symmetry can be slightly violated in an explicit way. In this case, the leading phenomenology can still be associated with the term described in Eq. (\ref{eq:coupling}), and the disformal scalar can be understood as a pseudo-Nambu-Godstone Boson (pNGB).
If the shift symmetry is not exact, interaction terms with a single scalar field are expected. If they are not present in the tree level Lagrangian, they will generally arise due to radiative corrections. However, Eq. (\ref{eq:coupling}) owns an additional discrete global symmetry. It is invariant under the parity transformation of the disformal field: $\pi(x^\mu)\rightarrow\pi'(x^\mu)\equiv -\pi(x^\mu)$. If this symmetry is imposed, terms with an odd number of disformal scalars can be forbidden. In this case, the scalars will be coupled in pairs to the SM particles. This symmetry implies a complete different phenomenology for disformal fields than for standard dilatons. In particular, this type of disformal scalars are stable and can play an important role in cosmology.
The first detailed framework where these ideas were developed was in flexible brane-world models. These models are characterized by the fact that Standard Model (SM) particles are restricted to propagate on a manifold of three spatial dimensions embedded in a higher dimensional space-time ($D=4+N$). Only the gravitational interaction has access to the whole bulk space.
The fundamental scale of gravity in $D$ dimensions, $M_D$, can be reduced in relation to Planck scale $M_P$, due to a large volume of the extra-space. In the original proposals, the value of $M_D$ was taken around the electroweak scale since they try to address the hierarchy problem
\cite{ADD}. However, the model suffers from important constraints, and $M_D$ has to be much larger \cite{Langlois,six}.
From an observer on the brane, the existence of extra dimensions introduces new degrees of freedoms, that can be studied within an
effective field theory at low energies. On the one hand, modes of fields propagating into the bulk space have associated a so-called Kaluza-Klein (KK) tower of states. In principle, this tower will be restricted to gravitons, but more complex models can have other degrees of freedom with access to the bulk space. On the other hand, the fluctuations of the brane will be parameterized by several $\pi^\alpha$ fields of spin zero.
The flexible character of the brane is quantified by the brane tension $\tau\equiv f^4$. These scalars are
called branons. If the translational invariance along the extra-dimensions is an exact symmetry, they can be understood as the massless
NGB arising from the spontaneous breaking of that symmetry induced by the location of the brane in a particular point of the extra-space \cite{Sundrum,DoMa}. In a more general case, a non-trivial energy content along the bulk space will explicitly break the translational invariance in the extra-space. In such a case, branons are expected to be massive \cite{BSky,Alcaraz:2002iu}.
It is interesting to note that flexible branes suppress exponentially the coupling of the SM particles to any KK mode \cite{GB}. Therefore, if the tension scale $f$ is much smaller than the fundamental scale of gravity $M_D$, the KK states decouple from the SM particles. In such a case, the constraints on the model \cite{KK} are strongly alleviated. In fact, for flexible enough branes, the only relevant new particles at
low energies are the branon fields.
Branons have the standard disformal coupling given by Eq. (\ref{eq:coupling}). The potential signatures in colliders have been studied in different works. The general massive case was first discussed in Ref. \citen{Alcaraz:2002iu}, whereas the massless one was studied previously in
Ref. \citen{strumia}.
The force mediated by disformal scalars can be found in Refs. \citen{thesis} and \citen{Kugo} for the massive and massless scalars.
Limits from supernovae and modifications of Newton's law at small distances in the massless case were obtained in Ref. \citen{Kugo}. Moreover, in Refs. \citen{CDM} and \citen{CDM2}, the interesting possibility that massive disformal branons could account for the observed Dark Matter (DM) of the universe was studied in detail. From a more general disformal model approach, the constraints have been also studied in different contexts \cite{Kaloper:2003yf}.
In particular, disformal interactions arise in galileon models and massive gravitational theories \cite{deRham:2010ik,deRham:2010kj}.
Within these frameworks, the work has focused on astrophysical and cosmological analyses \cite{Zumalacarregui:2010wj,Koivisto:2012za,Bettoni:2012xv,Brax:2013nsa,vandeBruck:2013yxa,Neveu:2014vua}, but laboratory \cite{Brax:2012ie} and fundamentally collider experiments \cite{Brax:2014vva} are presently the most constraining.
This manuscript is organized as follows: in
Section \ref{BraMo} we give a brief introduction to the branon model as an explicit example
of massive disformal fields.
Section \ref{Coll} contains a summary of the main disformal signatures at colliders.
Section \ref{Rel} is devoted to the standard calculation of relic abundance of
disformal scalars generated by the freeze-out phenomenon in an
expanding universe.
The constraints derived from this computation are presented in Section \ref{RelCon}.
In Section \ref{Dir} and \ref{Ind}, we analyze the possibility of detecting disformal DM
through direct and indirect searches respectively. On the other hand,
light disformal scalars suffer other types of astrophysical constraints.
In Section \ref{BBNs}, after reviewing the limits imposed by nucleosynthesis
on the number of relativistic species, we apply them to the disformal case.
Section \ref{Ste} contains an estimation for the rate of energy loss from stellar
objects in the form of disformal scalars and the corresponding constraints. In Section \ref{nonthermals} the possibility
for disformal DM to be produced non-thermally is explored.
Section \ref{Con} includes the main conclusions of our discussion. This
manuscript is completed with the addition of three appendices. \ref{Ver}
contains the disformal vertices with SM particles. \ref{Cross} includes the
formulas for the creation and annihilation cross-sections for disformal
scalar pairs. Finally, we give the results for the thermal averages of
disformal particle annihilation cross-sections into SM particles in \ref{Thermal}.
\section{Branon model}
\label{BraMo}
In this Section, we will describe in detail a particular model of disformal scalar
fields. We will introduce the main properties of massive brane fluctuations
in brane-world models \cite{DoMa,BSky,Alcaraz:2002iu}.
The original concept is associated with a single brane in the thin limit,
although the branon field can be more generally understood as a particular
coherent mode. The standard four-dimensional space-time $M_4$ is assumed to
be embedded in a bulk space of $D$ dimensions. For simplicity, we will assume it to
be of the form $M_D=M_4\times B$, i.e. we can define the extra space $B$,
as an independent $N$-dimensional compact manifold, so $D=4+N$. The brane will lie along
the $M_4$ space-time, and we will work in the probe brane limit
neglecting its contribution to the bulk gravitational field.
The coordinates $(x^{\mu},y^m)$ parameterize the points in the bulk space $M_D$,
where the indices run as $\mu=0,1,2,3$ and $m=1,2,...,N$. $M_D$ is endowed with a
metric $G_{MN}$ with signature $(+,-,-...-,-)$. For simplicity, we will
assume the separable geometry defined by the following ansatz:
\begin{eqnarray}
G_{MN}&=&
\left(
\begin{array}{cccc}
\tilde g_{\mu\nu}(x,y)&0\\ 0&-\tilde g'_{mn}(y)
\end{array}\right).
\end{eqnarray}
The position of the brane in the bulk space-time $M_D$ can be parameterized naturally with the
coordinates of $M_4$: $Y^M=(x^\mu, Y^m(x))$, with $M=0,\dots, 3+N$, so the first four
coordinates have been chosen to be identified with the space-time brane coordinates $x^\mu$.
In this way, the brane is located in a particular point in the extra space $B$,
i.e. $Y^m(x)=Y^m_0$. This position defines its ground state. We will consider
that the extra space is homogeneous, so that brane fluctuations can be written in terms
of properly normalized coordinates in $B$: $\pi^\alpha(x)=f^2 Y^\alpha(x)$, where
$\alpha=1,\dots, N$. The geometry that determines the dynamics on the brane is defined
by the induced metric. In the ground state, this metric is simply given by the four-dimensional
components of the bulk space-time metric: $g_{\mu\nu}=\tilde g_{\mu\nu}=G_{\mu\nu}$. However, in general,
brane fluctuations will modify it in the following way
\begin{eqnarray}
g_{\mu\nu}=\partial_\mu Y^M\partial_\nu Y^N G_{MN}(x,Y(x)) =\tilde
g_{\mu\nu}(x,Y(x))-\partial_{\mu}Y^m\partial_{\nu}Y^n\tilde
g'_{mn}(Y(x))\,.
\label{induced}
\end{eqnarray}
The induced metric (\ref{induced}) can be expanded around the ground state
in order to find explicitly the branon contributions \cite{DoMa,BSky,Alcaraz:2002iu} :
\begin{equation}
g_{\mu\nu}=
\tilde g_{\mu\nu}-\frac{1}{f^4}\delta_{\alpha\beta}\partial_{\mu}\pi^\alpha
\partial_{\nu}\pi^\beta
+\frac{1}{4f^4}\tilde g_{\mu\nu}M_{\alpha\beta}^2\pi^\alpha\pi^\beta
+\dots
\end{equation}
Branons can be defined as the mass eigenstates of the brane fluctuations within the extra-space directions.
The matrix $M_{\alpha\beta}$ determines the branon masses. It characterizes the local geometrical properties of the bulk space
where the brane is located. In the absence of an explicit model for the bulk dynamics, its elements should be
considered as free parameters (for instance, for a particular construction, see Ref. \citen{Andrianov}).
Branons have zero mass only in highly symmetric bulk spaces \cite{DoMa,BSky,Alcaraz:2002iu}.
There is also the possibility of having models with massless and massive branons.
It will mean that the extra space has an incomplete set of isometries,
since the isometries have associated zero eigenvalues of $M_{\alpha\beta}$, being
the massless branons, the fields that parameterize the corresponding flat directions.
In this review, we will not discuss the fundamental nature of the brane. On the contrary, we will assume
that its dynamics can be described by a low-energy effective action \cite{DoMa,BSky,Alcaraz:2002iu}.
In particular, we will consider that the kinetic term comes from the Nambu-Goto action and will take the
limit in which gravity decouples $M_D\rightarrow \infty$, since in such a case,
branon effects can be analyzed independently.
On the other hand, branon couplings to the SM particles
can be obtained from the standard action on a curved background
geometry given by the induced metric (\ref{induced}), which can be expanded
in branon fields. For example, the complete action up to second
order contains the SM terms, the kinetic term for branons and the
quadratic interaction term between branons and SM fields:
\begin{eqnarray}
S_B&=& \int_{M_4}d^4x\sqrt{g}[-f^4+ {\mathcal L}_{SM}(g_{\mu\nu})]
\nonumber\\
&=&\int_{M_4}d^4x\sqrt{\tilde g}\left[-f^4+ {\mathcal L}_{SM}
( \tilde g_{\mu\nu}) +
\frac{1}{2}\tilde g^{\mu\nu}\delta_{\alpha\beta}\partial_{\mu}
\pi^\alpha
\partial_{\nu}\pi^\beta-\frac{1}{2}M^2_{\alpha\beta}
\pi^\alpha\pi^\beta\right.
\nonumber\\
&+&
\left.\frac{1}{8f^4}(4\delta_{\alpha\beta}\partial_{\mu}\pi^\alpha
\partial_{\nu}\pi^\beta-M^2_{\alpha\beta}\pi^\alpha\pi^\beta
\tilde g_{\mu\nu})
T^{\mu\nu}_{SM}(\tilde g_{\mu\nu}) \right]
+\dots \label{lag}
\end{eqnarray}
Here, $T^{\mu\nu}_{SM}(\tilde g_{\mu\nu})$ is the conserved
energy-momentum tensor of the SM evaluated in the
background metric $\tilde g_{\mu\nu}$:
\begin{eqnarray}
T^{\mu\nu}_{SM}=-\left(\tilde g^{\mu\nu}{\mathcal L}_{SM}
+2\frac{\delta {\mathcal L}_{SM}}{\delta \tilde
g_{\mu\nu}}\right)\,.
\end{eqnarray}
The quadratic expression in (\ref{lag}) is general for any extra space $B$,
regardless of the form of the metric $\tilde g'_{mn}$.
Indeed the low-energy effective Lagrangian is model independent and is parameterized only by
the number of branon fields, their masses and the brane tension.
The particular geometry of the total bulk space only affects at higher orders.
Therefore, these effective couplings provide the necessary tools to analyze the phenomenology
of branons in terms of $f$ and the branons masses.
It is interesting to note that under a parity transformation in the bulk space,
the extra dimensions change sign. As they are parameterized by a the branon fields,
these fields changes sign as well. The symmetry under extra dimensional parity
is preserved by the above Lagrangian (\ref{lag}).
However, in a general case, this parity may be violated by higher orders
terms. As we have commented in the introduction, this violation may introduce
interacting terms in the Lagrangian with a single branon field. For this reason,
we will restrict our analysis to bulk geometries that are invariant under
parity transformations in the extra dimensions around the brane location.
This is the case of the most part of the brane-world models. In particular,
it is well known that in order to introduce quiral fermions on the bulk space,
the common $S^1$ symmetry should be promoted to a $S^1\times \mathbb{Z}_2$,
i.e. it has to be defined in an orbifold. This new symmetry, that we will call
brane-parity ensures the absence of terms in the effective Lagrangian with an odd number of
branons. In such a case, branons interact by pairs with the SM particles and are necessarily stable.
On the other hand, branon couplings are suppressed by the brane tension scale
$f$, which means that they may be weakly interacting. As we have discussed above,
they are generally massive. Therefore, their freeze-out temperature can be relatively high and their
relic abundances can be cosmologically important. We will discuss the phenomenology of
branons in laboratories and cosmology as a paradigmatic example of disformal scalar fields.
\section{Disformal fields at colliders}
\label{Coll}
Independently of the cosmological or astrophysical importance of disformal scalar fields,
they can be searched for in collider experiments. In fact, the parameters of the Lagrangian (\ref{lag})
suffer the constraints from present observations. On the other hand, they may be detected at the LHC
or in a future generation of colliders
\cite{Alcaraz:2002iu,Brax:2014vva,Coll,L3,Cembranos:2004jp,LHCDirect,Landsberg:2015pka,Khachatryan:2014rwa,BWRad}.
We will pay attention to the most sensitive searches that can be performed
at the LHC, but we will summarize a large amount of studies at the end
of this section and in Tables \ref{tabHad} and \ref{radcoll}.
The most sensitive production process for disformal particles in a proton-proton
collider, as the LHC, is gluon fusion giving a gluon in addition to
a disformal particle pair; and the quark-gluon interaction giving a
quark and the mentioned pair. These processes contribute to the
monojet $J$ plus transverse missing energy and momentum signature.
An additional and complementary process is the quark-antiquark annihilation giving a photon and a
pair of disformal particles.
In such a case, the signature is a single photon in addition to the transverse missing energy and momentum.
In order to show the constraints, we will restrict the study to a model with a degenerated spectrum of $N$ disformal fields with a common mass $M$. For simplicity, we will consider massless quarks except for the top case.
The cross-section of the subprocess $g g \rightarrow g \pi\pi$ was computed in
Refs. \citen{Cembranos:2004jp} and \citen{LHCDirect}:
\begin{eqnarray}
&&\frac{d\sigma(g g \rightarrow g\pi\pi)}{dk^2dt}=\frac{\alpha_s N
(k^2-4M^2)^2}{40960f^8\pi^2\hat s^3tu}\sqrt{1-\frac{4M^2}{k^2}} \nonumber \\
&&(\hat s^4+t^4+u^4-k^8+6k^4(\hat s^2+t^2+u^2)-4k^2(\hat
s^3+t^3+u^3))\,.
\end{eqnarray}
Here, $\hat s\equiv(p_1+p_2)^2$, $t \equiv(p_1-q)^2$, $k^2\equiv(k_1+k_2)^2$ and
$u\equiv(p_2-q)^2$. $p_1$ and $p_2$ are the initial gluon four-momenta, whereas
$q$ is the final gluon four-momentum. Finally, $k=k_1+k_2$ is the total branon four-momentum.
Thus, the contribution from this subprocess to the total cross-section for the process $p
p\rightarrow g\pi\pi$ can be written as
\begin{eqnarray}
\sigma_{gg}(p p\rightarrow g\pi\pi)= \int_{x_{min}}^1
dx\int_{y_{min}}^1 dy
g(y;\hat s) g(x;\hat s) \nonumber\\
\int_{k^2_{min}}^{k^2_{max}} dk^2 \int_{t_{min}}^{t_{max}}
dt\frac{d\sigma(g g\rightarrow g\pi\pi)}{dk^2dt},
\end{eqnarray}
where $g(x;s)$ is the gluon distribution function associated to the proton, $x$
and $y$ are the energy fractions of the protons carried by the two
initial gluons. The integration limits depend on the cuts used to define
the total cross-section. For example, the identification of a monojet forces to
consider a minimal value for its transverse energy $E_T$ and a pseudorapidity range:
$(\eta_{min},\eta_{max})$. It implies the limits
$k^2_{min}=4M^2$, $k^2_{max}=\hat s(1-2E_T/\sqrt{\hat s})$ and
$t_{min(max)}=-(\hat s-k^2)[1+\tanh{(\eta_{min(max)})}]/2$ . In addition, we can define
$x_{min}=s_{min}/s$ and $y_{min}=x_{min}/x$ where $s$ is the total center of mass energy
squared of the process. It has also a minimum value:
\begin{equation}
s_{min}=2E_T^2+4M^2+2E_T\sqrt{E_T^2+4M^2}.
\end{equation}
Similarly, the $q g \rightarrow q \pi\pi $ subprocess is characterized by the following cross-section:
\cite{Cembranos:2004jp,LHCDirect}
\begin{eqnarray}
&&\frac{d\sigma (qg\rightarrow q\pi \pi )}{dk^{2}dt} \nonumber \\
&=&-\frac{\alpha _{s}N}{2}\frac{(k^{2}-4M^{2})^{2}}{184320f^{8}\pi ^{2}\hat{s%
}^{3}tu}\sqrt{1-\frac{4M^{2}}{k^{2}}}(uk^{2}+4t\hat{s})(2uk^{2}+t^{2}+\hat{s}%
^{2})\,.
\end{eqnarray}%
In this case, $p_1$ and $p_2$ are the quark and the gluon four-momenta
respectively. $q$ is the four-momentum associated to the final quark.
$k_1$ and $k_2$ are again the branon four-momenta. The Mandelstam variables can
be defined as in the previous case.
The total cross-section for the reaction $p p\rightarrow q\pi\pi$ can be written as
\begin{eqnarray}
\sigma(p p\rightarrow q\pi\pi)= \int_{x_{min}}^1 dx\int_{y_{min}}^1
dy \sum_q
g(y;\hat s) q_p(x;\hat s) \nonumber\\
\int_{k^2_{min}}^{k^2_{max}} dk^2 \int_{t_{min}}^{t_{max}}
dt\frac{d\sigma(q g\rightarrow q\pi\pi)}{dk^2dt}\,.
\end{eqnarray}
Here, $x$ and $y$ are the energy fractions of the two protons
carried by the quark and the gluon associated to the subprocess,
and the limits of the integrals can be deduced as in the previous case
\cite{Cembranos:2004jp,LHCDirect}.
The above equations can be used to compute the total cross-section for
monojet production depending on the cut in the jet transverse energy $E_T$.
As we have commented, an interesting complementary signature is provided by the
single photon channel. The cross-section of the subprocess $q \bar q
\rightarrow \gamma \pi\pi$ was also computed in Refs. \citen{Cembranos:2004jp}
and \citen{LHCDirect}:
\begin{eqnarray}
&&\frac{d\sigma (qq\rightarrow \gamma \pi \pi )}{dk^{2}dt} \notag \\
&=&\frac{Q_{q}^{2}\alpha N(k^{2}-4M^{2})^{2}}{184320f^{8}\pi ^{2}\hat{s}%
^{3}tu}\sqrt{1-\frac{4M^{2}}{k^{2}}}(\hat{s}k^{2}+4tu)(2\hat{s}%
k^{2}+t^{2}+u^{2})\,.
\end{eqnarray}%
This analysis is simpler since it determines the only leading contribution:
\begin{eqnarray}
\sigma(p p\rightarrow \gamma\pi\pi)= \int_{x_{min}}^1
dx\int_{y_{min}}^1 dy \sum_q
\bar q_{ p}(y;\hat s) q_{ p}(x;\hat s) \nonumber\\
\int_{k^2_{min}}^{k^2_{max}} dk^2 \int_{t_{min}}^{t_{max}}
dt\frac{d\sigma(q q \rightarrow \gamma\pi\pi)}{dk^2dt}\,.
\end{eqnarray}
The above cross-sections allow to compute the expected number
of events produced at the LHC for the mentioned channels. This number
depends on the disformal coupling $f$, the disformal mass $M$, and the number of
disformal scalars $N$.
\begin{table
\tbl{
Summary of direct searches of disformal particles at colliders (results at the $95\;\%$ c.l.).
Monojet and single photon analyses are labeled by the upper indices ${1,2}$, respectively.
Present constraints and prospects for the LHC \cite{Landsberg:2015pka,Khachatryan:2014rwa,Cembranos:2004jp,LHCDirect}
are compared with current limits from LEP \cite{Alcaraz:2002iu,L3}, HERA and Tevatron \cite{Cembranos:2004jp}.
$\sqrt{s}$ means the center of mass energy associated with the total process;
${\mathcal L}$ denotes the total integrated luminosity;
$f_0$ is the limit on the coupling for one massless disformal particle ($N=1$)
and $M_0$ is the constraint on the disformal mass in the decoupling limit $f\rightarrow0$.
The effective disformal action (\ref{lag}) is not valid for energy scales
$\Lambda\gtrsim 4\,\pi^{1/2}f\,N^{-1/4}$ \cite{BWRad}. Thus, the $M_0$ is a limit that
cannot be reached but characterizes the sensitivity of the analysis for heavy disformal particles.
}
\begin{tabular}{||c|cccc||}
\hline Experiment
&
$\sqrt{s}$(TeV)& ${\mathcal
L}$(pb$^{-1}$)&$f_0$(GeV)&$M_0$(GeV)\\
\hline
HERA$^{\,1}
$& 0.3 & 110 & 16 & 152
\\
Tevatron-II$^{\,1}
$& 2.0 & $10^3$ & 256 & 902
\\
Tevatron-II$^{\,2}
$& 2.0 & $10^3$ & 240 & 952
\\
LEP-II$^{\,2}
$& 0.2 & 600 & 180 & 103
\\
LHC$^{\,2}
$& 8 & $19.6\times10^{3}$ & 440 & 3880
\\
\hline
LHC$^{\,1}
$& 14 & $10^5$ & 1075 & 6481
\\
LHC$^{\,2}
$& 14 & $10^5$ & 797 & 6781
\\
\hline
\end{tabular}
}\label{tabHad}
\end{table}
In addition to these channels, there are other constraints on the
disformal model parameters for tree-level
processes from analyses of other collider signatures. A summary of
these studies can be found in Table \ref{tabHad} \cite{BW2,Coll,Cembranos:2004jp,LHCDirect}.
In this Table, the present restrictions coming from HERA, LEP-II and Tevatron
are compared with the present bounds of the LHC running at a center of mass energy
(c.m.e.) of 8 TeV and the prospects for the LHC running at 14 TeV c.m.e.
with full luminosity. For the single photon channel, CMS has reported a
dedicated analysis for branons with a total integrated luminosity of 19.6 fb$^{-1}$,
whose results can be observed in Fig. \ref{CMSbranons}.
\cite{Khachatryan:2014rwa,Landsberg:2015pka}. Other missing energy and transverse momentum
processes, such as those associated with the monolepton channel analyzed in Ref. \citen{Brax:2014vva}, are
also potential signatures of disformal models. In the same reference, the authors
develop an interesting discussion of other different phenomenological signals of these models.
However, the important dependence of the interaction with the energy makes them subdominant with respect to the
collider constraints.
\begin{figure}[bt]
\begin{center}
\resizebox{8.5cm}{!}
{\includegraphics{CMSbranons.eps}}
\caption {\footnotesize Present constraints for one disformal scalar particle obtained by the CMS collaboration.
This figure is taken from a dedicated analysis for branon fields \cite{Khachatryan:2014rwa,Landsberg:2015pka}.
The results are compared with the restrictions coming from Tevatron \cite{Cembranos:2004jp} and LEP \cite{L3}.\label{CMSbranons}}
\end{center}
\end{figure}
On the other hand, it has been shown that disformal radiative corrections
generate new couplings among SM particles, which can be described by an
effective Lagrangian. The most relevant terms of these
effective interactions can be written as \cite{BWRad}:
\begin{eqnarray}\label{eff}
{\mathcal L}_{SM}^{(1)}\simeq \frac{N \Lambda^4}{192(4\pi)^2f^8}
\left\{2T_{\mu\nu}T^{\mu\nu}+T_\mu^\mu T_\nu^\nu\right\}\,.
\end{eqnarray}
The $\Lambda$ parameter is introduced when dealing with disformal radiative
corrections since the Lagrangian (\ref{lag}) is not renormalizable.
This parameter can be understood as a cutoff that limits
the energy range where the effective description of the disformal field
and SM particles is valid. In this case, from our approach, $\Lambda$ is
a phenomenological parameter, and $\Lambda/f$ parameterizes the intensity
of the quantum effects within the disformal theory. So, it quantifies the
relative importance of tree-level versus loop effects, which in principle, is unknown.
As it was shown in Ref. \citen{BWRad},
a perturbative treatment of the loop expansion is only consistent if $\Lambda\,<\,4\sqrt{\pi}fN^{-1/4}$.
In addition, one-loop effects computed
within the new effective four-fermion vertices coming from
(\ref{eff}) can be shown to be equivalent to the two-loop phenomenology
associated with the original action given by (\ref{lag}). For example,
this analysis can be used to obtain the contribution of disformal scalars
to the anomalous magnetic moment of the muon \cite{BWRad}:
\begin{equation}\label{gb}
\delta a_\mu \simeq \frac{5\, m_\mu^2}{114\,(4\pi)^4}
\frac{N\Lambda^6}{f^8}
\end{equation}
where $N$ is the number of disformal fields. On the other hand,
the most important disformal radiative effects in SM phenomenology
at colliders are related to four-fermion interactions or
different fermion anti-fermion pair annihilation into two gauge bosons
\cite{BWRad}. By considering current data, it is possible to constrain
the particular combination of parameters $f^2/(\Lambda N^{1/4})$.
Present results are shown in Table \ref{radcoll}, where it is also possible to find the
prospects for the LHC running at $14$ TeV \cite{BWRad} and current constraints obtained with data coming from HERA
\cite{Adloff:2003jm}, Tevatron \cite{d0} and LEP \cite{unknown:2004qh}.
An interesting conclusion of the full analysis is that the
present measurements of the anomalous magnetic moment define the
following preferred parameter region for the disformal model:
\begin{equation}
\text{6.0 GeV }\gtrsim \frac{f^{4}}{N^{1/2}\Lambda ^{3}}\gtrsim
\text{ 1.6 GeV (95\%\;c.l.)}\;,
\end{equation}
where we have updated the limits derived in Ref. \citen{BWRad} with the
current discrepancy between the measured and SM values
for the anomalous magnetic moment of the muon:
$\delta a_\mu=(26.1\pm 8.0)\times 10^{-10}$ \cite{muong2}.
As a consequence of this result and by taking into account the
prospects shown in Table \ref{radcoll}, the first signals of disformal
fields at colliders may be associated with radiative corrections \cite{BWRad}.
\begin{table}
\tbl{
Summary of virtual disformal field searches at colliders (at $95\;\%$ c.l.).
$\gamma\gamma$, $e^+e^-$ and $e^+p$ ($e^-p$) channels are denoted by The upper indices ${a,b,c}$,
respectively. The prospects for the LHC are compared with current constraints from HERA, LEP and Tevatron \cite{BWRad}.
The first two columns have the same information that in Table \ref{tabHad}. The third column shows the lower bound on
$f^2/(N^{1/4}\Lambda)$.}
{\begin{tabular}{||c|c c c||} \hline
Experiment & $\sqrt s$ (TeV) & ${\cal L}$ (pb$^{-1}$) &
$f^2/(N^{1/4}\Lambda)$ (GeV) \\ \hline
HERA$^{\,c}$ & 0.3 & 117 & 52 \\
Tevatron$^{\,a,\,b}$ & 1.8 & 127 & 69 \\
LEP$^{\,a}$ & 0.2 & 700 & 59 \\
LEP$^{\,b}$ & 0.2 & 700 & 75
\\ \hline
LHC$^{\,b}$ & 14 & $10^5$ & 383
\\ \hline
\end{tabular}}\label{radcoll}
\end{table}
\begin{figure
\begin{center}
\resizebox{10cm}{!}
{\includegraphics{forlilian2c.eps}} \caption {Parameter space limits for disformal radiative corrections for a model
with N = 1. In the (red) central area, the disformal model can account for the muon magnetic moment deficit,
being consistent with currente collider constraints (the most important one is associated with the the Bhabha scattering at LEP)
and electroweak precision observables. Prospects for the sensitivity of future colliders are also estimated.\cite{BWRad}
\label{FigRad}}
\end{center}
\end{figure}
\section{Relic abundances for disformal scalars}
\label{Rel}
We can calculate the thermal relic abundance associated to disformal scalars by
taking into account the disformal coupling given by Eq. (\ref{lag}) as it was
done in Refs. \citen{CDM,CDM2}. In this chapter, we will review the basic steps
of this computation by distinguishing
if the disformal field is relativistic (hot) or non-relativistic (cold) at decoupling.
The evolution of the number density $n_\alpha$ of the scalar $\pi^\alpha$,
$\alpha=1,\dots , N$ with $N$ the number of different types of disformal fields,
is given by the Boltzmann equation:
\begin{eqnarray}
\frac{dn_\alpha}{dt}=-3Hn_\alpha-\langle \sigma_A v\rangle
(n_\alpha^2 -(n_\alpha^{eq})^2)\,.
\label{Boltzmann}
\end{eqnarray}
Here,
\begin{eqnarray}
\sigma_A=\sum_X \sigma(\pi^\alpha\pi^\alpha\rightarrow X)
\end{eqnarray}
is the total annihilation cross-section of $\pi^\alpha$ into SM particles $X$.
$H$ is the Hubble parameter, and the $-3Hn_\alpha$ term takes into account the
dilution of the number density due to the expansion of the universe.
Under the disformal parity symmetry discussed in the introduction, they do not decay into other
particles. Therefore, these are the only terms which could change the number density
of disformal scalars. For simplicity, we will assume that all the disformal scalars
have the same mass. It implies that each disformal species evolves in an independent way,
and we will remove the $\alpha$ super-index.
The thermal average $\langle \sigma_A v\rangle$ of the total annihilation cross-section times the relative velocity can be written as
\begin{eqnarray}
\langle \sigma_A v\rangle=\frac{1}{n_{eq}^2}\int
\frac{d^3p_1}{(2\pi)^3}
\frac{d^3p_2}{(2\pi)^3} f(E_1) f(E_2)\frac{w(s)}{E_1 E_2}\,.
\end{eqnarray}
Here,
\begin{eqnarray}
w(s)=E_1 E_2 \sigma_A v_{rel}=\frac{s\sigma_A}{2}
\sqrt{1-\frac{4 M^2}{s}}\,.
\end{eqnarray}
In this case, the Mandelstam variable $s$ has the standard definition in terms of the four-momenta of the two branons $p_1$ and
$p_2$ as $s=(p_1+p_2)^2=2(M^2+E_1E_2-\vert \vec p_1\vert \vert \vec p_2\vert \cos\theta)$. We can assume a vanishing chemical potential
for the disformal scalars, whose distribution functions are given by the Bose-Einstein one:
\begin{eqnarray}
f(E)=\frac{1}{e^{E/T} + a}\,,
\end{eqnarray}
with $a=-1$. In the non-relativistic case $T\ll 3M$, the Maxwell-Boltzmann distribution ($a=0$) is a good approximation, which we will use.
The equilibrium abundance is just:
\begin{eqnarray}
n_{eq}=\int \frac{d^3p}{(2\pi)^3} f(E)\,.
\end{eqnarray}
The previous thermal average includes annihilations into all the SM particle-antiparticle pairs. If the temperature of the thermal bath is above the
QCD phase transition ($T>T_c$), we consider annihilations into quark-antiquark and gluons pairs. In the opposite case ($T<T_c$), we include annihilations into light hadrons. For numerical computations, we assume a critical temperature $T_c\simeq 170$ MeV although the final results are not very sensitive to the concrete value of this parameter.
To solve the Boltzmann equation is common to define $x=M/T$ and $Y=n/s_e$ with $s_e$ the entropy density of the entire thermal bath. We will assume that the total entropy is conserved, i.e. $S_e=a^3 s_e=\mbox{const}$, where $a$ is the scale factor. On the other hand, the Friedmann equation reads
\begin{eqnarray}
H^2=\frac{8\pi}{3M_P^2}\rho\,.
\end{eqnarray}
The energy density in a radiation dominated universe can be written as
\begin{eqnarray}
\rho=g_{eff}(T)\frac{\pi^2}{30}T^4\,,
\end{eqnarray}
whereas, the entropy density is given by
\begin{eqnarray}
s_e=h_{eff}(T)\frac{2\pi^2}{45}T^3\,.
\end{eqnarray}
Here, $g_{eff}(T)$ and $h_{eff}(T)$ mean the effective number of relativistic degrees of freedom contributing
to the energy density and to the entropy density respectively at a given temperature of the photon thermal bath $T$.
For $T\gtrsim 1$ MeV, $h_{eff}\simeq g_{eff}$. Using all these definitions:
\begin{eqnarray}
\frac{dY}{dx}=-\left(\frac{\pi M_P^2}{45}\right)^{1/2}
\frac{h_{eff}M}{g_{eff}^{1/2}x^2}\langle\sigma_A v
\rangle(Y^2-Y_{eq}^2)\,,
\label{YBoltz}
\end{eqnarray}
where we have neglected the contribution from derivative terms of the form $dh_{eff}/dT$. We can introduce the annihiliation rate as
$\Gamma_A=n_{eq}\langle\sigma_A v\rangle$. If $\Gamma_A$ is larger than the expansion rate of the universe $H$ at a given $x$, then
$Y(x)\simeq Y_{eq}(x)$, i.e., the abundance of disformal scalars is determined by the equilibrium one. On the contrary, as
$\Gamma_A$ decreases with the temperature, it eventually becomes similar to $H$ at some point $x=x_f$. From that
time on, disformal particles will be decoupled from the SM thermal bath and its abundance will remain frozen, i.e.
$Y(x)\simeq Y_{eq}(x_f)$ for $x\geq x_f$. For relativistic (hot) relics, the equilibrium abundance can be written as
\begin{eqnarray}
Y_{eq}(x)=\frac{45\zeta(3)}{2\pi^4}\frac{1}{h_{eff}(x)},
\;\;\;\;\; (x \ll 3)\,.\label{hot}
\end{eqnarray}
For non-relativistic (cold) particles:
\begin{eqnarray}
Y_{eq}(x)=\frac{45}{2\pi^4}\left(\frac{\pi}{8}
\right)^{1/2}x^{3/2}\frac{1}{h_{eff}(x)}\,e^{-x},
\;\;\;\;\; (x \gg 3)\,.\label{cold}
\end{eqnarray}
In the case of hot disformal fields, the equilibrium abundance is not very sensitive to the value of $x$. The situation is different for cold relics since $Y_{eq}$ decreases exponentially with the temperature. The sooner the decoupling takes place, the larger the relic abundance. First, we will discuss the relativistic decoupling. In such a case, the equilibrium abundance depends on $x_f$ only through $h_{eff}(x_f)$ and it is a good approximation to impose the condition $\Gamma_A=H$:
\begin{eqnarray}
H(T_f)=1.67\, g_{eff}^{1/2}(T_f) \frac{T_f^2}{M_P}=\Gamma_A(T_f)\,.
\label{Hubble}
\end{eqnarray}
It can be solved for $T_f$ by expanding $\Gamma_A(T_f)$ in the relativistic limit $T_f\gg M/3$.
Once $x_f$ is known, we can compute the current number density of disformal particles and its
corresponding energy density by taking into account Eq.(\ref{hot}) ($Y_{\infty}\simeq Y(x_f)$):
\begin{eqnarray}
\Omega_{D} h^2=7.83 \cdot 10^{-2} \frac{1}{h_{eff}(x_f)}
\frac{M}{\mbox{eV}}\,.
\label{eV}
\end{eqnarray}
For cold disformal fields, the computation of the decoupling temperature is a little more involved. We can use the well-known
semi-numerical result\cite{Kolb}:
\begin{eqnarray}
x_f=\ln\left(\frac{0.038\, c\,(c+2) M_P M \langle\sigma_A v
\rangle}{g_{eff}^{1/2}\,x_f^{1/2}}\right)\,,
\label{xf}
\end{eqnarray}
with $c\simeq 0.5$. The above equation can be solved iteratively with the result:
\begin{eqnarray}
\Omega_{D} h^2=8.77 \cdot 10^{-11} \mbox{GeV}^{-2}
\frac{x_f}{g_{eff}^{1/2}}\left(\sum_{n=0}^\infty
\frac{c_n}{n+1}x_f^{-n}\right)^{-1}\,.
\label{coldomega}
\end{eqnarray}
Here, we have explicitly used the standard expansion of $\langle \sigma_A v\rangle $ in powers of $x^{-1}$:
\begin{eqnarray}
\langle \sigma_A v\rangle =\sum_{n=0}^\infty c_n x^{-n}\,.
\end{eqnarray}
It means that the weaker the cross-section, the larger the relic abundance. This conclusion is expected since the sooner the decoupling takes place, the larger the relic abundance, and the decoupling occurs earlier for a weaker interaction. Therefore the cosmological bounds associated with the relic abundance are complementary to those coming from particle colliders. It implies that a constraint such as $\Omega_{D} < {\cal O}(0.1)$ means
a lower limit for the value of the cross-sections in contrast with the upper limit we discussed in the previous section from non observation effects in particle collisions.
The above formalism can be applied to obtain the relic abundance of disformal fields $\Omega_{D} h^2$, both for the relativistic and non-relativistic decoupling. For that purpose, we can use the thermal averages $\langle \sigma_A v\rangle $ summarized in \ref{Cross} and \ref{Thermal} and computed in Refs. \citen{CDM,CDM2}. The disformal coupling depends only on the nature of the concrete SM particle. We will take into account the annihilation of disformal fields into photons, massive $W^{\pm}$ and $Z$ gauge bosons, three massless Majorana neutrinos, charged leptons, quarks and gluons (or light hadrons depending on the temperature of the thermal bath) and a real scalar Higgs field, in terms of the brane tension $f$ and the generic disformal mass $M$.
\section{Constraints on disformal dark matter abundance}
\label{RelCon}
We shall discuss first the case of cold disformal fields. By taking into account the $c_n$ coefficients for the total cross-section summarized in \ref{Thermal}, the computation of the freeze-out value $x_f$ from Equation (\ref{xf}) is straightforward \cite{CDM,CDM2}. It allows to
evaluate the relic abundance $\Omega_{D}h^2$ from Eq. (\ref{coldomega}) in terms of $f$ and $M$. We can impose the observational limit on the total
cold DM (CDM) density from Planck: $\Omega_{D}h^2 < 0.126 - 0.114$ at the 95$\%$ C.L. \cite{Ade:2015xua},
and the results can be observed in Fig. \ref{coldplot}. \cite{CDM,CDM2}.
\begin{figure
\centerline{\psfig{file=coldlin2.eps,width=10.5cm}}
\caption{Exclusion limits for the abundance of disformal fields as cold DM.
The thick lines have associated an abundance of $\Omega_{D}h^2=0.126 - 0.114$ \cite{Ade:2015xua}
for $N=1$ and $N=7$. In this sense, the region above the curves ($\Omega_{D}h^2>0.126$) is excluded.
The area in between both lines contains the models with $1<N<7$.\cite{CDM,CDM2}\label{coldplot}}
\end{figure}
The validity of the cold (hot) relic approximation can be found also in Fig. \ref{Combined}, where we have plotted the $x_f=3$ line.
The excluded region by this argument is located between the two curves. On the other hand, the solid line shows explicitly
the values of the disformal model parameters where they can constitute the entire DM density of our universe.
For hot disformal particles, we can find the freeze-out temperature $T_f$ in terms of the coupling scale $f$
by using Equation (\ref{Hubble}). Indeed, there is an approximated power law relation between both quantities:
$\mbox{log}_{10} (f/1 \mbox{GeV}) \simeq(7/8)\mbox{log}_{10}(T_f/1 \mbox{ GeV}) + 2.8$ \cite{CDM,CDM2}. This relation is not very
sensitive to the number of disformal scalars and allows to obtain $\Omega_{D}h^2$ from (\ref{eV}).
We can consider two types of bounds. On the one hand, the constraints associated with the total DM of the universe
$\Omega_{D}h^2=0.126 - 0.114$ \cite{Ade:2015xua} (Fig. \ref{Totalhot}). On the other hand, there are more
constraining bounds on the amount of hot DM since the free-streaming effect associated to this type of DM reduces
the power of structures on small scales, modifying the shape of the matter power spectrum.
The present limit reads $\Omega_{D}h^2<0.0071$ at the 95$\%$C.L. and it is obtained
from a combined analysis of data coming from BAO, JLA and Planck \cite{Ade:2015xua}.
These limits are plotted in Fig.\ref{hot}. The important increase for $f\simeq 60$ GeV in Figs. \ref{Totalhot}
and \ref{hot} (and also in Fig. \ref{Combined})
is related to a decoupling temperature of $T\simeq 170$ MeV, that is the one we have assumed for the QCD phase transition.
Thus, the jump is associated with a sudden growth in the number of effective degrees of freedom
when entering in the quark-gluon plasma from the strong hadronic phase. The exact exclusion areas depend on the
number of disformal scalar fields. In Figs. \ref{Totalhot} and \ref{hot} is possible to see these limits for $N=1,2,3,7$.
\begin{figure
\centerline{\psfig{file=AB1237WM2.eps,width=10.5cm}}
\caption{Exclusion limits for $N=1,2,3,7$ disformal scalars from the total DM abundance corresponding to hot relics.
The shaded regions is associated with the total DM limit $\Omega_{D}h^2=0.126 - 0.114$ \cite{Ade:2015xua}
for a given $N$. The area above these region is excluded by overproduction of disformal DM.\cite{CDM,CDM2}\label{Totalhot}}
\end{figure}
The previous constraints assume that the disformal particles are relativistic
at freeze-out. If we require $x_f \ll 3$, the limits are not valid for $f<10^{-4}$ GeV.
The line $x_f=3$ is explicitly plotted in Fig. \ref{Combined} in the hot relic approximation.
In addition to $f\ll M_D$, the above discussion assume a standard cosmological evolution
up to a temperature around $f$. Indeed, the effective action (\ref{lag}) is only valid at low energies
in relation to $f$. This scale fixes the range of validity of the results. We have checked that the
previous computations are consistent with these assumptions. In particular, the decoupling temperature
is always smaller than $f$ in the allowed regions of the different figures.
\section{Disformal dark matter direct detection}\label{direct}
\label{Dir}
If we assume that the DM halo of the Milky Way is composed of disformal fields,
its flux on the Earth is of order $10^{5}(100$ GeV$/M )$ cm$^{-2}$s$^{-1}$. This
flux could be sufficiently large to be measured in direct detection
experiments such as DAMA, EDELWEISS-II, CoGeNT, XENON100 or CDMS II.
These experiments measure the rate $R$, and energy
$E_{R}$ of nuclear recoils \cite{LHCDirect}.
The differential counting rate for a nucleus with mass $m_{N}$ is given by
\begin{equation}
\frac{dR}{dE_{d}}=\frac{\rho _{0}}{m_{N} M}\int_{v_{\min
}}^{\infty }vf\left( v\right) \frac{d\sigma _{BrN}}{dE_{R}}\left(
v,E_{R}\right) dv\,.
\end{equation}
Here, $\rho_{0}$ is the local disformal density, $(d\sigma
_{DN}/dE_{R})(v,E_{R})$ is the differential cross-section for the
disformal-nucleus elastic scattering and $f(v)$ is the speed
distribution of the disformal particles in the detector frame normalized to unity.
\begin{figure
\centerline{\psfig{file=AN1237WM2.eps,width=10.5cm}}
\caption{Exclusion limits for hot relics in models with $N=1,2,3,7$ disformal scalar particles.
For different $N$, the curves shows the $\Omega_{D}h^2=0.0071$ DM abundance. The area above
the curves is excluded depending on $N$.\cite{CDM,CDM2}\label{hot}}
\end{figure}
The relative speed of DM particles is of order 100 km s$^{-1}$, so the elastic scattering is non-relativistic. Therefore,
the recoil energy of the nucleon in terms of the scattering angle in
the center of mass frame $\theta^{*}$, and the disformal-nucleus
reduced mass $\mu_{N}=Mm_{N}/(M+m_{N})$, can be written as \cite{LHCDirect}
\begin{equation}
E_{R}=\frac{\mu _{N}^{2}v^{2}\left( 1-\cos \theta ^{\ast }\right)
}{m_{N}}\,.
\end{equation}
On the one hand, the lower limit of the integral is given in terms of the minimum
disformal particle speed, which is able to produce a recoil of energy
$E_{R}$: $v_{min}=(m_{N}E_{R}/2\mu_{N}^{2})^{1/2}$. On the other hand,
the upper limit is not bounded, but it is interesting to note that a
value of $v_{esc}=$ 650 km s$^{-1}$ is standard for the local escape speed
$v_{esc}$ of WIMPs gravitationally captured by the Milky Way \cite{cerdeno}.
The total event rate of collisions of disformal particles with matter
per kilogram per day $R$, can be computed by integrating the differential event rate
over all the possible recoil energies. The threshold energy $E_{T}$ is
the smallest recoil energy that the detector is able to measure. In terms of $E_{T}$,
this total rate can be written as \cite{LHCDirect}:
\begin{equation}
R=\int_{E_{T}}^{\infty }dE_{R}\frac{\rho _{0}}{m_{N}M}
\int_{v_{\min }}^{\infty }vf\left( v\right) \frac{d\sigma
_{Br N}}{dE_{R}}\left( v,E_{R}\right) dv\,.
\end{equation}
The disformal-nucleus differential cross-section is determined by
the interaction described by Eq. (\ref{lag}). For a general DM candidate, its interaction with nucleus
is separated into a spin-independent (SI) and a spin-dependent (SD) contribution \cite{LHCDirect}:
\begin{equation}
\frac{d\sigma _{N}}{dE_{R}}=\frac{m_{N}}{2\mu _{N}^{2}v^{2}}\left(
\sigma _{0}^{SI}F_{SI}^{2}\left( E_{R}\right) +\sigma
_{0}^{SD}F_{SD}^{2}\left( E_{R}\right) \right)\,,
\end{equation}
where the form factors $F_{SI}(E_{R})$ and $F_{SD}(E_{R})$ account
for coherence effects. It includes a suppression in the event
rate for heavy WIMPs on nucleons and gives the dependence on the
momentum transfer $q=\sqrt{2m_{N}E_{R}}$. The spin-independent and spin-dependent cross-sections
at zero momentum transfer are $\sigma_{0}^{SI}$ and $\sigma_{0}^{SD}$ respectively.
These quantities depend on the nuclear structure and the isospin content, i.e.
the number of protons and neutrons \cite{kopp}. In the case of disformal particles,
the entire interaction is SI and is given by \cite{kopp}
\begin{equation}
\sigma^{\rm SI}=\frac{[Zf_p+(A-Z)f_n]^2}{f_p^2}\frac{\mu_{D
n}^2}{\mu_{D p}^2}\sigma_p^{\rm SI}\,,
\end{equation}
where $Z$ is the charge of the nucleus, $A$ is the atomic mass number,
$f_{p,n}$ is the SI DM coupling to proton and neutron respectively,
$\mu_{D p}$ is the reduced mass of the disformal particle-proton system,
and $\sigma_p^{SI}$ is the SI cross-section for scattering of the disformal
particle on a proton. In particular, disformal fields do not violate the isospin
symmetry if we neglect the proton-neutron mass difference. Within this
approximation: $f_{p}=f_{n}$, and $\mu\equiv\mu_{D n}=\mu_{D p}$.
Indeed, the disformal-nucleon cross-section $\sigma_{n}$ was computed in
Refs. \citen{CDM} and \citen{CDM2}:
\begin{equation}
\sigma_p^{\rm SI}=\sigma _{n}=\frac{9M^{2}{m_n}^{2}\mu ^{2}}{64\pi f^{8}}\,.
\end{equation}
Here, $m_n$ is the nucleon mass. Different direct search experiments
have reported possible candidate events associated with DM detection.
The DAMA/NaI and DAMA/LIBRA detectors at the Gran Sasso National Laboratory
found an annual modulation signature consistent with light WIMPs \cite{dama}.
Similar conclusions can be derived from data from the ultralow-noise germanium detector operated deep underground in
Soudan Underground Laboratory (CoGeNT) \cite{CoGeNT}. However, these measurements are in clear tension with constraints
of other experiments. For example, from detectors located in the same laboratories such as XENON100
(a liquid xenon detector at the Gran Sasso National Laboratory) \cite{xenon}, or CDMS II (a germanium and silicon detector at the
the Soudan Underground Laboratory) \cite{CDMSevents}.
The results for direct searches of disformal DM in terms of its mass
and the restrictions from present experiments are shown in Figure \ref{figJose}.
Curves of constant $f$ with 50 GeV separation are shown for reference. The regions on
the left of the $\Omega_{D}h^{2}=0.126 - 0.114$ lines are excluded
by overproduction of disformal particles. On the contrary, the right areas are compatible with
current observations. Such regions correspond to $f\gtrsim 120$ GeV and
$M\gtrsim 40$ GeV.
\begin{figure}
\begin{center}\includegraphics[width=9cm]{bfl2.EPS} \end{center}
\caption{Elastic nuclear cross-section for disformal scalars $\sigma_{n}$
in terms of its mass $M$. The two thick lines have associated the
$\Omega_{D}h^{2}=0.126 - 0.114$ abundances for cold disformal relics with $N=1$
(left) and $N=7$ (right). The shaded regions on the left are
excluded by colliders \cite{BW2,Cembranos:2004jp} for $N=1$ and $N=7$ respectively.
Current limits from direct searches are shown with solid lines.\cite{CDM,LHCDirect}\label{figJose}}
\end{figure}
\section{Indirect searches of disformal dark matter}
\label{Ind}
If DM is formed by disformal particles, two of them can annihilate into ordinary particles such as leptons, quarks and gauge bosons. Their annihilations from different astrophysical origins (galactic halo, Sun, Earth, etc.) produce fluxes of cosmic rays. Depending on the features of
these fluxes, they may be discriminated from the background through distinctive signatures. After the annihilation, the particle species that can be potentially detected by different experimental devices would be gamma rays, neutrinos and antimatter (fundamentally, positrons and antiprotons). In particular, gamma rays and neutrinos have the advantage of maintaining their original trajectory. On the contrary, analyses of charged cosmic rays are more involved due to galactic diffusion.
For example, the differential gamma-ray flux from annihilating DM particles is given, in general, by \cite{Vivi1}:
\begin{eqnarray}
\frac{\text{d}\,\Phi_{\gamma}^{\text{DM}}}{\text{d}\,E_{\gamma}} =
\frac{1}{4\pi M^2}\sum_i\langle\sigma_i v\rangle
\frac{\text{d}\,N_\gamma^i}{\text{d}\,E_{\gamma}}\, \times\, \frac{1}{\Delta\Omega}\int_{\Delta\Omega}\text{d}\Omega\int_{l.o.s.} \rho^2[(s)] \text{d}s\,.
\label{flux}
\end{eqnarray}
Here, the second term on the r.h.s. of this equation is commonly known as the astrophysical factor, which is defined by the integral of the DM mass density profile $\rho(r)$, along the line of sight ($l.o.s.$) between the source and the detector (divided by the detector solid angle). On the contrary, the first term depends on the nature of the DM particle, since $\langle\sigma_i v\rangle$ is the averaged annihilation cross-section times velocity of two DM particles into two SM particles (of a given type, labeled by the subindex $i$). In order to compute
the number of photons produced in each annihilating channel per energy $\text{d}\,N_\gamma^i/\text{d}\,E_{\gamma}$, it is necessary to take into account different SM decays and/or hadronization of unstable produced particles such as gauge bosons and quarks. Due to the non-perturbative QCD effects, the study of these particle chains requires Monte Carlo events generators such as PYTHIA \cite{PYTHIA}. In Ref. \citen{Ce10}, it was shown that the photon spectra for the SM particle-antiparticle channels can be written in terms of three different parametrizations. For leptons and light quarks, excluding the top, one can write:
\begin{eqnarray}
x^{1.5}\frac{\text{d}N_{\gamma}}{\text{d}x}\,&=&\, a_{1}\text{exp}\left(-b_{1} x^{n_1}-b_2 x^{n_2} -\frac{c_{1}}{x^{d_1}}+\frac{c_2}{x^{d_2}}\right) \nonumber\\
&+& q\,x^{1.5}\,\text{ln}\left[p(1-x)\right]\frac{x^2-2x+2}{x}\,;
\label{general_formula}
\end{eqnarray}
whereas for the top quark, the expression reads:
\begin{eqnarray}
x^{1.5}\frac{\text{d}N_{\gamma}}{\text{d}x}\,=\, a_{1}\,\text{exp}\left(-b_{1}\, x^{n_1}-\frac{c_{1}}{x^{d_1}}-\frac{c_{2}}{x^{d_2}}\right)\left\{\frac{\text{ln}[p(1-x^{l})]}{\text{ln}\,p}\right\}^{q}\,.
\label{general_formula_t}
\end{eqnarray}
On the other hand, for the electroweak gauge bosons ($W$ and $Z$):
\begin{eqnarray}
x^{1.5}\frac{\text{d}N_{\gamma}}{\text{d}x}\,=\, a_{1}\,\text{exp}\left(-b_{1}\, x^{n_1}-\frac{c_{1}}{x^{d_1}}\right)\left\{\frac{\text{ln}[p(j-x)]}{\text{ln}\,p}\right\}^{q}\,.
\label{general_formula_W_Z}
\end{eqnarray}
In these expressions, $x\equiv E_{\gamma}/M$ where $E_{\gamma}$ holds for the photon energy and $M$ is the mass of the disformal particle.
The particular values of the constants in the above equations can be found online \cite{Online} and in the original manuscript \cite{Ce10}.
For each channel, some of these parameters depend on $M$, whereas others are constant.
Finally, in order to calculate the gamma-ray spectra, we need to compute the total annihilation cross-section and its annihilation branching ratios into SM particle-antiparticle pairs. The annihilation cross-sections for disformal particles only depend on the spin and mass of the produced SM pairs. They can be found in Appendix C \cite{CDM,CDM2}. For instance, the leading term for non-relativistic disformal fields annihilating into a Dirac fermion $\psi$ with mass $m_\psi$ can be written as:
\begin{eqnarray}
\langle \sigma_{\psi} v\rangle\,=\,\frac{1}{16\pi^2f^8}M^2 m_\psi^2\left(M^2-m_\psi^2\right)\,\sqrt{1-\frac{m_\psi^2}{M^2}}\,;
\end{eqnarray}
for a massive gauge boson $Z$ (with mass $m_Z$):
\begin{eqnarray}
\langle \sigma_{Z} v\rangle\,=\,
\frac{1}{64\pi^2f^8}
M^2\,\left( 4\,M^4 - 4\,M^2\,{m_Z}^2 + 3\,
{m_Z}^4 \right)\,{\sqrt{1 - \frac{{m_Z}^2}{M^2}}}
\,;
\end{eqnarray}
whereas for a massless gauge field $\gamma$, the leading order is zero:
\begin{eqnarray}
\langle \sigma_{\gamma} v\rangle&=&0;
\end{eqnarray}
and, finally, for a (complex) scalar field $\Phi$ of mass $m_\Phi$:
\begin{eqnarray}
\langle \sigma_{\Phi} v\rangle\,=\,\frac{1}{32\pi^2f^8}
M^2\,{\left( 2\,M^2 + {m_\Phi}^2 \right) }^2\,
{\sqrt{1 - \frac{{m_\Phi}^2}{M^2}}}
\,.
\end{eqnarray}
It is interesting to note that disformal particles produce gamma-ray lines from direct annihilation into photons.
However, this annihilation is highly suppressed since it takes place in $d$-wave channel. Therefore, we will not search for the
monochromatic signal at an energy equal to the disformal mass. In the case of heavy disformal particles, the
leading annihilation channels are into $W^+ W^-$ and $ZZ$ (Fig. \ref{BR}) \cite{CDM,Vivi1}. In this case, the energy of the produced gamma rays could be in the range 30 GeV-10 TeV. These high-energy photons may be detectable by Atmospheric Cherenkov Telescopes (ACTs) such as MAGIC \cite{Mag, Mag11}. On the other hand, if the annihilation into electroweak gauge bosons is kinematically forbidden ($M<m_{W,Z}$), the leading annihilation channel is into the heaviest kinematically allowed quark-antiquark pair (Fig. \ref{BR}). For this case of light disformal DM, photon fluxes are more suitable to be detected by space-based gamma-ray observatories \cite{CDM,Vivi1,Ce10,indirect} such as EGRET \cite{EGRET} and FERMI \cite{Fer, FER}, with better sensitivities for an energy range between 30 MeV and 300 GeV.
\begin{figure}[bt]
\begin{center}
\resizebox{10cm}{!}
{\includegraphics{Fig11.eps}}
\caption {Branching ratios for disformal particles annihilatining into SM particle-antipaticle pairs. For heavy disformal scalars, the main contribution to the photon flux comes from the gauge bosons $W^+W^-$ and $ZZ$ annihilating channels. On the contrary, if
these channels are kinematically forbidden, the rest of contributions have to be computed.\cite{Vivi1}\label{BR}}
\end{center}
\end{figure}
With the above information, it is possible to estimate the current constraints and sensitivity for different targets and
detectors as shown in Figs. \ref{FER} and \ref{ACT} \cite{Vivi1}.
The resulting constraints on the total number of gamma-rays $N_\gamma \langle\sigma v\rangle$
does not depend on the number of disformal species since the proportional lower flux coming from the annihilation of a larger number of
disformal fields, is compensated by the associated higher abundance that a larger number of disformal species provides
(for a fixed coupling scale $f$).
By assuming the Planck relic density for disformal DM \cite{Ade:2015xua} and the standard astrophysical factors for different astrophysical sources such as dwarf galaxies or the galactic center, it is possible to obtain the straight lines in Figs. \ref{FER} and \ref{ACT},
which represent present constraints or expected sensitivity at $5\sigma$ for a particular target and detector. Fundamentally, present experiments (EGRET, FERMI or MAGIC) are unable to
detect signals from disformal DM annihilation. However, future experiments such as CTA may be able to detect gamma rays coming from such annihilations for disformal masses higher than 150 GeV for observations of the Galactic Center or Canis Major depending on the properties of the
DM density distribution.
\begin{figure}[t]
\begin{center}
\resizebox{10cm}{!}
{\includegraphics{Fig22.eps}}
\caption{Sensitivity and constraints on disformal DM annihilation into gamma rays coming from different targets.
Exclusion limits (at $5\sigma$ ) for satellite experiments (FERMI and EGRET) are shown with
straight lines. The (blue) thick line is associated to the fluxes assuming a DM abundance
$\Omega_{\text{CDM}} h^2 = 0.126 - 0.114$ \cite{Ade:2015xua}. The upper left corner is
excluded by collider searches, and figure shows the limit from $\text{LEP}$ and $\text{TEVATRON}$ experiments
for $N = 1$ and $N=7$ number of disformal particles.\cite{Vivi1}\label{FER}}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\resizebox{10cm}{!}
{\includegraphics{Fig33.eps}}
\caption {Similar to Fig. \ref{FER} but for current and future ground-based detectors.\cite{Vivi1}\label{ACT}}
\end{center}
\end{figure}
The above analysis about indirect searches of disformal DM are based on particular assumptions about DM distributions \cite{Cembranos:2005us}.
In particular, they assume profiles that are in good agreement with non-baryonic cold DM simulations, such as the standard NFW profile \cite{Navarro:1996gj}. However, when the baryonic gas is taken into account, it can modify the gravitational potential increasing the DM density
in the center of the halo. This conclusion has been reached in different studies \cite{Blumenthal, Prada:2004pi}, although there are
discrepancies \cite{Romano}. If this fact is correct, there are two important consequences for gamma-ray detection. On the one hand, the central accessible region is reduced to few tenths of a degree; on the other hand, the fluxes are enhanced by several orders of magnitude \cite{Prada:2004pi}. Indeed, the interpretation of very high energy (VHE) gamma rays from the Galactic Center (GC) is in good agreement with
disformal DM annihilation from these types of compressed dark halos \cite{ViviGC}. These fluxes have been observed by several collaborations, such as CANGAROO \cite{CANG}, VERITAS \cite{VER}, MAGIC \cite{MAG} or Fermi-LAT \cite{Vitale, ferm}, but we will discuss particularly on the data collected by the HESS collaboration from 2004 to 2006 relative to the HESS J1745-290 source \cite{Aha, HESS}.
\begin{table}
\tbl{Best fit parameters for the HESS J1745-290 data as gamma rays produced by disformal DM annihilation.
The dominant channels contributing are $W^+W^-$ and $ZZ$. The four fitted parameters can be found in this table.
The value of the disformal mass (TeV), the normalization factor of the signal
$C^2\equiv {\Delta\Omega\, \langle J_{(a)} \rangle_{\Delta\Omega}}/{(8\pi M^2)}$,
the normalitazion factor of the gamma-ray background $B\,(10^{-4}\, \text{GeV}^{-1/2} \text{cm}^{-1} \text{s}^{-1/2})$,
and its associated spectral index $\Gamma$. We also show the $\chi^2$ per degree of freedom (dof), and the astrophysical
factor in units of the one associated with a NFW profile: $b\equiv \langle J_{(2)} \rangle/\langle J^{\text{NFW}}_{(2)} \rangle$,
where $\langle J^{\text{NFW}}_{(2)} \rangle\simeq 280 \cdot 10^{23}\; \text{GeV}^2 \text{cm}^{-5}$.
$b$ is computed with the model fitted cross-section to Planck data
$\Omega_{\text{D}} h^2 = 0.126 - 0.114$ \cite{Ade:2015xua}:
$\langle \sigma v \rangle = (1.14\pm0.19)\cdot10^{-26}\; \text{cm}^{3} \text{s}^{-1}$.\cite{ViviGC}}
{\begin{tabular}{|c|c|c|}
\hline
\hline
$M$ (TeV) & $C\; (10^{-2}\, \text{GeV}\, \text{cm}^{-1} \text{s}^{-1/2})$ & $B\; (10^{-4}\, \text{GeV}^{-1/2} \text{cm}^{-1} \text{s}^{-1/2})$ \\
\hline
$50.6 \pm 4.5$ & $1.57 \pm 0.13$ &$5.27\pm2.32$ \\
\hline
\hline
$\Gamma$ & $\chi^2/\,$dof & $b$ \\
\hline
$2.80 \pm 0.15$ & 0.84 & $4843\pm1134$ \\
\hline
\hline
\end{tabular}}\label{DMBra}
\end{table}
The absence of temporal variability of these VHE events suggests a different emission mechanism than the IR and X-ray emission \cite{X}.
The source is very localized, in a region of few tenths of degree around the GC. On the other hand, the spectrum is characterized by
a cut-off at several tens of TeVs \cite{HESS}. The origin of these gamma rays is not clear. This flux may have been produced by particle
propagation close to the supermassive black hole Sgr A and the Sgr A East supernova remnant \cite{ferm,SgrA}, but the spectral features of
the data are perfectly consistent with the photons produced by the annihilation of disformal DM particles \cite{ViviGC} if it is complemented by
a background, which is well motivated by the radiative effects originated by particle acceleration in the vicinity of Sgr A East supernova and the supermassive black hole, as we have commented. Therefore, we can write the total flux as\cite{ViviGC}
\begin{equation}
\frac{d\Phi_{\text{Tot}}}{dE}=\frac{d\Phi_{\text{Bg}}}{dE}+\frac{d\Phi_{\text{DM}}}{dE}\,,
\label{gen}
\end{equation}
where we will assume a simple power-law
\begin{eqnarray}
\label{powerlaw}
\frac{d\Phi_{\text{Bg}}}{dE}=B^2 \cdot \left(\frac{E}{\mbox{GeV}}\right)^{-\Gamma}\;,
\end{eqnarray}
for the discussed background. This shape is motivated by the observations of the source
IFGL J1745.6-2900 by Fermi-LAT, that is spatially consistent with the HESS J1745-290 source \cite{Cohen,ferm}.
The background parameters, $B$ and $\Gamma$, can be also fitted from HESS data together with the disformal
DM mass $M$ and the astrophysical factor. We take into account a perfect efficiency and a experimental energy resolution
of $15\%$ ($\Delta E/E\simeq0.15$) \cite{ViviGC}. The disformal mass needed for fitting the data is around 50 TeV.
For this range of masses, as we have commented, the main contribution comes from the $ZZ$ and $W^+W^-$ annihilation channels,
producing a similar amount of $Z$, $W^+$ and $W^-$ bosons since
$\langle\sigma_{W^+W^-} v\rangle \simeq 2\langle\sigma_{ZZ} v\rangle \simeq M^6/(8\pi^2\, f^8)$
(we are assuming only one disformal species).
As soon as we know $M$, we can compute the coupling corresponding to the DM abundance in agreement with Planck data
\cite{Ade:2015xua}: $f=27.5 \pm 2.4$ TeV. Then, we can calculate the thermal averaged cross-section: $\langle \sigma v \rangle = \sum_{i=W,Z} \langle \sigma_i v\rangle= (1.14\pm0.19)\cdot10^{-26}\; \text{cm}^{3} \text{s}^{-1}$, and finally the astrophysical factor. In Table \ref{DMBra},
$\langle J_{(2)} \rangle$ is presented in units of the astrophysical factor associated with a standard NFW profile:
$b\equiv \langle J_{(2)} \rangle/\langle J^{\text{NFW}}_{(2)} \rangle$. The large mass of the disformal field in order to explain these data,
makes difficult to check its DM origin with collider or direct DM experiments. However, the study of other cosmic rays could be able to prove
or disprove the model \cite{ViviGCother}.
\begin{figure}[]
\begin{center}
\resizebo
{9cm}{!}
{\includegraphics{FigWZBraRIS2.eps}}
\caption {The total fit for the HESS J1745-290 data as gamma rays produced by disformal DM annihilation ($\chi^2/dof=0.84$).
The power law background with spectral index $\Gamma=2.80\pm0.15$ is shown with dotted line. The double-dotted line shows the
signal contribution with a $15\%$ of resolution uncertainity (R. U.) and normalization parameter $C=(1.57\pm0.24)\cdot10^{-7}\text{cm}^{-1}\text{s}^{-1/2}$.\cite{ViviGC}\label{BraGC}}
\end{center}
\end{figure}
\section{Nucleosynthesis constraints}
\label{BBNs}
On the other hand, there are astrophysical observations that can
restrict the parameter space of disformal scalars independently of
their abundance. For example,
one of the most successful predictions of the standard cosmological
model is the relative abundances of primordial elements. These abundances
are sensitive to several cosmological parameters and were used in Refs. \citen{CDM}
and \citen{CDM2}
in order to constrain the number of light branons. For instance, the production
of $^4$He increases with an increasing rate of the expansion $H$. From (\ref{Hubble}),
we can deduce that the Hubble parameter depends on the effective number of relativistic degrees of
freedom $g_{eff}$. Traditionally, this number has been parametrized in terms
of the effective number of neutrino species $N_\nu=3+\Delta N_\nu$ in the following way:
\begin{eqnarray}
g_{eff}(T\sim \mbox{MeV})= g_{eff}^{SM}+g_{eff}^{new}\leq
10.75+\frac{7}{4}\Delta N_\nu\,.
\label{neutrino}
\end{eqnarray}
Here $T\sim \mbox{MeV}$ means the thermal bath temperature
at nucleosynthesis. In the SM, $g_{eff}^{SM}(T\sim \mbox{MeV})\simeq 10.75$
corresponds to the degrees of freedom associated with the photon, the electron,
and the three neutrinos.
Recent Planck data (combined with BAO) implies $\Delta N_\nu < 0.71$ at the 95$\%$C.L \cite{Ade:2015xua}. If we include the contribution
of disformal particles, the number of relativistic degrees of freedom at a given temperature $T$ can be written as:
\begin{eqnarray}
g_{eff}(T)=g_{eff}^{SM}(T)+N\left(\frac{T_D}{T}\right)^4\,.
\label{geff}
\end{eqnarray}
Here $T_D$ is the temperature of the cosmic disformal brackground,
whereas $g_{eff}^{SM}(T)$ denotes the contribution from the SM particles.
We are assuming that there are no additional new particles.
If the disformal scalars are not decoupled at a given temperature $T$,
they share the same temperature as the photons: $T_D=T$. On the contrary,
if they are decoupled, its temperature will be in general lower
than the one of the radiation. We can compute it by assuming
that the expansion is adiabatic:
\begin{eqnarray}
h_{eff}(T)=h_{eff}^{SM}(T)+N\left(\frac{T_D}{T}\right)^3\,,
\end{eqnarray}
where $h_{eff}^{SM}(T)$ takes into account the SM contribution.
If at some time between the freeze-out of disformal fiels and nucleosynthesis,
some other particle species become non-relativistic while still
in thermal equilibrium with the photon background,
its entropy is transferred to the photons, but not
to the disformal particles which are already decoupled. Therefore,
the entropy transfer increases the SM bath temperature relative
to the disformal temperature. If the total entropy of particles in equilibrium with
the photons remains constant:
\begin{eqnarray}
h_{eff}^{eq} a^3 T^3=\mbox{constant}\,.
\end{eqnarray}
On the other hand, since the number of relativistic
degrees of freedom $h_{eff}^{eq}$ has decreased, then
$T$ should increase with respect to $T_D$.
Thus, we find:
\begin{eqnarray}
\frac{g_{eff}^{eq}(T_{f,\,D})}{g_{eff}^{eq}(T)}=\frac{T^3}{T_D^3}\,,
\end{eqnarray}
where $T_{f,\,D}$ is the disformal freeze-out temperature, and
for particles in equilibrium with the photons
$g_{eff}^{eq}=h_{eff}^{eq}$. The final constraint on the number of
massless disformal species $N$ can be set by using (\ref{geff}):
\begin{eqnarray}
\frac{7}{4}\Delta N_\nu\geq N\left(\frac{T_D}{T_{nuc}}\right)^4=
N\left(\frac{g_{eff}^{eq}(T_{nuc})}
{g_{eff}^{eq}(T_{f,\,D})}\right)^{4/3}\,.
\end{eqnarray}
If the disformal particles decouple after nucleosynthesis,
the bound can be written as:
\begin{eqnarray}
N\leq \frac{7}{4}\Delta N_\nu \label{N1}\,.
\end{eqnarray}
On the contrary, if they decouple before, we have $g_{eff}^{eq}(T_{nuc})=10.75$,
and we can rewrite the limit as:
\begin{eqnarray}
N\leq \frac{7}{4}\Delta N_\nu\left(\frac{g_{eff}(T_{f,\,D})}{10.75}\right)^{4/3}\,.
\label{N2}
\end{eqnarray}
By taking $\Delta N_\nu=0.71$, we can find a relation between the disformal freeze-out temperature $T_{f,\, D}$ and
the coupling scale $f$ that constrains the number of disformal fields $N$ (see Fig. 4).
For $f< 10$ GeV, the restrictions are really important: $N\leq 1$. This conclusion is derived by using Eq.(\ref{N1})
for $f<3$ GeV (corresponding to $T_{f,D}\lesssim 1$ MeV) or Eq. (\ref{N2}) otherwise. However the bounds are less restrictive
in the range $f\simeq 10-60$ GeV: $N \leq 3$. On the other hand, above the QCD phase transition, $f\simeq 60$ GeV,
the limit increases so much that the restrictions become extremely weak. In this case, we are assuming exclusively the SM
content: $g_{eff}^{SM}(T\gtrsim 300 \;\mbox{GeV})=106.75$.
\begin{figure
\centerline{\psfig{file=Nuclecd1.eps,width=10.0cm}}
\caption{Constraints on the number of massless disformal scalar fields $N$
from nucleosynthesis as a function of the disformal coupling $f$ \cite{CDM,CDM2}.\label{BBN}
}
\end{figure}
\section{Constraints from supernova SN1987A}
\label{Ste}
Astrophysical bounds on the coupling scale can be obtained from modifications of cooling
processes in stellar objects like supernovae \cite{Kugo,CDM,Brax:2014vva}. These processes take place by
energy loosing through light particles such as photons and neutrinos. However, if the
disformal mass is low enough, disformal particles are expected to carry a fraction of this energy,
depending on their coupling to the SM fields. In order to analyze the constraints on $f$ and $M$
imposed by the cooling process of the neutron star in supernovae explosions, we will estimate
the energy emission rate from the supernova core by studying the electron-positron pair annihilation.
Disformal scalars produced within the core can be scattered or absorbed again depending on the strength
of their interaction with SM particles. The disformal mean free path $L$ inside the neutron star
needs to be larger than the star size ($R\sim {\cal O} (10)$ Km) to be able to escape and contribute to
the cooling process. For a disformal scalar heavier than the supernova temperature $M\gg T_{SN}$,
we can estimate $L\sim (8\pi f^8)/(M^2 T_{SN}^4 n_e)$, where $n_e$ is the electron number density inside the star.
It means that the restrictions will apply only for $f\gtrsim 5$ GeV. The constraints arise because
the emitted energy in the form of disformal scalars could spoil the good agreement between the predictions for the
neutrino fluxes from supernova 1987A and the observations in Kamiokande II
\cite{kamiokandeII} and IMB \cite{IMB} detectors. The disformal emission could shorten
the duration of the neutrino signal with respect to the observed one, if the energy loss rate per
unit time and volume is $Q \gtrsim 5\times 10^{-30}$ GeV$^5$. For instance, the contribution of the mentioned electron-positron channel to the
volume emissivity can be written as \cite{CDM2}
\begin{eqnarray}
Q_{D}(f,M)&\equiv& \int\prod_{i=1}^2 \left\{\frac{d^3k_i}{(2\pi)^3
2E_i}2f_i \right\} (E_1+E_2)2s\,\,\sigma_{e^+e^-\rightarrow
\pi\pi}(s,f,M)\,.
\end{eqnarray}
Here, $i$ denotes the electron (1) and positron (2) particles (with negligible mass inside the supernova core).
The number density of electrons: $n_e\sim 1.4\times 10^{-3}\; \mbox{GeV}^3$, can be used to estimate the chemical
potential $\mu\sim (3 \pi^2 n_e)^{1/3}$, within the Fermi-Dirac distribution function $f_i=1/(e^{(E_i/T-\mu/T)}+1)$.
In such a case, $Q_{D}$ can be computed as \cite{CDM2}
\begin{eqnarray}
Q_{D}&=& \int_0^\infty dE_1\int_{M^2/E_1}^\infty dE_2\int_{-1}^{1
- 2 M^2/(E_1E_2)} d(cos)(E_1+E_2)
\nonumber\\
&&\frac{N[2E_1E_2\left(2E_1E_2(1-\cos)-4M^2\right)]^{5/2}\left(1-\cos\right)^{3/2}}
{{(2\pi)}^5\,7680\,f^8\,\left( 1 + e^{\frac{E_1-\mu}{T}} \right)
\,\left( 1 + e^{\frac{E_2+\mu}{T}} \right)}\,.
\label{Q}
\end{eqnarray}
\begin{figure
\centerline{\psfig{file=Tres3.eps,width=10.0cm}}
\caption{Contraints from supernova 1987A for a disformal scalar. The restrictions
are computed by estimating the disformal contribution to the cooling assuming
different supernova temperatures: $T_{SN}=10,\, 50,\, 70$ MeV. The solid lines
show the limits on the volume emissivity, whereas the dashed lines correspond
to the $L=10$ Km limits on the disformal mean free path \cite{CDM2}.\label{SN}}
\end{figure}
The integral in the angular variables can be performed analytically,
whereas the integral over the two energies is done numerically.
The final restrictions depend on the supernova temperature ($T_{SN}$)
and the number of disformal species ($N$). We have shown the limits
on $f$ and $M$ for $T_{SN}=30,\,50,\,70$ MeV and for $N=1$ in Fig. \ref{SN}
\cite{CDM2}. For disformal masses of the order of the GeV, the constraints on the coupling
disappear even for $T_{SN}=70$ MeV, due to the value of the mean free path.
\section{Non-thermal disformal dark matter}
\label{nonthermals}
In the previous sections we have considered disformal
scalars as thermal DM candidates, i.e. their
primordial abundances were generated by the thermal decoupling process
in an expanding universe. However, if the reheating temperature $T_{RH}$ after inflation was sufficiently low
then the disformal fields were never in thermal
equilibrium with the plasma. However, still there is
the possibility for them to be produced non-thermally, very much in the same ways as axions \cite{axions}
or other bosonic degrees of freedom \cite{Cembranos:2012kk}. Indeed,
if the disformal fields are understood as the pNGB associated to a global symmetry breaking from a
group $G$ to a subgroup $H$, then they will
correspond to the coordinates of the coset space $K=G/H$.
If $K$ is some compact space, we can denote its typical
size as $v$. In the axion case, $v$ would
correspond to the Peccei-Quinn scale $f_{PQ}$. On the
other hand, in the case of branons, it is possible to show \cite{nonthermal} that $v=f^2 R_B$ with $R_B$ the radius of the
compact extra space $B$.
Thus, if the reheating temperature was smaller than the
freeze-out temperature of the disformal fields, i.e.
$T_{RH}\ll T_f$, but larger compared to the explicit symmetry breaking scale $T_{RH}\gg (Mv)^{1/2}$, then the
disformal fields were essentially massless and
decoupled from the rest of matter fields.
In this case, we do not expect that after symmetry breaking, the initial value of the disformal field
would correspond to the minimum of the potential $\pi_0=0$,
but in general we would have $\pi_0\sim v$ within a region
of
size $H^{-1}$. The evolution of the $\pi$ fields
after symmetry breaking would then correspond to
that of a scalar field in an expanding universe, i.e., while $H\gg M$, the field is frozen in
its initial value $\pi=\pi_0$. When the temperature
falls below $T_i$ for which $3H(T_i)\simeq M$,
$\pi$ starts oscillating around the minimum with
a decaying amplitude. These oscillations correspond to a zero-momenum disformal condensate, whose energy
density scales precisely as that of CDM.
In principle, it would be possible that the disformal condensate could
transfer part of its energy to SM fields in $2\rightarrow 2$ process as those
discussed in previous sections. Asuming that $M\ll 1$ MeV and
that neutrinos are massless, it was shown in Ref. \citen{nonthermal} that
the condition to avoid the energy depletion is $H(T_{RH})\mbox{\raisebox{-.9ex}{~$\stackrel{\mbox{$>$}}{\sim}$~}} M$,
which translates into $T_{RH}\mbox{\raisebox{-.9ex}{~$\stackrel{\mbox{$>$}}{\sim}$~}} (MM_P)^{1/2}$ using the Friedmann equation in a radiation dominated universe. This condition automatically ensures $T_{RH}\gg (Mv)^{1/2}$.
\begin{figure*}[ht]
\centerline{\includegraphics[trim=0cm 8cm 2cm 8cm, clip,width=10cm]{disformal2.ps}}
\caption{Thermal vs. non-thermal disformal regions in the $f - M$ plane. The dashed (red) line separates the two regions
and corresponds to $T_i=T_f$ . The dotted (black) lines
correspond to
$\Omega_{D}h^2 = 0.126 - 0.114$ \cite{Ade:2015xua}
for different values of the $v$ scale.
The regions on the right of each dotted line would be excluded by
non-thermal disformal DM overproduction. }
\label{nonthermal}
\end{figure*}
Thus we see that in order for the condensate to form and survive until present, the reheating temperature should satisfy the condition $T_i\simeq (MM_P)^{1/2}<T_{RH}< T_f$. Therefore, if $T_i>T_f$ the interval disappears and only thermal relics are possible. In the opposite case $T_i<T_f$,
we can also have non-thermal production. As we show in Section 5, for light
disformal fields a good approximation for the freeze-out temperature
is $\mbox{log}(T_f/\text{GeV})\simeq (8/7)\mbox{log}(f/\text{GeV})-3.2$. In Fig. \ref{nonthermal} the $T_i=T_f$ line separating the thermal and non-thermal
regimes is plotted in the $(f,M)$ plane.
It is possible to obtain the present energy density of
the disformal field oscillations following the same
steps as in the axionic case. Assuming that $M$ does not depend on the
temperature, we find:
\begin{eqnarray}
\Omega_D h^2\simeq \frac{5 v^2 M T_0^3}{2M_P T_i\rho_c}\;,
\end{eqnarray}
with $T_0=2.75$ K the CMB temperature today and $\rho_c$ the
critical density. We can see in Fig. \ref{nonthermal}, that
for certain values of the $v$ scale, the above energy density can
be cosmologically relevant and in particular it could agree
with the measured value of the CDM abundance $\Omega_{D}h^2 = 0.126 - 0.114$ \cite{Ade:2015xua}.
\section{Conclusions}
\label{Con}
In this work, we have summarized the main phenomenology associated with disformal scalar particles.
Such a phenomenology is described by an effective action characterized by a dimension 8 interaction term with the SM
fields.
In order to keep the stability of the high dimensionality of the leading interaction against radiative corrections
it is necessary to introduce a distinctive pattern of symmetries. These disformal symmetries lead naturally to the
stability of the disformal scalars. We have started by showing these features within a particular model associated with
flexible brane worlds. In these scenarios, the brane tension scale $f$ is much smaller than the fundamental gravitational
scale in $D$ dimensions $M_D$. Within this framework, the relevant new low-energy phenomenology is associated with
new disformal scalars called branons, which are associated with the brane oscillations along the additional spatial dimensions.
\begin{figure}[h]
\centerline{\psfig{file=LogKeysoloplot1.EPS,width=11.0cm}}
\vspace*{8pt}
\caption{Combined exclusion plot for a model with a single disformal scalar from total and hot DM,
LEP-II \cite{Alcaraz:2002iu,L3} and
LHC \cite{Cembranos:2004jp,LHCDirect,Khachatryan:2014rwa,Landsberg:2015pka}
single photon events,
and supernovae cooling \cite{CDM2}.
The (blue) solid line on the right is associated with cold DM behaviour.
The two (red) solid lines on the left are associated with hot DM: the
thicker one corresponds to the total DM range $\Omega_{D}h^2=0.126 - 0.114$,
and the thin one is the hot DM limit
$\Omega_{D}h^2<0.0071$ \cite{Ade:2015xua}. The
dashed lines correspond to $x_f = 3$ for hot (upper curve) and cold (lower curve)
DM.
This figure is an updated version of the one obtained in Refs. \citen{CDM,CDM2}.
\label{Combined}}
\end{figure}
In this model, it is straightforward to deduce the corresponding effective action and the Feynman rules relative to the
couplings of branons with SM particles. They allow to compute the cross-sections and decay rates for different processes
relevant for disformal particles production and annihilation.
In particular, production rates are necessary to establish present constraints from different particle accelerators through the analysis of missing energy and transverse momentum.
For example, for electron-positron colliders, the most sensitive channel is the single-photon one. We have used the information
coming from LEP in order to get different exclusion plots on the disformal mass $M$, and coupling $f$.
We have also completed the study for future electron-positron colliders. In any case, the current most constraining limits come
from hadron colliders. We have taken into account data obtained by HERA and Tevatron, but recent observations by the LHC are the
most sensitive, in particular, the monophoton analysis by CMS.
There are complementary bounds on $f$ and $M$ coming from cosmology or astrophysics. It is interesting to notice that
the allowed range of parameters includes weak disformal coupling and large disfomal masses. Taking into account that disformal symmetries ensure the stability of the disformal scalars, they become natural DM candidates. Through an explicit calculation, we have shown that
the relic abundance relative to disformal particles can be cosmologically relevant and could account for the observed fraction of the abundance
in form of CDM \cite{CDM,CDM2}. From the commented effective low-energy action for massive disformal scalars, it is possible to compute the annihilation cross-sections of disformal pairs into SM particles. By solving the Boltzmann equation in an expanding universe, we can analyze the
disformal freeze-out and calculate the corresponding thermal relic abundances both for the hot and cold cases.
Comparing the results with the recent observational limits on the total and
hot DM energy densities, we have obtained exclusion plots in the
$f-M$ plane. Such plots are compared with the limits coming from collider
experiments. We conclude that there are essentially two allowed regions
in Fig. \ref{Combined}: one with
low disformal masses and weak disformal couplings corresponding
to hot relics, and a second region with large masses and not so weak couplings,
in which disformal particles behave as cold relics. In any case, the effective theory, which
describes the phenomenology of these disformal scalars are not valid for strongly coupled
fields. Indeed, disformal quantum effects can be parameterized by a cutoff $\Lambda$, that
limits the energy range of the model and establishes the importance of disformal loop effects
on SM phenomenology. It is interesting to note that disformal particles are able to improve the
agreement of the measured muon anomalous magnetic moment with the SM prediction (see Fig. \ref{forlilian1}).
In addition, there is an intermediate region where $f$ is comparable to $M$, which is precisely
the region studied in Refs. \citen{CDM} and \citen{CDM2}, and where disformal particles
could account for the measured cosmological DM.
This study can be also used to exclude different
regions of the parameter space of the model, where the disformal scalars are
overproduced or where they behave as hot DM avoiding a successful structure formation.
On the other hand, by using nucleosynthesis restrictions on the number of relativistic species,
we can impose an upper bound on the number of light disformal fields in terms of $f$.
If they decouple after the QCD phase transition ($f< 60$ GeV), the constraints are important
($N\leq 3$), but they become very weak otherwise. We have also discussed the possibility that
disformal scalars can contribute to the cooling of stellar objects. However, these restrictions
are not competitive with those coming from colliders.
Finally, apart from the thermal production, disformal fields can also be produced non-thermally by a similar mechanism to the axion misalignment. This fact allows to extend to lower masses the parameter space in which this kind of fields can play the role of DM.
\begin{figure}[bt]
\begin{center}
\resizebox{10cm}{!}
{\includegraphics{forlilian1d.EPS}}
\caption {The shaded area shows the parameter space of disformal DM with thermal relic in the range: $\Omega_{Br}h^2=0.126 - 0.114$,
and with a contribution to the muon anomalous magnetic moment: $\delta a_\mu=(26\pm 16)\times 10^{-10}$. The lower area is
excluded by single-photon processes at LEP together with monojet signals at Tevatron \cite{BW2,Cembranos:2004jp,LHCDirect}
and from the monophoton analysis at the LHC \cite{Khachatryan:2014rwa,Landsberg:2015pka} (intermediate area).
Prospects for the sensitivity at the LHC for real branon production are plotted also for the monojet analysis for a total integral
luminosity of ${\cal L}=10^5$ and total energy in the center of mass of the collision of $\sqrt{s}=14$ TeV. The explicit dependence on
the number of disformal fields $N$ is presented, since all these regions are plotted for the extreme values $N=1$ and $N=7$.\label{forlilian1}}
\end{center}
\end{figure}
\section*{Acknowledgments}
This work has been supported by the Spanish MICINNs Consolider-Ingenio 2010 Programme under grant MultiDark CSD2009-00064 and MINECO grant FPA2014-53375-C2-1-P and FIS2014-52837-P.
\newpage
|
1,116,691,500,769 | arxiv | \section{Introduction}
Over the past 15 years of the new century, TRACE, RHESSI, CORONAS-F, and
other spacecrafts have provided a wealth of data on nonstationary processes
in the Sun. The characteristics of X-ray sources and the relationship with
the microwave and optical emission of solar flares have been studied in
detail (Krucker et al., 2008). In the past 50 years it was believed that
these events occurred in the corona and were due to reconnection of the
magnetic field lines. However, the recent Hinode, SDO/HMI, and AIA
full-vector magnetic observations (Schou, et al., 2012; Lemen, et al.,
2012) and the theoretical extrapolation of photospheric magnetic fields to
the corona (Metcalf, et al., 2008) have shown that, in the course of
evolution of active regions, the magnetic-field energy is accumulated at
very low altitudes in the chromospheres, where the Lorentz forces operate,
and electric currents are amplified significantly. The free energy of the
currents is released at low altitudes giving rise to flares, coronal mass
ejections (CME), and ``sunquakes''. This makes us turn from the recently
introduced term ``coronal flare'' back to the notion of chromospheric
flares, which was widespread in the 1960-1970-ies. This opinion is
supported even by the advocates of the reconnection theory as a mechanism
of flares (Fletcher et al., 2011).
At the early stage of measurements of solar magnetic fields, A.B.Severny
(1988) showed that the sources of individual flare nodes in the
chromosphere are located in the vicinity of the polarity inversion line. He
and his co-authors showed also that these nodes arise in the areas of a
high-magnetic field gradient. The first full-vector measurements of AR
magnetic fields allowed them to study the relationship between the electric
currents and the process of evolution of solar flares. However, the
correlation between the distribution of the vertical magnetic field and the
flare nodes in the chromospheric H-alpha line proved to be very
complicated. Only now the problem discussed by A.B.Severny and D.Rust
(Rust, 1968) can be investigated more or less comprehensively using modern
observations of the magnetic-field dynamics, multi-wavelength observations
of flares, and new theoretical concepts of electromagnetic processes in the
Sun.
Recent studies of the relation between the evolution of magnetic fields and
the nonstationary processes are based on several earlier results. An
important finding was the emergence of new magnetic fields. The change in
the field configuration can result in a sudden release of energy or can
intensify the emission of impulsive flares along with the generation of
plasma motions (CME etc.) and EUV waves (sunquakes). The heating of plasma
in the chromosphere may be due to the dissipation of currents (Zaitsev and
Stepanov, 2008), to the Lorenz forces (Fisher, et al., 2012), or to the
energy loss of accelerated particles at the beginning of the gas-dynamic
response considered first by Kostyuk and Pikelner (1975). To understand the
origin of nonstationary processes, it is necessary to note that in some
cases before or during the flare a large-scale flux rope emerges usually in
the vicinity of the polarity inversion line leading to the occurrence of a
sigmoid flare (see (Hood et al., 2012, and references therein) and to
subsequent formation of post-eruption arcades.
Thus, the appearance of new theoretical and observational data required new
concepts of the development of MHD processes in the outer atmosphere of the
Sun. These concepts should take into account the direct effect of currents
on the heating and motion of plasmas in the chromosphere and its behavior
in the force-free and potential fields in the corona, including the
possibility of reconnection of the field lines and formation of thin
current sheets. Of primary importance in approaching this problem as a
whole is the simulation of the AR current system. In doing so, we must take
into account the topology of magnetic fields in the AR or in the complexes
of activity, which was studied first by Lee et al. (2010). A sudden
reconstruction of the magnetic configuration proves to be closely related
to the development of nonstationary phenomena. The role of shear motions in
such reconstruction was considered earlier (Matyukhin and Tomozov, 1991,
see following discussion in Prist and Forbes, 2000) usually for the
processes with reconnection in the corona. According to the numerical model
by Hood et al. (2012), the emergence of twisted flux ropes is indicative of
such reconstruction of the magnetic configuration and current system.
Of course, it is important to find observational evidence that would
support or disprove the new theoretical conjectures. Perhaps one of the
first publications in this context was the work by Sharykin and Kosovichev
(2015), where Figure 10 represents schematically a current system with
vertical currents $j_z$ of opposite signs on either side of the polarity
inversion line. Indeed, such a picture would be expected in the simplest
case of transition from the twisted flux rope to the flare post-eruption
phase. Now, however, there are numerous full-vector measurements in many
active regions where the distribution of the vertical current $j_z$ is
restored. As in the pioneering work by Severny, this distribution
throughout the AR does not correspond to the ideal pattern.
In this work we made an attempt to investigate the relationship between the
currents and the development of non-stationary processes on the example of
the events on 2012 May 10. The first flare with the maximum at 04:18~UT
(GOES) was unusual since the emergence of the new field in it was observed
in the umbra of the small spot with the simultaneous hard X-ray burst. This
event was accompanied by a sunquake. The second flare with the maximum at
05:10~UT was more typical, and the third flare (20:26~UT) was like the
first one, but less powerful. After the Introduction, in Section 2 we
analyse the emission of the first flare. In Section 3 the analysis of the
observations of the magnetic field emergence at 04:00~UT is given, in
Section 4 the behavior of the photospheric and coronal magnetic fields is
discussed, and in Section 5 the data are briefly compared with that on the
two subsequent flares. In the last section we discuss the question of the
origin and the possible model of sigmoid flares with the ejection of large
twisted magnetic ropes.
\section{Electromagnetic Emission in the Flare of May~10, 2012 (04:18~UT)}
AR 11476 appeared on the disk on May~5, 2012, and by May~10, its area
reached 1000 m.v.h. The corresponding sunspot group consisted of a large,
complex leading spot and minor spots at the center and in the tail part of
the group. On the day under discussion, the AR produced two large M-flares
and a series of weak flares of class C. Here, we consider the first flare
M5.7 (04:11--4:23, maximum at 4:18~UT, GOES) and then compare it with the
T7.9 flares that occurred at the decay of the first one and with the M1.7
flare observed in the same AR in the evening of that day (see Fig.~1).
\begin{figure}[!t]
\setcaptionmargin{5mm}
\begin{center}
\begin{tabular}{cc}
& \includegraphics[width=0.3\textwidth,clip=]{fig_1b.eps}\\
\raisebox{1cm}[0cm][0cm]{\includegraphics[width=0.7\textwidth,clip=]{fig_1a.eps}}
& \includegraphics[width=0.3\textwidth,clip=]{fig_1c.eps} \\
\end{tabular}
\end{center}
\caption{Left: diurnal GOES SXR time profiles (the lines show the flares
under examination). Right, at the top: the RHESSI HXR time profile-- count
rates above the background for the photon energy in the range 25 -- 50~keV
(black line) and 50 -- 100~keV (green line), at the bottom -- the flare
spectrum at 04:15:44~UT on May~10, 2012 where the background is shown as
magenda line. The flare emission is divided into the thermal (green line)
and nonthermal (orange line) components.}\label{fig1}
\end{figure}
The first flare was ordinary as to its X-ray and microwave emission. It
occurred in close proximity to a small spot at the center of the group near
the polarity inversion line, had impulsive nature, and was rather hard.
The left-hand part of Fig.~2 represents the entire AR 11476 in the white
light (SDO/HMI\textunderscore Ic). The rectangle on the right shows region
(A), where the flare was observed, and the squared region (B) is where a
change of sign of the signal was recorded on the line-of-sight magnetogram
(SDO/HMI\textunderscore LOS). The same flare in the 131 \AA\ line (SDO/AIA)
at the growth of its intensity is also shown on the right.
\begin{figure}
\setcaptionmargin{5mm}
\begin{center}
\begin{tabular}{cc}
&\includegraphics[width=0.45\textwidth,clip=]{fig_2b.eps}\\
\raisebox{0.1cm}[0cm][0cm]{\includegraphics[width=0.55\textwidth,clip=]{fig_2a.eps}}&
\includegraphics[width=0.45\textwidth,clip=]{fig_2c.eps}\\
\end{tabular}
\end{center}
\caption{Left: the sunspot group in AR 11476 at 04:12~UT
(SDO/HMI\textunderscore Ic), the rectangle is the selected region A. Right:
region A with the square region B for the same time (at the top); flare in
the 131 \AA\ line (SDO/AIA) in the vicinity of the maximum (at the bottom).
The scales show the distance from the disk center in arcsec.}\label{fig2}
\end{figure}
In general, judging by its X-ray emission recorded by GOES and RHESSI (Lin,
et al., 2002), this was a typical event somewhat harder than an average
flare of importance M5 as seen from Fig.~1 (the tilt of the photon spectrum
$\gamma\approx 3.0$ in the energy range of 20--80~keV, Fig.~1).
The microwave flux was also characteristic of the flares of this class with
a complex behavior of the polarized component (Stokes parameter V). The
time profiles of intensity (Stokes parameter I) and polarization (Stokes
parameter V) at the frequencies of 17.0, 9.4, 3.75 and 2.0~GHz based on the
Nobeyama polarimeter data (NoRP: Shibasaki et al., 1979;
Nakajima et al., 1985) are given on the left and at the center of Fig.~3.
The values and the dynamics of fluxes correspond to the majority of flares
of this intensity. As usual, there are two discernible footpoints of the
flare loop, located in the magnetic fields of the opposite signs (Fig.~3,
right).
\begin{figure}
\setcaptionmargin{5mm}
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth,clip=]{fig_3a.eps} &
\includegraphics[width=0.3\textwidth,clip=]{fig_3b.eps} &
\includegraphics[width=0.3\textwidth,clip=]{fig_3c.eps}
\end{tabular}
\end{center}
\caption{Evolution of the microwave flux from the M5.7 flare on May~10, 2012
(Stokes parameters I (left) and V (in the centre) at the frequencies of 2
GHz (red) 3.75~GHz (green), 9.4~GHz (blue) and 17~GHz (yellow) (Nobeyama,
NoRP).
Right -- a fragment of the LOS-magnetogram of the flare generation region
in the phase of maximum: the scales provide the distance from the disk
center in arcsec; the contours correspond to the emission at 34.0 GHz
(yellow line) and 17.0 GHz (red line) (Nobeyama, NoRH) at 04:15:30~UT;
these isophotes correspond 90\%, 70\%, 50\%, 30\% and 10\% of the
maximum at 17 GHz and 34 GHz. The sign X shows the flare occurrence site
according to HXR RHESSI data.}\label{fig3}
\end{figure}
On the other hand, the polarization demonstrates a rather complicated time
behavior. In particular, In particular, we have to be an inversion of the
polarized emission sign at the frequency of 9.4 GHz. Besides, a sudden
change of sign is observed at 2~GHz at 04:16~UT. Such a picture of
evolution of the polarized microwave emission may be associated with a
complex AR topology in the lower corona.
\section{Emergence of Magnetic Fields}
It is well known that in many cases, a relationship can be observed between
the occurrence of a flare and the emergence of a new magnetic field.
However these two processes do not usually coincide in time and space. In
other words, the emergence of a new magnetic field violates the stability
of the magnetic configuration. The flare in a given AR may occur far enough
from the emerging field, before or after the emergence or, even between its
separate episodes. In the case under consideration, the two processes
coincided in space and time. Besides that, though usually the new magnetic
field appears in the vicinity of the polarity inversion line separating the
fields of medium-high intensity, in our case, the emergence occurred in the
umbra of a small spot at the AR center (square B in Fig.~2). Here, the
change of sign of the signal (SDO/HMI\textunderscore LOS) is indicative of
the field emergence, but still it certainly does not mean the inversion of
the field sign.
We have analyzed the Stokes parameters recorded in the supposed emergence
area and obtained the Gaussian distribution. Of course, strong plasma
streams may also interfere with the field measurements. One of the
arguments for the reality of the field emergence is the fact that a similar
picture was observed during the flare recorded at the same site in the same
AR later in the evening at 20:26~UT.
The evolution of the emerging field in region B at some selected points of
time based on the \sloppy SDO/HMI\textunderscore LOS magnetograms is illustrated in
Fig.~4. The reversed sign of the signal was observed during 4 min., after
which the signal was restored to its former value during another 4 min.
Unfortunately, full-vector magnetic data are only available for the moments
before and after the emergence. Note that the upward directed motions are
registered according to the data of SDO/HMI in the frame 04:16:30~UT.
\begin{figure}
\setcaptionmargin{5mm}
\includegraphics[width=0.9\textwidth,clip=]{fig_4.eps}
\caption{Fragments of the image of the magnetic-field line-of-sight
component in the emergence region in AR NOAA 11476 (SDO/HMI\textunderscore
LOS) on May~10, 2012 at the selected moments during the flare. The
contours show the isolated local magnetic field along the polarity
inversion line. The scales provide the distance from the center of the disk
in arcsec.}\label{fig4}
\end{figure}
During the event under discussion, a sunquake was recorded in the vicinity
of the hard X-ray maximum (Buitrago-Cass, et al., 2015). Such a response to
the sudden energy release in the photosphere is projected onto the point at
the peak of a low flare loop. The asterisk (X) in the right-hand part of
Fig.~3 marks the position of the hard X-ray source at the maximum of the
burst. The isolines of radio brightness at 17.0 and 34.0 GHz superimposed
on the line-of-sight distribution of magnetic fields at the maximum of the
event are given according to the Nobeyama radio heliograph data (NoRH:
Nakajima et al., 1994). The footpoints of the low flare loop seen in
different SDO/AIA ranges are best pronounced at the high frequency of 34.0
GHZ. In this case, the sunquake is most likely bound to the time and site
of the primary energy release. This suggestion agrees with the sunquake
data available in (Sharykin and Kosovichev, 2015; Sharykin, Kosovichev, and
Zimovets, 2015)
\section{Extrapolation of Fields to the Corona and Distribution of Electric Currents in AR 11476}
In recent years, our understanding of the role of magnetic fields in the
development of activity in the outer atmosphere of the Sun has changed. It
was found that the magnetic beta (the ratio of the gas pressure to the
magnetic pressure), which is close to unity immediately above the
photosphere, becomes very small in the chromosphere and corona. As a
result, the plasma at the base of the active region is exposed to the
Lorenz forces, while higher the field becomes force-free. At the same time,
the evolution of fields in the chromosphere can lead to a significant
excess of the magnetic-field energy over the corresponding value of the
force-free (potential) field, and this free energy can be spent on the
development of nonstationary processes.
Full-vector magnetic observations in the photosphere allow us to
extrapolate the nonlinear force-free fields up to the corona. This method
called NLFFF (nonlinear force-free field) extrapolation is widely used in
heliophysics (e.g., see (Metcalf et al, 2008)).
In our work, for the calculation of the coronal magnetic field in the
non-linear force-free approximation an optimisation method (first proposed
in (Wheatland et al, 2000)) is employed in the implementation of
(Rudenko and Myshyakov, 2009). The core of the method is the subsequent transformation
of some initial (potential) distribution of the magnetic field toward the
force-free structure in accordance with the photospheric magnetogram. To do
this a functional of the following type is introduced:
We have performed calculations based on the solution of the
boundary-(Rudenko and Myshyakov, 2009) value problem for the nonlinear
force-free field using minimization of the functional introduced in:
\begin{eqnarray}
L=\int_V[B^{-2}\vert[\nabla\times\mathbf{B}]\times\mathbf{B}\vert^2+
\vert\nabla\cdot\mathbf{B}\vert^2\,] dV,
\end{eqnarray}
which equals $0$ if the field $B$ is force-free and is positive in the opposite case.
Thus, in the course of the solution of the functional (1) minimisation
problem, the field in the volume $V$ acquires the force-free configuration.
As the initial distribution we use a potential field calculated from the
normal component of the photospheric field via fast Fourier transform
(FFT). A remarkable feature of the optimisation method's realisation, which
we apply, is the use of the full system of the evolution equations of the
field (see (Rudenko and Myshyakov, 2009)).
The field lines in region A shown in Fig.~2 were calculated up to the
height of 25\,000~km. The result is represented in Fig.~5. One can see that
at the very beginning of the flare, the force lines around the field hill
in a small spot are clustering along the polarity inversion line, and the
flare nodes are formed in the region of their highest concentration (see
the EUV background on the negative image in Fig.~5). Generally speaking, a
bunch of force lines covering a significant part of the polarity inversion
line, particularly, where it separates the hills of strong magnetic fields,
can form in many flares. In our case, it can be supposed that a small bunch
was formed only on the south side of a small spot at the center of the
group. As in other sigmoid flares, such a shape of the field lines in AR
stimulates the development of sigmoid flare. In this case, the field
reconstruction occurred almost immediately after the formation of currents
over the neutral line.
\begin{figure}
\setcaptionmargin{5mm}
\includegraphics[width=0.95\textwidth,clip=]{fig_5.eps}
\caption{The background is a fragment of the inverted image of AR NOAA 11476
(SDO/AIA) in the EUV line 171 \AA\ obtained on May~10, 2012 at 04:12:00~UT.
The contour lines show the line-of-sight components of the magnetic field
of positive (white) and negative (black) sign in the flare region at 80\%,
60\%, 40\%, and 20\% of the maximum at the same time. The white line
presents force lines above the neutral line and the higher loops connecting
hills of the magnetic field of the opposite signs. The direct blue lines
show the projection cube with the height of 25\,000~km. The scales provide
the distance from the center of the disk in arcsec.}\label{fig5}
\end{figure}
Figure 5 shows only closed field lines in the selected region A in
projection on the plane of the sky. We can see a small spot of negative
polarity fringed with a bunch of field lines that continue further beyond
the neutral line. At present it is believed that this is where the main
force-free currents usually identified with the plasma flux rope are
flowing. It is important to note that this bunch of field lines coincides
with the position of the flare nodes forming a small sigmoid.
As shown by NLFFF-extrapolation, there are also field lines rising up to
5000 km and higher and connecting the small spot of negative polarity with
the hill of the field of opposite sign. They are located above the bunch of
field lines and perhaps prevent the ejection. In the course of the flare
evolution as inferred from SDO/AIA EUV data, the flare loops 10-20 thousand
km high were formed, but plasma did not escape into interplanetary space
(i.e., a CME did not occur as it follows from the radio data: there is no a
type II burst etc.).
Then at the photosphere level we have calculated the distribution of
vertical currents $j_z$ throughout the AR on May~10, 2012 for the time
interval from 03:48~UT to 05:48~UT using the full-vector magnetic data with
a step of 720 s. The analysis of the location of flares on that day
corroborates the correlation between the sites of occurrence of the flare
nodes and the areas of enhanced current (Grigor'eva et al., 2013). However,
no definite regularity was revealed in the behavior of the currents
throughout the AR and in the occurrence of flares. Namely, all currents are
concentrated along the neutral line, including the area near spots.
Therefore, the changes that may occur in the middle of the neutral line in
AR are not identified in the total current of one and the other sign.
Sharykin and Kosovichev (2015) drew attention to the fact that in an
``ideal'' current system, the mean currents of each sign must increase on
either side of the polarity inversion line. In our case, like in the
studies of the 1970-ies considering the relationship between the vertical
currents and flares, this effect is very weak in respect to the entire AR.
Since in the growth phase of the flare the largest changes in the magnetic
field are recorded at the site of occurrence of the sigmoid, we have
studied this effect in the selected region A (see Fig.~1). Figure 6
illustrates the distribution of the vertical currents in the photosphere
for two points of time. One can see that the currents of different sign
contact at the point $Y~256$, $X~-350$ arcsec. This area coincides with the
region of the field lines concentration in Fig.~5 above the point on the
polarity inversion line in the region of strong fields (see
Fig.~\ref{fig4}).
Generally, the evolution of the currents pattern is traced upon the set of
the $j_z$ maps constructed on the basis of the data on the full vector of
the field. The largest change of the currents takes place in the pointed
region of the sigmoid localization. The subsequent evolution of the current
system appears to be tightly bound with the development of numerous weak
flares in AR~11476. Note also that the substantial change of the currents
took place at 04:00--04:24~UT in the area $Y~250$, $X~-364$ arcsec which is
the footpoint of the loop connecting the place of the magnetic field
emergence with the hill of the positive polarity (see Fig.~\ref{fig5}).
Except the change of the currents pattern, the difference between two
images on left part of Fig.~\ref{fig6} is manifested in all the current
values, the fact which in this example is revealed even in the difference
of the scales.
\begin{figure}
\setcaptionmargin{5mm}
\begin{center}
\begin{tabular}{cc}
{\includegraphics[width=0.45\textwidth,clip=]{fig_6a.eps}}&\\
{\includegraphics[width=0.45\textwidth,clip=]{fig_6b.eps}}&
\raisebox{2cm}[0cm][0cm]{\includegraphics[width=0.55\textwidth,clip=]{fig_6c.eps}}\\
\end{tabular}
\end{center}
\caption{Left: (the background) fragments of the maps of vertical currents,
$j_z$, calculated for 04:12~UT and 04:24~UT, i.e., before and after the
flare maximum, respectively; (the contours) levels $j_z\pm0.05$~A/m$^2$ for
$j_z<0$ (red color) and $j_z>0$ (blue color). The scales show the distance
from the disk center in arcsec. Right: evolution of the mean current
$j_z>0$ and $j_z<0$ on May~10, 2012. For the magnetic field of negative
sign, the modulus of~$j_z$ is shown. The scale provides the time in the
form h:mm~(UT).}\label{fig6}
\end{figure}
To study this effect, we have determined the mean value of the vertical
current in the whole region (A). The averaging was done separately for the
pixels with the positive and negative fields. In the time interval from
04:00~UT to 05:48~UT in Fig.~\ref{fig6} one can observe the general trend
of the averaged vertical current $j_z$ of both signs. The whole trend
represents the fact that the flaring activity of this AR on that day was
gradually increasing.
The effect related to the flare in question is manifested as the increase
of the averaged currents during the flare 04:00~UT--04:24~UT. According to
the hard X-ray data, the maximum of the flare falls on 04:16:30~UT (see
Fig.~\ref{fig1}). In our case the maximum value corresponds to the time
04:24~UT and falls down afterwards. Such character of the change of $j_z$
is noticed in (Sharykin and Kosovichev, 2015) as well. The maximum value is small and
only slightly exceeds the value of 3~$\sigma$ (its $\sigma$ value of
approximately $2.3 \cdot 10^{-4}$~A/m$^2$ is determined from the change of the
currents in the area of the same active region in the quiescent state).
Note also that the flare maximum pointed out occured between 04:12~UT and
04:24~UT. Unfortunately, there is no freely accessed data on the full
magnetic field vector with better time resolution, therefore our effect of
the change of the currents' characteristics during this flare can be
somewhat underestimated.
\section{The Other Flares of May~10, 2012}
AR 11476, which we are discussing, displayed a high flare activity. Every
day since May 6 it produced a large number of weak C-flares and, on some
days until May 17, 1 or 2 M-flares.
In addition to the flare of importance M5.7 we are considering here, it
produced on the same day 16 C-flares and one flare of importance M1.7
recorded at 20:20~UT. A change in the signal intensity was recorded in the
umbra of the same spot as in the first M-flare according to the
line-of-sight field distribution (SDO/HMI\textunderscore LOS). The duration
of this effect was two times less than in the first flare (from ~20:24~UT
to 20:29~UT). However here, unlike the M5.7 flare, the signal on the
HMI\textunderscore LOS magnetogram in the emergence region did not change
the sign.
This M1.7 flare was also rather hard. The FERMI space mission recorded the
emission in the range of about 100 keV. At the same time (about 20:24~UT),
significant radiation fluxes in the energy range above 50 keV with a
maximum of more than 2000 pulses per second were recorded by the Suzaku
spacecraft (see the catalogue
\href{http://www.astro.isas.jaxa.jp/suzaku/HXD-WAM/WAM-GRB/solar/untrig/120510202227.html}{Suzaku Wide-band All-sky Monitor (WAM)}:
{\footnotesize{\underline{ http://www.astro.isas.jaxa.jp/suzaku/HXD-WAM/WAM-GRB/solar/untrig/120510202227.html }}}).
Thus, the conditions that lead to the events of the type of a ``fast''
sigmoid can arise in AR with a period from several hours to a day.
Most of the flares of class C on the day under discussion had a different
origin than the M-flares. Let us consider by way of example the flare T7.9,
which was observed at the decay of the soft X-ray emission from our
principal event. The flare occurred at 05:04~UT over the polarity inversion
line between the spots we are considering and the leading spots. Then, the
flare nodes extended from this point directly to the large spot. Such weak
events might be triggered by the activity in the latter. In some cases,
this activity is due to the emergence of a new field in the vicinity of the
spot or with its rotation, or simply with the formation of new magnetic
hills. Some of the weak events are rather hard, but they occur usually at
the decay of the soft X-ray emission from major flares. Their hardness is
due to the conditions favorable for particle acceleration in the traps
where a certain number of electrons with energies above 10 keV still remain
after the previous event (Vybornov et al., 2015).
\section{Results and Discussion}
Our conclusions are as follows:
1) In the event under examination, there actually occurred a pulse-like
energy release accompanied by the bursts of hard X-ray and microwave
emission. Simultaneously, a response of the photosphere was recorded in the
form of an sunquake. There is every reason to believe that at
the same time, the formation of a sigmoid flare began, but it was not
completed. Similar processes were repeated in the second M-flare at
20:20~UT, but were absent in many weak flares of class C in the other parts
of this AR.
2) Full-vector magnetic data were used to calculate the
vertical currents throughout the AR. A simple pattern of the current system
that consists of the loop currents above the neutral line (with allowance
for the shear) and the currents closed under the photosphere does not agree
with the results of calculations for first flare. However, if such
calculations are carried out in an area confined to the size of the flare
sigmoid, we will find local maxima in the time variation of both positive
and negative vertical currents. For the longer time interval, the time
variations of averaged vertical currents determines the overall evolution
of the flare activity in a given AR.
3) This example of moderate power
event has demonstrated at both the sunquake and particle acceleration are
more effective in sufficiently strong magnetic fields residing near the
polarity inversion line.
Let us briefly discuss the connection of our results to the gentral problem
of the development of non-stationary processes. The phenomenon of sigmoid
flares has been widely discussed in literature. A rich experience of
numerical simulation of these processes has been accumulated (e.g., see
(Hood et al., 2012, and references therein). Formerly, many authors adhered
to the idea of reconnection of magnetic-field lines. Now, it is believed
that the main factor is the dynamics of the currents, while the role of the
reconnection becomes significant at high altitudes in the corona. The
earlier conclusion that plasma flux ropes with the current are not ejected
directly from the top of the convection zone remains valid. Large-scale
sub-photospheric motions shear the feet of the field lines along the
polarity inversion line. In addition to that, the helicity is also carried
out from sub-photospheric layers.
So, the emergence of the shear and helicity creates conditions for the
formation of a magnetic flux rope. Both theoretical reasoning and
extrapolation of the nonlinear force-free fields to the corona suggest that
the evolution of the AR magnetic field described above must result in
accumulation of some amount of free energy in the chromosphere, which can
be spent on the development of nonstationary processes.
The very moment of a large impulsive flare is associated with the
reconstruction of the current system in AR. Namely, the bunch of force
lines and the currents along them change essentially approaching the ideal
configuration, i.e., currents along the loops, which are located in
projection onto the photosphere at a large angle to the neutral line.
After the flux rope is released, the magnetic configuration restores its
original state. If the previously existing large-scale sub-photospheric
motions continue, the formation of the flux rope is repeated, i.e., a
series of similar flares occur. These considerations agree with the results
of the latest numerical calculations (Savcheva, 2016).
One important issue is to figure out a mechanism of the flare. We note
that prior a flare the strong current in the magnetic rope has to be closed
under the photosphere. This suggests the toroid in the meridian plane,
partially risen above the photosphere. This geometry of the twisted
magnetic fiekd is given in Figure 2 in Titov and D\'emoulin (1999).
In fact, the flux ropes can exist for a long time even in active regions of
rather complex topology. However, in some cases, especially at geat
currents, a torus instability can arise, particularly in the presence of
strong currents (Shafranov, 1966). Conditions for the development of such
instabilities were determined in laboratory experiments (Myer et al.,
2015). Of course, for flares on the Sun and low-mass stars, we cannot
insist upon a specific type of instability, and more resonable to consider
a more general case of hydrodynamical instabilities in the plasma rope with
the strong current.
Simulation of the current system throughout the AR is an important problem.
Only now it has become clear that strong currents and large Lorenz forces
exist very low (up to 2-3 thousand km above the photosphere). Higher in the
corona, the fields rapidly become potential, and thin current sheets are
possibly present.
Another evidence in addition to the general considerations, is a complex
behavior of the polarization of microwave radiation. It is known that the
polarization changes its sign when passing from the disk center to the
limb. This is because the beam enters quasi-transverse magnetic field
(QT-region) (e.g., see Zheleznyakov, 1963; Peterova, 1973). In our case
(see Fig.~3) we see the inversion of the polarization sign at the frequency
of 9.4 GHz at the same time (without motion of the source).
In addition to this well-known effect, a secondary short-time inversion was
recorded at the frequency of 2 GHz with the maximum polarized emission at
04:16~UT. This, in turn, suggests a complex topology of the coronal
magnetic field, when a complex polarization picture is due both to the
projection effects, and to increasing complexity of the coronal topology.
Thus, an ``ideal'' current system can exist within a small volume. But
usually it is immersed into various independent magnetic fluxes separated
by separatrix surfaces. As a result, the particularities of the polarized
emission that appear at low frequencies in the microwave range indicate the
role of reconnection in the coronal layers in AR. Note that the foundations
for investigation of the topology of the solar corona were laid by Lee et
al. (2010).
Note also that two scenarios of events are possible in the Sun: (1) with
coronal mass ejection (CME), frequently observed, particularly, in major
flares and (2) with a rising loop without ejection (rarer events). The
latter was observed in the M-flares on May~10, 2012.
\phantomsection
\section*{Acknowledgments}
We are grateful to the reviewers and to V.I.~Vybornov for his assistance in
the course of the work. The SDO data are the courtesy of NASA and the HMI
and AIA science teams. We acknowledge the use of RHESSI, GOES and Fermi/GBM
data. We are grateful to the instrumental teams operating the Nobeyama
solar facilities and GOES satellites.
The work was supported by the Russian Foundation for Basic Research,
project nos.~14-02-00922, 16-32-00315, 15-32-20504, 15-02-03835,
15-02-01089, 15-02-01077, 16-02-00749, and particularly by Program no.~7 of
the Presidium of the Russian Academy of Sciences.
\phantomsection
\bibliographystyle{unsrt}
|
1,116,691,500,770 | arxiv | \section{\label{sec:Introduction} Introduction}
Optimization is integral to many scientific and industrial applications of applied mathematics including verification and validation, operations research, data analytics, and logistics, among others \cite{pardalos1987constrained, tsai2014optimization}. In many cases, exact methods of solution, including stochastic optimization and quadratic programming, are computationally intractable and novel heuristics are used frequently to solve problems in practice \cite{krentel1986complexity}. Quantum annealing (QA) offers a novel meta-heuristic that uses quantum mechanics for unconstrained optimization by encoding the problem cost function in a Hamiltonian \cite{farhi2000quantum,morita2008mathematical}. Recovery of the Hamiltonian ground state solves the original optimization problem and this approach has been mapped to a variety of application areas \cite{djidjev2018efficient, neukart2017traffic, stollenwerk2019quantum, martovnak2004quantum}. Several experimental efforts have realized quantum annealers \cite{johnson2011quantum, lanting2014entanglement, van_der_Ploeg_2007}, and application benchmarking of these systems has shown QA is capable of finding the correct result with varying probability of success \cite{katzgraber2014glassy, king2015benchmarking,zhu2016best,jarret2016adiabatic,o2018nonnegative,albash2018demonstration,ajagekar2020quantum}.
\par
QA performance depends implicitly on the complexity of the underlying problem instance as well as the controls that implement the heuristic \cite{venturelli2019reverse,quiroz2019robust}. Presently, there are multiple controls available to program quantum annealers that may each impact the observed probability of success. Notionally, the controls may be categorized as pre-processing, annealing, and post-processing methods. Whereas pre-processing controls define the encoded Hamiltonian and embedding onto the quantum annealer \cite{vinci2015quantum, bian2016mapping}, the annealing controls drive the time-dependent physics of the device and the underlying quantum state \cite{marshall2019power, venturelli2019reverse} while post-processing controls influence the read-out and decoding of the observed results \cite{pudenz2014error, pudenz2016parameter}. Collectively, the choice for each type of control may either enhance or impede the probability of reaching the encoded ground state and, therefore, impact the resulting solution state.
\par
Here we benchmark a selection of pre-processing and annealing controls available in a programmable quantum annealer \cite{johnson2011quantum} using a well-defined class of unconstrained optimization problems derived from the application of Markowitz portfolio theory \cite{markowitz1952portfolio}. As a variant of binary optimization, Markowitz portfolio optimization selects the subset of investment assets expected to yield the highest return value and minimal risk while staying within a total budget constraint \cite{markowitz1952portfolio, elsokkary2017financial}. We cast this problem which forms a complete graph as unconstrained optimization and benchmark the probability of success for QA to recover the global optimum. In particular, we benchmark the pre-processing and annealing controls available in the 2000Q, a programmable quantum annealer from D-Wave Systems \cite{johnson2011quantum}. This includes controls for mapping the logical problem onto hardware and scheduling the annealing process. We gather insight into the underlying dynamics using multiple measures of success tested across an ensemble of randomly generated instances of portfolio optimization.
\par
Previous research has benchmarked QA in comparison to classical heuristics for solving various optimization problems \cite{mcgeoch2013experimental, king2015benchmarking, steiger2015heavy}. In particular, several variations of portfolio optimization have been used to benchmark QA performance \cite{marzec2016portfolio, venturelli2019reverse, rosenberg2016solving}. Rosenberg et al.~demonstrated several encodings of a multi-period Markowitz portfolio optimization formulation to be solvable using a quantum annealer and found promising initial results in probability to find the optimal result \cite{rosenberg2016solving}. Venturelli et al.~benchmarked a similar mean-variance model of portfolio optimization using a hybrid solver that couples quantum annealing with a genetic algorithm \cite{venturelli2019reverse}. This hybrid algorithm was found to be 100x faster than forward annealing alone. In this work, we present a formulation of portfolio optimization to benchmark the behaviour of QA controls. We present studies focused on the variability in success with respect to available quantum annealing controls in an attempt to establish a methodology for finding an optimal set of controls which yield the highest solution quality \cite{pelofske2019optimizing, king2014algorithm}.
\par
The presentation is organized as follows. In Sec.~\ref{sec:Quantum Annealing}, we review quantum annealing and the the available controls. In Sec.~\ref{sec:Methods}, we provide an overview of the benchmarking methods and the use of Markowitz portfolio selection for problem specification. In Sec.~\ref{sec:Results}, we present the results from experimental testing using different quantum annealing controls with the 2000Q. We offer conclusions in Sec.~\ref{sec:level1}.
\section{Quantum Annealing \label{sec:Quantum Annealing}}
Under ideal conditions, forward annealing evolves a quantum state $\ket{\Psi(t)}$ under the time-dependent Schr\"{o}dinger equation
\begin{equation}
\label{Schrodinger Equation}
i \hbar \frac{\partial}{\partial t} \ket{\Psi(t)} = H(t) \ket{\Psi (t)} \hspace{1cm} t \in [0, T]
\end{equation}
where $T$ is the total forward annealing time and the time-dependent Hamiltonian is
\begin{equation}
\label{Adiabatic Evolution}
H(t) = A(s(t))H_{0} + B(s(t)) H_{1}.
\end{equation}
where $s(t) \in [0,1]$ is the control schedule and time-dependent amplitudes $A(s)$ and $B(s)$ satisfy the conditions $A(0)\gg B(0)$ and $A(1) \ll B(1)$. We consider the initial Hamiltonian $H_0 = - \sum_i^n \sigma_i^x$ as a sum of Pauli-$X$ operators $\sigma_i^x$ over $n$ spins. The final Hamiltonian $H_1$ represents the unconstrained optimization problem with a corresponding ground state that encodes the computational solution. We will consider below only problems represented using the Ising Hamiltonian
\begin{equation}
\begin{aligned}
\label{eq:Ising_Hamiltonian}
H_1 = \sum_{i} h_i \sigma_{i}^z + \sum_{i,j} J_{i,j} \sigma_{i}^z \sigma_{j}^z + \beta
\end{aligned}
\end{equation}
where $h_i$ is the bias on the $i^{th}$ spin, $J_{i,j}$ is the coupling strength between the $i^{th}$ and $j^{th}$ spin, $\sigma_{i}^z$ is the Pauli-$Z$ operator for the $i^{th}$ spin, and $\beta$ is a problem-specific constant. The Ising Hamiltonian is well known for representing a variety of unconstrained optimization problems \cite{lucas2014ising}.
\par
Instantaneous eigenstates at time $t$ are defined as
\begin{equation}
\label{instaneous_eigenstates}
H(t) \ket{\Phi_j(t)} = E_j (t) \ket{\Phi_j(t)}
\end{equation}
where $j$ ranges from $0$ to $N-1$ with $N=2^n$ the dimension of the Hilbert space.
For an initial quantum state prepared in the lowest-energy eigenstate at time $t=0$, i.e, $\ket{\Psi(0)} = \ket{\Phi_0(0)}$, adiabatic evolution under the Hamiltonian in Eq.~(\ref{Adiabatic Evolution}) to time $T$ will prepare the final state $\ket{\Psi(T)} = \ket{\Phi_0(T)}$ with high probability provided $T$ is sufficiently large. In particular, the evolution must be much longer than the inverse square of the minimum energy gap between the ground and first excited states \cite{farhi2000quantum}. At time $T$, the prepared quantum state is measured in the computational basis to generate a candidate solution for the encoded problem.
\par
Another variation of quantum annealing reverses the time-evolution process by beginning in an eigenstate of $H_1$. Known as reverse annealing, the initial quantum state evolves under Eq.~(\ref{Adiabatic Evolution}) in the reverse direction. The Hamiltonian starts as $H_1$ at time $t = 0$ and evolves backward to a point $s_{p}$ in the control schedule that corresponds to time $t_{1}$. The Hamiltonian then pauses for a time $t_p = t_{2} - t_{1}$ before evolving in the forward direction from the value $s_{p}$ at time $t_{2}$ back to the final Hamiltonian at time $T'$, where the latter time represent the reverse annealing time. The control schedule for reverse annealing is then defined as \cite{Yamashiro_2019, Passarelli_2020}
\begin{equation}
s'(t)= \begin{cases}
1 + \frac{(s_{p} - 1)}{t_1}t, & \ 0 \leq t \leq t_{1}\\
s_{p}, & t_{1} \leq t \leq t_{2}\\
s_{p} + \frac{(1 - s_{p})}{(T' - t_{2})} (t - t_{2}) & t_{2} \leq t \leq T'
\end{cases}
\label{eq: RA s(t)}
\end{equation}
\par
The differences in the control schedules of forward and reverse annealing are demonstrated in Fig.~\ref{fig:reverse_annealing}, where a linear reverse annealing schedule is compared to a linear forward annealing schedule using the amplitudes $A(s) = (1 - s)$ and $B(s) = s$. Notably, forward annealing controls increase monotonically with time whereas reverse annealing controls include a change in the direction of the control schedule where the ramp time from $s = 1$ to $s_p$ is $t_r = t_{1}$, the time paused at $s_p$ is $t_p$, and the quench time back from $s_p$ to $s = 1$ is $t_q = T^\prime - t_{2}$.
\begin{figure}[h!]
\centering
\includegraphics[width=85mm]{reverse_annealing.PNG}
\caption{The control schedule for reverse annealing (RA) compared to forward annealing (FA) plotted with respect to time. The control schedule for forward annealing starts at $t = 0, s = 0$ and anneals at a constant rate to $t = T, s = 1$, while the control schedule for reverse annealing starts at $t = 0$ with $s = 1$, decreases to a value $s_{p}$ at time $t_1$, pauses for time $t_p = t_2 - t_1$, and then increases to $s = 1$ at time $T'$.}
\label{fig:reverse_annealing}
\end{figure}
\subsection{Quantum Annealing Controls \label{sec:Controls}}
In practice, there are non-ideal behaviours that arise in practical implementations of quantum annealing. Equations (\ref{Schrodinger Equation})-(\ref{eq: RA s(t)}) represent quantum annealing under ideal adiabatic conditions that are difficult to realize in actual quantum devices. Real-world quantum annealers have limits in the ability to control the Hamiltonian and quantum dynamics \cite{pearson2019analog}. In addition, the presence of ill-characterized environmental couplings give rise to flux noise \cite{martinis2003decoherence}. The imperfect setting of the Hamiltonian parameters $(h, J_{i,j})$ by the analog control circuits gives rise to a small intrinsic control error \cite{king2014algorithm}. These errors undermine the accuracy of the physical hardware \cite{vinci2015quantum, pearson2019analog}. Finally, annealing too quickly may violate the essential adiabatic condition \cite{farhi2000quantum}, while annealing too slowly may lead to undesired thermal excitations of the quantum state due non-zero temperature fluctuations \cite{novikov2018exploring}. This multitude of effects complicates both the description of quantum annealing as well as the assessment of its performance.
\par
Given the implicit dependence on several competing factors, a variety of strategies have emerged for controlling quantum annealing to maximize probability of success in recovering the ground state and minimizing errors in the quantum computational solution. These control strategies include efficiently mapping the problem Hamiltonian onto the physical hardware Hamiltonian, tuning annealing schedule, applying variable transformations to mitigate control biases, and using reverse annealing to refine initial solutions \cite{king2014algorithm, Yamashiro_2019}.
\par
We investigate a subset of controls available in the D-Wave 2000Q, a programmable quantum annealer composed from an array of superconducting flux qubits operated at cryogenic temperatures \cite{Bunyk_2014}. The 2000Q consists of up to $2048$ physical qubits arranged in a sparsely connected array whose governing Hamiltonian is described by a time-dependent, transverse Ising Hamiltonian \cite{tichy2017quantum} for which with the Hamiltonian parameters in the device can be programmed individually. This enables a broad variety of computational problems, including portfolio optimization, to be realized. We briefly review some of the controls available to influence the success of solving these problems using quantum annealing.
\subsubsection{Problem Embedding \label{sec:Embedding}}
The Hamiltonian encoding the computational problem must be mapped into the physical hardware while satisfying the constraints of limited connectivity. The 2000Q hardware supports a sparse Chimera graph in which physical qubits are not fully connected but have average degree 6. A fully connected graph, like in Fig.~\ref{fig:embeddings}, must be mapped onto the more sparse Chimera graph. A single spin from the input Hamiltonian may be realized in hardware using multiple physical qubits that form a strongly interacting representative chain of spins. By judiciously choosing these chains and their interactions, the original input Hamiltonian may be constructed. This process, known as embedding, depends on the input problem as well as the target hardware connectivity. In general, embedding is NP-hard for arbitrary input graphs \cite{choi2008minorembedding}, and there are upper limits on the maximum graph that can be embedded \cite{klymko2012adiabatic}. For example, the largest fully connected problem that can be embedded onto the 2000Q has $\sim 60$ spins, while the limit in practice depends on the number of faulty/inactive physical qubits in the device.
\par
Embedding algorithms that optimize chain length may greatly reduce the number of physical qubits required by considering problem symmetry as well as the location of faults in the hardware. We highlight two embedding algorithms widely used in programming the 2000Q. The first method by Cai, Macready, and Roy is based on randomized placement and search for the weighted shortest path between spin chains \cite{cai2014practical}. This method, which we denote as CMR, applies to arbitrary input graphs but typically creates a distribution of chain lengths. By contrast, a second method by Boothby, King, and Roy based on a clique embedding typically generates shorter and uniform chain lengths of size
\begin{equation}
l_c = \frac{n}{4} + 1
\label{eq:clique_chain_length}
\end{equation}
for $n$ logical spins \cite{boothby2015fast}. A representative example of the output from these different methods is shown in Fig.~\ref{fig:embeddings} using a fully connected problem with $20$ logical spins. Both methods are available in the D-Wave Ocean software library \cite{embedding_tools}.
\begin{figure}[h!]
\centering
\includegraphics[width=60mm]{embeddings.PNG}
\caption{The embedding of a $20$ logical spin complete graph onto a Chimera graph structure. Figure $a)$ is complete $K_{20}$ graph which is fully connected with $20$ nodes and $190$ edges where each node represents a logical spin and each edge is a coupling between spins. Figure $b)$ is the CMR algorithm which requires the allocation of $23$ unit cells and $c)$ is the clique embedding algorithm which requires the allocation of $15$ unit cells. The nodes represent physical qubits, lines are the couplings between physical qubits, and each color is a different physical spin chain corresponding to a logic spin. }
\label{fig:embeddings}
\end{figure}
\par
Ensuring an embedded chain of qubits collectively represents a single logical variable requires an intra-chain coupling that is larger in magnitude than the the inter-chain couplings between chains. In other words, the chain of physical qubits must be strongly coupled to remain a single logical spin. However, it is possible for chains to become ``broken'' in so far as individual physical spins within the chain differ in their final state. In general, chain breaks arise from non-adiabatic dynamics that lead to local excitation out of the lowest energy state with longer chains more susceptible to these effects \cite{king2014algorithm, Dziarmaga_2005}.
\par
An additional control is required for decoding embedded chains to recover the computed logical spin state. In the absence of chain breaks, the logical value is inferred directly from the unanimous selection of a single spin state by every physical qubit. In the presence of chain breaks, several strategies may be employed to decide the logical value including majority vote \cite{king2014algorithm}, which selects the logical spin value as the value that occurs with the highest frequency in a chain.
\subsubsection{Spin Reversal \label{sec:Control_SpinReversal}}
Interactions between embedded chains arise from the required coupling between the logical spins. However, imperfections in the control of these spins lead to small biases that can become non-negligible for larger qubit chains and contribute to the complex dynamics describing the device. In turn, the probability for finding the expected ground state solution can decrease do to these bias errors. The influence of these errors on the computational result may be mitigated by using spin reversal transforms to average out biases.
\par
As a gauge transformation, spin reversal redefines the Hamiltonian by replacing the biases and couplings for a subset of spins with their negated value \cite{king2014algorithm, pelofske2019optimizing}. This transformation maintains the ground state of the logical problem. However, this transformation flips the sign of randomly selected qubits so that on average their bias is reduced. This strategy mitigates errors on individual spins by balancing the noise on the device prior to annealing \cite{pudenz2016parameter}. The number of selected spins as well as the parameter $g$ that defines the number of times to perform the transformation.
\subsubsection{Annealing Schedules \label{sec:Control_AnnealTime}}
Tailoring the annealing amplitudes $A(s)$ and $B(s)$ is perhaps the most direct method to control forward annealing. The annealing schedules control the rate of change of the $H(t)$, which must be sufficiently slow to approximate the adiabatic condition \cite{childs2001robustness}. An example of the amplitudes in a D-Wave 2000Q is shown in Fig.~\ref{fig:actual_schedules}. While forward annealing on the D-Wave 2000Q, $A(s(t)) >> B(s(t))$ at $t = 0$, $A(s(t))$ decreases and $B(s(t))$ increases for $0 < t < T$, and $B(s(t)) >> A(s(t))$ at $t = T$.
\begin{figure}[h!]
\centering
\includegraphics[width=85mm]{dwave_schedule.PNG}
\caption{The amplitudes of the D-Wave 2000Q over the range of control schedule as measured from $s = 0$ to $s = 1$ in increments of $0.001$.}
\label{fig:actual_schedules}
\end{figure}
\par
The optimal annealing time is problem dependent and inversely proportional to the minimum energy gap \cite{farhi2000quantum}, and, in general, the value and position of the minimum energy gap for a given $H(t)$ is typically unknown and hard to identify. Extending the annealing time $T$ arbitrarily long may not only be limited by hardware parameters but also be counter-productive due to competing thermal processes that depopulate the ground state \cite{pudenz2014error, albash2015decoherence}. There is an upper limit to the total job time $(N_s T \leq$ 1 s) as well as total annealing time $(T \leq 2$ s) on the D-Wave 2000Q.
\par
When reverse annealing, the three primary parameters for control are the initial state $e_{i}$, the pause point $s_p$, and the pause duration $t_p$. The times $t_r$ and $t_q$ can also be manipulated, but we keep these constant and symmetric for our experiments. Reverse annealing uses $e_{i}$ to set the initial quantum state, which may be based on the output of forward annealing, a heuristically computed candidate, a random state or other methods. Our experiments use a pre-computed initial state, e.g., using forward annealing. An iterative procedure is then used which replaces the $e_i$ of each subsequent reverse annealing sample with the output from previous reverse annealing iteration.
\subsection{Quantum Annealing Metrics}
We characterize quantum annealing using the probability of success
\begin{equation}
p_{s} = |\langle\Phi_{0}(T)|\rho|\Phi_{0}(T)\rangle|^2
\label{eq: p_s intro}
\end{equation}
defined as the overlap of the final, potentially mixed quantum state $\rho$ prepared by QA with the pure-state describing the expected computational outcome $\Phi_{0}(T)$. Empirically, the probability of success is estimated from the frequency with which the observed solution state matches the expected outcome. When the expected ground state solution is known, we define the statistic $\delta_{i} = 1$ if the $i$-th sample matches the known ground state and $\delta_{i} = 0$ if it does not. For the $k$-th problem Hamiltonian instance, the estimated probability of success is then defined as
\begin{equation}
\tilde{p}_{s}^{(k)} = \frac{1}{N_s}\sum_{i=1}^{N_s}{\delta_{i}}
\end{equation}
where $N_s$ is the total number of samples. The average over an ensemble of $N_{p}$ problem instances is defined as
\begin{equation}
\tilde{p}_{s} = \frac{1}{N_p}\sum_{k}^{N_p}{ \tilde{p}_{s}^{(k)}}.
\end{equation}
\par
A second metric for characterizing quantum annealing performance, and especially the non-adiabatic dynamics, is the number of chain breaks observed in the recovered solution samples. As noted above, a chain break is observed when the chain of physical qubits embedding a logical spin has more than one unique spin value. We estimate the probability of chain breaks for a problem instance
\begin{equation}
\begin{aligned}
\tilde{p}_{b}^{(k)} = \frac{1}{N_s}\sum_{i=1}^{N_s}{\epsilon_{i}}
\end{aligned}
\end{equation}
where the statistic $\epsilon_{i} = 1$ when the $i$-th sample solution contains at least one broken chain for any of the logical spins and $\epsilon_{i} = 0$ when no embedded chain is broken. The average probability of chain breaks over an ensemble of $N_{p}$ problem instances is then defined as
\begin{equation}
\begin{aligned}
\tilde{p}_{b} = \frac{1}{N_p}\sum_{k}^{N_p}{ \tilde{p}_{b}^{(k)}}.
\end{aligned}
\end{equation}
It is important to note that the effects of chain breaks can be mitigated by post-processing methods, such as majority vote, which make hard decisions on the logical spin value.
\par
While the above metrics quantify the probability with which quantum annealing recovers the correct solution, additional information about computational performance comes from the distribution of all solution samples obtained. In particular, the distribution over sample energies provides a representation for the weight of errors in the solution samples. A distribution concentrated around the lowest energy indicates a small number of errors in the computed solutions, while a broad or shifted distribution hints at a larger number of errors. We denote the energy computed from the $i$-th solution sample as $E(i)$ and we define the $j$-th energy bin as $h_j$. The bin $h_j$ counts the number of samples with an energy in the range $[j, j+1]\Delta$ where $\Delta$ controls the granularity of binning the energies. The resulting set $\{(j\Delta, h_{j})\}$ approximates the energy distribution of the sampled solutions.
\section{\label{sec:Methods} Benchmarking Methods}
We benchmark performance of the quantum annealing controls presented in Sec.~\ref{sec:Quantum Annealing} using a variant of constrained optimization derived from Markowitz portfolio theory. We recast this problem as unconstrained optimization before reducing to quadratic unconstrained binary optimization (QUBO) form. The latter form is easily translated to the classical Ising spin Hamiltonian and, subsequently, to the problem Hamiltonian defined by Eq.~(\ref{eq:Ising_Hamiltonian}).
\subsection{\label{sec:Markowitz}Markowitz Portfolio Selection}
Portfolio optimization selects the best allocation of assets to maximize expected returns while staying within the budget and minimizing financial risk. The Markowitz theory for portfolio selection focuses on diversification of the portfolio for risk mitigation \cite{markowitz1952portfolio}. Instead of allocating high percentages of a budget toward assets with the highest projected returns, the budget is distributed over assets that minimize correlation between the asset's historical prices. In this model, the covariance between purchasing prices serves as a proxy for risk in which positively correlated assets are considered to be more risky. We review the methods by which the benchmark problems are generated and solved in this section.
\par
We consider Markowitz portfolio optimization as a quadratic programming problem that determines the fraction of available budget $b$ to allocate toward purchasing assets with the goal of maximizing returns while minimizing risk. By selecting a partition number $w$, the fraction $p_w = \frac{1}{2^{(w -1)}}$ represents the granularity of the partition. The portfolio optimization problem selects how many of those partitions to allocate toward each asset with an integer $z_u$. Thus, the fraction of $b$ to invest in each $u^{th}$ asset is given by $p_w b z_u$, and portfolio optimization identifies how much of the $m$ assets to select given the budget $b$ and a risk threshold $c$. Thus, portfolio selection is cast as
\begin{maxi}
{z}{\sum_{u = 1}^{m} r_{u} z_{u}}{}{}
\addConstraint{\sum_{u = 1}^{m} p_w b z_{u} = b}
\addConstraint{\sum_{u,v = 1}^{m} c_{u, v} z_u z_v \leq c}
\label{eq:MPO_classic}
\end{maxi}
%
where for the $u^{th}$ asset $r_u$ is the expected return and $c_{u,v}$ is the historical price correlation between assets $u, v$.
\par
In Eq.~(\ref{eq:MPO_classic}), the first term represents maximization of the expected returns over the available assets. There are many methods for forecasting expected returns, e.g., based on market price, expert judgement, and historical price data \cite{huang2012mean, martin2017expected}. For simplicity, we model expected returns
as
\begin{equation}
r_{u} = p_w \bar a_u
\end{equation}
where $\bar a_u$ is the average of $a_{u}$, the history of price data for the $u^{th}$ asset. The first constraint in Eq.~(\ref{eq:MPO_classic}) places a hard constraint on the total allocation of assets to sum to $b$. We emphasize that this constraint penalizes portfolios that do not allocate the entire budget as well as those that over commit. Finally, the second constraint accounts for diversification by asserting that the sum of covariance between asset prices $c_{u,v}$ be less than or equal to the risk threshold $c$. The historical price covariance is calculated as the correlation between pairs of assets by comparing the $p_w$ fraction of each asset's historical price data. Here covariance is defined as
\begin{equation}
\begin{aligned}
c_{u, v} = \frac{p_w^2 \sum^{N_f}_{l = 1}( a_{u,l}- \bar a_u)( a_{v,l}- \bar a_v)}{N_f -1}
\end{aligned}
\end{equation}
where $a_{u,l}$ is the $l^{th}$ historical price value for asset $u$ and $N_f$ is the number of price points in the historical data.
\par
We solve this variation of Markowitz portfolio selection using quantum annealing by casting the formulation in Eq.~(\ref{eq:MPO_classic}) into quadratic unconstrained binary optimization (QUBO).
We express the integer variable $z_u$ as a $w$-bit binary expansion
\begin{equation}
z_u = \sum_{k = 1}^{w}{2^{k-1} x_{i(u,k)}}
\label{eq:convert}
\end{equation}
with $x_i \in \{0,1\}$ and the composite index $i(u,k) = (u-1)w + k$. The expected returns are then expressed as
\begin{equation}
{r_{u} z_{u}} = { \sum_{k=1}^{w} 2^{k - 1} r_u x_{i(u,k)}}
\end{equation}
while the allocation constraint becomes the penalty term
\begin{align}
-\big(\sum_{u=1}^{m} \sum_{k=1}^{w} 2^{k - 1} p_w b x_{i(u,k)} - b \big)^2
\label{eq:norm_prices}
\end{align}
We consider a correlation threshold $c=0$ such that the correlation constraint becomes
\begin{align}
\sum_{u,v}^{m} c_{u, v} z_u z_v = \sum_{u,v}^{m} \sum_{k, k^{\prime}}^{w} 2^{k - 1} 2^{k^{\prime} - 1} c_{u,v} x_{i(u,k)} x_{ j(v,k^{\prime})}.
\label{eq:norm_prices}
\end{align}
Our formulation of Markowitz portfolio selection as an unconstrained optimization problem then becomes
\begin{maxi}
{x}{{\theta_1 \sum^{n}_{i} r_{i} x_{i}}
\breakObjective{- \theta_2(\sum^{n}_{i}2^{k - 1}b p_w x_{i} - b)^2}
\breakObjective{- \theta_3 \sum^{n}_{i, j} c_{i, j} x_{i} x_{ j}}}{}{}
{\label{eq:MPO_unconstrained}}
\end{maxi}
where the problem size $n = mw$, $r_i = 2^{k - 1} r_u$, $c_{i, j} = 2^{k - 1} 2^{k^{\prime} - 1} c_{u,v}$, and $\theta_1, \theta_2$ and $\theta_3$ are Lagrange multipliers used to weight each term for maximization or penalization.
\par
For purposes of benchmarking, we generate an ensemble of problem instances by sampling from uniform random price data with a seed of ${b}/{5}$ . A random number is drawn as the initial price $a_{u,1}$ and every subsequent historical price point up to the purchasing price is $-25\%$ to $+25\%$ of the previous price $a_{u, l}$. The price range was set to be between $b/10$ and $b$ with $N_f = 100$ historical price points per asset. In addition, we normalize all $a_{u, l}$ by $a_{u, N_f}$ to keep all asset prices to a similar range.
\par
We set $\theta_1 = 0.3, \theta_2 = 0.5, \theta_3 = 0.2$ in the problem instances where $\theta_2$ is set higher to enforce the budget constraint. These weights were chosen after testing which combination stayed on budget and gave some diversity. By keeping $\theta_2$ constant and increasing $\theta_3$ while decreasing $\theta_1$, an investor could increase the diversity relative to the potential returns and vice versa when decreasing $\theta_3$ relative to $\theta_1$. We generate $1000$ problems for each problem size with $m = 2, 3, 4, 5$ assets and $w = 4$ slices.
\subsection{QUBO to Ising Hamiltonian \label{sec:qubo_ising}}
We formalize the unconstrained portfolio optimization problem in Eq.~(\ref{eq:MPO_unconstrained}) to quadratic unconstrained binary optimization (QUBO) as
\begin{mini}
{x}{\Big(\sum_{i}^{n} q_{i} x_{i} + \sum_{i, j}^{n} Q_{i, j} x_{i} x_{ j} + \gamma\Big)}{}{}
{\label{eq:QUBO}}
\end{mini}
where $q_i$ is the linear weight for the $i^{th}$ spin, $Q_{i,j}$ is the quadratic weight for interactions between the $i^{th}$ and $j^{th}$ bits, and $\gamma$ is a constant. Note that our definition of QUBO expresses optimization as minimization by switching the sign of Eq.~(\ref{eq:MPO_unconstrained}) to be consistent with the use of quantum annealing to recover the lowest-energy state. The corresponding relationships with the original problem instance are given as
\begin{equation}
\begin{aligned}
& q_i = -\theta_1 r_{i} - 2 \theta_2 b^2 p_w \\
& Q_{i,j} = \theta_2 b^2 p_{w}^2 + \theta_3 c_{i,j} \\
& \gamma = \theta_2 b^2
\end{aligned}
\end{equation}
Similarly, the quadratic binary form may be reduced to a classical Ising Hamiltonian
\begin{equation}
\label{Ising Hamiltonian}
H(s) = \sum_{i} s_i h_i + \sum_{i,j}s_i s_j J_{ij} + \beta
\end{equation}
where spin $s_i\in \{-1, 1\}$ is defined by $s_i = 2 x_1 - 1$ with $s = (s_1, s_2, \ldots, s_n)$ while $h_i$ is the spin weight, $J_{ij}$ is the coupling strength, and $\beta$ is a problem-specific constant. The parameters for the Ising Hamiltonian are given as
\begin{equation}
\begin{aligned}
& J_{i,j} = \frac{1}{4} Q_{i,j}\\
& h_i = \frac{q_i}{2} + \sum_{j} J_{i,j}\\
& \beta = \frac{1}{4} \sum_{i,j} Q_{i,j} + \frac{1}{2} \sum_{i} q_i + \gamma
\end{aligned}
\end{equation}
The classical Ising formulation is then converted into a corresponding quantum Ising Hamiltonian given by Eq.~(\ref{eq:Ising_Hamiltonian}) using the correspondence $s_i \rightarrow \sigma_{i}^{z}$.
\subsection{Computational Methods}
We used a D-Wave 2000Q quantum annealer for our experiments. We calculate the probability of success, the probability of chain breaks, and the energy distribution across each problem instance. For each instance, we estimated these metrics by collecting $N_s=1000$ samples of the computed solution. We used D-Wave's solver API (SAPI) with Python $2.7$ to solve each instance of Markowitz portfolio selection using the hardware controls outlined in Sec.~\ref{sec:Controls}. We ran $1,000$ samples per problem over a set of $1,000$ problems for forward annealing examples an $100$ problems for revere annealing examples. We implement the majority vote post-processing technique for any broken chains to interpret raw solutions returned by the $2000Q$. The program implementation and data sets collected from these experiments are available online \cite{repository}.
\par
For benchmarking purposes, we also solved each problem instance using brute force search for the minimal energy solutions of the QUBO formulation. We computed the complete energy spectrum for each portfolio instance. These energy spectrum and the corresponding states were then used as ground truth for testing the accuracy of results obtained from quantum annealing. By sorting the spectrum, we benchmarked the success of reverse annealing using initial states $e_{i}$ sampled from these different parts of the spectrum.
\section{Results \label{sec:Results}}
We benchmark quantum annealing controls by evaluating their influence on the probability of success and probability of chain breaks across problem instances. We first characterize how problem parameters influence the baseline performance by estimating the probability of success for forward annealing using $T = 15~\mu s$, $g = 0$, and a randomized embedding strategy. As shown in Fig.~\ref{fig:pos_slices}, we compare $\tilde{p}_{s}$ for two cases of $w = 1$ and $w = 4$ across increasing $n$. The estimated probability of success for problems with $w = 4$ is consistently higher for problems with no slicing.
\begin{figure}[h!]
\centering
\includegraphics[width=85mm]{POS_slices_maxVote.PNG}
\caption{The average probability of success over $1000$ problems each with $1000$ samples using CMR, $g= 0$, and $ T= 15$ $\mu$ s. The comparison is between a set of problems from problem sizes $8$ to $20$ for $w = 1$ (yellow) and $w = 4$ (blue). The problems set to slices $w = 1$ are much less complex and therefore have a much higher probability of success.}
\label{fig:pos_slices}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=85mm]{Slices_histogram.PNG}
\caption{Probability histogram ($100$ bins) of all possible energies for problem of size $20$ where $a)$ is of $w = 1$ and $b)$ is of $w = 4$. There is a higher density of states close to $e_{0}$ in figure $b)$ and therefore more opportunities to jump to an excited state throughout the sample.}
\label{fig:slices_hist}
\end{figure}
\par
These results are explained by the energy spectra for the different problem parameters, which indicate sharp differences in the density of states. As shown in Fig.~\ref{fig:slices_hist}, a typical problem instance with $w = 4$ has a much higher density of states than those with no slicing ($w=1$). Intuitively, the single-slice behavior results from the specification that the price for each asset is proportional to budget, and, therefore, only a single asset may be selected without penalty when $w = 1$. However, the number of satisfying solutions $v$ increases for arbitrary $w$ combinatorially and, as shown in Appendix~\ref{appx:num_solutions},
\begin{equation}
\begin{aligned}
v = \frac{(2^{w-1} + m - 1)!}{(2^{w-1})! (w - 2)!}.
\end{aligned}
\label{num_solutions}
\end{equation}
Consequently, the probability to recover the lowest-energy state competes with these closely spaced, higher energy solutions, which leads to a corresponding decrease in the probability of success. For the remaining benchmark tests below, we chose $w = 4$ as it represents a more challenging test for the quantum annealer as well as a greater interest to real-world financial applications.
\subsection{Benchmarking Forward Annealing Controls}
\subsubsection{Embedding \label{sec:Embedding}}
Embedding generates and places the physical spin chains for each logical spin on the quantum annealing hardware. We evaluated the CMR and clique embedding algorithms described in Sec.~\ref{sec:Embedding} by estimating the probability of success across problem sizes of $m = 8, 12, 16,$ and $20$ logical spins. For all problem instances of a same problem size, we use the same embedding because they require the same number of fully connected logical spins. We set the parameters of the embedded Ising Hamiltonian by scaling the inter-chain couplings $J_{i,j}$ to lie in the range $[-1,+1]$. We scale all $J_{i,j}$ using a rescale factor of $\frac{1}{j_{max}}$ where $j_{max}$ is the largest $J_{i, j}$ so all embedded $J_{i, j}= \frac{1}{j_{max}} J_{i, j}$. This scales all $J_{i, j }$ to be between $+- 1$. The intra-chain coupling strength is set to $-1$ to have a negative bias stronger than the $J_{i,j}$ values which range $-10^{-1} \leq J_{i,j} \leq 10^{-1}$ due to our data generation and normalization techniques.
\par
The average chain length $\langle l_c \rangle$ from CMR and clique embedding methods grows with the number of logical spins $n$. The average is computed with respect to all chains in an embedding and plotted with respect to $n$ in Fig.~\ref{fig:embedding_size}. As expected by Eq.~(\ref{eq:clique_chain_length}), the clique embedding method has a uniform chain length for each $n$. By contrast, the CMR method generates chains of variable length as indicated by the the average chain length and variance shown in the plot.
\begin{figure}[h!]
\centering
\includegraphics[width=85mm]{Embedding_size_compare.PNG}
\caption{The average chain length over all chains for a given embedding clique and CMR embedding as $n$ increases.}
\label{fig:embedding_size}
\end{figure}
\par
From each of the embedding methods, we estimate the probability of success and probability of broken chains. As shown in Fig.~\ref{fig:pos_embedding}, we observe very small differences in both metrics with increasing problem size. From fitting the resulting point to an exponential, we find $\tilde{p}_{s}$ decays sub-exponentially with respect to $n$ with rate $-0.523$ for the CMR embedding and rate $-0.528$ for the clique embedding. We find that $\tilde{p}_{b}$ grows at a sub-exponential rate of $0.1824$ for CMR embedding and $0.1656$ for clique embedding as $n$ increases. There is not a significant difference in the $\tilde{p}_{s}$ performance between CMR and clique embedding, but clique embedding requires a fewer number of spins as $n$ increases and shows a slight improvement in $\tilde{p}_{b}$. Therefore, we chose to use clique embedding for subsequent benchmarks.
\begin{figure}[h!]
\centering
\includegraphics[width=85mm]{POS_embedding_combined.PNG}
\caption{The $\tilde{p}_{s}$ (top) and $\tilde{p}_{b}$(bottom) on a log scale over $1,000$ samples for $1,000$ problems comparing CMR to clique embedding for parameter settings of $g = 0$ and $T = 100~\mu s$.}
\label{fig:pos_embedding}
\end{figure}
\subsubsection{Forward Annealing Time}
According to the adiabatic theorem, forward annealing more slowly should increase the probability of the system remaining in the ground state and thus increase the probability of success. We varied the forward annealing time $T$ from $1~\mu s$ to $999 \mu s$, which is the broadest range accessible on the D-Wave $2000Q$. As shown in the upper panel of Fig.~\ref{fig:anneal_times}, we observed statistically insignificant changes in the probability of success as annealing time increased at each problem size. Fitting the average probability of success with respect to problem size for the annealing time $T = 100~\mu s$, yields a sub-exponential decay rate for $\tilde{p}_{s}$ given by $-0.528$ and a sub-exponential growth rate for $\tilde{p}_{b}$ given by $0.1628$ as $n$ increases. We do observe a statistically significant difference in the estimated probability of chain breaks $\tilde{p}_{b}$ with respect to forward annealing time as shown in the lower panel of Fig.~\ref{fig:anneal_times}. For $T=100~\mu s$, we recover a growth rate of $0.1656$ for the probability of chain breaks with respect to problem size.
\begin{figure}[h!]
\centering
\includegraphics[width=\columnwidth]{annealtime_combined.PNG}
\caption{The average $\tilde{p}_{s}$ (top) and $\tilde{p}_{b}$(bottom) on a log scale over $1000$ samples for $1000$ problems at various annealing times for parameter settings of $g = 0$ and clique embedding.}
\label{fig:anneal_times}
\end{figure}
\subsubsection{Spin Reversal \label{sec:Spin_Reversal}}
As discussed in Sec.~\ref{sec:Embedding}, embedding maps a logical spin to many physical spins by creating strongly coupled chains. Coupling of these embedded spins via $J_{i,j}$ in Eq.~(\ref{eq:Ising_Hamiltonian}) can lead small biases that may be amplified by imperfections in the hardware. A spin reversal transform mitigates against bias errors by reversing the sign of a spin in the Ising Hamiltonian. This transform preserves the logical problem but reverses the bias error on the embedded spin chain. By randomly selecting a subset of spins to revise, we evaluate the influence of spin-reversal transform on the probability of success. We use $g$ transforms when estimating the probability of success for a given problem instance, such that there are ${N_s}/{g}$ samples per transform. We observed nominal improvements in Fig.~\ref{fig:sr} by using at least $g = 2$ with no advantage to using $g > 2$. For $g = 2$, we observe an sub-exponential decay rate of $-0.505$ for $\tilde{p}_{s}$ and a sub-exponential growth rate of $0.146$ for $\tilde{p}_{b}$ as problem size increases.
\begin{figure}[h!]
\centering
\includegraphics[width=85mm]{SpinReversal_combined.PNG}
\caption{The $\tilde{p}_{s}$ (top) and $\tilde{p}_{b}$ (bottom) on a log scale over $ N_s = 1000$ samples for $N_p = 1000$ problems at $g = 0 \rightarrow 10$) for parameter setting of $T = 100 \mu s$ and clique embedding.}
\label{fig:sr}
\end{figure}
\subsection{Benchmarking Reverse Annealing Controls}
From the reverse annealing controls listed in Sec.~\ref{sec:Control_AnnealTime}, we designed three experiments based on the $e_{i}$ for the reverse annealing heuristic that include (\textit{i}) starting in the known ground state $e_{0}$, (\textit{ii}) starting in the known first excited state $e_{1}$, and (\textit{iii}) starting in the lowest-energy state obtained from $1000$ forward annealing samples $e_{f}$. We then sweep over various schedules to find the optimal $s_p$ with a range of $[0.1, 0.9]$ and $t_p$ with a range of $[15 \rightarrow 800]\mu s$. The $t_r$ and $t_q$parameters were set to be constant and symmetric at $5 \mu s$ each. Thus, the total anneal time is $T^{\prime} = t_r + t_p + t_q$ where $t_p$ is the time parameter that we chose to analyze. For all experiments, we ran the reverse annealing iterative heuristic with $1000$ samples for $100$ random problems were also used in the forward annealing experiments. We estimated the probability of success for reverse annealing with respect to different choices for $e_{i}$, $s_p$, and $t_p$. We compared the combined heuristic of reverse annealing with forward annealing to forward annealing alone with $\tilde{p}_{s}$, $\tilde{p}_{b}$, as well as the frequency of finding energies in excited states to forward annealing alone \footnote[1]{After completing the majority of experiments on the D-Wave processor DW\_2000Q\_2\_1, the remaining experiments were performed on D-Wave processor DW\_2000Q\_5. This included the parametric tests of reverse annealing with respect to $s$ and $t_p$. Prior to testing, we confirmed computational consistency between the results generated using the first device and those using the second. We evaluated differences in $\tilde{p}_s$ and standard deviation between the processors by comparing a previous reverse annealing experiment on the DW\_2000Q\_2\_1 to the same experiment on the DW\_2000Q\_5. We found that the same $\tilde{p}_s$ using both devices and a standard deviation that was within $10^{-5}$ of the measurements on the previous D-Wave processor.}.
\par
By setting $e_{i}$ to the ground state, we tested for parameters $s_p$ and $t_p$ that decrease $\tilde{p}_s$ when the quantum annealer is fed the correct solution. For this experiment, $\tilde{p}_s$ can be thought of as the probability of staying in $e_{0}$
\begin{equation}
\tilde{p}_s(e_{0} \rightarrow e_{0}) = p_f * \tilde{p}_{s}
\end{equation}
\begin{equation}
p_f * \tilde{p}_s = \frac{\sum_i^{N_p} \alpha_i}{N_p} * \frac{\sum_i^{N_p}\sum_{j}^{N_s}\delta_{ij}}{N_s}
\end{equation}
\par
where $p_f$ is the probability that forward annealing found the ground state, $\alpha_i \in \{0, 1\}$ indicates whether forward annealing found the ground state for the $i^{th}$ problem prior to reverse annealing, and $\delta_j \in \{0, 1\}$ is a variable indicating whether the $j^{th}$ sample of the $i^{th}$ problem was measured to be the ground state with reverse annealing. By setting $e_{i} = e_{1}$, we tested whether reverse annealing enhances the probability to populate the ground state. For these tests, $\tilde{p}_s$ estimates the probability of moving from an excited state to the ground state
\begin{equation}
\tilde{p}_s(e_{e} \rightarrow e_{0}) = (1 - p_f) * \tilde{p}_{s}
\end{equation}
\begin{equation}
(1 - p_f) * \tilde{p}_{s} = \frac{\sum_i^{N_p} (1 - \alpha_i)}{N_p} * \frac{\sum_i^{N_p}\sum_{j}^{N_s}\delta_{ij}}{N_s}.
\end{equation}
In addition to testing reverse annealing at $e_{i} = e_{0}$ and $e_{1}$, We tested reverse annealing in combination with forward annealing for which $\tilde{p}_s$ estimates the cumulative probability of finding the correct solution state.
\begin{equation}
\tilde{p}_{s}(R) = \tilde{p}(e_{0} \rightarrow e_{0}) + \tilde{p}(e_{e} \rightarrow e_{0})
\end{equation}
For these experiments, we found it useful to primarily analyze $\tilde{p}_{s}(R) - \tilde{p}(e_{0} \rightarrow e_{0}) = \tilde{p}(e_{e} \rightarrow e_{0})$ to determine if reverse annealing improved upon the $\tilde{p}_{s}$ of forward annealing.
\par
The results from setting $e_{i} = e_{0}$ for each problem with a problem size of $n = 20$ where $m = 5$ and $w = 4$ is shown in Fig.~\ref{fig:pos_ground}. Because the computation begins in the correct solution state, this test measures the probability by which reverse annealing introduces errors into the correct solution. Ideally, $\tilde{p}_s$ will remain near unity for all $s_p$ and $t_p$. We observe that reverse annealing causes the system to leave the ground state with $\tilde{p}_{s}$ reducing to on the order of $10^{-5}$ by annealing back to at least $s = .6$ and increasing $t_p \geq 200 \mu s$
\begin{figure}[h!]
\centering
\includegraphics[width=85mm]{RA_size20_MaxVote_ISGround.PNG}
\caption{The $\tilde{p}_{s}$ (left) and $\tilde{p}_{b}$ (right) for reverse annealing where $e_{i} = e_{0}$ and as $s = [0.1 \rightarrow 0.9]$ and $t_p = [15\mu s \rightarrow 800 \mu s]$ for $n = 20$ with $m = 5$ assets and $w = 4$. }
\label{fig:pos_ground}
\end{figure}
\par
The results from setting $e_{i} = e_{1}$ with a problem size of $n = 20$ where $m = 5$ and $w = 4$ for each problem is shown in Fig.~\ref{fig:pos_excited}. A maximal value of $4.8\times 10^{-4}$ for $\tilde{p}_s$ is found with parameters $s = 0.7$ and $t_p = 800~\mu s$. This is is a $\tilde{p}_s$ one order of magnitude higher than what is observed with forward annealing. This suggests that if $e_{i}$ is very close to $e_{0}$, there may be some benefit to choosing reverse annealing over forward annealing.
\begin{figure}[h!]
\centering
\includegraphics[width=85mm]{RA_excited_MaxVote_5Assets.PNG}
\caption{The $\tilde{p}_{s}$ (left) and $\tilde{p}_{b}$ (right) for reverse annealing where $e_{i} = e_{1}$ for each problem, $s = [0.1 \rightarrow 0.9]$, and $t_p = [15 \mu s \rightarrow 800 \mu s]$ for problem size $20$ with $5$ assets and $4$ slices.}
\label{fig:pos_excited}
\end{figure}
\par
When solving optimization problems for applications in practice, the ground state and excited state will be unknown. However, one approach is to use reverse annealing in addition to forward annealing by using the lowest energy state found with $1,000$ forward annealing samples $e_{f}$ as $e_{i}$ for another $1,000$ samples of reverse annealing. The next experiment tests whether reverse annealing used in combination with forward annealing increases $\tilde{p}_s$ with a problem size of $n = 20$ where $m = 5$ and $w = 4$ . The experimental results from setting $e_{i} = e_{f}$ is shown in Fig.~\ref{fig:pos_no_ground}. These tests were constructed to determine when combining reverse annealing with forward annealing can improve upon forward annealing. Therefore, we removed the 6 problems forward annealing provided an $e_{i} = e_{0}$ and thus $\tilde{p}_{s}$ for this experiment is given by $\tilde{p}_s(R) - p(e_{0} \rightarrow e_{0})$ in this analysis. Similar to the previous experiment in Fig.~\ref{fig:pos_excited}, the $\tilde{p}_{s}$ is at best on the order of $10^{-4}$ at parameters $s = 0.7$ and $t_p = 400 \mu s$ which is one order of magnitude greater than the forward annealing experiments.
\begin{figure}[h!]
\centering
\includegraphics[width=85mm]{RA_forward_no_ground_chains.PNG}
\caption{The $\tilde{p}_{s}$ (left) and $\tilde{p}_{b}$ (right) for reverse annealing where $e_{i} = e_{f}$ for each problem, $s = [0.1 \rightarrow 0.9]$, and $t_p = [15 \mu s \rightarrow 800 \mu s]$ for problem size $20$ with $5$ assets and $4$ slices. The $6$ problems where $e_{f} = e_{g}$ were excluded. Thus, $\tilde{p}_s = p(e_{e} \rightarrow e_{0})$.} \label{fig:pos_no_ground}
\end{figure}
\par
Fig. ~\ref{fig:pos_no_ground} shows a potential for reverse annealing to improve upon results found with forward annealing in $\tilde{p}_{s}$. Therefore, we take a set of $100$ problems solved with reverse annealing and forward annealing and compare the $\tilde{p}_{s}$ of forward annealing (orange) alone to the $\tilde{p}_{s}$ of reverse annealing alone (blue) to the $\tilde{p}_{s}$ with a selection of either forward annealing or reverse annealing (green). If for a problem forward annealing found at least one ground state the forward annealing $\tilde{p}_{s}$ was plotted for that problem ($6$ problems) and otherwise the reverse annealing $\tilde{p}_{s}$ was plotted ($94$ problems). The $\tilde{p}_s$ is measured over $n$ ranging from $[8, 20]$. The reverse annealing parameters are set to have an $e_{i} = e_{f}$ , $s = .7$, and $t_p = 400 \mu s$. As shown in Fig.~\ref{fig:pos_track}, we observe that when taking the combination of best results from forward annealing and reverse annealing with $e_{i} = e_{f}$, we get a $\tilde{p}_{s}$ that improves by an order of magnitude over forward annealing alone for $n = [16, 20]$ with a sub-exponential decay at a rate of $-0.309$. Note that although the blue reverse annealing trend looks to perform the best, however this trend is artificially inflated because $6$ of the problems have $e_{i} = e_{0}$ which has been demonstrated in Fig.~\ref{fig:pos_ground} to yield a $\tilde{p}_s$ on the order of $10^{-2}$ at $s = .7$ and $t_p = 400 \mu s$.
\begin{figure}[h!]
\centering
\includegraphics[width=85mm]{TrackPOS.PNG}
\caption{The $\tilde{p}_s$ as a function of $n$ over a set of $100$ problems each with $1000$ samples. We compare reverse annealing (blue) with $e_{i} = e_{f}$, $s = .7$, and $t_p = 400 \mu s$ to forward annealing (orange) with clique embedding, $g = 0$, and annealing time $= 100$ $\mu s$. We also compare the combination of forward annealing and reverse annealing where the $\tilde{p}_{s}$ is chosen by problem (green). In this green trend, the $\tilde{p}_s$ is calculated using the forward annealing $\tilde{p}_{s}^{(k)}$ for the $6$ problems where forward annealing would have provided reverse annealing with an $e_{i} = e_{0}$ and the reverse annealing $\tilde{p}_{s}^{(k)}$ for the $94$ problems where $e_{i} \neq e_{0}$. }
\label{fig:pos_track}
\end{figure}
We next visualize a histogram, as seen in Fig.~\ref{fig:ra_vs_fa}, of all energies recorded from $1000$ samples returned for a set of $94$ problems where forward annealing did not find $e_{0}$ with $n = 20$. We compare forward annealing to reverse annealing where $e_{i} = e_{f}$. We observe even for problems where neither reverse annealing or forward annealing found $e_{0}$, reverse annealing still on average finds a lower energy solution than forward annealing.
\begin{figure}[h!]
\centering
\includegraphics[width=85mm]{FA_RA_histogram.PNG}
\caption{A probability histogram ($20$ bins) comparing all energies found with forward annealing and reverse annealing from all $1000$ samples for the $94$ problems where $e_i \neq e_0$ for problems with $m = 5$ assets and $w = 4$ .}
\label{fig:ra_vs_fa}
\end{figure}
\section{\label{sec:level1} Conclusions}
We have benchmarked quantum annealing using Markowitz portfolio selection to evaluate the effects of various controls on probability of success and chain breaks.
We have explored a variety of quantum annealing controls including the embedding algorithm, the forward annealing time $T$, and the number spin reversal transforms $g$. When comparing clique embedding against CMR embedding, we found little difference in the estimated probability of success $\tilde{p}_{s}$ as both techniques yielded a sub-exponential decay for $\tilde{p}_s$ with exponents of $-0.528$ and $-0.523$, respectively. We did observe that CMR demonstrated a slightly higher probability of chain breaks $\tilde{p}_{b}$, and we considered this a sufficient justification to use the clique embedding for studying the fully connected problems Markowitz portfolio selection problem.
\par
When varying the forward annealing time $T \in [1 \mu s, 999 \mu s]$, we found that $\tilde{p}_b$ was slightly higher in the range $T = [1 \mu s, 5 \mu s ] $ while increasing the annealing time further yielded little to no improvement. For this reason, we chose to continue all future forward annealing experiments using $T = 100 \mu s$ where the exponential decay rate in $p_s$ was $-0.528$. When varying $g = [0, 10]$, we found small improvements in $\tilde{p}_s$ between $g = 0$ and $g = 2$ where the exponential decay rate became $-0.505$ without much change from increasing the value of $g$ further, and there was no consistent difference in $\tilde{p}_b$.
\par
We benchmarked reverse annealing controls with respect to the parameters $e_{i}$, $s$, and $t_{p}$. We began by observing the results in $\tilde{p}_{s}$ and $\tilde{p}_b$ at $n=20$. We consistently observed that $\tilde{p}_b$ was the same order of magnitude as with the forward annealing experiments and $\tilde{p}_b$ was consistently highest for $s = 0.8$. By setting $e_{i} = e_{0}$, we observed that the $\tilde{p}_s$ decreases exponential as $s$ increased. By setting $e_{i} = e_{1}$, we observed that reverse annealing had a $\tilde{p}_s$ an order of magnitude higher than forward annealing. From these results, we conclude that when $e_{i}$ is close to the ground state, reverse annealing provided some advantage over forward annealing. Because in general the ground state won't be known for a problem, we developed a heuristic which sets $e_{i} = e_{f}$ where we again observed $\tilde{p}_s$ to be an order of magnitude higher than using forward annealing alone.
\par
We further evaluated $\tilde{p}_s$ as a function of $n$ to compare reverse annealing with $e_{i} = e_{f}$, $s = 0.7$, and $t_p = 400 \mu s $ to forward annealing with clique embedding, $T = 100 \mu s $, and $g =0$ alone. In particular, we used the $\tilde{p}_{s}^{(k)}$ of forward annealing for the $6$ problem instances in which $e_{i} = e_{0}$ and the $\tilde{p}_{s}^{(k)}$ of reverse annealing for the $94$ problems where $e_{i} \neq e_{0}$. We continued to observe reverse annealing demonstrate an order of magnitude increase in $p_s$ over forward annealing alone. Lastly, by creating a histogram which plots the lowest energies found across $1000$ samples for the $94$ problems where $e_{i} \neq e_{0}$, we found that reverse annealing$(e_{i} = e_{f})$ on average finds lower energy solutions as compared to forward annealing.
\par
In summary, the benchmarks presented here evaluate a variety of quantum annealing controls with respect to the baseline ground truth for portfolio selection. By comparing the observed influence of these controls on the performance of solution accuracy, we have developed insights into the best selections of controls for solving these problems with the highest accuracy which may help guide the future use of quantum annealing as a meta-heuristics for optimization.
\section*{\label{sec:ack} Acknowledgements}
This work is supported by the Department of Energy, Office of Science, Early Career Research Program. This research used quantum computing resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725. This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan. (http://energy.gov/downloads/doe-public-access-plan).
\bibliographystyle{unsrt}
\section{Introduction and features}
This Latex library provides a standard set of environments for writing optimization problems. The most important features are:
\begin{enumerate}
\item It references optimization problem using three different policies: no equation is referenced, the problem is referenced with a single label, each equation has an individual reference. For more details refer to Sections \ref{sec:syntax} and \ref{sec:environments}.
\item It defines two problem size formats: a long format and a short format. For more details refer to Sections \ref{sec:syntax} and \ref{sec:longshort}.
\item It allows four different outputs for the location of the constraints. For more details refer to Sections \ref{sec:syntax} and \ref{sec:format}.
\item It allows the definition of a limitless number of constraints. For more details refer to Section \ref{subsec:syntax}.
\item Four different type of problems: \textit{minimize}, \textit{maximize}, \textit{arg min} and \textit{arg max}. For more details refer to Sections \ref{sec:syntax} and \ref{sec:environments}.
\item The optimization problem can be broken in several pages without compromising the alignment or the structure of the problem. For more details refer to Section \ref{sec:breakpages}.
\item The objective function can be broken in several lines without compromising the alignment or the structure of the problem. For more details refer to Section \ref{sec:breakObj}.
\end{enumerate}
\section{Using the package}
The package can be imported by directly adding
\begin{lstlisting}
\usepackage{optidef}
\end{lstlisting}
to the document preamble. When importing the packages three options can be used, \verb|short|, \verb|nocomma|, and either \verb|c1|, \verb|c2|, or \verb|c3|:
\begin{lstlisting}
\usepackage[short,c1|c2|c3,nocomma]{optidef}
\end{lstlisting}
The first option changes the default long format of the optimization problems to a shorter format; for a better explanation (including examples) of the \verb|short| option check Section \ref{sec:longshort}.
The options \verb|c1|, \verb|c2|, and \verb|c3| change the default format of the constraints; the default format is format 0 (as defined in Section \ref{sec:format}); \verb|c1|, \verb|c2|, and \verb|c3| respectively change the default constraint arrangement to format 1, 2, and 3. For a better explanation of the four formats including examples, we refer to Section \ref{sec:format}.
For the \verb|nocomma| option check Section \ref{sec:comma}. For a detailed description of how to use the package keep reading the next section.
\section{Environment Syntax Definition}
\label{sec:syntax}
Considering that \verb|Const.i| stands for constraint $i$, \verb|LHS.i| stands for the left-hand-side of constraint $i$, and \verb|RHS.i| for the right-hand-side counterpart, the basic structure to define a general optimization problem with $N$ constraints is:
\begin{verbatim}
\begin{mini#}|sizeFormat|[constraintFormat]<break>
{optimizationVariable}
{objectiveFunction\label{objective}}
{\label{optimizationProblem}}
{optimizationResult}
\addConstraint{LHS.1}{RHS.1\label{Const1}}{extraConst1}
\addConstraint{LHS.2}{RHS.2\label{Const2}}{extraConst2}
.
.
\addConstraint{LHS.N}{RHS.N\label{ConstN}}{extraConstN}
\end{mini#}
\end{verbatim}
\subsection{Definition of Problem parameters}
\begin{enumerate}[label=(\roman*)]
\item \verb|mini#|: defines the type of environment and reference used. There are four environments: \verb|mini|, \verb|maxi|, \verb|argmini|, and \verb|argmaxi|. There are three types of referencing: \verb|mini|, \verb|mini*| and \verb|mini!|. Consult Section \ref{sec:environments} for more details.
\item (Optional) \verb|sizeFormat|: optional parameter to define the size format of the problem. The possible values are:
\begin{itemize}
\item l: for the long format as defined in Section \ref{sec:longshort}.
\item s: for the short format as defined in Section \ref{sec:longshort}.
\end{itemize}
\item (Optional) \verb|constraintFormat|: optional parameter to change the format of the constraints. The parameter \verb|constraintFormat| can take the following values:
\begin{itemize}
\item 0: for the Standard definition in Section \ref{sec:format}.
\item 1: for Alternative 1 in Section \ref{sec:format}.
\item 2: for Alternative 2 in Section \ref{sec:format}
\item 3: for Alternative 3 in Section \ref{sec:format}
\end{itemize}
\item (Optional) \verb|break|: optional parameter to allow the optimization problem to break across multiple pages. For details on this feature, check Section \ref{sec:breakpages}.
\item \verb|optimizationVariable|: variable to be optimizated in the problem, e.g. $w \in \Re^N$.
\item \verb|objectiveFunction\label{objective}|: function to be minimized/maximized as a function of the optimization variable, e.g. $\|w\|_2$. If required, the objective function label should also be included withing this term
\item \verb|\label{optimizationProblem}|: it defines the main and general reference for the optimization problem. It is used for the \verb|mini| and \verb|mini!| enviroments. In the \verb|mini*| environment should be left blank, i.e. \{\}, \textbf{not to be ommited}.
\item \verb|optimizationResult|: a term expressing the result of the optimization problem, e.g. $J(w^*)~=$. If not needed leave it blank, \textbf{not to be ommited}.
\end{enumerate}
The last two defined problem parameters, \verb|\label{optimizationProblem}| and \verb|optimizationResult|, could be made optional. However, in order to improve the problem readibility, line breaking between the 7 parametes was implemented; unfortunately, linea breaking and optional parameters are not compatible and these two parameters had to be made mandatory.
\subsection{Adding Constraints}
\label{subsec:syntax}
After the definition of the problem parameters, the environment accepts the definition of an infinite number of constraints. For this definitions the following command is used:
~\\
\verb|\addConstraint{LHS.k}{RHS.k\label{Const.k}}{extraConst.k}|
~\\
The command accepts three different parameters
\begin{enumerate}
\item \verb|LHS.k|: the left-hand side of the the constraint $k$, e.g. $3w^\top w$.
\item (Optional) \verb|RHS.k\label{Const.k}|: the right-hand side of the constraint k if the equations should be aligned in the equality or inequality signs, e.g. $\leq \|w\|_\infty$. If required, the constraint label should also be included in this term.
\item (Optional) \verb|extraConst.k|: optional parameter to add extra alignment point for additional constraint information. An example would be the constraint names. Look Example \ref{ex:extra} or the Section \ref{sec:extraAlign}.
\end{enumerate}
\subsubsection{Constraints referencing}
Notice that the label for the constraints is always included in the right hand side expression and it only makes sense for the case of using the \verb|mini!| enviroment. The label of the objective function can also be included in a similar way.
\section{Environment Types}
\label{sec:environments}
There are four basic environments depending on the type of referencing that should be used.
\begin{enumerate}
\item The \textbf{mini} environment for defining problems with a single reference label:
\begin{mini}
{w}{f(w)+R(w+6x)}
{\label{eq:Ex1}}{}
\breakObjective{+L(x)}
\addConstraint{g(w)}{=0}
\end{mini}
\item The \textbf{mini*} environment if the problem does not have to be referenced:
\begin{mini*}
{w}{f(w)+ R(w+6x)}
{}{}
\addConstraint{g(w)}{=0}
\end{mini*}
\item The \textbf{mini!} environment if each equation should be referenced:
\begin{mini!}
{w}{f(w)+ R(w+6x)\label{eq:Ex2}}
{\label{eq:Ex1}}{}
\addConstraint{g(w)}{=0}
\end{mini!}
\item The \textbf{minie} environment: same functionality as the \textbf{mini!} environment and it replaces \textbf{mini!} when using the \texttt{optidef} library with some languages in the babel package. For further details we refer to Section \ref{sec:babel}.
\end{enumerate}
\noindent Additionally, there are four basic definitions of optimization problems:
\begin{enumerate}
\item The \textbf{mini} environment:
\begin{mini}
{w}{f(w)+ R(w+6x)}
{}{}
\addConstraint{g(w)}{=0}
\end{mini}
\item The \textbf{maxi} environment:
\begin{maxi}
{w}{f(w)+ R(w+6x)}
{}{}
\addConstraint{g(w)}{=0}
\end{maxi}
\item The \textbf{argmini} environment:
\begin{argmini}
{w}{f(w)+ R(w+6x)}
{}{}
\addConstraint{g(w)}{=0}
\end{argmini}
\item The \textbf{argmaxi} environment:
\begin{argmaxi}
{w}{f(w)+ R(w+6x)}
{}{}
\addConstraint{g(w)}{=0}
\end{argmaxi}
\end{enumerate}
\section{Long and Short Output Formats}
\label{sec:longshort}
The library permits the definition of two different problem size: a long format and a short format.
\subsection{Long Format}
Selected by \verb|sizeFormat|=l. It makes use of \textit{subject to} and \textit{minimize/maximize}
\begin{mini*}|l|
{w}{f(w)+ R(w+6x)}{}{}
\addConstraint{g(w)}{=0}
\end{mini*}
\subsection{Short Format}
Selected by \verb|sizeFormat|=s. It uses instead the shorter \textit{s.t.} and \textit{min/max}
\begin{mini*}|s|
{w}{f(w)+ R(w+6x)}{}{}
\addConstraint{g(w)}{=0}
\end{mini*}
\noindent By the default the long format is used. To change the default to the short format the package must be imported with the \verb|short| option:
\begin{lstlisting}
\usepackage[short]{optidef}
\end{lstlisting}
\section{Output Formats for the Constraints}
\label{sec:format}
There are four basic output formats for the location of the constraints. They are controlled by the environment parameter \verb|constraintFormat|.
\subsection{Alternative 0}
In this format option, the constraints are located to the right of \textit{subject to} and aligned with the objective function. It also has a second alignment point at the $=,~\leq,~\geq$ signs:
\begin{mini}
{w}{f(w)+ R(w+6x)}
{\label{eq:Ex1}}{}
\addConstraint{g(w)+h(w)}{=0}
\addConstraint{t(w)}{=0.}
\end{mini}
\noindent It is the default format if no format option is provided. Alternatively, it can also be set by selecting \verb|constraintFormat|=0.
\subsection{Alternative 1}
Selected by \verb|constraintFormat|=1. It locates the constraints below \textit{subject to} and keeps them aligned at the inequality/equality signs:
\begin{mini}[1]
{w}{f(w)+ R(w+6x)}
{\label{eq:Ex1}}{}
\addConstraint{g(w)+h(w)}{=0}
\addConstraint{t(w)}{=0.}
\end{mini}
\subsection{Alternative 2}
Selected by \verb|constraintFormat|=2. It aligns all the constraints with the objective function.
\begin{mini}[2]
{w}{f(w)+ R(w+6x)}
{\label{eq:Ex1}}{}
\addConstraint{g(w)+h(w)}{=0}
\addConstraint{t(w)}{=0.}
\end{mini}
\subsection{Alternative 3}
Selected by \verb|constraintFormat|=3. It aligns all the constraints below \textit{subject to}:
\begin{mini}[3]
{w}{f(w)+ R(w+6x)}
{\label{eq:Ex1}}{}
\addConstraint{g(w)+h(w)}{=0}
\addConstraint{t(w)}{=0.}
\end{mini}
\begin{lstlisting}
\usepackage[c1|c2|c3]{optidef}
\end{lstlisting}
\subsection{Extra alignment alternative}
\label{sec:extraAlign}
By default, the constraints have 2 aligned elements. However, a third alignment point can be used to set some constraint features. A clear example could be the constraints names:
\begin{mini*}
{w}{f(w)+ R(w+6x)}{}{}
\addConstraint{g(w)+h(w)}{=0,}{\text{(Topological Constraint)}}
\addConstraint{l(w)}{=5w,\quad}{\text{(Boundary Constraint)}}
\end{mini*}
or the index of the constraints:
\begin{mini*}
{w,u}{f(w)+ R(w+6x)}{}{}
\addConstraint{g(w_k)+h(w_k)}{=0,}{k=0,\ldots,N-1}
\addConstraint{l(w_k)}{=5u,\quad}{k=0,\ldots,N-1}
\end{mini*}
This extra alignment point can be added using a third input parameter on the \verb|\addConstraint| parameter. An example using the last constraint of the previous example would be:
\begin{lstlisting}
\addConstraint{l(w_k)}{=5u,\quad}{k=0,\ldots,N-1}
\end{lstlisting}
\subsection{Default format}
The default format is alternative 0. To change the default format across the whole document, the package can be imported using one of the three options: \verb|c1|, \verb|c2|, \verb|c3|, i.e.:
\section{Breaking the optimization problem across multiple pages}
\label{sec:breakpages}
In several cases, people encounter the problem of having an optimization problem that is too long to fit in a single page. In those cases, optidef can automatically break the problem across multiple pages by simply using the optional argument \verb|<b>|. For example:
\begin{lstlisting}
\begin{mini*}<b>
{w,u}{f(w)+ R(w+6x)+ H(100w-x*w/500)}{}{}
\addConstraint{g(w_k)+h(w_k)}{=0,}{k=0,\ldots,N-1}
\addConstraint{l(w_k)}{=5u,\quad}{k=0,\ldots,N-1}
\end{mini*}
\end{lstlisting}
However, when using this option \verb|<b>|, it is important to note that labeling of equations is no longer automatic. To create the number/label, the command \verb|\labelOP{label}| should be used. In particular, in the equation/constraint of the optimization problem where the label/number should be located, simply add \verb|\labelOP{label}|. For example, the following code:
\begin{lstlisting}
\begin{mini}<b>
{w,u}{f(w)+ R(w+6x)+ H(100w-x*w/500)}{}{}
\addConstraint{g(w_k)+h(w_k)}{=0,}{k=0,\ldots,N-1 \labelOP{eq:label}}
\addConstraint{l(w_k)}{=5u,\quad}{k=0,\ldots,N-1}
\end{mini}
\end{lstlisting}
\noindent would display this:
\begin{mini}<b>
{w,u}{f(w)+ R(w+6x)+ H(100w-x*w/500)}{}{}
\addConstraint{g(w_k)+h(w_k)}{=0,}{k=0,\ldots,N-1 \labelOP{eq:label}}
\addConstraint{l(w_k)}{=5u,\quad}{k=0,\ldots,N-1}
\end{mini}
The option \verb|<b>| automatically breaks the optimization problem when the problem is too large to fit in one page (e.g.\ see an example in \ref{ex:break}). However, manual breaks at selected locations are also possible using the \verb|\displaybreak| command. Just add \verb|\displaybreak| between the two constraints that need to be broken, e.g.:
\begin{lstlisting}
\begin{mini}<b>
{w,u}{f(w)+ R(w+6x)+ H(100w-x*w/500)}{}{}
\breakObjective{-g(w^3-x^2*200+10000*w^5)}
\addConstraint{g(w_k)+h(w_k)}{=0,}{k=0,\ldots,N-1 \labelOP{eq:label}}
\displaybreak
\addConstraint{l(w_k)}{=5u,\quad}{k=0,\ldots,N-1}
\end{mini}
\end{lstlisting}
\noindent would display:
\begin{mini}<b>
{w,u}{f(w)+ R(w+6x)+ H(100w-x*w/500)}{}{}
\breakObjective{-g(w^3-x^2*200+10000*w^5)}
\addConstraint{g(w_k)+h(w_k)}{=0,}{k=0,\ldots,N-1 \labelOP{eq:label}}
\displaybreak
\addConstraint{l(w_k)}{=5u,\quad}{k=0,\ldots,N-1}
\end{mini}
\section{Breaking the objective across several lines}
\label{sec:breakObj}
In several cases, people encounter the problem of having an optimization problem which objective function is too long to be set in a single line. In such cases, a line breaking that respects the rest of the problem syntax would be desirable. To account for that, the command \verb|\breakObjective| can be used. The idea is that, if the objective function shall be split in $n$ different functions, e.g.~$f_1,\ldots,f_n$, the default objective parameter would include just $f_1$ and then, we would include $n-1$ statements \verb|\breakObjective|($f_k$), $\forall k=2,\ldots,n$ right before defining the \verb|\addConstraint| commands.
Let's illustrate this with an example. We could consider the example from before:
\begin{mini}
{w,u}{f(w)+ R(w+6x)}{}{}
\addConstraint{g(w_k)+h(w_k)}{=0,}{k=0,\ldots,N-1}
\addConstraint{l(w_k)}{=5u,\quad}{k=0,\ldots,N-1}
\end{mini}
If now the cost function were too long, i.e:
\[
f(w)+ R(w+6x)+ H(100w-x*w/500)-g(w^3-x^2*200+10000*w^5)
\]
We could split it as:
\begin{mini}
{w,u}{f(w)+ R(w+6x)+ H(100w-x*w/500)}{}{}
\breakObjective{-g(w^3-x^2*200+10000*w^5)}
\addConstraint{g(w_k)+h(w_k)}{=0,}{k=0,\ldots,N-1}
\addConstraint{l(w_k)}{=5u,\quad}{k=0,\ldots,N-1}
\end{mini}
by simpling using the following command:
\begin{lstlisting}
\begin{mini*}
{w,u}{f(w)+ R(w+6x)+ H(100w-x*w/500)}{}{}
\breakObjective{-g(w^3-x^2*200+10000*w^5)}
\addConstraint{g(w_k)+h(w_k)}{=0,}{k=0,\ldots,N-1}
\addConstraint{l(w_k)}{=5u,\quad}{k=0,\ldots,N-1}
\end{mini*}
\end{lstlisting}
It is important to notice the specific location of the \verb|\breakObjective| command. In order to work properly, it has to be defined right before \verb|\addConstraint| and right after the definition of the environment parameters; i.e.~in any case the command should be used right after defining the first part of the objective function and not finishing the definition of the mandatory environment parameters.
\section{Default comma at the end of the constraint}
\label{sec:comma}
By default, the algorithms adds a comma at the end of any constraint that is not the last one. This feature was implemented due to correctness of mathematical notation. However, this behavior can be removed by adding the option \verb|nocomma| when importing the package:
\begin{lstlisting}
\usepackage[nocomma]{optidef}
\end{lstlisting}
\section{Long Optimization Variables}
The standard appearance for long optimization variables is as follows:
\begin{mini!}
{x_0,u_0,x_1,\hdots,u_{N-1},x_N}
{\sum_{k=0}^{N-1} L(x_k,u_k)\!\!+\!\!E(x_N)\label{OCPobj}}
{\label{eq:OCP}}{}
\addConstraint{x_{k+1}-f(x_k,u_k)}{= 0, \label{dOCP:modelc}\quad k=0,\dots,N-1}
\addConstraint{h(x_k,u_k)}{\leq 0, \quad k=0,\dots,N-1}
\addConstraint{r(x_0,x_N)}{= 0. \label{dOCP:boundary}}
\end{mini!}
\noindent A possible way to reduce the large variable spacing is to stack them with the command: \begin{verbatim}
\substack{x_0,u_0,x_1,\hdots,\\u_{N-1},x_N}
\end{verbatim}
\begin{mini!}
{\substack{x_0,u_0,x_1,\hdots,\\ u_{N-1},x_N}}
{\sum_{k=0}^{N-1} L(x_k,u_k)\!\!+\!\!E(x_N)\label{OCPobj}}
{\label{eq:OCP}}{}
\addConstraint{x_{k+1}-f(x_k,u_k)}{= 0, \label{dOCP:modelc}\quad k=0,\dots,N-1}
\addConstraint{h(x_k,u_k)}{\leq 0, \quad k=0,\dots,N-1}
\addConstraint{r(x_0,x_N)}{= 0. \label{dOCP:boundary}}
\end{mini!}
\section{Compatibility issues with other packages}
Issues with three different packages have been reported: cleveref, babel, and mathabx.
\subsection{Cleveref}
When using the cleveref package in couple with the optidef package two measures have to taken for the packages to work properly:
\begin{enumerate}
\item As also indicated in the cleveref documentation, the optidef package has to be loaded before the cleveref package.
\item To avoid crashes, the \verb|\label| commands in the optidef environments have to be replaced by the protected counterparts \verb|\protect\label|. This is required because of the standard Latex issue of moving arguments and fragile commands\footnote{\url{goo.gl/wmKbNU}}.
\end{enumerate}
\noindent A code example taking into account both measures is the following:
\begin{verbatim}
\documentclass{article}
\usepackage{optidef}
\usepackage{cleveref}
\begin{document}
\begin{mini!}
{w}{f(w)+ R(w+6x) \protect\label{eq:ObjectiveExample1}}
{\label{eq:Example1}}{}
\addConstraint{g(w)}{=0 \protect\label{eq:C1Example3}}
\addConstraint{n(w)}{= 6 \protect\label{eq:C2Example1}}
\addConstraint{L(w)+r(x)}{=Kw+p \protect\label{eq:C3Example1}}
\end{mini!}
Example labels: \cref{eq:Example1} and \cref{eq:ObjectiveExample1}.
\end{document}
\end{verbatim}
As an alternative to the second step, i.e.~protecting the \verb|\label| command, the command can be robustify in the document preamble and then \verb|\protect| is not longer needed. To robustify the \verb|\label| command, the following has to be added to the preamble:
\begin{verbatim}
\usepackage{etoolbox}
\robustify{\label}
\end{verbatim}
\subsection{Babel}
\label{sec:babel}
When importing the package babel with some specific languages, e.g.~French, the \verb|mini!| environment clashes because of the exclamation mark.
This issue has been resolved starting from Optidef 2.7, where a working alternative to the \verb|mini!| environment is included: the \verb|minie| enviroment. Both environemnts have the same functionality, but when using the babel package it is recommended to use the \verb|minie| environment to avoid issues.
\subsection{Mathabx}
When using the mathabx package in couple with the optidef package, the optidef package must be loaded first in order to avoid malfunction of the mathabx package. In addition, the amsmath package should also be loaded before both of them. The preamble should look like:
\begin{verbatim}
\usepackage{amsmath}
\usepackage{mathabx}
\usepackage{optidef}
\end{verbatim}
\section{Examples}
\subsection{Example 1 - mini environment}
The code:
\begin{verbatim}
\begin{mini}
{w}{f(w)+ R(w+6x)}
{\label{eq:Example1}}{}
\addConstraint{g(w)}{=0}
\addConstraint{n(w)}{= 6}
\addConstraint{L(w)+r(x)}{=Kw+p}
\addConstraint{h(x)}{=0.}
\end{mini}
\end{verbatim}
\noindent outputs:
\begin{mini}
{w}{f(w)+ R(w+6x)}
{\label{eq:Ex11}}{}
\addConstraint{g(w)}{=0}
\addConstraint{n(w)}{= 6}
\addConstraint{L(w)+r(x)}{=Kw+p}
\addConstraint{h(x)}{=0.}
\end{mini}
\subsection{Example 2 - mini* environment}
On the other hand:
\begin{verbatim}
\begin{mini*}
{w}{f(w)+ R(w+6x)}
{}{}
\addConstraint{g(w)}{=0}
\addConstraint{n(w)}{= 6,}
\addConstraint{L(w)+r(x)}{=Kw+p}
\addConstraint{h(x)}{=0.}
\end{mini*}
\end{verbatim}
\noindent it is almost the same but removing the reference:
\begin{mini*}
{w}{f(w)+ R(w+6x)}
{}{}
\addConstraint{g(w)}{=0}
\addConstraint{n(w)}{= 6}
\addConstraint{L(w)+r(x)}{=Kw+p}
\addConstraint{h(x)}{=0.}
\end{mini*}
\subsection{Example 3 - mini! environment}
\noindent Finally, the multireferencing environment outputs:
\begin{verbatim}
\begin{mini!}
{w}{f(w)+ R(w+6x) \label{eq:ObjectiveExample1}}
{\label{eq:Example1}}{}
\addConstraint{g(w)}{=0 \label{eq:C1Example3}}
\addConstraint{n(w)}{= 6 \label{eq:C2Example1}}
\addConstraint{L(w)+r(x)}{=Kw+p \label{eq:C3Example1}}
\addConstraint{h(x)}{=0. \label{eq:C4Example1}}
\end{mini!}
\end{verbatim}
\begin{mini!}
{w}{f(w)+ R(w+6x)\label{eq:ObjectiveExample3}}
{\label{eq:Example3}}
{}
\addConstraint{g(w)}{=0 \label{eq:C1Example3}}
\addConstraint{n(w)}{= 6 \label{eq:C2Example3}}
\addConstraint{L(w)+r(x)}{=Kw+p \label{eq:C3Example3}}
\addConstraint{h(x)}{=0.\label{eq:C4Example3}}
\end{mini!}
\subsection{Example 4 - Problem Result}
\noindent Adding the problem result:
\begin{verbatim}
\begin{mini}
{w}{f(w)+ R(w+6x)}
{\label{eq:Example1}}
{J(w^*)=}
\addConstraint{g(w)}{=0}
\addConstraint{n(w)}{= 6}
\addConstraint{L(w)+r(x)}{=Kw+p}
\addConstraint{h(x)}{=0.}
\end{mini}
\end{verbatim}
\noindent outputs:
\begin{mini}
{w}{f(w)+ R(w+6x)}
{\label{eq:Ex1}}{J(w^*)~=~}
\addConstraint{g(w)}{=0}
\addConstraint{n(w)}{= 6}
\addConstraint{L(w)+r(x)}{=Kw+p}
\addConstraint{h(x)}{=0.}
\end{mini}
\subsection{Example 5 - Short Format}
\noindent Adding the short format parameter:
\begin{verbatim}
\begin{mini}|s|
{w}{f(w)+ R(w+6x)}
{\label{eq:Example1}}
{}
\addConstraint{g(w)}{=0}
\addConstraint{n(w)}{= 6}
\addConstraint{L(w)+r(x)}{=Kw+p}
\addConstraint{h(x)}{=0.}
\end{mini}
\end{verbatim}
\noindent outputs:
\begin{mini}|s|
{w}{f(w)+ R(w+6x)}
{\label{eq:Ex1}}{}
\addConstraint{g(w)}{=0}
\addConstraint{n(w)}{= 6}
\addConstraint{L(w)+r(x)}{=Kw+p}
\addConstraint{h(x)}{=0.}
\end{mini}
\subsection{Example 6 - Alternative 1 for Constraints}
\noindent If including a 1 as optional parameter, the first constraint will appear aligned to the left right below \textit{subject to}.
\begin{verbatim}
\begin{mini}[1]
{w}{f(w)+ R(w+6x)}
{\label{eq:Example1}}
{}
\addConstraint{g(w)}{=0}
\addConstraint{n(w)}{= 6}
\addConstraint{L(w)+r(x)}{=Kw+p}
\addConstraint{h(x)}{=0.}
\end{mini}
\end{verbatim}
\noindent outputs:
\begin{mini}[1]
{w}{f(w)+ R(w+6x)}
{\label{eq:Ex1}}{}
\addConstraint{g(w)}{=0}
\addConstraint{n(w)}{= 6}
\addConstraint{L(w)+r(x)}{=Kw+p}
\addConstraint{h(x)}{=0.}
\end{mini}
\subsection{Example 7 - Alternative 2 for Constraints}
\noindent If including a 2 as optional parameter, the constraint will appear to the right of \textit{subject to} but a single alignment point.
\begin{verbatim}
\begin{mini}[2]
{w}{f(w)+ R(w+6x)}
{\label{eq:Example1}}
{}
\addConstraint{g(w)}{=0}
\addConstraint{n(w)}{= 6}
\addConstraint{L(w)+r(x)}{=Kw+p}
\addConstraint{h(x)}{=0.}
\end{mini}
\end{verbatim}
\noindent outputs:
\begin{mini}[2]
{w}{f(w)+ R(w+6x)}
{\label{eq:Ex1}}{}
\addConstraint{g(w)}{=0}
\addConstraint{n(w)}{= 6}
\addConstraint{L(w)+r(x)}{=Kw+p}
\addConstraint{h(x)}{=0.}
\end{mini}
\subsection{Example 8 - Alternative 3 for Constraints}
\noindent If including a 3 as optional parameter, the first constraint will appear aligned to the left right below \textit{subject to} and with a single alignment point.
\begin{verbatim}
\begin{mini}[3]
{w}{f(w)+ R(w+6x)}
{\label{eq:Example1}}
{}
\addConstraint{g(w)}{=0}
\addConstraint{n(w)}{= 6}
\addConstraint{L(w)+r(x)}{=Kw+p}
\addConstraint{h(x)}{=0.}
\end{mini}
\end{verbatim}
\noindent outputs:
\begin{mini}[3]
{w}{f(w)+ R(w+6x)}
{\label{eq:Ex1}}{}
\addConstraint{g(w)}{=0}
\addConstraint{n(w)}{= 6}
\addConstraint{L(w)+r(x)}{=Kw+p}
\addConstraint{h(x)}{=0.}
\end{mini}
\subsection{Example 9 - Breaking a long objective}
\begin{lstlisting}
\begin{mini*}
{w,u}{f(w)+ R(w+6x)+ H(100w-x*w/500)}{}{}
\breakObjective{-g(w^3-x^2*200+10000*w^5)}
\addConstraint{g(w_k)+h(w_k)}{=0,}
\addConstraint{l(w_k)}{=5u,\quad}
\end{mini*}
\end{lstlisting}
outputs:
\begin{mini}
{w,u}{f(w)+ R(w+6x)+ H(100w-x*w/500)}{}{}
\breakObjective{-g(w^3-x^2*200+10000*w^5)}
\addConstraint{g(w_k)+h(w_k)}{=0}
\addConstraint{l(w_k)}{=5u.}
\end{mini}
\subsection{Example 10 - Extra Alignment in the Constraints}
\label{ex:extra}
Adding optional alignment to add constraint names:
\begin{verbatim}
\begin{mini*}
{w}{f(w)+ R(w+6x)}
{}{}
\addConstraint{g(w)}{=0,}{ \quad \text{(Dynamic constraint)}}
\addConstraint{n(w)}{= 6,}{ \quad \text{(Boundary constraint)}}
\addConstraint{L(w)+r(x)}{=Kw+p,}{ \quad \text{(Random constraint)}}
\addConstraint{h(x)}{=0,}{ \quad \text{(Path constraint).}}
\end{mini*}
\end{verbatim}
\subsection{Example 11 - The \textit{argmini} Environment}
Similar to the \verb|mini|, \verb|mini*| and \verb|mini!| environments, the environments \verb|argmini|, \verb|argmini*| and \verb|argmini!| are very similar environments that use the same syntax but the output is slightly different:
\begin{verbatim}
\begin{argmini}
{w}{f(w)+ R(w+6x)}
{\label{eq:Example1}}{w^*=}
\addConstraint{g(w)}{=0}
\addConstraint{n(w)}{= 6}
\addConstraint{L(w)+r(x)}{=Kw+p}
\addConstraint{h(x)}{=0.}
\end{argmini}
\end{verbatim}
\noindent outputs:
\begin{argmini}
{w}{f(w)+ R(w+6x)}
{\label{eq:Ex1}}{w^*~=~}
\addConstraint{g(w)}{=0}
\addConstraint{n(w)}{= 6}
\addConstraint{L(w)+r(x)}{=Kw+p}
\addConstraint{h(x)}{=0.}
\end{argmini}
\subsection{Example 12 - The \textit{maxi} and \textit{argmaxi} Environments}
Exactly the same syntax and definition as the previous environments, but now for defining maximization environments. The following code serves for illustration:
\begin{verbatim}
\begin{maxi}
{w}{f(w)+ R(w+6x)}
{g(w)}{=0}
{\label{eq:Example1}}{}
\addConstraint{g(w)}{=0}
\addConstraint{n(w)}{= 6}
\addConstraint{L(w)+r(x)}{=Kw+p}
\addConstraint{h(x)}{=0.}
\end{maxi}
\end{verbatim}
\noindent outputs:
\begin{maxi}
{w}{f(w)+ R(w+6x)}
{\label{eq:Example1}}{}
\addConstraint{g(w)}{=0}
\addConstraint{n(w)}{= 6}
\addConstraint{L(w)+r(x)}{=Kw+p}
\addConstraint{h(x)}{=0.}
\end{maxi}
\subsection{Example 13 - Breaking optimization problem}
\label{ex:break}
\begin{lstlisting}
\begin{mini}<b>
{w}{f(w)+ R(w+6x)}
{\label{eq:Example1}}{}
\addConstraint{g(w)}{=0}
\addConstraint{p(w)}{=0}
\addConstraint{q(w)}{=0}
\addConstraint{r(w)}{=0\labelOP{testLabel}}
\addConstraint{n(w)}{= 6}
\addConstraint{L(w)+r(x)}{=Kw+p}
\addConstraint{h(x)}{=0.}
\end{mini}
\end{lstlisting}
outputs:
\begin{mini}<b>
{w}{f(w)+ R(w+6x)}
{\label{eq:Example1}}{}
\addConstraint{g(w)}{=0}
\addConstraint{p(w)}{=0}
\addConstraint{q(w)}{=0}
\addConstraint{r(w)}{=0\labelOP{testLabel}}
\addConstraint{n(w)}{= 6}
\addConstraint{L(w)+r(x)}{=Kw+p}
\addConstraint{h(x)}{=0.}
\end{mini}
\subsection{Example 14 - All Possible Parameters}
\begin{verbatim}
\begin{mini!}|s|[2]<b>
{w}{f(w)+ R(w+6x)\label{eq:ObjectiveExample3}}
{\label{eq:Example3}}
{w^*=}
\addConstraint{g(w)}{=0 \label{eq:C1Example3}}
\addConstraint{n(w)}{= 6 \label{eq:C2Example3}}
\addConstraint{L(w)+r(x)}{=Kw+p \label{eq:C3Example3}}
\addConstraint{h(x)}{=0.\label{eq:C4Example3}}
\end{mini!}
\end{verbatim}
\begin{mini!}|s|[2]<b>
{w}{f(w)+ R(w+6x)\label{eq:ObjectiveExample3}}
{\label{eq:Example3}}
{w^*=}
\addConstraint{g(w)}{=0 \label{eq:C1Example3}}
\addConstraint{n(w)}{= 6 \label{eq:C2Example3}}
\addConstraint{L(w)+r(x)}{=Kw+p \label{eq:C3Example3}}
\addConstraint{h(x)}{=0.\label{eq:C4Example3}}
\end{mini!}
\section{Reporting bugs and feature requests}
To report any bug or request some feature please use the issue section in the github repository: \url{https://github.com/jeslago/optidef/issues}.
\end{document}
|
1,116,691,500,771 | arxiv | \section{Introduction}
\label{txt:introduction}
Low-metallicity galaxies have been found to exhibit high nebular \ion{He}{II} and \ion{Ne}{V} emission \citep[e.g.,][]{Garnett91,Izotov12}. The origin of this emission remains unknown, with multiple explanations in the recent literature: radiative shocks \citep[e.g.,][]{Plat19}, binary stars \citep[e.g.,][]{Gotberg17}, active galactic nuclei in dwarf galaxies \citep[e.g.,][]{Mezcua18}, XRBs \citep[e.g.,][]{Power13}, cluster winds \citep[e.g.,][]{Oskinova22}, etc. \citep[see discussion in][]{Olivier21}. Understanding ionizing sources is crucial for the study of the ISM \citep[e.g.,][]{Nanayakkara19}, and the heating of the IGM \citep[e.g.,][]{Mesinger13,Madau17}.
Observations show that XRB populations and their integrated luminosity are anticorrelated with the metallicity of the host galaxy \citep[e.g.,][]{Douna15}, also supported by theoretical work \citep[e.g.,][]{2013ApJ...764...41F,Fragos13}. This has motivated the study of the contribution of XRBs in the ionization of the ISM in low-metallicity galaxies, with contradicting results \citep[e.g.,][]{Schaerer19,Saxena20,Senchyna20,Umeda22}. In the case of the extremely low-metallicity galaxy I\,Zw\,18, exhibiting strong \ion{He}{II} emission, the X-ray luminosity is dominated by an ULX \citep[see review of][]{Kaaret17}, presenting an interesting case for the study of individual XRBs as ionizing sources. However, recent observational work is inconclusive: the source position and spectrum is not consistent with the \ion{He}{II} emission \citep[e.g.][]{Kehrig21}, unless a geometrical beaming effect is considered \citep[e.g.,][]{Rickards21}.
One of the most important difficulties in understanding the contribution of X-ray sources, which are typically observed at energy bands $0.2-10\,\rm keV$, is that their spectra are poorly constrained at the EUV regime (\ion{He}{II}: ${\sim}54\,\rm eV$, \ion{Ne}{V}: ${\sim}97.1\,\rm eV$; see \citealt{Lehmer22}). However, ULXs are strong candidates for four reasons: (i) ULXs exhibit softer spectra than typical XRBs and can potentially produce high rate of He-ionizing photons \citep[e.g.,][]{Simmonds21}, (ii) they have been shown to dominate the X-ray emission from normal galaxies due to the shallow slope of the high-mass XRB (HMXB) luminosity function \citep[XLF; e.g.,][]{Lehmer19}, (iii) they are abundant in low-metallicity environments \citep[e.g.,][]{Mapelli10}, and (iv) there is evidence of geometrical beaming \citep[e.g.,][]{2009MNRAS.393L..41K}, which indicates a larger underlying population of ULXs, and hence a stronger contribution in the ionization of the ISM than what is expected by the observed population \citep[e.g.,][]{Rickards21}.
An interesting case of a powerful ionizing source in the Galaxy is SS\,433 \citep{1977ApJS...33..459S}. It's high optical and UV emission \citep[e.g.,][]{2019A&A...624A.127W}, is believed to be powered by a super-critical accretion disk, despite its low observed X-ray luminosity \citep[e.g][]{2021MNRAS.506.1045M}. This can be explained through models of super-critical disks, which predict that close to the CO, the disk becomes geometrically thick, as strong optically-thick winds flow from its surface \citep[e.g.,][henceforth SS73]{1973A&A....24..337S}. The highly anisotropic radiation pattern results in face-on observers detecting collimated emission from the inner regions of the disk (beaming), and edge-on observers seeing \lq{}softer\rq{} emission escaping from the wind's photosphere extending to large radii with respect to the inner disk radius \citep[e.g.,][henceforth P07]{2007MNRAS.377.1187P}. Therefore, the common interpretation is that SS\,433 belongs to the general population of ULXs with observed luminosities $L_{\rm obs}{\gtrsim}10^{39}\,\rm erg\,s^{-1}$, but being viewed at a high inclination, its strong X-ray emission is invisible to us \citep[e.g.,][]{2006MNRAS.370..399B}. Consequently, a unified picture arises where the main difference between ULXs and soft ULXs is the viewing angle \citep[e.g.,][]{2013MNRAS.435.1758S,2017MNRAS.468.2865P}.
In this paper, we combine the properties of observed ULX populations and theoretical models describing their intrinsic spectra, and constrain the ionization power of ULXs under the super-Eddington accretion and beaming models.
\section{Methodology}
The contribution of ULX populations in the ionization of the ISM and the heating of the IGM is hampered by three observational challenges. Firstly, the hard EUV/soft--X-ray part of the spectrum cannot be detected directly due to absorption. Secondly, under the geometrical-beaming scenario, extragalactic edge-on ULXs are too faint to be detected by X-ray telescopes, and therefore, we may underestimate the ULX content of galaxies. Thirdly, the distributions of the physical properties of ULXs (e.g., accretor/donor masses, structure of the accretion disk, etc.) remain unknown.
To overcome these challenges in our analysis, we adopt models for super-critically accreting sources that provide the luminosity, geometrical beaming and spectral energy distribution (SED), given the CO mass and mass-transfer rate of the systems. Below, we show that by considering different CO masses, the beaming factor and the spectrum depend only on the luminosity, and therefore, the underlying ULX population and its integrated spectra can be anchored on empirical constraints of the HMXB XLF.
\subsection{Luminosity distribution and beaming in ULXs}
Super-critical disks are, by definition, encountered in systems with luminosities exceeding the Eddington limit:
\begin{equation}
L_{\rm Edd} = 1.26\times10^{38} m \ \rm erg\,s^{-1},
\label{eq:eddington}
\end{equation}
where $m{=}M/M_\odot$ is the mass of the accreting object in solar units \citep[e.g.,][]{1980ApJ...242..772A}. Based on the typical range of BH masses in XRBs (${\lesssim}15\,M_\odot$; \citealt{Remillard06}) sources with luminosities exceeding $L_{\rm lim}{=}2{\times}10^{39}\,\rm erg\,s^{-1}$ are likely to be super-Eddington. Although super-critically accreting COs with near-solar masses may exhibit lower luminosities than the aforementioned limit, when we anchor our results on XLFs we ignore the part below $L_{\rm lim}$ to avoid mixing under- and super-Eddington sources.
In \citetalias{1973A&A....24..337S} it is shown that when the mass-transfer rate $\dot{M}$ exceeds the rate corresponding to the Eddington limit, $\dot{M}_{\rm Edd}$, the bolometric luminosity is
\begin{equation}
L_{\rm bol} = L_{\rm Edd}\left(1 + \ln \dot{m}\right),
\label{eq:SS73}
\end{equation}
where $\dot{m}{=}\dot{M}/\dot{M}_{\rm Edd}$. This is the result of a fraction of accretion power being spent for driving strong outflows, which keep the accretion on the CO Eddington-limited.
Even at extreme mass-transfer rates (e.g., $\dot{m}{\sim}1000$), the bolometric luminosity cannot reach $10^{41}\,\rm erg\,s^{-1}$ without invoking COs of $M{>}100\,M_\odot$. In the beaming scenario of \citet{2001ApJ...552L.109K}, the most extreme ULXs are explained by the fact that their observed X-ray luminosities ($L_{\rm obs}$) are the isotropic-equivalent of moderately super-Eddington, beamed sources viewed face-on:
\begin{equation}
L_{\rm obs} = b^{-1} L_{\rm bol},
\label{eq:beaming}
\end{equation}
where $b$ is the beaming factor. Furthermore, \citet{2009MNRAS.393L..41K} hinted at a dependence of the beaming factor on the mass-transfer rate,
\begin{equation}
b = \begin{cases}
1 & \dot{m} \leq 8.5 \\
\left({8.5} / {\dot{m}}\right)^2 & \dot{m} > 8.5
\end{cases},
\label{eq:beamingfactor}
\end{equation}
which explains the observed anticorrelation between the luminosity and temperature of the soft X-ray emission in ULXs \citep[e.g.,][]{2009MNRAS.398.1450K,2016ApJ...831..117F}.
We use the $L_{\rm obs}{>}2\times10^{39}\rm\,erg\,s^{-1}$ part of the HMXB XLF from \citet{Lehmer19} to model the distribution of ULX luminosities:
\begin{equation}
\frac{dN}{dL_{38}} = {\rm SFR} \times K_{\rm HMXB} \times
\begin{cases}
L_{38}^{-\gamma} & 20{<}L_{38}{<}L_c \\
0 & \mbox{elsewhere}
\end{cases},
\label{eq:lehmer}
\end{equation}
where $L_{38}{=}L_{\rm obs}/10^{38}\,\rm erg\,s^{-1}$, SFR is the star-formation rate of the parent stellar population, and the normalization $K_{\rm HMXB}$, power-law index $\gamma$ and cut-off luminosity $L_c$ are fitted parameters.
Equations \eqref{eq:eddington}--\eqref{eq:lehmer} interconnect the observable $L_{\rm obs}$ with the mass-transfer rate $\dot{m}$ via the XLF, assuming a distribution of masses $m$.
However, ULXs are rare sources \citep[typically one per galaxy; e.g.,][]{Kovlakas20}, and their distances are prohibiting in constraining their parameters, such as the mass of the accretor. For this reason, we will consider three different accretor masses: $1.4$ (corresponding to neutron stars\footnote{the NS mass is used as a lower limit; we do not account for presence of magnetic field/alternative accretion modes \citep[e.g.,][]{2021MNRAS.504..701B}.}), $8$ and $20\,M_\odot$.
Using Eqs. \eqref{eq:SS73}-\eqref{eq:beamingfactor} and the fact that by definition at sub-Eddington luminosities $L_{\rm bol}{=}\dot{m} L_{\rm Edd}$, we infer $\dot{m}$ from the ratio $f{=}L_{\rm obs}/L_{\rm Edd}$:
\begin{equation}
\dot{m} = \begin{cases}
f & f \leq 1 \\
{\rm e}^{f-1} & 1 < f \leq{3.14} \\
8.5\left(\frac{f}{3.14}\right)^{\frac{4}{9}} & f > 3.14
\end{cases}
\label{eq:inverse}
,
\end{equation}
where the case for $f{>}1{+}\ln{8.5}{\simeq}3.14$ is an approximation of the inverse of $f{=}\left({\dot{m}}/{8.5}\right)^2\left(1{+}\ln\dot{m}\right)$ with accuracy ${<}2\%$ for $f{<800}$ (a sufficiently high value corresponding to $10^{41}\,\rm erg\,s^{-1}$ for $m{=}1$).
\subsection{The black-body component of the ULX spectrum}
The spectral properties of super-Eddington sources are highly dependent on the geometry of the disk, and the interplay between the outflowing gas and the radiation. Specifically, the wind's photosphere is expected to produce a soft component in the spectrum of ULXs \citep[e.g.,][]{2003MNRAS.345..657K}.
According to the \citetalias{1973A&A....24..337S} model, the local mass-transfer rate increases as we go towards the inner regions of the accretion disk. However, at super-Eddington rates, there is a radius, the spherization radius ($R_{\rm sp}$), at which the local Eddington limit is reached, initiating outflows. As a result, the mass-transfer rate decreases as we approach further the CO. At the $R_{\rm sp}$ the accretion disk is optically and geometrically thick, forming a nearly-spherical black-body source. The $R_{\rm sp}$ is computed by equalizing the accretion and Eddington luminosities \citep[cf.,][]{2002apa..book.....F}:
\begin{equation}
R_{\rm sp} = \frac{G M \dot{M}}{\eta \dot{M}_{\rm Edd} c^2} = \frac{G M_\odot}{\eta c^2} m \dot{m} = 1.5\times 10^{5} m \dot{m} {\rm\,cm},
\label{eq:Rsp}
\end{equation}
where $\eta$ is the accretion efficiency, for which we adopt the value of 0.1.
Using the Stefan-Boltzmann law we calculate the effective temperature at the $R_{\rm sp}$:
\begin{equation}
T_{\rm eff} = \left( \frac{L_{\rm Edd}}{4{\rm \pi} R_{\rm sp}^2 \sigma} \right)^{\frac{1}{4}}
\label{eq:Tsp}
.
\end{equation}
However, this model might underestimate the radius of the black body. Theoretical work considering various physical processes in super-critical disks (for example wind and advection; e.g, \citealt{1999AstL...25..508L}, \citetalias{2007MNRAS.377.1187P}) have provided more realistic prescriptions for the temperature and radius of the photosphere from which a soft black-body component originates in ULXs (see review in \citealt{Fabrika21}). In \citetalias{2007MNRAS.377.1187P} the radius of the photosphere is estimated as:
\begin{equation}
R_{\rm ph} = 3 \frac{e_w}{\zeta\beta} \dot{m}^{\frac{3}{2}} R_{\rm in},
\label{eq:Rph}
\end{equation}
where $R_{\rm in}$ is the inner disk radius, and $e_w$, $\zeta$, and $\beta$ are model parameters (see below). We adopt an inner radius equal to three Schwarzschild radii for BHs (assuming non-spinning BHs):
\begin{equation}
R_{\rm in} = \frac{6 G M_\odot}{c^2} m,
\end{equation}
and $11\rm\,km$ for NSs \citep{2016ApJ...820...28O}. The $T_{\rm eff}$ is
\begin{equation}
T_{\rm ph} = 9.28\times 10^6 {\rm\,K}
\left(\frac{\zeta \beta}{\epsilon_w}\right)^{\frac{1}{2}}
m^{-\frac{1}{4}} \dot{m}^{-\frac{3}{4}},
\label{eq:Tph}
\end{equation}
where $\zeta=\sqrt{\xi^2-1}$ is a function of the ratio of the perpendicular velocity of the ejected gas over the orbital velocity ($\xi{=}u_z/u_K$), and $\beta$ is the ratio of the mean wind radial velocity over the orbital velocity at the $R_{\rm sp}$. Due to energy constraints (cf. \citetalias{2007MNRAS.377.1187P}),
$\beta{\approx}1{\approx}\zeta$,
and therefore we vary only $e_w$, which is the fraction of the accretion energy powering the outflow. We adopt two values for $e_w$, 0.3 and 0.7 to study the effect of this parameter.
\subsection{The disk component of the ULX spectrum}
For all models, the soft black-body spectrum and its normalization is computed using Eqs. \eqref{eq:Rsp}, \eqref{eq:Tsp}, \eqref{eq:Rph}, and \eqref{eq:Tph}. The remaining emission is modeled by a multi-color disk black-body, using the \texttt{xspec} model \texttt{diskbb} from \texttt{sherpa} \citep{2007ASPC..376..543D}, which is parametrized by the inner temperature of the accretion disk, computed using the formula from \citetalias{2007MNRAS.377.1187P}:
\begin{equation}
T_{\rm in} = 1.6 \times m^{-\frac{1}{4}}\left(1 - 0.2\dot{m}^{-\frac{1}{3}}\right) \ \text{keV}.
\label{eq:Tin}
\end{equation}
\section{Results}
\subsection{Model predictions for the ionizing photons from ULXs}
\label{txt:resultssingle}
\begin{figure}
\centering
\includegraphics[width=0.97\columnwidth]{figures/Lobs_mdot.pdf}
\includegraphics[width=0.97\columnwidth]{figures/Lobs_b.pdf}
\includegraphics[width=0.97\columnwidth]{figures/mdot_Rph.pdf}
\includegraphics[width=0.97\columnwidth]{figures/mdot_Tph.pdf}
\caption{
Solutions of Eqs. \eqref{eq:eddington}-\eqref{eq:Tin} for different CO masses (see legend) with $L_{\rm obs}$ in the range $2{-}100\times10^{39}\rm\,erg\,s^{-1}$. From top to bottom:
\textbf{(i)}
The mass-transfer rate ($\dot{m}$) of a ULX as a function of the isotropic-equivalent luminosity when observed face-on ($L_{\rm obs}$).
\textbf{(ii)}
The beaming factor ($b$) as a function of $L_{\rm obs}$; the flattening occurs at $\dot{m}{=}8.5$.
\textbf{(iii)}
The radius of the region producing the soft component as a function of $\dot{m}$, for different models (line styles; see lower legend).
\textbf{(iv)}
The effective temperature of the black-body region as a function of the $\dot{m}$; the dotted-dashed line shows the inner temperature of the disk.
}
\label{fig:4plots}
\end{figure}
For a grid of observed luminosities ($2{\times}10^{39}$--$10^{41}\,\rm erg\,s^{-1}$), and for three different CO masses (1.4, 8, and $20\,M_\odot$), using Eqs. \eqref{eq:eddington}-\eqref{eq:Tin}, we compute the $\dot{m}$ and $b$, as well as the photosphere radius and temperature for three cases: the \citetalias{1973A&A....24..337S} model, and the \citetalias{2007MNRAS.377.1187P} model with two values for the parameter $e_{\rm w}$ (see Fig.~\ref{fig:4plots}). As expected, the $T_{\rm ph}$ from the \citetalias{2007MNRAS.377.1187P} model is lower than the $T_{\rm eff}$ from \citetalias{1973A&A....24..337S} because of the larger photosphere radius compared to the $R_{\rm sp}$. We find that the total energy of the soft component from \citetalias{2007MNRAS.377.1187P} is ${\sim}L_{\rm Edd}/3$.
For three different ionization potentials, $E_{\rm ion}$ (13.6\,eV for \ion{H}{I}, 54.4\,eV for \ion{He}{II} and 96.6\,eV for \ion{Ne}{V}), we compute the rate of ionizing photons
$
Q_{\rm ion}{=}\int_{E_{\rm ion}}^{E_{\rm max}} N_{\rm E}(E)\,dE,
$
where $N_E$ is the number of emitted photons with energy between $E$ and $E{+}dE$, and $E_{\rm max}{=}300\,\rm eV$. In Fig.~\ref{fig:Q_bb_disk} we show the number of \ion{He}{II}-ionizing photons as a function of the observed luminosity, and the relative contribution of the black-body component. The number of \ion{H}{I} and \ion{Ne}{V}-ionizing photons are higher by $0{-}5\%$, and lower by $5{-}20\%$, respectively, depending on the observed luminosity and the model.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{figures/Q_bb_disk_HeII.pdf}
\caption{\ion{He}{II} ionization from ULXs with different CO masses (colors; see top legends) and spectrum models (line styles; see bottom legends), as a function of the face-on luminosity of the source.
\textit{Top}:
The fraction of the photons emerging from the black-body (bb) component with respect to the total (sum of black body and multi-color disk).
\textit{Bottom}:
The rate of \ion{He}{II}-ionizing photons from the black-body and disk component, in the range $54.4{-}300\,\mathrm{eV}$.
}
\label{fig:Q_bb_disk}
\end{figure}
\subsection{Comparison between ULX and stellar populations}
\begin{figure*}
\centering
\includegraphics[width=\columnwidth]{figures/Q_comparison_HeII.pdf}
%
\includegraphics[width=\columnwidth]{figures/Q_comparison_NeV.pdf}
%
\caption{The rate of \ion{He}{II} (left) and \ion{Ne}{V} (right) ionizing photons from ULX populations in galaxies as a function of the gas-phase metallicity (solar value indicated with vertical magenta lines). The black lines show the prediction from stellar populations using \textit{BPASS} models. The colored solid lines depict the mean value from ULXs assuming three different combinations of CO masses and spectrum models (see legend), covering the full range of our (nine) estimates. The dashed lines show the upper 99\% limit to give a sense of the scatter due to the stochastic nature of ULXs.
\label{fig:BPASS_Q}
}
\end{figure*}
We compute the rate of ionizing photons from underlying stellar populations using \textit{BPASS} \citep{2017PASA...34...58E} spectra for different metallicities (see Fig.~\ref{fig:BPASS_Q}), assuming continuous star-formation for 100\,Myr. We adopt the \textit{BPASS} value for the solar metallicity, $Z_\odot{=}0.02$.
The ionizing power of a ULX population in a given galaxy depends on the number of ULXs and their individual spectra. Our results on the rate of ionizing photons from ULXs as a function of the observed X-ray luminosity (\S\ref{txt:resultssingle}), in combination with XLFs \citep[e.g.,][]{Lehmer19}, provide a handle on the ionizing power of ULX populations.
To get a representative galaxy sample, such as the one used in the recent ULX demographic study of \citet{Kovlakas20}, we select all star-forming galaxies within a distance of $40\rm\,Mpc$, with reliable SFR and metallicity estimates in the Heraklion Extragalactic Catalogue (\textit{HECATE}; \citealt{2021MNRAS.506.1896K}).
Since the \textit{HECATE} uses infrared SFR indicators and metallicities based on optical emission-line ratios, which are biased in the case of passive galaxies or in the presence of active galactic nuclei, we only select the 1,061 objects without nuclear activity, and \textit{SDSS} colors $g{-}r{<}0.65\,\rm mag$ (with a cut in the uncertainties in the photometry: $e_g, e_r{<}0.1\rm\,mag$; Kyritsis et al., in preparation). We use the \lq{}homogenized SFR indicator\rq{} which combines five different SFR indicators in the \textit{HECATE}. The SFR in this sample spans from $3.9\times 10^{-2}$ to $9.9\,M_\odot\,\rm yr^{-1}$, which is consistent with the range of SFRs in the calibration of the homogenized SFR indicator in the \textit{HECATE} (see fig. 5 in \citealt{2021MNRAS.506.1896K}), but most importantly falls in the region where the indicator is linearly correlated with SED-based SFR estimates for galaxies with $g{-}r{<}0.65$ (see fig. 6 in \citealt{2021MNRAS.506.1896K}). The metallicity\footnote{We convert the \textit{HECATE} gas-phase metallicities, $12{+}\log_{10}\left(O/H\right)$ to $Z$ using $12{+}\log_{10}\left(O/H\right)_\odot{=}8.69$ \citep{2009ARA&A..47..481A} and $Z_\odot{=}0.02$ \citep{2017PASA...34...58E}.} in our sample is in the range $\left[0.0025, 0.032\right]$.
For each galaxy, we sample the XLF (see Eq.~\ref{eq:lehmer}) above $2{\times}10^{39}\,\rm erg\,s^{-1}$. Each XLF is scaled for the SFR of the galaxy, and for its metallicity using the fit from \citet{Douna15}. However, we note that both the normalization and the shape of the HMXB XLF might depend on the metallicity \citep{Lehmer21}.
The XLFs are also corrected for the geometrical beaming. Under the beaming scenario, not all ULXs are observed due to the non-favorable angles towards us. While this has no effect on the integrated observed X-ray luminosity (the number of the sources is decreased by a factor of $b$, but their observed luminosity increases by $b$ as well), the intrinsic population of ULXs is underestimated. For a given CO mass, the beaming factor is a function of the luminosity (cf. Fig.~\ref{fig:4plots}) and therefore, the \lq{}unbeamed\rq{} XLF is:
$
\frac{dN_{\text{corr}}(L_{38})}{dL_{38}} = b^{-1}(L_{38}) \frac{dN(L_{38})}{dL_{38}}
$.
We sample each galaxy's XLF 1000 times to quantify the stochasticity characterizing the high-end of the XLF.
For each galaxy, we estimate the expected number of ULXs by integrating the XLF above $L_{\rm lim}$, and use this value as the mean of the Poisson distribution (typically $0.1{-}10$ ULXs depending on the properties of the galaxies) from which we sample the 1000 numbers of ULXs corresponding to each iteration. Then, for each iteration, we sample from the XLF to get a list of luminosities.
Using the results from \S\ref{txt:resultssingle} we sum the $Q_{\rm ion}$ of the ULXs (based on their observed luminosity; see bottom panel of Fig.~\ref{fig:Q_bb_disk}) in each iteration and galaxy, for all ionization potentials, CO masses, and models. In order to show the trend with the metallicity, we bin the galaxies by metallicity with bin edges at metallicity values that correspond to \textit{BPASS} results, 0.002, 0.003, 0.004, 0.006, 0.008, 0.01, 0.014, 0.02, 0.03, and 0.04, and compute the average rate of ionizing photons, as well as the 99\% percentile.
In Fig.~\ref{fig:BPASS_Q} we show the rate of ionizing photons for only three combinations of CO masses and models (everything else falls in-between), and only for the \ion{He}{II} and \ion{Ne}{V} (the Q for the \ion{H}{I} is smaller than the stellar one by ${\sim}4\,\rm dex$). The solid lines correspond to the mean value in the aforementioned metallicity bins, while the dashed ones, to the 99\% percentile showing how the stochastic nature of ULXs may lead to \ion{He}{II} emitters.
Only a small fraction of galaxies, and under a favorable scenario (massive CO, $e_w{=}0.7$), host ULX populations with \ion{He}{II}-ionizing power comparable to that from the stellar populations. Depending on the CO mass and the adopted model, the \ion{Ne}{V}-ionizing power of ULX populations can match, or even exceed by 2\,dex that of the stellar populations. The scatter of an order of magnitude is caused by the stochastic nature of ULXs and the shape of the XLF. Specifically, in low-metallicty galaxies, which in general are characterized by low SFRs, the ULX content is small (${\sim}0.1$ ULXs) and therefore they often do not host a ULX. On the other hand, actively star-forming galaxies, with higher metallicity, are expected to host ${\gtrsim}1$ ULXs, but due to the shallow slope of the XLF, their luminosities and ionizing power covers a wide range of values.
\subsection{Average spectrum of ULX populations}
\label{txt:averagespectrum}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{figures/avg_spectrum.pdf}
\caption{The average intrinsic spectrum from ULX populations assuming different CO masses (color) and spectrum models (line style), as well as stellar populations from \textit{BPASS} (black line for solar metallicity and gray for one-tenth of the solar metallicity). All spectra are normalized by the SFR, and the metallicity ($Z$); since we assume that the XLF scales with $Z_\odot/Z$, there is no variation in the ULX spectra, contrary to the stellar populations. The units in the $y$-axis are the same as the provided data (see Table~\ref{tab:average}). The ionization potentials of \ion{H}{I}, \ion{He}{II} and \ion{Ne}{V} are indicated with vertical lines. The shaded areas denote the regions in the spectrum that we do not consider as ionizing when calculating $Q_{\rm HI}$, $Q_{\rm HeII}$ and $Q_{\rm NeV}$ (see Figs.~\ref{fig:Q_bb_disk} and \ref{fig:BPASS_Q}).
}
\label{fig:avg_spectrum}
\end{figure*}
While in the previous section we focused on the ionizing power of stochastic ULX populations hosted in individual galaxies, here we construct a SFR-scaled and metallicity-dependent average spectrum of ULXs in star-forming galaxies that can be used as an input to IGM-heating and cosmic X-ray background studies \citep[e.g.,][]{2018ApJ...869..159U}.
To do so, we sum the XLF-weighted theoretical spectra for ULXs with $2{\times}10^{39}\,\rm erg\,s^{-1}{<}L_{\rm obs}{<}10^{41}\,\rm erg\,s^{-1}$. We also normalize the spectra for the metallicity since we assume that the XLF is linearly anticorrelated with the metallicity \citep[e.g.,][]{Douna15}. Fig.~\ref{fig:avg_spectrum} shows the spectra (see Table~\ref{tab:average} for the data) along with the stellar models from \textit{BPASS} for two different metallicity values.
\begin{table*}
\centering
\small
\caption{Average ULX spectra scaled by the SFR and metallicity ($Z$),
corresponding to all nine combinations of CO masses (1.4, 8, and $20\,M_\odot$), and black-body models (SS73, $e_w{=}0.3$, and $e_w{=}0.7$). The first and second columns are the photon energy ($E_{\rm ph}$) and wavelength ($\lambda$), respectively, while the rest are the decimal logarithm of the photon flux in the $(\lambda, \lambda+1\rm\AA$ bin for each combination of parameters. I.e., ${\rm SFR}\times(Z/Z_\odot)\times\sum_{i=1}^{10^4} 10^{y_i}{E_{\rm ph,i}}$, where $i$ is the row index and $y_i$ is a given model column, gives the total intrinsic luminosity (in eV) in the range $[1, 10^4]\rm \AA$ for a ULX population in an underlying stellar population of a given SFR and metallicity.
The full table with 100,000 entries is available in electronic form
at the CDS via anonymous ftp to \url{cdsarc.u-strasbg.fr} (130.79.128.5)
or via \url{http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/}.
}
\label{tab:average}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{2}{|r|}{CO mass ($M_\odot$)}
& 1.4 & 1.4 & 1.4
& 8.0 & 8.0 & 8.0
& 20.0 & 20.0 & 20.0 \\%\hline
\multicolumn{2}{|r|}{Model}
& SS73 & $e_w{=}0.3$ & $e_w{=}0.7$
& SS73 & $e_w{=}0.3$ & $e_w{=}0.7$
& SS73 & $e_w{=}0.3$ & $e_w{=}0.7$ \\\hline
$E_{\rm ph}$ & $\lambda$ &
\multicolumn{9}{|c|}{Decimal logarithm of photon flux in $\left(\lambda, \lambda+1\AA\right)$ bin, scaled by SFR and metallicity} \\
(eV) & (\AA) &
\multicolumn{9}{|c|}{$\rm (photon\,s^{-1}\,Myr^{-1}\,yr \times Z/Z_\odot)$}
\\\hline
0.123984 & 100000 & 32.11996 & 32.20639 & 32.21280 & 33.28700 & 33.39010 & 33.39158 & 33.84989 & 33.98321 & 33.98385 \\
0.123986 & 99999 & 32.11997 & 32.20640 & 32.21281 & 33.28701 & 33.39010 & 33.39159 & 33.84989 & 33.98321 & 33.98386 \\
0.123987 & 99998 & 32.11998 & 32.20640 & 32.21281 & 33.28702 & 33.39011 & 33.39160 & 33.84990 & 33.98322 & 33.98386 \\
0.123988 & 99997 & 32.12506 & 32.21149 & 32.21790 & 33.29210 & 33.39519 & 33.39668 & 33.85498 & 33.98830 & 33.98895 \\
0.123989 & 99995 & 32.11999 & 32.20642 & 32.21283 & 33.28703 & 33.39012 & 33.39161 & 33.84991 & 33.98323 & 33.98387 \\
\dotfill & \dotfill & \dotfill & \dotfill & \dotfill & \dotfill & \dotfill & \dotfill & \dotfill & \dotfill & \dotfill \\
\hline
\end{tabular}
\end{table*}
\subsection{Average effect on IGM heating}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{figures/softX.pdf}
\caption{The average soft X-ray luminosity ($L_{\rm 0.3-1\,keV}$) of ULX populations as a function of the host galaxy's metallicity (solar value indicated with a vertical magenta line), assuming CO masses of $20\,M_\odot$ (top, intense lines) and $1.4\,M_\odot$ (bottom, pale lines) and spectrum described by the \citetalias{2007MNRAS.377.1187P} model with $e_{\rm w}{=}0.3$. Black lines correspond to the intrinsic (unabsorbed) X-ray luminosity, while blue and orange lines account for the absorption in the galaxy's ISM, with $n_{\rm H}{=}10^{21}$ and $10^{22}\,\rm cm^2$ respectively. Dashed, dotted and solid lines indicate the black-body component, disk component and their sum, respectively. (For $20\,M_\odot$, the black dashed and dotted lines coincide, indicating equal component contributions.) The gray band denotes the $L_{\rm X{<}2\,keV}$ 68\% confidence interval constrained from the cosmic 21-cm signal for galaxies at $z{\sim}8$ \citep{HERA22}.}
\label{fig:softX}
\end{figure}
Using the average spectra in \S\ref{txt:averagespectrum}, we quantify the effect of ULX populations in IGM-heating at the epoch of cosmic heating. We focus on the $0.3{-}1\,\rm keV$ part of the intrinsic spectrum, since at lower energies the photons are absorbed by the ISM of the host galaxies, whereas higher energy photons are expected to penetrate the IGM and not contribute significantly in the heating. In addition, we study the effect of the metallicity-dependent photo-electric absorption of the ISM by applying the model \texttt{vphabs} from \texttt{xspec} for different values of hydrogen column density ($n_{\rm H}$).
For all CO masses, the soft X-ray luminosity as a function of metallicity depends weakly on the spectrum model (SS37, \citetalias{2007MNRAS.377.1187P}; within a factor two). In Fig.~\ref{fig:softX} we show the results for the two extreme CO masses (1.4 and $20\,M_\odot$) and for $e_w{=}0.3$. The normalization is roughly linear to the accretor mass, since the models predict harder spectra for lower CO masses (the black-body component is nearly Eddington-limited).
While the exact distribution of CO masses is important to constrain the soft X-ray part of average ULX spectra, we find that the black-body and disk components have comparable contribution, with a factor of $1.1$--$9.3$ underestimation if we neglect the black-body component.
\section{Discussion}
Assuming that the spectra of ULXs are the sum of a black body (using the \citetalias{1973A&A....24..337S} and \citetalias{2007MNRAS.377.1187P} models) and a disk component, we quantify the ionizing power of individual ULXs for different CO masses. We find a wide range in $Q_{\rm HeII}{\sim} 10^{45-48}\rm\,photon\,s^{-1}$, which is considerably lower than the rates from He-emitters such as the I\,Zw\,18 galaxy, ${\sim}1.33{\times}10^{50}\rm\,photon\,s^{-1}$ \citep{Kehrig15}. The ULX in this galaxy has been investigated as the origin of the ionizing radiation \citep[e.g.,][]{Schaerer19}, however difficulties in constraining the EUV part of the spectrum of a ULX \citep[e.g.,][]{Simmonds21} might result in stark differences in its ionization power. Our theoretical ULX SEDs are not capable of producing strong \ion{He}{II} ionization ($Q_{\rm HeII}/L_{\rm X}{\leq}4{\times}10^{7} \rm photon\, erg^{-1}$) as invoked in previous studies
(e.g., $2{\times}10^{10}\,\rm photon\,erg^{-1}$; \citealt{Schaerer19})
In Fig.~\ref{fig:compothers}, we compare the model spectra with previous studies, for a ULX with $L_{\rm obs}{=}10^{40}\,\rm erg\,s^{-1}$. Specifically, we show the spectrum of the disk and the black-body component of a ULX with observed (face-on) luminosity $10^{40}\rm\,erg\,s^{-1}$ for two different CO masses, and two different values of $e_{\rm w}$ in the \citetalias{2007MNRAS.377.1187P} model.
For the same observed, isotropic-equivalent luminosity, different CO masses correspond to different beaming factors. For this reason, in the case of the $1.4\,M_\odot$ accretor ($b{\simeq}0.076$), the bolometric luminosity of the source is lower than in the case of the $20\,M_\odot$ accretor ($b{\simeq}0.81$), leading to a lower normalization for the former.
While the shape of the black-body component is affected by the adopted value for $e_{\rm w}$, the disk component is the same for a given CO mass (the corresponding lines in Fig.~\ref{fig:compothers} are not repeated for different values of $e_{\rm w}$.)
We also show the models from \citet{Simmonds21} and \citet{Senchyna20}, scaled at the same luminosity, the normalization of which do not depend on the beaming factor, since geometrical collimation is not considered in these studies. We note that these models are overplotted to allow for qualitative comparisons against recent studies of the contribution of ULXs and XRBs in nebular emission.
Our individual ULX SEDs show that the photon flux at ${\sim}54.4\,\rm eV$ varies by one order of magnitude depending on the parameters of the models. However, they are weaker by many orders of magnitude compared to the literature models. We should stress again that no direct observations exist in this range, and in all cases, the ULX SEDs are extrapolated from theoretical or empirical models, calibrated to higher energies (${\gtrsim}300\,\rm eV$).
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{figures/comparison.pdf}
\caption{The disk (thick lines) and black-body (BB; thin lines) components of the intrinsic spectra of a ULX with $L_{\rm obs}{=}10^{40}\rm\,erg\,s^{-1}$ for two different CO masses, $1.4\,M_\odot$ (orange) and $20\,M_\odot$ (blue), and two different values of $e_{\rm w}$ for the \citetalias{2007MNRAS.377.1187P} model (line styles, see legend).
For comparison, we overplot in purple the DIS (dotted line) and BMC (dashed line) models from \citet{Simmonds21}, as well as the model for a BH of mass $10\,M_\odot$ (continuous line) from \citet{Senchyna20}, scaled for the same source luminosity.
We note that in our models, the normalization of the intrinsic spectra depends on the CO mass: the smaller the CO mass is, the smaller the beaming factor is for the same observed, face-on luminosity.
}
\label{fig:compothers}
\end{figure*}
When considering the ionizing power of ULX populations, the geometrical beaming model results into two opposing effects. On one hand, the bolometric luminosity of the sources, and consequently their ionization power is lower than what is inferred from the observed luminosities. On the other hand, a fraction of the ULXs cannot be observed in the X-rays, but nevertheless ionizes the ISM. While the X-ray budget is preserved, the picture is more complicated in the EUV part of the spectrum: its shape and the beaming factor are correlated through their dependence on the $\dot{m}$. Consequently, the use of observed spectra and population synthesis techniques \citep[e.g.,][]{Fragos13} might underestimate the ionizing power of ULXs. Using the beaming-corrected HMXB XLF to anchor our analysis, we derive the underlying distribution of $\dot{m}$ and calculate the ionizing power of ULX populations in a realistic sample of galaxies. We find that in only a small fraction (${\sim}1\%$) of galaxies the ULXs compete the underlying stellar populations in \ion{He}{II} emission, in agreement with recent studies \citep[e.g.,][]{Senchyna20}, while they are significant in the case of \ion{Ne}{V} \citep[e.g.,][]{Simmonds21}. On the other hand, observational studies may also put constraints on the contribution of the hot gas component in star-forming galaxies, which is found to be comparable to the stellar component in ionizing the \ion{He}{II} in low-metallicity galaxies \citep[e.g.,][]{Lehmer22}.
From our modeling, the \ion{Ne}{V}-ionizing power of ULXs points at the possibility of constraints for the EUV emission from ULXs with the use of emission lines associated with high-ionization potentials. We encourage the study of high-ionization emission line galaxies, their emission processes, and in general systematic studies constraining the ionizing power and spectra of ULXs \citep[e.g.,][]{Izotov21}.
In addition, we compute the average spectrum of the ULX population in a galaxy normalized by its SFR and metallicity, for the different theoretical spectra, and adopted CO masses. These spectra can be used as an input for IGM-heating studies, and be compared to observational constraints from the cosmic 21-cm signal \citep[e.g.,][]{HERA22}. As an example, we calculate the heating power of ULX populations via their soft X-ray emission. We find that the contribution from the black-body component is comparable to the disk component, highlighting the importance of the former in quantifying the effect of ULXs in the early Universe. However, the normalization in this case depends strongly on the CO mass, and only weakly in the spectrum model among the ones investigated here.
Concluding, the dependence of our estimates on the shape of the spectra, the stochastic nature of ULXs, and the CO mass distribution, result in a 2\,dex scatter in the ionizing and heating power of ULX populations. Despite the success of analytical models in interpreting key properties of accreting sources, hydrodynamical simulations are necessary for investigating the radiative and mechanical feedback in super-critical accretion disks \citep[e.g.,][]{Sadowski15}.
Due to the computational cost of these simulations, only a handful of fiducial systems has been investigated. We encourage such efforts to continue, especially in the case of NS accretors where the presence of magnetic fields is important, with some sources showing lack of beaming \citep[e.g.,][]{Binder18}. Combining accurate spectral models with detailed binary population synthesis techniques \citep[e.g.,][]{2022arXiv220205892F} will provide stringent constraints on the ionizing power of ULXs, and their contribution in the heating of the early Universe.
\begin{acknowledgements}
We would like thank the anonymous referee for critical comments which improved the paper.
This work was supported by the Swiss National Science Foundation Professorship Grant (PP00P2\_176868; PI Fragos). KK acknowledges support from the Federal Commission for Scholarships for Foreign Students for the Swiss Government Excellence Scholarship (ESKAS No. 2021.0277).
\end{acknowledgements}
\bibliographystyle{aa}
|
1,116,691,500,772 | arxiv | \section{Introduction}\label{sec:intro}
Nowadays we are witnessing a proliferation of industrial and research works in the field of autonomous small-size unmanned aerial vehicles (UAVs)~\cite{dunkley14iros,40gSTM32OpticFlow,palossi2017ultra,briod2013optic,7080923,palossi2017target,kang2019generalization,7487496}.
This considerable effort can be easily explained by the potential applications that would greatly benefit from intelligent miniature robots.
Many of these works, despite they refer to them self as ``autonomous'', are actually ``automatic'' but not independent from some external ad-hoc signal/computation (e.g., GPS, RFID signals, and ground-stations).
We belive that achieving full independence is the key condition to be truly autonomous.
\begin{figure}[t]
\centerline{\includegraphics[width=\columnwidth]{./imgs/drone2.png}}
\caption{Our prototype deployed on the filed. It is based on the \textit{Crazyflie 2.0} nano-quadrotor extended with our \textit{PULP-Shield}. The system can run the \textit{DroNet}~\cite{dronet} CNN for autonomous visual navigation up to \SI{18}{\hertz} using only onboard resources.}
\label{fig:prototype}
\end{figure}
Autonomous pocket-size drones can be particularly versatile and useful, acting as sensor nodes that acquire information, process and understand it, and use it to interact with the environment and with each other.
The ``ultimate'' will be capable of autonomously navigating the environment and, at the same time, sensing, analyzing, and understanding it~\cite{iotUAV_survey}.
In the context of wireless sensor networks (WSNs), such a miniaturized robotic-helpers can collect the data from a local WSN and bridge them towards the external world.
Moreover, a swarm of such intelligent and ubiquitous nano-drones can quickly build a remote sensing network in an emergency context, where their small size enables inexpensive and fast deployment on the field, as well as reaching location inaccessible for human operators or standard-size drones.
The tiny form-factor of nano-drones (i.e., featuring only few centimeters in diameter and few tens of grams in weight) is ideal both for indoor applications where they should safely operate near humans and for highly-populated urban areas, where they can exploit complementary sense-act capabilities to interact with the surroundings (e.g., smart-building, smart-cities, etc.).
To enable such ambitious scenarios many challenging problems must be addressed and solved.
Nano-scale commercial off-the-shelf (COTS) quadrotors still lack a meaningful level of autonomy, contrary to their larger counterparts~\cite{dronet, Lin_2017, loianno2018special}, since their tiny power envelopes and limited payload do not allow to host onboard adequate computing power for sophisticated workloads.
Of the total power available on a UAV (listed in Table~\ref{tab:taxonomy} for four classes of vehicles), Wood~et~al.~\cite{Wood2017} estimate that only up to 5\% is available for onboard computation, and payloads of maximum $\sim$25\% of the total mass can be allotted to the electronics.
\begin{table}[t]
\renewcommand{\arraystretch}{1.5}
\caption{Rotorcraft UAVs taxonomy by vehicle class-size.}
\label{tab:taxonomy}
\centering
\footnotesize
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c|c|c|c|}
\hline
Vehicle Class & $\oslash$ : Weight [cm:kg] & Power [W] & Onboard Device\\
\hline
\text{\textit{std-size} \cite{dronet}} & $\sim$ 50 : $\geq$ 1 & $\geq$ 100 & Desktop\\
\hline
\text{\textit{micro-size} \cite{conroy2009}} & $\sim$ 25 : $\sim$ 0.5 & $\sim$ 50 & Embedded\\
\hline
\text{\textit{nano-size} \cite{40gSTM32OpticFlow}} & $\sim$ 10 : $\sim$ 0.01 & $\sim$ 5 & MCU\\
\hline
\text{\textit{pico-size} \cite{Wood2017}} & $\sim$ 2 : $\leq$ 0.001 & $\sim$ 0.1 & ULP\\
\hline
\end{tabular}
}
\end{table}
The traditional approach to autonomous navigation of a UAV is the so-called \textit{localization-mapping-planning} cycle, which consists of estimating the robot motion using either off-board (e.g., GPS) or onboard sensors (e.g., visual-inertial sensors), building a local 3D map of the environment, and planning a safe trajectory through it~\cite{loianno2018special}.
These methods, however, are very expensive for computationally-constrained platforms.
Recent results have shown that much lighter algorithms, based on convolutional neural networks (CNNs), are sufficient for enabling basic reactive navigation of small drones, even without a map of the environment~\cite{dronet, giusti2016machine}.
However, their computational and power needs are unfortunately still above the allotted budget of current navigation engines of nano-drones, which are based on simple, low-power microcontroller units (MCUs).
In this work, we introduce several improvements over the state of the art of nano-scale UAVs.
First, we introduce the design of a low-power visual navigation module, the \textit{PULP-Shield}, featuring the high-efficiency \textit{GreenWaves Technologies GAP8 SoC}, a ULP camera and Flash/DRAM memory, compatible with the \textit{CrazyFlie} 2.0 nano-UAV.
The full system is shown in the field in Figure~\ref{fig:prototype}.
We propose a methodology for embedding the CNN-based \textit{DroNet}\cite{dronet} visual navigation algorithm, originally deployed on standard-sized UAVs with off-board computation, in a nano-UAV with fully onboard computation.
We demonstrate how this methodology yields comparable quality-of-results (QoR) with respect to the baseline, within a scalable power budget of \SI{64}{\milli\watt} at 6 frames per second (fps), up to \SI{272}{\milli\watt} at 18 fps.
We prove in the field the efficacy of this methodology by presenting a closed-loop fully functional demonstrator in the supplementary video material, showing autonomous navigation on \textit{i}) a \SI{113}{\meter} previously unseen indoor environment and \textit{ii}) collision robustness against the appearance of a sudden obstacle at a distance of \SI{2}{\meter} while flying at \SI{1.5}{\meter/\second}.
To the best of our knowledge, our design is the first to enable such complex functionality in the field on a nano-UAV consuming $<$\SI{100}{\milli\watt} for electronics.
To foster further research on this field, we release the PULP-Shield design and all code running on GAP8, as well as datasets and trained networks, as publicly available under liberal open-source licenses.
\section{Related Work}\label{sec:related}
The traditional approach to the navigation of nano-drones requires to offload the computation to a remote base-station~\cite{dunkley14iros,7080923,kang2019generalization}, demanding high-frequency video streaming, which lowers reliability and imposes constraints on maximum distance, introduces control latency and is poorly scalable.
On the other hand, COTS nano-size quadrotors, like the \textit{Bitcraze Crazyflie 2.0} or the \textit{Walkera QR LadyBug}, usually make use of very simple computing devices such as single-core microcontroller units (MCUs) like the \textit{ST Microelectronics} STM32F4~\cite{40gSTM32OpticFlow,dunkley14iros,7487496}.
Autonomous flying capabilities achievable on these platforms are, to the date, very limited.
In~\cite{40gSTM32OpticFlow} the proposed obstacle avoidance functionality requires favorable flight conditions (e.g., low flight speed of \SI{0.3}{\meter/\second}).
The solutions proposed in~\cite{briod2013optic,palossi2017ultra} are limited to hovering and do not reach the accuracy of computationally expensive techniques leveraged by powerful standard-size UAVs.
\cite{7487496} addresses only state estimation -- a basic building block of autonomous UAVs, but far from being the only required functionality.
An emerging trend in the evolution of autonomous navigation systems is the design and development of application-specific integrated circuit (ASIC) addressing specific navigation tasks~\cite{navion,CNN-SLAM}.
ASICs deliver levels of performance and energy efficiency for the specific tasks addressed that cannot be achieved by typical nano-UAV computing platforms for workloads such as visual odometry~\cite{navion} or simultaneous localization and mapping (SLAM)~\cite{CNN-SLAM}.
However, ASICs only accelerate a part of the overall functionality, requiring pairing with additional circuits for complementary onboard computation as well as for interacting with the drone's sensors.
Moreover, to date, systems based on these ASICs have not yet been demonstrated on board a real-life flying nano-UAV.
In this work, we demonstrate a sophisticated visual navigation engine that is entirely based on a general-purpose parallel, ultra-low power (PULP) computing platform, and works in closed-loop in the field within the power envelope and payload of nano-scale UAVs ($\sim$\SI{0.2}{\watt} and $\sim$\SI{15}{\gram}, respectively).
\section{Implementation}\label{sec:impl}
This section gives insight on DroNet, the key driving algorithm used by our visual navigation engine, on the hardware platform utilized in this work (the GAP8 SoC), and on how the algorithm was modified to fit within the constrained hardware platform while keeping the same original accuracy and the performance.
\subsection{The algorithm: DroNet}
\begin{figure}[t]
\centerline{\includegraphics[width=\columnwidth]{./imgs/dronet_original_V4.png}}
\caption{\textit{DroNet}~\cite{dronet} CNN topology.}
\label{fig:dronet-archi}
\end{figure}
The key driver for our proposed autonomous visual navigation engine is \textit{DroNet}: an algorithm proposed initially by Loquercio~et~al.~\cite{dronet} based on a convolutional neural network (CNN) whose topology is inspired on ResNet~\cite{he2016deep}.
The original DroNet was deployed on top of a commercial standard-size UAV streaming camera frames to an external laptop.
DroNet is trained to convert an unprocessed input image from a camera into two high-level pieces of information: \textit{i}) an estimation of the probability of \textit{collision} with an obstacle, which in turn can be used to determine the forward target velocity of the UAV; \textit{ii}) the desired \textit{steering} direction, following visual cues from the camera such as the presence of obstacles, white lines on the floor or in the street, etc.
Figure~\ref{fig:dronet-archi} reports the full topology of DroNet, which is shared between the two steering and collision tasks up to the penultimate layer.
To train the network\footnote{Following~\cite{dronet}, the steering and collision tasks were associated to mean squared error (MSE) and binary cross-entropy (BCE) losses, respectively. The Adam optimizer was used, with starting learning rate of $1^{-3}$ and learning rate decay per epoch equal to $1^{-5}$.
We refer to Loquercio~et~al.~\cite{dronet} for further details on the training methodology.} two openly available datasets were used -- \textit{Udacity}\footnote{https://www.udacity.com/self-driving-car}, a dataset designed to train self-driving cars, for the steering task, and the \textit{Z\"{u}rich bicycle} dataset\footnote{http://rpg.ifi.uzh.ch/dronet.html} for the collision task.
At inference time, the steering direction $\theta_{steer}$ and collision $P_{coll}$ outputs of the network are connected to the UAV control, influencing the target yaw rate $\omega_{yaw,target}$ and the target forward velocity $v_{x,target}$ through a simple low-pass filtering scheme:
\begin{align}
v_{x,target}[t] &= \alpha \cdot v_{\max} \cdot (1 - P_{coll}[t]) \\
&+ (1-\alpha) \cdot v_{x,target}[t-1]\notag\\
\omega_{yaw,target}[t] &= \beta \cdot \theta_{steer}[t] \\
&+ (1-\beta) \cdot \omega_{yaw,target}[t-1]\notag
\end{align}
where the parameters have default values $\alpha=0.3$ and $\beta=0.5$.
\subsection{The platform: GAP8 SoC}
While commercial off-the-shelf microcontrollers used in the most common nano-UAV platforms have acceptable computing capabilities of their own, these could not be enough to achieve autonomous flight functionality, which requires workloads in the order of 100 million -- 10 billion operations per second~\cite{dronet}.
Moreover, these microcontrollers are typically tasked with many computationally simple but highly critical real-time tasks to estimate the current kinematic state of the UAV, predict its motion and control the actuators.
To avoid tampering with this mechanism, we chose to execute our visual navigation engine on a different platform than the central nano-UAV microcontroller, acting as a specialized accelerator~\cite{contiDATE16} based on the GreenWaves Technologies GAP8 system-on-chip (SoC).
\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{./imgs/gap8_archi_V2.png}
\caption{Architecture of the PULP-GAP8 embedded processor.}
\label{fig:gap8_archi}
\end{figure}
GAP8 is a commercial embedded application processor based on the PULP open source architecture\footnote{http://pulp-platform.org} and the RISC-V open ISA.
Figure~\ref{fig:gap8_archi} shows the architecture of GAP8 in detail.
The GAP8 SoC is organized in two subsystems and power domains, a \textit{fabric controller} (FC) with one RISC-V core acting as an on-SoC microcontroller and a \textit{cluster} (CL) serving as an accelerator with 8 parallel RISC-V cores.
All the cores in the system are identical and support the RV32IMC instruction set with SIMD DSP extensions (e.g., fixed-point dot product) to accelerate linear algebra and signal processing.
The FC is organized similarly to a microcontroller system, featuring an internal clock generator, \SI{512}{\kilo\byte} of SRAM (L2 memory), a ROM for boot code, and an advanced I/O subsystem ($\mu DMA$) that can be programmed to autonomously move data between a wide set of I/O interfaces (including SPI, UART, I2C, L3 HyperRAM) and the L2 memory without the core's intervention.
The CL is meant to be used to accelerate parallel sections of the application code running on GAP8.
Its 8 RISC-V cores share a single shared cache for instructions and a shared L1 scratchpad memory of \SI{64}{\kilo\byte} for data; movement of data between the latter and the L2 is manually managed by the software running on the cluster using an internal DMA controller.
This enables us to achieve maximum efficiency and utilization on typical parallel kernels with regular, predictable access patterns for data while saving the area overhead of a shared data cache.
\subsection{Optimizations for embedded deployment}
\begin{figure}[t]
\centerline{\includegraphics[width=0.9\columnwidth]{./imgs/bn_folding.png}}
\caption{Batch-normalization layer folding methodology.}
\label{fig:bn_folding}
\end{figure}
Deploying a CNN algorithm developed in a high-level framework (such as TensorFlow in the case of DroNet) to a low-power application processor such as GAP8 involves several challenges, connected with the constraints imposed by the limited available resources.
First, the navigation algorithm must be able to execute the main workload ($\sim$ 41 million of multiply-accumulate operations for one inference pass-through of DroNet) at a frame rate sufficient to achieve satisfactory closed-loop performance in control.
Furthermore, while the embedded processor typically uses a lower precision to represent data and a lower resolution input camera, the quality-of-results must remain similar to the one of the original algorithm.
These constraints impose significant modifications to the original algorithm that in the case of DroNet can be grouped in two main categories.
\subsubsection{Dataset fine-tuning \& network quantization}
To improve the generalization capabilities of the original DroNet~\cite{Razavian2014transfer} with respect to the lower-quality images coming from the embedded camera, we collected an extension for the collision dataset using directly the camera available in the final platform: a grayscale QVGA-resolution HiMax.
We collected 1122 new images for training and 228 for test/validation, which we compounded with the openly available collision dataset.
We also replaced $3\times 3$ pooling layers with $2\times 2$ ones, which yield the same overall functionality (i.e., the reduction of the spatial size of feature maps in the CNN) while being smaller and generally easier to implement as each input pixel is projected to a single output one.
Finally, to adapt the network to execution on a low-power platform without support for floating-point numbers, we switched to fixed-point data representation.
Specifically, by analyzing the dynamic range of intermediate feature maps in the original DroNet, we found that a precision of $2^{-11}$ and a range $\pm 16$ was adequate to represent activation data after batch normalization (BN) layers.
Then, we replaced all activation ReLU layers with quantization-aware equivalents~\cite{HubaraQuantizedNeuralNetworks2016} using a 16-bit Q5.11 fixed-point format.
The entire network was retrained from scratch using the same framework of the original DroNet.
\subsubsection{Batch-norm folding}
During training, batch-normalization (BN) layers are essential to keep the dynamic range of feature map activations in check (hence helping with their quantization) and to regularize the learning process, which achieves a far better results in terms of generalization than an equivalent network, particularly for what concerns the regression task of computing the desired steering.
However, during inference, the BN layers are linear and can be merged with the preceding convolutional layer by \textit{folding} it inside its weights $W$ and biases $b$.
If $\gamma$, $\beta$, $\sigma$, and $\mu$ are the normalization parameters, then:
\begin{align}
\mathrm{BN}\big(\mathbf{W}\star\mathbf{x} + \mathbf{b}\big) &= \gamma/\sigma \cdot \left(\mathbf{W}\star\mathbf{x}+\mathbf{b} -\mu\right) + \beta \notag \\
&= \left(\gamma/\sigma \cdot \mathbf{W}\right) \star \mathbf{x} + \Big(\beta+\gamma/\sigma \left(\mathbf{b} -\mu\right)\Big) \notag \\
&\doteq \mathbf{W'} \star \mathbf{x} + \mathbf{b'} \label{eq:bn_folding}
\end{align}
In DroNet, the input of each RES block is normalized in the main branch, but non-normalized in the by-pass branch, making the direct application of Equation~\ref{eq:bn_folding} more difficult, as it is not possible to directly apply it to the convolution preceding those operations.
Therefore, we proceeded as follows: for each RES block, we first apply the folding ``as if'' the input of the entire RES block was normalized by using Equation~\ref{eq:bn_folding}.
This means that each BN is folded into the previous convolution layer, e.g., for RES block 1, in the initial convolutional layer of DroNet and in the first one of the main branch.
Second, we apply \textit{inverse folding} on the by-pass convolutional layer, to counteract the folding of BNs on its inputs:
\begin{align}
\mathrm{BN}^{-1}&\big(\mathbf{W'}\star\mathbf{x} + \mathbf{b'}\big) \doteq \mathbf{W''}\star\mathbf{x} + \mathbf{b''} \\
\mathbf{W''} &\doteq \sigma/\gamma \cdot \mathbf{W'} \notag \\
\mathbf{b''} &\doteq \mathbf{b'} + \sum_{ic}\Big(\mu \cdot \sum_{fs}\mathbf{W'}\Big) - \sum_{ic}\Big(\beta \cdot \sigma/\gamma \cdot \sum_{fs}\mathbf{W'}\Big) \notag
\end{align}
where $\sum_{ic}$ and $\sum_{fs}$ indicate marginalization along the input channels dimension and along the filter's spatial dimensions, respectively.
We apply this operation sequentially to each RES block as exemplified in Figure~\ref{fig:bn_folding}.
After this operation, the BN layers can be effectively removed from the network as other layers absorb their effects.
Finally, the new weights and bias values can be quantized according to their range requirements.
In the final DroNet deployment, we quantize weights for all layers at Q2.14, except for the first bypass layer, which uses Q9.7.
\section{The PULP-Shield}\label{sec:proto}
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{./imgs/PulpShield_Model_V4.png}
\caption{Interaction between the \textit{PULP-Shield} and the \textit{CrazyFlie 2.0} nano-drone.}
\label{fig:pulp-shield-model}
\end{figure*}
Our visual navigation engine is embodied, on its hardware side, in the so-called \textit{PULP-Shield}: a lightweight, modular and configurable printed circuit board (PCB) with a highly optimized layout.
We designed the PULP-Shield to be compatible/pluggable to the \textit{Crazyflie 2.0} (CF) nano-quadrotor\footnote{https://www.bitcraze.io/crazyflie-2}.
The CF has been chosen due to its reduced size (i.e., \SI{27}{\gram} of weight and \SI{10}{\centi\meter} of diameter), its open-source and open-hardware philosophy, and the availability of extra payload (up to \SI{15}{\gram}).
The PULP-shield features a PULP-based GAP8 SoC, two Cypress \textit{HyperBus Memories}\footnote{http://www.cypress.com/products/hyperbus-memory} enabling flexible configuration and an ultra-low-power gray-scale \textit{HiMax}\footnote{http://www.himax.com.tw/products/cmos-image-sensor/image-sensors} QVGA CMOS image sensor that communicates via the parallel camera interface (PCI) protocol.
On the two BGA memory slots we mounted a \SI{64}{\mega\bit} \textit{HyperRAM} (DRAM) chip and a \SI{128}{\mega\bit} \textit{HyperFlash} memory, embodying the system L3 and the external storage, respectively.
Two mounting holes, on the side of the camera connector, allow to plug a 3D-printed camera holder that can be set either in front-looking or down-looking mode, accounting for the most common visual sensors layouts and enabling a large variety of tasks like obstacle avoidance~\cite{dronet} and visual state estimation~\cite{palossi2017ultra}, respectively.
On the shield there are also a JTAG connector for debug purposes and an external I2C plug for future development.
Two headers, located on both sides of the PCB, grant a steady physical connection with the drone and at the same time they bring the shield power supply and allow communication with the CF's main MCU (i.e., \textit{ST Microelectronics STM32F405}\footnote{http://www.st.com/en/microcontrollers/stm32f405-415.html}) through SPI interface and GPIO signals.
The form factor of our final PULP-Shield prototype, shown in Figure~\ref{fig:pulp-shield-schematic}, is 30$\times$\SI{28}{\milli\meter} and it weighs $\sim$\SI{5}{\gram} (including all components), well below the payload limit imposed by the nano-quadcopter.
The PULP-Shield embodies the \textit{Host-Accelerator} heterogeneous architectural paradigm at the ultra-low power scale~\cite{contiDATE16}, where the CF's MCU offloads the intensive visual navigation workloads to the PULP accelerator.
As reported in Figure~\ref{fig:pulp-shield-model} the interaction starts from the host, which wakes up the accelerator with a GPIO interrupt \circled{1}.
Then, the accelerator fetches from its external HyperFlash storage the binary to be executed \circled{2}.
After the ULP camera is configured via I2C \circled{3} the frames can be transferred to the L2 shared memory through the $\mu$DMA \circled{4} and this can be performed in pipeline with the computation running on the \textsc{cluster} (i.e., in \textit{double buffering} fashion).
All additional data, like the weights used in our CNN, can be loaded from the DRAM/Flash memory \circled{5} and the parallel execution can start on the accelerator \circled{6}.
Once the computation is completed the results are returned to the drone's MCU via SPI \circled{7}.
Even if the PULP-Shield has been developed specifically to fit the CF quadcopter, its basic concept and the functionality it provides are quite general and portable to any drone based on an SPI-equipped MCU.
The system-level architectural template is meant for minimizing data transfers (i.e., exploiting locality of data) and communication overhead between the main MCU and the accelerator -- without depending on the internal microarchitecture of either one.
\begin{figure}[h]
\centering
\includegraphics[width=\columnwidth]{./imgs/PULP-Shield_bot_top_3.png}
\caption{The \textit{PULP-Shield} pluggable PCB.}
\label{fig:pulp-shield-schematic}
\end{figure}
\section{Experimental Results}\label{sec:results}
In this section we present the experimental evaluation of our visual navigation engine, considering three main metrics: \textit{i}) a QoR comparison with other CNNs for autonomous navigation of UAVs, \textit{ii)} the capability of performing all the required computations within the allowed power budget and \textit{iii)} a quantitative control accuracy evaluation of the closed-loop system when deployed on the field.
All the results are based on the PULP-Shield configuration presented in Section~\ref{sec:proto}.
\subsection{CNN Evaluation}
To assess the regression performance of our modifications to the original CNN, employing the testing sequence from the Udacity dataset, we present in Table~\ref{tab:dronet_pulp_accuracy} a comparison with the state-of-the-art.
We compare our version of the DroNet network, named \textit{PULP-DroNet}, against a set of other architectures from the literature~\cite{giusti2016machine,he2016deep,xu2017end} and also against the same original DroNet model~\cite{dronet}.
Note that, we report the same accuracy/performance previously presented in~\cite{dronet} for the same reference architectures.
Our regression and classification results are gathered analyzing the testing sequence on the official PULP simulator, that precisely models the behavior of the target architecture executing the same binary deployed on the PULP-Shield.
Performance results (e.g., processing time) are instead obtained running the PULP-DroNet CNN on the actual hardware.
In Table~\ref{tab:dronet_pulp_accuracy}, explained variance (EVA) and root-mean-square error (RMSE) refer to the regression problem (i.e., steering angle) whereas Accuracy and F1-score are related to the classification problem (i.e., collision probability).
\begin{table*}[t]
\renewcommand{\arraystretch}{1.3}
\caption{Results on regression and classification task.}
\label{tab:dronet_pulp_accuracy}
\centering
\begin{tabular}{c c c c c c c | c c}
\cline{1-9}
\textbf{Model} & \textbf{EVA} & \textbf{RMSE} & \textbf{Accuracy} & \textbf{F1-score} & \textbf{Num. Layers} & \textbf{Memory [MB]} & \textbf{Processing time [fps]} & \textbf{Device}\\
\cline{1-9}
Giusti et al.~\cite{giusti2016machine} & 0.672 & 0.125 & 91.2\% & 0.823 & 6 & 0.221 & 23 & Intel Core i7\\
ResNet-50~\cite{he2016deep} & 0.795 & 0.097 & 96.6\% & 0.921 & 50 & 99.182 & 7 & Intel Core i7\\
VGG-16~\cite{xu2017end} & 0.712 & 0.119 & 92.7\% & 0.847 & 16 & 28.610 & 12 & Intel Core i7\\
DroNet~\cite{dronet} & 0.737 & 0.109 & 95.4\% & 0.901 & 8 & 1.221 & 20 & Intel Core i7\\
\textbf{PULP-DroNet (Ours)} & 0.748 & 0.111 & 95.9\% & 0.902 & 8 & 0.610 & 18 & GAP8 SoC\\
\cline{1-9}
\end{tabular}
\end{table*}
From these results, we can observe that our modified design, even though 160 times smaller and running with two orders of magnitude lower power consumption than the best architecture (i.e., ResNet-50~\cite{he2016deep}), maintains a considerable prediction performance while achieving comparable real-time operation (18 frames per second).
Regarding the original DroNet, it is clear that the proposed modifications, like quantization and fixed-point calculation, are not penalizing the overall network's capabilities, quite the opposite.
In fact, both the regression and classification problems benefit from the fine-tuning, highlighting how the generalization of such models depends critically on the quantity and variety of data available for training.
\subsection{Performance and Power Consumption}
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{./imgs/perf_vs_freq_vs_power_V2.png}
\caption{\textit{DroNet} performance in frames per second (fps) in all tested configurations (coloring is proportional to total system power).}
\label{fig:perfVSfreqVSpower}
\end{figure}
We measured wall-time performance and power consumption by sweeping between several operating modes on GAP8.
We focused on operating at the lowest (\SI{1.0}{\volt}) and highest (\SI{1.2}{\volt}) supported core VDD voltages.
We swept the operating frequency between 50 and \SI{250}{\mega\hertz}, well beyond the GAP8 officially supported configuration\footnote{https://greenwaves-technologies.com/gap8-datasheet}.
In Figure~\ref{fig:perfVSfreqVSpower} we report performance as frame-rate and total power consumption measured on the GAP8 SoC.
Selecting a VDD operating point of \SI{1.2}{\volt} would increase both power and performance up to \SI{272}{\milli\watt} and \SI{18}{fps}.
We found the SoC to be working correctly @ \SI{1.0}{\volt} for frequencies up to $\sim$\SI{175}{\mega\hertz}; we note that as expected when operating @ \SI{1.0}{\volt} there is a definite advantage in terms of energy efficiency.
We identified the most energy-efficient configuration in VDD@\SI{1.0}{\volt}, FC@\SI{50}{\mega\hertz} and CL@\SI{100}{\mega\hertz}, that is able to deliver up to \SI{6}{fps}, with an energy requirement per frame of \SI{7.1}{\milli\joule}.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{./imgs/power_break.png}
\caption{Power envelope break-down of the entire cyber-physical system running at VDD@\SI{1.0}{\volt}, FC@\SI{50}{\mega\hertz}, CL@\SI{100}{\mega\hertz} with \textit{PULP-Shield} zoom-in.}
\label{fig:break_down}
\end{figure}
In Figure~\ref{fig:break_down}, we report the power break-down for the complete cyber-physical system and for the proposed PULP-Shield.
Our nano-quadcopter is equipped with a \SI{240}{\milli\ampere\hour} \SI{3.7}{\volt} LiPo battery enabling a flight time of $\sim$7 minutes under standard conditions, which results in an average power consumption of \SI{7.6}{\watt}.
The power consumption of all the electronics aboard the original drone amounts to \SI{277}{\milli\watt} leaving $\sim$\SI{7.3}{\watt} for the 4 rotors.
The electronics consumption is given by the 2 MCUs included in the quadrotor and all the additional devices (e.g., sensors, LEDs, etc.).
In addition to that, introducing the PULP-Shield, we increase the peak power envelope by \SI{64}{\milli\watt} (i.e., 0.8\% of the total) using the most energy-efficient configuration and accounting also for the cost of L3 memory access and the onboard ULP camera.
On the PULP-Shield break-down, visible on the right of Figure~\ref{fig:break_down}, we consider the worst-case envelope of the HyperRAM operating at full speed only for the time required for L3-L2 data transfers with an average power consumption of \SI{14}{\milli\watt}.
As onboard computation accounts for roughly 5\% of the overall power consumption (propellers, sensors, compute and control, \textit{cfr} Section~\ref{sec:intro}), our PULP-Shield enables the execution of the DroNet network (and potentially more) in all configurations within the given power envelope.
\begin{table}[h]
\renewcommand{\arraystretch}{1.3}
\caption{\textit{CrazyFlie 2.0} (CF) lifetime with and without \textit{PULP-Shield} (both turned off and running \textit{DroNet} at VDD@\SI{1.0}{\volt}, FC@\SI{50}{\mega\hertz}, CL@\SI{100}{\mega\hertz}).}
\label{tab:lifetime}
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{2}{*}{} & \multirow{2}{*}{Original CF} & \multicolumn{2}{c|}{CF + \textit{PULP-Shield}} \\
\cline{3-4}
& & \textit{PULP-Shield} (off) & \textit{PULP-Shield} (on) \\
\hline
Lifetime & $\sim$\SI{440}{\second} & $\sim$\SI{350}{\second} & $\sim$\SI{340}{\second} \\
\hline
\end{tabular}
\end{table}
Finally, we performed an experiment to evaluate the cost in terms of operating lifetime of carrying the physical payload of the PULP-Shield and of executing the DroNet workload.
To ensure a fair measurement, we decoupled the DroNet output from the nano-drone control and statically set it to \textit{hover} (i.e., keep constant position over time) at \SI{0.5}{\meter} from the ground.
We targeted three different configurations: \textit{i}) the original CrazyFlie without any PULP-Shield; \textit{ii}) PULP-Shield plugged but never turned on, to evaluate the lifetime reduction due to the additional weight introduced; \textit{iii}) PULP-Shield turned on and executing DroNet at VDD@\SI{1.0}{\volt}, FC@\SI{50}{\mega\hertz}, CL@\SI{100}{\mega\hertz}.
Our results are summarized in Table~\ref{tab:lifetime}, where as expected the biggest reduction in the lifetime is given by the increased weight.
Ultimately, the price for our visual navigation engine is $\sim22\%$ of the original lifetime.
This lifetime reduction can be curtailed through a number of optimization.
Starting from the straightforward redesign of the PCB and camera holder with lighter plastic materials (e.g., \textit{flexible substrate}), it is possible to integrate the entire electronics of the drone.
In this last case, we could either integrate the existing MCUs with the PULP SoC in the same PCB/frame or envision a PULP-based nano-drone, where the host MCU would be replaced by the PULP SoC, scheduling all the control tasks on the FC.
\subsection{Control Evaluation}
The figures of merit of this control accuracy evaluation are \textit{i}) the \textit{longest indoor traveled distance} the nano-drone is able to cover autonomously before stopping and \textit{ii}) its capability of \textit{collision avoidance} in presence of unexpected dynamic obstacles when flying at high speed.
In all the following experiments we use the most energy-efficient configuration of our visual navigation engine of VDD@\SI{1.0}{\volt}, FC@\SI{50}{\mega\hertz}, CL@\SI{100}{\mega\hertz}.
Note that all the control-loops and state estimation parameters running on the nano-drone are kept as they come with the official firmware\footnote{https://github.com/bitcraze/crazyflie-firmware}, leaving room for further improvements.
\begin{figure}[htbp]
\centerline{\includegraphics[width=\columnwidth]{./imgs/etz_j_path.png}}
\caption{Testing environment: indoor corridor.}
\label{fig:indoorpath}
\end{figure}
The first control accuracy experiment is conceived to assess the capability of our nano-UAV of autonomous navigation in a previously unseen indoor environment, particularly challenging due to the visual differences from training samples (dominantly outdoor).
As shown in Figure~\ref{fig:indoorpath}, our visual navigation engine enables an indoor traveled distance of $\sim$\SI{113}{\meter}, flying at a average speed of \SI{0.5}{\meter/\second}.
The path navigated is composed of two straight corridors ($\sim$\SI{50}{\meter} each), divided by two sharp 90\textdegree~turns, resulting in the central ``U'' turn.
As shown in the supplementary video material (\url{https://youtu.be/JKY03NV3C2s}), the straight corridors are traveled with minimal modification of the indoor environment, in contrast to the ``U'' turn that requires some more auxiliary white tape on the ground to enforce the correct understanding of the surrounding by the CNN.
The flight terminates due to the glossy paint at the end of the corridor, due to the interference of the light reflection with the CNN understanding, resulting in a constant high probability of collision.
In the second part of the control accuracy evaluation, we analyze the system's capability of preventing collisions.
Correctly identifying obstacles has been already implicitly demonstrated with the autonomous navigation test.
With the collision avoidance set of experiments we want to push our visual navigation engine to its limit, preventing collisions also under very unfavourable conditions -- i.e., high flight speed and small reaction space/time.
The setup of this experiment is represented by a straight path where, after the nano-drone has traveled the first \SI{8}{\meter} at full speed, an unexpected dynamic obstacle appear within only \SI{2}{\meter} of distance from it (i.e., at \SI{10}{\meter} from the starting point).
We performed multiple experiments, sweeping the flight speed, to identify the maximum one for which the nano-drone is still able to react promptly and prevent the collision.
Results, also shown in the supplementary video material (\url{https://youtu.be/JKY03NV3C2s}), demonstrate that our visual navigation engine enables safe flight up to $\sim$\SI{1.5}{\meter/\second}.
\begin{figure}[htbp]
\centerline{\includegraphics[width=\columnwidth]{./imgs/plot_log.png}}
\caption{Onboard real-time log of the collision avoidance experiment, paired with external events.}
\label{fig:log}
\end{figure}
Figure~\ref{fig:log} reports the real-time log of the relevant onboard information (i.e., probability of collision, estimated and desired velocity, and altitude), paired with external events (i.e., start, appearing of the obstacle, and braking), of this experiment.
The initial take-off is followed by $\sim$\SI{2}{\second} of in place hovering before the output of the CNN is used and the flight in the forward direction starts.
The altitude of this experiment is kept constant at \SI{0.5}{\meter}, as reported in Figure~\ref{fig:log}-A.
As soon as the probability of collision output from the CNN, shown in Figure~\ref{fig:log}-B, is higher of the \textit{critical probability of collision} threshold of 0.7, the target forward velocity is pushed to 0, resulting in a prompt obstacle avoidance mechanism.
The onboard state estimation of the current forward velocity (i.e., \textit{Vx estimated}) is reported in Figure~\ref{fig:log}-C paired with the desired velocity in the same direction, that is calculated on the basis of the probability of collision and bounded to the maximum forward velocity, i.e., \SI{1.5}{\meter/\second}.
If we would relax the experiment's constraints -- e.g., increasing the braking space/time -- we could enable a safe flight, avoiding collision, also at higher flight speed.
\section{Conclusions}\label{sec:conclusion}
Nano- and pico-sized UAVs are ideal ubiquitous nodes; due to their size and physical footprint, they can act as mobile sensor hubs and data collectors for tasks such as surveillance, inspection, etc.
However, to be able to perform these tasks, they must be capable of autonomous navigation of complex environments such as the indoor of buildings and offices.
In this work, we introduce the first vertically integrated visual navigation engine for autonomous nano-UAVs field-tested in closed loop demonstrations, as shown in supplementary video materials.
Our engine consumes \SIrange{64}{272}{\milli\watt} while running at \SIrange{6}{18}{fps}, enough \textit{i)} to enable autonomous navigation on a \SI{>100}{\meter} previously unseen indoor environment, and \textit{ii)} to ensure robustness against the appearance sudden obstacles at \SI{2}{\meter} distance while flying at \SI{1.5}{\meter/\second}.
To pave the way for a huge number of advanced use-cases of autonomous nano-UAVs as wireless mobile smart sensors, we release open-source our PULP-Shield design and all code running on it, as well as datasets and trained networks.
\section*{Supplementary Material}
Supplementary video at: \url{https://youtu.be/JKY03NV3C2s}.
The project's code, datasets and trained models are available at: \url{https://github.com/pulp-platform/pulp-dronet}.
\input{01-introduction.tex}
\input{02-related.tex}
\input{03-implementation.tex}
\input{04-proto.tex}
\input{05-results.tex}
\input{06-conclusions.tex}
\section*{Acknowledgment}
The authors thank Frank K. G\"urkaynak for his contribution in making the supplementary videos.
This work has been partially funded by projects EC H2020 OPRECOMP (732631) and ALOHA (780788).
\bibliographystyle{IEEEtran}
|
1,116,691,500,773 | arxiv | \section*{Introduction}
\label{se.intro}
This note was originally motivated by \cite[Theorem 1.1]{ess2} (or rather its earlier version, \cite[Theorem 1.2]{ess}):
\begin{theoremN}
For any field $k$, the inverse limit in the category of graded commutative rings of the diagram
\begin{equation}\label{eq:invlim}
\cdots\to k[x_1,x_2,x_3]\to k[x_1,x_2]\to k[x_1]
\end{equation}
of polynomial rings (obtained by annihilating the extra variable at each step) is again a polynomial ring.
\end{theoremN}
This is a curious and rather unexpected phenomenon: polynomial rings are the free objects in the category of commutative algebras (`free' in the sense of universal algebra, e.g. \cite[Definition 7.8.3]{berg-inv}), and hence expressible as coproducts. On the other hand, \Cref{eq:invlim} is a {\it limit} (rather than a {\it co}limit), so one would not necessarily expect compatibility between the two.
The natural question arises of which other varieties of algebras (apart from graded commutative algebras) exhibit the same type of freeness behavior: given free objects $F_n$ on sets of $n$ elements respectively, one can construct analogous diagrams
\begin{equation*}
\cdots\to F_3\to F_2\to F_1
\end{equation*}
provided the algebras in question are equipped with a distinguished element $e$: the extra free generator of $F_{n+1}$ can be sent to $e\in F_n$, giving the map $F_{n+1}\to F_n$. Examples include
\begin{itemize}
\item any number of ``linear'' varieties of algebras (commutative, associative, Jordan, etc.), with $0$ as the trivial element;
\item groups, monoids, and so on, with their respective trivial elements.
\end{itemize}
Groups, in particular, have been studied from this perspective by Higman: consider a limit
\begin{equation*}
\lim\left(\cdots\to G_1*G_2*G_3\to G_1*G_2\to G_1\right)
\end{equation*}
in the category of groups, where
\begin{itemize}
\item $G_i$ are groups;
\item `$*$' denotes the coproduct (or free product) of groups;
\item each connecting map annihilates the extra free factor and acts as the identity on the others.
\end{itemize}
In the language of \cite{hig-unr}, that limit is the {\it unrestricted free product} of the groups $G_i$. In particular, when all $G_i$ are isomorphic to $\bZ$, one obtains a kind of completion of a free group on countably-infinitely many generators, denoted here by $F$. The analogue of the question posed above is whether $F$ is again free. Higman shows that not only is this not the case, but in fact $F$ is in a sense at the opposite end of a freeness spectrum (\cite[Theorem 1]{hig-unr}):
\begin{theoremN}
Let $F_n$ be the free group on $n$ generators, and consider the limit
\begin{equation*}
F:=\lim\left(\cdots \to F_3\to F_2\to F_1\right)
\end{equation*}
in the category of groups. Then, any morphism from $F$ into a free group factors through one of the $F_n$.
\end{theoremN}
This shows that the freeness result recorded in \cite{ess,ess2} is far from being a given. Here, we prove a number of cognates. First, \Cref{th:gr-fr} is a direct non-commutative analogue of \cite[Theorem 1]{ess2}; the statement in the main text is more precise, but roughly, it reads:
\begin{theorem}
Let $k$ be a field and $A_n$ the free algebra on $n$ generators. Then, the limit
\begin{equation}\label{eq:assoclim}
\lim\left(\cdots\to A_3\to A_2\to A_1\right)
\end{equation}
in the category of graded associative algebras is free on a set of homogeneous elements.
\end{theorem}
On the other hand, if one were to instead take the limit \Cref{eq:assoclim} in the category of {\it plain} (as opposed to graded) associative algebras, a trace of this freeness behavior survives (\Cref{th.ungr-fr}):
\begin{theorem}
Let $k$ be a field and $A_n$ the free algebra on $n$ generators. Then, the limit
\begin{equation}\label{eq:assoclim}
\lim\left(\cdots\to A_3\to A_2\to A_1\right)
\end{equation}
in the category of associative algebras is a formal power series algebra on a set of homogeneous elements.
\end{theorem}
Finally, there is a graded-Lie-algebra version of the above (\Cref{th:lie-fr}):
\begin{theorem}
Let $k$ be a field and $L_n$ the free Lie algebra on $n$ generators. Then, the limit
\begin{equation}\label{eq:assoclim}
\lim\left(\cdots\to L_3\to L_2\to L_1\right)
\end{equation}
in the category of graded Lie algebras is free on a set of homogeneous elements.
\end{theorem}
\subsection*{Acknowledgements}
This work was partially supported through NSF grant DMS-2001128.
The material constitutes part of the second author's PhD thesis at the University at Buffalo.
\section{Preliminaries}
\label{se.prel}
We work over an arbitrary field $k$ (all additional properties, e.g. being perfect, will be specified if and when needed). The algebras under consideration will often be graded, meaning here $\bN$-graded where $\bN=\{0,1,\cdots\}$. As \cite{chn-fr} will be a central reference throughout, we follow it in denoting by $\nu$ the ``highest-degree'' function on a filtered algebra
\begin{equation*}
0=A_{(-\infty)}\subset A_{(0)}\subset A_{(1)}\subset \cdots, \ A=\cup_{n}A_{(n)}
\end{equation*}
i.e. $\nu(x)$ is the smallest $n$ such that $x\in A_{(n)}$. Our algebras will sometimes be graded, as in
\begin{equation*}
A=\bigoplus_{d\in \bN}A_d,
\end{equation*}
in which case we consider the corresponding filtration given by
\begin{equation*}
A_{(n)} = \bigoplus_{d=0}^n A_d.
\end{equation*}
For homogeneous elements in the graded case we also sometimes resort to $|x|$ for the degree of $x$. Our graded algebras will also often be {\it connected}: $A_0=k$, the ground field.
We will need some auxiliary material from \cite[Chapter 2]{chn-fr}, which we now present briefly. First, recall the following discussion from \cite[$\S$2.2]{chn-fr}.
\begin{definition}\label{def.dep}
A family $(a_i)_i$ of elements of $A$ is {\it right $\nu$-dependent} if one of the $a_i$ vanishes, or there exist $b_i\in A$, almost all zero, such that
\begin{equation*}
\nu\left(\sum a_ib_i\right) < \max_i(\nu(a_i)+\nu(b_i)).
\end{equation*}
An element $a\in A$ is {\it right $\nu$-dependent} on the family $(a_i)_i$ if $a=0$ or there are $b_i\in A$, almost all zero, such that
\begin{equation*}
\nu\left(a-\sum a_ib_i\right) < \nu(a),\quad \max_i(\nu(a_i)+\nu(b_i))\le \nu(a).
\end{equation*}
\end{definition}
The two properties above are ordered strength-wise: the right $\nu$-dependence of $a$ on $(a_i)$ entails the right $\nu$-dependence of the family $(a_i)\cup\{a\}$. Rings that in a certain sense satisfy the converse of this observation are the focus of \cite[Chapter 2]{chn-fr}, as they tend to have good ``freeness'' properties.
\begin{definition}\label{def.wk}
The filtered algebra $A$ {\it has (or satisfies) the weak algorithm} for $\nu$ if, for any $\nu$-dependent family $(a_i)$ in the sense of \Cref{def.dep} some $a_i$ is $\nu$-dependent on the family of those $a_j$ with $\nu(a_j)\le \nu(a_i)$.
\end{definition}
As hinted above, the importance of the concept for us is encapsulated by the following result (\cite[Proposition 2.4.2]{chn-fr}).
\begin{proposition}\label{pr.chn}
A filtered $k$-algebra $A$ with $A_{(0)}=k$ is free on some subset $X$ of positive-degree elements if and only if it satisfies the weak algorithm.
\end{proposition}
A parallel discussion can be carried out for algebras equipped with an {\it inverse filtration} as in \cite[$\S$2.9]{chn-fr}:
\begin{equation*}
A=A_{(0)}\supset A_{(1)}\supset \cdots\supset 0=A_{(\infty)}.
\end{equation*}
In this case for $x\in A$ we denote by $\nu(x)$ the largest $n$ such that $x\in A_{(n)}$. $\nu$-dependence can be defined as before by simply reversing the inequalities. We then have
\begin{definition}\label{def.inv-wk}
The inversely filtered algebra $A$ {\it has the inverse weak algorithm} for $\nu$ if for any $\nu$-dependent family $(a_i)$ in the sense of \Cref{def.dep} with reversed inequalities some $a_i$ is $\nu$-dependent on the family of those $a_j$ with $\nu(a_j)\le \nu(a_i)$.
\end{definition}
In the inverse-filtration setting the analogue of \Cref{pr.chn} reads as follows (see \cite[Proposition 2.9.8]{chn-fr}).
\begin{proposition}\label{pr.chn-inv}
A complete inversely filtered $k$-algebra $A$ with $A/A_{(1)}\cong k$ is a formal power series algebra on some subset $X$ if and only if it satisfies the inverse weak algorithm.
\end{proposition}
\section{Associative algebras}
\label{se.main}
\subsection{The graded case}
\label{subse.gr}
The setup is as follows. We fix a set $S$, and consider the free $k$-algebra $k\langle S\rangle$ on $S$. For an arbitrary finite subset $F\subset S$ consider the surjection $k\langle S\rangle\to k\langle F\rangle$ obtained by annihilating all generators in $S\setminus F$. These surjections form a co-filtered diagram in the category of graded algebras, and we can consider its limit
\begin{equation}\label{eq:diag}
A=A((S)):=\varprojlim_F \left(k\langle S\rangle \to k\langle F\rangle\right).
\end{equation}
The degree-$d$ component $A_d$ consists of formal $k$-linear combinations of the degree-$d$ monomials in the generators $x_s$, $s\in S$.
There is a canonical morphism $k\langle S\rangle\to A((S))$ that is evidently an isomorphism when $S$ is finite and only one-to-one when $S$ is infinite. In the latter case however, the algebra $A((S))$ is still free. This is the content of the main result of this section, which is a non-commutative analogue of \cite[Theorem 1.1]{ess2} (and of its precursor, \cite[Theorem 1.2]{ess}).
\begin{theorem}\label{th:gr-fr}
$A=A((S))$ defined by \Cref{eq:diag} is free as a $k$-algebra, on any set $X$ of homogeneous elements of $A$ forming a basis for $A_{>0}/A_{>0}^2$ is a free generating set for $A$.
\end{theorem}
\begin{proof}
We use \cite[\S 2.4, Theorem 4.1]{chn-fr}. To apply it, we have to prove that
\begin{enumerate}[(a)]
\item\label{item:1} $A$ satisfies the weak algorithm with respect to its grading;
\item\label{item:2} the monomials on any set $X$ as in the statement span $A$,
\item\label{item:3} and no element of $X$ is right-$\nu$-dependent on the rest.
\end{enumerate}
We handle these in turn.
{\bf \Cref{item:1}: $A$ satisfies the weak algorithm.} Suppose $a_i$, $1\le i\le n$ form a $\nu$-dependent family of non-zero homogeneous elements, ordered so that
\begin{equation*}
|a_1|\le \cdots\le |a_n|.
\end{equation*}
We will argue that one of the $a_i$ is $\nu$-dependent on $(a_j)_{j=1}^{i-1}$.
By homogeneity, the hypothesis proves the existence of (homogeneous) $b_i$ such that $\sum a_ib_i=0$; we then need to show that some $a_i$ is a right linear combination of $a_j$, $j<i$. We may as well assume all $a_ib_i$ have the same degree, so that
\begin{equation*}
|b_1|\ge \cdots\ge |b_n|.
\end{equation*}
We prove the statement by double induction on $n$ and then $|b_n|$, the base case being a simple exercise.
If $b_n=0$ then we may as well restrict attention to the family $(a_i)_{i=1}^{n-1}$, the inductive hypothesis taking care of the rest. We can thus assume that $b_n\ne 0$, moving over to the induction-by-$|b_n|$ branch of the argument. Once more, the base case $b_n\in k^\times$ is immediate (as there is then nothing to prove), so we in fact assume $|b_n|>0$.
Fix some $s$ such that $x_s$ appears as the rightmost generator in one of the monomials making up $b_n\ne 0$. In the relation $\sum a_ib_i=0$ the terms ending in $x_s$ still add up to zero, so we can ignore the summands of the $b_i$s that do {\it not} end in a rightmost $x_s$ and assume all $b_i$ belong to $Ax_s$. But then, if say $b_i=b_i'x_s$, we have
\begin{equation*}
\sum a_ib_i'=0,
\end{equation*}
allowing us to apply the inductive hypothesis to $|b'_n|<|b_n|$.
\vspace{.5cm}
{\bf \Cref{item:2}: The monomials on $X$ span $A$.} This is virtually automatic given the definition of $X$. Let $a\in A_d$ be a homogeneous element. It is then of the form
\begin{equation*}
\sum t_ix_i + a',\ t_i\in k,\ x_i\in X
\end{equation*}
for some $a'\in A_{>0}^2$, and we can use induction on $d$ applied to the factors $b_j,c_j\in A_{>0}$ in a decomposition
\begin{equation*}
a'=\sum_j b_jc_j.
\end{equation*}
\vspace{.5cm}
{\bf \Cref{item:3}: No element of $X$ is right $\nu$-dependent on the rest.} because our elements are homogeneous, a $\nu$-dependence relation would be of the form
\begin{equation*}
x=\sum x_ia_i,\ a_i\in A
\end{equation*}
for distinct elements $x,x_i\in X$. Modulo $A_{>0}^2$ this expresses $x$ as a $k$-linear combination of other elements of $X$, contradicting the fact that $X$ is a basis of $A_{>0}/A_{>0}^2$.
\end{proof}
\subsection{The ungraded case}
\label{subse.ungr}
The preceding discussion makes it natural to examine the structure of the limit \Cref{eq:diag} in the category of (plain, ungraded) $k$-algebras. We will denote the resulting algebra by $B=B(S)$, to distinguish it from the limit $A=A((S))$ in the category of graded algebras.
We have an inclusion $A((S))\subset B(S)$. In fact, the generic element of $B$ is a formal series of the form
\begin{equation*}
x=\sum_{d\ge 0}x_d,\ x_d\in A_d
\end{equation*}
with the property that for every finite subset $F\subset S$, the free algebra $k\langle F\rangle\subset A\subset B$ contains only finitely many of the monomials appearing in $x$.
$B$ admits a natural inverse filtration $\nu$ defined by
\begin{equation*}
x=\sum_{d\ge 0}x_d\in B_{(n)} \iff x_d=0\text{ for }d<n.
\end{equation*}
In other words,
\begin{equation*}
\nu(x) = \text{ smallest }d\text{ such that }x_d\ne 0.
\end{equation*}
We denote
\begin{equation*}
K=B_{(0)},\ B_+ = \{x\in B\ |\ \nu(x)>0\}.
\end{equation*}
Note that $K$ is a division ring.
We then have the following analogue of \Cref{th:gr-fr}.
\begin{theorem}\label{th.ungr-fr}
The algebra $B=B(S)$ is isomorphic to a formal power series $k$-algebra.
\end{theorem}
\begin{proof}
According to \Cref{pr.chn-inv} it suffices to argue that $B$ has the weak algorithm for its inverse filtration $\nu$ (as it is clear that $B$ is $\nu$-complete). Suppose , then, that we have
\begin{equation*}
\nu\left(\sum a_ib_i\right) > \min_i(\nu(a_i)+\nu(b_i))
\end{equation*}
for non-zero $a_i$ and $b_i$, $1\le i\le n$. We may as well assume that all $\min_i(\nu(a_i)+\nu(b_i))$ are equal and that $\nu(a_i)$ is non-decreasing in $i$.
If $\nu(b_n)=0$ then $b_n$ is invertible, and hence $a_n$ is a right $B$-linear combination of the other $a_i$ modulo $B_{(\nu(a_n)+1)}$, proving $\nu$-dependence.
Otherwise, the lowest-degree term of $b_n$ is a $k$-linear combination of monomials in the $x_s$, $s\in S$ of degree $\ge 1$. Let $x_t$ be the rightmost generator in one of these monomials. Denoting by a $t$ superscript the linear combination of those monomials that end in $x_t$, we then have
\begin{equation*}
\nu\left(\sum a_ib_i^t\right) > \min_i(\nu(a_i)+\nu(b_i^t)).
\end{equation*}
Since all $b_i^t$ are of the form $b'_i x_t$, we can eliminate $x_t$ to obtain
\begin{equation*}
\nu\left(\sum a_ib'_i\right) > \min_i(\nu(a_i)+\nu(b'_i)).
\end{equation*}
This is a $\nu$-dependence relation as before with $\nu(b'_n)<\nu(b_n)$, so we can repeat the procedure until the dependence relation has been reduced to the case $\nu(b_n)=0$.
\end{proof}
\section{Lie algebras}\label{se:lie}
The goal here is to prove an Lie-algebra analogue of \Cref{th:gr-fr}. Specifically, let $S$ be a set as before. If $S$ is finite then $L((S))$ will simply denote the free Lie algebra on $S$. On the other hand, if $S$ is infinite we set
\begin{equation}
\label{eq:1}
L=L((S)):=\varprojlim_F \left(L((S)) \to L(F)\right),
\end{equation}
precisely as in \Cref{eq:diag}:
\begin{itemize}
\item the limit is in the category of {\it graded} Lie algebras;
\item it is indexed by the filtered system of finite subsets $F\subseteq S$;
\item the connecting morphisms $L(F)\to L(F')$ for $F'\subset F$ simply annihilate all generators in $F\setminus F'$.
\end{itemize}
Furthermore, we assume we are working over a base field $k$ of characteristic zero. The advertised result is
\begin{theorem}\label{th:lie-fr}
Let $S$ be a set and $L=L((S))$ as in \Cref{eq:1}. Then, $L$ is freely generated as a Lie algebra by any set of homogeneous elements that projects to a basis of $L/[L,L]$.
\end{theorem}
We will make use of the material in \cite{reut}, which is a good reference for free Lie algebras in general. Following that source, we reserve the notation $L(S)$ for the free Lie algebra on the (finite or infinite) set $S$. In particular, for finite $S$ we have $L((S))=L(S)$. As explained in \cite[\S 1.2]{reut}, the free Lie algebra $L(S)$ can be identified with the set of {\it Lie polynomials} in its enveloping algebra $k\langle S\rangle$.
The proof given there for the celebrated theorem of Shirshov and Witt to the effect that Lie subalgebras of free Lie algebras are free \cite[Theorem 2.5]{reut} proceeds via a Lie-theoretic version of Cohen's theory of dependence for polynomials. The relevant appears in \cite[discussion preceding Theorem 2.3]{reut}:
\begin{definition}\label{def:lie-dep}
Let $p,p_i\in k\langle S\rangle$, $1\le i\le n$ be non-commutative polynomials. We say that $p$ is {\it Lie-dependent} on the $p_i$ if $p=0$ or there is a Lie polynomial $f$ in $n$ variables such that
\begin{itemize}
\item $\deg(p-f(p_1,\cdots,p_n))<\deg p$;
\item the degree of every monomial appearing formally in $f(p_i)$ is dominated by the degree of $p$.
\end{itemize}
\end{definition}
The main tool in the proof of the above-mentioned \cite[Theorem 2.5]{reut} is the following Lie analogue of Cohen's result \Cref{pr.chn} (see \cite[Theorem 2.3]{reut}).
\begin{proposition}\label{pr:lie-dep}
Let $p_i\in k\langle S\rangle$, $1\le i\le n$ be Lie polynomials. If the family is dependent then some $p_i$ is Lie-dependent on those $p_j$ of no-larger degree.
\end{proposition}
In order to state our first result, note that we have a natural embedding
\begin{equation}\label{eq:2}
L((S))\subset A((S))
\end{equation}
arising by taking the limit of the Lie-polynomial embeddings $L(F)\subset k\langle F\rangle$ over finite subsets $F\subset S$. We say that $L((S))$ consists of the {\it Lie elements} of $A((S))$.
Our first goal will be to prove the following version of \Cref{pr:lie-dep}.
\begin{theorem}\label{th:lie-is-dep}
Let $S$ be a set and $p_i\in A((S))$, $1\le i\le n$ be Lie elements. If the family is dependent then some $p_i$ is Lie-dependent on those $p_j$ of no-larger degree.
\end{theorem}
\begin{proof}
As in the proof of \cite[Theorem 2.3]{reut}, it will be enough to argue that, upon denoting the top homogeneous component of an element $q\in A((S))$ by $\overline{q}$, some $\overline{p_i}$ is expressible as a Lie polynomial of $\overline{p_j}$ for $j\ne i$.
To simplify the notation we assume from the start that the $p_i$ are homogeneous, as we well may: the hypothesis is that there are $b_i\in A((S))$, almost all zero, such that
\begin{equation}\label{eq:smldeg}
\deg \sum_i p_i b_i < \max_i \deg p_i b_i.
\end{equation}
Retaining only the top-degree components of the $p_i$ and $b_i$ and only those indices $j$ such that
\begin{equation*}
\deg p_j b_j = \max_i \deg p_i b_i,
\end{equation*}
we obtain
\begin{equation*}
\sum_i \overline{p_i} b_i=0,
\end{equation*}
i.e. the top components $\overline{p_i}$ are dependent. We henceforth drop the overlines and assume all $p_i$ are homogeneous, satisfying
\begin{equation*}
\sum_i p_i b_i=0
\end{equation*}
for some homogeneous $b_i$, almost but not all zero. The goal (sufficient, again as in the proof of \cite[Theorem 2.3]{reut}) will be to prove that some $p_i$ is a Lie polynomial in $p_j$, $j\ne i$. This follows from \Cref{pr:idlie} below.
\end{proof}
The following result is a version of \cite[Lemma 2.4]{reut}.
\begin{lemma}\label{pr:idlie}
Let $S$ be a set and $p$, $p_i\in A((S))$, $1\le i\le n$ be homogeneous Lie elements with
\begin{equation}\label{eq:lindep}
p = \sum_i p_i b_i
\end{equation}
for homogeneous $b_i\in A((S))$. Then,
\begin{equation}\label{eq:pisliepoly}
p = f(p_1,\cdots,p_n)
\end{equation}
for some Lie polynomial $f$.
\end{lemma}
\begin{proof}
For finite $S$ the argument appears in the course of the proof of \cite[Theorem 2.3]{reut}, so we are concerned with infinite $S$. For finite subsets $F\subset S$ we denote the images through the surjection $A((S))\to k\langle F\rangle$ with superscripts $F$, as in $p^F$, $p_i^F$, etc.
We assume throughout the discussion that the finite subsets $F$ are large enough to ensure that the projected elements $p^F$, $p_i^F$ and $b_i^F$ have the same degrees as their global counterparts $p$, $p_i$, etc. The relation \Cref{eq:lindep} projects to analogues
\begin{equation*}
p^F = \sum_i p^F_i b^F_i\in k\langle F\rangle,
\end{equation*}
and hence, according to the aforementioned proof of \cite[Theorem 2.3]{reut}, for each $F$ we have an expression
\begin{equation*}
p^F = f_F(p_1^F,\cdots,p_n^F)
\end{equation*}
for some Lie polynomial $f_F$. We will furthermore assume $f_F$ is chosen {\it minimally}, in the sense that its Lie monomials, evaluated at the $p_i^F$, produce linearly independent elements of $k\langle F\rangle$.
If $d$ and $d_i$ denote the degrees of the homogeneous elements $p$ and $p_i$ respectively, then every Lie monomial appearing in $f_F$ has some degree $e_i$ in $x_i$ so that
\begin{equation*}
d = \sum_i d_i e_i.
\end{equation*}
There are only finitely many choices of such monomials, so only finitely many monomials appearing in all $f_F$ collectively. This means that there is some {\it cofinal} collection $F_{\alpha}$ of $F$ (i.e. such that every finite subset $F\subset S$ is contained in some $F_{\alpha}$) for which
\begin{itemize}
\item the Lie monomials appearing in $f_{F_{\alpha}}$ are the same for all $\alpha$;
\item by minimality, the coefficients of those monomials are the same for all $\alpha$.
\end{itemize}
In other words, all Lie polynomials $f_{F_{\alpha}}$ coincide with some common Lie polynomial $f$. But this means that
\begin{equation*}
p^{F_{\alpha}} = f(p_1^{F_{\alpha}},\cdots,p_n^{F_{\alpha}}),\ \forall \alpha,
\end{equation*}
and hence we obtain the desired identity \Cref{eq:pisliepoly}.
\end{proof}
We first prove a weak version of \Cref{th:lie-fr}.
\begin{theorem}
\label{th:weak}
Let $S$ be a set and $L=L((S))$ as in \Cref{eq:1}. Then, $L$ is freely generated as a Lie algebra by some set of homogeneous elements.
\end{theorem}
\begin{proof}
$E_n$ is a subspace of $L$ defined by
\[
E_n=\{p\in L\subset A((S))|\deg(p)\leq n\}
\]
Let $\langle E\rangle$ denote the Lie subalgebra generated by $E\subset L$.
Let $E'_n$ be the subspace of $E_n$ defined by
\begin{displaymath}
E_n':=E_n \cap \langle E_{n-1}\rangle
\end{displaymath}
Let $X_n$ be a subset of $E_n$ which defines a basis of $E_n/E'_n$.
Let $X:=\cup_{n\geq 1}X_n$. In order to show $L$ is free on $X$, it is enough to show $L$ is isomorphic with $L(B)$, where $B$ is a set with bijection $b\mapsto x_b$, $B\to X$. We only need to show (i) $X$ generates $L$. (ii) for each nonzero Lie polynomial $f(b)_{b\in B}\in L(B)$, one has $f(x_b)_{b\in B}\neq 0$\\
For (i)\\
Let $p$ with $\deg(p)=n$. We will prove for each $n$, $E_n$ is generated by $X$. When $n=0$, clearly it is true. Suppose when $n=k-1$, $E_{k-1}$ generated by $X$. Since $X_k$ is a basis of $E_k/E'_{k-1}$, so for each $p\in E_k$, we have $\overline{p}=\sum \alpha_x \overline{x}$, here $\overline{p}$ is projection of $p$ in $E_k/E'_k$, so we have $$Q=P-\sum_{x\in X_k} \alpha_x x \in E'_k\subset \langle E_{k-1}\rangle.$$
Hence, $Q$ is generated by $E_{k-1}$. By induction $E_{k-1}$ is generated by $X$. This shows $E_k$ is generated by $X$, hence $X$ generates $L$.\\
For (ii)\\
Arguing by contradiction. Suppose $f(p_1\dots p_q)=0$, for some nonzero Lie polynomial $f(b_1\dots b_q)\in L(B)$ and some $p_1\dots p_q \in X$ with $\deg(p_1)\leq \cdots \leq \deg(p_q)$. Certainly, there exists a nonzero polynomial in $K(B)$ such that $f(p_1\dots p_q)=0$, take such polynomial with the least degree, we write $f$ as
$$f=\sum_{i=1}^q b_ig_i .$$
Some $r_i=g_i(p_1\dots p_q)$ is nonzero. Otherwise, suppose $g_i(p_1\dots p_q)=0$ for each $i$, at least one $g_i$ is nonzero polynomial so there is a polynomial $g_i$ satisfying $g_i(p_1\dots p_q)=0$, but its degree is less than $\deg(f)$, it contradict the minimality of $f$. Since $0=f(p_1\dots p_q)=\sum p_i r_i$, we deduce $p_1\dots p_q$ are dependent. By \Cref{th:lie-is-dep}, some polynomial $p_i$ is Lie dependent on $p_1\dots p_{i-1}$. Hence,$\deg(p_i-h(p_1\dots p_{i_1})<\deg(p_i)=n$, here $p_i-h(p_i \dots p_{i-1})\in E_{n-1}$, $h$ is a Lie polynomial. Hence, $p_i-h(p_1\dots p_{i-1})=\text{an element of } E_{n-1}$. This implies $p_i + \text{a linear combinations of those } p_j, j<i \text{ of the same degree as } p_i=\text{a Lie expression which are of degree less than } p_i + \text{an element of } E_{n-1}$ with $n=\deg(p_i)$. So we have polynomials in $X_n$ are not linearly independent in $E_n/E'_n$, which is a contradiction. Therefore, this proposition is proved.
\end{proof}
\pf{th:lie-fr}
\begin{th:lie-fr}
Since \Cref{th:weak} shows that we can choose a set $X$ of {\it homogeneous} free generators for $L((S))$, the conclusion follows from \Cref{pr:allhomog} below.
\end{th:lie-fr}
\begin{proposition}\label{pr:allhomog}
Let $L$ be a $\bZ_{>0}$-graded Lie algebra freely generated by some set of homogeneous elements. Then, $L$ is freely generated by {\it any} set of homogeneous elements projecting to a basis of $L/[L,L]$.
\end{proposition}
\begin{proof}
We abbreviate the phrase
\begin{equation*}
X \text{ projects to a basis of }L_{ab}:=L/[L,L].
\end{equation*}
to
\begin{equation*}
X \text{ is {\it relatively free}}.
\end{equation*}
Denote
\begin{itemize}
\item by $F$ a set of homogeneous elements generating $L$ freely (note that in particular $F$ is relatively free in the above sense);
\item by $X$ an arbitrary relatively free set of homogeneous elements;
\item by subscript degree decorations the respective homogeneous components of $F$ and $X$; for instance:
\begin{equation*}
X_d:=\{x\in X\ |\ |x| = d\};
\end{equation*}
\item by
\begin{equation*}
0<d_1<d_2<\cdots
\end{equation*}
the positive integers appearing as degrees of elements in $F\cup X$;
\item by overlines (e.g. $\overline{X}$) the images of sets through the projection $L\to L_{ab}$.
\end{itemize}
We will argue that there is an automorphism $\alpha$ of $L$ (as a graded Lie algebra) transforming $F$ into $X$; naturally, this will imply the desired conclusion.
We define $\alpha$ on $F_{d_n}$, inductively on $n$. First, note that because $\overline{X_{d_1}}$ and $\overline{F_{d_1}}$ are bases for the same subspace of $L_{ab}$, there is a linear automorphism of $\mathrm{span}(F_{d_1})$ that transforms $F_{d_1}$ into $X_{d_1}$ modulo $[L,L]$. Furthermore, because $[L,L]$ is spanned by commutators of elements of $F$ and such commutators are all of degree
\begin{equation*}
\ge \min(d_2,2d_1) > d_1,
\end{equation*}
said automorphism must in fact transform $F_{d_1}$ into $X_{d_1}$.
This would be the degree-$d_1$ component of $\alpha$; to simplify matters, we assume (as we can, by the argument in the preceding paragraph) that in fact $F_{d_1}=X_{d_1}$. This constitutes the base case in the recursive procedure we are outlining.
Next, consider degree $d_2$. Each element of $X_{d_2}$ lies in
\begin{equation*}
\mathrm{span}(F_{d_2}) + \mathrm{span}([F_{d_1},F_{d_1}])
\end{equation*}
and vice versa:
\begin{equation*}
F_{d_2}\subset \mathrm{span}(X_{d_2}) + \mathrm{span}([X_{d_1},X_{d_1}]),
\end{equation*}
because
\begin{itemize}
\item $F_{d_1}=X_{d_1}$, and
\item $\overline{X_{d_2}}$ and $\overline{F_{d_2}}$ constitute bases for the same subspace
\begin{equation}\label{eq:xfd2}
\mathrm{span}\left(\overline{X_{d_2}}\right) = \mathrm{span}\left(\overline{F_{d_2}}\right)
\end{equation}
of $L_{ab}$.
\end{itemize}
An automorphism of \Cref{eq:xfd2} that identifies $\overline{F_{d_2}}$ with $\overline{X_{d_2}}$ can thus be lifted to an automorphism of
\begin{equation*}
\mathrm{span}(F_{d_1})+\mathrm{span}(F_{d_2}) + \mathrm{span}([F_{d_1},F_{d_1}])
\end{equation*}
that
\begin{itemize}
\item is the identity on $F_{d_1}=X_{d_1}$,
\item respects the Lie bracket on that space,
\item and maps $F_{d_2}$ to $X_{d_2}$.
\end{itemize}
This procedure can similarly be continued recursively to higher degrees: we henceforth assume that $X_{d_i}=F_{d_i}$ for $i=1,2$, etc.
\end{proof}
We observe that homogeneity is crucial to \Cref{pr:allhomog}:
\begin{example}
Consider the free Lie algebra $L$ on two generators, $x$ and $y$. The elements
\begin{equation*}
x':=x+[x,[x,y]]\quad \text{and}\quad y':=y+[x,[x,[x,y]]]
\end{equation*}
surject to a basis for $L_{ab}$, but any non-trivial Lie polynomial in $x'$ and $y'$ will have degree at least 3. It follows that $[x,y]$, for instance, cannot lie in the (free) Lie algebra generated by $x'$ and $y'$.
\end{example}
\def\polhk#1{\setbox0=\hbox{#1}{\ooalign{\hidewidth
\lower1.5ex\hbox{`}\hidewidth\crcr\unhbox0}}}
|
1,116,691,500,774 | arxiv | \section{Introduction}
From a world where each application has its dedicated hardware support (high
performance chips for desktop computers, low power chips for mobile phones,
hardened chips for smartcards, \textit{etc.}), more and more applications
are today executed on a same device: the smartphone. They are powered by fast and
low-power components, the \glspl{SoC}, where a set of modules (modem, graphic
and sound card, Flash memory, \textit{etc.}) takes place on the same silicon
layout as the \gls{CPU}. With the democratization of smartphones, more sensitive
processes are done on the same component as the usual non sensitive processing. Aware of
this risk, vendors invest a lot of work into the software layer hardening
with numerous security mechanisms embedded into the \gls{OS} (such as Android or
iOS). However, the hardware layer misses the same attention: in particular it is
not protected against physical attacks, especially against fault injection
attacks.
Fault injection attacks is a well known class of attacks where an attacker modifies the
physical environment of the targeted chip in order to induce a fault during its
execution. The resulting failure can be used to extract sensitive information
(cryptographic keys) or bypass a PIN code verification for instance. More
generally, fault injection attacks give the ability to modify a program at runtime,
defeating static countermeasures that cannot foresee the failure (\textit{e.g.} secure
boot, access control).
The attacker's ability to subdue the system is highly dependant of its
experimental capacities: fault injection attacks can be performed with power glitches,
clock glitches, lasers, electromagnetic pulses, \textit{etc.} In this paper, we
propose an \gls{EMFI} attack against a \gls{SoC}. Electromagnetic pulses modify
the electric signals in the metallic chip wires. Faults are generated
when signals are modified during a small time window around the clock rising
edge. In this case, according to Ordas \textit{et al.}~\cite{OrdasGM15}, a
faulty value may be memorized.
Most previous hardware attacks target microcontrollers. Indeed, these chips are
slower and simpler than \glspl{SoC}. Therefore, an attacker can easily perturb
microcontrollers and exploit a fault.
\subsection{Motivation}
\label{sec:motivation}
The \gls{SoC} security model is focused almost entirely on software, so much
that hardware countermeasures such as TrustZone explicitly exclude hardware
security from their objectives. Fault injection attacks on \glspl{SoC} is a recent
research topic where some published articles
\cite{TimmersSW16,BADFET,MajericBB2016} focused on breaking software security
properties.
Fault injection is a versatile tool allowing to modify a program behaviour at
runtime, to inject vulnerabilities in a sound (and even proved)
software. The stake is the system security as a whole, as in controlling
what software is executed and what data can be accessed by whom. With software
security constantly improving and the costs and experimental difficulty of
performing fault injection attacks declining, we can surmise that the latter will become
a major threat in the future.
Understanding the \gls{EMFI} effects on a \gls{SoC} is required to understand
the threat and to design effective countermeasures. There is an extensive
literature on fault injection attacks on microcontrollers, and as a result, the most
secure devices against them are derived from microcontrollers (aka secure
components). The same work has to be done for \glspl{SoC}.
\subsection{Related work}
\label{sec:previous}
Since the seminal works of Boneh \textit{et al.}~\cite{boneh_faults} and Biham
and Shamir~\cite{biham_faults}, we know that fault injection attacks are a threat to the
security of cryptographic implementations. Researches on this topic has split in two
directions: on the one hand, the theoretical axis is interested in how a fault
can weaken the security of an application, mainly cryptographic algorithms
(\textit{e.g.}~\cite{piret_fa} and~\cite{giraud_fa} on AES).
On the other hand, the question is what faults can practically be performed in
systems. A formal description of these achievable faults is called a fault model
and it is this description that is used for theoretical analysis.
A fault model is always the interpretation of a physical behaviour at a
specific abstraction level. In other word, the same fault can be formalized
differently if we look at the transistor, the \gls{ISA} or the
software levels. Therefore, different studies looked at different levels for the
faults effects.
Several groups have developed fault models for
microcontrollers. Balasch \textit{et al.}~\cite{BalaschGV11} performed a clock
glitch attack on an 8-bit AVR microcontroller embedded in a smartcard. They show
that one can replace instructions in the execution flow by either targeting the
fetch or the execute stages of the 2-stage pipeline.
ARMv7-M microcontrollers have been thoroughly studied, in particular the Cortex-M3
\cite{MoroDHRE13} and Cortex-M4 \cite{RiviereNRDBS15}. These papers highlight
that \gls{EMFI} on these devices disturbs the correct behaviour of the pipeline.
Moro \textit{et al.}~\cite{MoroDHRE13} show that a fault can modify the fetched
opcode (at the time of fault injection) and that the new opcode has no side
effects with high probability. It is therefore equivalent to a \texttt{NOP} instruction
(no-operation). Another observed effect is the modification of the data in a
\texttt{LOAD} instruction. According to Rivière \textit{et al.}~\cite{RiviereNRDBS15},
the fault suppresses the instructions fetch\footnote{up to 4 instructions may be fetched
in one clock cycle}. The previously loaded instructions are instead executed
again before resuming correct execution, and the disrupted instructions are
never executed. These works only focus on microcontrollers.
But they are simple systems, they have in-order pipelines, most of
the time only one core, a simple memory hierarchy (only L1
cache if any) and no support for virtual memory. More recent works have been
invested in trying to fault more complex processors, mainly ARM \glspl{SoC}.
Timmers \textit{et al.}~\cite{TimmersSW16} show how to attack an ARMv7-A chip by
taking control of the \gls{PC} with fault injection attacks. Cui and Housley~\cite{BADFET}
demonstrate an \gls{EMFI} targeting the communication between the \gls{CPU} and
the external memory chips. The authors insist particularly on the difficulty to
achieve the required temporal and spatial resolution for \gls{EMFI} on modern
\gls{SoC}. Majéric \textit{et al.}~\cite{MajericBB2016} discuss how to find the
correct \gls{EMFI} parameters and setup a fault injection on the AES
co-processor to a Cortex-A9 core.
A critical feature of modern \glspl{SoC} is their complex micro-architecture,
increasing the attack surface. As a consequence, it is now possible to obtain
hardware faults from software execution. Tang~\textit{et al.}~\cite{TangSS17}
achieve a fault by taking control of the microcontroller in charges of
monitoring and managing the energy of the \gls{SoC}. By software means, they are
able to modify the power voltage and the clock frequency. DRAM memories also are
susceptible to Rowhammer attacks~\cite{VeenFLGMVBRG16}: with specific access
patterns, one can switch bits into the memory chip. The gap between the
\gls{ISA} abstraction and its real implementation (notably in Out-of-Order
processors) leads to the Meltdown~\cite{MELTDOWN} and Spectre~\cite{SPECTRE}
vulnerabilities.
Today, we understand the risks due to the complex micro-architecture of modern
\glspl{SoC}. Fault injection attacks have been demonstrated to be a threat to these
systems. Proy \textit{et al.}~\cite{proy2019} have proposed a fault model characterization on \gls{SoC} (with an \gls{OS}) at the \gls{ISA} level. But the faults effects on the \gls{SoC}
micro-architecture have still not been evaluated.
Our work intends to bring this missing piece to our understanding of
the security model of modern \glspl{SoC}.
\subsection{Contributions}
\label{sec:contribution}
In this article, we focus on an ARMv8 \gls{SoC}, namely the Broadcom BCM2837 chip
at the heart of the \gls{RPi3}. It is a widely successful low cost single
board computer. This quad-core Cortex-A53 \gls{CPU} runs at
\SI{1.2}{\giga\hertz} and features a modern ``smartphone class'' processor. We
are using \gls{EMFI} and observe the resulting failures to deduce their origins.
We observe radically new fault models that are neither described in other works
nor taken into account when discussing the modern embedded systems security.
In this paper, we demonstrate how we recovered these fault models and provide insights on the micro-architectural mechanisms leading to these models.
The consequences are dire: a \gls{SoC} must not be considered as a black box
with respect to security. It is not enough to work on the software side security
if it does not rely on solid hardware foundations.\newline
The goal is to provide a micro-architectural explanation of the observed
behaviour. To that end, we have to control the targeted system and limit its
complexity. It implies most notably that we use a single-core configuration and
setup an identity mapping for virtual memory. This simplification choice does
not imply necessarily a harder exploitation on more realistic systems, since the discovered fault models would still be present.
But on such systems, it becomes hard to isolate the effect of a fault and attribute it to one subsystem: we cannot propose a simple model explaining the observed behaviour.
We describe our setup in
\autoref{sec:setup}, both the experimental apparatus used to inject faults but
also the targeted hardware and software environment.
To stress out the software layer impact on the observed failures, we
compare the faults observability with and without an \gls{OS} in section~\ref{sec:os}.
Observed faults on a bare-metal setup are analysed in sections \ref{sec:icache},
\ref{sec:mmu} and \ref{sec:l2cache}. For each fault category, we will explain
the process that allows us to infer the cause of the failure. The possibility to
exploit these faults will be discussed as well as the experimental difficulties
to achieve them. We finish with propositions to protect \glspl{SoC} against
these attacks in \autoref{sec:countermeasures} and conclude in
\autoref{sec:conclusion}.
\section{Fault injection on embedded systems}
\label{sec:fault_injection}
\subsection{The targeted chip: The BCM2837 on the Raspberry Pi 3 B}
\label{sec:setup}
\subsubsection{Presentation}
The \acrfull{RPi3} is a low cost single board computer. It features a
complete system able to run a complex OS such as Linux or Windows and their
applications. The \gls{SoC} powering this board is the BCM2837 from Broadcom, a
quad-core Cortex-A53 \gls{CPU} running at \SI{1.2}{\giga\hertz} with the help of
a dual core VideoCore IV GPU at \SI{400}{\mega\hertz}.
Our experiments are performed with our own software stack\footnote{Released as
open-source software (MIT Licence). The git repository is available at
\textit{blinded for reviews}}, with only one core active. To control the
behaviour of the chip, we have implemented the bare minimum to run our
applications: initialization of \gls{JTAG}, UART, GPIO, \gls{CPU} caches and
\gls{MMU}. We want to stress out that no \gls{OS} is running during our
experiment in sections \ref{sec:icache}, \ref{sec:mmu} and \ref{sec:l2cache}, to
avoid interference that could hinder our ability to infer the fault models. In
particular, we want to avoid the effects of context switching due to preemptive
scheduling by the \gls{OS}, the error recovery mechanisms (if an error occurs,
we want to know) and the caches maintenance performed by the \gls{OS}.
In order to later explain the causes of the failures observed, we describe
in more details two important subsystems of this \gls{SoC}: the cache hierarchy
and the \gls{MMU}.
\subsubsection{Cache hierarchy}
In modern systems, memory accesses are a lot slower than the \gls{ALU}. To avoid
loosing too much performance to this latency difference, small and fast memories
called caches are used to mirror a part of the memory space.
In the targeted \gls{SoC}, each core has two L1 caches (the smallest and fastest
kind), one dedicated to instructions (L1I), one dedicated to data (L1D). These
caches are \SI{16}{\kilo\byte} with \SI{64}{\byte} line width.
Then a second layer of cache, the L2 cache, is common to all cores and thus
provides a unified view of the memory space. Its size is \SI{512}{\kilo\byte}
with \SI{64}{\byte} line width.
\begin{figure}[h]
\centering
\includegraphics[width=0.6\textwidth]{BCM2837_caches.png}
\caption{Memory hierarchy for the BCM2837.}
\label{fig:memhier}
\end{figure}
\subsubsection{Memory management unit}
The \gls{MMU} is a central component for every multi-applications system. It
aims at virtualizing the physical memory of the system into a virtual one.
Therefore, the CPU only works with virtual addresses and during a memory access
to one of these addresses, the \gls{MMU} translates it into the corresponding
physical address which is transmitted to the memory controller of the system.
The information required for the translation of an address is called a \gls{PTE}
and it is stored in the physical memory and cached in the \gls{TLB}. There is a
\gls{PTE} for every allocated pages in the physical memory. Our bare metal
implementation allocates the whole address space with an identity mapping
(virtual and physical addresses are the same) with \SI{64}{\kilo\byte} pages.
In modern systems, the translation phase does not only compute the physical
address but also realizes different checks. These checks are monitoring if the
page can be written or not, which kind of process (user or supervisor) can
access it or should the page be stored in cache or not.
Among all its roles, the \gls{MMU} is also a security mechanism. Ensuring that
a read-only page cannot be written to and ensuring that only authorized processes
can access their corresponding pages. This last security mechanism is the memory
partitioning. On multi-applications systems, it avoids a process to spy or
corrupt the memory used by another process.
On complex \glspl{OS}, the \gls{MMU} and the \gls{PTE} are setup by the
kernel and are critical assets.
\subsection{The Electromagnetic fault injection bench
To inject faults on the BCM2837, some apparatus is required. Our experimental
setup has been designed to be highly configurable and to work at higher
frequencies than most setups targeting microcontrollers. First, we use a
Keysight 33509B to control the delay between a trigger issued by the \gls{RPi3} board
before the instructions of interest. The Keysight 81160A generates the signal
for the EM pulse: one sinus period at \SI{275}{\mega\hertz} with a power of
\SI{-14}{\dBm}. A sinus is used instead of the usual pulse since it gives fewer
harmonics at the output of the signal generation chain. Then, this signal is
amplified with a Milmega 80RF1000-175 (\SI{80}{\mega\hertz} -
\SI{1}{\giga\hertz} bandwidth). Finally, the high power signal is connected to a
Langer RF U 5-2 near field probe. A part of this energy is therefore transmitted
into the metallic lines of the chip, which can lead to a fault.
The minimum latency between the initial trigger and the faulting signal reaching
the target is high: around \SI{700}{\nano\second}. As a consequence, the
targeted application must be long enough to be reachable by our fault injection
bench.
\subsection{Synchronization}
The main difficulty for fault injection is the synchronization: how to inject
precisely a fault on the targeted and vulnerable instructions.
To resolve this point, we need a temporal reference, given here by a GPIO: an
electrical rising edge is sent to a board pin by our application just before the
area of interest. Our setup is using the evaluator approach: the attacker can instrument the system to ease the experiments. In the case of a
real attack, the adversary would have to generate this trigger signal: it can be
done by monitoring communications, IOs, or EM radiation to detect patterns of
interest. In all cases, it is a tricky business highly application dependant.
But the trigger signal is just part of the problem: from this instant we must
wait the correct moment to inject the fault. To give a sense of the experimental
difficulty: for a chip running at \SI{1}{\giga\hertz}, a clock period lasts
\SI{1}{\nano\second}. In this lapse of time, light travels only for around
\SI{30}{\centi\meter}. Propagation times are not negligible. On a modern
\gls{SoC}, the matter is made more difficult by the memory hierarchy. Since
cache misses are highly unpredictable, they imply a corresponding jitter. It is
hard to precisely predict the duration of a memory access and therefore the time
to wait to inject the fault.
Synchronization is a problem, but not a hurdle that much. Indeed, the
attacker has only to inject faults until the correct effect is achieved. Because
of the jitter, for the same delay (time waited between fault injection and
trigger), different timing will be tested with respect to the running program.
If a fault with an interesting effect is possible, it will eventually be
achieved.
Additionally, as we will see in the next sections, memory transfers are
particularly vulnerable to \gls{EMFI}. They are also slower than the core
pipeline, allowing for a bigger fault injection timing window.
\subsection{How to change the fault effect?}
Fault effects are reproducible with a low ratio; meaning that if a fault has
been achieved, it will be achieved again with the same parameters but only for a
small ratio of the fault injections. In the other cases, no failure occurs or
another effect is observed (mostly due to jitter). To modify the fault effect,
the main parameters are the timing and the position of the probe over the
component. In particular, the signal parameters (shape, frequency, number of
periods) have an optimal value with respect to our requirements. The frequency
and the shape are chosen to maximize the EM coupling, the number of periods is
fixed to have the best timing precision.
\subsection{Forensic methodology}
As the targeted system is a closed box, we have a limited mean to
explore what is happening in the system, namely the \gls{JTAG}. With it, we are
able to halt the chip execution to read the register values and to read memory
as seen by a particular core (with a data viewpoint). Therefore, to pinpoint the
particular effects of a fault injection, we trade observability of the system
with controllability: we force the system state such that an observable change
gives us information on the fault mechanism. To maintain controllability, our
software footprint has to be minimal. As such we will not describe how to breach a
particular system with our faults since any exploit is highly application
dependant and our setup is not representative of a standard application
environment. Instead we will suggest exploit strategies: how such faults could
be used by a malicious attacker?
\newdate{date_raspbian}{18}{04}{2018}
\section{Impact of the operating system}
\label{sec:os}
To support our choice of bare metal applications to understand the fault model,
in this section we compare the faults observed with and without an \gls{OS} for
the same \gls{EMFI} parameters.
\subsection{Sensibility maps for the BCM2837}
Knowing where to place the probe for obtaining interesting effects is mandatory
for every perturbation experiments. Therefore, the first step consists in doing a
sensibility map of the component against EM perturbations.
During our experiments, two different setups were tested. The first one was
running the target program \autoref{lst:loop} on a bare metal system (one core
with only UART and \gls{JTAG} enabled). The second one was running the same
program as an application on a Linux-based OS\footnote{Raspbian Lite released on
\displaydate{date_raspbian} available here:
\url{https://downloads.raspberrypi.org/raspbian_lite/archive/2018-06-29-03:25/}}.
\lstinputlisting[language=Python,caption={Loop target
application},label=lst:loop]{loop.c}
The figures~\ref{fig:rpi3_carto_bare_metal} and~\ref{fig:rpi3_carto_linux} show the two sensibility maps with the number
of crashes induced by the perturbations for every probe location over the \gls{SoC} (for $27$ tries per location).
The area is divided in a $40$ per $40$ grid with a step of \SI{350}{\micro\meter}. This allows
us to cover the whole package of the \gls{SoC}. The first conclusion is that the
sensibility of the component under \gls{EMFI} depends on what is running
on it. The setup running with Linux has a wider sensitive area than the bare
metal one. However, the sensitive area of the bare metal setup is included in the
Linux one. This suggests that the two setups behave similarly under the
perturbations on this area. Since the Linux system embeds a far more complex
piece of software than the bare metal one, with more enabled interfaces, it may
explain that the Linux setup has a wider sensitive area.
\begin{figure}
\centering
\includegraphics[width=.7\textwidth]{baremetal_reboot_scatter.pdf}
\caption{Bare metal sensitivity map}
\label{fig:rpi3_carto_bare_metal}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.7\textwidth]{linux_reboot_scatter.pdf}
\caption{Linux sensitivity map}
\label{fig:rpi3_carto_linux}
\end{figure}
\subsection{Faults on bare metal versus faults on Linux}
The sensitive areas are not the only differences of behaviour between the two setups.
Another difference is the impacts of the perturbations.
In other words, the faults obtained are different on the bare metal setup and the Linux setup.
More precisely, the observable effect of a fault as seen by the same application is different whether it runs on a bare metal setup or on a Linux setup.
Evaluating the fault model on the Linux setup is a complex
analysis
With an \gls{OS}, faults are observable at the instruction level. This means the
effect induced by the perturbation is equivalent to a fault modifying one or
several instructions of the executed program.
On the bare metal setup, there are no observable faults at the instruction level
but instead at the micro-architectural level, as shown in the next sections. The
effect of the fault is equivalent to modifying the behaviour (signals or
configuration for instance) of subsystems like buses, \gls{MMU}, memory, caches,
etc.
This difference suggests that the usage of an \gls{OS} leads to a specific
perturbation behaviour. In this specific case, the fault model induced by the
\gls{OS} (instruction modification) is easier to exploit than the fault model on
bare metal setup: the \gls{OS} weakens the security of the system against fault
injection attacks.\newline
In this work, we will focus on the fault model on the bare metal setup. We will
show how to analyse and reconstruct the effect of a perturbation on the
micro-architectural elements of a processor.
However, we can suppose that the micro-architectural fault effect on the bare
metal setup explains the observed effect at the instruction level on the Linux
one.
\section{Fault on the instruction cache}
\label{sec:icache}
In this experiment, we achieve a fault in the L1 instruction cache (L1I).
\subsection{On the impossibility to fault the instruction execution flow}
Before reporting our positive results, we must report a negative one. Contrary
qto previous works on microcontrollers (where \textit{e.g.} instructions are
replaced with a \texttt{NOP} instruction), we are not able to prevent or modify directly
the execution of an instruction. In other words, in \autoref{lst:loop}, without
the \texttt{invalidate\_icache()} no faults are observed. Even if we cannot be
sure that no set of experimental parameters would achieve such a fault, we
thoroughly explored the parameters without success.
As we will see in this section and the following, all faults affect memory subsystems.
Probably because the buses involved in the memory transfers are easily coupled with our EM probe.
\subsection{The target}
The application targeted during this experiment is two nested loops shown on
\autoref{lst:loop}, executed after the L1I cache invalidation. No fault is
obtained without the invalidation. It is compiled without optimizations
(\verb|-O0|) since we do not want the compiler to optimize our code.
Since the instruction cache is invalidated before the loop execution, the
following instructions have to be (re)loaded in cache before their execution.
And it is this memory transfer that we will target with our fault injection. By
executing the same application with and without the cache invalidation and
measuring the duration of the high state of the trigger, we deduce that loading
instructions in cache has an overhead of \SI{2}{\micro\second}. Our bench has a
latency of \SI{700}{\nano\second}, so we can still hit this memory transfer. To
be able to observe the effect of a fault on the full timing range, a
\SI{2}{\micro\second} wait has been inserted between the trigger and the cache
invalidation.
\subsection{Forensic}
The fault is detected when the output sent back to the host is not $2500$ (the
value of \texttt{cnt} at the end of the program if everything went well). When a
fault is detected, we use the \gls{JTAG} to reexecute our application (in
\autoref{lst:loop}) by directly setting the \gls{PC} value at the start of the
loops. Then, we execute our program instruction by instruction while monitoring
the expected side effects (we do not inject faults anymore). All instructions are
correct except one, the counter incrementing instruction (the \texttt{add} instruction at
address \texttt{0x48a08} on \autoref{lst:loopasm}).
\lstinputlisting[language=C,caption={Loop target application assembly with -O0},label=lst:loopasm]{loop_short.asm}
By monitoring the \texttt{w0} register before and after the \texttt{cnt}
increment instruction, we observe that the value is kept unchanged: the
increment is not executed. Since the fault is still present after the EM
injection, we conclude that a wrong instruction value is memorized in L1I. We
confirm this fault model by executing a L1I cache invalidation instruction
\texttt{ic iallu} (we set the \gls{PC} value to the instruction location in
memory). Reexecuting our application, the fault has disappeared.
We can infer that the injected fault has modified a value in the L1I cache, but
it is impossible to read the new value. Since the fault happens during the cache
filling, we can suppose that it is the memory transfer that has been faulted.
\subsection{Exploits}
This fault is one of the easiest to exploit since it is similar to the classical
instruction skip model. Therefore, most exploits based on this classical model
apply here. Since the faulted value is still present in the cache, it will stay
faulted until the cache is invalidated: we can call this model ``sticky
instruction skip''. Bukasa \textit{et al.}~\cite{Bukasa2018} demonstrate
applications of this fault model: hijacking the control flow and initiating a
\gls{ROP} attack among others.
\section{Targeting the MMU}
\label{sec:mmu}
The \gls{MMU} is a critical component of \glspl{SoC}. It is in charge of the
virtual memory mechanism. In this section, the fault changes the virtual to
physical memory mapping, albeit in an uncontrolled manner. The targeted
application is the same as in section~\ref{sec:icache}, shown on
\autoref{lst:loop}.
\subsection{The configuration of a working MMU}
To understand the effect of the fault, we begin to explore the state of a
working application (without any fault). This state is a legitimate one.
\subsubsection{Page tables}
The page tables are used to memorize the mapping between virtual and physical
memory. In our configuration, we have 3 \glspl{PTD} (mapping
\SI{512}{\mega\byte} chunks) and
for each one, we have $8192$ \gls{PTE} pages of \SI{64}{\kilo\byte}. We show an
excerpt of the correct \glspl{PTE} on \autoref{fig:l3pt}.
\begin{figure}
\centering
\BVerbatimInput[fontsize=\small,commandchars=\\\{\}]{level3_page_tables_ok_half.dump}
\caption{Memory dump excerpt for \glspl{PTE} before fault.\label{fig:l3pt}}
\end{figure}
In the page tables, the most and least significant bits are used for the page
metadata (access rights, caches configuration, \textit{etc.}).
\subsubsection{TLB}
\glspl{TLB} (plural since there are several of them), are small buffers used to
speed up virtual to physical memory translation. As in a cache memory, the last
mappings are saved to be reused later without a full page tables walk by the
\gls{MMU}. In the targeted \gls{SoC}, \gls{TLB} hierarchy mirrors cache
hierarchy: the \gls{TLB} designates the unified Level2 buffer while
micro-\glspl{TLB} are dedicated to instructions or data in each core.
\subsubsection{Operating system}
In our bare metal application, all the pages are initialized in the page tables
with an identity mapping (virtual and physical addresses are identical). In a
system with an \gls{OS}, pages are allocated on-the-fly. On the one hand, this
simplifies the forensic analysis since we are sure that page tables are correct
prior to the fault. On the other hand, interesting faults may be missed if the
\gls{OS} page allocation is disrupted.
\subsection{Forensic}
To reconstruct the memory mapping, we use a pair of instruction computing the physical address (and the corresponding metadata) for a given virtual one.
A script has been designed to extract the memory mapping. By using the \gls{JTAG}, first the two instructions \verb|at s1e3r, x0; mrs x0, PAREL1| are written at a given address, then the
\texttt{x0} register is set to one virtual address, the two instructions are
executed and finally the \texttt{x0} register contains the corresponding
physical address.
With this method, we compare the memory mappings with (\autoref{fig:faulted_mapping}) and without (\autoref{fig:identity_mapping}) a fault.
\begin{figure}
\centering
\BVerbatimInput[fontsize=\small]{identity.txt}
\caption{Correct identity mapping}
\label{fig:identity_mapping}
\end{figure}
\begin{figure}
\centering
\BVerbatimInput[fontsize=\small]{working_fault.txt}
\caption{Mapping after fault}
\label{fig:faulted_mapping}
\end{figure}
Three different effects can be observed depending on the page:
\begin{itemize}
\item Pages are correct with an identity mapping up to \texttt{0x70000}.
Remarkably theses are all the pages used to map our application in memory.
Therefore, an hypothesis is that the corresponding translations are present in
caches and are not impacted by the fault.
\item Pages are incorrectly mapped to \texttt{0x0}. A read at \texttt{0x80000}
reads with success physical memory at \texttt{0x0}.
\item Pages are shifted. A read at \texttt{0xc0000} reads physical memory at
\texttt{0x800000}.
\end{itemize}
If we invalidate the \gls{TLB} after a fault, nothing changes: the mapping stays
modified.
We conclude that the fault does not affect the cache mechanism of address translation (at least what can be invalidated by software) but directly the \gls{MMU}.
To look for an explanation of the incorrect mapping,
we can look for the impact on page tables on \autoref{fig:l3xxxxpt}.
\begin{figure}
\centering
\BVerbatimInput[fontsize=\small,commandchars=\\\{\}]{level3_page_tables_fault_half2.dump}
\caption{Memory dump excerpt for \glspl{PTE} after a fault.\label{fig:l3xxxxpt}}
\end{figure}
The fault on the \gls{MMU} has shifted the page tables in memory, and has
inserted errors in it. Since the memory translation is still valid after the
fault, and do not correspond to the shifted page tables, this shift is not the only source of incorrect translation.
Either the page walk is done from physical addresses and/or some \gls{TLB} are
not properly invalidated when we try to.
\subsection{Exploit}
This fault shows that the cornerstone of the key security feature in any
\gls{SoC}, namely memory isolation, does not withstand fault injection. In
\cite{drammer}, the authors use the rowhammer attack to fault a \gls{PTE}. The
faulted \gls{PTE} accesses the kernel memory which allow the attacker to obtain a privilege escalation: by overwriting an userland \gls{PTE} for accessing all the
memory, by changing the user ID to root or by changing the entry point of an executable.
Additionally, this fault model is a threat to pointer authentication
countermeasures, as proposed in the recent ARMv8.3 ISA. This pointer protection
works by storing authentication metadata in the most significant bits (usually
useless) of a pointer value. To use a pointer, the chip first validates the
authentication metadata. In our case, the attacker does not need to alter the
pointer value, it can alter where it physically points to, at a coarse (page)
granularity.
\section{Shifting data chunks in L2}
\label{sec:l2cache}
Another interesting behaviour when faulting a modified version of the loop
target, \textit{cf} listing~\ref{lst:register_transfer}, was investigated with
\gls{JTAG}.
\lstinputlisting[language=C,caption={Targeted assembly},label=lst:register_transfer]{register_transfer.asm}
We observe that data are shifted in the L2 cache, as if addresses had been slightly modified in the memory transfer writing to L2.
\subsection{Forensic}
A quick step by step execution shows that we are trapped into an
infinite loop. A \gls{JTAG} memory dump at the instruction memory location shows
modified instructions as seen on \autoref{dump:L2unified}, to be compared to the unfaulted dump on \autoref{dump:L2ok}.
With the same parameters, several similar faults have been obtained with two
observed faulty memory dumps, here called F1 and F2. Similar because the same
infinite loop is obtained (as shown in step by step execution), but the memory
dumps obtained from \gls{JTAG} are slightly different. We will show that the
difference is due to discrepancies between caches for F2 and that invalidating
L1I cache restore coherence with the F1 results.
\begin{figure}[h]
\centering
\BVerbatimInput[fontsize=\small,commandchars=\\\{\}]{L2_unified_half.dump}
\caption{Memory dump showing the instructions in the infinite loop as seen
by the \gls{JTAG} for F1. The instructions in the infinite loop are
underlined. \label{dump:L2unified}}
\end{figure}
\begin{figure}[h]
\centering
\BVerbatimInput[fontsize=\small,commandchars=\\\{\}]{L2_ok_half.dump}
\caption{Memory dump showing correct (without fault) instructions for the same memory region.\label{dump:L2ok}}
\end{figure}
The step by step execution is coherent with the instructions shown by the JTAG
dump for F1 (registers are loaded, stored and incremented as specified by the
instructions in this dump, not the correct ones). Since \gls{JTAG} dump and
instructions execution are coherent, it seems that the fault is present in L2
cache (unified view). The reconstructed instructions can be seen in listing \ref{lst:l2faulted}.
\lstinputlisting[language=C,caption={Assembly reconstruction of the faulted instructions},label=lst:l2faulted]{l2faulted.asm}
\paragraph{MMU} After verification as in section~\ref{sec:mmu}, we observe that
the \gls{MMU} mapping is indeed modified but not for the memory region of
interest. In particular, memory addresses from \texttt{0x0} to \texttt{0x7FFFF}
are still correctly mapping identically virtual and physical addresses. The
fault on the \gls{MMU} cannot therefore explain our observations.
\paragraph{L1 fault} The L1I is of course modified by the fault since the
behaviour of the application is modified. But the execution is coherent with the
\gls{JTAG} dump (showing the L1D cache) hinting that L2 is impacted too. As seen
on figure~\ref{dump:L2nonunified}, the dump
after F2 shows different values in L1D memory with respect to F1 (for the first
two instructions), but the execution trace is coherent with F1 \gls{JTAG} dump
(on figure~\ref{dump:L2unified}), not with F2 \gls{JTAG} dump (on
figure~\ref{dump:L2nonunified}).
\begin{figure}[h]
\centering
\BVerbatimInput[fontsize=\small,commandchars=\\\{\}]{L2_nonunified_half.dump}
\caption{Memory dump showing the instructions in the infinite loop as seen
by the \gls{JTAG} for F2.\label{dump:L2nonunified}}
\end{figure}
As it turns out, \texttt{940087c1} encodes a branching instruction that is not
followed in the step by step execution. In the case of F2, the \gls{JTAG} dump
does not reflect the values in the L1I cache. If we invalidate L1I cache,
nothing change (either in the execution trace or in the \gls{JTAG} dump). But if
we invalidate to point of coherency at address \texttt{0x489f8}, then the new \gls{JTAG}
dump becomes the same as for F1.
\paragraph{Hypotheses} The effect of these faults seem to manifest in the L2
cache. We can observe that it consists in shifting groups of 4 instructions (128
bits or \SI{16}{\byte}, under the cache line size of \SI{64}{\byte}) at a nearby
memory location. 128-bit is the size of the external memory bus connected to L2. A fault model could be that the address corresponding to a cache transfer
toward L2 has been modified by a few bits.
\begin{figure}[h]
\centering
\includegraphics[width=0.8\textwidth]{mem_L2.png}
\caption{Summary of our fault model for the faults observed in \autoref{sec:l2cache}.}
\label{fig:l2sum}
\end{figure}
The fault model is summarized on \autoref{fig:l2sum}. But our model has some limits: the presence of F2 means that, with one fault, values in L2 and in L1D are modified simultaneously in an uncoherent way.
Is it due to the EM probe coupling with several buses or to a micro-architectural mechanism ?
\subsection{Exploits}
A fault in the L2 cache can impact either the instructions or the data giving
more power to the attacker. Yet she does not control in what way the memory
will be modified. Why these particular 16 bytes blocks have been shifted?
Nevertheless, just corrupting data or instruction randomly is often enough to
achieve the desired effect (as in the \texttt{NOP} fault model).
\section{Countermeasures}
\label{sec:countermeasures}
The vulnerabilities described above can be summarized as integrity and authenticity problems: instructions
(or data) are altered between their storage in memory and their reuse.
If we consider that only a legitimate entity can write to memory, then integrity is the only problem.
In the other case, we must also ensure the write authenticity.
\subsection{Ensuring integrity}
Ensuring the integrity of signals in a chip is a well known problem that had to be solved to allow the use of chips in harsh environments: space or nuclear reactors for example.
Yet one must account for a different threat model: when designing for security, the fault value must be considered intentional, not drawn from a uniformly random distribution.
To ensure integrity, designers use redundancy. We can duplicate the core, executing exactly the same things on both cores and verifying that results are identical. Redundancy can also be achieved using error detection codes. Of course, redundancy has an overhead but this is the cost of guaranteeing the integrity.
When dealing with the threat of physical attacks, ensuring integrity is often not enough. If we consider that the attacker can modify memory, she could bypass the error detection code or write the same error value in the duplicated memories.
In this case, we must ensure the data authenticity.
With cryptography, authenticity can be guaranteed by relying on \glspl{MAC}.
We can imagine a strategy based on this principle to be able, in hardware, to detect changes in data or instructions.
\subsection{MAC Generation}
A solution, to ensure authenticity has been proposed with SOFIA~\cite{SOFIA}.
For each data or instruction block (the block size has to be adapted to the
micro-architecture), the objective is to calculate and associate a \gls{MAC} to
detect any alteration. In addition to the data itself, the \gls{MAC} calculation
must also be address dependent to detect shifts between an address and the
corresponding data (as observed in the L2 cache in section \ref{sec:l2cache}).
\glspl{MAC} must be generated at the right time: in the case of instructions,
they can be computed at compile time. But for the data, it must be possible to
do the generation at the pipeline output (during memory access).
\subsection{MAC Verification}
Depending on the energy consumption/performance trade-off, two implementation strategies can then be considered on a
system-wide basis to perform the \gls{MAC} check.
\begin{itemize}
\item \textbf{Just-in-time:} To minimize the overall activity, this
strategy consists in bringing the data and its \gls{MAC} back to the \gls{CPU}
(or \gls{MMU}) in a classical way. The verification would then be performed by
the \gls{CPU} just before data consumption. In case of mismatch, a request has
to be made to the higher memory level (the L1 cache) to perform a
verification on its own data version. If the new verification is
successful, the data would be transmitted to the \gls{CPU} and if not, a
request to the next level (L2) has to be made. Thus, several checks are
performed only when necessary, but the cost of a mismatch (that can be due to
environmental radiations) is very high.
\item \textbf{Proactive:} A second strategy can be used to reduce the time
penalty in case of error. Integrity checks are automatically performed at each
level of the memory hierarchy. Thus, errors can be detected early, before they
reach the \gls{CPU}. Each cache level can ensure independently that it has
only valid data, but the energy consumption would be higher.
\end{itemize}
\section{Conclusion}
\label{sec:conclusion}
In this paper, we have demonstrated some vulnerabilities with respect to fault injection
attacks specific to \gls{SoC}. In particular, the memory hierarchy and the
\gls{MMU} can be altered which creates a mismatch between how the hardware
behave and what the software expect. Nowadays, the computing systems security
is focused on the software side, there are no efficient countermeasure against
hardware perturbations in modern \gls{SoC}. Pointer
authentication as proposed by the ARMv8.3 \gls{ISA}, for example, does not
resist the introduced fault model.
Exploitation depends heavily on the interaction between the hardware (the
specific device) and the software (including application and \gls{OS}). Therefore
the cautious developer cannot predict where vulnerabilities will occurs and as a
consequence cannot efficiently protect its application. Today, such attacks
using \gls{EMFI} are still quite hard to realize: they require expensive
apparatus, human resources to do the experiments, etc. But they are within reach
of small organizations and we can expect that the difficulty and cost of these
attacks will be lower in the future.
Actions must be taken to ensure that computing systems handle
sensitive information securely. The performances/energy consumption trade-off has been
settled by implementing the two kinds of cores in the same \gls{SoC}. In the
same way, we need \textit{secure} cores as well that compromise both on
performances and energy consumption but can offer much stronger security
guarantees.
\bibliographystyle{alpha}
|
1,116,691,500,775 | arxiv | \section{Introduction.}
More than twenty years' experience of the lattice gauge community has taught us
that it is always a good thing to have a bare action which respects symmetries,
because then no fine tunings are required to preserve the symmetries at distances
much greater than the cutoff scale. For this reason, essentially all lattice simulations of
gauge theories perform simulations with gauge invariant lattice actions,
and there is never any discussion about trading a small violation of gauge invariance in the simulation
for larger volume or an apparently more efficient simulation algorithm.
People have, however, always been willing to sacrifice chiral symmetry in their choice of lattice
discretization.
This seems somehow asymmetric:
Why not perform simulations with lattice actions which
preserve exact $SU(N_f)\times SU(N_f)$ chiral symmetry?
The advantages of this approach are obvious: One does not have to
separate the physical explicit
chiral symmetry breaking from a nonzero quark mass from the unphysical
chiral symmetry breaking induced by lattice artifacts.
The flavor content of the theory being
simulated is unambiguous. The index theorem is theoretically clean.
The topological charge can be measured to be exactly what the dynamical fermions
see during the simulation,
not something which is determined by some post-processing procedure. And because
the action preserves symmetries, correlation functions obey Ward identities which
considerably simplify their theoretical analysis. For example,
one does not have to spend any time measuring
(and trying to remove) lattice-artifact additive mass renormalization or operator mixing.
The way to do this is well known: use a lattice action which encodes
the Ginsparg-Wilson\cite{Ginsparg:1981bj} relation,
an overlap\cite{Neuberger:1997fp,Neuberger:1998my}
action. This article is a summary of our experiences with simulations
of two flavors of dynamical overlap fermions, using a version of the algorithm
of Fodor, et al\cite{Fodor:2003bh}.
It is a condensation of our two recent papers, Refs. \cite{DeGrand:2004nq,DeGrand:2005vb},
plus a little newer material.
\section{Algorithmic Issues}
Simulations with dynamical overlap have (at least) two problems:
\begin{itemize}
\item
They run so slowly
\item
Changing topology is hard
\end{itemize}
Our method of attack for the first problem is to replace the
usual link variable gauge connection by a fat link. It has been known
since at least 1998 \cite{DeGrand:1998pr}
that fat links improve the chiral properties of
non chiral fermion actions (and the flavor symmetry properties
of staggered fermions). The bottleneck has always been to find a
smearing method
which can be used in a molecular dynamics update, where the evaluation
of the force requires a fat link which is differentiable with respect to its
component thin links. Formulations like the Asqtad link solve this problem
by ``following the paths,'' but this does not give as much improvement
as one would like. Our solution was provided by
Peardon and Morningstar\cite{Morningstar:2003gk}
with the ``stout link'' (invented in a Dublin public house): a multilevel
blocking which is fully differentiable. In our runs, it pushes the thin link
plaquette ${\rm Tr} U_p \sim 1.7$ up to about 2.8 (with two levels of smearing with $\rho=0.15$).
The number of Dirac operator matrix times vector multiplies per trajectory is reduced by
about an order of magnitude compared to simulations with a thin link gauge connection.
The physics reason for this speedup is that fattening reduces the number of small eigenmodes
of the kernel operator, improving its conditioning number.
With a stout link, the fermions decouple from the UV fluctuations of the gauge field, and
the mean size of the fermion force is reduced to about an order of magnitude smaller than
the gauge force. This makes a multiple time scale integration algorithm very attractive.
We run with the Sexton-Weingarten\cite{Sexton:1992nu}
form of this updating, taking the integration
time step for the gauge fields to be 1/12 of that for the fermions.
The square of the Hermitian overlap operator projected on one chiral
sector is given by
\begin{equation}
H^2_\sigma(m)= 2 (R_0^2-\frac{m^2}{4})
P_\sigma\left[1+\sigma \sum_i \epsilon(\lambda_i)
|\lambda_i\rangle\langle \lambda_i| \right]P_\sigma+m^2
\label{eq:shift}
\end{equation}
with $R_0$ the radius of the Ginsparg--Wilson circle,
$P_\sigma=\frac{1}{2}(1+\sigma \gamma_5)$ the projector
on chirality $\sigma$ and $h(-R_0)$ the Hermitian kernel operator.
The sign function $\epsilon(h(-R_0))$ is
here given in its spectral representation.
Because of the sign function in its definition, the effective action of the overlap
operator has a discontinuity. It occurs when one
eigenvalue of the kernel $h(-R_0)$ changes sign during the
molecular dynamics evolution. These are the surfaces
in the space of the gauge fields
on which the topology as defined by the index theorem changes by one unit.
Ref.~\cite{Fodor:2003bh} gives a prescription of how
to account for this discontinuity
in the HMC algorithm. One essentially measures the height $\Delta S$
of the step in the action
(the potential of our Hamiltonian equations of motion)
and if the momentum perpendicular to the surface is large enough one reduces it
as one would do in classical mechanics. We will call this a ``refraction'' in
the following. If the perpendicular momentum is too small, we flip it, and thus
reflect the trajectory. With $N$ the vector normal to the surface
momenta $\pi$ are thus updated by
\begin{equation}
\Delta \pi =
\begin{cases}
-N \; \langle N |\pi \rangle + N \; {\rm sign} \langle N | \pi \rangle \;
\sqrt {\langle N | \pi \rangle^2-2 \Delta S_f}
& \text{if $\langle N | \pi \rangle^2>2 \Delta S_f $}\\
-2 N \langle N | \pi \rangle & \text{if $\langle N | \pi \rangle^2\leq 2 \Delta S_f$}
\end{cases}
\label{eq:ref}
\end{equation}
The discontinuity $\Delta S$ of the effective action is caused by one eigenvalue
changing sign, thus making the replacement
\begin{equation}
H^2_\sigma(m) \longrightarrow H_\sigma^2\pm (4R_0^2-m^2) P_\sigma|\lambda_0\rangle\langle \lambda_0|P_\sigma
\end{equation}
with $|\lambda_0\rangle$ the zero mode. The corresponding step in the effective action
can be evaluated using the Sherman--Morrison formula~\cite{Golub}
\begin{equation}
\Delta \left [ \langle \phi| P_\sigma \frac{1}{H_\sigma(m)^2}P_\sigma| \phi \rangle \right] =
\mp
\frac{(4R_0^2-m^2)}
{1\pm (4R_0^2-m^2) \langle \lambda_0|P_\sigma H^{-2}_\sigma(m) P_\sigma|\lambda_0 \rangle}
|\langle \phi |P_\sigma\frac{1}{H_\sigma(m)^2}P_\sigma| \lambda_0\rangle|^2 \ .
\label{eq:sherman}
\end{equation}
Interestingly, for the overlap not only can one compute the step in the effective action, but one can
also give a closed form expression
for the change in the fermionic determinant due to the change in
topology:
\begin{equation}
\frac{\det \tilde H^2_\sigma(m)}{\det H^2_\sigma(m)}
= 1\pm(4R_0^2-m^2) \langle \lambda_0 |P_\sigma \frac{1}{H^2_\sigma(m)} P_\sigma | \lambda_0\rangle \ .
\label{eq:stepdet}
\end{equation}
In the actual simulation one faces the problem that the trajectories reflect most of the
time off the zero eigenvalue surface and one never changes topology.
The reason is that the change in the determinant from the starting configuration
to the 'current' configuration in the MD evolution is only approximated well
as long as the Dirac operator is similar to the starting one.
The fluctuations are small as long as one has not changed topology.
However, this is definitely not
the case for the operator in a different topological sector. Since $\exp(-\phi^+1/H^2\phi)$
averages to the change in the determinant from the starting configuration to the end,
large fluctuations mean that most of the time $\exp(-\phi^+1/H^2\phi)\approx 0$ and
the effective action is thus large, whereas only a few times do we get small effective actions.
These
two observations combine to give large $\Delta S$ most of the time, and thus
a large auto-correlation time in the topological charge.
To reduce fluctuations, we used the method proposed in
Refs.~\cite{Hasenbusch:2001ne,Hasenbusch:2002ai},
which consists of rewriting the fermion determinant as
as
\begin{equation}
\det H^2(m) = \det H^2(m_{N_p})\prod_{i=1}^{N_p-1} \det \frac{H^2(m_i)}{H^2(m_{i+1})}
\label{eq:det}
\end{equation}
with $m_1=m$ and $m_i<m_{i+1}$ with suitably chosen larger masses.
In this method, only determinant ratios are evaluated
using pseudo-fermions for the light quark masses. The change in the spectra
while changing topological sector of the ratio $H(m)/H(m')$ is expected to be
less dramatic than the change of the spectrum of $H(m)$. Only the
determinant of $H(m_N)$ is evaluated directly.
However, for a large mass $m_N$ the spectrum of $H^2$ is confined to a smaller
region between $m_N^2$ and $4 R_0^2$
and the change in the spectrum therefore less drastic than for a smaller mass.
One or two extra pseudo-fermion fields ($N_p=2,3$ in Eq. \ref{eq:det}) help some, but do not
solve the problem.
To quantify our difficulty, we compare in Fig.~\ref{fig:1} the discontinuity of the effective
action with the physical step from the fermion determinant. We subtracted
the relevant quantity from the normal component of the momentum so that
positive values correspond to reflections whereas the topology changes for negative values.
We observe that the physical discontinuity would allow for significantly more changes in
topology than the step in the effective action does.
\FIGURE[tb]{
\includegraphics[width=0.3\textwidth, angle=-90, clip]{tunnel_0.03.eps}
\includegraphics[width=0.3\textwidth, angle=-90, clip]{tunnel_0.05.eps}
\caption{The stochastic estimate of the height of the step compared to the actual change
in the logarithm of the determinant from a subset of our ensemble.
We subtracted the normal component of the momentum squared (which is typically less than 10) such
that negative values mean refraction and positive ones reflection. For mass $m_q=0.03$ on
the left we have a number of events in the upper left quadrant that would have tunneled with the
exact change of the determinant and only a few that actually tunneled (in the two lower
quadrants). For $m_q=0.05$ the picture is similar, even though there are more tunneling events.
\label{fig:1}
}
}
The low correlation between the estimator and the physical step height Eq.~(\ref{eq:stepdet})
shows up in the large auto-correlation time of the topological charge. Even
though part of it is physics --- lighter quarks make it harder to get from
$\nu=0$ to $\nu=\pm 1$ --- the height of the step grows with $1/m^2$ instead
of the expected determinant ratio, $\log~m$. Since the normal component of the momentum
is roughly independent of the quark mass, it becomes more and more difficult to change topology.
Indeed, Fig.~\ref{fig:changevmq} shows that the mean time between
topological changes varies inversely with the square of the quark mass.
The large auto-correlation time for the topology is
a phenomenon that is also known with other fermions, e.g. improved staggered quarks.
To the extent that these formulations know about topology, the step in the fermion action
for the overlap might be replaced for them by a steep region which approximates the step.
The result is the same: if the approximation of the determinant is bad, the step is
overestimated most of the time and one does not change topology.
\begin{figure}
\begin{center}
\includegraphics[width=0.4\textwidth,clip]{change_vs_mq.eps}
\end{center}
\caption{Monte Carlo simulation time between topology changes versus quark mass.
\label{fig:changevmq}
}
\end{figure}
\section{The Topological Susceptibility}
In our second paper we made rough calculations of the topological
susceptibility and
chiral condensate using
eigenmodes of the Dirac operator. We made simulations on $8^4$ lattices, at a lattice
spacing of about $a\sim 0.16$ fm, with three quark masses, $am_q=0.03$, $0.05$, and $0.1$.
These lattices are really too small for physics, but they illustrate
the useful features of a calculation with overlap fermions.
We determined the string tension from the heavy quark potential and the Sommer parameter,
from Wilson loops of temporal extent $t=2$ and 3. The two measurements are not consistent,
but we performed tests on $8^4$ and $12^4$ quenched lattices which showed that the $t=3$
potentials were consistent with ones from further separations. So we used their fit values.
The lattice spacing varies by about ten percent as we change the quark mass.
One picture, Fig.~\ref{fig:topomr0},
illustrates our measurement of the topological susceptibility $\chi$.
We take our measurements of $r_0/a$ and the topological charge time history
to compute $\chi r_0^4$. We have computed the
lattice-to-$\overline{MS}$ matching factor in perturbation theory and use it to
convert the quark masses to their $\mu=2$ GeV
$\overline {MS}$ values.
D\"urr\cite{Durr:2001ty} has presented a phenomenological interpolating formula
for the mass dependence of the topological susceptibility,
in terms of the condensate $\Sigma$ and quenched topological susceptibility $\chi_q$,
\begin{equation}
\frac{1}{\chi}= \frac{N_f}{m_q\Sigma}+ \frac{1}{\chi_q}.
\label{eq:durr}
\end{equation}
Taking $\Sigma$ from our RMT analysis in the next section ( ${r_0}^3 \Sigma\sim0.43$)
produces the curve shown in the figure.
Most published measurements of the topological susceptibility present them as a function of the pseudoscalar
mass. Since we don't have spectroscopy, we can't do that. We can, however, use the D\"urr
curve as a fiducial, since most published measurements of the topological susceptibility
present it, too. Our data (as well as that of Ref. \cite{Fodor:2004wx}) lies below the D\"urr
curve.
Most measurements with nonchiral actions lie above it. (See, for example the figures in
Ref. \cite{Hasenfratz:2001wd} or \cite{Allton:2004qq}). Since our quenched results give
a value typical of simulations on larger lattices, $\chi \sim (190$ MeV$)^4)$, we don't think we are seeing
a finite volume effect.
\begin{figure}
\begin{center}
\includegraphics[width=0.4\textwidth,clip]{topo_mr0.eps}
\end{center}
\caption{Topological susceptibility versus quark mass, in units of
$r_0$.
The curved line is the D\"urr interpolating formula, Eq. \protect\ref{eq:durr}.
The three horizontal lines give the quenched value and its error.
}
\label{fig:topomr0}
\end{figure}
\section{The Condensate from Eigenmode Distributions}
It was proposed more than a decade ago that the distribution of the low-lying eigenvalues
of the QCD Dirac operator in a finite volume can be predicted by random matrix
theory (RMT)~\cite{Shuryak:1992pi,Verbaarschot:1993pm,Verbaarschot:1994qf}.
Since then this hypothesis has received impressive support from lattice calculations,
mainly quenched simulations
\cite{Berbenni-Bitsch:1997tx,Damgaard:1998ie,Gockeler:1998jj,Edwards:1999ra,Bietenholz:2003mi,Giusti:2003gf},
but also some dynamical ones using staggered quarks
\cite{Berbenni-Bitsch:1998sy,Damgaard:2000qt}.
Typically, the predictions are made in the so-called epsilon regime,
for which $1/\Lambda \ll L \ll 1/m_\pi$ with $\Lambda$ a typical hadronic scale.
However, it has been found that they describe the data in a wider range.
Two recent large scale studies, e.g., using the overlap operator on quenched
configurations~\cite{Bietenholz:2003mi,Giusti:2003gf}, needed lattices
with a length larger than $1.2-1.5~{\rm fm}$ for RMT predictions match the result of the
simulation. Our dynamical lattices have a spatial extent of about $1.3~{\rm fm}$.
As we will see, random matrix theory describes our low-lying Dirac spectra quite well.
Our analysis is based on
the distribution of the $k$-th eigenmode from RMT as presented
in Ref.~\cite{Damgaard:2000ah} and successfully compared to simulation results
in Ref.~\cite{Damgaard:2000qt}. The prediction is for the distribution of the
dimensionless quantity $\zeta=\rho \lambda_k \Sigma V$ in each topological sector, with
$\lambda_k$ the $k$-th eigenvalue of the Dirac operator,
$\Sigma$ the chiral condensate, and $V$ the volume
of the box.
The quantity $\rho$ is
the one-loop finite volume correction, $\rho = 1 + c/(f_\pi L)^2$ where $c$ is
a ``shape factor.''
These distribution are universal and depend only on
the number of flavors, the topological charge and the dimensionless
quantity $m_q \Sigma V$.
By comparing the distribution of the eigenmodes with the RMT
prediction one can thus measure the chiral condensate $\Sigma$.
The main advantage of this method is that it gives the
zero quark mass, infinite volume condensate directly. The validity of the approach
can be verified comparing the shape of the distribution for the various modes and topological
sectors. The main uncertainty comes from a too small volume which causes deviations in the
shape, particularly for the higher modes.
In Fig.~\ref{fig:rmt} we show the distribution of the two lowest eigenmodes of the overlap
operator (scaled by $\Sigma V$)
measured on the $\nu=0$ and $\nu=\pm1$ parts of the $am_q=0.03$ and $am_q=0.05$ ensembles.
We fit the RMT prediction from Ref.~\cite{Damgaard:2000ah} to these distributions.
The prediction agrees overall well with the measured distribution given the low statistics.
However, the distribution of the lowest mode in the $\nu=0$ sector seems to have a tail at larger
$\lambda \Sigma V$ that does not match the prediction. This could be an effect of the small
volume.
We also show the prediction for the distribution of the third mode from our fitted values
of $\Sigma V$ in the third column of Fig.~\ref{fig:rmt}. The RMT curve and the data, again,
agree quite well. However for the $|\nu|=1$ sector, the curve seems to be on the right of
the data.
This is probably a sign of the breakdown of RMT for eigenvalues larger than the Thouless
energy~\cite{Osborn:1998nm,Gockeler:1998jj}.
\FIGURE[tbh]{
\includegraphics[width=0.5\textwidth,clip,angle=-90]{rmt_0.03.eps}
\includegraphics[width=0.5\textwidth,clip,angle=-90]{rmt_0.05.eps}
\caption{\label{fig:rmt}Distribution of the lowest two eigenmodes of the
Dirac operator for our ensemble for
$\nu=0,\pm1$. The lines are fits of the random matrix
theory prediction to the data for the two lowest modes.
The lines for the third mode are predictions.}
}
\TABLE[htb]{
\begin{tabular}{c|c}
$am_q$ \ &\ $\rho \Sigma r_0^3$ \\
\hline
0.05 & 0.40(2) \\
0.03 & 0.44(2) \\
0.01 & 0.38(2) \\
\hline
\end{tabular}
\caption{\label{tab:tab1} Condensate versus quark mass.}
}
A combined fit to $\nu=0,1$, $n=1,2$ at each $m_q$ (four distributions fit simultaneously)
gives the results shown in Table \ref{tab:tab1}. In our small volumes,
and using the physical value for $f_\pi$
(93 MeV), $\rho \sim 1.4$, which is uncomfortably large. Dividing it out boldly
gives $\Sigma \sim (280$ MeV$)^3$.
After Ref. \cite{DeGrand:2005vb} appeared, we performed some simulations at lower
quark mass, $am_q=0.01$. We restricted the topological sector to $\nu=0$ by forbidding
refractions. It is unknown whether a particular topological sector is simply
connected or not (in the latter case, forbidding topology changes might make the simulation
non-ergodic).
We ran two separate molecular dynamics streams to look for any obvious discrepancies.
We did not see any:
In the two streams, the plaquette and string tension parameters were
consistent within statistical uncertainties; nothing looks unusual. We also did some
running at $am=0.005$ (or a 5 MeV quark mass), though not with enough statistics
to fit the condensate. In both cases the code ran stably and quietly.
Fig. \ref{fig:rmt01}
shows the distributions and fit to the condensate from this data set.
\FIGURE[tb]{
\includegraphics[width=0.33\textwidth,clip,angle=-90]{rmt_0.01.eps}
\caption{\label{fig:rmt01}Distribution of the lowest two eigenmodes of the
Dirac operator for our ensemble for
$\nu=0$ at $am_q=0.01$. The lines are fits of the random matrix
theory prediction to the data for the two lowest modes.
The lines for the third mode are predictions.}
}
\section{Conclusions}
Simulations with dynamical overlap fermions are poised to begin producing physics results.
We are presently simulating on larger volumes in order to make a more reliable calculation
of the condensate.
We also continue to groom our algorithms.
We believe that there are many more tricks to be found and encourage others to work
on dynamical simulations with overlap fermions.
The physics payoffs are potentially very high.
This work was supported by the US Department of Energy.
|
1,116,691,500,776 | arxiv | \section{Introduction}
\label{secintroduction}
Low rank perturbations of matrix pencils have been widely studied, and the problem has recently deserved the attention of several authors, as we will see in the next references. Given a matrix pencil $A(s)$ and a nonnegative integer $r$, the problem consists in characterizing the Kronecker structure of $A(s)+P(s)$, where $P(s)$ is a matrix pencil of bounded ($\mathop{\rm rank}\nolimits (P(s))\leq r$) or fixed rank ($\mathop{\rm rank}\nolimits (P(s))= r$).
Some authors focus their research on {\em generic} perturbations; it means that the perturbation pencil $P(s)$ belongs to an open and dense subset of the set of pencils of bounded or fixed rank (for this approach see for instance \cite{TeDo07, TeDo16, TeDoMo08, MoDo03, Sa02, Sa04} and the references therein).
In other papers the pencil $P(s)$ is allowed to be an {\em arbitrary} perturbation belonging to the whole set of pencils of bounded or fixed rank.
Within this framework and for bounded rank perturbations, the problem has been solved in \cite{Silva88_1, Za91} for pencils $A(s)=sI-A$, $P(s)=P$, with $A, P$ constant matrices (see Proposition \ref{propSiZa} and Corollary \ref{corless} below). In the earlier work \cite{Th80}, the same problem was solved for $r=1$. A solution for quasi-regular matrix pencils of the form $A(s)=\begin{matriz}{cc} sI_n-A_1 & A_2 \end{matriz}$ and constant perturbation pencils $P(s)=\begin{matriz}{cc} P_1 & P_2 \end{matriz}$ has been obtained in \cite{DoSt14}. For regular pencils $A(s)$ and $A(s)+P(s)$, a solution to the problem has recently been given in \cite{BaRo18} (see Proposition \ref{propless}) (see also \cite{BaRo18} for further references on the problem).
Concerning fixed rank perturbations, the problem has been solved in \cite{Silva88_1} when $A(s)=sI-A$ and $P(s)=P$ is a constant matrix. The result obtained holds over algebraically closed fields (see Proposition \ref{propSi2} below). When comparing the characterization of the solutions of the bounded (\cite{Silva88_1, Za91}) and fixed rank (\cite{Silva88_1}) perturbation problems, we observe that an extra condition appears in the fixed rank case, which proves that the two problems are of different nature.
In this paper we deal with regular matrix pencils and we require that $P(s)$ is a matrix pencil of fixed rank. More precisely, the first problem we solve is the following:
\begin{problem}\label{fixproblem}
Given two regular matrix pencils $A(s), B(s)\in \mathbb F[s]^{n \times n}$
and a nonnegative integer $r$, $r\leq n$,
find necessary and sufficient conditions for the existence of a
matrix pencil
$P(s)\in \mathbb F[s]^{n \times n}$ such that $\mathop{\rm rank}\nolimits (P(s))= r$ and
$A(s)+P(s)$ is strictly equivalent to $B(s)$.
\end{problem}
Recall that when $A(s)$ is a regular matrix pencil, the Kronecker structure of $A(s)$ is formed by its homogeneous invariant factors, and it is known as the Weierstrass structure of the pencil (see Theorem \ref{theoWei}).
A solution to Problem \ref{fixproblem} is given in Theorems \ref{theon1} and \ref{maintheorem}.
Unlike what happens when perturbing pencils of the form $sI-A$ with constant matrices,
in this case the solutions to the bounded and fixed rank perturbation problems are characterized in terms of the same conditions. This is due to the fact that as the perturbation matrix can be a matrix pencil, it introduces some more freedom that in the constant perturbation problem. But, the fact of being a more restrictive problem determines extra needs for achieving a solution, and in this case proofs are more demanding. To solve it under the same conditions of the bounded case, some specific technical lemmas must be introduced; nothing similar was required in the bounded case.
\medskip
The solution to Problem \ref{fixproblem} obtained allows us to solve the following eigenvalue placement problem:
\begin{problem}\label{prfixdet}
Given a regular matrix pencil $A(s)\in \mathbb F[s]^{n \times n}$, a nonnegative integer $r$, $r\leq n$, and a monic polynomial $0\neq q(s)\in \mathbb F[s]$ with $\mathop{\rm deg }\nolimits(q(s))\leq n$,
find necessary and sufficient conditions for the existence of a
matrix pencil
$P(s)\in \mathbb F[s]^{n \times n}$ such that $\mathop{\rm rank}\nolimits (P(s))= r$ and
$\det(A(s)+P(s))=kq(s)$, with $k\in \mathbb F$.
\end{problem}
A solution to Problem \ref{prfixdet} is given in Theorem \ref{thomainplacement} (see also Corollary \ref{corplacement}
and Remark \ref{rempln1}).
An analogous problem was solved in \cite{BaRo18} in the case that $\mathop{\rm rank}\nolimits (P(s))\leq r$. For $r=1$, see also \cite{GeTr17}.
\medskip
The paper is organized as follows. In Section \ref{secpreliminaries} we introduce the notation, basic definitions and preliminary results. In Section \ref{secmain} we solve Problem \ref{fixproblem}, first for pencils not having infinite elementary divisors and then for the general case. A solution to Problem \ref{prfixdet} is given in Section \ref{secplacement}. Finally, in Section \ref{conclusions} we summarize the main contributions of the paper.
\section{Notation and preliminary results}
\label{secpreliminaries}
The section contains three subsections, where we introduce notation and basic definitions (Subsection \ref{subsecnotation}), some results concerning matrix pencils (Subsection \ref{subsecpencils}), and previous results about matrix or pencil perturbations of bounded or fixed rank (Subsection \ref{subseclowrank}).
\subsection{Notation and basic definitions}
\label{subsecnotation}
Let $\mathbb F$ be a field. $\mathbb F[s]$ denotes the ring of polynomials in the indeterminate $s$ with coefficients in $\mathbb F$, $\mathbb F[s, t]$ the ring of polynomials in two indeterminates $s, t$ with coefficients in $\mathbb F$,
and $\mathbb F^{m\times n}$, $\mathbb F[s]^{m\times n}$ and $\mathbb F[s, t]^{m\times n}$ the vector spaces of $m\times n$ matrices with elements in $\mathbb F$, $\mathbb F[s]$ and $\mathbb F[s, t]$, respectively.
$\mathop{\rm Gl}\nolimits_n(\mathbb F)$ is the general linear group of invertible matrices in $\mathbb F^{n\times n}$.
\medskip
The number of elements of a finite set $I$ will be denoted by $\mid I \mid$.
If $G$ is a matrix in $\mathbb F^{m\times n}$, $I\subseteq \{1, \dots, m\}$, and
$J\subseteq \{1, \dots, n\}$, with $\mid I \mid=r$ and $\mid J \mid=s$, then
$G(I, J)$ denotes the $r\times s$ submatrix of $G$ formed by the rows in $I$ and the columns in $J$.
Similarly, $G(I, :)$ is the $r\times n$ submatrix of $G$ formed by the rows in $I$ and $G(:, J)$
is the $m \times s$ submatrix of $G$ formed by the columns in $J$.
If $\mid I \mid=\mid J \mid$ and $\det (G(I, J))\neq 0$, then the {\em Schur complement of $G(I, J)$ in $G$} is
$$ G/G(I,J)=
G(I^c, J^c)-G(I^c, J)G(I, J)^{-1}G(I, J^c),
$$
where $I^c=\{1, \dots, m\}\setminus I$ and $J^c=\{1, \dots, n\}\setminus J$ (see \cite{Ando87}).
It is satisfied that
$$\mathop{\rm rank}\nolimits (G)=\mathop{\rm rank}\nolimits (G(I, J)) + \mathop{\rm rank}\nolimits (G/G(I, J)),$$ and if $m=n$,
$$\det (G)=\pm \det (G(I, J)) \det(G/G(I, J)).$$
\medskip
Given a polynomial matrix $G(s)\in \mathbb F[s]^{m\times n}$, the {\em degree} of $G(s)$, denoted by $\mathop{\rm deg }\nolimits (G(s))$, is the maximum of the degrees of its entries. The {\em normal rank} of $G(s)$, denoted by $\mathop{\rm rank}\nolimits (G(s))$, is the order of the largest non identically zero minor of $G(s)$, i.e. it is the rank of $G(s)$ considered as a matrix on the field of fractions of $\mathbb F[s]$. If $\mathop{\rm rank}\nolimits (G(s))=\rho$, the {\em determinantal divisor of order $k$} of $G(s)$, denoted by $D_k(s)$, is the monic greatest common divisor of the minors of order $k$ of $G(s)$, $1\leq k \leq \rho$. The determinantal divisors satisfy
$D_{k-1}(s)\mid D_k(s)$, $1\leq k \leq \rho$ ($D_0(s):=1$) and the {\em invariant factors of $G(s)$} are the monic polynomials
$$
\gamma_k(s)=\frac{D_k(s)}{D_{k-1}(s)}, \quad 1\leq k \leq \rho.
$$
We will take $\gamma_i(s):=1$ for $i<1$ and $\gamma_i(s):=0$ for $i>\rho$.
\medskip
A matrix $U(s)\in \mathbb F[s]^{n\times n}$ is {\em unimodular} if $0\neq \det (U(s))\in \mathbb F$, equivalently $U(s)$ is a unit in the ring $\mathbb F[s]^{n\times n}$.
Two polynomial matrices $G(s), H(s)\in \mathbb F[s]^{m\times n}$ are {\em equivalent} ($G(s)\sim H(s)$) if there exist unimodular matrices $U(s)\in \mathbb F[s]^{m\times m}$, $ V(s) \in \mathbb F[s]^{n\times n}$ such that
$G(s)=U(s)H(s)V(s)$. A complete system of invariants for the equivalence of polynomial matrices is formed by the invariant factors, i.e. two polynomial matrices $G(s), H(s)\in \mathbb F[s]^{m\times n}$ are equivalent if and only if they have the same invariant factors.
\medskip
Given a square matrix $G\in \mathbb F^{n \times n}$, the {\em invariant factors of $G$} are the invariant factors of the polynomial matrix $sI_n-G$.
Two square matrices $G, H\in \mathbb F^{n\times n}$ are {\em similar} ($G\ensuremath{\stackrel{s}{\sim}} H$) if there exists an invertible matrix $Q\in \mathop{\rm Gl}\nolimits_n(\mathbb F)$, such that
$G=QHQ^{-1}$. It is well known that $G\ensuremath{\stackrel{s}{\sim}} H$ if and only if $sI_n-G\sim sI_n-H$, i.e. if and only if $G$ and $H$ have the same invariant factors (see, for instance, \cite[Ch. 6, Theorem 7]{Ga74}).
\subsection{Matrix pencils}
\label{subsecpencils}
We review now some basic definitions and results about matrix pencils.
For details see, for example, \cite[Ch. 12]{Ga74}.
\medskip
A {\em matrix pencil} is a polynomial matrix $G(s)\in \mathbb F[s]^{m\times n}$ with $\mathop{\rm deg }\nolimits(G(s))\leq1$. The pencil is {\em regular} if $m=n$ and $\det (G(s))$ is a non zero polynomial. Otherwise it is {\em singular}.
Two matrix pencils
$G(s)=G_0+sG_1, H(s)=H_0+sH_1\in \mathbb F[s]^{m\times n}$ are {\em strictly equivalent} ($G(s)\ensuremath{\stackrel{s.e.}{\sim}} H(s)$) if there exist invertible matrices $Q\in \mathop{\rm Gl}\nolimits_m(\mathbb F)$, $R\in \mathop{\rm Gl}\nolimits_n(\mathbb F)$ such that
$G(s)=QH(s)R$.
It is immediate that if $G(s)\ensuremath{\stackrel{s.e.}{\sim}} H(s)$ then $G(s)\sim H(s)$.
Moreover, if $n=m$, $\det (G_1)\neq 0$ and $\det (H_1 )\neq 0$, then $G(s)\ensuremath{\stackrel{s.e.}{\sim}} H(s)$ if and only if
$G(s)\sim H(s)$
(see, for instance, \cite[Ch.12, Theorem 1]{Ga74}).
\medskip
Given $G(s)=G_0+sG_1\in \mathbb F[s]^{m\times n}$, with $\rho = \mathop{\rm rank}\nolimits(G(s))$, the {\em homogeneous pencil associated to $G(s)$} is
$$G(s, t)=tG_0+sG_1\in \mathbb F[s, t]^{m\times n},$$
and the {\em homogeneous determinantal divisor of order $k$} of $G(s)$, denoted by $\Delta_k(s, t)$, is the greatest common divisor of the minors of order $k$ of $G(s, t)$,
$1\leq k \leq \rho$. We will assume that $\Delta_k(s, t)$ is monic with respect to $s$.
The homogeneous determinantal divisors of $G(s)$ are homogeneous polynomials and $\Delta_{k-1}(s, t)\mid \Delta_k(s,t)$, $1\leq k \leq \rho$ ($\Delta_0(s,t):=1$).
The {\em homogeneous invariant factors of $G(s)$} are the homogeneous polynomials
$$
\Gamma_k(s, t)=\frac{\Delta_k(s,t)}{\Delta_{k-1}(s, t)}, \quad 1\leq k \leq \rho.
$$
If $\gamma_1(s)\mid \dots\mid \gamma_\rho(s)$ are the invariant factors $G(s)$,
then
$$
\gamma_i(s)=\Gamma_i(s,1), \quad 1\leq i \leq \rho, $$
and
$$
\Gamma_i(s,t)=t^{m_i(\infty, G(s))}t^{\mathop{\rm deg }\nolimits(\gamma_i)}\gamma_i(\frac{s}{t}),\quad 1\leq i \leq \rho,
$$
for some integers $0\leq m_1(\infty, G(s)) \leq \dots\leq m_\rho(\infty, G(s)) $.
Hence $\Gamma_1(s,t)\mid \dots \mid\Gamma_{\rho}(s, t)$.
We take $\Gamma_i(s,t):=1$ for $i<1$ and
$\Gamma_i(s,t):=0$ for $i>\rho$.
If $m_i(\infty, G(s))>0$, then $t^{m_i(\infty, G(s))}$ is an {\em infinite elementary divisor} of $G(s)$.
The infinite elementary divisors of $G(s)$ exist if and only if $\mathop{\rm rank}\nolimits (G_1)< \mathop{\rm rank}\nolimits (G(s))$.
We denote by $\overline{ \mathbb F}$ the algebraic closure of $\mathbb F$.
The {\em spectrum} of
$G(s)=G_0+sG_1\in \mathbb F[s]^{m\times n}$ is defined as
$$
\Lambda(G(s))=\{\lambda\in \overline{\mathbb F}\cup\{\infty\}\; : \; \mathop{\rm rank}\nolimits (G(\lambda))< \mathop{\rm rank}\nolimits (G(s))\},
$$
where we agree that $G(\infty)=G_1$. The elements $\lambda\in \Lambda(G(s))$ are the {\em eigenvalues} of $G(s)$.
The invariant factors and the homogeneous invariant factors of $G(s)$ can be written as
\begin{equation} \label{gammas}
\gamma_i(s)=\prod_{\lambda\in \Lambda(G(s))\setminus\{\infty\}}(s-\lambda)^{m_i(\lambda, G(s))}, \quad 1\leq i \leq \rho,
\end{equation}
and
\begin{equation} \label{Gammas}
\Gamma_i(s, t)=t^{m_i(\infty, G(s))}\prod_{\lambda\in \Lambda(G(s))\setminus\{\infty\}}(s-\lambda t)^{m_i(\lambda, G(s))}, \quad 1\leq i \leq \rho.
\end{equation}
For $\lambda \in \Lambda(G(s))$, the integers
$0\leq m_1(\lambda, G(s))\leq \dots\leq m_\rho(\lambda, G(s))$ are called the
{\em partial multiplicities at $\lambda$ of $G(s)$}.
If $\lambda \in \overline{\mathbb F}\setminus \Lambda(G(s))$,
we put $m_1(\lambda, G(s))=\dots=m_\rho(\lambda, G(s))=0$.
For $\lambda \in \overline{\mathbb F}\cup \{\infty\}$, we will agree that $m_i(\lambda, G(s))=0$ for $i<1$ and $m_i(\lambda, G(s))=\infty$ for $i>\rho$.
\medskip
When a matrix pencil has infinite elementary divisors, we can perform a change of variable which turn it into a new pencil without infinite structure. This will be done in Section \ref{secmain}, and we will need the following results, which can be found in \cite{CaSil91}.
Let
$
X=
\begin{bmatrix}
x&y\\
z&w
\end{bmatrix} \in \mathop{\rm Gl}\nolimits_2(\mathbb F)
$.
For a matrix pencil $G(s)=sG_1+G_0\in \mathbb F[s]^{m \times n}$ and an homogeneous polynomial $\Phi(s,t)\in \mathbb F[s, t]$
we define:
$$P_X(sG_1+G_0)=s(xG_1+zG_0)+(yG_1+wG_0)\in \mathbb F[s]^{m \times n},$$
$$\Pi_X(\Phi)(s,t)=\Phi (sx+ty, sz+tw)\in \mathbb F[s, t].$$
\begin{lemma}
[\mbox{\cite[Lemma 6]{CaSil91}}]\label{lemmaCasi6}
The functions $P_X,
\Pi_X$ are invertible and
$$\left(P_X\right)^{-1}=P_{X^{-1}}, \quad
\left(\Pi_X\right)^{-1}=\Pi_{X^{-1}}.$$
\end{lemma}
\begin{lemma}
[\mbox{\cite[Lemma 7]{CaSil91}}]\label{lemmaCasi7}
Let $\Phi(s,t), \Psi(s,t)\in \mathbb F[s, t]$ be homogeneous polynomials.
Then,
$\Phi(s,t)\mid \Psi(s,t)$ if and only if $\Pi_X(\Phi)(s,t)\mid\Pi_X( \Psi)(s,t)$.
\end{lemma}
\begin{lemma}[\mbox{\cite[Lemma 9]{CaSil91}}]\label{lemmaCasi9}
Let
$G(s)=sG_1+G_0, H(s)=sH_1+H_0\in \mathbb F[s]^{m \times n}$. Then $G(s)\ensuremath{\stackrel{s.e.}{\sim}} H(s)$ if and only if $P_X(G(s))\ensuremath{\stackrel{s.e.}{\sim}} P_X(H(s)).$
\end{lemma}
\begin{lemma}[\mbox{\cite[Lemma 10]{CaSil91}}]\label{lemmaCasi10}
Let
$G(s)=sG_1+G_0\in \mathbb F[s]^{m \times n}$, $\rho=\mathop{\rm rank}\nolimits (G(s))$. Let $\Gamma_1(s,t) \mid \ldots \mid \Gamma_{\rho}(s,t)$ be the homogeneous invariant factors of $G(s)$. Then the homogeneous invariant factors of
$P_X(G(s))$ are $\Pi_X(\Gamma_1)(s,t) \mid \ldots \mid \Pi_X( \Gamma_{\rho})(s,t)$.
\end{lemma}
\begin{remark}\label{remrank}
Observe that
\begin{enumerate}
\item[(i)] $\mathop{\rm rank}\nolimits (P_X(G(s)))=\mathop{\rm rank}\nolimits (G(s)).$
\item[(ii)]
In Lemma \ref{lemmaCasi10},
$\Pi_X(\Gamma_i)(s,t)$ are not necessarily monic with respect to $s$. In fact, $\Pi_X(\Gamma_i)(s,t)$ are the homogeneous invariant factors of
$P_X(G(s))$ multiplied by a constant $0\neq k_i\in \mathbb F$.
\end{enumerate}
\end{remark}
\medskip
In this paper we deal with regular matrix pencils. The following theorem states that
the homogeneous invariant factors form a complete system of invariants for the strict equivalence of regular pencils. A proof can be found in
\cite[Ch. 12]{Ga74} for infinite fields and in \cite[Ch. 2]{Ro03} for arbitrary fields.
\begin{theorem}[\mbox{Weierstrass}]\label{theoWei}
Two regular matrix pencils are strictly equivalent if and only if they have the same homogeneous invariant factors.
\end{theorem}
\medskip
For regular matrix pencils, expressions (\ref{gammas}) and (\ref{Gammas}) allow us to write
$$
\det (G(s))=\prod_{i=1}^n\gamma_i(s)=
\prod_{\lambda\in \Lambda(G(s))\setminus\{\infty\}}(s-\lambda)^{\mu_a(\lambda, G(s))},
$$
$$
\det (G(s, t))
=\prod_{i=1}^n\Gamma_i(s, t)=
t^{\mu_a(\infty, G(s))}\prod_{\lambda\in \Lambda(G(s))\setminus\{\infty\}}(s-\lambda t)^{\mu_a(\lambda, G(s))},
$$
where, for $\lambda \in \overline{\mathbb F}\cup \{\infty\}$,
$\mu_a(\lambda, G(s))= \sum _{i=1}^n m_i(\lambda, G(s))$ is
the {\em algebraic multiplicity} of $\lambda$ in $G(s)$.
Notice that $\mathop{\rm deg }\nolimits(\det (G(s, t)))=n$ and $\mathop{\rm deg }\nolimits(\det (G(s)))=n-\mu_a(\infty, G(s)).$
\medskip
Finally, given an homogeneous polynomial $\Gamma(s, t)$,
we will use the following notation
$$\Lambda(\Gamma(s,t)):=\{\lambda \in \overline{\mathbb F}\cup \{\infty\}\; : \;
\Gamma(\lambda,1)=0\},$$
where $\Gamma(\infty, 1):=\Gamma(1, 0)$.
With this notation, if $G(s)\in \mathbb F[s]^{n\times n}$ is a regular matrix pencil with $\Gamma_1(s,t)\mid \dots \mid\Gamma_{n}(s, t)$ homogeneous invariant factors, then
$$
\Lambda(G(s))=\Lambda(\Gamma_n(s,t))=\Lambda(\Gamma_1(s,t) \dots \Gamma_{n}(s, t)).
$$
Also, for a polynomial $q(s)\in \mathbb F[s]$ with $\mathop{\rm deg }\nolimits(q(s))\leq n$, we define
$$\Lambda^n(q(s)):=\{\lambda \in \overline{\mathbb F}\; : \; q(\lambda)=0\}\mbox{ if } \mathop{\rm deg }\nolimits(q(s))=n,$$
$$\Lambda^n(q(s)):=\{\lambda \in \overline{\mathbb F}\; : \; q(\lambda)=0\}\cup \{\infty\}\mbox{ if } \mathop{\rm deg }\nolimits(q(s))<n.$$
\subsection{Rank perturbations of square matrices and regular matrix pencils}
\label{subseclowrank}
The problem of characterizing the Weierstrass structure of a regular matrix pencil obtained by a bounded rank perturbation of another regular matrix pencil
(i.e. Problem \ref{fixproblem} with the relaxed condition $\mathop{\rm rank}\nolimits (P(s))\leq r$) was solved in \cite{BaRo18}. The key point in the obtention of the solution was the next result. It was proven in \cite{Za91} and in \cite{Silva88_1} under another formulation. We present here the version of \cite{Za91}.
\begin{proposition}
[\mbox{\cite[Theorem 1]{Silva88_1}, \cite[Theorem 3]{Za91}}]
\label{propSiZa}
Let $A, B \in \mathbb F^{n\times n}$ and let $\alpha_1(s)\mid \dots \mid \alpha_n(s)$ and $\beta_1(s)\mid \dots \mid \beta_n(s)$ be the invariant factors of $A$ and $B$, respectively. Let $r$ be a nonnegative integer.
Then there exists a matrix
$P\in \mathbb F^{n\times n}$ such that $\mathop{\rm rank}\nolimits (P)\leq r$ and $A+P$ has $\beta_1(s)\mid \dots \mid \beta_n(s)$ as invariant factors if and only if
\begin{equation}\label{eqintif}
\beta_{i-r}(s)\mid \alpha_i(s)\mid\beta_{i+r}(s), \quad 1\leq i \leq n.
\end{equation}
\end{proposition}
Bearing in mind that
$$A+ P\ensuremath{\stackrel{s}{\sim}} B\Leftrightarrow sI_n+A+ P\sim sI_n+B\Leftrightarrow sI_n+A+ P\ensuremath{\stackrel{s.e.}{\sim}} sI_n+B,$$
we obtain the following corollary.
\begin{corollary}
\label{corless}
Let $A(s)=sI_n+A, B(s)=sI_n+B\in \mathbb F[s]^{n \times n}$.
Let $\alpha_1(s)\mid \dots \mid \alpha_n(s)$ and
$\beta_1(s)\mid \dots \mid \beta_n(s)$
be the invariant factors of
$A(s)$ and $B(s)$, respectively.
Let $r$ be a nonnegative integer.
Then there exists a matrix
$P\in \mathbb F^{n\times n}$ such that $\mathop{\rm rank}\nolimits (P)\leq r$ and $A(s)+P\ensuremath{\stackrel{s.e.}{\sim}} B(s) $
if and only if (\ref{eqintif}) holds.
\end{corollary}
The next proposition is the generalization of Proposition \ref{propSiZa} to regular matrix pencils obtained in \cite{BaRo18}.
\begin{proposition}
[\mbox{\cite[Theorem 4.12]{BaRo18}}]
\label{propless}
Let $A(s), B(s)\in \mathbb F[s]^{n \times n}$ be regular matrix pencils.
Let $\phi_1(s,t)\mid \dots\mid \phi_n(s, t)$ and $\psi_1(s,t)\mid \dots\mid \psi_n(s, t)$ be the homogeneous invariant factors of
$A(s)$ and $B(s)$, respectively, and assume that $ \mathbb F\cup \{\infty\}\not \subseteq \Lambda(A(s))\cup \Lambda(B(s)) $. Let $r$ be a nonnegative integer.
There exists a matrix pencil $P(s)\in \mathbb F[s]^{n \times n}$ such that $\mathop{\rm rank}\nolimits (P(s))\leq r$ and $A(s)+P(s)\ensuremath{\stackrel{s.e.}{\sim}} B(s)$ if and only if
\begin{equation}\label{interlacinghomogr1}
\phi_{i-r}(s, t)\mid \psi_i(s, t)\mid\phi_{i+r}(s, t), \quad 1\leq i \leq n.
\end{equation}
\end{proposition}
From this proposition we can derive the following result.
\begin{corollary}\label{corminrbounded2}
Let $A(s), B(s)\in \mathbb F[s]^{n \times n}$ be regular matrix pencils.
Let $\phi_1(s,t)\mid \dots\mid \phi_n(s, t)$ and $\psi_1(s,t)\mid \dots\mid \psi_n(s, t)$ be the homogeneous invariant factors of
$A(s)$ and $B(s)$, respectively, and assume that $ \mathbb F\cup \{\infty\}\not \subseteq \Lambda(A(s))\cup \Lambda(B(s)) $.
Let
$$
r_0=\min\{r\geq 0 \; : \; \phi_{i-r}(s,t)\mid\psi_{i}(s,t)\mid\phi_{i+r}(s,t), \quad 1\leq i \leq n\}.
$$
Then there exists a matrix pencil $P(s)\in \mathbb F[s]^{n \times n}$ such that $\mathop{\rm rank}\nolimits (P(s))= r_0$ and $A(s)+P(s)\ensuremath{\stackrel{s.e.}{\sim}} B(s)$.
\end{corollary}
In this paper we will show that for any $r$ , $r_0\leq r\leq n$, there exists a matrix pencil $P(s)\in \mathbb F[s]^{n \times n}$ such that $\mathop{\rm rank}\nolimits (P(s))= r$ and $A(s)+P(s)\ensuremath{\stackrel{s.e.}{\sim}} B(s)$ (see Corollary \ref{corrminfixed2}).
\medskip
When $\mathbb F$ is an algebraically closed field
the possible similarity class of a square matrix obtained by a fixed rank perturbation of another square matrix was characterized in \cite{Silva88_1}. The result is presented in the next proposition; the statement is different from the original one and more adapted to our problem.
\begin{proposition}
[\mbox{\cite[Theorem 2]{Silva88_1}}]
\label{propSi2}
Suppose that $\mathbb F$ is algebraically closed.
Let $A, B \in \mathbb F^{n\times n}$ and let $\alpha_1(s)\mid \dots \mid \alpha_n(s)$ and $\beta_1(s)\mid \dots \mid \beta_n(s)$ be the invariant factors of $A$ and $B$, respectively. Let $r$ be a nonnegative integer, $r\leq n$.
Then there exists a matrix
$P\in \mathbb F^{n\times n}$ with $\mathop{\rm rank}\nolimits (P)= r$ such that $A+P$ has $\beta_1(s)\mid \dots \mid \beta_n(s)$ as invariant factors if and only if
(\ref{eqintif}) is satisfied and
\begin{equation}\label{eq2Si}
r\leq \min \{\mathop{\rm rank}\nolimits (A-\lambda I_n)+\mathop{\rm rank}\nolimits (B-\lambda I_n)\; : \; \lambda \in \mathbb F\}.
\end{equation}
\end{proposition}
\medskip
As mentioned in the Introduction section, the aim of this paper is to solve an analogous problem to that solved in Proposition \ref{propSi2} for regular matrix pencils.
When $\mathbb F$ is algebraically closed, if $A(s)=sI_n+A$ and $B(s)=sI_n+B$, by Proposition \ref{propSi2} conditions (\ref{interlacinghomogr1}) and
(\ref{eq2Si}) are sufficient for the existence of a matrix pencil $P(s)\in \mathbb F[s]^{n\times n}$ such that $\mathop{\rm rank}\nolimits (P(s))= r$ and $A(s)+P(s)\ensuremath{\stackrel{s.e.}{\sim}} B(s)$.
Nevertheless, (\ref{eq2Si}) is not a necessary condition, as we can see in the next example.
\begin{example}
Let $c\in \mathbb F$ ($\mathbb F$ algebraically closed), $A=B=cI_n$, $r$ an integer, $0<r\leq n$ and $P(s)=\begin{bmatrix}I_r&0\\0&0\end{bmatrix}(sI_n+A)$. Then
$\mathop{\rm rank}\nolimits (P(s))=r$ and
$$
sI_n+A+P(s)=
\begin{bmatrix}2I_r&0\\0&I_{n-r}\end{bmatrix}
(sI_n+A)\ensuremath{\stackrel{s.e.}{\sim}} sI_n+A=sI_n+B,$$
but
$$\min\{\mathop{\rm rank}\nolimits(A-\lambda I_n)+ \mathop{\rm rank}\nolimits(B-\lambda I_n)\; : \; \lambda \in \mathbb F\}=0<r.$$
\end{example}
\section{Fixed rank perturbation for regular matrix pencils}
\label{secmain}
In this section we give a complete solution to Problem \ref{fixproblem} under the same restriction on the field $\mathbb F$ as in Proposition \ref{propless}.
According to this proposition, the interlacing conditions (\ref{interlacinghomogr1}) are necessary. We prove that they are also sufficient, except when $\mathbb F$ is a finite field with $\mid \mathbb F \mid=2$ and $r=n=1$.
Following the strategy of \cite{BaRo18}, we start analyzing the case when the pencils $A(s)$, $B(s)$ do not have infinite elementary divisors.
\subsection{Pencils $A(s)$, $B(s)$ without infinite elementary divisors}
\label{subsecsiasib}
First, we analyze the case when $r=n$, then when $r<n$.
Observe that conditions (\ref{interlacinghomogr1}) are trivially fulfilled for $r=n$.
We prove in Proposition \ref{proprn} that for regular pencils $A(s)=sI_n+A, B(s)=sI_n+B\in \mathbb F[s]^{n\times n}$, $n\geq 2$, there always exists a regular pencil $P(s)\in \mathbb F[s]^{n\times n}$ such that $A(s)+P(s)\ensuremath{\stackrel{s.e.}{\sim}} B(s)$. In order to do that we need the following technical lemma.
\begin{lemma}\label{lemmaei+e}
Let $n\geq 2$. Then there exists a matrix
$E_n\in \mathop{\rm Gl}\nolimits_n(\mathbb F)$ such that $I_n+E_n\in \mathop{\rm Gl}\nolimits_n(\mathbb F)$.
\end{lemma}
{\bf Proof.}
We prove the result by induction on $n$.
If $n=2$, put $E_2=\begin{bmatrix}1&1\\1&0\end{bmatrix}$. Then
$E_2, I_2+E_2\in \mathop{\rm Gl}\nolimits_2(\mathbb F)$.
Assume that there exists $E_p\in \mathop{\rm Gl}\nolimits_p(\mathbb F)$ such that $I_p+E_p\in \mathop{\rm Gl}\nolimits_p(\mathbb F)$ and let $n=p+1$.
Obviously, $E_p\neq I_p+E_p$. Therefore, if $R=E_p^{-1}-(I_p+E_p)^{-1}$, then
$R\neq 0$. Let $i, j\in \{1, \dots, p\}$ be such that $R(i,j)\neq 0$ and let $w=-1+e_i^tE_p^{-1}e_j\in \mathbb F$.
We define
$$
E_{p+1}=\begin{bmatrix}
E_p&e_j\\
e_i^t&w
\end{bmatrix}\in \mathbb F^{(p+1)\times (p+1)}.
$$
Then,
$$
\det (E_{p+1})=\det(E_{p+1}/E_p)\det (E_{p})=(w-e_i^tE_p^{-1}e_j)\det (E_{p})=-\det (E_{p})\neq 0,
$$
$$
\det(I_{p+1}+E_{p+1})=(1+w-e_i^t(I_p+E_p)^{-1}e_j)\det(I_p+E_{p})$$$$=
(1+w-e_i^t(E_p^{-1}-R)e_j)\det(I_p+E_{p})
=R(i,j)\det(I_p+E_{p})\neq 0,
$$
hence, $E_{p+1}, I_{p+1}+E_{p+1}, \in \mathop{\rm Gl}\nolimits_{p+1}(\mathbb F)$.
\hfill $\Box$
\begin{remark}
If $\mid\mathbb F\mid \neq 2$, Lemma \ref{lemmaei+e} is straightforward, and the result holds for $n\geq 1$. We can take, for example, $E_n=cI_n$, with $c\in \mathbb F$, $c\neq 0, -1$.
\end{remark}
\begin{proposition}\label{proprn}
Let $n\geq 2$ and
$A(s)=sI_n+A, B(s)=sI_n+B \in \mathbb F[s]^{n\times n}$. Then there exists a matrix pencil
$P(s)\in \mathbb F[s]^{n \times n}$ with $\mathop{\rm rank}\nolimits (P(s))=n$ such that $A(s)+P(s)\ensuremath{\stackrel{s.e.}{\sim}} B(s)$.
\end{proposition}
{\bf Proof.}
By Lemma \ref{lemmaei+e}, there exists $E_n\in \mathop{\rm Gl}\nolimits_n(\mathbb F)$ such that $I_n+E_n\in \mathop{\rm Gl}\nolimits_n(\mathbb F)$.
Let $P_0=(I_n+E_n)B-A \in \mathbb F^{n\times n}$
and $P(s)=E_ns+P_0\in \mathbb F[s]^{n\times n}$. Then $\mathop{\rm rank}\nolimits (P(s))=n$ and
$$
A(s)+P(s)=sI_n+A+E_ns+(I_n+E_n)B-A=(I_n+E_n)(sI_n+B)\ensuremath{\stackrel{s.e.}{\sim}} sI_n+B.
$$
\hfill $\Box$
When $r<n$, next lemma allows us to take advantage of a solution to the bounded case and out of it to built a solution for the fixed rank case. This is done in Proposition \ref{mainprop}.
\begin{lemma}\label{mainlemma}
Let $r_1, r, n$ be integers, $0\leq r_1<r< n$.
Let $I, J\subset \{1, \dots, n\}$ be such that $\mid I\mid=\mid J\mid =r_1\geq 0$. Then there exists a matrix $E\in \mathbb F^{n \times n}$ satisfying that
$\mathop{\rm rank}\nolimits (E)=r-r_1$, $I_n+E\in \mathop{\rm Gl}\nolimits_n(\mathbb F)$, $E(I, :)=0$, and $E(:, J)=0$.
\end{lemma}
{\bf Proof.}
First, let us see that there exist sets
$$
R_1=\{i_1, \dots, i_{x'}\}, \quad
R_2=\{i_{x'+1}, \dots, i_{x'+a'}\},\quad S_2=\{i_{x'+a'+1}, \dots, i_{x'+2a'}\}
$$
($x'\geq 0$, $a'\geq 0$) with $i_k\neq i_\ell$ for $k\neq \ell$, such that $R_1\dot {\cup} R_2 \subset I^c$, $R_1\dot{\cup} S_2 \subset J^c$, $x'+a'=r-r_1$, and $x'\neq 1$.
Let $$X=I^c\cap J^c,\quad Y=I^c\setminus X,\quad Z=J^c\setminus X$$
and let $x=\mid X\mid $, $a=n-r_1-x=\mid Y\mid =\mid Z\mid $.
\bigskip
\begin{itemize}
\item
If $a\geq r-r_1$, we put
$R_1=\emptyset$ and choose $R_2\subseteq Y$, $S_2\subseteq Z$ such that $\mid R_2\mid = \mid S_2 \mid =r-r_1$.
In this case, $x'=0$, $a'=r-r_1$.
\item
If $a<r-r_1$, then $x=n-r_1-a>n-r> 0$. Therefore
$x\geq 2$.
\begin{itemize}
\item
If $(r-r_1)-a\geq 2$ we put $R_2=Y$, $S_2=Z$ and choose $R_1\subset X$ with $\mid R_1\mid =r-r_1-a(<n-r_1-a=x)$. In this case, $x'=r-r_1-a\geq 2$, $a'=a$.
\item
If $(r-r_1)-a= 1$ and $a\geq 1$, we choose $R_1\subseteq X$ with $\mid R_1\mid =2$ and $R_2\subset Y$, $S_2\subset Z$ with $\mid R_2\mid =\mid S_2\mid =r-r_1-2=a-1$.
In this case, $x'= 2$, $a'=a-1$.
\item
If $(r-r_1)-a= 1$ and $a=0$, then $r-r_1=1< x$. We can choose $i, j\in X$ such that $i\neq j$. We put
$R_1=\emptyset$, $R_2=\{i\}$, $S_2=\{j\}$.
In this case, $x'= 0$, $a'=1$.
\end{itemize}
\end{itemize}
We have that
$
R_1\dot{\cup}R_2\dot{\cup}S_2\subseteq \{1, \dots, n\}
$, hence $x'+2a'\leq n$.
Let us denote
$(R_2\cup R_1\cup S_2)^c=\{i_{x'+2a'+1}, \dots, i_n\}$.
We have obtained that $x'= 0$ or $x'\geq 2$. If $x'\geq 2$, by Lemma \ref{lemmaei+e}
there exists
$E_{x'}\in \mathop{\rm Gl}\nolimits_{x'}(\mathbb F)$
such that $I_{x'}+E_{x'}\in \mathop{\rm Gl}\nolimits_{x'}(\mathbb F)$.
Let $\bar E\in \mathbb F^{n\times n}$ be the matrix having
$$
\bar E(\{1, \dots, x'\},\{1, \dots, x'\} )=E_{x'}, \;
\bar E( \{x'+1, \dots, x'+a'\}, \{x'+a'+1, \dots, x'+2a'\})=I_{a'},$$
and the rest of its entries equal to zero, i.e.
$$
\bar E=
\begin{bmatrix}
E_{x'}&0&0&0\\
0&0&I_{a'}&0\\
0&0&0&0\\
0&0&0&0\\
\end{bmatrix}\in \mathbb F^{(x'+a'+a'+(n-x'-2a'))\times (x'+a'+a'+(n-x'-2a'))}.
$$
(If $x'=0$ or $a'=0$, the corresponding block vanishes).
Obviously,
$$\mathop{\rm rank}\nolimits (\bar E)=\mathop{\rm rank}\nolimits (E_{x'})+\mathop{\rm rank}\nolimits (I_{a'})=x'+a'=r-r_1,$$
and
$$ I_n+\bar E=
\begin{bmatrix}
I_{x'}+E_{x'}&0&0&0\\
0&I_{a'}&I_{a'}&0\\
0&0&I_{a'}&0\\
0&0&0&I_{n-x'-2a'}
\end{bmatrix}
\in \mathop{\rm Gl}\nolimits_n(\mathbb F).$$
Let $P$ be the permutation matrix
$P=\begin{bmatrix}
e_{i_1}&\dots &e_{i_n}
\end{bmatrix}
$, where $e_k$ denotes
the $k$-th column of $I_n$. Then, $Pe_k=e_{i_k}$ for $1\leq k \leq n$; equivalently, $P^te_{i_k}=e_k$, and $e_{i_k}^tP=e_k^t$.
Let
$E=P\bar E P^t$.
Then,
$\mathop{\rm rank}\nolimits (E)=\mathop{\rm rank}\nolimits (\bar E)=r-r_1$, $I_n+ E=PP^t+P\bar E P^t=P(I_n+\bar E)P^t\in \mathop{\rm Gl}\nolimits_n(\mathbb F),$
$$
E((R_1\cup R_2)^c, :))=E(\{i_{x'+a'+1}, \dots, i_n\}, :)=\begin{bmatrix}
e_{i_{x'+a'+1}}^t\\\vdots \\e_{i_n}^t
\end{bmatrix}P\bar E P^t=\begin{bmatrix}
e_{x'+a'+1}^t\\\vdots \\e_n^t
\end{bmatrix}\bar E P^t$$$$=\bar E(\{x'+a'+1, \dots, n\}, :)P^t=0,
$$
and
$$
E(:, (R_1\cup S_2)^c)=
E(:,\{i_{x'+1}, \dots, i_{x'+a'}\}\cup\{i_{x'+2a'+1}, \dots, i_{n}\})$$$$=
P\bar E P^t
\begin{bmatrix}
e_{i_{x'+1}}&\dots &e_{i_{x'+a'}}&
e_{i_{x'+2a'+1}}&\dots &e_{i_n}
\end{bmatrix}$$$$=P\bar E \begin{bmatrix}
e_{x'+1}&\dots &e_{x'+a'}&
e_{x'+2a'+1}&\dots &e_n
\end{bmatrix}$$$$=P\bar E(:, \{x'+1, \dots, x'+a'\}\cup \{x'+2a'+1, \dots, n\})=0.
$$
Since $I\subseteq (R_1\cup R_2)^c$ and $J\subseteq (R_1\cup S_2)^c$, it results that
$
E(I, :)=0$ and $E(:, J)=0.
$
\hfill $\Box$
\begin{proposition}\label{mainprop}
Let $n\geq 2$ and
$A(s)=sI_n+A \in \mathbb F[s]^{n\times n}$. Let $P\in \mathbb F^{n \times n}$ be a matrix such that $\mathop{\rm rank}\nolimits (P)=r_1$
and
let $r$ be an integer, $r_1<r<n$. Then there exists a matrix pencil
$P(s)\in \mathbb F[s]^{n \times n}$ with $\mathop{\rm rank}\nolimits (P(s))=r$ such that $A(s)+P(s)\ensuremath{\stackrel{s.e.}{\sim}} A(s)+P$.
\end{proposition}
{\bf Proof.}
Since $\mathop{\rm rank}\nolimits (P)=r_1$, there exist $I, J\subset \{1, \dots, n\}$ such that
$\mid I\mid = \mid J\mid =r_1$ and $\det (P(I, J))\neq 0$ (if $r_1=0$, then
$I=J=\emptyset$).
By Lemma \ref{mainlemma}, there exists a matrix $E\in \mathbb F^{n \times n}$ such that
$\mathop{\rm rank}\nolimits (E)=r-r_1$, $I_n+E\in \mathop{\rm Gl}\nolimits_n(\mathbb F)$, $E(I, :)=0$, and $E(:, J)=0$.
Let $Q=I_n+E$. Then
$$
sI_n+A+P\ensuremath{\stackrel{s.e.}{\sim}} Q(sI_n+A+P)=sI_n+A+P+E(sI_n+A+P)=A(s)+P(s),
$$
where $P(s)=P+E(sI_n+A+P)$.
Let us see that $\mathop{\rm rank}\nolimits (P(s))=r$.
On one hand,
$$\mathop{\rm rank}\nolimits (P(s))\leq \mathop{\rm rank}\nolimits (P)+ \mathop{\rm rank}\nolimits (E(sI_n+A+P))\leq \mathop{\rm rank}\nolimits (P)+ \mathop{\rm rank}\nolimits (E)=r_1+r-r_1=r.$$
On the other one,
$$
P(s)(I, :)=P(I, :)+E(I,:)(sI_n+A+P)=P(I,:).
$$
Therefore,
$$\det (P(s)(I, J))=\det (P(I, J))\neq 0,$$
and
$$P(s)/P(s)(I, J)=P(s)(I^c, J^c)-P(s)(I^c, J) P(I, J)^{-1}P(I, J^c).$$
As
$$
P(s)(I^c, J^c)=P(I^c,J^c)+sE(I^c,J^c)+(E(A+P))(I^c, J^c),
$$
and
$$
P(s)(I^c,J)=P(I^c,J)+sE(I^c,J)+(E(A+P))(I^c, J)$$$$=P(I^c,J)+(E(A+P))(I^c, J)\in \mathbb F^{(n-r_1)\times r_1},
$$
we can write
$P(s)/P(s)(I, J)=sE(I^c, J^c)+P_0,$ with $P_0\in \mathbb F^{(n-r_1)\times (n-r_1)}$, from where
$$\mathop{\rm rank}\nolimits (P(s)/P(s)(I, J))\geq \mathop{\rm rank}\nolimits (E(I^c, J^c))=\mathop{\rm rank}\nolimits (E)=r-r_1.$$ Hence,
$$
\mathop{\rm rank}\nolimits (P(s))=\mathop{\rm rank}\nolimits (P(I, J))+\mathop{\rm rank}\nolimits (P(s)/P(s)(I, J))\geq r.
$$
\hfill $\Box$
\begin{theorem}\label{propsI+A}
Let $n\geq 2$ and $A(s)=sI_n+A, B(s)=sI_n+B\in \mathbb F[s]^{n \times n}$.
Let $\phi_1(s,t)\mid \dots\mid \phi_n(s, t)$ and $\psi_1(s,t)\mid \dots\mid \psi_n(s, t)$ be the homogeneous invariant factors of $A(s)$ and $B(s)$, respectively. Let $r$ be a nonnegative integer,
$r\leq n$. If (\ref{interlacinghomogr1}) is satisfied, then there exists
a matrix pencil
$P(s)\in \mathbb F[s]^{n \times n}$ such that $\mathop{\rm rank}\nolimits (P(s))= r$ and
$A(s)+P(s)\ensuremath{\stackrel{s.e.}{\sim}} B(s)$.
\end{theorem}
{\bf Proof.}
If $r=n$, we apply Proposition \ref{proprn}.
If $r<n$, let $\alpha_i(s)=\phi_i(s, 1)$ and $\beta_i(s)=\psi_i(s, 1)$,
$1\leq i \leq n$,
be
the invariant factors of $A(s)$ and $B(s)$, respectively.
Then, conditions
(\ref{interlacinghomogr1}) imply conditions (\ref{eqintif}).
By Corollary \ref{corless}, there exists $P\in \mathbb F^{n \times n}$ such that $\mathop{\rm rank}\nolimits (P)\leq r$ and
$A(s)+P\ensuremath{\stackrel{s.e.}{\sim}} B(s)$. If $\mathop{\rm rank}\nolimits (P)< r$, we apply Proposition \ref{mainprop}.
\hfill $\Box$
We show next an example of pencils $A(s)$ and $B(s)$ such that $B(s)$ cannot be obtained by a constant perturbation of rank $2$ of $A(s)$, but it does result as a pencil perturbation of rank $2$ of the pencil $A(s)$.
\begin{example}\label{exp1p}
Let $\mathbb F$ be an arbitrary field and $r=2$,
$$A(s)=\begin{bmatrix}s-1& 0&0\\0&s-1&0\\0&0&s-1 \end{bmatrix}, \quad
B(s)=\begin{bmatrix}s-1& 0&0\\0&s-1&0\\0&0&s \end{bmatrix}.$$
The homogeneous invariant factors of $A(s)$ and $B(s)$ are
$\phi_1(s, t)=\phi_2(s, t)=\phi_3(s, t)=(s-t)$ and
$\psi_1(s, t)=1, \psi_2(s, t)=(s-t), \psi_3(s, t)=s(s-t)$, respectively.
We have that
$$
\phi_{i-2}(s, t)\mid \psi_i(s, t)\mid\phi_{i+2}(s, t), \quad 1\leq i\leq 3.
$$
Therefore,
$$
\phi_{i-2}(s, 1)\mid \psi_i(s, 1)\mid\phi_{i+2}(s, 1), \quad 1\leq i\leq 3,
$$
hence, by Corollary \ref{corless}, there exists a matrix $P\in \mathbb F^{3\times 3}$ such that $\mathop{\rm rank}\nolimits P\leq 2$ and
$A(s)+P \ensuremath{\stackrel{s.e.}{\sim}} B(s)$.
In fact, taking $P= \begin{bmatrix}0& 0&0\\0&0&0\\0&0&1 \end{bmatrix}$, we have that $\mathop{\rm rank}\nolimits P =1$ and $A(s)+P = B(s)$.
Observe that
$$
\min\{\mathop{\rm rank}\nolimits A(\lambda)+ \mathop{\rm rank}\nolimits B(\lambda): \ \lambda \in \bar{\mathbb F}\}=1.
$$
By Proposition \ref{propSi2}, this means that there is no $P\in \bar{\mathbb F}^{3\times 3}$ such that $\mathop{\rm rank}\nolimits P= 2$ and $A(s)+P \ensuremath{\stackrel{s.e.}{\sim}} B(s)$.
Let $Q= \begin{bmatrix}1& 1&0\\0&1&0\\0&0&1 \end{bmatrix}\in \mathop{\rm Gl}\nolimits_3(\mathbb F)$.
Then
$$
B(s) \ensuremath{\stackrel{s.e.}{\sim}} QB(s)=Q(A(s)+P)=\begin{bmatrix}s-1& s-1&0\\0&s-1&0\\0&0&s \end{bmatrix}=A(s)+P(s),
$$
where $P(s)=\begin{bmatrix}0& s-1&0\\0&0&0\\0&0&1 \end{bmatrix}\in \mathbb F[s]^{3 \times 3}$, and $\mathop{\rm rank}\nolimits P(s)=2$.
\end{example}
\begin{corollary}\label{cornotid}
Let $n\geq 2$ and let $A(s)=A_0+sA_1, B(s)=B_0+sB_1\in \mathbb F[s]^{n \times n}$ be such that $\det (A_1)\neq 0$ and $\det (B_1) \neq 0$.
Let $\phi_1(s,t)\mid \dots\mid \phi_n(s, t)$ and $\psi_1(s,t)\mid \dots\mid \psi_n(s, t)$ be the homogeneous invariant factors of $A(s)$ and $B(s)$, respectively. Let $r$ be a nonnegative integer,
$r\leq n$. If (\ref{interlacinghomogr1}) is satisfied, then there exists
a matrix pencil
$P(s)\in \mathbb F[s]^{n \times n}$ such that $\mathop{\rm rank}\nolimits (P(s))= r$ and
$A(s)+P(s)\ensuremath{\stackrel{s.e.}{\sim}} B(s)$.
\end{corollary}
{\bf Proof.}
We have that $A(s)\ensuremath{\stackrel{s.e.}{\sim}} A_1^{-1}A_0+sI_n$ and $B(s)\ensuremath{\stackrel{s.e.}{\sim}} B_1^{-1}B_0+sI_n$.
Hence, the homogeneous invariant factors of $sI_n+A_1^{-1}A_0$ and $sI_n+B_1^{-1}B_0 $ are $\phi_1(s,t)\mid \dots\mid \phi_n(s, t)$ and $\psi_1(s,t)\mid \dots\mid \psi_n(s, t)$, respectively.
By Theorem \ref{propsI+A}, there exists a matrix pencil $P'(s)\in \mathbb F[s]^{n \times n}$ such that
$\mathop{\rm rank}\nolimits (P'(s))=r$ and $sI_n+A_1^{-1}A_0+P'(s)\ensuremath{\stackrel{s.e.}{\sim}} sI_n+B_1^{-1}B_0\ensuremath{\stackrel{s.e.}{\sim}} B(s)$.
Let $P(s)=A_1P'(s)$. Then, $\mathop{\rm rank}\nolimits (P(s))=\mathop{\rm rank}\nolimits (P'(s))=r$ and
$$
A(s)+P(s)=A_1(sI_n+A_1^{-1}A_0+P'(s))\ensuremath{\stackrel{s.e.}{\sim}} sI_n+A_1^{-1}A_0+P'(s)
\ensuremath{\stackrel{s.e.}{\sim}} B(s).
$$
\hfill $\Box$
\subsection{General case}
\label{subsecgeneral}
We analyze first the case $n=1$.
\begin{theorem} \label{theon1}
Let $a(s)=a_0+sa_1, b(s)=b_0+sb_1\in \mathbb F[s]$ be such that $a(s)\neq 0$ and $b(s)\neq 0$.
Let $\phi_1(s,t)$ and $\psi_1(s,t)$ be the homogeneous invariant factors of
$a(s)$ and $b(s)$, respectively. Let $r$ be an integer, $0\leq r\leq 1$.
\begin{enumerate}
\item If $\mid\mathbb F\mid >2$ or $r=0$, then there exists $p(s)=p_0+sp_1\in \mathbb F[s]$ such that $\mathop{\rm rank}\nolimits (p(s))=r$ and $a(s)+p(s)\ensuremath{\stackrel{s.e.}{\sim}} b(s)$ if and only if (\ref{interlacinghomogr1}) holds.
\item
If $\mid\mathbb F\mid =2$ and $r=1$, then there exists $p(s)\in \mathbb F[s]$ such that $\mathop{\rm rank}\nolimits (p(s))=1$ and $a(s)+p(s)\ensuremath{\stackrel{s.e.}{\sim}} b(s)$ if and only if $a(s)\neq b(s)$.
\end{enumerate}
\end{theorem}
{\bf Proof.}
\begin{enumerate}
\item
The necessity is an immediate consequence of Proposition \ref{propless}. Let us prove the sufficiency.
Since $n=1$, conditions (\ref{interlacinghomogr1}) reduce to
\begin{equation}\label{eqn1}
\psi_{1-r}(s,t)\mid\phi_1(s,t)\mid\psi_{1+r}(s,t).
\end{equation}
\begin{itemize}
\item
If $r=0$, then (\ref{eqn1}) implies $\psi_{1}(s)=\phi_1(s)$, hence $b(s)\ensuremath{\stackrel{s.e.}{\sim}} a(s)=a(s)+0$.
\item
If $r=1$, then (\ref{eqn1}) is trivially satisfied for any $a(s), b(s)$.
As $\mid\mathbb F\mid >2$, there exists $c\in \mathbb F\setminus\{ 0\}$ such that $a(s)\neq cb(s)$. Taking $p(s)=cb(s)-a(s)$, the sufficiency is proven.
\end{itemize}
\item It is enough to observe that if $\mid\mathbb F\mid =2$, there exists $p(s)\in \mathbb F[s]$ such that $a(s)+p(s)\ensuremath{\stackrel{s.e.}{\sim}} b(s)$ if and only if $a(s)+p(s)= b(s)$.
\end{enumerate}
\hfill $\Box$
Next theorem is our main result.
\begin{theorem}
\label{maintheorem}
Let $n\geq 2$. Let $A(s)=sA_1+A_0, B(s)=sB_1+B_0\in \mathbb F[s]^{n \times n}$ be regular matrix pencils.
Let $\phi_1(s,t)\mid \dots\mid \phi_n(s, t)$ and $\psi_1(s,t)\mid \dots\mid \psi_n(s, t)$ be the homogeneous invariant factors of $A(s)$ and $B(s)$, respectively, and assume that $ \mathbb F\cup \{\infty\}\not \subseteq \Lambda(A(s))\cup \Lambda(B(s)) $. Let $r$ be a nonnegative integer, $r\leq n$.
There exists a matrix pencil $P(s)\in \mathbb F[s]^{n \times n}$ such that $\mathop{\rm rank}\nolimits (P(s))=r$ and $A(s)+P(s)\ensuremath{\stackrel{s.e.}{\sim}} B(s)$ if and only if
(\ref{interlacinghomogr1}) holds.
\end{theorem}
{\bf Proof.}
The necessity is an immediate consequence of Proposition \ref{propless}.
Assume that (\ref{interlacinghomogr1}) holds.
As $ \mathbb F\cup \{\infty\}\not \subseteq \Lambda(A(s))\cup \Lambda(B(s)) $, there exists $c\in \mathbb F\cup \{\infty\}$ such that $c\not \in \Lambda(A(s))\cup \Lambda(B(s))$.
If $c=\infty$, we apply Corollary \ref{cornotid}.
If $c\neq \infty$, take
$$
X=\begin{bmatrix}
c&1\\
1&0
\end{bmatrix},
$$
and
$$A'(s)=P_X(sA_1+A_0)=s(cA_1+A_0)+A_1=sA_1'+A_0',$$
$$B'(s)=P_X(sB_1+B_0)=s(cB_1+B_0)+B_1=sB_1'+B_0'.$$
Then, $\det (A'_1)\neq 0$, $\det (B'_1)\neq 0$.
Let $\phi'_1(s, t), \dots, \phi'_n(s, t)$ and
$\psi'_1(s, t), \dots, \psi'_n(s, t)$ be the homogeneous invariant factors of $A'(s)$ and $B'(s)$, respectively.
By Lemma \ref{lemmaCasi10} and Remark \ref{remrank},
$$
\phi'_i(s, t)=c_i\Pi_X(\phi_i)(s,t), \quad \psi'_i(s, t)=d_i\Pi_X(\psi_i)(s,t),\quad 1\leq i \leq n,
$$
where $0\neq c_i\in \mathbb F$, $0\neq d_i\in \mathbb F$, $1\leq i \leq n$.
Applying Lemma \ref{lemmaCasi7}, from (\ref{interlacinghomogr1}) we obtain
$$
\phi'_{i-r}(s, t)\mid \psi'_i(s, t)\mid\phi'_{i+r}(s, t), \quad 1\leq i \leq n.
$$
By Corollary \ref{cornotid}, there exists a matrix pencil $P'(s)=sP_1'+P_0'\in \mathbb F[s]^{n \times n}$ such that $\mathop{\rm rank}\nolimits (P'(s))=r$ and
$$A'(s)+P'(s)\ensuremath{\stackrel{s.e.}{\sim}} B'(s).$$
Then, by Lemmas \ref{lemmaCasi6} and \ref{lemmaCasi9},
$$\left(P_X\right)^{-1}(A'(s))+\left(P_X\right)^{-1}(P'(s))=\left(P_X\right)^{-1}(A'(s)+P'(s)) \ensuremath{\stackrel{s.e.}{\sim}} \left(P_X\right)^{-1}(sB_1'+B_0').$$
Taking $P(s)=\left(P_X\right)^{-1}(P'(s))=P_{X^{-1}}(P'(s))$, we obtain that
$$
A(s)+P(s)\ensuremath{\stackrel{s.e.}{\sim}} B(s),
$$
and by Remark \ref{remrank}, $\mathop{\rm rank}\nolimits (P(s))=r$.
\hfill $\Box$
\begin{corollary}\label{corrminfixed2}
Let $n\geq 2$. Let $A(s)=sA_1+A_0, B(s)=sB_1+B_0\in \mathbb F[s]^{n \times n}$ be regular matrix pencils.
Let $\phi_1(s,t)\mid \dots\mid \phi_n(s, t)$ and $\psi_1(s,t)\mid \dots\mid \psi_n(s, t)$ be the homogeneous invariant factors of $A(s)$ and $B(s)$, respectively, and assume that $ \mathbb F\cup \{\infty\}\not \subseteq \Lambda(A(s))\cup \Lambda(B(s))$.
Let
$$
r_0=\min\{r\geq 0 \; : \; \phi_{i-r}(s,t)\mid\psi_{i}(s,t)\mid\phi_{i+r}(s,t), \quad 1\leq i \leq n\}.
$$
Then
there exists a matrix pencil $P(s)\in \mathbb F[s]^{n \times n}$ with $\mathop{\rm rank}\nolimits (P(s))=r$ and such that $A(s)+P(s)\ensuremath{\stackrel{s.e.}{\sim}} B(s)$ if and only if
$r_0\leq r\leq n$.
\end{corollary}
{\bf Proof.}
It is straigthtforward that $r\geq r_0$ if and only if conditions (\ref{interlacinghomogr1}) hold.
\hfill $\Box$
\begin{example}
Let $\mathbb F$ be an arbitrary field. Let $A(s), B(s)\in \mathbb F[s]^{5\times 5}$ be regular matrix pencils with homogeneous invariant factors
$$\phi_1(s,t)=\phi_2(s,t)=1, \quad \phi_3(s,t)=t, \quad \phi_4(s,t)=\phi_5(s,t)=t^2,$$
$$\psi_1(s,t)=\psi_2(s,t)=1, \quad \psi_3(s,t)=\psi_4(s,t)=s-t,\quad
\psi_5(s,t)=(s-t)^3,$$
respectively. Then
$$
\Lambda(A(s))=\{\infty\}, \quad \Lambda(B(s))=\{1\}, \quad 0\not \in \Lambda(A(s))\cup\Lambda(B(s)),
$$
and
$$
r_0=\min\{r\geq 0 \; : \; \phi_{i-r}(s,t)\mid\psi_{i}(s,t)\mid\phi_{i+r}(s,t), \quad 1\leq i \leq 5\}=3.
$$
Hence, for $3\leq r \leq 5$
there exist matrix pencils $P_r(s)\in \mathbb F[s]^{5\times 5}$ with
$\mathop{\rm rank}\nolimits(P_r(s))=r$ such that $A(s)+P_r(s)\ensuremath{\stackrel{s.e.}{\sim}} B(s)$.
Moreover, there is not any pencil $P(s)$ with $\mathop{\rm rank}\nolimits(P(s))\leq 2$ such that $A(s)+P(s)\ensuremath{\stackrel{s.e.}{\sim}} B(s)$.
\end{example}
The characterization of the solution given in Theorem \ref{maintheorem} can be stated in terms of the partial multiplicities of the elements of $ \Lambda(A(s))\cup \Lambda(B(s))$
(see \cite[Corollary 4.5]{BaRo18} for an analogous result when $\mathop{\rm rank}\nolimits (P(s))\leq r$; see also \cite[Proposition 4.2]{GeTr17} for $r=1$).
\begin{corollary}\label{cormainmult}
Let $n\geq 2$. Let $A(s), B(s)\in \mathbb F[s]^{n \times n}$ be regular matrix pencils.
Assume that $ \mathbb F\cup \{\infty\}\not \subseteq \Lambda(A(s))\cup \Lambda(B(s)) $.
Let $r$ be a nonnegative integer, $r\leq n$.
There exists a matrix pencil $P(s)\in \mathbb F[s]^{n \times n}$ such that $\mathop{\rm rank}\nolimits (P(s))= r$ and $A(s)+P(s)\ensuremath{\stackrel{s.e.}{\sim}} B(s)$ if and only if
\begin{equation}\label{eqnecmult}
m_{i-r}(\lambda, A(s))\leq m_i(\lambda, B(s))\leq m_{i+r}(\lambda, A(s)),
\, 1\leq i \leq n, \quad \lambda \in \overline{\mathbb F}\cup\{\infty\}.
\end{equation}
\end{corollary}
As pointed out in \cite[Remark 4.15]{BaRo18}, if $\#\mathbb F> 2n $,
the condition $\mathbb F\cup \{\infty\} \not \subseteq \Lambda(A(s))\cup \Lambda(B(s))$ is automatically satisfied. In the case that $\#\mathbb F\leq 2n $, Theorem \ref{maintheorem} can still be applied if there exists an element $c\in \mathbb F\cup \{\infty\}$ which is neither an eigenvalue of $A(s)$ nor of $B(s)$.
Moreover, we show in Corollary \ref{coreqmult} that the condition $ \mathbb F\cup \{\infty\} \not \subseteq \Lambda(A(s))\cup \Lambda(B(s))$ is not always necessary.
\begin{corollary}\label{coreqmult}
Let $A(s), B(s)\in \mathbb F[s]^{n \times n}$ be regular matrix pencils.
Let $\phi_1(s,t)\mid \dots\mid \phi_n(s, t)$ and $\psi_1(s,t)\mid \dots\mid \psi_n(s, t)$ be the homogeneous invariant factors of
$A(s)$ and $B(s)$, respectively, and assume that
for some $\lambda_0\in \mathbb F\cup \{\infty\}$,
$$
m_i(\lambda_0, A(s))=m_i(\lambda_0, B(s)), \quad 1\leq i \leq n.
$$
Let $r$ be a nonnegative integer, $r\leq n$.
There exists a matrix pencil $P(s)\in \mathbb F[s]^{n \times n}$ such that $\mathop{\rm rank}\nolimits (P(s))=r$ and $A(s)+P(s)\ensuremath{\stackrel{s.e.}{\sim}} B(s)$ if and only if
(\ref{interlacinghomogr1}) holds.
\end{corollary}
{\bf Proof.}
Analogous to the proof of \cite[Theorem 4.17]{BaRo18}. \hfill$\Box$
\begin{example}\label{exeqmult}
Let $\mathbb F=\mathbb Z_2$, $r=2$,
$$\hat A(s)=\begin{bmatrix}1&0&0&0\\0&s-1& 0&0\\0&0&s-1&0\\0&0&0&s-1 \end{bmatrix}, \quad
\hat B(s)=\begin{bmatrix}1&0&0&0\\0&s-1& 0&0\\0&0&s-1&0\\0&0&0&s \end{bmatrix}.$$
The homogeneous invariant factors of $\hat A(s)$ and $\hat B(s)$ are
$\phi_1(s, t)=1, \phi_2(s, t)=\phi_3(s, t)=(s-t),\phi_4(s, t)=t(s-t) $ and
$\psi_1(s, t)=\psi_2(s, t)=1, \psi_3(s, t)=(s-t), \psi_4(s, t)=ts(s-t)$, respectively.
Then
$$
\phi_{i-2}(s, t)\mid \psi_i(s, t)\mid\phi_{i+2}(s, t), \quad 1\leq i\leq 4,
$$
$$
\Lambda(\hat A(s))=\{1, \infty\}, \quad \Lambda(\hat B(s))=\{0, 1, \infty\},
$$
and $ \mathbb F\cup \{\infty\}= \Lambda(\hat A(s))\cup \Lambda(\hat B(s))=\{0,1, \infty\}$.
But
$$
(m_1(\infty,\hat A(s)), \dots, m_4(\infty,\hat A(s)))=(m_1(\infty, \hat B(s)), \dots, m_4(\infty, \hat B(s)))
=(0,0,0,1).
$$
We have that
$$
\hat A(s)=\begin{bmatrix}1&0\\0&A(s)\end{bmatrix}, \quad
\hat B(s)=\begin{bmatrix}1&0\\0&B(s)\end{bmatrix},
$$
where $A(s)$ and $B(s)$ are the pencils of Example \ref{exp1p} and we have seen that there exists a matrix pencil $P(s)\in \mathbb F[s]^{3\times 3}$ such that
$\mathop{\rm rank}\nolimits P(s)= 2$ and
$A(s)+P(s)\ensuremath{\stackrel{s.e.}{\sim}} B(s)$. Taking
$\hat P(s)= \begin{bmatrix}0&0\\0&P(s)\end{bmatrix}\in \mathbb F[s]^{(1+3)\times (1+3)}$, we have that
$\hat A(s)+\hat P(s)\ensuremath{\stackrel{s.e.}{\sim}} \hat B(s)$ and $\mathop{\rm rank}\nolimits \hat P(s)=2$.
\end{example}
\section{Eigenvalue placement for regular matrix pencils under fixed rank perturbations}
\label{secplacement}
In this section we give a solution to Problem \ref{prfixdet}.
\medskip
Recall that if $\Gamma(s, t)$ is an homogeneous polynomial,
$$\Lambda(\Gamma(s,t)):=\{\lambda \in \overline{\mathbb F}\cup \{\infty\}\; : \;
\Gamma(\lambda,1)=0\},$$
where $\Gamma(\infty, 1):=\Gamma(1, 0)$.
The following theorem is the main result in this section.
The proof is similar to that of Theorem 5.1 of \cite{BaRo18}.
\begin{theorem}\label{thomainplacement}
Let $n\geq 2$.
Let $A(s)\in \mathbb F[s]^{n \times n}$ be a regular matrix pencil and $\phi_1(s,t)\mid \dots\mid \phi_n(s,t)$ be its homogeneous invariant factors. Let $\Psi(s,t)\in \mathbb F[s,t]$ be a nonzero homogeneous polynomial, monic with respect to $s$, and such that $\mathop{\rm deg }\nolimits (\Psi(s, t))= n$.
Assume that $ \mathbb F\cup \{\infty\}\not \subseteq \Lambda(A(s))\cup
\Lambda(\Psi(s,t)) $.
Let $r$ be a nonnegative integer, $r\leq n$.
There exists a matrix pencil $P(s)\in \mathbb F[s]^{n \times n}$ with $\mathop{\rm rank}\nolimits (P(s))=r$
such that if $C(s, t)$ is the homogeneous pencil associated to $A(s)+P(s)$, then $\det (C(s,t))=k\Psi(s,t)$ with $0\neq k \in \mathbb F$ if and only if
\begin{equation}\label{desphi}
\phi_1(s,t)\dots\phi_{n-r}(s,t)\mid \Psi(s,t).
\end{equation}
\end{theorem}
{\bf Proof.}
{\it Necessity.}
Let $C(s)=A(s)+P(s)$ and let
$\psi_1(s,t)\mid \dots\mid \psi_n(s,t)$ be its homogeneous invariant factors.
Taking $\Psi(s,t)=\psi_1(s,t) \dots \psi_n(s,t)$, from Theorem \ref{maintheorem} condition (\ref{desphi}) is satisfied.
\medskip
{\it Sufficiency.}
Assume that (\ref{desphi}) holds.
Then, there exists an homogeneous polynomial
$\gamma(s, t)\in \mathbb F[s, t]$ such that
$$\Psi(s,t)=\phi_1(s,t)\dots\phi_{n-r}(s,t)\gamma(s, t).$$
We define
$$
\psi_i(s,t):=\phi_{i-r}(s,t), \; 1\leq i \leq n-1, \quad \psi_n(s,t):=
\phi_{n-r}(s, t)\gamma(s, t),
$$
then
$$\psi_1(s,t) \mid \dots \mid \psi_n(s,t) \ \text{ and } \
\sum_{i=1}^n\mathop{\rm deg }\nolimits(\psi_i(s, t))=\mathop{\rm deg }\nolimits (\Psi(s, t))=n.
$$
Let $B(s)$ be a pencil with homogeneous invariant factors
$\psi_1(s,t) \mid \dots \mid \psi_n(s,t)$.
Then, $B(s)$ is regular and condition (\ref{interlacinghomogr1}) is satisfied.
By Theorem \ref{maintheorem},
there exists a pencil $P(s)\in \mathbb F[s]^{n \times n}$ such that $\mathop{\rm rank}\nolimits (P(s))= r$ and $A(s)+P(s)\ensuremath{\stackrel{s.e.}{\sim}} B(s)$.
Let $B(s,t)$ be the homogeneous pencil associated to $B(s)$.
Then there exist $0\neq k_1, k_2 \in \mathbb F$ such that
$$
\det (C(s,t))=k_1\det (B(s,t)) =k_1k_2\psi_1(s,t)\dots \psi_n(s,t)$$$$=k_1k_2\phi_1(s,t)\dots\phi_{n-r}(s,t)\gamma(s,t)=k_1k_2\Psi(s,t),\quad 0\neq k_1k_2\in \mathbb F.
$$
\hfill $\Box$
Notice that Theorem \ref{thomainplacement} gives us a solution to Problem \ref{prfixdet} as we see in the following corollary (compare it with Theorem 5.4 in \cite{BaRo18}).
\begin{corollary}\label{corplacement}
Let $n\geq 2$.
Let $A(s)\in \mathbb F[s]^{n \times n}$ be a regular matrix pencil and $\alpha_1(s)\mid \dots\mid \alpha_n(s)$ be its invariant factors. Let $q(s)\in \mathbb F[s]$ be a nonzero monic polynomial with $\mathop{\rm deg }\nolimits (q(s))\leq n$.
Assume that $ \mathbb F\cup \{\infty\}\not \subseteq \Lambda(A(s))\cup \Lambda^n(q(s)) $.
Let $r$ be a nonnegative integer, $r\leq n$.
There exists a matrix pencil $P(s)\in \mathbb F[s]^{n \times n}$ such that $\mathop{\rm rank}\nolimits (P(s))= r$ and $\det(A(s)+P(s))=kq(s)$ with $0\neq k \in \mathbb F$ if and only if
\begin{equation}\label{eqdetpol}
\begin{array}{l}
\alpha_1(s)\dots\alpha_{n-r}(s)\mid q(s),\\
\sum_{i=1}^{n-r}m_i(\infty, A(s)) \leq n-\mathop{\rm deg }\nolimits(q(s)).
\end{array}
\end{equation}
\end{corollary}
{\bf Proof.}
Let $\phi_1(s,t)\mid \dots\mid \phi_n(s,t)$ be the homogeneous invariant factors of $A(s)$ and
let $\Psi(s,t)=t^{n}q(\frac{s}{t})$.
Then $\Psi(s,t)\in \mathbb F[s,t]$ is a nonzero homogeneous polynomial, $\mathop{\rm deg }\nolimits (\Psi(s, t))= n$ and
$ \mathbb F\cup \{\infty\}\not \subseteq \Lambda(A(s))\cup
\Lambda(\Psi(s,t)) $.
Take $\delta(s)=\alpha_1(s)\dots\alpha_{n-r}(s)$. Then
$$
\phi_1(s,t)\dots\phi_{n-r}(s,t)=t^{\sum_{i=1}^{n-r}m_i(\infty, A(s))}t^{\mathop{\rm deg }\nolimits(\delta)}\delta(\frac{s}{t}).
$$
Hence, (\ref{desphi}) is equivalent to (\ref{eqdetpol}).
Assume that there exists a matrix pencil $P(s)\in \mathbb F[s]^{n \times n}$ such that $\mathop{\rm rank}\nolimits (P(s))= r$ and $\det(A(s)+P(s))=kq(s)$ with $0\neq k \in \mathbb F$. Let
$C(s, t)$ be the homogeneous pencil associated to $C(s)=A(s)+P(s)$.
Then, $\mathop{\rm deg }\nolimits(\det (C(s, t)))=n$ and $\det (C(s, 1))=\det(A(s)+P(s))=kq(s)$, from where
$\det (C(s, t))=t^{n-\mathop{\rm deg }\nolimits(q)}t^{\mathop{\rm deg }\nolimits(q)}kq(\frac{s}{t})=k\Psi(s,t)$. By Theorem \ref{thomainplacement}, (\ref{desphi}) (equivalently,
(\ref{eqdetpol})) holds.
Conversely, assume that (\ref{eqdetpol}) (equivalently,
(\ref{desphi})) holds. Then by Theorem \ref{thomainplacement}, there exists a matrix pencil $P(s)\in \mathbb F[s]^{n \times n}$ with $\mathop{\rm rank}\nolimits (P(s))=r$ and
such that if $C(s, t)$ is the homogeneous pencil associated to $C(s)=A(s)+P(s)$, then $\det (C(s,t))=k\Psi(s,t)$ with $0\neq k \in \mathbb F$.
Therefore, $\det (A(s)+P(s))=\det (C(s,1)) =kq(s)$.
\hfill $\Box$
\begin{remark}\label{rempln1}
If $n=1$, given pencils $a(s), q(s), p(s)\in \mathbb F[s]$, then $\det(a(s)+p(s))=kq(s)$ with $0\neq k\in \mathbb F$ if and only if $a(s)+p(s)\ensuremath{\stackrel{s.e.}{\sim}} q(s)$, i.e. Problem \ref{prfixdet} is the same as Problem \ref{fixproblem}.
\end{remark}
\section{Conclusions}
\label{conclusions}
Given a regular matrix pencil, we have completely characterized the
Weiestrass structure of a regular pencil obtained by a perturbation of fixed rank. The characterization is stated in terms of interlacing conditions between the homogeneous invariant factors of the original an the perturbed pencils, except in a very particular case. This work completes the research carried out in \cite{BaRo18}, where the same type of problem was solved in the case that the perturbed pencil was of bounded rank. Surprisingly enough, both solutions are characterized in terms of the same interlacing conditions.
The necessity of the conditions holds over arbitrary fields and the sufficiency over fields with sufficient number of elements.
It is remarkable the fact that the characterization of the fix rank perturbation of a pencil of the form $A(s)=sI-A$ requires an extra condition when the perturbation is performed by a constant matrix (\cite{Silva88_1}), and that that extra condition disappears when the fixed rank perturbation is allowed to be a pencil of degree one.
We also solve an eigenvalue placement problem characterizing the assignment of the determinant to a regular matrix pencil obtained by a fixed rank pencil perturbation of another regular one.
\bibliographystyle{acm}
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1,116,691,500,777 | arxiv | \section{\label{S:Intro}Introduction}
In early 1990's, carbon nanotechnology revolution introduced a plethora of new advanced materials among which the carbon nanotube was notoriously attractive for making nanoscale field effects devices, including vacuum devices. Many labs studied effects associated with field emission from a single CNT or arrays with counted number of isolated CNTs.\cite{Motorola1, Thales, BrightCNT, HotCNT_PRL, CNT_arrays_apps1, CNT_arrays_apps2} Control over fabrication and emission of single CNT field emission devices was excellent and many field emission devices were demonstrated, e.g. field emission radio\cite{CNTradio} or field emission transistor\cite{CNT_FE_transistor}, amplifier\cite{CNT_FE_amp}, and many others.\cite{CNT_arrays_apps1,CNT_arrays_apps2}
\begin{figure}
\includegraphics[width=5.5cm]{Fig_nonuniform}
\caption{Typical micrograph showing large beam transverse spread and nonuniformity. The blue circle marks the cathode's position behind the imaging screen and its size. }\label{F:nonuniform}
\end{figure}
In order to increase the output power, macrosocopic large area CNT fibers, films, yarns and fabrics started to be used to increase the operating currents from pico-/nano- to many amperes. Here, CNTs were thought to replace legacy velvets.\cite{CNTvsVelvet} The multiple benefits of CNT fibers over legacy technology are low turn-on voltage and high emission current at relatively low operating electric field due to inherent high field enhancement factor, and high electrical and thermal conductivities.\cite{fabrication} It was conventionally assumed that emission would be uniform, i.e. uniformity would translate from previously studied arrays of counted CNTs to the large area CNT fibers. However, recent studies that employed field emission microscopy illustrated that emission is never uniform and moreover that the emission area is a function of the electric field (making it cumbersome for calculating current densities.) Fig.\ref{F:nonuniform}, reproduced from our past work,\cite{mypaper} highlights other important issue of the large transverse spread of the emitted beam where beam lands on the imaging screen millimeters away from the physical location of the cathode source (blue circle) after travelling only a millimeter between the cathode and the anode. This clearly points out a very large emittance and therefore very low brightness, making CNT fiber cathodes impractical for applications like rf or microwave traveling wave tubes (operating in GHz range), microscopy and bright X-ray sources for medicine or active scanning. Another issue arises from that---because all the current emerges from a few active spots, it leads to local heating, microbreakdowns\cite{mypaper} and short-lived cathodes.
After experimenting with many fiber arrangements, we found that it is tiny singular fibrils (comprising braided fibers) that set loose due to thermal and field related stress\cite{mypaper,morphology_dependent} and that eventually focus the field due to their high aspect ratio and become point-like randomized intense electron emitters eventually exploding (seen as micro-breakdowns) and re-populating surrounding areas with more new-born fibrils. This process repeats itself until the cathodes stops operating while emission always look like a family of single electron rays going in many directions that are not aligned with the desired main longitudinal propagation direction, such as in Fig.\ref{F:nonuniform}. To mitigate this issue, hypothesised to be the major problem, we study new cathode production technology where fibers are electroplated with Ni and laser cut; all to suppress the fibril occurrence and regeneration. Through experimental measurements and electrostatic and beam dynamics modeling, emission uniformity and beam brightness were analyzed.
\section{\label{S:experimental}Experimental}
To prepare the field emission CNT fiber cathodes, a commercially available CNTs fiber from DexMat, Inc was used. The fiber is made by a wet-spinning technology \cite{fabrication}---pre-grown arrays of CNTs are dissolved in an acid to form a spinnable liquid dope that is extruded through a spinneret into coagulant bath to remove acid, and then dried in an oven. The resulting product is highly aligned and densely packaged CNTs in a form of a fiber. DexMat fibers have high electrical and thermal conductivity. Such fibers were shown to feature anisotropic field emission\cite{unisotropic_emission} that is emission takes place along the fiber (not from side walls) which is a great property allowing for control over emittance. Raman spectroscopy shows the G peak positions at 1583 cm$^{-1}$ suggesting rich crystalline graphitic content as expected from high quality fibers (Fig.\ref{F:sample}A).
To mitigate the described stray fibril problem, few fibers of the described kind were placed side by side and electroplated with Ni in an electrochemical bath and flush cut from the top to the required length of about $5\,\text{mm}$ with a femtosecond micromachining laser beam. Then, it was welded on $1\times 1\,\text{inch}$ Ni base. The final fabricated structure can be seen in Fig.\ref{F:sample}B.
We tested two samples referred to as Sample A and Sample B through the rest of the paper. In Sample A fibers were twisted and in Sample B were not, i.e. simply place along each others. As scanning electron microscopy (SEM) demonstrates in Fig.\ref{F:sample}C, additional fiber twisting enabled a dense core in Sample A, while Sample B (Fig.\ref{F:sample}D) has visible voids between the individual fibers. Otherwise, both sample have fiber core diameter of $\sim 150\,\mu\text{m}$ and Ni shell thickness of $\sim50\,\mu\text{m}$. Sample A and Sample B have height of $4.8\,\text{mm}$ and $4.6\,\text{mm}$ respectively (see Fig.\ref{F:sample}B).
\begin{figure}
\centering
\includegraphics[scale=0.045]{Fig_sample}
\caption{(A) Raman spectra of the cathode surface showing a crystalline graphitic peak. (B) Electroplated CNT fiber welded on a Ni base; $\text{H}=4.8\,\text{mm}$ for Sample A and $\text{H}=4.6\,\text{mm}$ for Sample B. (C) and (D) SEM images of Sample A and Sample B, where scale bars are 50 $\mu$m.}
\label{F:sample}
\end{figure}
DC current tests and field emission microscopy were performed in our custom field emission microscope described in great detail in Ref.\onlinecite{exp_setup}. Images were processed by a custom image processing algorithm FEpic described elsewhere.\cite{mypaper2}
\section{\label{S:ivrelation}Field Emission Imaging and Conditioning}
After sample was installed and gap was tuned using a doublet of two orthogonal optical microscopes, the physical location of the fiber is determined and labeled. To do that, the test chamber is illuminated. Because the imaging anode YAG screen is semitransparent, the location of the fiber can be immediately seen and captured by photographing. The core of the fiber is marked with red circle for reference in Figs.\ref{F:sampleA}A and \ref{F:sampleB}A.
After that voltage is applied and field emission images are taken concurrently with I-V curves. Fig.\ref{F:sampleA}B shows emission micrograph of the Sample A. The improvement is immediately obvious when compared with Fig.\ref{F:nonuniform}. First, the emission spot appears exactly at the optical projection of the cathode. This means beam divergence angle is small, so emittance can be expected to be low. Second, there is only a single spot and its size is comparable to the size of the fiber core---this is an indication of uniformity and small angular spread of the electron beam. The same exact behaviour was observed for Sample B, as given in Fig.\ref{F:sampleB}. No evidence suggesting the stray fibril issue was observed for neither cathode. These results highlight that such a simple electroplating strategy is extremely effective at yielding emission uniformity and spatial coherence, thereby boosting the transverse beam brightness.
\begin{figure}
\includegraphics[scale=0.47]{Fig_sampleA}
\caption{A) Sample A seen through the YAG screen when the lights is on in the chamber. Its fiber core is marked with red circle. B) FE micrograph of the same region at the gap of $200\,\mu\text{m}$.}\label{F:sampleA}
\end{figure}
\begin{figure}
\includegraphics[scale=0.47]{Fig_sampleB}
\caption{A) Sample B seen through the YAG screen when the lights is on in the chamber. Its fiber core is marked with red circle. B) FE micrograph of the same region at the gap of $200\,\mu\text{m}$.}\label{F:sampleB}
\end{figure}
\begin{figure}
\includegraphics[width=7.cm]{Fig_Condition}
\caption{A) The conditioning scheme of Sample A. Both ramp up and down curves are shown. There is a clear decrease in performance. B) Conditioning scheme of Sample B. Only ramp-up curves are shown. There is no considerable change in performance.}\label{F:condition}
\end{figure}
Fig.\ref{F:condition} shows cathode conditioning I-V curves for Sample A and B at interelectrode gap of 200 $\mu$m. Conditioning\cite{tungsten_conditioning,diamond_conditioning,mypaper} or cycling, where the applied voltage is ramped up and down to a progressively higher number in every consecutive cycle until the desired operating current is achieved, is a crucial procedure to maximize field emission cathode performance and ensure stability at the operating point.
We found that the electroplated fibers should be conditioned with small incremental steps to avoid adverse effects such as sudden burn-down. Fig.\ref{F:condition}A demonstrates the case where the maximal field was doubled with respect to the previous conditioning cycle, such that the current went up from 3 to 22 $\mu$A. Next, after completing the ramp down Sample A stopped working completely, which was possibly due to applying electric power that exceed that in the previous run by more than an order of magnitude. Its operation could not be rejuvenated by applying higher electric fields. This is unlike a conditioning scheme that was used for Cathode B as shown in Fig.\ref{F:condition}B. The emission current was doubled at every conditioning cycle: up to 1 $\mu$A and down to 0, then to 2 $\mu$A, to 4 $\mu$A, to 8 $\mu$A, to 16 $\mu$A, and finally to 32 $\mu$A. By doing so, Cathode B was conditioned $softly$ (compared to Sample A) maintaining and enhancing its performance: $i$) the resulting operating field went up and doubled, reaching same exact value where Cathode A burned down; $ii$) turn-on field and field enhancement factor remained nearly the same meaning that Cathode B was conditioned to stably sustain higher local field.
When it is compared to our past cathode designs, detailed in Ref.\onlinecite{mypaper}, they emit less at any given field. This is an expected result because (with stray fibrils mitigated) the field enhancement is reduced. However, turn-on fields are still very low, between 1 and 2.5 V/$\mu$m. Because the beam was tight suggesting high current density we limited our measurements to between 10--100 $\mu$A as the power density deposition at the imaging screen could attain above 1 kW/cm$^2$ at the voltage source limit of 1100 V, thereby literally drilling holes in it.\cite{power-density-1, power-density-2} At 1100 V, Sample A maxed out at 20 $\mu$A and Sample B 30 $\mu$A, respectively. Again both cathodes had similar metrics. Having these metrics and qualitative results in mind, a step was taken to carry out more quantitative analysis and calculate cathodes' emittance and brightness. All detailed in the next section.
\section{\label{S:emittance}Emittance and Brightness}
In the phase space $(x,x')$, $x$ is spatial position and $x'=\frac{dx}{dz}=\frac{dx/dt}{dz/dt}=\frac{v_x}{v_z}$ is the slope of the trajectory from longitudinal centrosymmetric axis of each particle. Then, rms emittance $\tilde{\epsilon}_x$ is defined as
\begin{equation}\label{E:rmsemit}
\tilde{\epsilon}_x = \sqrt{\langle \Delta x^2 \rangle \langle \Delta x'^2 \rangle - \langle \Delta x \Delta x' \rangle}
\end{equation}
where $\Delta x=x-\langle x\rangle$ and $\Delta x'=x'-\langle x'\rangle$. For a beam with cylindrical symmetry in $(x,y)$ and $(x',y')$ centered around zero, $\langle x\rangle$ and $\langle x'\rangle$ are zero. The cross-corelation term, $\langle \Delta x \Delta x' \rangle$ can be removed with proper beam optics.\cite{jarvis_thesis} Then, Eq.\ref{E:rmsemit} reduces to
\begin{equation}\label{E:rmsemit2}
\tilde{\epsilon}_x = \sqrt{\langle x^2 \rangle \langle x'^2 \rangle}=\sigma_x\sqrt{\langle x'^2 \rangle}.
\end{equation}
Rms emittance is a function of the beam energy as $x'$ is changing under acceleration, and is not useful while comparing beam or beams at different energies. On the other hand, from Liouville's theorem, normalized emittance is a conserved quantity under acceleration as long as the beam is only subjected to conservative forces. The relation between rms emittance and normalized emittance is given by
\begin{equation}\label{E:normemit}
\epsilon_x^\text{N} = \gamma\beta\tilde{\epsilon}_x
\end{equation}
where $\gamma = \frac{1}{\sqrt{1-\beta^2}}$ is the Lorentz factor and $\beta = \frac{v}{\text{c}} \approx \frac{v_z}{\text{c}}$. In our case, $\gamma\approx1$ because energy is $<=$1 keV. Mean-transverse energy, $\text{MTE}$, is $\frac{1}{2}\text{m}_\text{e} \langle v^2 \rangle$, where $v^2=v_x^2+v_y^2$. Because of the cylindrical symmetry in $(v_x,v_y)$, $\text{MTE}\approx\frac{1}{2}\text{m}_\text{e} \langle 2v_x^2 \rangle=\text{m}_\text{e}\langle v_x^2\rangle$. Then, after substituting Eq.\ref{E:rmsemit2}, in terms of $\text{MTE}$, Eq.\ref{E:normemit} becomes
\begin{equation}\label{E:fundametal}
\epsilon_x^\text{N} = \frac{v_z}{\text{c}}\cdot\sigma_x\cdot\sqrt{\langle \frac{v_x^2}{v_z^2} \rangle}=\sigma_x\cdot\sqrt{\frac{\text{MTE}}{\text{m}_\text{e}\text{c}^2}}
\end{equation}
Practically, the normalized emittance at the cathode surface is calculated as follows. If the radius of the uniformly emitting surface of the cathode is $r_\text{i}$, $\sigma_x\approx r_\text{i}$ can be taken. Moreover, at the surface, MTE is due to statistical distribution of electrons inside the cathode itself. So, it is intrinsic, and is further redefined as $\text{MTE}_\text{i}$. Then, Eq.\ref{E:fundametal} becomes
\begin{equation}\label{E:ex_finalform}
\epsilon_x^\text{N}=r_\text{i}\cdot\sqrt{\frac{\text{MTE}_\text{i}}{\text{m}_\text{e}\text{c}^2}}
\end{equation}
By using normalized emittance, normalized transverse brightness, $B_\text{N}$, can be calculated as
\begin{equation}\label{E:brightness}
B_\text{N}= \frac{2\,I}{\epsilon_x^\text{N} \epsilon_y^\text{N}}
\end{equation}
where $I$ is the emitted current. $\epsilon_x^\text{N}=\epsilon_y^\text{N}$ can be taken in cylindrical symmetry.
\begin{figure}
\includegraphics[scale=0.24]{Fig_moveaway}
\caption{Series of field emission micrographs of Sample B with the screen moved away progressively from $200\,\mu\text{m}$ gap to $1600\,\mu\text{m}$ gap with $200\,\mu\text{m}$ steps. At each step, a micrograph is captured.}\label{F:moveaway}
\end{figure}
To determine $r_\text{i}$ and $\text{MTE}_\text{i}$ in Eq.\ref{E:ex_finalform}, we conducted experimental measurements in combination with beam dynamics in GPT (General Particle Tracer).\cite{gpt_ref} In the measurements, the imaging screen is moved away from the cathode progressively. The voltage is set accordingly to keep the current constant at 20 nA to enable strong beam image signal but avoid additional beam expansion due to vacuum space charge effect. A micrograph at each step is recorded (see Fig.\ref{F:moveaway}). Increase in the spot size due to larger time of flight is measured. As it is seen in Fig.\ref{F:moveaway}, the spots are Gaussian in nature with dense center and faint tails. Each spot can be modeled mathematically with a cylindrically symmetric Gaussian as
\begin{equation}\label{E:gauss_fit}
p=A\cdot \text{exp}\left( -\frac{(x-x_c)^2+(y-y_c)^2}{2\sigma_{\text{spot}}^2}\right)+C
\end{equation}
to extract projected transverse beam size. Here, $A$ is the amplitude, $\sigma_\text{spot}$ is the standard deviation, $C$ is the background offset, $p$ is the intensity, $(x,y)$ are the space dimensions, $(x_c,y_c)$ are the coordinates of the peak.\cite{mypaper2} The model parameters $A$, $\sigma_\text{spot}$, and $C$ for each spot are computed with least-square fitting method. After fitting, the emission spot diameter is taken as $2\sigma_\text{spot}$. An exemplary 3D fitting done by FEpic for the beam imaged at 200 $\mu$m gap is presented in Fig.\ref{F:gauss}; with the black mesh surface being the fitting surface. The resulting dependence of the spot size, first measured (Fig.\ref{F:moveaway}) and processed by FEpic, versus distance is shown in Fig.\ref{F:gapvsize} with black solid circles. The data in the figure is only for Sample B. Because Sample A burned down, studies similar to those presented in Fig.\ref{F:moveaway} could not be carried out.
\begin{figure}
\includegraphics[scale=0.2]{Fig_gauss}
\caption{The color surface shows the beam spot for $200\,\mu\text{m}$ gap in 3D. The black mesh surface shows its mathematical fit in Eq.\ref{E:gauss_fit}.}\label{F:gauss}
\end{figure}
\begin{figure}
\includegraphics[width=6.5cm]{Fig_gapvsize}
\caption{Comparison of experimental and computational final beam spot size as the screen is moving away from the cathode. $\text{MTE}_\text{i}$ of $250\,\text{eV}$ and $r_\text{i}$ of $75\,\mu\text{m}$ were used in GPT modeling.} \label{F:gapvsize}
\end{figure}
\begin{figure}
\includegraphics[width=6.5cm]{Fig_comsol}
\caption{Illustration of electric field computed in COMSOL for a $1\,\text{mm}$ gap. The color plot shows the field magnitude and contour. The dark region is the fiber core, and the gray region is the Ni shell.}\label{F:comsol}
\end{figure}
\begin{figure}
\includegraphics[width=8.5cm]{Fig_gpt}
\caption{In GPT: (A) Initial uniform beam distribution at the cathode surface in real-space, where $r_\text{i}$ is the radius of the beam. (B) The initial distribution in momentum-space, where $\beta_x=v_x/c$, $\beta_y=v_y/c$, and $r_\beta$ is the radius. Final distribution in (C) real space and (D) momentum space when the screen at is $1\,\text{mm}$.}\label{F:gpt}
\end{figure}
To calculate the phase space volume for emittance and brightness estimations and obtain $r_\text{i}$ and $\text{MTE}_\text{i}$, we switched to beam dynamics in GPT by comparing it with the experiment in Fig.\ref{F:moveaway}. To do that, a field map for each interelectrode gap was computed in COMSOL by solving Poisson's equation with given boundary conditions. An exemplary field distribution for 1 mm gap is shown in Fig.\ref{F:comsol}. Then, the field maps were imported to GPT. In GPT, the initial particle distribution in the real space $(x,y)$ (Fig.\ref{F:gpt}A) and momentum space $(\beta_x,\beta_y)=(v_x/\text{c},v_y/\text{c})$ (Fig.\ref{F:gpt}B) has to be set to propagate the beam. Because the cathode itself circular, we used a circular uniform distribution with the radius $r_\text{i}$ in position-space and the radius $r_\beta$ in $\beta$-space. $\text{MTE}_\text{i}$ of the fiber is expected to be $250\,\text{meV}$.\cite{mte_fairchild} To be used in GPT, $\text{MTE}_\text{i}$ is converted into $r_\beta$ as
\begin{equation}\label{E:mte_vs_maxB}
r_\beta=\sqrt{\frac{4\,\text{MTE}_\text{i}}{\text{m}_\text{e}\text{c}^2}}.
\end{equation}
$\text{MTE}_\text{i}$ of 250 meV translates to $r_\beta$ of $1.4\times10^{-3}$. At the same time, $r_\text{i}$ remains a free model parameter to be found by finding the best agreement between GPT with the experiment.
The beam was launched at a charge corresponding to 20 nA and allowed to drift through the distance corresponding to a specified interelectrode gap. Here, the beam dynamics is computed self-consistently taking the COMSOL calculated field. The final distribution in $(x,y)$ and $(\beta_x,\beta_y)$ was captured at a distance corresponding to the imaging YAG screen of the microscope and are shown in Fig.\ref{F:gpt}C and D. In the $(x,y)$ space, the standard deviation $\sigma_{gpt}$ of such projections were calculated for every cathode-anode gap, and $2\sigma_{gpt}$ was taken as the computed beam diameter (analogous to FEpic image processing of the experimental images). It was established that in GPT, when $r_\text{i}$ was set to the physical radius of the fiber core of $75\,\mu\text{m}$, and $r_\beta$ was set by MTE of 250 meV,\cite{mte_fairchild} then the final diameter (spot size $2\sigma_{gpt}$) of the resulting beam projection was in a very good quantitative agreement with the experiments (see Fig.\ref{F:gapvsize}). Note, the GPT results were fairly insensitive to MTE values set between 25 (typical Fermi level value for CNT) and 250 meV, and magnification was due to the radial field distribution. This points out that the divergence between the experiment and GPT (seen for the gap ranging between 1 and 1.6 mm) stems from the difference between the idealized computed and the actual field distribution in the gap. The summary of the results in Fig.\ref{F:gapvsize} confirms that the entire whole fiber surface actively and uniformly emitting with a small $\text{MTE}$.
Finally, using Eq.\ref{E:ex_finalform} and substituting $r_\text{i}=75\,\mu\text{m}$ and $\text{MTE}_\text{i}=250\,\text{eV}$, the upper limit of the normalized emittance on the fiber cathode surface can be estimated as
\begin{equation}\label{E:num_emit}
\epsilon_x^N=0.052\,\text{mm}\,\text{mrad} = 52\,\text{nm}.
\end{equation}
From this, taking the measured current (limited to 10--100 $\mu$A due to extremely high power density), as shown in Fig.\ref{F:condition}, the normalized brightness for 50 $\mu$A dc current is $B_N=3.7\times10^{10}\frac{\text{A}}{\text{m}^2\text{rad}^2}$. The same very fibers can draw currents of 1--10 A when operated in pulsed mode with a pulse length of 100--300 ns.\cite{fiber-IVEC2020} Using the estimated emittance of $52\,\text{nm\,rad}$, the brightness in the pulsed mode, preferable mode in most VED HPM applications, attains a notable value of $B_N=4.4\times10^{15}\frac{\text{A}}{\text{m}^2\text{rad}^2}$. This number is outstanding and is comparable with brightness metrics in the the state-of-the-art microwave/rf accelerator injectors.\cite{SOTA_Br} This (phase space) brightness can be converted into geometrical brightness, a definition of brightness commonly employed in the electron microscopy literature. The geometrical reduced brightness is defined as $B_r^G=\frac{\text{d}I}{\text{d}\Omega}\frac{1}{U}\frac{1}{S_{\text{cathode}}}$, where $\Omega$ is the solid angle, $U$ is the voltage at which the current $I$ is measured, and $S_{\text{cathode}}$ is the emission area of the cathode. Our calculations show that in pulsed mode it could attain $B_r^G=5.7\times 10^7$ A m$^{-2}$sr$^{-1}$V$^{-1}$. This number is within the range obtained for single CNT emitters.\cite{BrG_PRl2005}
\section{\label{S:conclusion}Conclusion}
In conclusion, we presented a simple and efficient field emission cathode design where CNT fiber core was plated with nickel shell. This design had two important functions. First, it compresses the core, provides mechanical strength thereby preventing stray fibril formation during conditioning and operation. Second, such design (while slightly reducing field enhancement and increasing turn-on field) reduces the fringing field on the CNT fiber and therefore the defocusing radial field.
As field emission microscopy directly demonstrated, both tested cathodes featured excellent spatially coherent emission. Field emission microscopy aided by image processing and beam dynamics simulations confirmed that the entire fiber core of 150 $\mu$m in diameter actively and uniformly emitted electrons, as well as enabled phase space analysis. All of these combined allowed to quantify the observed emission coherence through calculating emittance and brightness. The extremely low emittance resulting in record brightness highlight a simple and practical path forward for the CNT fiber technology that has long been expected to advance high frequency vacuum power devices but had limited success due to low brightness.
Finally and most importantly, it was demonstrated that the nanoscopic single CNT cathode technology can be translated to the macroscopic fiber CNT level in terms of emission uniformity. In other words, spatial coherence and uniformity (intrinsic to a single CNT emitter) can be achieved in a CNT fiber comprised out of billions of single CNT's. The obtained brightness figures of merit further confirm this technology translation in that ultimate single CNT emitter brightness is feasible to attain for CNT fiber cathodes.
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1,116,691,500,778 | arxiv | \section{Introduction}
\label{intro}
Voice-controlled virtual assistants (VVA) such as Siri and Alexa have experienced an exponential growth in terms of number of users and provided capabilities. They are used by millions for a variety of tasks including shopping, playing music, and even telling jokes. Arguably, their success is due in part to the emotional and personalized experience they provide. One important aspect of this emotional interaction is humor, a fundamental element of communication. Not only can it create in the user a sense of personality, but also be used as fallback technique for out-of-domain queries \cite{Bellegarda2014}. Usually, a VVA's humorous responses are invoked by users with the phrase \textit{"Tell me a joke"}. In order to improve the joke experience and overall user satisfaction with a VVA, we propose to personalize the response to each request. To achieve this, a method should be able to recognize and evaluate humor, a challenging task that has been the focus of extensive work. Some authors have applied traditional NLP techniques \cite{yan2017s}, while others deep learning models \cite{donahue2017humorhawk}. Moreover, \cite{yang2015humor} follows a semantic-based approach, while \cite{ruch1992assessment} and \cite{ahuja2018makes} tackle the challenge from a cognitive and linguistic perspective respectively.
To this end, we have developed two methods. The first one is based on traditional NLP techniques. Although relatively simple, it is robust, scalable, and has low latency, a fundamental property for real-time VVA systems. The other approaches combine multi-task learning \cite{caruana1997multitask} and self-attentional networks \cite{vaswani2017attention} to obtain better results, at the cost of added complexity. Both BERT \cite{devlin2018bert} and an adapted transformer \cite{vaswani2017attention} architecture are considered. This choice of architecture was motivated by the advantages it presents over traditional RNN and CNN models, including better performance \cite{liu2018generating}, faster training/inference (important for real-time systems), and better sense disambiguation \cite{tang2018self} (an important component of computational humor \cite{yang2015humor}).
The proposed models use binary classifiers to perform point-wise ranking, and therefore require a labelled dataset. To generate it, we explore two implicit user-feedback labelling strategies: five-minute reuse and one-day return. Online A/B testing is used to determine if these labelling strategies are suited to optimize the desired user-satisfaction metrics, and offline data to evaluated and compared the system's performance.
\section{Method}
\label{model}
\subsection{Labelling Strategies}
\label{labeling}
Generating labels for this VVA skill is challenging. Label generation through explicit user feedback is unavailable since asking users for feedback creates friction and degrade the user experience. In addition, available humor datasets such as \cite{yang2015humor,potash2017semeval} only contain jokes and corresponding labels, but not the additional features we need to personalize the jokes.
To overcome this difficulty, it is common to resort to implicit feedback. In particular, many VVA applications use interruptions as negative labels, the rationale being that unhappy users will stop the VVA. This strategy, however, is not suitable for our use-case since responses are short and users need to hear the entire joke to decide if it is funny. Instead, we explore two other implicit feedback labelling strategies: five-minute reuse and 1-day return. Five-minute reuse labels an instance positive if it was followed by a new joke request within five-minutes. Conversely, 1-day return marks as positive all joke requests that were followed by a new one within the following 1 to 25-hour interval. Both strategies assume that if a user returns, he is happy with the jokes. This is clearly an approximation, since a returning user might be overall satisfied with the experience, but not with all the jokes. The same is true for the implied negatives; the user might have been satisfied with some or all of the jokes. Therefore, these labels are noisy and only provide weak supervision to the models.
Table \ref{exLabels} shows an example of the labels' values for a set of joke requests from one user.
\begin{table}[t!]
\caption{\label{exLabels} Example of labelling strategies: five-minute reuse (label 1) and 1-day return (label 2)}
\begin{center}
\begin{tabular}{|c|c|c|
\hline
\bf Timestamp & \bf Label 1 & \bf Label 2 \\
\hline
2019/05/03-17:51:10& 1& 0\\
2019/05/03-17:53:10& 0& 0\\
2019/05/06-21:41:09& 1& 1\\
2019/05/06-21:44:19& 0& 1\\
2019/05/07-20:34:19& 0 & 0 \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Features}
\label{features}
All models have access to the same raw features, which we conceptually separate into user, item and contextual features. Examples of features in each of these categories are shown in Table \ref{featuresRaw}. Some of these are used directly by the models, while others need to be pre-processed. The manner in which each model consumes them is explained next.
\begin{table}
\caption{\label{featuresRaw} Examples of features within each category}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
\bf Feature& \bf Type & \bf Category \\
\hline
Country Code & Categorical & User\\
Joke Text & String& Item \\
Request Timestamp & Timestamp& Context\\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{NLP-based: LR-Model}
\label{prodSys}
To favor simplicity over accuracy, a logistic regression (LR) model is first proposed. Significant effort was put into finding expressive features. Categorical features are one-hot encoded and numerical ones are normalized. The raw Joke Text and Timestamp features require special treatment. The Joke Text is tokenized and the stop-words are removed. We can then compute computational humor features on the clean text such as sense combination \cite{yang2015humor} and ambiguity \cite{mihalcea2007characterizing}. In addition, since many jokes in our corpus are related to specific events (Christmas, etc), we check for keywords that relate the jokes to them. For example, if "Santa" is included, we infer it is a Christmas joke. Finally, pre-computed word embeddings with sub-word information are used to represent jokes by taking the average and maximum vectors over the token representations. Sub-word information is important when encoding jokes since many can contain out-of-vocabulary tokens. The joke's vector representations are also used to compute a summarized view of the user's past liked and disliked jokes. We consider that a user liked a joke when the assigned label is 1, an approximation given the noisy nature of the labels. The user's liked/disliked joke vectors are also combined with the candidate joke vector by taking the cosine similarity between them.
For the raw Timestamp feature, we first extract simple time/date features such as month, day and isWeekend. We then compute binary features that mark if the timestamp occurred near one of the special events mentioned before. Some of these events occur the same day every year, while others change (for example, the Super Bowl). In addition, many events are country dependent. The timestamp's event features are combined with the joke's event features to allow the model to capture if an event-related joke occurs at the right time of the year.
The LR classifier is trained on the processed features and one of the labels. The model's posterior probability is used to sort the candidates, which are chosen randomly from a pool of unheard jokes. Although useful (see Validation section), this model has several shortcomings. In particular, many of the used features require significant feature engineering and/or are country/language dependent, limiting the extensibility of the model.
\subsection{Deep-Learning-based: DL-Models}
\label{DL}
To overcome the LR-model's limitations, we propose the following model (see Figure \ref{architectureGlobal}). In the input layer, features are separated into context, item and user features. Unlike the LR-model, time and text features do not require extensive feature engineering. Instead, simple features (day, month and year) are extracted from the timestamp. After tokenization and stop-word removal, text features are passed through a pre-trained word embeding layer, and later, input into the joke encoder block.
\begin{figure}
\includegraphics[scale=0.5]{architectureV4.pdf}
\caption{Architecture of the transformer-based model}
\label{architectureGlobal}
\end{figure}
The basis of the joke encoder is a modified transformer. Firstly, only the encoder is needed. Moreover, since studies suggest that humor is subjective and conditioned on the user's context \cite{8e165aa5efb347fc937dc3189dbb4fb3}, we add an additional sub-layer in the transformer encoder that performs attention over the user's features. This sub-layer, inserted between the two typical transformer sub-layers at certain depths of the network, allows the encoder to adapt the representations of the jokes to different user contexts. Thus, the same joke can be encoded differently depending on the user's features. In practice, this additional sub-layer works like the normal self-attention sub-layer, except it creates its query matrix Q from the sub-layer below, and its K and V matrices from the user features. As an alternative, we also test encoding the jokes using a pre-trained BERT model.
Regardless of the used encoder, we average the token representations to obtain a global encoding of the jokes. The same encoder is used to represent the item's (the joke to rank) and the user's (liked and disliked jokes) textual features through weight sharing, and the cosine similarity between both representations are computed. The processed features are then concatenated and passed through a final block of fully connected layers that contains the output layers. Since experiments determined (see Validation section) that both labeling strategies can improve the desired business metrics, instead of optimizing for only one of them, we take a multi-task learning approach. Thus, we have two softmax outputs.
Finally, we use a loss function that considers label uncertainty, class imbalance and the different labeling functions. We start from the traditional cross-entropy loss for one labelling function. We then apply uniform label smoothing \cite{szegedy2016rethinking}, which converts the one-hot-encoded label vectors into smoothed label vectors towards $0.5$:
\begin{equation}
y_{ls}= y_{one-hot}*(1-\epsilon) + \frac{\epsilon}{2}
\end{equation}
with $\epsilon$ a hyper-parameter. Label smoothing provides a way of considering the uncertainty on the labels by encouraging the model to be less confident. We have also experimented with other alternatives, including specialized losses such as \cite{martinez2018taming}. However, they did not produce a significant increase in performance in our tests. To further model the possible uncertainty in the feedback, we apply sample weights calculated using an exponential decay function on the time difference between the current and the following training instance of the same customer:
\begin{equation}
w_i= a*b^{t_{i}}+1.0
\end{equation}
where $w_i$ is the weight of sample $i$, $t_i$ is the time difference between instances $i$ and $i+1$ for the same user, and $a,b$ are hyper-parameters such that $a>0$ and $0<b<1$. The rationale behind these weights is the following. If for example, we consider labeling function 1, and a user asks for consecutive jokes, first within 10 seconds and later within 4.9 minutes, both instances are labeled as positive. However, we hypothesize that there is a lower chance that in the second case the user requested an additional joke because he liked the first one. In addition, class weights are applied to each sample to account for the natural class imbalance of the dataset. Finally, the total loss to be optimized is the weighted sum of the losses for each of the considered labeling functions:
\begin{equation}
\mathcal{L}((f(x),\Theta),y) = \sum_{l=1}^2 w_l \mathcal{L}_{l}
\end{equation}
where $w_{l}$ are manually set weights for each label and $\mathcal{L}_{l}$ are the losses corresponding to each label, which include all the weights mentioned before.
\section{Validation}
\label{validation}
A two-step validation was conducted for English-speaking customers. An initial A/B testing for the LR model in a production setting was performed to compare the labelling strategies. A second offline comparison of the models was conducted on historical data and a selected labelling strategy. One month of data and a subset of the customers was used (approx. eighty thousand). The sampled dataset presents a fraction of positive labels of approximately 0.5 for reuse and 0.2 for one-day return. Importantly, since this evaluation is done on a subset of users, the dataset characteristic's do not necessarily represent real production traffic. The joke corpus in this dataset contains thousands of unique jokes of different categories (sci-fi, sports, etc) and types (puns, limerick, etc). The dataset was split timewise into training/validation/test sets, and hyperparameters were optimized to maximize the AUC-ROC on the validation set. As a benchmark, we also consider two additional methods: a non-personalized popularity model and one that follows \cite{kim2014convolutional}, replacing the transformer joke encoder with a CNN network (the specialized loss and other characteristics of the DL model are kept).
Hyperparameters were optimized using grid-search for the LR-Model. Due to computational constraints, random search was instead used for the DL-Model. In both cases, hyperparameters are selected to optimize the AUC-ROC on the validation set. Table \ref{paramsLR} lists some of the considered hyperparameter values and ranges for both models. The actual optimal values are sample specific.
\begin{table}[t!]
\caption{ \label{paramsLR} Hyperparameter values tuned over, LR (top) and DL models (bottom)}
\begin{center}
\begin{tabular}{|c|c|}
\hline
Name & Value \\
\hline
Elastic-Net param. & [0.01,0.5] \\
Regularization & [$10^{-3}$,10.0] \\
Fit intercept & True/False \\
\hline
Learning rate & [$10^{-3},10^{-5}$] \\
Batch size & [32,256] \\
Label smoothing & [0.1,0.3] \\
Keep probability & [0.5,0.8] \\
Num. heads & [2,6] \\
Num. transformer layers & [1,6] \\
F.C layers& [2,5] \\
F.C layer sizes& [16,256] \\
CNN filter sizes & [2,32] \\
CNN num. filters & [16,128]\\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Online Results: A/B Testing}
\label{ab}
Two treatment groups are considered, one per label. Users in the control group are presented jokes at random, without repetition. Several user-satisfaction metrics such as user interruption rate, reuse of this and other VVA skills, and number of active dialogs are monitored during the tests. The relative improvement/decline of these metrics is compared between the treatments and control, and between the treatments themselves. The statistical significance is measured when determining differences between the groups. Results show that the LR-based model consistently outperforms the heuristic method for both labeling strategies, significantly improving retention, dialogs and interruptions. These results suggest that models trained using either label can improve the VVA's joke experience.
\subsection{Offline Results}
\label{results}
One-day return was selected for the offline evaluation because models trained on it have a better AUC-ROC, and both labeling strategies were successful in the online validation. All results are expressed as relative change with respect to the popularity model.
\begin{table}[t!]
\caption{\label{res1} Relative change w.r.t popularity model of AUC-ROC and Overall Accuracy: transformer model (DL-T), BERT model (DL-BERT), transformer without special context-aware attention (DL-T-noAtt) and without both special attention and modified loss (DL-T-basic), CNN model (DL-CNN) and LR model (LR).}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Method & R. Ch. AUC-ROC & R. Ch. O.A. \\
\hline
DL-T & 0.31 & 0.24 \\
DL-BERT & 0.30 & 0.27 \\
DL-T-noAtt & 0.29 & 0.24 \\
DL-T-basic & 0.28 & 0.23 \\
DL-CNN & 0.28 & 0.13 \\
LR & 0.24 & 0.21 \\
\hline
\end{tabular}
\end{center}
\end{table}
We start by evaluating the models using AUC-ROC. As seen in Table \ref{res1}, the transformer-based models, and in particular our custom architecture, outperform all other approaches. Similar conclusions can be reached regarding overall accuracy. However, given the class imbalance, accuracy is not necessarily the best metric to consider. In addition, to better understand the effect to the original transformer architecture, we present the performance of the model with and without the modified loss and special attention sub-layer (see Table \ref{res1}). Results suggest both modifications have a positive impact on the performance. Finally, to further evaluate the ranking capabilities of the proposed methods, we use top-1 accuracy. Additional positions in the ranking are not considered because only the top ranked joke is presented to the customer. Results show that the DL based models outperform the other systems, with a relative change in top-1 accuracy of 1.4 for DL-BERT and 0.43 for DL-T, compared with 0.14 for the LR method.
Results show that the proposed methods provide different compromises between accuracy, scalability and robustness. On one hand, the relatively good performance of the LR model with engineered features provides a strong baseline both in terms of accuracy and training/inference performance, at the cost of being difficult to extend to new countries and languages. On the other hand, DL based methods give a significant accuracy gain and require no feature engineering, which facilitates the expansion of the joke experience to new markets and languages. This comes at a cost of added complexity if deployed in production. In addition, given the size of the BERT model (340M parameters), real-time inference using DL-BERT becomes problematic due to latency constraints. In this regard, the DL-T model could be a good compromise since its complexity can be adapted, and it provides good overall accuracy
\section{Conclusions and Future Work}
\label{conclusions}
This paper describes systems to personalize a VVA's joke experience using NLP and deep-learning techniques that provide different compromises between accuracy, scalability and robustness. Implicit feedback signals are used to generate weak labels and provide supervision to the ranking models. Results on production data show that models trained on any of the considered labels present a positive real-world impact on user satisfaction, and that the deep learning approaches can potentially improve the joke skill with respect to the other considered methods. In the future, we would like to compare all methods in A/B testing, and to extend the models to other languages
|
1,116,691,500,779 | arxiv | \section{Introduction}
The classification of Einstein metrics and Ricci solitons is a central problem in geometry. Gradient shrinking and steady Ricci solitons play a crucial role in the study of Ricci flow since they arise as blow-up limits of singularities. Recently, motivated by research in mathematical physics and complex geometry, a generalization of Ricci solitons was introduced.
A smooth manifold $M^n$ equipped with a Riemannian metric $g$, closed 3-form $H$ and function $f$ define a \emph{generalized Ricci soliton} if
\begin{align}\label{f:gk_soliton_intro}
\Rc - \tfrac{1}{4} H^2 + \nabla^2 f =&\ 0,\\
d^* H + i_{\nabla f} H =&\ 0,
\end{align}
where $\Rc$ is the Ricci curvature of $(M,g)$ and $(H^2)_{ij}:=H_{ikl}H_{j}^{\ kl}$. These equations generalize the Ricci soliton system, arising as the Euler-Lagrange equations for Einstein-Hilbert functional
\[
\mathcal{EH}(g,H,f):=\int_M\left(R-\frac{1}{12}|H|_g^2+|\nabla f|^2_g\right)e^{-f}dV_{g},
\]
where $R$ is the scalar curvature of $g$.
The generalized Ricci soliton system~\eqref{f:gk_soliton_intro} describes canonical geometric structures involving torsion (cf. \cite{gf-st-20}), and arises in physics as the type IIB string equations \cite{strings-85}. While in~\eqref{f:gk_soliton_intro} the metric $g$ and 3-form $H$ are unrelated, there many natural geometric settings in which $g$ and $H$ are tied together. One important special case occurs when $(M,g,I)$ is a complex Hermitian manifold with an integrable almost complex structure $I$ such that $H = -d^c \omega_I$, where $\omega_I:=g(I\cdot,\cdot)$. Since $H$ is closed, we necessarily have $dd^c\omega_I=0$, i.e. $(M,g,I)$ is a \textit{pluriclosed} manifold. There is a more refined case where $g$ is part of a generalized K\"ahler (GK) structure (cf.\,\S \ref{ss:GK_structures}), with the data then referred to as a \emph{generalized K\"ahler-Ricci soliton}, or simply GK soliton. In the context of string theory this corresponds to imposing supersymmetric constraints on the solution of~\eqref{f:gk_soliton_intro}, and the corresponding structures were first recognized in~\cite{ga-hu-ro-84}.
Mathematically, generalized K\"ahler structures were discovered by Gualtieri~\cite{gu-10,gu-14} in the context of Hitchin's \textit{generalized geometry}~\cite{hi-10}.
These extra geometric assumptions on $g$ and $H$ relate the classification of generalized Ricci solitons to deep questions in complex geometry. For instance, in~\cite{PCF} and~\cite{GKRF} Tian and the first-named author introduced natural geometric flows~--- pluriclosed flow and generalized K\"ahler-Ricci flow~--- extending the K\"ahler-Ricci flow to the non-K\"ahler setting. Similarly to the Ricci flow setup, pluriclosed and GK solitons appear as limits of the respective flows. Thus it is essential to classify all solutions to~\eqref{f:gk_soliton_intro} as the first step in understanding the long-term behavior of a relevant flow. In \cite{st-19-soliton}, \cite{SU} we gave a conjecturally exhaustive list of generalized K\"ahler-Ricci solitons on compact complex surfaces, by constructing GK solitons on all diagonal Hopf surfaces. Since the pointed limits of the above flows are not necessarily compact, we are forced to consider also solutions to the GK soliton system with a \textit{complete} background $(M,g)$. This is the primary motivation for the present paper.
In this work we give a partial classification of complete generalized K\"ahler-Ricci solitons in dimension four. The first step is to observe the presence of two commuting Killing fields associated to a given soliton. The case of primary interest is when these two Killing fields are aligned, and generate a biholomorphic $S^1$ action. Furthermore, associated to any generalized K\"ahler structure is a real Poisson tensor (cf.\,\S \ref{ss:Poisson}), which we will assume does not vanish identically. These structural hypotheses roughly speaking represent the generic case, while the other cases are more rigid, and will be treated elsewhere.
In the locus where the Poisson tensor in nondegenerate, it is possible to describe a GK structure in terms of a triple of symplectic forms which includes hyperK\"ahler triples as a special case \cite{AGG} (cf.\,Proposition \ref{p:holo_symplectic-acs}). In the hyperK\"ahler case, the (tri-Hamiltonian) $S^1$ action determines locally a moment map $\pmb \mu$ to $\mathbb R^3$, and we use the notation $\mathbb R^3_{\pmb \mu}$ to denote the image space of such a moment map. Using these moment map coordinates, the hyperK\"ahler metric has an explicit local description given by the famous Gibbons-Hawking ansatz \cite{GibbonsHawking}:
\begin{align*}
g = W \left( dx^2 + dy^2 + dz^2 \right) + W^{-1} \eta^2,
\end{align*}
where $W$ is a harmonic function on flat $\mathbb R^3_{\pmb\mu}$, and $\eta$ is a principal $S^1$ connection with curvature $d \eta = *d W$. This construction is reversible, and by choosing $W$ to be a linear combination of a constant and suitably normalized Green's functions based at points in $\mathbb R^3_{\pmb \mu}$, one obtains a hyperK\"ahler $4$-manifold admitting a free isometric circle action, which is incomplete due to the poles of $W$. These singularities are however removable, and by adding a single point for each pole of $W$, one obtains finally a complete hyperK\"ahler $4$-manifold admitting an effective $S^1$ action, with the added points being the fixed point set of the action.
As shown by Bielawski \cite{Bielawski}, the construction described above yields an exhaustive list of all possible simply connected hyperK\"ahler 4-manifolds admitting an effective circle action. A key point in the analysis, described in more detail below, is to determine the global structure of the moment map for an arbitrary hyperK\"ahler manifold with effective $S^1$ action. It is shown that the assumption of completeness implies that the image of the moment map, endowed with the \emph{background} horizontal metric, by which we mean the Euclidean metric on $\mathbb R^3_{\pmb\mu}$, is also \emph{complete}. This implies that the image must be all of $\mathbb R^3$, thus the harmonic function with poles $W$ is defined globally, and results from elliptic theory imply that it is a linear combination of a constant and Green's functions, as expected.
In the more general setting of generalized K\"ahler structures, the symplectic triple together with $S^1$ action still allows us to locally define a moment map $\pmb \mu$ to $\mathbb R^3_{\pmb \mu}$. We use this moment map to derive an explicit local description of the generalized K\"ahler structure. As above we let $W^{-1}$ denote the square length of the canonical vector field generating the $S^1$ action. Then one can express the GK metric locally as
\begin{align*}
g = W h + W^{-1} \eta^2,
\end{align*}
where $W$ satisfies an explicit linear second order elliptic PDE, and $\eta$ is a principal $S^1$ connection whose curvature is determined by $W$. The background horizontal metric $h$ now depends on an extra scalar function $p$, which is known as the \emph{angle function} associated to the GK structure. Specifically we have the following (cf. Theorem~\ref{t:nondegenerate_gk_description} below):
\begin{thm}[Generalized-K\"ahler Gibbons-Hawking ansatz] \label{t:mainGH}
Fix a smooth 3-dimensional manifold $N$ and consider
\begin{enumerate}
\item an open map $\iota\colon N\to \mathbb{R}^3_{\pmb\mu}$,
\item smooth functions
\[
p\colon N\to (-1,1),\quad W\colon N\to (0,+\infty)
\]
solving the equation
\begin{equation*}
W_{11}+W_{22}+W_{33}+2(pW)_{23}=0,
\end{equation*}
such that the closed differential form $\beta\in\Lambda^2(N,\mathbb{R})$
\[
\beta=(W_3+(pW)_2)d\mu_1\wedge d\mu_2-(W_2+(pW)_3)d\mu_1\wedge d\mu_3+W_1d\mu_2\wedge d\mu_3
\]
represents a class in $H^2(N,2\pi\mathbb{Z})$,
\item a connection form $\eta$ with curvature $\beta$ in the principal $S^1$-bundle $\pi\colon M\to N$ determined by $[\beta]$.
\end{enumerate}
Then the total space of the principal $S^1$-bundle $M$
admits a nondegenerate GK structure
\[
(M,g,I,J)
\]
with
\[
g=Wh+W^{-1}\eta^2,\quad h=(1-p^2)d\mu_1^2+d\mu_2^2+d\mu_3^2-2p\,d\mu_2d\mu_3,
\]
and $I$, $J$ the unique almost complex structures such that the complex-valued 2-forms
\[
\begin{split}
\Omega_{I}:=&
(-d\mu_1+\sqrt{-1}d\mu_2)\wedge(\eta+\sqrt{-1}W(d\mu_3-pd\mu_2)),\\
\Omega_{J}:=&
(-d\mu_1+\sqrt{-1}d\mu_3)\wedge(\eta+\sqrt{-1}W(-d\mu_2+pd\mu_3)),
\end{split}
\]
are holomorphic with respect to $I$ and $J$ respectively.
Conversely any nondegenerate GK manifold $(M,g,I,J)$ with a free isometric tri-Hamiltonian $S^1$ action arises via this construction.
\end{thm}
This is a local description of any nondegenerate GK structure with a tri-Hamiltonian $S^1$ action. If we further impose the soliton equations, then function $p$ is determined explicitly up to two real parameters $a_{\pm}$, yielding explicit choices of metric $h$ defined globally on $\mathbb R^3_{\pmb \mu}$. Crucially however, these metrics are not complete, with the completion points corresponding to the possible zeros of the background Poisson tensor. Furthermore, the $S^1$ action on the nondegeneracy locus will only be tri-Hamiltonian after lifting to a certain infinite covering space. Thus to determine the possible image spaces of the moment map for a complete GK soliton, we must determine the relevant deck transformation group $\Gamma$ and its action on $\mathbb R^3_{\pmb \mu}$, and understand the completions of the quotient spaces, i.e. $\bar{\mathbb R^3_{\pmb \mu} / \Gamma}$. We prove that there are three possibilities:
\begin{enumerate}
\item One has $a_- = 0$, $a_+ = \frac{2}{k_+}$, $k_+\in\mathbb{Z}$ and $\bar{\mathbb R^3_{\pmb \mu} / \Gamma} \cong N(a_+, 0)$ is an orbifold diffeomorphic to a global quotient of $\mathbb C \times \mathbb R$ by a linear $\mathbb Z_{|k_+|}$-action on $\mathbb C$. The metric $h$ extends as a smooth orbifold metric with cone angle $2\pi/ |k_+|$ along $\{0\} \times \mathbb R$.
\item One has $a_- = 0$, $a_+ = \frac{2}{k_+}$, and $\bar{\mathbb R^3_{\pmb \mu} / \Gamma} \cong N(a_+, 0) / \mathbb Z$, where $N(a_+, 0)$ is as in item (1) and the $\mathbb Z$ action is generated by a translation in the $\mathbb R$ factor and (possibly trivial) rotation in $\mathbb{C}_{|k_+|}$ factor.
\item One has $a_+=2/k_+$, $a_-=2/k_-$, $k_{\pm}\in\mathbb{Z}$ and $\bar{\mathbb R^3_{\pmb \mu} / \Gamma} \cong N(a_+, a_-)$ is an orbifold diffeomorphic to a product $S^2(k_+,k_-)\times \mathbb{R}$, where $S^2(k_+,k_-)$ is a \emph{spindle} 2-sphere with cone angles $2\pi/ |k_+|$ and $2\pi/|k_-|$.
\end{enumerate}
With this in hand we can now state the main existence theorem for GK solitons:
\begin{thm} \label{t:existence} The following hold:
\begin{enumerate}
\item Let $(k_+, l_+)$ be a pair of coprime integers, $0\leq l_+<|k_+|$ and denote $a_+ = \frac{2}{k_+}$. Fix a collection of points $\{z_1, \dots, z_n\}$ in the smooth locus of $N(a_+, 0)$. There exists a one-parameter family of complete rank one generalized K\"ahler-Ricci solitons $(M^4, g, I, J)$ admitting an isometric $S^1$ action, such that $M / S^1 \cong N(a_+, 0)$, and $M^{S^1} / S^1 = \{z_1, \dots, z_n\}$. The manifold $M \backslash M^{S^1}$ is a Seifert fibration over $N(a_+, 0) \backslash \{z_1, \dots, z_n\}$, and the lift of the orbifold locus in this domain consists of points of type $(k_+, l_+)$.
\item Let $(k_+,l_+)$ be a pair of coprime integers, $0\leq l_+<|k_+|$ and denote $a_+ = \frac{2}{k_+}$. There exists a two-parameter family of complete rank one generalized K\"ahler-Ricci solitons $(M^4, g, I, J)$ admitting an effective isometric $S^1$ action, such that $M / S^1 \cong N(a_+, 0) / \mathbb Z$. The manifold $M$ is a Seifert fibration over $N(a_+, 0) / \mathbb Z$, and the lift of the orbifold locus in this domain consists of points of type $(k_+, l_+)$.
\item Let $(k_+, l_+), (k_-, l_-)$ be two pairs of coprime integers, $0\leq l_\pm<|k_\pm|$ and denote $a_\pm = \frac{2}{k_\pm}$. Fix a collection of points $\{z_1, \dots, z_n\}$ in the smooth locus of $N(a_+, a_-)$.
There exists a two-parameter family of complete rank one generalized K\"ahler-Ricci solitons $(M^4, g, I, J)$ admitting an effective isometric $S^1$ action, such that $M / S^1 \cong N(a_+, a_-)$, and $M^{S^1} / S^1 = \{z_1, \dots, z_n\}$. The manifold $M \backslash M^{S^1}$ is a Seifert fibration over $N(a_+, a_-) \backslash \{z_1, \dots, z_n\}$, and the lift of the two orbifold loci in this domain consists of points of type $(k_{\pm}, l_{\pm})$, respectively.
\end{enumerate}
\end{thm}
The situation is summarized in Figures \ref{f:fig1} and \ref{f:fig2}. The quantity $\mu_1$ is a certain coordinate in $\mathbb R^3_{\pmb \mu}$ whose periods generate a primitive subgroup $\mathbb Z \subset \Gamma$, so that $\mu_1$ descends to an $S^1$-valued coordinate on $\mathbb{R}^3_{\pmb\mu}/\mathbb{Z}$.
\begin{figure}[ht]
\begin{tikzpicture}
\draw (-2.75,0) ellipse (0.18 and 0.6);
\draw [white,fill=white] (-2.75,1) rectangle (-2.5,-1);
\draw [dotted] (-2.75,0) ellipse (0.18 and 0.6);
\draw[->] (-3.5,0) .. controls (-2,1.5) and (0.25,1.85) .. (0.5,1.9);
\draw[->] (-3.5,0) .. controls (-2,-1.5) and (0.25,-1.85) .. (0.5,-1.9);
\draw[<->] (-4.1,0) to (-3.1,0);
\node at (-3.7175,0.1881) {\tiny{$\mathbb R$}};
\draw [->](-2.5,0.3) .. controls (-2.46,0.1) and (-2.46,-0.1) .. (-2.5,-0.3);
\node at (-2.25,0) {\tiny{$\mu_1$}};
\node (v1) at (-3.8537,-1.1406) {\tiny{$\mathbf{T_+} = \{p = 1\}$}};
\draw[dotted,->] (v1) to (-3.7386,-0.1001);
\draw [<->](-0.6084,1.0546) to (-0.9459,2.1143);
\node at (-1.0865,1.7506) {\tiny{$\mathbb R$}};
\node (v3) at (-3.8,1.3) {\tiny{$(D^2/\mathbb Z_{k_+})\times \mathbb{R}$}};
\draw[dotted,->] (v3) to (-3.2,0.4);
\node (v5) at (1.0,-0.8) {\tiny{$\{z_1, \dots, z_n\}$}};
\draw[fill=black] (-0.5,0.4) circle (0.03);
\draw[fill=black] (-1.1,-0.5) circle (0.03);
\draw[fill=black] (-2.2,0.5) circle (0.03);
\draw [dotted,->] (v5) to (-0.42,0.32);
\draw [dotted,->] (v5) to (-0.99,-0.52);
\draw [dotted,->] (v5) to (-2.1,0.5);
\end{tikzpicture}
\caption{Configuration space $N(a_+, 0)\simeq \mathbb{C}/{\mathbb{Z}_{k_+}}\times \mathbb{R}$}
\label{f:fig1}
\bigskip
\bigskip
\begin{tikzpicture}
\draw (0.4,0) ellipse (0.2 and 0.88);
\draw [white,fill=white] (0.4,1) rectangle (1,-1);
\draw [dotted] (0.4,0) ellipse (0.2 and 0.88);
\draw (-2.75,0) ellipse (0.18 and 0.6);
\draw [white,fill=white] (-2.75,1) rectangle (-2.5,-1);
\draw [dotted] (-2.75,0) ellipse (0.18 and 0.6);
\draw (-3.5,0) .. controls (-2,1.5) and (0.25,2) .. (1,0);
\draw (-3.5,0) .. controls (-2,-1.5) and (0.25,-2) .. (1,0);
\draw[<->] (-4.1,0) to (-3.1,0);
\node at (-3.7175,0.1881) {\tiny{$\mathbb R$}};
\draw [->](-2.5,0.3) .. controls (-2.46,0.1) and (-2.46,-0.1) .. (-2.5,-0.3);
\node at (-2.25,0) {\tiny{$\mu_1$}};
\draw[<->] (0.7,0) to (1.6,0);
\node at (1.2175,0.2092) {\tiny{$\mathbb R$}};
\draw [<-](0.1,0.3) .. controls (0.06,0.1) and (0.06,-0.1) .. (0.1,-0.3);
\node at (-0.1,0) {\tiny{$\mu_1$}};
\node (v1) at (-3.8537,-1.1406) {\tiny{$\mathbf{T_+} = \{p = 1\}$}};
\draw[dotted,->] (v1) to (-3.7386,-0.1001);
\node (v2) at (1.5216,-1.1054) {\tiny{$\mathbf{T}_- = \{p = -1\}$}};
\draw[dotted,->] (v2) to (1.2474,-0.0931);
\draw [<->](-0.8,0.8) to (-0.8,1.8);
\node at (-1.0251,1.5097) {\tiny{$\mathbb R$}};
\node (v3) at (-4.5,1.1) {\tiny{$(D^2/\mathbb Z_{k_+})\times \mathbb{R}$}};
\draw[dotted,->] (v3) to (-3.2,0.4);
\node (v4) at (2.3,1.3) {\tiny{$(D^2/\mathbb Z_{k_-})\times \mathbb{R}$}};
\draw[dotted,->] (v4) to (0.801,0.6521);
\node (v5) at (-2.5,2.0) {\tiny{$\{z_1,\dots,z_n\}$}};
\draw[fill=black] (-0.5,0.4) circle (0.03);
\draw[fill=black] (-1.1,-0.5) circle (0.03);
\draw[fill=black] (-2.2,0.5) circle (0.03);
\draw [dotted,->] (v5) to (-0.58,0.48);
\draw [dotted,->] (v5) to (-1.16,-0.42);
\draw [dotted,->] (v5) to (-2.22,0.58);
\end{tikzpicture}
\caption{Configuration space $N(a_+, a_-)\simeq S^2(k_+,k_-)\times \mathbb{R}$}
\label{f:fig2}
\end{figure}
\begin{rmk}
\begin{enumerate}
\item For the definition of points of type $(k, l)$ see Remark \ref{r:local_s1_quotient}. For the definition of rank of a soliton see Definition \ref{d:GKsoliton}.
\item One parameter in cases (1) and (2) corresponds to the weight on the constant function in the decomposition of $W$. This phenomenon also occurs in the original Gibbons-Hawking construction, where one may include an arbitrary nonnegative constant in the choice of $W$. Depending on whether this constant is zero or not, one obtains multi-Eguchi-Hanson or multi-Taub-NUT metrics.
\item The second parameter in case (2) corresponds to a choice of a connection $\eta$ on the Seifert bundle over $N(a_+,0)/\mathbb{Z}$ with a prescribed curvature form.
\item In case (3) there are two real parameters, given by weights on the constant function and an anomalous smooth solution, which further satisfy an integrality constraint (see~\eqref{f:W_quantization}).
\item The global topology of these examples is implicit in the construction, with the manifolds arising via completion of explicit Seifert bundles by adding finitely many points.
\end{enumerate}
\end{rmk}
Next, crucially, we show that our construction classifies all possible solitons in four dimensions with the \emph{smallest} possible isometry groups. Those with larger symmetry groups arise via constructions in toric geometry and will be treated elsewhere.
\begin{thm} \label{t:uniqueness} Let $(M^4, g, I, J)$ be a complete generalized K\"ahler-Ricci soliton with generically nondegenerate Poisson tensor, Ricci curvature bounded below and $\dim H^*(M,\mathbb{R})<\infty$. If $\dim \Isom(g) \leq 1$, then exactly one of the following holds.
\begin{enumerate}
\item $(M^4, g, I)$ is hyperK\"ahler.
\item $\Isom(g) = S^1$ and $(M^4, g, I, J)$ is isomorphic to one of the examples constructed in Theorem \ref{t:existence}.
\item $\Isom(g)=\mathbb{R}$ and $(M^4, g, I, J)$ is isomorphic to a $\mathbb{Z}$-cover of an example constructed in Theorem \ref{t:existence} with a trivial $S^1$-bundle structure.
\end{enumerate}
\end{thm}
\begin{rmk}
The hypothesis of a lower bound on Ricci curvature is needed for technical reasons described below, and it should be possible to remove this. The assumption on the finite topological type of $M$ is used to ensure that an $S^1$-action on $M$ necessarily has finitely many fixed points in the same way as in the work of Bielawski~\cite{Bielawski}. It should be noted that the existence statement of Theorem~\ref{t:existence} holds even for an infinite number of poles $\{z_\alpha\}\subset N(a_+,a_-)$ as long as $\{z_\alpha\}$ are sparse enough ensuring a uniform convergence of the sum of the corresponding Green's functions (see Proposition~\ref{p:elliptic_aneq0}) on compact sets. This observation is well-known in the hyperK\"ahler case, see e.g.,~\cite{an-kr-le-89}.
\end{rmk}
Let us give a brief description of the proofs of Theorems \ref{t:existence} and \ref{t:uniqueness}, expanding on the discussion above, which will also serve as an outline for the remainder of the paper. In \S \ref{s:background} we recall fundamental properties and examples of generalized K\"ahler structures, which are biHermitian triples $(g, I, J)$ satisfying certain integrability conditions. We recall the associated Poisson tensor $\sigma = \tfrac{1}{2} g^{-1} [I,J]$, and briefly indicate the geometric significance of the rank of this tensor. In the nondegeneracy locus we obtain a symplectic form $\Omega = \sigma^{-1}$, and recall that in dimension $4$ the entire GK structure is described in terms of $\Omega$, $I \Omega$ and $J \Omega$. We also recall some fundamental topological and geometric results on Seifert fibrations, specialized to the case of almost free $S^1$ actions on 4-manifolds.
In \S \ref{s:GKsymm} we study generalized K\"ahler 4-manifolds admitting a free $S^1$ action. As GK structures on 4-manifolds are described by a triple of symplectic forms, in \S \ref{ss:inv_gk_str} we are able to locally define a moment map generalizing that of the Gibbons-Hawking ansatz (Theorem~\ref{t:nondegenerate_gk_description}). In the image of the moment map, the horizontal geometry is determined by a single arbitrary function $p$, which determines the angle between the complex structures $I$ and $J$. The length of the circle fiber is determined by a function $W$ which solves a certain linear elliptic PDE depending on $p$. Turning to the case of effective $S^1$ actions in \S \ref{ss:NGKeff}, we first derive the local structure of the fibration near fixed points, which must be that of the standard Hopf fibration, then derive the local blowup rate of $W$ near these fixed points. Given this we prove removable singularities results in Propositions \ref{p:removable_singularity} and \ref{p:removable_singularity_GK} which say roughly that given the data of our ansatz on a punctured $3$-ball satisfying the necessary blowup rates, we can extend the total space of the fibration topologically by adding a point, with the generalized K\"ahler structure extending in $C^{1,1}$ sense across this point. Having established these basic structural results, we end \S \ref{s:GKsymm} by reviewing some known examples, including diagonal Hopf surfaces and a new description of LeBrun's generalized K\"ahler structures on parabolic Inoue surfaces \cite{LeBrun}.
So far the discussion has allowed for the angle function $p$ to be essentially arbitrary. In \S \ref{s:GKRS} we impose the generalized K\"ahler-Ricci soliton equation, and observe that it determines $p$. For K\"ahler-Ricci solitons, it is well-known that the associated soliton vector field $\nabla f$ is real holomorphic, with $J \nabla f$ a Killing field. Through a local analysis of the generalized K\"ahler-Ricci soliton equation, we show that aside from some exceptional and rigid cases, either the metric has a two-dimensional isometry group or there is a vector field $X$ preserving all the generalized K\"ahler structure, leading to the definition of a \emph{rank one} soliton (cf.\,Definition \ref{d:GKsoliton}). We next derive a kind of scalar reduction for the soliton equation, showing that the angle function $p$ must be determined by $\Omega$-Hamiltonian potential functions for the vector fields $I X$ and $JX$, and thus in the rank one case, in the image of the moment map we obtain an explicit formula for $p$ depending on two real parameters $a_{\pm}$.
The description of GK structures and solitons in sections \S \ref{s:GKsymm}--\S\ref{s:GKRS} is purely local and \textit{flexible}. To turn it into a reasonable classification problem, we need to introduce the key assumption of $(M,g)$ being \textit{complete}.
In \S \ref{s:completion} we provide a description of the topological and geometric structure of the possible completions of the quotient space $M/S^1$ for rank one solitons under natural geometric hypotheses on $M$. To begin we analyze the blowup behavior of the meromorphic $(2,0)$-form $\Omega_I$, showing in particular that it has a pole of order $1$ at the degeneracy locus. Using this we begin our analysis of the global properties of the moment map. We establish a key completeness property for the induced horizontal geometry in \S \ref{ss:compquot}. In particular, as derived earlier, the horizontal metric is naturally of the form $W h$, where $W^{-1}$ is the squared length of the circle fiber. Completeness of $M$ immediately implies completeness of $W h$, however, a priori we do not know the global structure of $h$. As discussed above, establishing the completeness of the domain with respect to the metric $h$ plays a key role in determining the global structure of the so-far only locally defined moment map. For hyperK\"ahler metrics with symmetry, this property was established by Bielawski \cite{Bielawski}, and plays a key role in proving their classification. Bielawski's proof exploits structural results of Schoen-Yau \cite{SchoenYau} on conformal immersions of manifolds of nonnegative scalar curvature, relying in a delicate way on the fact that $h$ is flat, which holds in the hyperK\"ahler setting. In our setup the metric $h$ is fairly explicit, but not flat, and thus we must rely on more robust elliptic theory. We establish the relevant completeness property in Proposition \ref{p:s1_bundle_complete}, relying on the gradient estimate of Cheng-Yau \cite{ChengYau} and certain ideas from the previously mentioned work of Schoen-Yau \cite{SchoenYau}. These arguments are where the technical condition of a lower Ricci curvature bound and finite topological type enter.
With this completeness property at hand we can analyze the global properties of the moment map. In particular, a priori the moment map is defined only after we remove the degeneracy loci of $\sigma$, and take an appropriate $\mathbb Z^k$ cover to render the $S^1$ action tri-Hamiltonian. Using the structure of $\Omega_I$ and our technical result on the completeness of the horizontal geometry, in Proposition \ref{p:deck_transform_mu} we analyze the $\mathbb Z^k$ covering action, and show that the moment map is naturally defined on the $\mathbb Z^k$ quotient, with the image space an explicit quotient of $\mathbb R^3_{\pmb \mu}$. Next in \S \ref{ss:metcomp} we explicitly describe the completions of these $\mathbb R^3_{\pmb \mu}$ quotients. Specifically we observe that by adding the degeneracy loci of $\sigma$, the metric extends smoothly in the orbifold sense, with cone angles determined by the parameters $a_{\pm}$. Given this, we finish in \S \ref{ss:globalmoment} by proving that the moment map is globally defined on $M$, with the images given by one of the spaces $N(a_+, 0), N(a_+, 0) / \mathbb Z$, or $N(a_+, a_-)$ described above.
Having now determined the possible geometry and topology of the images of the moment map, it remains to describe the possible choices of $W$. In \S \ref{s:construction} we first observe that solutions to the linear elliptic PDE for $W$ are in one-to-one correspondence with solutions to the Laplace equation for a metric conformal to the given background $h$. Using this interpretation together with a description of the spaces $N$ as global orbifold quotients, we show that the relevant solutions must be a linear combination of Green's functions, the constant function, and one exceptional solution in the case $a_- \neq 0$. Having established the characterization of entire solutions $W$ to the relevant PDE, we finish the proof of Theorem \ref{t:existence}. At this point, all that remains is to use the soliton equation to (1) show that the GK structure, which is only known to be $C^{1,1}$ at the poles of $W$, in fact extends smoothly, and (2) prove that the whole structure extends smoothly over the degeneracy loci. Finally, relying primarily on the prior classification of the moment map images and possible functions $W$, we finish the proof of Theorem \ref{t:uniqueness}.
\vskip 0.1in
\textbf{Acknowledgements:} The authors are grateful to anonymous referees for useful remarks and suggestions.
We would also like to thank Vestislav Apostolov and Connor Mooney for helpful discussions. The first author acknowledges support from the NSF via DMS-1454854.
\section{Background}\label{s:background}
The aim of this section is to provide the necessary background for the tools and notions used in this paper. In \S \ref{ss:GK_structures} we give a very brief overview of the biHermitian interpretation of generalized K\"ahler geometry with an emphasis on the $4$-dimensional case. We discuss the relation to Poisson geometry, and give an explicit pointwise description of a \emph{nondegenerate} generalized K\"ahler structure in terms of a preferred vector and \emph{angle function} $p$. We refer the reader to~\cite{ASNDGKCY} and references therein for more details.
In \S \ref{ss:s1_actions} we discuss the topology of $S^1$ actions on 4-dimensional manifolds, and review the notions of Seifert fibrations and orbifolds.
\subsection{Generalized K\"ahler structures}\label{ss:GK_structures}
Let $M$ be a smooth $2n$-dimensional manifold equipped with a Riemannian metric $g$ and two integrable almost complex structures $I$ and $J$ compatible with $g$. Denote by
\[
\omega_I:=g(I\cdot,\cdot)\quad \mathrm{and}\quad \omega_J:=g(J\cdot,\cdot)
\]
the associated fundamental 2-forms, and let
\[
d^c_I\colon \Lambda^k(M; \mathbb{R})\to \Lambda^{k+1}(M; \mathbb{R}),\quad
d^c_J\colon \Lambda^k(M; \mathbb{R})\to \Lambda^{k+1}(M; \mathbb{R})
\]
be the real differential operators, also known as \textit{twisted differentials}
\[
d^c_I=\sqrt{-1}(\bar\partial_{I}-\partial_I),\quad d^c_J=\sqrt{-1}(\bar\partial_{J}-\partial_J).
\]
\begin{defn} \label{def:gk structure}
The data $(M,g,I,J)$ is called a \emph{generalized K\"ahler} (GK) structure on a manifold $M$, if
\[
d_I^c\omega_I=-d_J^c\omega_J
\]
and the form $H:=- d_I^c\omega_I$ is closed:
\[
dH=0.
\]
In particular, the Hermitian manifolds $(M,g,I)$ and $(M,g,J)$ are \emph{pluriclosed} (sometimes also called SKT~--- \emph{strong K\"ahler with torsion}):
\[
dd^c_I\omega_I=dd^c_J\omega_J=0.
\]
\end{defn}
For a four-dimensional GK structure we furthermore define the two associated Lee forms $\theta_I$ and $\theta_J$ by the equations
\begin{align*}
d \omega_I = \theta_I \wedge \omega_I, \qquad d \omega_J = \theta_J \wedge \omega_J.
\end{align*}
Equivalently, in the four dimensional case, $\theta_I=-(*H)$. Thus in this case, when $I$ and $J$ induce the same orientation the generalized K\"ahler condition implies that $\theta_I=-\theta_J$.
In general there are two natural connections associated to a GK structure,
\begin{align*}
\nabla^{I} = D + \tfrac{1}{2} g^{-1} H, \qquad \nabla^{J} = D - \tfrac{1}{2} g^{-1} H
\end{align*}
where $D$ denotes the Levi-Civita connection. These are the Bismut connections associated to the Hermitian structures $(g, I)$ and $(g, J)$ \cite{Bismut}. Call the associated curvature tensors $R_I, R_J$ and then we obtain the associated Bismut-Ricci tensors
\begin{align*}
\rho_I = \tfrac{1}{2} \tr_{\omega_I} R_I, \qquad \rho_J = \tfrac{1}{2} \tr_{\omega_J} R_J.
\end{align*}
In \cite{PCF, GKRF} the first named author and Tian introduced a parabolic flow of generalized K\"ahler structures (generalized K\"ahler-Ricci flow) which can be expressed as
\begin{align*}
\dt \omega_I =&\ - \rho_I^{1,1}, \qquad \dt J = L_{\tfrac{1}{2} \left(\theta_J^{\#} - \theta_I^{\#}\right) } J.
\end{align*}
The equation for $\omega_I$ is the pluriclosed flow, which is defined for pluriclosed structures $(g, I)$. With generalized K\"ahler initial data $(g, I, J)$ and the evolution equation for $J$ above, the flow preserves the GK conditions. It turns out that pluriclosed flow is a gradient flow for a Perelman-type energy functional \cite{PCFReg}, and this energy is fixed on steady solitons:
\begin{defn}\label{d:soliton} A pluriclosed structure $(M^{2n}, g, I)$ is a \emph{steady soliton} if there exists a function $f\in C^\infty(M,\mathbb{R})$ such that
\begin{align*}
\Rc - \tfrac{1}{4} H^2 + \nabla^2 f =&\ 0,\\
d^* H + i_{\nabla f} H =&\ 0.
\end{align*}
\end{defn}
It follows from \cite{PCFReg} that these equations are satisfied by a solution to pluriclosed flow evolving purely by diffeomorphisms.
\subsubsection{Poisson geometry} \label{ss:Poisson}
Hitchin/Pontecorvo observed~\cite{HitchinPoisson, PontecorvoCS} that there is a real Poisson tensor associated to any generalized K\"ahler structure:
\begin{equation*}
\sigma = \tfrac{1}{2}g^{-1}[I, J].
\end{equation*}
\begin{rmk}
In order to match the common conventions in hyperK\"ahler geometry, our definition of the Poisson tensor associated to a GK structure differs by a factor of $2$ from the one in~\cite{ASNDGKCY}. This discrepancy will affect some of the general identities from~\cite{ASNDGKCY} which we will use in the present paper.
\end{rmk}
We call the generalized K\"ahler structure \emph{commuting} if $\sigma \equiv 0$, and \emph{nondegenerate} if $\sigma$ defines a nondegenerate pairing. In dimension $4$ these are the only two possibilities at a given point. In general the rank of $\sigma$ can vary at different points. Fix $(M,g,I,J)$ a generalized K\"ahler manifold with nondegenerate Poisson tensor, and define symplectic form
\begin{equation*}
\Omega = \sigma^{-1}=2[I,J]^{-1}g,
\end{equation*}
It is elementary to show that $\Omega$ is of holomorphic type $(2,0) + (0,2)$ with respect to both $I$ and $J$, and thus we can define the (respectively $I$- and $J$-) $(2,0)$-type holomorphic symplectic forms
\begin{align*}
\Omega_I:=\Omega-\sqrt{-1}I\Omega,\quad \Omega_J:=\Omega-\sqrt{-1}J\Omega,
\end{align*}
where $(I\Omega)(X,Y):=\Omega(IX,Y)$.
From here forward we will be dealing with 4-dimensional GK manifolds with generically nonzero Poisson tensor $\sigma$. In this case $\sigma$ is invertible on the complement of the locus
\[
\mathbf{T}=\{x\in M\ |\ I_x=\pm J_x\}.
\]
The set $\mathbf{T}$ is analytic with respect to both $I$ and $J$ and has complex dimension one. By definition the GK structure $(M\backslash\mathbf{T}, g, I, J)$ on the complement of $\mathbf{T}$ is nondegenerate. Furthermore, in this nondegeneracy locus, one can recover $I$ and $J$ from a pair of compatible complex-valued symplectic forms. Specifically we have the following:
\begin{prop}[{\cite[Thm.\,2]{AGG} and \cite[Lemma\,2.14]{ASNDGKCY}}]\label{p:holo_symplectic-acs}
Given a 4-dimensional manifold $M$ and two closed complex-valued forms $\Omega_I,\Omega_J\in\Lambda^2(M,\mathbb{C})$ satisfying
\[
\begin{split}
\Omega_I^2=\Omega_J^2=0,\\
\Re(\Omega_I)=\Re(\Omega_J),
\end{split}
\]
there are two unique integrable complex structures $I$ and $J$ such that $\Omega_I$ and $\Omega_J$ are holomorphic $(2,0)$-forms with respect to $I$ and $J$ respectively. Furthermore, if the symmetric pairing $g$ defined by
\[
g(X,X):=\Omega(JX,IX)
\]
is positive definite, then $(M,g,I,J)$ is a nondegenerate GK structure.
\end{prop}
\begin{ex}[Generalized K\"ahler structures on the standard Hopf surface]\label{ex:Hopf1}
Consider the standard Hopf surface
\begin{align*}
M^4 = \left(\mathbb C^2 - \{0\} \right) / \left< (z_1,z_2) \to (2 z_1, 2 z_2) \right> \cong S^3 \times S^1.
\end{align*}
Let $I$ denote the natural complex structure inherited from $\mathbb C^2$, and consider the \emph{round} metric
\begin{align*}
g=\Re\left(\frac{dz_1\otimes d\bar{z_1}+dz_2\otimes d\bar{z_2}}{|z_1|^2+|z_2|^2}\right).
\end{align*}
This is the natural cylindrical metric on $S^3 \times \mathbb R$, which is Hermitian, and descends to the quotient to give a direct sum of the round metric on $S^3$ with the standard metric on $S^1$. This is a pluriclosed structure, and we obtain a second complex structure $J$ by pulling it back via an orientation-preserving involution: $J:=j^*I$, where
\begin{equation*}
\begin{split}
j&\colon \mathbb{C}^2\backslash\{0\}\to \mathbb{C}^2\backslash\{0\}\\
j&(z_1,z_2)=\left(\frac{\bar{z_2}}{|z_1|^2+|z_2|^2}, \frac{z_1}{|z_1|^2+|z_2|^2}\right).
\end{split}
\end{equation*}
The corresponding holomorphic symplectic forms are
\begin{align*}
\Omega_I =&\
\frac{dz_1}{z_1} \wedge \frac{dz_2}{z_2},\\
\Omega_J =&\ \left(d\log(\brs{z_1}^2+\brs{z_2}^2)-\frac{d \bar z_2}{\bar z_2}\right)\wedge\left(d\log(\brs{z_1}^2+\brs{z_2}^2)-\frac{d z_1}{z_1}\right)
\end{align*}
and it is straightforward to check that $\Re(\Omega_I)=\Re(\Omega_J)$ and $\Omega(JX,IX)$ is given by metric $g$. Thus by Proposition~\ref{p:holo_symplectic-acs}, $(M,g,I,J)$ is a GK structure. In particular, Poisson tensor is given by a real part of a holomorphic Poisson tensor:
\begin{align*}
\sigma = -\Re\left(z_1 \frac{\partial}{\partial z_1} \wedge z_2\frac{\partial}{\partial z_2}\right).
\end{align*}
The degeneracy loci are the elliptic curves $\{z_1 = 0\}$, $\{z_2= 0\}$ where $I = - J$ and $I = J$, respectively. A schematic picture of the fundamental domain in $\mathbb C^2\backslash\{0\}$ is given in Figure \ref{fig:Hopf}.
\begin{figure}
\begin{tikzpicture} [scale=0.7]
\draw (0,-4) edge (0,4);
\draw (-4,0) edge (4,0);
\draw[line width=2pt] (3,0) edge (1.5,0);
\draw[line width=2pt] (0,3) edge (0,1.5);
\draw (0,0) circle (1.5);
\draw (0,0) circle (3);
\node at (0,4.5) {\small{$\{z_1 = 0\}$}};
\node at (5,0.4) {\small{$\{z_2 = 0\}$}};
\node (v1) at (-3.5,3.5) {\small{$\{I = -J\} \cong T^2$}};
\draw [dotted,->] (v1) edge (-0.1,2.20);
\node (v2) at (4.4,2.5) {\small{$\{I = J\} \cong T^2$}};
\draw [dotted,->] (v2) edge (2.25,0.1);
\draw[fill=white] (0,0) circle (0.1);
\node at (-1.5,-1,5) {\small{$\mathbb{C}^2\backslash\{0\}$}};
\end{tikzpicture}
\caption{Fundamental domain of standard Hopf surface with GK structure}
\label{fig:Hopf}
\end{figure}
The above GK structure $(M,g,I,J)$ on a standard Hopf surface has a large automorphism group: it admits an effective action of a 3-dimensional torus $(S^1)^3$, where the first two factors act via coordinate-wise complex multiplication, and the last factor $\mathbb{R}/\mathbb{Z}\simeq S^1$ acts by dilations: $t\cdot(z_1,z_2)=(e^tz_1,e^tz_2)$.
\end{ex}
\subsubsection{GK structures with a preferred vector}
Next we show that, given a preferred vector $Z\in T_xM$, it is possible to give an explicit description of a 4-dimensional GK structure in terms of one function. The key linear algebraic fact special to four dimensions is the pointwise identity~\cite[Lemma 3.2]{StreetsNDGKS}
\begin{equation*}
IJ+JI=-2p\Id,
\end{equation*}
where $p\colon M\to(-1,1)$, $p=-\frac{1}{4}\tr(IJ)$ is the \emph{angle function}. We also have a third natural endomorphism, which we can rewrite using $p$:
\begin{equation}\label{f:K}
K:=\tfrac{1}{2}[I,J]=IJ+p\Id=-JI-p\Id.
\end{equation}
Note that in the hyperK\"ahler setting, the function $p$ is identically zero and $K$ is another integrable complex structure. More generally, if $p\colon M\to(-1,1)$ is constant, then $I$ is a part of a hyperK\"ahler triple $(I,J',K')$ with $J'=\sqrt{1-p^2}I+pJ$, $K'=[I,J']$.
Thus, in the presence of a preferred vector $Z\in T_xM$, we can use the operators $I,J$ and $K$ to obtain a basis of $T_xM$, thus yielding a complete description of all structures in terms of the function~$p$. Specifically we have the following proposition:
\begin{prop}[Linear description of a GK structure on $M^4$] \label{p:matrices}
For $(M^4,g,I,J)$ as above, given a unit vector $Z\in TM$, the vectors $Z,IZ,JZ,KZ$ form a basis of $TM$ with $g,I,J,K$ and $\Omega$ given by the matrices
\begin{equation}\label{f:g_matrix}
g=\left[\begin{matrix}1 & 0 & 0 & 0\\0 & 1 & p & 0\\0 & p & 1 & 0\\0 & 0 & 0 & 1 - p^{2}\end{matrix}\right],
\end{equation}
\begin{equation}\label{f:I_matrix}
I=\left[\begin{matrix}0 & -1 & - p & 0\\1 & 0 & 0 & p\\0 & 0 & 0 & -1\\0 & 0 & 1 & 0\end{matrix}\right],
\end{equation}
\begin{equation}\label{f:J_matrix}
J=\left[\begin{matrix}0 & - p & -1 & 0\\0 & 0 & 0 & 1\\1 & 0 & 0 & - p\\0 & -1 & 0 & 0\end{matrix}\right],
\end{equation}
\begin{equation}\label{f:K_matrix}
K=\left[\begin{matrix}0 & 0 & 0 & p^{2} - 1\\0 & - p & -1 & 0\\0 & 1 & p & 0\\1 & 0 & 0 & 0\end{matrix}\right],
\end{equation}
\begin{equation}\label{f:Omega_matrix}
\Omega=\left[\begin{matrix}0 & 0 & 0 & -1\\0 & 0 & -1 & 0\\0 & 1 & 0 & 0\\1 & 0 & 0 & 0\end{matrix}\right].
\end{equation}
\end{prop}
\begin{proof}
The formulas for matrices of $g,I,J,K$ are straightforward from the compatibility of the metric with $I$ and $J$ and the identity~\eqref{f:K} defining $K$. Finally, for the matrix of $\Omega$ we have
\[
\Omega=(K^{-1})^t\cdot g
\]
which is given by the last matrix.
\end{proof}
\subsection{Manifolds with \texorpdfstring{$S^1$}{} action}\label{ss:s1_actions}
In this subsection we review basic facts about circle actions on manifolds, and discuss the related concepts of \emph{Seifert fibrations} and \emph{orbifolds}. We will usually think of the circle $S^1$ as the group of complex numbers of the unit norm.
\begin{defn}
Let $M$ be an $n$-dimensional manifold with an effective circle action
\[
S^1\times M\to M.
\]
Throughout this paper we keep the following notation
\begin{enumerate}
\item $\pi\colon M\to N$ is the projection onto the orbit space;
\item $M^{S^1}\subset M$ is the fixed point set;
\item $N_0:=(M\backslash M^{S^1})/S^1$ is the orbit space of the complement of the fixed point set.
\end{enumerate}
The action is called \emph{free} if the stabilizer $G_x\subset S^1$ of any point $x\in M$ is trivial. In this case $M\to N$ is the total space of a principal $S^1$ bundle.
The action $S^1\times M\to M$ is called \emph{almost free} if $M^{S^1}\subset M$ is empty, or, equivalently, the stabilizer of any point in $M$ is finite. In this case we call $M\to N$ a \emph{Seifert fibration}. The action is called \emph{effective} if no nontrivial element of $S^1$ acts as the identity on $M$.
\end{defn}
Given any point $x\in M$ there is a representation of its stabilizer $G_x\subset S^1$ in the normal space to the orbit $\nu_x:=T_xM/T_x(S^1x)$:
\[
\rho_x\colon G_x\to GL(\nu_x)
\]
If $M$ is oriented then the representation $\rho_x$ preserves the induced orientation on $\nu_x$. We have the following equivariant tubular neighbourhood theorem:
\begin{thm}[{Slice theorem \cite[\S B.2]{gu-gi-ka}}] \label{t:slice}
Let $M$ be a manifold with an effective $S^1$ action, $x\in M$. Then the representation $\rho_x$ in $\nu_x$ is faithful and there exists an $S^1$-invariant neighbourhood $U_x$ of $x$ and an equivariant diffeomorphism
\[
U_x\simeq S^1\times_{G_x} \nu_x:=(S^1\times \nu_x)/G_x.
\]
We will call $U_x$ a \emph{canonical neighbourhood} of $x\in M$.
\end{thm}
From now on we restrict our attention to a special case when $M$ is oriented, $\dim M=4$ and the action of $S^1$ has only isolated fixed points $M^{S^1}\subset M$. Note that every isotropy group $G_x$ is abelian, therefore its linear orientation-preserving representation is a sum of irreducible representations in $\mathbb{C}$ and a trivial representation. Considering separately the situations when $x\in M$ is fixed and when $x$ has a finite isotropy subgroup, we have the following.
\begin{cor}\label{c:slice}
Let $M$ be a 4-dimensional manifold with an effective $S^1$ action with isolated fixed points. Given $x\in M$, exactly one of the following holds:
\begin{enumerate}
\item $\nu_x\simeq \mathbb{C}\times\mathbb{R}$, $G_x=\mathbb{Z}_k\subset S^1$ and $\rho_x$ is given by
\[
\rho_x\colon \mathbb{Z}_k\to GL(\mathbb{C})\simeq \mathbb{C}^*,\quad
\rho_x\colon \exp\frac{2\pi\sqrt{-1}}{k}\mapsto \exp\frac{2\pi\sqrt{-1}l}{k},\quad \mathrm{gcd}(k,l)=1;
\]
\item $\nu_x\simeq \mathbb{C}^2$, $G_x=S^1$ and $\rho_x$ is given by
\[
\rho_x\colon S^1\to GL(\mathbb{C}^{2}),\quad \rho_x\colon z\mapsto \mathrm{diag}(z^{w_1},z^{w_{2}})
\]
where $\mathbf{w}=(w_1,w_2)\in \mathbb{Z}^{2}$ is the weight vector such that $w_1,w_2\neq 0$ and $\mathrm{gcd}(w_1,w_2)=1$.
\end{enumerate}
\end{cor}
\begin{rmk}\label{r:local_s1_quotient}
If $G_x\simeq \mathbb{Z}_k$ and $\rho_x$ is given by $l\in\mathbb{Z}_k^{\times}$, we will say that the point $x\in M$ is of type $(k,l)$. If we change the orientation of $\mathbb{C}$ in the identification $\nu_x\simeq \mathbb{C}\times \mathbb{R}$, then $l$ is substituted by $k-l$. In particular, if $M$ admits an $S^1$-invariant complex structure, $l$ is well-defined.
If $k=1$, then the stabilizer $G_x$ is trivial and we will say that point $x$ is \emph{regular}. Clearly the type is constant along the $S^1$ orbit, so we can also talk about the type of a point $[x]\in M/S^1$.
\end{rmk}
In general, when the action of $S^1$ on $M$ is not free, the orbit space $N=M/S^1$ is not a manifold. However, in our case when $\dim M=4$ and there are only isolated fixed points,
thanks to the slice theorem we have a good understanding of the geometry and local structure of $N$:
\begin{enumerate}
\item The set of regular points $M^{\mathrm{reg}}\subset M$ is open and dense, and the quotient $M^{\mathrm{reg}}/S^1$ is a smooth manifold, such that $M^{\mathrm{reg}}\subset M$ is the total space of a principal $S^1$-bundle over $N^{\mathrm{reg}}:=M^{\mathrm{reg}}/S^1$;
\item If $x\in M$ is a point of type $(k,l)$, then a neighbourhood of $[x]\in N$ is naturally isomorphic to $\mathbb{C}/\mathbb{Z}_k\times\mathbb{R}$. The latter can be identified with $\mathbb{C}\times\mathbb{R}$ via the map $\mathbb{C}\to\mathbb{C}$, $z\mapsto z^k$, providing a smooth structure near $[x]\in N$;
\item Given a fixed point $x\in M$ with weights $\mathbf{w}=(w_1,w_2)$ there are two disjoint subsets of points of types $(w_1,w_2)$ and $(w_2,w_1)$ in a punctured canonical neighbourhood $U_x$ of $x$. These subsets are given by complex coordinate planes in $U_x\simeq \mathbb{C}^2$
\item If a fixed point $x\in M$ has weights $\mathbf w=(\pm 1,\pm 1)$, then there is a neighbourhood $U_x$ such that the projection $U_x\to U_x/S^1$ is equivalent to the Hopf fibration projection $\mathbb{C}^2\to \mathbb{R}^3$. In particular there is a natural smooth structure in a neighbourhood of $[x]\in N$;
\end{enumerate}
Consider the complement of the fixed point set $M_0:=M\backslash M^{S^1}$. The action $S^1\times M_0\to M_0$ is almost free, and $M_0$ is a Seifert fibration. In this case the orbit space $N_0$ comes equipped with an extra structure:
\begin{enumerate}
\item The orbit space $N_0$ inherits a natural \emph{cyclic orbifold} structure. That is, given a point $x\in N_0$, its open neighbourhood $V_x$ can be realized as $V_x\simeq \nu_x/G_x\simeq \mathbb{C}/\mathbb{Z}_k\times \mathbb{R}$ and the triples $(\nu_x, G_x, V_x)$ satisfy the usual gluing compatibility properties~\cite[\S 1.3]{ad-le-ru-07}.
We will call a triple $(\nu_x,G_x,V_x)$ together with an identification $\nu_x/G_x\simeq V_x$ an \emph{orbifold chart} at a point $x\in N_0$. Note that in general an orbifold chart $\nu_x$ is not naturally embedded into the orbifold $N_0$.
\item Additionally, since in our case the orbifold structure on $N_0$ is very special, the space $N_0$ admits a canonical smoothing: every quotient $\mathbb{C}/\mathbb{Z}_k$ can be identified with $\mathbb{C}$. With this identification, the set of points of any given type $(k,l)$ forms a smooth submanifold $Z\subset N_0$ of codimension~2.
\end{enumerate}
If the action $S^1\times M\to M$ is free, then we can recover $M$ from the orbit space $N=M/S^1$ and a \emph{characteristic class}
\[
e(M\to N)\in H^2(N,\mathbb{Z})
\]
representing the Euler class of the bundle $M\to N$. Specifically, we have the following standard topological characterization of principal bundles:
\begin{prop}\label{p:s1_bundle_top}
Given a topological space $N$ and a class $e\in H^2(N,\mathbb{Z})$ there exists a unique (up to equivalence) principal $S^1$ bundle $M\to N$ such $e(M\to N)=e$.
\end{prop}
While the above proposition holds if $N$ is merely a topological space, if $N$ is a manifold there is a differential-geometric refinement.
\begin{prop}\label{p:s1_bundle_dg}
Given any manifold $N$ and a closed form $\beta\in \Lambda^2(N,\mathbb{R})$ such that $\beta/2\pi$ represents a class $e\in H^2(M,\mathbb{Z})$, let $\pi\colon M\to N$ be the principal $S^1$ bundle corresponding to the class $e$. Then there exists a principal connection $\eta$ in $M$ such that its curvature form is $\beta$. If $\eta'$ is another connection with the same curvature, then
\[
\eta-\eta'=\pi^* \alpha
\]
where $\alpha\in \Lambda^1(N,\mathbb{R})$ is a closed one-form.
\end{prop}
Any function $f\in C^\infty(N,S^1)\simeq C^\infty(N,\mathbb{R})\times H^1(N,\mathbb{Z})$ defines a gauge equivalence on $M$ such that $\eta \sim \eta+\pi^*df$. Hence, modulo gauge equivalence, the connection $\eta$ is defined up to an element in $H^1(N,\mathbb{R})/H^1(N,\mathbb{Z})$.
Remarkably, the above classical results can be generalized to the case of almost free actions and Seifert fibrations. To formulate the generalization of Proposition~\ref{p:s1_bundle_top} we need to extend the definition of the Euler class to Seifert fibrations.
\begin{defn}[Euler class]
Let $M\to N$ be a Seifert fibration with points of types $\{(k_i,l_i)\}$. Define $k:=\mathrm{lcm}(\{k_i\})$. Then there is a free action of $S^1/\mathbb{Z}_k$ on the quotient space $M/\mathbb{Z}_k$ with the orbit space $N$. Then we can define the usual Euler class $e(M/\mathbb{Z}_k\to N)\in H^2(N,\mathbb{Z})$ of the principal bundle $M/\mathbb{Z}_k\to N$. The \emph{Euler class} of the fibration $M\to N$ is
\[
e(M\to N):=\frac{1}{k}e(M/\mathbb{Z}_k\to N)\in H^2(N,\mathbb Q).
\]
\end{defn}
If $M\to N$ is a Seifert fibration, then points of its orbit space $N$ can be marked with a type $(k,l)$ according to the local structure provided by the slice theorem (see Remark~\ref{r:local_s1_quotient}). It turns out that, conversely, given a manifold $N$ with marked submanifolds of codimension two and a cohomology class $e\in H^2(N,\mathbb{Z})$, we can recover the Seifert fibration (uniquely if $H^2(N,\mathbb{Z})$ has no torsion). Namely, we have the following specialization of {\cite[Prop.30]{ko-05}} to the case $\dim N=3$.
\begin{prop}\label{p:s1_bundle_seifert_t}
Consider a manifold $N$, $\dim N=3$ and $\{Z_i\subset N\}$ a collection of cooriented pairwise non-intersecting one-dimensional submanifolds. Let $1\leq l_i<k_i$ be integers, $\mathrm{gcd}(l_i,k_i)=1$ and $e\in H^2(N,\mathbb{Z})$.
Then there exists a Seifert fiber bundle $\pi\colon M\to N$ such that $\pi^{-1}(Z_i)\subset M$ is the set of points of type $(k_i,l_i)$, all other points are regular, and
\[
e(M\to N)=e+\sum_i \frac{l_i}{k_i}[Z_i]\in H^2(N,\mathbb Q).
\]
If $H^2(N,\mathbb{Z})$ has no torsion, $M$ is uniquely defined.
\end{prop}
\begin{rmk}
Proposition~\ref{p:s1_bundle_seifert_t} states that the Seifert fibration $M\to N$ can be recovered from the marked orbit space $(N,\{Z_i,(k_i,l_i)\})$ and a cohomology class $e\in H^2(N,\mathbb{Z})$. If we realize $N$ as the orbit space $M/S^1$, then we can think of $N$ as an orbifold with orbifold charts $V_x$ of the form $\mathbb{C}\times\mathbb{R}\to\mathbb{C}/\mathbb{Z}_{k_i}\times\mathbb{R}\simeq V_x$, where $x\in Z_i$ and $Z_i\cap V_x=\{0\}\times \mathbb{R}$. This orbifold structure depends only on the underlying smooth manifold $N$ and the collection $\{Z_i, k_i\}$.
\end{rmk}
Our next goal is to formulate and prove an analogue of Proposition~\ref{p:s1_bundle_dg} for Seifert fibrations. To do that, we first need to define smooth tensors on orbifolds and introduce the notions of connections and basic differential forms in Seifert fibrations.
\begin{defn}[Smooth tensors on orbifolds]
Let $N$ be a smooth 3-dimensional manifold, and $\{Z_i\subset N\}$ a collection of pairwise non-intersecting one-dimensional manifolds, together with integers $\{k_i\}$, defining an orbifold structure $N_{\mathrm{orb}}$ on $N$. A \emph{smooth tensor field} $T$ on the orbifold $N_{\mathrm{orb}}$, is a collection of tensors $T_\alpha$ in the orbifold charts $(W_\alpha, G_\alpha, V_\alpha)$ such that $T_\alpha$ is a smooth $G_\alpha$-invariant tensor in $V_\alpha$, compatible with the gluing maps. In particular, $T$ defines a smooth tensor field in the usual sense on the regular open dense part $N^{\mathrm{reg}}=N\backslash \cup_i Z_i$.
\end{defn}
There is a natural one-to-one correspondence between the set of smooth Riemannian metrics on the orbifold $N_{\mathrm{orb}}$ as above and the set of Riemannian metrics on the underlying manifold $N$ with cone singularities along $\{Z_i\}$ of angles $2\pi/k_i$.
\begin{defn}[Basic differential forms]
Let $X\in \Gamma(M,TM)$ be the vector field generating an almost free $S^1$ action on $M$. A differential form $\alpha\in \Lambda^q(M,\mathbb{R})$ is \emph{basic} with respect to the vector field $X$ if
\[
i_X\alpha=\mathcal L_X\alpha=0.
\]
We will denote the space of basic $q$-forms by $\Lambda^q_b(M,\mathbb{R})$. The de Rham differential preserves the subcomplex $\Lambda^*_b(M,\mathbb{R})\subset \Lambda^*(M,\mathbb{R})$, thus we can define \emph{basic cohomology} $H^*_b(M,\mathbb{R})$ to be the cohomology of $(\Lambda^*_b(M,\mathbb{R}), d)$.
\end{defn}
The basic cohomology ring can be defined for any foliation on a manifold $M$. However, in our case the vector field $X$ generates an almost free $S^1$ action, and there is a natural isomorphism
\[
\pi^*\colon \Lambda^q(N_{\mathrm{orb}},\mathbb{R})\to \Lambda^q_b(M,\mathbb{R}).
\]
The de Rham cohomology of $(\Lambda^q(N_{\mathrm{orb}},\mathbb{R}), d)$ is isomorphic to the usual singular cohomology $H^*(N,\mathbb{R})$. Combining this observation with a general result of Koszul \cite{kosz} we see that map $\pi^*$ induces an isomorphism
\begin{equation*}
\pi^*\colon H^*(N,\mathbb{R})\to H_b^*(M,\mathbb{R}).
\end{equation*}
\begin{defn}
A \emph{connection} on a Seifert bundle $\pi\colon M\to N$ is an $S^1$-invariant differential form $\eta\in\Lambda^1(M,\mathbb{R})$ such that $\eta(X)=1$. The \emph{curvature} of a connection $\eta$ is the basic differential form $\beta:=d\eta$.
\end{defn}
Clearly, the difference of two connections is a basic one-form. Furthermore, we can always construct a connection $\eta$ by choosing an $S^1$-invariant Riemannian metric on $M$, and taking the orthogonal projection $TM\to \mathbb{R}\cdot X$ onto the vertical space spanned by $X$. Therefore, the set of connections on $M$ is an affine space modeled on $\Lambda^1_b(N,\mathbb{R})$. The orbifold version of Chern-Weil theory implies that the closed basic differential form $\beta/2\pi$ represents the Euler class $e(M\to N)$ in $H^2_b(M,\mathbb{R})\simeq H^2(N,\mathbb{R})$.
\begin{prop}\label{p:s1_bundle_seifert_dg}
Let $N$ be a smooth 3-dimensional manifold, and $\{Z_i\subset N\}$ a collection of cooriented pairwise non-intersecting one-dimensional manifolds. Let $1\leq l_i<k_i$ be integers, $\mathrm{gcd}(k_i, l_i)=1$ and $N_{\mathrm{orb}}$ be the corresponding orbifold. Then given a closed form $\beta\in \Lambda^2(N_{\mathrm{orb}},\mathbb{R})$ such that
\[
[\beta/2\pi]=e+\sum_i \frac{l_i}{k_i}[Z_i]\in H^2(N,\mathbb Q)
\]
where $e\in H^2(N,\mathbb{Z})$ is an integral class, there exists a Seifert fibration $\pi\colon M\to N$ with a connection $\eta$ and curvature $d\eta=\beta$.
The Seifert fibration $\pi\colon M\to N$ is unique if $H^2(N,\mathbb{Z})$ has no torsion. Modulo gauge equivalence, the connection $\eta$ is unique if $H^1(N,\mathbb{R})=0$.
\end{prop}
\begin{proof}
First, let us use Proposition~\ref{p:s1_bundle_seifert_t}, to construct a Seifert fibration $\pi\colon M\to N$ associated to the data $(N,\{Z_i,k_i,l_i\},e)$. Let $\eta'$ be any connection in the fibration, and denote by $\beta':=d\eta'$ its curvature form. The form $\beta'/2\pi$ represents the class $e(M\to N)=e+\sum_{i}l_i/k_i[Z_i]\in H^2(N,\mathbb Q)$. Then the basic forms $\beta$ and $\beta'$ represent the same cohomology class in $H^2_b(M,\mathbb{R})\simeq H(N,\mathbb{R})$. Therefore, there exists a basic 1-form $\alpha\in \Lambda^1_b(M,\mathbb{R})$ such that
\[
\beta=\beta'+d\alpha.
\]
Then $\eta:=\eta'+\alpha$ is the required connection with curvature $\beta$.
Uniqueness of $N$ follows from Proposition~\ref{p:s1_bundle_seifert_t}. If $\eta_0$ is another connection with curvature $\beta$, then $\eta_0=\eta+\alpha$, where $\alpha$ is a closed basic 1-form. Since $H^1(N,\mathbb{R})=0$, we have $\alpha=df$ where $f$ is a smooth function on the orbifold $N_{\mathrm{orb}}$. The function $f\in C^\infty(N_{\mathrm{orb}},\mathbb{R})$ generates a gauge equivalence on $M\to N$, transforming $\eta$ into $\eta_0$.
\end{proof}
\section{Generalized K\"ahler structures with \texorpdfstring{$S^1$}{} symmetries} \label{s:GKsymm}
In this section we will study generalized K\"ahler structures $(M^4,g,I,J)$ with a generically nonzero Poisson tensor $\sigma$, admitting an $S^1$ symmetry generated by a vector field $X$. First, we focus on the nondegeneracy locus, where the GK structure is recovered by a symplectic triple as in Proposition \ref{p:holo_symplectic-acs}. Where the action is free, it is locally tri-Hamiltonian, and we define the moment map $\pmb{\mu}$, which is a local submersion onto a domain in $\mathbb R^3_{\pmb \mu}$. The GK structure defines a metric $h$ on this domain, determined explicitly in terms of the angle function $p$, and furthermore the original GK structure is recovered in terms of $h$ and $W^{-1} = g(X,X)$, yielding Theorem \ref{t:mainGH} (cf. Theorem \ref{t:nondegenerate_gk_description}).
We then turn to addressing the case of an effective $S^1$ action. We derive the necessary asymptotics for $W$ at a fixed point of the action, then prove a removable singularity result, showing that a real analytic metric in our ansatz, defined on a punctured ball and satisfying the necessary asymptotics, will extend in a $C^{1,1}$ way across the missing point. Using the explicit form of the torsion we are able to then show that the entire GK structure extends in $C^{1,1}$ sense across the puncture. This leads to Theorem \ref{t:nondegenerate_gk_description_v2}, a version of Theorem \ref{t:nondegenerate_gk_description} which allows for fixed points of the action.
We finish this section with a discussion of some fundamental examples. First we recast the GK structures on the standard Hopf surface in this language, then do the same for the examples of GK structures on diagonal Hopf surfaces constructed in \cite{SU}. Lastly we recover the anti-self-dual metrics of LeBrun on parabolic Inoue surfaces~\cite{LeBrun} in this ansatz. This requires two key further steps, namely showing that the extensions across fixed points of the action are not just $C^{1,1}$ but smooth, and also extending across the degeneracy loci for the associated Poisson tensor. These points were treated explicitly by LeBrun, and we briefly indicate how to translate his setup to ours. We note that we are not able to give a general answer to these two points, namely the smooth extension across fixed points and the gluing in of degeneracy loci. However, starting in the next section, we will see that in the case of \emph{solitons}, we have further structure which allows for a complete answer to these questions.
\subsection{Nondegenerate GK structures with free \texorpdfstring{$S^1$}{} action}\label{ss:inv_gk_str}
Assume that a nondegenerate GK manifold $(M^4,g,I,J)$ admits a \emph{free} $S^1$-action preserving $g$, $I$, and $J$ and let $X$ be the vector field generating this action. In this case we have a smooth orbit space $N=M/S^{1}$ and $M$ is realized as a total space of a principal $S^1$-bundle:
\begin{equation}\label{f:pi_def}
\pi\colon M\to M/S^1 = N.
\end{equation}
Our first aim in this section is to give a local description for $(M,g,I,J)$ in terms of two scalar functions on the orbit space $N$. To this end we start with the construction of a moment map $\pmb\mu\colon M\to \mathbb{R}^3$ generalizing the moment map of the Gibbons-Hawking ansatz \cite{GibbonsHawking} in hyperK\"ahler geometry.
The vector field $X$ preserves the symplectic forms $\Omega$, $I\Omega$ and $J\Omega$, hence we have:
\[
d(i_X\Omega)=d(i_XI\Omega)=d(i_XJ\Omega)=0.
\]
If we furthermore assume the vanishing of the corresponding cohomology classes:
\begin{align*}
[i_X\Omega] = [i_XI\Omega] = [i_XJ\Omega] = 0 \in H^1_{dR}(M),
\end{align*}
then the vector field $X$ is \emph{Hamiltonian} with respect to each of three symplectic forms, and we can define the momentum map
\[
\pmb\mu\colon M\to\mathbb{R}^3,\quad \pmb\mu=(\mu_1,\mu_2,\mu_3),
\]
such tha
\[
d\mu_1=i_X\Omega, \quad d\mu_2=i_X I\Omega, \quad d\mu_3=i_X J\Omega.
\]
In this case we say that the action $S^1\times M\to M$ is \emph{tri-Hamiltonian}. Since the target of the moment map~--- the space $\mathbb{R}^3$ with distinguished coordinates $(\mu_1,\mu_2,\mu_3)$~--- will appear in this paper many times, we reserve for it the special notation $\mathbb{R}^3_{\pmb\mu}$.
If the action of $X$ is not tri-Hamiltonian, i.e., some of the forms $i_X\Omega, i_XI\Omega$ and $i_XJ\Omega$ are not exact, then one can take an appropriate $\mathbb{Z}^k$-cover $\til M\to M$, such that the pullbacks of these forms are exact and the action of $S^1$ on $M$ lifts to a tri-Hamiltonian action on $\til M$. From now on in this section, we assume that the action of $S^1$ on $M$ is tri-Hamiltonian.
\begin{prop}\label{p:principal_bundle}
Given a nondegenerate GK structure $(M,g,I,J)$ with a free tri-Hamiltonian $S^1$ action, there exists a local diffeomorphism
\begin{equation}\label{f:iota_def}
\iota\colon M/S^1\to \pmb\mu(M)\subset \mathbb{R}^3_{\pmb\mu}
\end{equation}
such that
\[
\iota\circ \pi=\pmb\mu.
\]
\end{prop}
\begin{proof}
Since $X\in\ker d\pmb\mu$, the moment map is constant on the $S^1$-orbits of $X$, hence the map $\pmb\mu\colon M\to \mathbb{R}^3_{\pmb\mu}$ descends to a map $\iota\colon M/S^1\to \mathbb{R}^3_{\pmb\mu}$. Since the symplectic forms $\Omega,I\Omega$ and $J\Omega$ are linearly independent, nondegenerate, and $X$ is nonzero, the moment map $\pmb\mu\colon M\to \pmb\mu(M)$ is a submersion with the kernel generated by $X$, therefore $\iota$ must be a local diffeomorphism.
\end{proof}
Consider, in addition to the angle function
\[
p\colon M\to (-1,1),\quad p:=-\tfrac{1}{4}\tr(IJ),
\]
a function $W\colon M\to (0,+\infty)$
\begin{equation}
W:= g(X, X)^{-1}.
\end{equation}
Both functions $p$ and $W$ are invariant under the $S^1$ action and thus descend to the orbit space $M/S^1$. Since the discussion below has a local nature, we will use $\iota\colon M/S^1\to \mathbb{R}^3_{\pmb\mu}$ to identify a neighbourhood of a point in $M/S^1$ with an open subset in $\pmb\mu(M)$. In particular in a neighbourhood of any point $x\in M/S^1$ functions $\{\mu_1,\mu_2,\mu_3\}$ provide local coordinates.
The principal bundle~\eqref{f:pi_def} carries with it a decomposition of the metric and complex structures according to the discussion of \S \ref{s:background}. Given a given point $x\in M$ we have the orthogonal decomposition of the tangent space into the vertical and the horizontal subspaces
\begin{equation}
T_xM=\mathbb{R} X\oplus\mathcal H_x,\quad \mathcal H_x=\langle IX,JX,KX\rangle.
\end{equation}
Thus we can define a connection 1-form $\eta \in \Lambda^{1}(M,\mathbb{R})$ by setting $\eta(X)=1$ and prescribing its kernel:
\begin{equation}
\eta \colon T_xM\to \mathbb{R},\quad \Ker\eta=\mathcal{H}_x.
\end{equation}
Now using Proposition \ref{p:matrices} we get explicit descriptions of the metric, almost complex structures and symplectic forms. In particular, using the definition of $W$, by~\eqref{f:g_matrix} the metric in this basis is given by
\begin{equation}\label{f:g_matrix_X}
g=W^{-1}\left[
\begin{matrix}
\w1 & \w0 & \w0 & \w0\\
0 & 1 & p & 0\\
0 & p & 1 & 0\\
0 & 0 & 0 & 1-p^2
\end{matrix}
\right].
\end{equation}
Note that we have two natural coframes
\[
\{X^*,(IX)^*,(JX)^*,(KX)^*\}\quad\mbox{and}
\quad\{\eta,d\mu_1,d\mu_2,d\mu_3\}.
\]
According to the matrix expression~\eqref{f:Omega_matrix} for $\Omega$ we have
\begin{align*}
\Omega=W^{-1}((KX)^*\wedge X^*+(JX)^*\wedge (IX)^*).
\end{align*}
Hence, by the definitions of $d\mu_i$, we find that
\begin{gather} \label{f:coframes}
\begin{split}
\eta&=X^*,\\
d\mu_1&=i_X\Omega=-W^{-1}(KX)^*,\\
d\mu_2&=i_{IX}\Omega=-W^{-1}(JX)^*,\\
d\mu_3&=i_{JX}\Omega=W^{-1}(IX)^*,\\
\end{split}
\end{gather}
Therefore, the metric $g$ can be expressed as
\begin{gather} \label{f:hdef}
\begin{split}
g =&\ W\pi^* h+W^{-1}\eta^2,\\
h=&\ (1-p^2)d\mu_1^2+d\mu_2^2+d\mu_3^2-2pd\mu_2d\mu_3.
\end{split}
\end{gather}
It remains to understand what restrictions on the functions $p$, $W$ and connection 1-form $\eta$ are imposed by the closedness of $\Omega,I\Omega,J\Omega$. We first express these forms in the basis $\{d\mu_1,d\mu_2,d\mu_3,\eta\}$ as
\begin{gather} \label{f:Omegaexplicit}
\begin{split}
\Omega&=W^{-1}\left((KX)^*\wedge X^*+(JX)^*\wedge(IX)^*\right) \\
&=-(d\mu_1\wedge \eta+Wd\mu_2\wedge d\mu_3),\\
\\
I\Omega&=W^{-1}\left(
(JX)^*\wedge X^*+((IX)^*+p(JX)^*)\wedge (KX)^*
\right)\\
&=-d\mu_2\wedge \eta+W(d\mu_3-pd\mu_2)\wedge (-d\mu_1),\\
\\
J\Omega&=W^{-1}\left(
-(IX)^*\wedge X^*+(p(IX^*)+(JX)^*)\wedge (KX)^*
\right)\\
&=-d\mu_3\wedge\eta+W(pd\mu_3-d\mu_2)\wedge (-d\mu_1).
\end{split}
\end{gather}
To determine the necessary condition we first recall that the curvature of the principal $S^1$-connection $\eta$ is a closed \emph{basic} 2-form in $\Lambda_{\mathrm{bas}}^2(M,\mathbb{R})$, which we identify with a pullback of a closed 2-form $\beta\in\Lambda^2(M/S^{1},\mathbb{R})$ under projection $\pi\colon M\to M/S^1$. In particular, we have that $\beta$ represents a class in $H^2(M/S^1,2\pi\mathbb{Z})\subset H^2(M/S^1,\mathbb{R})$. We omit the pullback in the notation and express
\begin{align} \label{f:betadef}
\beta:=d\eta=\beta_{ij}d\mu_i\wedge d\mu_j.
\end{align}
By taking the exterior derivative of the equations in (\ref{f:Omegaexplicit}), we obtain that in the coframe $d\mu_i$ the components of $\beta$ are given by
\begin{equation}\label{f:star_W}
\begin{split}
\beta_{23}&=W_1,\\
\beta_{31}&=W_2+(pW)_3,\\
\beta_{12}&=W_3+(pW)_2,
\end{split}
\end{equation}
where here and in the sequel $f_i$ means the partial derivative of a function $f$ with respect to $\mu_i$.
Now, if $W$ and $\eta$ satisfy system~\eqref{f:star_W}, then computing $d(d\eta)=0$ we find
\begin{equation}\label{f:W_laplace}
W_{11}+W_{22}+W_{33}+2(pW)_{23}=0.
\end{equation}
Furthermore the form $\beta$ defined by~\eqref{f:star_W} can be expressed as
\begin{equation}\label{f:beta0_def}
\beta=*_hdW+ W \beta_0,\quad \beta_0:=d\mu_1\wedge(p_2d\mu_2-p_3d\mu_3),
\end{equation}
where $*_h$ is computed with respect to the orientation of $\mathbb{R}^3_{\pmb\mu}$ given by $d\mu_1\wedge d\mu_2\wedge d\mu_3$.
The discussion above proves that any nondegenerate GK manifold $(M,g,I,J)$ with a free $S^1$ action has a principal $S^1$-bundle structure $\pi\colon M\to M/S^1$ together with an open map $\iota\colon M/S^1\to \mathbb{R}^3_{\pmb\mu}$ and the metric and complex structure are completely determined by the two scalar functions $W$ and $p$ solving equation~\eqref{f:W_laplace}. Crucially, the converse is also true: given an open map $\iota\colon N\to \mathbb{R}^3_{\pmb\mu}$, $\dim N=3$, functions $p,W$ on $N$ solving~\eqref{f:W_laplace} and a compatible connection 1-form $\eta$, one can construct a GK structure on the principal $S^1$ bundle $P\to N$ defined by the above data. Summarizing, we have the following theorem:
\begin{thm}[Generalized-K\"ahler Gibbons-Hawking ansatz]\label{t:nondegenerate_gk_description}
Fix a smooth 3-dimensional manifold $N$ and consider
\begin{enumerate}
\item an open map $\iota\colon N\to \mathbb{R}^3_{\pmb\mu}$,
\item smooth functions
\[
p\colon N\to (-1,1),\quad W\colon N\to (0,+\infty)
\]
solving the equation
\begin{equation*}
W_{11}+W_{22}+W_{33}+2(pW)_{23}=0,
\end{equation*}
such that the closed differential form $\beta\in\Lambda^2(N,\mathbb{R})$
\[
\beta=(W_3+(pW)_2)d\mu_1\wedge d\mu_2-(W_2+(pW)_3)d\mu_1\wedge d\mu_3+W_1d\mu_2\wedge d\mu_3
\]
represents a class in $H^2(N,2\pi\mathbb{Z})$,
\item a connection form $\eta$ with curvature $\beta$ in the principal $S^1$-bundle $\pi\colon M\to N$ determined by $[\beta]$.
\end{enumerate}
Then the total space of the principal $S^1$-bundle $M$
admits a nondegenerate GK structure
\[
(M,g,I,J)
\]
with
\[
g=Wh+W^{-1}\eta^2,\quad h=(1-p^2)d\mu_1^2+d\mu_2^2+d\mu_3^2-2p\,d\mu_2d\mu_3,
\]
and $I$, $J$ the unique almost complex structures such that the complex-valued 2-forms
\[
\begin{split}
\Omega_{I}:=&
(-d\mu_1+\sqrt{-1}d\mu_2)\wedge(\eta+\sqrt{-1}W(d\mu_3-pd\mu_2)),\\
\Omega_{J}:=&
(-d\mu_1+\sqrt{-1}d\mu_3)\wedge(\eta+\sqrt{-1}W(-d\mu_2+pd\mu_3)),
\end{split}
\]
are holomorphic with respect to $I$ and $J$ respectively.
Conversely any nondegenerate GK manifold $(M,g,I,J)$ with a free isometric tri-Hamiltonian $S^1$ action arises via this construction.
\end{thm}
\begin{proof}
The forms $\Omega_{I}$ and $\Omega_{J}$ satisfy the assumptions of Proposition~\ref{p:holo_symplectic-acs} defining the corresponding integrable complex structures $I$ and $J$. Furthermore, the tensors $\Omega=\Re(\Omega_I)$, $I$, $J$ and $g$ are chosen to satisfy the presentation in Proposition~\ref{p:matrices} in the basis $\{X,IX,JX,KX\}$. In particular,
\[
\Omega^{-1}=\frac{1}{2}g^{-1}[I,J]
\]
does not vanish identically, since $I$ and $J$ do not commute. Therefore $(M, g, I, J)$ is a nondegenerate GK structure with a free $S^1$ symmetry.
Conversely, given any nondegenerate GK structure $(M,g,I,J)$ with a free $S^1$ symmetry, we can identify $M$ with the total space of an $S^1$-bundle and express $g$, $\Omega_I$ and $\Omega_J$ through functions $W$ and $p$ as in equations~\eqref{f:hdef} and \eqref{f:Omegaexplicit}.
\end{proof}
\begin{rmk}
Given any smooth function $p\colon \mathbb{R}^3_{\pmb\mu}\to (-1,1)$ one can always solve the corresponding elliptic equation for $W$ at least locally. Therefore the germ of any function $p\colon \mathbb{R}^3_{\pmb\mu}\to (-1,1)$ can be realized on some nondegenerate GK manifold with $S^1$ symmetry.
\end{rmk}
\begin{rmk}[Relation to T-duality]
A surprising feature underlying both the Gibbons-Hawking ansatz and Theorem~\ref{t:nondegenerate_gk_description} is that a solution to a linear PDE yields a non-linear geometric structure on $M^4$. We would like to thank an anonymous referee for kindly bringing to our attention that this is not a coincidence, and can be explained through the notion of \textit{T-duality}. Specifically, as was observed by Cavalcanti and Gualtieri in~\cite{ca-gu-10}, given a non-degenerate GK structure with a free $S^1$-symmetry
\[
M^4\to M^4/S^1=N^3
\]
there is a natural T-dual $S^1$-bundle $\hat M^4\to N^3$ endowed with a \textit{commuting} (odd) GK structure $(\hat M^4, \hat g, \hat I,\hat J)$, $\hat I\hat J=\hat J\hat I$, and by a result of Apostolov and Gualtieri~\cite{ap-gu-07} any such structure determines a local holomorphic orthogonal splitting
\[
\hat M^4\simeq_{\mathrm{loc}}\mathbb{C}\times \mathbb{C}
\]
rendering a \textit{linear} description of the underlying GK structure, which can be translated back to a linear description of a nondegenerate GK structure on $M^4\to N^3$ via the inverse T-duality. We expect that one could push this idea even further and use the T-duality to derive a local ansatz for \textit{GK solitons} (see section \S\ref{s:GKRS} for the precise definitions) similar to an alternative description of the Gibbons-Hawking anstaz presented in~\cite[Ex.\,5.2]{ca-gu-10}. We, however, take a different approach in this paper and derive the soliton equations directly on a nondegenerate GK manifold $(M^4,g,I,J)$, see Proposition~\ref{p:invGKsoliton}.
\end{rmk}
To simplify further computations, we observe a useful change of coordinates in $\mathbb{R}^3_{\pmb\mu}$ suggested by the structure of $h$. \[
\mu_+:=\tfrac{1}{2}(\mu_2+\mu_3),\quad \mu_-:=\tfrac{1}{2}(\mu_2-\mu_3).
\]
In these coordinates the metric $h$ of (\ref{f:hdef}) diagonalizes and takes the form
\begin{equation}\label{f:h_diagonal}
h=(1-p^2)d\mu_1^2+2(1-p)d\mu_+^2+2(1+p)d\mu_-^2.
\end{equation}
Furthermore, the equation ~\eqref{f:W_laplace} for $W$ takes form
\begin{equation}\label{f:W_laplace2}
W_{11}+\tfrac{1}{2}((1+p)W)_{++}+\tfrac{1}{2}((1-p)W)_{--}=0.
\end{equation}
In the rest of this paper, we will use $(\mu_1,\mu_+,\mu_-)$ as the coordinates on $\mathbb{R}^3_{\pmb\mu}$.
\subsection{Nondegenerate GK structures with effective \texorpdfstring{$S^1$}{} action} \label{ss:NGKeff}
Theorem~\ref{t:nondegenerate_gk_description} provides a classification of nondegenerate GK structures with \emph{free} $S^1$ action. Our next step is to relax this assumption and allow the $S^1$ action to have nontrivial isotropy subgroups. The following lemma shows that in the nondegenerate case the only new possibility is the presence of fixed points.
\begin{lemma}\label{l:free_nondegenerate}
Let $(M,g,I,J)$ be a nondegenerate GK manifold, $\dim_\mathbb{R} M=4$. Then an effective action of $S^1$ on $(M,g,I,J)$ is free outside of a discrete set of points $M^{S^1}\subset M$. If $y\in M$ is a fixed point, then the isotropy representation of $S^1$ in $T_yM$ has weights $(\pm 1, \pm 1)$
\end{lemma}
\begin{proof}
Since the action of $S^1$ preserves $I$ and $J$, the fixed point set $M^{S^1}\subset M$ must be $I$ and $J$-holomorphic.
Note the complex structures $I$ and $J$ are compatible with $g$, induce the same orientation and $I\neq \pm J$ on $M$. Since $\dim_\mathbb{R} M=4$, this implies that $\Ker(I\pm J)=0$. Thus we conclude that $M^{S^1}$ must be zero-dimensional. To prove that the action of $S^1$ is free outside the fixed point set $M^{S^1}$, assume on the contrary that for some $x\in M\backslash M^{S^1}$ we have $\mathrm{Stab}(x)=\mathbb{Z}_k\subset S^1$. Then $Z:=M^{\mathbb{Z}_k}$ must be $I$ and $J$-holomorphic. On the other hand $Z$ is at least one-dimensional, since it contains $S^1\cdot x$. Therefore $Z$ must coincide with the whole $M$ and the action of $S^1$ on $M$ is not effective.
Now let $y\in M$ be a fixed point, and let $\mathbf w=(w_1,w_2)$ be the weights of the representation of $S^1$ in $T_yM$:
\begin{equation}\label{f:Li_decompoistion}
T_yM= L_1\oplus L_2,\quad L_i\simeq \mathbb{C}
\end{equation}
where the action of $S^1\simeq U(1)$ on $L_i$ is given by $t\cdot z=t^{w_i}z$. Since the action of $S^1$ on the complement of the fixed point set is free, we necessarily have $w_i=\pm 1$, as claimed.
\end{proof}
We next analyze the local behavior of $W$ near a fixed point. Fix $y\in M^{S^1}$. By Lemma~\ref{l:free_nondegenerate}, a small $S^1$-invariant open ball $U_y$ around $y$ is equivariantly diffeomorphic to a unit ball in $\mathbb{C}^2$ with the standard diagonal action of $S^1$ via coordinate-wise complex multiplication:
\[
e^{\sqrt{-1}t}\cdot (z_1,z_2)=(e^{\sqrt{-1}t}z_1,e^{\sqrt{-1}t}z_2).
\]
Therefore, there exist spherical coordinates $(s,\rho)\in S^3\times [0,1)_\rho$ on $U_y$ such that the projection onto the orbit space $U_y\to U_y/S^1$ is given by
\[
\mathrm{pr}\colon S^3\times [0,1)_\rho\to S^2\times [0,1)_r,\quad \mathrm {pr}(s,\rho)=(\chi(s),\rho^2/2)
\]
where $S^2\times [0,1)_r\simeq U_y/S^1$ and the map $\chi\colon S^3\to S^2$ is given by the Hopf fibration. Using this local model of the projection $M\to M/S^1$, we conclude that if the $S^1$ action on a 4-dimensional manifold is free outside the fixed point set, then the orbit space $M/S^1$ admits the structure of a smooth 3-dimensional manifold, such that the projection $\pi\colon M\to M/S^1$ is a smooth map with $d\pi$ surjective outside of the fixed point set.
The local model of the projection $M\to M/S^1$ near a fixed point $y\in M$ implies that the curvature form $\beta$ of the principal bundle
\[
U_y\backslash\{y\}\to (U_y\backslash\{y\})/S^1
\]
must pair to $-2\pi$ with the homology class of a 2 sphere enclosing the center of $(U_y\backslash \{y\})/S^1\simeq B_1(0;\mathbb{R}^3)$:
\begin{equation}\label{f:beta_pairing_s2}
\int_{S^2}\beta=-2\pi.
\end{equation}
Moreover, since the vector field $X$ has a simple zero at $y$, we have $W(x)=g(X,X)^{-1}\sim \rho^{-2}$ near $y\in U_y$, so that $W(x)$ descended to $U_y/S^1$ must blow up as $r^{-1}$ for $r\to 0$. Similarly $|dW|_{h}\sim r^{-2}$ as $r\to 0$.
Now we prove a partial converse of the above local observation about the behavior of $W$ and $\beta$ near the image of the fixed point.
\begin{prop}[Removable singularity result for the metric]\label{p:removable_singularity}
Let $B^3$ be the unit ball in $\mathbb{R}^3$ centered at the origin. Assume that metric $h$ and 2-form $\beta_0$ in $B^3$ are real analytic. Let $W$ be a positive solution to
\begin{equation}\label{f:removable_singualrity_W_eq}
d\beta=0,\quad \beta := *_hdW+W\beta_0
\end{equation}
in $B^3\backslash \{0\}$ such that $W(x)/r\to 1/2$ and $|dW|_h(x)=o(r^{-3})$ as $r:=d_h(0,x)\to 0$. Then
\begin{itemize}
\item $\beta$ pairs to $-2\pi$ with any 2-sphere enclosing the origin oriented by the outward normal vector;
\item the total space $P$ of $S^1$-bundle with the connection $\eta$ such that $d\eta=\beta$ over $B^3\backslash\{0\}$ is diffeomorphic to $B^4\backslash\{0\}$, where $B^4$ is a unit ball in $\mathbb{R}^4$;
\item the metric completion of $(P, g)$, $g=Wh+W^{-1}\eta^2$ is diffeomorphic to $B^4\supset B^4\backslash 0$, and the metric $g$ extends to a $C^{1,1}$ metric across $0\in B^4$.
\end{itemize}
\end{prop}
\begin{proof}
With our assumptions on the coefficients of equation $d\beta=0$ and on the solution $W$, we can apply a general result about fundamental solutions to elliptic equations with analytic coefficients~\cite{Fritz}. Specifically, using~\cite[\S 6]{Fritz} (see also p.\,275), we conclude that, in the notations of the cited paper, $W(x)=K(x,0)+A(x)$, where $A(x)$ is analytic in $B^3$ and $K(x,z)$ is the fundamental solution for the Dirichlet problem $d\beta=0$. Further, using the decomposition~\cite[eq.\,5.14]{Fritz} and the fact that the leading term in the decomposition is proportional to $1/r$ (see discussion on p.290), we find that $W$ admits a decomposition
\begin{equation}\label{eq:W_series}
W=\frac{1}{2r}+\sum_{k=0}^\infty \alpha_k(\zeta)r^k,
\end{equation}
where $(r,\zeta)\in (0,1)\times S^2$ are spherical coordinates, the functions $\alpha_k(\zeta)$ are analytic and the series is absolutely convergent with all its derivatives.
To prove the first claim, we note that since $\beta$ is closed, the pairing is independent of the choice of sphere. If we take $S_\epsilon:=\{r=\epsilon\}$, then using the decomposition~\eqref{eq:W_series}
\[
\int_{S_\epsilon^2}\beta=\lim_{\epsilon\to 0}\int_{S_\epsilon^2}\beta=\lim_{\epsilon\to 0}\int_{S_\epsilon^2}*_hdW=\lim_{\epsilon\to 0}\int_{S_\epsilon^2} \frac{\partial W}{\partial r}d\sigma_\epsilon=-2\pi.
\]
proving the first claim. The second claim follows from a simple topological observation. The principal bundle $P$ is uniquely determined by a class $[\beta]\in H^2(B^3\backslash\{0\},2\pi\mathbb{Z})\simeq 2\pi\mathbb{Z}$, and the bundle corresponding to $-2\pi$ is diffeomorphic to the total space of the standard Hopf fibration $B^4\backslash \{0\}\to B^3\backslash\{0\}$, which extends to a smooth map $B^4\to B^3$.
Our next goal is to prove the $C^{1,1}$ extension of $g$ in this chart, following the argument of LeBrun~\cite{LeBrun}. In the exponential coordinates $(r,\zeta)\in (0,1)\times S^2$, the metric $h$ can be written as
\[
h=dr^2+r^2h^{S}_r,
\]
where $h^S_r$ is an analytic family of metrics on $S^2$ such that $h^S_r\to h^S_0$, where $h^S_0$ is the standard round metric on the unit sphere. Decompose $h$ as
\[
h=h'+h'',\quad h':=dr^2+r^2 h^S_0,
\]
i.e., $h'$ is the flat metric in exponential coordinates, and $h''$ is the remainder. Since $h''$ is analytic, and $h'$ matches $h_1$ up to order 2 near the origin, we have
\[
h''=\sum_{k=2}^{\infty} r^k h''_k(\zeta),
\]
where $h''_k$ is a family of symmetric two-tensors on $\mathbb{R}^3$ parametrized by $\zeta\in S^2$.
Using the above two series expansions and inspecting the closedness of $\beta=*_hdW+W\beta_0$ in this decomposition, we find that
\[
\beta=*_{h'}d\left(\frac{1}{2r}\right)+d\gg, \qquad \gg=dr\sum_{k=0}^\infty r^k\gg'_k(\zeta)+\sum_{k=1}^\infty r^k \gg_k''(\zeta),
\]
where $\gg'_k(\zeta)$ and $\gg''_k(\zeta)$ are analytic functions and 1-forms on the sphere $S^2$. In particular, after a gauge transform $\eta=\eta_0+\gamma$, where $\eta_0$ is the connection form of the standard Hopf fibration $B^4\to B^3$ given by the flat metric on $B^4$.
Let, as above, $\mathrm{pr}\colon S^3\times [0,1)_\rho\to S^2\times [0,1)$ be the projection given by $\mathrm{pr}(x,\rho)=(\chi(s),\rho^2/2)$, where $\chi\colon S^3\to S^2$ is the Hopf fibration on a 3-sphere. In particular, $\mathrm{pr}^*dr=\rho d\rho$. By slight abuse of notation, we omit $\mathrm{pr}^*$ in front of $h,h',h''$ and $W$, below. As is well known from the geometry of the Hopf fibration $B^4\to B^3$, the metric
\[
g':= \frac{1}{\rho^2}h'+\rho^2 \eta_0^2=d\rho^2+\rho^2 (h_0^S+\eta_0^2)
\]
on $B^4\backslash\{0\}$ is flat and extends smoothly across $\{0\}\in B^4$. Now we observe that on $B^4\backslash\{0\}$
\begin{equation}
\begin{split}
g-g'&=\left(W(h'+h'')+W^{-1}(\eta_0+\gg)^2\right)-g'\\&=
Wh''+(W-\rho^{-2})h'+(W^{-1}-\rho^2)\eta_0^2+W^{-1}(2\gg\otimes \eta_0+\gg^2).
\end{split}
\end{equation}
Using the series expansion above we find that
\[
g-g'=\sum_{k=2}^\infty \rho^k g''_k(\xi),
\]
where $g''_k(\xi)$ is an analytic family of symmetric 2-tensors on $\mathbb{R}^4$ depending on $\xi\in S^3$. This difference extends to a $C^{1,1}$ symmetric tensor on $B^4$.
\end{proof}
\begin{rmk}
If $h$ and $\beta_0$ are additionally $SO(3)$-rotationally symmetric around $0$, then the $C^{1,1}$-regularity of $g$ can be upgraded to $C^\infty$ (analytic), since the coefficients in all the expansions are constant. This is the case with the original Gibbons-Hawking ansatz and LeBrun's construction of self-dual metrics~\cite{LeBrun}.
\end{rmk}
Proposition~\ref{p:removable_singularity} ensures that a metric $g$ of a certain form on the total space of a principal bundle $B^4\backslash\{0\}\to B^3\backslash\{0\}$ has a $C^{1,1}$-extension over the origin. If the principal bundle $B^4\backslash\{0\}\to B^3\backslash\{0\}\subset \mathbb{R}^3_{\pmb\mu}$ and the metric $g$ are given by Theorem~\ref{t:nondegenerate_gk_description} then we have a nondegenerate GK structure $(B^4\backslash\{0\}, g,I,J)$. The following proposition ensures that not only the metric, but the whole GK structure extends to a $C^{1,1}$ structure on $B^4$.
\begin{prop} \label{p:removable_singularity_GK} The following hold:
\begin{enumerate}
\item Suppose $(B^4\backslash\{0\},g,I,J)$ is a smooth GK structure such that $g$ has a $C^{1,1}$ extension to $B^4$, and $H$ has a $C^{0,1}$ extension to $B^4$. Then there exists a $C^{1,1}$ extension of $(g,I,J)$ to $B^4$.
\item If $B^4\backslash \{0\}\to B^3\backslash \{0\}$ is a Hopf fibration, and there is a GK structure $(g,I,J)$ on $B^4\backslash \{0\}$ given by Theorem~\ref{t:nondegenerate_gk_description}, then the torsion $H$ has a $C^{0,1}$ extension to $B^4$. In particular, the GK structure $(g,I,J)$ extends to a $C^{1,1}$ GK structure on $B^4$.
\end{enumerate}
\end{prop}
\begin{proof}
To show item (1), first note that by the hypotheses on the extensions of $g$ and $H$, the Christoffel symbols of the Bismut connections $\nabla^I$ and $\nabla^J$ have a $C^{0,1}$ extension to $B^4$. Thus we can define $I$ (resp.\,$J$) at the origin by the parallel $\nabla^I$ (resp.\,$\nabla^J$) transport. The $C^{0,1}$ regularity of $\nabla^I$ implies that the resulting almost complex structure does not depend on the choice of a path and is $C^{1,1}$.
Now we turn to item (2) and prove that the torsion 3-form $H$ of the GK structure provided by Theorem~\ref{t:nondegenerate_gk_description} in a principal $S^1$ bundle $B^4\backslash\{0\}\to B^3\backslash\{0\}$ has a $C^{0,1}$ extension. On the nondegenerate part of a GK manifold we have an identity (see~\cite[Lemma 3.8]{ASNDGKCY}) for $\theta_I^{\#}=-g^{-1}(*H)$,
\[
i_{\theta_I^{\#}}\Omega=-\frac{dp}{1-p^2}.
\]
In our case $\Omega$ is given by
\[
\Omega=-d\mu_1\wedge\eta+2Wd\mu_+\wedge d\mu_-
\]
hence
\[
\theta_I^\#=
\frac{1}{1-p^2}\left(
-p_1X+
\frac{1}{2}W^{-1}p_+\frac{\partial}{\partial \mu_-}-
\frac{1}{2}W^{-1}p_- \frac{\partial}{\partial \mu_+}
\right)
\]
or equivalently, since $g=W\left((1-p^2)d\mu_1^2+2(1-p)d\mu_+^2+2(1+p)d\mu_-^2\right)+W^{-1}\eta^2$, we find
\begin{equation}\label{f:theta_through_p}
\theta_I=-W^{-1}\frac{p_1}{1-p^2}\eta+\frac{p_+}{1-p}d\mu_--\frac{p_-}{1+p}d\mu_+.
\end{equation}
The construction of Proposition~\ref{p:removable_singularity} provides a smooth map $\mathrm{pr}\colon B^4\to B^3$ such that $X$ is a smooth vector field on $B^4$ vanishing at the origin and the metric $g$ has a $C^{1,1}$ extension. Therefore the pull back of a $C^{k,\alpha}$ function on $B^3$ is also $C^{k,\alpha}$ on $B^4$. This implies that
\[
\frac{p_1}{1-p^2}X,\quad
\frac{p_+}{1-p}d\mu_-,\quad \frac{p_-}{1+p} d \mu_+
\]
extend to $C^\infty$ vector field and differential forms on $B^4$. Therefore, since $g$ is $C^{1,1}$, tensors $\theta_I^\#, \theta_I$ and $H$ all have a $C^{1,1}$ extension over $B^4$.
\end{proof}
We can summarize the above observations as follows (compare with Theorem~\ref{t:nondegenerate_gk_description}).
\begin{thm}[Generalized-K\"ahler Gibbons-Hawking ansatz II]\label{t:nondegenerate_gk_description_v2}
Consider
\begin{enumerate}
\item an open map $\iota\colon N\to \mathbb{R}^3_{\pmb\mu}$,
\item a discrete subset $\{x_i\}\subset N$,
\item smooth functions
\[
p\colon N\to (-1,1),\quad W\colon N\backslash\{x_i\}\to (0,+\infty)
\]
solving equation
\begin{equation*}
W_{11}+W_{22}+W_{33}+2(pW)_{23}=0,
\end{equation*}
such that the closed differential form $\beta\in\Lambda^2(N\backslash\{x_i\},\mathbb{R})$
\[
\beta=-(W_3+(pW)_2)d\mu_1\wedge d\mu_2+(W_2+(pW)_3)d\mu_1\wedge d\mu_3-W_1d\mu_2\wedge d\mu_3
\]
represents a class in $H^2(N\backslash\{x_i\},2\pi\mathbb{Z})$, and for any $x_i\in\{x_i\}$ we have
$W(x)/r\to 1/2$ and $|dW|_h(x)=o(r^{-3})$ as $r:=d_h(x_i,x)\to 0$,
\item connection form $\eta$ with curvature $\beta$ in the principal $S^1$-bundle $\pi\colon P\to N\backslash\{x_i\}$ determined by $[\beta]$.
\end{enumerate}
Then there exists a unique GK structure $(M,g,I,J)$ with a tri-Hamiltonian $S^1$-action with fixed points $\{y_i\}$ such that
\[
(M\backslash\{y_i\},g,I,J)
\]
is the GK structure provided by Theorem~\ref{t:nondegenerate_gk_description}. The GK structure on $M$ is $C^\infty$ on $M\backslash\{y_i\}$ and is $C^{1,1}$ across $\{y_i\}$.
Conversely any smooth nondegenerate GK manifold $(M,g,I,J)$ with a tri-Hamiltonian $S^1$ action is isomorphic to a GK manifold as above.
\end{thm}
\subsection{Examples}
Before we have assumed that the Poisson tensor
\[
\sigma=\tfrac{1}{2}g^{-1}[I,J]
\]
is everywhere nondegenerate, in particular, there was a well defined symplectic form $\Omega=\sigma^{-1}$. Now we drop this assumption, and allow $\sigma$ to vanish along a \emph{proper} subset $\mathbf{T}\subset M$. As we mentioned in Section~\ref{s:background}, in this situation
\[
\mathbf{T}=\{x\in M\ |\ I_x=\pm J_x \}
\]
is the union of (possibly singular) complex one-dimensional jointly $I$ and $J$-holomorphic curves. Since the $S^1$ action on $(M,g,I,J)$ preserves all the structure, we see that $\mathbf{T}$ is $S^1$-invariant. In particular, $(M\backslash \mathbf{T},g,I,J)$ is a \emph{nondegenerate} GK manifold with an effective $S^1$ action. Therefore, we can invoke Theorem~\ref{t:nondegenerate_gk_description_v2} and conclude that an appropriate cover $\til{M\backslash\mathbf{T}}$ is given by the construction in the theorem. It remains to understand how and when one can glue back a degeneracy locus into a quotient of such a manifold.
We will not be able to give a complete answer to this question. Instead, in this section, we review several known examples of GK structures with $S^1$ symmetry and demonstrate how they arise via the ansatz of Theorem~\ref{t:nondegenerate_gk_description_v2}.
Later in the paper we will show how to extend GK structure across the degeneracy locus for GK solitons.
\begin{ex}[Generalized K\"ahler structure on the standard Hopf surface revisited]\label{ex:Hopf2}
Consider the standard Hopf surface $(M,g,I,J)$ with the GK structure of Example~\ref{ex:Hopf1}.
There is an $S^1$ action preserving the GK structure:
\[
e^{\sqrt{-1}t}\cdot(z_1,z_2):=(e^{\sqrt{-1}t}z_1, e^{\sqrt{-1}t}z_2).
\]
Let $\mathbb{C}^2\to(\mathbb{C}^*)^2/\mathbb{Z}\subset \mathbb{C}^2\backslash\{0\}/\mathbb{Z}=M$ be the universal cover of the nondegenerate part of the Hopf surface. Pick $w_1,w_2\in \mathbb{C}$ to be the coordinates in $\mathbb{C}^2$, $w_i=x_i+\sqrt{-1}y_i$ so that the holomorphic symplectic forms $\Omega_I$ and $\Omega_J$ are given by
\[
\begin{split}
&\Omega_I=(dx_1+\sqrt{-1}dy_1)\wedge (dx_2+\sqrt{-1}dy_2),\\
&\Omega_J = d\left(\log(e^{2x_1}+e^{2x_2})-x_2+\sqrt{-1}y_2\right)\wedge
d\left(\log(e^{2x_1}+e^{2x_2})-x_1-\sqrt{-1}y_1\right).
\end{split}
\]
In these coordinates, the vector field $X$ generating the $S^1$ action is given by $X=\partial_{y_1}+\partial_{y_2}$, and the moment map can be recovered from $i_X\Omega_I=d\mu_1-\sqrt{-1}d\mu_2$ and $i_X\Omega_J=d\mu_1-\sqrt{-1}d\mu_3$:
\[
\begin{split}
\mu_1&=y_1-y_2=\arg z_1-\arg z_2,\\
\mu_2&=x_1-x_2=\log\frac{|z_1|}{|z_2|},\\
\mu_3&=x_1+x_2-2\log(e^{2x_1}+e^{2x_2})=\log\frac{|z_1 z_2|}{(|z_1|^2+|z_2|^2)^2}.
\end{split}
\]
The functions $p$ and $W$ in this case are
\[
p=\frac{e^{2x_2}-e^{2x_1}}{e^{2x_2}+e^{2x_1}}=\frac{|z_2|^2-|z_1|^2}{|z_1|^2+|z_2|^2}, \qquad W^{-1}=g(X,X)=1.
\]
\end{ex}
\begin{ex}[Generalized K\"ahler structure on the diagonal Hopf surfaces]\label{ex:hopf_diagonal_gk}
Let $M$ be a diagonal Hopf surface with parameters $\alpha,\beta\in \{z\in\mathbb{C}\ |\ |z|>1\}$:
\[
M=(\mathbb{C}^2_{z_1,z_2}\backslash\{0\})/\Gamma,\quad \Gamma\colon (z_1,z_2)\mapsto (\alpha z_1, \beta z_2).
\]
In~\cite{SU} we have described a family of GK structures on $M$ determined by one scalar function. Let us review this construction and reinterpret it through the lens of Theorem~\ref{t:nondegenerate_gk_description}.
Let $\mathbb{C}^2\to(\mathbb{C}^*)^2/\Gamma\subset \mathbb{C}^2\backslash\{0\}/\Gamma=M$ be the universal cover of the nondegenerate part of the Hopf surface. Choose $w_i=\log z_i$ to be the coordinates in $\mathbb{C}^2$, $w_i=x_i+\sqrt{-1}y_i$ and denote $a=\log|\alpha|$, $b=\log|\beta|$. Given a function $p\colon \mathbb{R}\to (-1,1)$ of a real argument $2(\frac{b}{a} x_1-x_2)$, we can define a pair of complex structures $I$ and $J$ on $\mathbb{C}^2$ via a pair of complex-valued symplectic forms (in~\cite{SU} we used a function $k=(p+1)/2$ instead of $p$):
\[
\begin{split}
\Omega_I&=dw_1\wedge dw_2,\\
\Omega_J&=\left(dw_1-\frac{a}{b}d\bar w_2\right)\wedge \left(\frac{b(1+p)}{2a}d\bar{w_1}+\frac{1-p}{2}dw_2\right).
\end{split}
\]
By a direct computation, $\Omega_I,\Omega_J$ satisfy the assumptions of Proposition~\ref{p:holo_symplectic-acs}, thus we have a GK structure on $\mathbb{C}^2_{w_1,w_2}$. This structure descends to $(\mathbb{C}^*)^2_{z_1,z_2}$, and under certain assumptions on the asymptotic of $p(x)$ as $x\to\pm\infty$, extends to a smooth GK structure $(M,g,I,J)$ on our Hopf surface. The function $p$ provides the angle function of this GK structure and $g$ is given by
\[
g=\Re\left(\frac{b(1+p)}{2a}dw_1\otimes d\bar{w_1}+\frac{a(1-p)}{2b}dw_2\otimes d\bar{w_2}\right).
\]
To relate this construction to Theorem~\ref{t:nondegenerate_gk_description}, let us choose coprime $m,n\in \mathbb{Z}$ and introduce an $S^1$ action on $M$ preserving the entire GK structure:
\[
u\cdot (z_1,z_2)=(u^mz_1,u^nz_2).
\]
The vector field generating this action is given in coordinates $(w_1,w_2)$ by
\[
X=m\partial_{y_1}+n\partial_{y_2},
\]
so that
\[
W^{-1}=g(X,X)=\frac{m^2b^2}{2ab}(1+p)+\frac{n^2a^2}{2ab}(1-p).
\]
As in the previous example, we recover the moment map by computing $i_X\Omega_I$ and $i_X\Omega_J$:
\begin{equation}\label{f:diagonal_hopf_mu}
\begin{split}
\mu_1&=ny_1-my_2,\\
\mu_2&=nx_1-mx_2,\\
\mu_3&=-n\left(\frac{b}{a} x_1+\frac{1}{2}\chi\right)-m\left(\frac{a}{b}x_2+\frac{a}{2b}\chi\right),
\end{split}
\end{equation}
where $\chi\colon \mathbb{R}\to \mathbb{R}$ is an antiderivative of $p$ evaluated at $2(\frac{b}{a}x_1-x_2)$. We want to stress that this moment map $\pmb\mu$ is defined only on the universal cover of the nondegenerate part of $M$.
\end{ex}
\begin{rmk}
The key new feature of the above circle action on diagonal Hopf surfaces is the existence of nontrivial isotropy subgroups. Specifically, we have stabilizers
\[
G_{(0,z_2)}\simeq \mathbb{Z}_{n}\quad G_{(z_1,0)}\simeq \mathbb{Z}_{m}.
\]
Because of the nontrivial isotropy groups, the orbit space $M/S^1$ is not a manifold anymore, but rather an \emph{orbifold}. Geometrically, $M/S^1$ is the product of $S^1$ and a spindle 2-sphere $S^2(m,n)$ with two cone singularities of angles $2\pi/n$ and $2\pi/m$. In this case, $\pmb\mu$ identifies the universal cover of the smooth part of $S^1\times S^2(m,n)$ with $\mathbb{R}^3_{\pmb\mu}$.
The presence of points with nontrivial finite stabilizers is a new phenomenon compared to $S^1$ actions on hyperK\"ahler 4-dimensional manifolds. By Lemma~\ref{l:free_nondegenerate} only points on the degeneracy locus $\mathbf{T}$ might potentially have nontrivial stabilizers.
\end{rmk}
\begin{ex}[LeBrun's GK structure on parabolic Inoue surface]
In~\cite{LeBrun} LeBrun has constructed an explicit family of anti-self-dual metrics on blown-up Hopf surfaces and their deformations~--- parabolic Inoue surfaces. It was later observed by Pontecorvo that this Hermitian structure on Inoue surfaces naturally fits into a GK structure. Let us review this construction.
Let $\mathbb H^3$ denote hyperbolic 3-space modeled on the upper half-space
\[
\mathbb H^3=\{(x,y,z)\in\mathbb{R}^3\ |\ z>0\}, \quad \til h:=\frac{dx^2+dy^2+dz^2}{z^2}.
\]
It is well-known that given any point $p\in\mathbb H$ there exists a positive Green's function $G_p$~--- a solution to
\[
\Delta_{\til h}G_p=-2\pi \delta_p.
\]
Let $\{p_j\}_{j\in\mathbb{Z}}$ be a sequence of points $p_j=(0,0,\lambda^j)\in\mathbb H$, $\lambda>1$. Using the explicit form of $G_p$, one can check that the sum $V=1+\sum_j G_{p_j}$ is absolutely convergent and the limit solves Laplace equation $\Delta_{\til h}V=-2\pi\sum_j \delta_{p_j}$. Then we can define a 2-form $\beta=*_{\til h}dV$ representing a class in $H^2(\mathbb H\backslash\{p_j\},2\pi \mathbb{Z})$ and construct a principal $S^1$ bundle $M_0$ over $\mathbb H\backslash\{p_j\}$ with connection $\eta$ and curvature form $\beta$. One can then define a complex structure $I$ as the unique complex structure such that complex-valued symplectic form
\[
\Omega_I:=\frac{dw}{w}\wedge\left(V\frac{dz}{z}+\sqrt{-1}\eta\right),\quad w=x+\sqrt{-1}y
\]
is of type $(2,0)$. This complex structure is compatible with the metric
\[
g=\frac{z^2}{x^2+y^2+z^2}\left( V\til h+V^{-1}\eta^2\right).
\]
LeBrun proved that while the metric $g$ on $M_0$ is not complete, its completion is a complex manifold obtained from $M_0$ by gluing in points $\{p_j\}$ and a punctured plane $\mathbb{C}^*$ along $z=0$. This complete manifold admits a free isometric $\mathbb{Z}$ action generated by scalings $(x,y,z)\mapsto (\lambda^k x,\lambda^k y,\lambda^k z)$, and the quotient space $M$ is a compact complex surface, isomorphic to a \emph{parabolic Inoue surface}.
It was observed by Pontecorvo~\cite[\S 2.3]{PontecorvoCS} that inversion in the hemisphere $\{x^2+y^2+z^2=1\}$ composed with the reflection in $\{x=0\}$ plane yields an orientation preserving, isometric involution $j\colon M_0\to M_0$, and denoting by $J$ the pullback of $I$ under this involution, we obtain a GK structure $(M,g,I,J)$. The degeneracy locus of this GK structure consists of the disjoint union of an elliptic curve (the quotient of the \emph{glued in} copy of $\mathbb{C}^*$ along $z=0$) and a cycle of $k$ rational curves over $\{x=y=0\}\subset\mathbb H$.
Now, we can relate this construction to the Gibbons-Hawking ansatz by evaluating $i_X\Omega_I$ and $i_X\Omega_J=i_X j^*(\Omega_I)$ on the universal cover of the nondegenerate part of $M$:
\[
\begin{split}
\mu_1&=\arg w,\\
\mu_2&=\log|w|=\frac{1}{2}\log(x^2+y^2),\\
\mu_3&=\log(x^2+y^2+z^2)-\frac{1}{2}\log(x^2+y^2).
\end{split}
\]
Or equivalently $\mu_+=\frac{1}{2}\log(x^2+y^2+z^2)$ and $\mu_-=\frac{1}{2}(\log(x^2+y^2)-\log(x^2+y^2+z^2))$.
The range of the moment map is $\{\mu_-<0\}\subset \mathbb{R}^3_{\pmb\mu}$.
We can rewrite the metric $g$ as
\[
g=W\underbrace{\left(\frac{z^2}{x^2+y^2+z^2}\right)^2\til h}_{:=h}+ W^{-1}\eta^2,\quad W:=\frac{x^2+y^2+z^2}{z^2}V.
\]
With this change of perspective, the equation $\Delta_{\til h}V=0$ translates into the equivalent form ~\eqref{f:W_laplace} for $W$ and the curvature form $\beta=*_{\til h}dV$ matches the expression in ~\eqref{f:beta0_def}.
As before, the moment map is defined only on the universal cover of the nondgenerate set, and the entire parabolic Inoue surface is the completion of a quotient of a local model provided by Theorem~\ref{t:nondegenerate_gk_description}.
\end{ex}
\section{Generalized K\"ahler-Ricci solitons} \label{s:GKRS}
In this section we turn to analyzing generalized K\"ahler-Ricci solitons. We first establish the fundamental point that for a pluriclosed soliton, the vector field $V = \tfrac{1}{2} (\theta^{\#} - \nabla f)$ is holomorphic, and $IV$ is Killing. We thus have potentially \emph{two} Killing fields in the case of generalized K\"ahler-Ricci solitons. By further analysis we prove in Proposition \ref{p:IV_I_J_holomorphic} that either these vector fields both vanish, yielding hyperK\"ahler structure, one is a quotient of the standard Hopf surface, or one of these vector fields is biholomorphic. The last case of course is our main interest, yielding a structure in the setting of \S \ref{s:GKsymm}. A key point in this setting is to obtain a scalar reduction for the soliton equation, and in particular in Proposition \ref{p:invGKsoliton} we obtain an explicit local form for the angle function $p$ in $\mathbb R^3_{\pmb \mu}$ up to a choice of two real parameters. We finish by rederiving the solitons constructed in \cite{SU} in this ansatz, and show how it leads to a certain explicit solution $\til{W}$ to the elliptic equation \eqref{f:W_laplace2} depending on the real parameters determining $p$.
\subsection{Fundamental structure}
Recall that a GK structure $(M,g,I,J)$ is a (steady, gradient) soliton, if there exists a function $f\colon M\to \mathbb{R}$ such that
\begin{equation}\label{f:soliton}
\left\{
\begin{split}
\Rc - \tfrac{1}{4} H^2 + \nabla^2 f =&\ 0\\
d^* H + i_{\nabla f} H =&\ 0
\end{split}
\right.
\end{equation}
where $\Rc=\Rc^{M,g}$ is the Ricci curvature of $g$ and $H:=-d^c_I\omega_I=d^c_J\omega_J$.
We recall the well-known fact that the K\"ahler-Ricci soliton equations imply that the soliton vector field $\nabla f$ is holomorphic, with $I \nabla f$ a Killing field. It turns out that a similar phenomenon holds here for the modified vector field $\theta^{\#} - \nabla f$ (previously established in \cite{st-19-soliton} for the 4-dimensional case). Specifically, we have the following equivalent formulation of the soliton system. In this statement, $(M,g,I)$ is not necessarily a part of GK structure, but merely a pluriclosed Hermitian manifold.
\begin{prop} \label{p:solitonVF} Let $(M^{2n}, g, I)$ be a pluriclosed Hermitian manifold (i.e. $dd^c_I\omega_I=0$). Then $g$, $H=d^c_I\omega_I$ and $f\colon M\to \mathbb{R}$ solve the soliton system~\eqref{f:soliton} if and only if the vector field
\begin{align*}
V:=&\ \tfrac{1}{2} \left(\theta_I^{\#} - \nabla f \right).
\end{align*}
satisfies
\begin{equation}\label{f:solitonB}
\begin{cases}
\mathcal L_{V}I=0\\
\mathcal L_{IV}g=0\\
\mathcal L_{V}\omega_I=\rho_I^{1,1}
\end{cases}
\end{equation}
In particular $IV$ is holomorphic and Killing.
\begin{proof}
We claim that after splitting the soliton equations~\eqref{f:soliton} by holomorphic types we have the following equivalences:
\begin{equation}\label{f:soliton_equivalences}
\begin{split}
(\Rc - \tfrac{1}{4} H^2 + \nabla^2 f)^{2,0} &=0\
\Longleftrightarrow\ (\mathcal L_V g)^{2,0}=0\\
(d^* H + i_{\nabla f} H)^{2,0}&=0\ \Longleftrightarrow\ (\mathcal L_V\omega_I)^{2,0}=0\\
(\Rc - \tfrac{1}{4} H^2 + \nabla^2 f)^{1,1} &=0\
\Longleftrightarrow\ (\rho_I(I\cdot,\cdot)+\mathcal L_V g)^{1,1}=0\\
(d^* H + i_{\nabla f} H)^{1,1}&=0\ \Longleftrightarrow\ (\mathcal L_{IV}\omega_I)^{1,1}=0.
\end{split}
\end{equation}
Indeed, denote by $\nabla^C$ the Chern connection on $(M,g,I)$ and by $T_{ij}^k$ the components of its torsion tensor. Let $\theta_I=\theta_i dz^i+\theta_{\bar j} d\bar{z^j}$, $\theta_i=T_{ik}^k$ and $\theta_I^\#=g^{i\bar j}(\theta_i \partial_{\bar j}+\theta_{\bar j} \partial_i)$ be the local coordinate expressions for the Lee form and the Lee vector field respectively.
By the computations of~\cite[Prop.\,6.3]{PCFReg} we know that
\begin{equation}\label{f:ricci_bismut_identity}
\Rc-\tfrac{1}{4}H^2+\mathcal L_{\tfrac{1}{2}\theta_I^\#} g=-\rho_I^{1,1}(I\cdot,\cdot),
\end{equation}
Therefore,
\begin{equation*}
(\Rc-\tfrac{1}{4}H^2+\nabla^2 f)^{2,0}=\tfrac{1}{2}(-\mathcal L_{\theta_I^\#}g+\mathcal L_{\nabla f} g)^{2,0}=-(\mathcal L_V g)^{2,0}.
\end{equation*}
This establishes the first equivalence of~\eqref{f:soliton_equivalences}.
Furthermore, by an explicit computation we have
\begin{equation*}
\begin{split}
(d^*H)^{2,0}&=
-\sqrt{-1}\,\bar{\partial}^*\partial \omega=
\sqrt{-1}[\nabla^C_k T_{ij}^k+\theta_k T_{ij}^k]dz^i\wedge dz^j =
\sqrt{-1}(\mathcal L_{\theta_I^\#}\omega)^{2,0}.
\end{split}
\end{equation*}
Next we have
\[
(i_{\nabla f}H)^{2,0}=
-\sqrt{-1}(i_{\nabla f}d\omega)^{2,0}=
-\sqrt{-1}(i_{\nabla f}d\omega+dd^cf)^{2,0}=
-\sqrt{-1}(\mathcal L_{\nabla f}\omega)^{2,0}.
\]
Collecting the last two identities together we find:
\begin{equation*}
(d^*H+i_{\nabla f} H)^{2,0}=2\sqrt{-1}(\mathcal L_V\omega)^{2,0}.
\end{equation*}
This establishes the second equivalence of~\eqref{f:soliton_equivalences}.
The two quantities $(\mathcal L_V\omega)^{2,0}$ and $(\mathcal L_V g)^{2,0}$ represent the $g$-symmetric and $g$-antisymmetric parts of the operator $\mathcal L_V I$, thus the first two identities of~\eqref{f:soliton_equivalences} are equivalent to
\[
\mathcal L_V I=0.
\]
Using again identity~\eqref{f:ricci_bismut_identity} we conclude that
\[
(\Rc - \tfrac{1}{4} H^2 + \nabla^2 f)^{1,1}=-(\rho_I(I\cdot,\cdot)+\mathcal L_V g)^{1,1}.
\]
This is the third identity of~\eqref{f:soliton_equivalences}. If we also assume that $V$ is holomorphic, we can equivalently rewrite it as
\[
\rho_I^{1,1}=\mathcal L_V\omega_I.
\]
For the last identity we observe that
\begin{equation*}
\begin{split}
(d^*H)^{1,1}=&
\sqrt{-1}(\bar{\partial^*}\bar{\partial}-\partial^*\partial)\omega=
g^{m\bar n}(
\nabla^C_m T_{\bar n\bar j i}
+\theta_m T_{\bar n \bar j i}
-\nabla^C_{\bar n}T_{mi\bar j}
-\theta_{\bar n} T_{mi\bar j}
)
dz^i\wedge d\bar{z^j}\\=&
g^{m\bar n}(
\nabla^C_{\bar j}T_{im\bar n}
-\nabla^C_i T_{\bar j\bar n m}
+\theta_m T_{\bar n\bar j i}
-\theta_{\bar n}T_{mi\bar j}
)
=(\mathcal L_{I\theta^\#}\omega)^{1,1},
\end{split}
\end{equation*}
where we have used the following corollary of the differential Bianchi identity:
\[
g^{m\bar n}(
\nabla^C_m T_{\bar n \bar j i}-\nabla^C_{\bar n} T_{m i \bar j})
=
g^{m\bar n}(
\nabla^C_{\bar j}T_{im\bar n}-\nabla^C_i T_{\bar j\bar n m}
).
\]
At the same time
\[
(i_{\nabla f}H)^{1,1}=-(i_{I \nabla f}d\omega+\underbrace{di_{I\nabla f}\omega}_{d^2f=0})^{1,1}=-(\mathcal L_{I\nabla f}\omega)^{1,1}.
\]
Combining the above two equations gives
\[
(d^*H+i_{\nabla f}H)^{1,1}=(\mathcal L_{IV}\omega)^{1,1},
\]
as claimed. This finishes the proof of~\eqref{f:soliton_equivalences}. It remains to notice that the right hand side equations of~\eqref{f:soliton_equivalences} are immediately equivalent to~\eqref{f:solitonB}.
\end{proof}
\end{prop}
The interpretation of the soliton equations \eqref{f:solitonB} becomes particularly useful on a non-degenerate GK manifold, since the Bismut-Ricci tensors $\rho_I$ and $\rho_J$ can be expressed through the associated \emph{Ricci potential}, as
derived in \cite{ASNDGKCY}. We recall the basic elements, referring to \cite{ASNDGKCY} for further detail. Given a nondegenerate generalized K\"ahler structure $(M^{2n}, g, I, J)$, let
\begin{align*}
\Phi = \frac{1}{2}\log \frac{\det (I - J)}{\det (I + J)}.
\end{align*}
In dimension $4$ this is determined by $p$, in particular
\begin{align*}
\Phi = \log \frac{1-p}{1 + p}.
\end{align*}
The function $\Phi$ is in a sense analogous to the usual Ricci potential in K\"ahler geometry. In particular, one has
\begin{align*}
\rho_I = - \tfrac{1}{2} d J d \Phi, \qquad \rho_J = - \tfrac{1}{2} d I d \Phi.
\end{align*}
Furthermore, (\cite{ASNDGKCY} Lemma 3.8), the difference of Lee vector fields is $\Omega$-Hamiltonian with potential function $\Phi$, i.e.
\begin{equation}\label{f:thetaPhi}
\begin{split}
\left(\theta_I^{\#} - \theta_J^{\#}\right) = \sigma d \Phi.
\end{split}
\end{equation}
\begin{defn} \label{d:GKsoliton} If a generalized K\"ahler manifold $(M^{2n}, g, I, J)$ is a soliton, then both pluriclosed structures $(M^{2n},g,I)$ and $(M^{2n},g,J)$ must satisfy equations~\eqref{f:solitonB}. Thus we obtain vector fields
\begin{align*}
V_I = \tfrac{1}{2} \left( \theta_I^{\#} - \nabla f \right), \qquad V_J = \tfrac{1}{2} \left( \theta_J^{\#} - \nabla f \right),
\end{align*}
which are holomorphic with respect to the corresponding complex structures, as well as Killing fields $I V_I$ and $J V_J$. We say the \emph{rank} of the soliton is
\begin{align*}
\sup_{p \in M} \dim \spn \{I V_I, J V_J \} \subset T_p M.
\end{align*}
\end{defn}
\begin{ex}[GK solitons the standard Hopf surface and its universal cover]\label{ex:hopf_round_soliton}
Let $(M,g,I,J)$ be the standard Hopf surface with the round metric and GK structure of Example~\ref{ex:Hopf1}. This structure solves~\eqref{f:soliton} with $f=0$. Since $g$ is conformally K\"ahler with a factor $(|z_1|^2+|z_2|^2)^{-1}$, we find that $\theta=-d\log(|z_1|^2+|z_2|^2)$. Hence $V_I$ and $V_J$ are nowhere zero and are given by $\pm \tfrac{1}{2}\log(|z_1|^2+|z_2|^2)$. In particular $IV_I$ and $JV_J$ are not proportional to each other, so $(M,g,I,J)$ with $f=0$ is a GK soliton of rank two.
In~\cite{GauduchonIvanov} Gauduchon and Ivanov proved that this is the only non-K\"ahler compact complex surface admitting a GK soliton with $f=0$.
Let $\til M$ be the universal cover of $M$ with the GK structure inherited from $M$. Clearly, we still can think of $(\til M,g,I,J)$ and a function $f_0=0$ as a GK soliton of rank two. Consider now a new function $f_1:=-\tfrac{1}{2}\log(|z_1|^2+|z_2|^2)$. The gradient flow of this function preserves the entire GK structure, therefore $f_1$ (and any of its multiples) is also a solution to the soliton system. For this choice of the soliton function, we will have $V_J=0$ and $V_I=2\nabla f_1$. In particular, $(\til M,g,I,J)$ with a function $f_1$ is now a rank \emph{one} soliton. However, since $f_1$ does not descend to the quotient $M$, this solution makes sense only on the universal cover of $M$.
The observed phenomenon~--- different soliton functions compatible with the same GK structure~--- can potentially occur only on non-compact manifolds with a gradient vector field preserving the entire GK structure. In Example~\ref{ex:hopf_diagonal_soliton} below we review our construction from~\cite{SU} and show that the universal cover of a diagonal Hopf surface admits a rank one GK soliton.
\end{ex}
Now we observe further fundamental properties of the vector fields $V_I, V_J$ on the generalized K\"ahler Ricci solitons.
\begin{lemma} \label{l:jointholo} Let $(M^{2n}, g, I, J)$ be a generalized K\"ahler-Ricci soliton. Then either
\begin{enumerate}
\item $g$ is compatible with a one-parameter family of distinct complex structures,
\item $I V_I$ is $J$-holomorphic and $J V_J$ is $I$-holomorphic. In this case $[I V_I, J V_J] = 0$.
\end{enumerate}
\begin{proof} Suppose $I V_I$ is not $J$-holomorphic. Since $I V_I$ is an $I$-holomorphic Killing field, we can pull back the GK structure $(g, I, J)$ by the one-parameter family of diffeomorphisms $\phi_t$ generated by $I V_I$ to see that $(\phi_t^* g, \phi_t^* I, \phi_t^* J) = (g, I, \phi_t^* J)$ is GK. In particular, $g$ is compatible with the one-parameter family of complex structures $\phi_t^* J$, which are distinct since $I V_I$ is not $J$-holomorphic. An identical argument holds if $J V_J$ is not $I$-holomorphic.
For the second claim, we prove that if $IV_I$ is $J$-holomorphic, then $[I V_I, J V_J] = 0$. Indeed, since $IV_I$ is Killing and $I$-holomorphic, we find $\mathcal L_{IV_I}\theta_I=\mathcal L_{IV} \left(*d^c_I\omega_I\right)=0$, thus
\begin{align*}
\mathcal L_{I V_I} df=\mathcal L_{IV_I} (-\theta_I+df)=-2g([IV_I,V_I],\cdot)=0.
\end{align*}
Since $I V_I$ is Killing it also follows that $\mathcal L_{IV_I}(g^{-1}df)=[I V_I, \nabla f] = 0$. Using that $I V_I$ is both $I$ and $J$-holomorphic we then obtain
\begin{align*}
[I V_I, J V_J] = J [I V_I, V_J] =
- J [I V_I, V_I] = 0,
\end{align*}
where we have used the identity $V_J=-V_I-\nabla f$.
\end{proof}
\end{lemma}
Using structures special to four dimensions, we can show in this case that without loss of generality, $IV_I$ is $J$-holomorphic, unless $(M,g,I,J)$ has a very special geometry.
\begin{prop}\label{p:IV_I_J_holomorphic} Let $(M^4, g, I, J)$ be a generalized K\"ahler-Ricci soliton. Up to exchanging the roles of $I$ and $J$, at least one of the following holds:
\begin{enumerate}
\item Both $V_I = 0$ and $V_J = 0$, and $(M^4, g, I)$ is hyperK\"ahler.
\item $I V_I$ is nonzero and $J$-holomorphic.
\item $I V_I$ is nonzero, and is not $J$-holomorphic, and $J V_J$ is nonzero, and is not $I$-holomorphic. In this case $(M^4, g, I)$ and $(M^4, g, J)$ are isometric to a quotient of the universal cover of the standard Hopf surface.
\end{enumerate}
\begin{proof} To address case (1), assume that both $\theta_I^{\#}-\nabla f$ and $\theta_J^{\#}-\nabla f$ vanish. Since on a $4$-dimensional GK manifold one has $\theta_I=-\theta_J$, we have $\theta_I^{\#} =\nabla f=0$, i.e., $(M,g,I)$ and $(M,g,J)$ are K\"ahler, Ricci flat and in fact $(g, I)$ is part of a hyperK\"ahler structure as discussed in \S \ref{s:background}.
Now assume without loss of generality that $V_I$ is nonzero. We first suppose that $V_J$ is zero. Using that $\theta_I = - \theta_J$ and $\theta_J^{\#} = \nabla f$ it follows that
\begin{align*}
\sigma d \Phi = (\theta_I - \theta_J)^{\#} = -2 (\theta_J)^{\#} = -2 \nabla f.
\end{align*}
Noting that now $V_I = -\nabla f$, we have that $I \nabla f$ is $I$-holomorphic and Killing. Also, since $V_J = 0$, by \eqref{f:solitonB} we have $\rho_J^{1,1} = 0$. We also know that $\rho_J^{2,0}=0$, since Bismut-Ricci curvature is conformally invariant, and $(M,g,J)$ is conformally equivalent to a K\"ahler metric $(M,e^{-f}g,J)$. Thus
\begin{align*}
-2\mathcal L_{I \nabla f} \Omega = d i_{I \sigma d \Phi} \Omega = 2d I d \Phi = -2 \rho_J = 0.
\end{align*}
Also, since $\sigma$ is type $(2,0) + (0,2)$ with respect to $I$,
\begin{align*}
\mathcal L_{I \nabla f} \Phi = I d \Phi (\nabla f) = \tfrac{1}{2} \sigma( d \Phi, I d \Phi) = 0.
\end{align*}
To summarize, we have shown that the vector field $I V_I = - I \nabla f$ preserves $g$, $\sigma$, and $\Phi = \log \frac{1-p}{1+p}$. By rearranging formula \eqref{f:K}, it is possible to express $J$ in terms of $\sigma, g$, and $p$, and thus
\begin{align*}
\mathcal L_{I V_I} J = \mathcal L_{\nabla f} J = 0,
\end{align*}
yielding case (2).
Now we address the remaining case, where $I V_I$ is nonzero and not $J$-holomorphic and $J V_J$ is nonzero and not $I$-holomorphic. Using first the hypothesis that $I V_I$ is not $J$-holomorphic, by Lemma \ref{l:jointholo} it follows that $g$ is compatible with a one-parameter family of complex structures, given by $J_t = \phi_t^* J$ where $\phi_t$ is generated by $I V_I$. It follows from (\cite{PontecorvoCS} Corollary 1.6) that $(M^4, g)$ is antiself-dual, so $W_+ = 0$. As is well-known, in the K\"ahler setting $W_+$ is determined by the scalar curvature. In the Hermitian setting $W_+$ is determined by the scalar curvature and $d \theta$, and in particular by (\cite{BoyerConformalDuality} Lemma 1), the vanishing of $W_+$ implies that $d \theta_I$ is antiself-dual (noting that the proof of the equivalence of items 1-3 in that statement is local and does not need the compactness hypothesis and hence applies in this setting). Taking the Hodge dual of the second soliton equation, we get an equivalent identity
\begin{align*}
d (e^{-f} \theta_I) = e^{-f} \left( d \theta_I - df \wedge \theta_I \right)=0.
\end{align*}
It follows that $df \wedge \theta_I$ is also antiself-dual, but as a wedge product of one-forms this can only be antiself-dual if it vanishes, hence we conclude $d \theta_I = 0$. By passing to the universal cover we obtain a smooth function $\phi$ such that $\theta_I = d \phi$. Now note that we have a one-parameter family of generalized K\"ahler structures $(g, I, J_t)$, all of which have generically nondegenerate associated Poisson tensor. It follows that
\begin{align*}
\theta_{J_t} = - \theta_I = - d \phi.
\end{align*}
Hence we obtain a one-parameter family of (possibly incomplete) K\"ahler structures $(g_0 = e^{- \phi} g, J_t)$ with fixed Riemannian metric. Thus the reduced holonomy group of $(M,g_0)$ commutes with $J_t$, so must be a subgroup of $SU(2)$, and $g_0$ is hyperK\"ahler with complex structures $(I', J, K)$ (note that the original complex structure $I$ is \emph{not} part of this hyperK\"ahler structure, otherwise the original structure is already hyperK\"ahler). Note that, by the conformal invariance of the Bismut-Ricci tensor in dimension $4$, it then follows that $\rho_J = 0$, and thus $V_J$ is a Killing field.
Now we use that $J V_J$ is nonzero, and also not $I$-holomorphic. The same line of arguing above holds with the roles of $I$ and $J$ reversed, leading to the conclusion that $\rho_I = 0$ and $V_I$ is a Killing field. It follows that $\theta_I = V_I - V_J$ is a nonzero Killing field, and thus also is parallel. The equation $(\rho_I)^{1,1} = 0$ now takes the simplified form
\begin{gather}
\Rc - \tfrac{1}{4} H^2 = 0.
\end{gather}
The universal cover of $(M^4, g, I, J)$ now splits according to the leaves of $\theta_I$, and by the last equation and the fact that $H = * \theta_I$ it follows that the transverse metric has constant positive sectional curvature, and is thus a quotient of $S^3$. It follows that the universal cover of $(M^4, g)$ is a standard cylinder $S^3 \times \mathbb R$, and the complex structure $I$ and $J$ must be that of the lift of a standard Hopf surface by \cite{GauduchonWeyl}.
\end{proof}
\end{prop}
\begin{rmk}\label{r:rank1_s1}
Proposition~\ref{p:solitonVF} implies that the dimension of the isometry group $\Isom(g)$ of rank one soliton $(M,g,I,J)$ is at least 1. Going further, it is possible to reduce the study of the `least symmetric' rank one solitons to the study of the rank one solitons among GK manifolds with $S^1$ symmetry as in Section~\ref{s:GKsymm}.
\end{rmk}
\begin{prop}\label{p:one-diml_isometry}
Let $(M,g,I,J)$ be a complete GK soliton of rank one. Assume that the isometry group $\Isom(g)$ is one-dimensional. Then $\Isom(g)$ preserves GK structure and we have two possibilities
\begin{enumerate}
\item $G\simeq \mathbb{R}$ acts freely and properly on $M$. In particular, there is a free $S^1\simeq\mathbb{R}/\mathbb{Z}$ action on a GK soliton $(M/\mathbb{Z},g,I,J)$;
\item $G\simeq S^1$.
\end{enumerate}
\end{prop}
\begin{proof}
First we note that we can assume $IV_I$ is both $I$ and $J$ holomorphic, since otherwise by Proposition~\ref{p:IV_I_J_holomorphic} $(M,g)$ is isometric to a quotient of the standard Hopf surface, and in particular $\dim\Isom(g) > 1$.
Without loss of generality we assume that $IV_I$ is a nonzero vector field. Then $IV_I$ generates a one-dimensional subgroup $G$ of $\Isom(g)$. Since the isometry group is one-dimensional itself, $\Isom(g)=G$. Thus either $G\simeq \mathbb{R}$ or $G\simeq \mathbb{R}$. In either case the action of $G$ must be proper~\cite[Prop.\,1]{YauIsometries}, and if $G\simeq \mathbb{R}$, then the action must be also free, since the isotropy subgroups of an isometric action are compact.
\end{proof}
\subsection{Invariant solitons}
Let $(M,g,I,J)$ be a GK soliton of rank one. Then there is a generically nonzero vector field $X$ preserving the structure, and we can use the construction of \S\ref{s:GKsymm} to give a local description of $(M,g,I,J)$ in terms of $p$ and $W$. In this subsection we show that on a rank one soliton $p$ can be completely described locally in the nondegenerate locus. The key ingredients of this description are the simple expression of the Bismut-Ricci curvature and the fact that the difference of the Lee vector fields is a Hamiltonian vector field as discussed at the start of this section.
\begin{prop} \label{p:invGKsoliton} Suppose $(M^{2n}, g, I, J)$ is a nondegenerate generalized K\"ahler-Ricci soliton. Then the soliton vector fields $V_I$ and $V_J$ preserve $\Omega$, and
\begin{align*}
\Phi =&\ 2\psi_{V_I} - 2\psi_{V_J}+\mbox{const},
\end{align*}
where $\psi_{V_I}, \psi_{V_J}$ denote the associated local Hamiltonian potentials.
In particular, in the case $n=2$, if one has a rank one soliton with $V_I = {a_I} IX$, $V_J = {a_J} JX$ then locally $\Phi$ satisfies
\begin{equation}\label{f:Phi_soliton}
\Phi = a_+ \mu_+ + a_- \mu_-+ \mbox{const},\quad a_+=-2a_I+2a_J,\ a_-:=-2a_I-2a_J
\end{equation}
with the associated soliton function satisfying
\begin{align*}
df &= \frac{1}{2}\left(p(a_+d\mu_++a_-d\mu_-)- a_+d\mu_+ + a_-d\mu_-\right).
\end{align*}
Conversely, if $\Phi$ is given by~\eqref{f:Phi_soliton}, then $ \frac{1}{2}\left(p(a_+d\mu_++a_-d\mu_-)- a_+d\mu_+ + a_-d\mu_-\right)$ is closed, and if it is exact with potential function $f$, then $(M,g,I,J)$ is a rank one soliton with soliton potential function $f$.
\begin{proof}
Consider the solution to GKRF in the $I$-fixed gauge with initial data $(g, I, J)$. By (\cite{ASNDGKCY} Lemma 5.2) we know that $\dt \Omega = 0$. On the other hand all of the data evolves by pullback by the family of diffeomorphisms generated by $V_I$. Thus
\begin{align*}
0 =&\ \dt \Omega = \mathcal L_{V_I} \Omega = d \left( i_{V_I} \Omega \right).
\end{align*}
The same argument using GKRF in the $J$-fixed gauge implies that vector field $V_J$ also preserves symplectic structure $\Omega$. Let $\psi_{V_I}$ and $\psi_{V_J}$ be local Hamiltonian potentials for vector fields $V_I$ and~$V_J$:
\[
i_{V_I}\Omega=d\psi_{V_I},\quad i_{V_J}\Omega=d\psi_{V_J}.
\]
At the same time, by definition of $V_I$ and $V_J$ we have $V_I = \tfrac{1}{2} \left(\theta_I^{\#} - \nabla f \right)$, $V_J = \tfrac{1}{2} \left( \theta_J^{\#} - \nabla f \right)$. Thus by \eqref{f:thetaPhi} we see that $2(V_I - V_J)$ is $\Omega$-Hamiltonian with potential function $\Phi$, therefore:
\begin{align*}
\Phi =&\ 2\psi_{V_I} - 2\psi_{V_J}+\mbox{const}.
\end{align*}
If $n=2$ and $(M,g,I,J)$ is a rank one soliton, then there exists a vector field $X$ preserving the GK structure and constants $a_I,a_J$ such that $V_I=a_IIX$ and $V_J=a_JJX$. By the definition of $\mu_2$ and $\mu_3$, we have
\[
i_{V_I}\Omega=a_Id\mu_2,\quad i_{V_J}\Omega=a_Jd\mu_3.
\]
Hence, by the first part of the proposition, $\Phi =2{a_I} \mu_2 - 2{a_J} \mu_3+\mbox{const}$. Introducing new constants $a_+:=2a_I-2a_J$, $a_-:=2a_I+2a_J$ we can rewrite $\Phi$ as
\begin{align*}
\Phi = {a_+} \mu_+ + {a_-} \mu_-+\mbox{const}.
\end{align*}
To find the soliton function $f$, we observe that $- \nabla f = V_I + V_J$, therefore
\begin{align*}
i_{\nabla f} \Omega = -{a_I} d \mu_2 -{a_J} d \mu_3=-\tfrac{1}{2}(a_-d\mu_++a_+d\mu_-),
\end{align*}
where the second identity is the definition of constants $a_-$ and $a_+$. Referring to the explicit form of $\Omega$ in (\ref{f:Omegaexplicit}) then gives the formula for $\nabla f$. Since $g=Wh+W^{-1}\eta^2$, the latter yields the claimed expression for $df$.
Conversely, assume that in the Gibbons-Hawking ansatz of Theorem~\ref{t:nondegenerate_gk_description_v2} we have locally
\[
\Phi=a_+\mu_++a_-\mu_-+\mbox{const}
\]
for some constants $a_+$ and $a_-$. Then $p$ is also a function of $a_+\mu_++a_-\mu_-$, forcing the 1-form $-a_+d\mu_++a_-d\mu_-+p(a_+d\mu_++a_-d\mu_-)$ to be closed. Assume further that this form is exact and equals $df$ for some function $f$. We claim that the corresponding GK structure $(M,g,I,J)$ is a GK soliton solving~\eqref{f:solitonB} with the soliton function $f$.
First we note that starting with the exact expressions for $\Phi$ and $df$ and following the above computations backwards, one concludes that $V_I=\tfrac{1}{2}(\theta_I-\nabla f)$ and $V_J=\tfrac{1}{2}(\theta_J-\nabla f)$ are given by
\[
V_I=a_IIX,\quad V_J=a_J JX.
\]
Since $IV_I$ is Killing and holomorphic, it remains to prove that
\[
\rho_I^{1,1}=\mathcal L_{V_I}\omega_I.
\]
Using the relation between $\omega_I=g(I\cdot,\cdot)$ and $\Omega=2[I,J]^{-1}g$ we find:
\[
\omega_I=\tfrac{1}{2}I[I,J]\Omega=-(J\Omega)^{1,1}.
\]
Together with the identity $\rho_I=-\tfrac{1}{2}dJd\Phi$ this gives
\[
\begin{split}
\rho_I^{1,1}-\mathcal L_{V_I}\omega_I&=
-\left(\tfrac{1}{2}dJd\Phi+\mathcal L_{V_I}(J\Omega)\right)^{1,1}=
\left(-dJi_{V_I-V_J}\Omega+d i_{V_I}J\Omega\right)^{1,1}\\&=
\left(-dJi_{V_I}\Omega+di_{V_I}J\Omega \right)^{1,1}=0
\end{split}
\]
where in the third identity we used the fact that $Ji_{V_J}\Omega=-a_Ji_X\Omega$ is closed. This proves the claim that $(M,g,I,J)$ with $\Phi$ locally given by $a_+\mu_++a_-\mu_-+\mbox{const}$ is a rank one GK soliton, as long as the closed form $-a_+d\mu_++a_-d\mu_-+p(a_+d\mu_++a_-d\mu_-)$ is exact.
\end{proof}
\end{prop}
\begin{ex}[Rank one solitons on diagonal Hopf surfaces]\label{ex:hopf_diagonal_soliton}
Let $(M,g,I,J)$ be a GK structure on a diagonal Hopf surface as in Example~\ref{ex:hopf_diagonal_gk} defined by a single scalar function $p$. Assume additionally that the parameters $a,b\in\mathbb{R}$ satisfy $a/b=m^2/n^2$, where $m,n$ are coprime integers, and consider the $S^1$ action on $M$ induced by the action on $\mathbb{C}^2$:
\[
u\cdot (z_1,z_2)=(u^mz_1,u^nz_2).
\]
Then using the expression~\eqref{f:diagonal_hopf_mu} for the moment map $\mu$, we find
\begin{equation}
\begin{split}
\frac{2}{n}\mu_++\frac{2}{m}\mu_-=\left(1+\frac{a}{b}\right)\left(\frac{b}{a}x_1-x_2\right)+\left(1-\frac{a}{b}\right)\frac{\chi}{2},
\end{split}
\end{equation}
where, as in Example~\ref{ex:hopf_diagonal_gk}, $\chi$ is the antiderivative of $p$ evaluated at $2(\tfrac{b}{a}x_1-x_2)$.
The above calculation makes sense for any function $p$ defining the GK structure on $M$. However, as we proved in~\cite{SU}, $(M,g,I,J)$ is a soliton if and only if $p$ solves an ODE,
\[
\left(\log\frac{1-p}{1+p}\right)'=\frac{1}{2}\left(1-\frac{a}{b}\right)p+\frac{1}{2}\left(1+\frac{a}{b}\right)
\]
which is equivalent to the identity:
\[
\Phi = \frac{2}{n}\mu_++\frac{2}{m}\mu_-+\mbox{const},
\]
in accordance with Proposition~\ref{p:invGKsoliton}. In particular, the solitons constructed in~\cite{SU} can be viewed as rank one solitons. (Strictly speaking we have established this claim only in the case $a/b=m^2/n^2$, with $n,m\in\mathbb{Z}$. In general, if $a/b$ is arbitrary, we still can run the same argument, keeping in mind that $X=m\partial_{y_1}+n\partial_{y_2}$ only generates an $\mathbb{R}$ action).
\end{ex}
The above example provides a function $W$ solving equation~\eqref{f:W_laplace2} with $a_+=2/n$ and $a_-=2/m$. Motivated by this example, it is easy to construct an explicit solution $\til{W}$ to the equation~\eqref{f:W_laplace2} with any given constants $a_-,a_+$ determining function $p$. In what follows, we will use this \emph{baseline} solution to modify the GK structure on a given soliton. A key point is that $\til W$ is bounded below by a positive constant and on the set $\{|p|<1-\delta\}$ we have bounds on $\til W$ and $|d\til W|_h$.
\begin{lemma}\label{l:W0_solution} Given constants $a_+,a_-\in\mathbb{R}$ and setting $\Phi=\log\frac{1-p}{1+p} = a_+ \mu_+ + a_- \mu_-$, the function
\begin{equation}\label{f:W0_def}
\til W=(a_+^2(1+p)+a_-^2(1-p))^{-1}
\end{equation}
satisfies equation~\eqref{f:W_laplace2}.
\end{lemma}
\begin{proof}
As both $p$ and $\til W$ are functions of the same linear combination of $\mu_+$ and $\mu_-$, we obtain
\begin{align*}
\til W_{11} + \tfrac{1}{2} \left( (1+p) \til W \right)_{++} + \tfrac{1}{2} \left( (1-p) \til W \right)_{--} =&\ \tfrac{1}{2} \left[ a_+^2 ((1 + p) \til W)'' + a_-^2 ((1-p) \til W)'' \right]\\
=&\ \tfrac{1}{2} \left[ \left( a_+^2 (1 + p) + a_-^2 (1 - p) \right) \til W \right]''=0,
\end{align*}
as claimed.
\end{proof}
\section{Analysis of the completion} \label{s:completion}
Let $(M,g,I,J)$ be a GK soliton of rank one. Following Remark~\ref{r:rank1_s1} we assume that one of the vector fields $IV_I$ or $JV_J$ generates an $S^1$-action on $M$ preserving the GK structure. We impose a lower bound on the Ricci curvature $\Rc^{M,g}$, and finiteness of the set of isolated fixed points. The assumptions imposed on $(M,g,I,J)$ in the rest of the paper are summarized in the following definition.
\begin{defn}[Complete regular rank one solitons]\label{d:regular_rank_one}
Let $(M^4,g,I,J)$ be a generalized K\"ahler manifold such that
\begin{enumerate}
\item $(M,g)$ is complete;
\item The Poisson tensor $\sigma$ of $(M,g,I,J)$ is not identically zero;
\item $(M,g,I,J)$ is a \emph{rank one} soliton, i.e. it satisfies the equations of Definition~\ref{d:GKsoliton} and $\mathrm{span}\{IV_I,JV_J\}$ is generically one-dimensional;
\item Either $IV_I$ or $JV_J$ generates an $S^1$ action preserving the GK structure $(M,g,I,J)$ with finitely many fixed points. The latter is automatically satisfied if $\dim H^*(M,\mathbb{R})<\infty$;
\item On the nondegeneracy locus of $\sigma$,
\begin{align*}
\Phi=\log\frac{1-p}{1+p}=a_+\mu_++a_-\mu_-+\mbox{const},
\end{align*} where $\mu_+$ and $\mu_-$ are defined locally up to an additive constant, and without loss of generality we assume that $a_+\neq 0$;
\item There exists a constant $k$ such that $\Rc^{M,g}>-k^2$;
\end{enumerate}
Then we call $(M,g,I,J)$ a \emph{complete regular rank one soliton}.
\end{defn}
Our ultimate goal is to give an exhaustive classification of such GK manifolds. So far we have obtained an explicit local description in the nondegeneracy locus of $\sigma$ in terms of the functions $W$ and $p$, where now, thanks to Proposition \ref{p:invGKsoliton}, $p$ is determined by the choice of constants ${a_+}, {a_-}$. To address the global structure of complete rank one solitons we have to understand how these local models can be patched together, and to determine their behavior and possible extension to the degeneracy locus. In the end we will obtain a global definition of the moment map together with a complete classification of the possible images.
\subsection{Local structure near the locus \texorpdfstring{$\{I=\pm J\}$}{}}
In this subsection we determine the local structure of a GK soliton near the degeneracy locus for the Poisson tensor $\sigma$. Since $\sigma$ does not vanish identically, $I$ and $J$ induce the same orientation, and $\sigma$ will vanish precisely when $I = \pm J$. Define
\[
\mathbf T_+:=\{x\in M\ |\ I_x= J_x \},\quad \mathbf T_-:=\{x\in M\ |\ I_x=- J_x \}.
\]
The sets $\mathbf T_{\pm}$ are one-dimensional complex analytic subsets of $M$ with respect to either complex structure. Our aim is to understand the local structure of the generator of the $S^1$ action $X$ and the symplectic form $\Omega$ in the image of $\pmb{\mu}$, near points in $\mathbf{T}=\mathbf{T}_+\cup \mathbf{T}_-$.
Note that the complex-valued form $\Omega_I:=\Omega-\sqrt{-1}I\Omega$, initially defined on $M \backslash\ \mathbf T$, extends to an $I$-meromorphic $(2,0)$ form on $M$, and similarly for $\Omega_J$. In this section we let
\begin{align*}
\mathbf X = \tfrac{1}{2} \left(X - \i I X \right)
\end{align*}
denote the complex holomorphic vector field associated to $X$.
\begin{prop} \label{p:Omegapoles} Given $(M^4, g, I, J)$ a complete regular rank one soliton, the forms $\Omega_{I}, \Omega_J$ have poles of order $1$ at $\mathbf{T}$, and $\mathbf{X}$ does not vanish identically along $\mathbf{T}$.
\end{prop}
The proof will consist of several lemmas. Our analysis is local, and we can choose local $I$-holomorphic coordinates $z,w$ on a neighborhood $U$ of $p \in \mathbf{T}_+$ such that $g_{i\bar{j}}(p) = \delta_i^j$ and
\[
\Omega_I=w^{-k}dz\wedge dw + \mbox{higher order terms},\quad k\geq 1,
\]
and $\mathbf T_+\cap U=\{w=0\}$.
\begin{lemma} \label{l:deglocuslemma1}
If $X\neq 0$ at $x_0\in\mathbf T_+$, then $k=1$
\end{lemma}
\begin{proof}
For two real-valued functions $f_1,f_2$ we will write $f_1 \sim f_2$ if there is a positive constant $C>0$ such that $C^{-1}f_1<f_2<Cf_1$.
Let $d$ denote the distance from the divisor $\mathbf T_+$:
\[
d(x):=\mathrm{dist}_g(x,\mathbf T_+).
\]
In the $I$-holomorphic coordinates $(z,w)$ as above centered at $x_0$ we have $d(x) \sim \brs{w}$, thus in a neighbourhood of $\mathbf T_+$ we will have
\[
\frac{\Omega_I \wedge \bar{\Omega}_I}{dV_g}\sim d^{-2k},
\]
where $dV_g$ is the volume form associated with $g$.
Fix an $S^1$-invariant neighborhood $U$ of $x_0$. By~\eqref{f:Omega_matrix} and \eqref{f:coframes} at $x\in U\backslash\mathbf T_+$ we have
\[
\Omega=W^{-1}\left(X^*\wedge (KX)^*+(IX)^*\wedge(JX)^*\right),
\]
so that in the local coordinates $\{\mu_i\}$,
\[
\Omega_I \wedge \bar{\Omega}_I = \Omega\wedge\Omega=C W\eta\wedge d\mu_1\wedge d\mu_+\wedge d\mu_-.
\]
Now, since the differentials
\[
W^{-1/2}\eta, \quad (1-p^2)^{1/2}W^{1/2}d\mu_1, \quad (1-p)^{1/2}W^{1/2}d\mu_+, \quad (1+p)^{1/2}W^{1/2}d\mu_-
\]
form an orthonormal basis of $T^*_xM$, we have
\[
\Omega\wedge\Omega=C(1-p^2)^{-1}dV_g.
\]
In a neighbourhood of $\mathbf{T_+}$ function $1+p$ is bounded from above and away from zero, therefore, it remains to estimate $d(x)$ in terms of $1-p$. Since $X\neq 0$ at $x_0$, we can shrink $U$ to ensure that $X \neq 0$ in $U$, and obtain a constant $C>0$ such that $W:=g(X,X)^{-1}$ satisfies
\[
C^{-1}<W<C
\]
in $U$. Then
metric $g$ is uniformly equivalent in $U$ to the metric
\[
\til g=h+\eta^2.
\]
Let $\gamma\subset U$ be a horizontal geodesic with respect to $\til g$ connecting $x_0\in \mathbf{T}_+$ to a point $x_1\in U$ which realizes the shortest distance between $\mathbf{T}_+$ and $x_1$. In particular, $d(\gamma(t))\sim t$. By Theorem~\ref{t:nondegenerate_gk_description} the universal cover of ${U\backslash\mathbf{T}_+}$ admits a well-defined moment map
\[
\pmb\mu\colon \til{U\backslash\mathbf{T}_+}\to\mathbb{R}^3_{\pmb\mu}.
\]
Under this map the geodesic $\gamma$ is isometrically mapped to an $h$-geodesic $\gg_0:=\pmb\mu(\gg)$. Since $\mathbf T_{+}=\{p=1\}$, we have $p(\gg_0(t))\to 1$ as $t\to 0$. By definition of $\Phi$ we have
\[
p=\frac{1-e^\Phi}{1+e^\Phi}
\]
thus it follows from Proposition \ref{p:invGKsoliton} that along $\gg_0(t)$ we have $\Phi=a_+\mu_++a_-\mu_-\to -\infty$ as $t\to 0$. Given the explicit form of the metric~\eqref{f:h_diagonal}, we conclude that $a_+\neq 0$, since otherwise $\gg_0(t)$ would have an infinite length. So without loss of generality we assume $a_+>0$.
Then the length of $\gg_0$ with respect to $h$ is bounded from above by the length of the curve $\gamma_1$ with $\mu_1=\mbox{const}$, $\mu_-=\mbox{const}$ and $\mu_+\to -\infty$. Thus we estimate
\[
d(x_1)\sim L_{h}\gg_0\leq L_h\gg_1=
\int_{\gg_1} (1-p)^{1/2}d\mu_+
\sim
\int_{-\infty}^{\mu_+}\left(\frac{e^{a_+s}}{1+e^{a_+s}}\right)^{1/2}ds\sim e^{a_+\mu_+/2}\sim(1-p)^{1/2}.
\]
Since $\til g$ and $g$ are equivalent, we have $d(x) \leq C (1-p)^{1/2}$, so that along $\gamma$ we have
\[
d(x)^{-2k} \sim \frac{\Omega\wedge\Omega}{dV_g}=C(1-p^2)^{-1}\leq C' d(x)^{-2}.
\]
Thus $k = 1$, as claimed.
\end{proof}
\begin{lemma} \label{l:deglocuslemma2}
If $X$ vanishes identically along $\mathbf T_+$, then $k=1$.
\end{lemma}
\begin{proof}
Given that $X$ vanishes on $\mathbf T_+$, then at any point $x\in \mathbf T_+$ there is a local chart $U\simeq \mathbb{C}^2$ provided by the slice theorem such that the action of $S^1$ on $\mathbb{C}^2$ has weights $\mathbf{w}=(\alpha,0)$. We claim that $\alpha\neq 0$. Indeed, if $\alpha=0$ then the vector field $X$ would act trivially in $U\simeq \mathbb{C}^2$, and being Killing, $X$ must preserve all geodesics emanating from $x$. Thus $X$ will be identically zero entirely on $M$, which contradicts our basic assumptions on $X$.
Given the weights of $X$ at $x$ and knowing that $X$ vanishes on $\mathbf{T}_+$, we can pick local $I$-holomorphic coordinates $(z,w)$ near $x$ such that $\mathbf{T_+}=\{w=0\}$, $x=(0,0)$ and in $T_xM$,
\[
[X,\partial_z]=0,\quad [X,\partial_w]=\sqrt{-1}\alpha \partial_w.
\]
Then locally we can write the holomorphic vector field $\mathbf{X}=X-\sqrt{-1}IX$ as
\[
\mathbf X=wf_1(z)\partial_w+(\mbox{higher order terms in $w$}),\quad f_1(0)=\sqrt{-1}\alpha\neq 0.
\]
In these coordinates we still have
\[
\Omega_I = w^{-k} dz \wedge dw+(\mbox{higher order terms in }w),
\]
and thus
\[
i_X\Omega_I=\frac{1}{w^{k-1}}(f_1(z)dz+f_2(z)dw)+(\mbox{higher order terms in }w)
\]
Since $f_1(0)\neq 0$, this form cannot be closed, unless $k = 1$.
\end{proof}
\begin{lemma}\label{l:X_nonvanish} $\mathbf{X}$ does not vanish identically along $\mathbf{T}_+$.
\begin{proof} It follows from Lemmas \ref{l:deglocuslemma1} and \ref{l:deglocuslemma2} that $\Omega_I$ has a pole of order one. As above we choose local $I$-holomorphic coordinates so that
\[
\Omega_I = w^{-1} dz \wedge dw + (\mbox{higher order terms in $w$}),
\]
If $\mathbf{X}$ vanishes identically along $\mathbf{T}_+$ then we can express
\[
\mathbf X= w \left( f_1 \partial_z + f_2 \partial_w \right), \qquad f_i \in C^{\infty}.
\]
It follows that
\begin{align*}
i_{\mathbf{X}} \Omega_I = f_1(z) dw - f_2(z) dz + (\mbox{higher order terms in $w$})
\end{align*}
is smooth across $\mathbf{T}_+$. Therefore $\int_\gamma i_{\mathbf X}\Omega_I$ is finite, where $\gamma$ is any path connecting $x_0\in\mathbf{T}_+$ and $x_1\in M\backslash\mathbf{T}_+$. Using a similar argument for $\Omega_J$, we conclude that the integrals
\[
\int_\gamma i_XI\Omega,\quad \int_{\gamma} i_XJ\Omega
\]
are also finite. On the other hand, the pullbacks of $i_XI\Omega, i_XJ\Omega$ to $\til{M\backslash \mathbf{T}_+}$ are $d \mu_2$ and $d\mu_3$. Since $p$ is bounded away from $1$ on any compact subset of $\mathbb{R}^3_{\pmb\mu}$, the integral of $a_Id\mu_2+a_Jd\mu_3$ over any curve $\gamma_0\subset \mathbb{R}^3_{\pmb\mu}$ escaping to the locus $\{p=1\}$ is infinite, giving a contradiction.
\end{proof}
\end{lemma}
\begin{proof}[Proof of Proposition \ref{p:Omegapoles}] Lemmas \ref{l:deglocuslemma1} and \ref{l:deglocuslemma2} gives that $\Omega_I$ has a pole of order $1$, and then Lemma \ref{l:X_nonvanish} gives that $\mathbf{X}$ does not vanish identically.
\end{proof}
\subsection{Completeness properties of the quotient} \label{ss:compquot}
Now we prove a general statement about Riemannian 4-manifolds with $S^1$ action, which might be of independent interest.
\begin{prop}\label{p:s1_bundle_complete}
Let $(M,g)$ be a connected smooth complete Riemannian 4-dimensional manifold with boundary admitting
an isometric $S^1$ action with isolated fixed points $\{y_i\}$.
Let
\[\pi\colon M\to N\] be the projection onto the orbit space, and set $z_i=\pi(y_i)$. Assume that the orbifold with boundary $N_0=N\backslash\{z_i\}$ is
equipped with a Riemannian metric $h$ and a smooth 2-form $\beta_0$ such that
\begin{enumerate}
\item the curvature 2-form of the $S^1$-bundle $M\backslash\{y_i\}\to N_0$ is given by a closed 2-form
\[
\beta=*_hdW+W\beta_0,
\]
where $W$ is a smooth function on $N_0$;
\item there are constants $c_W,C_{\beta_0} >0$ such that $W>c_W$ and $|\beta_0|_h+|d\beta_0|_h<C_\beta$;
\item $g$ on $M\backslash\{y_i\}$ is given by
\[
g=Wh+W^{-1}\eta^2
\]
where $\eta$ is the connection 1-form with curvature $\beta$;
\item $\{z_i\}\subset N$ is complete with respect to the distance function $d_h$ induced by $h$;
\item $|\Rm^{N_0,h}|<b^2$ for some constant $b$;
\item $\Rc^{M,g}>-k^2$ for some constant $k$.
\end{enumerate}
Then any closed subset $K\subset N$ such that $d_h(K,\partial N_0)>0$ is complete with respect to $d_h$
\end{prop}
\begin{rmk}
The assumptions of the above proposition seem rather restrictive. However, as we will show later, aside from the lower Ricci curvature bound, they are automatically satisfied on complete regular rank one solitons.
\end{rmk}
\begin{proof}
Let $\partial N_0$ be the usual boundary of an orbifold $N_0$, and for a subset $K\subset N$ denote by $\partial_h K$ its $d_h$-completion boundary, i.e.,
\[
\partial_h K:=\bar{(K,d_h)}\backslash K,
\]
where $\bar{(K,d_h)}$ denotes the metric completion of $(K,d_h)$.
We will prove the completeness by contradiction. The proof is rather technical, so we start with a brief overview. The first step is to get a gradient estimate for $W$. The function $W$ satisfies a second order elliptic PDE on a complete manifold $(M,g)$ which allows for the application of the standard method of Cheng-Yau \cite{ChengYau} using the lower bound on $\Rc^{M,g}$. With the gradient estimates at hand, we exploit completeness of $(M,g)$ and use an argument of Schoen-Yau~\cite{SchoenYau} to get a lower bound on the blow up rate of $W$ near $\partial_h N$. Finally, relying on the control over the geometry of $(N_0,h)$ we prove that averages of $W$ over geodesic spheres in $(N_0,h)$ blow up as the boundary of the sphere approaches $\partial_h N$, and obtain a contradiction with the mean value inequality. This implies that $K$ must be complete.
\medskip
\noindent\textbf{Step 1} (Gradient estimate for $W$)
\noindent
Assume that some closed $K\subset N$ with $d_h(K,\partial N_0)>0$ is not complete so that $\partial_h K$ is not empty. Since $d_h(K,\partial N_0)>0$, there is a point in $x_\infty\in \partial_h K$ such that $d_h(x_\infty,\partial N_0)>0$, and as $K\subset N$ is closed, $x_\infty\in \partial_h N$.
The set $\{z_i\}$ is complete, therefore we can find $\epsilon>0$ such that
\[
d_h(x_\infty,\{z_i\}\cup \partial N_0)>\epsilon.
\]
Pick a point $x_0\in N_0$ such that $d_h(x_0,x_\infty)<\epsilon/3$. Then $d_h(x_0,\{z_i\}\cup \partial N_0)>2\epsilon/3$ and $d_h(x_0,\partial_h N)<\epsilon/3$. Since $d_h(x_0, \{z_i\}\cup \partial N_0)>2\epsilon/3$ and $h$ is a Riemannian metric on orbifold $N_0$, we can choose the largest $h$-geodesic ball around $x_0$ not containing points in $\{z_i\}\cup \partial N_0\cup \partial_h N$, and call it $B(x_0)$. Since $d_h(x_0,\partial_h N)<\epsilon/3$ and $d_h(x_0, \{z_i\}\cup \partial N_0)>2\epsilon/3$, there exists a point $x^*\in \partial_h N$ such that $x^*\in \bar{B(x_0)}$. Thus, as $d_h(x_\infty, x^*)<2\epsilon/3$, we have that
\[
d_h(x^*,\{z_i\}\cup \partial N_0)>\epsilon/3.
\]
Let $r=\epsilon/6$. We are going to prove a gradient estimate for $W$ in $U_r(x^*)=\{x\in N\ |\ d_h(x,x^*)\leq r\}$. Our plan is to apply the Cheng-Yau local gradient estimate~\cite{ChengYau} to $W$ defined on $\pi^{-1}(U_r(x^*))\subset M$, where
\[
\pi\colon M\to N
\]
is the natural projection onto the orbit space. Recall that the metric $g$ on $M$ is given by $Wh+W^{-1}\eta^2$ with $W>c_W>0$. Furthermore, $d_h(x^*,\{z_i\}\cup\partial N_0)\geq \epsilon/3=2r$. Therefore for any $y\in \pi^{-1}(U_r(x^*))$, the $g$-geodesic ball of radius $c_Wr$ centered at $y$ does not intersect the boundary $\partial M$ and the fixed point set $\{y_i\}$
To apply the local Cheng-Yau estimate to $W$ we first note that for any function $f\colon M\to \mathbb{R}$ invariant under the $S^1$-action we have
\[
\Delta_{g}f=W^{-1}\Delta_{h}f,
\]
therefore $W$ solves
\[
\Delta_{g}W=W^{-1}\Delta_h W=-W^{-1}(\langle dW, *\beta_0\rangle_h+(*_hd\beta_0)W).
\]
We know that $W$ is bounded from below by $c_W>0$ and $|\beta_0|_h+|d\beta_0|_h<C_{\beta_0}$. It is straightforward now to check that there exists $C'>0$ such that $W$ satisfies the differential inequalities
\begin{equation}
\begin{split}
\Delta_{g}W&\leq C'(|dW|_{g}+W),\\
|\nabla_{g}(\Delta_{g}W)|_{g}&\leq C'
\left(|dW|_{g}+W^{-1}|d W|^2_{g}+|\nabla^2_{g}W|_{g}+W\right),
\end{split}
\end{equation}
which allow to run the proof of~\cite[Theorem 6]{ChengYau}. Therefore there exists a constant $C>0$ such that
\[
|d W|_{g}<CW
\]
in $\pi^{-1}(U_r(x^*))$. Translating this inequality back to $h$ we find that in $U_r(x^*)$
\begin{equation}\label{f:W_gradient_estimate}
|dW|_h<CW^{3/2}.
\end{equation}
\medskip
\noindent\textbf{Step 2} (Lower bound on the growth rate of $W(x)$ as $x\to x^*$)
Next we exploit the upper bound of ~\eqref{f:W_gradient_estimate} together with the completeness of $(M,g)$ to obtain a lower bound on the growth of $W$ as its argument approaches $x^*$. The manifold $(M,g)$ is complete, therefore the distance function $d_{Wh}$ on $N$ induced by the metric $Wh$ on $N\backslash\{z_i\}$ is complete. Let $\gg\colon [0,1)\to N$ be any smooth curve of finite $h$-length, avoiding $\{z_i\}$, such that $\lim_{t\to 1}\gg(t)=x^*$. Since $(N,d_{Wh})$ is complete, and $x^*\not\in N$, we know that
\[
\ell_{Wh}(\gg)=\int_{0}^{1} W^{1/2}|\gg'(t)|_{h}dt=\infty.
\]
This implies that $W\to +\infty$ subsequentially along any curve approaching $x^*$. We now use the gradient estimate for $W$ to upgrade this subsequential blow up to a uniform lower bound on the growth rate.
Consider any curve $\gg$ in ${U_r}(x^*)$ approaching $x^*$. Pick two points $x,x'$ on $\gg$. Then
\[
W^{-\frac{1}{2}}(x)-W^{-\frac{1}{2}}(x')=\int_x^{x'} h\left(\nabla W^{-\frac{1}{2}}, \gg'(t)\right) dt\leq \int_x^{x'}\frac{1}{2}|\nabla W|_h W^{-\frac{3}{2}}dt.
\]
Using the gradient estimate~\eqref{f:W_gradient_estimate}, we conclude
\[
W^{-\frac{1}{2}}(x)-W^{-\frac{1}{2}}(x')\leq C'' \ell_h(\gg).
\]
Picking $x'$ along a sequence $x_i\in\gg$ such that $W(x_i)\to \infty$, we conclude that
\[
W(x)\geq \frac{C}{\ell^2_h(\gg)}.
\]
As the curve $\gg\subset U_r(x^*)$ above is arbitrary, we conclude that for any $x\in U_r(x^*)$
\begin{equation}\label{f:W_blowup_bound}
W(x)\geq \frac{C}{d_{h}^2(x,x^*)}.
\end{equation}
\medskip
\noindent\textbf{Step 3} (Contradiction with the mean value theorem)
Finally, we show that the function $W$ satisfying the elliptic equation $d(*_hdW+W\beta_0)=0$ can not have a boundary blow up as in~\eqref{f:W_blowup_bound} using the bound $|\Rm^{N,h}|<b^2$. Recall that we have a ball $B(x_0)\subset N$ such that $x^*\in\partial B(x_0)$. Let $r_0:=d_h(x_0,x^*)$ be the radius of $B(x_0)$. Pick a new center $x_1$ along the $h$-geodesic segment $[x_0, x^*]$ such that
\[
r_1:=d_h(x_1,x^*)<\min(r_0, \pi/4b).
\]
Denote by $B_{r_1}(x_1)\subset N$ the $h$-geodesic ball of radius $r_1$ centered at $x_1$ and by $B^T_{r_1}(x_1)\subset T_{x_1}N$ the ball of radius $r_1$ in the tangent space $(T_{x_1}N, h_{x_1})$. By Rauch's comparison theorem, the exponential map
\[
\exp_{x_1}^h\colon B^T_{r_1}(x_1) \to B_{r_1}(x_1)
\]
is an immersion and there is a constant $\delta_b>0$ such that the norm of the differential of $\exp_{x_1}^h$ satisfies bounds:
\[
C_0^{-1} < |D(\exp_{x_1}^h)| < C_0,\quad C_0=C_0(b)>0.
\]
We use $\exp_{x_1}^h$ to pull back $W$, $\beta_0$ and $h$ to $B^T_{r_1}(x_1)\subset T_{x_1}N$. The estimate for $D(\exp_{x_1}^h)$ implies that the metric $h_{x_1}$ in $B^T_{r_1}(x_1)\subset T_{x_1}N$ is uniformly equivalent to the pull back of $h$. The estimates to follow are conducted inside $B^T_{r_1}(x_1) \subset T_{x_1} N$ with respect to the metric $(\exp_{x_1}^h)^* h$ which we still denote $h$.
Let $\rho$ be the $h$-distance function from $x_1$, and let $G=G(\rho)$ be a function to be chosen later. Recall that $W$ solves
\[
L(W):=*_hd(*_hdW+ W\beta_0)=0.
\]
Then by Green's identity, for any $R<r_1$,
\begin{equation}
\begin{split}
\int_{B_R(x_1)} (L(W)G-WL^*(G))\,d\mu_h=
\int_{\partial B_R(x_1)} \left((*_hdW+W\beta_0)G-W\frac{\partial G}{\partial\nu}\right)\,d\sigma_h,
\end{split}
\end{equation}
where
\[
L^*(G):=*_h(d*_hdG-dG\wedge\beta_0)
\]
is the dual operator. Using that $L(W)=0$ and
\[
G\Big|_{\partial B_R(x_1)}=G(R),\quad \frac{\partial G}{\partial \nu}\Big|_{\partial B_R(x_1)}=G'(R),
\]
we find
\[
\int_{\partial B_R(x_1)} (*_h dW+W\beta_0)Gd\sigma_h=
G(R)\int_{\partial B_R(x_1)} (*_h dW+W\beta_0)d\sigma_h=0.
\]
Therefore
\[
G'(R)\int_{\partial B_R(x_1)}W\,d\sigma_h=\int_{B_R(x_1)} W L^*(G)\,d\mu_h.
\]
In particular for any $0<R_1<R_2<r_1$ we have
\begin{equation}\label{f:mean_value_ineq}
G'(R_1)\int_{\partial B_{R_2}(x_1)}W\,d\sigma_h-G'(R_1)\int_{\partial B_{R_2}(x_1)}W\,d\sigma_h=\int_{B_{R_2}(x_1)-B_{R_1}(x_1)} W L^*(G)\,d\mu_h.
\end{equation}
Now consider a function
\[
G(\rho)=1-e^{-\lambda \rho}.
\]
By the Laplacian comparison theorem, $\Delta \rho $ is bounded from above for $\rho\in (r_1/2, r_1)$. Then given a bound on $|\beta_0|_h$ it is easy to check that for a large constant $\lambda>0$,
\[
L^*(G)<0
\]
in the annulus $B_{r_1}(x_1)-B_{r_1/2}(x_1)$. Therefore identity~\eqref{f:mean_value_ineq} implies that for any $R\in(r_1/2,r_1)$ we have a mean value inequality
\begin{equation}\label{f:mean_value_ineq_2}
\int_{\partial B_{R}(x_1)}Wd\sigma_h<C \int_{\partial B_{r_1/2}(x_1)}W\,d\sigma_h
\end{equation}
where $C$ depends only on $r_1$ and operator $L$. In particular the integral on the left hand side is bounded from above as $R\to r_1$.
Fix $R<r_1$ and let $x_N\in \partial B_{R}(x_1)$ be the point which is $h$-closest to $x^*$. Pick spherical coordinates $(\phi,\psi)\in [0;2\pi)\times (-\pi/2;\pi/2)$ such that $\phi$ is a longitude, $\psi$ is an latitude $x_N$ is the northern pole. Then there exists a constant $C_1$ independent of $R\in(r_1/2,r_1)$ such that for a point $p\in \partial B_{R}(x_1)$ with coordinates $(\phi,\psi)$ we have
\[
d_h(p,x^*)<C_1\sqrt{(r_1-R)^2+(\pi/2-\psi)^2}.
\]
Now we estimate
\begin{equation}
\begin{split}
\int\limits_{\partial B_{R}(x_1)}Wd\sigma_h&\geq C\int\limits_{\partial B_{R}(x_1)}Wd\sigma_{h_1}=CR^2 \int_{-\pi/2}^{\pi/2}\left(\int_{0}^{2\pi} W \,d\phi\right) \cos\psi\,d\psi\\ &\geq
C'\int_{-\pi/2}^{\pi/2} \frac{\pi/2-\psi}{(r_1-R)^2+(\pi/2-\psi)^2}d\psi\geq -C''\log(r_1-R),
\end{split}
\end{equation}
where in the second inequality we used the blow up estimate~\eqref{f:W_blowup_bound} and uniform equivalence of $h$ and $h_{x_1}$ in $B^T_{r_1}(x_1)$. The final term is unbounded as $R\to r_1$, contradicting the mean value inequality~\eqref{f:mean_value_ineq_2}. Thus we proved that $K\subset N$ is complete for any closed $K$ such that $d_h(K,\partial N_0)>0$.
\end{proof}
\subsection{Global definition of the moment map on the non-degeneracy locus of \texorpdfstring{$M$}{}}
Suppose $(M^4,g,I,J)$ is a complete regular rank one GK soliton. Considering the nondegenerate domain $M\backslash \mathbf{T}$, $\mathbf{T}=\mathbf T_+\cup \mathbf T_-$, the construction of Theorem~\ref{t:nondegenerate_gk_description_v2} defines a moment map
\[
\pmb\mu\colon \til{M\backslash \mathbf{T}}\to \mathbb{R}^3_{\pmb\mu}.
\]
on an appropriate covering space $\mathbb{Z}^k\to\til{M\backslash \mathbf{T}}\to M\backslash \mathbf{T}$. Our goal in this section is to describe the structure of the covering space $\til{M\backslash \mathbf{T}}$, and show that the moment map $\pmb{\mu}$ descends to the $\mathbb{Z}^k$-quotient space.
\begin{prop} \label{p:deck_transform_mu} Given the setup above, the map $\pmb{\mu}$ descends to a map
\begin{equation}\label{f:proj_nondegenerate}
M\backslash \mathbf{T}\to (M\backslash \mathbf{T})/S^1\simeq \mathbb{R}^3_{\pmb\mu}/\Gamma,
\end{equation}
where $\Gamma\simeq \mathbb{Z}^k$ acts on $(\mathbb{R}^3_{\pmb\mu},h)$ by isometric translations freely, and properly discontinuously. There are two possibilities for the action of $\Gamma$ on $\mathbb{R}^3_{\pmb\mu}$:
\begin{enumerate}
\item $\Gamma_1\simeq \mathbb{Z}$ is generated by $g_1$ such that
\[
g_1(\mu_1,\mu_+,\mu_-)=(\mu_1+c_1,\mu_+,\mu_-),\quad c_1\neq 0.
\]
\item $\Gamma_2\simeq \Gamma_1\times \mathbb{Z}\simeq \mathbb{Z}^2$ and $a_-=0$, where $\Gamma_2$ is generated by $g_1$ and $g_2$ such that
\begin{align*}
g_1&(\mu_1,\mu_+,\mu_-)=(\mu_1+c_1,\mu_+,\mu_-),\quad c_1\neq 0\\
g_2&(\mu_1,\mu_+,\mu_-)=(\mu_1+c_1',\mu_+,\mu_-+c),\quad c\neq 0.
\end{align*}
\end{enumerate}
\begin{proof}
First, let us describe the covering space
\[
\til{M\backslash \mathbf{T}}\to M\backslash \mathbf{T}.
\]
On $M\backslash \mathbf{T}$ we have three closed forms
\[
\alpha_1=-i_X\Omega, \quad \alpha_2=-i_X I\Omega, \quad \alpha_3=-i_X J\Omega.
\]
Let $K:=\{a\in H_1(M\backslash \mathbf{T}; \mathbb{Z})\ |\ \langle\alpha_i,a\rangle=0, i=1,2,3\}\subset H_1(M\backslash \mathbf{T}; \mathbb{Z})$. Then we have a map
\[
\pi_1(M\backslash \mathbf{T})\to \Gamma\to 1,
\]
where $\Gamma=H_1(M\backslash \mathbf{T}; \mathbb{Z})/K\simeq \mathbb{Z}^k$. The action $S^1\times M\to M$ is tri-Hamiltonian if and only if $\Gamma=0$. In general, $\Gamma$ is nontrivial, and in order to construct the moment map of Section~\ref{s:GKsymm} we first need to `kill' $\Gamma$ by taking a cover. Specifically, the kernel of the map $\pi_1(M\backslash \mathbf{T})\to \Gamma$ corresponds to a covering space
\[
\mathrm{pr}\colon \til{M\backslash \mathbf{T}}\xrightarrow{\Gamma} M\backslash \mathbf{T}
\]
with the deck transformation group $\Gamma$. On $\til{M\backslash \mathbf{T}}$ the forms $\mathrm{pr}^*\alpha_i$ represent the zero class in de Rham cohomology, so we can write $\mathrm{pr}^*\alpha_i=d\mu_i$ and recover the moment map $\pmb \mu$ in the construction of Section~\ref{s:GKsymm}: $\pmb\mu\colon \til{M\backslash \mathbf{T}}\to\mathbb{R}^3_{\pmb\mu}$. We also note that the $S^1$ action on $M\backslash \mathbf{T}$ lifts to an $S^1$ action on the covering space $\til{M\backslash \mathbf{T}}$. Indeed, for any free loop $\gamma\subset M\backslash \mathbf{T}$ represented by an orbit of $S^1$, we have $[\gamma]\in K$, since $\alpha_i(X)=0$. Therefore, the lift of $\gamma$ to $\til{M\backslash \mathbf{T}}$ is still a closed loop. It follows that the lift of the vector field $X$ to $\til{M\backslash \mathbf{T}}$ has periodic orbits.
Next we claim that the image $\pmb\mu(\til{M\backslash\mathbf{T}})$ is the whole $\mathbb{R}^3_{\pmb\mu}$. We want to apply Proposition~\ref{p:s1_bundle_complete} to $\til{M\backslash \mathbf{T}}$. However, we can not do it directly, since, first, this manifold is not necessarily complete, and, second, $W$ might not be bounded from below by a positive constant. Thus we first modify $\til{M\backslash \mathbf{T}}$.
Recall that $\mathbf{T}=\{p= \pm1\}$. For $\delta>0$ denote $\mathbf{T}_\delta=\{|p|>1-\delta\}$ and consider the complement $M\backslash \mathbf{T}_\delta\subset M\backslash \mathbf{T}$. For $\delta>0$ small enough $M\backslash \mathbf{T}_\delta$ is a connected manifold with boundary. Let $M_\delta\subset \til{M\backslash\mathbf{T}}$ be the inverse image of $M\backslash\mathbf{T}_\delta$ under the covering map $\til{M\backslash\mathbf{T}}\to M\backslash\mathbf{T}$. By Proposition~\ref{p:invGKsoliton} characterizing rank one GK solitons, locally $\Phi=a_+\mu_++a_-\mu_-+\mbox{const}$. Since $\Phi$ and $\mu_i$ are globally defined on a connected manifold $M_\delta$, the same identity holds globally. We can absorb the constant term by redefining $\mu_i$ and assume that $\Phi=a_+\mu_++a_-\mu_-$. In particular, $p$, $h$ and $\beta_0$ all descend to $\mathbb{R}^3_{\pmb\mu}$.
Next we need to modify the function $W$, and as a consequence the geometry of $M$, to ensure that $W$ is bounded from below by a positive constant. Let $\til W$ be the baseline solution provided by Lemma \ref{l:W0_solution}. The equation for $\til W$ implies that the 2-form $\til \beta=*_h d \til W+\til W\beta_0\in \Lambda^2(M/S^1)$ is closed, and exact since it is defined on $\mathbb R^3_{\pmb \mu}$, with an antiderivative $\alpha\in \Lambda^1(N)$. Thus we can choose a new GK structure on the manifold $M$ by setting
\[
W_t:=W+t\til W,\quad \eta_t:=\eta+t\alpha,\quad t\in\mathbb{R}.
\]
\begin{lemma}\label{l:ricci_bound}
Let $g_t=W_th+W_t^{-1}\eta_t^2$. If $(M,g)$ is complete and has $\Rc^{M,g}>-k^2$, then $(M,g_t)$ is also complete. If additionally $|\beta_0|_h, |\nabla_h\beta_0|_h, \til W, |d\til W|_h$ are bounded on $M/S^1$, then for $t>0$ large enough we have $\Rc^{M,g_t}>-k^2$.
\end{lemma}
\begin{proof}
Since the fibers of the map $M\to M/S^1$ are compact, $(M,g_t)$ is complete if and only if $M/S^1$ with the distance function induced by the metric $W_th$ is complete. We know that metric $Wh$ induces a complete distance function on $M/S^1$ and since $W_t>W$ the same must be true for $W_t h$, implying that $(M,g_t)$ is complete.
Now we claim that for some $t>0$ we still have
\begin{equation}\label{eq:pf_ricci_Mt}
\Rc^{M,g_t}>-k^2g_t.
\end{equation}
It is enough to prove that bound on the subset of $M$ where $S^1$ acts freely. Let $\{E^1,E^2,E^3\}$ be an $h$-orthonormal frame of $T_xN$, $\{E_t^1,E_t^2,E_t^3\}$ are the horizontal lifts of $E_i$ normalized to the $g_t$-unit length, and $E_t^4$ is the unit vertical vector field. In order to prove the bound~\eqref{eq:pf_ricci_Mt}, we use the exact formula for $\Rc^{M,g_t}$ in terms of $\Rc^{N,h}$, $W_t$ and $\beta_t$ (see \cite{Gilkey-98}) and conclude that for any $1\leq i, j\leq 4$,
\begin{equation}
\Rc^{M,g_t}(E_t^i,E_t^j)=W_t^{-2} \langle dW_t,\Psi_1 \rangle_h+W_t^{-1}\Psi_2,
\end{equation}
where $\Psi_1$ and $\Psi_2$ are a covector and a function independent of $t$ with the $h$-norms bounded by $C(|\beta_0|^2_h+|\nabla_h\beta_0|_h)$, where $C>0$ is a universal constant. Notably, there are no second order terms in $W$ here~--- a special artifact of this ansatz recovering that in the case $h$ is Euclidean, the metric on the total space $M$ is Ricci flat for an arbitrary harmonic function $W$. We know that
\begin{equation}\label{f:proof_ricci_bound}
W^2\Rc^{M,g}(E_0^i,E_0^j)=\langle dW,\Psi_1\rangle_h+\Psi_2 W>-k^2W^2.
\end{equation}
If we fix
\[
t>\max\left(0, \sup_{\mathbf{\mathbb{R}^3_{\pmb\mu}}} \left\{ -\til W^{-2}\langle d\til W,\Psi_1\rangle_h-\Psi_2\til W^{-1} \right\} \right),
\]
then
\[
\langle d(t\til W),\Psi_1\rangle_h+\Psi_2 (t\til W)>-2k^2tW\til W-k^2t^2\til W^2,
\]
and using~\eqref{f:proof_ricci_bound} we find
\[
\langle dW_t,\Psi_1\rangle+\Psi_2W_t>-k^2W_t^2,
\]
which is equivalent to the required Ricci bound on $(M_t,g_t)$.
\end{proof}
Furthermore, by a direct computation, on $\{|p|\leq 1-\delta\}$ all the norms $|\beta_0|_h, |\nabla_h\beta_0|_h, \til W, |d\til W|_h$ are bounded.
Hence we can apply Lemma~\ref{l:ricci_bound} and modify the metric on $(M_\delta, g)$ obtaining $(M_\delta,g_t)$ with $\Rc^{M_\delta,g_t}>-k^2$ and $W_t>t\inf_{\mathbb{R}^3_{\pmb\mu}} \til W>0$. We claim that $(M_\delta, g_t)$ satisfies the assumptions of Proposition~\ref{p:s1_bundle_complete}:
\begin{enumerate}
\item $(M\backslash\mathbf{T}_\delta, g)$ is a closed subset of a complete manifold, therefore it is itself complete. $M_\delta$ is a covering space over $M\backslash\mathbf{T}_\delta$, therefore it is also complete with respect to $g$. By Lemma~\ref{l:ricci_bound} $M_\delta$ is also complete with respect to $g_t$
\item $S^1$ has only isolated fixed points by Lemma~\ref{l:free_nondegenerate}.
\item By Theorem~\ref{t:nondegenerate_gk_description_v2} and our construction of $(M_\delta,g_t)$, metric $g_t$ is given by $g=W_th+W_t^{-1}\eta^2$ and the curvature form is given by $\beta=*_hdW_t+W_t\beta_0$.
\item $M$ has finitely many fixed points. Since $M_\delta\to M\backslash \mathbf{T}_\delta\supset M$ is a covering map, the set of fixed points on $M_\delta$ is complete with respect to the pull back of any distance function on $M$, in particular with respect to the distance function induced by the metric $h+\eta^2$ on $M_\delta\backslash M^{S^1}$.
\item $\Rm^{\mathbb{R}^3_{\pmb\mu},h}$ has a two-sided bound and $\beta_0$, $\nabla_h\beta_0$ are both bounded with respect to $h$ on $d_h$-bounded subsets. Indeed, as we observe below (see Remark~\ref{r:Norbifold1}), the metric completion of an isometric quotient $(\mathbb{R}^3_{\pmb\mu}/\mathbb{Z}, h)$ is a smooth orbifold, such that $h$ and $\beta_0$ extend to smooth tensors. Therefore all geometric quantities naturally attached to $\beta_0$ and $h$ have a bounded norm on $d_h$-bounded subsets.
\item $\Rc^{M,g_t}$ is bounded from below by our standing assumption on $M$ and by Lemma~\ref{l:ricci_bound}.
\end{enumerate}
Given $\delta'>0$, Proposition~\ref{p:s1_bundle_complete} implies that for any $\delta>\delta'$ a closed subset
$M_{\delta}/S^1\subset M_{\delta'}/S^1$ is complete with respect to the metric $h$. Therefore the immersion $\iota$ of Theorem~\ref{t:nondegenerate_gk_description_v2}
\[
\iota\colon M_\delta/S^1\to \{x\in\mathbb{R}^3_{\pmb\mu}\ |\ |p(x)| \leq 1-\delta\}
\]
is a local diffeomorphism of $h$-complete manifolds. It follows that map $\iota$ satisfies the \textit{curve lifting property}: for any smooth curve $\til{\gg}\subset \{x\in\mathbb{R}^3_{\pmb\mu}\ |\ |p(x)| \leq 1-\delta\}$ starting at $\til\gg(0)=\iota(x_0)$ there exists its lift $\gg\subset M/S^1$, such that $\gg(0)=x_0$. A general result~\cite[Prop.\,6, \S5-6]{Carmo} implies that $\iota$ is a covering map.
Now, since $\{x\in\mathbb{R}^3_{\pmb\mu}\ |\ |p(x)| \leq 1-\delta\}$ is simply connected, $\iota$ must be a diffeomorphism. Letting $\delta\to 0$, we conclude that there is a diffeomorphism
\[
(\til{M\backslash \mathbf{T}})/S^1\simeq \mathbb{R}^3_{\pmb\mu}
\]
so that $\pmb\mu \colon \til{M\backslash \mathbf{T}}\to \mathbb{R}^3_{\pmb\mu}$ is just the projection on the orbit space.
\begin{lemma}\label{l:T_nonempty}
The set $\mathbf{T}_+=\{p=1\}$ is nonempty as long as $a_+\neq 0$. Similarly $\mathbf{T}_-=\{p=-1\}$ is nonempty as long as $a_-\neq 0$.
\end{lemma}
\begin{proof}
Assume that $a_+\neq 0$ yet $\mathbf{T}_+$ is empty. Then $\mathbf T_{+,\epsilon}:=\{y\in M\ |\ p(y)\geq 1-\epsilon\}$ is complete. By the same argument as above invoking Proposition~\ref{p:s1_bundle_complete}, we have an isomorphism of $h$-complete manifolds with boundary
\[
\mathbf{T}_{+,\epsilon}/S^1\simeq\{x\in\mathbf \mathbb{R}^3_{\pmb\mu}\ |\ p(x)\geq 1-\epsilon\}.
\]
Now consider a curve $\gamma\subset \mathbb{R}^3_{\pmb\mu}$ such that $\mu_1$ and $\mu_-$ are constant and $a_+\mu_+\to -\infty$. Along $\gamma$ we have $p\to 1$ and by a direct computation $\gamma$ has a finite $h$-length, giving a contradiction with the $h$-completeness of $\mathbf{T}_{+,\epsilon}/S^1$.
\end{proof}
The action of the deck transformation group $\Gamma$ on $\til{M\backslash \mathbf{T}}$ commutes with the flow of $X$ and therefore descends to the action on $\pmb\mu(\til{M\backslash \mathbf{T}})=\mathbb{R}^3_{\pmb\mu}$. By construction, the action of $\Gamma$ on $\til{M\backslash \mathbf{T}}$ must preserve $(\Omega,I,J,X)$, therefore $\Gamma$ preserves the 1-forms $d\mu_i$. Hence, the descended action of $\Gamma$ on $\mathbb{R}^3_{\pmb\mu}$ must be via translations in $\mu_i$ coordinates.
Now we prove that the action of $\Gamma$ on $\mathbb{R}^3_{\pmb\mu}$ is free. Assume on the contrary that the deck transformation, corresponding to a loop $\gg\subset M\backslash \mathbf{T}$ fixes a point $x_0\in \mathbb{R}^3_{\pmb\mu}$. Take any $y_0\in\pmb\mu^{-1}(x_0)$, and let $\til{\gg}$ be a path between $y_0$ and some $y_1$, which corresponds to the lift of $\gg$ to $\til{M\backslash \mathbf{T}}$. Since the element corresponding to $\gg$ fixes $x_0$, we have that $\pmb\mu(y_0)=\pmb\mu(y_1)=x_0$. The fibers of $\pmb\mu$ are connected, so we can ``close up'' $\til{\gg}$ by joining $y_0$ to $y_1$ within $\pmb\mu^{-1}(x)$. The loop $\hat{\gg}$ obtained this way satisfies:
\[
0=\int_{\hat{\gg}}d\mu_i=\int_{\til{\gg}}d\mu_i=\int_{\gg} \alpha_i,
\]
where the first identity is just Stokes' Theorem and the second identity uses the fact that $d\mu_i$ vanishes on $\pmb\mu^{-1}(x)$. We see that $\langle\alpha_i,[\gg]\rangle=0$, so $[\gg]\in K\subset H_1(M\backslash \mathbf{T};\mathbb{Z})$ and $\gg$ corresponds to the trivial element of $\Gamma$. Thus the action is indeed free. Next, since the action of $\Gamma$ on $\til{M\backslash \mathbf{T}}$ is properly discontinuous (this is true for any covering space), and the fibers of $\pmb\mu$ are compact, the action of $\Gamma$ on $\mathbb{R}^3_{\pmb\mu}$ also must be properly discontinuous. With this in place we can take the fiberwise quotient of the moment map $\til{M\backslash \mathbf{T}}\to \mathbb{R}^3_{\pmb\mu}$ to obtain a map
\[
\pmb\mu\colon M\backslash \mathbf{T}\to \mathbb{R}^3_{\pmb\mu}/\Gamma,
\]
which by abuse of notation we still denote $\pmb\mu$.
Next, the action of $\Gamma$ on $\til{M\backslash\mathbf{T}}$ preserves GK structure, hence the action of $\Gamma$ descends to an \emph{isometric} action on $(\mathbb{R}^3_{\pmb\mu}, h)$. Given the precise form of the metric $h$, and using the fact that $p$ is a function of $a_+\mu_++a_-\mu_-$, we conclude that $\Gamma\simeq \mathbb{Z}^k$ has rank $k\leq 2$ and is a subgroup of the group $\mathbb{R}^2\subset \Isom(h)$ generated by the translations in $\mu_i$-coordinates, preserving the linear function $a_+\mu_++a_-\mu_-$.
Our final goal is to determine which subgroups $\Gamma\subset\Isom(h)$ could occur for a given complete GK soliton $(M,g,I,J)$. By Lemma~\ref{l:T_nonempty} we know that $\mathbf{T}$ is nonempty. For an irreducible component $T_i\subset \mathbf{T}$ of the degeneracy divisor take a small neighbourhood $U_i=U(y_i)$ of a smooth point $y_i\in T_i$ and choose a loop $\gg_i\subset U_i$ with winding number $1$ with respect to $T_i$. By Proposition~\ref{p:Omegapoles}, $i_X\Omega_I$ is a meromorphic one-form with a pole along $T_i$, therefore $\int_{\gamma_i} i_X\Omega_I\neq 0$, and the corresponding deck transformation $g_i\in\Gamma$ is nontrivial.
Since we can choose $\gg_i$ in its free-loop homotopy class to be arbitrarily short with respect to the metric $g$, element $g_i\in\Gamma$ acting on $\til{M\backslash \mathbf{T}}$ satisfies the following property:
\[
\inf_{y\in \til{M\backslash \mathbf{T}}} d_g(y,g_i y)=0.
\]
By Lemma~\ref{l:X_nonvanish} we can assume that $X$ does not vanish at $y_i$ so $W$ is bounded from above and away from zero in $U_i$. Then metric $Wh$ on $\pmb\mu(U_i)\subset\mathbb{R}^3_{\pmb\mu}$ is uniformly equivalent to $h$. In particular, for the induced action of $g_i\in\Gamma$ on $\mathbb{R}^3_{\pmb\mu}$ we also must have
\[
\inf_{x\in \mathbb{R}^3_{\pmb\mu}} d_h(x,g_i x)=0.
\]
The only such nontrivial translation preserving $a_+\mu_++a_-\mu_-$ is
\[
g_i(\mu_1,\mu_+,\mu_-)=(\mu_1+c,\mu_+,\mu_-),\quad c\neq 0.
\]
We have proved that such a translation always belongs to $\Gamma$. Now, since $\Gamma\simeq \mathbb{Z}^k$ is a proper discrete subgroup of a two-dimensional translation group, either $\Gamma\simeq \mathbb{Z}$ generated by a translation in $\mu_1$ coordinate, or $\Gamma\simeq \mathbb{Z}^2$ generated by translations preserving the linear form $a_+\mu_++a_-\mu_-$. In the latter case $\Gamma$ can be generated by translations
\begin{equation}\label{f:deck_proof}
\begin{split}
g_1&(\mu_1,\mu_+,\mu_-)=(\mu_1+c_1,\mu_+,\mu_-),\quad c_1\neq 0\\
g_2&(\mu_1,\mu_+,\mu_-)=(\mu_1+c_1',\mu_++ca_-,\mu_--ca_+),\quad c\neq 0.
\end{split}
\end{equation}
At the same time, by Proposition~\ref{p:invGKsoliton}, the closed one-form
\[
\alpha=-a_+d\mu_++a_-d\mu_-+p(a_+d\mu_++a_-d\mu_-)
\]
must be exact on $M$ and therefore on $\mathbb{R}^3_{\pmb\mu}/\Gamma$. Since $g_2$ preserves $a_+\mu_++a_-\mu_-$, for any $x\in \mathbb{R}^3_{\pmb\mu}$ we have
\[
\int_{x}^{g_2x}\alpha=\int_x^{g_2x}(-a_+d\mu_++a_-d\mu_-).
\]
Assuming $a_+\neq 0$, the latter integral vanishes if and only if $a_-=0$. In this case, redefining the constant $c$ in~\eqref{f:deck_proof} we arrive at the second possibility in the statement of the proposition.
\end{proof}
\end{prop}
\begin{rmk}[Scaling convention]
After scaling the metric $g$, we can assume that the action of $\Gamma$ in Proposition~\ref{p:deck_transform_mu} is such that there is a primitive subgroup $\mathbb{Z}\subset \Gamma$ acting on $\mathbb{R}^3_{\pmb\mu}$ via
\[
\mu_1\mapsto \mu_1+2\pi m,\quad m\in \mathbb{Z}.
\]
This will be our default scaling through the rest of the paper.
\end{rmk}
%
\subsection{The metric completion of \texorpdfstring{$(\mathbb{R}^3_{\pmb\mu}/\Gamma ,h)$}{}} \label{ss:metcomp}
%
%
On the complement of the fixed point set $M\backslash\{y_i\}$ we have a well-defined horizontal metric $h$, which defines a Riemannian metric and a distance function $d_h$ on the orbit space $(M\backslash\{y_i\})/S^1$. The local asymptotic of $W$ near each fixed point $y_i$ implies that $d_h$ extends to a distance function on the whole $M/S^1$. In particular, there is a natural $d_h$-isometric map
\begin{equation}\label{f:proj_orbit_extension}
M/S^1\to \bar{(\mathbb{R}^3_{\pmb\mu}/\Gamma,h)}
\end{equation}
to the metric completion of $(\mathbb{R}^3_{\pmb\mu}/\Gamma,h)$, which extends the isomorphism~\eqref{f:proj_nondegenerate} of Proposition~\ref{p:deck_transform_mu}:
\[
(M\backslash\mathbf{T})/S^1 \to \mathbb{R}^3_{\pmb\mu}/\Gamma.
\]
The goal of this subsection is to prove that $\bar{(\mathbb{R}^3_{\pmb\mu}/\Gamma,h)}$ has a natural orbifold structure, and then in the next subsection we will show the map~\eqref{f:proj_orbit_extension} is a diffeomorphism of orbifolds.
We just have proved that the orbit space of the nondegenerate part $M\backslash\mathbf{T}$ of $(M,g,I,J)$ is isomorphic to one of the
\[
\mathbb{R}^3_{\pmb\mu}/\Gamma_1,\quad \mathbb{R}^3_{\pmb\mu}/\Gamma_2,
\]
where $\Gamma_1\simeq \mathbb{Z}$, $\Gamma_2\simeq \mathbb{Z}^2$ are groups described in Proposition~\ref{p:deck_transform_mu}. Our next goal is to describe the metric completions of $(\mathbb{R}^3_{\pmb\mu}/\Gamma_i, h)$. Since $\mathbb{R}^3_{\pmb\mu}/\Gamma_2$ is an isometric quotient of $\mathbb{R}^3_{\pmb\mu}/\Gamma_1$ it suffices to determine the completion of $(\mathbb{R}^3_{\pmb\mu}/\Gamma_1,h)$.
The metric $h$ on $\mathbb{R}^3_{\pmb\mu}/\Gamma_1$ is characterized by two constants $a_+,a_-\in\mathbb{R}$ which do not vanish simultaneously (if $a_+=a_-=0$ then the underlying GK soliton is hyperK\"ahler). In what follows, we assume that $a_+\neq 0$. It turns out that the completions $(\mathbb{R}^3_{\pmb\mu}/\Gamma_1,h)$ are essentially different in the cases $a_-=0$ and $a_-\neq 0$ which we describe separately. To highlight the dependence of $h$ on the real parameters $a_+$ and $a_-$, we will write $h=h_{a_+,a_-}$.
\subsubsection{Case $a_-=0$}
Introduce new coordinates on $\mathbb{R}^3_{\pmb\mu}/\Gamma_1$:
\[
(\mu_1,\rho,\mu_-)\in S^1\times \mathbb{R}_{>0}\times \mathbb{R},
\]
where
\begin{equation*}
\begin{split}
\rho=\exp(a_+\mu_+/2).
\end{split}
\end{equation*}
In these coordinates, $h_{a_+,0}=(1-p^2)d\mu_1^2+2(1-p)d\mu_+^2+2(1+p)d\mu_-^2$ takes form
\begin{equation}\label{f:h_a0}
h_{a_+,0}=\frac{4}{1+\rho^2}
\left(
\frac{\rho^2}{1+\rho^2}d\mu_1^2+\frac{4}{a_+^2}d\rho^2+4d\mu_-^2.
\right)
\end{equation}
\begin{prop}\label{p:completion_a0}
The metric completion of $(\mathbb{R}^3_{\pmb\mu}/\Gamma_1, h_{a_+,0})$ is isomorphic to
\[
S^1\times \mathbb{R}_{>0}\times \mathbb{R}\simeq \mathbb{R}^3
\]
with the metric~\eqref{f:h_a0} extending to a metric with the cone singularity of angle $\pi |a_+|$ along the codimension two subset $\{\rho=0\}$.
\end{prop}
\begin{proof}
We will construct the metric completion of $(\mathbb{R}^3_{\pmb\mu}/\Gamma_1, h_{a_+,0})$ explicitly by hand. Pick a positive number $k\in\mathbb{R}$ and consider $\mathbb{R}^2$ and $\mathbb{R}^2\backslash\{0\}\simeq \mathbb{R}\times S^1$ equipped respectively with a cone metric and a regular cylinder metric given in the polar coordinates by
\[
\begin{split}
g_1&=k^2\, d\rho^2+\rho^2\, d\phi_1^2,\\
g_2&=4d\mu_-^2+d\phi_2^2.
\end{split}
\]
Consider $M=\mathbb{R}^2\times \mathbb{R}\times S^1$ with a metric $g=\cfrac{g_1\oplus g_2}{1+\rho^2}$. The metric space $(M,g)$ is evidently complete, and admits a free isometric $S^1$ action induced by the vector field $\partial_{\phi_1}+\partial_{\phi_2}$. The orbit space $M/S^1$ is naturally isomorphic to $\mathbb{R}^2\times \mathbb{R}$ and the horizontal component of metric $g$ descends to a complete metric $h_0$ with a cone singularity along $\{0\}\times \mathbb{R}$.
The metric $h_0$ is regular on the open part $(\mathbb{R}^2\backslash\{0\})\times \mathbb{R}$, and to express it, we choose coordinates $(\phi_1-\phi_2,\rho,d\mu_-)$ on this regular locus. Computing $|d(\phi_1-\phi_2)|^2_g$, $|d\rho|^2_g$ and $|d\mu_-|^2_g$, we find that:
\begin{equation}\label{f:h_a0_pf}
h_0=\frac{1}{1+\rho^2}\left(
\frac{\rho^2}{1+\rho^2}{d(\phi_1-\phi_2)^2+k_1^2d\rho^2+4d\mu_-^2}
\right).
\end{equation}
If we set $k=2/|a_+|$ and denote $\mu_1=\phi_1-\phi_2$, then the metric in~\eqref{f:h_a0} will coincide with ~\eqref{f:h_a0_pf} up to a constant factor $4$. Therefore, $(M/S^1, 4h_0)$ is the metric completion of $(\mathbb{R}^3_{\pmb\mu}/\Gamma_1, h_{a_+,0})$ as claimed.
\end{proof}
\subsubsection{Case $a_-\neq 0$}
The argument in this case is very similar to the case $a_-=0$. Introduce new coordinates on $\mathbb{R}^3_{\pmb\mu}/\Gamma_1$:
\[
(\mu_1,\rho_1,\rho_2)\in S^1\times \mathbb{R}_{>0}\times \mathbb{R}_{>0}
\]
where
\begin{equation*}
\begin{split}
\rho_1=\frac{e^{-a_-\mu_-/2}}{a_-^2e^{a_+\mu_+}+a_+^2e^{-a_-\mu_-}},\\
\rho_2=\frac{e^{a_+\mu_+/2}}{a_-^2e^{a_+\mu_+}+a_+^2e^{-a_-\mu_-}}.
\end{split}
\end{equation*}
In these coordinates, the metric $h_{a_+, a_-}=(1-p^2)d\mu_1^2+2(1-p)d\mu_+^2+2(1+p)d\mu_-^2$ takes the form
\begin{equation}\label{f:h_a_neq0}
h_{a_+, a_-}=\frac{4}{\rho_1^2+\rho_2^2}
\left(
\frac{\rho_1^2\rho_2^2}{\rho_1^2+\rho_2^2}d\mu_1^2+\frac{4}{a_-^2}d\rho_1^2+\frac{4}{a_+^2}d\rho_2^2
\right).
\end{equation}
\begin{prop}\label{p:completion_aneq0}
The metric completion of $(\mathbb{R}^3_{\pmb\mu}/\Gamma_1, h_{a_+, a_-})$ is isomorphic to
\[
(S^1\times \mathbb{R}_{\geq 0}\times \mathbb{R}_{\geq 0})\backslash (S^1\times \{0\}\times \{0\})\simeq S^2\times \mathbb{R}
\]
with the metric~\eqref{f:h_a_neq0} extending to a metric with cone singularities of angles $\pi |a_-|$ and $\pi |a_+|$ along the codimension two subsets $\{\rho_1=0\}$ and $\{\rho_2=0\}$ respectively.
\end{prop}
\begin{rmk}
The completion in the above proposition is isomorphic to a direct product $S^2\times \mathbb{R}$, where $S^2$ has a `spindle' metric with two cone singularities of angles $\pi a_-$ and $\pi a_+$.
\end{rmk}
\begin{proof}
We will construct the metric completion of $(\mathbb{R}^3_{\pmb\mu}/\Gamma_1, h)$ explicitly by hand. Pick two positive numbers $k_1,k_2\in\mathbb{R}$ and consider two copies of $\mathbb{R}^2$ equipped with cone metrics given in polar coordinates by
\[
\begin{split}
g_1&=k_1^2\, d\rho_1^2+\rho_1^2\, d\phi_1^2,\\
g_2&=k_2^2\, d\rho_2^2+\rho_2^2\, d\phi_2^2.
\end{split}
\]
Each copy $(\mathbb{R}^2,g_i)$ has a cone angle $2\pi/k_i$ at the origin. Consider $M=(\mathbb{R}^2\times \mathbb{R}^2)\backslash\{(0,0)\}\simeq S^3\times \mathbb{R}$ with a metric $g=\cfrac{g_1\oplus g_2}{\rho_1^2+ \rho_2^2}$. The metric space $(M,g)$ is evidently complete, and admits a free isometric $S^1$ action induced by the vector field $\partial_{\phi_1}+\partial_{\phi_2}$. The orbit $M/S^1$ space is naturally isomorphic to $S^2\times \mathbb{R}$ and the horizontal component of metric $g$ descends to a complete metric $h_0$ with cone singularities along $((\mathbb{R}^2\backslash\{0\})\times \{0\})/S^1$ and $(\{0\}\times (\mathbb{R}^2\backslash\{0\}))/S^1$.
The metric $h_0$ is regular on the open part $((\mathbb{R}^2\backslash\{0\})\times (\mathbb{R}^2\backslash\{0\}))/S^1$, and to express it, we choose coordinates $(\phi_1-\phi_2,\rho_1,\rho_2)$ on this regular locus. Computing $|d(\phi_1-\phi_2)|^2_g$, $|d\rho_1|^2_g$ and $|d\rho_2|^2_g$, we find that:
\begin{equation}\label{f:h_a_neq0_pf}
h_0=\frac{1}{\rho_1^2+\rho_2^2}\left(
\frac{\rho_1^2\rho_2^2}{\rho_1^2+\rho_2^2}{d(\phi_1-\phi_2)^2+k_1^2d\rho_1^2+k_2^2\rho_2^2}
\right).
\end{equation}
If we set $k_1=2/|a_-|$, $k_2=2/|a_+|$ and denote $\mu_1=\phi_1-\phi_2$, then the metric in~\eqref{f:h_a_neq0} will coincide with~\eqref{f:h_a_neq0_pf} up to a constant factor $4$. Therefore, $(M/S^1, 4h_0)$ is the metric completion of $(\mathbb{R}^3_{\pmb\mu}/\Gamma_1, h_{a_+,a_-})$ as claimed.
\end{proof}
We have just described the metric completions of $(\mathbb{R}^3_{\pmb\mu}/\Gamma_1,h)$ in the cases $a_-=0$ and $a_-\neq 0$, where group $\Gamma_1\simeq \mathbb{Z}$ acts on $\mathbb{R}^3_{\pmb\mu}$ via translations in $\mu_1$ direction. By Proposition~\ref{p:deck_transform_mu}, in the case $a_-=0$, we also have to deal with the completion $(\mathbb{R}^3_{\pmb\mu}/\Gamma_2,h)$, where $\Gamma=\Gamma_2\simeq \Gamma_1\times \mathbb{Z}$, with $\mathbb{Z}$ acting on $\mathbb{R}^3_{\pmb\mu}$ via
\begin{equation}\label{f:Z_action}
(\mu_1,\mu_+,\mu_-)\mapsto (\mu_1+c_1',\mu_+,\mu_-+c),\quad c\neq 0
\end{equation}
translations preserving the linear form $a_+\mu_++a_-\mu_-$. In the latter case, $\mathbb{R}^3_{\pmb\mu}/\Gamma_2$ is an isometric $\mathbb{Z}$-quotient of $\mathbb{R}^3_{\pmb\mu}/\Gamma_1$, thus the same is true for the completions:
\[
\bar{(\mathbb{R}^3_{\pmb\mu}/\Gamma_2,h)}\simeq \bar{(\mathbb{R}^3_{\pmb\mu}/\Gamma_1,h)}/\mathbb{Z},
\]
where the action of the factor $\mathbb{Z}$ of $\Gamma_2$ is given by Proposition~\ref{p:deck_transform_mu}. In either case, the completion $\bar{(\mathbb{R}^3_{\pmb\mu}/\Gamma,h)}$ has a natural smooth structure, and the metric $h$ extends to a metric with cone singularities along codimension two submanifolds.
\subsection{The global definition of the moment map on \texorpdfstring{$M$}{}} \label{ss:globalmoment}
\begin{defn}\label{d:N(a+,a-)}
Define $N(a_+,a_-)$ to be the metric completion $\bar{(\mathbb{R}^3_{\pmb\mu}/\Gamma_1,h_{a_+,a_-})}$. The metric $h_{a_+,a_-}$ induces a smooth metric with cone singularities on $N(a_+,a_-)$.
\end{defn}
Using the explicit description of $N(a_+,a_-)$ above, we can rewrite $p$ as
\[
p=\frac{1-\rho^2}{1+\rho^2},\quad p=\frac{\rho_1^2-\rho_2^2}{\rho_1^2+\rho_2^2}
\]
in the cases $a_-=0$ and $a_-\neq 0$ respectively. It follows that function $p$ defined initially on the open dense part has a continuous extension to the whole space, and the completion locus $N(a_+,a_-)\backslash (\mathbb{R}^3_{\pmb\mu}/\Gamma_1)$ consists of disjoint subsets $\{p=1\}$ and $\{p=-1\}$ (the latter possibly empty if $a_-=0$). The metric $h_{a_+,a_-}$ extends to a metric with cone singularities along $\{p=1\}$ and $\{p=-1\}$. Note that since $(\mathbb{R}^3_{\pmb\mu}/\Gamma_2,h_{a_+,a_-})$ is an isometric $\mathbb{Z}$-quotient of $(\mathbb{R}^3_{\pmb\mu}/\Gamma_1,h_{a_+,a_-})$, the same is true for the completion. Furthermore, with the explicit description of the completion, it is easy to see that the induced action of $\mathbb{Z}$ on $N(a_+,a_-)$ is free proper and isometric.
Let $(M,g,I,J)$ be a regular compete rank one soliton. By our assumption $a_+\neq 0$ and by Lemma~\ref{l:T_nonempty} the degeneracy set $\mathbf{T_+}$ is nonempty. Consider any point $x\in \mathbf{T_+}$ such that $X\neq 0$ at $x$. Let $G_x\simeq \mathbb{Z}_{k_+}$ be the stabilizer of $x$. Then the orbit space $\pi\colon M\to N$ has an orbifold structure in a neighourhood $U$ of $[x]\in N$ and the horizontal metric $Wh$ as well as $h$ have a cone singularity of angle $2\pi/{k_+}$ along the codimension two subset $(\pi^{-1}(U)\cap \mathbf{T}_+)/S^1$. On the other hand, there is an isometric embedding $N\to N(a_+,a_-)$ (or $N\to N(a_+,a_-)/\mathbb{Z}$) (see~\eqref{f:proj_orbit_extension}), and by the above propositions $N(a_+,a_-)$ has cone singularity of angle $\pi |a_+|$ along $\{p=1\}$. Therefore $2/k_+=|a_+|$. Arguing similarly in the case $a_-\neq 0$, we obtain the following \emph{quantization} result for $a_+,a_-\in \mathbb{R}$.
\begin{prop}[Quantization of parameters $a_+$ and $a_-$]\label{p:quantization}
If $a_+\neq 0$, then $\mathbf{T_+}$ is nonempty, and
\[
a_+=2/k_+,\ k_+\in \mathbb{Z},
\] where $|k_+|$ is the order of the stabilizer of any point $x\in\mathbf{T_+}\backslash M^{S^1}$.
If $a_-\neq 0$, then $\mathbf{T_-}$ is nonempty, and \[
a_-=2/k_-,\ k_-\in \mathbb{Z}\]
where $|k_-|$ is the order of the stabilizer of any point $x\in\mathbf{T_-}\backslash M^{S^1}$.
\end{prop}
\begin{rmk} \label{r:Norbifold1}
$N(a_+,a_-)$ has a natural manifold structure and a metric with cone angles of value $2\pi/k, k\in \mathbb{Z}$ along its codimension two submanifolds. Therefore we can think of $N(a_+,a_-)$ as an orbifold, and of $h$ as an orbifold metric. It is also straightforward to check that $\beta_0=d\mu_1\wedge(p_+ d\mu_-+p_-d\mu_+)$ defined on $\mathbb{R}^3_{\pmb\mu}/\Gamma$ extends to a smooth 2-form on the orbifold $N(a_+,a_-)$.
\end{rmk}
\begin{rmk}[$N(a_+,a_-)$ as a base space of a Seifert fibration]\label{r:N_as_quotient}
So far, in Propositions~\ref{p:completion_a0} and~\ref{p:completion_aneq0} we have obtained $(N(a_+,a_-), h)$ as a global quotient of a smooth manifold equipped with a cone metric. With the quantization of parameters $a_+$ and $a_-$ established above, there is an alternative description of $(N(a_+,a_-), h)$ as a global quotient of a smooth manifold equipped with a smooth metric:
\begin{enumerate}
\item Case $a_+=2/k_+, a_-=0$. Consider $\hat M=\mathbb{C}\times \mathbb{C}^*$ with coordinates $z,w\in \mathbb{C}$ and define an almost free circle action $S^1\times \hat M\to \hat M$ as $u\cdot (z,w)=(uz,u^{k_+}w)$. Then there is an isomorphism between the smooth parts of $\hat M/S^1$ and $N(2/k_+,0)$ given by the coordinate change
\begin{equation*}
\begin{split}
k_+&\arg z-\arg w=\mu_1,\\
|z|&=\exp(\mu_+/k_+),\\
|w|&=\exp (2\mu_-).
\end{split}
\end{equation*}
As in the proof of Proposition~\ref{p:completion_a0}, this isomorphism transforms the horizontal component $\hat g_{\mathrm{hor}}$ of the metric
\[
\hat g=\frac{4}{1+|z|^2}\Re\left(
k_+^{2} dz\otimes d\bar z + \frac{dw\otimes d\bar w}{|w|^2}
\right)
\]
to $h$ and extends to an isometry of complete Riemannian orbifolds
\begin{equation*}
(\hat M/S^1,\hat g_{\mathrm{hor}})\simeq (N(2/k_+,0), h).
\end{equation*}
\item Case $a_+=2/k_+, a_-=2/k_-$. Assume first that $\mathrm{gcd}(k_+,k_-)=1$. Consider $\hat M=\mathbb{C}^2\backslash\{0\}$ with coordinates $z,w$ and define an almost free circle action $S^1\times \hat M\to \hat M$ as $u\cdot (z,w)=(u^{k_+}z,u^{k_-}w)$. Then there is an isomorphism between the smooth parts of $\hat M/S^1$ and $N(2/k_+,2/k_-)$ given by the coordinate change
\begin{equation*}
\begin{split}
k_-&\arg z-k_+\arg w=\mu_1,\\
|z|&=\frac{e^{-\mu_-/k_-}}{4(k_-^{-2}e^{2\mu_+/k_+}+k_+^{-2}e^{-2\mu_-/k_-})},\\
|w|&=\frac{e^{\mu_+/k_+}}{4(k_-^{-2}e^{2\mu_+/k_+}+k_+^{-2}e^{-2\mu_-/k_-})}.
\end{split}
\end{equation*}
As in the proof of Proposition~\ref{p:completion_aneq0}, this isomorphism transforms the horizontal component $\hat g_{\mathrm{hor}}$ of the metric
\[
\hat g=\frac{4}{|z|^2+|w|^2}\Re\left(k_-^2dz\otimes d\bar z+k_+^2dw\otimes d\bar w\right)
\]
into $h$, and extends to an isomorphism of the underlying orbifolds:
\begin{equation*}
(\hat M/S^1,\hat g_{\mathrm{hor}})\simeq (N(2/k_+,2/k_-), h).
\end{equation*}
Finally, if $\mathrm{gcd}(k_-,k_+)=d>1$, then orbifold $(N(2/k_+,2/k_-), h)$ is a $\mathbb{Z}_d$ quotient of $(N(2d/k_+,2d/k_-), h)$, where $\mathbb{Z}_d$ acts on $N(2d/k_+,2d/k_-)$ via rotations in $\mu_1$ coordinate.
\end{enumerate}
\end{rmk}
Finally we are ready to show the surjectivity of the moment map. Assume for concreteness that in Proposition~\ref{p:deck_transform_mu} we have group $\Gamma=\Gamma_1$. We aim to show that the natural isometric embedding of the orbit space $N\to N(a_+,a_-)$ is an isometry. We know that the map $(M\backslash\mathbf T)/S^1\to \mathbb{R}^3_{\pmb\mu}/\Gamma$ is a diffeomorphism, and the metric $h$ can be intrinsically defined both on $N_0=(M\backslash M^{S^1})/S^1$ and on $N(a_+,a_-)$. Since $N\to N(a_+,a_-)$ is an isometric embedding, smooth on an open dense part, it must be smooth on the whole $N_0$. This can be seen, e.g., by considering exponential charts centered outside of the locus $\{p=\pm 1\}$.
To prove surjectivity of $N\to N(a_+,a_-)$, we are going to invoke Proposition~\ref{p:s1_bundle_complete} once again. Consider $\mathbf{T}_{+,\ge}:=\{p\geq 1-\ge\}\subset M$. For a generic constant $\ge>0$, this is a connected complete manifold with boundary. As in Remark \ref{r:Norbifold1}, $\beta_0$ and $h$ are well-defined smooth tensors on $N_0$. Using the same argument as before in the proof of Proposition~\ref{p:deck_transform_mu}, we add a multiple of the baseline solution $\til W$ to function $W$, and modify the metric $g$ and connection $\eta$ accordingly. Repeating the argument in the proof of Proposition~\ref{p:deck_transform_mu}, we conclude that $N\cap \{p\geq 1-\ge\}$ is complete with respect to the distance function $d_h$ induced by the metric $h$. Similarly the pieces $N\cap \{p\leq -1+\ge\}$ and $N\cap \{|p|\leq 1-\ge\}$ are $d_h$-complete. Therefore $N$ is also complete with respect to $d_h$, hence $N\to N(a_+,a_-)$ is an isometry.
The orbifold $N(a_+,a_-)$ comes equipped with a distinguished metric $h=h_{a_+,a_-}$, function $p$ and 2-form $\beta_0$ on an open dense part. Using the coordinates of Proposition~\ref{p:completion_a0} and~\ref{p:completion_aneq0}, it is straightforward to check that $h$, $p$, $\beta_0$ (see equation~\eqref{f:beta0_def}), and $\theta_I$ (see equation~\eqref{f:theta_through_p}) extend to smooth (in the orbifold sense) tensors on $N(a_+,a_-)$.
Collecting Theorem~\ref{t:nondegenerate_gk_description}, Proposition~\ref{p:deck_transform_mu} and Proposition~\ref{p:quantization}, we obtain the following.
\begin{thm}\label{t:M_orbifold_quotient}
Let $(M,g,I,J)$ be a complete regular rank one soliton. Then there exist constants $a_+,a_-\in \{2/k\ |\ k\in\mathbb{Z} \}\cup \{0\}$, $(a_+,a_-)\neq (0,0)$ such that either
\[
M/S^1\simeq N(a_+,a_-),
\]
or
\[
M/S^1\simeq N(a_+,0)/\mathbb{Z},
\]
where $\mathbb{Z}$ acts on $N(a_+,0)$ by translations in $(\mu_1, \mu_-)$ coordinates preserving $a_+\mu_++a_-\mu_-$.
Furthermore on the complement of the fixed point set $M^{S^1}$ there exists an $S^1$-invariant function $W$ such that the Seifert fibration
\[
M\backslash M^{S^1}\to(M\backslash M^{S^1})/S^1
\]
has curvature $\beta=*_hdW+W\beta_0$ with connection $\eta$ and the underlying GK structure is given by
\begin{equation*}
\begin{split}
g=&\ Wh+W^{-1}\eta^2\\
\Omega_{I}=&
(-d\mu_1+\sqrt{-1}d\mu_2)\wedge(\eta+\sqrt{-1}W(d\mu_3-pd\mu_2)),\\
\Omega_J=&
(-d\mu_1+\sqrt{-1}d\mu_3)\wedge(\eta+\sqrt{-1}W(-d\mu_2+pd\mu_3)).
\end{split}
\end{equation*}
\end{thm}
\begin{rmk}
Assume as before $a_+\neq 0$. The topology of the orbit space $N(a_+,a_-)/H$ depends on $a_+,a_-$ and $H$ as follows:
\begin{enumerate}
\item $H=\{\mathrm{id}\}$, $a_+=2/k_+$, $a_-=0$. In this case orbifold $N(a_+,a_-)$ is diffeomorphic to a global quotient of $\mathbb{C}\times \mathbb{R}$ by a linear $\mathbb{Z}_{k_+}$-action in the first coordinate. The underlying smooth manifold is diffeomorphic to $\mathbb{R}^3$;
\item $H=\mathbb{Z}$, $a_+=2/k_+$, $a_-=0$. In this case $N(a_+,a_-)$ is diffeomorphic to a quotient $(\mathbb{C}/\mathbb{Z}_{k_+}\times \mathbb{R})/\mathbb{Z}$, where $\mathbb{Z}$ acts via rotation on the first factor and via nontrivial translation on the second. The underlying smooth manifold is diffeomorphic to $\mathbb{R}^2\times S^1$;
\item $H=\{\mathrm{id}\}$, $a_+=2/k_+$, $a_-=2/k_-$. In this case orbifold $N(a_+,a_-)$ is diffeomorphic to a product $S^2(k_+,k_-)\times \mathbb{R}$, where $S^2(k_+,k_-)$ is a \emph{spindle} 2-sphere with cone angles $2\pi/k_+$ and $2\pi/k_-$.
The underlying smooth manifold is diffeomorphic to $S^2\times \mathbb{R}$;
\end{enumerate}
\end{rmk}
\begin{cor}\label{c:fixed_smooth}
Let $(M,g,I,J)$ be a complete regular rank one soliton, and denote by $\pi\colon M\to N(a_+,a_-)/H$ the projection onto the orbit space. If $y\in M$ is a fixed point of the $S^1$ action, then the representation of $S^1$ in $T_yM$ has weights $(\pm 1,\pm 1)$ and $\pi(y)$ is a smooth point of orbifold $N(a_+,a_-)/H$.
\end{cor}
\begin{proof}
We already know by Lemma~\ref{l:free_nondegenerate} that any fixed point $y$ in the nondegenerate part $M\backslash\mathbf T$ must have weights $(\pm 1,\pm 1)$. Now assume that $y\in\mathbf{T}\subset M$ is a fixed point with weights $\mathbf{w}=(w_1,w_2)$, $\mathrm{gcd}(w_1,w_2)=1$. Let $y\in U\simeq \mathbb{C}^2$ be a neighbourhood of $y$ provided by the slice theorem. Then $(U\backslash\{y\})/S^1$ is an orbifold isomorphic to a product $\mathbb{R}\times S^2(|w_1|,|w_2|)$ of $\mathbb{R}$ and a spindle $S^2$ with two orbifold points of orders $|w_1|$ and $|w_2|$.
On the other hand, given a point $z\in N(a_+,a_-)$ there are two options. Either $z$ is a smooth point, and its punctured neighbourhood is smooth. Or a punctured neighbourhood of $z$ is isomorphic as an orbifold to a product $\mathbb{R}\times S^2(k,k)$, $k\in\{k_+,k_-\}$. Hence at $y\in M$ we must have $|w_1|=|w_2|$ which is possible only if $w_i=\pm 1$.
\end{proof}
The above corollary implies that if there is a fixed point on the degeneracy divisor $\mathbf T_+$ (resp.\,$\mathbf T_-$), then $k_+=\pm 1$ (resp.\,$k_-=\pm 1$).
\section{Construction and classification of solutions} \label{s:construction}
In this section we give a construction of complete rank one generalized K\"ahler-Ricci solitons, proving Theorem \ref{t:existence}. Having already determined the function $p$ using Proposition \ref{p:invGKsoliton}, our task is to find suitable functions $W$ solving (\ref{f:W_laplace2}), which extend naturally over the degeneracy loci. Note that in Lemma \ref{l:W0_solution} we already identified the baseline solution $\til{W}$ to equation (\ref{f:W_laplace2}). We first observe that for any other solution $W$, the function $V = W/\til{W}$ satisfies a Laplace equation for a certain metric conformal to to the induced metric on the quotient space. Using this fact and analyzing the explicit constructions of the spaces $N(a_+, a_-)$ in \S \ref{s:completion}, we classify the possible solutions to equation (\ref{f:W_laplace2}). With the explicit construction of the horizontal geometry and the classification of possible functions $W$ in hand, we give the proof of the main existence theorem (Theorem \ref{t:existence}), and the classification theorem (Theorem \ref{t:uniqueness}).
\subsection{Reduction of the equation for \texorpdfstring{$W$}{} to a Laplace equation} \label{ss:Wlaplace}
The first step in our analysis is to better understand the structure of the equation for $W$ in the case of solitons. Recall that by Lemma~\ref{l:W0_solution} the function
\[
\til W:=\left(a_+^2(1+p)+a_-^2(1-p)\right)^{-1}
\]
solves equation~\eqref{f:W_laplace2} provided $p$ corresponds to a rank one GK soliton as in Proposition~\ref{p:invGKsoliton}. With the use of this baseline solution we can reduce the equation for $W$ to the Laplace equation for a metric $\til h$ conformally related to $h$.
\begin{prop} \label{p:laplace_reduction} Equation~\eqref{f:W_laplace2} for $W$ is equivalent to the Laplace equation
\[
\Delta_{\til h}V=0
\]
for the function $V:=W/\til W$, where
\[
\til h:= \left(\frac{2\til W^2}{e^{a_+\mu_+}+e^{-a_-\mu_-}} \right)^2 h.
\]
\end{prop}
\begin{proof}
Using that $\til W$ is a solution, we can rewrite equation~\eqref{f:W_laplace2} as an equation for $V = W/\til W$:
\begin{equation}\label{f:V_laplace2}
\begin{split}
\til W\bigl(V_{11}+\tfrac{1}{2}(1+p)V_{++}+\tfrac{1}{2}(1-p)V_{--}\bigr)
+(\til W(1+p))_+V_++(\til W(1-p))_-V_-=0.
\end{split}
\end{equation}
Now let
\[
\psi = \frac{2\til W^2}{e^{a_+\mu_+}+e^{-a_-\mu_-}} = \til W^2(1-p)e^{-a_+\mu_+} = \til W^2(1+p)e^{a_-\mu_-},
\]
so that the conformal metric is $\til h=\psi^2h$. Then we find that
\begin{equation*}
\begin{split}
\Delta_{\til h}V&=\frac{1}{2\psi^3(1-p^2)}
\left(
2\psi V_{11}+(\psi(1+p)V_{+})_{+}+(\psi(1-p)V_{-})_{-}
\right)\\
&=\frac{1}{\psi^2(1-p^2)}\left(V_{11}+\tfrac{1}{2}(1+p)V_{++}+\tfrac{1}{2}(1-p)V_{--}+
\frac{(\psi(1+p))_{+}}{2\psi}V_{+}+
\frac{(\psi(1-p))_{-}}{2\psi}V_{-}\right).
\end{split}
\end{equation*}
Using the different expressions for $\psi$ above it is straightforward to check that
\[
\frac{(\psi(1+p))_{+}}{\psi}=2\frac{(\til W(1+p))_+}{\til W}, \qquad \frac{(\psi(1-p))_{-}}{\psi}=2\frac{(\til W(1-p))_-}{\til W},
\]
which implies that \eqref{f:V_laplace2} is equivalent to $\Delta_{\til h}V=0$.
\end{proof}
The function $\psi:=2\til W^2/(e^{a_+\mu_+}+e^{a_-\mu_-})$ is defined initially on $\mathbb{R}^3_{\pmb\mu}$, and extends to a smooth function on the orbifold $N(a_+,a_-)$. Thus $\til h$ can be thought of as an orbifold metric on $N(a_+,a_-)$. If $\{a_+,a_-,W\}$ corresponds to a complete rank one regular GK soliton $(M,g,I,J)$, then $V=W/\til W$ is a positive smooth solution to the Laplace equation $\Delta_{\til h} V=0$ on $N(a_+,a_-)\backslash\{z_i\}$, where $\{z_i\}$ is the set of fixed point orbits. Furthermore, since the vector field $X$ generating the $S^1$ action has a simple zero at each $z_i$, we have $V(x)\sim d_h(x,z_i)^{-1}$ in a neighbourhood of $z_i$.
Thus, to proceed with the construction and classification of GK solitons, we need to construct and classify positive solutions to the Laplace equation $\Delta_{\til h}V=0$ on $N(a_+,a_-)$ with poles on a finite set $\{z_i\}$. There are two difficulties with answering this question. The first one is technical: $N(a_+,a_-)$ is an orbifold rather than a manifold, therefore we have to be somewhat cautious in developing the elliptic theory on $N(a_+,a_-)$. The second one is more substantial~--- $N(a_+,a_-)$ equipped with the metric $\til h$ is not necessarily complete, its completion is not an orbifold, and we do not have any a priori knowledge about the behavior of $V$ near the completion. Thus a large part of the general well-developed elliptic theory is not available to us.
Luckily, thanks to a very special structure of orbifold $N(a_+,a_-)$ and metric $\til h$, we are able to surpass these difficulties by `lifting' the Laplace equation $\Delta_{\til h}V=0$ on an incomplete 3-dimensional orbifold to an equivalent Laplace equation on a complete flat 4-dimensional manifold. This idea works equally well if either $a_-=0$ or $a_-\neq 0$, however, since the details differ slightly, we consider the two cases separately.
\subsection{Equation \texorpdfstring{$\Delta_{\tilde h}V=0$}{} in the case \texorpdfstring{$a_- = 0$}{}}
\begin{prop}\label{p:elliptic_a0}
Let $N(a_+,0)$ be the orbifold from Definition~\ref{d:N(a+,a-)}.
\begin{enumerate}
\item Given any point $z\in N(a_+,0)$ there exists a unique smooth positive function
\[G_z(x)\colon N(a_+,0)\backslash\{z\}\to (0;\infty)\] such that:
\begin{enumerate}
\item $\Delta_{\til h}G_z(x)=-2\pi\delta_z(x)$ in the sense of distributions;
\item $\inf_{x\in N(a_+,0)} G_z(x)=0$;
\end{enumerate}
\item If $V$ is a positive solution to the equation $\Delta_{\til h}V=0$ on the complement of a finite set $\{z_i\}$, then there exist unique positive constants $c_V,c_i$ such that
\[
V(x)=c_V+\sum_{i} c_i G_{z_i}(x).
\]
\item Any positive superharmonic function on $N(a_+,0)/\mathbb{Z}$ is constant.
\end{enumerate}
\end{prop}
\begin{rmk}
Borrowing terminology from the standard elliptic theory on smooth complete manifolds, we can say that $(N(a_+,0),\til h)$ is \emph{non-parabolic}, i.e., admits a positive Green's function and satisfies a \emph{Liouville property}, i.e., any positive harmonic function is constant; while $(N(a_+,0)/\mathbb{Z},\til h)$ is \emph{parabolic}~--- any positive superharmonic function on it is constant.
\end{rmk}
\begin{proof}
We start by recalling an observation from the proof of Proposition~\ref{p:s1_bundle_complete}. If $M\to N$ is a principal $S^1$ bundle with a connection one-form $\eta$, and the metric $g$ on $M$ is of the form
\[
g=Wh+W^{-1}\eta^2,
\]
where $W\colon M\to N$ is a smooth function and $h$ is a metric on $N$, then $f\colon N\to \mathbb{R}$ solves the Laplace equation $\Delta_h f=0$ on $M$ if and only if its lift to $M$ solves equation $\Delta_g f=0$, and the same is true for inequality. Hence there is a bijection:
\begin{equation}\label{f:laplace_equivalence}
\{\mbox{harmonic functions on }(N,h)\}\longleftrightarrow \{S^1\mbox{ invariant harmonic functions on }(M,g)\}.
\end{equation}
We use the constructions of Remark~\ref{r:N_as_quotient} and realize $N(a_+,0)$ as a quotient of $\hat M=\mathbb{C}\times \mathbb{C}^*$ by a circle action $u\cdot (z,w)=(uz,u^{k_+}w)$. Under this identification the conformal factor $\psi^2$ of Proposition~\ref{p:laplace_reduction} becomes
\[
\psi^2=\frac{(1+|z|^2)^2}{a_+^8}.
\]
Let $X\in \Gamma(T\hat M)$ be the vector field generating the $S^1$ action, and denote by $\eta$ the connection induced by the metric $\hat g$. Then $|X|^2_{\hat g}=4k_+^2=16a_+^{-2}$, and on $\hat M$ we can express the metric $\hat g$ as
\[
\hat g=h+\frac{16}{a_+^2}\eta^2.
\]
Thus the flat metric on $\hat M$,
\[
g_f:=\frac{1+|z|^2}{4}\hat g=\Re(k_+^2dz\otimes d\bar z+\frac{dw\otimes d\bar w}{|w|^2}),
\]
can be expressed as
\[
g_f=\frac{1+|z|^2}{4}h+\frac{4(1+|z|^2)}{a_+^2}\eta^2=\frac{a_+^2}{4(1+|z|^2)} \underbrace{\frac{(1+|z|^2)^2}{4a_+^2}h}_{\mbox{const} \cdot \til h}\ +\ \frac{4(1+|z|^2)}{a_+^2}\eta^2.
\]
Now, using observation~\eqref{f:laplace_equivalence} we find that $V$ on $N(a_+,0)$ solves equation $\Delta_{\til h}V=0$ if and only if its lift to $\hat M\simeq \mathbb{R}^3\times S^1$ solves the Laplace equation
\[
\Delta_{g_f}V=0
\]
with respect to the flat metric $g_f$.
Now we use the well-developed elliptic theory on the flat $\mathbb{R}^3\times S^1$ to conclude the proof. It is known $\mathbb{R}^3\times S^1$ is non-parabolic, i.e., given any point $y\in \mathbb{R}^3\times S^1$ there exists an everywhere positive Green's function $\Gamma_y(x)$ solving the equation
\[
\Delta_{g_f}\Gamma_y(x)=-2\pi\delta_y(x).
\]
Such $\Gamma_y(x)$ can be either constructed explicitly as a convergent sum of Green's functions of $\mathbb{R}^4$, or its existence can be proved using a general result, e.g.,~\cite[\S 3]{GrigoryanParabolicity}. Furthermore the flat $\mathbb{R}^4$ satisfies the Liouville property~--- any entire positive harmonic function must be constant, therefore its isometric quotient $\mathbb{R}^3\times S^1$ also satisfies the Liouville property.
Let $z=[S^1y]\in \hat M/S^1$ be the corresponding point in the orbit space. Averaging $\Gamma_y(x)$ with respect to the $S^1$ action
\[
G_{z}(x):=\frac{1}{2\pi}\int_{S^1} \Gamma_{y}(u\cdot x)du
\]
we obtain an $S^1$-invariant function $G_{z}(x)$ which descends to the orbifold quotient $\hat M/S^1$ and solves $\Delta_{g_f} G_z(x)=-2\pi\delta_{S^1y}(x)$ in $\hat M$ or equivalently
\[
\Delta_{\til h} G_z(x)=-2\pi\delta_z(x),
\]
in $\hat M/S^1$. Moreover $\lim G_z(x)=0$ as $d_{g_f}(x,z)\to \infty$. If $G'_z(x)$ is another function solving the same equation with $\inf G'_z(x)=0$, then the difference $G'_z(x)-G_z(x)$ is an entire harmonic function on $\hat M$. By the maximum principle, the difference is bounded from below by $0$, therefore by the Liouville property the two functions must differ by a constant. Since $\inf G'_z(x)=0$, the constant is zero.
Let $V$ be any $S^1$-invariant positive harmonic function on the complement of finitely many $S^1$-orbits $\hat M\backslash \{S^1z_i\}$. Using the Riesz representation theorem and the $S^1$ invariance of $V$, we can decompose $V$ as
\[
V=V_0+\sum c_i G_{z_i},
\]
where $c_i$ are constants and $V_0$ is an entire harmonic function. Since $V$ is positive, $c_i>0$. Hence by the maximum principle $V_0$ is positive, and must be constant.
Now consider $N(a_+,0)/\mathbb{Z}$. Action of $\mathbb{Z}$ on $N(a_+,0)$ lifts to a $g_f$-isometric action on $\hat M$. Using the same argument as above we have a bijection between superharmonic functions with poles at $\{z_i\}$ on $(N(a_+,0)/\mathbb{Z},\til h)$ and $S^1$-invariant superharmonic functions on $(\hat M/\mathbb{Z}, g_f)$ with poles along $S^1$ orbits. It remains to note that manifold $\hat M/\mathbb{Z}=(\mathbb{C}\times \mathbb{C}^*)/\mathbb{Z}$ with metric $g_f$ has quadratic volume growth thus by a result of Cheng and Yau~\cite{ChengYau} does not admit any non-constant superharmonic functions.
\end{proof}
\subsection{Equation \texorpdfstring{$\Delta_{\tilde h}V=0$}{} in the case \texorpdfstring{$a_- \neq 0$}{}}
\begin{prop}\label{p:elliptic_aneq0}
Let $N(a_+,a_-)$ be the orbifold from Definition~\ref{d:N(a+,a-)}. Consider a finite collection of points $\{z_i\}\subset N(a_+,a_-)$.
\begin{enumerate}
\item Given any point $z\in N(a_+,a_-)$ there exists a unique smooth positive function \[G_z(x)\colon N(a_+,a_-)\backslash\{z\}\to (0;\infty)\] such that:
\begin{enumerate}
\item $\Delta_{\til h}G_z(x)=-2\pi\delta_z(x)$ in the sense of distributions;
\item $\inf_{x\in N(a_+,a_-)} G_z(x)=0$;
\end{enumerate}
\item If $V$ is a positive solution to the equation $\Delta_{\til h}V=0$ on the complement of a finite collection of points $\{z_i\}$, then there exist unique positive constants $c'_V,c_V'',c_i$ such that
\[
V(x)=c_V'+c_V''G_0(x)+\sum_{i} c_i G_{z_i}(x),
\]
where $G_0(x)=k_+^2e^{2\mu_+/k_+}+k_-^2e^{-2\mu_-/k_-}$.
\item Any positive superharmonic function on $N(a_+,a_-)/\mathbb{Z}$ is constant.
\end{enumerate}
\end{prop}
\begin{proof}
The proof is essentially the same as the proof in the case $a_-=0$. For simplicity, we assume that $\mathrm{gcd}(k_+,k_-)=1$. The proof in the general case is analogous.
As before, we use Remark~\ref{r:N_as_quotient} to realize $N(a_+,a_-)$ as the quotient of $\hat M=\mathbb{C}^2\backslash\{0\}$ by a circle action $u\cdot (z,w)=(u^{k_+}z,u^{k_-}w)$. Under this identification, the conformal factor $\psi^2$ becomes
\[
\psi^2=\frac{(|z|^2+|w|^2)^2}{4}.
\]
Let $X$ be the vector field generating the $S^1$ action on $\hat M$, and denote by $\eta$ the principal connection induced by $\hat g$. Then $|X|_{\hat g}=2k_+k_-$ and on $\hat M$ we can express metric $\hat g$ as
\[
\hat g=h+(2k_+k_-)^2\eta^2.
\]
Thus the flat metric on $\hat M$,
\[
g_f:=\frac{|z|^2+|w|^2}{4}\hat g=\Re\left(k_-^2dz\otimes d\bar z+k_+^2dw\otimes d\bar w\right),
\]
can be expressed as
\[
\begin{split}
g_f&=\frac{|z|^2+|w|^2}{4}h+k_+^2k_-^2(|z|^2+|w|^2)\eta^2\\&=
\frac{1}{k_+^2k_-^2(|z|^2+|w|^2)} \underbrace{\frac{k_+^2k_-^2(|z|^2+|w|^2)^2}{4} h}_{\mbox{const}\cdot \til h}\ +\ k_+^2k_-^2(|z|^2+|w|^2)\eta^2
\end{split}
\]
Using the same observation~\eqref{f:laplace_equivalence}, we find that $V$ on $N(a_+,a_-)$ solves $\Delta_{\til h}V=0$ if and only if its lift to $\hat M$ solves the Laplace equation with respect to the flat metric $g_f$.
At this point we can repeat the argument of Proposition~\ref{p:elliptic_a0} with one difference~--- $\hat M=\mathbb{C}^2\backslash\{0\}$ is not complete with respect to the flat metric $g_f$. To this end, we have to consider $V$ on its completion $\mathbb{C}^2\supset \mathbb{C}^2\backslash\{0\}$. We observe that $V$ is a positive harmonic function (possibly with poles along a finite collection of $S^1$ orbits), undefined at the origin. Therefore we can decompose it as a linear combination of the Green's function of $\mathbb{C}^2$ with the pole at $\{0\}$ and a positive harmonic function (with the same poles as $V$) well-defined at $\{0\}$. The rest of the proof proceeds verbatim as the proof of Proposition~\ref{p:elliptic_a0}. Thus we have to add an extra term $G_0(x)$ in the decomposition of $V(x)$. It remains to note that up to a constant multiple the Green's function on $\mathbb{R}^4$ is given by $G_0(x)=d_{g_f}(x,0)^{-2}= (k_-^2|z|^2+k_+^2|w|^2)^{-1}$, which (again up to a constant multiple) equals $k_+^2e^{2\mu_+/k_+}+k_-^2e^{-2\mu_-/k_-}$ as in the statement of the proposition.
In the second case of $N(a_+,a_-)/\mathbb{Z}$, the action of group $\mathbb{Z}$ lifts to a $\mathbb{Z}$-action on $\mathbb{C}^2\backslash \{0\}$ via contractions fixing the origin:
\[
|z|\mapsto \lambda|z|, \quad |w|\mapsto \lambda |w|,\quad \arg z\mapsto \arg z+\alpha_z, \quad \arg w\mapsto \arg w+\alpha_w
\]
It remains to note if a positive superharmonic function on $\mathbb{R}^4$ is invariant under these contractions, then it attains a minimum at an interior point and must be constant by the maximum princple.
\end{proof}
\begin{rmk}
In either case, whether $a_-=0$ or $a_-\neq 0$, let $G_{z_i}$ be a Green's function on $N(a_+,a_-)\backslash\{z_i\}$ constructed in Propositions~\ref{p:elliptic_a0} and~\ref{p:elliptic_aneq0}. For $W=\til W G_{z_i}$ we have
\[
\beta=*_hdW+W\beta_0=\frac{\psi}{\til W}*_{\til h}dG_{z_i}+G_{z_i}(*_hd\til W+\til W\beta_0).
\]
Consider $z_i\in N(a_+,a_-)$~--- a smooth point, and a sphere $S^2_\ge$ of $\til h$-radius $\ge$ enclosing $z_i$ oriented using the outer normal. Then using the closedness of $\beta$ and the standard estimate $G_{z_i}(z)\sim d_{\til h}(z,z_i)^{-1}$ near $z_i$ we compute
\[
\int_{S^2_\ge}\beta=\lim_{\ge\to 0}\int_{S^2_\ge}\beta=\lim_{\ge\to 0}\int \frac{\psi}{\til W}*_{\til h}dG_{z_i}=-2\pi \frac{\psi(z_i)}{\til W(z_i)}.
\]
In particular, $c_{z_i}\beta\in H^2(N(a_+,a_-)\backslash\{z_i\},2\pi\mathbb{Z})$ is a generator, where the normalization constant is given by
\begin{equation}\label{f:c_zi_normalization}
c_{z_i}:=\til W(z_i)/\psi(z_i).
\end{equation}
\end{rmk}
\subsection{Existence and uniqueness}
We conclude with the proofs of the main existence and classification results. As above our discussion splits into the cases where $a_-$ is either vanishing or nonvanishing. Furthermore, the case $a_- = 0$ splits into the cases according to the possible quotient spaces $N(a_+, 0)$ and $N(a_+, 0) / \mathbb Z$ as in Theorem \ref{t:nondegenerate_gk_description}. We point to Figures \ref{f:fig1} and \ref{f:fig2} for a brief summary.
\begin{prop} \label{p:gensoln1} Let $(k_+,l_+)$ be a pair of coprime integers, $0\leq l_+<|k_+|$ and denote $a_+ = \frac{2}{k_+}$. Given $\lambda\geq0$ and a finite collection of points $\{z_1, \dots, z_n\}$ in the smooth locus of $N(a_+, 0)$, let
\[
W=\til{W} \Bigl( \lambda + \sum_{i=1}^nc_{z_i}G_{z_i} \Bigr),
\]
where $G_{z_i}$ is the Green's function centered at $z_i$ constructed in Proposition \ref{p:elliptic_a0} and $c_{z_i}>0$ is the normalization constant as in~\eqref{f:c_zi_normalization}.
There exists a unique complete regular rank one soliton $(M^4, g, I, J)$ such that
\begin{enumerate}
\item the orbit space $\pi\colon M\to M/{S^1}$ is homeomorphic to $N(a_+, 0)$,
\item $\pi^{-1} \{z_1, \dots, z_n\} = M^{S^1}$,
\item $\mathbf{T_+}\backslash \pi^{-1}\{z_1, \dots, z_n\}$ is the set of points of type $(k_+, l_+)$ (cf.\,Remark \ref{r:local_s1_quotient}),
\item In the image of the nondegeneracy locus we have
\begin{align*}
\Phi = a_+ \mu_+,
\end{align*}
\item With the above choice of functions $W$ and $p$, locally in the nondegeneracy locus the GK structure on $(M,g,I,J)$ is given by the construction of Theorem~\ref{t:nondegenerate_gk_description_v2}.
\end{enumerate}
\begin{proof}
Condition (5) uniquely prescribes the curvature form $\beta=*_hdW+W\beta_0$ on a dense open part of $N(a_+,0)$, and given the explicit form of $\beta_0$ (see Remark \ref{r:Norbifold1}) and since $W$ is a smooth function on orbifold $N(a_+,0)$, the curvature form also extends to a smooth two-form on $N(a_+,0)$. Since $H^2(N(a_+,0),\mathbb{Z})=0$, by our normalization of $G_{z_i}$, $\beta$ represents a class in
\[
H^2(N(a_+,0)\backslash\{z_1,\dots,z_n\},2\pi\mathbb{Z}).
\]
Thus the assumptions of Proposition~\ref{p:s1_bundle_seifert_dg} are satisfied, and there exists a unique Siefert fibration
\[
\pi : M_0 \to N(a_+,0)\backslash\{z_1,\dots,z_n\}
\]
together with a smooth connection form $\eta$ with curvature $\beta$. Since $N(a_+,0)$ is simply connected, $\eta$ is also unique up to a gauge transform.
Let $\Sigma = \{p = 1\} = \{\rho = 0\} \subset N(a_+, 0)$. We set $\Phi = a_+ \mu_+$, which in turn defines a smooth function $p$ on $N(a_+,0)$:
\[
p=\frac{1-e^{a_+\mu_+}}{1+e^{a_+\mu_+}}.
\]
In the case $k_+ = \pm 1$, $\Sigma$ consists of smooth points, and by reordering we assume $\{z_1, \dots, z_m\} \in \Sigma^c, \{z_{m+1}, \dots, z_n\} \in \Sigma$. Otherwise if $|k_{\pm}|>1$, we have $m=n$.
By definition, the complement $N(a_+, 0) \backslash \Sigma$ is isomorphic to $\mathbb{R}^3_{\pmb\mu}/\Gamma_1$, where $\Gamma_1$ acts by translations in $\mu_1$ coordinate.
In particular, we can identify a simply connected open subset $U$ of $N(a_+, 0) \backslash \Sigma$ with an open subset in $\mathbb{R}^3_{\pmb\mu}$. By our choice of function $W$ and the corresponding form $\beta$, we can apply Theorem~\ref{t:nondegenerate_gk_description_v2} and endow $\pi^{-1}(U)$ with a GK structure determined by $p$ and $W$. Since $p$ does not depend on $\mu_1$, this GK structure is independent of the identification between $U$ and a subset of $\mathbb{R}^3_{\pmb\mu}$, yielding a GK structure on
\begin{align*}
M_1 := \left(M_0 \backslash \pi^{-1} (\Sigma) \right) \cup \{y_1, \dots, y_m\},
\end{align*}
where $\{y_i\}$ are the fixed points of the natural $S^1$ action on $M_1$. The orbit space of this action is homeomorphic to $N(a_+, 0) \backslash \Sigma$ with $\pi(y_i)=z_i$, $1\leq i\leq m$. The constructed GK structure is smooth away from the points $\{y_1, \dots, y_m\}$ and is $C^{1,1}$ globally.
We now prove that $(g, I, J)$ extends smoothly across $\pi^{-1}(\Sigma)$. As explained above, we know that $W, h$, and $\beta$ are smooth on the orbifold $N(a_+, 0) \backslash \{z_1, \dots, z_n\}$, so the metric $g$ will extend smoothly across $\pi^{-1} \left(\Sigma \backslash \{z_{m+1}, \dots, z_n\} \right)$. Next for our specific choice of $p$, the Lee form $\theta_I$ (see~\eqref{f:theta_through_p}) descends to $N(a_+,0)$ and extends smoothly across $\Sigma$, thus $H=-*_g\theta_I$ is a smooth 3-form on $M_0$. Arguing using parallel transport as in Proposition \ref{p:removable_singularity_GK}, it follows that the complex structures $I$ and $J$ extend smoothly (in fact real analytically) across this locus as well. This gives a possibly incomplete smooth GK structure on
\[
M_0\cup\{y_1,\dots,y_m\}
\]
with an orbit space $N(a_+,0)\backslash\{z_{m+1},\dots, z_n\}$.
Next, we prove that we can glue in fixed points $\{y_{m+1},\dots y_n\}$ such that
\[
M_0\cup \{y_1,\dots,y_{n}\}
\]
is a complete $C^{1,1}$ GK manifold with orbit space $N(a_+,0)$. Using the normalization of the Green's functions at $\{z_{m+1}, \dots, z_{n}\}$, and the fact that the locus $\Sigma\subset N(a_+,0)$ must be smooth as long as $m<n$ (see Corollary~\ref{c:fixed_smooth}), we observe that preimage $\pi^{-1}(B^3\backslash \{z_i\})$ of a small punctured ball around $z_i$ is diffeomorphic to $B^4\backslash \{0\}$. The function $W$ evidently satisfies the required local behavior of Proposition~\ref{p:removable_singularity} near each $x_i\in\{x_1,\dots,x_m\}$, thus we can extend the metric across $B^4$ after gluing in a point $y_i$.
Therefore $H=-*_g\theta_I$ admits a $C^{1,1}$ extension on $M_0\cup \{y_1,\dots,y_{n}\}$, and we can apply the first part of Proposition \ref{p:removable_singularity_GK} to show that the GK structure extends in $C^{1,1}$ sense. Thus we have constructed a $C^{1,1}$ GK structure $(M,g,I,J)$ with an orbit space
\[
\pi\colon M\to N(a_+,0)
\]
which is smooth outside of a fixed point set $\{y_i\}$.
We now show that this GK structure extends smoothly across $\{y_i\}$. Given our choice of function $\Phi$, Proposition~\ref{p:invGKsoliton} implies that $(M,g,I,J)$ is a GK soliton on the nondegerate locus. Since the nondegenerate locus is open and dense, $(M,g,I,J)$ is a soliton globally. Given the explicit form of $df$ in Proposition \ref{p:invGKsoliton}, we know that the extension of $f$ is smooth. With $C^{1,1}$ regularity for the metric, we can construct a harmonic coordinate system for $g$, in which the soliton system takes the form
\begin{align*}
0 =&\ g^{kl} \frac{\partial^2 g_{ij}}{\partial x^k \partial x^l} + Q(g, \partial g, H) + \partial^2 f + \partial g \star \partial f,\\
0 =&\ g^{kl} \frac{\partial^2 H_{ijk}}{\partial x^k \partial x^l} + \partial^2 g \star H + \partial g \star \partial H + d (i_{\nabla f} H).
\end{align*}
The first equation is a strictly elliptic equation for $g$ with $C^{1,\alpha}$ coefficients and a $C^{\alpha}$ inhomogeneous term, so by Schauder estimates we conclude $C^{2,\alpha}$ estimates for $g$. The second equation is then a strictly elliptic equation for $H$ with $C^{2,\alpha}$ coefficients and a $C^{\alpha}$ inhomogeneous term, thus we conclude $C^{2,\alpha}$ estimates for $H$. Differentiating this system and applying a standard bootstrap argument gives $C^{\infty}$ regularity for both $g$ and $H$. By the parallel transport argument for extending $I$ and $J$ from Proposition \ref{p:removable_singularity_GK}, it follows that $I$ and $J$ are smooth as well. Thus $(M,g,I,J)$ is a smooth regular rank one soliton.
It remains to address the completeness of $(M,g)$. The projection onto the orbit space
\[
\pi\colon M\to N(a_+,0)
\]
has compact fibers, therefore $(M,g)$ is complete if and only if $N(a_+,0)$ with the distance function induced by the horizontal metric is complete. By the construction, in the complement of the finite set $\{z_i\}$ the horizontal metric is given by
\[
g_{\mathrm{hor}}=Wh.
\]
Since $h$ is complete, an appropriate lower bound on $W$ will imply the completeness of $Wh$. Let us identify $(N(a_+,0), h)$ with a quotient of $(\mathbb{C}\times \mathbb{C}^*, \hat g)$ by a circle action (see Remark~\ref{r:N_as_quotient}) and denote coordinates on $\mathbb{C}\times \mathbb{C}^*$ by $z$ and $w$. Then
\[
\hat g=\frac{4}{1+|z|^2}\Re\left(
k_+^{2} dz\otimes d\bar z + \frac{dw\otimes d\bar w}{|w|^2}
\right)
\]
and
\[
\til W= C(|z|^2+1).
\]
On the other hand, using the explicit form of the Green's function on $\mathbb{C}\times \mathbb{C}^*$ with respect to a flat metric $g_f$ (see the proof of Proposition~\ref{p:elliptic_a0}) we observe that for any $z_i\in N(a_+,0)$, there is a constant $C'>0$ such that outside a compact subset $K$
\[
G_{z_i}\geq C'(|z|^2+\log|w|^2)^{-1/2}.
\]
This implies that
\[
W\hat g=C(\lambda+c_i\sum G_{z_0})\Re\left(
k_+^{2} dz\otimes d\bar z + \frac{dw\otimes d\bar w}{|w|^2}
\right)
\]
is a complete metric on $\mathbb{C}\times \mathbb{C}^*$ as long as at least one of the numbers $(\lambda,c_1,\dots,c_n)$ is positive. Since $Wh$ on $N(a_+,0)$ corresponds to the horizontal component of $W\hat g$ under the projection onto the orbit space $\mathbb{C}\times \mathbb{C}^*\to N(a_+,0)$, this implies the completeness of $Wh$, which in turn is equivalent to the completeness of $(M,g)$.
\end{proof}
\end{prop}
\begin{rmk} \label{r:polesonT} In the result above, it is necessary to restrict the points $\{z_1, \dots, z_n\}$ to the smooth locus of $N(a_+, 0)$. Given a point $z$ on the orbifold locus, the Seifert fibration in the complement of $z$, near $z$, is not locally trivial, thus no neighborhood of $p$ can be homeomorphic to a punctured ball, and thus the metric completion is not a smooth manifold. The same restrictions apply to the construction in the case $a_- \neq 0$ below individually on the two components $\mathbf{T_{\pm}}$.
\end{rmk}
Fix an action $\mathbb{Z}\colon N(a_+,0)\to N(a_+,0)$ induced by~\eqref{f:Z_action}. Now we are going to describe the set of GK solitons fibered over $N(a_+,0)/\mathbb{Z}$.
\begin{prop} \label{p:gensoln2}
Let $(k_+,l_+)$ be a pair of coprime integers, $0\leq l_+<|k_+|$, and denote $a_+ = \frac{2}{k_+}$. Given $\lambda>0$ let
\[
W=\lambda \til{W}.
\]
There exists an $S^1$ worth of complete regular rank one solitons $(M^4, g, I, J)$ such that
\begin{enumerate}
\item the orbit space $\pi\colon M\to M/{S^1}$ is homeomorphic to $N(a_+, 0)/\mathbb{Z}$,
\item the action of $S^1$ on $M$ is free,
\item $\mathbf{T_+}$ is the set of points of type $(k_+, l_+)$ (cf.\,Remark \ref{r:local_s1_quotient}),
\item in the image of the nondegeneracy locus we have
\begin{align*}
\Phi = a_+ \mu_+,
\end{align*}
\item with the above choice of functions $W$ and $p$, locally in the nondegeneracy locus the GK structure on $(M,g,I,J)$ is given by the construction of Theorem~\ref{t:nondegenerate_gk_description_v2}.
\end{enumerate}
\begin{proof}
By Proposition~\ref{p:gensoln1}, we know that there is a unique GK soliton $(\til M,g,I,J)$ such that
\begin{enumerate}
\item the orbit space $\pi\colon \til M\to \til M/{S^1}$ is homeomorphic to $N(a_+, 0)$,
\item the action of $S^1$ on $\til M$ is free,
\item $\mathbf{T_+}$ is the set of points of type $(k_+, l_+)$,
\item In the image of the nondegeneracy locus we have
\begin{align*}
\Phi = a_+ \mu_+,
\end{align*}
\item With the above choice of functions $\til W$ and $p$, locally in the nondegeneracy locus the GK structure on $(\til M,g,I,J)$ is given by the construction of Theorem~\ref{t:nondegenerate_gk_description_v2}.
\end{enumerate}
Now, since the action of $\mathbb{Z}$ on $N(a_+,0)$ preserves $\Phi$, the curvature form $\beta$ is also invariant under $\mathbb{Z}$, and there is a family of lifts of this action to an action
\begin{equation}\label{f:Z_action_lift}
\mathbb{Z}\times \til M\to \til M
\end{equation}
parametrized by $\mathrm{Hom}(\pi_1(N(a_+,0)/\mathbb{Z}), S^1)\simeq S^1$. Using again the invariance of $\Phi$ and $W$ under $\mathbb{Z}$ we see that the lifted action~\eqref{f:Z_action_lift} preserves $(g,I,J)$, thus induces a GK structure on $M:=\til M/\mathbb{Z}$, which we denote by the same symbols $(g,I,J)$. Finally, since the one-form
\[
\tfrac{1}{2}\left(p a_+ d\mu_+-a_+d\mu_+\right)
\]
from Proposition~\ref{p:invGKsoliton} is exact on $N(a_+,0)/\mathbb{Z}$, we conclude that $(M,g,I,J)$ is a GK soliton.
\end{proof}
\end{prop}
Now we focus on the case $a_-\neq 0$. Recall that $N(a_+,a_-)$ is isomorphic to the quotient of $\mathbb{C}^2\backslash\{0\}$ by a linear circle action with weights $k_{\pm}=2/a_\pm$. In particular, $N(a_+,a_-)$ is homeomorphic to $S^2\times \mathbb{R}$. Let $\{z_1,\dots,z_n\}\subset N(a_+,a_-)$ be a finite set of points in the smooth locus, and choose a two-cycle $S_0\in C_2(N(a_+,a_-),\mathbb{Z})$, $S_0=S^3/S^1$, where $S^3\subset \mathbb{C}^2$ is a sphere of small radius enclosing the origin. As before, let $W$ be a function on $N(a_+,a_-)\backslash\{z_1,\dots, z_n\}$ such that
\[
\beta=*_hdW+W\beta_0
\]
is closed and denote
\[
S(W):=\frac{1}{2\pi}\int_{S_0}\beta\in \mathbb{R}.
\]
Let $\Sigma_{\pm}=\{p=\pm 1\}\subset N(a_+,a_-)$ be a disjoint union of two curves. We orient each of the components so that $S_0\cap \Sigma_{\pm}=+1$.
\begin{prop} \label{p:gensoln3}
Let $(k_+,l_+)$, $(k_-,l_-)$ be two pairs of coprime integers, $0\leq l_\pm<|k_\pm|$ and denote $a_{\pm} = \frac{2}{k_{\pm}}$. Given $\lambda>0$, $\lambda_0\geq 0$ and a finite collection of points $\{z_1, \dots, z_n\}$ in the smooth locus of $N(a_+, a_-)$, let
\[
W=\til W\left(\lambda+\lambda_0G_0+\sum_{i=1}^n c_{z_i}G_{z_i}\right),
\]
where $G_{z_i}$ is the Green's function centered at $z_i$ constructed in Proposition~\ref{p:elliptic_aneq0} and $c_{z_i}>0$ is the normalization constant as in~\eqref{f:c_zi_normalization}. Assume that
\begin{equation}\label{f:W_quantization}
S(W)-\frac{l_+}{k_+}-\frac{l_-}{k_-}\in \mathbb{Z}.
\end{equation}
Then there exists a unique complete regular rank one soliton $(M^4, g, I, J)$ such that
\begin{enumerate}
\item the orbit space $\pi\colon M\to M/{S^1}$ is homeomorphic to $N(a_+, a_-)$,
\item $\pi^{-1} \{z_1, \dots, z_n\} = M^{S^1}$,
\item $\mathbf{T_{\pm}}\backslash \pi^{-1}\{z_1, \dots, z_n\}$ is the set of points of type $(k_{\pm}, l_{\pm})$ (cf.\,Remark \ref{r:local_s1_quotient}),
\item in the image of the nondegeneracy locus we have
\begin{align*}
\Phi = a_+ \mu_+ + a_- \mu_-,
\end{align*}
\item with the above choice of functions $W$ and $p$, locally in the nondegeneracy locus the GK structure on $(M,g,I,J)$ is given by construction of Theorem~\ref{t:nondegenerate_gk_description_v2}.
\end{enumerate}
\begin{proof}
As in the proof of Proposition~\ref{p:gensoln1} we start by constructing the smooth manifold $M$. We are given an orbifold $N(a_+,a_-)$ with cooriented submanifolds $\Sigma_{\pm}=\{p=\pm 1 \}$ of codimension two. Each submanifold $\Sigma_{\pm}$ comes equipped with a pair of integers $(k_\pm,l_\pm)$. We claim that under assumption~\eqref{f:W_quantization},
\[
[\beta/2\pi]-\frac{l_+}{k_+}[\Sigma_+]-\frac{l_-}{k_-}[\Sigma_-]\in H^2(N(a_+,a_-)\backslash\{z_1,\dots,z_n\},\mathbb{Z}).
\]
Indeed, $H_2(N(a_+,a_-)\backslash\{z_1,\dots,z_n\},\mathbb{Z})\simeq \mathbb{Z}^{n+1}$ is generated by $S_0$ and a collection of $n$ small spheres $\{S_i\}_{i=1}^n$ each enclosing one point $z_i$. If $z_i\not\in \Sigma_{\pm}$, we can assume that the curves $\Sigma_{\pm}$ do not intersect the spheres $S_i$. Otherwise, if $z_i\in \Sigma_+$ ($z_i\in \Sigma_-$), we necessarily have $k_+=\pm 1$, $l_+=0$ (resp.\,$k_-=\pm 1$, $l_-=0$). In either case, by our choice of normalization constants $c_{z_i}$, we have
\[
\left\langle [\beta/2\pi]-\frac{l_+}{k_+}[\Sigma_+]-\frac{l_-}{k_-}[\Sigma_-], S_i \right\rangle=\frac{1}{2\pi}\int_{S_i} \beta=-1.
\]
It remains to check that
\[
\left\langle [\beta/2\pi]-\frac{l_+}{k_+}[\Sigma_+]-\frac{l_-}{k_-}[\Sigma_-], S_0 \right\rangle\in \mathbb{Z}
\]
but this is exactly the statement of equation~\eqref{f:W_quantization}.
Thus we have all the ingredients in place to apply Proposition~\ref{p:s1_bundle_seifert_dg} and construct a smooth manifold $M_0$ and a Seifert fibtration
\[
\pi\colon M_0\to N(a_+,a_-)\backslash\{z_1,\dots,z_n\},
\]
with connection $\eta$ and curvature form $\beta$ (if either of $k_\pm$ is negative, we reverse the coorientation of the corresponding submanifold $\Sigma_{\pm}$). The rest of the construction of the GK soliton $(M,g,I,J)$ goes exactly as in the proof of Proposition~\ref{p:gensoln1}.
Up to this point the construction works for any constants $\lambda,\lambda_0\geq 0$. It remains to understand if the constructed manifold $(M,g)$ is complete. Repeating the argument in Proposition~\ref{p:gensoln1}, completeness of $(M,g)$ is equivalent to the completeness of $\mathbb{C}^2\backslash\{0\}$ equipped with the metric $W\hat g$, where
\[
\hat g=\frac{4}{|z|^2+|w|^2}\Re\left(k_-^2dz\otimes d\bar z+k_+^2dw\otimes d\bar w\right)
\]
is a complete metric on $\mathbb{C}^2\backslash \{0\}$ (see part (2) of Remark~\ref{r:N_as_quotient}). Since $\til W$ is bounded below by a positive constant, function
\[
W=\til W\left(\lambda+\lambda_0G_0+\sum_{i=1}^n c_{z_i}G_{z_i}\right)
\]
is also bounded from below by a positive constant, as long as $\lambda>0$. Thus, for $\lambda>0$ the metric $W\hat g$ induces a complete distance function on $\mathbb{C}^2\backslash\{0\}$, and $(M,g)$ is also complete.
We claim that if $\lambda=0$, then $g$ is not complete. To this end, we estimate $W$ from above. Choose a compact set $K\subset \mathbb{C}^2\supset \mathbb{C}^2\backslash\{0\}$ containing preimages of all the points $\{z_i\}$. Using the explicit form of Green's function on $\mathbb{C}^2$ equipped with a flat metric, we observe that there exists a constant $C=C(K)>0$ such that in the complement of $K$,
\[
G_{z_i}<C(|z|^2+|w^2|)^{-1}.
\]
Therefore, as long as $\lambda=0$
\[
W<(n+1)C(|z|^2+|w^2|)^{-1}.
\]
Thus $(\mathbb{C}^2\backslash\{0\}, W\hat g)$ is incomplete, as any radial curve $(tz_0,tw_0)$, $t\in[1,+\infty)$ has a finite length.
\end{proof}
\end{prop}
With the above propositions we have Theorem~\ref{t:existence}:
\begin{proof}[Proof of Theorem~\ref{t:existence}]
Propositions~\ref{p:gensoln1},~\ref{p:gensoln2} and~\ref{p:gensoln3} give explicit existence statements for complete regular rank one solitons as stated in Theorem~\ref{t:existence}.
\end{proof}
Finally we give the proof of the classification statement of Theorem~\ref{t:uniqueness}:
\begin{proof}[Proof of Theorem~\ref{t:uniqueness}]
Let $(M,g,I,J)$ be a 4-dimensional complete generalized K\"ahler-Ricci soliton, such that its Poisson tensor does not vanish identically. Assume that the Ricci curvature of $(M,g)$ is bounded from below and $\dim\Isom(g)\leq 1$. As before, denote by $V_I=\tfrac{1}{2}(\theta_I^\#-\nabla f)$ and $V_J=\tfrac{1}{2}(-\theta_I^\#-\nabla f)$ the corresponding soliton vector fields. By Proposition~\ref{p:solitonVF}, $IV_I$ and $JV_J$ are Killing.
If $\dim\Isom(g)=0$, then both Killing vector fields $IV_I$ and $JV_J$ must vanish, thus $\theta_I^\#-\nabla f=-\theta_I^\#-\nabla f=0$. Hence $\theta_I=0$ and $\nabla f=0$, so $(M,g,I,J)$ is a usual Calabi-Yau metric with respect to either of complex structure $I$ and $J$. Since $I$ and $J$ are linearly independent, this forces the holonomy of $g$ to be contained in $SU(2)$, so $(M,g,I)$ is indeed hyperK\"ahler.
From now on we can assume that $\mathrm{span}(IV_I, JV_J)$ is generically one-dimensional, so $(M,g,I,J)$ is a rank one soliton. By Proposition~\ref{p:one-diml_isometry}, we have two possibilities:
\begin{enumerate}
\item $\Isom(g)\simeq S^1$,
\item $\Isom(g)\simeq \mathbb{R}$.
\end{enumerate}
In the first case there is a vector field $X$ generating an $S^1$ action on $(M,g,I,J)$ preserving the GK structure, such that $IV_I$ and $JV_J$ are constant multiples of $X$. Furthermore, since $M$ has finite topology~--- $\dim H^*(M,\mathbb{R})<\infty$~--- the action of $S^1$ has finitely many fixed points. Thus $(M,g,I,J)$ is a complete regular rank one soliton (cf.\,Definition~\ref{d:regular_rank_one}). Therefore, we can invoke Theorem~\ref{t:M_orbifold_quotient} and conclude that the orbit space $\pi\colon M\to N$ is given by one of the spaces $N(a_+,0)$, $N(a_+,0)/\mathbb{Z}$ or $N(a_+,a_-)$ and in the complement of the degeneracy locus the GK structure is locally given by Theorem~\ref{t:nondegenerate_gk_description_v2}. Let $\pi\colon M_0\to N_0$ be the Seifert fibration in the complement of the isolated fixed points. By Propositions~\ref{p:elliptic_a0} and~\ref{p:elliptic_aneq0}, the function $W$ solving the equation $d(*_hdW+W\beta_0)=0$ must be a linear combination of $\til W$ and $\til W G_{z_i}$, and if $N=N(a_+,0)/\mathbb{Z}$, $W$ must be a multiple of $\til W$. Since $G_{z_i}(x)\to +\infty$ as $x\to z_i$, and $W$ is positive, the constants $c_{z_i}$ in front of $\til W G_{z_i}$ in the expansion of $W$ must be positive. Furthermore, the local structure of $(M,g,I,J)$ near fixed points (Lemma~\ref{l:free_nondegenerate} and Corollary~\ref{c:fixed_smooth}) uniquely determines the exact values of constants $c_{z_i}$, $z_i\in N$. If $N=N(a_+,a_-)$, $a_-\neq 0$, there is also a necessary integrality condition
\[
\frac{1}{2\pi}\int_{S^2_0}\beta - \frac{l_+}{k_+} -\frac{l_-}{k_-}\in \mathbb{Z}
\]
where $S^2_0$ is a 2-sphere generating $H^2(N(a_+,a_-),\mathbb{Z})$. Due to Theorem~\ref{t:M_orbifold_quotient}, the choice of $W$ and the prescribed form of $\Phi$ uniquely determine the GK structure $(M,g,I,J)$ up to an isomorphism generated by translations in $\mu$-coordinates.
In the second case, the $\mathbb{Z}$-quotient $M/\mathbb{Z}$ of $M$ admits a complete regular rank one GK soliton structure $(M/\mathbb{Z},g,I,J)$ with a free $S^1$ action such that the bundle $M/\mathbb{Z}\to (M/\mathbb{Z})/S^1$ is trivial. Thus $(M/\mathbb{Z}, g,I,J)$ is given by one of the constructions of Propositions~\ref{p:gensoln1}, $\ref{p:gensoln2}$ and $\ref{p:gensoln3}$ with $k_{\pm 1}=\pm 1$, $\{z_i\} = \emptyset$, and $[\beta]$ representing the zero class in the second cohomology. The initial manifold $(M,g,I,J)$ is then a $\mathbb{Z}$ cover of $(M/\mathbb{Z},g,I,J)$.
\end{proof}
|
1,116,691,500,780 | arxiv | \section{Introduction}
The result of a classic frequentist hypothesis test is a dichotomous test decision. However, scientific research questions are very versatile and there is not always the demand to guide a decision. By the their nature, frequentist hypothesis tests prohibit a statistical hypothesis-based analysis without making decisions. In that sense, a statistical framework that provides results without requiring an underlying (potentially artificially constructed) decision problem seems to be advantageous.
The Bayes factor -- a Bayesian quantity that is used for hypothesis comparisons \citep{Jeffreys1961,Kass1995,Goenen2005,Rouder2009} -- is argued to do so, as it is typically interpreted as evidence quantification w.r.t.\ the contrasted hypotheses \citep[see e.g.][]{Morey2016} without requiring a decision to be made. In this regard, \citet{Rouder2018} state that ``[r]efraining from making decisions strikes [them] as advantageous in most contexts.''
Naturally, the evidence (as quantified by the Bayes factor) might then be used to update beliefs in the considered hypotheses and subsequently to guide a respective decision. The essential point, however, is that the researcher might stop the analysis after calculating a Bayes factor without guiding a decision, e.g.\ if merely the evidence quantification is of interest. Then the result of the Bayesian analysis is the Bayes factor itself and not a decision.
Yet, for those research situations that do indeed aim at guiding a decision, the Bayes factor might naturally be used to do so. The aim of this elaboration is to outline the decision theoretic framework in which Bayes factors are involved.
Further, it shall be acknowledged that the specification of the relevant quantities within such a decision theoretic framework as precise values might not always be possible for an applied scientist, as the available relevant information might be scarce, vague, partial, and ambiguous. To tackle this issue, also a robust version of the framework shall be outlined in which the applied researcher is allowed to specify the essential quantities less precisely as sets of values, such that the partial nature of the relevant information might be captured more accurately. Although such robust specifications might be possible for all essential quantities \citep[see e.g.][]{Schwaferts2019,Schwaferts2021}, the present elaboration is restricted to a robustly specified inverval-valued loss function, as it is this quantity which characterizes the difference between a decision-theoretic and a non-decision-theoretic analysis, yet its precise specification is expected to bear serious difficulties for applied scientists.
The elaborations within this paper are structured as follows: After delineating the general (Section~\ref{sec:decision}) and the hypothesis-based (Section~\ref{sec:hypo_decision}) framework of Bayesian decision theory, its relation with Bayes factors is depicted (Section~\ref{sec:BF}). To facilitate a more user-friendly employment of the hypothesis-based Bayesian decision theoretic framework, a robust interval-valued specification of the loss function was allowed (Section \ref{sec:robust_loss}) and the restriction of the prior distributions to be proper can be alleviated (Section~\ref{sec:improper_priors}). Both the resulting framework (Secion~\ref{sec:guide1}) and how to derive optimal actions from existing Bayes factor values (Secion~\ref{sec:guide2}) are presented in respective step-by-step guides.
\section{Bayesian Decision Theory}\label{sec:decision}
Within the framework of Bayesian decision theory \citep[e.g.][]{Berger1985,Robert2007}, the objective is to decide between different actions. In accordance with the context of Bayes factors, only two actions shall be considered, namely $a_0$ and $a_1$, being comprised within the action space $\mathcal{A}= \{ a_0 , a_1 \}$.
The researcher plans to conduct an investigation that yields data $\boldsymbol{x}$, which is characterized by a parametric sampling distribution with parameter $\theta \in \Theta$, where $\Theta$ is the parameter space. Accordingly, the density of the data is $f(\boldsymbol{x}|\theta)$.
In a Bayesian setting, a prior distribution on the parameter $\theta$ with density $\pi(\theta)$ needs to be specified. This prior reflects the information (or belief or knowledge or uncertainty) about the parameter before the investigation is conducted.
In addition, also a loss function $L: \Theta \times \mathcal{A} \to \mathbb{R}_0^+ : (\theta,a) \mapsto L(\theta,a)$ needs to be specified, which quantifies the ``badness'' of the consequences of deciding for the action $a \in \mathcal{A}$ if the parameter value $\theta \in \Theta$ is true. Usually, the exact shape of this loss function is inaccessible and hypothesis-based analyses are able to tackle this issue. These are depicted within the next section, but first -- to delineate the ideal Bayesian solution -- assume $L$ is fully known.
Now, after specifying the parametric sampling distribution, the prior, as well as the loss function, the investigation can be conducted and the data $\boldsymbol{x}$ are observed.
This allows to update the prior distribution via Bayes rule to the posterior distribution with density
\begin{equation}
\pi(\theta|\boldsymbol{x}) = \frac{f(\boldsymbol{x}|\theta) \, \pi(\theta)}{f(\boldsymbol{x})} = \frac{f(\boldsymbol{x}|\theta) \, \pi(\theta)}{\int_\Theta f(\boldsymbol{x}|\theta) \, \pi(\theta) \, d\theta} \, .
\end{equation}
There are plenty of resources available about how to obtain this posterior \citep[e.g.][]{Gelman2013,Kruschke2015}, which reflects the information (or belief or knowledge or uncertainty) about the parameter after the investigation was conducted.
Based on this posterior distribution, it is possible to calculate the expected posterior loss $\rho:\mathcal{A} \to \mathbb{R}_0^+$ for each action by integrating the loss function $L$ over the posterior density:
\begin{equation}
\rho(a) = \int_\Theta L(\theta,a) \, \pi(\theta|\boldsymbol{x}) \, d\theta \, .
\end{equation}
The optimal action $a^*$ has minimal expected posterior loss:
\begin{equation}
a^* = \argmin_{a \in \mathcal{A}} \rho(a) \, .
\end{equation}
\section{Hypothesis-Based Bayesian Decision Theory}\label{sec:hypo_decision}
As mentioned, typically, the loss function $L$ is not fully accessible as the essential information about it might be scarce, vague, partial, and ambiguous. A commonly employed solution is a hypothesis-based analysis: The researcher considers each possible parameter value $\theta$ and assesses which action should be preferred if this parameter value would be true.
These considerations lead to two sets of parameters $\Theta_0$ and $\Theta_1$ for which the actions $a_0$ and $a_1$ should be preferred, respectively. These sets define the hypotheses
\begin{equation} \label{eq:hypo}
H_0: \theta \in \Theta_0 \quad \text{vs.} \quad H_1: \theta \in \Theta_1
\end{equation}
employed in conventional analyses, such as hypothesis tests or Bayes factors.
From the posterior density $\pi(\theta|\boldsymbol{x})$ it is possible to determine the posterior probabilities of the parameters sets $\Theta_0$ and $\Theta_1$, i.e.\ of the hypotheses $H_0$ and $H_1$, by
\begin{equation} \label{eq:post_belief_hypo}
P(H_0|\boldsymbol{x}) = \int_{\Theta_0} \pi(\theta|\boldsymbol{x}) \, d\theta
\quad \quad \text{and} \quad \quad
P(H_1|\boldsymbol{x}) = \int_{\Theta_1} \pi(\theta|\boldsymbol{x}) \, d\theta \, ,
\end{equation}
respectively.
The ratio of these beliefs $P(H_0|\boldsymbol{x}) /P(H_1|\boldsymbol{x})$ is referred to as posterior odds.
The underlying assumption of hypothesis-based analyses is that the loss values within these sets $\Theta_0$ and $\Theta_1$ are constant, respectively (see Figure~\ref{fig:loss}). This assumption shall be referred to as \textit{simplification assumption} and is inherent to a statistical analysis which considers statistical hypotheses and derives applied conclusions based on respective (hypothesis-based) results.
In addition (without loss of generality), the loss values for deciding correctly (i.e.\ for $a_0$ if $\theta \in \Theta_0$ or for $a_1$ if $\theta \in \Theta_1$) can be set to $0$. The resulting loss function is in regret form (depicted in Table~\ref{tab:loss}) and has only two values to specify: $k_0 := L(a_0, \theta)$ if $\theta \in \Theta_1$ and $k_1 := L(a_1, \theta)$ if $\theta \in \Theta_0$.
\begin{table}[]
\centering
\caption{Simplified Hypothesis-Based Loss Function.}
\label{tab:loss}
\begin{tabular}{c|cc}
$L(\theta,a)$ & $\theta \in \Theta_0$ & $\theta \in \Theta_1$ \\ \hline
$a = a_0$ & $0$ & $k_0$ \\
$a = a_1$ & $k_1$ & $0$
\end{tabular}
\end{table}
\begin{figure}[ht]
\centering
\includegraphics[width=1\textwidth]{loss.png}
\caption{Hypothesis-Based Loss Function. Assume $\Theta = \mathbb{R}$, $\Theta_0 = [-1,1]$, and $\Theta_1 = (-\infty,-1) \cup (1,\infty)$. The hypothesis-based loss function $L$ (y-axis) in regret form (see Table~\ref{tab:loss}) in dependence of the parameter $\theta$ (x-axis) and the actions $a_0$ (left) and $a_1$ (right) is assumed to be constant within the sets $\Theta_0$ and $\Theta_1$, respectively. This is an assumption (\textit{simplification assumption}) inherent to a hypothesis-based statistical analysis which -- at least implicitly -- considers an underlying applied decision problem.}
\label{fig:loss}
\end{figure}
With this simplified loss function (Table \ref{tab:loss}), the expected posterior loss of each action can be calculated as
\begin{align}
\rho(a_0) &= \int_\Theta L(\theta,a_0) \, \pi(\theta|\boldsymbol{x}) \, d\theta = k_0 \cdot P(H_1|\boldsymbol{x}) \\
\rho(a_1) &= \int_\Theta L(\theta,a_1) \, \pi(\theta|\boldsymbol{x}) \, d\theta = k_1 \cdot P(H_0|\boldsymbol{x}) \,
\end{align}
and the action with minimal expected posterior loss shall be selected.
Only the ratio $k := k_1/k_0$ is required to determine this optimal action. This ratio $k$ states how much worse it would be to decide for $a_1$ if $\theta \in \Theta_0$ is true (type-I-error) than to decide for $a_0$ if $\theta \in \Theta_1$ is true (type-II-error), if deciding correctly has loss $0$. With the ratio of expected posterior losses
\begin{equation}
\varrho(k) := \frac{\rho(a_1)}{\rho(a_0)} = k \cdot \frac{P(H_0|\boldsymbol{x})}{P(H_1|\boldsymbol{x})}
\end{equation}
the optimal action is
\begin{equation}
a^* =
\begin{cases}
a_0 \quad \text{if} \quad \varrho(k) > 1 \\
a_1 \quad \text{if} \quad \varrho(k) < 1
\end{cases} \, .
\end{equation}
For $\varrho(k) = 1$ any action might be chosen.
\section{Bayes Factors}\label{sec:BF}
By assessing even the prior distribution in the light of the hypotheses, it is possible to obtain the prior probabilities in the hypotheses (illustrated in Figure~\ref{fig:prior_decomposition}, left):
\begin{equation}
P(H_0) = \int_{\Theta_0} \pi(\theta) \, d\theta
\quad \quad \text{and} \quad \quad
P(H_1) = \int_{\Theta_1} \pi(\theta) \, d\theta \, ,
\end{equation}
Analogously, the ratio of these beliefs $P(H_0) /P(H_1)$ is referred to as prior odds.
In addition, the prior distribution can be restricted to each of the hypotheses, referred to as within-hypothesis priors (illustrated in Figure~\ref{fig:prior_decomposition}, middle and right), and the corresponding densities are
\begin{align}
\pi(\theta|H_0) &= \frac{1}{P(H_0)} \, \pi(\theta) \cdot \mathsf{1}(\theta \in \Theta_0) \label{eq:hypo_prior_0}\\
\pi(\theta|H_1) &= \frac{1}{P(H_1)} \, \pi(\theta) \cdot \mathsf{1}(\theta \in \Theta_1) \, ,\label{eq:hypo_prior_1}
\end{align}
where $\mathsf{1}(s) = 1$ if the statement $s$ is true and $\mathsf{1}(s) = 0$ if $s$ is false.
The overall prior distribution can be decomposed \citep[cp.][]{Rouder2018} into the prior probabilities of the hypotheses and the within-hypothesis priors (Figure~\ref{fig:prior_decomposition}):
\begin{equation}
\pi(\theta) = P(H_0) \, \pi(\theta|H_0) + P(H_1) \, \pi(\theta|H_1) \, .
\end{equation}
\begin{figure}[ht]
\centering
\includegraphics[width=1\textwidth]{prior_decomposition.png}
\caption{Prior Decomposition. Assume $\Theta = \mathbb{R}$, $\Theta_0 = [-1,1]$, $\Theta_1 = (-\infty,-1) \cup (1,\infty)$ and a standard normal distribution for $\theta \sim N(0,1)$. Left: The prior density $\pi(\theta)$ is depicted as solid line. $P(H_0)$ and $P(H_1)$ can be calculated as respective areas under this density, depicted as light gray and dark gray, respectively. Middle: The within-hypothesis density $\pi(\theta|H_0)$ as in equation~(\ref{eq:hypo_prior_0}) is depicted as solid line. Right: The within-hypothesis density $\pi(\theta|H_1)$ as in equation~(\ref{eq:hypo_prior_1}) is depicted as solid line. }
\label{fig:prior_decomposition}
\end{figure}
Instead of considering the overall prior distribution together with the hypotheses (which leads to the posterior odds, as in Section~\ref{sec:hypo_decision}), the Bayes factor is obtained by considering only the within-hypothesis priors together with the hypotheses:
\begin{equation} \label{eq:BF}
B\!F := \frac{\int_{\Theta_0} f(\boldsymbol{x}|\theta) \, \pi(\theta|H_0) \, d\theta}{\int_{\Theta_1} f(\boldsymbol{x}|\theta) \, \pi(\theta|H_1) \, d\theta} \, .
\end{equation}
The posterior odds can then be calculated from the Bayes factor and the prior odds:
\begin{equation}\label{eq:post_odds}
\frac{P(H_0|\boldsymbol{x})}{P(H_1|\boldsymbol{x})} = B\!F \cdot \frac{P(H_0)}{P(H_1)} \, .
\end{equation}
The optimal decision can now be obtained as in the previous section (Section~\ref{sec:hypo_decision}) by considering the loss function.
\section{Robust Loss Function}\label{sec:robust_loss}
However, a precise specification of the value $k$ is typically not accessible, as essential information about the ``badness'' of the consequences of the decision are scarce, vague, partial, and ambiguous.
Yet, this partial information needs to be included into the analysis, as ignoring it facilitates suboptimal decisions. A decision cannot be guided properly without considering its consequences.
This partial information about the loss can be captured less arbitrarily and more robustly by an interval $[\underline{K},\overline{K}]$ than by a precise value $k$ \citep[cp.\ e.g.][]{Walley1991, Augustin2014}. To do so, the researcher has to determine a lower bound $\underline{K}$ and an upper bound $\overline{K}$ for reasonable $k$ values (i.e.\ for the ratio of how much worse the type-I-error is compared to the type-II-error, if deciding correctly has a loss of $0$).
To perform a robust analysis \citep[cp.\ also][]{RiosInsua2012} with this interval-valued specification, it is possible to obtain and consider the optimal action for each value within this interval $[\underline{K},\overline{K}]$.
If the optimal action is the same for each $k$ within $[\underline{K},\overline{K}]$, then this action should be chosen.
If not, the decision should be withheld, because the data or the information about the decision problem are not sufficient to unambiguously guide the decision.
Formally \citep[see also][]{Schwaferts2019, Schwaferts2020, Schwaferts2021}, the ratios of expected posterior losses need to be calculated for both the lower and upper bound, respectively:
\begin{equation} \label{eq:ratio_imprecise}
\varrho(\underline{K}) = \underline{K} \cdot \frac{P(H_0|\boldsymbol{x})}{P(H_1|\boldsymbol{x})}
\quad \quad \text{and} \quad \quad
\varrho(\overline{K}) = \overline{K} \cdot \frac{P(H_0|\boldsymbol{x})}{P(H_1|\boldsymbol{x})} \, .
\end{equation}
Then, the optimal action is
\begin{equation} \label{eq:optimal_decision_imp}
a^* =
\begin{cases}
a_0 \quad \text{if} \quad \varrho(\underline{K}) \geq 1 \\
a_1 \quad \text{if} \quad \varrho(\overline{K}) \leq 1
\end{cases} \, .
\end{equation}
For $\varrho(\underline{K}) < 1 < \varrho(\overline{K})$, the decision should be withheld.
\section{Improper Priors}\label{sec:improper_priors}
Furthermore, the calculation of Bayes factors comes along with a restriction \citep[][]{Jeffreys1961} on the prior distribution: It must be a proper distribution, i.e.\ the density has to integrate to $1$:
\begin{equation} \label{eq:proper_density}
\int_\Theta \pi(\theta) \, d \theta = 1 \, .
\end{equation}
In contrast, an improper prior distribution is characterized by a non-integrable function (e.g.\ $\pi(\theta) \propto c$ with $c >0$ being a constant, see Figure~\ref{fig:improper_prior}, dotted line) and, technically, this prior distribution is no proper probability distribution. However, these improper priors are frequently allowed within Bayesian prior specifications, because they might lead to proper posterior distributions (see Figure~\ref{fig:improper_prior}, solid line). In this case, the posterior odds $P(H_0|\boldsymbol{x})/P(H_1|\boldsymbol{x})$ can be calculated reasonably and a decision can be guided consistently.
The prior odds, however, might not be reasonable (e.g.\ with $P(H_0)/P(H_1) = 0$ as in Figure~\ref{fig:improper_prior}). Accordingly, the Bayes factor (calculated via equation~(\ref{eq:post_odds}))
\begin{equation}
B\!F = \left. \frac{P(H_0|\boldsymbol{x})}{P(H_1|\boldsymbol{x})} \middle/ \frac{P(H_0)}{P(H_1)} \right.
\end{equation}
cannot be calculated reasonably due to its dependence on the prior odds. Therefore, Bayes factors require -- in contrast to a Bayesian analysis in general -- proper prior distributions. This is truly a limitation, as improper priors are frequently employed for representing non-knowledge or for letting the data speak for themselves \citep[e.g.][]{Gelman2013}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.9\textwidth]{improper_prior.png}
\caption{Improper Prior. Assume the model $X_i \overset{iid}{\sim} N(\mu,\sigma^2)$ for $i=1,\dots,n$, with known variance $\sigma^2=1$ and unknown parameter $\mu \in \mathbb{R}$, the hypotheses $\Theta_0 = [-1,1]$, $\Theta_1 = (-\infty,-1) \cup (1,\infty)$. The function $\pi(\mu) = c$, with $c=0.2$ being an arbitrary constant, characterizes the improper prior distribution for $\mu$ (dotted line). For a sample of size $n=10$ with in-sample mean $\bar{x} = 0.5$, the posterior distribution is proper (solid line), such that its density integrates to $1$. The prior ``beliefs'' into the hypotheses are with $P(H_0) = 2c$ and $P(H_1) = \infty$ not reasonably interpretable (light gray and dark gray, respectively).}
\label{fig:improper_prior}
\end{figure}
This issue is alleviated in hypothesis-based Bayesian decision theoretic accounts, as improper priors typically yield proper posterior odds. Accordingly, a researcher who is interested in guiding a decision might employ the decision theoretic framework directly without explicitly calculating the Bayes factor. Then, also improper priors might be employed.
Please note that it is still an ongoing debate whether non-knowledge can be formalized by a precise improper prior distribution and if so, which improper prior distribution shall be employed. Although the authors of this paper doubt that non-knowledge can be formalized by a precise prior distribution, even if it is improper \citep[cp.\ e.g.][]{Augustin2014}, this issue shall not be addressed here. In general, it is important that the employed prior distribution matches with the available information (or non-information) about the phenomenon of interest, and this applies to every point of view within this debate. In this regard, the present elaboration emphasizes only that it is mathematically possible to employ improper priors if decisions are of interest, which is an advantage of the (more general) hypothesis-based Bayesian decision theoretic account over Bayes factors.
\section{Step-By-Step Guides}\label{sec:guides}
\subsection{Hypothesis-Based Bayesian Decision Theory}\label{sec:guide1}
In order to apply this hypothesis-based Bayesian decision theoretic framework with robust loss function, a researcher might follow the following steps.
\textbf{Step 1: Actions.} First of all, the researcher needs to specify the actions. It is recommended to explicitly state and report these actions, e.g.\ by \citep[this example is taken from][]{Bartolucci2011}
\begin{itemize}
\item[$a_0$:] do not administer aspirin to prevent myocardial infarction
\item[$a_1$:] administer aspirin to prevent myocardial infarction
\end{itemize}
If the researcher has difficulties stating the actions, maybe there is no decision to guide and a descriptive analysis might suffice \citep[cp.\ also][]{Cumming2014,Kruschke2018b}.
\textbf{Step 2: Sampling Distribution.} Next, the researcher should provide a detailed description of the investigation and how it is characterized (i.e.\ the sampling distribution). It is recommended to also explicitly state the employed parameter $\theta$ and its interpretation. This is the basis for specifying the hypotheses.
\textbf{Step 3: Prior Distribution.} In the Bayesian setting, it is possible to include prior information (or belief or knowledge or uncertainty) into the analysis. In that, the researcher has to specify a prior distribution on the parameter. It is recommended to fully report the available prior information about the parameter $\theta$ and why this leads to the prior density $\pi(\theta)$.
Of course, this specification is far from being unambiguous. However, this is a fundamental issue inherent to every Bayesian analysis (not only Bayesian decision theoretic accounts) and solving this issue is not the intention of this elaboration. Nevertheless, solutions, such as sensitivity analyses \citep[found in almost every Bayesian textbook, e.g.][]{Gelman2013}, exist. It is recommended at this step of the analysis to also state all other possible prior densities that are in accordance with the available prior information, as these serve as basis for a subsequent sensitivity analysis.
Naturally, also non-informative priors might be specified and they might also be improper (as long as they lead to proper posterior distributions).
\textbf{Step 4: Assumption.} If the researcher is unable to specify the loss function $L$, then a hypothesis-based simplification as in Section \ref{sec:hypo_decision} might be a solution. This simplification is an assumption on the loss function, namely that the loss function is constant within each of two parameter sets.
If this assumption is not appropriate, it might lead to errors (which are inherent to every hypothesis-based analysis) and the researcher needs to be aware of this consequence. It is recommended to explicitly report that this assumption was made. Transparency is one of the basic principles in science \citep[cp.][]{Gelman2017}.
\textbf{Step 5: Hypotheses.} Now, the researcher has to consider each possible parameter value $\theta$ and assess which action should be preferred if this parameter value would be true.
All parameters for which $a_0$ or $a_1$ should be preferred are comprised within the sets $\Theta_0$ or $\Theta_1$, respectively.
Certainly, there are parameter values that define the border between both sets $\Theta_0$ and $\Theta_1$. It is recommended to explicitly state what these values mean in real-life and why they define reasonable borders between $\Theta_0$ and $\Theta_1$.
\textbf{Step 6: Errors.}
Deciding for $a_1$ if $\theta \in \Theta_0$ is the type-I-error and deciding for $a_0$ if $\theta \in \Theta_1$ is the type-II-error. Both errors should be delineated, as they serve as basis for specifying the ratio $k$. It is recommended to explicitly state these errors and their consequences, e.g.\ by
\begin{itemize}
\item[] Type-I-error: administer aspirin to prevent myocardial infarction, but the effect is negligible. Consequence: patients unnecessarily suffer side effects of aspirin.
\item[] Type-II-error: do not administer aspirin to prevent myocardial infarction, although it would have an effect. Consequence: some patients suffer a myocardial infarction, which could have been prevented.
\end{itemize}
Of course, this is only a schematic illustration and in real empirical studies these elaborations will be more comprehensive.
\textbf{Step 7: Loss Magnitude.} The researcher has to imagine that the ``badness'' of deciding correctly is $0$. In this context, the researcher has to determine how much worse the type-I-error is compared to the type-II-error. This is the value $k$. As a precise value for $k$ is difficult to determine, it might be easier to specify a range $[\underline{K},\overline{K}]$ of plausible values for $k$. It is recommended to report all considerations that lead to this specification.
\textbf{Step 8: Investigation.} Now, the investigation can be conducted and it is recommended to preregister\footnote{Study designs can be preregistered e.g.\ at \url{www.cos.io/initiatives/prereg}.} the previous specifications, the design of the experiment, and the planned (decision theoretic) analysis of the data \citep[cp.][]{Nosek2018,Klein2018}. Registered reports\footnote{Information about registered reports can be found e.g.\ at \url{www.cos.io/rr}.} even allow to obtain a peer-review prior to collecting the data.
\textbf{Step 9: Posterior Distribution.} The observed data are used to obtain the posterior distribution as well as the posterior beliefs in the hypotheses $P(H_0|\boldsymbol{x})$ and $P(H_1|\boldsymbol{x})$. There are countless references on how to do this \citep[e.g.][]{Gelman2013,Kruschke2015}.
\textbf{Step 10: Optimal Action.} The researcher has to calculate $\varrho(\underline{K})$ and $\varrho(\overline{K})$ as in equation~(\ref{eq:ratio_imprecise}) to find the optimal action as in equation ~\ref{eq:optimal_decision_imp}).
For $\varrho(\underline{K}) < 1 < \varrho(\overline{K})$, the decision should be withheld, because the data or the information about the decision problem are not sufficient to unambiguously guide the decision. In this case, a reasonable strategy might be to collect more data or to gather more information about the decision problem, especially about the consequences of the errors, to narrow down $[\underline{K},\overline{K}]$. However, it is recommended to transparently report that a decision was withheld at first and which subsequent steps were taken to obtain more information.
\textbf{Step 11: Publish Data.} Of course, other researchers might need the data to guide their decisions. It is to expect that they have different prior knowledge and that their decisions employ different hypotheses. Without having access to the data set (but only to the reported analysis), it might be difficult, or even impossible, for them to guide their decisions properly, emphasizing the importance of open science\footnote{Comprehensive information about open science are provided e.g.\ by the LMU Open Science Center: \url{www.osc.uni-muenchen.de}.}.
\subsection{From Bayes Factors to Decisions}\label{sec:guide2}
Sometimes, a researcher wants to use the results of a previous study to guide a decision. Assume a Bayes factor $B\!F$ was already calculated and shall now be used to guide this decision.
\textbf{Step A: Applicability of the Sampling Distribution.} Confirm that the interpretation of the parameter $\theta$ is actually relevant for the decision of interest. If this is not the case, the available data (or Bayes factor) can hardly be used to guide the decision of interest.
\textbf{Step B: Applicability of the Hypotheses.} Certain specific hypotheses were assumed in order to calculate the Bayes factor. These need to match with the decision problem of interest. To assess this, the potential actions of the decision problem of interest need to be delineated as in \textit{Step~1} and the parameter sets $\Theta_0$ and $\Theta_1$ need to be obtained as in \textit{Step~5}. These sets have to be equivalent to the hypotheses that were employed in the calculation of the Bayes factor. If this is not the case, it is recommended to not use this Bayes factor value and restart the decision theoretic account within the previous section (Section~\ref{sec:guide1}). In this regard, it is helpful if the data set, that was used to calculate the original Bayes factor, is fully accessible.
\textbf{Step C: Applicability of the Prior Distribution.} Confirm that the employed within-hypothesis prior distributions for calculating the Bayes factor match with the available information about the phenomenon of interest. If this is not the case, it is recommended to not use this Bayes factor value and restart the decision theoretic account within the previous section (Section~\ref{sec:guide1}). Again, to do so it is helpful if the data set, that was used to calculate the original Bayes factor, is fully accessible.
\textbf{Step D: Prior Odds.} As the calculation of the Bayes factor does not require the prior odds, only the within-hypothesis prior distributions, former need to be specified to guide a decision. In this regard, the researcher has to specify the prior probabilities of the hypotheses. Analogue to \textit{Step~3}, as this is part of the Bayesian prior specification, it is recommended to fully report the available information about the parameter and why it leads to the prior probabilities of the hypotheses.
\textbf{Step E: Loss Function.} Specify the (interval-valued) loss function by following \textit{Steps~4, 6, and 7}.
\textbf{Step F: Posterior Odds.} Use the available Bayes factor $B\!F$ to calculate the posterior odds via equation~(\ref{eq:post_odds}).
\textbf{Step G: Optimal Action.} The optimal action can be derived as in \textit{Step~10}.
\section{Concluding Remarks}\label{sec:conclusion}
Statisticians and methodologists do -- in general -- not know all the different fields of applications and research contexts a statistical method will eventually be employed in. The scientific endeavor is extremely versatile and research problems might arise that have not been thought of before. In that, versatility of research methods is of paramount importance. While it might be considered as disadvantageous that frequentist hypothesis tests are restricted to a dichotomous decision context, it might similarly be considered as disadvantageous if Bayes factors are restricted to only an evidential, non-decision context. Fortunately, the mathematics underlying Bayes factors suggest their involvement in guiding decisions. In this regard, Bayes factors might be seen as evidential quantification or as a quantity in the context of guiding decisions, depending on the goals of the scientific investigation.
In order to use Bayes factors correctly when guiding decision, their embedding within the framework of Bayesian decision theory has to be considered. It is important that the research context as well as the decision problem are formalized appropriately. If misspecified, results inform past the research question. Naturally, the specification of essential quantities (such as the prior distribution, the hypotheses, or the loss function) is an applied problem and might be rather difficult for the applied scientist. In order to alleviate these issues, these quantities might be specified robustly as interval-valued or set-valued quantities. Then the researcher might consider a range or a set of plausible values, avoiding the necessity to (arbitrarily) commit to one single precise value. Within this elaboration only an interval-valued loss value was considered, as it keeps the calculations simple (compare Section~\ref{sec:robust_loss}) yet allows to include essential loss information (about the consequences of the decision) into the analysis. Naturally, also the prior distribution and the hypotheses might be included into the analysis as set-valued quantities \citep[see e.g.][]{Ebner2019}. How to deal with set-valued quantities on a formal level is extensively elaborated on within the field of imprecise probabilities \citep[see e.g.][]{Walley1991, Augustin2014, Huntley2014}.
In summary, this elaboration delineates the decision theoretic embedding of Bayes factors by outlining the framework of hypothesis-based Bayesian decision theory, supplemented by considerations about robust loss specifications and straightforward step-by-step guides. These guides try to help those applied scientists who want to guide decisions with Bayes factors.
\bibliographystyle{diss-style}
|
1,116,691,500,781 | arxiv | \section{\label{sec:Introduction}Introduction}
High intensity particle accelerators are vital in many applications, from spallation neutron sources to the transmutation of nuclear waste. However, in high intensity machines the Coulomb interactions between particles, known as space charge forces, cannot be neglected. The magnitude of these space charge forces impose the ultimate limit on beam intensity in recirculating accelerators.
Coulomb forces between charged particles in the beam of an accelerator are repulsive, acting as a lens which defocuses in both transverse planes \cite{Schindl}. As the intensity of a beam is increased the strength of this defocusing becomes greater, resulting in a shift in the oscillation frequency, called the tune, of circulating particles in high intensity beams.
As the transverse beam distribution in an accelerator does not generally lead to linear defocusing (except when a KV distribution is assumed \cite{Kapchinski1959}), space charge forces are a nonlinear perturbation to the linear focusing force produced by the magnets. Each particle therefore experiences a different tune shift, resulting in an overall tune spread within the beam. A greater spread in tune makes it more likely that some particles will have a tune lying within the stopband of a resonance, where they can be resonantly excited, leading to increased beam loss.
It is known that a single particle experiences a resonance when its tune agrees with the frequency of a periodic driving force. This is described by the resonant condition
\begin{equation}
\label{eq:IncoherentResonance}
Q_{\rm 0} = \frac{n}{m},
\end{equation}
\noindent
where $Q_{\rm 0}$ represents the bare tune of the particle (not including any tune shift from space charge forces) and $n$ and $m$ the harmonic and order of the perturbing driving force respectively, so that $m=1$ corresponds to a dipole perturbation, $m=2$ to a quadrupole perturbation and so on.
Including the tune shift due to space charge forces, $\Delta Q$, equation (\ref{eq:IncoherentResonance}) becomes
\begin{equation}
\label{eq:IncoherentResonance2}
Q_{\rm 0} + \Delta Q= \frac{n}{m},
\end{equation}
\noindent
where $ \Delta Q$ is negative, known as the tune depression.
In the past accelerators were designed so that the incoherent tunes of the particles in the beam avoided this resonant condition \cite{Morin1962}. However, the increased space charge forces at high intensity can result in the beam responding as a whole to a perturbation. This can lead to a collective motion of the beam, known as coherent oscillation \cite{Baartman1998, Lapostolle1963, Smith1963}.
It is the coherent oscillation of a beam that leads to the excitation of the greatest number of particles and therefore the greatest beam loss \cite{Sacherer1968}. In the case of coherent oscillation a self-consistent solution to the equations of motion of the beam is necessary to describe the resonant response. For a quadrupole ($m$=2) coherent oscillation this is done by constructing a set of envelope equations to describe the beam motion along the trajectory $s$, where $s$ the path length in the beam direction \cite{Sacherer1968},
\begin{eqnarray}
\label{eq:EnvelopeEquations}
\frac{\rmd^{2}a}{\rmd s^{2}} + K_{\rm a}(s)a - \frac{\epsilon_{\rm a}^{2}}{a^{3}} - \frac{K_{\rm sc}}{2(a + b)} = 0, \\
\frac{\rmd^{2}b}{\rmd s^{2}} + K_{\rm b}(s)b - \frac{\epsilon_{\rm b}^{2}}{b^{3}} - \frac{K_{\rm sc}}{2(a + b)} = 0.
\end{eqnarray}
In these envelope equations $a$ and $b$ are the root mean square (rms) sizes of the beam in the transverse directions and $\epsilon_{a}$ and $\epsilon_{b}$ the rms emittance. The periodic focusing is described by $K(s)$ and the space charge defocusing is given by
\begin{equation}
K_{\rm sc} = \frac{N q^{2}}{2 \pi \epsilon_{\rm 0} m \gamma^{3} v^{2}},
\end{equation}
\noindent
also known as the perveance. Here $q$ is the charge of the particle, $m$ its rest mass, $N$ the line density of the beam, $\gamma$ the Lorentz factor and $v$ the beam velocity.
By applying the smooth approximation and solving the envelope equations the new resonant condition is found to be
\begin{equation}
\label{eq:CoherentResonance2}
Q_{\rm0} + C_{2}\Delta Q = \frac{n}{2}.
\end{equation}
\noindent
where $C_{2} = \frac{3}{4}$ when the tunes in both transverse directions are equal \cite{Sacherer1968}. The condition can be generalised to higher order $m$ by the linearised Vlasov-Poisson equation, giving
\begin{equation}
\label{eq:CoherentResonance}
Q_{\rm 0} + C_{m}\Delta Q = \frac{n}{m}.
\end{equation}
The $C_{m}$ factor now present in the resonant condition depends on the mode of coherent oscillation that the beam is experiencing \footnote{$C_{mk}$ is also commonly used, where $m$ denotes the azimuthal mode of oscillation and $k$ the radial mode. Here $k$ is dropped as $m=k$ is assumed.}.
It has been analytically shown that the $C_{m}$ factors all lie in the range $0<C_{m}<1$, with the special case of $C_{1} = 0$ for a dipole perturbation \cite{Sacherer1968}. As the order of the coherent oscillation increases so does the $C_{m}$ factor, so that $C_{1}<C_{2}<C_{3}...$ etc. As the $C_{m}$ factors are less than $1$ this gives a small increase in the high intensity beam limit, meaning that the intensity of the beam can be increased further before the coherent resonant condition is met.
This work was later extended by Okamoto and Yokoya in reference \cite{Okamoto2002}, where no smooth approximation is applied. The theory presented in \cite{Okamoto2002} shows that, when alternating gradient focusing is considered, space charge driven resonances emerge at twice the density in tune space, according to the resonant condition
\begin{equation}
\label{eq:SpacechargeResonance}
Q_{\rm 0} + C_{m}\Delta Q = \frac{1}{2}\big(\frac{n}{m}\big).
\end{equation}
The growth rate and stop-band width of these space charge driven resonances were found to be proportional to the perveance, and therefore the intensity of the beam. Such resonances, also referred to as structure space charge instabilities or coherent parametric instabilities \cite{Hofmann2017}, are therefore neglected at low intensities.
Analytical values of $C_{m}$ have been calculated assuming a Kapchinski-Vladimirskij (KV) distribution, where the space charge forces provide linear defocusing in both planes. These values can then be used to describe different particle distributions assuming rms equivalent beams. These $C_{m}$ values are presented in Table \ref{table:TheoreticalCm} \cite{Sacherer1968}.
\begin{table}
\caption{\label{table:TheoreticalCm}Table showing the analytically calculated $C_{m}$ values for a 2D KV beam \cite{Sacherer1968}.}
\begin{indented}
\lineup
\item[]\begin{tabular}{@{}*{5}{l}}
\br
Order of coherent resonance ($m$)& 1 & 2 & 3 & 4\\
\mr
$C_{m}$ & 0 & $\frac{3}{4}$ & $\frac{11}{12}$ & $\frac{31}{32}$ \\
\br
\end{tabular}
\end{indented}
\end{table}
Additionally, the transverse distribution of the beam itself is thought to determine whether a beam is excited coherently. Hofmann, in reference \cite{Hofmann2017-2}, shows that KV distributions can be excited coherently by higher order modes, but that for more realistic beam models, such as waterbag or Gaussian distributions, the spread in particle tunes may lead to Landau damping of coherent modes higher than $m=2$. For these higher order modes the significant overlap between the incoherent tunes in the distribution and the tune of the coherent resonance can lead to energy transfer from the coherent excitation of the distribution to the incoherent excitation of particles within the beam, so that coherent oscillations never form \cite{Landau1946}.
The resonances occurring at a cell tune of $\frac{1}{4}$ have been studied in simulation a number of times \cite{Hofmann2015, Jeon2009, Li2014}. Simulations show clearly the competition between the fourth order incoherent excitation of the beam and the second order coherent resonance at this tune.
Accurate knowledge of the location of resonances in high intensity beams is desirable so that accelerators can be designed to avoid such resonances, minimising beam loss in high intensity machines. However, due to the non-linear space charge forces present in high intensity beams even particle motion in an accelerator with purely linear elements cannot be analytically described, it can only be approximated. Further complication is introduced by the emittance change due to particle loss near a resonance, which is not accounted for in the analytical calculation of the $C_{m}$ factors; the potential for Landau damping of coherent excitation; and the competition between resonances of different orders at the same tune.
Simulating a high intensity accelerator numerically requires a number of simplifying assumptions and is computationally intensive, making the study of high intensity resonances challenging. Instead, a Linear Paul Trap (LPT) can be used to recreate the alternating gradient (FODO) lattice of an accelerator experimentally. A LPT can study high intensity beam dynamics in such lattices by trapping a large number of argon ions, which have a transverse Gaussian distribution, in an electrical quadrupole potential. By varying the tune of the trapped ions, ion loss due to resonance can be studied.
The Simulator of Particle Orbit Dynamics (S-POD) at Hiroshima University, Japan, and the Intense Beam Experiment (IBEX) at the Rutherford Appleton Laboratory, UK, are LPTs designed for high intensity accelerator physics. A significant amount of previous work exists using S-POD to show the overlap in high intensity beams of external field driven resonances (satisfying the resonant condition in equation (\ref{eq:CoherentResonance})) and space charge driven resonances (meeting the resonant condition in equation (\ref{eq:SpacechargeResonance})) \cite{Ito2017}.
Furthermore, the splitting between the location of the externally driven coherent dipole resonance and self driven coherent quadrupole resonance at integer resonances, due to the differing $C_{m}$ factors, has been shown qualitatively \cite{Moriya2016}. As expected, this splitting becomes more pronounced as intensity is increased.
In reference \cite{Ohtsubo2010} Ohtusubo et al. applied a quadrupole error to the S-POD trap to excite resonances and ion loss was studied at three different intensities. They qualitatively draw the conclusion that the $C_{m}$ factors for $m = 2$, $m = 3$ and $m = 4$ ``appear to be less than unity'' at high intensities. However, the underlying assumptions of this study do not allow the result to be quantified. In this paper we study resonances in high intensity Gaussian beams over longer time scales and quantitatively for the first time, a significant advancement in the use of Paul ion traps to study high intensity effects in particle accelerators.
In this paper we build on this previous work to study resonances in high intensity Gaussian beams over longer time scales and quantitatively for the first time, a significant advancement in the use of Paul ion traps to study high intensity effects in particle accelerators. To start, we do not assume that resonances are either coherent or incoherent, we only assume that the resonance condition for maximal beam loss can be defined as
\begin{equation}
\label{eq:A}
Q_{\rm 0} + A_{m}\Delta Q = \frac{1}{2}\big(\frac{n}{m}\big).
\end{equation}
\noindent
We use the S-POD LPT to extract a value for $A_{m}$, which is of practical use in the design of high intensity accelerators.
We varied the cell tune (tune per focusing period) in the S-POD LPT over a wide range and studied the location of the resonances experienced at a range of intensities. We do not apply any external error fields to the trap, resonances are either driven by space charge forces or by small multipole fields from slight trap misalignments and in the high intensity regime in which this experiment is conducted space charge driven resonances dominate \cite{Ohtsubo2010} . From this data we extracted a numerical value for the location of four resonances of different orders at different ion numbers. By fitting to this we extract $A_{m}$, which we then compare to the $C_{m}$ factors predicted for purely coherent resonances.
To understand the experiment further we simulated the setup in the particle-in-cell (PIC) code Warp.
We first briefly describe the experimental setup of the S-POD and IBEX LPTs in section \ref{sec:Linear Paul Traps}. Our experimental procedures are shown in section \ref{sec:Experimental procedure and results}. We then present our simulation of the system in section \ref{sec:Simulation}, as the results helped to guide the analysis of the experimental data. The analysis of the experimental data is described in section \ref{sec:Analysis}.
\section{Linear Paul Traps\label{sec:Linear Paul Traps}}
A LPT stores ions using an alternating voltage applied to four cylindrical rods, which creates an electrical quadrupole potential of the form
\begin{equation}
\label{eq:potential}
U = \frac{V(t)}{2r_{\rm 0}^{2}}(x^{2} - y^{2}).
\end{equation}
\noindent
Here $r_{0}$ is the inscribed radius of the LPT rods, $x$ and $y$ are the usual transverse coordinates and $V(t)$ is the time dependent voltage applied to the rods.
The resultant LPT has a transverse Hamiltonian equivalent to an alternating gradient lattice in an accelerator, meaning that transverse particle motion in the two systems is the same \cite{Okamoto2002Trap, Davidson2000}.
The Hamiltonian for a LPT is expressed as
\begin{equation}
\label{eq:PaulHamiltonian}
H_{\rm Paul} = \frac{(p_{x}^{2} + p_{y}^{2})}{2} + \frac{1}{2}K_{\rm P}(\tau)(x^{2} - y^{2}) + \frac{q}{mc^{2}}\phi_{\rm sc},
\end{equation}
\noindent
where the focusing term is given by
\begin{equation}
\label{eq:PaulFocusing}
K_{\rm P}(\tau) = \frac{2qV(\tau)}{mc^2r_{0}^{2}}.
\end{equation}
\noindent
Here $p_{x}$ and $p_{y}$ are the particle momenta in the transverse directions, $q/m$ is the ion charge to mass ratio and $c$ the speed of light. The potential due to the space charge forces is denoted by $\phi_{\rm sc}$ and $\tau = ct$.
The Hamiltonian in equation (\ref{eq:PaulHamiltonian}) can be directly compared to the Hamiltonian of particle in a FODO cell of an accelerator,
\begin{equation}
\label{eq:FODOHamiltonian}
H_{\rm beam} = \frac{(p_{x}^{2} + p_{y}^{2})}{2} + \frac{1}{2}K(s)(x^{2} - y^{2}) + \frac{q}{p_{\rm 0}\beta_{\rm 0}c\gamma_{\rm 0}^{2}}\phi,
\end{equation}
\noindent
where the focusing is now described by
\begin{equation}
\label{eq:FODOFocusing}
K(s) = -\frac{q}{p_{\rm 0}}\frac{\rmd B_{z}}{\rmd x} = -\frac{1}{B\rho}\frac{\rmd B_{z}}{\rmd x}.
\end{equation}
Equation (\ref{eq:FODOFocusing}) describes the focusing due to a quadrupole magnet, where $B\rho$ is the magnetic rigidity and $\rmd B_{z}/ \rmd x$ the quadrupole gradient.
The third term on the right hand side of equation (\ref{eq:FODOHamiltonian}) (describing the space charge forces) now includes the Lorentz factor, $\gamma_{\rm 0}$, the ratio of the particle velocity to the speed of light, $\beta_{\rm 0}$ and the forward momentum of the particle, $p_{\rm 0}$. These terms are present as the interactions between relativistic line currents compete with static Coulomb interactions.
The LPT is a compact experimental system which allows a large number of cell tunes to be accessed by changing the amplitude of the voltage applied to the four confining rods. Equation (\ref{eq:PaulFocusing}) shows the effect of changing the voltage on the focusing term of the Hamiltonian. The number of ions stored can also be varied over a large range, allowing space charge effects to be investigated at different intensities. Additionally, a large ion loss does not damage a LPT or irradiate it, making it a useful tool in the study of beam loss at a resonance. It should be noted that dispersion due to momentum spread is not modelled in a LPT as the confining quadrupole is electric, however, such effects are not relevant to this study.
\subsection{\label{sec:Emittance and Temperature in a linear Paul trap}Emittance and Temperature in a linear Paul trap}
To understand the transverse dynamics of a LPT fully the emittance of the trapped ions must be known. The temperature and therefore the emittance in a Paul trap can be estimated from the envelope equations.
Particle motion in a LPT is non-relativistic and so magnetic forces can be neglected. Considering only the horizontal direction and changing the independent variable from $s$ to $\tau$ through $s = \beta_{0}\tau$, equation (\ref{eq:EnvelopeEquations}) becomes
\begin{equation}
\label{eq:EnvelopePaul}
\frac{\rmd^{2}a}{\rmd \tau^{2}} + K_{\rm a}(\tau)a - \frac{\epsilon_{\rm a}^{2}}{a^{3}} - \frac{N_{\rm tot}r_{\rm p}}{2a} = 0,
\end{equation}
\noindent
in the non-relativistic approximation, where $N_{\rm tot}$ is the total line density of the ion cloud, $\epsilon_{\rm a}$ is the rms emittance and $r_{\rm p}$ is the classical particle radius of the trapped ions.
Following the analysis in reference \cite{Okamoto2002Trap} the rms emittance can be expressed as
\begin{equation}
\label{eq:emit}
\epsilon_{\rm a} = a \sqrt{\frac{K_{\rm B} T}{m c^{2}} },
\end{equation}
\noindent
where $T$ is the ion distribution temperature and $K_{\rm B}$ the Boltzmann constant.
Using this definition and assuming a stationary ion distribution ($\rmd ^{2}a/\rmd\tau^{2} = 0$) equation (\ref{eq:EnvelopePaul}) can be rearranged to give an expression for the rms size of the beam in terms of transverse temperature,
\begin{equation}
\label{eq:size}
a = \frac{c}{\omega_{\rm q}}\sqrt{\frac{K_{\rm B} T}{m c^{2}} + \frac{N_{\rm tot}r_{\rm p}}{2}}.
\end{equation}
\noindent
Here $\omega_{\rm q}$ is the angular frequency of the betatron oscillations of the trapped ions.
Rearranging this equation and defining the effective value of the incoherent phase advance as
\begin{equation}
\label{eq:phaseAdvance}
\sigma^{2} = \sigma_{\rm 0}^{2} - \frac{N_{\rm tot} r_{\rm p}}{2} \bigg( \frac{\lambda}{a}\bigg)^{2},
\end{equation}
\noindent
where $\sigma_{\rm 0}$ is the bare phase advance and $\sigma$ is the shifted (depressed) phase advance, the following expression for transverse temperature in terms of tune depression can be derived,
\begin{equation}
\label{tuneDepression}
T = \frac{N_{\rm tot} r_{\rm p} m c^{2}}{2 K_{\rm B}} \bigg( \frac{1}{1 - \eta^{2}} - 1 \bigg).
\end{equation}
\noindent
The tune depression is defined to be $\eta = \sigma/\sigma_{\rm 0}$.
\subsection{\label{sec:Structure of a Paul trap}Structure of a Paul trap}
The experimental setups of both the S-POD and IBEX LPTs have been described in detail in a number of previous publications \cite{Okamoto2002Trap, Davidson2000,Sheehy2017} and so will only be briefly outlined here.
\begin{figure}
\makebox[\columnwidth][c]{\includegraphics[width=0.7\columnwidth]{figure1.pdf}}
\caption{Diagram showing the layout of the S-POD LPT. Here the region labeled IS is the Interaction Section, where the ions are stored, and ER the Experimental Region. During this study the ER was used only during the extraction of ions. IBEX is identical in construction except that the ER section and gate are removed, allowing direct extraction onto the Faraday cup. Image adapted from \cite{Ito2015a}.}
\label{fig:SPODSetup}
\end{figure}
Figure \ref{fig:SPODSetup} shows a diagram of the trapping region in both S-POD and IBEX. The trapping region sits inside a larger vacuum vessel which maintains an Ultra High Vacuum (UHV) of $\sim 10^{-10}$ mbar. Before performing experiments, argon gas is introduced to the vessel through a leak valve.
The argon gas is ionised by an electron gun directed into the space between the four cylindrical trapping rods. It is the $^{40}$Ar$^{+}$ ions created in this electron bombardment process that are confined by the rods and used in the high intensity beam experiments.
\begin{figure}
\makebox[\columnwidth][c]{\includegraphics[width=0.4\columnwidth]{figure2.pdf}}
\caption{Diagram showing geometry of and the voltage applied to the confining rods in the S-POD and IBEX Paul traps. Image adapted from \cite{Okamoto2002Trap}.}
\label{fig:SPODRods}
\end{figure}
Ions are confined longitudinally using the endcaps, sections of shorter cylindrical rod not electrically connected to the main confining rods (End A, End B and Gate in figure \ref{fig:SPODSetup}). A positive DC offset is applied to these rods, added to the alternating voltage by an Arbitrary Wavefunction Generator (AWG) before amplification. This creates a longitudinal potential well, trapping the ions. As the confining rods are much longer than the end caps an almost rectangular potential well is produced, leading to an approximately homogeneous longitudinal ion distribution.
The trap confines the ions transversely using an alternating voltage applied to the four cylindrical rods, as shown in figure \ref{fig:SPODRods}. Ideally hyperbolic electrodes would be used to create the perfect quadrupole potential, however, cylindrical rods are much easier to machine accurately and give a very good approximation to a perfect quadrupole in the centre of the trap, with the addition of some small higher order modes \cite{Ohtsubo2010}. The transverse confining potential (equation (\ref{eq:potential})) is created by applying either a sinusoidal voltage or a step function to the rods. The rf trapping voltage is supplied using two synchronised channels from the AWG (created separately) with the rf components 180 degrees out of phase, which are then amplified. The rods are powered in pairs, with opposing rods electrically connected. The IBEX and S-POD systems are operated with a sinusoidal voltage of 1 MHz and the transverse focusing is controlled by varying the amplitude of the applied voltage. The cell tune ($Q$) in the trap, not including the negligible term due to the endcap voltage, is approximately
\begin{equation}
\label{eg:tune}
Q \approx \frac{q V_{\rm rf}}{m \omega_{\rm rf}^{2} r_{\rm 0}^{2}},
\end{equation}
\noindent
where $V_{\rm rf}$ is the amplitude of the applied radio frequency voltage, $\omega_{\rm rf}$ is its angular frequency and $r_{\rm 0}$ the inscribed radius of the trap \cite{Okamoto2002}.
After an experiment is performed, i.e. after the ions have been stored for a given time under certain conditions, the ions are extracted either onto a Faraday cup or a Multi Channel Plate (MCP). Both of these detectors are entirely destructive. To extract ions from the trap the DC component of one of the endcap voltages is switched off so that ions are no longer confined longitudinally. Throughout the experiment a positive DC potential offset is applied to the central rods, this ensures that ions are forced out of the trap when an endcap voltage is removed.
The MCP contains a number of channels which amplify the signal through an electron cascade. The amplified electrons hit a phosphor screen, emitting light which is photographed by a CCD camera. The MCP can be used to give an integrated image of the ion distribution or, when calibrated, act as an ion counter. The Faraday cup on the other end of the trapping region is used as an ion counter.
\section{\label{sec:Experimental procedure and results}Experimental procedure and results}
\begin{figure*}
\makebox[\columnwidth][c]{\includegraphics[width=\columnwidth]{figure3.pdf}}
\caption{Measured ion number extracted from the S-POD trap against cell tune. Each point on the line represents a separate experiment where the ions are created, stored and extracted.}
\label{fig:Results}
\end{figure*}
\begin{figure}
\makebox[\columnwidth][c]{\includegraphics[width=0.7\columnwidth]{figure4.pdf}}
\caption{Tune shift against ion number for the four resonances at lowest tune in figure \ref{fig:Results}. Error bars are determined from the width of the resonance minima.}
\label{fig:Analysisplot}
\end{figure}
The experimental data was taken using the S-POD II trap, one of three LPTs at Hiroshima university. To study space charge driven resonances we conducted a number of experiments, each using a different cell tune between the values 0.1 and 0.4. This ensured that a wide range of resonances could be investigated. We then increased the number of ions stored in the trap and repeated the process. As coherent oscillations arise due to space charge forces in high intensity beams large numbers of ions were stored in the trap for these experiments. Measurements were taken for eight different ion numbers between $10^{6}$ and $10^{7}$.
Identical waveforms were applied to each rod pair (180 degrees out of phase) so that the transverse tunes were always the same.
At each tune value we performed a new experiment with following steps:
\begin{itemize}
\item Argon gas was introduced into the vessel.
\item A sinusoidal voltage was applied to the confining rods with an amplitude that gave a cell tune of 0.15, at this tune the largest possible number of ions are captured by the trap.
\item The electron gun was switched on for less than 1 s to ionise the argon gas.
\item The captured ions were stored in the trap for 50 $\mu$s to reach equilibrium.
\item The amplitude of the voltage applied to the confining rods was altered smoothly over 100 focusing periods until the tune at which the experiment should take place was reached.
\item At this tune the ions were stored for 100ms, equivalent to $10^{5}$ focusing periods.
\item Ions were extracted by setting to zero the DC voltage on the endcap closest to the MCP.
\item The signal from the MCP, taken from before the phosphor screen, was recorded to calculate the number of ions extracted.
\end{itemize}
We changed the number of ions stored in the trap and repeated these steps over the same tune range. The number of ions stored in the trap is altered by changing the length of time which the electron gun is on, or by increasing the pressure of argon in the vessel.
In total $\sim$ 330 experiments were performed within the range of tunes, each for a given beam intensity. S-POD is automated and capable of taking one measurement approximately every 10 s and so recording the ion loss over this wide range of tunes took only $\sim$ 1.5 hours.
The relationship between applied voltage and cell tune in the trap is calibrated using data taken at a very low ion number ($\sim 5\times10^{3}$). At this ion number any resonances will not be shifted due to space charge forces. Any offset of the resonances from their expected tune values allows a calibration factor to be calculated. This factor accounts for any systematic errors, such as the actual inscribed radius of the trap differing from 5 mm due to misalignments.
Figure \ref{fig:Results} shows the raw data from the experiments at large ion number, with the calibration applied.
When the beam has encountered a resonance the resultant beam loss will lead to a reduction in the number of ions extracted. Resonances are represented by local minima in the plot of extracted ion number against tune, as shown in figure \ref{fig:Results}.
Figure \ref{fig:Analysisplot} clearly shows that as the number of particles stored in the trap is increased the location of the resonance shifts to a higher bare tune, as expected due to the increased space charge tune depression. The ion loss observed as the cell tune approaches 0.1 and 0.4 is due to an increase in the maximum beta function at these tunes, reducing the number of ions that can be stored. Furthermore, the depth of the pseudo-potential created by the trap changes with the amplitude of the rf voltage, together, these effects characterise the maximum number of ions that can be stored at a given tune \cite{March2005}.
Ion-neutral collisions are found to lead to beam loss over the 100\,ms time scale of this experiment. However, the resonance is experienced immediately after the tune ramp, just 1\,ms after the ions have been left to stabilise. At this point in time, very few ion-neutral collisions will have occurred. As ion-neutral collisions are a non-conservative process they are not included in the Hamiltonian of the LPT. However, similar collisions will also take place in an accelerator, where Intra Beam Scattering (IBS) can also lead to emittance growth. In this paper we describe the effect of the resonant excitation only.
\section{\label{sec:Simulation}Simulation}
\begin{figure}
\makebox[\columnwidth][c]{\includegraphics[width=0.7\columnwidth]{figure5.pdf}}
\caption{Simulated rms emittance growth at the cell tunes $\frac{1}{6}$, $\frac{1}{4}$ and $\frac{1}{3}$. Left and right insets show enlarged versions of the $\frac{1}{6}$ and $\frac{1}{3}$ resonances respectively.}
\label{fig:emittanceGrowth}
\end{figure}
We use the PIC code Warp \cite{Warp} to simulate the experimental setup in 2D. The simulation applies a voltage to four cylindrical rods, the resultant poisson equation is solved on a grid in the trapping region and when ions are introduced their space charge contribution to the confining potential is included. The simulation is of an ideal LPT, with no misalignments in the trapping rods. Any misalignments in the real system will create additional higher order fields in the trapping region.
The simulation is designed to mimic the experiment as closely as possible, except for the total storage time. A matched distribution of argon ions is injected into the trap at the starting cell tune of 0.15. The ions are trapped at this tune for 50 focusing periods before the tune is ramped to the tune of interest by changing the amplitude of the voltage applied to the confining rods. As in the experiment, this change in tune occurs over 100 focusing periods regardless of the final tune. The ions are then stored in the trap for 550 focusing periods at the final tune, so that the simulation runs for 700 focusing periods in total.
In the simulation ions are stored for only a fraction of the $10^{5}$ focusing periods used in the experiment. This is due to the significant computational time required to simulate even 550 focusing periods, highlighting the importance of LPTs in understanding resonant effects over longer time scales.
We used this method to study the increase in the emittance of the distribution at the resonances at cell tunes $\sim \frac{1}{6}$, $\sim \frac{1}{4}$ and $\sim \frac{1}{3}$, focusing in greater detail on the resonance at $\sim \frac{1}{4}$. In each simulation the growth in emittance was recorded as well as the evolution of the phase space and any particle loss due to scraping on the rods. Emittance growth is studied instead of particle loss as little, if any, loss occurs on such short timescales. Furthermore, any loss that occurs after only 550 focusing periods is due to the loss of halo ions, and therefore provides very little information on the ion distribution as a whole.
\begin{figure}
\makebox[\columnwidth][c]{\includegraphics[width=0.8\columnwidth]{figure6.pdf}}
\caption{(a) shows the rms emittance growth in the horizontal (blue) and vertical (red) directions at the resonance at cell tune $\frac{1}{4}$, with the four tunes studied in greater detail highlighted by vertical lines. Further information on these tunes is provided in figure \ref{fig:phaseSpace}. (b) shows the rms emittance growth as a function of time for each of the four highlighted tunes, vertical lines represent the start and end of the voltage ramp.}
\label{fig:025Res}
\end{figure}
Figure \ref{fig:emittanceGrowth} shows the rms emittance growth at the 3 resonances for $2\times10^{6}$ trapped argon ions. From the phase space distributions extracted from each simulation we found that even after such a short time the $\frac{1}{4}$ resonance shows signs of coherent motion (see the inset phase space plots in figure \ref{fig:phaseSpace}(b)). However, at cell tunes of $\frac{1}{6}$ and $\frac{1}{3}$ this was not observed. This may be due to the relatively short timescale of the simulation or due to the Landau damping of coherent higher order modes, as predicted by Hofmann in reference \cite{Hofmann2017-2}.
The Warp simulations provided the opportunity to study the effect of the ramp in tune on the distribution of the trapped ions. We found that when trapped ions were ramped to a cell tune of $\frac{1}{4}$ or below the change in emittance during the ramp was negligible. However, when the ions are ramped through the $\frac{1}{4}$ resonance the emittance is increased, the magnitude of this increase depends on the final tune and therefore the rate of resonance crossing. This information was then used to guide the analysis of the data presented in section \ref{sec:Analysis}.
\subsection{The effect of fourth order resonance on the envelope instability}
In the case of the resonance at a cell tune of $\frac{1}{4}$ we chose four tunes across the resonance to study in greater detail. At each of these tunes we tracked the ions in the Warp simulation, and from the positional information we used Numerical Analysis of Fundamental Frequencies (NAFF) \cite{Laskar1992} to extract the tunes of individual macro particles. We also calculated the kurtosis, a measure of the shape of the distribution \cite{Press1988}, at every focusing period.
We chose to study the resonance at cell tune $\frac{1}{4}$ as the competition between the fourth order incoherent resonance and the second order coherent resonance (also known as the envelope instability) at this tune is the subject of much previous work. Simulations by Hofmann and Boine-Frankenheim in reference \cite{Hofmann2015} show that these two resonances overlap, with the fourth order first exciting particles and the stronger envelope instability taking over as the simulation progresses. In reference \cite{Jeon2009} the fourth order resonance was found to dominate on smaller timescales and its width was measured at GSI.
\begin{figure}[!htb]
\makebox[\columnwidth][c]{\includegraphics[width=\columnwidth]{figure7.pdf}}
\caption{Emittance growth (rms) in the horizontal direction at bare cell tunes (a) 0.253, (b) 0.255, (c) 0.258 and (d) 0.26. Emittance growth in the vertical direction is very similar in each case. Insets show the x x' phase space at points of interest.}
\label{fig:phaseSpace}
\end{figure}
Once the emittance growth due to either the coherent or incoherent resonance is so great that the tune of the beam no longer falls within the stop band of the envelope instability, the distribution can no longer be excited \cite{Li2014}. This leads to the characteristic shape of the coherent resonance, with a sharp increase in particle loss on the lower tune side of the resonance and a gentler gradient on the high tune side.
On the other hand, the fourth order resonance has the potential to excite particles over much longer time scales, trapping them in the stop band.
Our simulations agree very well with these previous results. Figure \ref{fig:025Res} shows that the simulated resonance has the characteristic shape of a coherent resonance with the addition of areas of emittance growth outside of this coherent region (for example the region between bare cell tunes of 0.251 and 0.253 in figure \ref{fig:025Res}(a)), where the emittance growth is significantly reduced, but not yet negligible.
In figure \ref{fig:025Res}(a) a small difference in the emittance growth between the horizontal and vertical directions indicates that the second order resonance dominates at those tunes. The second order resonance is a strong instability and small, random, differences in initial conditions influence in which plane the emittance grows first \cite{Hofmann2017}. In figure \ref{fig:025Res}(a) the four tunes that were studied in greater detail are highlighted with vertical lines and the emittance growth as a function of time at each of these tunes is shown in the lower panel.
Figure \ref{fig:phaseSpace} shows the evolution of the phase space at the four tunes studied in detail. Together figures \ref{fig:025Res}, \ref{fig:phaseSpace}(a) and \ref{fig:phaseSpace}(d) show that the outer regions of emittance growth (at the higher and lower tunes within the resonance) are due to the excitation of particles through the fourth order resonance, with this excitation at the lower tune side of the resonance leading to the formation of a large halo, which can lead to particle loss over a shorter timescale. In the centre of the resonance the phase space plots in figures \ref{fig:phaseSpace}(b) and \ref{fig:phaseSpace}(c), clearly show a fourth order excitation followed by the stronger envelope (second order) instability, leading to an initial growth in the emittance, a plateau, and then further emittance growth.
The kurtosis of the distributions is shown in figure \ref{fig:kurtosis}. The kurtosis represents the peakedness of the ion distribution with respect to a Gaussian, which has a kurtosis of 0. Larger positive values indicate a more pointed shape, whereas larger negative values indicate that the distribution is more square. The incoherent fourth order excitation of particle leads to a positive value of the kurtosis as a small number of particles are excited to large amplitudes. In the case of halo formation at a bare cell tune of 0.253 (figure \ref{fig:kurtosis}(a)) the kurtosis remains large, showing that the halo remains an important feature over a large number of cells. In the case of the incoherent excitation at a bare cell tune of 0.26 (figure \ref{fig:kurtosis}(d)) the kurtosis returns to 0, showing that although the emittance increases the distribution is able to redistribute and return to a roughly Gaussian shape.
In the regions of coherent excitation (figures \ref{fig:kurtosis}(b) and \ref{fig:kurtosis}(c)) the kurtosis first returns to 0 and then becomes negative as the beam is excited as a whole.
The tune footprint for the four tunes studied is shown in figure \ref{fig:tune}. The tunes are calculated over 50 focusing periods, firstly at the start of the simulation (at cell tune of 0.15 in each case), then immediately after the tune ramp and finally at the end of the simulation. For each tune footprint the rms tune and the theoretical location of the coherent envelope instability is also plotted.
Figure \ref{fig:tune} shows that the maximum emittance growth does not occur at the theoretical location of the coherent resonance. The maximum emittance growth occurs at a bare cell tune of 0.255 (figure \ref{fig:tune}(c)), instead of the predicted theoretical value of closer to 0.259. This is due to the initial fourth order excitation of the ions, which alters the ion distribution, in turn reducing the tune depression. This, coupled with the finite width of the resonance, means that the coherent resonance does not occur at the tune predicted by the $C_{m}$ factor. At 0.255, fourth order excitation pushes the distribution onto the coherent envelope instability, the ions spend the maximum time within the coherent stopband and therefore the emittance growth is maximised. At 0.258, the distribution is moved towards the exit of the stop band of the envelope instability by the fourth order incoherent excitation, the time spent within the coherent stopband is reduced and so is the emittance growth.
Through further simulation we confirmed that this remains true even when simulations are extended to a total length of 1350 focusing periods, after which time the coherent emittance growth has plateaued. Simulations which introduce a matched distribution to the trap directly at the tune of interest, without the voltage ramp, verified that this effect is not due to the slow ramp onto the resonance.
Returning now to the experimental work with the S-POD LPT, we therefore expect our experimental results to show evidence of this effect, with the extracted value of $A_{m}$ (equation (\ref{eq:A})) at a cell tune of $\frac{1}{4}$ smaller than $C_{2} = \frac{3}{4}$.
\begin{figure}[!htb]
\makebox[\columnwidth][c]{\includegraphics[width=\columnwidth]{figure8.pdf}}
\caption{Kurtosis in the horizontal direction at bare cell tunes of (a) 0.253, (b) 0.255, (c) 0.258 and (d) 0.26. Kurtosis in the vertical direction is very similar in each case.}
\label{fig:kurtosis}
\end{figure}
\begin{figure*}
\makebox[\columnwidth][c]{\includegraphics[width=0.7\columnwidth]{figure9.pdf}}
\caption{Tune footprints for bare cell tunes of (b) 0.253, (c) 0.255, (d) 0.258 and (e) 0.26. Plot (a) shows the initial distribution in space (left) and the starting tune footprint before the ramp (right). For subplots (b), (c), (d) and (e) the left plot shows the tune footprint immediately after the voltage ramp and the right hand plot shows the tune footprint at the end of the simulation. Distributions are coloured to highlight the relationship between particle location in space and tune. Red dots represent theoretical bare tune based on the applied voltage and solid black lines highlight the location of a tune of $\frac{1}{4}$. Broken red lines show the rms of each tune footprint and solid red lines show the theoretical location of the coherent resonance, assuming $C_{2} = \frac{3}{4}$.}
\label{fig:tune}
\end{figure*}
\section{\label{sec:Analysis}Analysis of experimental results and discussion}
When analysing the experimental data we chose to focus on only the four resonances at the lower tune values. These resonances are clearly distinct and it is possible to locate the cell tune where the fewest ions are extracted from the trap for each resonance. Furthermore, the simulations presented in section \ref{sec:Simulation} suggest that at a cell tune higher than $\frac{1}{4}$ the emittance will be increased by passing through the strong envelope instability, complicating the data analysis for these resonances.
We located the point of maximum beam loss for the four resonances studied. The tune corresponding to this point was extracted from the data by a minimum finding algorithm. The change in the tune at which the resonance is located is plotted against the ion number to give figure \ref{fig:Analysisplot}.
A number of steps are required to extract $A_{m}$ (from equation (\ref{eq:A})) from the data in figure \ref{fig:Analysisplot}. Firstly the expected single particle tune shift at each intensity should be calculated. To do this a value of the emittance of the ion cloud and the longitudinal length of the distribution in the trap are required. Then, a linear fit can be applied to a plot of the tune shift of each resonance multiplied by the emittance, against the ion number in the trap (figure \ref{fig:Analysisplot2}). Any deviation from the expected single particle tune shift indicates that $A_{m}$ deviates from 1. Final values for $A_{m}$ are presented in figure \ref{fig:cm}.
To calculate the expected single particle tune shift, $\Delta Q$, the ions are assumed to occupy a cylinder with a transverse Gaussian distribution and length $L$ \cite{Schindl}. As ion motion in the Paul trap is non-relativistic magnetic forces can be neglected and the rms tune shift of the equivalent beam is calculated to be
\begin{equation}
\label{eq:deltaQ}
\Delta Q = \frac{\lambda_{\rm rf}r_{\rm p}}{8 \pi}\frac{N}{L \epsilon_{\rm a}},
\end{equation}
\noindent
where $N$ is the number of ions stored in the trap, $\epsilon_{\rm a}$ is the rms emittance, $r_{\rm p}$ is the classical particle radius and $\lambda_{\rm rf}$ the wavelength of the applied rf voltage.
We assume a Gaussian distribution based on significant evidence from previous S-POD experiments in which the MCP is used to image the ion distribution \cite{Moriya2015, Ito2009}. Calculating $\Delta Q$ from equation (\ref{eq:deltaQ}) gives a result that agrees very well with the rms tune shift calculated from the Warp simulations through NAFF.
The ion distribution length $L$ was determined experimentally using IBEX. As both traps are designed to be longitudinally identical and the same electron gun is used, the result applies to both S-POD and IBEX (see figure \ref{fig:SPODSetup}). Further details of this experiment can be found in \ref{Effective plasma length measurement}.
Using the equations presented in section \ref{sec:Emittance and Temperature in a linear Paul trap} and the experimentally determined value for the longitudinal distribution length, the emittance was calculated for each ion number. A single emittance value cannot be assumed as the increased space charge forces at higher intensity will distort the potential experienced by the ions. The tune depression for the resonance at a cell tune of $ \frac{1}{6}$ was used to calculate the rms tune depression, at first assuming $A_{m} = 1$. This was then used to calculate the transverse temperature. The temperature was used to find a value for the rms beam size and finally the rms emittance.
The data in figure \ref{fig:Analysisplot} was combined with the results of the emittance calculation at each ion number and a linear fit was applied, as shown in figure \ref{fig:Analysisplot2}. The gradient of the fit gives $A_{m}$ for the resonance, as
\begin{equation}
\label{eq:deltaQ2}
\frac{\epsilon_{\rm a} \Delta Q }{N} = \frac{ \lambda_{\rm rf} r_{\rm p }}{8 \pi} \frac{A_{m}}{ L }.
\end{equation}
This process was repeated iteratively for the $\frac{1}{6}$ resonance until the factor of $A_{m}$ included in the calculation of the rms tune depression equaled the factor extracted from the fit. The emittance values for this case are presented in Table \ref{table:emittance}.
\begin{table}
\caption{\label{table:emittance}Table showing the transverse temperature, rms beam size and rms emittance calculated for each ion number studied. }
\begin{indented}
\lineup
\item[]\begin{tabular}{@{}*{4}{l}}
\br
Ion number & Transverse temperature & rms beam size & rms emittance \\
($*10^{6}$) & (eV) & (mm) & ($*10^{-3}$ mm mrad) \\
\mr
1.897 & 0.144 $\pm$ 0.003 & 0.670 $\pm$ 0.008 & 1.318 $\pm$ 0.030\\
3.026 & 0.190 $\pm$ 0.004 & 0.779 $\pm$ 0.008 & 1.760 $\pm$ 0.036\\
4.223 & 0.211 $\pm$ 0.005 & 0.837 $\pm$ 0.009 & 1.998 $\pm$ 0.045\\
5.245 & 0.244 $\pm$ 0.006 & 0.906 $\pm$ 0.010 & 2.320 $\pm$ 0.052\\
6.316 & 0.252 $\pm$ 0.006 & 0.934 $\pm$ 0.010 & 2.430 $\pm$ 0.054\\
7.192 & 0.275 $\pm$ 0.006 & 0.981 $\pm$ 0.010 & 2.664 $\pm$ 0.056\\
8.554 & 0.291 $\pm$ 0.007 & 1.024 $\pm$ 0.011 & 2.886 $\pm$ 0.064\\
9.917 & 0.297 $\pm$ 0.006 & 1.051 $\pm$ 0.011 & 2.967 $\pm$ 0.063\\
\br
\end{tabular}
\end{indented}
\end{table}
This analysis relies on the further assumption that the tune change from a cell tune of 0.15 (where the ions are collected) to the operating point, is adiabatic and that during this process no ions are lost. This assumption is well supported by the Warp simulations.
\begin{figure}
\makebox[\columnwidth][c]{\includegraphics[width=0.7\columnwidth]{figure10.pdf}}
\caption{Tune shift multiplied by the ion cloud emittance for the four resonances studied against the ion number. Here error bars include the error on the emittance values and the solid lines show the gradient of a linear fit to the data.}
\label{fig:Analysisplot2}
\end{figure}
\begin{figure}
\makebox[\columnwidth][c]{\includegraphics[width=0.7\columnwidth]{figure11.pdf}}
\caption{Values of $A _{m}$ from the fitting process in figure \ref{fig:Analysisplot2} with error bars based on fit in figure \ref{fig:Analysisplot2} and the calculation of the ion cloud emittance at each ion number. The red points represent the theoretical $C_{m}$ values for the lowest order space charge driven resonance at each cell tune.}
\label{fig:cm}
\end{figure}
In the experimental data in figure \ref{fig:Results} the resonance at a cell tune of $\frac{1}{4}$ shows the characteristic shape of a coherent resonance (i.e. the dip in ion number in tune space is asymmetric, with a much steeper slope for tunes lower than the point where the ion number reaches a minimum), with the addition of ion loss at higher and lower tunes. This additional ion loss is explained by the Warp simulation as due to the excitation of a subset of ions through the fourth order resonance. The shape of the experimental ion loss data agrees well with the emittance growth in the Warp simulation, supporting the idea that emittance growth is a good predictor for ion loss. This also suggests that a simulation of only 1350 focusing periods may be sufficient to predict ion loss over longer time scales. The resonance shows a similar shape as intensity is increased.
The other resonances studied experimentally ($\frac{1}{8}$, $\frac{1}{6}$ and $\frac{1}{5}$) do not have the sharp increase in ion loss that usually indicates a coherent resonance, suggesting that ion loss may not be due to coherent oscillations.
The fitting process is performed for all four resonances and the results of this fit are shown in figure \ref{fig:cm}, alongside the $C_{m}$ factors for purely coherent resonances.
The values of $A_{m}$ extracted from the resonances at tunes of $\frac{1}{8}$ and $\frac{1}{6}$ are consistent with 1, rather than the $C_{m}$ factor for the coherent resonance which is expected to occur at that tune (also plotted in figure \ref{fig:cm}), again suggesting that these resonances may not be coherent. As it cannot be claimed that the excitation of a subset of particles is simply an incoherent effect, as the motion of these particles effects the rest of the beam through the Coulomb interaction, it is interesting to note that the peak ion loss still occurs at roughly the tune with the highest particle density.
A value of $A_{m} = 0.517 \pm 0.091$ is extracted for the $\frac{1}{4}$ resonance. This is smaller than the other resonances and indeed smaller than the analytically predicted value of $\frac{3}{4}$. This agrees with the Warp simulations, we expect to see a greater coherent advantage due to the fourth order resonance first increasing the emittance of the beam.
Any resonances present in a LPT are clearly complex and the product of competing sources of perturbation, as in an accelerator. Whether these resonances are coherent or incoherent, the values of $A_{m}$ presented in figure \ref{fig:cm} show the relative location of the beam loss for a resonance of a given order in a Gaussian beam. These factors help us understand the real behaviour of an accelerator and are directly relevant in future accelerator design.
Further work is possible to build on this study. The stronger resonances at cell tunes of $\sim\frac{1}{4}, \sim\frac{1}{6}$ and $\sim\frac{1}{8}$ can be studied with a lower storage time to verify that the values of $A_{m}$ remain unchanged. There is also the potential to study anisotropic beams using a LPT, as non-circular beams are expected to have different $C_{m}$ factors \cite{Hofmann1998}.
Further simulation work to study the $\frac{1}{4}$ resonance at increasing intensities is also desirable to determine the effect of intensity on the relationship between the fourth order resonance and the envelope instability.
To understand fully the mechanism behind beam loss at these resonances further diagnostics are required in the experimental setup. Detection of the coherent oscillations via the induced current on the confining rods should be attempted. Further diagnostics would also allow for an emittance measurement to be made at each ion number. This would improve the accuracy of the extracted values of $A_{m}$.
\section{\label{sec:Conclusion}Conclusion}
This paper presents the first quantitative study of the interaction and difference between coherent and incoherent resonances in a Paul trap, and furthermore uses simulation to explain why the experimental results differ from those theoretically predicted. A LPT has previously been used to study the behaviour of a beam only qualitatively, showing the shift in resonance location with beam intensity. Here we extract numerical values describing the locations of resonances at high intensity.
We have shown through simulation using the PIC code Warp that in a LPT, as in an accelerator, we expect to see coherent oscillation at a cell tune of $\frac{1}{4}$. Here the coherent resonance is in competition with the fourth order resonance, which has a wider stopband. For $2\times10^{6}$ ions, simulations show that, even at tunes where coherent oscillations occur, ions are first excited by the incoherent fourth order resonance. By analysing the tunes of trapped ions through NAFF the tune footprint at a number of tunes across the resonance was extracted. This showed that the peak in emittance growth did not occur at the location predicted by the $C_{m}$ factor for coherent resonances. We identified that this was due to both the finite width of the envelope instability and the fourth order excitation.
The shape of the ion loss at the $\frac{1}{4}$ resonance agrees well with shape of increased emittance growth in the Warp simulation, suggesting that a short simulation may be in some cases sufficient to predict beam loss over longer timescales. The experimental results show that the $C_{m}$ factor for a second order coherent resonance does not accurately predict the location of the resonance. Instead we find that the factor describing the shift of the resonance with intensity is less than the theoretical $C_{2}=\frac{3}{4}$, due to the interaction between the fourth order incoherent resonance and the second order coherent resonance at this tune, agreeing with Warp simulation predictions.
Simulation over 700 focusing periods did not show any signs of coherent oscillation for the resonances at cell tunes $\frac{1}{6}$ and $\frac{1}{3}$.
The experimental ion loss for the resonances at cell tunes $\frac{1}{8}$, $\frac{1}{6}$ and $\frac{1}{5}$ did not show the shape of coherent resonance and the location of the peak in ion loss occurred in a location consistent with the rms of the tune distribution, suggesting that at this tune coherent oscillation may indeed be Landau damped for a Gaussian beam.
\ack
The Authors would like to thank the beam physics group at Hiroshima university for the experimental time on the SPOD experiment and for their advice and comments.
We would like to thank Dr Kiersten Ruisard for her advice on the use of Warp.
S.L. Sheehy gratefully acknowledges the support of The Royal Society.
\section*{References}
|
1,116,691,500,782 | arxiv | \section{Preliminaries}
In a drawing of a graph in the plane,
vertices are represented by points, edges are represented by curves
connecting the points, which correspond to adjacent vertices.
The points (curves) are also
called vertices (edges).
We assume that an edge does not go through any vertex,
and three edges do not cross at the same point.
A drawing is {\em simple} if
any two edges have at most one point in common, which is either a common endpoint or a crossing.
In particular, there is no self-crossing.
A graph together with its drawing is a {\em topological graph}.
In this paper, we assume the underlying graph has
neither loops nor multiple edges.
For any $k\ge 0$, a topological graph is {\it $k$-plane} if each edge contains at most $k$ crossings.
A graph $G$ is {\it $k$-planar}
if it has a simple {\it $k$-plane}
drawing in the plane.
A $k$-plane drawing is {\em saturated} if no edge can be added
so that the obtained drawing is also $k$-plane.
The $0$-planar graphs are the well-known planar graphs.
A plane graph of $n$ vertices has at most $3n-6$ edges.
If it has exactly $3n-6$ edges, then it is a triangulation of the plane.
If it has fewer edges, then we can add some edges so that it
becomes a triangulation with $3n-6$ edges.
That is, saturated plane graphs have $3n-6$ edges.
Pach and Tóth \cite{pt} proved that the maximum number of edges of an $n$-vertex
$1$-planar graph is $4n-8$.
Brandenburg et al. \cite{brandenburg} noticed that
saturated $1$-plane graphs can have much fewer edges, namely
$\frac{45}{17}n + O(1)\approx 2.647n$.
Bar\'at and T\'oth \cite{bt} proved that a
saturated $1$-plane graph has at least
$\frac{20n}{9}-O(1)\approx 2.22n$ edges.
For any $k, n$, let $s_k(n)$ be the minimum number of edges of an $n$-vertex saturated simple $k$-plane drawing.
With these notations,
$\frac{20n}{9}-O(1)\le s_1(n)\le \frac{45}{17}n + O(1)$.
For $k>1$, the best bounds known for $s_k(n)$ are shown by
Auer et al \cite{auer} and by Klute and Parada \cite{kp}.
Interestingly for $k\ge 5$ the bounds are very close.
In this note, we concentrate on $2$-planar graphs on $n$ vertices. Pach and Tóth \cite{pt} showed
the maximum number of edges of a $2$-planar graph is $5n-10$.
Auer et al \cite{auer} and Klute and Parada \cite{kp} proved
$\frac{4n}{3}\ge s_2(n)\ge \frac{n}{2}$.
We improve the lower bound.
\begin{theorem}\label{n-1}
For any $n>0$, $s_2(n)\ge n-1$.
\end{theorem}
A drawing is {\em $l$-simple} if
any two edges have at most $l$ points in common.
By definition a simple drawing is the same as a $1$-simple drawing.
Let $s_k^l(n)$ be the minimum number of edges of a saturated $l$-simple $k$-plane drawing.
In \cite{kp} it is shown that
$\frac{4n}{5}\ge s_2^2(n)\ge \frac{n}{2}$ and
$\frac{2n}{3}\ge s_2^3(n)\ge \frac{n}{2}$.
We make the following improvements:
\begin{theorem}\label{2-3-simple}
(i)
$s_2^2(3)=3$, and for $n\neq 3$
$\lfloor3n/4\rfloor \ge s_2^2(n)\ge \lfloor2n/3\rfloor$,\\
(ii) $s_2^3(3)=3$, and for $n\neq 3$
$s_2^3(n)=\lfloor2n/3\rfloor$.
\end{theorem}
\iffalse
\begin{tabular}{|l|c|c|c|}
\hline
& self-crossing & multi-crossing & simple\\
\hline
lower bound & $\frac{1}{3}n$ & $\frac{1}{2}n$ & $\frac{1}{2}n$\\
\hline
upper bound & $\frac{1}{3}n$ & $\frac{2}{3}n$& $\frac{4}{3}n$\\
\hline
\end{tabular}
2-plane, $1$-simple: previous lower bound: $n/2$, upper: $4n/3$ KP21, ABGH12
2-plane, 2-simple: previous lower bound: $n/2$, upper: $4n/5$ KP21
2-plane, 3-simple: previous lower bound: $n/2$, upper: $2n/3$ KP21
\fi
The saturation problem for $k$-planar graphs has many different settings,
we can allow self-crossings, parallel edges, or we can consider non-extendable {\em abstract} graphs.
See \cite{kozos} for many recent results and a survey.
\section{Proofs}
\begin{definition}
Let $G$ be a topological graph.
A degree 1 vertex $u$ is a {\em leaf}, that is $d(u)=1$.
Let $v$ be the only neighbor of $u$.
The pair $(u, uv)$ is a {\em flag}.
That is, $u$ and the incident edge $uv$ together.
If there is no crossing on $uv$, then $(u, uv)$ is an {\em empty flag}.
\end{definition}
\begin{definition}
Let $G$ be a $2$-plane, $l$-simple topological graph.
If an edge contains two crossings, then its piece between the two crossings is a {\em middle segment}.
The edges of $G$ divide the plane into cells.
A cell $C$ is {\em special} if it is bounded only by middle segments and isolated vertices.
Equivalently, $C$ is {\em special},
if there is no vertex on its boundary, apart from isolated vertices.
An edge that bounds a special cell is also {\em special}.
\end{definition}
Let $G$ be a saturated $2$-plane $l$-simple topological graph, where $1\le l\le 3$.
Suppose that a cell $C$ contains an isolated vertex $v$.
Since $G$ is saturated, $C$ must be a special cell and
there is no other isolated vertex in
$C$.
Now suppose that $C$ is an empty special cell.
All boundary edges contain two crossings.
Therefore, if we put an isolated vertex in $C$, then the topological graph remains saturated.
So if we want to prove a lower bound on the number of edges,
we can assume without loss of generality that each special cell contains an isolated vertex.
\bigskip
\begin{claim} \label{only1}
A special edge can bound at most one special cell.
\end{claim}
\begin{proof}
Suppose that $uv$ is a special edge and let $pq$ be its middle segment.
If $uv$ bounds more than one special cell, then there is a special cell on both sides of $pq$, $C_1$ and $C_2$ say.
Let $p$ be a crossing of the edges $uv$ and $xy$.
There is no crossing on $xy$ between $p$ and one of the
endpoints, $x$ say.
Therefore, one of the cells $C_1$ and $C_2$ has $x$ on its boundary, a
contradiction.
\end{proof}
\begin{proof}[Proof of {Theorem \ref{n-1}}]
Suppose $G$ is a saturated 2-plane
simple topological graph of $n$ vertices and $e$ edges. We assume that each special cell contains an isolated vertex.
\begin{claim} \label{emptyflag}
All flags are empty in $G$.
\end{claim}
\begin{proof}
Let $(u, uv)$ be a flag.
Suppose to the contrary there is at least one crossing on $uv$.
Let $p$ be the crossing on $uv$ closest to $u$, with edge $xy$. Since it is a 2-plane drawing, there is no crossing on $xy$ between $p$ and one of the endpoints, $x$ say.
In this case, we can connect $u$ to $x$ along $up$ and $px$.
Since the drawing was saturated, $u$ and $x$ are adjacent in $G$, and $x\neq v$, that contradicts to $d(u)=1$.
\end{proof}
Remove all empty flags from from $G$.
Observe that the resulting topological graph $G'$ is also saturated.
If we can add an edge to $G'$, then we could have added the same edge to $G$.
Suppose to the contrary that $G'$ contains a flag $(v, vw)$.
Since $G'$ is saturated, it is empty by Claim~\ref{emptyflag}.
In $G$, vertex $v$ had degree at least $2$, so $v$ had some other neighbors,
$u_1, \ldots, u_m$ say, in clockwise order. The flags $(u_i,u_iv)$ were all empty.
However, $u_1$ can be connected to $w$, which is a contradiction.
Therefore, there are no flags in $G'$.
On the other hand, the graph $G'$ may contain isolated vertices.
Let $n'$ and $e'$ denote the number of vertices and edges of $G'$.
Since $n-n'=e-e'$, it suffices to show that $e'\ge n'-1$.
If there are no isolated vertices in $G'$,
then $e'\ge n'$ is immediate.
We assign weight $1$ to each edge.
If $G'$ has no edge, then it has one vertex and we are done.
We discharge the weights to the vertices so that each
vertex gets weight at least $1$.
If $uv$ is not a special edge, then
it gives weight $1/2$ to both endpoints $u$ and $v$.
Suppose now that $uv$ is a special edge.
It bounds the special cell $C$ containing the isolated vertex $x$.
If $d(u)=2$, then $uv$ gives weight $1/2$ to $u$, if $d(u)\ge 3$, then
it gives weight $1/3$ to $u$. We similarly distribute the weight to vertex $v$.
We give the remaining weight of $uv$ to $x$.
We show that each vertex gets weight at least 1.
This holds immediately for all vertices of positive degree.
We have to show the statement only for isolated vertices.
Let $x$ be an isolated vertex in a special cell $C$ bounded by
$e_1, e_2, \ldots, e_m$ in clockwise direction.
Let $e_i=u_iv_i$, such that the oriented curve $\overrightarrow{u_iv_i}$ has $C$ on its right.
See Figure 1. Let $p_i$ be the crossing of $e_i$ and $e_{i+1}$. Indices are understood modulo $m$.
In general, it may happen that some of the points in
$\{\ u_i, v_i\ | \ i=1, \ldots, m\ \}$ coincide.
For each vertex $u_i$ or $v_i$ of degree at least $3$, the corresponding boundary edge of $C$ has a remainder charge at least $1/6$.
We have to prove that (with multiplicity) at least $6$ of the vertices
$u_i$, $v_i$ have degree at least $3$.
Consider vertex $v_i$.
\smallskip
Case 1: $v_i=u_{i+2}$.
The vertex $v_i=u_{i+2}$ can be connected to $u_{i+1}$ along the segments $v_ip_i$ and $p_iu_{i+1}$,
that are crossing-free segments of the corresponding edges.
Similarly, $v_i=u_{i+2}$ can be connected to $v_{i+1}$ along $v_ip_{i+1}$ and $p_{i+1}v_{i+1}$.
Since the drawing was simple and saturated,
$u_i$, $u_{i+1}$, $v_{i+1}$, $v_{i+2}$ are all different and they are already connected to $v_i=u_{i+2}$,
so it has degree at least $4$.
\smallskip
\begin{figure}\label{specialdisch}
\begin{center}
\scalebox{0.25}{\includegraphics{2planar1.eps}\hspace{3cm} \includegraphics{2planar2.eps}\hspace{3cm}\includegraphics{2planar3.eps} }
\caption{Case 1, $d(v_1)\ge 4$, \hfil Case 2, $d(v_1)\ge 3$ and Case 2, $u_1=u_3$.}
\end{center}
\end{figure}
Case 2: $v_i\neq u_{i+2}$.
The vertex $v_i$ can be connected to $u_{i+1}$ as before, and to
$u_{i+2}$ along
$v_ip_i$, $p_ip_{i+1}$ and $p_{i+1}u_{i+2}$.
Since the drawing was saturated, $v_i$ is already adjacent to $u_i$, $u_{i+1}$, $u_{i+2}$.
Unless $u_i=u_{i+2}$, vertex $v_i$ has degree at least 3.
Note that $u_{i+1}\neq u_i$ and $u_{i+1}\neq u_{i+2}$,
since the drawing was $1$-simple.
We can argue analogously for $u_i$.
We conclude that $v_i$ has degree $2$ only if $u_i=u_{i+2}$, and
$u_i$ has degree $2$ only if $v_{i}=v_{i+2}$.
For $m=3$, both of these are impossible.
For $i=1, 2, 3$ all six vertices $u_i$, $v_i$ have degree at least $3$.
Let $m>3$, and suppose $v_1$ has degree $2$, consequently
$u_1=u_3$. In this case, we prove that
$u_1$, $u_2$, $u_3$, all have degree at least $3$.
We show it for $u_2$, the argument is the same for the vertices $v_2$, $u_m$, and $v_m$.
Let $C$ be the cell formed by the segments $u_1p_1$, $p_1p_2$ and $p_2u_3$.
Notice that $u_1=u_3$.
Suppose that $d(u_2)=2$. By the previous observations, $v_m=v_2$.
However, $v_m$ and $v_2$ lies on different sides of $C$, therefore cannot coincide.
The vertex $u_1=u_3$ is adjacent to $v_1$ and $v_3$ and can be connected to $u_2$, therefore has degree at least 3.
Notice that $u_2$ and $v_1$ might coincide.
In that case they have degree at least 4.
Therefore, there are always at least six vertices $u_i$, $v_i$, with multiplicity,
which
have degree at least $3$, so the isolated vertex $x$ gets weight at least $1$.
This concludes the proof.
\end{proof}
We recall that $s^3_2(n)$ denotes the the minimum number of edges of a saturated 3-simple 2-plane $n$-vertex drawing.
\medskip
\begin{proof}[Proof of {Theorem~\ref{2-3-simple}}]
We start with the lower bounds.
Let $$f(n)=
\left\{ \begin{array}{rl}
3 & \mbox{if} \ \ n=3 \\
\lfloor 3n/4\rfloor & \mbox{otherwise}
\end{array}\right.
$$
\begin{figure}\label{propel}
\begin{center}
\scalebox{0.25}{\includegraphics{2planar4.eps}\hspace{3cm} \includegraphics{2planar5.eps}}
\caption{A $3$-propeller and a $2$-propeller.}
\end{center}
\end{figure}
First we construct a saturated $2$-plane, $2$-simple topological graph
with $n$ vertices and $f(n)$ edges, for every $n$.
Let $k\ge 3$. A $k$-{\em propeller} is isomorphic to a star with $k$ edges as an abstract graph,
drawn as in Figure~\ref{propel}.
Clearly it is a saturated $2$-plane, $2$-simple topological graph with $k+1$ vertices,
$k$ edges and the unbounded cell is special.
For $n=1, 2, 3$, a complete graph of $n$ vertices satisfies the statement.
For $n\ge 4$, $n\equiv 0 \bmod 4$, consider $n/4$ disjoint $3$-propellers such that
each of them is in the unbounded cell of the others.
For $n\ge 4$, $n\equiv 1, 2, 3 \bmod 4$,
replace one of the propellers by an isolated vertex, a $K_2$, and a $4$-propeller, respectively.
This implies that $s_2^2(n)\le f(n)$.
Now we construct a saturated $2$-plane, $3$-simple topological graph
with $n$ vertices and $\lfloor 2n/3\rfloor$ edges, for every $n$.
A $2$-{\em propeller} is isomorphic to a path of $2$ edges as an abstract graph,
drawn as in Figure~\ref{propel}.
Clearly it is a saturated $2$-plane, $3$-simple topological graph with $3$ vertices,
$2$ edges and the unbounded cell is special.
For $n\equiv 0 \bmod 3$, take $n/3$ disjoint $2$-propellers such that
each of them is in the unbounded cell of the others.
For $n\equiv 1, 2 \bmod 3$, add an isolated vertex or an independent edge.
This implies that $s_2^3(n)\le \lfloor 2n/3\rfloor$.
\smallskip
We prove by induction on $n$ that $s_2^2(n)\ge \lfloor2n/3\rfloor$ and
$s_2^3(n)\ge \lfloor2n/3\rfloor$.
It is trivial for $n\le 4$. Let $n>4$ and assume that
$s_2^2(m), s_2^3(m)\ge \lfloor2m/3\rfloor$ for every $m<n$.
Let $G$ be a saturated $2$-plane,
$2$-simple or
$3$-simple drawing with $n$ vertices and $e$ edges.
We may assume again that every special cell contains an isolated vertex.
Suppose that $(u,uv)$ is an empty flag.
We remove $u$ from $G$.
Analogous to the proof of Theorem~\ref{n-1},
the obtained topological graph is saturated, it has
$n-1$ vertices and $e-1$ edges.
By the induction hypothesis, $e-1\ge \lfloor2(n-1)/3\rfloor$,
which implies that $e\ge\lfloor2n/3\rfloor$.
Therefore, we assume for the rest of the proof that
$G$ does not contain empty flags.
\begin{claim} \label{deg3}
If $(u, uv)$ is a flag,
then either $d(v)\ge 3$ or $u$ and $v$ are included in a $2$-propeller.
\end{claim}
\begin{proof}
Since $G$ does not contain empty flags,
there is a crossing on $uv$.
Let $p$ be the crossing on $uv$ closest to $u$, with edge $xy$.
There is no crossing on $xy$ between $p$ and one of the
endpoints, $x$ say, and $x\neq u$ by the assumptions.
We can connect $u$ to $x$ along the segments $up$ and $px$.
Since the drawing was saturated, $u$ and $x$ are adjacent in $G$.
Since $u$ has degree 1, $x=v$.
This implies $d(v)\ge 2$. We exclude parallel edges, so $y\neq u$.
Suppose that $d(v)=2$.
There is a crossing on the segment $py$ of $vy$, otherwise we could connect
$u$ to $y$ along the segments $up$ and $py$ contradicting the degree assumption of $u$.
Let $q$ be the crossing of $vy$ with edge $ab$.
There is no crossing on $ab$ between $q$ and one of the
endpoints, $a$ say.
If $a$ and $u$ are on the same side of edge $vy$ (that is, the directed
edges $\overrightarrow{ab}$ and $\overrightarrow{uv}$ cross the directed edge $\overrightarrow{vy}$ from the same side),
then we can connect $u$ to $a$ along the segments $up$, $pq$, $qa$.
Therefore $a=v$, so either $d(v)\ge 3$, or $b=u$, and edges $uv$ and $vy$ form a $2$-propeller. Note that this case is possible only if $G$ is $3$-simple.
So we may assume that $a$ is on the other side.
If $a=v$, then $d(v)\ge 3$, so we also assume that $a\neq v$.
Consider now the edge $uv$.
If there was no crossing on the segment $pv$ of $uv$, then we can connect $u$ to $a$ along $up$,
the segment $pv$ of $yv$, the segment $vp$ of $uv$, $pq$, and $qa$.
Therefore, there is a crossing on the segment $pv$ of $uv$.
Let $r$ be this crossing of $uv$ with edge $cd$,
and we can assume there is no crossing on the segment $cr$. (Here, $c$ or $d$ might coincide with $a$.)
If $c$ and $y$ are on the same side of $uv$ (that is, the directed
edges $\overrightarrow{vy}$ and $\overrightarrow{dc}$ cross the directed edge $\overrightarrow{vu}$ from the same side),
then we can connect $u$ to $c$ along $up$, $px$, $xr$, $rc$, which means that $c=v$, so $d(v)\ge 3$.
If $c$ and $y$ are on the opposite sides of $uv$, then
we can connect $c$ to $v$, so they are already connected.
Therefore, $c=y$.
However, we assumed that
$\overrightarrow{vy}$ and $\overrightarrow{dc}$ cross the directed edge $\overrightarrow{vu}$ from the opposite sides, so there is another
crossing of $uv$ and $vy$. If $G$ is $2$-simple, this is impossible and we are done. If $G$ is $3$-simple, then this crossing can only be $r$, so $c=y$ and $d=x$.
Now the edges $uv$ and $vy$ form a $2$-propeller.
\end{proof}
In a graph $G$, a connected component with at least two vertices is an {\em essential component}.
If $G$ has only one essential component, then $G$ is {\em essentially connected}.
\begin{claim} \label{ess}
We can assume without loss of generality that $G$ is essentially connected.
\end{claim}
\begin{proof}
Suppose to the contrary that $G$ has at least two essential components.
We define a partial order on the essential components of $G$:
$G_i\prec G_j$ if and only if $G_i$
lies in a bounded cell of $G_j$.
Let $G_1$ be a minimal element with respect to $\prec$ and let $G_2$
be the union of all other essential components.
There is a cell $C$ of $G$, which is bounded by both $G_1$ and $G_2$. Let $C$ correspond
to the cell $C_1$ of $G_1$ and cell $C_2$ of $G_2$.
By the definition of $G_1$, $C_1$ is the unbounded cell of $G_1$.
Since $G$ is saturated, at least one of $C_1$ or $C_2$ is a special cell, otherwise $G_1$ and $G_2$ can be connected.
For $i=1, 2$, let $H_i$ be the topological graph $G_i$ together with an isolated vertex in every special cell.
Let $n_i$ denote the number of vertices and $e_i$ the number of edges in $H_i$.
We notice $e=e_1+e_2$ and $n=n_1+n_2-1$ if exactly one of $C_1$ and $C_2$ is a special cell.
Also $n=n_1+n_2-1$ if both of them are special cells, since we can add 1 isolated vertex instead of 2.
By the induction hypothesis, we have $e_i\ge \lfloor2n_i/3\rfloor$,
so $e\ge\lfloor2n_1/3\rfloor+\lfloor2n_2/3\rfloor$,
and it is easy to check, that
for any $n_1, n_2\ge 2$,
$\lfloor2n_1/3\rfloor+\lfloor2n_2/3\rfloor\ge \lfloor2(n_1+n_2-1)/3\rfloor$.
Therefore,
$e\ge\lfloor2n_1/3\rfloor+\lfloor2n_2/3\rfloor\ge \lfloor2(n_1+n_2-1)/3\rfloor= \lfloor2n/3\rfloor$.
So, if $G$ is not essentially connected, then we reduce the problem and proceed by induction.
\end{proof}
Assume the $3$-simple $2$-planar drawing $G$ has a flag $(u, uv)$.
If $d(v)=1$, then $G$ is isomorphic to $K_2$ and the theorem holds.
If $d(v)=2$, then $G$ contains a $2$-propeller $u,v,w$ by Claim \ref{deg3}.
Since $G$ is essentially connected, but there is an isolated vertex in every special cell, there is
an isolated vertex $x$ in the special cell of the $2$-propeller.
Therefore, if $d(v)=2$ and $d(w)=1$, then $G$ is isomorphic to a $2$-propeller plus an isolated vertex and we are done.
If $d(v)=2$ and $d(w)>1$, then remove the $2$-propeller on $u,v,w$ and $x$ and use induction
\footnote{The vertex $w$ remains in the drawing with one incident edge removed.}, since we removed
3 vertices and 2 edges.
In the rest of the proof,
we assume that every leaf of $G$ is adjacent to a vertex of degree at least $3$,
and there is no $2$-propeller subgraph in $G$.
We give weight $3/2$ to every edge.
We discharge the weights to the vertices
and show that either every vertex gets weight at least 1,
or we can prove the lower bound on the number of edges by induction.
Let $uv$ be an edge.
Vertex $u$ gets $1/d(u)$ weight and $v$ gets $1/d(v)$ weight from $uv$.
Every edge has a non-negative remaining charge.
If $uv$ is a special edge, then it gives the remaining charge to the isolated vertex lying in its special cell
\footnote{here we use the proof of Claim~\ref{only1}}.
After the discharging step any vertex $x$ with $d(x)>0$ gets charge at least 1.
Now let $x$ be an isolated vertex, its special cell being $C$.
We distinguish several cases.
\smallskip
Case 1. The special cell $C$ has two sides.
Let $u_1v_1$ and $u_2v_2$ be the bounding edges.
They cross twice, in $p$ and $q$ say,
so there are no further crossings on $u_1v_1$ and $u_2v_2$.
The four endpoints are either distinct,
or two of them $u_1$ and $u_2$ might coincide, if $G$ was $3$-simple.
Suppose the order of crossings on the edges is $u_ipqv_i$, for $i=1,2$.
If the vertices $u_1$ and $u_2$ are distinct, then they can be connected along $u_1p$ and $pu_2$. Therefore, $u_1$ and $u_2$ are either adjacent or coincide in $G$.
Similarly, $v_1$ and $v_2$ are also adjacent.
Therefore, all four endpoints have degree at least $2$, and both $u_1v_1$ and $u_2v_2$ give
at most charge $1/2$ to its endpoints.
Their remaining charges are at least $1/2$, so $x$ gets at least charge $1$.
\smallskip
Case 2. The special cell $C$ has $3$ sides.
Let $u_1v_1, u_2v_2, u_3v_3$
be the bounding edges in clockwise order, and the $u$- and $v$-vertices alternate.
If none of the bounding edges is a flag,
then we are done since each
of those edges give weight at least $1/2$ to $x$.
Suppose that $u_1$ is a leaf.
We can connect $u_1$ to $v_2$ along segments of the edges $u_1v_1$ and $u_2v_2$.
Since $u_1$ is a leaf and the drawing was saturated, $u_1$ and $v_2$ are adjacent, consequently
$v_1=v_2$. Similarly,
we can connect $u_1$ to $v_3$, so $v_1=v_2=v_3$.
If $u_2$ is not a leaf, then $u_1v_1$ and $u_3v_3$ both give at least $1/6$
to $x$, and $u_2v_2$ gives at least $2/3$, so we have charge at least $1$ for $x$.
The same applies if $u_3$ is not a leaf.
So assume that $u_1$, $u_2$ and $u_3$ are all leaves.
If there are no other edges in $G$, then $G$ is a $3$-propeller and an isolated vertex.
That is, $n=5$ and $e=3$ and the required inequality holds.
Suppose that there are further edges. By Claim~\ref{ess}, $G$ is essentially connected. Since
$u_1$, $u_2$, $u_3$ are leaves, $v_1$ is a cut vertex.
Let
$H_1=G\setminus \{x, u_1, u_2, u_3\}$.
The induced subgraph $H_1$ has $n-4$ vertices and $e-3$ edges, and it is saturated.
Therefore, by the induction hypothesis, $e-3\ge f(n-4)$. Notice that $f(n)\le f(n-4)+3$,
consequently $e\ge f(n)$.
\smallskip
Case 3. The special cell $C$ has at least $4$ sides.
Each edge gives at least $1/6$ charge to $x$ by Claim~\ref{deg3}.
If an edge is not a flag, then it gives at least $1/2$ charge to $x$.
If there is at least one non-flag bounding edge, we are done.
For $k\ge 4$, let $u_1v_1, \ldots, u_kv_k$ be the bounding edges of $C$ in clockwise order.
Suppose that each edge $u_iv_i$ is a flag
(that is, $d(u_i)$ or $d(v_i)$ is $1$).
We may also assume that $u_1$ is a leaf.
Now, as in the previous case, we can argue that $v_3=v_2=v_1$.
It implies $u_2$ and $u_3$ are leaves, and by the same argument, $v_5=v_4=v_3=v_2=v_1$.
We can continue and finally we obtain that
all $v_i$ are identical and all $u_i$ are leaves. So the vertices
$u_i, v_i$ $1\le i\le k$ form a star, and they have the same crossing pattern as a $k$-propeller.
Therefore, $u_i, v_i$ $1\le i\le k$ span a $k$-propeller.
We can finish this case exactly as Case 2.
If there are no further edges in $G$, then the graph is a $k$-propeller and an isolated vertex.
That is, $n=k+2$ and $e=k$ and the inequality holds.
If there are further edges,
then $v_1$ is a cut vertex, and we can apply induction.
This concludes the proof of Theorem~\ref{2-3-simple}.
\end{proof}
|
1,116,691,500,783 | arxiv | \section{Introduction}
Observations of standard candles and of standard rulers allow us to
learn how the universe has evolved in recent times. Both sets of observations favour
a universe containing a large component of dark energy but the
cosmologies preferred by the two sets of data are not consistent; in
fact at higher redshifts they appear to diverge. This indicates that
there is still something missing in our understanding of the
accelerated expansion of the universe. In this paper I will show how
the discrepancy between the two sets of observations can be explained
if photon number is not conserved because photons are allowed to mix
with chameleon particles.
In metric theories of gravity where photons
travel on unique null geodesics there is a well-defined notion of distance as
long as photon number is conserved.
The luminosity distance, $d_L(z)$, to an object at redshift $z$ and the angular diameter
distance, $d_A(z)$, to the same object are related by the reciprocity
relation \cite{Etherington}
\begin{equation}
\label{duality}
d_L(z)=d_A(z)(1+z)^2
\end{equation}
If photon number is not conserved then this relation no longer
holds and a discrepancy in the
distance measures should be observed. There are a number of current
proposals which do not conserve photon number; the light from distant objects could be scattered
by dust or free electrons in the intergalactic medium, photons may
decay, or the photon
could mix with another light state such as the axion.
It had been thought that loss of photons in the intergalactic medium could explain the observed
dimming of type 1a supernova without the need for dark
energy \cite{Csaki:2001yk,Song:2005af,Kaloper:2007gq}. Photon number non-conservation changes the luminosity distance
to an object but does not affect distance measurements from standard
rulers. Therefore, if loss of photons is the explanation for the
dimming of supernova, angular diameter distance measurements should be consistent with an $\Omega_{\Lambda}=0$ universe.
This is not the case; observations of standard rulers imply a universe with a significant dark energy
component.
\begin{figure}
\begin{center}
\includegraphics[width=14cm,height=8cm]{notchameleon.eps}
\caption{Distance modulus [$m-M=5\log(d_L)-25=5\log(d_A(1+z)^2)-25$]
compared to a $\Lambda$CDM universe plotted against redshift,
showing the best fits to observations of standard candles and
standard rulers.}
\label{fig:DeltaM}
\end{center}
\end{figure}
When luminosity distances and angular diameter distances are compared there is a noticeable
disagreement at $z \gtrsim0.5$ \cite{Bassett:2003vu,Bassett:2004}. This is shown in Figure
\ref{fig:DeltaM} where distance modulus (compared to a $\Lambda$CDM
universe) is plotted against redshift for measurements
coming from standard rulers and standard candles. The standard
candles used are type 1a supernovae
\cite{Riess:2004nr,Tonry:2003zg,Barris:2003dq,Knop:2003iy} and the
standard rulers used are FRIIb radio galaxies \cite{Daly:2003iy,Daly:2002kn}, compact radio
sources \cite{Gurvitis,Jackson:2006ib,Jackson:2003jw} and X-ray
clusters \cite{Allen:2002eu}. Considering the two sets of data
separately the best fit
to the angular diameter distance data is a universe with $\Omega_m=0.22$,
$\Omega_{\Lambda}=0.79$ but the best fit to the luminosity distance data is a very closed universe
with $\Omega_m=0.46$, $\Omega_{\Lambda}=0.98$, such a universe is already ruled out by
measurements of the CMB. If the explanation for this apparent
violation of the reciprocity relation is that photon number is not
conserved, then computed luminosity distances need to be modified.
Photon number non-conservation does not affect the angular diameter
distance to an object so the cosmology which best fits observations of
standard rulers is the correct one. If this is the case Figure \ref{fig:DeltaM}
implies that non-conservation of photon number actually brightens the
image of a supernova. This is in contrast to all previously studied
mechanisms for non-conservation of photon number which all predict
that photons should be lost in the intergalactic medium, and thus that
the supernova is dimmed \cite{Bassett:2003vu,Bassett:2004}.
In this paper I show that the images of supernova can be brightened if
photons are allowed to mix with chameleons.
The chameleon is a scalar particle which arises in certain models of
scalar-tensor gravity \cite{Khoury:2003rn,Khoury:2003aq}. In the
Einstein frame the action is
\begin{eqnarray}
S&=&\int d^4x\sqrt{-g}\left(\frac{M_P^2R}{2}-\frac{1}{2}(\partial
\phi)^2-V(\phi)\right)\nonumber\\
& &-\int d^4x \mathcal{L}_m(\psi_m^{(i)},g_{\mu,
\nu}^{(i)})
\end{eqnarray}
where $\psi_m^{(i)}$ are matter fields which couple to the metric $g^{\mu
\nu}_{(i)}=e^{-2\phi/M_i}g^{\mu \nu}$. Assuming that the
chameleon couples to all forms of matter in the same way so that
$M_i=M$ $\forall i$ the current bound on $M$ is $10^6\mbox{
GeV}<M$ \cite{Mota:2006fz}. I assume a runaway potential of the form
\begin{equation}
\label{pot}
V(\phi)=\Lambda^4e^{\Lambda^n/\phi^n}
\end{equation}
but the chameleon feels an effective potential
\begin{equation}
V_{eff}(\phi)=V(\phi)+\rho e^{\phi/M}
\end{equation}
which depends on the local matter density $\rho$. The chameleon sits
in the minimum of the effective potential with a mass which depends
on the local matter density. In high
density regions such as on earth the chameleon becomes very massive
and so evades current experimental bounds, but in low density regions
it can become almost massless. The way in which the scalar field can
change its mass is known as the `chameleon mechanism' and this
allows the scalar field to evade all current searches for
fifth-force effects or violations of the equivalence principle if $\Lambda \sim 10^{-3}\mbox{ eV}$. In addition it has been shown that the chameleon is compatible with recent searches for
axion like particles at PVLAS and CAST
\cite{Brax:2007hi,Brax:2007ak}.
As the universe evolves the local energy density falls and the
chameleon changes its properties as the universe cools. In particular
at early times the chameleon behaves like
dust but as the universe cools below temperatures $T \approx 10\mbox{
MeV}$ the chameleon begins to behave like a fluid with equation of state
$\omega=-1$ \cite{Brax:2004qh,Brax:2004px}.
In this way the chameleon is a natural
explanation for dark energy.
The paper is organised as follows. Section \ref{homogeneous} shows
how photons mix with
chameleon particles and gives the probability of a photon converting into a
chameleon in a homogeneous magnetic field. Section \ref{varying} discusses the
probability of conversion as the particles pass through a magnetic
field made up of many randomly oriented domains and section
\ref{supernova} explains how photons can be converted into chameleons
in the interior of a supernova, resulting in a flux of chameleons at
the surface. Then Section \ref{sec:brightening} describes how
this can account for the brightening of the image of the
supernova. In Section \ref{CMB} I show that chameleon-photon mixing is in agreement with observations of the CMB. I
conclude in section \ref{conc}.
\section{Photon-Chameleon Mixing}
\label{homogeneous}
In
the presence of a magnetic field the chameleon couples to
photons. Assuming that the coupling of the chameleon to matter is
universal, the interaction term in the Lagrangian is
\begin{equation}
\mathcal{L}_{int} =\frac{\phi B^2}{M}
\end{equation}
In a homogeneous
magnetic field where the particles propagate along the $x$ axis which is aligned in the direction of the magnetic field the equations governing
the evolution of the fields \cite{Raffelt:1987im} are
\begin{equation}
\left[\omega^2 +\partial^2_x+\left(\begin{array}{ccc}
-\omega_p^2 & 0 & 0\\
0 & -\omega_p^2 & \frac{B\omega}{M}\\
0 & \frac{B\omega}{M} & -m_c^2
\end{array}\right)\right]\left(\begin{array}{c}
A_{\parallel}\\
A_{\perp}\\
\phi
\end{array}\right)=0
\end{equation}
where $\omega$ is the frequency of the photons and $B$ is
the magnitude of the magnetic field. As photons and chameleons propagate through the
intergalactic medium they pass through a plasma of ionised electrons
\cite{Csaki:2001jk}. $\omega_p$ is the
frequency of this plasma; $\omega_p^2=4\pi \alpha n_e/m_e\approx
10^{-47}\mbox{ GeV}^2$, where $n_e$ is
the electron number density and $m_e$ is the electron mass.
$A_{\parallel}$ and $A_{\perp}$ are the polarisations of the photon
parallel and perpendicular to the magnetic field and $\phi$ is the chameleon
field. Note that the
chameleon only mixes with the polarisation of the photon which is
orthogonal to the magnetic field. $m_c$ is the chameleon mass which, given the potential (\ref{pot}), is
\begin{equation}
m_c^2=n(n+1)\frac{\Lambda^{4+n}}{\phi_{min}^{2+n}}
\end{equation}
where
\begin{equation}
\phi_{min}=\left(\frac{n\Lambda^{4+n}M}{\rho}\right)^{\frac{1}{1+n}}
\end{equation}
is the value of $\phi$ at the minimum of the effective potential. In
what follows I will assume $n=\mathcal{O}(1)$.
The probability of an orthogonally polarised photon converting into a chameleon whilst
travelling a distance $x$ through this homogeneous field is
\begin{eqnarray}
P(x)&=&\frac{4\omega^2B^2}{M^2(\omega_p^2-m_c^2)^2+4\omega^2B^2}\nonumber\\
& &\times\sin^2\left(\frac{x\sqrt{M^2(\omega_p^2-m_c^2)^2+4\omega^2B^2}}{4\omega
M}\right)
\label{PIGM}
\end{eqnarray}
The observed dimming of type 1a
supernova is achromatic. hence if chameleon-photon mixing modifies
observations of supernovae the effect must also be achromatic. If the
particles pass through a magnetic domain of size $L_{dom}$ there are two regimes in which the conversion
probability (\ref{PIGM}) is independent of frequency: In the limit of high energy photons $M|\omega_p^2-m^2| \ll B\omega$ the
mixing is maximal and independent of the photon energy.
\begin{equation}
\label{probIGM}
P\approx \sin^2\left(\frac{L_{dom}B}{2M}\right)
\end{equation}
Alternatively if the oscillation length
\begin{equation}
L_{osc}=\frac{4\pi \omega
M}{\sqrt{M^2(\omega_p^2-m_c^2)^2+4\omega^2B^2}}
\end{equation}
is much greater that the size of a magnetic domain $2\pi L_{dom} \ll
L_{osc}$ then the probability of conversion (\ref{PIGM}) is
\begin{equation}
\label{probS}
P \approx \frac{B^2L_{dom}^2}{4M^2}
\end{equation}
which is independent of photon energy. It should be noted that both
(\ref{probIGM}) and (\ref{probS}) are also independent of the mass of the
chameleon.
Current observations suggest that the
intergalactic magnetic field has coherence length $L_{dom} \sim
1\mbox{ Mpc}$ and magnitude $B\lesssim
10^{-9}\mbox{ G}$
\cite{Kronberg:1993vk,Blasi:1999hu,Jedamzik:1999bm}. The
density of the intergalactic medium is $\rho_{IGM} \sim
10^{-44}\mbox{ GeV}^4$ so the mass of the chameleon is
$m_c^2 \lesssim 10^{-45}\mbox{ GeV}^2$. An optical photon $\omega \approx
10\mbox{ eV}$ passing through the
intergalactic medium is in the high energy regime if
\begin{equation}
M\lesssim 10^{10}\mbox{ GeV}
\end{equation}
so that the
probability of conversion is given by (\ref{probIGM}).
For this probability to be small requires $BL_{dom}<2M$.
\subsection{Conversion in a Varying Background}
\label{varying}
The intergalactic magnetic field is not homogeneous on large scales; it is made up of many randomly orientated
magnetic domains. To calculate the probability of a photon converting into
a chameleon whilst travelling through the intergalactic medium I
assume that: Particles
traverse $N$ domains of equal length, $B$ is homogeneous in each
domain and there is a discrete change in $B$ from one domain to
another. The component of the magnetic field parallel to the
direction of flight has a random orientation but equal size in each
domain.
The initial state is
\begin{equation}
\alpha_1(0)|\gamma_1\rangle +\alpha_2(0)|\gamma_2\rangle +\alpha_c(0)|c\rangle
\end{equation}
where $|\gamma_i\rangle$ are the photon states parallel and perpendicular to
the magnetic field in the first domain and $|c\rangle$ is the
chameleon. The initial photon and
chameleon fluxes are
\begin{eqnarray}
I_{\gamma}(0) &\sim& |\alpha_1(0)|^2+|\alpha_2(0)|^2\\
I_{c}(0) &\sim& |\alpha_c(0)|^2
\end{eqnarray}
In the n-th domain the magnetic field is tilted by an angle $\theta_n$
compared to the first domain, so that
\begin{eqnarray}
|\gamma^n_{\parallel}\rangle&=&\cos\theta_n|\gamma_1\rangle+\sin\theta_n|\gamma_2\rangle\\
|\gamma^n_{\perp}\rangle&=& -\sin\theta_n|\gamma_1\rangle+\cos\theta_n|\gamma_2\rangle
\end{eqnarray}
and the transition probability $P$ in each domain is given by (\ref{PIGM}).
Assuming
that $P$ is small and that
$\theta_n$ is a random variable so that on average $\cos^2\theta_n \sim
\sin^2\theta_n \sim 1/2$ then at the
end of the n-th domain
\begin{eqnarray}
I_c(y) &=&
\frac{1}{3}(I_c(0)+I_{\gamma}(0))+\frac{Q(y)}{3}(2I_c(0)-I_{\gamma}(0))\label{Ic}\\
I_{\gamma}(y) &=&
\frac{2}{3}(I_c(0)+I_{\gamma}(0))+\frac{Q(y)}{3}(I_{\gamma}(0)-2I_c(0))\label{Igamma}
\end{eqnarray}
where
\begin{equation}
\label{Q}
Q(y)=\left( 1-\frac{3P}{2}\right)^{y/L_{dom}}
\end{equation}
and $y(z)$ is the
proper particle distance to the astronomical object. If $P$ is small then at large distances $Q(y)$ becomes
exponentially small and the system reaches an equilibrium
configuration with a third of the initial flux in chameleons and two
thirds in photons.
The probability of a single photon converting to a chameleon in a distance
$y$ is
\begin{eqnarray}
P_{\gamma \rightarrow c}(y)&=&\frac{1}{3}(1-Q(y))\\
&\lesssim & \frac{y^2B^2}{8M^2N}
\label{totalprob}
\end{eqnarray}
It can be seen from (\ref{Ic}), (\ref{Igamma}) that if the ratio of the initial chameleon flux to the initial photon
flux is large enough more photons will be received than were emitted; the image of the supernova is brightened.
\subsection{Conversion in the Supernova}
\label{supernova}
A flux of chameleons is emitted by a supernova if some photons are
converted into chameleons in the interior of the supernova.
A type 1a supernova is thought to be the thermonuclear explosion of
a white dwarf whose mass is close to the Chandrasekhar limit. To
compute the probability of conversion between photons and chameleons
inside the supernova I consider a simple model: The supernova is a sphere of uniform density with initial
radius $R_0 \sim 10^9\mbox{ cm}$. The supernova expands with outer velocity
$v=c/30 \sim 10^9 \mbox{ cm/s}$. I also
assume that the size of a magnetic domain in the
supernova is roughly
equal to the length of the mean free path of the photons $L_{dom}
\approx L_{mfp}$. I assume that only photons are produced by the reactions driving the
explosion of the supernova and that the photons are emitted uniformly
throughout the volume of the supernova. Peak luminosity occurs about
10 days after the start of the explosion.
The explosion of a supernova is
homologous and so the magnetic field obeys
\begin{equation}
\frac{B_{SN}(t)}{B_{WD}}=\left(\frac{R_{WD}}{R_{SN}(t)}\right)^2
\end{equation}
where $B_{SN}(t)$ and $R_{SN}(t)$ are the magnetic field and radius of the
supernova at time $t$ after the start of the explosion and $B_{WD}$ and
$R_{WD}$ are the magnetic field and radius of the initial white dwarf.
Models of the magnetic field of a white dwarf vary, but all
predictions lie in the range $10^5\mbox{ G} \lesssim B_{WD} \lesssim
10^{11}\mbox{ G}$. The size of
the mean free path of photons at peak luminosity of the supernova is
also not well known but is expected to satisfy $10^6\mbox{ cm}
\lesssim L_{mfp} \lesssim 10^{14}\mbox{ cm}$.
At peak luminosity the mean free path of photons is much smaller than
the radius of the supernova and the path of the photons can be modelled
as a random walk so that it takes the photon
$N=3R^2/L_{mfp}^2$ steps to escape from a region of radius $R$.
Inside the supernova the chameleon is more massive than in the
intergalactic medium, however because
the oscillation length of optical
photons is much greater than the coherence length of the magnetic
field the oscillations are still independent of frequency (\ref{probS}). The probability of
a photon converting into a chameleon inside the supernova is thus
\begin{eqnarray}
P_{\gamma \rightarrow c}(R_{SN}) &\lesssim& \frac{3B_{SN}^2R_{SN}^2}{8M^2}\\
& \lesssim& 9.4\times 10^{32}
\left(\frac{B_{WD}^2}{\mbox{GeV}^2M^2}\right)
\end{eqnarray}
Therefore even though only photons are produced by the
thermonuclear reactions in the supernova, the relatively high
probability for a photon to
convert into a chameleon in the interior of the supernova means
that there is a significant flux of chameleons at the surface of the supernova.
In \cite{Grossman:2002by} the possibility of producing a flux of axions from the
supernova was considered in a similar way to that described above. In the photon-axion coupling
model $M\approx 10^{11}\mbox{ GeV}$ which means that the probability
of conversion in the supernova is negligible. The chameleon
mechanism, which changes the mass of the scalar field, means that the
experimental constraints on $M$ in the chameleon model are less
severe. $M$ can be much smaller than in the axion model and therefore
the probability of conversion can be much higher.
\section{Supernova Brightening}
\label{sec:brightening}
If there is a flux of chameleons at the surface of the supernova, the
ratio of the flux of photons received on earth to the flux of photons
leaving the supernova is
\begin{eqnarray}
P_{\gamma \rightarrow
\gamma}(y)&=&\frac{I_{\gamma}(y)}{I_{\gamma}(0)}\nonumber\\
&=&\frac{2}{2+(1-\frac{3}{2}P_{SN})^N}+Q(y)\left(\frac{(1-\frac{3}{2}P_{SN})^N}{2+(1-\frac{3}{2}P_{SN})^N}\right)\label{brightening}
\end{eqnarray}
where $y=0$ is now the surface of the supernova. $P_{SN}$ is the
probability of conversion in one domain in the supernova and $Q(y)$ is
given in (\ref{Q}) with $P=P_{IGM}$ the probability of conversion in
one domain in the intergalactic magnetic field.
If photon number is not conserved in the intergalactic medium then the reciprocity relation between
luminosity distance and angular diameter distance
(\ref{duality}) must be modified by sending $d_L
\rightarrow d_L/\sqrt{P_{\gamma \rightarrow \gamma}}$.
Writing the photon survival probability as
\begin{equation}
\label{form}
P(z)=A+(1-A)e^{-y(z)H_0/c}
\end{equation}
where $A$ and $c$ are real constants. I consider the effect of
chameleon-photon mixing on predictions for supernova observations. As
chameleon-photon mixing has no effect on the angular diameter
distance I assume that measurements of standard rulers give the correct relation between distance and
redshift. It is possible to fit the
observed supernova well if $c<0$. The example in Figure
\ref{fig:brightening2} has $1-A=7\times 10^{-5}$ and $c=-0.056$. However
$c<0$ corresponds to $\ln|1-3P_{IGM}/2|>0$ which is not possible
within this model. If $c>0$ the tension between the $d_A$ and $d_L$
measurements can be eased by photon-chameleon mixing. Suitable values
of the parameters are $A=1.271$ and $c=1.10$ and the resulting
prediction for the supernova is shown in Figure \ref{fig:brightening}. For $z\lesssim 0.5$
the prediction is close to a $\Lambda$CDM universe and for larger $z$
there is a constant brightening from the radio galaxy data. Clearly
more observations of high redshift supernova
or smaller error bars would significantly improve the constraints on
our model.
\begin{figure}
\begin{center}
\includegraphics[width=14cm,height=8cm]{chameleon2.eps}
\caption{ Assuming standard rulers give the correct relation between
$\Delta(m-M)$ and $z$, the solid line shows the prediction for
observations of type 1a supernova if the probability of photon
survival has the form (\ref{form}) with
$A= 0.99993$ and $c= -0.056$.}
\label{fig:brightening2}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=14cm,height=8cm]{chameleon.eps}
\caption{Assuming standard rulers give the correct relation between
$\Delta(m-M)$ and $z$, the solid line shows the prediction for
observations of type 1a supernova if the probability of photon
survival has the form (\ref{form}) with
$A= 1.271$ and $c= 1.10$.}
\label{fig:brightening}
\end{center}
\end{figure}
The values of $A$ and $c$ used in Figure \ref{fig:brightening} correspond to $P_{SN}N \approx 0.95$
and $P_{IGM} \approx 10^{-4}$ where
\begin{equation}
P_{SN}N \approx \frac{3B_{WD}^2R_{WD}^4}{4M^2R^2_{SN}}, \;\;\;\;\;
P_{IGM} \approx \frac{B_{IGM}^2L_{dom}^2}{4M^2}
\end{equation}
This implies the following relations between
parameters
\begin{equation}
B_{WD}\sim 10^3\left(\frac{M}{\mbox{GeV}}\right)\mbox{ G},\;\;\;\;\;
B_{IGM}\sim 10^{-20}\left(\frac{M}{\mbox{GeV}}\right)\mbox{ G}
\end{equation}
which are consistent with all current experimental bounds.
In these calculations I have neglected the effect of the galactic magnetic field. The
probability of conversion in one domain (\ref{probIGM}) depends on the combination
$BL_{dom}$. The galactic magnetic field has strength $B_G \approx
10^{-5}\mbox{ G}$ and coherence length $L_{dom}=100\mbox{ pc}$ so
that $B_GL_{dom}\approx 10^9\mbox{ GeV}$. This is
the same order of magnitude for $BL_{dom}$ as for an intergalactic magnetic field
with $B_{IGM}\approx10^{-9}\mbox{ G}$ and coherence length
$L_{dom}\approx 1\mbox{ Mpc}$. So in this case passing from the
intergalactic magnetic field to the galactic magnetic field does not
affect the probability of conversion. If $B_{IGM} \ll 10^{-9} \mbox{
G}$ then there will be a change in the
probability of conversion when the photons move into the galaxy,
however the distance travelled through the galaxy is small compared to
the total distance from the supernova so the effects of the galaxy
will at most be a small correction on the above result.
\section{CMB Photons}
\label{CMB}
If
microwave photons from the CMB oscillate into chameleon states in the intergalactic magnetic
field their anisotropy could be large due to variations in the
magnetic field and thus disagree with observations. The frequency of a photon from the CMB is $\omega \sim
10^{-4}\mbox{ eV}$ so microwave photons fall into the low energy
regime $M|\omega_p^2-m^2| \gg B\omega$ where the mixing is small
and the probability of chameleon-photon mixing is bounded by
\begin{eqnarray}
P &\leq&
4B^2\omega^2/M^2(\omega_p^2-m^2)^2\\
&\lesssim& 10^{-6}
\end{eqnarray}
for all allowed
values of $M$. The anisotropy sourced by one magnetic domain is less
that the primordial CMB anisotropy $\Delta T/T \sim 10^{-5}$.
If intergalactic magnetic fields are a relatively
recent phenomenon and only exist out to redshifts of a few then the photons from the
CMB have not travelled far enough through a magnetic field to reach their
equilibrium configuration. The probability of conversion is not significantly
enhanced by travelling through the many magnetic domains in the
intergalactic medium and therefore chameleon-photon mixing does not conflict with
observations of the CMB. The effect on the CMB of axion-photon mixing
was considered in \cite{Mirizzi:2005ng} but because we do not require a
large probability of conversion in the intergalactic medium we are
able to avoid the strict bounds that observations of the black body
spectrum of the CMB put on the axion photon mixing model.
An observation of a primordial magnetic field would put
severe constraints on the chameleon model because then photons from
the CMB would travel for large distances through a magnetic field.
Conversely if the existence of chameleons is demonstrated, observations of the CMB would put strict bounds on the existence of a primordial magnetic field.
\section{Conclusions}
\label{conc}
The chameleon model provides an explanation for the observed
accelerated expansion of the universe. Observations of
standard candles and standard rulers do not give a consistent picture
of this acceleration which implies that the reciprocity relation
(\ref{duality}) does not hold. This can be explained if the chameleon couples to photons in the presence of
a magnetic field. The strong magnetic field inside a supernova means
that chameleons and photons mix in the interior of the
supernova and so there is a flux of chameleon particles at the surface of the
supernova. This is in contrast to axion-photon coupling which does
not allow for a flux of axions at the surface of the supernova. Photons and chameleons also mix as they travel to earth
through the intergalactic magnetic field.
This requires $M\lesssim10^{10}\mbox{ GeV}$ to ensure the effect of
the mixing
is achromatic for optical photons. A chameleon model with a coupling
of this strength should be detectable in future experiments looking for
chameleonic afterglow \cite{Gies:2007su,Ahlers:2007st} or Casimir
forces.
The overall effect of mixing between photons and chameleons
is that observers on earth see a brightened image of the supernova.
The brightening of supernovae eases the tension between
observations of standard candles and standard rulers. Future observations of high redshift
supernova will significantly improve the constraints on this model.
\section*{Acknowledgements}
I would like to thank A.C. Davis, P. Brax and D. Shaw for very useful
discussions. This work was supported by STFC.
|
1,116,691,500,784 | arxiv | \section{Introduction}
Tversky and Shafir (1992)~\cite{tversky1992choice} discovered a phenomenon called the {\it disjunction effect}, while following the process of testing a rational axiom of decision theory. It is also called as the sure thing principle (Savage, 1954)~\cite{savage1954foundations}. First consider the states $A$ and $B$ that belong to the state of the world $X$. This principle states that, if action A over B is preferred, and under the complementary state of the world, again, action A over B is preferred, it is expected that one should prefer action A over B even when the state of the world is not known. Symbolically, this can be expressed as
If $${P(A\cap X) > P(B \cap X)} $$ and $${P(A\cap X^C) > P(B \cap X^C)}$$
Then $$ P(A) = P(A \cap (X\cup X^C) > P(B \cap (X \cup X^C) = P(B)$$
i.e. $$ P (A) > P (B) $$ occurs always.
With the aim of testing this principle, in their experiment, Tversky and Shafir (1992)~\cite{tversky1992choice} performed the test considering a two stage gamble by presenting $98$ students.
They adopted a two stage gamble, i.e., it is possible to play the gamble twice. The gamble is done under the following two conditions:
\begin{itemize}
\item The students are informed that they lost the first gamble.
\item The students remained unaware of the outcome of the first gamble.
\end{itemize}
The gamble to be played, had an equal stake, i.e., of wining ${200}$ or loosing ${100}$ for each stage of taking decision, i.e., whether to play or not to play the gamble.
Interestingly, the results of these experiments can be described into the following manner:
\begin{itemize}
\item The students who won the first gamble 69\% choose to play at the second stage;
\item The students who lost then 59 \% choose to play again;
\item The students who are unaware whether they won or lost 36\% of them (i.e., less of the majority of the students) choose to play again.
\end{itemize}
Explaining the findings in terms of choice based on reasons, Tversky and Shafer (1992)~\cite{tversky1992choice} did raise some questions for these surprising results: whenever, the persons, related to the play, knew that if they win, then as they would have extra house money, they can play again, because if they lose, they can play again to recover the loss. Now the students who did not know the outcome, then the main issue is why sizable fraction of the students want to play again the game since either they win or lose and cannot be anything else? Thus they arrived at the key result, but faced the problem of explaining the outcome just as either a win or loss. Busemeyer, Wang and Townsand (2006)~\cite{busemeyer2006quantum} originally suggested that this finding could be an example of an interference effect, similar to that found in the double slit experiments conducted in modern particle physics.
Let us consider the following analogy between the disjunction experiment and the classic double slit type of experiment in physics:
Both the cases involve two possible paths: here, in the disjunction experiment, the two paths are inferring the outcome of either a {\it win} or {\it a loss} with the first gamble; for the double split experiment, the two paths are splitting the photon into the upper or lower channel applying a beam splitter. The path taken can be known (observed) or unknown (unobserved), in both the experiments. Finally in both the cases, the fact is that when the case of gambling for disjunction experiment and hence, the detection at a location for the two slit experiments are considered for the chosen unknown, i.e., unobserved conditions, the resultant probability, meant for observing interference phenomena, are found to be much less than each kind of the probabilities which is observed for the known (observed) cases. Under these circumstances, we can speculate that during the disjunction experiment, under the unknown condition, instead of being definite, so far as the win or loss state is concerned, the student enters a superposition state. In fact, this state prevents finding a reason for choosing the gamble. In double-slit experiments, the law of additivity of probabilities of occurring two mutually exclusive events (particle aspect or wave aspect) is violated i.e. total probability
$$P_{AB} = P_A + P_B$$
for two mutually exclusive events A and B.
This is due to the existence of interference effects, known as Formula of Total Probability (FTP). It has already been established fact that the two slit (interference) experiment is the basic experiment which violates FTP. Feynman (1951)~\cite{feynman1966feynman}, in many of his works in physics, presented his points with detailed arguments about this experiment. There, the results, i.e., the appearance of interference fringes appeared to him, not at all surprising phenomena. He explained it as follows:
In principle, interaction with slits placed on the screen may produce any possible kind of distribution of points on the registration screen. Now, let us try to explain following quantum probabilistic features which appear only when one considers following three kind of different experiments~\cite{conte2009mental}:
\begin{itemize}
\item When only the first slit open i.e., the case $B=+1$, in an equivalent manner.
\item When only the second slit is open i.e., $B= -1$, in this case.
\item In the particular case, both slits being open, it is the random variable B determining the slit to pass through.
\end{itemize}
At this stage, let us now choose any point at the registration screen. Then resultant scenario will be as follows:
\begin{itemize}
\item the random variable A if $A= +1$.
\item the opposite case happens if a particle hits the screen at this point, i.e., $A= -1$.
\end{itemize}
Now for classical particles, FTP should predict the probability for the experiment (both slits are open), supposed to be provided by the (1) and (2) experiments. But, it has already been mentioned that, in case of quantum particles, FTP is violated: for the additional cosine-type term appearing in the right-hand side of FTP, it is the interference effect in probabilities which is responsible. Feynman characterized this particular characteristic feature of the two slit experiment as the most profound violation of laws of classical probability theory. He explained it the following way:
In an ideal experiment, where there is no presence of any other external uncertain disturbances, the probability of an event, called probability amplitude is the absolute square of the complex quantity. But, when there is possibility of having the event in many possible ways, the probability amplitude is the sum of the probability amplitude considered separately. Following their experimental results, Tversky and Shafir (1992)~\cite{tversky1992choice} demand that this violation of classical probability is also possible to be present, happening in their experiments with cognitive systems. Though, due to the possible restrictions present in quantum mechanics, we could not start from Hilbert formalism at the start, in the laboratory, this formalism was justified by experiments.
Let us now try to interpret the meaning of interference effect within the context of the experiments on gambling, described above. We will follow, here, Busemayer's formulation (2011)~\cite{busemeyer2011quantum} which is as follows:
Two different judgment tasks A and B are considered in this case. The task A is considered having $j$ (taking $j= 2$, binary choice) different levels of response variable and B, with $k$ (say, $k= 7$ points of confidence rating) levels of a response measure. Two groups, being randomly chosen, out of the total participants we have:
\begin{itemize}
\item Group A gets task A only.
\item Group BA gets task B followed by task A.
\end{itemize}
The response probabilities can be estimated as follows:
\begin{itemize}
\item From the group A, let $p( A = j)$ be estimated; This denotes the probability of choosing level j out of the response to task A .
\item And then from the group B, let $p(B=k)$ be estimated; the corresponding probability denoted by choosing level k from the task B.
\item Now, it is possible to estimate the conditional probability $p (A=j|B=k)$; which can be stated as the probability of responding with level j from the task A, given the person responded with level k on earlier task B.
\end{itemize}
So, we can write the estimated interference for level j to task A (produced, when responding to task B) as,
$$ A (j) = p (A = j) â p_T (A = j)$$,
where, $p_T (A = j)$ denotes the total probability for the response to task A.
In defense of using Quantum formalism in case of human judgments, Busemeyer and Truebold did put four following reasons, beautifully, in their famous paper (Busemeyer et al., 2011~\cite{busemeyer2011quantum}):
{\it (a) judgment is not a simple readout from a preexisting or recorded state, but instead it is constructed from the current context and question; from this first point it then follows that (b) making a judgment changes the context which disturbs the cognitive system; and the second point implies that (c) changes in context produced by the first judgment affects the next judgment producing order effects, so that (d) human judgments do not obey the commutative rule of classic probability theory(Busemeyer et al; 2011~\cite{busemeyer2011quantum})}.
\noindent
In fact, the existence of interference term for microscopic entities or quantum entities clearly indicates the existence of three valued or non-Boolean logic. This is popularly known as Quantum logic. It is mathematically shown that a set of propositions which satisfies the different axiomatic structures for the non-Boolean logic generates Hilbert space structures. The quantum probability associated with this type of Quantum logic can be applied to decision making problems in cognitive domain. It is to be noted that, up till now, no quantum mechanical framework is taken as valid description of the anatomical structures and function of the brain. This framework of quantum probability is very abstract and devoid of any material content. So it can be applied to any branch of knowledge like Biology, Social science etc. Of course, it is necessary to understand the issue of contextualization, for example, here, in case of decision making in brain. It is worth mentioning that the decision making may depend also on culture. For example, the students participated in the above mentioned gamble are mainly taken from the west. So far as the authors' knowledge, concern of this kind of experiment has not been performed taking the subjects from the east. We presume that this is an important factor one should consider in this kind of experiment involving gambling since the very concept of gambling may depend on culture. While experimenting to find the effect of culture, we have found an additional interesting fact. It is not only culture but also gender that might play an important role on disjunction effect. It is natural to expect and consider that the gender dependency is closely dependent on culture. Recently, we performed the above mentioned two stage gambling considering almost the same sample size in India. The results clearly show that the violation of classical probability rule depends very much on the variation of gender. In the next section we will describe the experiment and the results.
\section{Materials and Methods}
In order to get an insight of the above discussion we have performed a gambling experiment in the line of Tvaersky and Shafir (1952). Our main objective is to see whether there is any disjunction effect, especially in Indian context.
\subsection{Participants}
As we believe that there might be a cultural effect, we carefully select the participants. The participants should consist of both males and females as cultural effect with respect to gender is sometimes significantly observable. We have selected the population of experimental objects as a homogeneous groups with respect to age so that there would not be any effect that can mask our objectives. From Raja Rajeswari Engineering College, Bengaluru, India, we select $101$ college students randomly. It is seen that there are $50$ female and $51$ male students participating the experiment. The students belong 1st, 2nd, or 3rd year of their engineering curriculum and naturally belong to a very homogeneous age group. Prior permission is taken from the principal of the college and the studetns gave consents to this experiment.
\subsection{Design and procedure}
In the experiment we toss a coin and the experiment depends on the outcome. We first divide students into two groups: one consisting $70$ students and the other $31$ students. We perform the experiment similar to that described in Busemeyer. If a students wins the first toss, he/she will receive Rs. $100$; if loses, he/she will lose Rs. $50$. After the first toss, we ask a student whether he/she wants to play again. If he/she wants to play again, he/she will win Rs. $200$ if his/her guess about he outcome matches with the outcome of the second toss; otherwise will lose Rs. $100$. We play this game under two different schemes.
In scheme $1$, each student belonging to the first group ($70$ students), will declare his/her guess about the result of a coin tossing experiment and will be informed the result after the toss. We then ask that student whether he/she wants to play again and record his/her response. In scheme $2$, we perform this experiment for the second group of students ($31$ students) slightly differently than the first scheme. Here, each student will declare his/her guess about the result if the first toss, but he/she will not be informed the result. However, we then ask him/her whether he/she wants to play again and record his/her response. We have used only one coin throughout experiment. Before the beginning of this experiment, we checked whether the coin is unbiased using a binomial experiment and confirmed that it is unbiased. We also make sure that the student who has perfumed the experiment has no way to disclose the result or his/her guess and attitude towards this experimental result. Moreover, there is no exchange of information between the two groups of students for two different schemes.
\begin{itemize}
\item We have selected randomly 101 college students from Raja Rajeswari Engineering College, Bengaluru, India. The students belong 1st, 2nd, or 3rd year of their engineering curriculum and naturally belong to a very homogeneous age group.
\item We used an unbiased coin for the experiment. Before the experiment we have checked that the coin used is unbiased.
\item We then divide them into two groups: one consisting $70$ students and the other $31$ students.
\item We performed the experiment similar to that described in Busemeyer. If a students wins the first toss, he/she will receive Rs. $100$; if loses, he/she will lose Rs. $50$.
\item Each student belonging to the first group ($70$ students), will declare his/her guess about the result of a coin tossing experiment and will be informed the result after the toss. We record if he/she wants to play again.
\item Each student belonging to the second group ($31$ students) will declare his/her guess about the result if the first toss, but he/she will not be informed the result. We then record if he/she wants to play again.
\end{itemize}
\noindent
{\it {\bf Probability theoretic explanation:}}
We have already discussed that $P(A) > P(B)$ is true always under the conditions that $P(A\cap X) > P(B \cap X)$ and $P(A\cap X^C) > P(B \cap X^C)$.
Thus, if you prefer A when the event $X$ is known, and if you prefer B when the complementary event $X^C$ is known, then it will imply that you will always prefer A over B irrespective of the events $X$ or $X^C$. Any violation of this is termed as disjunction effect.
We have performed experiment similar to that given in Busemeyer and we observed marked violation of this probabilistic claim and its explanation depends on several factors, not reported or discussed in literature. Not only the psychological factors, but also other factors like sex, culture, geographical region etc, might have important role to play.
\section{Results}
We have observed a few interesting facts from the experiment. Instead on looking at the numerical figures of the outcomes of the experiment only as in Tversky and Shafir (1952), we have done a detailed statistical analysis in order to strengthen the interpretation of our observations. As described in the previous section, our experiment consists of two different schemes. Under the first scheme, the students were informed the outcome of the first toss. If they know that they won the first gamble, $76.47\%$ want to play again, i.e. majority want to play again. However we would like to be sure that the result is statistically significant. For this, we perform a test of null hypothesis $H_0:p=0.5$ against an alternative hypothesis $H_1:p>0.5$. The p-value associated with this test is $0.0004$ indicating that the majority wants too play again. On the other hand, if they did not know, whether they won or lost, $58.33\%$ want to play again. Although it seems that majority wants to play again, but the result is not statistically significant (p-value$=0.1215$). Under scheme $2$, when the students did not know the outcome of the first toss, i.e. when they did not know whether they won or lost, $54.84\%$ want to play again. So, in this case, although it seems that majority wants to play again, but the result is not statistically significant since the associated p-value is $0.3601$.
In each case majority wants to play again. But in Busemeyer the corresponding figures are $69\%$, $59\%$, and $36\%$ respectively, while in our experiment the figures are $76.47\%$, $58.33\%$, and $54.84\%$ respectively. The last figure differs widely while the second figure matches surprisingly. We think that probably p-value corresponding to the second figure (i.e. $59\%$) is not significant. So Busemeyer inference needs to be revisited. However, the idea of paying again when they know that whether they won or lost still remains valid if we go only by the actual figures compared to $50\%$, which indicates the state of indifference. So we did an exact test of hypothesis that $H_0: p=0.5$ against $H_1: p>0.5$ i.e. to see absence of disjunction effect. The p-value is $0.3601$ indicating that they are indifferent to the decision. So either disjunction effect is not observed here, or very weak, but they do not prefer to play when they do not know the state.
It seems clear that the observations by Busemeyer is {\it{not}} matching with ours. We conjecture that this discrepancy may be due to the effect of different cultural settings in which the experiments are conducted. However, we proceed further to examine whether there exists any other factor that might play an implicit role in the experimental results. Indian culture and gender are intermingled always. Hence it would be a wise idea to revisit the experimental results incorporating gender factor. We observed that among females, $75\%$ (p-value$=0.0106$) females want to play again if they know that they won, $52.94\%$ (p-value$=0.3145$) females want to play again if they know that they lost. This result is not at all significant indicating that once lost, females do not want to take one more risk. Moreover, $41.18\%$ (p-value$=0.6855$) females want to play again if they do not the first result.
This picture is markedly different among males. $77.78\%$ (p-value$=0.0038$) males want to play again if they know that they won, $63.16\%$ (p-value$=0.0835$) males want to play again if they know that they lost. However, this result is marginally significant, although not strong. Moreover, $71.43\%$ (p-value$=0.0288$) males want to play again when they do not know the outcome of the first experiment. Since majority wants to play again whether they won or lost or uninformed, there is no disjunction effect observed among males.
\noindent
\vskip5pt
\section{Discussions}
The above results of our experiment raise the following important issues:
\begin{itemize}
\item If we consider only absolute values, males do not show any disjunction effect, but females show strong disjunction effect. Now we have to explain this in Indian context, if possible.
\item However, if we go by the p-values of the corresponding tests of significance, interesting observations can be made using the combination of testsâ results. For all individuals, males and females taken together, let us first combine the results of the conditions that first result is known to be win and that to be loss. The combined p-value for this is 0.00053 indicating that majority of the individuals want to play again if they know the result. But the p-value for the result when they do not know the first result is 0.2366 indicating that there is disjunction effect.
\item For males, disjunction effect is not observed if irrespective of whether we go by the actual values or by the p-values; combined p-value is 0.0029 whereas p-value when the first result is not known is 0.0288. However, the general effect is moving towards the disjunction effect although not established statistically or by observations.
\item For females, although actual observations suggest only weak disjunction effect, but comparing p-values for the combined p-value (0.0223) to the p-value (0.6855) when the first result is unknown shows clear disjunction effect.
\item Disjunction effect is although a clear concept and is realized in a number of experiments, cultural effects are strong especially among males and females. In Indian context, probably disjunction effect for females is so strong that they overcome the absence of disjunction effect among males.
\item Busemeyer's experimental results are based only on the actual values and not tested statistically. This is highly influenced by specific cases and specific scenarios considered in the experiment. It is not wise to declare that the effect is present or absent whenever the number of observations is less or greater than $50\%$; statistical test must be employed to validate and confirm the findings.
\item The missing part of all other previous experiments is the appropriate statistical analysis of the results. Based on our analysis, we propose different kinds of disjunction effect, e.g. strong and weak.
\end{itemize}
This can be stated in the following manner.
\begin{table}[!h]
\caption{Categorisation of disjunction effect based on actual observation and statistical tests.}\centering
\vspace{2mm}
\begin{tabular}{|c|c|c|}
\hline
\diaghead{\theadfont ColumnmnHead}
{}{Actual observation} & significant & Not significant\\
{Statistical test}{} & & \\
\hline
Significant & Strong effect & --- \\
(p-value is small) & & \\
\hline
Not significant & Weak effect & No effect \\
(p-value is large) & & \\
\hline
\end{tabular}
\end{table}
The above categorization triggers to rethink the classification of disjunction effect with respect toothed factor like gender etc. Here we propose a general categorization based on actual observations and results of statistical significance tests in presence of another factor `gender'. This is given in Table 2.
\begin{table}[!h]
\caption{A general categorization of disjunction effect based on actual observation and statistical tests.}\centering
\vspace{2mm}
\begin{tabular}{|c|c|c|c|}
\hline
{}{Actual observation} & Significant in both & Significant in males & Not significant \\
\diaghead{\theadfont ColumnmnHead}
{}{} & males and females & but not in females & in both \\
{Statistical test}{} & & or vice versa & \\
\hline
Significant in both & & & \\
males and females & Strong effect & --- & --- \\
(p-value is small) & & & \\
\hline
Significant in males but & & & \\
not in females or vice & Moderate effect & Moderate effect & --- \\
versa (large p-value) & & & \\
\hline
Not significant & & & \\
in both & Weak effect & Weak effect & No effect \\
(p-value is large) & & & \\
\hline
\end{tabular}
\end{table}
In this manner we may come up with more logical categorization.
\noindent
\vskip5pt
\section*{Acknowledgements}
The authors acknowledge the Rector, RR Group of institutions, the principal R.R.Engineering College and the participating students from R.R.Engineering college for their help and cooperation. We would also like to thank Mr. Nepal Banerjee for his help in conducting the experiments. This work is under the project SB/S4/MS:844/2013 approved by SERB, DST, Government of India.
\section*{References}
|
1,116,691,500,785 | arxiv |
\section{Introduction}
Hybrid quantum classical variational algorithms, including those of the
variational quantum eigensolver (VQE) type \cite{
peruzzo2014variational,
mcclean2016theory},
are among the leading candidates for quantum algorithms that may yield quantum
advantage in areas such as computational chemistry or machine learning already
in the era of noisy intermediate scale quantum (NISQ) computing
\cite{preskill2018quantum}. A foundational issue in VQE \cite{
peruzzo2014variational,
mcclean2016theory},
and in many of its extensions and alternatives
\cite{
mcclean2017hybrid,
parrish2019quantum,
nakanishi2019subspace,
urbanek2020chemistry,
ollitrault2020quantum,
huggins2020non,
parrish2019qfd,
stair2020multireference},
is finding a ``good'' definition of the entangler circuit. Here the qualifier
``good'' has many facets, possibly including:
(1) Providing an efficient approximate representation of the target quantum
states in the limit of an intermediate (ideally polynomial-scaling) depth
(2) Consisting of a low number of distinct physically realizable gate elements
(3) Exhibiting a simple pattern of how these gate elements are applied
(4) Exhibiting sparse spatial locality that is further compatible with device connectivity
(5) Exhibiting simple analytical gradient recipes and robust numerical convergence behavior during optimization of the
VQE energy, e.g., by mitigating the effects of barren plateaus \cite{mcclean2018barren}
(6) Respecting exactly the natural particle and spin quantum number symmetries
of the target quantum states, e.g., as notably explored in
\cite{gard2020efficient}
(7) Providing an exact representation of the target quantum states in the limit
of sufficient (usually exponential-scaling) depth.
Especially within the context of VQE for spin-$1/2$ fermions governed by real,
spin-free Hamiltonian operators (e.g., electrons in molecules and materials, the
prime application of VQE), a variety of compelling VQE entangler circuit recipes
have been discussed in the literature. Prominent examples include
UCCSD \cite{peruzzo2014variational,PhysRevX.6.031007,Ryabinkin2018},
$k$-UpCCGSD \cite{lee2018generalized,o2019generalized},
Jastrow-Factor VQE \cite{matsuzawa2020jastrow},
the symmetry-preserving ans\"atze \cite{gard2020efficient},
the hardware efficient ans\"atze \cite{Kandala2017,Bian2019},
ADAPT-VQE \cite{Grimsley2019},
pUCCD \cite{elfving2020simulating},
and additional methods discussed below \cite{
evangelista2019exact,
ganzhorn2019gate,
xia2020qubit,
yordanov2020efficient,
salis2019short,
khamoshi2020correlating}.
Each of these generally satisfies a subset of the ``good'' facets listed above,
though no extant ansatz that we are aware of obtains all of them, with the
notable exception of the generalized swap network form of $k$-UpCCGSD
of \cite{o2019generalized}.
In this work we develop a VQE entangler circuit recipe for fermions in the
Jordan-Wigner representation and show, or at least provide evidence, that it
obtains all facets, with facet (5) partially left to future numerical studies.
Perhaps the most notable property of our fabrics is the exact preservation of
all relevant quantum numbers the individual gate elements of the fabric,
which is why we refer to them as quantum number
preserving (QNP). This property may be critical for employment of VQE in larger
systems, where contaminations from or even variational collapse onto states
with different particle or spin quantum numbers can severely degrade the
quality of the VQE wavefunction.
Note that after we posted the first version of our manuscript, we became aware
of the generalized swap network reformulation of $k$-UpCCGSD of
\cite{o2019generalized}. This paper refactors $k$-UpCCGSD to use
nearest-neighbor connectivity, yielding a circuit fabric that could be written
in terms of four-qubit gates containing diagonal pair exchange and orbital
rotation elements in a very similar manner as our $\hat Q$-type QNP gate fabric
discussed below. There are some tactical differences in the qubit ordering and
the generalized swap network paper does not emphasize the role of quantum number
symmetry as much as the present manuscript. Moreover, the origin of the $\hat
Q$-type QNP gate fabric as a simplification of our more-complete $\hat F$-type
QNP gate fabric of Appendix \ref{appendix:other_qnp_gate_fabrics} provides a
markedly different approach to developing this gate fabric. In any case, we
encourage any readers interested in the present manuscript to also explore
\cite{o2019generalized}.
\section{Gate fabrics}
Our VQE entangler circuit recipe draws inspiration from
the well known fact that the qubit Hilbert space (without any fermionic symmetries)
$\mathcal{SU}(2^{N})$ can be spanned by a tessellation of 2-qubit gates
universal for $\mathcal{SU}(4)$ in alternating layers (see Figure~\ref{fig:SU4}).
\begin{figure}[t]
\centering
\begin{equation*}
\label{eq:SU2N}
\begin{array}{l}
\Qcircuit @R=0.3em @C=0.3em @!R {
& \multigate{1}{\mathit{SU}(4)}
& \qw
& \multigate{1}{\mathit{SU}(4)}
& \qw
& \qw \\
& \ghost{\mathit{SU}(4)}
& \multigate{1}{\mathit{SU}(4)}
& \ghost{\mathit{SU}(4)}
& \multigate{1}{\mathit{SU}(4)}
& \qw \\
& \multigate{1}{\mathit{SU}(4)}
& \ghost{\mathit{SU}(4)}
& \multigate{1}{\mathit{SU}(4)}
& \ghost{\mathit{SU}(4)}
& \qw \\
& \ghost{\mathit{SU}(4)}
& \multigate{1}{\mathit{SU}(4)}
& \ghost{\mathit{SU}(4)}
& \multigate{1}{\mathit{SU}(4)}
& \qw \\
& \multigate{1}{\mathit{SU}(4)}
& \ghost{\mathit{SU}(4)}
& \multigate{1}{\mathit{SU}(4)}
& \ghost{\mathit{SU}(4)}
& \qw \\
& \ghost{\mathit{SU}(4)}
& \qw
& \ghost{\mathit{SU}(4)}
& \qw
& \qw \\
}
\end{array}
\ldots
\phantom{}
\cong
\phantom{}
\begin{array}{l}
\Qcircuit @R=0.3em @C=0.3em @!R {
& \multigate{5}{\mathit{SU}(2^6)}
& \qw \\
& \ghost{\mathit{SU}(2^6)}
& \qw \\
& \ghost{\mathit{SU}(2^6)}
& \qw \\
& \ghost{\mathit{SU}(2^6)}
& \qw \\
& \ghost{\mathit{SU}(2^6)}
& \qw \\
& \ghost{\mathit{SU}(2^6)}
& \qw \\
}
\end{array}
\end{equation*}
\begin{equation*}
\mathit{SU}(4)
\coloneqq
\exp(\hat X) : \hat X = -\hat X^\dagger, \mathrm{Tr} (\hat X) = 0, \hat X \in \mathbb{C}^{4} \times
\mathbb{C}^{4}
\end{equation*}
\begin{equation*}
\cong
\begin{array}{l}
\Qcircuit @R=0.3em @C=0.3em @!R {
& \gate{SU(2)}
& \ctrl{1}
& \gate{SU(2)}
& \ctrl{1}
& \gate{SU(2)}
& \ctrl{1}
& \gate{SU(2)}
& \qw \\
& \gate{SU(2)}
& \ctrl{-1}
& \gate{SU(2)}
& \ctrl{-1}
& \gate{SU(2)}
& \ctrl{-1}
& \gate{SU(2)}
& \qw \\
}
\end{array}
\end{equation*}
\caption{Sketch for $N=6$ of a gate fabric universal for $\mathcal{SU}(2^N)$ providing
inspiration for the fermionic quantum number preserving gate fabrics developed here.
The gate fabric is a 2-local-nearest-neighbor tessellation of alternating
15-parameter, 2-qubit $SU(4)$ gates. The $SU(2)$ gate
in the $SU(4)$ gate decomposition on the bottom line is the 3-parameter universal
gate for the 1-qubit Bloch sphere. The indicated 24-parameter decomposition of
$SU(4)$ is overcomplete for the 15-parameter $\mathcal{SU}(4)$ group.}
\label{fig:SU4}
\end{figure}
This tessellation can formally be
repeated to infinite depth. However, one finds that after some finite,
$N$-dependent critical depth of order $\mathcal{O}(2^{2N})$, additional gate
layers do not increase the expressiveness of the circuit, as formal completeness
(denoted ``universality'') in $\mathcal{SU}(2^{N})$ is achieved.
In practice usually shorter circuit depths are of interest.
For instance, one may consider the case where the tessellation is restricted to
be polynomial scaling in $N$, in which case universality cannot be exactly
achieved. However, a good approximation of specialized (e.g., physically relevant) parts of some
subgroup may still be achievable in a way that is tractable to compute even on a
NISQ computer but intractable to compute with a classical device.
Particularly striking in Figure \ref{fig:SU4} is the locality
(alternating nearest neighbor connectivity) and simplicity (single gate element) of the
circuit, properties of what we call a ``gate fabric.'' More precisely,
throughout this manuscript, we define a gate fabric for a subgroup of
$\mathcal{SU}(2^{N})$ to be a tessellation of gates over $N$-qubits with the
following properties:
\begin{compactenum}
\item Simplicity: Composed of a single type of
$k$-qubit, $l$-parameter gate element (with a known decomposition into
elementary gates), where $k$ and $l$ are independent of $N$.
\item Linear Locality: When the qubits are thought of as arranged on a vertical line
the gate elements are arranged in layers and connect up to $k$ contiguous qubits.
\item Universality: Achieving universality within the target subgroup
of $\mathcal{SU}(2^{N})$ within a finite number of layers depending on $N$.
\item Symmetry: commuting with all symmetry operators used to define the
subgroup of $\mathcal{SU}(2^{N})$, i.e., $[\hat U, \hat N] = 0$, where $\hat U$
is the circuit unitary for any set of parameters and $\hat N$ is the
symmetry operator.
\end{compactenum}
Depending on the subgroup of $\mathcal{SU}(2^{N})$ of interest it can be more or
less difficult to find fabrics akin to the one shown in Figure~\ref{fig:SU4}.
In Appendix~\ref{sec:subgroups} we discuss the trivial restriction to
$\mathcal{SO}(2^{N})$ and the less-trivial restriction to subspaces of definite
Hamming weight.
\section{Gate fabrics for fermions under the Jordan-Wigner mapping}
The focus of this work is the construction of gate fabrics for the subgroup
$\mathcal{F}(2^{2M}) \in \mathcal{SU}(2^{N})$ constrained to spin-restricted fermionic symmetry under
the Jordan-Wigner representation.
To make this more precise, we define $M$ real orthogonal spatial
orbitals $\{| \phi_{p} \rangle\}_{p=0}^M$. For each spatial orbital, we define
corresponding $\alpha$($\beta$) spin orbitals $| \psi_{p} \rangle \coloneqq |\phi_{p}
\rangle | \alpha \rangle$ ($| \psi_{\bar p} \rangle \coloneqq |\phi_{p} \rangle | \beta
\rangle$) for a total of $N \coloneqq 2M$ spin orbitals in a spin-restricted
formalism. We associate the occupation numbers of these spin orbitals with the
occupation numbers of $N$ qubits. We number the qubits in
``interleaved'' ordering
$\ldots|1_{\beta}\rangle|1_{\alpha}\rangle|0_{\beta}\rangle|0_{\alpha}\rangle$.
The fermionic creation/annihilation operators are defined in terms of the qubit
creation/annihilation operators via the Jordan-Wigner mapping in
``$\alpha$-then-$\beta$'' ordering,
$p^{\pm} \coloneqq (\hat X_{p} \mp i
\hat Y_{p}) / 2 \bigotimes_{q = 0}^{p-1} \hat Z_{q}$
and
$\bar p^{\pm} \coloneqq (\hat X_{\bar p} \mp i
\hat Y_{\bar p}) / 2 \bigotimes_{q = 0}^{p-1} \hat Z_{\bar q}
\bigotimes_{q = 0}^{M-1} \hat Z_{q}$.
We note that for the majority of applications in the space of
spin-$1/2$ fermions, the governing Hamiltonians are real (e.g., for
non-relativistic electronic structure theory), and so we restrict
from complex to real unitary operators, i.e., $\mathcal{SU}(2^{N}) \rightarrow
\mathcal{SO}(2^{N})$. The spin-restricted fermionic subgroup is then defined as
the subgroup of $\hat U \in \mathcal{SO}(2^{N})$ that respect the commutation
relations
$[ \hat U, \hat N_{\alpha} ] = 0$,
$[ \hat U, \hat N_{\beta} ] = 0$, and
$[ \hat U, \hat S^2] = 0$.
Here the $\alpha$($\beta$) number operator is
$\hat N_{\alpha} \coloneqq \sum_{p} p^\dagger p = \sum_{p} (\hat I - \hat Z_p) /
2$
[
$\hat N_{\beta} \coloneqq \sum_{p} \bar p^\dagger \bar p = \sum_{p} (\hat I - \hat
Z_{\bar p}) /
2$
].
\cite{ntnote}
The spin-squared operator is $\hat S^2 \coloneqq \sum_{pq} p q^\dagger \bar
p^\dagger \bar q + (\hat N_{\alpha} - \hat N_{\beta}) / 2 + (\hat N_{\alpha} -
\hat N_{\beta})^2 / 4$, and does not admit a local description in terms of Pauli
operators in the Jordan-Wigner basis (we provide further details in Appendices~\ref{appendix:jw_details} and \ref{appendix:two_mode_fermions}).
We denote this real subgroup, preserving $\hat
N_{\alpha}$, $\hat N_{\beta}$, and $\hat S^2$, as $\mathcal{F}(2^{2M})$.
Naively one might expect that there should not be any local gate fabric exactly
preserving all three fermionic quantum numbers, since the $\hat{S}^2$ operator
is non-local. The crux of this work is thus the simple
quantum-number-preserving gate fabric of Figure~\ref{fig:F}. This gate fabric is
composed of 2-parameter 4-qubit gate elements $\hat Q$, each composed of a
1-parameter 4-qubit spin-adapted spatial orbital rotation gate
$\mathrm{QNP_{OR}} (\varphi)$ and a 1-parameter 4-qubit diagonal pair exchange
gate $\mathrm{QNP_{PX}} (\theta)$. We describe further related
quantum-number-preserving gate fabrics for $\mathcal{F}(2^{2M})$ in
Appendix~\ref{appendix:other_qnp_gate_fabrics} - these were the progenitors of
the simpler gate fabrics shown in the main text, and may have advantageous
properties in specific realizations of VQE entangler circuits.
Facets (2-4) of gate fabrics are manifestly fulfilled for all these proposals
and facet (6) holds by construction as all gates individually preserve all
quantum numbers. For facets (1) and (7) we provide numerical evidence below and
in Appendix~\ref{sec:additional numerics}. It is worth noting that these tests
numerically indicate that our gate fabrics are universal in the vast bulk of
quantum number irreps (with the exception of a few high-spin edge cases for the
$\hat Q$-type gate fabrics of the main text, see Appendix~\ref{sec:additional
numerics}), i.e., that they may be used for cases where $S \neq 0$ and/or where
$N_{\alpha} \neq N_{\beta}$ (including both even and odd spin cases).
We believe that the methods from
\cite{brandao2016local,oszmaniec2020epsilon} or \cite{evangelista2019exact} can
be used to rigorously show universality in the (bulk of the) quantum number
sectors of $\mathcal{F}(2^{2M})$ in all cases (as well as that our circuits are
polynomial depth $\epsilon$-approximate unitary $t$-designs and form
$\epsilon$-nets). Working out the details of a rigorous proof, which we believe
has to be done spin sector by spin sector in some cases, is however beyond the
scope of this work.
For $\mathrm{QNP_{OR}}$, $\mathrm{QNP_{PX}}$ (and all other parametrized quantum
number preserving gates introduced in
Appendix~\ref{appendix:other_qnp_gate_fabrics}) we provide explicit
decompositions into elementary gates in Appendix~\ref{sec:qnp_decompositions}.
We further provide generalized parameter shift rules
\cite{Mitarai_Fujii_18,farhi2018classification, Schuld2019, PennyLane} for
theses gates in Appendix~\ref{sec:four-term-rule}, enabling a computation of the
gradients with respect to their circuit parameters with a maximum of four
distinct circuits and without increasing circuit depth, gate count, or qubit
number. We compare this gradient recipe to the generalization presented in
\cite{Kottmann_Guzik_20}, extend the variance minimization technique from
\cite{Mari_Killoran_20} to the $\mathrm{QNP}$ gate gradients and note that our
new rule can be applied to a large variety of other gates.
\begin{figure*}[tb]
\centering
\begin{equation*}
\begin{array}{llll}
\begin{array}{l}
\Qcircuit @R=0.3em @C=0.3em @!R {
& \multigate{3}{Q}
& \qw
& \qw \\
& \ghost{Q}
& \qw
& \qw \\
& \ghost{Q}
& \multigate{3}{Q}
& \qw \\
& \ghost{Q}
& \ghost{Q}
& \qw \\
& \multigate{3}{Q}
& \ghost{Q}
& \qw \\
& \ghost{Q}
& \ghost{Q}
& \qw \\
& \ghost{Q}
& \qw
& \qw \\
& \ghost{Q}
& \qw
& \qw \\
}
\end{array}
\ldots
&
\cong
\begin{array}{l}
\Qcircuit @R=0.3em @C=0.3em @!R {
& \multigate{7}{F(2^{2 * 4})}
& \qw \\
& \ghost{F(2^{2 * 4})}
& \qw \\
& \ghost{F(2^{2 * 4})}
& \qw \\
& \ghost{F(2^{2 * 4})}
& \qw \\
& \ghost{F(2^{2 * 4})}
& \qw \\
& \ghost{F(2^{2 * 4})}
& \qw \\
& \ghost{F(2^{2 * 4})}
& \qw \\
& \ghost{F(2^{2 * 4})}
& \qw \\
}
\end{array}
&
\begin{array}{l}
\Qcircuit @R=0.3em @C=0.3em @!R {
& \multigate{3}{\mathrm{QNP_{OR}} (\varphi)}
& \qw \\
& \ghost{\mathrm{QNP_{OR}} (\varphi)}
& \qw \\
& \ghost{\mathrm{QNP_{OR}} (\varphi)}
& \qw \\
& \ghost{\mathrm{QNP_{OR}} (\varphi)}
& \qw \\
}
\end{array}
&
\coloneqq
\begin{array}{l}
\Qcircuit @R=0.3em @C=0.3em @!R {
& \gate{G (\varphi)}
& \qw
& \qw \\
& \qw \qwx[1] \qwx[-1]
& \gate{G (\varphi)}
& \qw \\
& \gate{G (\varphi)}
& \qw \qwx[1] \qwx[-1]
& \qw \\
& \qw
& \gate{G (\varphi)}
& \qw \\
}
\end{array}
\\
\\
\begin{array}{l}
\Qcircuit @R=0.3em @C=0.3em @!R {
& \multigate{3}{Q (\varphi, \theta)}
& \qw \\
& \ghost{Q (\varphi, \theta)}
& \qw \\
& \ghost{Q (\varphi, \theta)}
& \qw \\
& \ghost{Q (\varphi, \theta)}
& \qw \\
}
\end{array}
&
\coloneqq
\begin{array}{l}
\Qcircuit @R=0.3em @C=0.3em @!R {
& \multigate{3}{\Pi}
& \multigate{3}{\mathrm{QNP_{PX}} (\theta)}
& \multigate{3}{\mathrm{QNP_{OR}} (\varphi)}
& \qw \\
& \ghost{\Pi}
& \ghost{\mathrm{QNP_{PX}} (\theta)}
& \ghost{\mathrm{QNP_{OR}} (\varphi)}
& \qw \\
& \ghost{\Pi}
& \ghost{\mathrm{QNP_{PX}} (\theta)}
& \ghost{\mathrm{QNP_{OR}} (\varphi)}
& \qw \\
& \ghost{\Pi}
& \ghost{\mathrm{QNP_{PX}} (\theta)}
& \ghost{\mathrm{QNP_{OR}} (\varphi)}
& \qw \\
}
\end{array}
\phantom{QISFTW}
&
\begin{array}{l}
\Qcircuit @R=0.3em @C=0.3em @!R {
& \multigate{3}{\mathrm{QNP_{PX}} (\theta)}
& \qw \\
& \ghost{\mathrm{QNP_{PX}} (\theta)}
& \qw \\
& \ghost{\mathrm{QNP_{PX}} (\theta)}
& \qw \\
& \ghost{\mathrm{QNP_{PX}} (\theta)}
& \qw \\
}
\end{array}
&
\coloneqq
\begin{array}{l}
\Qcircuit @R=0.3em @C=0.3em @!R {
& \targ
& \ctrl{1}
& \targ
& \qw \\
& \ctrlo{-1}
& \multigate{1}{G (\theta)}
& \ctrlo{-1}
& \qw \\
& \ctrlo{+1}
& \ghost{G (\theta)}
& \ctrlo{+1}
& \qw \\
& \targ
& \ctrl{-1}
& \targ
& \qw \\
}
\end{array}
\\
\end{array}
\end{equation*}
\begin{align*}
\begin{array}{l}
\Qcircuit @R=0.3em @C=0.3em @!R {
& \multigate{1}{G (\lambda)}
& \qw \\
& \ghost{G (\theta)}
& \qw \\
}
\end{array}
&\coloneqq
\begin{array}{l}
\Qcircuit @R=0.3em @C=0.3em @!R {
& \gate{R_{y} (+\pi / 2)}
& \ctrl{1}
& \gate{R_{y} (+\lambda)}
& \ctrl{1}
& \gate{R_{y} (-\pi / 2)}
& \qw \\
& \gate{R_{y} (+\pi / 2)}
& \ctrl{-1}
& \gate{R_{y} (-\lambda)}
& \ctrl{-1}
& \gate{R_{y} (-\pi / 2)}
& \qw \\
}
\end{array}
&=
\begin{array}{l}
\Qcircuit @R=0.3em @C=0.3em @!R {
& \gate{H}
& \ctrl{1}
& \gate{R_{y} (\lambda / 2)}
& \ctrl{1}
& \gate{H}
& \qw \\
& \qw
& \targ
& \gate{R_{y} (\lambda / 2)}
& \targ
& \qw
& \qw \\
}
\end{array}
=
\left [
\begin{array}{rrrr}
1 & & & \\
& c & +s & \\
& -s & c & \\
& & & 1 \\
\end{array}
\right ]
:
\
\begin{array}{l}
c \coloneqq \cos(\lambda/2)
\\
s \coloneqq \sin(\lambda/2)
\\
\end{array}
\end{align*}
\caption{Proposed gate fabric for $\mathcal{F}(2^{2M})$
(sketched for $M=4$). The spin orbitals in Jordan-Wigner representation are
in ``interleaved'' ordering with even (odd) qubit indices
denoting $\alpha$ ($\beta$) spin orbitals. The Jordan-Wigner strings are taken to be
in ``$\alpha$-then-$\beta$'' order.
The gate fabric is a 4-local-nearest-neighbor- tessellation of alternating even
and odd spatial-orbital-pair 2-parameter, 4-qubit $\hat Q$ gates.
Each $\hat Q$ gate has two independent parameters and contains a 1-parameter, 4-qubit spatial orbital rotation
gate $\mathrm{QNP_{OR}}(\varphi)$ and a 1-parameter, 4-qubit diagonal pair
exchange gate $\mathrm{QNP_{PX}}(\theta)$. The order of $\mathrm{QNP_{OR}}$
and $\mathrm{QNP_{PX}}$ (note $[\mathrm{QNP_{OR}}, \mathrm{QNP_{PX}}]
\neq 0$) does not seem to substantially change expressiveness at intermediate depths.
$\mathrm{QNP_{OR}} (\varphi)$
implements the spatial orbital Givens rotation
$|\phi_{0} \rangle = c |\phi_{0}\rangle + s |\phi_{1}\rangle$
and
$|\phi_{1} \rangle = -s |\phi_{0}\rangle + c |\phi_{1}\rangle$, with the same
orbital rotation applied in the $\alpha$ and $\beta$ spin orbitals.
$\mathrm{QNP_{PX}}(\theta)$ implements the diagonal pair Givens rotation,
$|0011\rangle = c |0011\rangle + s |1100\rangle$
and
$|1100\rangle = -s |0011\rangle + c |1100\rangle$.
The real 1-parameter, 1-qubit rotation gate is $\hat R_{y} (\lambda) \coloneqq e^{-i
\lambda \hat Y/2}$.
In $\hat Q$, we include the optional constant $\hat \Pi$ gate, for which natural choices include
the 4-qubit identity gate, i.e., $\hat \Pi = \hat I$, or the fixed spin-adapted
orbital rotation gate $\hat \Pi = \mathrm{QNP_{OR}} (\pi)$. In the latter
case, the gate fabric with all parameters $\{ \theta = 0\}$ and $\{\varphi = 0\}$ promotes exchange of orbitals.
We find that the choice of $\hat \Pi \in \{\hat I,
\mathrm{QNP_{OR}} (\pi)\}$ does not appear to affect the expressiveness of the quantum
circuit, but the latter choice has turned out to be advantageous during gradient-based parameter
optimization. Regardless of the choice of $\hat \Pi$,
this gate fabric exactly preserves the real nature of the subgroup,
exactly commutes with the $\hat N_{\alpha}$, $\hat N_{\beta}$, and $\hat S^2$
symmetry operators, and numerically appears to provide universality at
sufficient parameter count.
}
\label{fig:F}
\end{figure*}
\section{Numerical demonstrations}
To numerically investigate the properties of the gate fabric from
Figure~\ref{fig:F} and to collect evidence that it satisfies all facets of a
``good'' entangler circuit, we consider two prototypical examples of highly
correlated molecular ground states: The first is p-benzyne, which exhibits a
biradical open-shell singlet ground state, with two unpaired electrons indicated
by significant deviations from Hartree-Fock natural orbital occupation numbers,
and four other moderate deviations from Hartree-Fock natural orbital occupation
numbers. We use the geometry from \cite{keller2015selection}, build the orbitals
at RHF/cc-pVDZ, and construct a (6e, 6o) active space Hamiltonian with the
orbitals ranging from HOMO-2 to LUMO+2. This corresponds to a case of $M=6$
spatial orbitals (i.e., $N=12$ qubits), and we focus on the ground state
irreducible representation $(N_{\alpha} = 3, N_{\beta} = 3, S = 0)$. For a
larger test case, we consider naphthalene, which while not intrinsically
biradical, has multiple natural orbitals with significant deviations from
Hartree-Fock natural orbital occupation numbers. We build the orbitals at
RHF/STO-3G, and then construct a (10e, 10o) active space Hamiltonian consisting
of the $\pi$ and $\pi^*$ orbitals. This corresponds to a case of $M=10$ spatial
orbitals (i.e., $N=20$ qubits), and we focus on the ground state irreducible
representation $(N_{\alpha} = 5, N_{\beta} = 5, S = 0)$. In both cases, we
consider VQE gate fabrics of the form of Figure \ref{fig:F}.
\begin{figure*}[ht!]
\centering
\includegraphics[width=\linewidth]{figures/p-benzyne/result_final.pdf}
\caption{Results of the discussed VQE fabric for representative molecular test
cases:
(a) Convergence of the VQE energy relative to the exact ground state energy
$E_\mathrm{FCI}$ of the $12$-spin-orbital active space of p-benzyne as a
function of the number of parameters in the fabric,
(b) Occupation $\left|\braket{I|\Psi}\right|^2$ of each computational basis
state $\langle I|$ in the optimized VQE state $|\Psi\rangle$ at the
color-indicated parameter counts in (a). Blue area indicates the computational
basis states of the FCI ground state in the active space.
Each set of computational basis states is sorted in descending order, we show
a figure with consistent ordering between sets in see Appendix~\ref{sec:additional numerics},
(c) Convergence of the VQE energy to the exact ground state energy
$E_\mathrm{FCI}$ in the 20-spin-orbital active space of naphthalene as a
function of the number of parameters an for the two different initialization schemes,
(d) Convergence under the L-BFGS optimizer for the color coded parameter
counts indicated in (c), dotted (solid) convergence lines correspond to a data
point from the dotted (solid) curve in (c), Inset highlights the plateaus encountered
with initialization method B during the first 180 epochs.}
\label{fig:experiments}
\end{figure*}
Our final VQE circuit starts with the preparation of an uncorrelated product state by
applying local Pauli $\hat X$ gates to appropriate qubits of an all-zero state depending on the
number of alpha and beta electrons.
The qubits are chosen such that for all parameters equal to zero in the following fabric
the state is transformed to the state with the energetically lowest orbitals occupied.
We then consider two parameter initialization strategies:
(A) The fabric is initialized with all $\theta = 0$ and all $\varphi = \pi/2$ and
$\hat \Pi = \mathrm{QNP_{OR}} (\pi)$ (solid lines in \ref{fig:experiments})
(B) The fabric is initialized with all $\theta = 0$ and all $\varphi = \pi$ and
$\hat \Pi = \hat I$ (dotted lines in \ref{fig:experiments}).
In both cases we optimize the VQE ground state energy with respect to the
VQE gate fabric parameters via L-BFGS. As the purpose of this study is
to explore the expressive power of this gate fabric, we consider neither shot
noise nor decoherence. This restriction permits the use of
analytical expressions for the Hamiltonian expectation values and VQE parameter
gradients thereof, greatly accelerating the classical statevector simulation of
the VQE.
Figure~\ref{fig:experiments} shows the salient results of this study. For the
case of p-benzyne, Figure~\ref{fig:experiments}a shows the VQE ground state
energy vs. full configuration interaction (FCI) with respect to the depth of the
gate fabric. The first notable point is that the fabric is able to
provide higher accuracy than either Hartree-Fock (HF) (a fabric of
$\mathrm{QNP_{OR}}$ gates - redundant here due to the use of Hartree-Fock
orbitals in the active space) or doubly-occupied configuration interaction
(DOCI) (a fabric of $\mathrm{QNP_{PX}}$ gates - equivalent to the pUCCD ansatz
from \cite{elfving2020simulating}). Focusing on the early convergence
behavior on the left side of the plot, even with only a few layers of the VQE
gate fabric, e.g. $\sim 50-80$ parameters, absolute accuracy of $1$ kcal
mol$^{-1}$
is achieved, which is commonly referred to as chemical accuracy. As the gate fabric depth is increased, roughly
geometric (exponential) convergence of the absolute energy is achieved, modulo some minor
aberrations due to difficulties in tightly converging the L-BFGS-based numerical
optimizations of the VQE gate fabric parameters. Focusing on the later
convergence behavior on the right side of the plot, as the number of parameters
in the VQE gate fabric approaches the number of parameters in the FCI problem
(note that in this irrep there are $175$ configuration state functions (CSFs), see Appendix \ref{appendix:two_mode_fermions}.3), the error convergence turns
sharply downward. At 180 parameters we are able to achieve very tight convergence
to errors of $\sim 10^{-10}$ $E_{\mathrm{h}}$ relative to FCI, numerically
indicating the onset of universality. Figure~\ref{fig:experiments}b shows the
sorted power spectra of the computational basis state (determinant) amplitudes
of the various VQE gate fabrics and the FCI state. The exact zeros in
the FCI state amplitudes are an artifact of the $D_{\mathrm{2h}}$ spatial point
group symmetry of this molecule, which our VQE gate fabric was not optimized to
capture. Even for low VQE gate fabric circuit depths, we see that all
determinants are populated by nonzero amplitudes, with a compromise apparently
being made to allow for some nonzero error in all amplitudes to provide for the
best variational energy. As more layers are added to the gate fabrics, the
precision of the amplitude spectra increases, as indicated by, e.g., significant
attenuation of the symmetry-driven zero block of the amplitude spectrum.
The tail of amplitudes exactly zero in FCI is exactly extinguished in the VQE state only when numerical
universality is achieved at a 180-parameter VQE gate fabric. This behavior is
reminiscent of the nonzero but structured tensor factorized representation of
the determinant amplitudes in coupled cluster theories, where here the tensor
structure is provided by the local quantum gate fabric.
Moving to the larger test case of naphthalene, Figure~\ref{fig:experiments}c
tells a similar story as the corresponding plot for
p-benzyne. Here we see similar and roughly geometric convergence of energy error
vs. VQE gate fabric depth and parameter count, albeit with a smaller prefactor. As with p-benzyne,
the VQE gate fabric rather quickly outstrips both the HF and DOCI ans\"atze, which
its primitive gates are constituted from, and achieves chemical accuracy of
$\sim 1$ kcal mol$^{-1}$ in absolute energy at just $\sim 800-1000$ parameters (there
are $19404$ CSFs in this irrep, so universality is not reached for any of the depth explored here).
Figure~\ref{fig:experiments}d considers
convergence of the energy with respect to the L-BFGS epoch for a number of different
VQE gate fabric depths.
A first key finding is that making the gate fabric deeper
decreases the epoch count needed to converge to chemical accuracy.
A second key insight is that, while initialization strategy (B)
has shallower circuits and ultimately achieves lower energy error at very high
epoch count, plateaus are visible during the optimization with the L-BFGS
optimizer (Figure~\ref{fig:experiments}). Strategy (A), which exchanges
orbitals by means of the non trivial choice $\hat \Pi = \mathrm{QNP_{OR}}
(\pi)$, appears to circumvent the plateaus entirely and for deeper circuits
speeds up (the power-law like) convergence to below chemical accuracy.
The fabric presented here has favorable properties for implementation on NISQ hardware:
The $12$ qubit ansatz at $110$ parameters is without (with) $\hat \Pi$ gates
decomposable into elementary gates (2-qubit controlled Pauli and 1-qubit gates)
with resulting depth of $507$ ($617$). The $20$ qubit ansatz at $1080$
parameters without (with) $\hat \Pi$ gates has depth $2761$ ($3361$) in such
decomposition. To put this into perspective, a single trotter step of a naive
UCCSD circuit has gate depth $\approx 6600$ ($12$ qubits), respectively $\approx
57600$ ($20$ qubits). Another considerable advantage is that only $N-2$ unique
$4$-qubit gates have to be calibrated on hardware as the structure is repetitive
after the first two layers.
\section{Comparison with other entangler circuits}
Having numerically demonstrated the features of the VQE gate fabric, it is worth
considering the relationship of this gate fabric to other proposed VQE entangler
circuits. There has been substantial prior work along these lines in the past
few years.
For one instance, the hardware efficient ansatz \cite{Kandala2017,Bian2019} is
manifestly a local gate fabric, using essentially $\mathcal{SU}(4)$ entangler
elements or subsets thereof from the native gate set of the underlying
quantum circuit architecture. However, this gate fabric does not respect the
particle or spin quantum number symmetries, and therefore is likely to encounter
substantial difficulties in locating low-lying states within a target quantum
number irrep, particularly in larger active spaces.
In another direction, there are myriad proposed entangler circuit
constructions which are either already explicitly or in principle could be
adapted to real amplitudes and strict commutation with the number and/or
spin-squared symmetry operators, but which are either nonlocal circuits or
composed of heterogeneous gate layers. For instance, UCCSD
\cite{peruzzo2014variational,PhysRevX.6.031007,Ryabinkin2018}, (here referring
to the Trotterized version thereof) and its sparse and/or low-rank cousins
$k$-UpCCGSD \cite{lee2018generalized,huggins2020non}, ADAPT-VQE \cite{Grimsley2019}, and Jastrow-Factor VQE
\cite{matsuzawa2020jastrow} all may have the power to achieve
universality at sufficient depth, e.g., as proved in a recent analysis of
distangled UCC \cite{evangelista2019exact} and have been either partially or
completely symmetrized already. However, as written, all of these ans\"atze
require nonlocal gate elements that, e.g., mediate excitations among non-adjacent
spin orbitals in UCCSD, and thus are not gate fabrics. Moreover, many of these
constructions involve heterogeneous gate layers. For a canonical instance,
Jastrow-factor VQE \cite{matsuzawa2020jastrow} involves alternating circuit
layers of orbital rotations and substitutions (with the last of these being
nonlocal and complex-valued in the usual formulation). Of
all methods discussed in the prior literature, $k$-UpCCGSD is likely closest
to our proposed method, with products of single and diagonal double substitution
operators comprising the method. $k$-UpCCGSD as described in \cite{lee2018generalized} involves
nonlocal pair substitutions and therefore does not yield a local gate fabric.
Note however that the generalized swap network reformulation of $k$-UpCCGSD
described in and around Equation 12 and Figure 7 of \cite{o2019generalized}
(noticed after the first version of this manuscript was posted) appears to
realize $k$-UpCCGSD by means of a local circuit composed of 4-qubit gates that is
essentially a gate fabric.
Yet another interesting direction to consider is previously proposed true gate
fabrics that preserve quantum number symmetry, but do not achieve
universality. Orbital rotation fabrics \cite{
reck1994experimental,
wecker2015solving,
google2020hartree},
(i.e., Hartree-Fock) are clearly one example here, but so too is doubly
occupied configuration interaction (DOCI), for which a gate fabric was developed
with the pUCCD ansatz \cite{elfving2020simulating}. Both of these ans\"atze have the
interesting property that they can be mapped into gate fabrics requiring only
$M$ qubits, but both fail to reach FCI universality as the parameter depth is
increased.
Another interesting gate fabric construction is the ``gate-efficient ansatz''
presented in \cite{ganzhorn2019gate}, which presents as a gate fabric that
preserves total particle number $\hat N_{\alpha} + N_{\beta}$, but does not
appear to respect high-spin particle number $\hat N_{\alpha} - \hat N_{\beta}$
or $\hat S^2$ symmetry.
Yet another interesting entangler is the ``qubit coupled cluster'' approach
presented in \cite{xia2020qubit}, which essentially implements a partial spin
adaption of UCCSD to preserve $\hat N_{\alpha}$ and $\hat N_{\beta}$ symmetry
within the single and double excitation operations, but neither preserves $\hat
S^2$ symmetry nor attains the structure of a local gate fabric.
Another related approach presented in \cite{yordanov2020efficient} constructs
fermion-adapted excitation operators which preserve particle number symmetry
but not spin symmetry, and additionally are aimed at optimizing the number of
CNOT gates in non-gate-fabric UCCSD methods.
An entangler that has the potential to preserve all quantum number symmetries
with additional spin-adaption work is the QAOA-inspired Pauli-term approach,
presented in \cite{salis2019short}, but this approach yields highly nonlocal
circuits which do not resemble gate fabrics.
Another approach which has some intersection with the present work is the
correlating antisymmetric geminal power (AGP) approach explored in
\cite{khamoshi2020correlating}, which first implements classically-tractable
APG to provide state preparation in quantum circuits and then augments AGP with
an anti-Hermitian pair hopping entangler which resembles our pair exchange gate.
However, the correlating AGP is not written in the form of a local gate fabric.
Another interesting direction to explore that we propose here is alternative local gate fabrics that
fully preserve quantum number symmetry, but which exhibit different gate
constructions than the $\mathrm{QNP_{PX}}$ and $\mathrm{QNP_{OR}}$ gates used in
Figure~\ref{fig:F}. Examples of such gate fabrics using generic 4-qubit 5-parameter
FCI gates and decompositions of these gates into $\mathrm{QNP_{PX}}$,
$\mathrm{QNP_{OR}}$, 1-hole/particle substitution, and pair-break up/down gates
are described in Appendix~\ref{appendix:other_qnp_gate_fabrics}.
One additional interesting direction is the ``symmetry preserving state
preparation circuits'' of \cite{gard2020efficient}. This work primarily focuses
on total number symmetry, but does introduce a four-qubit gate that preserves
particle and spin quantum numbers via a hyperspherical parametrization.
Note that an alternative approach to the exact
symmetry preservation explored in this manuscript is symmetry projection
\cite{tsuchimochi2020spin,lacroix2020symmetry}, which often requires ancilla
qubits and extra measurements due to the necessarily non-unitary nature of the
projection operation.
\section{Summary and Outlook}
In this work, we set out to construct doppelg\"angers of the well known gate
fabric (i.e., a potentially infinitely repeatable, simple and geometrically
local pattern of gate elements that span the parent group at sufficient depth)
for the unrestricted qubit Hilbert space
$\mathcal{SU}(2^{N})$
consisting of simple 2-qubit gate elements $SU(4)$.
Our major result is the construction of a gate fabric for the
important special case of spin-1/2 fermionic systems in the Jordan-Wigner representation
$\mathcal{F} (2^{2M})$
consisting of simple 4-qubit gate elements $\hat Q$.
Each 2-parameter $\hat Q$ gate comprises a
1-parameter spatial orbital rotation gate $\mathrm{QNP_{OR}} (\varphi)$
and a 1-parameter diagonal pair exchange gate $\mathrm{QNP_{PX}}
(\theta)$.
A fabric made of either of these gate elements alone does not achieve FCI
universality with sufficient parameter depth, but our VQE gate fabric, being an amalgamation
of the two appears to be able to do so.
Moreover, at intermediate depths, the VQE gate fabric appears to be
pragmatically expressive as evidenced by tests of the ground state energy
convergence in strongly correlated molecular systems.
It is worth emphasizing that these properties seem to hold in the vast bulk of
quantum number irreps, i.e., that these fabric circuits can be applied for cases
where $S \neq 0$ (including even or odd spin cases) and/or where $N_{\alpha}
\neq N_{\beta}$ (see Appendix~\ref{sec:additional numerics} for details on
specific high-spin edge cases that are not universal with the $\hat Q$-type QNP
gate fabrics of the main text, but that can be addressed with elements of the
$\hat F$-type QNP gate fabrics of
Appendix~\ref{appendix:other_qnp_gate_fabrics}).
Many important questions remain
regarding our quantum number preserving gate fabrics. These include:
(1) How does the numerical optimization of parameters for such gate fabrics behave in the
presence of shot and/or decoherence noise?
(2) How can numerical optimization algorithms be adapted to exploit the
knowledge that the VQE entangler circuit is a gate fabric?
(3) Is the fixed $\hat \Pi$ gate construction or an extension thereof an
effective way to mitigate barren plateaus during numerical optimization?
(4) How does the VQE gate fabric perform for relative properties, for properties
at different nuclear geometries, and for properties in different quantum number
irreps?
(5) Is the construction of the VQE gate fabric in terms of $\hat Q$ gates
optimal, or do more elaborate constructions, e.g., using the $\hat F$ gates of
Appendix \ref{appendix:other_qnp_gate_fabrics} provide additional benefits?
(6) What is the scaling behavior of the error in absolute and/or relative
properties as a function of parameter depth for representative interesting
molecular systems?
(7) Can the gate fabric be adapted to additionally exploit external symmetries
such as spatial point group symmetries, e.g., as explored in
\cite{setia2020reducing}? Taken together, the results of this work might
provide an interesting guide for the required symmetries and limiting
simplicities when constructing more elaborate VQE entanglers for fermionic
systems.
\textbf{Acknowledgements:} RMP is grateful to Dr.~Edward Hohenstein for many
discussions on the structure of the $\hat S^2$ operator. The authors further
thank Fotios Gkritsis for discussions. QC Ware Corp. acknowledges generous
research funding from Covestro Deutschland AG for this project. Covestro
acknowledges funding from the German Ministry for Education and Research (BMBF)
under the funding program quantum technologies as part of project HFAK
(13N15630). DW acknowledges funding by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Strategy Cluster of
Excellence Matter and Light for Quantum Computing (ML4Q) EXC2004/1 390534769.
\textbf{Conflict of Interest:} The VQE gate fabrics described in this work are
elements of two US provisional applications for patents both filed jointly
by QC Ware Corp. and Covestro Deutschland AG. RMP owns stock/options in QC Ware
Corp.
\bibliographystyle{unsrturl}
|
1,116,691,500,786 | arxiv | \section{Introduction}
\label{intro}
Weak gravitational lensing refers to small but coherent shape distortions (cosmic shear) of background galaxies caused by gravity. It provides a direct way of probing the cosmic structures on large scales \citep{bs01,refregier03,hj08,kilbinger15}. A number of ongoing galaxy surveys are focusing on the measurement of weak lensing statistics with a large ensemble of galaxy images, for the purpose of better understanding the cosmic evolution history and the nature of dark matter and dark energy (e.g. , DES\footnote{http://www.darkenergysurvey.org/} , HSC\footnote{http://www.naoj.org/Projects/HSC/}, KIDs\footnote{http://www.astro-wise.org/projects/KIDS/}, LSST\footnote{http://www.lsst.org/lsst}, WFIRST\footnote{http://wfirst.gsfc.nasa.gov/} ).
Currently, a great deal of efforts in the field of weak lensing are on constructing unbiased cosmic shear estimators. This is challenging due to a number of facts involved in the image formation process of modern CCD cameras, including the point spread function (PSF), the pixelation effect, the photon noise, etc.. Many different algorithms have been proposed, tested in recent open tests, and used on real galaxy data \citep{mandelbaum15}. At this stage, it is timely to raise a related question: what is the best way of taking the ensemble average of the shear estimators, or their products (for shear-shear correlation)?
Cosmic shear is typically estimated with galaxy ellipticities. It is known that when the ensemble average of shear estimators is taken, the statistical uncertainty on the shear signal can be suppressed if larger ellipticities are weighted less, as they contribute larger shape noises than average. \citealt{bj02} (BJ02 hereafter) shows that a weighting scheme based on the probability-distribution-function (PDF) of the galaxy ellipticities can be used to achieve optimal statistical uncertainty, or the Cram\'{e}r-Rao bound (called C-R bound hereafter), given that the shear response function is properly calculated and multiplied on the ensemble average. Nevertheless, a successful application of the BJ02 idea relies on accurate measurement of galaxy ellipticities, which is unfortunately difficult in practice. For example, ellipticities estimated in the model-fitting methods generally contain biases due to noise and underfitting of galaxy morphologies \citep{vb10,bernstein10,refregier12}.
Zhang et al.(2015) (ZLF15 hereafter) proposes an alternative form of shear estimators using the multipole moments of the galaxy power spectrum. The new method does not make assumptions on the morphologies of the galaxy or the PSF, therefore does not have the underfitting problem. The contribution of background noise to the shear estimator can be removed statistically using a neighboring noise image, and the Poisson noise contribution can also be removed directly in Fourier space. These features motivate us to understand how to approach the C-R bound with the ZLF15 shear estimators.
Instead of taking a weighted sum, we find that one can recover the shear signal by symmetrizing the PDF of the ZLF15 shear estimators, or the joint PDF of the shear estimator pairs for measuring shear-shear correlations. It turns out that the new method allows us to approach the C-R bound without incuring systematic errors. This is realized under very general observational conditions, and without prior knowledge of the PDF. In \S\ref{MLE}, we introduce the C-R bound, and a way of realizing it through nulling of the PDF asymmetry. \S\ref{realizing_MLE_estimator} shows how to apply the PDF symmetrization method on shear estimators of ZLF15, thereby to approach the C-R bound in shear statistics, including the recovery of constant shear and shear-shear correlation. Numerical examples/proves are shown in \S\ref{numerical} using mock galaxies of very general conditions. We conclude and discuss the application of this new method in \S\ref{summary}.
\section{PDF Symmetrization Method}
\label{MLE}
\subsection{The Cram\'{e}r-Rao Bound}
\label{example}
For simplicity, let us consider $N$ random numbers $x_i$ ($i=1, 2, \cdots, N$) with an intrinsically symmetric PDF denoted as $P(x)$. Each random number is shifted by a small amount $g (\ll \sqrt{\langle x_i^2\rangle })$. Note that this situation is very similar to the case of shear estimator: $x_i$ is analogous to the galaxy ellipticity, and $g$ can be regarded as the cosmic shear signal. According to the Maximum Likelihood Estimation, an estimator ${\hat{g}}$ of $g$ is given by:
\begin{equation}
\label{MLE_ave}
0=\frac{d}{d{\hat{g}}}\sum_i\ln P(x_i-{\hat{g}})
\end{equation}
The C-R bound for the variance of ${\hat{g}}$ is then given by:
\begin{equation}
\label{MLE_error}
\sigma_{{\hat{g}}}^{-2}=-\sum_i\frac{\partial^2\ln P(x_i-{\hat{g}})}{\partial {\hat{g}}^2}
\end{equation}
As a result, eq.(\ref{MLE_ave}) yields an estimate of $g$ as:
\begin{equation}
\label{MLE_ave3}
{\hat{g}}=\frac{\sum_i P'(x_i)P(x_i)^{-1}}{\sum_i\left[P''(x_i)P(x_i)^{-1}-P'(x_i)^2P(x_i)^{-2}\right]}
\end{equation}
It is straightforward to show that eq.(\ref{MLE_ave3}) is unbiased. Meanwhile, the C-R bound for the variance can be derived from eq.(\ref{MLE_error}) as:
\begin{eqnarray}
\label{MLE_error2}
\sigma_{{\hat{g}}}^{-2}&=&-\sum_i\frac{P''(x_i)P(x_i)-P'(x_i)^2}{P(x_i)^2}\\ \nonumber
&=&-N\int dx \frac{P''(x)P(x)-P'(x)^2}{P(x)}\\ \nonumber
&=&N\int dx \frac{P'(x)^2}{P(x)}
\end{eqnarray}
Note that here and in the rest of the paper, to simplify the notation, an integration symbol without the lower and upper limits refers to integration from negative infinity to positive infinity. The numerator in eq.(\ref{MLE_ave3}) can be regarded as a weighted sum of the data $x_i$ with the weighting function given by $P'(x_i)[P(x_i)x_i]^{-1}$, and the denominator as the sum of the weighting function multiplied by a correction factor (or response function), similar to the discussion in BJ02. However, to reach the optimal statistical uncertainty given in eq.(\ref{MLE_error2}) and an unbiased estimate of the signal $g$ simultaneously, it requires an accurate knowledge of the PDF of the data $x_i$, which is difficult if the amount of data is not large enough. More importantly, if one thinks of $x_i$ as the galaxy ellipticity plus some additional measurement errors, eq.(\ref{MLE_ave3}) becomes a biased estimator of $g$ even when the measurement error has zero mean. Due to the nonlinearity of eq.(\ref{MLE_ave3}), the correction of such a bias must be complicated. It is therefore interesting to ask if there is a way to approach the C-R bound with less stringent requirements.
\subsection{Nulling of the PDF Asymmetry}
\label{nulling}
In the example of \S\ref{example}, $g$ causes asymmetric distribution of the measured values of $x_i$ with respect to zero. This fact suggests that one can estimate $g$ by asking how much we shall shift each $x_i$ to symmetrize the PDF of the data. To be more specific, let us set up bins for the data that are symmetrically placed around zero. We define $u_{j-1}$ and $u_j$ as the two boundaries of the $j^{th}$ bin. For convenience, we let half of the bin indices to take negative values, with $u_j=\Delta*j$ for $j=-(M-1), \cdots, -1, 0, 1, \cdots, M-1$, and $u_{\pm M}=\pm\infty$, where $\Delta$ is the bin size, and $2M$ is the total bin number. Assuming the shifted amount is ${\hat{g}}$, the number of data $x_i-{\hat{g}}$ that fall to the $j^{th}$ bin on the right side of $0$ is defined as:
\begin{equation}
n_j=\sum_i H(x_i-{\hat{g}}-u_{j-1})H(u_j-x_i+{\hat{g}}) \,\,\;(j> 0)
\end{equation}
where we have used the Heaviside step function $H$, and the subindex $i$ covers all data ID's. The bins of negative indices satisfy:
\begin{equation}
n_{-j}=\sum_i H(x_i-{\hat{g}}-u_{-j})H(u_{-j+1}-x_i+{\hat{g}}) \,\,\;(j> 0)
\end{equation}
Note that to avoid confusion, we explicitly write out the formulae for bins of positive and negative indices respectively.
To estimate the value of ${\hat{g}}$ that can maximally symmetrize the distribution of $x_i-{\hat{g}}$ with respect to zero, we form the $\chi^2$ as follows:
\begin{equation}
\label{chi2}
\chi^2=\frac{1}{2}\sum_{j> 0}\frac{(n_j-n_{-j})^2}{n_j+n_{-j}}
\end{equation}
where we assume that the fluctuation of $n_j$ obeys Poisson statistics, so that $\langle (n_j-n_{-j})^2\rangle \approx n_j+n_{-j}$. ${\hat{g}}$ is estimated by minimizing $\chi^2$. Let us now show that such an estimator is unbiased. For this purpose, we assume that the number of measurements is large, so that $n_j$ and $n_{-j}$ can be written as integrations ($j> 0$):
\begin{eqnarray}
&&n_j=N_T\int_{u_{j-1}+\Delta g}^{u_j+\Delta g}dx P(x)\\ \nonumber
&&n_{-j}=N_T\int_{u_{-j}+\Delta g}^{u_{-j+1}+\Delta g}dx P(x)
\end{eqnarray}
where $\Delta g={\hat{g}}-g$, $N_T$ is the total number of data points, and $P(x)$ is the original (symmetric) PDF of $x$ when $g=0$. With Taylor expansion, and keep terms up to the second order in $\Delta g$, we get:
\begin{eqnarray}
\frac{n_{j}}{N_T}&=&\int_{u_{j-1}}^{u_j}dx P(x)+\left[P(u_j)-P(u_{j-1})\right]\Delta g\\ \nonumber
&+&\frac{1}{2}\left[P'(u_j)-P'(u_{j-1})\right]\Delta g^2\\ \nonumber
\frac{n_{-j}}{N_T}&=&\int_{u_{-j}}^{u_{-j+1}}dx P(x)+\left[P(u_{-j+1})-P(u_{-j})\right]\Delta g\\ \nonumber
&+&\frac{1}{2}\left[P'(u_{-j+1})-P'(u_{-j})\right]\Delta g^2
\end{eqnarray}
Since $P(x)$ is a symmetric function, we must have $P(u_j)=P(u_{-j})$ and $P'(u_j)=-P'(u_{-j})$. Therefore,
\begin{equation}
n_j-n_{-j}=2N_T\left[P(u_j)-P(u_{j-1})\right]\Delta g
\end{equation}
Consequently, we have:
\begin{equation}
\chi^2=2N_T^2\sum_{j>0}\frac{\left[P(u_j)-P(u_{j-1})\right]^2({\hat{g}}-g)^2}{n_j+n_{-j}}
\end{equation}
which shows that when ${\hat{g}}=g$, $\chi^2$ reaches its minimum, meaning that the best fit value of ${\hat{g}}$ is an unbiased estimator of $g$. $\chi^2$ can be rewritten as:
\begin{equation}
\chi^2=\frac{({\hat{g}}-g)^2}{2\sigma_{{\hat{g}}}^2}
\end{equation}
where
\begin{equation}
\sigma_{{\hat{g}}}^{-2}=4N_T^2\sum_{j> 0}\frac{\left[P(u_j)-P(u_{j-1})\right]^2}{n_j+n_{-j}}
\end{equation}
In the limit of small bin size $\Delta$, we have $n_j\approx n_{-j}\approx N_TP(u_j)\Delta$, therefore,
\begin{equation}
\frac{\sigma_{{\hat{g}}}^{-2}}{N_T}\approx 2\sum_{j> 0}\frac{\left[P(u_j)-P(u_{j-1})\right]^2}{P(u_j)\Delta}\approx\int dx\frac{P'(x)^2}{P(x)}
\end{equation}
which recovers the C-R bound given in eq.(\ref{MLE_error2}) in the limit of small bin size. The above calculation shows that one can approach the C-R bound by symmetrizing the PDF of the data. It only requires binning the data symmetrically with respect to zero, and a reasonably large bin number.
As examples, we consider three different types of PDF:
\begin{eqnarray}
\label{Ps}
P_1(x)&=&\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{x^2}{2}\right)\\ \nonumber
P_2(x)&=&\frac{2}{\pi}(1+x^2)^{-2}\\ \nonumber
P_3(x)&=&\frac{\vert x\vert^{-2/3}}{3\sqrt{2\pi}}\exp\left(-\frac{\vert x\vert^{2/3}}{2}\right)
\end{eqnarray}
The C-R bounds (labelled as 'CR') and the variances $\sigma^2_{1,2,3}$ using the direct averaging method (labelled as 'Ave') can be worked out for the three cases respectively as:
\begin{eqnarray}
\label{sigmas}
N_T\sigma_1^2(\mathrm{Ave})&=&1,\quad N_T\sigma_1^2(\mathrm{CR})=1,\\ \nonumber
N_T\sigma_2^2(\mathrm{Ave})&=&1,\quad N_T\sigma_2^2(\mathrm{CR})=0.5,\\ \nonumber
N_T\sigma_3^2(\mathrm{Ave})&=&15,\quad N_T\sigma_3^2(\mathrm{CR})\rightarrow 0.
\end{eqnarray}
To test the PDF symmetrization method (called 'PDF-SYM' hereafter), we set the signal $g=0.01$, and use $10^7$ data points to recover the signal in each example. The results are shown in table \ref{result_test}. Note that the number in the parentheses at the end of each result refers to the statistical error on the last digit. We use this format for the notation of statistical uncertainty all through this paper. The corresponding variances $N_T\sigma^2_{1,2,3}$ in the two methods are listed in table \ref{sigma_test}. The results in the tables agree with our theoretical expectations. PDF-SYM can indeed make the statistical uncertainty approach the C-R bound when the bin number is large. In practice, $8-10$ bins are usually good enough for the purpose, unless the PDF has a number of nonmonotonic features. The figure shows that even $2$ bins can be used in PDF-SYM. This is useful when the number of data points is small ($\lsim 100$).
Note that in making the bins, one should guarantee that each bin to have more than roughly $100$ samples, so that $\chi^2$ has a smooth dependence on the assumed value of the signal ${\hat{g}}$, leading to a reliable determination of the $\chi^2$ minimum and the uncertainty of the recovered signal. The boundaries between the bins can be determined by sorting the samples according to their absolute values, and making the sample number in each bin on the positive side of zero roughly the same. The bins on the negative side are then symmetrically set up.
It is interesting to note that when the PDF has a singular behavior at the origin, such as $P_3(x)$ defined in eq.(\ref{Ps}), the C-R bound approaches zero. The results in table \ref{result_test} \& \ref{sigma_test} confirm this fact. Indeed, according to eq.(\ref{MLE_error2}), this phenomenon can occur whenever the intrinsic PDF contain sharp peaks, located either at zero, or symmetrically at the two sides of zero. This phenomenon has been previously mentioned in BJ02. It is a very useful feature in signal recovery.
As shown next, the PDF symmetrization procedure allows us to approach the C-R bound in shear measurement with shear estimators defined in ZLF15. The abovementioned advantages of the new method can be achieved under very general observational conditions.
\begin{table*}[!htb]
\centering
\caption{\textnormal{The results of signal recovery (input value is 0.01) for $10^7$ data points of three types of PDF's defined in eq.(\ref{Ps}). }}
\begin{tabular}{llllll}
\hline\hline
Results: & Averaging & PDF-SYM (2 bins) & PDF-SYM (8 bins) & PDF-SYM (16 bins) & PDF-SYM (32 bins) \\
\hline\hline
$P_1$ & 0.0102(3) & 0.0104(4) & 0.0101(3) & 0.0100(4) & 0.0102(3) \\
\hline
$P_2$ & 0.0099(3) & 0.0101(2) & 0.0101(2) & 0.0100(2) & 0.0101(2)\\
\hline
$P_3$ & 0.011(1) & 0.0099999998(2) & 0.0099999998(1) & 0.0099999998(1) & 0.0099999999(2) \\
\hline\hline
\end{tabular}
\footnote{The number in the parentheses at the end of each result refers to the statistical error on the last digit.}
\label{result_test}
\end{table*}
\begin{table*}[!htb]
\centering
\caption{\textnormal{The measured average variances $N_T\sigma^2$ of three types of PDF's defined in eq.(\ref{Ps}).} }
\begin{tabular}{llllll}
\hline\hline
$N_T\sigma^2$: & Averaging & PDF-SYM (2 bins) & PDF-SYM (8 bins) & PDF-SYM (16 bins) & PDF-SYM (32 bins) \\
\hline\hline
$P_1$ & 1.0 & 1.6 & 1.1 & 1.2 & 0.96 \\
\hline
$P_2$ & 0.99 & 0.61 & 0.52 & 0.50 & 0.57\\
\hline
$P_3$ & 15 & $5\times 10^{-13}$ & $2\times 10^{-13}$ & $2\times 10^{-13}$ & $3\times 10^{-13}$\\
\hline\hline
\end{tabular}
\label{sigma_test}
\end{table*}
\section{PDF-SYM in Shear Measurement}
\label{realizing_MLE_estimator}
\subsection{Shear Estimator}
\label{estimator}
Let us define the intrinsic galaxy surface brightness distribution before lensing as $f_I(\vec{x}^I)$, the lensed galaxy (before being processed by the PSF) as $f_L(\vec{x}^L)$, and the observed image as $f_O(\vec{x}^O)$, where $\vec{x}^S$ is the coordinate in the source plane, and $\vec{x}^L$ and $\vec{x}^O$ are the positions in the image plane \citep{jz11}. We have the following relations:
\begin{eqnarray}
&&f_L(\vec{x}^L)=f_I(\vec{x}^I), \quad\quad \vec{x}^I={\mathbf M}\vec{x}^L, \nonumber \\
&&f_O(\vec{x}^O)=\int d^2\vec{x}^L W_{\beta}(\vec{x}^O-\vec{x}^L)f_L(\vec{x}^L),
\label{define1}
\end{eqnarray}
where $W_{\beta}$ is the isotropic Gaussian PSF defined as:
\begin{equation}
\label{beta}
W_{\beta}(\vec{x})=\frac{1}{2\pi\beta^2}\exp\left(-\frac{\left\vert\vec{x}\right\vert^2}{2\beta^2}\right).
\end{equation}
For now, let us only consider the case of isotropic Gaussian PSF, and no noise. ${\mathbf M}$ is the lensing distortion matrix typically defined as: ${\mathbf M}_{ij}=\delta_{ij}-\phi_{ij}$ with $\phi_{ij}=\delta_{ij}-\partial x^I_i/\partial x^L_j$ being the spatial derivatives of the lensing deflection angle. $\phi_{ij}$ is often replaced by the convergence $\kappa$ [$=(\phi_{11}+\phi_{22})/2$] and the two shear components $\gamma_1$ [$=(\phi_{11}-\phi_{22})/2$] and $\gamma_2$ ($=\phi_{12}$). The reduced shears are defined as $g_{1,2}=\gamma_{1,2}/(1-\kappa)$. The multipole moments of the galaxy power spectrum are defined as:
\begin{eqnarray}
\label{defineP}
P_{ij}&=&\int d^2\vec{k}k_1^ik_2^j\left\vert\widetilde{f_O}(\vec{k})\right\vert^2,\nonumber \\
D_n&=&\int d^2\vec{k}\left\vert\vec{k}\right\vert^n\left\vert\widetilde{f_O}(\vec{k})\right\vert^2.
\end{eqnarray}
The dependence of $P_{ij}$ on the cosmic shear can be worked out directly as:
\begin{eqnarray}
\label{pldl}
P_{ij}&=&\vert \mathrm{det} (\mathbf{M}^{-1})\vert^2\int d^2\vec{k}k_1^ik_2^j\left\vert\widetilde{W}_{\beta}(\vec{k})\widetilde{f_I}(\mathbf{M}^{-1}\vec{k})\right\vert^2\\ \nonumber
&=&\vert\mathrm{det} (\mathbf{M}^{-1})\vert\int d^2\vec{k}(\mathbf{M}\vec{k})_1^i(\mathbf{M}\vec{k})_2^j\left\vert\widetilde{W}_{\beta}(\mathbf{M}\vec{k})\widetilde{f_I}(\vec{k})\right\vert^2.
\end{eqnarray}
The last step is achieved by re-defining $\mathbf{M}^{-1}\vec{k}$ as $\vec{k}$. For convenience in the rest of our calculation, we define the galaxy multipole moments in the absence of lensing as:
\begin{eqnarray}
\label{pldl_I}
P_{ij}^I&=&\int d^2\vec{k}k_1^ik_2^j\left\vert\widetilde{W}_{\beta}(\vec{k})\widetilde{f_I}(\vec{k})\right\vert^2\\ \nonumber
D_n^I&=&\int d^2\vec{k}\left\vert \vec{k}\right\vert^n\left\vert\widetilde{W}_{\beta}(\vec{k})\widetilde{f_I}(\vec{k})\right\vert^2
\end{eqnarray}
Expanding eq.(\ref{pldl}) up to the first order in shear/convergence, we get:
\begin{eqnarray}
&&P_{20}-P_{02}\\ \nonumber
&=&P_{20}^I-P_{02}^I-2g_1D_2^I+\beta^2\left[g_1D_4^I+2\kappa(P_{40}^I-P_{04}^I)\right.\\ \nonumber
&+&\left.g_1(P_{40}^I-6P_{22}^I+P_{04}^I)+4g_2(P_{31}^I-P_{13}^I)\right]
\end{eqnarray}
\begin{eqnarray}
&&2P_{11}\\ \nonumber
&=&2P_{11}^I-2g_2D_2^I+\beta^2\left[g_2D_4^I+4\kappa(P_{13}^I+P_{31}^I)\right.\\ \nonumber
&-&\left.g_2(P_{40}^I-6P_{22}^I+P_{04}^I)+4g_1(P_{31}^I-P_{13}^I)\right]
\end{eqnarray}
The shear estimators can therefore be defined as:
\begin{eqnarray}
\label{shearFourier}
\frac{1}{2}\frac{\left\langle P_{20}-P_{02}\right\rangle }{\left\langle D_2-\beta^2D_4/2\right\rangle }&=&-g_1,\nonumber \\
\frac{\left\langle P_{11}\right\rangle }{\left\langle D_2-\beta^2D_4/2\right\rangle }&=&-g_2,
\end{eqnarray}
The formulae are generalized in ZLF15 to take into account the conversion of the PSF form and the correction of the noise contribution. Three components ($G_1$, $G_2$, $N$) are defined as the multipole moments of the power spectrum of the galaxy image in Fourier space:
\begin{eqnarray}
\label{shear_estimator}
G_1&=&-\frac{1}{2}\int d^2\vec{k}(k_x^2-k_y^2)T(\vec{k})M(\vec{k})\\ \nonumber
G_2&=&-\int d^2\vec{k}k_xk_yT(\vec{k})M(\vec{k})\\ \nonumber
N&=&\int d^2\vec{k}\left[k^2-\frac{\beta^2}{2}k^4\right]T(\vec{k})M(\vec{k})
\end{eqnarray}
where
\begin{eqnarray}
\label{TM}
&&T(\vec{k})=\left\vert\tilde{W}_{\beta}(\vec{k})\right\vert^2/\left\vert\tilde{W}_{PSF}(\vec{k})\right\vert^2\\ \nonumber
&&M(\vec{k})=\left\vert\tilde{f}^S(\vec{k})\right\vert^2-F^S-\left\vert\tilde{f}^B(\vec{k})\right\vert^2+F^B
\end{eqnarray}
\begin{equation}
\label{shear_estimator_dis_para4}
F^S=\frac{\int_{\vert\vec{k}\vert > k_c} d^2\vec{k}\left\vert\tilde{f}^S(\vec{k})\right\vert^2}{\int_{\vert\vec{k}\vert > k_c} d^2\vec{k}}, \;\;\; F^B=\frac{\int_{\vert\vec{k}\vert > k_c} d^2\vec{k}\left\vert\tilde{f}^B(\vec{k})\right\vert^2}{\int_{\vert\vec{k}\vert > k_c} d^2\vec{k}}
\end{equation}
and $\tilde{f}^S(\vec{k})$ and $\tilde{f}^B(\vec{k})$ are the Fourier transformations of the galaxy image and a neighboring image of background noise respectively. The two additional terms $F^S$ and $F^B$ are estimates of the Poisson noise power spectra on the source and background images respectively. The critical wave number $k_c$ is chosen to be large enough to avoid the regions dominated by the source power. The factor $T(\vec{k})$ is used to convert the form of the PSF to the desired isotropic Gaussian function for correcting the PSF effect. $\beta$ should be somewhat larger than the scale radius of the original PSF to avoid singularities in the conversion. It is shown in ZLF15 that the ensemble averages of the shear estimators defined above do recover the shear values to the second order in accuracy (assuming that the intrinsic galaxy images are statistically isotropic), {\it i.e. },
\begin{equation}
\label{shear_measure}
\frac{\left\langle G_1\right\rangle }{\left\langle N\right\rangle }=g_1+O(g_{1,2}^3),\;\;\;\frac{\left\langle G_2\right\rangle }{\left\langle N\right\rangle }=g_2+O(g_{1,2}^3)
\end{equation}
Note that the ensemble averages are taken for $G_1$, $G_2$, and $N$ separately \citep{zk11}.
\subsection{Constant Shear Recovery with PDF-SYM}
\label{const_shear}
Eq.(\ref{shear_measure}) uses unweighted sums of the unnormalized galaxy moments. This is far from optimal, as the measurement is dominated by bright galaxies \citep{bernstein16}. Even if the shear estimators are normalized by the galaxy flux (squared), as we will show later, there is still space to further improve the statistical uncertainty. We now show that the C-R bound can be approached by symmetrizing the PDF's of $G_1$ and $G_2$.
Intuitively, according to eq.(\ref{shear_measure}), one may think that the PDF's of $G_1-g_1N$ and $G_2-g_2N$ are symmetric with respect to zero. However, as shown below, $\langle G_i-g_iN\rangle =0$ does not guarantee that the PDF of $G_i-g_iN$ is symmetric. It is therefore necessary and interesting to dig out some details in the shear estimators of ZLF15. To do so, we should first understand the parity properties of $P_{20}-P_{02}+g_1(2D_2-\beta^2D_4)$ and $2P_{11}+g_2(2D_2-\beta^2D_4)$.
The quantities of our interests can be worked out as:
\begin{eqnarray}
\label{wrong1}
&&P_{20}-P_{02}+g_1(2D_2-\beta^2D_4)\\ \nonumber
&=&P_{20}^I-P_{02}^I+\beta^2\left[2\kappa(P_{40}^I-P_{04}^I)\right.\\ \nonumber
&+&g_1(P_{40}^I-6P_{22}^I+P_{04}^I)+\left.4g_2(P_{31}^I-P_{13}^I)\right]
\end{eqnarray}
\begin{eqnarray}
\label{wrong2}
&&2P_{11}+g_2(2D_2-\beta^2D_4)\\ \nonumber
&=&2P_{11}^I+\beta^2\left[4\kappa(P_{31}^I+P_{13}^I)\right.\\ \nonumber
&-&g_2(P_{40}^I-6P_{22}^I+P_{04}^I)+\left.4g_1(P_{31}^I-P_{13}^I)\right]
\end{eqnarray}
The above equations indicate that the PDF of $P_{20}-P_{02}+g_1(2D_2-\beta^2D_4)$ and $2P_{11}+g_2(2D_2-\beta^2D_4)$ are not exactly symmetric with respect to zero. This is due to the presence of the $P_{40}^I-6P_{22}^I+P_{04}^I$ term in both equations. The rest terms on the right sides of the equations have symmetrized PDF assuming the intrinsic galaxy images have parity symmetry statistically. It is straightforward to show that the PDF's of the following terms are symmetric:
\[P_{20}-P_{02}+g_1[2D_2-\beta^2(D_4+P_{40}-6P_{22}+P_{04})]\] and
\[2P_{11}+g_2[2D_2-\beta^2(D_4-P_{40}+6P_{22}-P_{04})]\]. In Appendix A, we show that the PDF's of these two quantities remain symmetric to the second order in shear/convergence. Note that $[P_{40}-6P_{22}+P_{04},4(P_{31}-P_{13})]$ form a pair of spin-4 quantities under spatial rotation. Their presence does not affect the ensemble average, but modifies the parity property of the PDF.
Based on the above calculation, we conclude that to use PDF-SYM, we need to define two more terms in addition to those in eq.(\ref{shear_estimator}):
\begin{eqnarray}
\label{def_U}
U&=&-\frac{\beta^2}{2}\int d^2\vec{k}\left(k_x^4-6k_x^2k_y^2+k_y^4\right)T(\vec{k})M(\vec{k})\\ \nonumber
V&=&-2\beta^2\int d^2\vec{k}\left(k_x^3k_y-k_xk_y^3\right)T(\vec{k})M(\vec{k})
\end{eqnarray}
It is straightforward to show that the PDF's of $G_1-g_1(N+U)$ and $G_2-g_2(N-U)$ are symmetric with respect to zero. Note that $V$ is kept for transforming $U$ in case of coordinate rotation in shear measurement.
Let us show how to measure, e.g., the first component of shear, with PDF-SYM. For convenience, let us define $B=N+U$, $G_1^S=G_1-g_1B$, and the PDF as $P_S(G_1^S,B)$, with $P_S(G_1^S,B)=P_S(-G_1^S,B)$. Note that $B$ is an observable, but $G_1^S$ is not. An observable can be defined as ${\hat{G}}_1=G_1-{\hat{g}}_1B$, in which ${\hat{g}}_1$ is the assumed (pseudo) value of the shear component. ${\hat{G}}_1$ is related to $G_1^S$ through:
\begin{equation}
{\hat{G}}_1=G_1^S+(g_1-{\hat{g}}_1)B
\end{equation}
Define the PDF of ${\hat{G}}_1$ as $P({\hat{G}}_1)$. We have the following relation:
\begin{eqnarray}
&&P({\hat{G}}_1)\\ \nonumber
&=&\int dB\int dG_1^SP_S(G_1^S,B)\delta_D\left[{\hat{G}}_1-G_1^S-(g_1-{\hat{g}}_1)B\right]\\ \nonumber
&=&\int dBP_S\left[{\hat{G}}_1-(g_1-{\hat{g}}_1)B,B\right]
\end{eqnarray}
which is not symmetric as long as ${\hat{g}}_1\ne g_1$. This fact allows us to find the unbiased estimate of $g_1$ by searching for the value of ${\hat{g}}_1$ that can best symmetrize $P({\hat{G}}_1)$.
For this purpose, we can set up bins for ${\hat{G}}_1$'s of the galaxies that are symmetrically placed with respect to zero. The number of data that fall to the $i^{th}$ bin on the right side of zero is defined as:
\begin{equation}
n_i=\sum_j H({\hat{G}}_1^j-u_{i-1})H(u_i-{\hat{G}}_1^j) \,\,\;(i> 0)
\end{equation}
where $u_{i-1}$ and $u_i$ are the two boundaries of the $i^{th}$ bin, and the upper index $j$ covers all galaxy ID's. Similar to the setup of \S\ref{nulling}, we let half of the bin indices to take negative values, with $u_i=-u_{-i}$, and $i=-(M-1), \cdots, -1, 0, 1, \cdots, M-1$, and $u_{\pm M}=\pm\infty$, assuming there are $2M$ bins in total. Bins of negative indices satisfy:
\begin{equation}
n_{-i}=\sum_j H({\hat{G}}_1^j-u_{-i})H(u_{-i+1}-{\hat{G}}_1^j) \,\,\;(i> 0)
\end{equation}
To estimate the value of ${\hat{g}}_1$ that can maximally symmetrize the distribution of ${\hat{G}}_1^j$ with respect to zero, we form the $\chi^2$ as follows:
\begin{equation}
\label{chi2_const}
\chi^2=\frac{1}{2}\sum_{i> 0}\frac{(n_i-n_{-i})^2}{n_i+n_{-i}}
\end{equation}
${\hat{g}}_1$ is estimated by minimizing $\chi^2$. In Appendix B, we show that {\it minimizing $\chi^2$ defined in eq.(\ref{chi2_const}) leads to an unbiased estimate of cosmic shear, with a statistical uncertainty that approaches the C-R bound in the limit of small bin sizes}.
\subsection{Shear-Shear Correlation with PDF-SYM}
\label{correlation}
Shear-shear correlation here refers to the correlation between the shear components of two galaxies defined along the line of their connection. For convenience, we use the indices $1$ and $2$ to refer to the tangential $(+)$ and cross $(\times)$ components of the shear. To use PDF-SYM, one may consider symmetrizing the PDF of the products of two shear estimators. For example, one may define the following quantity:
\begin{equation}
\xi_{11}=G_1(\vec{x})G_1(\vec{x}+\Delta\vec{x})-\hat{\xi}_{11}B(\vec{x})B(\vec{x}+\Delta\vec{x}),
\end{equation}
and use $\hat{\xi}_{11}$ that can best symmetrize the PDF of $\xi_{11}$ to infer the correlation of $g_1(\vec{x})$ and $g_1(\vec{x}+\Delta\vec{x})$ as a function of $\Delta\vec{x}$. However, it turns out that this is not a correct way, because the PDF of $g_1(\vec{x})g_1(\vec{x}+\Delta\vec{x})-\langle g_1(\vec{x})g_1(\vec{x}+\Delta\vec{x})\rangle $ is generally not symmetric with respect to zero. It turns out that we need to consider the joint PDF of the shear estimators of two galaxies.
In the measurement of shear-shear correlation, the shear components of galaxy pairs all have random (but correlated) values. This is different from the constant shear problem discussed in the last section. The solution is to examine the joint distribution of the correlated shear estimators, which exhibits certain asymmetric pattern. For example, fig.\ref{joint_PDF} shows the distribution of $[G_1(1),G_1(2)]$ measured from many pairs of galaxies, whose underlying shear components $[g_1(1),g_1(2)]$ satisfy a given joint Gaussian distribution with a positive correlation. The index in the parentheses refers to the galaxy ID. Note that the input shear correlation and amplitude have been amplified here for the purpose of illustration.
\begin{figure}[htbp]
\centering
\includegraphics[width=8cm,height=6cm]{joint_PDF.pdf}
\caption{The joint PDF of shear estimators $[G_1(1),G_1(2)]$ of two galaxies whose underlying tangential shear-components are positively correlated.}\label{joint_PDF}
\end{figure}
We find that a way to apply PDF-SYM in the measurement of shear correlation is to apply a set of pseudo shear values generated in pairs according to an assumed Gaussian PDF of cross-correlation $\hat{\xi}$, and to find the value of $\hat{\xi}$ that can bring the joint PDF of $[G_1(1),G_1(2)]$ back to a symmetric state. We can show that the resulting value of $\hat{\xi}$ is an unbiased estimate of the opposite of the original shear-shear correlation, as it brings back the symmetry of the joint PDF. We give the details of the prove below. For convenience, we only consider the measurement of the correlation between the tangential shear components.
Let us follow the notation of the last section. Suppose the shear estimators of two galaxies are $G_1,B,G_1',B'$, and the true underlying shear values are $g_1,g_1'$. Let us also assume that the pseudo shear values are ${\hat{g}}_1,{\hat{g}}_1'$, and therefore the shear estimators can be modified as:
\begin{eqnarray}
&&{\hat{G}}_1=G_1-{\hat{g}}_1B=G_1^S+(g_1-{\hat{g}}_1)B \\ \nonumber
&&{\hat{G}}_1'=G_1'-{\hat{g}}_1'B'={G'}_1^S+(g_1'-{\hat{g}}_1')B'
\end{eqnarray}
where $G_1^S$ and ${G'}_1^S$ are the unlensed quantities that enjoy a symmetric joint PDF\footnote{Note that this point may not be true due to the presence of intrinsic alignment of galaxy shapes. In this case, one should correct the recovered shear-shear correlation by removing the intrinsic alignment contribution estimated using either close galaxy pairs or computer simulations. These topics are beyond the scope of this work.}. Define $P_S(G_1^S,B,{G'}_1^S,B')$ as the joint PDF of the unlensed quantities, which satisfy:
\begin{eqnarray}
&&P_S(G_1^S,B,{G'}_1^S,B')\\ \nonumber
&=&P_S(G_1^S,B,-{G'}_1^S,B')=P_S(-G_1^S,B,{G'}_1^S,B')
\end{eqnarray}
We can then relate the PDF of the modified shear estimators $P({\hat{G}}_1,{\hat{G}}_1')$ to $P_S$ as:
\begin{eqnarray}
\label{PG1G1}
&&P({\hat{G}}_1,{\hat{G}}_1')\\ \nonumber
&=&\int dg_1dg_1'\phi(g_1,g_1')\int d{\hat{g}}_1d{\hat{g}}_1'\hat{\phi}({\hat{g}}_1,{\hat{g}}_1')\\ \nonumber
&\times&\int dB\int dB'\int dG_1^S\int d{G'}_1^SP_S(G_1^S,B,{G'}_1^S,B')\\ \nonumber
&\times&\delta_D\left[{\hat{G}}_1-G_1^S-(g_1-{\hat{g}}_1)B\right]\\ \nonumber
&\times&\delta_D\left[{\hat{G}}_1'-{G'}_1^S-(g_1'-{\hat{g}}_1')B'\right]\\ \nonumber
&=&\int dg_1dg_1'\phi(g_1,g_1')\int d{\hat{g}}_1d{\hat{g}}_1'\hat{\phi}({\hat{g}}_1,{\hat{g}}_1')\int dB\int dB'\\ \nonumber
&\times&P_S\left[{\hat{G}}_1-(g_1-{\hat{g}}_1)B,B,{\hat{G}}_1'-(g_1'-{\hat{g}}_1')B',B'\right]
\end{eqnarray}
where $\phi(g_1,g_1')$ is the PDF of $g_1$ and $g_1'$, and $\hat{\phi}({\hat{g}}_1,{\hat{g}}_1')$ is the presumed PDF of the pseudo shears ${\hat{g}}_1$ and ${\hat{g}}_1'$. As we will show, the form of $\hat{\phi}$ is not important (does not have to be the same as $\phi(g_1,g_1')$, but usually chosen to be Gaussian) in terms of determining the shear-shear correlation. As the shear values are small, we can Taylor expand $P_S$ to the second order in shear as:
\begin{eqnarray}
&&P_S\left[{\hat{G}}_1-(g_1-{\hat{g}}_1)B,B,{\hat{G}}_1'-(g_1'-{\hat{g}}_1')B',B'\right]\\ \nonumber
&=&P_S({\hat{G}}_1,B,{\hat{G}}_1',B')-(g_1-{\hat{g}}_1)B\partial_{{\hat{G}}_1}P_S\\ \nonumber
&-&(g_1'-{\hat{g}}_1')B'\partial_{{\hat{G}}_1'}P_S+\frac{1}{2}(g_1-{\hat{g}}_1)^2B^2\partial^2_{{\hat{G}}_1}P_S\\ \nonumber
&+&\frac{1}{2}(g_1'-{\hat{g}}_1')^2B'^2\partial^2_{{\hat{G}}_1'}P_S+(g_1-{\hat{g}}_1)(g_1'-{\hat{g}}_1')BB'\partial_{{\hat{G}}_1}\partial_{{\hat{G}}_1'}P_S
\end{eqnarray}
Integrating over all possible values of shear, Eq.(\ref{PG1G1}) can be rewritten as:
\begin{eqnarray}
\label{PG1G1_s}
&&P({\hat{G}}_1,{\hat{G}}_1')\\ \nonumber
&=&\int dB\int dB'\left[P_S({\hat{G}}_1,B,{\hat{G}}_1',B')\right.\\ \nonumber
&+&\frac{1}{2}(\langle g_1^2\rangle +\langle {\hat{g}}_1^2\rangle )\left(B^2\partial^2_{{\hat{G}}_1}P_S+B'^2\partial^2_{{\hat{G}}_1'}P_S\right)\\ \nonumber
&+&\left.(\langle g_1g_1'\rangle +\langle {\hat{g}}_1{\hat{g}}_1'\rangle )BB'\partial_{{\hat{G}}_1}\partial_{{\hat{G}}_1'}P_S\right]
\end{eqnarray}
in which we have set $\langle g_1\rangle $, $\langle g_1'\rangle $, $\langle {\hat{g}}_1\rangle $, $\langle {\hat{g}}_1'\rangle $, $\langle g_1{\hat{g}}_1\rangle $, $\langle g_1'{\hat{g}}_1'\rangle $, $\langle g_1{\hat{g}}_1'\rangle $, $\langle g_1'{\hat{g}}_1\rangle $ to zero, and $\langle g_1^2\rangle =\langle g_1'^2\rangle $, $\langle {\hat{g}}_1^2\rangle =\langle {\hat{g}}_1'^2\rangle $. On the right side of eq.(\ref{PG1G1_s}), it is clear that only the last term has odd parity, which breaks the symmetry of the joint PDF of the shear estimators. Therefore, to remove the asymmetry, we must have $\langle {\hat{g}}_1{\hat{g}}_1'\rangle =-\langle g_1g_1'\rangle $, which allows us to achieve an unbiased estimate of the shear-shear correlation.
\begin{figure}[!htb]
\centering
\includegraphics[width=8cm,height=6cm]{bins.pdf}
\caption{The configuration of bins used for symmetrization of the joint PDF in shear-shear correlation measurement.}\label{bins}
\end{figure}
For this purpose, we can set up bins that are symmetrically placed in the four quadrants in the plane of $[{\hat{G}}_1,{\hat{G}}_1']$, and each bin is labelled with two integers, as shown in fig.\ref{bins}. The number of data that fall to bin $(i,j)$ is denoted as $n_{i,j}$, and can be calculated as:
\begin{eqnarray}
n_{i,j}&=&\sum_k H({\hat{G}}_1^k-u_{i-1})H(u_i-{\hat{G}}_1^k)\\ \nonumber
&\times&H({\hat{G}}_1'^k-u_{j-1})H(u_j-{\hat{G}}_1'^k) \,\,\;(i,j> 0)
\end{eqnarray}
where $u_{i-1}$, $u_i$, $u_{j-1}$, $u_j$ are the boundaries of bin $(i,j)$, and the upper index $k$ is the index of galaxy pair. Again, we let half of the bin indices to take negative values, with $u_i=-u_{-i}$, and $i=-(M-1), \cdots, -1, 0, 1, \cdots, M-1$, and $u_{\pm M}=\pm\infty$, similar to the case of the last section. The number of data in bins of negative indices can be similarly defined.
To estimate the value of $\langle {\hat{g}}_1{\hat{g}}_1'\rangle $ that can maximally symmetrize the distribution of the joint PDF of $[{\hat{G}}_1,{\hat{G}}_1']$, we form the $\chi^2$ as follows:
\begin{equation}
\label{chi2_shear_shear}
\chi^2=\frac{1}{2}\sum_{i,j> 0}\frac{(n_{i,j}+n_{-i,-j}-n_{-i,j}-n_{i,-j})^2}{n_{i,j}+n_{-i,-j}+n_{-i,j}+n_{i,-j}}
\end{equation}
Appendix C shows that {\it minimizing $\chi^2$ leads to an unbiased estimate of $\langle g_1g_1'\rangle $ (as $-\langle {\hat{g}}_1{\hat{g}}_1'\rangle $), with a statistical uncertainty approaching the C-R bound in the limit of small bin sizes}.
Note that in minimizing $\chi^2$, one should fix the variances $\langle {\hat{g}}_1^2\rangle$ and $\langle {\hat{g}}_1'^2\rangle$ at the same value, the choice of which could be somewhat arbitrary without affecting $\chi^2$ as long as it is larger than the absolute value of the covariance $\langle {\hat{g}}_1{\hat{g}}_1'\rangle $. The later can be roughly estimated by the direct averaging method. Another thing to mention is that since ${\hat{g}}_1$ and ${\hat{g}}_1'$ are drawn randomly for a given value of $\langle {\hat{g}}_1{\hat{g}}_1'\rangle$, one may repeat it for several times for each $n_{i,j}$, so that the resulting $\chi^2$ is less noisy, particularly when the galaxy pair number is not large.
Finally, we shall point out that eq.(\ref{PG1G1_s}) can be expanded to include higher order terms in shear. It is not hard to show that the next-leading-order terms that can affect the PDF symmetry of our interest is on the order of shear to the fourth power, which has been neglected here.
\section{Numerical Examples}
\label{numerical}
In this section, we present numerical examples to show the accuracy and certain characteristics of PDF-SYM. The general setup of our simulations are given in \S\ref{setup}. We discuss the case of constant shear measurement in \S\ref{constant_shear}, and shear-shear correlation in \S\ref{corr}.
\subsection{General Setup}
\label{setup}
Each of our mock galaxies is made of a number of point sources \citep{jz08}. There are a few advantages of this setup: 1) the lensing effect can be added by simply changing the positions of the point sources; 2) convolution with PSF is straightforward; 3) the richness of galaxy morphologies can be modified by changing the intrinsic distribution and the number of the point sources; 4) the image generation pipeline is very fast, suitable for testing shear recovery accuracy with a large mock galaxy ensemble. Each galaxy is placed at the center of a square grid. The pixel size of the grid is set to be the length unit in this paper. The stamp size is $48\times 48$.
The PSF has a truncated Moffat profile used in the GREAT08 project \citep{bridle09}:
\begin{equation}
\label{PSFs}
W_{PSF}(r)\propto\left[1+\left(\frac{r}{r_d}\right)^2\right]^{-3.5}{\mathrm H}(r_c-r)
\end{equation}
The FWHM of this PSF is very close to $r_d$. We set $r_c=3r_d$ and $r_d=3$ in the simulations of this paper. In our shear measurement method, the PSF is transformed into the isotropic Gaussian form through reconvolution in Fourier space. The scale radius ($\beta$) of the target PSF is set to $r_d$, so that the size of the target PSF is somewhat larger than that of the original PSF.
\subsection{Constant Shear}
\label{constant_shear}
In this section, we study the recovery of a constant shear from a large ensemble of galaxies.
\subsubsection{Ring Galaxies}
\label{ring_gal}
In our first example, all galaxies are generated as circular rings, each of which is made of $100$ point sources (of a fixed luminosity) homogeneously placed at a fixed distance ($4$ in unit of the pixel size) from the galaxy center. The positions of the points of a galaxy are projected onto the source plane with a random projection angle, followed by the lensing and PSF effect. We generate $10000$ such galaxies, with $g_1=-0.018$ and $g_2=0.011$ ($\kappa=0$). The shear estimators are defined in eq.(\ref{shear_estimator}) and eq.(\ref{def_U}).
\begin{table}[!htb]
\centering
\caption{\textnormal{The recovered shear values in different methods.} }
\begin{tabular}{lll}
\hline\hline
Method & $g_1 (-0.018)$ & $g_2 (0.011)$ \\
\hline\hline
Direct Averaging & $-0.017(2)$ & $0.011(2)$ \\
\hline
PDF-SYM (2 bins) & $-0.01798(3)$ & $0.01100(2)$ \\
\hline
PDF-SYM (4 bins) & $-0.01799(2)$ & $0.01102(2)$ \\
\hline
PDF-SYM (8 bins) & $-0.01800(2)$ & $0.01101(2)$ \\
\hline\hline
\end{tabular}
\label{result_ring}
\end{table}
Table \ref{result_ring} shows the results for four different ways of achieving the shear signals from the ensemble of shear estimators. The first method is to take the direct averages of the shear estimators, as those defined in eq.(\ref{shear_measure}). The result is shown on the first row of the table. The other three rows show the results from PDF-SYM introduced in \S\ref{const_shear}, with three different choices of bin number. In every case, the bins are symmetrically placed on the two sides of zero. The boundaries between the bins are determined by evenly dividing the galaxies according to the absolute values of $G_1$ or $G_2$, making the galaxy number in different bins roughly the same. The results in the table indicate that the PDF-SYM greatly reduces the statistical uncertainty of the recovered shear signals. Fig.\ref{perfect_ring} shows the PDF's of ${\hat{G}}_1$ before (blue) and after (green) symmetrization. It is clear that with the recovered shear value, the PDF of ${\hat{G}}_1$ is indeed symmetrized with respect to zero.
\begin{figure}[!htb]
\centering
\includegraphics[width=8cm,height=6cm]{example2.pdf}
\caption{The PDF's of ${\hat{G}}_1$ before (blue) and after (green) symmetrization for ring galaxies.}\label{perfect_ring}
\end{figure}
Note that in this example, the tiny statistical uncertainty in PDF-SYM is caused by the sharp peak of the PDF at zero, as shown in \S\ref{nulling}. This is consistent with the prediction of BJ02, which discussed galaxies of pure 2D disks with random projection angles, similar to our ring galaxies. Even for the 2-bin case, the shear uncertainties in the new technique are much smaller than those from direct averaging.
\subsubsection{Mixed Types of Galaxies}
\label{mixed_gal}
In shear measurement, among the ensemble of galaxies, certain types of galaxies may be unusually sensitive to cosmic shear, such as the ring/disk galaxies with random projection angles shown in \S\ref{ring_gal}. Assuming they really exist in nature, the question is how to maximally utilize their advantages in shear recovery when their shear estimators are mixed with those of other galaxies. We now show that PDF-SYM provides a way.
Another type of galaxies we consider are generated by 2D random walks (called RW galaxies hereafter). Each galaxy is made of $100$ point sources (of constant luminosity), the positions of which are determined by a series of random walks with random directions and step sizes (between 0 and 1). When the point's position from the grid center is larger than 4, the random walk restarts from the grid center and continues from there.
The input shear values are $g_1=0.02277$, $g_2=-0.01386$ ($\kappa=0.01$). To recover shear, we generate $10^5$ RW galaxies. The results from the averaging method and PDF-SYM (8 bins) are shown in table \ref{result_mix}. For RW galaxies, PDF-SYM does not seem to improve on the statistical uncertainties with respect to the averaging method. This is very different from the case of ring galaxies. The reason is that the shear estimators' PDF shapes of the two types of galaxies are significantly different. The PDF's of ${\hat{G}}_1$ of the RW galaxies before or after symmetrization are similar to Gaussian functions, as shown in fig.\ref{pdf_rw}. They are much less peaked in the neighbourhood of zero than those of the ring galaxies shown in fig.\ref{perfect_ring}. In this case, as discussed in \S\ref{nulling}, direct averaging has a similar performance as PDF-SYM (see the example of the Gaussian PDF).
It becomes interesting when the two types of galaxies are mixed. In the sample of RW galaxies, if $10\%$ are replaced by $10000$ ring galaxies, we find that shear recovery accuracy of the averaging method almost does not change, according to table \ref{result_mix}. In contrast, in PDF-SYM, the shear uncertainties are reduced by almost a factor of $10$, implying that the accurate shear information contained in the ring galaxies is significantly utilized. Note that this is achieved without separating the two types of galaxies.
\begin{table*}[!htb]
\centering
\caption{\textnormal{The recovered shear values in two different methods. The input shear values are: $g_1=0.02277$, $g_2=-0.01386$.} }
\begin{tabular}{lll}
\hline\hline
Results of $[g_1,g_2]$: & $10^5$ RW Galaxies & $9\times 10^4$ RW+$10^4$ Ring \\
\hline\hline
Averaging & $[0.0226(6),-0.0130(6)]$ & $[0.0231(6),-0.0132(6)]$ \\
\hline
PDF-SYM (8 bins) & $[0.0225(7),-0.0129(6)]$ & $[0.02278(8),-0.01392(7)]$ \\
\hline\hline
\end{tabular}
\label{result_mix}
\end{table*}
\begin{figure}[!htb]
\centering
\includegraphics[width=8cm,height=6cm]{PDF_rw_gals.pdf}
\caption{The PDF's of ${\hat{G}}_1$ before (blue) and after (green) symmetrization for RW galaxies.}\label{pdf_rw}
\end{figure}
\subsubsection{Noise and Misidentified Stars}
\label{noise_and_mis}
According to ZLF15, direct averaging of our shear estimators as defined in eq.(\ref{shear_measure}) is accurate in the presence of noise (both background noise and Poisson noise). It is also interesting to note that the accuracy is immune to possible misidentifications of stars as galaxies, because on average, point sources contribute zero values to both the numerators and the denumerators of eq.(\ref{shear_measure}). This is a very useful feature for handling faint sources. It turns out that these good characters of the shear estimators of ZLF15 remain valid in PDF-SYM. The noise and the misidentified stars make the symmetrized PDF's of the shear estimators more noisy, but do not bias the best-fit shear values. We give numerical examples below.
We still use the ring and RW galaxies defined in the previous two sections. We add Poisson noise of a constant amplitude to each galaxy stamp. The total galaxy flux is random, leading to a distribution of the signal-to-noise-ratios (SNR's) of the galaxies shown in fig.\ref{SNR_dist}. $20\%$ of the sources are actually set to be stars (single-point sources). We let the stars to have the same flux distribution as the galaxies'. The input shear values are $g_1=-0.01008$, $g_2=-0.02016$ ($\kappa=-0.008$). The shear recovery results are shown in table \ref{result_noise}.
\begin{figure}[!htb]
\centering
\includegraphics[width=8cm,height=6cm]{SNR_dist.pdf}
\caption{The distribution of SNR for the galaxy images with noise used in the test of shear recovery in \S\ref{noise_and_mis}.}\label{SNR_dist}
\end{figure}
\begin{table*}[!htb]
\centering
\caption{\textnormal{Shear recovery with noise and misidentified stars. The input shear values are $g_1=-0.01008$, $g_2=-0.02016$.}}
\begin{tabular}{lllll}
\hline\hline
Results of $[g_1,g_2]$: & $10^6$ RW Galaxies & $9\times 10^5$ RW+$10^5$ Ring \\
\hline\hline
Averaging & $[-0.0091(5),-0.0201(5)]$ & $[-0.0100(5),-0.0208(5)]$ \\
\hline
PDF-SYM (8 bins) & $[-0.0101(3),-0.0207(3)]$ & $[-0.0100(1),-0.0203(1)]$ \\
\hline
Averaging (flux normalized) & $[-0.0102(2),-0.0203(2)]$ & $[-0.0100(2),-0.0204(2)]$ \\
\hline
PDF-SYM (flux normalized) & $[-0.0102(2),-0.0204(3)]$ & $[-0.01003(9),-0.02030(9)]$ \\
\hline\hline
\end{tabular}
\label{result_noise}
\end{table*}
The results of table \ref{result_noise} indicate that through either direct averaging or PDF-SYM, shear recovery is accurate in the presence of noise and misidentified stars. The PDF method once again shows an advantage over the averaging method for the ring galaxies, and when galaxies of different types are mixed. Moreover, in this example, even for pure RW galaxies, the PDF method yields a smaller error than the averaging method. The reason is that the galaxies have a range of SNR's, and the corresponding shear estimators have very different amplitudes. Direct averaging of the shear estimators therefore under-represents the contribution from faint galaxies. One can try to weaken this problem by normalizing the shear estimators by the galaxy flux squared\footnote{Assuming the sky background has been subtracted, our galaxy flux is measured by summing over {\it the absolute values} of all the pixel readouts. This is for stablizing the total flux of the faint sources, which can be arbitrarily close to zero in principle due to the presence of noise.}. The results are shown in table \ref{result_noise} as well. Normalization does help in reducing the statistical error in the averaging method of RW galaxies, but not so much in PDF-SYM, implying that normalization of the shear estimators is not quite necessary in PDF-SYM.
\subsubsection{GalSim Galaxies}
\label{galsim}
In this section, we further test the new method with galaxy images generated by GalSim, which is a collaborative and open-source project aiming at providing a software library for image simulations in astronomy \citep{rowe2015}. We adopt LSST like observing condition for our galaxy simulation. We set the PSF size to be 0.7", typical at the site of Cerro Tololo. For simplicity we set the PSF ellipticity fixed for the whole galaxy sample with $e_1=0.03$ and $e_2=0.02$. We simulate LSST r band disk galaxies with Sersic index equals 2, and pixel scale of 0.2".
Two sets of simulated images are created. The first one contains $10,000$ galaxies without noise. The intrinsic galaxy ellipticities are generated so that they mimic randomly oriented disks. The shear recovery results are shown in table \ref{galsim_no_noise}. The results again confirms the advantage of the PDF-SYM method.
In another example, we generate $10^6$ galaxies with SNR in the range of $20 - 100$ to check the performance of the new method for noisy galaxy images. The noise is simulated using the Exposure Time Calculator (ETC; http://lsst.org/etc). The distribution of SNR is shown in fig.\ref{SNR_galsim}. The results are shown in table \ref{galsim_with_noise}, indicating that for galaxies of smaller SNR, the performance of the PDF-SYM method becomes comparable to direct averaging. This is not surprising, as information is lost due to the existence of noise. Numerically, this is because the singular feature in the PDF of the shear estimators is smeared out by noise.
\begin{table}[!htb]
\centering
\caption{\textnormal{Shear recovery with 10000 GalSim-generated noiseless galaxies. The input shear values are $g_1=0.016$, $g_2=-0.003$.}}
\begin{tabular}{lllll}
\hline\hline
& $g_1$ & $g_2$ \\
\hline\hline
Averaging & $0.017(1)$ & $-0.003(1)$ \\
\hline
PDF-SYM (8 bins) & $0.0162(2)$ & $-0.0029(2)$ \\
\hline\hline
\end{tabular}
\label{galsim_no_noise}
\end{table}
\begin{table}[!htb]
\centering
\caption{\textnormal{Shear recovery with $10^6$ GalSim-generated galaxies with SNR in the range of $20-100$. The input shear values are $g_1=0.007$, $g_2=-0.008$.}}
\begin{tabular}{lllll}
\hline\hline
& $g_1$ & $g_2$ \\
\hline\hline
Averaging & $0.0072(2)$ & $-0.0078(2)$ \\
\hline
PDF-SYM (8 bins) & $0.0070(2)$ & $-0.0077(2)$ \\
\hline\hline
\end{tabular}
\label{galsim_with_noise}
\end{table}
\begin{figure}[!htb]
\centering
\includegraphics[width=8cm,height=6cm]{SNR_galsim.pdf}
\caption{The distribution of SNR for the galaxy images generated using GalSim in the test of shear recovery in \S\ref{galsim}.}\label{SNR_galsim}
\end{figure}
\subsection{Shear-Shear Correlation}
\label{corr}
To test the recovery of shear-shear correlation, we generate a large number of galaxy pairs whose underlying shear values are correlated. To avoid cosmic variance, the shear values of each galaxy pair are not drawn from a shear field, but are generated with a coupled Gaussian distribution directly. The direction of the line connecting the two galaxies is taken to be random with respect to the grid axes. The tangential and cross components of the shear pairs are generated according to the following covariance matrix: \\
\[ \left[ \begin{array}{cc}
\langle g_t^{(1)}g_t^{(1)}\rangle & \langle g_t^{(1)}g_t^{(2)}\rangle \\
\langle g_t^{(2)}g_t^{(1)}\rangle & \langle g_t^{(2)}g_t^{(2)}\rangle \end{array} \right]
= \left[ \begin{array}{cc}
2\times 10^{-4} & 10^{-4} \\
10^{-4} & 2\times 10^{-4} \end{array} \right],\] \\
\[ \left[ \begin{array}{cc}
\langle g_{\times}^{(1)}g_{\times}^{(1)}\rangle & \langle g_{\times}^{(1)}g_{\times}^{(2)}\rangle \\
\langle g_{\times}^{(2)}g_{\times}^{(1)}\rangle & \langle g_{\times}^{(2)}g_{\times}^{(2)}\rangle \end{array} \right]
= \left[ \begin{array}{cc}
2\times 10^{-4} & -10^{-4} \\
-10^{-4} & 2\times 10^{-4} \end{array} \right].\] \\
Table \ref{result_corr} shows the results for the recovery of shear-shear correlations. The first three rows show the results achieved with $4\times 10^7$ galaxy pairs, and different types of galaxies and statistical methods. In these simulations, we do not add point sources as misidentified galaxies, neither any noise to the galaxy images. The last row shows the results of another simulation, in which we use $1.6\times 10^8$ galaxy pairs to recover the shear-shear correlations, with Poisson noise in every galaxy images (with SNR distribution similar to that shown in fig.\ref{SNR_dist}), and $10\%$ point sources as misidentified galaxies. We only use ring galaxies in this experiment. The results confirm the robustness of our shear measurement under general conditions, and the advantage of PDF-SYM.
\begin{table*}[!htb]
\centering
\caption{\textnormal{The recovered shear-shear correlations. The inputs are $\langle g_t^{(1)}g_t^{(2)}\rangle=10^{-4}$ and $\langle g_{\times}^{(1)}g_{\times}^{(2)}\rangle=-10^{-4}$.}}
\begin{tabular}{lll}
\hline\hline
Results of $[\langle g_t^{(1)}g_t^{(2)}\rangle ,\langle g_{\times}^{(1)}g_{\times}^{(2)}\rangle ]$($10^{-4}$) : & Averaging & PDF-SYM (8$\times$8 bins) \\
\hline\hline
$4\times 10^7$ RW Gal. Pairs & $ [1.09(8),-1.00(8)]$ & $ [1.09(8),-1.01(9)] $ \\
\hline
$4\times 10^7$ Ring Gal. Pairs & $ [1.05(7),-1.08(7)]$ & $ [1.002(5),-1.002(5)]$ \\
\hline
$4\times 10^7$ Gal. Pairs with 90\% RW and 10\% Ring & $ [1.09(8),-1.02(8)]$ & $ [0.99(3),-1.00(3)]$ \\
\hline
$1.6\times 10^8$ Ring Gal. Pairs with noise and 10\% stars & $ [0.97(4),-1.05(4)]$ & $ [1.000(3),-1.001(3)] $ \\
\hline\hline
\end{tabular}
\label{result_corr}
\end{table*}
\section{Summary and Discussions}
\label{summary}
Weak lensing statistics, such as mean shear or shear-shear correlation, are typically evaluated as the weighted sum of the shear estimators or their products. Traditionally, galaxy ellipticities are used as shear estimators. As discussed in BJ02, the weighting factor can be designed as a function of the galaxy ellipticities, so that the resulting statistical error can approach the Cram\'{e}r-Rao bound. This weighting scheme is however hard to realize in practice given the presence of noise and bias in the measurement of galaxy ellipticities and their PDF.
Based on the shear estimators of ZLF15, we propose to evaluate shear statistics by symmetrizing the PDF of the shear estimator or the joint PDF of shear estimator pairs (for shear-shear correlation). We find that this is a way to approach the C-R bound without introducing systematic errors, as shown analytically in \S\ref{realizing_MLE_estimator}.
In \S\ref{numerical}, we test shear recovery accuracy with large ensembles of galaxies or galaxy pairs (for the measurement of shear-shear correlation). We find that {\it both direct averaging and PDF symmetrization can recover shear or shear-shear correlation accurately in the presence of noise and stars that are misidentified as galaxies}, proving the robustness of both methods in practice. Note that the allowance of stars in the galaxy ensemble is a quite unusual and useful feature of the ZLF15 shear estimators. It is mostly due to the linearity of the shear estimator form.
In our numerical experiment, we use two different types of mock galaxies: 1. RW galaxies that are made of point sources connected by 2D random walks; 2. Ring galaxies that are made of point sources evenly distributed on a circle, and projected to the plane of the sky with random angles. With the method of PDF-SYM, we find that on average, every ring galaxy contains much more shear information than each RW galaxy. This is consistent with the theoretical expectation regarding the C-R bound, and agrees with BJ02, who concerns thin-disk galaxies that are similar to our ring galaxies. The advantage of ring galaxies in shear recovery cannot be easily exploited in direct averaging of the shear estimators. When these two types of galaxies are mixed in an ensemble, we find that PDF-SYM can maximally utilize the shape information in all galaxies, typically generating a much smaller statistical uncertainty than that by direct averaging. This fact is again consistent with the theory regarding the C-R bound.
It is interesting to note that PDF-SYM allows us to approach the C-R bound in shear measurements without prior knowledge of the PDF form of the shear estimator. This is because of the monotonic dependence of the shear estimator on its corresponding shear component in the weak shear/convergence limit, implying that restoring the PDF is equivalent to symmetrizing the PDF. This fact has been proven useful for recovering the 1-point and 2-point shear statistics. Similar ideas may be developed for the measurement of n-point shear statistics in a future work.
It is straightforward to apply the new method in several areas of weak lensing to optimize the statistical uncertainty of the results, including: 1. shear-shear correlation measurement binned in both angular separation and redshift; 2. galaxy-shear cross correlation at a fixed 3D relative position; 3. testing shear recovery accuracy. The results of these measurements all correspond to explicit theoretical expectation values, therefore are easy to interpret. In measuring shear-shear correlation ({\it e.g. }, for a survey of size larger than or comparable to CFHTlens \citep{erben2013}), we find that there are typically a large number of galaxy pairs for a given angular separation and two redshift bins, making the PDF-SYM method an ideal tool to use. Note that when the number of galaxy pairs is too large, it is better to take the average of the shear estimators in each spatial bin first, and then measure the shear correlation with bin pairs of given spatial separations. We will report the application of the new method on the CFHTlens galaxy images in separate papers\footnote{If the purpose is only to make shear-map, either 2D or 3D, PDF-SYM may not be as good as direct averaging (with flux normalized shear estimators of ZLF15), because in each grid, the galaxy number may be too few ($\sim 10$ galaxies /$\mathrm{arcmin}^2$) for PDF-SYM to use more than 2 bins in shear recovery within the grid ({\it e.g. }, table \ref{result_ring})}.
If the source galaxies in the ensemble cover a broad redshift range, {\it e.g. }, in the recovery of 2D shear map, or in the measurement of 2D shear-shear correlation, using PDF-SYM becomes more complicated. Indeed, in this case, the PDF of the shear estimators cannot be symmetrized at all in principle, though we can still use the $\chi^2$ formalism developed in \S\ref{const_shear} \& \S\ref{correlation}, which would result in a weighted sum of the shear signal or the shear-shear correlation within the redshift range. The weighting function can be calculated, though it requires the PDF of the shear estimator as a function of redshift. Appendix D shows a derivation of the weighting function for shear recovery. This problem will be studied more carefully in a future work.
Overall, the performance of PDF-SYM depends on the PDF form/galaxy type, distribution of galaxy SNR, and other image distortion effects \citep{rhodes10,gruen2015}. We have only examined a few cases. More work will be done to quantify the improvement with real observational data.
\acknowledgments{The authors thank Eiichiro Komatsu for motivating this work, and Zuhui Fan, Liping Fu, Guoliang Li, Chenggang Shu for useful discussions. This work is supported by the national science foundation of China (Grant No. 11273018, 11433001, 11320101002), the national basic research program of China (973 Program 2015CB857001, 2013CB834900), the national “Thousand Talents Program” for distinguished young scholars, a grant (No.11DZ2260700) from the Office of Science and Technology in Shanghai Municipal Government.}
|
1,116,691,500,787 | arxiv | \section{Deviation from the dipole-ice model in the new spinel spin-ice candidate, MgEr$_2$Se$_4$,\\
Supplemental Materials}
\subsection{Monte Carlo simulation}
We performed Monte Carlo simulations on a 2048-site cluster with periodic boundary conditions in all directions. Our Hamiltonian (Eq.~\ref{eq:hamiltonian}) includes nearest and next nearest neighbor Ising interactions and long range dipole-dipole interactions:
\begin{multline}
H = -3J_{nn}\sum_{\langle i,j \rangle}\mathbf{S}_i^{z_i}\cdot\mathbf{S}_j^{z_j}
-3J_{nnn}\sum_{\langle\langle i,j \rangle\rangle}\mathbf{S}_i^{z_i}\cdot\mathbf{S}_j^{z_j}\\
+ \frac{3D_{nn}}{5}r_{nn}^3\sum_{i<j}\left(\frac{\mathbf{S}_i^{z_i}\cdot\mathbf{S}_j^{z_j}}{|\mathbf{r}_{ij}|^3} - \frac{3(\mathbf{S}_i^{z_i}\cdot\mathbf{r}_{ij})(\mathbf{S}_j^{z_j}\cdot\mathbf{r}_{ij})}{|\mathbf{r}_{ij}|^5}\right),
\label{eq:hamiltonian}
\end{multline}
To efficiently compute dipole terms we precompute effective couplings between pairs of spins by summing terms arising from nearest 40 periodic images.
We find that to be an accurate alternative to an Ewald summation used in \cite{denhertog2000}.
We use parallel tempering with a range of temperatures from $T=0.5$ K to $T=6$ K with a grid $\Delta T=0.02$ K.
This is done to improve the mixing of the Markov chain.
The energy $E(T)$ is computed naively in the Monte Carlo and is fit to a spline.
The specific heat,
\begin{equation}
C_{V}(T) = \frac{dE(T)}{dT}
\end{equation}
is computed by taking the derivative of this spline.
Spin configurations were fed into the program SPINVERT \cite{paddison2013} to obtain predicted neutron scattering cross-sections, I(Q).
For a given choice of $J_{nn}$, it was found that sufficiently large $|J_{nnn}|$ drove a transition into a long-range ordered state.
Figure~\ref{fig:MC_order} shows the neutron $I(Q)$ and real space pattern of spins for the case of both ferromagnetic and antiferromagnetic $J_{nnn}$.
Both ordered states preserve the two-in-two-out constraint of the spin-ice ground state.
The ferromagnetic state prefers a state which preserves the symmetries of the Fd$\bar{3}$m space-group, and thus demonstrated spin-spin correlations at locations consistent with a Q=0 state. Real space spin configurations indicate a similar state as preferred by Ho$_2$Ti$_2$O$_7$ and Dy$_2$Ti$_2$O$_7$ for applied fields $\mathbf{H} \parallel [0 0 1]$\cite{fennell2005}, or by ferrimagnetic spinels\cite{macdougall2012}. This is shown in Fig.~\ref{fig:MC_order}(a) for the I(Q) and Fig.~\ref{fig:MC_order}(c) for a real space respresentation.
The antiferromagnetic interaction case preferred a state which broke Fd$\bar{3}$m symmetry, and demonstrated distinct anti-correlations between chains of spins along the [1 1 0] which are antiparallel to neighbouring chains. Neutron intensity indicates a Q=X phase, in that it shows correlations at the cubic (1 0 0) and equivalent Bragg positions\cite{harris1997}. Although similar to the Q=X phase favored by $\mathbf{H} \parallel [1 1 0]$ fields in that both have antiparallel chains of spins along the [1 1 0]\cite{fennell2005}, the current phase is actually distinct, in that there is no net moment. These plots are shown in Fig.~\ref{fig:MC_order}(b) and (d) for I(Q) and real space respectively.
\begin{figure}[hbt]
\centering
\includegraphics[width=\columnwidth]{SM_figure_MC_order.pdf}
\caption{I(Q) for simulated temperature subtracted NPD pattens for the case of ferromagnetic (a) and antiferromagnetic (b) J$_{nnn}$. Real space spin configurations for ferromagnetic (c) and antiferromagnetic (d) ordered phases as viewed along the cubic [0,0,1] axis.}
\label{fig:MC_order}
\end{figure}
\subsection{Sample Preparation and Characterization of Structure}
Polycrystalline samples of MgEr$_2$Se$_4$ were prepared via a two-step solid state reaction, following the method laid out in Ref.~\onlinecite{flahaut1965}.
The precursors MgSe and Er$_2$Se$_3$ were prepared by the direct reaction of a stoichiometric amount of the elements at 650$^{\circ}$C. Stoichiometric quantities of the two precursors were then combined, pelletized and reacted in vacuum at 1000$^{\circ}$C for two days, this step was repeated at least one more time for the precursors to fully react.
Purity and structure were studied with both x-ray diffraction (XRD) and neutron powder diffraction (NPD), and diffraction patterns on our primary sample are shown in Fig.~\ref{fig:diffraction}, along with the results of FULLPROF refinements. Impurity peaks in both patterns are denoted by crosses, and were largely accounted for by the orthorhombic phase of Er$_2$Se$_3$, which is consistent with some amounts of Mg evaporating during synthesis. In addition to Er$_2$Se$_3$, refinements also showed small amounts of E$_2$O$_2$Se and elemental Er, as well as some small unindexed impurity peaks which are not consistent with any known compound. The best fit refinement implies that the sample had purity of 94.66(62)~$\%$ and, 91.4(1.5)~$\%$ by mass from the NPD and XRD fits respectively. Quantities of impurity phases from refinements are shown in Table~\ref{table:XRD_impurity}.
A model independent estimate of the impurity fraction of 7.8(1.2~)$\%$ by mass was obtained by comparing the integrated intensity of XRD Bragg peaks associated with the majority phase and all impurity phases, and this is the value we cite in the main text of the paper.
Both NPD and XRD show that the MgEr$_2$Se$_4$ phase is of high quality and shows no observable defects in the structure.
To test for any structural defects, we allowed Se occupancy, Mg occupancy as well as Er and Mg site inversion to vary, with best fits showing no such defects, as shown in Table~\ref{tab:struct}. In this way, the level of cation inversion and point defects is limited to $<$1$\%$.
As an additional model independent check for point defects, we performed Le Bail refinements \cite{lebail1988} of our data and compared the $\chi^2$ of those fits to our best Rietveld refinement. In Lebail fits, the structure factor is not calculated, and instead every peak height is allowed to vary and fit independently-- effectively identifying the ideal description for peaks associated with a single phase in a mixed powder\cite{toby2006}. In current case, we see that the $\chi^2$ achieved through a Le Bail peak-by-peak fitting of the majority phase is identical to that achieved via the above Rietveld refinement. This is a powerful result, which effectively eliminates the existence of cation inversion, off-stoichiometry on the Se-sublattice, or any other point defect which has the capacity to change the height in a neutron scattering pattern.
\begin{figure}[hbt]
\centering
\includegraphics[width=\columnwidth]{SM_figure_diffraction.pdf}
\caption{NPD (left) and XRD (right) of the MgEr$_2$Se$_4$ sample used with data points in blue, he best fit Rietveld refinement in black, and the difference shown in red. Tick marks show positions of MgEr$_2$Se$_4$ peaks, while crosses show position of peaks from fit impurity phases.}
\label{fig:diffraction}
\end{figure}
\begin{table}[h]
\begin{ruledtabular}
\begin{tabular}{l c c}
\multicolumn{3}{c}{Rietveld refinement phases} \\
& XRD & NPD \\
\hline
MgEr$_2$Se$_4$ & 94.6(6) & 91.4(1.6) \\
Er & 1.14(9) & 1.3(3) \\
Er$_2$O$_2$Se & 2.40(8) & 2.3(2) \\
Er$_2$Se$_3$ & 1.79(9) & 5.0(2) \\
\end{tabular}
\end{ruledtabular}
\caption{Mass percentages of phases from Rietveld refinements of the XRD and NPD ( at 38 K) data.}
\label{table:XRD_impurity}
\end{table}
\begin{table}[htb]
\begin{ruledtabular}
\begin{tabular}{ l c c c }
\multicolumn{4}{c}{MgEr$_2$Se$_4$ lattice parameters} \\
\multicolumn{4}{c}{space group Fd$\bar{3}$m}\\
& XRD 300K & NPD 38K & NPD 470mK \\
\hline
a & 11.5207(14) & 11.4999(42) & 11.5048(81) \\
$\chi^{2}$ & 10.41 & 6.39 & 8.63 \\
$\chi^{2}$ Lebail & 11.40 & 6.83 & 9.53 \\
Se deficiency (\%) & 0.00(70) & 0.00(98) & 0.0(1.2) \\
Site inversion (\%) & 0.00(47) & 0.0(3.7) & 0.0(4.5) \\
\end{tabular}
\begin{tabular}{ l c c c c }
\multicolumn{5}{c}{MgEr$_2$Se$_4$ atom positions} \\
& x & y & z & $B_{\text{iso}} ($\AA$^2)$ \\
\hline
Mg & 0.375 & 0.375 & 0.375 & 0.2(1) \\
Er & 0.000 & 0.000 & 0.000 & 0.39(5)\\
Se & 0.2456(9) & 0.2456(9) & 0.2456(9) & 0.40(3) \\
\end{tabular}
\caption{Structural parameters obtained from the XRD and NPD refinements of MgEr$_2$Se$_4$ data.
The structural parameters in the lower part of the table are from the 38K NPD refinement.}
\label{tab:struct}
\end{ruledtabular}
\end{table}
\subsection{Magnetic Properties of the Impurity Phase}
In Fig.~\ref{fig:NPD_impurity} we show the low temperature NPD patterns at T = 1.5 K and 0.47 K, without the T=38 K pattern subtracted. In addition to broad features associated with spin-ice correlations in the main text and Bragg peaks associated with the lattice, the data shows a series of weak Bragg magnetic peaks which appear only at the lower temperature. These peaks were not indexable in the Fd$\bar{3}$m spacegroup of the spinel structure and had total weight of $0.55(5)~\%$ of the diffuse correlations. We thus associate them with the same impurity phase discussed above and subtracted these peaks from Figure 3 in the main text, for cosmetic purposes only. Neither the position nor the weight of the magnetic impurity peaks are capable of accounting for the large peak predicted by MC simulations using best fits parameters for heat capacity data, and the presence or absence of these peaks in the NPD data do not change the conclusion of this work in any way.
\begin{figure}[hbt]
\centering
\includegraphics[width=\columnwidth]{SM_figure_NPD_impurity.pdf}
\caption{NPD data at 1.5 K and 470 mK, with new peaks marked by arrows.}
\label{fig:NPD_impurity}
\end{figure}
\subsection{Additional Inelastic Neutron Scattering Data}
Figure 1 of the main text showed inelastic neutron scattering (INS) data taken with E$_i$=30 meV neutrons at T = 2 K and 40 K. We combined this data with equivalent measurements taken with E$_i$=50 meV to extract CEF data. We present the additional INS data and fits here in Fig. 4, including T = 150 K spectra for both incident energies and T = 2 and 40 K spectra for $E_i$ = 50 meV. None of the data shown in Fig. 4 was included in the main text. These plots show the level of detail captured by our fits of the CEF scheme, scheme and indicate that the data is very well described by the fit for both incident energies at all measured temperatures.
\begin{figure*}[!hbt]
\centering
\includegraphics[width=\linewidth]{SM_figure_INS_plots.pdf}
\caption{ Constant Q (Q = $[2,2.5]~$\AA{}$^{-1}$) cuts of the INS data shown as (blue squares) with best fit line shown in black. Marks below the curve show transition energies. }
\label{fig:seq_SM}
\end{figure*}
\subsection{INS fitting method and results}
The resulting wavefunctions for all CEF levels are shown in table~\ref{tab:CEF}.The code for fitting the INS data to the CEF model was written in MATLAB.
The assumptions for the refinement are as follows: the CEF levels are thermally populated according to the partition function $Z = \sum_n \text{exp}\left( -\beta E_n \right) $, where $E_n$ is the energy of the $n^{\text{th}}$ CEF level.
Excitation energies are determined by diagonalizing the Hamiltonian \begin{equation}
H = \sum_{nm} B_n^m O_n^m,
\label{eq_ham2}
\end{equation}
where $B_n^m$ are the crystal field parameters and $O_n^m$ are Steven's operators.
In order to isolate the unique solution where the wavefunctions are maximally parallel/antiparallel to the $\langle 1 1 1 \rangle$ directions, a small field of $10^{-7}$~T is added to the potential; we note that this field is too small to change the energy of the calculated CEF levels.
Peak intensities are given by $p_n \bra{\psi_n} J_{\alpha} \ket{\psi_m}^2 $ where $p_n$ is the probability of an Er$^{3+}$ ion being in the $\psi_n$ state, and $J_\alpha = J_- + J_+ + J_z$.
The fitting was done using a Monte Carlo refinement method that explores the six dimensional phase space of the six non-zero $B_n^m$ coefficients.
The variation within this phase space was done by projecting random vectors in the phase space and then finding the least squares minima of the simulated pattern along those vectors.
The overall minimum is then taken as the next starting point, and the process is repeated until convergence.
The program was initially run with the lowest-lying CEF excitation fixed to $E = 4.1$~meV, thereby limiting the search to vectors in the five dimensional manifold that satisfied this condition.
After initial convergence, this condition was relaxed.
To check against false minima, the program was run eight separate times, and it was confirmed to converge to the same values.
\begin{table*}[bht]
\begin{ruledtabular}
\begin{tabular}{ l l l l }
\multicolumn{4}{c}{MgEr$_2$Se$_4$ crystal field levels} \tabularnewline
\hline
$n$ & $\delta E$ & $I_{relative}$ & $\Psi_{n}$ \tabularnewline
\hline
0 & 0 & - & $ \pm 0.920 \ket{\pm15/2} - 0.354\ket{\pm9/2} \pm 0.152 \ket{\pm3/2} + 0.075 \ket{\mp3/2} \pm0.004 \ket{\mp9/2} - 0.003 \ket{\mp15/2}$ \tabularnewline
1 & 4.155 & 1 & $ 0.730 \ket{\pm13/2} \mp 0.481 \ket{\pm7/2} + 0.450 \ket{\pm1/2} \pm 0.176 \ket{\mp5/2} + 0.046 \ket{\mp11/2}$ \tabularnewline
2 & 6.279 & 0.0517 & $ \pm 0.490 \ket{\pm13/2} - 0.078 \ket{\pm7/2} \mp 0.863 \ket{\pm1/2} - 0.058 \ket{\mp5/2} \pm 0.077 \ket{\mp11/2}$ \tabularnewline
3 & 9.193 & 0.0862 & $\pm 0.258 \ket{\pm15/2} + 0.249 \ket{\pm9/2} \mp 0.920 \ket{\pm3/2} - 0.133 \ket{\mp3/2} \pm 0.042 \ket{\mp9/2} - 0.071 \ket{\mp15/2} $ \tabularnewline
4 & 10.133 & 0.2846 & $\pm 0.662 \ket{\pm11/2} - 0.727 \ket{\pm5/2} \mp 0.083 \ket{\mp1/2} - 0.006 \ket{\mp7/2} \mp 0.163 \ket{\mp13/2} $ \tabularnewline
5 & 27.273 & 0.0329 & $\mp0.693 \ket{\pm11/2} - 0.567 \ket{\pm5/2} \mp 0.186 \ket{\mp1/2} + 0.346 \ket{\mp7/2} \mp 0.208 \ket{\mp13/2}$ \tabularnewline
6 & 29.91 & 0.0188 & $ + 0.287 \ket{\pm15/2} \pm 0.901 \ket{\pm9/2} + 0.310 \ket{\pm3/2} \pm 0.104 \ket{\mp3/2} + 0.011 \ket{\mp9/2} \mp 0.001 \ket{\mp15/2} $ \tabularnewline
7 & 29.945 & 0.0055 & $\pm0.398 \ket{\pm13/2} + 0.801 \ket{\pm7/2} \pm 0.106 \ket{\pm1/2} + 0.340 \ket{\mp5/2} \mp 0.270 \ket{\mp11/2}$ \tabularnewline
\end{tabular}
\end{ruledtabular}
\caption{The full CEF scheme of MgEr$_2$Se$_4$ as calculated from the best fit to the data. The energy levels, relative neutron scattering intensity at 0 K, and wavefunctions are presented for the 8 CEF doublets associated with the ground state manifold..}
\label{tab:CEF}
\end{table*}
\subsection{Magnetization calculation}
Magnetization curves in the main text were obtained through calculations which generalized methods of Bramwell\cite{bramwell2000} and Xu \cite{Xu2015} to include the both the effects of higher energy CEF levels and the first order perturbation shifts to wavefunctions themselves. This generalization allowed us to describe the moment of the material at both higher temperatures, where multiple CEF levels are occupied, and at higher fields, where mixing of the states leads to an increased moment.
In order to calculate the moment, we ignored interactions between moments and treated the problem in the single ion picture. We then set up a partition function using the wavefunctions $\psi_n$ we got as a result of the CEF fits to calculate the g-tensor and the susceptibility of the Er$^{3+}$ ions in MgEr$_2$Se$_4$ to calculate the moment at a given field and temperature. For the $n^{\mathrm{th}}$ doublet we have two wavefunctions $\psi_{na} ~ \mathrm{and} ~ \psi_{nb}$, and use degenerate perturbation theory with a field applied along an arbitrary direction to find the energy and proper linear combination of the doublets
\begin{equation}
g_j
\begin{pmatrix}
\bra{\psi_{na}} H_i J_i \ket{\psi_{na}} & \bra{\psi_{na}} H_i J_i \ket{\psi_{nb}} \\
\bra{\psi_{nb}} H_i J_i \ket{\psi_{na}} & \bra{\psi_{nb}} H_i J_i \ket{\psi_n^b}
\end{pmatrix}
\begin{pmatrix}
\psi_{n+} \\
\psi_{n-}
\end{pmatrix} =
\begin{pmatrix}
E_{n+}^1 \\
E_{n-}^1
\end{pmatrix}
\end{equation}
where $g_j$ is the Land\'e g factor, and the repeated indices indicate summation.
Thus we can find the response of each doublet to a field at an arbitrary direction.
To find the first order shift the in wavefunctions we use
\begin{equation}
\psi_{n\alpha}^1 = \sum_{n \neq m, \alpha, \beta} \frac{\bra{\psi_{n\alpha}^0} H_i J_i \ket{\psi_{m\beta}^0} }{ E_n^0 - E_m^0 } \ket{\psi_{m\beta}}
\end{equation}
We can take the inner product of these wavefunctions with $\frac{1}{|\mathbf{H}|} \mathbf{H}\dot J$ to find the contribution to the moment from any specific wavefunction $\psi_{n\alpha}$
\begin{widetext}
\begin{equation}
\frac{m_{n\alpha}}{\mu_b} = \frac{1}{|H|} g_l \left( \bra{\psi_{n\alpha}^0} H_i J_i \ket{\psi_{n\alpha}^0}
+ \bra{\psi_{n\alpha}^0} H_i J_i \ket{\psi_{n\alpha}^1} + \bra{\psi_{n\alpha}^1} H_i J_i \ket{\psi_{n\alpha}^0} \right)
\end{equation}
\end{widetext}
Armed with the contribution of the moment from each wavefunction $m_{n\alpha}$ and the energy of each wavefunction, $E_{n\alpha},$ we find the partition function
\begin{equation}
Z(\mathbf{H}) = \sum_{n,\alpha} \exp \left( - \frac{E^0_{n\alpha} + E^1_{n\alpha}(\mathbf{H})}{k_bT} \right)
\end{equation}
And the occupation of each doublet is
\begin{equation}
p_{n\alpha}(\mathbf{H}) = \frac{\exp\left(- \frac{E_{n\alpha}^1(\mathbf{H})}{k_b T} \right)}{Z(\mathbf{H})}
\end{equation}
This leads to the equation for average moment per $Er^{3+}$ ion in a field $\mathbf{H}$ to be
\begin{equation}
m(\mathbf{H}) = \sum_{n\alpha} m_{n\alpha}(\mathbf{H}) p_{n\alpha} (\mathbf{H})
\end{equation}
We can then integrate the above function over the whole sphere to get the magnetic moment of a powder in a field.
\begin{equation}
M = \frac{1}{4 \pi} \int_0^{2\pi} \int_0^\pi m(\mathbf{H}) \sin (\theta) d\theta d\phi
\end{equation}
We confirmed that the states are cylindrically symmetric, and are symmetric to inversion $m\left(\mathbf{H}\left(\theta, \phi\right)\right) = m\left(\mathbf{H}\left(-\theta, \phi'\right)\right)$, which allowed us to limit the integration to simply
\begin{equation}
M = \int_0^{\frac{\pi}{2}} m \left(\mathbf{H}\left(\theta,0\right)\right) \sin (\theta) d\theta
\end{equation}
This gives the full powder averaged magnetization per Er atom including the effects of first order perturbation of the wavefunction, which we integrated numerically.
\par
The effects of higher energy wavefunctions, as well as the perturbation of the wavefunctions themselves, were found to be particularly important to the calculation of magnetization. To highlight this fact, Fig. \ref{fig:mag_comparison} shows a comparison of the data for this material calculated with and without these effects considered.
At any fields higher than 1 T, the difference between the calculated magnetization when ignoring the shifts of the wavefunctions themselves is considerable.
In Fig.~\ref{fig:mag_comparison}(b) the same calculation is repeated but with $+15.845~$meV artificially added to all of the excited energy levels, in order to bring the overall first excited energy to 20 meV. This value is much closer to the energy of the 227 rare earth pyrochlores studied, and we can see that the effect now becomes far less important; the resulting curves are quite similar to the moment calculated considering only the ground state doublet.
\begin{figure}[hbt]
\centering
\includegraphics[width=\linewidth]{SM_figure_mac_calc_compare}
\caption{Plots showing the importance of including the perturbation of wavefunctions and higher energy doublets for this material. The left panel presents the measured data at 2 and 40 K as red and blue marks respectively. The calculated magnetization is superimposed on the data, for cases where perturbations of the wavefunctions were considered (solid curves) and not considered (dashed curves). The right panel compares our exact moment calculation (black) to predictions of the effective spin-1/2 model (green), which neglects perturbation effects and effects of higher older multiplets. The red curve shows the exact calculation again, but where the excited mulitplets have been artificially moved 20meV higher in energy, showing the presence of these modes to be the dominant effect.}
\label{fig:mag_comparison}
\end{figure}
\subsection{The Effects of Disorder}
In order to explore the effects of disorder on the material properties of MgEr$_2$Se$_4$, we performed a series of thermodynamic measurements on a second, less pure powder sample with a demonstrably higher level of local disorder. In Fig.~\ref{fig:sampleB_XRD}, we show the XRD pattern from this sample (which we call ``sample-B''), fit using a standard Rietveld refinement (a) and using a model Le Bail analysis (b) as described above. As compared to our primary sample (henceforth referred to as ``sample-A''), the XRD data on sample-B revealed a marginally higher fraction of impurities ($\approx 10 \%$ total), and a Le Bail fit which improved $\chi^2$ considerably over refinement values. Though the peak positions are consistent with a cubic Fd$\bar{3}$m space group, the inability of the refinement to describe peak heights within the spinel model, even allowing for variations in site occupancy, reveals the presence of a significant level of structural disorder.
Fig.~\ref{fig:sampleB_magnetization} shows the magnetization of sample-B over a range of fields and temperatures, which are interesting to compare to measurements on sample-A presented in the main text. Since INS measurements were not performed on sample-B, solid lines in Fig. 7 represent predictions based on the CEF level scheme determined for sample-A. Parallel INS measurements were not performed, and sold lines Fig.~\ref{fig:sampleB_magnetization} represent predictions based on the CEF structure determined for sample-A in the main text. The agreement between these predictions and the sample-B magnetization data is notable and suggests that conclusions about the CEF level scheme, and associated inferences about Ising symmetry and multipoles, is robust against significant disorder effects.
In contrast, heat capacity is modified significantly by disorder effects, as revealed by Fig.~\ref{fig:sampleB_HC}. In Fig.~\ref{fig:sampleB_HC}(a), we show a comparison of the heat capacity of the two samples, again with the best fit MC curve for sample-A. The peak in the heat capacity for sample-B is s reduced by almost a factor of two, and we have confirmed that there is no spin configuration in the next-nearest-neighbor dipole-ice model capable of reproducing this behavior. Further inspection reveals that the heat capacity of sample-B is not uniformly reduced, but rather that the peak is skewed to higher temperature. This leads to a long high-temperature tail wherein the curve for sample-B lies above sample-A. Integrating the area under these curves leads to the associated entropy curves in Fig.~\ref{fig:sampleB_HC}(b). Quite surprisingly, we find that sample-B recovers full Pauling residual entropy as T$\rightarrow$0 K, in direct contrast to the conclusions of the main text for sample-A. This suggests that disorder acts to hinder the mechanisms which are leading to the reduction of residual entropy in relatively pure samples of MgEr$_2$Se$_4$
\begin{figure}[!hbt]
\centering
\includegraphics[width=\columnwidth]{SM_figure_sampleB_XRD_pattern.pdf}
\caption{X-ray diffraction pattern from the disordered sample-B MgEr$_2$Se$_4$, with fits performed using a best-fit refinement to the spinel structure (a) and using a model-independent Le Bail analysis (b).}
\label{fig:sampleB_XRD}
\end{figure}
\begin{figure}[!hbt]
\centering
\includegraphics[width=\columnwidth]{SM_figure_sampleB_magnetization.pdf}
\caption{Magnetization of sample-B versus field at several temperatures, compared to the predictions based on the CEF level scheme determined from INS data for sample-A. The strong agreement here confirms that spins in MgEr$_2$Se$_4$ remain strongly Ising-like, even in the presence of significant disorder.}
\label{fig:sampleB_magnetization}
\end{figure}
\begin{figure}[!hbt]
\centering
\includegraphics[width=\columnwidth]{SM_figure_sampleB_HC.pdf}
\caption{A comparison of the magnetic heat capacity (a) and associated entropy (b) of high quality and disordered powder samples of MgEr$_2$Se$_4$.}
\label{fig:sampleB_HC}
\end{figure}
\pagebreak
\pagebreak
\par
|
1,116,691,500,788 | arxiv | \section{Introduction}
Let $(M,L)$ be a polarized \kahler manifold of dimension $m$. We endow $L$ with a Hermitian metric $h$ with positive curvature. And we use $\omega=\frac{i}{2}\Theta_h$ as the \kahler form. By abuse of notation, we still use $h$ to denote the induced metric on the $k$-th power $L^k$. Then we have a Hermitian inner product on $H^0(M,L^k)$, defined by
$$<s_1,s_2>=\int_M h(s_1,s_2)\frac{\omega^m}{m!}$$
Let $\{s_i\}$ be an orthonormal basis of $H^0(M,L^k)$. Then the on-diagonal Bergman kernel $$\rho_k(z)=\sum |s_i(z)|_h^2$$ has very nice asymptotic expansion by
the results of Tian, Zelditch, Lu, etc. \cite{Tian1990On, Zelditch2000Szego, Lu2000On, Catlin, MM}.
Recall that
$$\rho_k(z)=\frac{k^m}{\pi^m}[1+\frac{S(z)}{2k}+O(\frac{1}{k^2})]$$
where $S(z)$ is the scalar curvature of the Riemannian metric associated to $\omega$, and the other coefficients are all functions of the derivatives of the Riemannian curvature tensor.
Let $V$ be a subvariety(subscheme) of $M$, we denote by $\hcal_{k,V}$ be the subspace of $H^0(M,L^k)$ consisting of sections that vanish along $V$. We will call the Bergman kernel of $\hcal_{k,V}$ the k-th logarithmic Bergman kernel of $(M,L,V)$, denoted by $\rho_{k,V}$.
\
The asymptotics of on- and off-diagonal Bergman kernel have been extensively used in the value distribution theory of sections of line bundles by Shiffman-Zelditch and others\cite{bsz0,bsz1,bsz2,bsz3,sz1,sz2,sz3,sz4,sz6,sz7,dsz1,dsz2,dsz3,FengConditional,Feng2019}. Of closest relation to this article are \cite{bsz0,bsz1,bsz2,bsz3,FengConditional}. In \cite{bsz0,bsz1,bsz2,bsz3}, the n-point correlation functions of Gaussian random holomorphic sections of $H^0(M,L^k)$ were calculated, together with the scaling limits of these correlation functions, which very interestingly have universality among manifolds of the same dimension. In particular, in the case of Riemann surfaces, they showed that the scaling limit of pair correlation depends only on the distance of the pair of points. In \cite{FengConditional}, in the case of Riemann surfaces, Feng calculated the conditional expectation of density of critical points given a fixed zero and the conditional expectation of zeros given a fixed critical point of Gaussian random holomorphic sections of $H^0(M,L^k)$. He also calculated the rescaling limits, which also exhibit universality. In particular, the universal scaling limit of conditional expectation of density of zeros with a fixed critical point depend also only on the distance between a point and the given critical point. It was first shown in \cite{Shiffman1999} by Shiffman and Zeldtich that the mean of the zero currents $[Z_s]$ for $s\in H^0(M,L)$ endowed with the Gaussian measure is the pull back of the Fubini-Study form $\omega_{FS}$, whose difference from $\omega$ is just the $i\ddbar\log $ of the Bergman kernel. So it is not surprising that the conditional expectation of zeros is very closely related to the logarithmic Bergman kernel.
\
In this article, we will study the asymptotics of logarithmic Bergman kernels. And as an application, we will calculate the conditional expectation of density of zeros of Gaussian random holomorphic sections of $H^0(M,L^k)$ which vanishes along a fixed smooth subvariety $V$ of $M$. The logarithmic Bergman kernel behaves like Bergman kernel for the singular metric \cite{SunSun,Punctured, Sun2019} in the sense that it behaves very much like the smooth case ``away" from the subvariety $V$, while it exhibit very different nature ``around" $V$. We use the notation $\epsilon(k)$ to mean a term that is bounded by $Ck^{-r}$ for all $r$, which becomes invisible in any asymptotic expansion in inverse powers of $k$.
\
``Away" from $V$, we have
\begin{theo}\label{main1}
For $z\in M$, let $r=d(z,V)$ be the distance. Then
when $r\geq \frac{\log k}{\sqrt{k}}$, we have $$\rho_{k,V}(z)=\rho_k(z)-\epsilon(k).$$
In particular, $\rho_{k,V}(z)$ has the same asymptotic expansion as $\rho_k(z)$
\end{theo}
And ``around" $V$, we have
our main result
\begin{theo}\label{main2}
For $z\in M$, let $r=d(z,V)$ be the distance. Then
\begin{itemize}
\item[$\bullet$] when $r\leq \frac{\sqrt{\log k}}{\sqrt{2k}}$, we have $$\frac{\rho_{k,V}}{\rho_k}(z)=(1-e^{-kr^2})(1+R_k(z))$$
where $|R_k(z)|\leq C_\epsilon\frac{kr^2}{k^{-1/2+\epsilon}}$ for any $\epsilon>0$.
\item[$\bullet$]When $r$ satisfies
$$ \frac{\sqrt{\log k}}{\sqrt{2k}}<r< \frac{\log k}{\sqrt{k}}.$$
Then we have $$\frac{\rho_{k,V}}{\rho_k}(z)=1-O(e^{-kr^2})$$
\end{itemize}
\end{theo}
We would like to comment that like the asymptotic of the Bergman kernel in the smooth case, the asymptotic around $V$ also depend on mainly on the geometry of the manifolds $M$ and $V$. But unlike the smooth case where the geometry only kicks in from the second term, the dependence on the geometry appears from the first term in the asymptotic of the logarithmic Bergman kernel near $V$. When $V$ has singularities, the asymptotic should be more interesting near the singularities.
Our proof depends on two important tools: the Ohsawa-Takegoshi-Manivel extension theorem and the asymptotic of the off-diagonal Bergman kernel developed by Shiffman-Zelditch etc.
\
It is necessary to compare our logarithmic Bergman kernel to the partial Bergman kernel studied by Zelditch-Zhou \cite{ZelditchZhou-Interface,ZelditchZhou2019}, Ross-Singer\cite{Ross2017} and Coman-Marinescu\cite{ComanMarinescu}, etc. The partial Bergman kernel is for the space of sections that vanishes along a subset to a order than grows with the power $k$, while our logarithmic Bergman kernel vanishes to order 1(or one can study other fixed orders). So it appears to the author that partial Bergman kernel is more ``analytic", while our logarithmic Bergman kernel is more ``algebraic". Also, so far, the author have not been able to find results for partial Bergman kernel that vanishes a long a submanifold of codimension $\geq 1$.
\
Recall that the expectation of density of zeros of sections of a line bundle is defined as a $(1,1)$-current. More precisely, given a line bundle $L\to M$, with $\dim M=m$, and a Hermitian inner product $H$ on $H^0(M,L)$, we have a complex Gaussian measure $d\mu$ on $H^0(M,L)$. Let $Z_s$ denote the zero variety of $s\in H^0(M,L)$. Then the expectation is defined as
$$\E(Z_s)(f)=\int_{H^0(M,L)}d\mu(s)\int_{Z_s}f$$
for any smooth $(m-1,m-1)$-form $f$ with compact support. Given a subset $V$ of $M$, the conditional expectation of density of zeros of sections of $L^k$ is denoted by $\Zb_k(z|V)$, defined by
$$\Zb_k(z|V)(f)=\E_{(H^0(M,L^k),d\mu_k)}(\int_{Z_s}f|\quad s|_V=0), $$
for any smooth $(m-1,m-1)$-form $f$ with compact support, where $d\mu_k$ is the complex Gaussian measure on $H^0(M,L^k)$ corresponding to the Hermitian inner product on this space. The inner product on $\hcal_{k,V}$ is inherited from that on $H^0(M,L^k)$. We denote by $d\gamma_k$ the induced complex Gaussian measure on $\hcal_{k,V}$, then we have the following
$$\E_{(H^0(M,L^k),d\mu_k)}(\int_{Z_s}f|\quad s|_V=0)=\E_{(\hcal_{k,V},d\gamma_k)}(\int_{Z_s}f)$$
Then it follows from \cite{Shiffman1999} and \cite{Sun2019GA} that
\begin{prop}
$$\Zb_k(z|V)=\frac{i}{2\pi}\ddbar \log \rho_{k,V}+k\omega.$$
\end{prop}
Therefore, as a corollary of theorem \ref{main1}, we have
\begin{cor}\label{cor1}
For $z\in M\backslash V$, we have
$$\Zb_k(z|V)=k\omega(1+O(\frac{1}{k^2})).$$
\end{cor}
And as a corollary of theorem \ref{main2}, we have
\begin{cor}\label{cor2}
For $z_0\in V$, we fix a normal coordinates $(w_1,\cdots,w_m)$ so that $V$ is given by $w_{n+1}=\cdots=w_m$ and $\omega(z_0)=\sqrt{-1}\sum_{i=1}^{m}dw_i\wedge d\bar{w}_i$. Then for $z\in \C^m$ have the scaling limit
$$\Zb_{\infty}(z|V)=\lim_{k\to \infty}\Zb_k(z_0+\frac{z}{\sqrt{k}}|V)=\sqrt{-1}\sum_{i=1}^{n}dz_i\wedge d\bar{z}_i+\sqrt{-1}\ddbar\log (e^{\sum_{i=n+1}^{m}|z_i|^2}-1)$$
\end{cor}
\
The structure of this article is as follows. We will first do the calculations in the complex projective space, which gives us important insight for the general picture. Then we use the Ohsawa-Takegoshi-Manivel extension theorem to prove lemma \ref{theoR}, which is very important for this article, and theorem \ref{main1}. Then we use the asymptotic of the off-diagonal Bergman kernel to prove theorems \ref{main2Part1} and \ref{main2Part2} which form theorem \ref{main2}. Then we quickly prove corollaries \ref{cor1} and \ref{cor2}.
\
\textbf{Acknowledgements.} The author would like to thank Professor Bernard Shiffman for his continuous and unconditional support. The author would also like to thank Professor Chengjie Yu and Professor Song Sun for many very helpful discussions.
\iffalse
\fi
\section{Complex projective space and its geometry}
Let $[Z_0,\cdots,Z_N]$ be the homogeneous coordinates of $\CP^N$. $U_0=\{[1,z],z\in \C^N\}$ is a coordinate patch with $z_i=\frac{Z_i}{Z_0}$. The $Z_i$'s can be identified as generating sections in $H^0(\CP^N,\ocal(1))$. In particular, $Z_0$ is a local frame in $U_0$. Then on $U_0$, the Fubini-Study form $\omega=\frac{i}{2}\ddbar\log (1+|z|^2)$ has the following explicit form $$\omega=\frac{i}{2}\frac{(1+|z|^2)\sum dz^i\wedge d\bar{z}_i-(\sum \bar{z}_idz_i)(\sum z_id\bar{z}_i)}{(1+|z|^2)^2}$$
and the point-wise norm of $Z_0$ is given by
$$|Z_0|^2_{FS}(z)=\frac{1}{1+|z|^2}=e^{-\varphi}$$
For simplicity, we use the volume form $\frac{\omega^N}{\pi^N}$ instead of $\frac{\omega^N}{N!}$. Then the total volume of $M=\CP^N$ is $1$. With the Riemannian metric associated to $\omega$, the distance between two points $[Z]$ and $[W]$ in $M$ is given by $\arccos \frac{|<Z,W>|}{|Z||W|}$.
\
Sections in $H^0(\CP^N,\ocal(k))$ are represented by homogeneous polynomials of variables $Z_0,$$\cdots,Z_N$. Endowed with the inner product
$$<s_1,s_2>=\int_{\CP^N}s_1\bar{s}_2e^{-k\phi}\omega^N$$
$H^0(\CP^N,\ocal(k))$ is then a Bergman space, denoted by $\hcal_k$. And the Bergnan kernel $\rho_k(z)$, by $U(N+1)$-invariance, is constant. So we have
$\rho_k(z)=\dim H^0(\CP^N,\ocal(k))=N_k$. Therefore, we can read out an orthonormal basis of $\hcal_k$ from the binomial expansion of $\rho_k(z)e^{k\varphi}=N_k(1+|z|^2)^k$.
\
Let $V\subset \CP^N$ be a linear subspace of codimension $m$. But a $U(N+1)$-change of coordinates, we can assume that in the coordinate patch $U_0=\{[1,z],z\in \C^N\}$, $V$ is defined by $z_1=\cdots=z_m=0$. Then the sections of $H^0(\CP^N,\ocal(1))$ that vanish along $V$ are generated by $Z_i$, $i=1,\cdots,m$, where $Z_i$ is represented by $z_i$ in $U_0$. More generally, we consider the space $\hcal_{k,V}$, consisting of sections of $H^0(\CP^N,\ocal(k))$ that vanishes along $V$. By a quick calculation, the Bergman kernel of $\hcal_{k,V}$ on $U_0$ is
$$\rho_{k,V}=N_k\frac{(1+|z|^2)^k-(1+\sum_{i=m+1}^{N}|z_i|^2)^k}{(1+|z|^2)^k}$$
In particular $$\rho_{1,V}=(N+1)\frac{\sum_{i=1}^{m}|z_i|^2}{1+|z|^2}$$
We will need the following lemma:
\begin{lem}\label{projectivedistance}
For each point $[Z]\in M$, the number $\arcsin\sqrt{\frac{\rho_{1,V}([Z])}{N+1}}$ is just the distance of $[Z]$ to $V$ under the Fubini-Study distance.
\end{lem}
\begin{proof}
This is straightforward, since the distance of $[1,z]\in U_0$ to $[1,0]$ is just $\arccos\frac{1}{1+|z|^2}$.
\end{proof}
We clearly have
\begin{eqnarray*}
\lim_{k\to \infty}\sqrt{-1}\ddbar\log [(1+|z|^2/k)^k&-&(1+\sum_{i=m+1}^{N}|z_i|^2/k)^k]\\
&=&\sqrt{-1}\ddbar\log (e^{|z|^2}-e^{\sum_{i=m+1}^{N}|z_i|^2})\\
&=&\sqrt{-1}\sum_{i=m+1}^{N}dz_i\wedge d\bar{z}_i+\sqrt{-1}\ddbar\log(e^{\sum_{i=1}^{m}|z_i|^2}-1)
\end{eqnarray*}
which gives us a hint on why corollary \ref{cor2} should be true.
\section{Setting-up for the general case}
Recall that the on-diagonal Bergman kernel can also be defined as
$$\rho_k(z)=\sup_{\parallel s\parallel=1} |s(z)|^2_h$$
And at each point $p\in M$, the supremum is achieved by an unique(up to a multiple of $e^{i\theta}$) unit section, denoted by $s_p$, called the peak section at $p$. $s_p$ can also be characterized as the unit section that is orthogonal to all holomorphic sections that vanishes at $p$.
The techniques of peak sections have been very useful in the calculation of asymptotics of Bergman kernel, for example \cite{Lu2000On}. For reader's convenience and for later use, let us quickly copy some details of the construction of the peak sections.
\
First of all, we can choose local holomorphic coordinates $\{w_a\}$ centered at a given point $p$ and local \kahler potential $\phi$ for the \kahler form $\omega$
so that $$\phi(w)=\sum_aw_a\bar{w}_a+O(w^3)$$
Then by a careful change of coordinates we can make $\phi$ be of the form
$$\phi(w)=\sum_a w_a\bar{w}_a+\sum P_{abcd}w_aw_b\bar{w}_c\bar{w}_d+\textit{higher order terms}$$
Then by the rescaling of coordinates $z_a=\sqrt{k}w_a$, $\phi$ becomes
$$\Phi(z)=|z|^2+\frac{1}{k}P(z)+k^{-3/2}Q(z)+O(k^{-2})$$
And the volume form $(i\ddbar \Phi)^m$ is of the form
$$J=1+k^{-1}p(z)+k^{-3/2}q(z)+O(k^{-2})$$
Then one choose local frame $\sigma_0$ for the line bundle $L^k$ on the ball $|z|\leq k^{1/4}$ for example so that $|\sigma_0|^2_h=e^{-\Phi}$. We will mention $\sigma_0$ as a "normal frame".
One then modify $\sigma_0$ to get the peak section. More precisely, $\sigma_0$, when regarded as a global discontinuous section of $L^k$, extending by zero outside our ball, is almost orthogonal to all holomorphic sections vanishing at $p$. In fact, by the symmetry of $e^{-|z|^2}$, we have $$|<\sigma_0,\tau>|\leq Ck^{-1}\parallel \tau\parallel$$
and $$\parallel\sigma_0\parallel^2=\pi^m(1+ak^{-1}+O(k^{-2}))$$
Then by H\"{o}rmander's $L^2$-techniques, we can modify $\sigma_0$ to get a global section, by introducing an error of the size $\epsilon(k)$, due to the fact that $e^{-|z|^2}$ decays very fast near the boundary of our ball. Therefore, under the local frame $\sigma_0$, the peak section $s_p$ is represented by a holomorphic function of the form $\sqrt{\frac{k^m}{\pi^m}}(1+O(k^{-1}))$. We will be using the important property of $s_p$ that
$$|s_p(z)|_h=\epsilon(k),$$
for $z$ whose distance to $p$ is $\geq \frac{\log k}{\sqrt{k}}$, since $e^{(\log k)^2}=\epsilon(k)$.
\
Let $V$ be a subvariety of $M$, then for $k$ large enough, we have the exact sequence
$$0\to H^0(M,L^k\otimes I_V)\to H^0(M,L^k)\to H^0(V,L^k)\to 0$$
where $I_V$ is the ideal sheaf of $V$. So $\hcal_{k,V}=H^0(M,L^k\otimes I_V)$, and
the orthogonal complement $\hcal_{k,V}^{\perp}$ is isomorphic to $H^0(V,L^k)$.
We have the following lemma.
\begin{lem}\label{span}
$\hcal_{k,V}^{\perp}$ is spanned by the peak sections $s_p$ for $p\in V$.
\end{lem}
\begin{proof}
We denote by $W$ the linear space spanned by the peak sections $\{s_p\}_{p\in V}$. Then for any $s\in \hcal_{k,V}$, and each $p\in V$, $s$, vanishing at $p$, is orthogonal to $s_p$. Therefore $s_p\in \hcal_{k,V}^{\perp}$, namely $W\subset \hcal_{k,V}^{\perp}$. On the other hand, if $s\in \hcal_k$ is orthogonal to $W$, then $s$ has to vanish at each $p\in V$, so $s\in \hcal_{k,V}$.
\end{proof}
We denote by $\pi$ the restriction map $H^0(M,L^k)\to H^0(V,L^k)$. The restriction of $\pi$ on $\hcal_{k,V}^{\perp}$ is an isomorphism, and is denoted by $R$.
\
When $V$ is of pure dimension $n$, $H^0(V,L^k)$ is also endowed with a Hermitian inner product by integrating over the smooth part of $V$. We want to show that $\frac{R}{\sqrt{k^r}}$ is a quasi-isometry. For this purpose, we need to use the Ohsawa-Takegoshi-Manivel extension theorem. There are several versions of this theorem, for example, \cite{OhsawaDemailly, Manivel1993,Varolin2015}. The version that is most useful for our purpose is the one from \cite{Ohsawa5}. In order to state the theorem, we copy the setting-up from \cite{Ohsawa5}.
\
Let $M$ be a complex manifold of dimension $m$ with continuous measure $d\mu_M$ and let $(E,h) $ be a holomorphic Hermitian vector bundle over $M$. Let $S$ be a closed complex submanifold of dimension $n$.
Consider a class of continuous functions $\Psi:M\to [-\infty,0)$ such that
\begin{itemize}
\item[(1)] $S\subset\Psi^{-1}(-\infty)$
\item[(2)] If $S$ is $n$-dimensional around a point $x$, there exists a local coordinate $(z_1,\cdots,z_m)$ on a neighborhood $U$ of $x$ such that $z_{n+1}=\cdots=z_m=0$ on $S\cap U$ and
$$\sup_{U\backslash S}|\Psi(z)-(m-n)\log \sum_{n+1}^{m}|z_j|^2|<\infty$$
\end{itemize}
The set of such functions $\Psi$ will be denoted by $\sharp(S)$. Clearly, the condition (2) does not depend on the choice of local coordinate. For each $\Psi\in \sharp(S)$, one can associate a positive measure $dV_M[\Psi]$ on $S$ as the minimum element of the partially ordered set of positive measure $d\mu$ satisfying
$$\int_{S}fd\mu\geq \overline{\lim_{t\to \infty}}\frac{2(m-n)}{\sigma_{2m-2n-1}}\int_Mfe^{-\Psi} \chi_{R(\Psi,t)}dV_M$$
for any nonnegative continuous function $f$ with supp$f\subset\subset M$
. Here $\sigma_m$ denotes the volume of the unit sphere in $\R^{m+1}$, and $\chi_{R(\Psi,t)}$ denotes the characteristic function of the set
$$R(\Psi,t)=\{x\in M|-t-1<\Psi(x)<-t \}.$$
Let $\Theta_h$ be the curvature form of the fiber metric $h$. Let $\Delta_h(S)$ be the set of functions $\tilde{\Psi}$ in $\sharp(S)$ such that, for any point $x\in M$, $e^{-\tilde{\Psi}}h=e^{-\hat{\Psi}}\hat{h}$ around $x$ for some plurisubharmonic function $\hat{\Psi}$ and some fiber metric $\hat{h}$ whose curvature form is semipositive in the sense of Nakano.
\begin{theo}[Ohsawa-Takegoshi-Manivel,\cite{Ohsawa5}]
Let $M$ be a complex manifold with a continuous volume form $dV_M$, let $E$ be a holomorphic vector bundle over $M$ with a $C^{\infty}$ fiber metric $h$, let $S$ be a closed complex submanifold of $M$, let $\Psi\in \sharp(S)$ and let $K_M$ be the canonical line bundle of $M$. If the following are satisfied,
\begin{itemize}
\item[1)] There exists a closed subset $X\subset M$ such that \begin{itemize}
\item[(a)] $X$ is locally negligible with respect to $L^2$ holomorphic functions, i.e., for any local coordinate neighborhood $U\subset M$ and for any $L^2$ holomorphic function $f$ on $U\backslash X$, there exists a holomorphic function $\tilde{f}$ on $U$ such that $\tilde{f}|_{U\backslash X}=f$.
\item[(b)] $M\backslash X$ is a Stein manifold with intersects with every component of $S$.
\end{itemize}
\item[2)] $\Theta_h\geq 0$ in the sense of Nakano.
\item[3)]$(1+\delta)\Psi\in \Delta_h(S)\cap C^{\infty}(M\backslash S)$ for some $\delta>0$.
\end{itemize}
then there exists a constant $C$ such that for any $f\in H^0(S,E\otimes K_M|_S)$ such that
$$\int_S |f|^2_{h\otimes (dV_M)^{-1}}dV_M[\Psi]<\infty,$$
there exists $F\in H^0(M,E\otimes K_M)$ such that
$$\int_M|F|^2_{h\otimes (dV_M)^{-1}}dV_M\leq (C+\delta^{-3/2})^2\int_S |f|^2_{h\otimes (dV_M)^{-1}}dV_M[\Psi].$$
If $\Psi$ is plurisubharmonic, the constant $(C+\delta^{-3/2})^2$ can be chosen to be less than $256\pi $.
\end{theo}
In our situation, $V$ is the $S$ in the theorem. The volume form is $dV_M=\frac{\omega^m}{m!}$. The vector bundle $E$ is the line bundle $L^k-K_M$ with a twisting of the metric $e^{-k\phi}\otimes dV_M=e^{-k\phi+\kappa}$. Since our manifold $M$ is projective, the $X$ in the theorem exists. Let $N_V$ denote the normal bundle of $V$, with the metric induced by $\omega$ on $T_M$. Let $r$ denote the length of vectors in $N_V$. Denote by $N_V(\rho)$ the subset of vectors with length $r<\rho$, the for $\rho$ small enough, the exponential map $$\exp: N_V(\rho)\to M$$
is a diffeomorphism of $N_V(\rho)$ with its image. Then $r$ is a function in a neighborhood of $V$ in $M$, we then choose a nonnegative smooth function $\chi$ on $[0,\infty)$, which is concave and satisfies the following conditions
\begin{itemize}
\item[(1)]$\chi(x)=x$ for $x\leq \frac{(\log k)^2}{k}$;
\item[(2)]$\chi(x)$ is constant for $x\geq \frac{(10\log k)^2}{k}$
\end{itemize}
So $\chi(r^2)$ can be seen as a smooth function on $M$, which is constant away from $V$. Then we twist the metric on $L^k-K_M$ by $e^{\beta_k \chi(r^2)}$, for $\beta_k$ to be determined.
Then we let $\Psi=(m-n)\log r^2$ and extend it smoothly to be defined on $M$. Clearly this function $\Psi$ satisfies the two conditions in the definition of $\sharp(V)$. Also $(1+1)\Psi\in \Delta_h(V)\cap C^{\infty}(M\backslash S)$ for $h=e^{-k\phi+\kappa+\beta_k \chi(r^2)}$ when $\beta_k$ not too big. We want to make $\beta_k$ as large as possible. So we calculate
$$\ddbar \chi(r^2)=\chi'\ddbar r^2+\chi^{''}\partial r^2\wedge\bar{\partial}r^2$$
By our construction of $\chi$, we have $0\leq \chi'\leq 1$ and $\chi^{''}\leq 0$. So one can see that for $k$ large, we can allow $\beta_k$ to be of the size $k-O(\frac{\log k}{\sqrt{k}})$. Finally, we calculate the measure $dV_M[\Psi]$. By integrating along fibers, one sees easily that for our $\Psi$, the measure $dV_M[\Psi]$ is just the smooth measure $\frac{\omega^n}{n!}$.
\
Now we can apply the Ohsawa-Takegoshi-Manivel extension theorem to get that for any $f\in H^0(V,L^k)$, one can find $F\in H^0(M,L^k)$ satisfying the inequality:
\begin{equation}
\int_M |F|^2e^{-k\phi+\beta_k\chi(r^2)}\frac{\omega^m}{m!}\leq C\int_V |f|^2e^{-k\phi}\frac{\omega^n}{n!}
\end{equation}
for some constant $C$ independent of $k$.
For simplicity, we assume that $\int_V |f|^2e^{-k\phi}\frac{\omega^n}{n!}=1$.
\
First of all, since $e^{-\beta_k\chi(r^2)}=\epsilon(k)$ for $r\geq \rho=\frac{\log k}{\sqrt{k}}$,
we see that
$$\int_{M\backslash N_V(\rho)}|F|^2e^{-k\phi}dV_{M,\omega}=\epsilon(k)$$
So we can see that the mass of $F$ is concentrated within a small tubular neiborhood of $V$ with radius $\rho=\frac{\log k}{\sqrt{k}}$. To estimate the integral within the small neighborhood. We only need to notice that
$$\int_{\C}|z|^{2a}e^{-k|z|^2}dV=\frac{\pi a!}{k^{a+1}}$$
Since $\beta_k=k-O(\frac{\log k}{\sqrt{k}})$, by integrating along fibers of $N_V$ first, one can see that
$$\int_{N_V(\rho)}|F|^2e^{-k\phi}dV_{M,\omega}\approx \frac{1}{k^{m-n}} \int_{N_V(\rho)} |F|^2e^{-k\phi+\beta_k\chi(r^2)}\frac{\omega^m}{m!}$$
Therefore, we have the following
\begin{theo}\label{theoR}
The restriction map $R: \hcal_{k,V}^{\perp}\to H^0(V,L^k)$ has norm satisfying
$$\parallel R\parallel^2=O(\frac{1}{k^{m-n}})$$
\end{theo}
\begin{rem}
This theorem is very important for our subsequent arguments. If one wishes to be more precise than the big O, one need to have optimal constant in the Ohsawa-Takegoshi-Manivel extension theorem \cite{ZhouZhu1,ZhouZhu2}.
\end{rem}
As a direct application, we have theorem \ref{main1}:
\begin{theo}
For $z\in M$, let $r=d(z,V)$ be the distance. Then
when $r\geq \frac{\log k}{\sqrt{k}}$, we have $$\rho_{k,V}(z)=\rho_k(z)-\epsilon(k).$$
In particular, $\rho_{k,V}(z)$ has the same asymptotic expansion as $\rho_k(z)$
\end{theo}
\begin{proof}
Consider the peak section $s_z$. We know that for any point $w$ with distance $d(z,w)\geq \frac{\log k}{\sqrt{k}}$, the length
$|s_z(w)|_h=\epsilon(k)$. So, the $L^2$ norm of $s_z|_V$ is $\epsilon(k)$. So by the the theorem above, if we write $s_z=s_1+s_2$ with
$s_1\in \hcal_{k,V}$ and $s_2\in \hcal_{k,V}^{\perp}$, then $\parallel s_2\parallel^2=\epsilon(k)$. Therefore $\parallel s_1\parallel^2=1-\epsilon(k)$ and
$$|s_1(z)|_h=|<s_1,s_z>||s_z(z)|_h=|<s_1,s_1>||s_z(z)|_h=(1-\epsilon(k))|s_z(z)|_h$$
So $\rho_{k,V}(z)\geq \rho_k(z)-\epsilon(k)$, on the other hand, we clearly have $\rho_{k,V}(z)< \rho_k(z)$, so the theorem is proved.
\end{proof}
\section{Calculations near $V$}
Next, we study the asymptotic of $\rho_{k,V}$ around $V$. For this,
we need the off-diagonal asymptotics for the Bergman kernel, about which we now recall some details from \cite{sz1,sz2,sz3}.
Let $\pi:X\to M$ be the circle bundle of unit vectors in the dual bundle $L^*\to M$ with respect to $h$. Sections of $L^k$ lift to equivariant functions on $X$. Then $s\in H^0(M,L^k)$ lifts to a CR holomorphic function on $X$ satisfying $\hat{s}(e^{i\theta }x)=e^{ik\theta}\hat{s}(x)$. We denote the space of such functions by $\hcal_k^2(X)$. The Szeg\"{o} projector is the orthogonal projector
$$\Pi_k:L^2(X)\to \hcal_k^2(X),$$
which is given by the Szeg\"{o} kernel(Bergman kernel)
$$\Pi_k(x,y)=\sum \hat{s}_j(x)\overline{\hat{s}_j(y)} \quad (x,y\in X). $$
(Here the functions $\hat{s}_j$ are the lifts to $\hcal_k^2(X)$ of the orthonormal sections $\{s_j\}$; they provide an orthonormal basis for $\hcal_k^2(X)$.)
The covariant derivative $\nabla s$ of a section $s$ lifts to the horizontal derivative $\nabla_h\hat{s}$ of its equivariant lift $\hat{s}$ to $X$; the horizontal derivative is of the form
$$\nabla_h\hat{s}=\sum_{j=1}^{m}(\frac{\partial \hat{s}}{\partial z_j}-A_j\frac{\partial \hat{s}}{\partial \theta}dz_j)$$
For $z=\pi(x),w=\pi(y)\in M$, we will write
$$|\Pi_k(z,w)|=|\Pi_k(x,y)|,$$
in particular, on the diagonal $\Pi_k(z,z)>0$ is the same as our previous notation $\rho_k(z)$. For each point $z_0\in M$, we choose a neighborhood $U$, a local coordinate chart $\rho: (U,z_0)\to (\C^m,0)$, and a preferred local frame at $z_0$, which is a local frame $e_L$ such that
$$\parallel e_L(z)\parallel_h=1-\frac{1}{2}|\rho(z)|^2+\cdots$$
For $u=(u_1,\cdots,u_m)\in \rho(U)$, $\theta\in (-\pi, \pi)$, let
$$\tilde{\rho}(u_1,\cdots,u_m,\theta)=\frac{e^{i\theta}}{|e_L^*(\rho^{-1}(u))|_h}e_L^*(\rho^{-1}(u))\in X$$
so that $(u_1,\cdots,u_m,\theta)\in \C^m\times \R$ give local coordinates on $X$. We then write
$$\Pi_k^{z_0}(u,\theta;v,\varphi)=\Pi_k(\tilde{\rho}(u,\theta),\tilde{\rho}(v,\varphi))$$
\begin{theo}[\cite{sz1,sz2,sz3}]
Let $(L,h)$ be a positive Hermitian holomorphic line bundle over a compact m-dimensional \kahler manifold $M$. $\omega=\frac{i}{2}\Theta_h$ is the \kahler form. Let $z_0\in M$, then
\begin{itemize}
\item[(i)]\begin{eqnarray*}
&&\pi^m k^{-m}\Pi^{z_0}_k(\frac{u}{\sqrt{k}},\frac{\theta}{\sqrt{k}};\frac{v}{\sqrt{k}},\frac{\phi}{\sqrt{k}})\\
&=&e^{i(\theta-\phi)+u\cdot v-\frac{1}{2}(|u|^2+|v|^2)}[1+\sum_{r=1}^{l}k^{-\frac{r}{2}}p_r(u,v)+k^{-\frac{l+1}{2}}R_{kl(u,v)}],
\end{eqnarray*}
where the $p_r$ are polynomials in $(u,v)$ of degree $\leq 5r$(of the same parity as $r$), and
$$|\nabla^jR_{kl}(u,v)|\leq C_{jl\epsilon b}k^{\epsilon}\quad \textit{for}\quad |u|+|v|<b\sqrt{\log k},$$
for $\epsilon, b\in \R^+, j,l\geq 0$. Furthermore, the constant $C_{jl\epsilon b}$ can be chosen independent of $z_0$.
\item[(ii)] For $b>\sqrt{j+2l+2m}, j, l\geq 0$, we have
$$|\nabla_h^j\Pi_k(z,w)|=O(k^{-l})$$
uniformly for $\textit{dist}(z,w)\geq b\sqrt{\frac{\log k}{k}}$.
\end{itemize}
\end{theo}
The so called normalized Bergman kernel $P_k(v,z_0)$ was also defined in \cite{sz1} as
$$P_k(v,z_0)=\frac{|\Pi_k(v,z_0)|}{\sqrt{\Pi_k(v,v)}\sqrt{\Pi_k(z_0,z_0)}}$$
which contains plenty of information of the geometry of the image of $M$ under the Kodaira embedding\cite{sun1}. Recall that it was proved in \cite{sz1} the following estimations.
\begin{theo}
$$P_k(z_0+\frac{u}{\sqrt{k}},z_0+\frac{v}{\sqrt{k}})=e^{-\frac{1}{2}|u-v|^2}(1+R_N(u,v))$$
where the remainder satisfies the following \begin{eqnarray*}
|R_k(u,v)|&\leq& \frac{C_1}{2}|u-v|^2k^{-1/2+\epsilon}\\
|\nabla R_k(u,v)|&\leq& C_1|u-v|k^{-1/2+\epsilon}\\
|\nabla^jR_k(u,v)|&\leq& C_jk^{-1/2+\epsilon}\\
\end{eqnarray*}
for $|u|+|v|<b\sqrt{\log k}$, where the $C_i's$ all depend on $b$.
\end{theo}
Locally, the orthonormal basis $\{s_i\}$ are represented by holomorphic functions $\{f_i\}$, so the Kodaira embedding $\Phi_K$ is given by
$$\Phi_k(z)=[f_0(z),\cdots,f_{N}(z)]$$
We denote by $$Q_k(z,w)=\sum f_i(z)\bar{f}_i(w)$$
So we have $$Q_k(z,z)=\frac{N^m}{\pi^m}(1+O(\frac{1}{k}))e^{k\phi}$$
For simplicity, we first work on the case of codimension 1 to illustrate the idea of calculations. One immediately realizes that this idea works for codimension $\geq 1$. Now we assume $V$ is a smooth divisor $D$.
Notice that the property required for the choice of $e_L$ by the theorem above is only that
$\phi=|z|^2+$higher order terms. So we are allowed to choose local coordinates $z$ so that $D$ is defined by $\{z_m=0\}$.
We use the notation $f_{i,m}=\frac{\partial}{\partial z_m}f_i$.
Then by taking the derivatives $\frac{\partial}{\partial \bar{z}_m}P(z,z)$ and $\frac{\partial^2}{\partial z_m\partial \bar{z}_m}P(z,z)$, and since $\partial \varphi(z_0)=0$ and $\ddbar \varphi(z_0)=\sum dz_i\wedge d\bar{z}_i$, we get the following estimations
\begin{eqnarray}
\sum f_i(z_0)\overline{f_{i,m}(z_0)}&=&O(k^{m-1})\\
\sum f_{i,m}(z_0)\overline{f_{i,m}(z_0)}&=&\frac{k^{m+1}}{\pi^m}(1+O(k^{-1}))
\end{eqnarray}
We denote by $f=(f_0,\cdots,f_N)$ and $f_{,m}=(f_{1,m},\cdots,f_{N,m})$. Then we define an unit section of $L^k$ as
$$\alpha_{z_0}(z)=\frac{\sum \overline{f_{i,m}(z_0)}s_i(z)}{|f_{,m}(z_0)|}$$
The estimations above implies that
$$|\alpha_{z_0}(z_0)|_h^2=O(k^{m-3})$$
We need also to estimate the norm of $\alpha_{z_0}(z)$ for points $z\neq z_0$. For $z,w\in U$, by the asymptotics for the off-diagonal Bergman kernel, we have the following
$$\sum f_i(z/\sqrt{k})\overline{f_i(w/\sqrt{k})}=\frac{k^m}{\pi_m}e^{z\cdot w}[1+\sum_{r=1}^{l}k^{-r/2}p_r(z,w)+k^{-\frac{l+1}{2}}R_{kl}(z,w)]$$
where $p_r$ and $R_{kl}$ are slightly different from those in the theorem, but enjoy similar estimations. Therefore, we have
$$\sum f_i(z)\overline{f_{i,m}(z_0)}=O(k^{m})$$
for $z\in D$ satisfying $|z|<\sqrt{2m+3}\frac{\sqrt{\log k}}{\sqrt{k}}$, since $z_m=0$. This implies that
$$|\alpha_{z_0}(z)|_h^2=O(k^{m-1})e^{-k|z|^2},$$
for these $z$.
Finally, when $\textit{dist}(z,z_0)\geq \sqrt{2m+3}\sqrt{\frac{\log k}{k}}$, we can use part (ii) of Shiffman-Zelditch's theorem to get that
$$|\alpha_{z_0}(z)|_h^2=O(k^{-1})$$
Therefore we can estimate the $L^2$ norm of $\alpha_{z_0}$ on $D$ to get
$$\int_D |\alpha_{z_0}(z)|_h^2\frac{\omega^{m-1}}{(m-1)!}=O(1+\frac{1}{k})=O(1)$$
as $\int_{\C^{m-1}}^{}e^{-k|z|^2}dV=O(k^{1-m})$. So we have proved the following
\begin{lem}\label{almostortho}
$\alpha_{z_0}$ is almost orthogonal to the space $\hcal_{k,D}^{\perp}$. More precisely, if we write $$\alpha_{z_0}=s_1+s_2,$$
with $s_1\in \hcal_{k,D}$ and $s_2\in \hcal_{k,D}^{\perp}$, then $\parallel s_2\parallel^2=O(k^{-1})$.
\end{lem}
Before proceeding, we want to explain the meaning of this lemma in the sense of complex projective geometry. A complex vector space equipped with a Hermitian inner product can be identified with its dual space by a conjugate linear map. In our case, let $W=H^0(M,L^k)$, then the Kodaira map $\Phi_k$ maps $M$ to $\PP W^*$. Fixing an orthonormal basis $(s_0,\cdots,s_N)$, and local frame $e_L$, then $\Phi_k(z)=(f_0(z),\cdots,f_N(z)$, where $s_i=f_ie_L^k$. Then by taking the complex conjugate $(\overline{f_0(z)},\cdots,\overline{f_N(z)})$, we get a conjugate-holomorphic embedding $\overline{\Phi_k}:M\to \PP W$, with $\overline{\Phi_k}(z)=[\sum \overline{f_i(z)}s_i]$. What interesting is that $\overline{\Phi_k}(p)$ is just the complex line in $W$ containing the peak section $s_p$. And lemma \ref{span} implies that $\overline{\Phi_k}(D)$ linearly span $\PP \hcal_{k,D}^{\perp}$. So the preceding lemma, in this setting, says that the image of $\frac{\partial}{\partial z_m}(z_0)$ under the tangent map of $\overline{\Phi_k}$ is almost orthogonal to the linear subspace $\PP \hcal_{k,D}^{\perp}$.
We can apply similar calculation for the point in the $z_m$-disk passing through $z_0$, which have coordinates $(0,\cdots,0,z_m)$ with $z_m$ small. We use $v$ to denote the points on this disk. We again define
$$\alpha_v=\frac{\sum \overline{f_{i,m}(v)}s_i}{|f_{,m}(v)|}$$
as an unit section of $L^k$. Then we estimate the point-wise norm of $\alpha_v$ along $D$ by differentiating the function $Q_k(z,w)$. We have
\begin{eqnarray}
\sum f_i(z)\overline{f_{i,m}(v)}&=&O(k^{m})|e^{kv\cdot z}|=O(k^{m})\\
\sum f_{i,m}(v)\overline{f_{i,m}(v)}&=&\frac{k^{m+1}}{\pi^m}(1+O(k^{-1}))(1+k|v|^2)e^{k|v|^2}
\end{eqnarray}
for $z\in D$ satisfying $|z|<\sqrt{2m+3}\frac{\sqrt{\log k}}{\sqrt{k}}$. Therefore, for these $z$, we have
$$|\alpha_v(z)|^2_h=O(k^{m-1})e^{-k|z|^2-k|z|^2}$$
So the integral over this patch of $z$ is $O(1)$ as $\alpha_{z_0}$.
And for the remaining $z\in D$, we still have $|\alpha_v(z)|^2_h=O(1/k)$, so in conclusion, we have
$$\int_D |\alpha_v|_h^2\frac{\omega^{m-1}}{(m-1)!}=O(1)$$
Again, this implies that $\alpha_v$ is almost orthogonal to $\hcal_{k,D}^{\perp}$ with the same estimation as in lemma \ref{almostortho}. Now we put all these ideas together to get
\begin{theo}We have the following estimation for the logarithmic Bergman kernel along the disk:
$$1-\frac{P_k^2(v,z_0)}{1-\beta(r)^2}\leq \frac{\rho_{k,D}}{\rho_k}(v)\leq 1-P_k^2(v,z_0),$$
where $r=|v|=|v_m|$ and $\beta(r)=C\int_{0}^{r}\sqrt{1+kx^2}e^{kx^2/2}dx$ with $C$ independent of $k$.
\end{theo}
\begin{proof}
We decompose $\overline{f}(v)-\bar{f}(z_0)=b+c$, with $b\in \hcal_{k,D}$ and $c\in \hcal_{k,D}^{\perp}$. Let $d$ denote the distance in the Fubini-Study metric. Then
$$\cos d( \bar{f}(v),\bar{f}(z_0))=\frac{|< \bar{f}(v),\bar{f}(z_0)>|}{|f(v)||f(z_0)|}=\frac{<\bar{f}(z_0)+c,\bar{f}(z_0)>}{|f(v)||f(z_0)|}$$
$$\cos d(\bar{f}(v),\hcal_{k,D}^{\perp})=\frac{|< \bar{f}(v),\bar{f}(z_0)+c>|}{|f(v)||f(z_0)+c|}=\frac{|\bar{f}(z_0)+c|}{|f(v)|}$$
So $$\frac{\cos d( \bar{f}(v),\bar{f}(z_0))}{\cos d(\bar{f}(v),\hcal_{k,D}^{\perp})}=\frac{<\bar{f}(z_0)+c,\bar{f}(z_0)>}{|\bar{f}(z_0)+c||f(z_0)|}$$
Then
$$|c(r)|\leq \int_0^r O(\frac{1}{\sqrt{k}})\sqrt{\frac{k^{m+1}}{\pi^m}(1+kx^2)}e^{kx^2/2}dx $$
So $$\frac{|c(r)|}{|f(z_0)|}\leq C\int_{0}^{r}\sqrt{1+kx^2}e^{kx^2/2}dx=\beta(r)$$
When $\frac{|c(r)|}{|f(z_0)|}<1$, we have
$$1\leq \frac{\cos d(\bar{f}(v),\hcal_{k,D}^{\perp})}{\cos d( \bar{f}(v),\bar{f}(z_0))}\leq (1-(\frac{|c(r)|}{|f(z_0)|})^2)^{-1}$$
By lemma \ref{projectivedistance}, we are interested in $\lambda_v=\sin^2 d(\bar{f}(v),\hcal_{k,D}^{\perp})=\frac{\rho_{k,D}}{\rho_k}$, which satisfies
$$1-(1-(\frac{|c(r)|}{|f(z_0)|})^2)^{-1}\cos^2 d( \bar{f}(v),\bar{f}(z_0))\leq \lambda_v\leq 1-\cos^2 d( \bar{f}(v),\bar{f}(z_0))$$
We have $\frac{|< \bar{f}(v),\bar{f}(z_0)>|}{|f(v)||f(z_0)|}=\frac{|\Pi_k(v,z_0)|}{\sqrt{\Pi_k(v,v)}\sqrt{\Pi_k(z_0,z_0)}}$ is just the normalized Bergman kernel $P_k(v,z_0)$.
So
\end{proof}
Notice that the term $\beta(r)$ is small when $r$ is small. With the substitution $R=\sqrt{k}r$, it becomes $$\frac{C}{\sqrt{k}}\int_{0}^{R}\sqrt{1+x^2}e^{x^2/2}dx$$
So for fixed $R$, $\beta=O(\frac{1}{\sqrt{k}})$.
More generally, when $V$ is a smooth subvariety, we can choose local coordinates $(z_1,\cdots,z_m)$ so that $V$ is defined as $z_m=z_{m-1}\cdots=z_{m-r+1}=0$. Then we can repeat our calculations for the divisor case without any difficulties. More precisely, in each normal direction at a point of $V$, we can apply an unitary change of coordinates, so that that direction is contained in the space spanned by $\frac{\partial}{\partial z_m}$
So the conclusions for the divisor case hold for this more general case.
Notice that when $|z|$ is small, $|z|$ is about the distance of $z$ to $V$, since $\omega(z)=\sum \delta_{ij}dz_i\wedge d\bar{z}_i+O(|z|)$. More precisely, we have
$$d(z,D)=|z|(1+O(|z|)),$$ so we can talk about the asymptotics without going local. In particular, when $|z|\leq \frac{\log k}{\sqrt{k}}$,
$$e^{-k|z|^2+kd^2(z,V)}=(1+k|z|^2\frac{\log k}{\sqrt{k}})$$
When $r\leq \frac{\sqrt{\log k}}{\sqrt{2k}}$, we have
$$\beta^2(r)=O(\frac{\log k}{\sqrt{k}}),$$ so we have the following theorem.
\begin{theo}\label{main2Part1}
For $z\in M$, let $r=d(z,V)$ be the distance. Then
when $r\leq \frac{\sqrt{\log k}}{\sqrt{2k}}$, we have $$\frac{\rho_{k,V}}{\rho_k}(z)=(1-e^{-kr^2})(1+R_k(z))$$
where $|R_k(z)|\leq C_\epsilon\frac{kr^2}{k^{-1/2+\epsilon}}$ for any $\epsilon>0$.
\end{theo}
\begin{rem}
If the reader is careful enough, he/she must have noticed there is a gap area between our estimations around $V$ and away from $V$, namely when $$ \frac{\sqrt{\log k}}{\sqrt{2k}}<r< \frac{\log k}{\sqrt{k}},$$
which, very interestingly, have also been seen in the case of Bergman kernel for Poincar\'{e} type metrics \cite{SunSun,Punctured, Sun2019}, where it was called the "neck". Luckily, unlike the Poincar\'{e} type metrics, the "neck" in our case is not very difficult.
\end{rem}
\
Let $z_0\in M$ with distance $r$ satisfying
$$ \frac{\sqrt{\log k}}{\sqrt{2k}}<r<\frac{\log k}{\sqrt{k}}.$$
We consider the peak section $s_{z_0}$. Using the normal coordinates centered at $z_0$, and the normal frame, we know that $$|s_{z_0}(z)|_h^2=(1+O(\frac{1}{k}))\frac{k^m}{\pi^m}e^{-k|z|^2}$$
Let $w\in V$ be the point that is closest to $z_0$. Then for $z\in V$ in this coordinates patch, we have
$$|z|^2\approx |w|^2+|z-w|^2,$$
since we are looking at geodesics of very small scales, meaning things work like Euclidean spaces.
Therefore
$$\int_V |s_{z_0}(z)|_h^2\frac{\omega^{m-r}}{(m-r)!}=O(k^r)e^{-k|r|^2}$$
Therefore, the image of the orthogonal projection of $s_{z_0}$ onto $\hcal_{k,V}^{\perp}$ has $L^2$ norm $O(e^{-kr^2})$. If we decompose $s_{z_0}=s_1+s_2$, with $s_1\in \hcal_{k,V}$ and $s_2\in \hcal_{k,V}^{\perp}$, then $\parallel s_1\parallel^2=1-O(e^{-kr^2})$ and
$|s_1(z_0)|_h=|<s_1,s_{z_0}>||s_{z_0}|_h=<s_1,s_1>|s_{z_0}|_h=(1-O(e^{-kr^2}))|s_{z_0}|_h$. Therefore
$$\rho_{k,V}(z_0)>|s_1(z_0)|_h^2=(1-O(e^{-kr^2}))\rho_k(z_0)$$
So we have proved:
\begin{theo}\label{main2Part2}
Let $z_0\in M$ with distance to $V$ $r$ satisfies
$$ \frac{\sqrt{\log k}}{\sqrt{2k}}<r< \frac{\log k}{\sqrt{k}}.$$
Then we have $$\frac{\rho_{k,V}}{\rho_k}(z)=1-O(e^{-kr^2})$$
\end{theo}
\
\begin{proof}[Proof of corollary \ref{cor1}]
We have $\rho_{k,V}=\frac{k^m}{\pi^m}(1+O(\frac{1}{k}))$, so $i\ddbar \log \rho_{k,V}=O(\frac{1}{k})\omega$.
\end{proof}
\begin{proof}[Proof of corollary \ref{cor2}]
In this normal coordinates, let $r=d(w,V)$, then we have $r^2=\sum_{i=n+1}^{m}|w_i|^2(1+o(\frac{1}{k^{1/4}}))$ for $w$ satisfying $|w|^2\leq \frac{\log k}{k}$, by Toponogov's compparison theorem. We also choose local frame so that $\phi=|w|^2+\textit{higher order terms}$.
Therefore by theorem \ref{main2Part2}, we have
$$(\rho_{k,V}e^{k\phi})(\frac{z}{\sqrt{k}})=\frac{k^m}{\pi^m}(e^{|z|^2}-e^{\sum_{i=1}^{n}|z_i|^2})(1+o(\frac{1}{k^{1/4}}))$$
Then by taking the limit of $\sqrt{-1}\ddbar\log (\rho_{k,V}e^{k\phi})(\frac{z}{\sqrt{k}})$, we get the conclusion.
\end{proof}
\bibliographystyle{plain}
|
1,116,691,500,789 | arxiv | \section{Introduction}
The term super-shell was observationally defined
by \cite{heiles1979}
as holes in the H\,I-column density distribution of our Galaxy.
The dimensions of these objects span from 100~\mbox{pc}
to 1700~\mbox{pc}
and present elliptical shapes.
These structures are commonly explained through introducing
theoretical objects named bubbles or Superbubbles (SB);
these are created by mechanical energy input from stars
(see for example \cite{PikelNer1968}; \cite{weaver}).
The name Fermi bubbles starts to appear in the literature with
the observations of Fermi-LAT
which revealed two large gamma-ray bubbles, extending
above and below the Galactic center, see
\cite{Su2010}.
Detailed observations of the Fermi bubbles
analyzed
the all-sky radio region, see \cite{Jones2012},
the Suzaku X-ray region, see
\cite{Kataoka2013,Tahara2015,Kataoka2015},
the ultraviolet absorption-line spectra, see \cite{Fox2015,Bordoloi2017},
and
the very high-energy gamma-ray emission, see \cite{Abeysekara2017}.
The existence of the Fermi bubbles suggests
some theoretical processes on how they are formed.
We now outline some of them:
processes connected with the
galactic super-massive black hole in Sagittarius A,
see \cite{Cheng2011,Yang2012,Fujita2013}.
Other studies try to explain the non thermal radiation
from Fermi bubbles in the framework of the following
physical mechanisms:
electron's acceleration inside the bubbles,
see \cite{Thoudam2013},
hadronic
models,
see \cite{Fujita2014,Cheng2015a,Sasaki2015,Keshet2017,Abeysekara2017},
and leptonic models,
see \cite{Crocker2014,Keshet2017}.
The previous theoretical efforts allow to build a
dynamic model for the Fermi bubbles for which the physics remains
unknown.
The layout of the paper is as follows.
In Section \ref{section_profiles} we analyze
three profiles in vertical density
for the Galaxy.
In Section \ref{section_motion} we review two
existing equations of motion for the Fermi Bubbles
and we derive a new equation of motion
for an inverse square law in density.
In Section \ref{section_results} we discuss
the results for the three equations of motion
here adopted in terms of reliability of the model.
In Section \ref{section_image} we derive
two results for the Fermi bubbles:
an analytical model for the cut in intensity
in an elliptical framework
and a numerical map for the
intensity of radiation based on the numerical section.
\section{The profiles in density}
\label{section_profiles}
This section reviews the gas distribution in the galaxy.
A new inverse square dependence for the gas
in introduced.
In the following we will use
the spherical coordinates which are defined by
the radial distance $r$,
the polar angle $\theta$,
and the azimuthal angle $\varphi$.
\subsection{Gas distribution in the galaxy}
\label{sectiongas}
The vertical density distribution
of galactic neutral atomic hydrogen (H\,I)
is well-known; specifically, it has the following
three component behavior as a function of
{\it z}, the distance from the galactic plane in pc:
\begin{equation}
n(z) =
n_1 e^{- z^2 /{H_1}^2}+
n_2 e^{- z^2 /{H_2}^2}+
n_3 e^{- | z | /{H_3}}
\,.
\label{equation:ism}
\end{equation}
We took \cite{Lockman1984,Dickey1990,Bisnovatyi1995}
$n_1$=0.395 ${\mathrm{particles~}}{\mathrm{cm}^{-3}}$, $H_1$=127
\mbox{pc},
$n_2$=0.107 $\mathrm{particles~}{\mathrm{cm}^{-3}}$, $H_2$=318
\mbox{pc},
$n_3$=0.064 $\mathrm{particles~}{\mathrm{cm}^{-3}}$, and $H_3$=403
\mbox{pc}.
This distribution of galactic H\,I is valid in the range
0.4 $\leq$ $r$ $\leq$ $r_0$, where $r_0$ = 8.5 \mbox{kpc}
and $r$ is the
distance from the center of the galaxy.
A recent evaluation for galactic H\,I quotes:
\begin{equation}
n_H = n_H(0) \exp{-\frac{z^2}{2 h^2}}
\quad ,
\end{equation}
with $ n_H(0)=1.11\mathrm{particles~}{\mathrm{cm}^{-3}}$,
$h=75.5\,pc$, and $z<1000\,pc$ see \cite{Zhu2017}.
A density profile of a thin
self-gravitating disk of gas which is characterized by a
Maxwellian distribution in velocity and distribution which varies
only in the $z$-direction (ISD) has the following
number density distribution
\begin{equation}
n(z) = n_0 sech^2 (\frac{z}{2\,h})
\quad ,
\label{sech2}
\end{equation}
where $n_0$ is the density at $z=0$,
$h$ is a scaling parameter,
and $sech$ is the hyperbolic secant
(\cite{Spitzer1942,Rohlfs1977,Bertin2000,Padmanabhan_III_2002}).
\subsection{The inverse square dependence}
The density is assumed to have the following
dependence on $z$
in Cartesian coordinates,
\begin{equation}
\rho(z;z_0,\rho_0) =\rho_0\left( 1+{\frac {z}{{\it z_0}}} \right) ^{-2}
\quad .
\label{squareprofile}
\end{equation}
In the following we will adopt the following
density profile in spherical coordinates
\begin{equation}
\rho(z;z_0,\rho_0) =
\cases{\rho_{{0}}&$r<r_{{0}}$\cr {\rho_{{0}} \left( 1+{\frac {r\cos \left(
\theta \right) }{z_{{0}}}} \right) ^{-2}}&$r_{{0}}<r$\cr}
\end{equation}
where the parameter $z_0$ fixes the scale and $\rho_0$ is the
density at $z=z_0$.
Given a solid angle $\Delta \Omega$
the mass $M_0$ swept
in the interval $[0,r_0]$
is
\begin{equation}
M_0 =
\frac{1}{3}\,\rho_{{0}}\,{r_{{0}}}^{3} \,\Delta \Omega
\quad .
\end{equation}
The total mass swept, $M(r;r_0,z_0,\theta,\rho_0,\Delta \Omega) $,
in the interval $[0,r]$
is
\begin{eqnarray}
M(r;r_0,z_0,\theta,\rho_0,\Delta \Omega)=
\nonumber \\
\Biggr (
\frac{1}{3}\,\rho_{{0}}{r_{{0}}}^{3}+{\frac {\rho_{{0}}{z_{{0}}}^{2}r}{
\left( \cos \left( \theta \right) \right) ^{2}}}-2\,{\frac {\rho_{{0
}}{z_{{0}}}^{3}\ln \left( r\cos \left( \theta \right) +z_{{0}}
\right) }{ \left( \cos \left( \theta \right) \right) ^{3}}}
\nonumber \\
-{\frac {
\rho_{{0}}{z_{{0}}}^{4}}{ \left( \cos \left( \theta \right) \right) ^
{3} \left( r\cos \left( \theta \right) +z_{{0}} \right) }}-{\frac {
\rho_{{0}}{z_{{0}}}^{2}r_{{0}}}{ \left( \cos \left( \theta \right)
\right) ^{2}}}+2\,{\frac {\rho_{{0}}{z_{{0}}}^{3}\ln \left( r_{{0}}
\cos \left( \theta \right) +z_{{0}} \right) }{ \left( \cos \left(
\theta \right) \right) ^{3}}}
\nonumber \\
+{\frac {\rho_{{0}}{z_{{0}}}^{4}}{
\left( \cos \left( \theta \right) \right) ^{3} \left( r_{{0}}\cos
\left( \theta \right) +z_{{0}} \right) }}
\Biggl )
\Delta \Omega
\quad .
\label{mass_square}
\end{eqnarray}
The density $\rho_0$ can be obtained
by introducing the number density expressed in particles
$\mathrm{cm}^{-3}$,
$n_0$,
the mass of hydrogen, $m_H$,
and a multiplicative factor $f$,
which is chosen to be 1.4, see \cite{Dalgarno1987},
\begin{equation}
\rho_0 = f m_H n_0
\quad .
\end{equation}
An astrophysical version of the total swept mass,
expressed in solar mass
units, $M_{\hbox{$\odot$}}$, can be obtained
introducing
$z_{0,pc}$, $r_{0,pc}$ and $r_{0,pc}$
which are $z_0$, $r_0$ and $r$ expressed in pc units.
\section{The equation of motion}
\label{section_motion}
This section reviews the equation of motion
for a thermal model
and for a recursive cold model.
A new equation of motion for a thin layer
which propagates in a medium
with an inverse square dependence for the density is analyzed.
\subsection{The thermal model}
\label{secthermal}
The starting equation for the evolution
of the SB \cite{Dyson1997,mccrayapj87,Zaninetti2004}
is momentum conservation
applied to a pyramidal section.
The parameters of the thermal model are
$N^*$,
the number of SN explosions in $5.0 \cdot 10^7$ \mbox{yr},
$z_{\mathrm{OB}}$,
the distance of the OB associations from the galactic plane,
$E_{51}$,
the energy in $10^{51}$ \mbox{erg},
$v_0$,
the initial velocity which is fixed by the bursting phase,
$t_0$,
the initial time in $yr$ which is equal to the bursting time,
and $t$ the proper time of the SB.
The SB evolves in a standard three component medium,
see formula (\ref{equation:ism}).
\subsection{A recursive cold model}
\label{secrecursive}
The 3D expansion that starts at the origin
of the coordinates;
velocity and radius are given
by a recursive relationship,
see \cite{Zaninetti2012g}.
The parameters are the same of
the thermal model and the SB evolves in a
self-gravitating medium as given by equation (\ref{sech2}).
\subsection{The inverse square model}
\label{secinversesquare}
In the case of an inverse square density profile
for the interstellar medium ISM
as given by equation (\ref{squareprofile}),
the differential equation
which models momentum conservation
is
\begin{eqnarray}
\Biggl ( \frac{1}{3}\,\rho_{{0}}{r_{{0}}}^{3}+{\frac {\rho_{{0}}{z_{{0}}}^{2}r
\left( t \right) }{ \left( \cos \left( \theta \right) \right) ^{2}}}
-2\,{\frac {\rho_{{0}}{z_{{0}}}^{3}\ln \left( r \left( t \right) \cos
\left( \theta \right) +z_{{0}} \right) }{ \left( \cos \left( \theta
\right) \right) ^{3}}}
\nonumber \\
-{\frac {\rho_{{0}}{z_{{0}}}^{4}}{ \left( \cos
\left( \theta \right) \right) ^{3} \left( r \left( t \right) \cos
\left( \theta \right) +z_{{0}} \right) }}
-{\frac {\rho_{{0}}{z_{{0}}}
^{2}r_{{0}}}{ \left( \cos \left( \theta \right) \right) ^{2}}}
\nonumber \\
+2\,{
\frac {\rho_{{0}}{z_{{0}}}^{3}\ln \left( r_{{0}}\cos \left( \theta
\right) +z_{{0}} \right) }{ \left( \cos \left( \theta \right)
\right) ^{3}}}+{\frac {\rho_{{0}}{z_{{0}}}^{4}}{ \left( \cos \left(
\theta \right) \right) ^{3} \left( r_{{0}}\cos \left( \theta \right)
+z_{{0}} \right) }}
\Biggr)
{\frac {\rm d}{{\rm d}t}}r \left( t
k \right)
\nonumber \\
-\frac{1}{3}\,\rho_{{0}}{r_{{0}}}^{3}v_{{0}}=0
\quad ,
\end{eqnarray}
where the initial conditions
are $r=r_0$ and $v=v_0$
when $t=t_0$.
We now briefly review that
given a function $f(r)$, the Pad\'e approximant,
after \cite{Pade1892},
is
\begin{equation}
f(r)=\frac{a_{0}+a_{1}r+\dots+a_{p}r^{o}}{b_{0}+b_{1}%
r+\dots+b_{q}r^{q}}
\quad ,
\end{equation}
where the notation is the same of \cite{NIST2010}.
The coefficients $a_i$ and $b_i$
are found through Wynn's cross rule,
see \cite{Baker1975,Baker1996}
and our choice is $o=2$ and $q=1$.
The choice of $o$ and $q$ is a compromise between
precision, high values for $o$ and $q$, and
simplicity of the expressions to manage,
low values for $o$ and $q$.
The inverse of the velocity
is
\begin{equation}
(\frac{1}{v(r)}) = \frac{NN}{DD}
\quad ,
\end{equation}
where
\begin{eqnarray}
NN=
\left( \cos \left( \theta \right) \right) ^{5}{r_{{0}}}^{4}r+
\left( \cos \left( \theta \right) \right) ^{4}{r_{{0}}}^{4}z_{{0}}+
\left( \cos \left( \theta \right) \right) ^{4}{r_{{0}}}^{3}rz_{{0}}
\nonumber \\
+
\left( \cos \left( \theta \right) \right) ^{3}{r_{{0}}}^{3}{z_{{0}}}
^{2}
-3\, \left( \cos \left( \theta \right) \right) ^{3}{r_{{0}}}^{2}r
{z_{{0}}}^{2}+3\, \left( \cos \left( \theta \right) \right) ^{3}r_{{0
}}{r}^{2}{z_{{0}}}^{2}
\nonumber \\
-6\, \left( \cos \left( \theta \right) \right)
^{2}\ln \left( r\cos \left( \theta \right) +z_{{0}} \right) r_{{0}}r{
z_{{0}}}^{3}
\nonumber \\
+6\, \left( \cos \left( \theta \right) \right) ^{2}\ln
\left( r_{{0}}\cos \left( \theta \right) +z_{{0}} \right) r_{{0}}r{z_
{{0}}}^{3}-3\, \left( \cos \left( \theta \right) \right) ^{2}{r_{{0}}
}^{2}{z_{{0}}}^{3}
\nonumber \\
+3\, \left( \cos \left( \theta \right) \right) ^{2}
{r}^{2}{z_{{0}}}^{3}-6\,\cos \left( \theta \right) \ln \left( r\cos
\left( \theta \right) +z_{{0}} \right) r_{{0}}{z_{{0}}}^{4}
\nonumber \\
-6\,\cos
\left( \theta \right) \ln \left( r\cos \left( \theta \right) +z_{{0}
} \right) r{z_{{0}}}^{4}+6\,\cos \left( \theta \right) \ln \left( r_{
{0}}\cos \left( \theta \right) +z_{{0}} \right) r_{{0}}{z_{{0}}}^{4}
\nonumber \\
+6
\,\cos \left( \theta \right) \ln \left( r_{{0}}\cos \left( \theta
\right) +z_{{0}} \right) r{z_{{0}}}^{4}-6\,\cos \left( \theta
\right) r_{{0}}{z_{{0}}}^{4}+6\,\cos \left( \theta \right) r{z_{{0}}}
^{4}
\nonumber \\
-6\,\ln \left( r\cos \left( \theta \right) +z_{{0}} \right) {z_{{0
}}}^{5}+6\,\ln \left( r_{{0}}\cos \left( \theta \right) +z_{{0}}
\right) {z_{{0}}}^{5}
\end{eqnarray}
and
\begin{eqnarray}
DD= \nonumber \\
{r_{{0}}}^{3}v_{{0}} \left( \cos \left( \theta \right) \right) ^{3}
\left( rr_{{0}} \left( \cos \left( \theta \right) \right) ^{2}+\cos
\left( \theta \right) r_{{0}}z_{{0}}+\cos \left( \theta \right) rz_{{0
}}+{z_{{0}}}^{2} \right)
\quad .
\end{eqnarray}
The above result allows deducing a solution $r_{2,1}$
expressed through the Pad\`e approximant
\begin{equation}
r(t)_{2,1} = \frac{AN}{AD}
\quad ,
\label{rtpadesquare}
\end{equation}
with
\begin{eqnarray}
AN=3\, \left( \cos \left( \theta \right) \right) ^{2}{r_{{0}}}^{3}+2\,r_
{{0}}tv_{{0}}z_{{0}}\cos \left( \theta \right) -2\,r_{{0}}{\it t_0}\,v_
{{0}}z_{{0}}\cos \left( \theta \right)
\nonumber \\
+10\,\cos \left( \theta
\right) {r_{{0}}}^{2}z_{{0}}+2\,tv_{{0}}{z_{{0}}}^{2}-2\,{\it t_0}\,v_
{{0}}{z_{{0}}}^{2}-2\,r_{{0}}{z_{{0}}}^{2}
\nonumber \\
-\Bigg ( \left( r_{{0}}\cos
\left( \theta \right) +z_{{0}} \right) ^{2} \bigg ( 9\, \left( \cos
\left( \theta \right) \right) ^{2}{r_{{0}}}^{4}-12\,\cos \left(
\theta \right) {r_{{0}}}^{2}tv_{{0}}z_{{0}}
\nonumber \\
+12\,\cos \left( \theta
\right) {r_{{0}}}^{2}{\it t_0}\,v_{{0}}z_{{0}}+4\,{t}^{2}{v_{{0}}}^{2}
{z_{{0}}}^{2}-8\,t{\it t_0}\,{v_{{0}}}^{2}{z_{{0}}}^{2}+4\,{{\it t_0}}^{
2}{v_{{0}}}^{2}{z_{{0}}}^{2}
\nonumber \\
+18\,\cos \left( \theta \right) {r_{{0}}}^
{3}z_{{0}}+42\,r_{{0}}tv_{{0}}{z_{{0}}}^{2}-42\,r_{{0}}{\it t_0}\,v_{{0
}}{z_{{0}}}^{2}+9\,{r_{{0}}}^{2}{z_{{0}}}^{2} \bigg ) \Bigg )^{1/2}
\quad ,
\end{eqnarray}
and
\begin{equation}
AD= z_{{0}} \left( 4\,r_{{0}}\cos \left( \theta \right) -5\,z_{{0}}
\right)
\quad .
\end{equation}
A possible set of initial values is reported
in Table \ref{datafitsquare} in which
the initial value of radius and velocity
are fixed by the bursting phase.
\begin{table}
\caption
{
Numerical values of the parameters
for the simulation
in the case of the inverse square model.
}
\label{datafitsquare}
\[
\begin{array}{lc}
\hline
\hline
n_0 [\frac{particles}{cm^3}] & 1 \\
E_{51} & 1 \\
N^* & 5.87 \,10^8 \\
r_0 & 220 \,pc \\
v_0 & 3500 \frac{km}{s} \\
z_0 & 12 \\
t & 5.95 \,10^7 \,yr \\
t_0 & 36948 \,yr \\
\noalign{\smallskip}
\hline
\hline
\end{array}
\]
\end {table}
The above parameters allows to obtain an approximate
expansion law as function of time and polar angle
\begin{equation}
r(t)_{2,1} = \frac{BN}{BD}
\quad ,
\label{rtpadesquareastro}
\end{equation}
with
\begin{eqnarray}
BN=
31944000\, \left( \cos \left( \theta \right) \right) ^{2}+
18.8632\,t\cos \left( \theta \right) + 5111040\,\cos \left(
\theta \right)
\nonumber \\
+ 1.0289\,t- 101376
- \Bigg ( { ( 220
\,\cos ( \theta ) +12 ) ^{2} ( 21083040000\,
( \cos ( \theta ) ) ^{2}
} \nonumber \\
{
- 24899.49\,t\cos
( \theta ) + 3219955200\,\cos ( \theta ) +
0.0073517\,{t}^{2}
}
\nonumber \\
{
+ 4210.27\,t- 102871295 ) } \Bigg )^{1/2}
\quad ,
\end{eqnarray}
and
\begin{equation}
BD=
10560 \,\cos \left( \theta \right) - 720
\quad .
\end{equation}
\section{Astrophysical Results}
\label{section_results}
This section introduces a test for
the reliability of the model,
analyzes the observational details of the Fermi bubbles,
reviews the results for the two models of reference
and reports the results of the inverse square model.
\subsection{The reliability of the model}
An observational
percentage reliability, $\epsilon_{\mathrm {obs}}$,
is introduced over the whole range
of the polar angle $\theta$,
\begin{equation}
\epsilon_{\mathrm {obs}} =100(1-\frac{\sum_j |r_{\mathrm {obs}}-r_{\mathrm{num}}|_j}{\sum_j
{r_{\mathrm {obs}}}_{,j}})
\quad many\,directions
\quad,
\label{efficiencymany}
\end{equation}
where
$r_{\mathrm{num}}$ is the theoretical radius,
$r_{\mathrm{obs}}$ is the observed radius,
and
the index $j$ varies from 1 to the number of
available observations.
\subsection{The structure of the Fermi bubbles}
The exact shape of the Fermi bubbles is a
matter of research and
as an example in \cite{Miller2016}
the bubbles are modeled with
ellipsoids centered
at 5 kpc up and below the Galactic plane
with semi-major axes
of 6 kpc and minor axes of 4 kpc.
In order to test our models
we selected the image of the Fermi bubbles
available at
\url{https://www.nasa.gov/mission_pages/GLAST/news/new-structure.html}
which is reported in Figure~\ref{figbubbles}.
\begin{figure*}
\begin{center}
\includegraphics[width=7cm]{f01.eps}
\end {center}
\caption
{
A gamma - X image of the Fermi bubbles in 2010 as given by the NASA.
}
\label{figbubbles}
\end{figure*}
A digitalization of the above advancing surface
is reported in Figure~\ref{fermisb_obs}
as a 2D section.
This allows to fix the observed
radii to be inserted in equation (\ref{efficiencymany}).
\begin{figure*}
\begin{center}
\includegraphics[width=7cm]{f02.eps}
\end {center}
\caption
{
A section of the Fermi bubbles digitalized by the author.
}
\label{fermisb_obs}
\end{figure*}
The actual shape of the bubbles in galactic coordinates is shown
in Figure 3 and 15
of \cite{Su2010}
and Figure 30 and Table 3 of by \cite{Ackermann2014}.
\subsection{The two models of reference}
The thermal model is outlined in Section \ref{secthermal}
and Figure \ref{efficiency_fermib} reports the numerical
solution as a cut in the $x-z$ plane.
\begin{figure*}
\begin{center}
\includegraphics[width=7cm]{f03.eps}
\end {center}
\caption
{
Section of the Fermi bubbles
in the $x-z$ plane with a thermal model
(green points)
and observed profile (red stars).
The bursting parameters
$N^*$= 113000,
$z_{\mathrm{OB}}$=0 pc,
$E_{51}$=1,
$t_0=0.036 \,10^7\,yr$
when $t=90\,10^7yr$
give
$\epsilon_{\mathrm {obs}}=73.34\%$.
}
\label{efficiency_fermib}
\end{figure*}
The cold recursive model is outlined in Section \ref{secrecursive}
and Figure \ref{section_auto} reports the numerical
solution as a cut in the $x-z$ plane.
\begin{figure*}
\begin{center}
\includegraphics[width=7cm]{f04.eps}
\end {center}
\caption
{
Section of the Fermi bubbles
in the $x-z$ plane with the cold recursive model
(green points)
and observed profile (red stars).
The bursting parameters
$N^*$= 79000, $z_{\mathrm{OB}}$=2 pc,
$t_0 =0.013\,10^7 $ yr and
$E_{51}$=1
gives
$r_0=90.47$\ pc
and
$v_0\,=391.03$ km s$^{-1}$.
On inserting
$h=90$\ pc,
$t=13.6\,10^7$\ yr
the reliability
is
$\epsilon_{\mathrm {obs}}=84.70\%$.
}
\label{section_auto}
\end{figure*}
\subsection{The inverse square model}
The inverse square model is outlined
in Section \ref{secinversesquare}
and Figure \ref{fermisb_theo_obs_square} reports the numerical
solution as a cut in the $x-z$ plane.
\begin{figure*}
\begin{center}
\includegraphics[width=7cm]{f05.eps}
\end {center}
\caption
{
Section of the Fermi bubbles
in the $x-z$ plane with the inverse square model
(green points)
and observed profile (red stars).
The parameters are
reported in Table \ref{datafitsquare}
and
the reliability
is
$\epsilon_{\mathrm {obs}}=90.71\%$.
}
\label{fermisb_theo_obs_square}
\end{figure*}
A rotation around the
$z$-axis of the above theoretical section allows
building a 3D surface, see
Figure \ref{fermisb3d}.
\begin{figure*}
\begin{center}
\includegraphics[width=7cm]{f06.eps}
\end {center}
\caption
{
3D surface of the Fermi bubbles
with parameters as in Table \ref{datafitsquare},
inverse square profile.
The three Euler angles are $\Theta=40$, $\Phi=60$ and
$ \Psi=60 $.
}
\label{fermisb3d}
\end{figure*}
The temporal evolution of the advancing surface is reported
in Figure \ref{square_fermisb_molti} and
a comparison should be done
with Fig.~6 in \cite{Sofue2017}.
\begin{figure*}
\begin{center}
\includegraphics[width=7cm]{f07.eps}
\end {center}
\caption
{
Sections of the Fermi bubbles
as function of time
with parameters as in Table \ref{datafitsquare}.
The time of each section is
0.0189$\,10^7\,yr$,
0.059$\,10^7\,yr$,
0.189$\,10^7\,yr$,
0.6$\,10^7\,yr$,
1.89 $\,10^7\,yr$,
and
6$\,10^7\,yr$.
}
\label{square_fermisb_molti}
\end{figure*}
\section{Theory of the image}
\label{section_image}
This section reviews
the transfer equation and
reports a new analytical result for the intensity of radiation
in an elliptical framework
in the non-thermal/thermal case.
A numerical model for the image formation
of the Fermi bubbles is reported.
\subsection{The transfer equation}
The transfer equation in the presence of emission only
in the case of optically thin layer
is
\begin{equation}
j_{\nu} \rho =K C(s)
\quad ,
\end{equation}
where
$K$ is a constant,
$j_{\nu}$ is the emission coefficient,
the index $\nu$ denotes the frequency of
emission and
$C(s)$ is the number density of particles,
see for example \cite{rybicki}.
As an example the synchrotron emission,
as described in sec. 4 of \cite{Schlickeiser},
is often used in order to model the radiation from a SNR,
see for example
\cite{Yamazaki2014,Tran2015,Katsuda2015}.
According to the above equation the increase in intensity
is proportional to the number density
integrated along the line of sight, which for constant density,
gives
\begin{equation}
I_{\nu}=K^{\prime} \times l
\quad ,
\label{equationintensity}
\end{equation}
where $K^{\prime}$ is a constant and $l$ is the length
along the line of sight interested in the emission;
in the case of synchrotron emission see formula (1.175)
in \cite{lang}.
\subsection{Analytical non thermal model}
A real ellipsoid, see \cite{Zwillinger2018},
represents a first approximation of the Fermi bubbles,
see \cite{Miller2016},
and has equation
\begin{equation}
\frac{z^2}{a^2} + \frac{x^2}{b^2} + \frac{y^2}{d^2}=1
\quad ,
\label{ellipsoid}
\end{equation}
in which the polar axis of the Galaxy is the z-axis.
Figure \ref{ellipsoid_fermisb} reports
the astrophysical application of the ellipsoid
in which due to the symmetry about the azimuthal
angle $b=d$.
\begin{figure*}
\begin{center}
\includegraphics[width=7cm]{f08.eps}
\end {center}
\caption
{
Fermi bubbles approximated by an ellipsoid
when $a=6\,kpc,b=4\,kpc and d=4\,kpc$.
}
\label{ellipsoid_fermisb}
\end{figure*}
We are interested in the section of the ellipsoid $y=0$
which is defined by the following external ellipse
\begin{equation}
\frac{z^2}{a^2} + \frac{x^2}{b^2} =1
\quad .
\label{ellipse}
\end{equation}
We assume
that the emission takes place in a thin layer comprised between
the external ellipse
and the internal ellipse defined by
\begin{equation}
\frac{z^2}{(a-c)^2} + \frac{x^2}{(b-c)^2} =1
\quad ,
\label{ellipseint}
\end{equation}
see Figure \ref{int_ext_ellipses}.
\begin{figure*}
\begin{center}
\includegraphics[width=7cm]{f09.eps}
\end {center}
\caption
{
Internal and external ellipses
when $a=6\,kpc$,$b=4\,kpc$ and $c=\frac{a}{12}\,kpc$.
}
\label{int_ext_ellipses}
\end{figure*}
We therefore
assume that the number density $C$ is constant and in particular
rises from 0 at (0,a) to a maximum value $C_m$,
remains constant
up to (0,a-c) and then falls again to 0.
The length of sight, when
the observer is situated at the infinity of the $x$-axis, is the
locus parallel to the $x$-axis which crosses the position $z$ in
a Cartesian $x-z$ plane and terminates at the external
ellipse.
The locus length
is
\begin{eqnarray}
l_I = 2\,{\frac {\sqrt {{a}^{2}-{z}^{2}}b}{a}}
\\
when \quad (a-c) \leq z < a \nonumber \\
l_{II} =
2\,{\frac {\sqrt {{a}^{2}-{z}^{2}}b}{a}}-2\,{\frac {\sqrt {{a}^{2}-2\,
ac+{c}^{2}-{z}^{2}} \left( b-c \right) }{a-c}}
\\
when \quad
0 \leq z < (a-c) \quad .
\nonumber
\label{length}
\end{eqnarray}
In the case of optically thin medium,
according to equation (\ref{equationintensity}),
the intensity is split in two cases
\begin{eqnarray}
I_I(z;a,b) = I_m \times 2\,{\frac {\sqrt {{a}^{2}-{z}^{2}}b}{a}}
\\
when \quad (a-c) \leq z < a \nonumber \\
I_{II}(z;a,,c) = I_m \times
\Big (
2\,{\frac {\sqrt {{a}^{2}-{z}^{2}}b}{a}}-2\,{\frac {\sqrt {{a}^{2}-2\,
ac+{c}^{2}-{z}^{2}} \left( b-c \right) }{a-c}}
\Big )
\\
when \quad
0 \leq z < (a-c) \quad ,
\nonumber
\label{intensitycut}
\end{eqnarray}
where $I_m$ is a constant which allows to compare the theoretical
intensity with the observed one.
A typical profile in intensity along the z-axis
is reported in Figure \ref{cut_ellipse}.
\begin{figure*}
\begin{center}
\includegraphics[width=7cm]{f10.eps}
\end {center}
\caption
{
The intensity profile along the z-axis when
when $a=6\,kpc$,$b=4\,kpc$, $c=\frac{a}{12}\,kpc$
and $I_m$=1.
}
\label{cut_ellipse}
\end{figure*}
The ratio, $r$, between the theoretical intensity at the maximum, $(z=a-c)$,
and at the minimum, ($z=0$),
is given by
\begin{equation}
\frac {I_I(z=a-c)} {I_{II}(z=0)} =r=
{\frac {\sqrt {2\,a-c}b}{\sqrt {c}a}}
\quad .
\label{ratioteorrim}
\end{equation}
As an example the values $a=6\,kpc$,$b=4\,kpc$, $c=\frac{a}{12}\,kpc$
gives $r=3.19$.
The knowledge of the above ratio from the observations
allows to deduce $c$
once $a$ and $b$ are given by the observed morphology
\begin{equation}
c =
2\,{\frac {a{b}^{2}}{{a}^{2}{r}^{2}+{b}^{2}}}
\quad .
\end{equation}
As an example in the
inner regions of the northeast Fermi bubble we have $r=2$,
see \cite{Kataoka2013},
which coupled with $a=6\,kpc$ and $b=4\,kpc$
gives $c=1.2\,kpc$.
The above value is an important astrophysical result because
we have found the
dimension of the advancing thin layer.
\subsection{Analytical thermal model}
A thermal model for the image is characterized by a
constant temperature in the internal region of the advancing
section which is approximated by an ellipse, see
equation~(\ref{ellipse}).
We therefore
assume that the number density $C$ is constant and in particular
rises from 0 at (0,a) to a maximum value $C_m$, remains constant
up to (0,-a) and then falls again to 0.
The
length of sight, when the observer is situated at the infinity of
the $x$-axis, is the locus parallel to the $x$-axis which
crosses the position $z$ in a Cartesian $x-z$ plane and
terminates at the external ellipse in the point (0,a).
The locus length is
\begin{eqnarray}
l = 2\,{\frac {\sqrt {{a}^{2}-{z}^{2}}b}{a}} \quad ; -a \leq z < a
\quad .
\end{eqnarray}
The number density $C_m$ is constant in the ellipse
and therefore the intensity of radiation is
\begin{eqnarray}
I(z;a,b,I_m) =I_m\times 2\,{\frac {\sqrt {{a}^{2}-{z}^{2}}b}{a}}
\quad ; -a \leq z < a \quad .
\label{ithermal}
\end{eqnarray}
A typical profile in intensity along the z-axis for
the thermal model
is reported in Figure \ref{cut_ellipse_therm}.
\begin{figure*}
\begin{center}
\includegraphics[width=7cm]{f11.eps}
\end {center}
\caption
{
The intensity profile along the z-axis for the thermal model
when $a=6\,kpc$, $b=4\,kpc$ and $I_m$=1.
}
\label{cut_ellipse_therm}
\end{figure*}
\subsection{Numerical model}
The source of luminosity is assumed here to be
the flux of kinetic energy, $L_m$,
\begin{equation}
L_m = \frac{1}{2}\rho A V^3
\quad,
\label{fluxkineticenergy}
\end{equation}
where $A$ is the considered area, $V$ the velocity
and $\rho$ the density,
see formula (A28)
in \cite{deyoung}.
In our case $A=R^2 \Delta \Omega$,
where $\Delta \Omega$ is the considered solid angle along
the chosen direction.
The observed luminosity along a given direction
can be expressed as
\begin{equation}
L = \epsilon L_{m}
\label{luminosity}
\quad ,
\end{equation}
where $\epsilon$ is a constant of conversion
from the mechanical luminosity to the
observed luminosity.
A numerical algorithm which allows us to
build a complex image
is
outlined in Section 4.1 of \cite{Zaninetti2013c}
and
the orientation of the object is characterized by
the
Euler angles $(\Phi, \Theta, \Psi)$
The threshold intensity can be
parametrized to $I_{max}$,
the maximum value of intensity
characterizing the map.
The image of the Fermi bubbles
is shown in Figure \ref{fermisb_heat}
and
the introduction of a threshold intensity
is visualized in Figure \ref{fermisb_heat_hole}.
\begin{figure}
\includegraphics[width=6cm]{f12.eps}
\caption
{
Map of the theoretical intensity of
the Fermi bubbles
for the
inverse square model
with parameters as
in Table \ref{datafitsquare}.
The three Euler angles
characterizing the orientation
are $ \Phi $=0$^{\circ }$,
$ \Theta $=90 $^{\circ }$
and $ \Psi $=90 $^{\circ }$.
}
\label{fermisb_heat}
\end{figure}
\begin{figure}
\includegraphics[width=6cm]{f13.eps}
\caption
{
Map of the theoretical intensity of the Fermi bubbles
as in Figure \ref{fermisb_heat}.
In this map $I_{tr}= I_{max}/2$.
}
\label{fermisb_heat_hole}
\end{figure}
\section{Conclusions}
{\bf Law of motion}
We have compared two existing models for the
temporal evolution of the Fermi bubbles,
a thermal model, see Section \ref{secthermal},
and an autogravitating model, see Section
\ref{secrecursive},
with a new model which conserves the momentum
in presence of an inverse square law for the density of the ISM.
The best result is obtained by the inverse square
model which produces a reliability
of $\epsilon_{\mathrm {obs}}=90.71\%$ for the expanding radius
in respect to a digitalized section of the Fermi bubbles.
A semi-analytical law of motion as function
of polar angle and time is derived
for the inverse square model,
see equation (\ref{rtpadesquareastro}).
{\bf Formation of the image}
An analytical cut for the intensity of radiation
along the z-axis is derived in the framework of
advancing surface characterized by an internal
and an external ellipses.
The analytical cut in theoretical intensity
presents a characteristic "U" shape which has
a maximum in the external ring and a
minimum at the center, see equation~(\ref{intensitycut}).
The presence of a hole in the intensity of radiation
in the central region of the elliptical Fermi bubbles
is also confirmed by a numerical algorithm for the
image formation, see Figure \ref{fermisb_heat_hole}.
The theoretical prediction of a hole in the intensity
map explains the decrease in intensity
for the 0.3 kev plasma by ~= $50\%$
toward
the central region of the northeast Fermi bubble, see
\cite{Kataoka2013}.
The intensity of radiation for the
thermal model conversely presents a maximum of the intensity at the
center of the elliptical Fermi bubble,
see equation (\ref{ithermal})
and this theoretical prediction does not agree
with the above observations.
\section*{Acknowledgments}
Credit for
Figure \ref{figbubbles} is given
to NASA.
\leftline{\bf References}
\providecommand{\newblock}{}
|
1,116,691,500,790 | arxiv | \section{Introduction}
In light of current collider data \cite{Aad:2012tfa, Chatrchyan:2012ufa}, the Standard Model (SM) of elementary particle interactions remains as an accurate
description of nature \cite{Azatov:2012bz, Giardino:2013bma, Alanne:2013dra}. Nevertheless, observational evidence for the dark matter abundance, lack of a mechanism for generating the matter-antimatter asymmetry, and no understanding of the dynamics underlying the flavor patterns, continue to provide impetus for development and analysis of beyond-the-Standard-Model scenarios. The need for such scenarios is further augmented by the hierarchy and naturality problems associated with a fundamental scalar field \cite{'tHooft:1979bh, Farina:2013mla, Heikinheimo:2013fta,Craig:2013xia,Antipin:2013exa}.
A traditional model building paradigm to address the naturality problem is to replace the scalar sector with new fermions charged under
new gauge dynamics \cite{Weinberg:1975gm, Susskind:1978ms}. These technicolor (TC), or composite Higgs, models also provide dark matter
candidates, and affect the electroweak phase transition to make electroweak baryogensis possible. Extended technicolor
models (ETC)\cite{ Dimopoulos:1979es, Eichten:1979ah} provide ways to understand generational hierarchies \cite{Appelquist:2003hn}. However, TC model building is challenging due to our limited
understanding of strong dynamics from first principles. Consequently, a large amount of common wisdom concerning the
phenomenological details of dynamical electroweak symmetry breaking rests on the naive scaling arguments from what we
know about QCD. The most profound example of this is the scalar spectrum: if we consider the sigma meson, i.e. the
resonance denoted $f_0(500)$ by the particle data group \cite{Beringer:1900zz}, and scale from QCD with $f_\pi\simeq 100$ MeV to a
technicolor theory with $F_\pi\simeq 250$ GeV, we find that the corresponding scalar has mass $M_\sigma\simeq 1250$ GeV.
Small refinements arise from taking into account also possibly different number of colours and different fermion
representations. But still one is typically bound to the conclusion that the scalar mass is in the TeV range, thus much heavier than the recently observed scalar boson at 125 GeV \cite{Aad:2012tfa, Chatrchyan:2012ufa}. Refinements to naive scaling from QCD arise in theories which are nearly conformal, also called walking tehcnicolor theories. Various arguments for the existence of a light scalar in these theories have been provided in the literature \cite{Yamawaki:1985zg,Bando:1986bg,Miransky:1989nu, Dietrich:2005jn,Chacko:2012sy,Elander:2010wd,Alho:2013dka}.
Whichever is the case, one should be careful when comparing the estimates of the dynamical mass directly with the mass
of the Higgs boson observed at the LHC experiments. An element which is often overlooked is that the scaling described
above applies to strongly interacting theories in isolation. When coupled with the full electroweak sector, there can be
important contributions affecting the conclusions \cite{Foadi:2012bb}. Here again, QCD provides a guide to orient our thinking: the mass
difference between charged and neutral pions arises from the electromagnetic interaction, and is explicitly and rigorously
given in the chiral limit by
\begin{eqnarray}
\Delta m_\pi^2 &\equiv& m^2_{\pi^+}-m_{\pi0}^2 \nonumber \\
&=& -\frac{3\alpha_{\rm{EM}}}{4\pi f_\pi^2}\int_0^\infty ds\, s\, \ln s (\rho_V(s)-\rho_A(s)),
\end{eqnarray}
where $\rho_{V,A}(s)$ are the vector and axial vector spectral functions. Experimentally, this mass difference is determined
to be (35 MeV)$^2$ for QCD pions. The relative smallness, $\Delta m_\pi^2/m_\pi^2\simeq 0.06$, is entirely due to the
weakness of the electroweak interaction. If instead we had $\alpha_{\rm{EM}}\sim 1/(4\pi)$, the relative magnitude
would be 0.7, i.e. of ${\cal O}(1)$ relative to the dynamical mass from strong interactions. Here the key element is that in the case of QCD the electromagnetic interaction is completely external to the
strong dynamics, while in the case of technicolor it is the electroweak theory itself that plays the role of an external interaction. For the
electroweak sector the couplings, especially the Yukawa coupling of the top quark, are larger than the electromagnetic coupling.
In the ETC framework the coupling between the technicolor sector and SM matter fields is modelled at
low energies via four-fermion operators. In this paper we consider the lightest isosinglet scalar boson arising from a strongly interacting
sector, and compute the effect on its mass from the four-fermion operator responsible for the top mass. We find that the corrections can be
large and potentially reduce the scalar resonance mass from ${\cal O}(1)$ TeV to $\sim$125 GeV. We work with a simple setup, consisting of a
chiral model for the TC sector, and a single four-fermion coupling. This is sufficient for exhibiting how the correction to the
dynamical mass arises and what its expected magnitude is. To account for the fermion mass generation as well as to
address the precision electroweak data, a more detailed sector of four-fermion interactions would be needed. We leave such
investigation for future work and concentrate here only on the determination of the Higgs mass in this framework, which is
currently the most important feature in light of current and future LHC data.
The paper is structured as follows: In section~\ref{Sec:model} we introduce the model and in section~\ref{Sec:EWmatch} we
discuss the matching with the electroweak theory, the top mass and the Fermi coupling constant. The main result, i.e. the analysis of the mass of the scalar
isosinglet, is presented in section~\ref{Sec:scalarmass}. In section~\ref{Sec:ST} we briefly discuss the oblique electroweak parameters and the implications of our analysis on the underlying strong dynamics. In section~\ref{Sec:conclusions} we present our conclusions and outlook
for future work.
\section{Chiral techniquark model}
\label{Sec:model}
We consider TC theories in which the lightest resonances are composed of fermions belonging to
one weak technidoublet $Q\equiv (U,D)$. This does not imply that there could not exist additional technifermions,
as these can be heavy enough to be decoupled from the lightest resonances. In QCD, for instance, the sigma meson contains the $u$ and $d$ quarks but not the $s$ quark, which is heavier and decoupled.
To discuss phenomenology at the energies explored by the LHC, we consider the effective Lagrangian
\begin{equation}
{\cal L}={\cal L}_{\overline {\rm SM}} + {\cal L}_{\rm TC} + {\cal L}_{\rm ETC}\ ,
\label{Eq:lagfull}
\end{equation}
where ${\cal L}_{\overline {\rm SM}} $ is the SM Lagrangian without Higgs and Yukawa terms, ${\cal L}_{\rm TC}$ is a model Lagrangian accounting for the non-perturbative TC dynamics, and ${\cal L}_{\rm ETC}$ contains interactions mediated by exchanges of heavy ETC gauge bosons, which we assume to exceed the energy reach of the LHC. The full Lagrangian is invariant under the SM gauge group, and the electroweak symmetry is spontaneously broken by the TC force to electromagnetism.
The ${\cal L}_{\rm TC}$ part of the full Lagrangian is given by a symmetry-breaking chiral-techniquark model, containing both composite resonances and the techniquarks. Among the resonances, we assume the lightest ones to be the massless technipion isospin triplet $\Pi^a$, which becomes the longitudinal component of the $W$ and $Z$ bosons, and the scalar singlet $H$. Our main goal is to determine whether the latter can be a candidate for the recently observed 125 GeV resonance. In order to account for compositeness, we take the kinetic terms for $\Pi^a$ and $H$ to be radiatively generated. Explicitly,
\begin{equation}
{\cal L}_{\rm TC} = \overline{Q}_L\ i\slashed{D} Q_L + \overline{U}_R\ i\slashed{D} U_R + \overline{D}_R\ i\slashed{D} D_R
-M\left(1+\frac{y}{v}H+\cdots\right)\left(\overline{Q}_L\Sigma Q_R+\overline{Q}_R\Sigma^\dagger Q_L\right) -\frac{m^2}{2} H^2 +\cdots\ ,
\label{eq:TC}
\end{equation}
where $\Sigma\equiv\exp\left(i\Pi^a\tau^a/v\right)$, and $\tau^a$, $a=1,2,3$, are the Pauli matrices.
The dynamical constituent mass of the techniquarks is denoted by $M$. The ellipses denote higher-order interactions, which we assume to give negligible contribution to the masses and decay constants.
Note that we have suppressed the TC gauge index in the techniquarks, and that the covariant derivatives are with respect to the SM gauge interactions.
In the ETC sector we only consider the four-fermion operator which allows the top quark to interact with the techniquark condensate and acquire mass,
\begin{equation}
{\cal L}_{\rm ETC} = 2 G\left(\overline{q}_L^i t_R \overline{U}_R Q_{iL} + \overline{Q}_L^i U_R \overline{t}_R q_{iL}\right) \ ,
\label{eq:ETC}
\end{equation}
where $q\equiv (t,b)$ is the top-bottom doublet, and $i$ is a weak isospin index. The inclusion of this single interaction term
is sufficient for the study of the generic effect of four-fermion interactions on the scalar mass.
A more complete discussion of fermion mass patterns and precision electroweak observables would require additional operators. We do not consider such refinements in this paper.
\subsection{Cutoff and confinement}
Using the Lagrangian of (\ref{Eq:lagfull}), we can compute observables such as the mass of the scalar singlet $H$ and the technipion decay constant. We do so in the large-$N$ and large-$N_c$ limit, with $N/N_c$ finite: here $N$ is the dimension of the techniquark representation under TC, whereas $N_c=3$ is the dimension of the quark representation under QCD. We assume that the loop integrals are finite, and that their absolute size has a physical meaning. Practically, this means that we use a physical cutoff. However, there are two relevant mass scales, the mass of the lightest ETC gauge boson, ${M_{\rm ETC}}$, and the scale of compositeness, ${\Lambda_{\rm TC}}$, involved. It is not clear which of these should be used as a cutoff. A reasonable approach would consist in using the smaller mass scale, which is expected to be ${\Lambda_{\rm TC}}$. This, however, implies losing information from the dynamics occurring between ${\Lambda_{\rm TC}}$ and ${M_{\rm ETC}}$. Furthermore, it is well know that making the techniquark loop integrals finite with a sharp cutoff does not account for confinement, as the fermion propagators go on-shell for sufficiently large external momenta. A solution to both problems is provided by models of confinement \cite{Efimov:1993zg}. To briefly review these, let $S(x_1-x_2)$ be the Green function of a confined techniquark in the vacuum technigluon field. This can be represented as
\begin{equation}
S(x_1-x_2)=-\int\frac{d^4 p}{(2\pi)^4}\, i\, \rho(\slashed{p})\, e^{-i\, p\cdot (x_1-x_2)}\ .
\label{Eq:Confinement}
\end{equation}
Confinement means that the techniquark field has no asymptotically free states which describe non-interacting free particles. This, in turn, implies that the function $\rho(\slashed{p})$ is everywhere holomorphic. Therefore, we can use the Cauchy representation
\begin{equation}
\rho(\slashed{p}) = -\int_L \frac{dM}{2\pi i}\, \frac{\rho(M)}{\slashed{p}-M}\ ,
\label{Eq:Cauchy}
\end{equation}
where $L$ is a closed contour around $\slashed{p}=0$. Inserting (\ref{Eq:Cauchy}) in (\ref{Eq:Confinement}) gives
\begin{equation}
S(x_1-x_2)=\int_L \frac{dM}{2\pi i}\, \rho(M)\, S(x_1-x_2,M)\ ,
\label{Eq:S}
\end{equation}
where $S(x_1-x_1,M)$ is the propagator of a free techniquark of mass $M$. The techniquark condensate is
\begin{equation}
\langle \overline{Q}Q \rangle = -\, {\rm Tr}\, S(0) = \int_L \frac{dM}{2\pi i}\, \rho(M)\, \langle \overline{Q}Q \rangle_M\ ,
\end{equation}
where $\langle \overline{Q}Q \rangle_M\equiv -\, {\rm Tr}\, S(0,M)$ is the condensate of the free techniquark field. This means that $\rho(M)$ can be interpreted as a distribution density of techniquark masses in the vacuum technigluon field. We can generalize (\ref{Eq:S}) to Green functions of $n$ external technihadrons,
\begin{equation}
{\cal T}_n(x_1,x_2,\dots,n)\, =\,
\int_L \frac{dM}{2\pi i}\, \rho(M)\, {\rm Tr}\, \Big( \Gamma_1\, S(x_1-x_2,M)\cdots \Gamma_n\, S(x_n-x_1,M) \Big)
\end{equation}
where $\Gamma_k$ are matrices in Dirac space. In each propagator the techniquark constituent mass is "smeared" by the distribution density $\rho(M)$, which playes a double role: First, it provides convergence of the integrals in momentum space, within a domain the size of which is
determined by the
mass scale $\Lambda_{\rm TC}$. Second, it prevents the techniquark propagators to go on-shell, thus removing unwanted production of free techniquarks. The function $\rho(M)$ cannot be computed exactly,
and an ansatz must be made based on first principles and phenomenological constraints.
If we use a distribution density $\rho(M)$ to smear the integrals over techniquarks, we may cutoff the ${\cal L}_{\rm TC} + {\cal L}_{\rm ETC}$ theory at ${M_{\rm ETC}}$. The integrals over SM quarks are cutoff at ${M_{\rm ETC}}$, whereas the integrals over techniquarks are naturally finite and of the order of ${\Lambda_{\rm TC}}$. No techniquark propagator can go on-shell, and confinement is therefore guaranteed. Clearly we must choose an appropriate function $\rho(M)$, and integrals are unavoidably more difficult to evaluate than the standard loop integrals. However in our analysis we are only interested in small external momenta, and no internal techniquark propagator can go on-shell, even without a distribution density.
If a distribution density was used, the consequence would be that is that it makes the integrals finite and of the order of ${\Lambda_{\rm TC}}$. Therefore, we make the approximation of using a sharp cutoff
${\Lambda_{\rm TC}}$ for the loop integrals over techniquark momenta, rather than a distribution density, while still cutting off the SM-fermion loop integrals at ${M_{\rm ETC}}$. This approach makes the dynamics between ${\Lambda_{\rm TC}}$ and ${M_{\rm ETC}}$ contribute to the low-energy observables, and, as we shall see, preserves the global symmetries of the Lagrangian. Of course, working with a cutoff carries some ambiguity when the external momentum $q^2$ is non-zero.
However, for the observables we consider in our analysis, $q^2$ is either exactly zero or very small in comparison to the techniquark mass squared. The standard loop integrals used in our computations are explicitly given in the appendix.
\section{Matching with the electroweak theory} \label{Sec:EWmatch}
\begin{figure}[htb]
\includegraphics[width=4.5in]{Gap.pdf}
\caption{Diagrams contributing to the fermion masses at leading order in $N$ and $N_c$. $M_D$ is entirely given by the tree-level term, whereas $M_U$ receives both tree-level and loop contributions. Note that the bottom quark is massless in our approximation.}
\label{Fig:Gap}
\end{figure}
In our approximation the top quark acquires mass, whereas the bottom quark is massless. In the large-$N$, $N_c$ limit, with $N/N_c$ finite, the fermion masses are given by the diagrams of Fig.~\ref{Fig:Gap}. These correspond to the coupled gap equations
\begin{eqnarray}
&& M_U = M + 4\, N_c\, G\, M_t\, I_1^{M_{\rm ETC}}(M_t)\ , \nonumber \\
&& M_t = 4\, N\, G\, M_U\, I_1^{\Lambda_{\rm TC}}(M_U)\ ,
\label{eq:Gap}
\end{eqnarray}
and $M_D=M$, where $I_1^\Lambda(M)$ is defined in (\ref{Eq:I1}). As motivated in the last section, here and in the computations below, the integrals over techniquarks are cutoff at
${\Lambda_{\rm TC}}$, whereas the integrals over SM fermions are cutoff at the ETC mass scale ${M_{\rm ETC}}$. We define a measure of isospin mass splitting in the techniquark sector as
\begin{equation}
\delta\equiv \frac{M_U-M}{M}\ .
\label{Eq:Delta}
\end{equation}
The gap equations can be solved numerically for $G$ and $\delta$.
The charged and neutral technipion are the Goldstone bosons "eaten" by the longitudinal components of the $W$ and $Z$ boson, respectively. To ensure that the cutoff scheme respects chiral symmetry, we need to check that these absorbed Goldstone bosons are massless.
Let us start with the neutral technipion. Its self-energy $\Sigma_{\Pi^0\Pi^0}$ is given by the infinite sum of diagrams shown in Fig.~\ref{Fig:NeutralPi}.
This leads to the expression
\begin{equation}
\Sigma_{\Pi^0\Pi^0} = -\frac{4\, N\, M}{v^2}\left(M_U\, I_1^{\Lambda_{\rm TC}}(M_U)+M\, \ I_1^{\Lambda_{\rm TC}}(M)\right)
+\frac{N\, M^2}{v^2}\left(\chi_D + \chi_U\right)
+\frac{N\, M^2}{v^2}\frac{N\, N_c\, G^2\, \chi_U^2\, \chi_t}{1-N\, N_c\, G^2\, \chi_U\, \chi_t}\ ,
\end{equation}
where
\begin{equation}
i\ \chi_X\equiv -\int\frac{d^4k}{(2\pi)^4} {\rm Tr}\ \gamma_5 \frac{i\Big(\slashed{k}+M_X\Big)}{k^2-M_X^2}\gamma_5 \frac{i\Big(\slashed{k}+\slashed{q}+M_X\Big)}{(k+q)^2-M_X^2}\ .
\end{equation}
Assuming $q^2\ll M_X^2$ the integral is given by
\begin{eqnarray}
&& \chi_X = 2\, q^2\, I_2^{\Lambda_{\rm TC}}(M_X)+4\, I_1^{\Lambda_{\rm TC}}(M_X)\ ,\ \ X=U,D \\
&& \chi_t = 2\, q^2\, I_2^{M_{\rm ETC}}(M_t) + 4\, I_1^{M_{\rm ETC}}(M_t)\ ,
\end{eqnarray}
where $I^\Lambda(M)$ is defined in (\ref{Eq:I2}).
Using these expressions, $\Sigma_{\Pi^0\Pi^0}$ becomes
\begin{equation}
\Sigma_{\Pi^0\Pi^0} = \frac{ N\, M^2}{v^2}\left(2\, I_2^{\Lambda_{\rm TC}}(M) q^2
-4 \frac{M_U}{M}\ I_1^{\Lambda_{\rm TC}}(M_U)
+\frac{\chi_U}{1-N\, N_c\, G^2\, \chi_U\, \chi_t}\right)\ .
\end{equation}
Using the gap equations it is rather straightforward to show that this expression vanishes at $q^2=0$,
proving that the neutral technipion is massless.
\begin{figure}[!t]
\includegraphics[width=6.2in]{NeutralPi.pdf}
\caption{Diagrams contributing to the neutral technipion self-energy at leading order in $N$ and $N_c$.}
\label{Fig:NeutralPi}
\end{figure}
\begin{figure}[!t]
\includegraphics[width=6.7in]{ChargedPi.pdf}
\caption{Diagrams contributing to the charged technipion self-energy at leading order in $N$ and $N_c$.}
\label{Fig:ChargedPi}
\end{figure}
Then let us turn to the charged technipion. The computation showing that it is massless is similar to the one for the neutral technipion. The charged-technipion self-energy is given by the diagrams of Fig.~\ref{Fig:ChargedPi}, and summing the infinite series gives
\begin{equation}
\Sigma_{\Pi^-\Pi^+} = -\frac{4\, N\, M}{v^2}\left(M_U\, I_1^{\Lambda_{\rm TC}}(M_U)+M\, \ I_1^{\Lambda_{\rm TC}}(M)\right)
+\frac{2\, N\, M^2}{v^2}\left(\chi_{UD}
+\frac{N\, N_c\, G^2\, \chi_{UD}^2\, \zeta_{tb}}{1-4 N\, N_c\, G^2\, \zeta_{UD}\, \zeta_{tb}}\right)\ ,
\end{equation}
where
\begin{eqnarray}
&& i\ \chi_{UD}\equiv -\int\frac{d^4k}{(2\pi)^4} {\rm Tr}\ \gamma_5 \frac{i\Big(\slashed{k}+M\Big)}{k^2-M^2}\gamma_5 \frac{i\Big(\slashed{k}+\slashed{q}+M_U\Big)}{(k+q)^2-M_U^2}\ , \nonumber \\
&& i\ \zeta_{XY}\equiv \int\frac{d^4k}{(2\pi)^4} {\rm Tr}\ P_L \frac{i\Big(\slashed{k}+M_Y\Big)}{k^2-M_Y^2} P_R \frac{i\Big(\slashed{k}+\slashed{q}+M_X\Big)}{(k+q)^2-M_X^2}\ .
\end{eqnarray}
Evaluating the integrals gives
\begin{eqnarray}
&& \chi_{UD} = 2\left(q^2-\delta^2 M^2\right) I_2^{\Lambda_{\rm TC}}(M_U,M)+2 \left(I_1^{\Lambda_{\rm TC}}(M_U)+I_1^{\Lambda_{\rm TC}}(M)\right)\ ,
\end{eqnarray}
and
\begin{eqnarray}
&& \zeta_{UD} = \left(q^2-M_U^2-M^2\right) I_2^{\Lambda_{\rm TC}}(M_U,M)+I_1^{\Lambda_{\rm TC}}(M_U)+I_1^{\Lambda_{\rm TC}}(M)\ ,\nonumber \\
&& \zeta_{tb} = \left(q^2-M_t^2\right) I_2^{M_{\rm ETC}}(M_t,0)+ I_1^{M_{\rm ETC}}(M_t)+I_1^{M_{\rm ETC}}(0)\ .
\end{eqnarray}
The function $I_2^\Lambda(M_1,M_2)$ is defined in Eq. (\ref{Eq:I2b}).
Using these expressions, $\Sigma_{\Pi^-\Pi^+}$ at zero momentum becomes
\begin{equation}
\Sigma_{\Pi^+\Pi^-}\left(q^2=0\right) = \left.
\frac{ N\, M^2}{2 v^2}\frac{ \chi_{UD}}{\zeta_{UD}}\left(
-4 \frac{M_U}{M}\ I_1^{\Lambda_{\rm TC}}(M_U)
+\frac{\chi_{UD}}{1-4\,N\, N_c\, G^2\, \zeta_{UD}\, \zeta_{tb}}\right)\right\vert_{q^2=0}\ .
\end{equation}
Again, applying the gap equations it can be straightforwardly shown that this expression vanishes, proving that the charged technipion is massless.
\begin{figure}[!t]
\includegraphics[width=6.0in]{GF.pdf}
\caption{Diagrams contributing to the mixing between charged technipion and axial current, at leading order in $N$ and $N_c$. At zero momentum this gives $F_\Pi$ times the square root of the charged-technipion wave-function renormalization.}
\label{Fig:GF}
\end{figure}
Finally, in order to match this chiral-techniquark model to the electroweak theory, we need to impose that the charged-technipion decay constant is equal to $F_\Pi\equiv 246$ GeV. The decay constant can be extracted from the mixing term between the charged technipion and the charged component of the axial current. This is given by the sum of diagrams shown in Fig.~\ref{Fig:GF}, and leads to the expression
\begin{equation}
\mu_{\Pi {\cal A}} = \frac{4\, N\, M}{v}\left[
\mu^{\Lambda_{\rm TC}}(M_U,M)\left(1+\frac{N\, N_c\, G^2\, \chi_{UD}\, \zeta_{tb}}{1-4\, N\, N_c\, G^2\, \zeta_{UD}\, \zeta_{tb}}\right)
+\frac{\mu^{M_{\rm ETC}}(M_t,0)}{2}\frac{N_c\, G\, \chi_{UD}}{1-4\, N\, N_c\, G^2\, \zeta_{UD}\, \zeta_{tb}}\right]\ .
\end{equation}
The function $\mu^\Lambda(M_1,M_2)$ is defined in Eq. (\ref{Eq:mu}).
Using the gap equations, the expression for $\mu_{\Pi {\cal A}}$ can be brought into the compact form
\begin{equation}
\mu_{\Pi {\cal A}} = 2\left[N\, (M_U+M)\, \mu^{\Lambda_{\rm TC}}(M_U,M)+N_c\, M_t\, \mu^{M_{\rm ETC}}(M_t,0)\right]\ .
\end{equation}
To obtain the decay constant we must divide the latter by the square root of the charged-technipion wave-function renormalization constant:
\begin{equation}
F_\Pi = \displaystyle{\lim_{q^2\to 0}}\displaystyle{\frac{\mu_{\Pi {\cal A}}(q^2)}{\sqrt{\Sigma_{\Pi^-\Pi^+}^\prime(q^2)}}}\ .
\label{Eq:FP}
\end{equation}
It is not difficult to show that, in the limit $G\to 0$, this becomes the Pagels-Stokar relation:
\begin{equation}
F_\Pi^2\, \substack{\ \\ \ \\ {\displaystyle =}\\ G\to 0}\,
4\, N\, M^2\, I_2^{\Lambda_{\rm TC}}(M)\ .
\end{equation}
Once $F_\Pi$ is fixed, this implies that small variations of $M$ lead to exponentially large variations of ${\Lambda_{\rm TC}}$, as the ${\Lambda_{\rm TC}}$ dependence in $I_2^{\Lambda_{\rm TC}}(M)$ is essentially logarithmic. Therefore, requiring ${\Lambda_{\rm TC}}$ to be above 2-3 TeV and well below 10 TeV, as expected if ${\Lambda_{\rm TC}}$ is of the order of the heavy technihadron masses, forces $M$ to be within a very narrow interval.
\section{The dynamical mass of the scalar singlet}\label{Sec:scalarmass}
\begin{figure}[!t]
\includegraphics[width=6.0in]{TechniHiggs.pdf}
\caption{Diagrams contributing to the TC-Higgs self-energy at leading order in $N$ and $N_c$.}
\label{Fig:TechniHiggs}
\end{figure}
Next, we will consider the subject of our main interest in this paper, which is the mass of the scalar singlet. Because of the interaction with the top quark, the {\em physical} mass $M_H$ of the scalar singlet may be very different from its {\em dynamical} mass $M_{H0}$: the latter is defined as the mass solely due to the strong TC dynamics, and is obtained by setting $G=0$. Taking also the iteraction with the top quark into account, the scalar-singlet self-energy is given by the diagrams of Fig.~\ref{Fig:TechniHiggs}. Summing the series gives
\begin{equation}
\Sigma_{HH} = -m^2+\frac{N\, y^2\, M^2}{v^2}\left(\xi_D + \xi_U\right)
+\frac{N\, y^2\, M^2}{v^2}\frac{N\, N_c\, G^2\, \xi_U^2\, \xi_t}{1-N\, N_c\, G^2\, \xi_U\, \xi_t}\ ,
\label{higgsself}
\end{equation}
where
\begin{equation}
i\ \xi_X\equiv -\int\frac{d^4k}{(2\pi)^4} {\rm Tr}\ \frac{\slashed{k}+M_X}{k^2-M_X^2} \frac{\slashed{k}+\slashed{q}+M_X}{(k+q)^2-M_X^2}\ .
\end{equation}
Evaluating the integral gives
\begin{eqnarray}
&& \xi_X = 2\left(q^2-4M_X^2\right) I_2^{\Lambda_{\rm TC}}(M_X)+4 I_1^{\Lambda_{\rm TC}}(M_X)\ ,\ \ X=U,D \\
&& \xi_t = 2\left(q^2-4M_t^2\right) I_2^{M_{\rm ETC}}(M_t) + 4 I_1^{M_{\rm ETC}}(M_t)\ .
\end{eqnarray}
We eliminate $m^2$ by requiring the dynamical mass to be $M_{H0}$, i.e. we set $\Sigma_{HH}$ of Eq. (\ref{higgsself}) equal to zero at $q^2=M_{H0}$ at $G=0$. This leads to the expression
\begin{equation}
\Sigma_{HH}=\frac{N y^2 M^2}{v^2} \left(4 I_2^{\Lambda_{TC} } (M) \left(q^2-M_{H0}^2\right)-\xi_D+\frac{\xi_U}{1-N N_c G^2 \xi_U \xi_t}\right)\ .
\end{equation}
We would like to test whether the scalar singlet $H$ can be interpreted as the recently observed 125 GeV resonance. Therefore, we set $\Sigma_{HH}=0$ at $q^2=M_H^2=(125\ {\rm GeV})^2$, and solve for $M_{H0}$. We can then compare the latter with independent estimates for the dynamical mass, such as those obtained from scaling up the QCD spectrum.
Let us first take the limit $M^2\ll \Lambda_{\rm TC}^2\ll M_{\rm ETC}^2$. Solving for $M_{H0}$ gives, at leading logartithmic order,
\begin{equation}
M_{H0}^2\simeq
\frac{1}{\log \Lambda_{\rm TC}^2/M^2}\frac{
\displaystyle {\frac{N_c}{N}\frac{M_t^2}{M_U^2}}}
{1-
\displaystyle{\frac{N_c}{N}\frac{M_t^2}{M_U^2}\frac{M_{\rm ETC}^2}{\Lambda_{\rm TC}^2}}}M_{\rm ETC}^2
\,.
\label{Eq:Approx}
\end{equation}
This shows that the dynamical mass required to give a 125 GeV physical mass grows with $M_{\rm ETC}$. Ignoring $M_H^2$ gives a linear growth, as shown in Fig.~\ref{Fig:MH} (left) for $N=3$, and (right) for $N=6$. Note that $M$ and $M_{H0}$ are expected to scale like $1/\sqrt{N}$. We see that $M_{H0}$ may indeed be much larger than 125 GeV.
\begin{figure}[!t]
\includegraphics[width=3.0in]{MHN3.pdf}
\includegraphics[width=3.0in]{MHN6.pdf}
\caption{The dynamical mass of the scalar singlet, $M_{H0}$, which is required to give a 125 GeV scalar, after the radiative corrections from the top quark are included. For large enough TC scale $\Lambda_{\rm TC}$, $M_{H0}$ grows with the ETC mass $M_{\rm ETC}$, as shown by (\ref{Eq:Approx}). The left plot is for $N=3$, whereas the right plot is for $N=6$, where $N$ is the dimension of the techniquark representation under the TC gauge group.}
\label{Fig:MH}
\end{figure}
\section{Results and constraints}\label{Sec:ST}
The results of our analysis can be related to the underlying gauge dynamics of some simple technicolor models. Setting $N=3$ would correspond to the QCD like dynamics of SU(3) gauge theory with two fermions in the fundamental representation of the gauge group. On the other hand $N=6$ would correspond to the dynamics of SU(3) gauge theory with two fermions in the sextet representation. The $N=6$ case has been introduced \cite{Sannino:2004qp} as one of the minimal models with walking dynamics. Its phenomenological viability has been studied e.g. in \cite{Belyaev:2008yj,Dietrich:2005jn,Dietrich:2005wk,Alanne:2013dra} and its nonperturbative properties have been also recently studied on the lattice \cite{DeGrand:2013uha,Fodor:2012ty}
With these concret models in mind, in Fig.~\ref{Fig:MH} we have shown the dynamical mass of the isosinglet scalar resonance which is required to give a 125 GeV scalar after radiative corrections from the top quark. The values obtained are within several hundreds of GeVs and below $\sim$ 1 TeV. This is a range which can be reasonably expected from a TC theory, especially in the case of non-QCD-like dynamics. The mass scale $\Lambda_{\rm TC}$ is determined by the underlying strong dynamics, and is ignored in this model computation. The ETC mass scale is a free parameter, and Fig.~\ref{Fig:MH} shows that larger values of $M_{\rm ETC}$ lead to larger radiative corrections to the scalar mass. However $M_{\rm ETC}$ cannot be arbitrarily large: the first of (\ref{eq:Gap}) shows that the isospin splitting $\delta$ grows with $M_{\rm ETC}$, which should therefore be constrained by the $T$ parameter. In a TC theory with one weak technidoublet the $S$ and $T$ parameters, to leading order in a a large-$N$ expansion, can be written as
\begin{eqnarray}
S &=& \frac{N}{6\pi} + \Delta S_{\rm ETC} + \Delta S_{\rm vectors} \ , \\
T &=& \frac{N}{16\pi\, s_W^2\, c_W^2\, M_Z^2}\, \left[M_U^2+M_D^2-\frac{2\, M_U^2\, M_D^2}{M_U^2-M_D^2}\, \log\frac{M_U^2}{M_D^2}\right] + \Delta T_{\rm ETC} + \Delta T_{\rm vectors} \ , \label{Eq:T}
\end{eqnarray}
where, in the equation for $T$, $M_U$ and $M_D$ are defined to satisfy the gap equations (\ref{eq:Gap}). The first terms in the above equations are the usual one-loop contributions from heavy techniquarks, \cite{Peskin:1990zt,Peskin:1991sw}. The terms $\Delta S_{\rm ETC}$ and $\Delta T_{\rm ETC}$ are corrections due to ETC operators other than (\ref{eq:ETC}). As argued above, these are expected to be of comparable magnitude, and do certainly contribute to low-energy observables (for instance, a four-techniquark operator contributing more to $M_D$ than $M_U$ would reduce the isospin mass splitting introduced by (\ref{eq:ETC})). Finally, the terms $\Delta S_{\rm vectors}$ and $\Delta T_{\rm vectors}$ are the contributions from the spin-one resonances, which are also expected to be relevant. Also, sub-leading contributions in $1/N$ may be important when $N$ is not too large. It should be noted that additional ETC operators and sub-leading contributions are also expected to affect $M_{H0}$.
It is therefore not possible to impose
strict constraints from oblique corrections
solely based on the Lagrangian $L_{\rm TC}+L_{\rm ETC}$ of (\ref{eq:TC}) and (\ref{eq:ETC}). Nonetheless, it is useful to explore the $(\Lambda_{\rm TC},M_{\rm ETC})$ parameter space. In Fig.~\ref{Fig:Viable} (left) we consider the case $N=3$. Inside the lower-right
triangular region the TC cutoff $\Lambda_{\rm TC}$ is larger than $M_{\rm ETC}$, and our parametrisation of the ETC sector in terms of four-fermion operators breaks down. This means that in this region we should employ the full ETC theory to evaluate the correction to the mass of the scalar singlet, and the results are necessarily model-dependent. We may take the values of $M_{H0}$ in this region as an estimate for the correct $M_{H0}$. In the upper-left region, above the dashed line, the $T$ parameter evaluated using the first term of (\ref{Eq:T}) is greater than the experimental 95\% C.L. upper bound, $T_{\rm up}=0.23$ derived from the results of the gfitter group \cite{Baak:2012kk}. Finally, the contours correspond to fixed values of $M_{H0}$ as shown by the contour labels in the figure. In Fig.~\ref{Fig:Viable} (right) we show the case $N=6$. We see the that additional contributions to the $T$ parameter should be negative, or else the ETC scale associated to the top mass would be forced to be smaller than a few TeVs.
\begin{figure}[h]
\includegraphics[width=3.0in]{N3.pdf}
\includegraphics[width=3.0in]{N6.pdf}
\caption{The dynamical scalar mass $M_{H0}$ is shown here as contour lines in the $(\Lambda_{\rm TC},M_{\rm ETC})$ parameter space. Inside the triangualr region in the lower-right corner of the plot, the TC cutoff $\Lambda_{\rm TC}$ is larger than $M_{\rm ETC}$, and our parametrisation of the ETC sector in terms of four-fermion operators breaks down. In the upper-left regions, above the dashed line, the $T$ parameter evaluated using the first term of (\ref{Eq:T}) is greater than its 95\% C.L. upper bound, $T_{\rm up}=0.23$. The left plot is for $N=3$, whereas the right plot is for $N=6$.}
\label{Fig:Viable}
\end{figure}
\section{Conclusions and outlook}
\label{Sec:conclusions}
It is a novel observation that the dynamical mass of the isosinglet scalar in the spectrum of a strongy coupled gauge theory acquires nontrivial contributions from interactions external to the strong dynamics. A familiar example of such a contribution is the one, due to the electromagnetic interaction, which generates the mass splitting between charged and neutral pions in QCD. In technicolor theories such external interactions are provided by the electroweak sector itself and by any extension aimed at the generation of the masses for the SM matter fields. The profound consequence of this observation, then, is that the lightness of the Higgs boson observed at the LHC experiments can be well compatible with the composite nature of the scalar boson.
In this paper we have carried out a quantitative analysis of this issue within a model of strong dynamics where the technicolor sector is described by a chiral-meson model and the extended technicolor interactions are modelled by an effective four-fermion interaction between the technifermions and the top quark. Restricting the interaction terms to a single four-fermion operator is a simple but sufficient approximation which serves to illustrate our point: we find that 125 GeV scalar can easily result from underlying strong dynamics corresponding to gauge dynamics of SU(3) gauge theory with two Dirac fermions in the sextet representation.
Our results should be applicable to a wide variety of models utilizing strong dynamics and four-fermion couplings to explain both the electroweak symmetry breaking and the large mass splitting of top and bottom quarks via a dynamical mechanism \cite{Miransky:1988xi,Miransky:1989ds,Bardeen:1989ds,Dobrescu:1997nm,Chivukula:1998wd,He:2001fz,Fukano:2012qx,Fukano:2013kia}. Furthermore, our results together with the first principle determinations of the dynamical mass of the scalar from the lattice \cite{Fodor:2014pqa} will provide constraints on the dynamical models.
The results we have obtained in this paper could be refined by considering a more detailed set of four-fermion operators. Then one could also describe the generation of fermion mass patterns and attempt a more precise determination of the oblique corrections, i.e. $S$ and $T$ parameters. The effect of additional four-fermion interactions on the scalar mass are expected to be qualitatively similar to the case we have considered here.
\section*{Acknowledgements}
This work was financially supported by the Academy of Finland project 267842.
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1,116,691,500,791 | arxiv | \section{#1} \setcounter{equation}{0}}
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\begin{document}
\title{Entropic trade--off relations for quantum operations}
\author{Wojciech Roga$^{1,2}$, Zbigniew~Pucha{\l}a$^3$, {\L}ukasz Rudnicki$^4$, Karol {\.Z}yczkowski$^{2,4}$}
\affiliation{
$^1$Universit\`a degli Studi di Salerno, Via Ponte don Melillo, I-84084 Fisciano (SA), Italy \\
$^2$Institute of Physics, Jagiellonian University, ul.\ Reymonta 4, 30-059 Krak\'ow, Poland\\
$^3$Institute of Theoretical and Applied Informatics, Polish Academy
of Sciences, Ba{\l}tycka 5, 44-100 Gliwice, Poland \\
$^4$Center for Theoretical Physics, Polish Academy of Sciences,
al.\ Lotnik\'ow 32/46, 02-668 Warszawa, Poland}
\date{05--02--2013, ver. 19.2}
\begin{abstract}
Spectral properties of an arbitrary matrix can be characterized by the entropy
of its rescaled singular values. Any quantum operation can be described by the
associated dynamical matrix or by the corresponding superoperator. The entropy
of the dynamical matrix describes the degree of decoherence introduced by the
map, while the entropy of the superoperator characterizes the a
priori knowledge of the receiver of the outcome of a quantum channel $\Phi$.
We prove that for any map acting on a $N$--dimensional quantum system the sum of
both entropies is not smaller than $\ln N$. For any bistochastic map this lower
bound reads $2\ln N$. We investigate also the corresponding R{\'e}nyi entropies,
providing an upper bound for their sum and analyze entanglement
of the bi-partite quantum state associated with the channel.
\end{abstract}
\pacs{03.67.Hk 02.10.Ud 03.65.Aa}
\maketitle
\section{Introduction}
From the early days of quantum mechanics the uncertainty principle was one
of its the most significant features,
as it shows in what respect the quantum theory differs from its
classical counterpart.
It was manifested that
the variances of the two noncommuting observables cannot be simultaneously arbitrarily small.
Therefore, if we prepare the quantum state as an eigenstate of one observable
we get the perfect knowledge about the corresponding physical quantity, however, we loose
ability to predict the effect of measurement of the second, noncommuting observable.
Preparing a state one shall always consider some trade--off
regarding the observables which will be specified in the experiment.
Limits for such a trade--off have been formulated as different uncertainty relations
\cite{heisenberg,robertson,BBM75,De83,maassen}. However, we shall point out that
these uncertainty relations are not necessarily related to noncommuting observables,
but may describe the trade--off originating from different descriptions
of the same quantum state.
As an example consider the entropic uncertainty relation \cite{BBM75} derived
for two probability distributions related to the same quantum state in position and momentum representations.
The original formulation of Heisenberg \cite{heisenberg} of the uncertainty principle
concerns the product of variances of two non-commuting observables.
Assume that a physical system is described by a quantum state $|\psi\>$, and
several copies of this state are available. Measuring an observable $A$ in each
copy of this state results with the standard deviation $\Delta_{\psi} A
=\sqrt{\<\psi|A^2|\psi\>-\<\psi|A|\psi\>^2}$, while $\Delta_{\psi} B$ denotes
an analogous expression for another observable $B$. According to the approach of
Robertson \cite{robertson} the product of these deviations is bounded from
below,
\begin{equation}
\Delta_{\psi}A\ \Delta_{\psi} B\geq\frac{1}{2}\left|\<\psi|[A,B]|\psi\>\right|,
\end{equation}
where $[A,B]=AB-BA$ denotes the commutator. If the operators $A$ and $B$ do not
commute it is thus impossible to specify simultaneously precise values of both
observables.
Uncertainty relations can also be formulated for other quantities characterizing
the distributions of the measurement outcomes. One possible choice is to use
entropy which leads to entropic uncertainty relations of Bia{\l}ynicki--Birula
and Mycielski \cite{BBM75}. This formulation can be considered as a
generalization of the standard approach as it implies the relations of
Heisenberg.
In the case of a finite dimensional Hilbert space the uncertainty relation can
be formulated in terms of the Shannon entropy. Consider a non-degenerate
observable $A$, the eigenstates $|a_i\rangle$ of which determine an orthonormal
basis. The probability that this observable measured in a pure state
$|\psi\rangle$ gives the$i$th outcome reads $a_i=|\<a_i|\psi\>|^2$. The
non-negative numbers $a_i$ satisfy $\sum_{i=1}^N a_i=1$, so this distribution
can be characterized by the Shannon entropy, $H(A)=-\sum_i a_i\ln a_i$. Let
$H(B)$ denotes the Shannon entropy corresponding the the probability vector
$b_i=|\<b_i|\psi\>|^2$ associated with an observable $B$. If both observables do
not commute the sum of both entropies is bounded from below, as shown by Deutsch
\cite{De83}. His result was improved by Maassen and Uffink \cite{maassen}, who
proved that
\begin{equation}
\label{masuuf}
H(A) + H(B) \ \geq \ -2\ln{c},
\end{equation}
where $c^2=\max_{j,k}|\<a_j|b_k\>|^2$ denotes the maximal overlap between the
eigenstates of both observables. Note that this bound depends solely on the
choice of the observables and not on the state $|\psi\rangle$. Recent reviews
on entropic uncertainty relations can be found in \cite{WW10,BR11}, while a link
to stabilizer formalism was discussed in \cite{NKG12}. Certain generalizations
of uncertainty relations for more than two spaces can be based on the strong
subadditivity of entropy \cite{FL12}.
Relation (\ref{masuuf}) describes a bound for the information which can be
obtained in two non-complementary projective measurements.
Entropic uncertainty relations formulated for a pair of arbitrary measurements,
described by positive operator valued measures (POVM),
were obtained by Krishna and Parthasarathy \cite{KP02}.
A more general class of inequalities was derived later
by Rastegin \cite{Ra10,Ra11}.
Related recent results \cite{BCCRR10,TR11,CYGG11,CCYZ12}
concerned sum of two conditional entropies
characterizing two quantum measurements
described in terms of their POVM operators.
The so called {\it collapse of wave function} during the measurement
is often considerd as another characteristic feature of quantum mechanics.
This postulate of
quantum theory implies that the measurement disturbs the quantum state
subjected to the process of quantum measurement.
In general, describing a quantum operation performed on an arbitrary state
one can consider a kind of trade--off relations between the efficiency of the measurement
and the disturbance introduced to the measured states.
Even though the trade--off relations were investigated from the beginnings of quantum mechanics,
this field became a subject of a considerable scientific interest in the recent
decade \cite{Fuchs2001,DAriano2003,Buscemi2009}.
The notion of {\sl disturbance}
of a state introduced by Maccone \cite{Maccone2006,Maccone2007},
can be related to the average fidelity
between an initial state of the system and and the state after the measurement \cite{Maccone2006,Buscemi2009}.
Another version of disturbance can be defined as a difference between the initial entropy of a quantum state
and the coherent information between the system and the measuring apparatus \cite{Maccone2007}.
In this work we will investigate
a single measurement process described by a quantum operation: a complete
positive, trace preserving linear map which acts on an input state of size $N$.
We attempt to compare the information loss introduced by the map
(disturbance) with the
information the receiver knows about the outgoing state before the measurement
(the information gained by the apparatus).
The former quantity can be characterized \cite{ZB04} by the entropy of a map
$S^{\map}(\Phi)$, equal to the von Neumann entropy of the quantum
state which corresponds to the considered map by the Jamio\l{}kowski isomorphism
\cite{jamiolkowski,BZ06}. The latter quantity will be described
by the \emph{singular quantum entropy} $S^{\rec}(\Phi)$
of Jumarie \cite{Ju00}, given by the
Shannon entropy of the normalized vector of singular values of the superoperator
matrix. We are going to show that the sum of these two entropies is bounded
from below by $\ln N$.
Note that our approach concerns a given quantum map $\Phi$,
but it does not depend on the particular choice of the
Kraus operators (or POVM operators) used to represent the
quantum operation. We also derive an upper bound for the sum of entropies
$S^{\map}(\Phi)$ and $S^{\rec}(\Phi)$ and analyze entanglement properties
of the corresponding Jamio\l{}kowski-Choi state.
Our paper is organized as follows. In Section II we review basic concepts on
quantum maps, define entropies investigated and present a connection with
trade--off
relations for quantum measurements. A motivation for our study
stems from investigations of the one--qubit maps presented in Section III.
General entropic inequalities for arbitrarily reordered matrices
are formulated in Section IV.
Main results of the work are contained in Section~V, in
which the trade--off
relations for quantum channels are derived
and entanglement of the corresponding states is analyzed.
Discussion of some other properties of the dynamical matrix
and a bound for the entropy of a map are
relegated to Appendices.
\section{Quantum operations and entropy}\label{secII}
A quantum state is described by a density matrix -- a~Hermitian, positive
semi-definite matrix of trace one. A~density matrix of dimension $N$ represents
the operator acting on $\c H_N$. The set of density matrices of dimension $N$ is
denoted as:
\begin{equation}
\c M_N=\{\rho: \rho=\rho^\dagger, \rho\geq0, \tr\rho=1\}.
\end{equation}
A \emph{quantum operation} $\Phi$, also called a \emph{quantum channel}, is
defined as a completely positive (CP) and trace preserving (TP) quantum map which
acts on the set of density matrices:
\begin{equation}
\Phi:\c M_N\rightarrow\c M_N.
\end{equation}
Complete positivity means that any extended map acting on an enlarged quantum
system
\begin{equation}
\Phi\otimes{\1}_d:\c M_{Nd}\rightarrow\c M_{Nd}
\end{equation}
transforms positive matrices into positive matrices for any extension of
dimension $d$. Due to the Choi theorem, see e.g. \cite{BZ06}, to verify whether a
given quantum map $\Phi^A$ acting on a quantum $N$--level system $A$ is
completely positive it is necessary and sufficient that the following operator
on the Hilbert space $\c H_{N}^A\otimes\c H_{N}^B$ of a composed subsystems $A$
and $B$
\begin{equation}
D_{\Phi^A}:= N(\Phi^A\otimes\mathbbm{1}^{B})\big(|\phi_+^{AB}\left.\right\rangle\left\langle\right. \phi_+^{AB}|\big)\geq 0,
\label{choi}
\end{equation}
is non-negative. Here
$|\phi_+^{AB}\left.\right\rangle=\frac{1}{\sqrt{N}}\sum_{i=1}^N
|i^A\left.\right\rangle\otimes|i^B\left.\right\rangle\in \c H_N^A\otimes\c
H_N^B$ denotes the maximally entangled state in the extended space. The above
relation, called the \emph{Jamio{\l}kowski isomorphism}, implies a
correspondence between quantum maps $\Phi$ and quantum states
$\omega_{\Phi}=\frac{1}{N}D_{\Phi}$. The operator $\omega_\Phi$ is called the
Jamio{\l}kowski--Choi state, whereas the matrix $D_{\Phi}$ is called the
\emph{dynamical matrix} associated with the map $\Phi$.
Any quantum channel acting on a quantum system $A$ can be represented by a
unitary transformation $U^{AB}$ acting on an enlarged system and
followed by the partial trace over the ancillary subsystem $B$:
\begin{equation}
\Phi(\rho^A)={\rm Tr}_B\left[U^{AB}(\rho^A\otimes \ket{1^B}\bra{1^B})(U^{AB})^{\dagger}\right].
\label{repenv}
\end{equation}
This formula is called the environmental representation
of a quantum channel.
Another useful representation of a quantum channel
is given by a set of operators $K_i$ satisfying an
identity resolution, $\sum_iK_i^{\dagger}K_i=\idty$,
which implies the trace preserving property.
The Kraus operators $K_i$ define the {\sl Kraus representation}
of the map $\Phi$,
\begin{equation}
\Phi(\rho)=\sum_iK_i\rho K_i^{\dagger} .
\label{repkraus}
\end{equation}
Since $\Phi:\rho\rightarrow\rho'$ acts on an operator $\rho$, it is sometimes
called a \emph{superoperator}. If we reshape a density matrix into a vector of
its entries $\vec{\rho}$, the superoperator $\Phi$ is represented by a matrix
of size $N^2$. It is often convenient to write the discrete dynamics
$\vec{\rho'}=\Phi \vec{\rho}$ using the four--index notation
\begin{equation}
\label{super}
\rho'_{k\ \!\!l}=\Phi_{\!\!\stackidx{k}{l}{m}{n}}
\rho_{m\ \!\!n},
\end{equation}
where the sum over repeating indices is implied and
\begin{equation}
\Phi_{\!\!\stackidx{k}{l}{m}{n}}
=\<k\ l|\Phi|m\ n\> .
\end{equation}
The dynamical matrix $D=D_{\Phi}$
is related to the superoperator matrix $\Phi$
by reshuffling its entries,
\begin{equation}
D_{\stackidx{k}{m}{l}{n}}
=\Phi_{\!\!\stackidx{k}{l}{m}{n}}
\label{reshuffling}
\end{equation}
written $D_{\Phi}=\Phi^R$ or $\Phi= D_{\Phi}^R$.
The von Neumann entropy of the Jamio{\l}kowski-Choi state was studied in
\cite{verstaete,roga0,ziman,RFZ11,Ra12} and it is also investigated in this work. We
are going to compare the spectral properties of the Jamio{\l}kowski-Choi state
$\omega_{\Phi}$ and the spectral properties of the corresponding superoperator
matrix $\Phi$.
\subsection{Entropy of a map}
Entropy $S^{\map}(\Phi)$ is defined \cite{ZB04} as the von
Neumann entropy of the corresponding Jamio{\l}kowski-Choi state
$\omega_{\Phi}=\frac{1}{N}D_\Phi$,
\begin{equation}
\label{smapphi}
S^{\map}(\Phi)\; :=\; -\tr \omega_{\Phi} \ln \omega_{\Phi}.
\end{equation}
This quantity can be interpreted as the special case of the \emph{ exchange entropy}
\cite{szuma}
\begin{equation}
S^{\,\rm exchange}\left(\Phi^A,\rho^A\right)\equiv S\left(\Phi^A\otimes\idty^B(|\phi^{AB}_{\rho^A}\>
\<\phi^{AB}_{\rho^A}|)\right),
\label{exchange}
\end{equation}
where $|\phi^{AB}_{\rho^A}\>\in\c H_{N_A}^A\otimes\c H_{N_B}^B$ is a
purification of $\rho^A$, that is such a pure state of an enlarged system which has the partial trace
given by ${\rm Tr}_B|\phi^{AB}_{\rho^A}\>\<\phi^{AB}_{\rho^A}|=\rho^A$.
The exchange entropy characterizes the information exchanged during a quantum
operation between a principal quantum system $A$ and an environment $B$, assumed
to be initially in a pure state. Under the condition that an initial state of the
quantum system $A$ is maximally mixed, $\rho^A_*=\frac{1}{N}\idty$, the exchange
entropy $S^{\,\rm exchange}\left(\Phi^A,\rho^A_*\right)$ is equal to the entropy
of a channel $S^{\map}\left(\Phi^A\right)$.
We will treat the entropy of a map as a measure of disturbance caused by a measurement performed on the quantum system.
The work \cite{Maccone2007} contains a list of the properties expected from a good measure of disturbance.
Among them there is the requirement that the disturbance measure should be equal to zero if and only if
the measuring process is invertible. For unitary transformations of the quantum state
the dynamical matrix given in Eq. (\ref{choi}) has rank one,
so the related entropy of the map is equal to zero as expected.
Moreover, if the map preserves identity,
the entropy of a map is equivalent to the
state independent disturbance analyzed in \cite{Maccone2007}.
It is useful to generalize the von Neumann entropy and to
introduce the family of the R{\'e}nyi entropies
\begin{equation}
\label{ren1}
S_q(\rho)=\frac{1}{1-q}\ln\tr\rho^{q},
\end{equation}
as they allow to formulate a more general
class of uncertainty relations \cite{BB06}.
Here $q\geq 0$ is a free parameter
and in the limit $q\rightarrow 1$
the generalized entropy tends to the von Neumann entropy,
$S_q(\rho)\rightarrow S_1(\rho)\equiv S(\rho)$.
For any classical probability vector and any
quantum state $\rho$ the R{\'e}nyi entropy $S_q(\rho)$
is a monotonously decreasing function of the
R{\'e}nyi parameter $q$ \cite{beck}.
The generalized entropy (\ref{smapphi})
of a quantum map $\Phi$, obtained by applying the above form of R{\'e}nyi
to the state $\omega_{\Phi}$ will be denoted by $S_q^{\map}(\Phi)$.
\subsubsection{Connection with the uncertainty principle for measurements}
Let $\Phi$ be a CP TP map with Kraus operators $\{A_i\}_i$, we define, $P_i =
A_i^\dagger A_i$ and note that the operators $\{P_i\}_i$ form a POVM~\cite{BZ06},
i.e. are positive semidefinite and
\begin{equation}
\sum_i P_i = \1.
\end{equation}
If $\rho$ is a state of a given system, the probability of the outcome
associated with a measurement of the operator $P_i$ reads
\begin{equation}
p_i = \tr P_i \rho.
\end{equation}
The uncertainty involved in a described measurement can be quantified by the entropy~\cite{KP02}
\begin{equation}
H_q(P,\rho) = S_q(p).
\end{equation}
Let us consider an uncertainty involved in the measurement in the case when the state of a given
system is maximally mixed, i.e.
\begin{equation}
p_i = \tr \left(P_i \frac1N \1\right) = \frac1N \tr P_i.
\end{equation}
We have the following corollary
\begin{corollary}\label{cor:min-POVM}
If the state of a given system is maximally mixed, then
\begin{equation}
\min_{P} H_q\left(P,\frac1N \1 \right) = S_q^{\mathrm{map}} (\Phi),
\end{equation}
where the minimum is taken over all possible POVM's such that
\begin{equation}
P_i = A_i^\dagger A_i
\end{equation}
and $A_i$ are the Kraus operators of the quantum channel $\Phi$.
\end{corollary}
\begin{proof}
Let $A_i$ be a Kraus representation of the channel $\Phi$ and denote by $\ket{res(A_i)}$ a vector obtained from the matrix $A_i$ by putting its elements in the lexicographical order i.e. rows follow one after another. We introduce
\begin{equation}
\begin{split}
\kappa_i &= \scalar{res(A_i)}{res(A_i)} = \tr A_i^\dagger A_i = \tr P_i, \\
\ket{a_i} &= \frac{1}{\sqrt{\kappa_i}} \ket{res(A_i)}.
\end{split}
\end{equation}
Assume that we put the $\kappa_i$ coefficients in a decreasing order such that $\kappa_1$ has the largest value. We have~\cite{BZ06}
\begin{equation}
D_{\Phi} = \sum_{i=1}^{l} \ketbra{res(A_i)}{res(A_i)},
\end{equation}
where $l\leq N^2$. Using the variational characterization of eigenvalues~\cite{HJ}, for the
Hermitian matrix $D_{\Phi}$, we get for each $k\leq N^2$ the following expression for the sum of $k$ largest eigenvalues
\begin{eqnarray}
\sum_{i=1}^{k}\lambda_{i}\left(D_{\Phi}\right) & = & \max_{U_{k}}\textrm{Tr}\left(U_{k}^{\dagger}D_{\Phi}U_{k}\right)\\
& = & \max_{U_{k}}\sum_{i=1}^{l}\kappa_{i}\textrm{Tr}\left(U_{k}^{\dagger}\left|a_{i}\right\rangle \left\langle a_{i}\right|U_{k}\right), \nonumber
\end{eqnarray}
where $U_{k}$ is a matrix of size $N^{2}\times k$ fulfilling the
relation $U_{k}^{\dagger}U_{k}=\1_k$, and $\1_k$ denotes the $k\times k$ identity.
For a specific choice $\tilde{U}_k$ of the matrix $U_k$, such that the vectors
$\ket{res(A_1)},\ldots,\ket{res(A_k)}$ belong to the subspace spanned by all $k$
columns of $\tilde{U}_k$, we have $\textrm{Tr} \left(\tilde{U}_{k}^{\dagger}
\left|a_{i}\right\rangle \left\langle a_{i}\right|\tilde{U}_{k}\right)=1$, for
$i=1,\ldots,k$. This property implies
\begin{equation}
\begin{split}
\sum_{i=1}^{k}\lambda_{i}\left(D_{\Phi}\right)&=
\max_{U_{k}}\sum_{i=1}^{l}\kappa_{i}\textrm{Tr}\left(U_{k}^{\dagger}\left|a_{i}\right\rangle \left\langle a_{i}\right|U_{k}\right)\\
&\geq\sum_{i=1}^{l}\kappa_{i}\textrm{Tr}\left(\tilde{U}_{k}^{\dagger}\left|a_{i}\right\rangle \left\langle a_{i}\right|\tilde{U}_{k}\right) \\
&=\sum_{i=1}^{k}\kappa_{i}+\sum_{i=k+1}^{l}\kappa_{i}\textrm{Tr}\left(\tilde{U}_{k}^{\dagger}\left|a_{i}\right\rangle \left\langle a_{i}\right|\tilde{U}_{k}\right)\\
&\geq\sum_{i=1}^{k}\kappa_{i} .
\end{split}
\end{equation}
In the last inequality we neglected the remaining non-negative terms
labeled by $i>k$.
The above set of inequalities imply
the majorization relation, $\kappa \prec \lambda(D_\phi)$. Using the fact that
R{\'e}nyi entropies are Schur--concave, we arrive at the desired inequality,
\begin{equation}
\begin{split}
H_q\left(P,\frac1N \1 \right) &= S_q\left(\left\{\tr P_i \frac1N \1\right\}\right) = S_q(\kappa/N)\\
& \geq S_q(\lambda(D_\Phi)/N) = S_q^{\mathrm{map}}(\Phi).
\end{split}
\label{eq21}
\end{equation}
\end{proof}
Using the monotonicity of the R{\'e}nyi entropies, we get
\begin{equation}
S_q^{\mathrm{map}} (\Phi) \geq S_\infty^{\mathrm{map}} (\Phi) = - \log (\lambda_1(D_{\Phi})/N).
\end{equation}
For $q=1$ this inequality combined with (\ref{eq21}) resembles the
uncertainty principle for a single quantum measurement \cite{KP02,CYGG11,CCYZ12},
since for an optimal POVM the lower bound obtained in these papers
depends on $c=\max_j \tr P_j = \lambda_1/N$.
\subsection{Receiver entropy}
Since a superoperator matrix $\Phi$ is in general not Hermitian, we characterize
this matrix by means of the entropy of the normalized vector of its singular
values
\begin{equation}
S^{\rec}(\Phi)=-\sum_i \mu_i \ln \mu_i .
\label{receiver}
\end{equation}
Here $\mu_i=\frac{\sigma_i}{\sum_k\sigma_k}$ and $\sigma_i$ denote the singular
values of $\Phi$, so that $\sigma_i^2$ are eigenvalues of the positive matrix
$\Phi\Phi^{\dagger}$. This quantity characterizes an arbitrary matrix $\Phi$ and
depends only on its singular values, so it was called \emph{singular quantum
entropy} by Jumarie \cite{Ju00}.
In the case of one--qubit states entire set $\c M_2$ can be represented as a
three-dimensional ball of the radius one - Bloch ball. For one qubit
bistochastic channels, which preserve the center of the Bloch ball, the
entropy $S^{\rec}(\Phi)$ characterizes the vector $\{1,\eta_1,\eta_2,\eta_3\}$
after normalization to unity, where $\{\eta_1,\eta_2,\eta_3\}$ denote the
lengths of semiaxes of an ellipsoid obtained as the image of the Bloch ball
under a quantum operation $\Phi$ -- see \cite{fuji,RSW02}
and Fig. \ref{elipsoida}.
As demonstrated with a few examples presented below
the entropy $S^{\rec}$ characterizes
the a priori knowledge of the receiver of the outcome of a quantum channel $\Phi$. Therefore the
quantity (\ref{receiver}) will be called the \emph{receiver entropy}.
\begin{figure}
\ellipsoidInBall
\caption{(Color online) An image of the Bloch ball under an exemplary
one--qubit bistochastic map $\Phi$.}
\label{elipsoida}
\end{figure}
To illustrate the meaning of the receiver entropy let us consider the
following examples of one-qubit channels. In the case of completely
depolarizing channel $\Phi_*:\rho\rightarrow\rho_*=\frac{1}{N}\idty$ the entire
Bloch ball is transformed into a single point. All three semiaxes vanish,
$\eta_i=0$, so $\mu=(1,0,0,0)$ and
the receiver entropy is equal to $0$.
This value characterizes the perfect knowledge of the receiver of the states
transmitted through the channel, since having the information about this channel
the receiver knows that every time he or she obtains
the very same output state.
In the case of a coarse graining channel $\Phi_{CG}$, which
sets all off--diagonal elements of a density matrix to zero, and preserves the
diagonal populations unaltered,
the Bloch ball is transformed into an interval of unit length.
This means that the information about
the possible output state missing to the receiver
can be described by a single variable only.
In this case one has $\mu=(1/2,1/2,0,0)$,
so the entropy reads $S^{\rec}(\Phi_{CG})= \ln 2$.
Consider now an arbitrary unitary channel, which only rotates the entire Bloch ball.
Then the receiver has no knowledge in which part of the Bloch ball the output state will
appear, and the corresponding receiver entropy is maximal, $S^{\rec}=2\ln 2$.
In general, the receiver entropy is bounded by the logarithm of the rank of the
superoperator characterizing the channel $S^{\rec}(\Phi)\leq \ln{\rm rank}
(\Phi)$. Every quantum state can be represented by a real vector in the basis of
generalized Pauli matrices (see for instance \cite{pittenger}). Therefore, in
this basis the superoperator is a real matrix and its rank characterizes the
dimensionality of the vector space accessible for the outcomes from the channel.
Consider any orthonormal basis $\{K_i\}$ with respect to the Hilbert-Schmidt scalar product
which includes rescaled identity.
Such a set of matrices satisfies normalization condition $\sum_iK_i^{\dagger}K_i=\idty$,
therefore can define the POVM measurement.
During the measurements of a quantum state $\rho$
the outcomes $K_i\rho K_i^{\dagger}/(\tr K_i\rho K_i^{\dagger})$
are observed with probabilities $p_i=\tr K_i\rho K_i^{\dagger}$.
The receiver entropy is related to the probability
distribution characterizing frequency of different outcomes of the measurement apparatus.
If the entropy is low the receiver may expect
that only a small amount of outcomes of the measuring apparatus will occur.
High values of the entropy imply that several different results of the measurement
will appear.
Hence the receiver entropy $S^{\rec}$ characterizes the number of
measurement operators needed to obtain a complete information about the measured state.
\section{One qubit examples}
To analyze discrete dynamics of a one--qubit system let us define two matrices
of order four:
\begin{equation}
G=
\left(\begin{smallmatrix}
1 &0 &0 &0 \\
0 &1 &0 &0 \\
0 &0 &1 &0 \\
0 &0 &0 &1
\end{smallmatrix}
\right)
=
\left(\begin{smallmatrix}
G_1 & | & G_2 \\
-- & + & -- \\
G_3 & | & G_4 \\
\end{smallmatrix}
\right),
\label{blo}
\end{equation}
and
\begin{equation}
C=G^R=
\left(\begin{smallmatrix}
1 &0 &0 &1 \\
0 &0 &0 &0 \\
0 &0 &0 &0 \\
1 &0 &0 &1
\end{smallmatrix}
\right).
\end{equation}
Note that the first row of $C$ is obtained by reshaping the block $G_1$ into a
vector, the second row of $C$ contains the reshaped block $G_2$, etc. Such
transformation of a matrix is related to the fact that in any linear, one--qubit
map, $\rho'=\Phi\rho$, the $2 \times 2$ matrix $\rho$ is treated as a vector of
length $4$.
Normalizing the spectra of both matrices to unity we get the entropies
$S(G)=\ln{4}$ and $S(C)=0$. Making use of the above notation we can represent
the identity map $\Phi_{\idty}=G$ and the corresponding dynamical matrix
$D_{\idty}=\Phi_{\idty}^R=C$. Moreover, the completely depolarizing channel
which maps any state $\rho$ into the maximally mixed state
$\Phi_{*}:\rho\rightarrow\frac{1}{2}\idty$ can be written as
$\Phi_{*}=\frac{1}{2}C$ while $D_{*}=\Phi_{*}^R=\frac{1}{2}G$. In both cases the
sum of the entropy of a dynamical matrix $S^{\map}\equiv S(\frac{1}{2}D)$ and
the entropy of normalized singular values of superoperator $S^{\rec}\equiv
S(\frac{|\Phi|}{\tr{|\Phi|}})$ reads $S^{\map}+S^{\rec}=2\ln{2}$. Thus, both maps
$\Phi_{\idty}$ and $\Phi_{*}$ are in a sense distinguished, as they occupy
extreme positions at both entropy axes. It is easy to see that the above
reasoning can be generalized for an arbitrary dimension $N$. For the identity
map acting on $\c M_N$ and the maximally depolarizing channel $\Phi_{*}$ one
obtains $S^{\map}+S^{\rec}=2\ln{N}$. For these two maps the above relation holds
also for the R{\'e}nyi entropies, $S_q^{\map}+S_q^{\rec}=2\ln{N}$.
Investigations of one--qubit quantum operations enabled us to specify the set of
admissible values of the channel entropy $S^{\map}(\Phi)$ and the receiver
entropy $S^{\rec}(\Phi)$.
We analyzed the images of the set of one--qubit quantum maps
on to the plane $(S^{\map}, S^{\rec})$.
This problem was first analyzed numerically
by constructing random one-qubit maps \cite{frob}
and marking their position on the plane.
A special care was paid to the case of bistochastic maps, i.e. maps preserving the identity,
which form a tetrahedron spanned by the identity $\sigma_0=\idty$ and the three Pauli
matrices $\sigma_i$ (see e.g. \cite{BZ06})
\begin{equation}
\Phi_{bist}(\rho)=\sum_{i=0}^3p_i\sigma_i\rho \sigma_i.
\label{paulichan}
\end{equation}
Fig. \ref{fig:sketch} can thus be interpreted as a
non--linear projection of the set of all one--qubit channels onto the plane
$(S^{\map},S^{\rec})$,
in which bistochastic maps correspond to the dark stripped region.
\begin{figure}[ht]
\centering
\scalebox{.4}{\includegraphics{nieozn3.eps}}
\caption{(Color online)
Striped region denotes the allowed set of points representing one--qubit quantum
operations characterized by the entropy of the channel $S^{\map}(\Phi)$ and the
receiver entropy $S^{\rec}(\Phi)$. Gray (colored) region represents
bistochastic channels, while white stripped region corresponds to interval
channels. Distinguished points of the allowed set represent: $a$ --- completely
depolarizing channel, $b$ --- identity channel, $c$ --- coarse graining channel
and $d$ --- channel completely contracting the entire Bloch ball into a given
pure state. The action of these four channels on the Bloch ball is
schematically shown on the auxiliary circular plots.
} \label{fig:sketch}
\end{figure}
\medskip
The distinguished points of the allowed region in the plane
$(S^{\map},S^{\rec})$ correspond to:
\begin{itemize}
\item $a)$ completely depolarizing channel:
$\Phi_*:\rho\rightarrow\rho_*=\frac{1}{2}{\1}_2$ ,
\item $b)$ identity channel $\Phi_{\idty}={\1}$,
\item $c)$ coarse graining channel $\Phi_{CG}$, which sets all off--diagonal
elements of a density matrix to zero, and preserves the diagonal populations
unaltered,
\begin{equation}
\Phi_{CG}=
\left(
\begin{smallmatrix}
1 &0 &0 &0 \\
0 &0 &0 &0 \\
0 &0 &0 &0 \\
0 &0 &0 &1
\end{smallmatrix}
\right),
\end{equation}
\item $d)$ spontaneous emission channel sending any state into a certain pure
state (e.g. the ground state of the system),
$\Phi_{SE}:\rho\rightarrow|0\rangle\langle0|$.
\end{itemize}
Basing on the numerical analysis the following curves are recognized as the limits of the region
\begin{itemize}
\item the curve $ab$ given by the depolarizing channels
$\Phi_{\alpha}=\alpha\idty+(1-\alpha)\Phi_*$, for $0\leq \alpha\leq 1$.
This family of states provides
the upper bound for the entire region of $(S^{\map},S^{\rec})$ available for the one-qubit quantum maps.
Because of the importance of this curve we provide its parametric expression
\begin{eqnarray*}
\ \ \ \ \ S^{\map}\!&=&\!-\frac{3}{4}(1\!-\!\alpha)\ln\!\left[\frac{1}{4}(1\!-\!\alpha)\right]\!-\!(1\!+\!3\alpha)\ln(1\!+\!3\alpha),\\
S^{\rec}&=&\ln(1+3\alpha)-\frac{3\alpha\ln\alpha}{(1+3\alpha)}.
\end{eqnarray*}
\item the curve $bc$ which represents the combination of identity and the
coarse graining,
\item the interval $ad$ which represents completely contracting channels:
linear combinations of the completely
depolarizing channel and the spontaneous emission,
\item the interval $cd$ which includes the maps of the form:
\begin{equation}
\Phi_{cd}=
\left(
\begin{smallmatrix}
\alpha &0 &0 &\beta \\ \
\sqrt{\alpha(1-\alpha)}e^{i\phi_1} &0 &0 &\sqrt{\beta(1-\beta)}e^{i\phi_2} \\
\sqrt{\alpha(1-\alpha)}e^{-i\phi_1} &0 &0 &\sqrt{\beta(1-\beta)}e^{-i\phi_2} \\
1-\alpha &0 &0 & 1-\beta
\end{smallmatrix}
\right)
\label{czysty}
\end{equation}
\end{itemize}
with $\alpha, \beta \in (0,1)$
and two arbitrary phases $\phi_1$ and $\phi_2$.
The above maps belong to a broader family of one--qubit operations:
\begin{equation}
\Phi_I=
\left(
\begin{smallmatrix}
\alpha &0 &0 &\beta \\ \
\gamma_1 &0 &0 &\gamma_2 \\
\bar{\gamma}_1 &0 &0 &\bar{\gamma}_2 \\
1-\alpha &0 &0 & 1-\beta
\end{smallmatrix}
\right),
\end{equation}
with complex numbers $\gamma_1$ and $\gamma_2$,
such that the first column
reshaped into a matrix of order two forms a positive state $\rho_1$,
while the reshaped
last column corresponds to a state $\rho_2$.
These operations can be called \emph{interval channels}, as they transform the
entire Bloch ball into an interval given by the convex combination of the states
$\rho_1$ and $\rho_2$. The dynamical matrix corresponding to an interval map can
be transformed by permutations into a block diagonal form.
Fig. \ref{fig:sketch} representing all one-qubit channels
distinguishes two regions. Bistochastic quantum
operations
correspond to the dark region.
As the set of one--qubit bistochastic maps forms a tetrahedron (a convex set given in Eq. (\ref{paulichan}))
spanned by three Pauli matrices and identity (see \cite{BZ06}),
to justify this observation that the bistochastic maps cover the dark region of Fig. \ref{fig:sketch} it is sufficient to analyze the images
of the edges of the antisymmetric part of the tetrahedron onto the plane
$(S^{\map},S^{\rec})$.
The white striped region $acd$ contains for instance interval maps, which will be
shown in Proposition \ref{propozition1}. Note that there exist several maps
which correspond to a given point in Fig. \ref{fig:sketch}.
A further insight into the interpretation of the receiver entropy is due to the
fact that for any completely contractive channel (interval $ad$ in the plot),
which sends any initial state
into a concrete, selected state, $\Phi_{\xi}:\rho\rightarrow\xi$, the receiver
entropy is equal to zero. This is implied by the fact that the dynamical matrix of
such an operation reads $D_{\Phi_{\xi}}=\xi\otimes\idty$. After reshuffling of
this matrix we obtain the superoperator matrix of rank one, since all non-zero
column are the same, therefore it has only one nonzero singular value.
Normalization of the vector of singular values sets this number to unity so that
$S^{\rec}(\Phi_{\xi})=0$. This observation supports an interpretation of
$S^{\rec}$ as the amount of information missing to the receiver of the output
$\rho'$ of a quantum channel, who knows the operation $\Phi$, but does not
know the input state~$\rho$.
\section{Entropic inequalities for reordered matrices}
Before we establish several
trade--off
relations for quantum channels we shall introduce a framework concerning matrices
(in general non--hermitian) together with their reordered counterparts.
An arbitrary $d\times d$ matrix $X$ has $d^2$ independent matrix elements. A matrix $Y_\pi$ can be called a reordering of $X$ if $Y_\pi=X^\pi$, where $\pi$ denotes some permutation of matrix entries. Thus, for each matrix $X$ we can consider $\left(d^2\right)!$ reordered matrices $Y_\pi$.
Denote by $x_{i}$ the singular values of the matrix $X$ and introduce the following $q$-norms:
\begin{equation}\label{qnorms}
\left\Vert X\right\Vert _q=\left(\textrm{Tr}\left[XX^{\dagger}\right]^{q/2}\right)^{1/q}=\left(\sum_{i}x_{i}^{q}\right)^{1/q}.
\end{equation}
Moreover, by $x_{1}\equiv\left\Vert X\right\Vert _{\infty}$ denote the
greatest singular value of the matrix $X$ and by
\begin{equation}
\Lambda_{x}=\left\Vert X\right\Vert _{1}=\sum_{i}x_{i},
\end{equation}
the trace norm of $X$, i.e. the sum of all singular values $x_i$. Finally, define the R{\'e}nyi
entropy
\begin{equation}
S_{q}\left(X\right)=\frac{1}{1-q}\ln\sum_{i}\left(\frac{x_{i}}{\left\Vert X\right\Vert _{1}}\right)^{q}.
\end{equation}
The first result holds in general. \begin{lemma}\label{lemma2}
For an arbitrary matrix $X$ and $1\leq q<\infty$ we have
\begin{equation}
\ln\left(\frac{\Lambda_{x}}{x_{1}}\right)\leq S_{q}\left(X\right)\leq\frac{q}{q-1}\ln\left(\frac{\Lambda_{x}}{x_{1}}\right),\label{boundy1}
\end{equation}
\end{lemma}
The second inequality relates matrices $X$ and $Y_\pi$. \begin{lemma}\label{lemma3}
If $Y_\pi=X^{\pi}$ where the transformation $\pi$ is an arbitrary permutation
of matrix entries, then we have for $1\leq q<\infty$
\begin{equation}
F_{\textrm{min}}\ln\left(\frac{\Lambda_{y}}{\sqrt{x_{1}\Lambda_{x}}}\right)\leq S_{q}\left(Y_\pi\right)\leq F_{\textrm{max}}\ln\left(\frac{\Lambda_{y}}{x_{1}}\right),\label{boundyb}
\end{equation}
where $F_{\textrm{min}}=\min\left(\frac{q}{q-1};2\right)$ and $F_{\textrm{max}}=\max\left(\frac{q}{q-1};2\right)$.\end{lemma}
The symbol $\Lambda_y$ inside Lemma \ref{lemma3} denotes the trace norm of $Y_\pi$. Both lemmas are proven in Appendix \ref{algebraic lemmas}.
\section{Trade--off relations for quantum channels}\label{uncer}
The structure of the set of allowed values of both entropies $S^{\map}$ and
$S^{\rec}$ describing all one--qubit stochastic maps shown in Fig.
\ref{fig:sketch} suggests that their sum is bounded from below. For the smaller
class of bistochastic maps the bound looks to be more tight. Indeed we are going
to prove the following trade--off relation for the sum of two von Neumann entropies
\begin{subequations}
\begin{equation}
S^{\map}(\Phi)+ S^{\rec}(\Phi)\geq \ln{N},
\label{gen_1}
\end{equation}
and a sharper inequality
\begin{equation}
S^{\map}(\Phi)+ S^{\rec}(\Phi)\geq 2\ln{N},
\label{bist_1}
\end{equation}
\end{subequations}
which holds for any bistochastic map acting on a $N$ dimensional system. Note that
the second expression can be interpreted as a kind of entropic trade--off
relation for unital quantum channels: if the map entropy $S^{\map}(\Phi)$, which
quantifies the interaction with the environment during the operation or the degree of disturbance of a quantum state, is small,
the receiver entropy $S^{\rec}(\Phi)$ cannot be small as well. This implies that
the results of the measurement could be very diverse. Conversely, a small value
of the receiver entropy implies that the map $\Phi$ is strongly contracting,
so the map entropy is sufficiently large and a lot of information escapes from the
system to an environment and the
disturbance of the initial state is strong.
\medskip
Instead of proving directly the bounds (\ref{gen_1}) and (\ref{bist_1}) for the von Neumann entropy
$S\equiv S_1$ we are going to prove a more general inequalities formulated for the
R{\'e}nyi entropies $S_q$ with $q \in [1,\infty[$. All bounds in the limiting case $q=2$,
related to the Hilbert--Schmidt norm of a matrix, are shown in Fig.
\ref{fig:sketch2} for one--qubit quantum operations.
In the case of the $N^2 \times N^2$ matrices $\Phi$ and $D_{\Phi}$, let us denote by
$\sigma_{1}$ the greatest singular value of $\Phi$, by $d_{1}$
the greatest eigenvalue of $D_{\Phi}$, and by $\Lambda_{\Phi}=\left\Vert \Phi\right\Vert _{1}$
the sum of all singular values of $\Phi$. Since the Jamio{\l}kowski--Choi state $\omega_\Phi$ is normalized we have
$\left\Vert D_{\Phi}\right\Vert _{1}\equiv N$.
If we apply Lemma \ref{lemma2} to both matrices we obtain the following bounds:
\begin{subequations}
\begin{equation}
\ln\left(\frac{\Lambda_{\Phi}}{\sigma_{1}}\right)\leq S_q^{\rec}(\Phi)\leq\frac{q}{q-1}\ln\left(\frac{\Lambda_{\Phi}}{\sigma_{1}}\right),\label{boun1}
\end{equation}
\begin{equation}
\ln\left(\frac{N}{d_{1}}\right)\leq S_q^{\map}(\Phi)\leq\frac{q}{q-1}\ln\left(\frac{N}{d_{1}}\right).\label{boun2}
\end{equation}
Since $\Phi=D_\Phi^R$ and $D_\Phi=\Phi^R$,
where the reshuffling operation $R$ is a particular example of reordering we have
two additional bounds originating from Lemma \ref{lemma3}:
\begin{equation}
F_{\textrm{min}}\ln\left(\frac{\Lambda_{\Phi}}{\sqrt{Nd_{1}}}\right)\leq S_q^{\rec}(\Phi)\leq F_{\textrm{max}}\ln\left(\frac{\Lambda_{\Phi}}{d_{1}}\right),\label{boun3}
\end{equation}
\begin{equation}
F_{\textrm{min}}\ln\left(\frac{N}{\sqrt{\sigma_{1}\Lambda_{\Phi}}}\right)\leq S_q^{\map}(\Phi)\leq F_{\textrm{max}}\ln\left(\frac{N}{\sigma_{1}}\right).\label{boun4}
\end{equation}
\end{subequations}
The bounds (\ref{boun3}, \ref{boun4}) are in fact implied by the equality of Hilbert--Schmidt norms $\left\Vert \Phi\right\Vert _2=\left\Vert D_\Phi\right\Vert _2$, what is a consequence of the reshuffling relation $D_\Phi=\Phi^R$ .
Inequalities (\ref{boun1}---\ref{boun4}) provide individual limitations for ranges of the entropies $S_q^{\rec}$ and $S_q^{\map}$. However, if we consider a particular inequality we can always recover a full range $[0,2\ln N]$. The above inequalities can be combined in four different ways: (\ref{boun1}) with (\ref{boun2}), (\ref{boun3}) with (\ref{boun4}), (\ref{boun1}) with (\ref{boun4}) and (\ref{boun2}) with (\ref{boun3}) in order to obtain upper and lower bounds for the sum $S_q^{\map}+S_q^{\rec}$. These bounds shall depend on the three parameters: $\sigma_1$, $d_1$ and $\Lambda_\Phi$, thus, without an additional knowledge about these parameters, they do not lead to a trade--off relation. In particular, for $d_1=\sigma_1=\Lambda_\Phi=N$ we find from (\ref{boun1}---\ref{boun4}) that $S_q^{\rec}=0$ and $S_q^{\map}=0$. This case would correspond
to a pure, separable Jamio{\l}kowski--Choi state $\omega_\Phi$.
In order to show that the above example cannot be realized by a CP TP map we shall prove the following theorem which provides an upper bound on the greatest singular value $\sigma_1$.
\begin{theorem} \label{th:super-bound}
Let $\Phi$ be a CP TP channel acting on a set of density operators of size $N$.
Its superoperator $\Phi$ is a $N^2 \times N^2$ matrix. The greatest singular
value $\sigma_1$ is:
\begin{enumerate}
\item given by the expression
\begin{equation}
\label{singq1}
\sigma_1(\Phi) = \max_{\rho \in \mathcal{M}_N} \sqrt{\frac{ \tr \Phi(\rho)^2}{\tr \rho^2}},
\end{equation}
\item bounded
\begin{equation}
\sigma_{1}\left(\Phi\right)\leq\sqrt{N\tau_1}\leq\sqrt{N},\label{1}
\end{equation}\end{enumerate}where $\tau_1$ denotes the greatest eigenvalue of the density matrix $\Phi\left(\frac{1}{N}\1\right)\in \c M_N$.
\end{theorem}
The bound $\sigma_1\leq\sqrt{N}$ is saturated for quantum channels which transform the maximally
mixed state onto a pure state (only in that case $\tau_1=1$).
In the case of a bistochastic map all eigenvalues of
$\Phi\left(\frac{1}{N}\1\right)$ are equal to $1/N$ and therefore
$\sigma_1(\Phi)\leq 1$.
The proof of Theorem \ref{th:super-bound} is presented in Appendix \ref{greatestsingular}.
Some other bounds on singular values of reshuffled
density matrices have been studied in \cite{chikwongli}. In particular, there was shown that for $\rho \in \c M_N$ the largest singular value of the matrix $\rho^R$ is greater than $N^{-1}$. Because $\Phi=N\omega_\Phi^R$ we immediately find that $\sigma_1\geq 1$. Thus, for bistochastic maps we have the equality $\sigma_1=1$.
We are now prepared to prove the following theorem which establishes the entropic trade--off relations between
$S_q^{\map}$ and $S_q^{\rec}$.
\begin{theorem} \label{th:general-uncertainty}
For a CP TP map $\Phi$ acting on a system of an arbitrary dimension $N$ the
following relations hold:
\begin{enumerate}
\item For an arbitrary map $\Phi$
\begin{equation}\label{stoch_q}
S_q^{\map}(\Phi)+S_q^{\rec}(\Phi)\geq \frac{F_\textrm{min}}{2}\ln{N},
\end{equation}
\item If the quantum channel $\Phi$ is bistochastic
\begin{equation}\label{bistoch_q}
S_q^{\map}(\Phi)+ S_q^{\rec}(\Phi)\geq F_\textrm{min}\; \ln{N}.
\end{equation}
\end{enumerate}
\end{theorem}
Since for $q=1$ the coefficient $F_\textrm{min}=2$,
from Theorem \ref{th:general-uncertainty} we recover the particular
bounds (\ref{gen_1}, \ref{bist_1}) for the von Neumann entropies.
\begin{proof}[Proof of Theorem \ref{th:general-uncertainty}] In a first step we shall add two lower bounds present in (\ref{boun1}) and (\ref{boun4}) to obtain
\begin{equation}\label{bound genL}
S_q^{\map}(\Phi)+S_q^{\rec}(\Phi)\geq F_\textrm{min}\ln\left(\frac{N}{\sigma_1}\right)+\left(1\!-\!\frac{F_\textrm{min}}{2}\right)\!\ln\!\left(\frac{\Lambda_\Phi}{\sigma_1}\right)\!.
\end{equation}
Since $F_\textrm{min}\leq2$ and the greatest singular value $\sigma_1$ is less than the sum $\Lambda_\Phi$ of all singular values, the second term is always nonnegative. Thus, due to the upper bound (\ref{1}) we have
\begin{equation}\label{bound genL2}
S_q^{\map}(\Phi)+S_q^{\rec}(\Phi)\geq \frac{F_\textrm{min}}{2}\ln\left(\frac{N}{\tau_1}\right).
\end{equation}
The first statement of Theorem \ref{th:general-uncertainty} follows immediately, when instead of $\tau_1$ we put its maximal value $1$ into the inequality (\ref{bound genL2}).
The second statement is related to the fact that bistochastic
quantum channels preserve the identity i.e. $\Phi\left(\frac{1}{N}\1\right)=\frac{1}{N}\1$. The greatest eigenvalue $\tau_1$ is in this case equal to $\frac{1}{N}$, thus the value of $N^2$ appears inside the logarithm and cancels the factor of $2$ in the denominator.
\end{proof}
In fact, the inequality (\ref{bound genL2}) quantifies the deviation from the set of bistochastic maps, with the greatest eigenvalue of $\Phi\left(\frac{1}{N}\1\right)$ playing the role of the interpolation parameter.
\subsection{Additional upper bounds}
The receiver entropy $S_q^{\rec}(\Phi)$ is upper bounded due to the relations (\ref{boun1}) and (\ref{boun3}). However, these bounds diverge in the limit $q\rightarrow1$. Since the greatest singular value $\sigma_1$ is not less than $1$ we can derive another upper bound which gives a nontrivial limitation valid for all values of $q$.
\begin{theorem} \label{th: recq}
For a CP TP map $\Phi$ acting on a system of an arbitrary dimension $N$ the
following relation holds:
\begin{equation}\label{recq}
S_q^{\rec}(\Phi)\leq
\frac{1}{1-q}\ln\left(\Lambda_\Phi^{-q}+\frac{\left(\Lambda_\Phi-1\right)^{q}}
{\Lambda_\Phi^q\left(N^2-1\right)^{q-1}}\right).
\end{equation}
\end{theorem}
\begin{proof}
Since the map $\Phi$ is CP TP the greatest singular value $\sigma_1\geq1$. Thus, the vector $\boldsymbol{\sigma}$ of the singular values of the $N^2\times N^2$ matrix $\Phi$ majorizes ($\boldsymbol{\sigma}\succ\boldsymbol{\sigma}_0$) the vector:
\begin{equation}
\boldsymbol{\sigma}_{0}=\left(1,\underset{N^{2}-1}{\underbrace{\frac{\Lambda_{\Phi}-1}{N^{2}-1},\frac{\Lambda_{\Phi}-1}{N^{2}-1},\ldots,\frac{\Lambda_{\Phi}-1}{N^{2}-1}}}\right).
\end{equation}
Since $S_q^{\rec}(\Phi)= S_q(\boldsymbol{\sigma}/\Lambda_\Phi)$ and the R{\'e}nyi entropy is Schur concave we obtain the inequality $S_q^{\rec}(\Phi)\leq S_q(\boldsymbol{\sigma}_0/\Lambda_\Phi)$ which is equivalent to (\ref{recq}).
\end{proof}
As a limiting case of Theorem \ref{th: recq} we have the corollary
\begin{corollary}\label{upperB}
The von Neumann entropy $S^{\rec}(\Phi)$ is bounded
\begin{equation}
S^{\rec}(\Phi)\leq \frac{\Lambda_\Phi-1}{\Lambda_\Phi}\ln\left(\frac{N^2-1}{\Lambda_\Phi-1}\right)+\ln\Lambda_\Phi\leq2\ln N.
\end{equation}
\end{corollary}
\begin{figure}[ht]
\centering
\scalebox{.4}{\includegraphics{rys2_a.eps}}
\caption{(Color online)
Stripped region represents the set of one--qubit operations projected into the
plane spanned by the linear entropy of the map, $S_2^{\map}(\Phi)$, and the
linear receiver entropy $S_2^{\rec}(\Phi)$, i.e. the R{\'e}nyi entropies of order
$q=2$. Dark region represents the bistochastic quantum operations. Dashed
antidiagonal line represents the lower bound (\ref{bistoch_q}) which holds for
bistochastic operations, while solid antidiagonal line denotes the weaker bound
(\ref{stoch_q}) which holds for all quantum operations.
Dotted antidiagonal line represents the upper bound (\ref{upp}) applied for $N=2$.
} \label{fig:sketch2}
\end{figure}
The relation between the matrices $\Phi$ and $D_\Phi$ allows us to derive an upper bound for the sum of the R{\'e}nyi entropies $S_2^{\map}(\Phi) + S_2^{\rec}(\Phi)$.
\begin{proposition}\label{upperQ1}
The following relation holds:
\begin{equation}\label{upp}
S_2^{\map}(\Phi)+S_2^{\rec}(\Phi)\leq 2\ln\left(\frac{N(N+1)}{2}\right).
\end{equation}
\end{proposition}
\begin{proof}
Since $\left\Vert \Phi\right\Vert _2=\left\Vert D_\Phi\right\Vert _2$ we have an easy relation between both entropies:
\begin{equation}\label{S2}
S_2^{\map}(\Phi)=S_2^{\rec}(\Phi)+2\ln N-2\ln\Lambda_\Phi.
\end{equation}
According to (\ref{recq}) we are able to estimate
\begin{equation}
S_2^{\map}(\Phi)+S_2^{\rec}(\Phi)\leq 2\ln (N\Lambda_\Phi)-2\ln\left(1+\frac{\left(\Lambda_\Phi-1\right)^{2}}{N^2-1}\right).
\end{equation}
In order to complete the proof of Proposition \ref{upperQ1} we shall perform the maximization of the above upper bound over the parameter $\Lambda_\Phi\in\left[1,N^2\right]$.
\end{proof}
The bound presented in Proposition~\ref{upperQ1} can be saturated by a
quantum channel, which is a mixture of the identity channel and the maximally
depolarizing channel, i.e.
\begin{equation}
\Phi = \frac{1}{N+1} \1 + \frac{N}{N+1}\Phi_{*}.
\end{equation}
In fact, we are able to generalize the relation (\ref{S2}) to the case of all $1\leq q\leq\infty$.
\begin{proposition}\label{upperQ2}
The following relation holds:
\begin{equation}\label{upp2}
S_q^{\map}(\Phi)\geq F_\textrm{min}\ln \frac{N}{\Lambda_\Phi} + G_\textrm{min} S_q^{\rec}(\Phi),
\end{equation}
where $G_\textrm{min}=\min\left(\frac{q}{2(q-1)};\frac{2(q-1)}{q}\right)$.
\end{proposition}
\begin{proof}
Assume that $q\leq2$. In that case we have the following monotonicity properties for the R{\'e}nyi entropy: $S_q\geq S_2\geq 2\left(\frac{q-1}{q}\right) S_q$. These relations together with Eq. (\ref{S2}) provide a chain of inequalities:
\begin{eqnarray}
S_{q}^{\textrm{map}}\left(\Phi\right) & \geq & S_{2}^{\textrm{map}}\left(\Phi\right)\nonumber \\
& = & 2\ln\left(\frac{N}{\Lambda_{\Phi}}\right)+S_{2}^{\textrm{rec}}\left(\Phi\right)\\
& \geq & 2\ln\left(\frac{N}{\Lambda_{\Phi}}\right)+2\left(\frac{q-1}{q}\right)S_{q}^{\textrm{rec}}\left(\Phi\right),\nonumber
\end{eqnarray}
The same method applied for $q\geq2$ with associated monotonicity relations $2\left(\frac{q-1}{q}\right) S_q\geq S_2\geq S_q$
completes the proof of inequality (\ref{upp2}).
\end{proof}
\medskip
We can also show in which region of the plot $(S^{\map}, S^{\rec})$ the interval
maps are located. Notice that the classical maps, which transform the set of
$N$--point probability vectors into itself, also satisfy these inequalities.
\begin{proposition}\label{propozition1}
The interval maps satisfy the following inequalities
$S^{\rec}(\Phi)\leq \ln{N} \leq S^{\map}(\Phi)$.
\end{proposition}
\begin{proof}
The left inequality concerning the receiver
entropy follows from the fact that the entire set of states
is mapped into an interval. To show the right inequality
observe that the dynamical matrix corresponding to
an interval channel is block diagonal or can be
transformed to this form by a permutation.
Due to the trace preserving condition
every block of the normalized dynamical matrix can be interpreted as
$\frac{1}{N}\rho_i$ where $\rho_i$ is some density matrix. Therefore,
up to a permutation $P$
the normalized dynamical matrix has the structure
\begin{equation}
\omega=\frac{1}{N}P^{\dagger}D_{\Phi}P=\sum_{i=1}^{N}\frac{1}{N}\rho_i\otimes|i\>\<i|.
\end{equation}
Hence the entropy of the normalized dynamical matrix reads
\begin{equation}
\begin{split}
S(\omega)&= S\left(\frac{D_{\Phi}}{N}\right)=-\sum_i\tr\frac{1}{N}\rho_i\ln{\frac{1}{N}\rho_i}\\
&= \ln{N}+\sum_i\frac{1}{N}S(\rho_i).
\end{split}
\end{equation}
This implies the desired inequality for
the entropy of a map, $S^{\map}(\Phi)$.
\end{proof}
The last string of equations exemplifies the Shannon rule known as \emph{the
grouping principle} \cite{shannon, hall} that the information of expanded
probability distribution should be the sum of a reduced distribution and
weighted entropy of expansions. Notice that the grouping rule does not hold for
all dynamical matrices corresponding to generic quantum operations. As an
example take a maximally entangled state which is a purification of the maximally
mixed state.
\subsection{Super--positive maps and separability of the Jamio{\l}kowski--Choi state}
The aim of this part is to answer the question: How the
separability (entanglement) of the state $\omega_\Phi$
can be described in terms of the entropies $S_q^{\map}$ and $S_q^{\rec}$?
In other words we wish to identify the class of {\sl superpositive maps}
(also called entanglement breaking channels -- see \cite{BZ06}),
for which $\omega_\Phi$ is separable on the plane $(S_q^{\map}, S_q^{\rec})$.
Furthermore, we will determine the region on this plane
where no such maps can be found. The method to answer
these questions is based on the previously given uncertainty relations and
the realignment separability criteria \cite{chenWu,HorRealign}. These criteria
state that if $\omega_\Phi$ is separable then the sum of all singular values of
the matrix $\frac{1}{N}\Phi$ cannot be greater than $1$, what straightforwardly
implies $\Lambda_\Phi\leq N$.
\begin{figure}[ht]
\centering
\scalebox{.4}{\includegraphics{entanglement2.eps}}
\caption{(Color online) One--qubit maps projected onto the entropy plane (stripped set)
with superimposed bounds given in Proposition \ref{separable} concerning the separability of the
dynamical matrix.
Region $A$ contains no superpositive channels,
while region $B$ contains both classes of the maps. Region $C$ (determined by PPT criteria) contains only superpositive channels (all corresponding states are separable).
Lower (\ref{stoch_q}) and upper bounds (\ref{upp}) imply that
there are no one--qubit quantum operations projected into region $D$.
Diagonal of the figure contains the reshuffling--invariant channels,
in particular, the coarse graining channel ($c$)
and the {\sl transition depolarizing channel}
$\Phi_{1/3}=\frac{1}{3}\idty+\frac{2}{3}\Phi_*$,
located at the boundary of the set of superpositive maps ($e$).}
\label{fig:entanglement}
\end{figure}
We shall prove the following proposition
\begin{proposition}
\label{separable}
If $\omega_\Phi$ is separable, then:
\begin{enumerate}
\item $S_{q}^{\textrm{map}}\left(\Phi\right)\geq\frac{F_{\textrm{min}}}{4}\ln N$, \;and
\item $S_{q}^{\textrm{rec}}\left(\Phi\right)\leq\frac{1}{1-q}\ln\left(\frac{\left(N+1\right)^{q}+N^{2}-1}{N^{q}\left(N+1\right)^{q}}\right)$, \;and
\item $S_{q}^{\textrm{map}}\geq G_{\textrm{min}}S_{q}^{\textrm{rec}}\left(\Phi\right)$.
\end{enumerate}
\end{proposition}
\begin{proof}
In order to prove the statements 1--3 we apply
the separability criteria $\Lambda_\Phi\leq N$ directly to the inequalities (\ref{boun4}), (\ref{recq}) and (\ref{upp2}) respectively. In the case 1 we also include the bound $\sigma_1\leq\sqrt{N}$.
\end{proof}
The above result leads immediately to the separability criteria. If at least
one inequality from Proposition \ref{separable} is violated,
then the state $\omega_\Phi$ is entangled,
so the map $\Phi$ is not superpositive -- see Fig. \ref{fig:entanglement}.
The last inequality in Proposition \ref{separable} is saturated for the channels for which $S_2^{\map}=S_2^{\rec}$. They are located at
the diagonal of Fig. \ref{fig:entanglement}. This class contains maps
with dynamical matrix symmetric with respect to the reshuffling,
$D=D^R=\Phi$. This condition implies that the superoperator $\Phi$
is hermitian so its spectrum is real.
The following proposition characterizes
the set of one--qubit channels invariant with respect
to reshuffling.
\begin{proposition}
The following one--qubit bistochastic channels
$\Phi_{R-inv}$ are reshuffling--invariant
\begin{equation}
\Phi_{R-inv}=\Phi_U\Phi_{\eta_1,\eta_2}\Phi_{U^{\dagger}},
\label{forma}
\end{equation}
where
\begin{eqnarray}
& &\Phi_{\eta_1,\eta_2}=\frac{1}{2}\begin{pmatrix}1+\eta_3&0&0&1-\eta_3\\
0&\eta_1+\eta_2&\eta_1-\eta_2&0\\
0&\eta_1-\eta_2&\eta_1+\eta_2&0\\
1-\eta_3&0&0&1-\eta_3\label{eta1eta2}\\
\end{pmatrix}\\
& &\qquad\qquad\qquad{\rm and}\quad 1=\eta_1+\eta_2+\eta_3.\nonumber
\end{eqnarray}
The map $\Phi_U=U\otimes \bar{U}$
describes an arbitrary unitary channel,
as $U$ is a unitary matrix of order two
and $\bar{U}$ denotes its complex conjugation.
\end{proposition}
\begin{proof}
To justify the above statement we use the following general property of the reshuffling operation, which can be easily verified by checking the matrix entries of both sides
\begin{eqnarray}
& &\!\!\!\!\!\!\!\!\Big[\left(X^1_{n}\otimes X^2_n\right)\ Y_{n^2}\ \left(X^3_{n}\otimes X^4_{n}\right)\Big]^R=\label{przetas}\\
&=&\left(X^1_{n}\otimes \left(X^3_n\right)^T\right)\ Y_{n^2}^R \ \left(\left(X^{2}\right)^T_{n}\otimes X^4_n\right),\nonumber
\end{eqnarray}
where lower indices denote the dimensionalities of square matrices.
Since (\ref{eta1eta2}) is a reshuffling--invariant matrix, using (\ref{przetas})
we see that (\ref{forma}) is preserved after reshuffling.
\end{proof}
Two extreme examples of the reshuffling--invariant maps
are distinguished in Fig. \ref{fig:entanglement}:
the coarse graining channel ($c$)
for which $\eta_1=\eta_2=0$ and $\eta_3=1$,
and the transition depolarizing channel ($e$) at the boundary of super--positivity,
$\Phi_{1/3}=\frac{1}{3}\idty+\frac{2}{3}\Phi_*$,
for which $\eta_1=\eta_2=\eta_3=1/3$.
\section{Concluding remarks}
In this work an entropic trade--off relation analogue of the entropic uncertainty relation characterizing a
given quantum operation (\ref{stoch_q}) was established. We have shown that
for any stochastic quantum map
the sum of the map entropy, characterizing the decoherence introduced to the
system by the measurement process, and the receiver entropy, which describes the
knowledge on the output state without any information on the input, is bounded
from below. The more one knows a priori concerning the outcome state, the more
information was exchanged between the principal subsystem and the environment
due to the quantum operation. A stronger bound (\ref{bistoch_q}) is obtained
for the class of bistochastic maps, for which the maximally mixed state is
preserved. Entanglement properties of a Jamio\l{}kowski--Choi state were
investigated in terms of the entropies $S_q^{\map}$ and $S_q^{\rec}$.
Dynamical entropic trade--off relations were obtained also
for the R{\'e}nyi entropies of an arbitrary order $q$.
From a mathematical perspective this result is
based on inequalities relating the spectrum of a positive hermitian matrix
$X=X^{\dagger}$ and the singular values of the non--hermitian reshuffled matrix
$X^R$. Related algebraic results were recently obtained in \cite{chikwongli}
and applied to the separability problem. It is tempting to believe that further
algebraic investigations on the spectral properties of a reshuffled matrix will
lead to other results applicable to physical problems motivated by the quantum
theory.
\begin{acknowledgments}
The authors would like to thank P. Gawron for his help
with the preparation of the figures.
We are grateful to M.~Zwolak and A.E.~Rastegin for helpfull correspondence
and appreciate encouraging discussions with I.~Bia{\l}ynicki--Birula, J.~Korbicz
and R.~Horodecki. W.R. acknowledges financial support
from the EU STREP Projects HIP, Grant Agreement No. 221889.
Z.P. was supported by MNiSW under the project number IP2011~044271. {\L}.R. acknowledges financial support by MNiSW research grant, number IP2011~046871, for years 2012-2014.
K.{\.Z}. acknowledges financial support by
the Polish NCN research grant, decision number DEC-2011/02/A/ST2/00305.
\end{acknowledgments}
|
1,116,691,500,792 | arxiv | \section{Introduction}
In the context of single--winner elections, a \emph{spoiler} is usually defined as a losing candidate whose removal would affect the outcome by changing the winner \citep{Borgers10,Kaminski18} (although other definitions have also been put forward -- see, e.g., \citep{HollidayPacuit21}).
We are particularly interested in investigating spoiler effects in political elections to multi--member representative bodies (e.g., parliaments). Such elections are distinguished from other types of multi--winner elections primarily by their character as \emph{party elections}. By party election we mean such an election that all candidates are affiliated with parties, and that the allocation of the winning positions (``seats'') among those parties, rather than the identity of the winners themselves, is the most important outcome of the election. Thus, spoiler effects should also be considered in terms of spoiler parties and party (rather than individual) results.
In this paper, we seek first to extend the definition of a spoiler to multi-winner party elections, and then to compare multi-winner social choice rules according to their degree of spoiler susceptibility. Such an extension involves a number of issues. First, there is no longer a single natural definition of winners and losers. Most measures of election outcome (such as seat shares or voting power) are non--binary, and many parties are likely to have non-zero seat shares. Second, since the probabilistic models commonly used in computational social choice have been developed for non-party elections, we need to extend them to be able to analyze party elections probabilistically.
\subsection{Contribution}
To address the issues described above, we propose a more general definition of spoilership. A party $j$ is considered a spoiler if there exists a set of parties such that if $j$ is eliminated, that coalition's total share in the election outcome exceeds $j$'s original share. The measure of such excess can be considered to be a measure of the ``degree of spoilership''.
This approach allows us to compare multi-winner social choice rules according to their degree of spoiler susceptibility. We analyze seven well--known rules: $k$--Borda, Chamberlin--Courant, Harmonic Borda, Jefferson--D'Hondt, $k$--PAV, SNTV, and STV. We only consider anonymous rules in which the election outcome can be unequivocally determined from the ordinal preferences alone. We focus on experimental results under several probabilistic models extending the statistical cultures commonly used in non-party elections: Impartial Culture, 1-D and 2-D spatial models, single--peaked models, and the Mallows model. In the Appendix, we also analyze a toy model with three parties and single-peaked preference profiles.
\subsection{Related Work}
While spoiler effects have long been a familiar subject in the field of voting theory, there have been relatively few attempts to formally define spoilers or to measure the immunity of electoral systems to spoilers. However, spoiler effects have been tangentially considered in classical social choice theory primarily in the context of stronger postulates such as \emph{independence of irrelevant alternatives} \citep{Arrow50,Ray73,BordesTideman91} or \emph{candidate stability} \citep{DuttaEtAl01,EhlersWeymark03,EraslanMcLennan04,RodriguezAlvarez06}
or otherwise distinct though related postulates such as the \emph{independence of clones} \cite{Tideman87}.
In computation social choice, spoilers have been addressed from the point of view of \emph{electoral control problems} \citep{MeirEtAl08,LiuEtAl09,ChevaleyreEtAl10,FaliszewskiRothe16,NevelingEtAl20,ErdelyiEtAl21}, in particular of the problems of \textsc{Constructive-Control-by-Adding-Candidates} (CCAC) and \textsc{Constructive-Control-by-Deleting-Candidates} (CCDC), and their destructive control counterparts (DCAC and DCDC). In constructive control problems we seek to alter the election outcome by making a specific candidate a unique winner, while in destructive control problems we seek to make that candidate a non-winner. In CAC problems that objective is to be achieved by adding spoiler candidates, while in CDC problems we restrict the set of candidates by deleting some. While closely related to our subject, those problems are distinct primarily because they treat the election outcome in binary terms of being a winner / non-winner which are inapposite in party elections. Moreover, research on control problems considers vulnerability to spoilers in terms of computational complexity of solving the control problems, while we focus on the expected impact of a spoiler.
\citet{Kaminski18} has been the first to propose a generalized definition of a spoiler applicable to party elections. He focused primarily on the problem of defining the election outcome by distinguishing ways in which a potential spoiler can affect seat payoff in a politically significant manner.
Thus, apart from \emph{classical spoilers} (which affect the outcome by turning a majority winner, i.e., the player with a majority of seats, into a majority loser, while making another player a majority winner), he notes the existence of \emph{kingmaker spoilers} (who turn a majority loser in an election with no majority winner into a majority winner), \emph{kingslayer spoilers} (who turn a majority winner into a majority loser, but make no other player a majority winner), and \emph{valuegobbler spoilers} (who affect the seat payoff of one player by an amount greater than their own seat payoff).
Nonetheless, his approach, while representing an important step, has two substantial limitations: first, his enumeration of politically significant ways for a spoiler to affect the election outcome is by no means exhaustive, and second, distinguishing minor and major players, as well as identifying feasible vote redistributions, all require qualitative judgments.
\section{Preliminaries}
Traditional computational social choice models have focused on $n$ voters electing a $k$--member committee out of $m$ candidates. In our models, each candidate is also affiliated with one of $p$ parties, and we focus on the allocation of committee members among those parties. Moreover, there are $c$ electoral districts, with an election held in each district in parallel. The overall election outcome is usually obtained by aggregation of district--level outcomes. We further assume in our experimental models that the number of candidates affiliated with each party equals $k$, i.e., $m = p k$, and that the numbers $n$, $m$, and $k$ are constant across districts.
\subsection{Notation}
\begin{description}[labelwidth=3em, leftmargin=\dimexpr\labelwidth+\labelsep\relax]
\item[{$[t]$}]
For $t \in \mathbb{Z_{+}}$, let $[t]$ denote the set $\{1,\dots,t\}$.
\item[{$[S]^{k}$}]
For a set $S$, let $[S]^{k}$ denote the set of $k$-subsets of $S$.
\item[{$\Delta_{n}$}]
Let $\Delta_{n}$ denote an $n-1$-dimensional unit simplex, i.e., $\Delta_{n} := \{ \mathbf{x} \in \mathbb{R}_{+}^{n} : \sum_{i=1}^{n} x_i = 1 \}$.
\item[{$\langle x_i \rangle_{i=a}^{b}$}]
Let $\langle x_i \rangle_{i=a}^{b}$ denote the average of $x_i$ over $i = a, \dots, b$, i.e., $$\langle x_i \rangle_{i=a}^{b} := \frac{1}{b-a+1} \sum_{i=a}^{b} x_i.$$
\item[{$v_{(k)}$}]
For a strict preference order $v$ on a set $C$, let $v_{(k)}$ denote the $k$-th highest ranked element of $C$ in $v$.
\item[{$x_{k}^{\downarrow}$}]
For $\mathbf{x} \in \mathbb{R}^n$, let $x_{k}^{\downarrow}$ denote the $k$-th largest coordinate of $\mathbf{x}$.
\item[{$\mathbf{u}_{k,l}$}]
Let the $l$-dimensional \emph{$k$--approval vector} $\mathbf{u}_{k,l}$, $l \ge k$, be a vector of $k$ ones followed by $l-k$ zeroes:
$$\mathbf{u}_{k,l} := (\overbrace{1, \dots, 1}^{k}, \overbrace{0, \dots, 0}^{l-k}).$$
\item[{$\mathbf{b}_{k}$}]
Let the $k$-dimensional \emph{Borda vector} $\mathbf{b}_{k}$ be defined as $$\mathbf{b}_{k} := (k-1, k-2, \dots, 0) / ({k-1}).$$
\item[{$\mathbf{h}_{k}$}]
Let the $k$-dimensional \emph{harmonic vector} $\mathbf{h}_{k}$ be defined as $$\mathbf{h}_{k} := \left(1, \frac{1}{2}, \dots, \frac{1}{k}\right).$$
\end{description}
\subsection{Definitions}
\begin{definition}[Party election]
Let $C$ be a finite set of \emph{candidates}, $m := |C|$, and let $L(C)$ be a set of linear orders on $C$. Let $P := \{\rho_1, \dots, \rho_p\}$ be a finite set of \emph{parties}, and let $a: C \rightarrow P$ be a \emph{party affiliation function}. Finally, let the \emph{preference profile} $V$ be an $n$-element sequence of \emph{votes}, $v_1, \dots, v_n \in L(C)$, $n \in \mathbb{N}$. We refer to the quadruple $E := (C, P, a, V)$ as a \emph{party election}.
\end{definition}
\begin{definition}[Multi-winner social choice function]
A $k$-winner \emph{social choice function} is a function $\mathcal{R}$ that maps a party election to a set of \emph{winning} $k$--subsets (\emph{$k$--committees}) of $C$. An election is \emph{tied} under the social choice function $\mathcal{R}$ if $|\mathcal{R}(V)| > 1$.
\end{definition}
\begin{definition}[Allocation rule]
An \emph{allocation rule} is a function that maps a party election into the probability simplex $\Delta_{p}$. Each $k$-winner social choice function $\mathcal{R}$ naturally induces an allocation rule $s_{\mathcal{R}, k}$ such that the $i$-th coordinate of $s_{\mathcal{R}, k}(V)$ equals $\langle |\{y \in S : a(y) = \rho_i\}|/k \rangle_{S \in \mathcal{R}(V)}$, where $i \in [p]$. Note this assumes that ties are resolved by choosing a winning committee randomly with a uniform distribution on $\mathcal{R}(V)$.
\end{definition}
\begin{definition}[OWA--based scoring rule]
An \emph{OWA--based scoring rule}\footnote{OWA stands for \emph{ordered weighting averaging}.} induced by an \emph{OWA vector} $\mathbf{z} \in [0, 1]^{k}$ and a \emph{scoring vector} $\mathbf{w }\in [0, \infty)^{m}$, $\mathcal{R}^{\mathrm{OWA}}_{\mathbf{w}, \mathbf{z}}$, is a social choice function that maps a party election to the set of all $k$--subsets of $C$ maximizing over $S \in C^{[k]}$ the expression $\sum_{v \in V} \sum_{i=1}^k z_i w_{\mathrm{pos}(v, S)_i}$, where $\mathrm{pos}(v, S)$ is a $k$--dimensional vector such that $\mathrm{pos}(v, S)_i$ is the position of the $i$--th highest ranked member of $S$ in $v$.
\end{definition}
\begin{definition}[Multi-district party election]
A $c$-district party election is a sequence $\boldsymbol{E} := \{E_{1}, \dots, E_{c}\}$, where $E_{j} := (C_j, P, a_j, V_j)$, $j \in [c]$.
\end{definition}
\begin{definition}[Multi-district allocation rule]
A $c$-district allocation rule is a function that maps a multi-district party election into the probability simplex $\Delta_{p}$.
\end{definition}
\subsection{Voting and Allocation Rules}
We analyze seven allocation rules. Five of them are induced by OWA--based scoring rules \citep{FaliszewskiEtAl18a,FaliszewskiEtAl18d}:
\smallskip
\begin{definition}[SNTV (plurality)]
Let $\mathcal{R}_{k}^{SNTV} := \mathcal{R}^{\mathrm{OWA}}_{\mathbf{u}_{1,k}, \mathbf{u}_{1,m}}$ be an OWA--based rule induced by the $1$-approval scoring vector $\mathbf{u}_{1,m}$ and the $1$-approval OWA--vector $\mathbf{u}_{1,k}$, i.e., such rule that maps a profile $V$ to a set of $k$--subsets of the set of candidates with scores not lesser than the $k$--th best scoring candidate according to the plurality scoring rule (where only the first position counts).
\end{definition}
\begin{definition}[$k$--Borda]
Let $\mathcal{R}_{k}^{Borda} := \mathcal{R}^{\mathrm{OWA}}_{\mathbf{u}_{k,k}, \mathbf{b}_{m}}$ be an OWA--based rule induced by a Borda scoring vector $\mathbf{b}_{m}$ and a $k$-approval OWA--vector $\mathbf{u}_{k,k}$, i.e., such rule that maps a profile $V$ to a set of $k$--subsets of the set of candidates with scores not lesser than the $k$--th best scoring candidate according to the Borda scoring rule (with uniformly decreasing position scores) \citep{deBorda81,FaliszewskiEtAl17}.
\end{definition}
\begin{definition}[Chamberlin--Courant (CC)]
Let $\mathcal{R}_{k}^{CC} := \mathcal{R}^{\mathrm{OWA}}_{\mathbf{u}_{1,k}, \mathbf{b}_{m}}$ be an OWA--based rule induced by a Borda scoring vector $\mathbf{b}_{m}$ and a $1$--approval OWA--vector $\mathbf{u}_{1,k}$, i.e., such rule that maps a profile $V$ to a set of $k$--subsets of the set of candidates maximizing the sum of Borda scores of the highest--ranking committee member \citep{ChamberlinCourant83}.
\end{definition}
\begin{definition}[Harmonic Borda]
Let $\mathcal{R}_{k}^{HB} := \mathcal{R}^{\mathrm{OWA}}_{\mathbf{h}_{k}, \mathbf{b}_{m}}$ be an OWA--based rule induced by a Borda scoring vector $\mathbf{b}_{m}$ and a harmonic OWA--vector $\mathbf{h}_{k}$, i.e., such rule that maps a profile $V$ to a set of $k$--subsets of the set of candidates maximizing the sum of committee scores defined as the sum of the Borda scores of committee members with harmonically decreasing weights \citep{FaliszewskiEtAl17}.
\end{definition}
\begin{definition}[Proportional $k$--Approval Voting]
Let $\mathcal{R}_{k}^{PAV} := \mathcal{R}^{\mathrm{OWA}}_{\mathbf{h}_{k}, \mathbf{u}_{k,m}}$ be an OWA--based rule induced by an $k$-approval scoring vector, $\mathbf{u}_{k,m}$, and a harmonic OWA--vector $\mathbf{h}_{k}$, i.e., such rule that maps a profile $V$ to a set of $k$--subsets of the set of candidates maximizing the sum of committee scores defined as the $j$-th harmonic number, $H_j$, where $j$ is the number of committee members included within the voter's $k$ highest--ranked candidates \citep{Kilgour10,Thiele95}. Note this rule differs from classical PAV by requiring each voter to approve exactly $k$ candidates.
\end{definition}
\noindent We also consider two other allocation rules that are frequently encountered in real--life political elections:
\smallskip
\begin{definition}[STV]
Let $\mathcal{R}_{k}^{STV}$ be a social choice function defined as follows: let $q := {n}/{(k+1)}$ be the \emph{fractional Droop quota}, let initial vote weights be $1$, and let the committee $S$ and the set of discarded candidates $D$ be initially defined as empty sets. We iteratively rank the candidates according to the weighted number of votes in which they have been ranked highest of all candidates $c \in C \setminus (S \cup D)$, denoted for the $i$-th candidate as $t_i$. Let $Q := \{c \in C \setminus (S \cup D): t_c \ge q\}$ be the set of the candidates for whom the number of such votes exceeds the quota. If $Q$ is non-empty, we add those candidates to the committee and multiply the weight of all votes cast for each candidate $c \in Q$ by $q / t_c$. If $Q$ is empty, we add the last--ranked candidate to $D$. The process is repeated until $|S| \ge k$, and the result is the set of all $k$--subsets of $S$ \citep{Hare59,TidemanRichardson00}.
\end{definition}
\begin{definition}[Jefferson--D'Hondt]
Let $r_{k}^{JDH}$ be an allocation rule defined as follows: let $w_i$, $i \in [p]$, be the share of votes such that the $i$-th party's candidate is ranked first, and let $\mathcal{M}$ be such interval that for each $M \in \mathcal{M}$, $\sum_{i \in [p]} \lfloor M w_i\rfloor = k$. Then the $i$-th coordinate of $r_{k}^{JDH}$ is given by $\lfloor M w_i\rfloor / k$, where $M \in \mathcal{M}$ (the rule is independent of the choice of $M$) \citep{Jefferson92,DHondt82,DHondt85,BalinskiYoung78,Pukelsheim17}.
\end{definition}
\begin{remark}
If $\mathcal{M}$ is empty, an electoral tie occurs. For consistency with tie resolution in other voting rules, we can extend our definition of $r_{k}^{JDH}$ as follows: let $k_{+} \in \{j \in \mathbb{N}: j > k\}$ be the smallest such integer that there exists a non-empty interval $\mathcal{M}_{+}$ such that for each $M \in \mathcal{M}_{+}$, $\sum_{i \in P} \,\lfloor M w_i\rfloor = k_{+}$, and let $k_{-} \in \{j \in \mathbb{N}: j < k\}$ be the greatest such integer that there exists a non-empty interval $\mathcal{M}_{-}$ such that for each $M \in \mathcal{M}_{-}$, $\sum_{i \in P} \lfloor M w_i\rfloor = m_{-}$. Then the $i$-th coordinate of $r_{k}^{JDH}$ is given by:
$$
\frac{k - k_{-}}{k_{+} - k_{-}} \frac{\lfloor M_{+} w_i\rfloor}{k_{+}} +
\frac{k_{+} - k}{k_{+} - k_{-}} \frac{\lfloor M_{-} w_i\rfloor}{k_{-}},
$$ where $M_{+} \in \mathcal{M}_{+}$ and $M_{-} \in \mathcal{M}_{-}$ (the rule is independent of the choice of $M_{+}$ and $M_{-}$).
\end{remark}
\begin{remark}
To simplify theoretical calculations, we can approximate the $i$--th coordinate of the multi--district Jefferson--D'Hondt allocation rule using the following formula \citep{FlisEtAl20}: Let parties be ordered degressively by $w_{i}$. Then
\begin{equation} \label{eq:potLadle}
s_{i}\approx \widehat{w}_{i}\left( 1+\frac{r}{2k}\right) -\frac{1}{2k}
\end{equation}
for $i = 1, \dots, r$, where,
\begin{equation}
r:=\max \left\{ l \in [p]:\frac{w_{l}}{\sum_{j=1}^{l}w_{j}}>\frac{1}{2k+l}%
\right\} ,
\end{equation}
$\widehat{w}_{i}$ is the renormalized vote share of the $i$-th party:%
\begin{equation}
\widehat{w}_{i}:= {w_{i}} / \left({\sum{}_{j=1}^{r}w_{j}}\right).
\end{equation}
\end{remark}
\bigskip
Any family of allocation rules $r := (r_i, \dots, r_c)$ induces a multi-district allocation rule $s$ given by the weighted average $s(\boldsymbol{E}) := (\sum_{i=1}^{c} k_i r_i(V_i)) / (\sum_{i=1}^{c} k_i)$, where $k_i \in \mathbb{N}_{+}$ is the number of seats allocated in the $i$-th district.
\section{Measuring Spoiler Susceptibility}
Intuitively, a \emph{restriction of an election} to a subset $P'$ is one in which candidate sets are restricted to candidates affiliated with parties in $P'$, and others are eliminated from all votes.
\begin{definition}[Restriction of an election]
Formally, let $\boldsymbol{E} := \{E_{1}, \dots, E_{c}\}$ be a $c$--district party election. We define a \emph{restriction of $\boldsymbol{E}$ to $P'$}, $\boldsymbol{E}_{P'} := \{E^{\sim}_{1}, \dots, E^{\sim}_{c}\}$, with $P' \subseteq P$, as a $c$--district party election such that $E^{\sim}_{j} := \{C'_j, P', a'_j, V'_j\}$, $j \in [c]$, where:
\begin{enumerate}
\item $C'_j$ is the preimage of $P'$ under $a_j$, i.e., the set of candidates restricted to those affiliated with parties in $P'$,
\item $a'_j := a_j\restriction_{C'_j}$ is the restriction of $a$ to $C'_j$,
\item $V'_j := (v_{j,1} \restriction_{C'_j}, \dots, v_{j,n} \restriction_{C'_j})$.
\end{enumerate}
\noindent We will denote $\boldsymbol{E}_{P \setminus \{i\}}$ by $\boldsymbol{E}_{-i}$.
\end{definition}
\smallskip
\begin{definition}[Redistribution region]
Let $\boldsymbol{E}$ be a $c$--district party election. A \emph{redistribution region} of the $i$-th party, $R_i \subset \Delta_{p}$, is a set of possible redistributions of the $i$--th party's share in the election outcome $\mathbf{s}$:
\begin{equation}
R_i := \left\{ x \in \Delta_p : {\lVert\mathbf{x} - \mathbf{s}\rVert}_1 \le s_i \right\}.
\end{equation}
\noindent Note that $R_i$ is a convex hull of $\{\mathbf{s} + s_i (\mathbf{e}_{j} - \mathbf{e}_{i}) : j \neq i\}$, where vectors $\mathbf{e}_1, \dots, \mathbf{e}_p$ are vertices of $\Delta_{p}$. Thus, $R_i$ is a regular $(p-2)$--dimensional simplex embedded in the facet $\{\mathbf{x} \in \Delta_{p} : x_i = 0 \}$.
\end{definition}
\smallskip
\begin{figure}[t]
\resizebox{0.75\columnwidth}{!}{
\begin{tikzpicture}[scale=5]
\draw (0,0) -- (0.5,0.866025) -- (1,0) -- cycle;
\filldraw[black] (0.361111,0.336788) circle (0.2pt) node[anchor=north]{$s(\boldsymbol{E})$};
\filldraw[black] (0.805556,0.336788) circle (0.2pt) node[anchor=west]{$s(\boldsymbol{E}) + s(\boldsymbol{E})_i (\mathbf{e}_1 - \mathbf{e}_i)$};
\filldraw[black] (0.583333,0.721688) circle (0.2pt) node[anchor=west]{$s(\boldsymbol{E}) + s(\boldsymbol{E})_i (\mathbf{e}_2 - \mathbf{e}_i)$};
\filldraw[black] (0.875,0.216506) circle (0.2pt) node[anchor=west]{$s(\boldsymbol{E}_{-i})$};
\filldraw[black] (0,0) circle (0pt) node[anchor=north east]{$\mathbf{e}_i$};
\filldraw[black] (1,0) circle (0pt) node[anchor=north west]{$\mathbf{e}_1$};
\filldraw[black] (0.5,0.866025) circle (0pt) node[anchor=south]{$\mathbf{e}_2$};
\draw[gray,ultra thick] (0.805556,0.336788) -- (0.583333,0.721688);
\draw[black] (0.694444,0.529238) node[anchor=west]{$R_i$};
\draw[very thick] (0.805556,0.336788) -- (0.875,0.216506);
\draw[black] (0.847222, 0.254619) node[anchor=east]{$\lambda_i$};
\draw[thin,dashed] (0.361111,0.336788) -- (0.805556,0.336788);
\draw[thin,dashed] (0.361111,0.336788) -- (0.583333,0.721688);
\end{tikzpicture}
}
\vspace*{-0.5cm}
\caption{Geometric interpretation of excess electoral impact.}
\label{fig:elecImpact}
\end{figure}
\begin{definition}[Excess electoral impact]
Let $\boldsymbol{E}$ be a $c$--district party election. We define the \emph{excess electoral impact} of the $i$-th party as the $L_1$ distance between the outcome of the restriction $\boldsymbol{E}_{-i}$ and the redistribution region of $R_i$, see Fig. \ref{fig:elecImpact}:
\begin{align}
\lambda _{i}^s (\boldsymbol{E}) &:= \min_{\mathbf{x} \in R_i} {\left\lVert\mathbf{x} - s(\boldsymbol{E}_{-i}) \right\rVert}_1 \\&= {\left\lVert s (\boldsymbol{E}) - s(\boldsymbol{E}_{-i}) \right\rVert}_1 - 2 s (\boldsymbol{E})_{i}.
\end{align}
\end{definition}
Let us consider the set of parties that gain shares in the election outcome when party $i$ is eliminated, i.e., $L_i := \{j \in [p]: s(\boldsymbol{E}_{-i})_{j} > s(\boldsymbol{E})_{j}\}$. Party $i$ has a non-zero excess electoral impact if and only if the total gains of parties in $L_i$ exceed $s(\boldsymbol{E})_{i}$. In such a case, we describe $i$ as a \emph{spoiler} and parties $j \in L_i$ as \emph{spoilees}. Note that the existence of spoilees implies that there also exist parties which lose shares in the election outcome when $i$ is eliminated.
Finally, we define spoiler susceptibility of a multi-district allocation rule as the expected maximum excess electoral impact, where the maximum is taken for each election over the set of parties, and the expectation is over some probability distribution on the set of profiles.
\begin{definition}[Spoiler susceptibility]
\label{def:spoilSuscept}
Let $s$ be a multi-district allocation rule, and let $\mathcal{D}$ be a probability distribution on the set of elections with fixed $p$ and $c$. Finally, let $\boldsymbol{E} \sim \mathcal{D}$. Then the \emph{spoiler susceptibility} of $s$ is given by:
\begin{equation}
\Phi(\mathcal{D}, s) := \mathbb{E} \, \max \limits_{i \in P} \lambda_i^s (\boldsymbol{E}).
\end{equation}
\end{definition}
To illustrate the definition, we present a real-life example.
\section{Real-life Example (Polish election of 2015)}
In the Polish general election of 2015, the Jefferson--D'Hondt rule has been used to allocate a total of 460 seats in 41 districts with the district magnitude parameter $m$ varying between $7$ and $20$. In addition, a statutory threshold has been set at $5\%$ of the total number of valid votes for parties and $8\%$ of the total number of valid votes for electoral coalitions. Only parties and coalitions whose vote shares exceeded the appropriate threshold participated in the seat allocation. There have been eight major contenders (see Table \ref{tbl:plParties}).
Using party position data from the Chapel Hill Expert Survey \citep{BakkerEtAl15, PolkEtAl17, BakkerEtAl21} we have mapped each party to a point $(x_i, y_i)$. We assume that if the $i$-th party were removed, its votes would have been redistributed among other parties, $j \neq i$, inversely proportionally to the Euclidean distances between points $(x_j, y_j)$ and $(x_i, y_i)$. Under this assumption, we have computed for each party $i = 1, \dots, 8$ the excess electoral impact $\lambda_i^s$ under the Jefferson--D'Hondt rule.
\begin{table}[ht]
\newcolumntype{Z}{>{\centering\arraybackslash}X}
\begin{tabularx}{\columnwidth}{lZZZZZ} \toprule
\emph{name} & $w_i$ & seats & $x_i$ & $y_i$ & $\lambda^s_i$ \\ \midrule
PiS & $.376$ & $235$ & $.295$ & $.914$ & $.000$ \\
PO & $.241$ & $138$ & $.657$ & $.467$ & $.061$ \\
Kukiz & $.088$ & $42$ & $.675$ & $.761$ & $.097$ \\
Nowoczesna & $.076$ & 28 & $.767$ & $.261$ & $.114$ \\
Lewica (*) & $.076$ & 0 & $.391$ & $.357$ & $.124$ \\
PSL & $.051$ & 16 & $.433$ & $.710$ & $.131$ \\
Korwin & $.048$ & 0 & $.906$ & $.953$ & $.144$ \\
Razem & $.036$ & 0 & $.129$ & $.122$ & $.184$ \\
\hline
\end{tabularx}
\caption{Parties in the Polish general election of 2015, where $w_i$ is the vote share. Asterisk (*) denotes electoral coalitions.}
\label{tbl:plParties}
\end{table}
We find that the left-wing party Razem has attained the highest excess electoral impact in our example. This is in accord with a common intuition that Razem has indeed acted as a spoiler, preventing its nearest neighbor Lewica from crossing the $8\%$ electoral threshold for coalitions and thus enabling the winning PiS party to attain a standalone majority. Lewica has been the main spoilee for Razem in this example, followed by Korwin and Nowoczesna.
\section{Toy Models}
Let us consider a following class of toy models: there are three parties, $P = \{X, Y, Z\}$, with $k$ candidates each, ordered as $X_1 \ll \dots \ll X_k \ll Y_1 \ll \dots \ll Y_k \ll Z_1 \ll \dots \ll Z_k$. We assume that the preference profile is single-peaked with respect to $\ll$, and that the candidates of a single party are grouped together within each vote. Thus, there are $4\cdot2^{k-1}$ possible votes -- four admissible orderings on parties and, for each of them, $2^{k-1}$ admissible orderings on the candidates of the party that is ranked first (``peak party''). For other parties, candidate orders are determined by the choice of the peak party. We assign probabilities to those orders by first choosing the probabilities of the four party orderings, $\mathrm{Pr}_{X \prec Y \prec Z}$, $\mathrm{Pr}_{Z \prec Y \prec X}$, $\mathrm{Pr}_{X \prec Z \prec Y}$, and $\mathrm{Pr}_{Z \prec X \prec Y}$, and then fix the probabilities for each candidate ordering.
If the middle party were the largest one, or if all parties were of approximately equal size, the probability of spoiler effects would be relatively small, making the model uninteresting. Thus, we assume that one of the extreme parties -- $Z$, without loss of generality -- is the large party, i.e., is ranked first by the largest number of voters. We also assume that the probabilities of all orderings are unequal. Those two assumptions leave us with six configurations of parties, each of which is defined by an ordering on the coordinates of the vector $\mathbf{Q} := (\mathrm{Pr}_{X \prec Y \prec Z}, \mathrm{Pr}_{Z \prec Y \prec X}, \mathrm{Pr}_{X \prec Z \prec Y}, \mathrm{Pr}_{Z \prec X \prec Y})$ such that $\mathrm{Pr}_{X \prec Y \prec Z}$ is always the largest one.
An ordering of four probabilities corresponds to the division of a unit simplex $\Delta_4$ into $24$ asymmetric simplices. However, since $\mathrm{Pr}_{X \prec Y \prec Z}$ is always the largest of those four probabilities, only six of the asymmetric simplices will be of interest to us. For each configuration, we draw the vector of probabilities $\mathbf{Q}$ from the uniform distribution on the corresponding asymmetric simplex, thus ensuring the consistency of the ordering of the probabilities with the chosen configuration.
Ordering of the peak party's candidates is fixed independently of the choice of the said party. We choose a fixed vector of probabilities $\mathbf{u}_k := (u_1^k, \dots, u_{2^{k-1}}^k)$, where
$$u_i^k := \frac{1}{2^{k-1}} \sum_{j=i}^{2^{k-1}} \frac{1}{j},$$
and $i = 1, \dots, 2^{k-1}.$
For any OWA--based rule, we can express the expected score for each $k$--committee $S$ as a linear combination of the probabilities assigned to each possible vote, and thus easily determine, for each of the enumerated configurations, seat allocations under the $k$--Borda, Chamberlin-Courant, Harmonic Borda, PAV, and SNTV rules for random single-district party elections $E_{\mathbf{Q}} := (\{X,Y,Z\} \times [k], \{X,Y,Z\}, a, V)$, where $a(x) = x_1$ (see the attached Mathematica notebook for source code). The random preference profile $V$ is drawn according to the distribution of votes determined by vector $\mathbf{Q}$ as described above. We likewise determine, for each such profile, seat allocations for $E_{\mathbf{Q}}$ to $[\{X,Y,Z\}]^2 := \{\{X, Y\}, \{X, Z\}, \{Y, Z\}\}$. Finally, using Definition 4 we obtain an expected maximum excess electoral impact for each rule under consideration.
\begin{table}[htb]
\begin{tabular}{cccccc} \toprule
\multicolumn{6}{c}{$\mathrm{Pr}_{X \prec Y \prec Z} > \mathrm{Pr}_{Z \prec Y \prec X} > \mathrm{Pr}_{X \prec Z \prec Y} > \mathrm{Pr}_{Z \prec X \prec Y}$} \\ \midrule
\multirow{2}{*}{$k=2$} & PAV & Borda & HB & SNTV & CC \\
& .220 & .166 & .072 & .066 & .003 \\ \hline
\multirow{2}{*}{$k=3$} & Borda & PAV & HB & SNTV & CC \\
& .176 & .127 & .043 & .027 & .000 \\ \bottomrule
\\ \toprule
\multicolumn{6}{c}{$\mathrm{Pr}_{X \prec Y \prec Z} > \mathrm{Pr}_{Z \prec Y \prec X} > \mathrm{Pr}_{Z \prec X \prec Y} > \mathrm{Pr}_{X \prec Z \prec Y}$} \\ \midrule
\multirow{2}{*}{$k=2$} & Borda & HB & PAV & SNTV & CC \\
& .141 & .085 & .040 & .013 & .002 \\ \hline
\multirow{2}{*}{$k=3$} & Borda & HB & PAV & SNTV & CC \\
& .138 & .072 & .040 & .001 & .000 \\ \bottomrule
\\ \toprule
\multicolumn{6}{c}{$\mathrm{Pr}_{X \prec Y \prec Z} > \mathrm{Pr}_{X \prec Z \prec Y} > \mathrm{Pr}_{Z \prec Y \prec X} > \mathrm{Pr}_{Z \prec X \prec Y}$} \\ \midrule
\multirow{2}{*}{$k=2$} & PAV & Borda & SNTV & HB & CC \\
& .221 & .079 & .063 & .030 & .005 \\ \hline
\multirow{2}{*}{$k=3$} & PAV & Borda & HB & SNTV & CC \\
& .127 & .085 & .023 & .005 & .000 \\ \bottomrule
\\ \toprule
\multicolumn{6}{c}{$\mathrm{Pr}_{X \prec Y \prec Z} > \mathrm{Pr}_{X \prec Z \prec Y} > \mathrm{Pr}_{Z \prec X \prec Y} > \mathrm{Pr}_{Z \prec Y \prec X}$} \\ \midrule
\multirow{2}{*}{$k=2$} & PAV & Borda & SNTV & HB & CC \\
& .087 & .065 & .040 & .032 & .001 \\ \hline
\multirow{2}{*}{$k=3$} & PAV & Borda & HB & SNTV & CC \\
& .130 & .071 & .026 & .003 & .000 \\ \bottomrule
\\ \toprule
\multicolumn{6}{c}{$\mathrm{Pr}_{X \prec Y \prec Z} > \mathrm{Pr}_{Z \prec X \prec Y} > \mathrm{Pr}_{Z \prec Y \prec X} > \mathrm{Pr}_{X \prec Z \prec Y}$} \\ \midrule
\multirow{2}{*}{$k=2$} & Borda & HB & PAV & SNTV & CC \\
& .252 & .153 & .087 & .042 & .003 \\ \hline
\multirow{2}{*}{$k=3$} & Borda & PAV & HB & SNTV & CC \\
& .254 & .128 & .118 & .004 & .000 \\ \bottomrule
\\ \toprule
\multicolumn{6}{c}{$\mathrm{Pr}_{X \prec Y \prec Z} > \mathrm{Pr}_{Z \prec X \prec Y} > \mathrm{Pr}_{X \prec Z \prec Y} > \mathrm{Pr}_{Z \prec Y \prec X}$} \\ \midrule
\multirow{2}{*}{$k=2$} & Borda & HB & PAV & SNTV & CC \\
& .250 & .166 & .040 & .014 & .003 \\ \hline
\multirow{2}{*}{$k=3$} & Borda & HB & PAV & SNTV & CC \\
& .244 & .137 & .040 & .002 & .000 \\ \bottomrule
\end{tabular}
\caption{Voting rules ordered from the most to the least susceptible to spoiler effects for six toy models. Values below the name of the voting rule are the spoiler susceptibility indices of such voting rule.}
\end{table}
The performance of the five social choice rules under consideration with respect to their susceptibility to spoiler effects depends strongly on the choice of the model, but we can note a number of regularities: Chamberlin--Courant is always most resistant to spoilers, followed usually by SNTV. Harmonic Borda is always more resistant to spoilers than $k$--Borda. Finally, $k$--Borda is the most spoiler-susceptible method, except where $\mathrm{Pr}_{X \prec Z \prec Y}$ is the second largest probability, in which case $k$--PAV takes its place.
\section{Experiments}
In this section we provide experimental results regarding spoilerity-resistance of different voting rules in practice. We start by describing probabilistic models that we use for generating election. Then, we present the results.
\subsection{Probabilistic Models}
The probabilistic models commonly used in computational social choice (see generally \citep{BergLepelley94,SzufaEtAl20}) have been developed for single-district non-party elections. In order to measure spoiler susceptibility of social choice rules used in multi-district party elections, we need to introduce appropriate extensions of those rules, addressing two basic challenges:
\begin{enumerate}
\item grouping of candidates into parties, and
\item existence of multiple electoral districts.
\end{enumerate}
\emph{Grouping of candidates into parties} is based on a latent assumption of intra-party candidate clustering: candidates of the same party, while not identical, are assumed to be perceived by voters as, on average, more similar than candidates of different parties. Accordingly, we would assume them to be usually (though not necessarily) clustered together in votes. Without that assumption, parties would tend to obtain roughly equal seat shares, leading to an overall electoral tie.
\emph{Existence of multiple electoral districts} reflects another latent assumption -- one of intra-district voter clustering. Voters within a single district are assumed not to constitute an unbiased sample of the population, but to share preferences to a greater extent. Without that assumption, the $c$-district allocation rule would (per the central limit theorem) converge (as $O(1/\sqrt{c})$) to the expected value of the allocation rule applied to the preference profile of the full population.
We consider four classes of probabilistic models:
\smallskip
\begin{definition}[Spatial Models]
In a $d$-dimensional Euclidean model each party, voter, and candidate is assigned an ideal point in $\mathbb{R}^{d}$ \citep{EnelowHinich84,Merrill84}. First, party ideal points are drawn from the uniform distribution on $(0, 1)^{d}$, and then candidate ideal points for each district are drawn from the multivariate normal distribution with location at the party's ideal point and the correlation matrix $\mathbf{\Sigma} := \sigma I_{d}$, where $I_{d}$ is a $d \times d$ identity matrix and $\sigma \in \mathbb{R}_{+}$. Voter ideal points are drawn independently from the uniform distribution on $(0, 1)^{d}$, and then shifted in each district independently by a shift vector drawn from a uniform distribution on $(-1/4, 1/4)^{d}$ (to account for intra-district voter clustering). Finally, each vote is obtained by sorting candidates according to the increasing Euclidean ($L_2$) distance between the candidate's and the voter's ideal points.
\end{definition}
\begin{definition}[Single-Peaked Models]
We consider two models for the generation of single-peaked profiles. In the model proposed by \citet{wal:t:generate-sp} we are given an ordering on the set of candidates, and each vote is drawn from the uniform distribution on the set of all single-peaked votes consistent with that ordering. In the model proposed by \citet{con:j:eliciting-singlepeaked} the peak is drawn from a uniform distribution on candidates, and the remainder of the vote is obtained through a random walk. In both models, candidate ordering in each district is obtained in the same manner as in the $1$-dimensional Euclidean model described above, and there is no mechanism accounting for intra-district voter clustering.
\end{definition}
\begin{definition}[Mallows Model]
The Mallows model \citep{Mallows57,CritchlowEtAl91} is parametrized by a single parameter $\phi \in [0,1]$, and a (central) vote $v_c \in L(C)$. The probability of generating a vote $v$ is proportional to $\phi^{f(v_c, v)}$, where $f(v_c, v)$ is the Kendall tau distance \citep{Kendall38} between $v_c$ and $v$, i.e., the minimum number of swaps of adjacent candidates needed to transform the vote $v$ into the central vote $v_c$. We apply the Mallows model in two stages, first generating a central vote for each district, $v_c^i$ with parameter $\phi_{1}$ and an overall central vote $v_c^0 := [n]$, then generating votes within each district with parameter $\phi_{2}$ and a district--wide central vote $v_c^i$. Intra-party clustering is achieved by grouping party candidates together in the overall central vote. On sampling from the Mallows model, see \citep{LuBoutilier14}.
For our simulations, we use a novel Mallows model parameterization proposed by \citet{boe-bre-fal-nie-szu:t:compass}, which instead of classical $\phi$ uses a normalized dispersion parameter ${{\mathrm{norm}\hbox{-}\phi}}$.
\end{definition}
\begin{definition}[Impartial Culture (IC)]
Under the Impartial Culture model, each vote is drawn randomly from the uniform distribution on $L(C)$, the set of linear orders on the set of candidates $C$, and there is no dependence between votes within or across districts \citep{CampbellTullock65}. This model is unable to account for intra-party and intra-district clustering, and is generally considered a poor approximation of real-life elections \citep{RegenwetterEtAl06,TidemanPlassmann12}, thus being only tested as a reference point.
\end{definition}
\subsection{Results}
To compare voting methods, we analyze their performance under 11 probabilistic models: four spatial models ($d = 1, 2$ and $\sigma = 0.05, 0.2$), four single-peaked models (Walsh and Conitzer, $\sigma = 0.05, 0.2$), two Mallows models ($\phi_1 = 0.75$, $\phi_2 = 0.25, 0.75$) and Impartial Culture. For every voting method, model, committee size $k \in \{1, 5, 15\}$, and number of parties $p \in \{3,\dots,10\}$ we run 600 simulations, with the number of districts $c = 100$, and the number of voters per district $n = 100$. For Jefferson--D'Hondt we use the Pot \& Ladle approximation (see Appendix, Eq.~1), for SNTV, $k$-Borda, and STV we use the optimal algorithms, while for CC, HB, and $k$-PAV we use greedy approximation algorithms.
\begin{figure*}[p]
{\centering
\includegraphics[height=\textheight,width=\textwidth]{plots1.pdf}
\caption{Spoiler susceptibility of electoral rules.}
\label{fig:results}
}
\end{figure*}
Experimental results plotted on Figure \ref{fig:results} demonstrate several regularities. First, spoiler susceptibility of social choice rules depends strongly on the choice of the probabilistic model. Under spatial models, which are most likely to approximate political elections, we can distinguish several classes of rules. Those least susceptible to spoilers are STV, Chamberlin--Courant (only in multi-member district models), and Harmonic Borda (only in multi-member district models with $d > 1$). The ordering on them depends on the number of parties -- CC outperforms STV for large values of $p$. SNTV and Jefferson-D'Hondt are the middle performers, with JDH becoming the more resistant rule as the number of parties increases. On the other hand, $k$--Borda performs poorly against spoilers in models with high degree of party clustering ($\sigma = 0.05$), but is more spoiler--resistant than JDH and SNTV in models with greater degree of candidate dispersion. In single-member districts, Borda rule is the most spoiler-susceptible one for three-party models, but for large values of $p$, it outperforms even STV. Finally, $k$--PAV in multi-member districts and FPTP in single-member districts are almost consistently the worst performers. For virtually all rules and models, switching from single--member to multi--member districts substantially improves spoiler resistance.
The Conitzer model is generally most susceptible to spoilers, especially when single-member districts are employed. Again, STV and Chamberlin--Courant are the best performers in multi-member districts, followed by Jefferson--D'Hondt, $k$--Borda, and Harmonic Borda. SNTV performs very poorly in 5-member districts, and much better -- but still worse than all but one of the alternatives -- in 15-member districts. Finally, $k$--PAV is again most susceptible to spoilers. In single-member districts STV and Borda perform similarly, except that for small values of the number of parties $p$ Borda's spoiler--susceptibility depends on the parity of $p$, while FPTP exhibits very strong susceptibility to spoilers, especially for large values of $p$.
The Mallows model exhibits high resistance to spoiler effects, especially for multi-member districts. In single-member districts, STV is the best performer, followed by Borda, and finally by FPTP. In multi-member districts (where spoiler susceptibility is altogether very small), Jefferson--D'Hondt and Harmonic Borda are the best performers, $k$--Borda is usually the worst, while the spoiler susceptibility of the remaining four rules depends on the number of seats per district: $k$--PAV and STV are highly spoiler--resistant for $k = 15$, but quite susceptible for $k = 5$, while the reverse is true for Chamberlin--Courant and SNTV (which even tie as the worst--performing rules for $k = 15$ and $\phi = 0.75$). Finally, spoiler susceptibility under Mallows increases (convexly) with $p$, and decreases with $\phi$ (for $k = 15$ and $\phi = 0.75$, we have observed no spoilers under the four best-performing rules).
We treat Walsh and Impartial Culture models as essentially reference points, since they are considered unlikely to correspond to any real-life party elections. Nevertheless, we note that for both, spoiler susceptibility is a significant issue only in single-member district cases. Under Walsh, FPTP is more susceptible to spoilers than STV, while the spoiler-susceptibility of Borda depends on the parity of the number of parties $p$. Under IC, Borda is the best performer, followed by STV and FPTP. In multi-member districts under the Walsh model, Harmonic Borda and STV are consistently spoiler--resistant, Chamberlin--Courant, SNTV and Jefferson--D'Hondt are generally spoiler-susceptible, and Borda performs well for dispersed parties and badly for clustered parties. Under IC, SNTV and CC are most spoiler--susceptible, STV, $k$--PAV, $k$--Borda, and Harmonic Borda -- only for $k = 5$, and JDH is fully spoiler--resistant.
\section{Power Indices}
In many classes of political elections, such as elections to representative bodies like parliaments, seat allocation fails to capture a number of important features of election outcome. Most importantly, it does not account for the fact that parties in a representative body do not share power in proportion to their respective seat shares. Instead, power is primarily wielded by the majority coalition of parties, and within such a coalition -- by the pivotal parties, i.e., those without which the coalition would no longer command the requisite majority. This concept has been independently formalized in the theory of \emph{weighted voting games} leading to the development of a number of \emph{voting power indices} \cite{Penrose46,Banzhaf64,ShapleyShubik54}, expressing the probability that a player is \emph{decisive} in the formation of a winning coalition.
\begin{definition}[Weighted voting game]
Let $\mathbf{s}\in \Delta _{p}$ be the vector of voting weights (in our case, usually a value of some multi--district allocation rule), and let $q\in (0.5, 1] $ be the \emph{qualified majority quota}. A coalition $K\in \mathcal{P}(P) $ is winning if and only if $\sum_{i\in K} s_{i}\geq q$.
\end{definition}
\begin{definition}[Penrose--Banzhaf index]
Let us denote the set of all winning coalitions by $\mathcal{W}$. The $i$-th player, $i \in [p]$, is \emph{pivotal for coalition $K$} if and only if $i\in K$ and $K\setminus \left\{ i\right\} \notin \mathcal{W}$ or $i\notin K$ and $K\cup \{i\} \in \mathcal{W}$. The \emph{absolute Penrose--Banzhaf power index} of the $i$-th player, $\psi_{i}^q$, is the probability that the $i$-th player is pivotal assuming that each coalition $K\in \mathcal{P}(P) $ is equiprobable. The \emph{normalized Penrose--Banzhaf power index} \citep{DubeyShapley79} of the $i$-th player, $\beta_{i}^q$, is obtained by normalizing $\psi_{i}^q$:%
\begin{equation}
\beta_{i}^q:=\frac{\psi_{i}^q}{\sum_{j=1}^{p}\psi_{j}^q}.
\end{equation}
\end{definition}
\begin{definition}[Shapley--Shubik index]
The $i$-th player, $i \in [p]$, is \emph{pivotal for permutation $\sigma$} if and only if $\sum_{j=1}^{\sigma(i)-1} s_{\sigma^{-1}(j)}<q$ and $\sum_{j=1}^{\sigma(i)} s_{\sigma^{-1}(j)}\geq q$. The \emph{Shapley--Shubik power index} of the $i$-th player, $S_{i}^q$, is the probability that the $i$-th player is pivotal assuming that each permutation $\sigma$ is equiprobable. Note that $( S_{1}^q,\dots ,S_{p}^q) \in \Delta _{p}$, making any further normalization unnecessary.
\end{definition}
We can easily extend our definition of excess electoral impact by treating power indices normalized to map onto the probability simplex $\Delta_p$ as a special class of allocation rules.
From those definitions, for each multi-district allocation rule $s$, we easily obtain two additional multi-district allocation rules: $\beta_{s}$, which maps a vector of profiles to a vector of normalized Penrose--Banzhaf power indices, $\beta(\boldsymbol{E})_i := \lim_{q\rightarrow 1/2} \beta_{i}^{q},$ and $S_{s}$, which maps a vector of seat shares to a vector of Shapley--Shubik power indices, $S(\boldsymbol{E})_i := \lim_{q\rightarrow 1/2} S_{i}^{q}$, with weight vectors in both cases being equal to $s(\boldsymbol{E})$.
\begin{remark}
Power indices are known to be difficult to compute. It has been established by \citet{PrasadKelly90}, and more generally by \citet{MatsuiMatsui01}, that the problem of computing either of the two standard power indices is NP-complete in a generic case. However, in typical political elections the values of $p$ are small enough that the direct enumeration of all winning coalitions, although running in exponential time, is feasible.
\end{remark}
Preliminary results of experiments for the Penrose--Banzhaf index are plotted on Figure \ref{fig:results2}. While interpretation of the results is more difficult due to number-theoretic anomalies arising from the small number of parties, certain regularities emerge. The ordering of the social choice rules does not differ significantly from that obtained for seat shares. However, it is readily apparent that power indices are substantially more susceptible to spoiler effects than seat shares.
\begin{figure*}[p]
\centering\includegraphics[height=\textheight,width=\textwidth]{plots2.pdf}
\caption{Spoiler susceptibility of electoral rules with respect to the normalized Penrose--Banzhaf power index.}
\label{fig:results2}
\end{figure*}
\section{Summary}
We have introduced a novel approach to defining spoilers in multi-district party elections, alongside with modified probabilistic models that serve for randomly generating such elections. We compare various voting methods via simulations. The results depend strongly on a given distribution of preferences, although there are some clear patterns. An electoral system designer acting behind a veil of ignorance who is interested in maximizing spoiler resistance would do well to choose STV, CC, or Harmonic Borda, since they perform best under spatial and Conitzer models (which are most susceptible to spoilers), and under other models differences are negligible. On the other hand, for the same reason he should avoid $k$--PAV and FPTP (which is equivalent to $1$--PAV). The performance of $k$--Borda depends strongly on the degree of party clustering, while of that of Jefferson--D'Hondt -- on the single peakedness of the model. Finally, single-member districts are nearly always more susceptible to spoilers than multi-member ones.
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1,116,691,500,793 | arxiv | \section{Introduction}
\vspace{-0.1cm}
At the end of the asymptotic giant branch (AGB) intermediate mass stars go through an intense mass-loss phase when 50-90\% of their mass is expelled and expands in a nebula surrounding the core of the star. After the loss of the envelope, the AGB star contracts and heats up, ionising the nebula ejected during the AGB phase, which shines as a planetary nebula (PN). The reason for the dramatic increase in mass-loss during the late AGB phase is not well known. AGB stars appear to lose mass spherically, but the PNe that result from the mass-loss primarily have non-spherical morphologies. It is likely that the mechanism that causes the heavy AGB mass loss is related to the mechanism that dictates the ejecta's departure from sphericity. Binarity may be an effective way to break spherical symmetry and to stimulate mass-loss.
Some fundamental and rapid change in structure must occur between the spherical AGB mass-loss phase and the non-spherical, collimated PN phase (\cite[Iben 1995, Balick and Frank 2002, De Marco 2009]{1995PhR...250....2I, 2002ARA&A..40..439B}).
Disks around evolved stars have been observed and are suspected to play a fundamental role in the shapes of PNe (e.g. Huggins 2007).
Theoretical work envisions their role as the collimating agent for non-spherical mass-loss (e.g. Soker \& Livio 1994, Blackman et al. 2001).
By studying the characteristics of these disks we can understand the engine that is forming the PNe. Only with the Very Large Telescope Interferometer (VLTI) can we reach high enough spatial resolution to measure the parameters of the inner circumstellar environment of post-AGB stars, in transition between the AGB and the PN phase. These parameters will meaningfully constrain single-star as well as binary models of AGB mass-loss and nebulae collimation.
\vspace{-0.1cm}
\section{Previous Observations}
\vspace{-0.1cm}
\subsection{Compact disks around planetary nebulae central stars}
\vspace{-0.1cm}
Some PNe are found to harbour compact disks. For example, disks have been observed with the VLTI in two young PNe, M2-9 and Mz 3
(Lykou et al. 2011, Chesneau et al. 2007).
These disks are compact toroids with an inner radius $\sim$ 10 AU, but contain much less mass than their surrounding PNe, with the total dust mass in the disk $\sim$ 10$^{-5}$~M$_\odot$
(Lykou et al. 2011, Corradi et al 2011, Chesneau et al. 2007).
The disks found in these PNe contain amorphous silicate grains, which implies a young age. In addition, M2-9 has a binary companion with a period of $\sim$ 90 years (Lykou et al. 2011, Corradi et al. 2011). Although a binary has not been detected in Mz 3, it is believed to exist with a similar period, due to its overall similarity to M2-9. Finally, it must be noted that both of these PNe may actually be PNe mimics
(Frew and Parker et al. 2010).
Another two young PNe, CPD-56$^\circ$8032 (\cite[De Marco et al. 2002]{2002ApJ...574L..83D}) and M2-29, also had disks detected in their cores. However, their properties are very different than those of M2-9 and Mz 3. The compact disks have an inner radius $\sim$100 AU. The disk mass is $\sim$ 10$^{-3}$~M$_\odot$ for CPD-56$^\circ$8032 and $\sim$ 10$^{-6}$~M$_\odot$ for M2-29 (Chesneau et al. 2006, Gesicki et al. 2010, Miszalski et al. 2011). In addition, the disks found in these PNe both contain dual-dust chemistry (O and C rich).
No known binary exists for CPD-56$^\circ$8032, but a binary in M2-29 with a period of $\sim$ 17 years is likely (Hajduk et al. 2008).
\vspace{-0.1cm}
\subsection{Compact disks around Post-AGB stars}
\vspace{-0.1cm}
Looking to the post-AGB objects, which are closer on the evolutionary phase to when the disks were likely formed, we find two categories of objects: 1) naked post-AGB objects and 2) pre-PNe.
\vspace{-0.1cm}
\subsubsection{Naked post-AGB objects}
\vspace{-0.1cm}
Naked post-AGB stars tend to not have a reflection nebulae, and as such are not thought to evolve into PNe (the exceptions are HD44179 and HR 4049, see Section 2.2.2). While we call them ``naked" because they lack nebulosity, the SEDs of these stars actually show a larger IR-excess at near/mid infrared wavelengths, indicating the presence of a compact disk (De Ruyter et al. 2006). Several of these disks have been observed with the VLTI and have an inner radius of $\sim$15 AU at 8 $\mu$m and with dust masses of $\sim$1x10$^{-2}$~M$_\odot$
(e.g. Deroo et al 2007, Deroo et al 2006).
No expansion of the dust has been detected, implying that the dust is gravitationally bound in a Keplarian rotation. The Keplerian rotation indicates that it is likely a relic of a strong interaction phase when the primary was an AGB giant and indicates that a binary companion to the post-AGB star is likely present. Such companions were eventually found with orbital periods between 100 and 2000 days
(Van Winckel et al. 2009).
Finally, the disks in the naked post-AGB stars exhibit strong crystalline silicate features. This suggests that the disks found around naked post-AGB stars are actually older than the disks in some young PNe as the dust is more processed.
\vspace{-0.1cm}
\subsubsection{Post-AGB stars with a nebula (pre-PNe)}
\vspace{-0.1cm}
Some post-AGB stars, known as pre-PNe, have collimated nebulae shining in the optical by reflected light or shock ionisation which are thought to become PNe. It is unknown why some, otherwise similar, post-AGB stars have resolved nebulae while others do not. Pre-PNe are very similar to young PNe and are extremely likely to be their immediate predecessors. The SEDs observed for many pre-PNe are double peaked, indicative of a detached shell, others have a near-IR excess due to hot dust in the core (Lagadec et al. 2011). Pre-PNe do not appear to harbour close binaries although wider binaries may be present (e.g. IRAS 22272+5435 appears to have $>$ 22 year period, Hrivnak et al. 2011). HD 44179 and HR 4049 are the only known pre-PNe that have compact disks detected. However, they are unique in that they have both a naked post-AGB-style disk (Keplarian rotation, $\sim$10 AU in diameter, with dual-dust chemistry, and a close binary companion with P $\sim$ 300 - 400 days;
Men'shchikov et al. 2002, Dominik et al. 2003, Waelkens et al. 1991) and a reflection nebula, thereby classifying them as pre-PNe. Therefore, we do not class HD 44179 or HR 4049 as typical pre-PN, but as ``cross-over" objects.
We believe that the immediate circumstellar environments of pre-PNe, primarily the disks that are found there, hold the key to the breaking of AGB mass-loss symmetry which leads to collimated PN morphology. The basic questions that drive our research are: What type of disks, if any, exist inside typical pre-PNe? What is the connection between the older, post-AGB Keplerian disks and the lower-mass, toroidal disks around young PNe? Are the disks a cause or a consequence of the asymmetric mass-loss
\vspace{-0.1cm}
\section{New and Future Observations with the VLTI}
\vspace{-0.1cm}
The VLTI has a spatial resolution up to 10 mas in the mid-IR (using MIDI) and up to 2 mas in the near-IR (using AMBER), making it a powerful tool to observe the inner circumstellar regions of post-AGB stars. The VLTI allows us to determine the geometry and mass of a disk or any other structure present on the 10 -100~AU scale.
We have carefully selected eight pre-PNe to observe with the VLTI. Each target has a bright, compact, unresolved or slightly resolved (at the 0.3$''$ scale) core in the mid-IR as seen from VISIR observations (Lagadec et al. 2011). Therefore, the chances of obtaining useful interferometric observations with the VLTI are very high. All targets also have observed non-spherical nebulae around them. We expect to find disks that have affected the shaping of the pre-PNe or that are a remnant of the shaping mechanism. Likely, these disks will have different characteristics from the naked post-AGB stars without a nebula. Four of the eight pre-PNe have already been observed (from April - June 2011).
Another four targets
will be observed during November 2011 - April 2012.
We report preliminary results on IRAS 16279-4757. IRAS 16279-4757 is a complex axis-symmetric nebula located $\sim$ 2 kpc away, with intermediate inclination to the line of sight. It is classified as a post-AGB object based on its SED (van der Veen, Habing and Geballe 1989). Its optical spectrum suggests an inner star with spectral type of G5 (Hu et al. 1993). It has PAHs and crystalline silicates similar to HD 44179 (Matsuura et al 2004). We obtained 2 baselines of data with MIDI
and 3 baselines with AMBER
The resulting MIDI visibility curves show a sinusoidal pattern, indicating the presence of a ring-like disk (see Figure 1). In addition, the visibility is high ($\sim$0.4 - 0.6) indicating the presence of a compact source in the centre. Preliminary analysis of the visibility curves suggest the presence of a disk. One baseline observed a disk with an inner radius of $\sim$ 150 AU, while the second baseline observed a disk with an inner radius of $\sim$ 70 AU. This implies that the disk around IRAS 16279-4757 has an inner radius of $\sim$ 150 AU and we are viewing it at an inclination of $\sim$60$^\circ$ to the line of sight. This size is similar to the disks seen in young PNe and larger than those observed around naked post-AGB stars. Additionally, the sinusoidal pattern seen in the visibility curves is very similar to the visibility curves observed by the VLTI for the disk seen in CPD-56$^\circ$8032 (see Section 2.1) which has an inner radius of 97 $\pm$ 11 AU and an inclination of 28 $\pm$ 7$^\circ$ (Chesneau et al. 2006).
The preliminary analysis of the AMBER observations indicates a 2-component source inside the disk found with the MIDI observations. The 2 components are likely a stellar source and a possible second disk inside the first disk with a radius of $\sim$ 12 AU. This second, smaller disk is similar in size to those around naked post-AGB stars.
IRAS 16279-4757 has shown that promising results can be found by searching for disks inside pre-PNe with the VLTI. More detailed radiative transfer models will be conducted to derive disk parameters such as inner and outer radii, scale-height, mass, and inclination. These models will be similar to the models for Mz~3 and M2-9 (e.g. Lykou et al. 2010, Chesneau et al. 2007). In conclusion, observations with the VLTI coupled with detailed modelling of the disks will
lead to a quantified comparison between the physical properties of the disks which will allow us to study the evolutionary connection between the compact disks of naked post-AGBs and the tenuous, larger disks inside young PNe.
\vspace{-0.1cm}
\begin{figure}
\begin{center}
\includegraphics[width=4.2in]{S_Bright_figure.pdf}
\caption{MIDI visibility curves for IRAS 16279-4757 displaying a sinusoidal patter, indicating a ring-like disk.}
\label{fig1}
\end{center}
\end{figure}
\vspace{-0.1cm}
|
1,116,691,500,794 | arxiv | \section{Introduction, heuristics, and main result}
\subsection{The context}
We consider a class of birth-and-death processes $(X^{{\scriptscriptstyle K}}_t)_{t\geq 0}$ with state space $\mathds{Z}_{{\scriptscriptstyle \geq 0}}$\footnote{We denote by $\mathds{Z}_{{\scriptscriptstyle \geq 0}}$ the set of non-negative integers, and by $\mathds{Z}_{{\scriptscriptstyle >0}}$ the set of
positive integers.} which describes how the size of a single population evolves according to birth and death rates of the form
\begin{equation}\label{def-lambdaKnmuKn}
\lambda^{{\scriptscriptstyle (K)}}_{n}=K\, b\left(\frac{n}{K}\right)\quad\text{and}\quad \mu^{{\scriptscriptstyle (K)}}_{n}=K\,d\left(\frac{n}{K}\right)
\end{equation}
where $n\geq 1$, and $K\in \mathds{Z}_{{\scriptscriptstyle >0}}$ is a scaling parameter, often called `carrying capacity'. We suppose that $b(0)=d(0)=0$, implying that $0$ is an absorbing
state for the process, modelling extinction, and our assumptions are such that the probability to reach this state is equal to one.
The unique stationary distribution is the Dirac measure at $0$, so a relevant distribution to look for is a quasi-stationary distribution.
A probability measure $\nu^{{\scriptscriptstyle (K)}}$ on the positive integers is a quasi-stationary distribution if, for all $t>0$ and for all subsets $A\subset \mathds{Z}_{{\scriptscriptstyle >0}}$, one has
${\mathbb P}_{\nu^{{\scriptscriptstyle (K)}}}(X^{{\scriptscriptstyle K}}_t\in A \,|\, T^{{\scriptscriptstyle (K)}}_0>t)=\nu^{{\scriptscriptstyle (K)}}(A)$, where $\TK$ is the extinction time, that is, the smallest $t>0$ such that
$X^{{\scriptscriptstyle (K)}}_t=0$.
In other words, a quasi-stationary distribution plays the role of a stationary distribution when conditioning upon non-extinction.
We refer to \cite{CMSM,MV} for more informations about quasi-stationary distributions.
When $K\to\infty$, the trajectories of the rescaled process $(K^{-1}X^{{\scriptscriptstyle (K)}}_t)_{t\geq 0}$ converge in probability, in any fixed time-window, to the solutions of the differential equation
\begin{equation}\label{the-edo}
\frac{\mathrm{d} x}{\mathrm{d} t} =b(x)-d(x)
\end{equation}
if the initial condition state is for instance of the form $\lfloor Kx_0\rfloor$ for a given $x_0>0$.
We assume that the functions $b$ and $d$ only vanish at $0$, and that
\[
d'(0) - b'(0)<0
\]
meaning that the fixed point $0$ is repulsive. We also assume that there is a unique attractive fixed point $x_{*}>0$, that is
\[
b(x_{*})=d(x_{*})\quad\text{and}\quad b'(x_{*})-d'(x_{*})<0.
\]
(We will give the complete set of assumptions on the functions $b$ and $d$ later on.)
A famous example is the so-called logistic process for which $b(x)=\lambda x$, $d(x)=x(\mu +x)$, where $\lambda$ and $\mu$ are positive real numbers.
We assume that $\lambda>\mu$ and we have $x_{*}=\lambda-\mu$.
In \cite{CCM1} we obtained the precise asymptotic behaviour of the first eigenvalue of the generator $L_{{\scriptscriptstyle K}}$ of the process killed at $0$, and also of the law of the
extinction time starting from the quasi-stationary distribution (among other results).
Here we go further and obtain the complete spectrum of the generator of the killed process, in the limit $K\to\infty$.
In particular, the knowledge of the spectral gap allows us to obtain the time of relaxation for the process conditioned on non-extinction to
obey the quasi-stationary distribution.
\subsection{Notations for basic function spaces}
We denote by
\begin{itemize}
\item
$\mathscr{D}$ the space of $C^\infty$ $\mathds{C}$-valued functions with compact support on $\mathds{R}$;
\item
$c_{00}$ the space of $\mathds{C}$-valued sequences with finitely many nonzero values;
\item
$\ell^{2}$ the space of square-summable $\mathds{C}$-valued sequences equipped with the standard scalar product.
\item
$L^{2}$ the space of square-integrable $\mathds{C}$-valued functions with respect to Lebesgue measure on $\mathds{R}$.
\end{itemize}
We will define several operators on $c_{00}$ and will consider their closure on $\ell^{2}$.
For simplicity, we will use the same notation for an operator and its closure. As we will see later, there is no ambiguity on the extensions.
\subsection{Heuristics}
The fundamental object in this paper is the spectrum of the following operator that we momentarily define on $c_{00}$:
\begin{equation}\label{pre-LK}
\big(L_{{\scriptscriptstyle K}} v\big)(n)=\lambda^{{\scriptscriptstyle (K)}}_n\big(v(n+1)-v(n)\big)+\mu^{{\scriptscriptstyle (K)}}_n \big(v(n-1){\mathds{1}}_{\{n>1\}}-v(n)\big)
\end{equation}
for $n\in\mathds{Z}_{{\scriptscriptstyle >0}}$.
The idea is to `localize' this operator either around $n=\lfloor Kx_{*}\rfloor$ or $n=1$, which corresponds in the dynamical system to the fixed point $x_{*}$
or the fixed point $0$.
A natural idea would be to `cut' the operators in order to differentiate these two dynamics.
However the main difficulty is that the two different pieces involve different scales and different function spaces.
Since we don't know how to cope with this problem at the level of operators, we work at the level of the eigenfunctions
that we will split on different regions, and study their asymptotic dependence on $K$ in each region.
To have an idea of the different scales involved in the problem, let us first study the asymptotic behaviour of the birth and death rates.
By Taylor expansion around $Kx_{*}$ we have
\[
\lambda^{{\scriptscriptstyle (K)}}_{n}=K\,b\big(x_{*}\big)+(n-K\xf)\,b'\big(x_{*}\big)+\mathcal{O}\left(\frac{(n-K\xf)^{2}}{K}\right)
\]
and
\[
\mu^{{\scriptscriptstyle (K)}}_{n}=K\,d\big(x_{*}\big)+(n-K\xf)\,d'\big(x_{*}\big)+\mathcal{O}\left(\frac{(n-K\xf)^{2}}{K}\right).
\]
Taking $v^{{\scriptscriptstyle (K)}}(n)=u\big((n-K\xf)/\sqrt{K}\big)$ with $u\in\mathscr{D}$, we get
\[
\big(L_{{\scriptscriptstyle K}} v^{{\scriptscriptstyle (K)}}\big)(n)=\EuScript{OU}_{*} u\left(\frac{n-K\xf}{\sqrt{K}}\right)+\mathcal{O}\left(\frac{1}{\sqrt{K}}\right)
\]
where
\begin{equation}\label{def-OU-generator}
\EuScript{OU}_{*} f(x)= b(x_{*}) f''(x)+ \,\big(b'(x_{*})-d'(x_{*})\big)\,x f'(x)
\end{equation}
is the generator of an Ornstein-Uhlenbeck process on $\mathds{R}$ which satisfies the stochastic differential equation
$\mathrm{d} X_t=\big(b'(x_{*})-d'(x_{*})\big)X_t \mathrm{d} t + \sqrt{2b(x_{*})}\, \mathrm{d} B_t$, where $(B_t)_{t\geq 0}$ is a one-dimensional Brownian motion.
It is well known (see Remark \ref{rem:OUHO}) that the spectrum of $\EuScript{OU}_{*}$ is $-S_1$ where
\begin{equation}\label{def-S1}
S_1=\big(d'(x_{*})-b'(x_{*})\big)\mathds{Z}_{{\scriptscriptstyle \geq 0}}
\end{equation}
in the space $L^2\Big(\sqrt{\frac{d'(x_{*})-b'(x_{*})}{2\pi b(x_{*})}}\,\mathrm{e}^{-\frac{(d'(x_{*})-b'(x_{*})}{2b(x_{*})}x^2}\mathrm{d} x\Big)$.
Now we look at $n$ near $1$. By Taylor expansion, we have
\[
\lambda^{{\scriptscriptstyle (K)}}_{n}=n\left(b'(0)+\mathcal{O}\left(\frac{n}{K}\right)\right)\quad\text{and}\quad \mu^{{\scriptscriptstyle (K)}}_{n}=n\left(b'(0)+\mathcal{O}\left(\frac{n}{K}\right)\right).
\]
If $v\in c_{00}$, then we get
\[
\left\| L_{{\scriptscriptstyle K}} v -\EuScript{Q}_{0} v\right\|_{\ell^{2}}\leq \frac{\Oun}{K}
\]
where
\[
\big(\EuScript{Q}_{0} v\big)(n)=b'(0)\,n\,\big(v(n+1)-v(n)\big)+d'(0)\, n\,\big(v(n-1)\,{\mathds{1}}_{\{n>1\}}-v(n)\big)
\]
which is the generator of a (continuous-time) binary branching process killed at $0$.
We shall prove later on that, in a weighted $\ell^2$ space defined below, the spectrum of $\EuScript{Q}_{0}$ is $-S_2$ where
\begin{equation}\label{def-S2}
S_2=\big(b'(0)-d'(0)\big)\mathds{Z}_{{\scriptscriptstyle >0}}.
\end{equation}
The previous observations suggest that the limit of the spectrum of the generator of the birth-and-death process $(X^{{\scriptscriptstyle (K)}}_t)_{t\geq 0}$, in an appropriate space, is
\[
\big(d'(0)-b'(0)\big)\mathds{Z}_{{\scriptscriptstyle >0}} \bigcup \big(b'(x_{*})-d'(x_{*})\big)\mathds{Z}_{{\scriptscriptstyle \geq 0}}.
\]
Notice that all the elements of this set are negative and this is not a disjoint union in general. The logistic model is an example illustrating this since
$d'(0)-b'(0)=b'(x_{*})-d'(x_{*})=\mu-\lambda$, so we will have asymptotic double eigenvalues in this case.
We will prove that the limit of the spectrum of $L_{{\scriptscriptstyle K}}$ is obtained from the explicit spectra of the above two operators.
Notice that one is differential operator and the other one is a finite-difference operator.
This is a reverse situation with respect to numerical analysis where the spectrum of limiting differential operators are obtained
from the knowledge of the spectrum of finite-difference operators. See for instance \cite{Chatelin}.
\subsection{Main result}
For each $K\in \mathds{Z}_{{\scriptscriptstyle >0}}$, the sequence of numbers
\[
\pi^{{\scriptscriptstyle (K)}}_n:=\frac{\lambda^{{\scriptscriptstyle (K)}}_1\cdots \lambda^{{\scriptscriptstyle (K)}}_{n-1}}{\mu^{{\scriptscriptstyle (K)}}_1\cdots \mu^{{\scriptscriptstyle (K)}}_n}, \, n\geq 2\quad\text{and}\quad \pi^{{\scriptscriptstyle (K)}}_1:=\frac{1}{\mu^{{\scriptscriptstyle (K)}}_1}
\]
naturally shows up in the study of birth-and-death processes.
We will give below a set of assumptions on the functions $b$ and $d$, defining the differential equation \eqref{the-edo}, ensuring that the process
reaches $0$ in finite time with probability one, that the mean-time to extinction is finite, and that the quasi-stationary distribution exists and is unique.
Let $\ell^2(\pi^{{\scriptscriptstyle (K)}})$ be the space of $\mathds{C}$-valued sequences $(v_n)_{n\geq 1}$ such that
\[
\sum_{n\geq 1} |v_n|^2\pi^{{\scriptscriptstyle (K)}}_n<\infty\,.
\]
This is a Hilbert space when endowed with the scalar product $\langle v,w\rangle_{\pi^{{\scriptscriptstyle (K)}}}:=\sum_{n\geq 1} \bar{v}_nw_n \pi^{{\scriptscriptstyle (K)}}_n$.
We know from \cite{CCM1} that the operator $L_{{\scriptscriptstyle K}}$ is closable in $\ell^2(\pi^{{\scriptscriptstyle (K)}})$.
The closure (which we denote by the same symbol) is self-adjoint and has a compact resolvent, hence its spectrum is discrete, composed of simple eigenvalues which are
negative real numbers, and the corresponding eigenvectors are orthogonal. We normalize these eigenvectors and we can assume that they are real (since they are
defined by a second-order real recurrence relation whose solution is determined by choosing the first element).
We write
\begin{equation}\label{def-eigen-LK}
L_{{\scriptscriptstyle K}}\psi^{{\scriptscriptstyle (K)}}_j=-\rho^{{\scriptscriptstyle (K)}}_j \psi^{{\scriptscriptstyle (K)}}_j
\end{equation}
where we order the eigenvalues $-\rho^{{\scriptscriptstyle (K)}}_j$ in decreasing order as $j$ increases.
To emphasize that all operators considered in this paper are negative, we have decided to write their eigenvalues under the form $-\rho$, with $\rho>0$.
As shown in \cite{CCM1}, the quasi-stationary distribution exists, is unique, and given by
\begin{equation}
\label{QSD}
\nu^{{\scriptscriptstyle (K)}}(\{n\})=\frac{\pi^{{\scriptscriptstyle (K)}}_n\psi^{{\scriptscriptstyle (K)}}_0(n)}{\langle \psi^{{\scriptscriptstyle (K)}}_0,\mathbf{1}\rangle_{\pi^{{\scriptscriptstyle (K)}}}},\;n\in\mathds{Z}_{{\scriptscriptstyle >0}}
\end{equation}
where $\mathbf{1}=(1,1,\ldots)$.
Note that it also follows from Theorem 3.2 and Lemma 9.3 in \cite{CCM1} that there exists $D>1$ such that for all $K\in\mathds{Z}_{{\scriptscriptstyle >0}}$, we have
$D^{-1}\leq \psi_0^{{\scriptscriptstyle (K)}}\leq D$. Therefore, the Hilbert spaces $\ell^{2}(\nu^{{\scriptscriptstyle (K)}})$ and $\ell^{2}(\pi^{{\scriptscriptstyle (K)}})$ are isomorphic.
Let $S^{{\scriptscriptstyle (K)}}_t f(n):={\mathds{E}}_n\Big( f(X^{{\scriptscriptstyle (K)}}_t) {\mathds{1}}_{\{\TK>t\}}\Big)$ ($t\geq 0$) be the semigroup of the killed process, where $n\in\mathds{Z}_{{\scriptscriptstyle >0}}$ and $f\in \ell^\infty$. The following result justifies that we look for the spectrum of $L_{{\scriptscriptstyle K}}$ in $\ell^{2}(\pi^{{\scriptscriptstyle (K)}})$ .
\begin{proposition}
The semigroup $(S^{{\scriptscriptstyle (K)}}_t)_{t\geq 0}$, defined on $\ell^\infty$, extends to a $C_0$-contraction semigroup on $\ell^{2}(\nu^{{\scriptscriptstyle (K)}})$.
\end{proposition}
We refer to \cite{yosida} for definitions and properties of $C_0$-contraction semigroups.
\begin{proof}
We follow the argument of Proposition 8.1.8 p. 162 in \cite{BL}. Since $\nu^{{\scriptscriptstyle (K)}}$ is a quasi-stationary distribution, for $f\in c_{00}$ and $t\geq 0$, we have
\begin{align*}
\int |S^{{\scriptscriptstyle (K)}}_t f|^2 \mathrm{d}\nu^{{\scriptscriptstyle (K)}}
& \leq \int \mathrm{d}\nu^{{\scriptscriptstyle (K)}}(n)\, {\mathds{E}}_n \Big( \big|f(X^{{\scriptscriptstyle (K)}}_t)\big|^2 {\mathds{1}}_{\big\{\TK>t\big\}}\Big)\\
& = \mathrm{e}^{-\rho^{{\scriptscriptstyle (K)}}_0 t} \int |f|^2 \mathrm{d}\nu^{{\scriptscriptstyle (K)}}.
\end{align*}
Therefore
\[
\| S^{{\scriptscriptstyle (K)}}_t f\|_{\ell^{2}(\nu^{{\scriptscriptstyle (K)}})}\leq \mathrm{e}^{-\frac{\rho^{{\scriptscriptstyle (K)}}_0 \mathlarger{t}}{2}} \|f\|_{\ell^{2}(\nu^{{\scriptscriptstyle (K)}})},\;t\geq 0.
\]
Since $c_{00}$ is dense in $\ell^{2}(\nu^{{\scriptscriptstyle (K)}})$, we get
\[
\| S^{{\scriptscriptstyle (K)}}_t \|_{\ell^{2}(\nu^{{\scriptscriptstyle (K)}})}\leq \mathrm{e}^{-\frac{\rho^{{\scriptscriptstyle (K)}}_0 \mathlarger{t}}{2}}, \; t\geq 0.
\]
This implies that $(S^{{\scriptscriptstyle (K)}}_t)_{t\geq 0}$ extends to a contraction semigroup in $\ell^{2}(\nu^{{\scriptscriptstyle (K)}})$.
Now, since ${\mathbb P}_n(X^{{\scriptscriptstyle (K)}}_t=m,\TK>t)\to \delta_{n,m}$, as $t\to 0$, then for any $f\in c_{00}$, $S^{{\scriptscriptstyle (K)}}_t f\to f$ pointwise,
hence by dominated convergence we obtain $S^{{\scriptscriptstyle (K)}}_t f \to f$ in $\ell^{2}(\nu^{{\scriptscriptstyle (K)}})$ as $t\to 0$.
The proposition follows from the contraction property obtained above and the fact that $c_{00}$ is dense in $\ell^{2}(\nu^{{\scriptscriptstyle (K)}})$.
\end{proof}
Observe that the same result holds in fact for any $1\leq p <\infty$ (with a similar proof).
Recall from \cite{CCM1} that we also have
\[
{\mathbb P}_{\nu^{{\scriptscriptstyle (K)}}}(\TK>t)=\mathrm{e}^{-\rho^{{\scriptscriptstyle (K)}}_0 t},\;t>0
\]
and the mean-time to extinction starting from $\nu^{{\scriptscriptstyle (K)}}$ is
\begin{align*}
\MoveEqLeft {\mathds E}_{\nu^{{\scriptscriptstyle (K)}}}\big[\TK\big]\\
&=\frac{1}{\rho^{{\scriptscriptstyle (K)}}_0}= \frac{\sqrt{2\pi}\,\exp\bigg(K\mathlarger{\int}_{0}^{x_{*}} \log\frac{b(x)}{d(x)}\,\mathrm{d} x\bigg)}{b(x_{*})\left(\sqrt{\frac{b(1/{\scriptscriptstyle K})}{d(1/{\scriptscriptstyle K})}}-\sqrt{\frac{d(1/{\scriptscriptstyle K})}{b(1/{\scriptscriptstyle K})}}\right)\!
\sqrt{K H''(x_{*})}}
\!\left(\!1\!+\!\mathcal{O}\!\left(\frac{(\log K)^3}{\sqrt{K}}\right)\!\!\right).
\end{align*}
(See the next section for the definition of $H$.)
In the logistic model this gives (recall that $x_*=\lambda-\mu$)
\[
{\mathds E}_{\nu^{{\scriptscriptstyle (K)}}}\big[\TK\big]=
\frac{\sqrt{2\pi}\,\mu\,\exp\Big(K\big(\lambda-\mu+\mu\log\frac{\mu}{\lambda}\big)\Big)}{(\lambda-\mu)^2\sqrt{K}}
\!\left(\!1\!+\!\mathcal{O}\!\left(\frac{(\log K)^3}{\sqrt{K}}\right)\!\!\right).
\]
Thus $\rho^{{\scriptscriptstyle (K)}}_0$ is exponentially small in $K$, and we also proved in \cite{CCM1} that the `spectral gap' satisfies
(see Theorem 3.3 in \cite{CCM1})
\begin{equation}
\label{lower-bound-spectral-gap}
\rho^{{\scriptscriptstyle (K)}}_1-\rho^{{\scriptscriptstyle (K)}}_0 \geq \frac{\Oun}{\log K}, \; K\in\mathds{Z}_{{\scriptscriptstyle >1}}.
\end{equation}
This lower bound goes to $0$ as $K\to+\infty$. A noteworthy consequence of the main results of this paper (see Corollary \ref{trou}) is
that the spectral gap does not close when $K$ tends to infinity, contrary to what could have been suspected from the lower bound in \eqref{lower-bound-spectral-gap}.
In fact we will fully describe the asymptotics of all eigenvalues, which is the content of our main theorem. To state it,
we need to order $S_1\cup S_2$ to take care of possible multiplicities. Recall that $S_{1}$ and $S_{2}$ have been defined in \eqref{def-S1} and \eqref{def-S2}
and that they are positive sequences.
This is done through the definition of a non-decreasing infinite (positive) sequence $(\eta_n)_{n\geq 0}$.
Let $\eta_0=0$. We construct this sequence recursively as follows.
\begin{itemize}
\item[$\diamond$]
If $\eta_n\in S_1\Delta S_2$, then $\eta_{n+1}=\min\{\eta: \eta \in S_1\cup
S_2: \eta>\eta_n\}$.
\item[$\diamond$]
If $\eta_n\in S_1\cap S_2$, then
\begin{itemize}
\item[$\bullet$]
If $\eta_{n-1}=\eta_n$, then $\eta_{n+1}=\min\{\eta: \eta \in S_1\cup S_2: \eta>\eta_n\}$.
\item[$\bullet$]
If $\eta_{n-1}<\eta_n$, then $\eta_{n+1}=\eta_n$.
\end{itemize}
\end{itemize}
The main result of this paper, whose assumptions will be stated in Section 2, is the following.
\begin{theorem}[Convergence of the spectrum]\label{main-theorem}
\leavevmode\\
The spectrum of $L_{{\scriptscriptstyle K}}$ in $\ell^2(\pi^{{\scriptscriptstyle (K)}})$ converges pointwise to $(-\eta_n)_{n\geq 0}$ when $K$ tends to infinity. In other words
\[
\lim_{K\to+\infty}\rho^{{\scriptscriptstyle (K)}}_j = \eta_j, \;\forall j\in\mathds{Z}_{{\scriptscriptstyle \geq 0}}.
\]
\end{theorem}
\begin{corollary}\label{trou}
The spectral gap $\rho^{{\scriptscriptstyle (K)}}_1-\rho^{{\scriptscriptstyle (K)}}_0$ converges to
\[
\min\big\{b'(0)-d'(0),\,d'(x_{*})-b'(x_{*})\big\}\;.
\]
\end{corollary}
Let us give some examples.
In the logistic model, we have $d'(0)-b'(0)=b'(x_{*})-d'(x_{*})=\mu-\lambda$ (asymptotic double eigenvalues), and the spectral gap is equal to $\lambda-\mu$.
Another example is the Ayala-Gilpin-Ehrenfeld model \cite{AGE} defined by $b(x)=\lambda x$, $d(x)=x(\mu+x^\theta)$ where $\theta\in (0,1)$ is a parameter,
and $\lambda>\mu$.
In this case, $d'(0)-b'(0)=\mu-\lambda$, $x_{*}=(\lambda-\mu)^{1/\theta}$, and $b'(x_{*})-d'(x_{*})=\theta(\mu-\lambda)$, so the spectral gap is $\theta(\lambda-\mu)$.
Yet another example is Smith's model \cite{smith} defined by $b(x)=\lambda x/(1+x)$, $d(x)=(x(\mu+x))/(1+x)$, where $\lambda>\mu$. One easily finds
$d'(0)-b'(0)=\mu-\lambda$, $x_{*}=\lambda-\mu$, and $b'(x_{*})-d'(x_{*})=(\mu-\lambda)/(1+\lambda-\mu)$, so the spectral gap is $(\lambda-\mu)/(1+\lambda-\mu)$.
\subsection{Consequences on relaxation times}
Recall that the spectral gap is the inverse of the relaxation time to the quasi-stationary distribution, namely
for $\psi\in\ell^2(\pi^{{\scriptscriptstyle (K)}})$, $t>0$ and $K\in\mathds{Z}_{{\scriptscriptstyle >0}}$ we have
\[
\left\|\mathrm{e}^{\rho^{{\scriptscriptstyle (K)}}_0 t} \, S^{{\scriptscriptstyle (K)}}_t \psi - \psi_0^{{\scriptscriptstyle (K)}}\langle \psi_0^{{\scriptscriptstyle (K)}},\mathbf{1} \rangle_{\pi^{{\scriptscriptstyle (K)}}} \nu^{{\scriptscriptstyle (K)}}(\psi)\right\|_{\ell^2(\pi^{{\scriptscriptstyle (K)}})}
\leq \|\psi\|_{\ell^2(\pi^{{\scriptscriptstyle (K)}})}\,\mathrm{e}^{-\big(\rho^{{\scriptscriptstyle (K)}}_1-\rho^{{\scriptscriptstyle (K)}}_0\big)t}.
\]
From Corollary \ref{trou}, it turns out that the relaxation time converges to a finite limit as $K$ tends to infinity.
We can also characterize the decay of correlations for the so-called $Q$-process, namely the birth-and-death process conditioned on survival. Recall that
the $Q$-process is the irreducible Markov process with state space $\mathds{Z}_{{\scriptscriptstyle >0}}$, defined by the semigroup
\[
R^{{\scriptscriptstyle (K)}}_t g=\mathrm{e}^{\rho_0^{{\scriptscriptstyle (K)}}t} \frac{1}{\psi_0^{{\scriptscriptstyle (K)}}} \, S^{{\scriptscriptstyle (K)}}_t\big(g\psi_0^{{\scriptscriptstyle (K)}}\big).
\]
It satisfies $R^{{\scriptscriptstyle (K)}}_t {\mathds{1}}_{\{n\geq 1\}}= {\mathds{1}}_{\{n\geq 1\}}$, and its unique invariant distribution $\mathfrak{m}^{{\scriptscriptstyle (K)}}$, defined by
${\big(R_t^{{\scriptscriptstyle (K)}}\big)}^\dagger\mathfrak{m}^{{\scriptscriptstyle (K)}}=\mathfrak{m}^{{\scriptscriptstyle (K)}}$,
is related to $\nu^{{\scriptscriptstyle (K)}}$ by $\mathfrak{m}^{{\scriptscriptstyle (K)}}(g)=\nu^{{\scriptscriptstyle (K)}}(\psi_0^{{\scriptscriptstyle (K)}} g)$ where $\nu^{{\scriptscriptstyle (K)}}$ has been defined in \eqref{QSD}.
Indeed we have $g \in \ell^{2}(\mathfrak{m}^{{\scriptscriptstyle (K)}})$ if and only if $\psi_0^{{\scriptscriptstyle (K)}}g \in \ell^{2}(\pi^{{\scriptscriptstyle (K)}})$, and
\[
\| g\|^2_{\ell^{2}(\mathfrak{m}^{{\scriptscriptstyle (K)}})}=\frac{\| \psi_0^{{\scriptscriptstyle (K)}}g\|^2_{\ell^{2}(\pi^{{\scriptscriptstyle (K)}})}}{\langle \psi_0^{{\scriptscriptstyle (K)}},\mathbf{1}\rangle_{\pi^{{\scriptscriptstyle (K)}}}}.
\]
Hence, we get the following result.
\begin{proposition}
Let $g\in \ell^{2}(\mathfrak{m}^{{\scriptscriptstyle (K)}})$. Then for all $t> 0$
\[
\big\| R^{{\scriptscriptstyle (K)}}_t g -\langle \psi_0^{{\scriptscriptstyle (K)}},\mathbf{1}\rangle_{\pi^{{\scriptscriptstyle (K)}}} \mathfrak{m}^{{\scriptscriptstyle (K)}}(g)\big\|_{\ell^{2}(\mathfrak{m}^{{\scriptscriptstyle (K)}})}
\leq \|g\|_{\ell^{2}(\mathfrak{m}^{{\scriptscriptstyle (K)}})}\, \mathrm{e}^{-(\rho^{{\scriptscriptstyle (K)}}_1-\rho^{{\scriptscriptstyle (K)}}_0)\,t}.
\]
Furthermore, for $g_1,g_2\in\ell^{2}(\mathfrak{m}^{{\scriptscriptstyle (K)}})$ and for all $t>0$, we have
\begin{align*}
\MoveEqLeft[6] \bigg|\int R^{{\scriptscriptstyle (K)}}_t g_1\cdot g_2 \, \mathrm{d}\mathfrak{m}^{{\scriptscriptstyle (K)}}- \int g_1\, \mathrm{d}\mathfrak{m}^{{\scriptscriptstyle (K)}}\int g_2\, \mathrm{d}\mathfrak{m}^{{\scriptscriptstyle (K)}}\bigg|\\
& \leq \|g_1\|_{\ell^{2}(\mathfrak{m}^{{\scriptscriptstyle (K)}})}\|g_2\|_{\ell^{2}(\mathfrak{m}^{{\scriptscriptstyle (K)}})} \,\mathrm{e}^{-\big(\rho^{{\scriptscriptstyle (K)}}_1-\rho^{{\scriptscriptstyle (K)}}_0\big)t}.
\end{align*}
\end{proposition}
As before, the rate of decay of correlations converges when $K$ goes to infinity.
\subsection{Organization of the paper}
The proof of the Theorem \ref{main-theorem} relies on two results stated in Section \ref{preuvemain}. Theorem \ref{liminf} ensures that the
set $S_{1}\cup S_{2}$ is contained in the set of accumulation points of the eigenvalues of $\L_{{\scriptscriptstyle K}}$ when $K$ tends to infinity.
The proof is based on the construction of quasi-eigenvectors and is given in Section \ref{proof-maintheorem}.
The second result is Theorem \ref{prsquelafin} which ensures that all the previous accumulation points are contained in $S_{1}\cup S_{2}$ taking care of
eventual multiplicities. Its proof, given in section \ref{proof-maintheorem}, relies on two propositions. The first one (Proposition \ref{strucvp}) is the splitting of the
eigenvectors of $\mathcal{L}_{{\scriptscriptstyle K}}$ into two dominant parts, one localised near the origin, the other one near $\lfloor Kx_{*}\rfloor$.
The second (Proposition \ref {decoupe}) relies on compactness arguments of each piece of the previous splitting. Section \ref{auxiliaires} collects various
auxiliary results (some of more general nature).
One of the main difficulties of the proof is that the two pieces of the spectrum correspond to limiting operators which are obtained at
different scales and leave in different function spaces.
\section{Standing assumptions}\label{sec:assumptions}
We work under the assumptions of \cite{CCM1} which we recall for convenience.
The functions $b,d:\mathds{R}_+\to\mathds{R}_+$ defining the differential equation \eqref{the-edo} are supposed to be such that
\[
b(0) = d(0)=0
\]
and the functions $x\mapsto b(x)/x$ and $x\mapsto d(x)/x$ are defined on $\mathds{R}_+$ and assumed to be positive, twice differentiable and increasing (in particular
the sequences $(\lambda^{{\scriptscriptstyle (K)}}_{n})_n$ and $(\mu^{{\scriptscriptstyle (K)}}_{n})_n$ defined in \eqref{def-lambdaKnmuKn} are increasing for each $K$).
We start by the biologically relevant assumptions:
\begin{itemize}
\renewcommand{\labelitemi}{$\centerdot$}
\item
$\lim_{x\to+\infty} \frac{b(x)}{d(x)}=0$ (deaths prevail over births for very large densities).
\item
$b'(0)>d'(0)>0$ (at low density births prevail).
\item
There is a unique $x_{*}>0$ such that $b(x_{*})=d(x_{*})$, so $x_{*}$ is the only positive fixed point of \eqref{the-edo}.
\end{itemize}
We assume that $ b'(x_{*})\neq d'(x_{*})$ (genericity condition). The remaining (technical) assumptions are the following:
\begin{itemize}
\renewcommand{\labelitemi}{$\centerdot$}
\item
$\mathlarger{\int}_{\frac{x_{*}}{2}}^{+\infty} \frac{\mathrm{d} x}{d(x)}<+\infty$ and $\sup_{x\in\mathds{R}_+} \big(\frac{d'(x)}{d(x)}-\frac{1}{x}\big)<+\infty$.
\item
The function $x\mapsto \log\frac{d(x)}{b(x)}$ is increasing on $\mathds{R}_+$.
\item
The function $H:\mathds{R}_+\to\mathds{R}$ defined by $H(x)=\int_{x_{*}}^x \log\frac{d(s)}{b(s)}\mathrm{d} s$ is three times differentiable, and $\sup_{x\in\mathds{R}_+}(1+x^2)|H'''(x)|<+\infty$.
\end{itemize}
The assumptions imply that $0$ is a repulsive (or unstable) fixed point of \eqref{the-edo}, whereas $x_{*}$ is an attractive (or stable) one, that is, $b'(x_{*})<d'(x_{*})$. It also follows that $H''(x_{*})>0$.
These assumptions are satisfied for many classical examples.
As explained in \cite{CCM1}, the above conditions imply the following properties:
\begin{itemize}
\renewcommand{\labelitemi}{$\centerdot$}
\item $\sum_{n\geq 1} (\lambda^{{\scriptscriptstyle (K)}}_n \pi^{{\scriptscriptstyle (K)}}_n)^{-1}=+\infty$, which implies that the process reaches $0$ in finite time with probability one.
\item
$\sum_{n\geq 1} \pi^{{\scriptscriptstyle (K)}}_n<+\infty$, which implies finiteness of the mean time to extinction.
\item $\sum_{n\geq 1} (\lambda^{{\scriptscriptstyle (K)}}_n \pi^{{\scriptscriptstyle (K)}}_n)^{-1} \sum_{i\geq n+1} \pi^{{\scriptscriptstyle (K)}}_i<+\infty$, which is a necessary and sufficient condition for existence and uniqueness of the quasi-stationary distribution.
\end{itemize}
We add a last condition to the previous ones, namely
\begin{equation}\label{hipopo}
\lim_{x\to\infty}\frac{\log b(x)}{x}=\lim_{x\to\infty}\frac{\log d(x)}{x}=0.
\end{equation}
We could avoid it makes our life easier and we don't have any natural example which does not satisfy it.
\section{Proof of Theorem \ref{main-theorem}}\label{preuvemain}
\subsection{Some useful operators}
Instead of working with $L_{{\scriptscriptstyle K}}$ on the weighted Hilbert space $\ell^2(\pi^{{\scriptscriptstyle (K)}})$, we find more convenient to work on the `flat' Hilbert space $\ell^{2}$.
We introduce the conjugated operator
\begin{equation*}
\mathcal{L}_{{\scriptscriptstyle K}}=(\Pi^{{\scriptscriptstyle (K)}})^{\frac{1}{2}}L_{{\scriptscriptstyle K}}\,(\Pi^{{\scriptscriptstyle (K)}})^{-\frac{1}{2}}
\end{equation*}
where $\Pi^{{\scriptscriptstyle (K)}}$ denotes the mutiplication operator
\[
\Pi^{{\scriptscriptstyle (K)}} v(n)=\pi^{{\scriptscriptstyle (K)}}_{n}v(n)
\]
for $v\in c_{00}$ and $n\in\mathds{Z}_{{\scriptscriptstyle >0}}$. One can check that
\begin{align*}
& \big(\mathcal{L}_{{\scriptscriptstyle K}} v\big)(n)\\
& =\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n}\,\mu^{{\scriptscriptstyle (K)}}_{n+1}}\,v(n+1)+\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n-1}\,\mu^{{\scriptscriptstyle (K)}}_{n}}\,v(n-1)\,{\mathds{1}}_{\{n>1\}}-\big(\lambda^{{\scriptscriptstyle (K)}}_{n}+\mu^{{\scriptscriptstyle (K)}}_{n}\big)\,v(n)
\end{align*}
for $n\in\mathds{Z}_{{\scriptscriptstyle >0}}$.
We denote also by $\mathcal{L}_{{\scriptscriptstyle K}}$ its closure in $\ell^{2}$ and by $\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle K}})$ its domain, and we have
\begin{equation*}
\mathcal{L}_{{\scriptscriptstyle K}} \phi^{{\scriptscriptstyle (K)}}_j=-\rho^{{\scriptscriptstyle (K)}}_j \phi^{{\scriptscriptstyle (K)}}_j
\end{equation*}
where the eigenvalues $-\rho^{{\scriptscriptstyle (K)}}_j$ are the same as for $L_{{\scriptscriptstyle K}}$ (cf. \eqref{def-eigen-LK}), and $\phi^{{\scriptscriptstyle (K)}}=\big(\Pi^{{\scriptscriptstyle (K)}}\big)^{\frac{1}{2}}\psi^{{\scriptscriptstyle (K)}}$.
To capture the behavior of the eigenvectors of $\mathcal{L}_{{\scriptscriptstyle K}}$ near $\lfloor Kx_{*}\rfloor$ at scale $\sqrt{K}$, we are going to embed $\ell^{2}$ into $L^{2}$.
For this purpose we define for each $K\in\mathds{Z}_{{\scriptscriptstyle >0}}$ the functions
\[
\eK{n}(x)=K^{\frac{1}{4}}\,{\mathds{1}}_{I^{{\scriptscriptstyle (K)}}_{n} }(x),\; x\in\mathds{R},\;n\in\mathds{Z}_{{\scriptscriptstyle >0}}
\]
where
\[
I^{\scriptscriptstyle (K)}_{n}=\left[\frac{n-0.5}{\sqrt{K}}-x_{*}\sqrt{K}, \frac{n+0.5}{\sqrt{K}}-x_{*}\sqrt{K}\right[.
\]
The functions $\eK{n}$ are orthogonal and of norm one in $L^{2}$. They form a basis of a sub-Hilbert space $\mathscr{H}_{{\scriptscriptstyle K}}$ of piecewise constant
functions in $L^{2}$.
We define two maps denoted by $Q_{{\scriptscriptstyle K}}$ and $P_{{\scriptscriptstyle K}}$ as follows:
\begin{equation*}
Q_{{\scriptscriptstyle K}}: \ell^{2}\toL^{2},\quad
Q_{{\scriptscriptstyle K}} u(x)=\sum_{n\geq 1}u(n)\,\eK{n}(x)\
\end{equation*}
and
\begin{equation}\label{PP}
P_{{\scriptscriptstyle K}}:L^{2} \to \ell^{2},\quad
P_{{\scriptscriptstyle K}}f(n)=\int f(x)\,\eK{n}(x)\,\mathrm{d} x\;, n\in\mathds{Z}_{{\scriptscriptstyle >0}}.
\end{equation}
We will use the following properties of $P_{{\scriptscriptstyle K}}$ and $Q_{{\scriptscriptstyle K}}$ stated as two lemmas.
\begin{lemma}\label{isometry}
For each $K\in\mathds{Z}_{{\scriptscriptstyle >0}}$, the map $Q_{{\scriptscriptstyle K}}$ is an isometry between $\ell^{2}$ and $\mathscr{H}_{{\scriptscriptstyle K}}$.
\end{lemma}
The proof of this lemma is left to the reader.
\begin{lemma}\label{isomQP}
Let $f\in C^{1}(\mathds{R})$ and assume that there exists $a>0$ and $A>0$ such that
\[
\big|f(x)\big|+\big|f'(x)\big|\le A\,\mathrm{e}^{-a\,|x|},\, x\in\mathds{R}.
\]
Then
\begin{itemize}
\item[\textup{(i)}] $\lim_{K\to\infty}\big\|f-Q_{{\scriptscriptstyle K}}P_{{\scriptscriptstyle K}}f\big\|_{L^{2}}=0$\,.
\item[\textup{(ii)}] $\lim_{K\to\infty}\big\|P_{{\scriptscriptstyle K}}f\big\|_{\ell^{2}}=\big\|f\big\|_{L^{2}}$\,.
\end{itemize}
\end{lemma}
\begin{proof}
Let us prove (i).
We have $Q_{{\scriptscriptstyle K}}P_{{\scriptscriptstyle K}}f(x)=e^{{\scriptscriptstyle (K)}}_{n(x)}(x)\;\int e^{{\scriptscriptstyle (K)}}_{n(x)}(y)\,f(y) \,\mathrm{d} y$
for $(n(x)-0.5)/\sqrt{K}-x_{*}\le x \le (n(x)+0.5)/\sqrt{K}-x_{*}$.
We get from our hypothesis
\[
Q_{{\scriptscriptstyle K}}P_{{\scriptscriptstyle K}}f(x)=f(x)+\mathcal{O}\big(K^{-\frac{1}{2}}\big)\,\mathrm{e}^{-a|x|}
\]
and the result follows.\newline
We now prove (ii).
From the isometric property of $Q_{K}$ on the space of piecewise functions $\mathscr{H}_{{\scriptscriptstyle K}}$, we get
\[
\big\|P_{{\scriptscriptstyle K}}f\big\|_{\ell^{2}}=\big\|Q_{{\scriptscriptstyle K}}P_{{\scriptscriptstyle K}}f\big\|_{L^{2}}
\]
and the result follows from (i).
\end{proof}
We now introduce the operator
\begin{equation*}
\mathscr{L}_{{\scriptscriptstyle K}}= Q_{{\scriptscriptstyle K}}\mathcal{L}_{{\scriptscriptstyle K}} P_{{\scriptscriptstyle K}}
\end{equation*}
and we keep the same notation for its closure in $L^{2}$. Since $\mathscr{L}_{{\scriptscriptstyle K}}$ when acting on $\mathscr{H}_{{\scriptscriptstyle K}}$ is conjugated to $\mathcal{L}_{{\scriptscriptstyle K}}$, we have
\[
\mathscr{L}_{{\scriptscriptstyle K}}\varphi^{{\scriptscriptstyle (K)}}_j=-\rho^{{\scriptscriptstyle (K)}}_j \varphi^{{\scriptscriptstyle (K)}}_j.
\]
We will prove in the next proposition that the operator $\mathscr{L}_{{\scriptscriptstyle K}}$ converges weakly, when $K\to+\infty$, to the operator
\begin{align}
\label{harm}
\MoveEqLeft[4] \EuScript{H}_{*} f(x)=\\
& b(x_{*})\,\frac{\mathrm{d}^{2}f(x)}{\mathrm{d} x^{2}}-\frac{\big(d'(x_{*})-b'(x_{*})\big)^{2}}{4b(x_{*})}\,x^{2}f(x)+\frac{d'(x_{*})-b'(x_{*})}{2}f(x) .
\nonumber
\end{align}
\begin{proposition}\label{convS}
Let $f\in C^{3}(\mathds{R})$ and assume that there exist $a>0$ and $A>0$ such that
\[
\sum_{j=0}^{3}\big|f^{(j)}(x)\big|\le A\, \mathrm{e}^{-a\,|x|},\, x\in\mathds{R}.
\]
Then
\[
\lim_{K\to\infty}\big\|\mathscr{L}_{{\scriptscriptstyle K}} f-\EuScript{H}_{*} f\big\|_{L^{2}}=0\,.
\]
\end{proposition}
\begin{proof}
By the assumption made on $f$, it follows easily that
\[
\lim_{K\to\infty}\big\|{\mathds{1}}_{\{|\,\cdot\, |>(\log K)^{2}\}}\;\EuScript{H}_{*} f\big\|_{L^{2}}=0.
\]
We have
\begin{align}\label{LKL}
& \mathscr{L}_{{\scriptscriptstyle K}} f(x)=
\sum_{n\geq 1} e^{{\scriptscriptstyle (K)}}_{n}(x)\,\left(\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n}\,\mu^{{\scriptscriptstyle (K)}}_{n+1}}\,\int e^{{\scriptscriptstyle (K)}}_{n+1}(y)\,f(y)\,\mathrm{d} y\right. \\
\nonumber
& \left.+\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n-1}\,\mu^{{\scriptscriptstyle (K)}}_{n}}\,{\mathds{1}}_{\{n>1\}} \,\int e^{{\scriptscriptstyle (K)}}_{n-1}(y)\,f(y)\,\mathrm{d} y -\big(\lambda^{{\scriptscriptstyle (K)}}_{n}+\mu^{{\scriptscriptstyle (K)}}_{n}\big)\,\int e^{{\scriptscriptstyle (K)}}_{n}(y)\,f(y)\,\mathrm{d} y\right).
\end{align}
It follows easily from the assumption made on $f$ and assumption \eqref{hipopo} that
\[
\lim_{K\to\infty}\big\|{\mathds{1}}_{\{|\,\cdot\, |>(\log K)^{2}\}}\mathscr{L}_{{\scriptscriptstyle K}} f\big\|_{L^{2}}=0\,.
\]
Therefore we only have to consider $|x|\le (\log K)^{2}$.
Note also that for such an $x$, the sum in \eqref{LKL} reduces to one element for $K$ large enough, namely $n=n(x) = \big\lfloor Kx_{*} +\sqrt{K} x +\frac{1}{2}\big\rfloor$.
For $x\in\mathds{R}$, we have
\begin{align*}
\mathscr{L}_{{\scriptscriptstyle K}} f(x)
& = \mathrm{e}^{{\scriptscriptstyle (K)}}_{n(x)}(x)\left(\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n(x)}\mu^{{\scriptscriptstyle (K)}}_{n(x)+1}}\,\int e^{{\scriptscriptstyle (K)}}_{n(x)+1}(y)f(y)\,\mathrm{d} y\right. \\
& \left.\hspace{2.1cm}+\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n(x)-1}\,\mu^{{\scriptscriptstyle (K)}}_{n(x)}} \int e^{{\scriptscriptstyle (K)}}_{n(x)-1}(y)f(y)\,\mathrm{d} y\right.\\
& \left. \hspace{2.1cm}-\,\big(\lambda^{{\scriptscriptstyle (K)}}_{n(x)}+\mu^{{\scriptscriptstyle (K)}}_{n(x)}\big)\int e^{{\scriptscriptstyle (K)}}_{n(x)}(y) f(y)\,\mathrm{d} y\right)\\
&= e^{{\scriptscriptstyle (K)}}_{n(x)}(x)\left(\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n(x)}\,\mu^{{\scriptscriptstyle (K)}}_{n(x)+1}}\int e^{{\scriptscriptstyle (K)}}_{n(x)}(y)f\left(y-\frac{1}{\sqrt{K}}\right)\mathrm{d} y\right.\\
& \left.\hspace{2cm}+\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n(x)-1}\,\mu^{{\scriptscriptstyle (K)}}_{n(x)}} \int e^{{\scriptscriptstyle (K)}}_{n(x)}(y) f\left(y+\frac{1}{\sqrt{K}}\right)\mathrm{d} y\right.\\
& \left. \hspace{2cm}-\big(\lambda^{{\scriptscriptstyle (K)}}_{n(x)}+\mu^{{\scriptscriptstyle (K)}}_{n(x)}\big)\int e^{{\scriptscriptstyle (K)}}_{n(x)}(y)f(y)\,\mathrm{d} y\right).
\end{align*}
Now we have
\begin{align*}
\MoveEqLeft \int e^{{\scriptscriptstyle (K)}}_{n(x)}(y)\,f(y)\,\mathrm{d} y \\
& =K^{\frac{1}{4}}\int_{-\frac{1}{2\sqrt{K}}}^{\frac{1}{2\sqrt{K}}}f\left(\frac{n(x)-K\xf}{\sqrt{K}}-s\right)\,\mathrm{d} s\\
&=K^{-\frac{1}{4}}\,f\left(\frac{n(x)-K\xf}{\sqrt{K}}\right)+\frac{K^{-\frac{5}{4}}}{24}f''\left(\frac{n(x)-K\xf}{\sqrt{K}}\right)+\mathcal{O}\Big(K^{-\frac{7}{4}}\Big)
\end{align*}
where the error term is unifom in $x$. Similarly
\begin{align*}
\MoveEqLeft \int e^{{\scriptscriptstyle (K)}}_{n(x)}(y)\,f\left(y\pm \frac{1}{\sqrt{K}}\right)\mathrm{d} y \\
&= \int e^{{\scriptscriptstyle (K)}}_{n(x)}(y)\,f(y)\,\mathrm{d} y \pm\frac{1}{\sqrt{K}}\int e^{{\scriptscriptstyle (K)}}_{n(x)}(y)\,f'(y)\,\mathrm{d} y \\
&\quad +\frac{1}{2K}\int e^{{\scriptscriptstyle (K)}}_{n(x)}(y)\,f''(y)\,\mathrm{d} y+ \mathcal{O}\Big(K^{-\frac{7}{4}}\Big)\\
& =K^{-\frac{1}{4}}\,f\left(\frac{n(x)-K\xf}{\sqrt{K}}\right) \pm K^{-\frac{3}{4}} \,f'\left(\frac{n(x)-K\xf}{\sqrt{K}}\right)\\
& \quad +\frac{13\,K^{-\frac{5}{4}}}{24} f''\left(\frac{n(x)-K\xf}{\sqrt{K}}\right)+\mathcal{O}\big(K^{-\frac{7}{4}}\big).
\end{align*}
Recall that
\[
\lambda^{{\scriptscriptstyle (K)}}_{n}=K b\left(\frac{n}{K}\right)\quad\text{and}\quad \mu^{{\scriptscriptstyle (K)}}_{n}=K d\left(\frac{n}{K}\right)
\]
hence
\[
\lambda^{{\scriptscriptstyle (K)}}_{n}=K b\big(x_{*}\big)+(n-K\xf) \,b'\big(x_{*}\big)+\mathcal{O}\left(\frac{(n-K\xf)^{2}}{K}\right)
\]
and
\[
\mu^{{\scriptscriptstyle (K)}}_{n}=K d\big(x_{*}\big)+(n-K\xf)\,d'\big(x_{*}\big)+\mathcal{O}\left(\frac{(n-K\xf)^{2}}{K}\right).
\]
After a tedious but straightforward computation, we obtain that
\begin{align*}
\mathscr{L}_{{\scriptscriptstyle K}} f(x)
&=\EuScript{H}_{*} f\left(\frac{n(x)-K\xf}{\sqrt{K}}\right)+\mathcal{O}\big(K^{-\frac{1}{4}}\big)\\
&=\EuScript{H}_{*} f(x)+\mathcal{O}\big(K^{-\frac{1}{4}}\;(\log K)^{4}\big)
\end{align*}
and the error term is uniform in $|x|\le (\log K)^{2}$.
We get
\[
\big\|{\mathds{1}}_{\{|\,\cdot\, |\leq (\log K)^{2}\}}\;\big(\mathscr{L}_{{\scriptscriptstyle K}} f-\EuScript{H}_{*} f\big)\big\|_{L^{2}}=\mathcal{O}\big(K^{-\frac{1}{4}}\;(\log K)^{6}\big)
\]
and the result follows.
\end{proof}
\begin{remark}\label{rem:OUHO}
Let us recall the relationship between the generator of the Ornstein-Uhlenbeck process \eqref{def-OU-generator} and \eqref{harm} which is, up to a minus sign and a shift, the Schr\"odinger operator
for the quantum harmonic oscillator. We refer to {\em e.g.} \cite[Chapter 3]{babusci-et-al} or \cite[Sections 4.4 and 4.9]{pavliotis}.
In $L^{2}$, the eigenvalues of\, $\EuScript{H}_{*}$ are $-\big(d'(x_{*})-b'(x_{*}))n$, $n\in\mathds{Z}_{{\scriptscriptstyle \geq 0}}$, and the corresponding eigenfunctions are
\begin{align}\label{eigenstates-HO}
&\psi_n(x)=\\
&\frac{1}{\sqrt{2^n n!}} \left(\frac{\big(d'(x_{*})-b'(x_{*})\big)^2}{2\pi b(x_{*})}\right)^{\frac{1}{4}} \mathrm{e}^{-\frac{d'(x_{*})-b'(x_{*})}{4b(x_{*})}x^2}H_n\left(\sqrt{\frac{d'(x_{*})-b'(x_{*})}{2b(x_{*})}}x\right)
\nonumber
\end{align}
where $(H_n)_n$ is the family of the physicists' Hermite polynomials defined by
\[
H_n(x)=(-1)^n \mathrm{e}^{x^2} \frac{\mathrm{d}^n}{\mathrm{d} x^n}\mathrm{e}^{-x^2}.
\]
One can check that $\EuScript{H}_{*}$ is conjugated to the generator of the Ornstein-Uhlenbeck process \eqref{def-OU-generator}
acting on $L^2\Big(\sqrt{\frac{d'(x_{*})-b'(x_{*})}{2\pi b(x_{*})}}\,\mathrm{e}^{-\frac{(d'(x_{*})-b'(x_{*})}{2b(x_{*})}x^2}\mathrm{d} x\Big)$ in the following way:
$\frac{1}{\psi_0}\EuScript{H}_{*}(\psi_0 f)=\EuScript{OU}_{*} f$.
\end{remark}
In Proposition \ref{convM0} we prove that the operator $\mathcal{L}_{{\scriptscriptstyle K}}$ converges weakly, when $K$ tends to infinity, to the operator $\EuScript{M}_{0}$ defined for $v\in c_{00}$ by
\begin{align}
\label{le-M0}
& \big(\EuScript{M}_{0} v\big)(n)=\\
\nonumber
& \sqrt{b'(0)\,d'(0)\,n\,(n+1)} \,v(n+1)+\sqrt{b'(0)\,d'(0)\,n\,(n-1)}\,v(n-1)\,{\mathds{1}}_{\{n>1\}}\\
\nonumber
& -n\,(b'(0)+d'(0)) v(n)).
\end{align}
Here again we denote the operator on $c_{00}$ and its closure by the same letter.
\begin{proposition}\label{convM0}
Let $u\in c_{00}$. Then
\[
\lim_{K\to\infty}\mathcal{L}_{{\scriptscriptstyle K}} u=\EuScript{M}_{0} u
\]
where $\EuScript{M}_{0}$ is defined in \eqref{le-M0}.
\end{proposition}
\begin{proof}
Follows from the fact that for each fixed $n$
\[
\lim_{K\to\infty}\lambda^{{\scriptscriptstyle (K)}}_{n}=b'(0)\,n
\qquad
\mathrm{and} \qquad
\lim_{K\to\infty}\mu^{{\scriptscriptstyle (K)}}_{n}=d'(0)\,n\;.
\]
\end{proof}
\subsection{Steps of the proof of Theorem \ref{main-theorem}}
The proof of Theorem \ref{main-theorem} relies on the following two theorems whose proofs are postponed to Section \ref{proof-maintheorem}.
Recall that for any fixed $K$, the spectrum $\mathrm{Sp}(\mathcal{L}_{{\scriptscriptstyle K}})$ is discrete, and let
\[
G=\bigcup_{j=0}^\infty\big(\rhoK{j}\big)^{\mathrm{{\scriptscriptstyle acc}}}
\]
where $\big(\rhoK{j}\big)^{\mathrm{{\scriptscriptstyle acc}}}$ is the set of accumulation points of $\big(\rhoK{j}\big)$ when $K\to+\infty$.
\begin{theorem}\label{liminf}
We have
\[
S_1\cup S_2 \subset G
\]
where $S_1$ and $S_2$ are defined in \eqref{def-S1} and \eqref{def-S2}.
\end{theorem}
This theorem is proved in Section \ref{proof-liminf}.
\begin{corollary}\label{cfini}
For every fixed $j$ we have
\[
\limsup_{{\scriptscriptstyle K}\to+\infty} \rho_j^{{\scriptscriptstyle (K)}}<+\infty.
\]
\end{corollary}
\begin{proof}
We proceed by contradiction.
Assume that there exists $j_0$ such that
\[
\limsup_{{\scriptscriptstyle K}\to+\infty} \rho_{j_0}^{{\scriptscriptstyle (K)}}=+\infty.
\]
Let $j_{c}=\min\{0<\ell\leq j_0:\limsup_{K\to+\infty} \rho_{\ell}^{{\scriptscriptstyle (K)}}=+\infty\}$.
Hence there exists $\alpha<+\infty$ such that $\limsup_{K\to+\infty} \rho_{j_c-1}^{{\scriptscriptstyle (K)}}=\alpha$.
By definition of $j_c$, there exists a diverging sequence $(K_p)_p$ such that $\lim_{p\to+\infty} \rho_{j_{c}}^{{\scriptscriptstyle (K_p)}}=+\infty$.
Let $\rho\in S_1\cup S_2$ such that $\rho>\alpha$.
If $j\leq j_c-1$, we have $\limsup_{p\to+\infty} \rho_{j}^{{\scriptscriptstyle (K_p)}}\leq \limsup_{p\to+\infty} \rho_{j_c-1}^{{\scriptscriptstyle (K_p)}}\leq \alpha<\rho$.
For all $j\geq j_c$, we have $\liminf_{p\to+\infty} \rho_{j}^{{\scriptscriptstyle (K_p)}}\geq \liminf_{p\to+\infty} \rho_{j_c}^{{\scriptscriptstyle (K_p)}}=+\infty$.
This implies $\rho\notin G$, contradicting Theorem \ref{liminf}.
\end{proof}
\begin{theorem}\label{prsquelafin}
We have
\[
S_1\cup S_2\supset G.
\]
Moreover, for each $j\in\mathds{Z}_{{\scriptscriptstyle \geq 0}}$, let $(K_{p})_p$ be a diverging sequence such that
\[
\lim_{p\to\infty}\rhoKp{j}{p}=\rho_{*}
\]
where $\rho_*$ is finite by Corollary \ref{cfini}.
Then
\begin{enumerate}[1)]
\item
If $\rho_{*}\in S_{1}\Delta S_{2}$ then
\begin{equation}\label{difsym}
\liminf_{p\to\infty}\; \min\left\{\bigg|\rhoKp{j+1}{p}-\rho_{*}\bigg|,\; \bigg|\rhoKp{j-1}{p}-\rho_{*}\bigg|\right\}>0\;.
\end{equation}
Moreover there are only two cases:
\begin{enumerate}
\item
If $\rho_*\in S_1$ then there exists a diverging sequence of integers $(p_\ell)$ such that $Q_{{\scriptscriptstyle K_{p_\ell}}} \phi^{{\scriptscriptstyle(K_{p_{\ell}}})}_j \xrightarrow[]{L^{2}} \varphi_*$,
where $\rho_*$ and $\varphi_*$ are such that $\EuScript{H}_{*} \varphi_*= - \rho_* \varphi_*$.
\item
If $\rho_*\in S_2$ then $\phi^{{\scriptscriptstyle(K_{p})}}_j\xrightarrow[]{\ell^{2}} \phi_*$, where $\rho_*$ and $\phi_*$ are such that $\EuScript{M}_{0} \phi_*= - \rho_* \phi_*$.
\end{enumerate}
\item If $\rho_{*}\in S_{1}\cap S_{2}$ then we have the following two assertions:
\begin{enumerate}
\item
There exists a diverging sequence of integers $(p_{\ell})$ such that:
\[
\text{either}\quad \lim_{\ell\to\infty}\rhoKp{j+1}{p_{\ell}}=\rho_{*}\quad\text{or}\quad\lim_{\ell\to\infty}\rhoKp{j-1}{p_{\ell}}=\rho_{*}.
\]
\item
We have
\begin{equation}\label{madmin}
\liminf_{p\to\infty}\; \min\left\{\bigg|\rhoKp{j+1}{p}-\rho_{*}\bigg|,\; \bigg|\rhoKp{j-1}{p}-\rho_{*}\bigg|\right\}=0
\end{equation}
and
\begin{equation}\label{madmax}
\liminf_{p\to\infty}\; \max\left\{\bigg|\rhoKp{j+1}{p}-\rho_{*}\bigg|,\; \bigg|\rhoKp{j-1}{p}-\rho_{*}\bigg|\right\}>0\;.
\end{equation}
\end{enumerate}
\end{enumerate}
\end{theorem}
Note that \eqref{difsym} means that if $\rho_{*}\in S_{1}\Delta S_{2}$ then $-\rho_{*}$ is a simple asymptotic eigenvalue, and either $-\rho_{*}$
is an eigenvalue of $\EuScript{H}_{*}$ if $\rho_{*}\in S_{1}$, or of $\EuScript{M}_{0}$ if $\rho_{*}\in S_{2}$. In addition, \eqref{madmin} and \eqref{madmax} mean that if
$\rho_{*}\in S_{1}\cap S_{2}$, then $-\rho_{*}$ is a double asymptotic eigenvalue which is an eigenvalue of both $\EuScript{H}_{*}$ and $\EuScript{M}_{0}$.
\noindent\textbf{Proof of Theorem \ref{main-theorem}.} The proof is recursive.
For $j=0$ it follows from \cite{CCM1} that $\lim_{{\scriptscriptstyle K}\to+\infty} \rho_0^{{\scriptscriptstyle (K)}}=0$.
Let $j\geq 0$ and assume that for $\ell\leq j$ (if any) $\lim_{{\scriptscriptstyle K}\to+\infty} \rho_\ell^{{\scriptscriptstyle (K)}}=\eta_\ell$.
We now prove that $\lim_{{\scriptscriptstyle K}\to+\infty} \rho_{j+1}^{{\scriptscriptstyle (K)}}=\eta_{j+1}$. There are several cases to consider.
\begin{itemize}
\item
If $\eta_j\in S_1\Delta S_2$, we claim that $\liminf_{{\scriptscriptstyle K}\to+\infty}\rho_{j+1}^{{\scriptscriptstyle (K)}}\geq \eta_{j+1}$. Otherwise, by Theorem \ref{prsquelafin} and the recursive hypothesis,
there would exist $K_p\to+\infty$ such that $\rho_{j+1}^{{\scriptscriptstyle (K_p)}}\to \eta_*<\eta_{j+1}$. Since by the recursive hypothesis
$\lim_{{\scriptscriptstyle K}\to+\infty} \rho_j^{{\scriptscriptstyle (K_p)}}=\eta_j$, we have $\eta_*\geq \eta_j$. From the first statement of Theorem \ref{prsquelafin} it follows that $\eta_*= \eta_j$.
This contradicts 1) of Theorem \ref{prsquelafin}.\newline
We now claim that $\limsup_{{\scriptscriptstyle K}\to+\infty}\rho_{j+1}^{{\scriptscriptstyle (K)}}\leq \eta_{j+1}$. Otherwise, by Theorem \ref{prsquelafin} and the recursive hypothesis,
there would exist $K_p\to+\infty$ such that $\rho_{j+1}^{{\scriptscriptstyle (K_p)}}\to \eta_*>\eta_{j+1}$. This implies that $\eta_{j+1}\not\in G$, contradicting Theorem \ref{liminf}.\newline
Hence, in this case, $\lim_{{\scriptscriptstyle K}\to+\infty}\rho_{j+1}^{{\scriptscriptstyle (K)}}= \eta_{j+1}$.
\item
If $\eta_j\in S_1\cap S_2$ (which implies $j>0$), we have two cases:
\begin{itemize}
\item
If $\eta_{j-1}=\eta_j$, then we claim that $\liminf_{{\scriptscriptstyle K}\to+\infty}\rho_{j+1}^{{\scriptscriptstyle (K)}}\geq \eta_{j+1}$. Otherwise, by the same argument as before,
there would exist $K_p\to+\infty$ such that $\rho_{j+1}^{{\scriptscriptstyle (K_p)}}\to \eta_{j}$, contradicting \eqref{madmax} in Theorem \ref{prsquelafin}.\newline
We now claim that $\limsup_{{\scriptscriptstyle K}\to+\infty}\rho_{j+1}^{{\scriptscriptstyle (K)}}\leq \eta_{j+1}$. Otherwise, as before, this would contradict that $\eta_{j+1}\in G$.
\newline
Hence, in this case, $\lim_{{\scriptscriptstyle K}\to+\infty}\rho_{j+1}^{{\scriptscriptstyle (K)}}= \eta_{j+1}$.
\item
If $\eta_{j-1}<\eta_j$, then we obviously have $\liminf_{{\scriptscriptstyle K}\to+\infty}\rho_{j+1}^{{\scriptscriptstyle (K)}}\geq \eta_j$.
If $\limsup_{{\scriptscriptstyle K}\to+\infty}\rho_{j+1}^{{\scriptscriptstyle (K)}}> \eta_j$, then there exists $K_p\to+\infty$ such that $\rho_{j+1}^{{\scriptscriptstyle (K_p)}}\to \eta_*>\eta_j$,
contradicting \eqref{madmin} in Theorem \ref{prsquelafin}.\newline
Hence, in this case, $\lim_{{\scriptscriptstyle K}\to+\infty}\rho_{j+1}^{{\scriptscriptstyle (K)}}= \eta_{j+1}$.
\end{itemize}
\end{itemize}
Therefore we $\lim_{{\scriptscriptstyle K}\to+\infty}\rho_{j+1}^{{\scriptscriptstyle (K)}}= \eta_{j+1}$. As announced, the proof of Theorem \ref{main-theorem} follows recursively.
\section{Properties of the eigenvectors}
Our aim in this part is to prove that for $K$ large enough, the eigenvectors $\phi^{{\scriptscriptstyle (K)}}_{j}$ of $\mathcal{L}_{{\scriptscriptstyle K}}$ (see \eqref{def-eigen-LK}) are functions whose representation is
sketched in the figure.
An eigenvector is `negligible' outside the union of a neighborhood of $1$, and a neighborhood of $Kx_{*}$. It is `non-negligible' in at least one of these neighborhoods.
\begin{figure}[htb!]
\centering
\includegraphics[scale=.75]{sketch}
\legend{{\small Figure: Schematic representation of one of the three possible `shapes' of the eigenvectors $\phi^{{\scriptscriptstyle (K)}}$ (with distortion).}}
\end{figure}
To separate the different behaviors, we introduce a `potential' defined by
\begin{equation}
\label{potential}
V_{n}(K)=\lambda^{{\scriptscriptstyle (K)}}_{n}+\mu^{{\scriptscriptstyle (K)}}_{n}-\,\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n}\mu^{{\scriptscriptstyle (K)}}_{n+1}}-\,\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n-1}\mu^{{\scriptscriptstyle (K)}}_{n}}\,{\mathds{1}}_{\{n>1\}}.
\end{equation}
For $\eta>0$, let $n_{\mathrm{g}}({\scriptstyle K},\eta)$ and $n_{\mathrm{d}}({\scriptstyle K},\,\eta)$ be integers such that $\llbracketn_{\mathrm{g}}({\scriptstyle K},\eta),n_{\mathrm{d}}({\scriptstyle K},\,\eta)\rrbracket$ is the maximal interval containing $K\xf/2$ such that
\[
\inf_{n\,\in\, \llbracketn_{\mathrm{g}}({\scriptstyle K},\eta),\,n_{\mathrm{d}}({\scriptstyle K},\,\eta)\rrbracket}\big(V_{n}(K)-\eta\big)>0\;.
\]
Let $n_{\ell}{\scriptstyle (K)}=\big\lfloor (\log K)^{2}\big\rfloor$ and $n_{r}{\scriptstyle (K)}=\big\lfloorK\xf-K^{\frac{2}{3}}\, \log K\big\rfloor$. It follows from our assumptions that for $K$ large enough
\[
1<n_{\mathrm{g}}({\scriptstyle K},\eta)<n_{\ell}{\scriptstyle (K)}\ll \frac{K\xf}{2}<n_{r}{\scriptstyle (K)}<n_{\mathrm{d}}({\scriptstyle K},\,\eta)\;.
\]
\begin{proposition}\label{strucvp}
For any $\eta>0$ there exists $a_{\eta}>0$, and $K_{\eta}>0$ such that, if $K>K_{\eta}$ and $\phi$ of norm one in $\ell^{2}$ satisfies
\[
\mathcal{L}_{{\scriptscriptstyle K}}\phi=-\rho\,\phi\quad\text{where}\quad\rho<\eta
\]
then
\[
\sup_{n_{\ell}{\scriptstyle (K)}\le n\le n_{r}{\scriptstyle (K)}}\big|\phi(n)\big| \le \mathrm{e}^{-a_{\eta}(\log K)^{2}}\;.
\]
\end{proposition}
\begin{proof} Let us consider an eigenvector $\phi$ of norm one in $\ell^{2}$ satisfying
$\
\mathcal{L}_{{\scriptscriptstyle K}}\,\phi= - \rho\,\phi$.
If $\phi({n_{\mathrm{g}}({\scriptstyle K},\eta)})\neq 0$, define $\tilde n_{\mathrm{g}}({\scriptstyle K},\eta)=n_{\mathrm{g}}({\scriptstyle K},\eta)$ and if needed, change the sign of $\phi$ such that $\phi({\tilde n_{\mathrm{g}}({\scriptstyle K},\eta)})>0$. If $\phi({n_{\mathrm{g}}({\scriptstyle K},\eta)})=0$, define
$\tilde n_{\mathrm{g}}({\scriptstyle K},\eta)=n_{\mathrm{g}}({\scriptstyle K},\eta)+1$ and if needed, change the sign of $\phi$ such that $\phi({\tilde n_{\mathrm{g}}({\scriptstyle K},\eta)})>0$ (note that $\phi({n_{\mathrm{g}}({\scriptstyle K},\eta)})=\phi({n_{\mathrm{g}}({\scriptstyle K},\eta)+1})=0$ contradicts the normalisation since
$\phi$ solves a second-order recurrence relation). Then, changing the definition of $n_{\mathrm{g}}({\scriptstyle K},\eta)$ if necessary, we can assume that $\phi({n_{\mathrm{g}}({\scriptstyle K},\eta)})> 0$.
Thanks to the local maximum-minimum principle (see Proposition \ref{principe}) we only have four cases.
\begin{enumerate}[1)]
\item
$\phi({n_{\mathrm{g}}({\scriptstyle K},\eta)+1})\ge \phi({n_{\mathrm{g}}({\scriptstyle K},\eta)})$ and $\phi({n})$ is increasing on $\llbracketn_{\mathrm{g}}({\scriptstyle K},\eta),n_{\mathrm{d}}({\scriptstyle K},\,\eta)\rrbracket$.
\item
$\phi({n_{\mathrm{g}}({\scriptstyle K},\eta)+1})< \phi({n_{\mathrm{g}}({\scriptstyle K},\eta)})$ and $\phi({n})$ is decreasing and stays nonnegative on $\llbracketn_{\mathrm{g}}({\scriptstyle K},\eta), \,n_{\mathrm{d}}({\scriptstyle K},\,\eta)\rrbracket$.
\item $\phi({n_{\mathrm{g}}({\scriptstyle K},\eta)+1})< \phi({n_{\mathrm{g}}({\scriptstyle K},\eta)})$ and $\phi({n})$ has a minimum in the interval $n_{\mathrm{min}}{\scriptstyle (K)}$ in $\llbracketn_{\mathrm{g}}({\scriptstyle K},\eta)-1, \,n_{\mathrm{d}}({\scriptstyle K},\,\eta)-1\rrbracket$ and $\phi({n_{\mathrm{min}}{\scriptstyle (K)}})\ge0$. Note that
$\phi({n})$ is decreasing on $\llbracketn_{\mathrm{g}}({\scriptstyle K},\eta), \,n_{\mathrm{min}}{\scriptstyle (K)}\rrbracket$ and increasing on $\llbracketn_{\mathrm{min}}{\scriptstyle (K)}, \,n_{\mathrm{d}}({\scriptstyle K},\,\eta)\rrbracket$.
\item
$\phi({n})$ is decreasing on $\llbracketn_{\mathrm{g}}({\scriptstyle K},\eta), \,n_{\mathrm{d}}({\scriptstyle K},\,\eta)\rrbracket$ and $\phi({n_{\mathrm{d}}({\scriptstyle K},\,\eta)})<0$.
\end{enumerate}
We first observe that since $b'(0)>d'(0)$ we have
\begin{align*}
\MoveEqLeft[10] \lim_{K\to\infty}\sup_{n\,\in \big\llbracket \frac{n_{\ell}{\scriptstyle (K)}}{2},\,n_{\ell}{\scriptstyle (K)}\big\rrbracket}\frac{\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n-1}\,\mu^{{\scriptscriptstyle (K)}}_{n}}}{\lambda^{{\scriptscriptstyle (K)}}_{n}+\mu^{{\scriptscriptstyle (K)}}_{n}-\eta-\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n}\mu^{{\scriptscriptstyle (K)}}_{n+1}}} \\
& =\frac{\sqrt{b'(0)\,d'(0)}}{b'(0)+d'(0)-\sqrt{b'(0)\,d'(0)}}<1.
\end{align*}
We also observe that there exists $c>0$ such that for $K$ large enough, and any $n\in\llbracketn_{r}{\scriptstyle (K)},\, n_{\mathrm{d}}({\scriptstyle K},\,\eta)\rrbracket$ we have
\[
\frac{\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n}\mu^{{\scriptscriptstyle (K)}}_{n+1}}}
{\lambda^{{\scriptscriptstyle (K)}}_{n}+\mu^{{\scriptscriptstyle (K)}}_{n}-\eta-\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n-1}\,\mu^{{\scriptscriptstyle (K)}}_{n}}}\le\frac{1}{1+\frac{c\,(n-K\xf)^{2}}{K^{2}}}\;.
\]
The result follows by inspecting the monotonicity in the different cases and using the last part of Proposition \eqref{locmax}.
\end{proof}
\begin{theorem}\label{thm-eig}
Let $\phi\in\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle K}})\subset \ell^{2}$ of norm $1$, satisfying
\[
\mathcal{L}_{{\scriptscriptstyle K}} \phi=-\rho\, \phi
\]
for some real $\rho$.
Then there exist $C(\phi)>0$ and an integer $r(\phi)$ such that for all $n\geq r(\phi)$ we have
\[
|\phi(n)| \leq C(\phi) \, 2^{-n}.
\]
Moreover, $(\phi(n))_n$ does not vanish and is of constant sign.
\end{theorem}
\begin{proof}
It follows from our hypothesis that for any $K>1$ there exists an integer $r_0{\scriptstyle (K)}$ such that, for all $n\geq r_0{\scriptstyle (K)}$, we have
\[
0<\frac{\sqrt{\lambda^{{\scriptscriptstyle (K)}}_n\mu^{{\scriptscriptstyle (K)}}_{n+1}}}{\lambda^{{\scriptscriptstyle (K)}}_n+\mu^{{\scriptscriptstyle (K)}}_{n+1}+\rho-\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n-1}\,\mu^{{\scriptscriptstyle (K)}}_n}}\leq\frac12.
\]
We can assume that $(\phi(n))_n$ is a sequence of real numbers and $\phi(r_0{\scriptstyle (K)})> 0$.
We start by proving that $(\phi(n))_n$ is positive and decreasing for $n\ge r_0{\scriptstyle (K)}$.
There are only the following four possibilities.
\begin{enumerate}
\item
$\phi(r_0{\scriptstyle (K)}+1)\geq \phi(r_0{\scriptstyle (K)})$. It follows from Proposition \ref{pouet} that $\phi$ is increasing for
$n\geq r_0{\scriptstyle (K)}$, contradicting that $\phi$ has norm $1$.
\item
$\phi(r_0{\scriptstyle (K)}+1)< \phi(r_0{\scriptstyle (K)})$, and there exists $r'>r_0{\scriptstyle (K)}$ such that $\phi(r')<0$ and $\phi$ decreases on
$\llbracket r_0{\scriptstyle (K)},r'\rrbracket$, and $\phi\geq 0$ on $\llbracket r_0{\scriptstyle (K)},r'-1\rrbracket$.
Then by Proposition \ref{pouet}, $\phi$ is decreasing for $n\geq r'$, contradicting that $\phi$ is normalized.
\item
$\phi(r_0{\scriptstyle (K)}+1)< \phi(r_0{\scriptstyle (K)})$, and there exists $r'>r_0{\scriptstyle (K)}$ such that $\phi(r')<0$ and $\phi$ is not monotonous on
$\llbracket r_0{\scriptstyle (K)},r'\rrbracket$, and $\phi\geq 0$ on $\llbracket r_0{\scriptstyle (K)},r'-1\rrbracket$.
Then there exists $r''<r'$ such that $\phi$ is decreasing on $\llbracket r_0{\scriptstyle (K)},r''\rrbracket$, and such that $\phi(r''+1)\geq \phi(r'')$. If $ \phi(r'')>0$, then we are in
case 1. If $\phi(r''+1)>0$, it follows from Proposition \ref{pouet} that $\phi$ is increasing for $n\geq r''+1$, contradicting that $\phi$ has norm $1$. If $\phi(r'''+1)= \phi(r''')=0$ then $\phi$ is the null sequence
as solution of a second order equation, which leads to a contradiction.
\item
$\phi(r_0{\scriptstyle (K)}+1)< \phi(r_0{\scriptstyle (K)})$, $\phi\geq 0$. Suppose that there exists a local minimum at $r'''$ (finite). Then if $\phi(r'''+1)\geq \phi(r''')>0$, we are in case 1.
If $\phi(r'''+1)>\phi(r''')=0$, then it follows from Proposition \ref{pouet} that $\phi$ is increasing for $n\geq r'''+1$, contradicting that $\phi$ has norm $1$.
If $\phi(r'''+1)= \phi(r''')=0$ then $\phi$ is the null sequence, which leads to a contradiction.
\end{enumerate}
Therefore $\phi$ is striclty positive and monotone decreasing. The result then follows by using the last part of Proposition \ref{locmax}.
\end{proof}
Let us now prove two key lemmas.
\begin{lemma}\label{convergenceS2}
Let $\phi_{{\scriptscriptstyle K}}\in \mathrm{Dom}\big(\mathcal{L}_{K}\big)$ be a normalized sequence such that
\[
\mathcal{L}_{{\scriptscriptstyle K}}\phi^{{\scriptscriptstyle (K)}}=-\rho^{{\scriptscriptstyle (K)}}\phi^{{\scriptscriptstyle (K)}}.
\]
Assume that there exists a diverging sequence $(K_{p})$ such that
\[
0\leq \lim_{p\to\infty} \rho^{{\scriptscriptstyle (K_p)}}=\rho_{*}<+\infty
\]
and
\[
\limsup_{p\to\infty}\big\|\phi^{{\scriptscriptstyle (K_p)}}{\mathds{1}}_{\{\cdot\,\le n_l(K_p)\}}\big\|_{\ell^{2}}>0.
\]
Then $\rho_{*}\in S_{2}$ and there exists a diverging subsequence
$\big(K_{p_{\ell}}\big)$ such that the limit
\[
\lim_{\ell\to\infty} \phi^{(K_{p_{\ell}})}=\phi_{*}
\]
exists in $\ell^{2}$, $\|\phi_{*}\|_{\ell^{2}}>0$ and $\phi_{*}$ is an eigenvector of\, $\EuScript{M}_{0}$ with eigenvalue $-\rho_{*}$.
\end{lemma}
\begin{proof}
Let $(p_{\ell})$ be a diverging sequence of integers such that
\[
\lim_{\ell\to\infty} \big\|\phi^{(K_{p_{\ell}})}\;{\mathds{1}}_{\{\cdot\,\le n_l(K)\}}\big\|_{\ell^{2}}
=\limsup_{p\to\infty}\big\|\phi^{(K_{p})}\;{\mathds{1}}_{\{\cdot\,\le n_l(K_p)\}}\big\|_{\ell^{2}}\;.
\]
We define for each $\ell$ a normalized sequences in $\ell^{2}$ by
\[
\psi_{\ell}(n)=\frac{\phi^{(K_{p_{\ell}})}(n)\; {\mathds{1}}_{\{n\,\le n_l(K_{p_{\ell}})\}} }{
\big\|\phi^{(K_{p_{\ell}})}\;{\mathds{1}}_{\{\cdot\,\le n_l(K_{p_\ell})\}}\big\|_{\ell^{2}}}\,.
\]
It is easy to verify using Proposition \ref{strucvp} that $\psi_{\ell}\in \mathrm{Dom}(\EuScript{M}_{0})$ and
\[
\big\|\EuScript{M}_{0} \psi_{\ell}+\rho^{(K_{p_{\ell}})}\psi_{\ell}\|_{\ell^{2}}\le \Oun\;
\frac{(\log K_{p_{\ell}})^{2} \,\mathrm{e}^{-a\,(\log K_{p_{\ell}})^{2}}}{\big\|\phi^{(K_{p_{\ell}})}\;
{\mathds{1}}_{\{\cdot\,\le n_l(K_{p_{\ell}})\}}\big\|_{\ell^{2}}}\;.
\]
The first result follows from Proposition \ref{yaspectre} since the r.h.s. tends to zero.
The second result follows from Proposition \ref {approxvp} since
the spectrum of $\EuScript{M}_{0}$ is discrete and
simple by Theorem \ref{specdeux}.
\end{proof}
\begin{lemma}\label{convergenceS1}
Let $\phi^{(K)}\in \mathrm{Dom}\big(\mathcal{L}_{K}\big)$ be a normalized sequence such that
\[
\mathcal{L}_{{\scriptscriptstyle K}}\phi^{{\scriptscriptstyle (K)}}=-\rho^{{\scriptscriptstyle (K)}}\;\phi^{{\scriptscriptstyle (K)}}.
\]
Let us assume that there exists a diverging subsequence $(K_{p})$ such that
\[
0\leq \lim_{p\to\infty} \rho^{(K_{p})}=\rho_{*}<+\infty
\]
and
\[
\lim_{p\to\infty}\big\|\phi^{(K_{p})}\;{\mathds{1}}_{\{\cdot\,\ge n_r(K_p)\}}\big\|_{\ell^{2}}>0.
\]
Then $\rho_{*}\in S_{1}$ and there exists a diverging subsequence of integers $(p_{\ell})$ such that
\[
\lim_{\ell\to\infty}\frac{ Q_{K_{p_{\ell}}}\phi^{(K_{p_{\ell}})}}{\big\| Q_{K_{p_{\ell}}}\phi^{(K_{p_{\ell}})}\big\|_{L^{2}}}=\psi_{*}
\]
exists in $L^{2}$, $\|\psi_{*}\|_{L^{2}}=1$, and $\psi_{*}$ is an eigenvector of\, $\EuScript{H}_{*}$ with eigenvalue $-\rho_{*}$.
\end{lemma}
\begin{proof}
We define for each $p$ a normalized sequence in $\ell^{2}$ by
\[
\psi^{(K_{p})}(n)=\frac{\phi^{(K_{p})}(n)\; {\mathds{1}}_{\{n\,\ge\, n_r(K_p)\}} }{\big\|\phi^{(K_{p})}\;{\mathds{1}}_{\{\cdot\,\ge\, n_r(K_p)\}}\big\|_{\ell^{2}}}\;.
\]
It is easy to verify using Proposition \ref{strucvp} that $\psi^{(K_{p})}\in \mathrm{Dom}\big(\mathcal{L}_{K_p})$ and
\begin{equation}\label{quasierreur}
\big\|\mathcal{L}_{K_{p}} \psi^{(K_{p})} +\rho^{(K_{p})}\psi^{(K_{p})}\|_{\ell^{2}}\le
\Oun\;\frac{K_{p} \,\mathrm{e}^{-a\,(\log K_{p})^{2}}}{ \big\|\phi^{(K_{p})}\;{\mathds{1}}_{\{\cdot\,\ge\, n_r(K_p)\}}\big\|_{\ell^{2}}}\;.
\end{equation}
We apply Proposition \ref{sobmodif}, Lemma \ref{unifint}, Lemma \ref{regu}, and Theorem \ref{KFR} to conclude that there exists a diverging sequence
of integers $(p_{\ell})$ such that the sequence of functions $Q_{K_{p_{\ell}}}\psi^{(K_{p_\ell})}$ converges in $L^{2}$ to a normalised fonction $\psi_{*}$.
Let $u\in \mathscr{D}$. We have from \eqref{quasierreur}
\[
\lim_{\ell\to\infty}\big\langle P_{K_{p_{\ell}}}u\,,\,\mathcal{L}_{K_{p_{\ell}}}\, \psi^{(K_{p_{\ell}})}+\rho^{(K_{p_{\ell}})}\,\psi^{(K_{p_{\ell}})}\big\rangle_{\ell^{2}}=0
\]
hence (since $P_{K_{p_{\ell}}}\,u\in \mathrm{Dom}\big(\mathcal{L}_{K_{p_{\ell}}}\big)$)
\[
\lim_{\ell\to\infty}\big\langle \mathcal{L}_{K_{p_{\ell}}}\,P_{K_{p_{\ell}}}u\,,\,\psi^{(K_{p_{\ell}})}\big\rangle_{\ell^{2}}=
-\rho_{*}\lim_{\ell\to\infty}\big\langle P_{K_{p_{\ell}}}u\,,\,\psi^{(K_{p_{\ell}})}\big\rangle_{\ell^{2}}\;.
\]
From the isometric property of $Q$ (see Lemma \ref{isometry}) we get
\begin{align*}
\MoveEqLeft \lim_{\ell\to\infty}\big\langle Q_{K_{p_{\ell}}}\mathcal{L}_{K_{p_{\ell}}}\,P_{K_{p_{\ell}}}u\,,\,Q_{K_{p_{\ell}}}\psi^{(K_{p_{\ell}})}\big\rangle_{L^{2}}\\
&=-\rho_{*}\lim_{\ell\to\infty}\big\langle Q_{K_{p_{\ell}}} P_{K_{p_{\ell}}}u\,,\,Q_{K_{p_{\ell}}} \psi^{(K_{p_{\ell}})}\big\rangle_{L^{2}}\;.
\end{align*}
In other words
\[
\lim_{\ell\to\infty}\big\langle \mathscr{L}_{K_{p_{\ell}}}u,Q_{K_{p_{\ell}}}\psi^{(K_{p_{\ell}})}\big\rangle_{L^{2}}
=-\rho_{*}\lim_{\ell\to\infty}\big\langle Q_{K_{p_{\ell}}}P_{K_{p_{\ell}}}u,Q_{K_{p_{\ell}}}\psi^{(K_{p_{\ell}})}\big\rangle_{L^{2}}.
\]
Using the convergence in $L^{2}$ of $\big(\,Q_{K_{p_{\ell}}}\psi^{(K_{p_{\ell}})}\big)_\ell$ to $\psi_{*}$,
Proposition \ref{convS}, and Lemma \ref{isomQP}, we get for all $u\in \mathscr{D}$,
\[
\big\langle
\EuScript{H}_{*} u,\psi_{*}\big\rangle_{L^{2}}=-\rho_{*}\big\langle u\,,\, \psi_{*}\big\rangle_{L^{2}}\;.
\]
Since $\mathscr{D}$ is dense in the domain of the self-adjoint operator $\EuScript{H}_{*}$, we conclude that $\psi_{*}$ is an eigenvector of $\EuScript{H}_{*}$.
\end{proof}
We now state three key propositions.
\begin{proposition}\label{decoupe}
Let $j$ be fixed and let $(K_{p})$ be a diverging sequence such that
\[
\lim_{p\to\infty} \rhoKp{j}{p}=\rho_{*}\;.
\]
Let $\phiKp{j}{p}$ be a normalized eigenvector of $\mathcal{L}_{{\scriptscriptstyle K}_p}$ with eigenvalue $-\rhoKp{j}{p}$.
Then there exists an infinite sequence of integers $(p_\ell)$ such that
\begin{itemize}
\item
$
\phiKp{j}{p_\ell}\,{\mathds{1}}_{\{\cdot\,<\,n_l({\scriptscriptstyle K}_{p_\ell})\}}\xrightarrow[]{\ell^{2}} \phi_*
$, where $\phi_*$ is either the null sequence or an eigenvector of \,$\EuScript{M}_{0}$ with eigenvalue $-\rho_*$, namely $\rho_*\in S_{2}$.
\item
$
Q_{K_{p_{\ell}}}\phiKp{j}{p_\ell}\,{\mathds{1}}_{\{\cdot\,\ge\, n_r({\scriptscriptstyle K}_{p_\ell})\}} \xrightarrow[]{L^{2}} \varphi_*
$, where $\varphi_*$ is either the null function or an eigenvector of \,$\EuScript{H}_{*}$ with eigenvalue $-\rho_*$, namely $\rho_*\in S_{1}$.
\end{itemize}
Moreover, $\|\phi_*\|_{\ell^{2}}^2+\|\varphi_*\|_{L^{2}}^2=1$.
\end{proposition}
\begin{proof}
We have either
\[
\lim_{p\to\infty}\Big\|\phiKp{j}{p}\,{\mathds{1}}_{\{\cdot\,<n_l(K_p)\}}\Big\|_{\ell^{2}}=0
\]
or there exists an infinite sequence of integers $(p_{\ell})$ such that
\[
\lim_{\ell\to\infty}\Big\|\phiKp{j}{p_\ell}\,{\mathds{1}}_{\{\cdot\,<n_l(K_{p_{\ell}})\}}\Big\|_{\ell^{2}}>0\,.
\]
The first statement of the proposition follows from Lemma \ref{convergenceS2} applied to the diverging sequence $(K_{p_\ell})$.
Similarly, either
\[
\lim_{p\to\infty}\Big\|\phiKp{j}{p}\,{\mathds{1}}_{\{\cdot\,\ge\,n_{r}{\scriptstyle (K)}\}}\Big\|_{\ell^{2}}=0
\]
or there exists an infinite sequence of integers $(p_{\ell})$ such that
\[
\lim_{\ell\to\infty}\Big\|\phiKp{j}{p_\ell}\,{\mathds{1}}_{\{\cdot\,\ge\, n_r(K_{p_\ell})\}}\Big\|_{\ell^{2}}>0.
\]
The second statement of the proposition follows from Lemma \ref{convergenceS1} applied to the diverging sequence $(K_{p_\ell})$.
The last statement follows from the normalisation $\phiKp{j}{p}$ and Proposition \ref{strucvp}.
\end{proof}
\begin{proposition}\label{prop-quasi-S2}
Let $\rho\in S_2$.
Then there exists a normalized vector $v_\rho\in\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle K}})$ with eigenvalue $-\rho$ such that
\[
\lim_{K\to\infty} \|\mathcal{L}_{{\scriptscriptstyle K}} v_\rho +\rho\, v_\rho\|_{\ell^2}=0.
\]
Moreover
\[
\lim_{K\to+\infty} \big\| v_\rho {\mathds{1}}_{\{\cdot\,>\,n_l({\scriptscriptstyle K})\}}\big\|_{\ell^{2}}=0.
\]
\end{proposition}
\begin{proof}
Let $v_\rho$ be a normalized eigenvector corresponding to the eigenvalue $-\rho$ for the operator $\EuScript{M}_{0}$ in the space $\ell^{2}$.
From the assumptions we have for $n\leq \lfloor \log K\rfloor$
\[
\lambda^{{\scriptscriptstyle (K)}}_n=b'(0) + \mathcal{O}\left(\frac{\log K}{K}\right)\quad\text{and}\quad\mu^{{\scriptscriptstyle (K)}}_n=d'(0)+ \mathcal{O}\left(\frac{\log K}{K}\right).
\]
Since $\EuScript{M}_{0} v_\rho+\rho\, v_\rho=0$, the reader can easily check that
\[
\|\mathcal{L}_{{\scriptscriptstyle K}} v_\rho +\rho\, v_\rho\|_{\ell^2(\mathds{Z}_{{\scriptscriptstyle >0}}\cap \{ 1,\ldots,\lfloor\log K\rfloor\})} \leq \mathcal{O}\left(\frac{\log K}{K}\right).
\]
Now using Theorem \ref{specdeux} we have
\[
v_{\rho}(n)=\sqrt{n}\,\left(\frac{d'(0)}{b'(0)}\right)^{\frac{n}{2}}P(n)
\]
where $P$ is certain polynomial. Hence there exists $c_v>0$ such that for any $n\in \mathds{Z}_{{\scriptscriptstyle >0}}$,
$|v_{\rho}(n)|\leq c_v \left(\frac{d'(0)}{b'(0)}\right)^{\frac{n}{4}}$.
By \eqref{hipopo} there exists $c>0$ such $b(x)+d(x)\leq c\, \mathrm{e}^x$ for all $x\geq 0$.
This implies that $\lambda_n^{{\scriptscriptstyle (K_p)}}\leq c K_p\, \mathrm{e}^{n/K_p}$ and $\mu_n^{{\scriptscriptstyle (K_p)}}\leq c K_p \,\mathrm{e}^{n/K_p}$.
The reader can easily check that
\[
\|\mathcal{L}_{{\scriptscriptstyle K}} v_\rho +\rho\, v_\rho\|_{\ell^2(\mathds{Z}_{{\scriptscriptstyle >0}}\cap \{\lfloor \log K\rfloor+1,\ldots,\infty\})} \leq \Oun \left(\frac{d'(0)}{b'(0)}\right)^{\frac{\log K}{4}}.
\]
It follows that $v_\rho\in \mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle K}})$, and (remember that $b'(0)>d'(0)$)
\[
\lim_{K\to\infty} \|\mathcal{L}_{{\scriptscriptstyle K}} v_\rho +\rho \, v_\rho\|_{\ell^{2}}=0.
\]
The other statement follows at once from the exponential decay of $v$.
\end{proof}
\begin{proposition}\label{prop-quasi-S1}
Let $\rho\in S_1$.
Then there exists a sequence of normalized vectors $(\psi_\rho^{(K)})_K\subset \ell^{2}$
such that $\psi_\rho^{{\scriptscriptstyle (K)}}\in\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle K}})$, and
\[
\lim_{K\to\infty}\big\| \mathcal{L}_{{\scriptscriptstyle K}} \psi_\rho^{{\scriptscriptstyle (K)}}+\rho\, \psi_\rho^{{\scriptscriptstyle (K)}}\big\|_{\ell^{2}}=0.
\]
Moreover
\[
\lim_{K\to\infty}\big\| \psi_\rho^{{\scriptscriptstyle (K)}} {\mathds{1}}_{\{\cdot\,<n_r{\scriptscriptstyle (K)}\}}\big\|_{\ell^{2}}=0.
\]
\end{proposition}
\begin{proof}
Let $\varphi_\rho$ be a real normalized eigenvector of $\EuScript{H}_{*}$ corresponding to the eigenvalue $-\rho$ (in $L^{2}$).
Since $\varphi_\rho$ is a (rescaled) Hermite function (see \cite{babusci-et-al} or Remark \ref{rem:OUHO}), it satisfies the hypothesis of Proposition \ref{convS}, hence
\[
\lim_{K\to\infty}\big\|\mathscr{L}_{{\scriptscriptstyle K}} \varphi_\rho+\rho\,\varphi_\rho\big\|_{L^{2}}=0.
\]
Using Lemma \ref{isomQP} we get
\[
\lim_{K\to\infty}\big\|\mathscr{L}_{{\scriptscriptstyle K}} \varphi_\rho+\rho\;Q_{{\scriptscriptstyle K}}P_{{\scriptscriptstyle K}}\varphi_\rho\big\|_{L^{2}}=0
\]
and then
\[
\lim_{K\to\infty}\big\|\mathcal{L}_{{\scriptscriptstyle K}} \, P_{{\scriptscriptstyle K}}\varphi_\rho+\rho\,P_{{\scriptscriptstyle K}}\varphi_\rho\big\|_{\ell^{2}}=0\;
\]
and
\[
\lim_{K\to\infty}\big\|P_{{\scriptscriptstyle K}}\varphi_\rho\big\|_{\ell^{2}}=1.
\]
The first statement follows by letting $\psi_{\rho}^{{\scriptscriptstyle (K)}}(n)=(P_{{\scriptscriptstyle K}}\varphi_\rho)(n)$ where $P_{{\scriptscriptstyle K}}$ is defined in \eqref{PP}.
The other statement follows from an exponential bound on the decay of $\varphi_\rho$.
\end{proof}
\section{Proof of Theorems \ref{liminf} and \ref{prsquelafin}}\label{proof-maintheorem}
\subsection{Proof of Theorem \ref{liminf}}\label{proof-liminf}
The proof of Theorem \ref{liminf} is an immediate consequence of the
following two propositions.
\begin{proposition}
We have $\ S_2\subset G$.
\end{proposition}
\begin{proof}
The proof is by contradiction.
Let $\rho\in S_2$ and assume
$\rho\notin G$.
Then there exists $\eta>0$ be such that for $K$ large enough, $\left[\,\rho-\eta,\rho+\eta\,\right]\cap G=\emptyset$.
It follows from Proposition \ref{prop-quasi-S2} that there exists a normalized vector $v$ in $\ell^{2}$, such that $v\in\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle K}})$, and
\[
\lim_{K\to\infty} \|\mathcal{L}_{{\scriptscriptstyle K}} v +\rho\, v\|_{\ell^{2}}=0.
\]
Therefore, using Proposition \ref{yaspectre}, we obtain a contradiction, hence the proof is finished.
\end{proof}
\begin{proposition}
We have $\ S_1\subset G$.
\end{proposition}
\begin{proof}
The proof is by contradiction.
Let $\rho\in S_1$ and assume $\rho\notin G$.
Then there exists $\eta>0$ be such that for $K$ large enough, $[\rho-\eta,\rho+\eta]\cap G=\emptyset$.
It follows from Proposition \ref{prop-quasi-S1} that there exists a sequence of normalized vectors $(\psi^{(K)})_K\subset \ell^{2}$
such that $\psi^{(K)}\in\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle K}})$ and
\[
\lim_{K\to\infty}\big\| \mathcal{L}_{{\scriptscriptstyle K}} \psi^{{\scriptscriptstyle (K)}} +\rho\, \psi^{{\scriptscriptstyle (K)}}\big\|_{\ell^{2}}=0.
\]
Therefore, using Proposition \ref{yaspectre}, we obtain a contradiction, hence the proof is finished.
\end{proof}
\subsection{Proof of Theorem \ref{prsquelafin}
The first statement follows at once from Proposition \ref{decoupe}.\\
The proof of \eqref{difsym} is by contradiction.
Assume
\[
\liminf_{p\to\infty}\; \min\left\{\bigg|\rhoKp{j+1}{p}-\rho_{*}\bigg|, \bigg|\rhoKp{j-1}{p}-\rho_{*}\bigg|\right\}=0.
\]
Assume $\rho_*\in S_1\backslash S_2$ and let $(p_\ell)$ be a diverging sequence of positive integers such that
$\rho_i^{({\scriptscriptstyle K}_{p_\ell})}\to \rho_*$ and $\rho_{i+1}^{({\scriptscriptstyle K}_{p_\ell})}\to \rho_*$ (where $i=j$ or $i=j-1$).
Let $\phi_i^{({\scriptscriptstyle K}_{p_\ell})}$ be a normalized eigenvector of $\mathcal{L}_{K_{p_{\ell}}}$ corresponding to the eigenvalue $-\rho_i^{({\scriptscriptstyle K}_{p_\ell})}$.
We define $\phi^{({\scriptscriptstyle K}_{p_\ell})}_{i+1}$ similarly.
We claim that
\[
\limsup_{\ell\to+\infty} \Big\|\phiKp{i}{p_\ell}\,{\mathds{1}}_{\{\cdot\,<\,n_l({\scriptscriptstyle K}_{p_\ell})\}}\Big\|_{\ell^{2}}=0.
\]
Otherwise Proposition \ref{decoupe} would imply that $\rho_*\in S_2$, a contradiction.
By Proposition \ref{strucvp} we have
\[
\lim_{\ell\to+\infty} \Big\|\phiKp{i}{p_\ell}\,{\mathds{1}}_{\{\cdot\,>\,n_r({\scriptscriptstyle K}_{p_\ell})\}}\Big\|_{\ell^{2}}=1.
\]
This implies 1-a). \\
By a similar argument, we have
\[
\lim_{\ell\to+\infty} \Big\|\phiKp{i+1}{p_\ell}\,{\mathds{1}}_{\{\cdot\,>\,n_r({\scriptscriptstyle K}_{p_\ell})\}}\Big\|_{\ell^{2}}=1.
\]
By Proposition \ref{decoupe}, there exists a diverging sequence of integers $(\ell_r)$ such that
\[
\Big\| Q_{K_{p_{\ell_r}}}\Big(\phi_i^{(K_{p_{\ell_r}})}\,{\mathds{1}}_{\{\cdot\,>\,n_r({\scriptscriptstyle K}_{p_{\ell_r}})\}}\Big)-\varphi_*\Big\|_{L^{2}}\to 0
\]
where $\varphi_*\in\mathrm{Dom}(\EuScript{H}_{*})$ is a normalized eigenfunction of $\EuScript{H}_{*}$ corresponding to the eigenvalue $-\rho_*$.
By Proposition \ref{decoupe} again, there exists a diverging sequence of integers $(r_s)$ such that
\[
\Big\| Q_{K_{p_{\ell_{r_s}}}}\Big(\phi_{i+1}^{(K_{p_{\ell_{r_s}}})}\,{\mathds{1}}_{\{\cdot\,>\,n_r({\scriptscriptstyle K}_{p_{\ell_{r_s}}})\}}\Big)-\varphi'_*\Big\|_{L^{2}}\to 0
\]
where $\varphi'_*\in\mathrm{Dom}(\EuScript{H}_{*})$ is a normalized eigenfunction of $\EuScript{H}_{*}$ corresponding to the eigenvalue $-\rho_*$.
Since $\phi_{i}^{(K_{p_{\ell_{r_s}}})}$ and $\phi_{i+1}^{(K_{p_{\ell_{r_s}}})}$ are orthogonal in $\ell^{2}$, it follows from the previous estimates that
\[
\lim_{s\to+\infty}
\left\langle \phi_{i}^{(K_{p_{\ell_{r_s}}})}\,{\mathds{1}}_{\{\cdot\,>\,n_r({\scriptscriptstyle K}_{p_{\ell_{r_s}}})\}},\phi_{i+1}^{(K_{p_{\ell_{r_s}}})}\,{\mathds{1}}_{\{\cdot\,>\,n_r({\scriptscriptstyle K}_{p_{\ell_{r_s}}})\}}\right\rangle_{\ell^{2}}=0.
\]
By Lemma \ref{isometry} we have
\[
\left\langle \!Q_{K_{p_{\ell_{r_s}}}}\!\!\Big(\phi_{i}^{(K_{p_{\ell_{r_s}}})}{\mathds{1}}_{\{\cdot\,>\,n_r({\scriptscriptstyle K}_{p_{\ell_{r_s}}})\}}\Big),
Q_{K_{p_{\ell_{r_s}}}}\!\!\Big(\phi_{i+1}^{(K_{p_{\ell_{r_s}}})}{\mathds{1}}_{\{\cdot\,>\,n_r({\scriptscriptstyle K}_{p_{\ell_{r_s}}})\}}\Big)\!\right\rangle_{\!\!L^{2}}
\xrightarrow[s\to\infty]{}0.
\]
In other words, $\langle \varphi_*,\varphi'_*\rangle_{L^{2}}=0$. This
is a contradiction since $\varphi_*$ and $\varphi'_*$ are normalized eigenfunctions of $\EuScript{H}_{*}$ corresponding to the same
eigenvalue $-\rho_*$ which is simple.
The case $\rho_*\in S_2\backslash S_1$ is similar (using again Proposition \ref{decoupe}), so it is left to the reader.
\smallskip
Let us now assume that $\rho_* \in S_{1}\cap S_{2}$. We now prove \eqref{madmin} by contradiction. So we assume that there exists $\delta>0$ such that
\[
\liminf_{p\to\infty}\; \min\left\{\bigg|\rhoKp{j+1}{p}-\rho_{*}\bigg|, \bigg|\rhoKp{j-1}{p}-\rho_{*}\bigg|\right\}>\delta.
\]
Since $\rho_* \in S_{1}$ and from Proposition \ref{prop-quasi-S1}, there exists a sequence of normalized vectors $(\psi_{\rho_*}^{(K_p)})_p\subset \ell^{2}$
such that $\psi_{\rho_*}^{(K_p)}\in\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle K}_p})$ for all $p$, and
\[
\lim_{p\to\infty}\Big\| \mathcal{L}_{{\scriptscriptstyle K}_p} \psi_{\rho_*}^{(K_p)} -\rho_* \psi_{\rho_*}^{(K_p)}\Big\|_{\ell^{2}}=0.
\]
We also have (since $\rho^{({\scriptscriptstyle K}_p)}_j \to \rho_*$) that
\[
\lim_{p\to\infty}\Big\| \mathcal{L}_{{\scriptscriptstyle K}_p} \psi_{\rho_*}^{(K_p)} -\rho^{({\scriptscriptstyle K}_p)}_j \psi_{\rho_*}^{(K_p)}\Big\|_{\ell^{2}}=0.
\]
For each $p$, let $\phi^{({\scriptscriptstyle K}_p)}_j$ be a normalized eigenvector of $\mathcal{L}_{{\scriptscriptstyle K}_p}$ corresponding to the eigenvalue $-\rho^{({\scriptscriptstyle K}_p)}_j$. By using Lemma \ref{approxvp} with $A=\mathcal{L}_{{\scriptscriptstyle K}_p}$, we deduce that there exists a sequence of real numbers $(\theta_p)_p$ such that
\begin{equation}
\label{pierre1}
\lim_{p\to\infty}\Big\| \psi_{\rho_*}^{(K_p)} -\mathrm{e}^{\mathrm{i} \theta_p}\phi^{({\scriptscriptstyle K}_p)}_j\Big\|_{\ell^{2}}=0.
\end{equation}
Since $\rho_* \in S_{2}$ and from Proposition \ref{prop-quasi-S2}, there exists a normalized vector $v_{\rho_*}\in\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle K}_p})$ for all $p$ such that
\[
\lim_{p\to\infty} \|\mathcal{L}_{{\scriptscriptstyle K}_p} v_{\rho_*} +\rho_* v_{\rho_*}\|_{\ell^{2}}=0.
\]
We also have
\[
\lim_{p\to\infty}\Big\| \mathcal{L}_{{\scriptscriptstyle K}_p} v_{\rho_*} +\rho^{({\scriptscriptstyle K}_p)}_j v_{\rho_*}\Big\|_{\ell^{2}}=0.
\]
By using Lemma \ref{approxvp} with $A=\mathcal{L}_{{\scriptscriptstyle K}_p}$, we deduce that there exists a sequence of real numbers $(\theta'_p)_p$ such that
\begin{equation}
\label{pierre2}
\lim_{p\to\infty}\Big\| v_{\rho_*} -\mathrm{e}^{\mathrm{i} \theta'_p} \phi^{({\scriptscriptstyle K}_p)}_j\Big\|_{\ell^{2}}=0.
\end{equation}
Moreover, from Propositions \ref{prop-quasi-S1} and \ref{prop-quasi-S2}, it follows that
\[
\lim_{p\to\infty}\Big\| \psi_{\rho_*}^{(K_p)} {\mathds{1}}_{\{\cdot\,<\,n_r({\scriptscriptstyle K}_p)\}}\Big\|_{\ell^{2}}=0\quad\text{and}\quad
\lim_{K\to+\infty} \big\| v_{\rho_*} {\mathds{1}}_{\{\cdot\,>\,n_l({\scriptscriptstyle K})\}}\big\|_{\ell^{2}}=0
\]
which implies (using Proposition \ref{strucvp}) that
\[
\lim_{p\to\infty}\langle \psi_{\rho_*}^{(K_p)},v_{\rho_*}\rangle=0.
\]
This is a contradiction with \eqref{pierre1} and \eqref{pierre2}.
Finally, we prove \eqref{madmax} by contradiction. So we assume that there exists a diverging sequence of integers $(p_\ell)$ such that
\[
\lim_{\ell\to+\infty}\Big|\,\rhoKp{j+1}{p_\ell}-\rho_{*}\Big|+ \Big|\,\rhoKp{j-1}{p_\ell}-\rho_{*}\Big|=0.
\]
We now apply Proposition \ref{decoupe} with $j$ and $K_{p_\ell}$.
Hence there exists a diverging sequence of integers $(\ell_s)$ such that
\[
\phiKp{j}{p_{\ell_s}}\,{\mathds{1}}_{\{\cdot\,<\,n_l({\scriptscriptstyle K}_{p_{\ell_s}})\}}\xrightarrow[]{\ell^{2}} \phi_{*,1}
\]
and
\[
Q_{K_{p_{\ell_s}}}\phiKp{j}{p_{\ell_s}}\,{\mathds{1}}_{\{\cdot\,\ge\, n_r({\scriptscriptstyle K}_{p_{\ell_s}})\}} \xrightarrow[]{L^{2}} \varphi_{*,1}
\]
and we have $\|\phi_{*,1}\|_{\ell^{2}}^2+\|\varphi_{*,1}\|_{L^{2}}^2=1$.
We now apply Proposition \ref{decoupe} with $j-1$ and $K_{p_{\ell_s}}$.
Hence there exists a diverging sequence of integers $(s_r)$ such that
\[
\phiKp{j-1}{p_{\ell_{s_r}}}\,{\mathds{1}}_{\{\cdot\,<\,n_l({\scriptscriptstyle K}_{p_{\ell_{s_r}}})\}}\xrightarrow[]{\ell^{2}} \phi_{*,2}
\]
and
\[
Q_{K_{p_{\ell_{s_r}}}}\phiKp{j-1}{p_{\ell_{s_r}}}\,{\mathds{1}}_{\{\cdot\,\ge\, n_r({\scriptscriptstyle K}_{p_{\ell_{s_r}}})\}} \xrightarrow[]{L^{2}} \varphi_{*,2}
\]
and we have $\|\phi_{*,2}\|_{\ell^{2}}^2+\|\varphi_{*,2}\|_{L^{2}}^2=1$.
We now apply Proposition \ref{decoupe} with $j+1$ and $K_{p_{\ell_{s_r}}}$.
Hence there exists a diverging sequence of integers $(r_q)$ such that
\[
\phiKp{j+1}{p_{\ell_{s_{r_q}}}}\,{\mathds{1}}_{\{\cdot\,<\,n_l({\scriptscriptstyle K}_{p_{\ell_{s_{r_q}}}})\}}\xrightarrow[]{\ell^{2}} \phi_{*,3}
\]
and
\[
Q_{K_{p_{\ell_{s_{r_q}}}}}\phiKp{j+1}{p_{\ell_{s_{r_q}}}}\,{\mathds{1}}_{\{\cdot\,\ge\, n_r({\scriptscriptstyle K}_{p_{\ell_{s_{r_q}}}})\}} \xrightarrow[]{L^{2}} \varphi_{*,3}
\]
and we have $\|\phi_{*,3}\|_{\ell^{2}}^2+\|\varphi_{*,3}\|_{L^{2}}^2=1$.
Moreover, since $\phiKp{j-1}{p_{\ell_{s_{r_q}}}}, \phiKp{j}{p_{\ell_{s_{r_q}}}}, \phiKp{j+1}{p_{\ell_{s_{r_q}}}}$ are pairwise orthogonal, we have
\begin{equation}\label{lestrois}
\left\langle \phi_{*,m}, \phi_{*,m'}\right\rangle_{\ell^{2}}
+
\left\langle \varphi_{*,m}, \varphi_{*,m'}\right\rangle_{L^{2}}
=0,\quad\forall m\neq m'.
\end{equation}
The linear subspace of $\ell^{2}$ spanned by $\phi_{*,1}, \phi_{*,2}, \phi_{*,3}$ is of dimension at most one because they are eigenvectors of
$\EuScript{M}_{0}$ for the same simple eigenvalue $-\rho_*$.
The linear subspace of $L^{2}$ spanned by $\varphi_{*,1}, \varphi_{*,2}, \varphi_{*,3}$ is of dimension at most one because they are eigenfunctions of
$\EuScript{H}_{*}$ for the same simple eigenvalue $-\rho_*$.
Therefore the subspace of $\ell^{2}\oplus L^{2}$ spanned by the three vectors $(\phi_{*,1},\varphi_{*,1})$, $(\phi_{*,2},\varphi_{*,2})$,
$(\phi_{*,3},\varphi_{*,3})$ is of dimension at most two. However, these three vectors are normalized and pairwise orthogonal by \eqref{lestrois}.
We thus arrive at a contradiction.
\section{Fr\'echet-Kolmogorov-Riesz compactness criterion and Dirichlet form}\label{auxiliaires}
\subsection{Fr\'echet-Kolmogorov-Riesz compactness criterion}
We recall the Fr\'echet-Kolmogorov-Riesz compactness criterion in $L^{2}$.
\begin{theorem}\label{KFR}
Let $(f_{p})_p$ be a normalized sequence in $L^{2}$ such that the following two conditions are satisfied.
\begin{enumerate}[(i)]
\item
There exists $\epsilon_0>0$ such that there exists a function $R(\epsilon)>0$ on $(0,\epsilon_0)$ such that
\[
\sup_{p}\int_{\{|x|>R(\epsilon)\}}\big|f_{p}(x)\big|^{2}\mathrm{d} x\le \epsilon.
\]
\item
There exits a positive function $\alpha$ on $]0,1]$ satisfying
\[
\lim_{y\searrow\, 0}\alpha(y)=0
\]
and such that for any $p$ and $y\in \left]-1,1\right]$
\[
\int\big|f_{p}(x+y)-f_{p}(x)\big|^{2}\mathrm{d} x\le \alpha(|y|).
\]
\end{enumerate}
Then one can extract from $(f_{p})_p$ a convergent subsequence in $L^{2}$.
\end{theorem} We refer to Remark \textbf{5} on page 387 in \cite{HH}. The following
Lemmas provide expressions for $R(\epsilon)$ and $\alpha(y)$ in our case. Recall that the potential $V_{n}(K)$ and $n_{r}(K)$ have been defined in Section 4 (see \eqref{potential}).
\begin{lemma}\label{unifint}
Let $C>0$.
Let $\mathscr{F}_{C,\,K}$ be the set of normalized sequences $(\phi(n))_n$ in $\ell^{2}$ such that $\phi(n)=0$ for any $n<n_{r}(K)$ and
\[
\sum_{n=n_{r}(K)}^{\infty}\big(1\vee V_{n}(K)\big)\big)\,\phi^{2}(n)\le C\;.
\]
Then, for any $\phi\in\mathscr{F}_{C,K} $, the function $Q_{{\scriptscriptstyle K}}\phi(x)$ satisfies condition \textup{(i)} in Theorem \ref{KFR} with
\[
R(\epsilon)=\frac{C}{\epsilon}
\]
for any $0<\epsilon<1$.
\end{lemma}
\begin{proof}
We are going to prove that there exist $R_{0}>1$ and $K_{0}>4$ such that for any $K>K_{0}$, any $R>R_{0}$ and any $\phi\in \mathscr{F}_{C,K}$
\[
\int_{\{|x|>R\}}\big(Q_{{\scriptscriptstyle K}}\phi(x)\big)^{2}\mathrm{d} x\le \frac{C}{R}.
\]
We observe that
\[
\int_{\{|x|>R\}}\big(Q_{{\scriptscriptstyle K}}\phi(x)\big)^{2}\mathrm{d} x\le \sum_{n:|n-K\xf|>R\,\sqrt{K}-1}\phi^{2}(n).
\]
Our aim is to prove that the right-hand side of the above inequality is bounded above by $C/R$.
It follows from the hypotheses on $\lambda^{{\scriptscriptstyle (K)}}_{n}$ and
$\mu^{{\scriptscriptstyle (K)}}_{n}$ that there exist constants $K_{0}>4$, $1>C_{2}>0$, $\Gamma>0$ and $\zeta>0$, such that for any $K>K_{0}$ there exists an integer
$\Gamma\, K>m_{*}{\scriptstyle (K)}>2\,K\xf$ (hence of order $K$) such that $\mu^{{\scriptscriptstyle (K)}}_{n}>\zeta\,n$ for any $n\gem_{*}{\scriptstyle (K)}$ and
\begin{align*}
\sum_{n=n_{r}(k)}^{m_{*}{\scriptstyle (K)}}(1\veeV_{n}(K))\;\phi^{2}(n)
&\ge C_{2}\, K^{-1}\sum_{n=n_{r}(K)}^{m_{*}{\scriptstyle (K)}}(n-K\xf)^{2} \phi^{2}(n)\\
\sum_{n=m_{*}{\scriptstyle (K)}+1}^{\infty}(1\veeV_{n}(K))\;\phi^{2}(n)
& \ge C_{2}\, \sum_{n=m_{*}{\scriptstyle (K)}+1}^{\infty}n\,\phi^{2}(n)\;.
\end{align*}
These estimates imply the following bounds for any integer $L>0$
\begin{align}
\label{zone2}
\sum_{\{n_{r}(K)\leq n\leq m_{*}{\scriptstyle (K)}\}\cap \{|n-K\xf|>L\}}\phi^{2}(n) & \le \frac{K C}{C_{2}\, L^{2}}\, {\mathds{1}}_{\{L<m_{*}{\scriptstyle (K)}\}}\\
\sum_{\{n>m_{*}{\scriptstyle (K)}\}\cap \{|n-K\xf|>L\}}\phi^{2}(n) & \le \frac{C}{C_{2}(L\vee m_{*}{\scriptstyle (K)})}
\label{zone3}\;.
\end{align}
We now replace $L$ with $R\,\sqrt K-1$ in the above estimates.
Let $K>4$ be fixed. We distinguish two cases according to the value of $R$.
\begin{enumerate}[1)]
\item
$1\le R<m_{*}{\scriptstyle (K)}/\sqrt K$. Then $\sqrt{K}-1\le L<m_{*}{\scriptstyle (K)}\le \Gamma K$. Since $L=R\sqrt{K}-1>R\sqrt{K}/2$ (because $K>4$), we have
\[
\sum_{\{|n-K\xf|>R\,\sqrt{K}-1\}}\phi^{2}(n)\le \frac{4\,\,C}{R^{2}}+\frac{C}{C_{2}m_{*}{\scriptstyle (K)}}
\le \frac{4C}{C_{2}R^{2}}+ \frac{C\, \Gamma}{C_{2}R^{2}}\le \frac{C}{R}
\]
if $R>C_{2}^{-1}\big(4+\,\Gamma)$.
\item $R\ge m_{*}{\scriptstyle (K)}/\sqrt K$. Then $L\ge m_{*}{\scriptstyle (K)}$. We get
\[
\sum_{\{|n-K\xf|>R\,\sqrt{K}-1\}}\phi^{2}(n)\le
\frac{C}{C_{2}L}\le \frac{2C}{R\,C_{2}\sqrt{K}}\le \frac{C}{R}
\]
if $C_{2}^{2} K>4$.
\end{enumerate}
We define $K_{0}=5+4\,C_{2}^{-2}$ and $R_{0}=1+C_{2}^{-1}\big(4+\,\Gamma)$.
The result follows.
\end{proof}
\begin{lemma}\label{regu}
Let $(\phi^{{\scriptscriptstyle (K)}})_K$ be a sequence of normalized elements of $\ell^{2}$ such that $\phiK{(n)}=0$ for $n\le n_{r}(K)$.
Assume also that there exists $C>0$ such that
\begin{align*}
\MoveEqLeft[10]
\sup_{K>1} \left\{\sum_{n=n_{r}{\scriptscriptstyle (K)}}^{\infty}\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n}\mu^{{\scriptscriptstyle (K)}}_{n+1}} \big(\phiK{(n+1)}-\phiK{(n)}\big)^{2} \right.\\
& \left. +\sum_{n=n_{r}{\scriptscriptstyle (K)}}^{\infty}(1\veeV_{n}(K))\,\big(\phiK{(n)}\big)^{2}\right\}
\le C.
\end{align*}
Then there exits a positive constant $\tilde C$ such that for any $|h|\le 1$ and any $K>1$
\[
\int \big[Q_{{\scriptscriptstyle K}}\phiK{}(x+h)-Q_{{\scriptscriptstyle K}}\phiK{}(x)\big]^{2}\mathrm{d} x\le\alpha(h)=\tilde C\, |h|\,.
\]
Hence, for any $(\phiK{})_K$ satisfying the above assumptions, the sequence of functions $(Q_{{\scriptscriptstyle K}}\phiK{})_K$ satisfies condition \textit{ii)} in Theorem \ref{KFR} with
\[
\alpha(y)=\tilde C y\;.
\]
\end{lemma}
\begin{proof}
It is enough to consider the case $0<h<1$.
We first consider the case $0<h\le 1/\sqrt{K}$.
We have
\[
\int\! \big[Q_{{\scriptscriptstyle K}}\phiK{}(x+h)-Q_{{\scriptscriptstyle K}}\phiK{}(x)\big]^{2}\mathrm{d} x=
\sum_{q\geq 1}\int_{I^{(K)}_{q}} \!\big[Q_{{\scriptscriptstyle K}}\phiK{}(x+h)-Q_{{\scriptscriptstyle K}}\phiK{}(x)\big]^{2}\mathrm{d} x.
\]
Since
\[
Q_{{\scriptscriptstyle K}}\phiK{}(x)=K^{\frac{1}{4}}\,\sum_{q\geq 1}\phiK{(q)} \,{\mathds{1}}_{I^{{\scriptscriptstyle (K)}}_{q}}(x)
\]
and since the intervals $I^{{\scriptscriptstyle (K)}}_{q}$ are disjoint, we get
\begin{align*}
\MoveEqLeft \int \big[Q_{{\scriptscriptstyle K}}\phiK{}(x+h)-Q_{{\scriptscriptstyle K}}\phiK{}(x)\big]^{2}\mathrm{d} x\\
& =K^{\frac{1}{2}} \sum_{q\geq 1}\int_{I^{{\scriptscriptstyle (K)}}_{q}}
\Big(\phiK{(q)}\,{\mathds{1}}_{I^{(K)}_{q}}(x+h) \\
& \hspace{2.8cm}+\phiK{(q+1)}\,{\mathds{1}}_{I^{{\scriptscriptstyle (K)}}_{q+1}}(x+h)-\phiK{(q)}\,{\mathds{1}}_{I_{q}^{{\scriptscriptstyle (K)}}}(x)\Big)^{2}\mathrm{d} x\\
& =K^{\frac{1}{2}} \sum_{q\geq 1}\int_{I^{{\scriptscriptstyle (K)}}_{q}} \Big(\big(\phiK{(q)}\big)^{2}\,{\mathds{1}}_{I^{{\scriptscriptstyle (K)}}_{q}}(x+h)+\big(\phiK{(q+1)}\big)^{2}\,{\mathds{1}}_{I^{{\scriptscriptstyle (K)}}_{q+1}}(x+h)\\
& \hspace{2.8cm}+\big(\phiK{(q)}\big)^{2}\,{\mathds{1}}_{I^{{\scriptscriptstyle (K)}}_{q}}(x)
-2\, \big(\phiK{(q)}\big)^{2}\,{\mathds{1}}_{I_{q}}(x){\mathds{1}}_{I^{{\scriptscriptstyle (K)}}_{q}}(x+h)\\
& \hspace{2.8cm} -2\, \phiK{(q)}\,\phiK{(q+1)}\,{\mathds{1}}_{I^{{\scriptscriptstyle (K)}}_{q}}(x){\mathds{1}}_{I^{{\scriptscriptstyle (K)}}_{q+1}}(x+h)\Big)\mathrm{d} x
\end{align*}
Let us consider each term separately. Since $\int_{I^{(K)}_{q}} \mathrm{d} x = K^{-\frac{1}{2}}$, we have
\[
K^{\frac{1}{2}} \sum_{q\geq 1}\int_{I_{q}^{{\scriptscriptstyle (K)}}} \big(\phiK{(q)}\big)^{2}\,\mathrm{d} x=\sum_{q\geq 1}\,\big(\phiK{(q)}\big)^{2}.
\]
Then we have
\begin{align*}
\MoveEqLeft K^{\frac{1}{2}}\sum_{q\geq 1}\int_{I_{q}^{{\scriptscriptstyle (K)}}}\big(\phiK{(q)}\big)^{2}\,{\mathds{1}}_{I_{q}^{{\scriptscriptstyle (K)}}}(x+h)\,\mathrm{d} x\\
&=K^{\frac{1}{2}}\sum_{q\geq 1}\,\big(\phiK{(q)}\big)^{2}\int_{\frac{q}{\sqrt{K}}-\frac{1}{2\sqrt{K}}-x_{*}}^{\frac{q}{\sqrt{K}}+\frac{1}{2\sqrt{K}}-x_{*}-h}\,\mathrm{d} x\\
&=\big(1-h K^{\frac{1}{2}}\big)\,\sum_{q\geq 1}\big(\phiK{(q)}\big)^{2}.
\end{align*}
We also have
\begin{align*}
\MoveEqLeft K^{\frac{1}{2}}\sum_{q\geq 1}\int_{I_{q}^{{\scriptscriptstyle (K)}}} \big(\phiK{(q+1)}\big)^{2}\,{\mathds{1}}_{I_{q+1}^{{\scriptscriptstyle (K)}}}(x+h)\,\mathrm{d} x\\
&=K^{\frac{1}{2}}\sum_{q\geq 1} \big(\phiK{(q+1)}\big)^{2}\int_{\frac{q}{\sqrt{K}}+\frac{1}{2\sqrt{K}}-x_{*}-h}^{\frac{q}{\sqrt{K}}+\frac{1}{2\sqrt{K}}-x_{*}} \mathrm{d} x\\
&=K^{\frac{1}{2}}\sum_{q\geq 1} \big(\phiK{(q+1)}\big)^{2}\,h\\
&=K^{\frac{1}{2}}\,h\,\sum_{q\geq 1} \big(\phiK{(q)}\big)^{2}.
\end{align*}
Similarly
\begin{align*}
\MoveEqLeft
-2\, K^{\frac{1}{2}} \sum_{q\geq 1}\int_{I_{q}^{{\scriptscriptstyle (K)}}}\big(\phiK{(q)}\big)^{2}\,{\mathds{1}}_{I_{q}^{{\scriptscriptstyle (K)}}}(x){\mathds{1}}_{I_{q}}(x+h)\, \mathrm{d} x\\
& =-2\,\big(1-h K^{\frac{1}{2}}\big)\,\sum_{q\geq 1} \big(\phiK{(q)}\big)^{2}
\end{align*}
and
\begin{align*}
\MoveEqLeft -2 K^{\frac{1}{2}} \sum_{q\geq 1} \int_{I_{q}^{{\scriptscriptstyle (K)}}} \phiK{(q)}\,\phiK{(q+1)}\,{\mathds{1}}_{I_{q}^{{\scriptscriptstyle (K)}}}(x){\mathds{1}}_{I_{q+1}^{{\scriptscriptstyle (K)}}}(x+h) \,\mathrm{d} x\\
&=-2 h K^{\frac{1}{2}} \sum_{q\geq 1}\,\phiK{(q)}\,\phiK{(q+1)}\,.
\end{align*}
We rewrite the last term:
\begin{align*}
\MoveEqLeft -2\,h\,\sqrt{K}\, \sum_{q\geq 1}\,\phiK{(q)}\,\phiK{(q+1)}\\
& =-2\,h\,\sqrt{K}\, \sum_{q\geq 1}\,\big(\phiK{(q)}\big)^{2}+2\,h\,\sqrt{K}\, \sum_{q\geq 1}\,\phiK{(q)}\,\big(\phiK{(q+1)}-\phiK{(q)}\big).
\end{align*}
Summing up, we get
\begin{align*}
\MoveEqLeft \int \big[Q_{{\scriptscriptstyle K}}\phiK{}(x+h)-Q_{{\scriptscriptstyle K}}\phiK{}(x)\big]^{2}\mathrm{d} x\\
& = \sum_{q\geq 1}\,\big(\phiK{(q)}\big)^{2}+\big(1-h\,\sqrt{K}\big)\,\sum_{q\geq 1}\big(\phiK{(q)}\big)^{2}+K^{\frac{1}{2}}\,h\,\sum_{q\geq 1} \big(\phiK{(q)}\big)^{2}\\
& \quad - 2\,\big(1-h\,\sqrt{K}\big)\,\sum_{q\geq 1} \big(\phiK{(q)}\big)^{2}-2\,h\,\sqrt{K}\, \sum_{q\geq 1}\,\big(\phiK{(q)}\big)^{2}\\
& \quad + 2\,h\,\sqrt{K}\, \sum_{q\geq 1}\,\phiK{(q)}\,\big(\phiK{(q+1)}-\phiK{(q)}\big)\\
& = 2\,h\,\sqrt{K}\, \sum_{q\geq 1}\,\phiK{(q)}\,\big(\phiK{(q+1)}-\phiK{(q)}\big).
\end{align*}
By Cauchy-Schwarz inequality we get
\begin{align*}
\MoveEqLeft \int \big[Q_{{\scriptscriptstyle K}}\phiK{}(x+h)-Q_{{\scriptscriptstyle K}}\phiK{}(x)\big]^{2}\mathrm{d} x\\
& \le 2 h\sqrt{K}\,\left(\sum_{q\geq 1}\,\big(\phiK{(q+1)}-\phiK{(q)}\big)^{2}\right)^{\frac{1}{2}}\big\|\phiK{}\big\|_{\ell^{2}}.
\end{align*}
From the assumption of the lemma and using
\[
\inf_{n\,\ge \big\lfloor \frac{Kx_{*}}{3}\big\rfloor-1}\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n}\mu^{{\scriptscriptstyle (K)}}_{n+1}}> \zeta K
\]
for some $\zeta>0$ independent of $K$, we get
\[
K^{\frac{1}{2}}\left(\sum_{q\geq 1}\,\big(\phiK{(q+1)}-\phiK{(q)}\big)^{2}\right)^{1/2}\le \sqrt{{C\over \zeta}}.
\]
Therefore, since the sequence $\phiK{}$ is normalized, we get
\[
\int \big[Q_{{\scriptscriptstyle K}}\phiK{}(x+h)-Q_{{\scriptscriptstyle K}}\phiK{}(x)\big]^{2}\mathrm{d} x\le 2 h\; \sqrt{{C\over \zeta}}.
\]
We now consider the case $1>h>1/\sqrt{K}$. Let $r=\lfloor h\sqrt{K}\rfloor$ and $h'=h-r/\sqrt{K}$. Note that $0\le h'\le 1/\sqrt{K}$.
We have
\begin{align*}
\MoveEqLeft \int \big[Q_{{\scriptscriptstyle K}}\phiK{}(x+h)-Q_{{\scriptscriptstyle K}}\phiK{}(x)\big]^{2}\mathrm{d} x\\
& =\int\bigg[\sum_{j=1}^{r-1}\left(Q_{{\scriptscriptstyle K}}\phiK{}\Big(x+\frac{j+1}{\sqrt{K}}\Big)-Q_{{\scriptscriptstyle K}}\phiK{}\Big(x+\frac{j}{\sqrt{K}}\Big)\right)\\
&\hspace{1.2cm} +Q_{{\scriptscriptstyle K}}\phiK{}\Big(x+\frac{r}{\sqrt{K}}+h'\Big)-Q_{{\scriptscriptstyle K}}\phiK{}\Big(x+\frac{r}{\sqrt{K}}\Big)\bigg]^{2}\mathrm{d} x\\
& \le 2 \int \bigg[\sum_{j=1}^{r-1}\left(Q_{{\scriptscriptstyle K}}\phiK{}\Big(x+\frac{j+1}{\sqrt{K}}\Big)-Q_{{\scriptscriptstyle K}}\phiK{}\Big(x+\frac{j}{\sqrt{K}}\Big)\right)\bigg]^{2} \mathrm{d} x\\
&\hspace{1.4cm} + 2 \int \bigg[Q_{{\scriptscriptstyle K}}\phiK{}\Big(x+\frac{r}{\sqrt{K}}+h'\Big)-Q_{{\scriptscriptstyle K}}\phiK{}\Big(x+\frac{r}{\sqrt{K}}\Big)\bigg]^{2}\mathrm{d} x.
\end{align*}
We have already estimated the last term. For the first term we now observe that $e^{{\scriptscriptstyle (K)}}_{n}\Big(x+\frac{j}{\sqrt{K}}\Big)=e^{{\scriptscriptstyle (K)}}_{n-j}(x)$. Therefore we can write
\begin{align*}
\MoveEqLeft\sum_{j=1}^{r-1}\left(Q_{{\scriptscriptstyle K}}\phiK{}\Big(x+\frac{j+1}{\sqrt{K}}\Big)-Q_{{\scriptscriptstyle K}}\phiK{}\Big(x+\frac{j}{\sqrt{K}}\Big)\right) \\
&=\sum_{j=1}^{r-1}\left(\sum_{n}\phiK{(n)}\, e^{{\scriptscriptstyle (K)}}_{n-j-1}(x)-\sum_{n}\phiK{(n)}\, e^{{\scriptscriptstyle (K)}}_{n-j}(x)\right)\\
&=\sum_{j=1}^{r-1}\left(\sum_{p}e^{{\scriptscriptstyle (K)}}_{p}(x)\,\big(\phiK{(p+j+1)}-\phiK{(p+j)}\right)\\
&=\sum_{p}e^{{\scriptscriptstyle (K)}}_{p}(x)\,\sum_{j=1}^{r-1}\big(\phiK{(p+j+1)}-\phiK{(p+j)}\big).
\end{align*}
This implies
\begin{align*}
\MoveEqLeft \int \bigg[\sum_{j=1}^{r-1}Q_{{\scriptscriptstyle K}}\phiK{}\Big(x+\frac{j+1}{\sqrt{K}}\Big)-Q_{{\scriptscriptstyle K}}\phiK{}\Big(x+\frac{j}{\sqrt{K}}\Big)\bigg]^{2} \mathrm{d} x\\
& =\sum_{p}\left(\sum_{j=1}^{r-1}\big(\phiK{(p+j+1)}-\phiK{(p+j)}\big)\right)^{2}\\
& \le r\, \sum_{p}\sum_{j=1}^{r-1}\big(\phiK{(p+j+1)}-\phiK{(p+j)}\big)^{2}\\
&=r^{2}\, \sum_{p}\big(\phiK{(p+1)}-\phiK{(p)}\big)^{2}\le \frac{C }{\zeta } \;\frac{r^{2}}{K}
\end{align*}
as we have seen before. We observe that $r^{2}/K\le h^{2}\le h$ since $h\le 1$, and the result follows by taking $\tilde C= 2 \sqrt{\frac{C}{\zeta } } + \frac{C}{\zeta }$.
\end{proof}
\subsection{Dirichlet form for the operator \texorpdfstring{$\mathcal{L}_{{\scriptscriptstyle K}}$}{LKI}}
We need an estimate on the decay at infinity of the eigenfunctions. Note that since the eigenvalues are real, we can assume that the eigenfunctions are real.
\begin{proposition}\label{diri}
If $\phi$ is a normalized sequence in $\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle K}})$ decaying exponentially fast at infinity, then
\begin{align*}
&-\langle\phi,\mathcal{L}_{{\scriptscriptstyle K}}\phi\rangle \\
& =\sum_{n=1}^{\infty}\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n}\mu^{{\scriptscriptstyle (K)}}_{n+1}}\,
\big(\phi(n+1)-\phi(n)\big)^{2}
+\sum_{n=1}^{\infty}V_{n}(K)\,{\phi(n)}^{2}-\frac{1}{2}\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{1}\mu^{{\scriptscriptstyle (K)}}_{2}}\,\phi(1)^{2}.
\end{align*}
\end{proposition}
\begin{proof}
For any fixed positive integer $N$ we have
\begin{align*}
\MoveEqLeft \sum_{n=1}^{N}\phi(n)\,(\mathcal{L}_{{\scriptscriptstyle K}}\phi)(n)\\
&= \sum_{n=1}^{N}\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n}\mu^{{\scriptscriptstyle (K)}}_{n+1}}\,\phi(n)\,\phi(n+1)
+\sum_{n=2}^{N}\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n-1}\,\mu^{{\scriptscriptstyle (K)}}_{n}}\,\phi(n)\,\phi(n-1)\\
&\quad -\sum_{n=1}^{N}\big(\lambda^{{\scriptscriptstyle (K)}}_{n}+\mu^{{\scriptscriptstyle (K)}}_{n}\big)\,\phi(n)^{2}\\
& =-\frac{1}{2}\sum_{n=1}^{N}\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n}\mu^{{\scriptscriptstyle (K)}}_{n+1}}\,(\phi(n)-\phi(n+1))^{2}
+\frac{1}{2}\sum_{n=1}^{N}\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n}\mu^{{\scriptscriptstyle (K)}}_{n+1}}\,\phi(n)^{2}\\
& \quad +\frac{1}{2}\sum_{n=1}^{N}\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n}\mu^{{\scriptscriptstyle (K)}}_{n+1}}\,\phi(n+1)^{2}-\frac{1}{2}\sum_{n=2}^{N}\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n-1}\,\mu^{{\scriptscriptstyle (K)}}_{n}}\,(\phi(n)-\phi(n-1))^{2}\\
& \quad +\frac{1}{2}\sum_{n=2}^{N}\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n-1}\,\mu^{{\scriptscriptstyle (K)}}_{n}}\,\phi(n)^{2}
+\frac{1}{2}\sum_{n=2}^{N}\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n-1}\,\mu^{{\scriptscriptstyle (K)}}_{n}}\,\phi(n-1)^{2}\\
& \quad -\sum_{n=1}^{N}\big(\lambda^{{\scriptscriptstyle (K)}}_{n}+\mu^{{\scriptscriptstyle (K)}}_{n}\big)\,\phi(n)^{2}\\
& =-\sum_{n=1}^{N-1}\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n}\mu^{{\scriptscriptstyle (K)}}_{n+1}}\,(\phi(n)-\phi(n+1))^{2}\\
& \quad -\sum_{n=1}^{N}\left(\lambda^{{\scriptscriptstyle (K)}}_{n}+\mu^{{\scriptscriptstyle (K)}}_{n}-\, \sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n}\mu^{{\scriptscriptstyle (K)}}_{n+1}}-\, \sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n-1}\,\mu^{{\scriptscriptstyle (K)}}_{n}}\,{\mathds{1}}_{\{n>1\}}\right)\,\phi(n)^{2}\\
&\quad -\frac{1}{2}\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{N}\mu^{{\scriptscriptstyle (K)}}_{N+1}}\,(\phi(N)-\phi(N+1))^{2}+\frac{1}{2}\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{N}\mu^{{\scriptscriptstyle (K)}}_{N+1}}\,\phi(N+1)^{2}\\
& \quad -\frac{1}{2}\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{1}\mu^{{\scriptscriptstyle (K)}}_{2}}\,\phi_{1}^{2}-\frac{1}{2}\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{N}\mu^{{\scriptscriptstyle (K)}}_{N+1}}\,\phi(N)^{2}.
\end{align*}
Since $\phi \in \mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle K}})$, the functions $\phi\, {\mathds{1}}_{\{n\le N\}}$ and $\mathcal{L}_{{\scriptscriptstyle K}}( \phi {\mathds{1}}_{\{n\le N\}})$ converge to $\phi$, respectively $\mathcal{L}_{{\scriptscriptstyle K}} \phi$, in $\ell^{2}$ when $N$ tends to
infinity. The result follows by letting $N$ tend to infinity, since $V_{n}(K)$ is positive for $n$ large enough, and since $\lambda_{N}(K)$ and $\mu_{N}(K)$ are exponential in
$N$ and $\phi$ decays exponentially fast by assumption.
\end{proof}
\begin{lemma}\label{borneVn}
There exists $\xi>0$ such that for all $K\in\mathds{Z}_{{\scriptscriptstyle >0}}$, $\inf_{n\geq 1} V_{n}(K)\geq -\xi$.
\end{lemma}
\begin{proof}
Since $(\lambda^{{\scriptscriptstyle (K)}}_{n})_n$ is an increasing sequence we have
\begin{align*}
V_{n}(K)
&=\lambda^{{\scriptscriptstyle (K)}}_{n}+\mu^{{\scriptscriptstyle (K)}}_{n}-\,\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n}\mu^{{\scriptscriptstyle (K)}}_{n+1}}-\,\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n-1}\,\mu^{{\scriptscriptstyle (K)}}_{n}}\,{\mathds{1}}_{\{n>1\}}\\
& \geq \frac{\lambda^{{\scriptscriptstyle (K)}}_{n}+\mu^{{\scriptscriptstyle (K)}}_{n}}{2}-\,\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n}\mu^{{\scriptscriptstyle (K)}}_{n+1}}.
\end{align*}
It follows from the general assumptions (see Section \ref{sec:assumptions}) that there exists $\tilde{x}\geq 1$ such that all $K\in\mathds{Z}_{{\scriptscriptstyle >0}}$ and for all
$n\geq K\tilde{x}$ we have
\[
\frac{\lambda^{{\scriptscriptstyle (K)}}_n}{\mu^{{\scriptscriptstyle (K)}}_n}\leq \frac{1}{5}\quad\text{and}\quad \frac{\lambda^{{\scriptscriptstyle (K)}}_{n+1}}{\mu^{{\scriptscriptstyle (K)}}_n}\leq \frac{5}{4}.
\]
For all $n\geq K\tilde{x}$ we have $V_{n}(K)\geq 0$. When $n\leq K\tilde{x}$, we write
\begin{align*}
\MoveEqLeft[5] \frac{\lambda^{{\scriptscriptstyle (K)}}_{n}+\mu^{{\scriptscriptstyle (K)}}_{n}}{2}-\,\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n}\mu^{{\scriptscriptstyle (K)}}_{n+1}} \\
&= \frac{\big(\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n}}-\sqrt{\mu^{{\scriptscriptstyle (K)}}_{n}}\big)^2}{2}+\frac{\lambda^{{\scriptscriptstyle (K)}}_{n}}{\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n}}+\sqrt{\mu^{{\scriptscriptstyle (K)}}_{n+1}}}\big(\mu^{{\scriptscriptstyle (K)}}_n-\mu^{{\scriptscriptstyle (K)}}_{n+1}\big).
\end{align*}
Now observe that
\[
\mu^{{\scriptscriptstyle (K)}}_n-\mu^{{\scriptscriptstyle (K)}}_{n+1}=K\left(d\left(\frac{n}{K}\right)-d\left(\frac{n+1}{K}\right)\right)\geq -\sup_{0\,\leq\, x\,\leq \,\tilde{x}+1} d'(x).
\]
The rest of the proof is obvious.
\end{proof}
\begin{proposition}\label{sobmodif}
Let $\delta>0$ and $\phi$ be a real normalized sequence in $\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle K}})$ decaying exponentially fast, such that
$\big\|\mathcal{L}_{{\scriptscriptstyle K}}\phi+\rho\,\phi\big\|_{\ell^{2}}\le \delta.$
Then
\begin{align*}
\MoveEqLeft \sum_{n=1}^{\infty}\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{n}\mu^{{\scriptscriptstyle (K)}}_{n+1}}\,\big(\phi(n+1)-\phi(n)\big)^{2}
+\sum_{n=1}^{\infty}(1\veeV_{n}(K))\,\big(\phi(n)\big)^{2}\\
& \le 1+\rho+\xi+\delta+\frac{1}{2}\sqrt{\lambda^{{\scriptscriptstyle (K)}}_{1}\mu^{{\scriptscriptstyle (K)}}_{2}}.
\end{align*}
\end{proposition}
The proof is left to the reader. It is a direct consequence of Proposition \ref{diri} and Lemma \ref{borneVn}.
\section{Spectral theory of \texorpdfstring{$\EuScript{M}_{0}$}{PO}}
Recall that (cf. \eqref{le-M0})
\begin{align*}
& \big(\EuScript{M}_{0} v\big)(n)=\\
\nonumber
& \sqrt{b'(0)\,d'(0)\,n\,(n+1)} \,v(n+1)+\sqrt{b'(0)\,d'(0)\,n\,(n-1)}\,v(n-1)\,{\mathds{1}}_{\{n>1\}}\\
\nonumber
& -n\,(b'(0)+d'(0)) v(n)).
\end{align*}
\begin{theorem}\label{specdeux}
The operator $\EuScript{M}_{0}$ defined on $c_{00}$ is symmetric for the scalar product of $\ell^{2}$.
We denote by $\EuScript{M}_{0}$ its closure which is self-adjoint and bounded above.
The spectrum of\, $\EuScript{M}_{0}$ is discrete, all eigenvalues are simple, and we have
\[
\mathrm{Sp}(\EuScript{M}_{0})=\big(d'(0)-b'(0)\big)\mathds{Z}_{{\scriptscriptstyle >0}}=-S_2.
\]
The eigenvector $v_{m}$ corresponding to the eigenvalue $(d'(0)-b'(0)\big)m$, where $m\in\mathds{Z}_{{\scriptscriptstyle >0}}$, is given (up to a multiplicative factor) by
\begin{equation}
\label{fatigue}
v_{m}(n)=\sqrt{n}\,\left(\frac{d'(0)}{b'(0)}\right)^{\frac{n}{2}} P_{m}(n)
\end{equation}
where $P_{m}$ is the monic orthogonal polynomial of degree $m-1$ associated with the measure $q$ on $\mathds{Z}_{{\scriptscriptstyle >0}}$ defined by
\begin{equation}
\label{q}
q(n)=n\left(\frac{d'(0)}{b'(0)}\right)^{n}.
\end{equation}
\end{theorem}
\begin{proof}
It is easy to verify that $\EuScript{M}_{0}$ is a symmetric operator on $c_{00}$, which is bounded above since from $b'(0)>d'(0)$, we have
\[
\inf_{n}\big(n\,(b'(0)+d'(0))-\sqrt{b'(0)\,d'(0)\,n\,(n+1)} - \sqrt{b'(0)\,d'(0)\,n\,(n-1)}>-\infty.
\]
It is easy to verify that $\EuScript{M}_{0}$ is closable and we denote by $\EuScript{M}_{0}$ its closure.
Since for any $m\in \mathds{Z}_{{\scriptscriptstyle >0}}$, the sequence $v_{m}(n)$ defined by \eqref{fatigue} decays exponentially fast with $n$, it
is easy to verify that $v_{m}\in \mathrm{Dom}(\EuScript{M}_{0})$. Note also that if $m\neq
m'$, $v_{m}$ is orthogonal to $v_{m'}$ in $\ell^{2}$.
By a direct computation one checks that $\EuScript{M}_{0} v_{1}=(d'(0)-b'(0))v_{1}$ (recall that $P_{1}(n)=1$).
It is left to the reader to check that
\[
\EuScript{M}_{0} v_{m}(n)=\sqrt{n}\left(\frac{b'(0)}{d'(0)}\right)^{\frac{n}{2}}\,Q_{m}(n)
\]
where $Q_{m}$ is a polynomial in $n$ in which the coefficient of $n^{m-1}$ is
\[
m (d'(0)-b'(0))\;.
\]
To check that the $v_{m}$ are eigenvectors, we use a recursive argument. Assume that $m\ge 2$ and for $1\le k\le m-1$
\[
\EuScript{M}_{0} v_{k}=k\,(d'(0)-b'(0))\,v_{k}.
\]
We can write
\[
\EuScript{M}_{0} v_{m}=m\,(d'(0)-b'(0))\,v_{m}+r_{m}
\]
with
\[
r_{m}=\sqrt{n}\left(\frac{b'(0)}{d'(0)}\right)^{\frac{n}{2}}R_{m}
\]
where $R_{m}$ is a polynomial in $n$ of degree at most $m-2$.
Therefore
\[
r_{m}\in\text{Span}\big\{v_{1},\ldots, v_{m-1}\big\}\;.
\]
From our recursive assumption, the symmetry of $\EuScript{M}_{0}$, and the orthogonality of the $v_{k}$ (following from the orthogonality of the $P_{k}$), we get that for any $1\le k\le m-1$
\[
0=\langle v_{k}, \EuScript{M}_{0} v_{m}\rangle_{\ell^{2}}=\langle v_{k},r_{m}\rangle_{\ell^{2}}.
\]
Therefore $r_{m}=0$. Hence $\EuScript{M}_{0} v_{m}=m\,(d'(0)-b'(0))\,v_{m}$, and we can proceed with the recursion.
We now prove that the $v_{m}$ form a basis of $\ell^{2}$.
Assume the contrary, namely there exists $u\in\ell^{2}$ of norm one such that for any $m$
\[
\sum_{n=1}^{\infty}\overline u(n)\,v_{m}(n)=0.
\]
We observe that the sequence
\[
w(n)=\frac{1}{\sqrt{n}}\left(\frac{b'(0)}{d'(0)}\right)^{\frac{n}{2}}u(n)
\]
belongs to $\ell^{2}(q)$, whith $q$ defined in \eqref{q}. Therefore our assumption on $u$ implies that $w$ is orthogonal to all the polynomials in $\ell^{2}(q)$. Let us
show that the set of polynomials is dense in $\ell^{2}(q)$. It is sufficient to prove that the measure $q$ is the solution of a determinate moment problem,
see \cite[Corollary 2.50, p. 30]{Deift}. Following \cite[Proposition 1.5, p. 88]{Simon}, it is enough to prove that the moments of order $m$, denoted by
$\gamma_{m}$ of $q$, satisfy the following property: there exists $C>0$ such that, for any $m\in\mathds{Z}_{{\scriptscriptstyle >0}}$,
\[
\gamma_{m}=\sum_{n=1}^{\infty}n^{m+1} \left(\frac{d'(0)}{b'(0)}\right)^{n}\le C^{m}\,m! \,.
\]
The proof is left to the reader. Therefore the set of all polynomials is dense in $\ell^{2}(q)$ implying $w=0$ and we get a contradiction with the existence of a $u$ nonzero orthogonal
to all the $v_{m}$ in $\ell^{2}$. Therefore, the $v_{m}$ form a basis of $\ell^{2}$.
\smallskip We now observe that $\EuScript{M}_{0}$ is bounded above. The proof is similar to that of Proposition \ref{diri} and left to the reader.
Since the $v_m$'s form a basis of $\ell^{2}$, for any $B>0$ we have $\text{ker}(\EuScript{M}_{0}^{\dagger}-B)=\{0\}$.
Hence $\EuScript{M}_{0}$ is self adjoint (see for instance \cite[Prop. 3.9, p. 43]{KS}) and the spectrum is given by
\[
\mathrm{Sp}(\EuScript{M}_{0})=(d'(0)-b'(0))\mathds{Z}_{{\scriptscriptstyle >0}}.
\]
This ends the proof.
\end{proof}
\section{Local maximum principle and consequences thereof}
We will state and prove a maximum/minimum principle in a form which is well suited for our purposes.
We start with a proposition giving elementary inequalities following from the order on the real line.
\begin{proposition}\label{locmax}
Assume $a>0$, $c>0$ and $b>a+c$. Let $u,w\in\mathds{R}$.\newline
If $v>0$ is such that $a\, u+c\,w-b\,v\ge0$, then $v< \max\{u,w\}$.\newline
If $v<0$ is such that $a\, u+c\,w-b\,v\le0$, then $v> \min\{u,w\}$.\newline
Moreover, if $u\ge v\ge w$ are such that $a\, u+c\,w-b\,v\ge0$, then $v\le \frac{a}{b-c}\,u$.
\end{proposition}
\begin{proof}
If $v>0$ we have $0< (b-a-c)\,v\le a\,(u-v)+c\,(w-v)$ leads to a contradiction if $v\ge\max\{u,w\}$. The case $v<0$ is similar.
The last statement is trivial since $b\,v\le a\,u+c\,w\le a\,u+c\,v$.
\end{proof}
\begin{proposition}\label{principe}
Let $1<n_1<n_2$ be integers such $n_2> n_1+1$. Let $(\alpha_n)$ be a finite sequence of strictly positive real numbers defined for
$n_1-1,\ldots,n_2$. Let $(\beta_n)$ be a finite sequence of strictly positive real numbers defined for $n_1,\ldots,n_2$.
Let $(u_n)$ be a finite sequence of real numbers defined for $n_1-1,\ldots,n_2+1$. Assume that, for all $n_1\leq n\leq n_2$, we
have $\beta_n> \alpha_n+\alpha_{n-1}$.
\newline If $\alpha_n u_{n+1}+\alpha_{n-1}u_{n-1}-\beta_n u_n\geq 0$, then the sequence $(u_n)$ has no positive local maxima for
$n \in\{n_1+1,\ldots,n_2-1\}$. Moreover, if there exists some $u_n>0$ then the maximum is attained only at the boundary, that
is, on the set $\{n_1,n_2\}$.
\newline If $\alpha_n u_{n+1}+\alpha_{n-1}u_{n-1}-\beta_n u_n\leq 0$, then the sequence $(u_n)$ has no positive local minima for
$n \in\{n_1+1,\ldots,n_2-1\}$, and if there exists some $u_n<0$ then the minimum is attained only at the boundary, that is, on the
set $\{n_1,n_2\}$.
\end{proposition}
\begin{proof}
It follows from Proposition \ref{locmax}.
\end{proof}
\begin{proposition}\label{pouet}
Let $1<n_1<n_2$ be integers such $n_2> n_1+1$. Let $(\alpha_n)$ be a finite sequence of strictly positive real numbers defined for
$n_1-1,\ldots,n_2$. Let $(\beta_n)$ be a finite sequence of strictly positive real numbers defined for $n_1,\ldots,n_2$.
Let $(u_n)$ be a finite sequence of real numbers defined for $n_1-1,\ldots,n_2+1$. Assume that, for all $n_1\leq n\leq n_2$, we
have $\beta_n> \alpha_n+\alpha_{n-1}$. \newline
If $\alpha_n u_{n+1}+\alpha_{n-1}u_{n-1}-\beta_n u_n\geq 0$, $u_{n_1+1}>0$ and $u_{n_1+1}\geq u_{n_1}$, then the sequence $(u_n)$
is increasing.
\newline If $\alpha_n u_{n+1}+\alpha_{n-1}u_{n-1}-\beta_n u_n\leq 0$, $u_{n_1+1}<0$ and $u_{n_1+1}\leq u_{n_1}$, then the sequence $(u_n)$
is decreasing.\newline Finally, if $(u_n)$ is a positive sequence then there cannot be two local (positive) minima separated by a distance larger than one.
\end{proposition}
\begin{proof}
It follows recursively from Proposition \ref{locmax}.
\end{proof}
|
1,116,691,500,795 | arxiv | \section{Introduction}
The AdS/CFT correspondence \cite{Maldacena:1997re,Witten:1998qj}
relates a theory of gravity in AdS space to a conformal field
theory on the boundary.
One consequence of the correspondence is that bulk quantum fields
can be expressed as CFT operators.
In the large N limit the bulk fields are free and can be written
as smeared CFT operators.
The explicit construction, called the HKLL (Hamilton, Kabat,
Lifschytz, and Lowe)
bulk reconstruction, was accomplished in a series of papers
\cite{Hamilton:2005ju,Hamilton:2006az,Kabat:2012hp}.
In the simplest case a massive free scalar field operator $\Phi(Y)$
is considered in ${\rm AdS}_{d+1}$.
The HKLL bulk reconstruction represents it in terms of the
boundary CFT primary of weight $\Delta$, $O(x)$, as
\begin{eqnarray}
\Phi(Y) &=& \int_{\Sigma_Y} {\rm d}x\, K(Y,x) O(x),
\label{eq:HKLL}
\end{eqnarray}
where $K(Y,x)$ is a smearing function, and the integration at the
boundary should be performed in a region $\Sigma_Y$ space-like separated
from the bulk point $Y$. We refer to \cite{DeJonckheere:2017qkk,
Harlow:2018fse,Kajuri:2020vxf} for recent reviews.
See also \cite{Bhowmick:2019nso} for an alternative derivation based
on Gel'fand-Graev-Radon transforms.
Later the reconstruction has been extended to higher spins
as well\cite{Kabat:2012hp,Kabat:2012av,Heemskerk:2012np,
Kabat:2013wga,Sarkar:2014dma,Foit:2019nsr}. After having constructed the
free case the next step is to study bulk interactions \cite{Kabat:2011rz}.
An elegant way to introduce interactions as well as to reproduce the bulk
reconstruction for free fields is the method based on space-like Green's functions
\cite{Hamilton:2006az,Heemskerk:2012mn}.
In the original papers it was not explicitly stated that
\eqref{eq:HKLL} holds only for $\Delta > d-1$, due to the convergence for the integral.
This restriction is not essential for applications of the AdS/CFT correspondence in the case of
supersymmetric gauge theories, in particular in the prime example of the ${\cal N}=4$ SUSY U$(N)$
gauge theory in $d=4$ dimensions,
since the conformal dimensions of
physically relevant operators are typically (much) larger
than this lower bound $\Delta > d-1$. See however \cite{Klebanov:1999tb} for some explicit examples for
small $\Delta$ primaries in the AdS/CFT context.
More importantly there exists an other family of models often
used in the AdS/CFT context, namely
O$(N)$ vector models and their holographic
duals, higher spin theories in the bulk\cite{Klebanov:2002ja,Sezgin:2002rt}.
In the most interesting $d=3$ case, the simplest singlet operator
has $\Delta=1$ ($d-2$).
Furthermore, its square, an operator which can be used as a relevant deformation, has
$\Delta=2$ ($d-1$).
The HKLL formula \eqref{eq:HKLL} can not be used to relate
these singlet scalar operators in the free $O(N)$ vector model to bulk operators.
It turned out\cite{Kabat:2012hp} that for the special case $\Delta=d-1$
the smearing function in Poincare coordinates is supported on the intersection of the light-cone of the bulk point and the
boundary. In \cite{DelGrosso:2019gow}
the range of allowed $\Delta$ values was extended down to $\Delta>d/2$ by analytic
continuation.
While the bulk-boundary relation remains linear,
the smearing kernel in (\ref{eq:HKLL}) is replaced by a suitable distribution.
In \cite{Aoki:2021ekk} we found, in some special cases mainly concentrating
on the (simpler) case of even AdS spaces, a
generalized HKLL formula for $\Delta$ values below the original lower
bound $d-1$ by a direct derivation, without using analytic continuation.
When we explicitly evaluated the results of
\cite {DelGrosso:2019gow}, we found that they precisely agree with the results
of the direct calculation in the range where they overlap. We also discussed the interesting special cases
$\Delta = d-s$, where $s$ is a positive integer only
limited by the requirement that the conformal weight satisfies the unitarity
bound $\Delta \ge( d-2)/2$ (equality holds for the free scalar theory).
In these integer $\Delta$ cases the bulk operator $\Phi(Y)$ is expressed
in terms of CFT operators living on
$\Sigma_Y^{(0)}$ (boundary points light-like separated from $Y$).
In this paper we carefully re-analyse the HKLL bulk reconstruction, both for
the case of even and odd AdS spaces (odd and even boundary manifolds),
paying special attention to the range of conformal dimensions where the
construction is valid (not emphasized in the original HKLL papers).
After a setup for the HKLL bulk reconstruction in section~\ref{sec:setup}, we consider the case of even and odd ${\rm AdS}_{d+1}$
(odd and even $d$) in sections \ref{sec:odd_d} and \ref{sec:even_d}, respectively and first recall the very well-known HKLL bulk
reconstruction \cite{Hamilton:2005ju,Hamilton:2006az} for a massive
free scalar boson field with conformal weight $\Delta>d-1$ in each section.
The purpose of this
review is to introduce our notations and conventions,
which will be needed later in the paper when we extend the validity
of the construction to smaller values of $\Delta$.
We then recall some pertinent results from \cite{Aoki:2021ekk} in both sections~\ref{sec:odd_d} and \ref{sec:even_d} before discussing
explicit reconstruction formulas for the regions $d-1>\Delta>d-2$ and
$d-2>\Delta>d-3$.
Some detailed calculations and necessary properties are summarized in several appendices.
In addition, in appendices G and H, we discuss the use of space-like Green's functions in the bulk reconstruction.
We demonstrate that this alternative method correctly reproduces the same results but in the extended range also
the singular part of the Green's function, omitted in the original paper \cite{Hamilton:2006az}, has to be included.
\section{Setup for the HKLL bulk reconstruction in AdS$_{d+1}$}
\label{sec:setup}
The HKLL bulk reconstruction\cite{Hamilton:2005ju,Hamilton:2006az} starts with a free scalar operator $\Phi(t,\rho, n)$ on the $d+1$ dimensional global AdS spacetime, whose metric is given by
\begin{eqnarray}
ds^2 &=& R^2 d\rho^2 -R^2 \cosh^2\rho\, dt^2 +R^2\sinh^2\rho\, dn^i d n^i,
\end{eqnarray}
where $R$ is the AdS radius and $Y = (t,\rho, n^i)$ with $n\cdot n=1$ (or $Y=(t,\rho,\Omega)$) are the standard global coordinates of AdS$_{d+1}$.
The value of $\Phi$ at the middle of the AdS, $Y_o = (0,0, n)$, is expressed as (see appendix~\ref{appA})
\begin{eqnarray}
\Phi(Y_o) &=& D(1) + D_1(1), \quad D(z) = \sum_{n=0}^\infty d_n z^n, \ D_1(z) = \sum_{n=0}^\infty d_n^\dagger z^n,
\label{eq:bulk_op_middle}
\end{eqnarray}
where $d_n$ and $d_n^\dagger$ are (rescaled) annihilation and creation operators, and $\Delta$ is related to the mass of the free scalar $m$ as
$m^2 R^2 =\Delta( \Delta-d)$.
On the other hand, using the BDHM relation
\begin{eqnarray}
O(x) &=& \lim_{\rho\to\infty} (\sinh\rho)^\Delta \Phi(\tilde t,\rho,\tilde n),
\end{eqnarray}
where $O(x)$ is a CFT operator with conformal dimension $\Delta$ at the AdS boundary $x=(\tilde t, \tilde n)$ with $\tilde n \cdot\tilde n=1$,
we have
\begin{eqnarray}
{\cal C}(\tilde t) &:=&\int{\rm d}\tilde\Omega\, O(\tilde t,\tilde \Omega) = e^{-i\Delta \tilde t} B(-e^{-2i\tilde t}) + e^{i\Delta \tilde t} B_1(-e^{2i\tilde t}),
\label{eq:CFT_op} \\
&&B(z) := \sum_{n=0}^\infty b_n z^n, \quad B_1(z) := \sum_{n=0}^\infty b_n^\dagger z^n,\
\end{eqnarray}
and $b_n, b_n^\dagger$ are related to $d_n,d_n^\dagger$ as
\begin{eqnarray}
b_n&:=& \Omega_d {P_n(1+\alpha)\over P_n(d/2)} d_n, \quad b_n^\dagger:= \Omega_d {P_n(1+\alpha)\over P_n(d/2)} d_n^\dagger,
\end{eqnarray}
where
$\alpha := \Delta-d/2$, $P_n(z):=\Gamma(z+n)/\Gamma(z)$ is the Pochhammer symbol, and $\Omega_d=\dfrac{2\pi^{d/2}}{\Gamma(d/2)}$
is the volume of the $d$ dimensional unit sphere.
The HKLL bulk reconstruction goes as follows. First a relation between $\Phi(Y_o)$ and $O(x)$ is derived, then $\Phi(Y)$ is obtained using the AdS
isometry $g$ and associated unitary operator $U(g)$ as $\Phi(Y) = U^\dagger (g) \Phi( Y_o) U(g)$, where $Y= g^{-1} Y_o$ is a generic point in the AdS space.
\section{Bulk reconstruction for odd $d$}
\label{sec:odd_d}
We first consider the bulk reconstruction for the odd $d$ case.
\subsection{Results of the original HKLL bulk reconstruction}
In papers by HKLL\cite{Hamilton:2005ju,Hamilton:2006az}, a relation between $\Phi(Y_o)$ and $O(x)$ has been derived for odd $d$
(see also appendix~\ref{appX})
\begin{eqnarray}
\Phi(Y_o)&=& \xi \int {\cal D} x\, k_0(\tilde t)
\Theta( \pi/2 -\tilde t) \Theta(\tilde t +\pi/2) O(x),
\label{eq:odd_HKLL}
\end{eqnarray}
where
\begin{eqnarray}
\xi := {1\over \pi\Omega_d}{\Gamma(1-d/2) \Gamma(1+\alpha)\over \Gamma(\nu+1)}, \quad
k_0(u):= (2 \cos u)^\nu, \quad \nu:=\Delta -d,
\label{eq:xi}
\end{eqnarray}
The convergence of the $\tilde t$ integral near $\tilde t=\pm\pi/2$ implies $\nu > -1$. Thus the condition $\Delta > d-1$ is (implicitly) assumed for the original HKLL bulk reconstruction.
A relation at a generic point $Y=(t,\rho,n)$ was given as
\begin{eqnarray}
\Phi (Y) &=& \xi \int {\cal D}x\, I^\nu(Y,x) T(Y,x) O(x),
\label{eq:odd_HKLL_generic}
\end{eqnarray}
where
\begin{eqnarray}
I(Y, x) &=& 2[\cosh(\rho) \cos(t-\tilde t) -(\sinh\rho) n^i \tilde n^i]\quad
T(Y,x) = \Theta(T_2-\tilde t) \Theta(\tilde t - T_1) ,
\end{eqnarray}
where $ T_1=t-\omega, T_2= t+\omega$ with $\omega=\arccos[(\tanh\rho) n\cdot\tilde n]$ and $ 0<\omega<\pi$. Note that
\begin{eqnarray}
\lim_{\rho,t\to 0} I^\nu(Y, x) &=& k_0 (\tilde t),\quad \lim_{\rho,t\to 0} T(Y,x) = \Theta( \pi/2 -\tilde t) \Theta(\tilde t +\pi/2).
\end{eqnarray}
\subsection{Results of Ref. \cite{Aoki:2021ekk} for smaller $\Delta$}
\label{sec2-2}
In our previous paper\cite{Aoki:2021ekk} we have derived (see appendix~\ref{appY})
\begin{eqnarray}
\Phi(Y_o) &=&{\eta\over 2\Omega_d} \left[ {\cal C}(\pi/2) + {\cal C}(-\pi/2) \right] + \xi \int_{\rm (sub)}{\rm d}\tilde t\,k_0(\tilde t) {\cal C}(\tilde t),
\label{eq:odd_ours}
\end{eqnarray}
where
\begin{eqnarray}
\eta &=& {\Gamma(1-d/2)\Gamma(1+\alpha)\over \Gamma^2(1+\nu/2)},
\end{eqnarray}
and the subtracted integral is defined by
\begin{eqnarray}
\int_{\rm (sub)}{\rm d}\tilde t\,K(\tilde t) f(\tilde t) &=& \int^0_{-\pi/2}{\rm d}\tilde t K(\tilde t)\left[f(\tilde t) -f(-\pi/2)\right]
+ \int^{\pi/2}_{0}{\rm d}\tilde t K(\tilde t)\left[f(\tilde t) -f(\pi/2)\right],~~~~~
\label{eq:int_subt}
\end{eqnarray}
which converges for $\nu > -2$ thanks to subtractions.
Thus \eqref{eq:odd_ours} is valid for $\Delta > d-2$ and it reduces to \eqref{eq:odd_HKLL} for $\Delta> d-1$.
For $\Delta = d-s$ with an integer $s$, simple relations without integral have been given \cite{Aoki:2021ekk}:
\begin{eqnarray}
\Phi(Y_o) &=&\xi_o{\prod_{k=1}^\ell \left\{{\partial^2\over \partial t^2} +(2k-1)^2\right\}\over \prod_{k=1}^{2\ell} (d-2k)} C_+(t)\big\vert_{t=0},\quad
\xi_o := {(-1)^{{d-1\over 2}}\over 2\Omega_d}
\label{eq:ours_special}
\end{eqnarray}
for $\Delta=d-(2\ell+1)$, where $C_+(t) ={\cal C}(t+{\pi\over 2})+{\cal C}(t-{\pi\over 2})$, and
\begin{eqnarray}
\Phi(Y_o) &=&\xi_o{\prod_{k=1}^\ell \left\{{\partial^2\over \partial t^2} +4k^2\right\}\over \prod_{k=1}^{2\ell+1} (d-2k)}
\frac{\partial}{\partial t}C_-(t)\big\vert_{t=0}
\end{eqnarray}
for $\Delta=d-2(\ell+1)$, where $C_-(0) ={\cal C}(t+{\pi\over 2})-{\cal C}(t-{\pi\over 2})$.
In the previous paper we have not derived a formula for $\Phi(Y)$ at a generic point $Y$ for the whole extended range.
Results at the special points $\Delta = d-1$ and $d-2$ only were given, which are as follows.
\begin{eqnarray}
\Phi(Y) &=&\xi_o \int{\rm d}\tilde\Omega\, {1\over {\cal R}(Y,x)}\left[O(T_1,\tilde\Omega) +O(T_2,\tilde\Omega)\right]
\label{eq:d-1}
\end{eqnarray}
for $\Delta=d-1$, where ${\cal R}(Y,x) = \sqrt{\cosh^2\rho -(n\cdot \tilde n)^2\sinh^2\rho}$, and
\begin{eqnarray}
\Phi(Y) &=&\tilde \xi_o \int {d\tilde\Omega\over {\cal R}^2(Y,x)}\left[\dot O(T_2,\tilde\Omega) -\dot O(T_1,\tilde\Omega)
- \cot \omega\{O(T_2,\tilde\Omega) +O(T_1,\tilde\Omega)\}\right] ~~~
\label{eq:d-2}
\end{eqnarray}
for $\Delta = d-2$,
where $\tilde\xi_o := {(-1)^{{d-1\over 2}}\over 2(d-2)\Omega_d}$, and $\dot O(x) :=\partial_{\tilde t} O(\tilde t, \tilde n)$.
\subsection{Bulk reconstruction for the extended range $\Delta > d-3$ for odd $d$}
In this subsection we derive a bulk reconstruction formula for a generic bulk point for $d-1 \geq \Delta > d-3$.
\subsubsection{Formula at a generic point by partial integration}
Since it is not easy to transform \eqref{eq:odd_ours} to a generic bulk point by the AdS isometry, we take a different strategy and we start from \eqref{eq:odd_HKLL_generic}, which is first rewritten by partial integration as
\begin{eqnarray}
\Phi(Y) = \eta_o (2\cosh\rho)^\nu \int{\rm d}\tilde\Omega&& \left\{ - {1\over \Gamma(\nu+1)} \int_{-\omega}^\omega{\rm d}\tilde t\, \phi_1(\tilde t) \dot O(t+\tilde t,\tilde\Omega)\right. \nonumber \\
&&+\left. K_1(\nu, \omega) \left[O(T_1,\tilde\Omega) + O(T_2,\tilde\Omega) \right] \right\},
\label{eq:generic_1}
\end{eqnarray}
where $\eta_o :=\Gamma(\nu+1)\xi = {\Gamma(1-d/2)\over \pi \Omega_d} \Gamma(1+\alpha)$,
\begin{eqnarray}
\phi_1(u) &=&\int_0^u {\rm d}v\, \phi_0(v), \ \phi_0(u) := (\cos u -\cos \omega)^\nu,\
K_1(\nu,\omega) = {\phi_1(\omega)\over \Gamma(\nu+1)} .~~~~
\end{eqnarray}
Since $\phi_1(u) \sim (\omega-\vert u\vert)^{\nu+1}$ at $u\sim \pm \omega$, the $\tilde t$ integral is convergent for $\nu>-2$.
Performing a second integration by parts, \eqref{eq:generic_1} becomes
\begin{eqnarray}
\Phi(Y) &=& \eta_o (2\cosh\rho)^\nu \int{\rm d}\tilde\Omega \left\{ {1\over \Gamma(\nu+1)} \int_{-\omega}^\omega{\rm d}\tilde t\, \phi_2(\tilde t) \ddot O(t+\tilde t,\tilde\Omega)\right. \nonumber \\
&+&\left.
K_2(\nu,\omega)\left[\dot O(T_1,\tilde \Omega) -\dot O(T_2,\tilde \Omega)\right]
+ K_1(\nu, \omega) \left[O(T_1,\tilde\Omega) + O(T_2,\tilde\Omega) \right] \right\},
\label{eq:generic_2}
\end{eqnarray}
where
\begin{eqnarray}
\phi_2(u) &=& \int_0^u\,{\rm d}v (u-v)\phi_0(v), \ \phi_2^\prime(u) = \phi_1(u), \
K_2(\nu,\omega)= {\phi_2(\omega)\over \Gamma(\nu+1)}.
\end{eqnarray}
The $\tilde t$ integral in this expression is convergent for $\nu > -3$.
Although we do not need to go further for later analysis,
we can repeat the procedure to obtain
\begin{eqnarray}
\Phi(Y) &=& \eta_o (2\cosh\rho)^\nu \int{\rm d}\tilde\Omega \left\{ {(-1)^k\over \Gamma(\nu+1)} \int_{-\omega}^\omega{\rm d}\tilde t\, \phi_k(\tilde t) O^{(k)}(t+\tilde t,\tilde\Omega)\right. \nonumber \\
&+&\left.
\sum_{\ell=1}^k {\phi_\ell(\omega)\over \Gamma(\nu+1)}\left[ O^{(\ell-1)}(T_1,\tilde \Omega) +(-1)^{\ell-1} O^{(\ell-1)}(T_2,\tilde \Omega)\right]\right\}
\label{eq:generic_k}
\end{eqnarray}
for an arbitrary positive integer $k$, where
\begin{eqnarray}
\phi_\ell(u) :={1\over (\ell-1)!}\int_0^u {\rm d}v \,(u-v)^{\ell-1}\phi_0(v), \quad O^{(\ell)}(u,\tilde \Omega) := {\partial^\ell\over \partial u^\ell} O(u,\tilde\Omega).
\end{eqnarray}
The $\tilde t$ integral is convergent for $ \nu > -(k+1)$.
\subsubsection{Analytic continuation of $K_1(\nu,\omega)$}
While the $\tilde t$ integral in \eqref{eq:generic_1} is convergent for $ \nu > -2$,
we must show that $K_1(\nu,\omega)$ is convergent for $\nu > -2$.
Using a limiting case of (3.663-1) in the table of integrals by Gradshteyn and Ryzhik, $K_1(\nu,\omega)$ can be evaluated for $\nu > -1$ as
\begin{eqnarray}
K_1(\nu, \omega) &=&\sqrt{\pi}2^\nu \left( \sin{\omega\over 2}\right)^{2\nu+1} {1\over \Gamma(\nu+3/2)}{}_2F_1\left({1\over 2},{1\over 2};\nu+{3\over 2};\sin^2{\omega\over 2}\right).
\label{eq:K1}
\end{eqnarray}
Since the Gamma function in the denominator of \eqref{eq:K1} regularizes the hypergeometric function, \eqref{eq:K1}
can be analytically continued to all $\nu$.
The integral part is convergent for $\nu>-2$ and therefore \eqref{eq:generic_1} provides the analytic extension of the bulk reconstruction to the range $\nu > -2$.
\subsubsection{Analytic continuation of $K_2(\nu,\omega)$}
Since the $\tilde t$ integral in \eqref{eq:generic_2} is convergent for $\nu>-3$ and we have already seen that $K_1(\nu,\omega)$ is analytic for all $\nu$,
we now concentrate on the integral
\begin{eqnarray}
K_2(\nu,\omega) =\omega K_1(\nu, \omega) -J_1(\nu,\omega),
\label{eq:K2}
\end{eqnarray}
where
\begin{eqnarray}
J_1(\nu,\omega) &:=& {1\over \Gamma(\nu+1)}\int_0^\omega {\rm d}u\, u (\cos u -\cos \omega)^\nu ,
\label{eq:Jint}
\end{eqnarray}
which, unfortunately, can not be found in integral tables.
In the absence of an explicit formula for \eqref{eq:Jint} we have
derived (see appendix \ref{appD}) a
recursion relation for $J_1(\nu,\omega)$, which can also be used for
analytic continuation:
\begin{equation}
J_1(\nu,\omega)=\frac{1}{\sin^2\omega}\Big\{(\nu+2)^2\,
J_1(\nu+2,\omega)+(2\nu+3)\cos\omega \, J_1(\nu+1,\omega)+\frac
{(1-\cos\omega)^{\nu+2}}{\Gamma(\nu+3)}\Big\}.
\label{eq:recursion}
\end{equation}
The integrals related to the right hand side of (\ref{eq:recursion}) are convergent
for $\nu>-2$ and so this relation can be used to extend the left hand side
to $\nu>-2$ too. After this extension the right hand side will be defined
to $\nu>-3$ and it defines the left hand side also to $\nu>-3$. In this
way we can extend, step by step, $J_1(\nu,\omega)$ for all $\nu$.
Therefore, \eqref{eq:K2} implies that $K_2(\nu,\omega)$ is analytic for all $\nu$.
Since the $\tilde t$ integral is convergent for $\nu>-3$, and $K_{1,2}(\nu,\omega)$ are analytic for all $\nu$,
$\Phi(Y)$ in \eqref{eq:generic_2} provides the analytic extension of the bulk reconstruction to the range $\nu> -3$ ($\Delta > d-3$), as promised.
\subsubsection{Comparisons with previous results}
Since the starting formula in \eqref{eq:odd_HKLL_generic} is valid only for $\nu > -1$,
results \eqref{eq:generic_1} for $\nu > -2$ and \eqref{eq:generic_2} for $\nu > -3$
are not regarded as direct derivations but should be considered as analytic continuations to $\nu > -2$ and $\nu > -3$.
Therefore, it is useful to compare \eqref{eq:generic_1} and \eqref{eq:generic_2} for special cases with previous results obtained without using
analytic continuation.
For this purpose, using the hypergeometric identities ${}_2F_1(a,b;c;z) =(1-z)^{-a} {}_2F_1(a,c-b;c;z/(z-1) )$,
we rewrite \eqref{eq:K1} as
\begin{eqnarray}
K_1(\nu, \omega) &=&\sqrt{\pi}2^\nu {\left( \sin{\omega\over 2}\right)^{2\nu+1}\over \cos{\omega\over 2}} {1\over \Gamma(\nu+3/2)}{}_2F_1\left({1\over 2},\nu+1;\nu+{3\over 2};-\tan^2{\omega\over 2}\right),
\label{eq:K1_alt}
\end{eqnarray}
which gives
\begin{eqnarray}
K_1(-1,\omega) ={1\over \sin \omega} , \quad K_1(-2,\omega) = -{\cos \omega\over \sin^3 \omega}.
\end{eqnarray}
To start the recursion of $J_1(\nu,\omega)$ for a negative integer $\nu$,
we calculate
\begin{equation}
J_1(0,\omega)=\frac{\omega^2}{2}, \quad
J_1(1,\omega)=\int_0^\omega{\rm d}u\,u(\cos u-\cos \omega)=
-\frac{\omega^2}{2}\cos\omega+\cos\omega+\omega\sin\omega-1,
\end{equation}
which, through the recursion \eqref{eq:recursion}, lead to
\begin{equation}
J_1(-1,\omega)=\frac{\omega}{\sin\omega},\qquad
J_1(-2,\omega)=\frac{1}{\sin^2\omega}-\frac{\omega\cos\omega}
{\sin^3\omega}.
\end{equation}
Thus, \eqref{eq:K2} leads to
\begin{eqnarray}
K_2(-1,\omega)=0, \quad K_2(-2,\omega)= -{1\over \sin^2 \omega} .
\end{eqnarray}
Using these, \eqref{eq:generic_1} for $\nu=-1$ and \eqref{eq:generic_2} for $\nu=-2$ reduce to
\begin{eqnarray}
\Phi(Y) &=&\xi_o \int {d\tilde\Omega\over {\cal R}(Y,x)} \left[ O(T_1,\tilde\Omega) + O(T_2,\tilde\Omega) \right],\\
\Phi(Y)&=&\tilde\xi_o \int \frac{{\rm d}\Omega}{{\cal R}^2(Y,x)}\Big\{
[{\dot O}(T_2,\Omega)-{\dot O}(T_1,\Omega)]-\cot\omega [O(T_2,\Omega)+O(T_1,\Omega)]\Big\},
\end{eqnarray}
which reproduce our previous results from the direct evaluation, \eqref{eq:d-1} and \eqref{eq:d-2}.
By applying a (backward) partial integration to the integral in \eqref{eq:generic_1}, we obtain
\begin{eqnarray}
\Phi(Y)&=&\eta_o(2\cosh\rho)^\nu\int{\rm d}\Omega\Big\{\frac{1}{\Gamma(\nu+1)}
\int_0^\omega{\rm d}u\,\phi(u)[O(t+u,\Omega)-O(T_2,\Omega)] + \frac{1}{\Gamma(\nu+1)}\nonumber \\
&\times
\int_{-\omega}^0{\rm d}u\,\phi(u)[O(t+u,\Omega)-O(T_1,\Omega)]
+K_1(\nu,\omega)[O(T_2,\Omega)+O(T_1,\Omega)]\Big\}.
\end{eqnarray}
For the middle point $Y_o$ this reduces to
\eqref{eq:odd_ours}, which was obtained by direct calculation for $\Delta > d -2$.
The above results show that the analytic continuation and the direct calculation without analytic continuation lead to the same formula
(at least in these special cases) for the extended range of $\Delta$.
\section{Bulk reconstruction for even $d$}
\label{sec:even_d}
We now consider a more difficult task, the extension of the bulk reconstruction for even $d$ to smaller $\Delta$.
\subsection{Results of the original HKLL bulk reconstruction}
For the even $d$ case, the result at the middle point has been obtained by HKLL\cite{Hamilton:2005ju,Hamilton:2006az}:
\begin{eqnarray}
\Phi(Y_o) = \tilde\xi \int {\cal D}x\, T(Y_o,x) I^\nu(Y_o,x) \ln[I(Y_o,x)] O(x), \quad \tilde\xi:=\left(-{1\over \pi}\right)^{d/2+1} {\Gamma(1+\alpha)\over \Gamma(\nu+1)} .
\label{eq:HKLL_eve_mid}
\end{eqnarray}
Using transformation properties under the AdS isometry $g$
\begin{eqnarray}
U^\dagger(g) \Phi(Y_o) U(g) = \Phi(g^{-1} Y_o), \quad U^\dagger(g) O(x) U(g) = H^\Delta(g^{-1},x) O(g^{-1}x),
\end{eqnarray}
$\Phi$ at a generic bulk point $Y=g^{-1} Y_o$ becomes
\begin{eqnarray}
\Phi(Y) &=& \tilde\xi \int {\cal D}y\,T(Y_o,y) I^\nu(Y_o,y) \ln[I(Y_o,y)] H^\Delta(g^{-1},y) O(g^{-1}y)\nonumber \\
&=& \tilde\xi \int {\cal D}x\, T(g^{-1} Y_o,x) I^\nu(g^{-1} Y_o,x) \ln[I(g^{-1} Y_o,x) H(g,x) ] O(x)\nonumber \\
&=& \Phi^{\rm HKLL}(Y) +\tilde\xi \hat\Phi(g),
\end{eqnarray}
where
\begin{eqnarray}
\Phi^{\rm HKLL}(Y) &=&\tilde\xi \int {\cal D}x\, T(Y,x) I^\nu(Y,x) \ln[I(Y,x) ] O(x),
\label{eq:even_HKLL} \\
\hat\Phi(g) &=& \int {\cal D}x\, T(g^{-1}Y_o,x) I^\nu(g^{-1}Y_o,x) \ln[H(g,x) ] O(x).
\end{eqnarray}
In the above derivation we used results in appendix \ref{appB} ((\ref{DD77}), (\ref{DD88}), (\ref{DD13}))
in the form
\begin{eqnarray}
I(Y_o,y) H(g^{-1},y) &=& I(g^{-1}Y_o, g^{-1} y), \quad
T(Y_o,y) = T(g^{-1}Y_o, g^{-1} y),
\end{eqnarray}
and
\begin{eqnarray}
{\cal D} (gx) H^d(g^{-1}, gx) = {\cal D} x, \quad H(g,x) ={1\over H(g^{-1}, gx)}.
\end{eqnarray}
It has been claimed in the HKLL papers\cite{Hamilton:2005ju,Hamilton:2006az} that the bulk field at a generic point is given by $ \Phi^{\rm HKLL}(Y)$, which is true if
\begin{eqnarray}
\hat\Phi(g)\equiv 0
\label{Phig}
\end{eqnarray}
for all group elements $g$.
Note that from the derivation it follows that $\hat\Phi(g)$ only depends on $Y=g^{-1}Y_o$.
Although this was already discussed in appendix B of \cite{Hamilton:2006az},
an elementary proof of (\ref{Phig}) is presented in appendix \ref{appC} for the sake of completeness.
\subsection{Previous results for integer $\Delta=d-s$}
In \cite{Aoki:2021ekk} we have derived results at the middle point for $\Delta = d -s$ with a positive integer $s$, which are summarized as
\begin{eqnarray}
\Phi(Y_o) &=&\left. {(-1)^{d/2}\over \pi \Omega_d}{\prod_{k=1}^\ell \left\{ {\partial^2\over \partial t^2} +(2k-1)^2\right\}\over \prod_{k=1}^{2\ell} (d-2k)}
{\partial \over \partial\Delta} C_+(t)\right\vert_{t=0,\Delta=d-(2\ell+1)},
\label{eq:even1}\\
\Phi(Y_o) &=&\left. {(-1)^{d/2}\over \pi \Omega_d}{\prod_{k=1}^\ell \left\{ {\partial^2\over \partial t^2} +4k^2\right\}\over \prod_{k=1}^{2\ell+1} (d-2k)}
{\partial \over \partial t} {\partial \over \partial\Delta} C_-(t)\right\vert_{t=0,\Delta=d-2(\ell+1)}.
\label{eq:even2}
\end{eqnarray}
\subsection{New results for smaller $\Delta$}
\subsubsection{Bulk reconstruction at the middle point}
\label{sec3-3-1}
For even $d$, the bulk reconstruction at the middle point $Y_o$ is given by
\begin{equation}
\Phi(Y_o)=\tilde\xi\int_{({\rm sub})}{\rm d}\tilde t\,k_1(\tilde t){\cal C}(\tilde t)
+\tilde\xi\,g^\prime(\nu)[{\cal C}(\pi/2)+{\cal C}(-\pi/2)].
\label{finaleven}
\end{equation}
where
\begin{eqnarray}
k_1(u) = (2\cos u)^\nu \ln (2\cos u), \quad g(\nu)=\int_0^{\pi/2} du\, (2\cos u)^\nu ={\pi\over 2}{\Gamma(\nu+1)\over \Gamma^2(\nu/2+1)}.
\end{eqnarray}
Details of the derivation are presented in appendix \ref{appY}.
The derivative of the identity (\ref{extiden}) with respect to $\nu$ gives
\begin{eqnarray}
\int_{({\rm sub})}{\rm d}\tilde t\,k_1(\tilde t){\cal C}(\tilde t) &+&
\int_{({\rm sub})}{\rm d}\tilde t\,k_0(\tilde t){\cal C}_\Delta(\tilde t)\nonumber \\
&+&g^\prime(\nu)[{\cal C}(\pi/2)+{\cal C}(-\pi/2)]
+g(\nu)[{\cal C}_\Delta(\pi/2)+{\cal C}_\Delta(-\pi/2)]=0,~~
\end{eqnarray}
which leads to an alternative form of the bulk reconstruction as
\begin{equation}
\Phi(Y_o)=-\tilde\xi\int_{({\rm sub})}{\rm d}\tilde t\,k_0(\tilde t){\cal C}_\Delta(\tilde t)
-\tilde\xi g(\nu)[{\cal C}_\Delta(\pi/2)+{\cal C}_\Delta(-\pi/2)],
\label{evenext}
\end{equation}
where we define
\begin{eqnarray}
{\cal C}_\Delta(t) :={\partial \over \partial \Delta} {\cal C}(t)
\end{eqnarray}
\subsubsection{Analytic continuation}
As in the odd $d$ case, analytic continuation is employed to obtain the bulk reconstruction at a generic point $Y$ also for even $d$.
We start with the formula $\Phi(Y) =\Phi^{\rm HKLL}(Y)$ in \eqref{eq:even_HKLL} rewritten as
\begin{eqnarray}
\Phi(Y) &=& \tilde\eta (2\cosh\rho)^\nu \int{\rm d}\tilde\Omega\, H(t,\omega,\tilde\Omega), \quad \tilde \eta := \left(-{1\over \pi}\right)^{d/2+1} \Gamma(1+\alpha) ,\label{eq:Phi_from_H}
\end{eqnarray}
where
\begin{eqnarray}
H(t,\omega,\tilde\Omega) &:=& {1\over \Gamma(\nu+1)}\int_{-\omega}^\omega{\rm d}\tilde t\, g(\tilde t, \omega) O(\tilde t+t,\tilde\Omega),
\end{eqnarray}
with $g(u,\omega) :=(\cos u -\cos \omega)^\nu \ln (\cos u -\cos \omega)$.
A partial integration gives
\begin{eqnarray}
H(t,\omega,\tilde\Omega) &=& - {1\over \Gamma(\nu+1)}\int_{-\omega}^\omega{\rm d}\tilde t\, g_1(\tilde t, \omega) \dot O(\tilde t+t,\tilde\Omega)
+P_1(\nu,\omega) \left[ O(T_1,\tilde \Omega) + O(T_2,\tilde \Omega)\right],~~~~~~~~
\label{eq:H}
\end{eqnarray}
where
\begin{eqnarray}
g_1(u,\omega) &:=& \int_0^u dv\, g(v,\omega), \quad P_1(\nu,\omega): ={ g_1(\omega,\omega)\over \Gamma(\nu+1)},
\end{eqnarray}
The $\tilde t$ integral in \eqref{eq:H} is convergent for $\nu > -2$. Although the integral defining $P_1(\nu,\omega)$ is convergent only for $\nu > -1$,
it can be analytically continued using the recursion relation
\begin{eqnarray}
P_1(\nu,\omega) &=&{1\over \sin^2 \omega} \Bigl\{ (2\nu+3) \cos\omega P_1(\nu+1,\omega) +(\nu+2)^2 P_1(\nu+2,\omega) \nonumber \\
&+& 2\cos\omega K_1(\nu+1,\omega) +(\nu+2) K_1(\nu+2,\omega)\Bigr\} -{K_1(\nu,\omega)\over \nu+1},
\label{eq:recursionP}
\end{eqnarray}
which is derived in appendix \ref{appD}.
The right hand side of the recursion relation is defined for $\nu> -2$ except a pole at $\nu=-1$, whose residue is given by
\begin{eqnarray}
-K_1(-1,\omega) = -{1\over \sin \omega} .
\end{eqnarray}
The recursion relation allows to extend $P_1(\nu,\omega)$ step by step to all $\nu$ but there will be poles for all negative integer values of $\nu$.
\subsubsection{Comparison at the middle point for $\Delta=d -1$}
While the presence of poles at negative integer $\nu$ prevents us from extending $\Phi(Y)$ to $\nu=-1$ by analytic continuation,
we can proceed for the special case $Y=Y_o$ (the middle point) starting from
\begin{eqnarray}
\Phi(Y_o) &=& 2^\nu \tilde \eta \left\{ -{1\over \Gamma(\nu+1)}\int_{-{\pi\over 2}}^{\pi\over2}d\tilde t\, g_1\left(\tilde t,{\pi\over 2}\right) \dot {\cal C}(\tilde t)
+ P_1\left(\nu,{\pi\over 2}\right)\left[ {\cal C}\left({\pi\over 2}\right) + {\cal C}\left(-{\pi\over 2}\right) \right]
\right\}.~~~~~
\label{eq:Phi_middel}
\end{eqnarray}
We then explicitly evaluate the integral (convergent for $\nu>-1$)
\begin{eqnarray}
g_1\left({\pi\over 2},{\pi\over 2}\right)=\frac{\partial}{\partial\nu}[2^{-\nu}g(\nu)]=
{\pi\over 2^{\nu+1}}{\Gamma(\nu+1)\over \Gamma^2(\nu/2+1)}\left\{ -\ln 2 +\psi(\nu+1) -\psi(\nu/ 2+1)\right\},
\end{eqnarray}
giving
\begin{eqnarray}
P_1\left(\nu,{\pi\over 2}\right) &=& {\pi\over 2^{\nu+1}}{1\over \Gamma^2(\nu/2+1)}\left\{ -\ln 2 +\psi(\nu+1) -\psi(\nu/ 2+1)\right\}.
\label{eq:P1}
\end{eqnarray}
Since the digamma function in \eqref{eq:P1} has a pole at $\nu = -1$ we see that
\begin{eqnarray}
P_1 \left(\nu,{\pi\over 2}\right) = - {1\over \nu + 1} + O(1).
\end{eqnarray}
On the other hand, since ${\cal C}(\tilde t)$ in \eqref{eq:CFT_op} is anti-periodic in $\tilde t$ with a period $\pi$ for an odd integer $\Delta$,
we have
\begin{eqnarray}
{\cal C}\left(t+{\pi\over 2}\right) + {\cal C}\left(t-{\pi\over 2}\right) &=& (\nu +1) \left[{\cal C}_\Delta\left(t+{\pi\over 2}\right) + {\cal C}_\Delta\left(t-{\pi\over 2}\right)\right] + O\left( (\nu+1)^2\right)
\end{eqnarray}
near $\Delta = d+1$ for even $d$. Thus we conclude that $\Phi(Y_o)$ is finite in the $\nu \to -1$ limit as
\begin{eqnarray}
\Phi(Y_o) &=& {(-1)^{d/2}\over \pi \Omega_d} \left[{\cal C}_\Delta\left({\pi\over 2}\right) + {\cal C}_\Delta\left(-{\pi\over 2}\right)\right] ,
\label{eq:even_middle_int}
\end{eqnarray}
which reproduce the previous result in \eqref{eq:even1} for $\ell=0$, obtained by a different method without analytic continuation.
\subsubsection{An alternative expression at a generic point}
We can avoid difficulties arising from poles of $P_1(\nu, \omega)$ by considering an alternative expression of $\Phi$.
We start from the fact that for even $d$ and $\nu>-1$ (see (\ref{BB14}))
\begin{eqnarray}
\int {\cal D} x\, I^\nu(Y_0,x) T(Y_o,x) O(x) =0,
\end{eqnarray}
which can be transformed to a generic point by the isometry transformation as
\begin{eqnarray}
\int {\cal D} x\, I^\nu(Y,x) T(Y,x) O(x) =0 .
\end{eqnarray}
By taking its derivative with respect to $\nu$ and comparing to (\ref{eq:even_HKLL}), we arrive at an alternative expression (valid for $\nu>-1$):
\begin{eqnarray}
\Phi(Y) &=& -\tilde\xi \int{\cal D} x\, I^\nu(Y,x) T(Y,x) O_\Delta(x), \quad O_\Delta(x):={\partial \over \partial \Delta} O(x).
\end{eqnarray}
Applying the partial integration to this alternative expression, $H$ in \eqref{eq:Phi_from_H} becomes
\begin{eqnarray}
H(t, \omega,\tilde\Omega) &=& {1\over\Gamma(\nu+1)}\int_{-\omega}^\omega{\rm d}\tilde t\, \phi_1(\tilde t) \dot O_\Delta(\tilde t+t,\tilde\Omega) -K_1(\nu,\omega)\left[O_\Delta(T_1,\tilde\Omega) +O_\Delta(T_2,\tilde\Omega) \right].~~~~~~~~~
\label{eq:even_H}
\end{eqnarray}
Since the $\tilde t$ integral is convergent for $\nu>-2$ and $K_1(\nu,\omega)$ is analytic for all $\nu$, $\Phi(Y)$ can be analytically continued to
$\nu > -2$ as
\begin{eqnarray}
\Phi(Y) &=&\tilde\eta (2\cos\rho)^\nu \int{\rm d}\tilde\Omega\, H(t,\omega,\tilde \Omega)
\label{eq:generic_even}
\end{eqnarray}
with $H(t,\omega,\tilde \Omega)$ in \eqref{eq:even_H}.
In particular for $\nu=-1$, we have
\begin{eqnarray}
\Phi(Y) &=& {(-1)^{d/2}\over \pi \Omega_d} \int{\rm d}\tilde \Omega {1\over {\cal R}(Y,x)} \left[O_\Delta(T_1,\tilde\Omega) +O_\Delta(T_2,\tilde\Omega) \right].
\end{eqnarray}
For $Y_o$ this agrees with \eqref{eq:even_middle_int}, and thus with \eqref{eq:even1} for $\ell=0$.
For $Y_o$, but for generic $\nu>-2$ \eqref{eq:generic_even} reduces to
\begin{eqnarray}
\Phi(Y_o) &=& -2^\nu\tilde\xi \int_{\rm (sub)}{\rm d}\tilde t\, (\cos (\tilde t))^\nu {\cal C}_\Delta(\tilde t)
-2^\nu \tilde\eta K_1(\nu,\pi/2)\left[{\cal C}_\Delta(\pi/2) +{\cal C}_\Delta(-\pi/2) \right]\nonumber \\
&=&- \tilde\xi \int_{\rm (sub)}{\rm d}\tilde t\, k_0(\tilde t) {\cal C}_\Delta(\tilde t)
-\tilde\xi g(\nu) \left[{\cal C}_\Delta(\pi/2) +{\cal C}_\Delta(-\pi/2) \right],
\end{eqnarray}
reproducing \eqref{evenext}, which has been obtained without analytic continuation.
\if0
\input sec1A
\input sec1e
\section{Bulk reconstruction for $\Delta>d-1$ (generic point)}
\label{sec2}
\input sec2o
\input sec2e
\section{Extension of the HKLL bulk reconstruction for $\Delta>d-2$}
\label{sec3}
\input sec3A
\input sec3e
\section{Bulk reconstruction for $\Delta>d-2$}
\label{sec4}
\input sec4o
\input sec7
\section{Bulk reconstruction for $\Delta>d-3$}
\label{sec5}
\input sec5
\fi
\vspace{5ex}
\section*{Acknowledgements}
We thank Dr. S. Terashima for useful discussions and for partially checking
our results.
J.B. acknowledges support from the International Research Unit of Quantum
Information (QIU) of Kyoto University Research Coordination Alliance,
and also would like to thank the Yukawa Institute for Theoretical Physics
at Kyoto University, where most of this work has been carried out, for
support and hospitality by the long term visitor program.
This work has been supported in part by the NKFIH grant K134946,
by the Grant-in-Aid of the Japanese Ministry of Education, Sciences and
Technology, Sports and Culture (MEXT) for Scientific Research
(Nos.~JP16H03978, JP18H05236).
\par\bigskip
|
1,116,691,500,796 | arxiv | \section{Introduction}
Spectral observations in radio astronomy are aimed to detect emission or absorption line features that bring us rich information such as intensity, velocity, line width, etc.
To acquire a spectrum, signals received by a radio telescope are processed in a spectrometer.
The performance of a spectral observation is characterized by the spectral resolution, the bandwidth, the sensitivity, and the stability.
A certain bandpass calibration procedure is needed to obtain a desired performance of spectral observations.
An acquired spectrum is a summation of unwanted noise, $T_{\mathrm{sys}}$, and the signal from the target source that is denoted as the antenna temperature, $T_{\mathrm{a}}(\nu)$.
The spectral shape is affected by the bandpass response, $H(\nu)$, that is the transfer function of the receiving system as a function of frequency, $\nu$.
The acquired ON-source spectrum will be then $T_{\mathrm{ON}}(\nu) = H(\nu) (T_{\mathrm{sys}} + T_{\mathrm{a}}(\nu))$.
Bandpass calibration is necessary to estimate $T_{\mathrm{a}}(\nu)$ from the acquired spectrum, $T_{\mathrm{ON}}(\nu)$.
In most of radio observations, the acquired spectrum is dominated by the system noise, i.e., $T_{\mathrm{sys}} \gg T_{\mathrm{a}}(\nu)$.
To relieve the source spectrum from the system noise, a spectrum of OFF-source blank sky adjacent to the target source is subtracted from the ON-source spectrum.
This position-switching scan yields the source spectrum as
\begin{eqnarray}
T_{\mathrm{a}}(\nu) = \frac{T_{\mathrm{ON}}(\nu) - T_{\mathrm{OFF}}(\nu)}{H(\nu)}, \label{eqn:onoff_subtraction}
\end{eqnarray}
where $T_{\mathrm{OFF}}$ is the spectrum taken from the OFF-source scan.
In conventional position-switching observations, the bandpass $H(\nu)$ is also acquired from the OFF-source scans because the blank sky emits a featureless flat spectrum. For $\displaystyle H(\nu) = \frac{T_{\mathrm{OFF}}(\nu)}{T_{\mathrm{sys}}}$, equation \ref{eqn:onoff_subtraction} will be
\begin{eqnarray}
T_{\mathrm{a}}(\nu) = T_{\mathrm{sys}} \left( \frac{T_{\mathrm{ON}}(\nu)}{T_{\mathrm{OFF}}(\nu)} - 1\right). \label{eqn:onoff_subtraction2}
\end{eqnarray}
The sensitivity of spectal observations is evaluated by a SD (standard deviation), $\sigma$, of the acquired spectrum in line-free channels.
It is derived from equation \ref{eqn:onoff_subtraction2} as
\begin{eqnarray}
\left( \frac{\sigma}{T_{\mathrm{a}}(\nu) + T_{\mathrm{sys}}} \right)^2 = \left( \frac{\sigma_{\mathrm{ON}}}{T_{\mathrm{ON}}(\nu)} \right)^2 + \left( \frac{\sigma_{\mathrm{OFF}}}{T_{\mathrm{OFF}}(\nu)} \right)^2, \label{eqn:sensitivity1}
\end{eqnarray}
where $\sigma_{\mathrm{ON}}$ and $\sigma_{\mathrm{OFF}}$ are SDs of the ON- and OFF-source spectra, respectively.
For weak sources for which $T_{\mathrm{ON}}(\nu) \simeq T_{\mathrm{OFF}}(\nu)$, $\sigma$ is given by a root of sum squared (RSS) of $\sigma_{\mathrm{ON}}$ and $\sigma_{\mathrm{OFF}}$.
When $\sigma_{\mathrm{ON}}$ and $\sigma_{\mathrm{OFF}}$ are dominated by thermal noise,
according to \citet{2009tra..book.....W},
they are expected to relate to the integration time, $t_{\mathrm{ON}}$ and $t_{\mathrm{OFF}}$, the spectral resolution, $\nu_{\mathrm{res}}$, and $T_{\mathrm{sys}}$ as followed,
\begin{eqnarray}
\sigma_{\mathrm{ON}} = \frac{T_{\mathrm{sys}} + T_{\mathrm{a}}}{\sqrt{\nu_{\mathrm{res}} t_{\mathrm{ON}}}}, \ \ \sigma_{\mathrm{OFF}} = \frac{T_{\mathrm{sys}}}{\sqrt{\nu_{\mathrm{res}} t_{\mathrm{OFF}}}}. \label{eqn:sigma}
\end{eqnarray}
The total telescope time is the sum of $t_{\mathrm{ON}}$, $t_{\mathrm{OFF}}$, and other overhead time such as setup, system noise measurement, gaps between scans, etc.
Optimized scan pattern is designed to maximize the sensitivity, i.e., to minimize $\sigma$, within the minimal telescope time.
Conventionally $t_{\mathrm{ON}}$ and $t_{\mathrm{OFF}}$ are equally allocated in position-switching observations to have $\sigma_{\mathrm{ON}} \simeq \sigma_{\mathrm{OFF}}$.
This manner makes the total telescope time longer than twice of $t_{\mathrm{ON}}$ and lets $\sigma \simeq \sqrt{2} \sigma_{\mathrm{ON}}$.
If there is an efficient way to reduce $\sigma_{\mathrm{OFF}}$ less than $\sigma_{\mathrm{ON}}$, we can save the total telescope time and get better sensitivity.
For an ideal receiving system, whose bandpass response were known {\it a priori} and very stable, OFF-source observations would be unnecessary.
Modern systems offer better spectral stability in the trend in replacing analog devices with a high-speed digital sampling system in an upper stream of signal transfer.
The bandpass response of a digital system, including a digital filter and a digital spectrometer, is determined by a fixed algorithm and is not affected by environmental variations.
Utility of digital devices brings us a possibility to reduce the integration time and the SD of OFF-source scans.
In this paper we propose the Smoothed Bandpass Calibration (SBC) method where the OFF-source spectrum is smoothed across the bandwidth to reduce the integration time of the OFF-source scans and the SD of the spectrum.
Section 2 describes the concept of the SBC method.
Section 3 reports test experiments that verify the SBC and that quest for the optimal parameters of a smoothing window and a scan pattern.
In section 4 we discuss the behavior of the spectral stability and estimate sensitivity improvements expected in some scan patterns.
\section{Method}\label{sec:method}
We propose relevant spectral smoothing along frequency for OFF-source scans to reduce required integration time that achieves a desired SD.
Before the spectral smoothing, a bandpass correction is employed using a template bandpass response that is presumed to be relevantly static and taken by long-time integration toward blank sky.
The bandpass-corrected OFF-source spectrum will be almost flat with slight warp due to fluctuations of the receiving system and smeared by thermal noise.
In the following subsections we formularize the spectral behavior and establish a strategy to reduce spectral unevenness.
\subsection{Description and Assumption}
As described in section 1, the purpose of OFF-source scans is to subtract $T_{\mathrm{sys}}$ and calibrate the bandpass, $H(\nu)$, which is a frequency-dependent system response.
Since the bandpass can be time-variable, we denote it as $H(\nu, t) = H_0(\nu) + H_1(\nu, t)$, where $H_0(\nu)$ is a static template bandpass and $H_1(\nu, t)$ is a fluctuation from $H_0(\nu)$.
The observed OFF-source spectrum is composed of not only $H(\nu, t)$ but also thermal noise, $\epsilon(\nu, t)$, which is assumed to be attributed to a stochastic random process following the normal distribution with the mean, $\mu = 0$, and the variance,
$\displaystyle \sigma^2_{\mathrm{noise}} = \frac{1}{\nu_{\mathrm{res}} t_{\mathrm{integ}}}$,
where $t_{\mathrm{integ}}$ is the integration time.
Thus the observed OFF-source spectrum will be $T_{\mathrm{OFF}}(\nu, t) = \left( H_0(\nu) + H_1(\nu, t) \right) T_{\mathrm{sys}}(1 + \epsilon(\nu, t))$, and its expectation will be $\displaystyle \left< T_{\mathrm{OFF}}(\nu, t) \right> = \left( H_0(\nu) + H_1(\nu, t) \right) T_{\mathrm{sys}}$.
Sufficiently long integration time is required for conventional OFF-source scans to reduce $\sigma_{\mathrm{noise}}$.
We assume that $H_1(\nu, t)$ is relatively smaller than $H_0(\nu)$, almost flat and smooth along the frequency, with a typical frequency scale, $\nu_{\mathrm{var}}$ of variation.
Relevant spectral smoothing for OFF-source spectra efficiently reduces $\sigma_{\mathrm{noise}}$
by a factor of $\displaystyle \sqrt{\frac{\nu_{\mathrm{res}}}{\nu_{\mathrm{var}}}}$
and keeps the expectation of OFF-source spectrum.
Since the spectral smoothing will not be applied for ON-source spectra, desired spectral resolution will be kept.
\subsection{Allan Variance Decomposition}
The flatness of $H_1(\nu, t)$ is evaluated by the spectral Allan variance (SAV) analysis.
The SAV in defined as
\begin{eqnarray}
\sigma^2_{\mathrm{y}}(\Delta \nu) &=& \left< \frac{[H(\nu + \Delta \nu) - 2H(\nu) + H(\nu - \Delta \nu)]^2}{2\Delta \nu^2} \right>. \label{eqn:def_av}
\end{eqnarray}
For a small frequency span ($\Delta \nu \ll \nu_{\mathrm{var}}$) the SAV is dominated by thermal noise and follows the characteristics of $\sigma^2_{\mathrm{y}}(\Delta \nu) \propto \Delta \nu^{-2}$.
At a larger frequency span ($\Delta \nu \gg \nu_{\mathrm{var}}$), $H_1(\nu, t)$ exceeds the thermal noise and the SAV shows different characteristics from that of the thermal noise.
SAV analysis allows us to decompose $H_1(\nu, t)$ from the thermal noise.
An example of SAV for an OFF-source spectrum is shown in figure \ref{fig:quantization_diagram},
as a function of the channel spacing, $\nu_{\mathrm{sp}}$\footnote{As described in section \ref{sec:tests}, we used an FX spectrometer whose spectral resolution function is a squared sinc function and $\nu_{\mathrm{sp}} = \nu_{\mathrm{res}}$ by 2048-point FFT to produce 1024 ch.}.
The SAV in the 1-min integrated raw spectrum ($a$) is decomposed into two components; (1) thermal noise, $\epsilon(\nu, t)$, that dominates
in the range of $\frac{\Delta \nu}{\nu_{\mathrm{sp}}} < 5$ and whose SAV has a dependence,
$\Delta \nu^{-2}$
and (2) bandpass characteristics, $H(\nu, t)$, that dominates
in the range of $\frac{\Delta \nu}{\nu_{\mathrm{sp}}} > 5$.
When components (1) and (2) show approximately symmetric power-law characteristics, the bottom of the SAV appears at the frequency span where those two components are almost equivalent (see results in subsectino 3.2 and discussion in subsection 4.1).
After bandpass correction using a template spectrum, $H_0(\nu)$, component (2) is considerably eliminated as ($b$).
The rest fluctuation, $H_1(\nu, t)$, dominates the SAV
in $\frac{\Delta \nu}{\nu_{\mathrm{sp}}} > 120$.
As the thermal noise will be reduced via time integration in ($c$), the bottom of the SAV shifts to smaller $\Delta \nu$
(e.g. $\frac{\Delta \nu}{\nu_{\mathrm{sp}}} \sim 40$ for 546-min integration).
Relevant spectral smoothing with the smoothing window given by the bottom in SAV, as described in the next subsection, will yield efficient reduction of the thermal noise keeping the bandpass response as shown in ($d$).
\subsection{Strategy for Adequate Smoothing and Scans}\label{subsec:strategy}
The relevant smoothing window and ON--OFF scan pattern should be determined to minimize the SD in the OFF-source-subtracted spectrum within the given telescope time, or to minimize the total telescope time that achieves the given SD.
The best smoothing window,
$N_{\mathrm{sw}} = \frac{\Delta \nu}{\nu_{\mathrm{res}}}$
is determined by the bottom of the SAV.
After we apply spectral smoothing to the OFF-source spectra, their variance is expected to be reduced to $\frac{\sigma^2_{\mathrm{OFF}}}{N_{\mathrm{sw}}}$.
Because of inequality in variances of ON- and OFF-source spectra, we need to re-design the best duty cycles of $t_{\mathrm{ON}}$ and $t_{\mathrm{OFF}}$ under the condition where total telescope time is constant.
Let an ON-to-OFF-source integration time ratio set to $1-x:x$ in each of the duty cycle and the total telescope time of $t_{\mathrm{tot}}$\footnote{Here, we omit overhead and scan gaps for the simplicity.}.
Substituting $t_{\mathrm{ON}} = (1-x) t_{\mathrm{tot}}$, $t_{\mathrm{OFF}} = x t_{\mathrm{tot}}$, and assuming that $T_{\mathrm{a}} \ll T_{\mathrm{sys}} \simeq T_{\mathrm{ON}} \simeq T_{\mathrm{OFF}}$ in equations \ref{eqn:sensitivity1} and \ref{eqn:sigma}, we have
\begin{eqnarray}
\left( \frac{\sigma}{T_{\mathrm{sys}}} \right)^2 = \frac{1}{\nu_{\mathrm{res}} t_{\mathrm{tot}}} \left( \frac{1}{1-x} + \frac{1}{N_{\mathrm{sw}} x} \right). \label{eqn:smoothed_sigma}
\end{eqnarray}
The variance is minimized to
\begin{eqnarray}
\left( \frac{\sigma_{\mathrm{min}}}{T_{\mathrm{sys}}} \right)^2 = \frac{1}{\nu_{\mathrm{res}} t_{\mathrm{tot}}} \frac{(1 + \sqrt{N_{\mathrm{sw}}})^2}{N_{\mathrm{sw}}}, \label{eqn:min_sigma}
\end{eqnarray}
when we have $\displaystyle x = \frac{1}{1 + \sqrt{N_{\mathrm{sw}}}}$.
When $N_{\mathrm{sw}} = 45$ for instance, the best ratio is $0.13$.
The interval between adjacent OFF-source scans should be shorter than the timescale of spectral stability in the receiving system.
The spectral stability is evaluated by the time-based Allan Variance (TAV).
As we'll see in figure \ref{fig:caledSpec}, a typical spectral shape of $H_1(\nu, t)$ show an {\cal S}-shaped feature with a peak and a bottom.
Since we focus on stability of the bandpass shape, the TAV is calculated by the difference between levels of the peak and the bottom, i.e. $\Delta H_1(t) = H_1(\nu_{\mathrm{peak}}, t) - H_1(\nu_{\mathrm{bottom}}, t)$. Thus, the TAV is derived as
\begin{eqnarray}
\sigma^2_{\mathrm{y}}(\tau) &=& \left< \frac{[ \Delta H_1(t + \tau) - 2 \Delta H_1(t) + \Delta H_1(t - \tau)]^2}{2\tau^2} \right>, \label{eqn:def_time_av}
\end{eqnarray}
where $\tau$ is the time lag.
The stability timescale is determined by the time lag where the TAV follows $\sigma^2_{\mathrm{y}}(\tau) \propto \tau^{-2}$.
In summary, our strategy to determine the optimal spectral smoothing window and the scan pattern will be:
\begin{enumerate}
\item
Acquire the template bandpass response, $H_0(\nu)$, and its dependence on a frequency band by sufficiently long integration before programed observations (e.g. during seasonal maintenance).
Relevant spectral smoothing can be applied to the template bandpass to reduce the thermal noise in it.
\item
Set the target SD for the observation and estimate approximate integration time.
\item Acquire test scan data to take SAV and TAV to decide $N_{\mathrm{sw}}$ at the bottom of the SAV.
\item Derive the ON--OFF duty cycle of $1-x : x$ basing on $N_{\mathrm{sw}}$.
\item Estimate the timescale of stability and determine the OFF-source-scan interval.
\item Design the optimal ON--OFF scan pattern basing on the time stability of the bandpass and the ON-OFF duty cycle.
\item Apply bandpass calibration of $H_0(\nu)$ for observed ON- and OFF-source spectra before integration. Employ spectral smoothing for the OFF-source spectra with the smoothing window of $N_{\mathrm{sw}}$.
\item Apply the bandpass calibration of $H_1$ for ON-source spectra using the adjacent smoothed OFF-source spectra. Time-integrate the bandpass-calibrated ON-source spectra.
\item Apply
baseline subtraction,
if necessary, to the OFF-source-subtracted spectrum and get the final result.
\end{enumerate}
\section{Tests and Results} \label{sec:tests}
Field tests have been carried out for the SBC to investigate how does it work efficiently.
We used the VERA \citep{2003ASPC..306..367K} Iriki 20-m antenna in the single-dish mode.
The antenna pointed to the zenith throughout our tests.
The 22-GHz HEMT receiver was tuned at 22.235 GHz with the bandwidth of 512 MHz.
The received signal was downconverted with the first LO of 16.8 GHz, and the second LO of 5179 MHz before digitized at 1024 Msps 2-bit quantization by the digital sampler ADS-1000 \citep{2001ExA....11...57N}.
Digital filtering was applied using the digital filter unit \citep{2005PASJ...57..259I} to split the signal into sixteen 16-MHz streams where we used only the first stream.
Spectroscopy was employed using the software spectrometer, VESPA (VEra SPectrum Analyzer; \cite{VCON2011}), that produces
1024 ch by 2048-point FFT.
This produces a squared sinc spectral resolution function whose first null appears at the adjacent ch, i.e., $\nu_{\mathrm{sp}} = \nu_{\mathrm{res}}$.
The template bandpass, $H_0(\nu)$, was obtained on 2010 Apr. 15 for 230-min integration.
A 10-hour observation was conducted on 2010 Oct. 5, half-a-year later than acquiring the template bandpass to include seasonal variation.
Pseudo ON--OFF scan pattern was produced from the 10-hour continuous observation; OFF-source scans was pilfered from the continuous zenith observations.
Data quality was checked via monitoring the total power of every spectrum.
The first 50-min data were not used because of unexpected operations of hot-load insertion.
We also flagged out 3-sec data because of obviously irregular spectra probably due to data transmission error.
Finally, we obtained 32768-sec continuous spectra.
For calculation of SAV, we did not use the first spectral channel that includes direct current (DC) component affected by voltage bias offset at the digital sampler.
In this test, we set the target SD of the antenna temperature of $50$ mK, which is the standard of regular monitoring program in the VERA Iriki single-dish observations.
This corresponds to $\frac{\sigma}{T_{\mathrm{sys}}} = 5 \times 10^{-4}$ for $T_{\mathrm{sys}} = 100$ K.
We employed 3rd-order B-spline smoothing for OFF-source spectra in step 7 described in subsection \ref{subsec:strategy}.
Node intervals are set to be equal to the smoothing window, $N_{\mathrm{sw}}$.
The statistical package R was used for data analyses of integration, bandpass calibration, AV calculation, and B-spline smoothing.
All of source codes for these procedures are presented in the GitHub\footnote{https://github.com/kamenoseiji/BPsmooth}.
\subsection{Bandpass Spectra}
The template bandpass, $H_0(\nu)$ is shown in figure \ref{fig:BP}.
The SD of its random noise was $1.97 \times 10^{-5}$, which was estimated by $H_0(\nu) / \bar{H}_0(\nu) - 1$ where $\bar{H}_0(\nu)$ was a B-spline-smoothed spectrum.
We used $\bar{H}_0(\nu)$ for the following tests, though the SD was one order of magnitude smaller than the target noise level, to quest the ultimate performance.
Bandpass-corrected spectra, $\displaystyle \frac{T_{\mathrm{OFF}}(\nu, t)}{\bar{T}_{\mathrm{OFF}} \bar{H}_0(\nu)} - 1$, are shown in figure \ref{fig:caledSpec}.
Here, $\bar{T}_{\mathrm{OFF}}$ is the mean value of the OFF-source spectra that represents $T_{\mathrm{sys}}$.
Thus the bandpass-corrected spectrum indicates $\displaystyle \frac{H_1(\nu, t)}{\bar{H}_0(\nu)} (1 + \epsilon(\nu, t)) + \epsilon(\nu, t)$.
The solid line in figure \ref{fig:caledSpec} was computed by the 3rd-order B-spline smoothing with the node intervals, $N_{\mathrm{sw}} = 45$ and represent $\displaystyle \frac{\bar{H}_1(\nu, t)}{\bar{H}_0(\nu)}$.
Departures from the solid line were dominated by the thermal noise, $\displaystyle \left( 1+ \frac{H_1(\nu, t)}{H_0(\nu)} \right) \epsilon(\nu, t) \simeq \epsilon(\nu, t)$.
The smoothed spectrum was time variable. Its time variability is described in subsection \ref{subsec:timeVariability}.
\subsection{Spectral Allan Variance}\label{subsec:SAV}
The SAVs of bandpass-corrected spectra defined as equation \ref{eqn:def_av} are shown in figure \ref{fig:spactralAV}.
At every frequency channel separation, the SAV decreased as was integrated for longer time.
While the SAV was decreasing as a function of frequency channel separation at shorter integration than 16 s, it appeared a local bottom and a top for longer integration than 16 s.
They appeared at $< 200$ ch and $\sim 259$ ch, respectively.
Figure \ref{fig:SAV_fit} shows the SAV of a 64-min-integrated spectrum as an example.
The profile was composed of two power-law components.
For narrower channel separations than the bottom, the SAVs was dominated by a power-law profile with the index of $-2.003 \pm 0.001$.
The index above the bottom was $1.29 \pm 0.02$.
The bottom of the SAV appeared near the frequency span where the two power-law components were equivalent.
At longer time integration, the bottom shifted towards narrower spectral separation in longer integration time, while the top stayed at almost the same separation.
These behavior is summarized in table \ref{tab:spectralAV}.
\subsection{Time Stability} \label{subsec:timeVariability}
Figure \ref{fig:SpectralVariation} displays variability of $\displaystyle \frac{\bar{H}_1(\nu, t)}{\bar{H}_0(\nu)}$.
The smoothed spectrum shows time variation in terms of not only the power level but also its shape such as slope, curvature, and local bumps.
Variation of the bandpass was evaluated by the standard deviation of the smoothed spectrum, $\sigma_t(\nu)$, defined as $\displaystyle \sigma^2_t(\nu) = \left< \left(\frac{\bar{H}_1(\nu, t)}{\bar{H}_0(\nu)}\right)^2 \right> - \left< \frac{\bar{H}_1(\nu, t)}{\bar{H}_0(\nu)} \right>^2$.
Here, the expectation is taken by time average.
Figure \ref{fig:SpectralSD} shows the standard deviation of the smoothed spectrum.
Two clear peaks appeared at 71 ch and 920 ch, near the shoulders of the bandpass, $H_0(\nu)$.
The standard deviation of $\sim 2$\% was significantly greater than the target accuracy of $\frac{\sigma}{T_{\mathrm{sys}}} = 5 \times 10^{-4}$.
Hence, it is necessary to calibrate the bandpass using the smoothed OFF-source spectrum within the timescale while $\displaystyle \frac{\bar{H}_1(\nu, t)}{\bar{H}_0(\nu)}$ is stable enough.
The timescale of stability was evaluated using the TAV defined in equation \ref{eqn:def_time_av}.
Since the top and bottom channels of the smoothed bandpass were also time variable, we instead used the twin peak channels in $\sigma_t(\nu)$, i.e.
we substituted $\Delta H_1(t) = H_1(71 {\rm ch}, t) - H_1(921 {\rm ch}, t)$.
The TAV is shown in figure \ref{fig:TimeAV}.
The first bottom of 30-sec-integrated spectrum appeared at 300 s.
The power-law index of the TAV was $\sim -2$ for $\tau < 60$ s and was $-1.7$ at $\tau = 80$, which we regulate as the spectral stability timescale in following processes.
\subsection{Scan Patterns and Performance}
Basing on above results we set the new scan pattern and compared with the conventional ON--OFF scans without spectral smoothing.
The conventional pattern (case 1) consists of pairs of 30-s ON and 30-s OFF scans.
The new scan pattern (case 2) was a set of 70-s ON and 10-s OFF with the SBC.
The pattern was designed as the cycle period to be shorter than the spectral stability timescale of 80 s and the optimal ON-to-OFF integration time ratio described in equation \ref{eqn:smoothed_sigma} with $N_{\mathrm{sw}} = 45$.
To reach the targeted noise level of $\sigma/T_{\mathrm{sys}} = 5 \times 10^{-4}$, the conventional pattern required 20 sets (1200 s) of the total telescope time.
Its resultant spectrum is shown in figure \ref{fig:resultSpectra} (A1).
The case-2 spectrum with the telescope time of 1200 s (15 sets of 80-s scans) is shown in figure \ref{fig:resultSpectra} (C1).
The SD was $(2.8^{+0.3}_{-0.2}) \times 10^{-4}$ (median, minimum, and maximum of 27 samples), $\times \frac{1}{1.74}$ that of the conventional scan pattern.
To achieve the targeted noise level, case 2 required 5 sets (400 s). Its resultant spectrum is shown in figure \ref{fig:resultSpectra} (B1).
Since the resultant spectrum appeared wiggled features other than random noise, we attempted baseline fitting and subtraction using the 3rd-order B-spline function with the node interval of 45 ch.
After the baseline subtraction, the SDs of cases 1, 2 (1200 s), and 2 (400 s) became $(4.7^{+0.3}_{-0.2}) \times 10^{-4}$, $(2.6^{+0.2}_{-0.2}) \times 10^{-4}$, and $(4.5^{+0.3}_{-0.3}) \times 10^{-4}$, respectively, and shown in figures \ref{fig:resultSpectra} (A2, C2, and B2).
The performance of SD as functions of total telescope time is summarized in table \ref{tab:rms_integ} and figure \ref{fig:residual_rms}.
At any total telescope time, the SBC exhibited better performance than conventional scans. While the SD of the conventional scans followed $\sigma \propto t^{-0.5}_{\mathrm{integ}}$ dependence, as was expected, the SBC resulted in a shallower slope with the power index of $-0.46$.
\subsection{Smoothing Window}
The optimal spectral smoothing window, $\Delta \nu$, was determined by the bottom in the SAV listed in table \ref{tab:spectralAV}.
For the timescale longer than 16 min, the bottom appeared
at $\Delta \nu / \nu_{\mathrm{sp}} \sim 32 - 60$ ch.
We chose the window of 45 ch for time stability tests and spectral performance tests in previous subsections.
We also tested the spectral performance with various windows of 2, 3, 4, $\dots$, 191 ch for the scan pattern of case 2 with the total telescope time of 400 and 1200 s.
Figure \ref{fig:residual_SD} shows the results.
The median values of SDs in the resultant spectrum before baseline subtraction recorded a minimum at the 64-ch window for both telescope times of 400 s and 1200 s.
After baseline subtraction, the SD became significantly lower and the minimum appeared a flat bed for wider window than 45 ch.
In all cases the bottom was smaller than the targeted noise level.
\section{Discussion}
\subsection{Bandpass Flatness}
As is shown in figure 1, the SAV across bandwidth is composed of the thermal noise and fluctuation of bandpass characteristics.
The crossover point, where the thermal noise and the bandpass fluctuation is equivalent, appears at the bottom of the SAV.
Relevant smoothing window, $\Delta \nu$, should be set around the crossover point.
For longer integration the bottom of SAV shifts toward narrower channel separation.
The regression of the crossover point indicates that the channel separation of the bottom relates to the integration time as $\Delta \nu_{\mathrm{bottom}} \propto t^{-0.273\pm 0.005}_{\mathrm{integ}}$ for $t_{\mathrm{integ}} < 128$ min.
This behavior is explained as following consideration.
The thermal noise component is stochastic which does not depend on time and frequency.
Thus the SAV of thermal noise component $\propto \Delta \nu^{-2} t^{-1}_{\mathrm{integ}}$.
The bandpass fluctuation SAV $\propto \Delta \nu^{1.29 \pm 0.02}$.
This power-law index is slightly steeper than SAV $\propto \Delta \nu$ which is expected for a random-walk process along frequency.
Since the bandpass characteristics is frozen during short timescale, the SAV of bandpass fluctuation is independent of $t_{\mathrm{integ}}$.
Therefore, the crossover point appear under the condition of $\Delta \nu^{-2} t^{-1}_{\mathrm{integ}} = \alpha \Delta \nu^{1.29}$ where $\alpha$ is a proportional coefficient.
Thus, the SAV bottom will appear at $\Delta \nu_{\mathrm{bottom}} \propto t^{-\frac{1}{3.29}}_{\mathrm{integ}}$.
The behavior of the test observation, which indicates $\Delta \nu_{\mathrm{bottom}} \propto t^{-0.273\pm 0.005}_{\mathrm{integ}}$, is consistent with the theoretical consideration.
Although a different combination of a receiving system and a spectrometer other than the VERA system may have different properties of bandpass flatness, measuring SAVs at various integration time allows us to decompose it into thermal-noise and random-walk components.
\subsection{Expected Time Reduction}
As shown in subsection 3.4, the SBC (case 2) allows us to shorten the total telscope time to $\frac{1}{3}$ to achieve the targeted noise level, compared with the conventional method (case 1).
In other words, the SBC reduces the SD by a factor of $\sqrt{3}$ within the same total telescope time.
These results demonstrate that the telescope time efficiency can be tripled by the SBC method with the optimal scan pattern.
Since our analysis did not take scan gaps time into consideration, the efficiency can be different in realistic observations.
As shown in figure \ref{fig:resultSpectra}, systematic undulation remains in the case-2 spectrum before baseline subtraction.
The undulation can be ascribed to OFF-scan time interval of 70 s that may be longer than the stability timescale ({\bf see figure \ref{fig:TimeAV}}).
OFF-scan intervals should be shorter to reduce the undulation, however, this modification increases the time loss in scan gaps.
Optimal scan pattern should be tested in detail under realistic conditions of scan gaps and time stability.
As shown in figure \ref{fig:residual_rms}, the slope of SD reduction by time integration for the SBC was shallower than that for conventional scans.
The shallower slope indicates that the SBC underperforms at a long integration time.
This underperformance can be caused by time instability of the bandpass.
The power index of the TAV of $-1.7$ at $\tau = 80$ s, with a small departure from $-2$, indicates that bandpass fluctuation remains with the level of $\sigma_{\mathrm{y}} \sim 10^{-5}$ ({\bf see figure \ref{fig:TimeAV}}).
This level of fluctuation is not tangible in the conventional scans even at the longest integration ($\frac{\sigma}{T_{\mathrm{sys}}} = 9.3 \times 10^{-5}$) but can affect in the SBC ($\frac{\sigma}{T_{\mathrm{sys}}} = 6.3 \times 10^{-5}$).
The bandpass stability can be the principal component to determine the sensitivity at a longer integration time then 32400 s.
To pursue the sensitivity at the longer integration, shorter scan pattern should be considered in spite of time loss in scan gaps.
\subsection{Advantages in a Dual Beam System}
Dual beam systems are equipped in some radio telescopes such as the GBT 100-m \citep{2002ursi.confE...3J}, the Nobeyama 45-m \citep{2010ASJM...V45a}, and the VERA \citep{2003ASPC..306..367K}, to double the time efficiency by interchanging ON- and OFF-source scans between two beams.
The duty cycle for ON and OFF scans in dual beam systems should be 1:1 (or $x=0.5$ in equation \ref{eqn:min_sigma}), if performance of two receivers is uniform, and different from the optimal duty cycle for the SBC.
In this case the variance, $\sigma^2$, will be reduced by a factor of $\displaystyle \frac{1}{2} \left(1+\frac{1}{N_{\mathrm{sw}}}\right)$ when we employ the SBC.
Taking $N_{\mathrm{sw}} = 45$ for instance, the variance is reduced by the factor of $0.51$ and the telescope time efficiency is almost doubled, 3.9 times better than conventional ON--OFF scans with a single beam system, or 1.3 times better than the SBC with a single beam system.
Consideration in this subsection implies that the SBC for single-beam with conventional ON--OFF scans (1:1 duty cycles) also doubles SNR.
Re-analysis with the SBC for previous observations can offer an opportunity to enhance SNR, if ON- and OFF-source spectra are separately recorded, and the efficiency depends on the spectral stability of the system.
\subsection{Advantages in OTF Scans}
ON-the-fly (OTF) mapping shares the OFF-source pointing among a continuous scan pass \citep{2007A&A...474..679M}.
\citet{2008PASJ...60..445S} argued that
the noise level achieved in the unit observing time
of the OTF map is written as
\begin{eqnarray}
\Delta T^*_{\mathrm{A}}(0) &=& \frac{T_{\mathrm{sys}}}{\sqrt{B}} \sqrt{\left( \frac{1}{t^{\mathrm{ON}}_{\mathrm{cell}}} + \frac{1}{t^{\mathrm{OFF}}_{\mathrm{cell}}} \right) } \nonumber \\
& \times & \sqrt{\left( t_{\mathrm{scan}}+t_{\mathrm{OH}} + \frac{t_{\mathrm{OFF}}}{N^{\mathrm{SEQ}}_{\mathrm{scan}}} \right) N_{\mathrm{row}} f_{\mathrm{cal}}}, \label{eqn:sawada2008}
\end{eqnarray}
and derived the optimal OFF-source integration time to map a rectangular area with the row span of $l_1$ was given $\displaystyle t_{\mathrm{OFF}} = \sqrt{(t_{\mathrm{scan}} + t_{\mathrm{OH}}) \frac{\eta d t_{\mathrm{scan}}}{l_1}} \sqrt{N^{\mathrm{SEQ}}_{\mathrm{scan}}}$, where $t_{\mathrm{scan}} = l_1 / v_{\mathrm{scan}}$ is the ON-source scan time for a row by the scan speed of $v_{\mathrm{scan}}$, $t_{\mathrm{OH}}$ is the overhead time between row scans, $\eta$ is the efficiency determined by the gridding convolution function, $d$ is the grid spacing, and $N^{\mathrm{SEQ}}_{\mathrm{scan}}$ is the number of row scans between OFF-source integrations.
In the practical case of $\eta = 4.3$, $l_1 = 600^{\prime \prime}$, $\Delta l = 5^{\prime \prime}$, $d = 7^{\prime \prime}.5$, $t_{\mathrm{OH}} = 25$ s, and $N^{\mathrm{SEQ}}_{\mathrm{scan}} = 1$, according to \citet {2008PASJ...60..445S}, the ON-source scan time of $t_{\mathrm{scan}} =20$, $40$, and $60$ s yields $t^{\mathrm{optimal}}_{\mathrm{OFF}} = 7$, 12, and 17 s, respectively.
The optimization is modified when the SBC is applied.
Equation \ref{eqn:sawada2008} is modified as
\begin{eqnarray}
\Delta T^*_{\mathrm{A}}(0) &=& \frac{T_{\mathrm{sys}}}{\sqrt{B}} \sqrt{\left( \frac{1}{t^{\mathrm{ON}}_{\mathrm{cell}}} + \frac{1}{N_{\mathrm{sw}} t^{\mathrm{OFF}}_{\mathrm{cell}}} \right) } \nonumber \\
& \times & \sqrt{\left( t_{\mathrm{scan}}+t_{\mathrm{OH}} + \frac{t_{\mathrm{OFF}}}{N^{\mathrm{SEQ}}_{\mathrm{scan}}} \right) N_{\mathrm{row}} f_{\mathrm{cal}}}, \label{eqn:OTF_BP}
\end{eqnarray}
when the smoothing window is $N_{\mathrm{sw}}$ ch.
The optimal OFF-source integration will be $\displaystyle t_{\mathrm{OFF}} = \sqrt{(t_{\mathrm{scan}} + t_{\mathrm{OH}}) \frac{\eta d t_{\mathrm{scan}}}{N_{\mathrm{sw}} l_1}} \sqrt{N_{\mathrm{scan}}^{\mathrm{SEQ}}}$.
For comparison to \citet{2008PASJ...60..445S}, $t_{\mathrm{scan}} =20$, $40$, and $60$ s yields $t^{\mathrm{optimal}}_{\mathrm{OFF}} = 1$, 1.8, and 2.5 s when we take $N_{\mathrm{sw}} = 45$, and the noise level decreases by factors of 0.886, 0.874, and 0.856, respectively.
Thus, the total telescope time that achieves the same SD will be reduced by a factor of $\sim 1.3 - 1.4$.
The effect of the SBC is less than simple ON-OFF scans, nevertheless, the SBC offers somewhat better results.
Consideration in this subsection does not involve bandpass fluctuation.
More realistic optimization and noise level should be estimated basing on time-stability of receiving systems.
\subsection{Vulnerability against Spurious Signals}
SBC yields weakness against unwanted spurious emissions such as RFI, contamination of sampling clocks and reference signals in a receiving system, artificial pattern caused by numerical errors of spectral calculation, etc.
These spurious signals usually show spiky features in both ON- and OFF-scan spectra.
They are suppressed in the smoothed OFF-scan spectra while kept in the unsmoothed ON-scan spectra.
Thus they will appear in the final results, though they would be subtracted by the conventional ON - OFF scans if they were stable.
In our test observations, spurious emissions appeared at 704 ch and 832 ch (see figure \ref{fig:resultSpectra}) that correspond to 11/16 and 13/16 of the bandwidth.
Their line width is narrower than the spectral resolution.
Digital noise such as harmonics of the sampling clock can cause such spurious emissions whose frequencies are simple fractions of the bandwidth.
These spurious frequencies should be masked in spectral data reduction.
Observers need to pay deeper attention to spurious when they apply SBC.
\section{Summary}
It is presented that the proposed SBC method offers significantly better performance than conventional ON--OFF scans, especially when a stable spectrometer such as digital system was equipped.
Our tests showed that the total telescope time was reduced to $\frac{1}{3}$ to attain the same SNR, or $\times 1.7$ better SNR was obtained in the same telescope time for a single-pointing ON-OFF scans.
The SBC method was also efficient for dual-beam systems, ON-the-fly mapping.
The optimal analysis scheme of the SBC is presented in this paper.
To apply this method, radio observatories should offer
\begin{itemize}
\item Long-time integrated bandpass to obtain good-enough $H_0(\nu)$. This can be done in off seasons.
\item SAV to show flatness of $H_1(\nu, t)$ to allow observers to compute the optimal smoothing window.
\item TAV to estimate stability timescale of $H_1(\nu, t)$ to design the optimal scan pattern and total telescope time required.
\item Spectral outputs of ON- and OFF-source, separately. Automated ON-OFF subtraction makes it impossible to OFF-source spectral smoothing in post-observation analysis.
\end{itemize}
Finally, re-analysis of previous spectral data with the SBC can also offer an opportunity to enhance SNR, if ON- and OFF-source spectra are separately recorded.
\bigskip
The VERA Iriki 20-m telescope is a part of the VERA VLBI network which is operated by the National Astronomical Observatory of Japan and staffs and students of the Kagoshima University.
|
1,116,691,500,797 | arxiv | \section{Introduction}
Axions are hypothetical particles introduced to solve the Strong $CP$ problem in QCD \cite{PQ1,PQ2,Weinberg,Wilczek,DFS,Zhitnitsky,Kim,Shifman}. Axions are self-adjoint bosons, with no conserved discrete quantum numbers to guarantee particle number conservation. The axion potential can be written in terms of an angular variable with a $2\pi$ shift symmetry. Axion-like scalar particles also appear in a variety of models beyond QCD, especially in low-energy limits of string theories \cite{Dimopoulos,ChoiKim,Dine}. Those axions have properties similar to QCD axions, but their mass scales and decay constants are vastly different.
Axion-like particles are among the prime candidates for the composition of dark matter~\cite{Preskill,Sikivie1,Davidson,DF,Holman,Sikivie2}. Axions, being scalar bosons, can condense. Axion condensates have been discussed by numerous groups, with condensate sizes ranging from galaxy or galaxy cluster scale \cite{SikivieYang,Witten}, down to stellar size and smaller (termed ``axion stars'') \cite{KolbTkachev,Iwazaki,BarrancoNS,TkachevFRB,ESVW,Guth}, all the way to radii of a few meters \cite{BB,Braaten}. Also of interest is the cosmological evolution of the axion field, which has been worked on extensively in \cite{Khlopov1,Khlopov2,Khlopov3,Khlopov4,Khlopov5}, but this will not be disussed further here.
Uncondensed QCD axions are not stable, as they can decay to photons through a process $a\to 2\,\gamma$, but the decay rate is slow enough such that most axions created after the Big Bang would survive many Hubble times~\cite{TkachevPossibility}. Axion condensates, however, may also decay slowly due to the self-interaction of axions \cite{ESW,Braaten2016}. The self-interaction term of their Lagrangian (for both the instanton~\cite{nonchiral,Vecchia} and chiral~\cite{Vecchia,Cortona} cases) have only terms containing an even number of axion fields. Thus, disregarding the rare decay into photons, the axion number is conserved only modulo 2.
In a recent paper~\cite{ESW} we have investigated the decay of weakly bound axion stars due to the self-interaction of axions. The decay proceeds mostly through a sequence of processes,
\be\label{theprocess}
{\cal{A}}_N\to{ \cal{A}}_{N-3}+a_p,
\ee
where ${\cal{A}}_k$ is an axion star, which is a condensate containing $k$ axions, and $a_p$ denotes an axion in a scattering state with the magnitude of the momentum $p$.
The process (\ref{theprocess}) is the simplest of many possible decay modes responsible for the decay of axion stars.\footnote{The decay rates via processes ${\cal{A}}_N\to{ \cal{A}}_{N-5}+a_p$ or ${\cal{A}}_N\to{ \cal{A}}_{N-4}+a_{p_1}+a_{p_2}$ are significantly lower and unlikely to have any cosmological significance \cite{ESW}.} This process is allowed by energy-momentum conservation, provided the binding energy of a bound axion is small enough that a relativistic particle can be produced: $\delta E < 2\,m\,/\,3$, where $m$ is the mass of a free axion.
In~\cite{ESW} we used an axion field operator, which was the generalization of the field proposed by Ruffini and Bonazzola~\cite{RB}. To facilitate the decay process, terms creating and annihilating axions in the continuum of scattering states were included in the quantum field of axions, in addition to terms creating and annihilating bound axions~\cite{RB}. The Ruffini-Bonazzola method is based on taking the expectation value of the quantum Einstein and Klein-Gordon equations in the condensate to derive equations of motion for the metric components and the scalar field.
We solved the equations of motions numerically in the weak gravity and weak binding ($\delta E\ll m$) limits, to find the wave function of axions in the condensate~\cite{ESVW}.
The bound axions {\em are not in momentum eigenstates}. They have definite energies, but their wave functions extend over the size of the axion star. Accordingly, the bound axions have an extended momentum distribution as well. In the recent publications~\cite{Braaten2016,Braaten2}, the authors questioned the validity of the decay mechanism proposed in~\cite{ESW}, arguing that momentum is not conserved in (\ref{theprocess}), and that the decay rate through this process is exactly zero by the Optical Theorem. These authors have further suggested that one can show the rate to be zero by the classical equation of motion for the condensate. We will address these issues and explain our response in Appendix A.
In the present paper we will apply our method of discussing the decay of dilute axion stars \cite{ESW} to condensates of cosmological size; such models have been referred to previously as Fuzzy Dark Matter (FDM) \cite{Hu}. Condensates of galactic sizes have been considered by a number of authors, and typically correspond to a scalar particle mass of $m\sim 10^{-22}-10^{-21}$ eV \cite{Turner,Press,Sin,Hu,Goodman,Peebles,Amendola,Shapiro,Schive,Marsh,Witten}.\footnote{For a brief but recent review of ultralight scalar field dark matter models, see \cite{Lee} and references therein.} When such condensates are formed from real scalars, a version of the decay analysis of \cite{ESW} applies, and we will investigate whether interesting bounds can be placed on these models by taking decays into account. As we will explain in the next section, we will utilize the axion potential with a cosine dependence on the field; other proposals, for example a $\cosh$ potential \cite{Sahni,Alcubierre}, have been investigated in the context of ultralight scalars as well.
It is also an aim of this paper to emphasize the inclusion of axion self-interactions in investigations of axion condensates. Although the self coupling $\lambda \sim m^2/f^2 \sim 10^{-95} \lll 1$ for typical FDM models ($f$ is the axion decay constant), the astronomical number of axions in a condensate participating in these interactions could lead to large corrections to certain important physical quanties. We investigate the macroscopic properties of these condensates using the fully self-interacting analysis and emphasize the differences from the non-interacting limit.
In the next section, we give a more detailed explanation of how axion star decay through the process (\ref{theprocess}) can be calculated. In Section \ref{WavefunctionSec}, we will outline the calculation of the wavefunction, following largely \cite{ESVW}. In Section \ref{DecayRateSec}, we apply the formulas for the macroscopic properties and decay rates to condensates formed from ultralight axion-like particles. We conclude in Section \ref{ConclusionSec}.
We will use natural units throughout, where $\hbar = c = 1$.
\section{Decay through self interaction} \label{OverviewSec}
There is a variety of methods for the quantitative investigation of axion condensates, as classical~\cite{TkachevPossibility,Iwazaki,KolbTkachev,Kouvaris2,Lee2}, quantum mechanical~\cite{ChavanisMR,ChavanisMR2,Guth,Kouvaris,Guzman1,Guzman2,Guzman3}, and field theoretic~\cite{BB,ESVW,Braaten}. Our field theoretic discussion of the decay of an axion condensates into relativistic axions \cite{ESW} was based on an extension of the Ruffini-Bonazzola operator~\cite{RB}, by the addition of scattering state contributions.\footnote{Appendix B we will discuss why a continuous spectrum of scattering state solutions can be added to the boson field operator. Furthermore, we will also discuss why using free spherical wave scattering states is quite sufficient in our calculations.} Thus, we proposed to extend the expansion of the boson field using the form \cite{ESW}
\be\label{field2}
\Phi (r,t) = R(r)\,e^{-i\,E_0\,t}\,a_0 + R(r)\,e^{i\,E_0\,t}\,a_0^\dagger
+ \psi_f(r,t) + \psi_f^\dagger(r,t),
\ee
where $E_0$ is the energy eigenvalue of a single bound axion, and where $R(r)$ and $a_0$ are the wave function and annihilation operator of the axions in the condensate (respectively). $\psi_f(r,t)$ is the annihilation part of a complete system of free axion operators expanded in scattering states,
\be \label{free}
\psi_f(r,t) =\frac{1}{2\,\pi^2}\sum_{l,m}Y_{l}^m(\hat{r})
\int_0^\infty \frac{dp\,p}{2\,\omega_p}\,
j_l(p\,r)\,e^{-i\,\omega_p\,t}a_{lm}(p),
\ee
where $\omega_p$ and $a_{lm}(p)$ are the energy eigenvalue and the annihilation operator of the scattering state axion, with quantum numbers $l$ and $m$, respectively. The functions $j_l(x)$ and $Y_{l}^m(\hat{x})$ are spherical Bessel functions and spherical harmonics, respectively.
The annihilation operator in the spherical wave basis is defined by
\be
a_{lm}(p)= i^l\, p\int d\Omega_p\,Y_{l}^{m*}(\hat{p})\,a (\vec p),
\ee
where $a(\vec p)$ is the annihilation operator for a particle which is the eigenstate of the momentum operator with eigenvalue $\vec p$ (i.e. a plane wave). This annihilation operator and its adjoint, the creation operator, satisfy the commutation relation
\be
[a_{lm}(p),a_{l'm'}^\dagger(p')] =
2\,\omega_p\, (2\,\pi)^3 \,\delta(p-p')\,\delta_{ll'}\delta_{mm'}.
\ee
Note that (\ref{free}) is exactly equal to the negative frequency part of a complete system of free axion states given by
\be
\frac{1}{(2\,\pi)^3}\int\frac{d^3p}{2\,\omega_p}
e^{i(\vec{p}\cdot\vec{r}-\omega_p\,t)}a(\vec p),
\ee
which was used in~\cite{ESW} to investigate the decay of QCD axion stars.
We will see later that the bosons produced by the decay of a weakly bound boson condensate are relativistic. Therefore, we have chosen to use free particle states to approximate the scattering states. For the purposes of this calculation, this choice is admissible, because the energy level of produced axions is sufficiently high compared to the effective depth of the potential created by gravitation and self-interactions. This is explained in greater detail in Appendix B. In a future work we will take into account corrections to this approximation.
Let us consider now (\ref{theprocess}) in the Born approximation. The axion self-interaction potential can be approximated by the so-called instanton potential \cite{nonchiral,Vecchia}
\be \label{instanton}
V(\Phi) = m^2\,f^2\,\Big[1 - \cos\Big(\frac{\Phi}{f}\Big)\Big],
\ee
where $m$ and $f$ are the axion mass and decay constant (respectively). In the Ruffini-Bonazzola paradigm, one finds the expectation value of eq. (\ref{instanton}) transforms the cosine into a Bessel function $J_0$ \cite{ESVW}.\footnote{More generally, the annihilation process for $k$ bound axions generates an effective potential proportional to the Bessel function $J_k$, as explained in the Appendix of \cite{ESW}.} One also finds that the transition matrix element for the process (\ref{theprocess}) is
\begin{align}\label{matrix1}
{\cal M}_3 &= \int dt\,d^3r\,\langle N\vert \,V(\Phi) \vert N-3,\,\phi(p)\rangle \nonumber \\
&= -i\, m^2\,f\,\int dt\,dr\,r^2\, J_3\left[X(r)\right]\,
e^{3\,i\,E_0\,t} \int d\Omega_r\langle 0\vert\psi_f(r ,t)\vert \phi(p)\rangle
\end{align}
where $X(r)=2\,\sqrt{N}\,R(r)\,/\,f$ is the rescaled wave function of the condensate which, as we will see in Section \ref{WavefunctionSec}, can be obtained by solving the equations of motion~\cite{ESVW}. We are considering transitions of the form (\ref{theprocess}), where $\langle N \vert$ is the initial $N$ particle condensate (the left hand side of (\ref{theprocess})), and $\vert N-3,\,\phi(p)\rangle$ is the direct product of the final state $N-3$ particle condensate and a scattering state $\phi(p)$ of momentum magnitude $p$ (the right hand side of (\ref{theprocess})).
We restrict this work to non-rotating axion condensates; the reason for this is twofold. First, for the potential in eq. (\ref{instanton}) and the parameters we use here, only the inner cores of galaxies composed of axion particles can be described as a condensate; outside of this inner region, the dark matter halo is described by a virialized gas of particles \cite{Witten} and cannot participate in the decay processes we describe here.\footnote{For ultralight bosons with \emph{repulsive} self-interaction, like those presented in e.g. \cite{Goodman,Shapiro}, the condensates can be much larger, and can even constitute the entire dark matter halo.} Because the condensed core is small compared to the full radius of the halo, it carries at most a tiny fraction of the angular momentum of the galaxy, so as a first approximation we believe restricting to $\ell=0$ angular momentum states is appropriate. The second reason is that a full treatment of rotating axion condensates has not yet been done, though slowly rotating condensates were analyzed in a particular limit by \cite{Davidson}. This is a topic we hope to return to in the near future.
Because we work in the limit of zero angular momentum, annihilation processes of the form $a\,a\to G$, where $G$ is a spin-2 graviton \cite{SuperRad1,SuperRad2}, have a rate of zero. Such an interaction would require the participating axions to have at least $\hbar$ of angular momentum each; and even if we accounted for the nonzero rotation speed of the galaxy, by our estimation the vast majority of axions in a galactic condensate would have far less angular momentum than what would be necessary for this process to occur.
For static condensates, note that the integration over $\Omega_r$ in eq. (\ref{matrix1}) vanishes for all but $s$-wave axions.\footnote{Should we consider rotating axion stars, higher angular momentum scattering states would also contribute. } In that case, the scattering state axion is described by the zero angular momentum contribution only, $\vert \phi(p)\rangle = a_{00}^\dagger(p)\vert 0\rangle$. Then the wave function of the emitted axion is
\be
\phi(r,t)=\langle 0\vert \psi_f(r ,t)\vert \phi(p)\rangle
=\sqrt{4\,\pi}\,\frac{e^{-i\,\omega_p\,t}\sin(p\,r)}{r}.
\ee
The integration over $t$ also fixes the energy of the outgoing axion to $\omega_p=3\,E_0$. The matrix element takes the form~\cite{ESW}
\be\label{matrixr}
{\cal M}_3 =-4\,\pi^2\sqrt{4\pi}\,f\,
\delta(3\,E_0-\omega_p)\,I_3(p),
\ee
where the dimensionless integral is
\begin{align} \label{I3}
I_3(p)
&= m^2\,\int_{-\infty}^{\infty} dr\,r\,e^{i\,p\,r}\,J_3\left[X(r)\right] \nonumber \\
&\approx \frac{m^2}{48}\,\int_{-\infty}^{\infty} dr\,r\,e^{i\,p\,r}\,X(r)^3.
\end{align}
The symmetry of the integrand for the substitution $r\to-r$ has been used to extend the integration region to all real values of $r$, and to switch from $\sin(p\,r)$ to $e^{i\,p\,r}$ in the integrand. In the second equality, we expanded the Bessel function $J_3$ to leading order, an appropriate limit for dilute axion stars.
Now observe that for dilute axion stars the radius of the star $R$ is very large. In other words, $X(r)$ has a large coordinate uncertainty, $\delta r\sim R \sim (m\,\Delta)^{-1}$, where $\Delta=\sqrt{1-(E_0\,/\,m)^2}\ll1$ \cite{ESVW,ESW}. As a result, the range of $p$, as represented by the momentum uncertainty $\delta p\sim m\,\Delta \ll m$, is very small. Then due to the delta function in eq. (\ref{matrixr}), enforcing energy conservation, the emitted axion has a momentum peaked at a very large value, $p\simeq \sqrt{8}\,m$; as a result, the matrix element (\ref{matrixr}) is very small for weak binding. However, as the binding energy $\delta E$ increases, ${\cal M}_3$ will take larger values and the decay rate $\Gamma\sim |{\cal M}_3|^2$ also increases.
Now to bring out issues related to momentum conservation, we can define the momentum representation wave function as
\be\label{fourier}
\Xi(q)=\frac{1}{(2\,\pi)^3}\int d^3r\,X(r)\,e^{i\, \vec{q}\cdot\vec{r}}.
\ee
Then we can rewrite eq. (\ref{matrixr}) as
\be\label{matrixp}
{\cal M}_3=- \frac{\pi^2\,\sqrt{4\,\pi}\,m^2\,f}{12}\,\delta(3\,E_0-\omega_p)\,
\int \delta^3(\vec{p}-\vec{q}_1-\vec{q}_2-\vec{q}_3)
\prod_{k=1}^3\,\Xi(q_k)\,d^3q_k.
\ee
Since for weakly bound condensates $p\simeq \sqrt{8}\,m$, the magnitude of the transition matrix depends crucially on the large $q$ tail of momentum distribution $\Xi(q)$. However, for calculational purposes, it is still advantageous to use (\ref{matrixr}) rather than (\ref{matrixp}). One can rely on the numerical solution of the equations of motion, as explained below, using a simple approach to estimate approximate behavior of $\Xi(q)$ at large $q$~\cite{ESW}.
Note that the process (\ref{theprocess}) is not the only possible channel through which decay can proceed; however, it is by far the dominant process. First, we have shown previously \cite{ESW} that processes of the form ${\cal A}_N\to {\cal A}_{N-(2\,j+1)}+a_p,$ are suppressed by higher powers of $\Delta$ for each higher $j>1$. In the weak-binding limit, where $\Delta \ll 1$, these corrections are completely negligible. On the other hand, this argument breaks down for dense axion stars \cite{Braaten,ELSW}, where $\Delta = \mathcal{O}(1)$; we will return to this case in a future publication.
Second, there are processes of the form $ {\cal A}_N\to {\cal A}_{N-k}+\mu\,a_p,$ where $\mu > 1$ axions are emitted at once. Unlike the process (\ref{theprocess}), the emission of $\mu>1$ axions from a condensate can proceed on-shell, implying that the corresponding decay rate has a weak dependence on $\Delta$. Nonetheless, as shown in \cite{ESW}, these processes are suppressed by the very small factor $m^2/f^2$ for each additional axion in the final state. Since in FDM $m^2/f^2 \sim 10^{-95} \lll 1$, we can safely neglect these corrections as well. We conclude that (\ref{theprocess}) is by far the dominant contribution to the decay of axion condensates.
\section{The calculation of the wave function of the condensate} \label{WavefunctionSec}
We review here the calculation of the condensate wavefunction $X(r)$ \cite{ESVW}. The matrix elements of the $rr$ and $tt$ components of the Einstein equation, along with the Klein-Gordon equation, form a closed set of equations for the metric and the axion field, $X(r)$:
\begin{eqnarray}
\frac{A'}{A}&=&\frac{1-A}{r}+2\,\pi\,r\,\delta\,A\left\{\frac{E_0{}^2\,X^2}{B}
+\frac{X'^2}{A}+m^2X^2-\frac{m^2}{16}X^4\right\},\label{Ett}\\
\frac{B'}{B}&=&\frac{A-1}{r}+2\,\pi\,r\,\delta\,A\left\{\frac{E_0{}^2 \, X^2}{B}
+\frac{X'^2}{A}-m^2X^2+\frac{m^2}{16}X^4\right\},\label{Err}\\
X''&=&-\left[\frac{2}{r}+\frac{B'}{2\,B}-\frac{A'}{2\,A}\right]X'
-A\left[\frac{E_0{}^2\,X}{B}-m^2X+\frac{m^2}{8}\,X^3\right]\label{KG},
\end{eqnarray}
where the metric is
\be \label{metric}
ds^2=-B(r)\,dt^2+A(r)\,dr^2+r^2\,d\Omega^2,
\ee
with $\delta = f^2/M_P{}^2$ and $M_P =1\,/\,\sqrt{G}= 1.22\times10^{19}$ GeV (the Planck mass). As above, we have taken only the leading contribution to the Bessel function which represents the self-interaction potential; doing so preserves the leading attractive $X(r)^4$ interaction term.
Assuming that $\delta\ll1$, a condition satisfied in applications where gravity is weak (Newtonian limit), we can write $A = 1 + \delta \,a$ and $B = 1 + \delta\,b$, where $a,\,b=\mathcal{O}(1)$. Furthermore using the large radius approximation and the definition of the scale parameter $\Delta=\sqrt{m^2-E_0{}^2}/m$, we can introduce dimensionless radial coordinate as $x= m\,r\,\Delta$. In that case the axion field also scales with its engineering dimension, such that $X(r)=\Delta\,Y(x)$, leading to the following system of equations for $a$, $b$, and $Y$ in leading order of $\Delta$ and $\delta$~\cite{ESVW}:
\begin{eqnarray}\label{eomY}
Y''(x)&=&[1+\kappa \,b(x)]Y(x)-\frac{2}{x}\,Y'(x)-\frac{1}{8}\,Y(x)^3,\nonumber\\
a'(x)&=&\frac{x}{2}\,Y(x)^2-\frac{1}{x}\,a(x),\nonumber\\
b'(x)&=&\frac{1}{x}\,a(x),
\end{eqnarray}
where\footnote{This definition of $\kappa$ appears to differ by a factor of $8\pi$ compared with \cite{ESW}, because in that work we wrote $\delta$ in terms of the reduced Planck mass. In fact, the two definitions of $\kappa$ are equivalent.} $\kappa=8\pi\delta\,/\Delta^2$. Since $b(x)$ is proportional to the Newtonian gravitational potential, $\kappa\sim G$ is the effective coupling constant of the field $Y(x)$ to gravity. Further details, and a more comprehensive justification of this double expansion of the equations, can be found in \cite{ESVW}.
Solutions of the equations of motion (\ref{eomY}) correspond to ground-state configurations of axions, which can be stable or metastable. In \cite{ESVW}, we solved these equations and found a spectrum of solutions which were parameterized by $\kappa$ (or, equivalently, by $\Delta$). When applied to QCD axion parameters $m=10^{-5}$ eV and $f=6\times10^{11}$ GeV, we found a maximum gravitationally stable mass of $M_c \sim 10^{19}$ kg.\footnote{The maximum masses for attractive interactions were discussed by Stoof \cite{Stoof2} in the context of condensed matter BECs, and in the context of boson stars by Chavanis and Delfini \cite{ChavanisMR,ChavanisMR2}.} By rescaling these solutions to values of $m$ and $f$ appropriate for FDM, we can analyze the properties of galactic-size condensates in a way that includes the self-interaction term in the potential. The physical interpretation of these condensates is that they form the cores of FDM halos; they are surrounded by a virialized distribution of axions which extend to the outer edge of the dark matter halo.
To analyze the decay of these condensates through processes like (\ref{theprocess}), we investigate the singularity structure of solutions of (\ref{eomY}). Now, (\ref{eomY}) is a system of equations with two singular points, $x=0$ and $x=\infty$. Using boundary conditions we require that the solutions are regular at $x=0$ and decrease exponentially at $x\to\infty$. The solutions are even functions of $x$, so they also approach zero at $x\to -\infty$. Then they are real analytic functions at $-\infty<x<\infty$ and can be continued into the complex plane of $x$. As they fast vanish at infinity, the contour of integration can be moved up along the imaginary axis in the rescaled version of the integrals in eqs. (\ref{matrixr}) and (\ref{I3}), until we reach a singularity in the complex plane. The contribution of that singularity dominates the decay rate integrals at large momentum.
It is easy to show that the Klein-Gordon equation (\ref{eomY}), in which the the leading order singular terms are retained, is\footnote{In fact, this expression contains the next to leading order term proportional to $Y'(x)$.}
\be
Y''(x)+\frac{2}{x}Y'(x)+\frac{1}{8}Y(x)^3 = 0.
\ee
Near the singular point $x=i\,\rho$, this has a solution of the form
\be\label{singular}
Y(x)=\frac{8\,\rho}{x^2+\rho^2}-\frac{2}{3\,\rho}-\frac{1}{18\,\rho^3}(x^2+\rho^2)+\mathcal{O}([x^2+\rho^2]^2).
\ee
The parameter $\rho$ is an integration constant, having a one-to-one relationship with the rescaled central density of the axion field $Y(0)^2$, and in turn, with mass and the radius of the condensate. In fact, high order Taylor series expansion of equations around $x=0$ show that the singularity closest to the origin is indeed of the form (\ref{singular}), connecting the value of $Y(0)$ with $\rho$ \cite{ESW}. In principle, gravitational interactions have an effect on the solutions $Y(x)$. In practice, however, the term in the equations of motion (\ref{eomY}) which couple $Y(x)$ to gravity give a subleading contribution to the singularity. We can therefore solve the equations in the limit $\kappa\ll 1$, i.e. where gravity decouples. In that limit, the nontrivial solution has $Y(0)=12.268$, which implies a fixed value $\rho=.603156$.
Finally, we can rewrite the Fourier transform of eq. (\ref{fourier}) as
\be
\Xi(q) = \frac{1}{(2\,\pi)^2\,i\,q\,m^2\,\Delta}\int_{-\infty}^\infty dx\, x \,\exp\Big(\frac{i\,q\,x}{m\,\Delta}\Big)\,Y(x).
\ee
In this form, it is clear that at small $\Delta$ the singular term of (\ref{singular}) term dominates the integral.
To calculate the decay rate, we follow the procedure of \cite{ESW}: we take the leading order solution for $Y(x)$ near the singularity $x = i\,\rho$, given by eq. (\ref{singular}), and evaluate $I_3(p)$ in the matrix element of eq. (\ref{matrixr}). The result is
\be
I_3(p_0)\simeq\frac{32\,i\,\pi}{3}\frac{\rho}{\Delta}
\exp\left(-\frac{2\sqrt{2}\,\rho}{\Delta}\right),
\ee
where $p_0=\sqrt{9\,E_0{}^2-m^2}\simeq \sqrt{8}\,m$.
The decay rate for the process (\ref{theprocess}) is then
\begin{equation}
\Gamma_3 =\frac{1}{T} \int\,\frac{dp}{(2\pi)^32\,\omega_p}\,
\Big|\mathcal{M}_3\Big|^2
= \frac{2\,\pi\,f^2}{p_0}\Big|I_3(p_0)\Big|^2,
\end{equation}
where $T$ is the duration of the decay process. Then the lifetime of the condensate through this decay process is
\begin{equation} \label{dtaudN}
\frac{d\tau}{dN} \simeq m\,\frac{d\tau}{dM}
\simeq -\frac{1}{3\,\Gamma_3}.
\end{equation}
Further details regarding the evaluation of eq. (\ref{dtaudN}) can be found in \cite{ESW}; the result for the process (\ref{theprocess}) is
\begin{equation} \label{lifetime}
\tau = \frac{3\,y_M\,\Delta^2}{4096\,\pi^3\,\rho^3\,m}
\exp\left(\frac{4\,\sqrt{2}\,\rho}{\Delta}\right),
\end{equation}
where $y_M\simeq 75.4$ is determined by the relationship between $M$ and $\Delta$ in the large $\Delta$ region \cite{ESVW}. The lifetime is a monotonically decreasing function of $\Delta$ in the relevant range; in the case of QCD axions, we found in \cite{ESW} that above a value $\Delta\simeq .05-.06$, axion condensates become very unstable to decay to relativistic axions, their lifetimes becoming shorter than the age of the universe. We will examine the consequences of this fact in the context of ultralight axions in the next section.
\section{Stable spectrum of ultralight axion condensates} \label{DecayRateSec}
Very light axion fields can have de Broglie wavelengths as large as entire dark matter halos, possibly implying a connection between these two scales. Ultralight bosons appear often in theories of physics beyond the Standard Model, including those requiring compactification of extra dimensions. Such models, termed ``Fuzzy Dark Matter'' (FDM) \cite{Hu}, have been written about extensively \cite{Turner,Press,Sin,Hu,Goodman,Peebles,Amendola,Shapiro,Schive,Marsh,Witten}. While there are significant constraints on these models\footnote{While this work was being finalized, a paper appeared suggesting a strong tension between the preferred mass scale for FDM, $m\sim 10^{-22}-10^{-21}$ eV, and data from Lyman-$\alpha$ forest simulations \cite{Irsic}. We will not comment here about whether such constraints could rule out FDM as a viable paradigm.}, they remain a viable alternative to WIMP or QCD axion models of dark matter.
\begin{figure}[t]
\includegraphics[scale=1]{Mass_FDM.pdf}
\caption{The allowed masses for condensates of axion particles in FDM, as a function of the binding energy parameter $\Delta$; these condensates constitute the cores of FDM halos. Axion condensates in the shaded region are unstable to decay to relativistic axions with a very short lifetime. Here we have used the model parameters $m=10^{-22}$ eV, and $f$ in the range between $10^{14}$ and $10^{18}$ GeV; increasing the particle mass $m$ merely shifts these curves down proportionally to $1/m$.}
\label{Mass_FDM}
\end{figure}
\begin{figure}[t]
\includegraphics[scale=1]{Radius_FDM.pdf}
\caption{The allowed radii for condensates of axion particles in FDM, as a function of the binding energy parameter $\Delta$; these condensates constitute the cores of FDM halos. Axion condensates in the shaded region are unstable to decay to relativistic axions with a very short lifetime. Here we have used the model parameters $m=10^{-22}$ eV, and $f$ in the range between $10^{14}$ and $10^{18}$ GeV; increasing the particle mass $m$ merely shifts these curves down proportionally to $1/m$.}
\label{Radius_FDM}
\end{figure}
We will consider an ultralight axion in this context, using the potential of eq. (\ref{instanton}). The mass of the ultralight axion in question will be taken to be $m \sim 10^{-22}$ eV, which gives the right approximate scale for the size of dark matter halos \cite{Witten},
\begin{equation}
\frac{\lambda_{dB}}{2\pi} = \frac{1}{m\,v}
= 1.92 \text{ kpc}\Big(\frac{10^{-22}\text{ eV}}{m}\Big)
\Big(\frac{10\text{ km/sec}}{v}\Big),
\end{equation}
where $v$ is the velocity in the halo. This choice is also consistent with the known epoch of matter-radiation equality. Further, a decay constant of $f \sim 5\times10^{16}$ GeV naturally leads to the correct relic density, and can thus account for the observed dark matter abundance \cite{Witten}; however, to remain as general as possible, we allow $f$ to deviate from this value by a few orders of magnitude. At the upper limit of what we consider, $f=10^{18}$ GeV is still below the Planck scale, implying that the parameter $\delta = f^2/M_P^2 \approx .007 \ll 1$; thus, the weak-gravity approximation holds reasonably well over our entire range.
In \cite{ESVW}, we found numerically the solutions to the system (\ref{eomY}) over a wide range of $\kappa$. In the FDM picture, these solutions correspond to the cores of FDM halos discussed (most recently) in \cite{Witten}. We found that there exists a maximum mass at $\kappa \approx .34$, above which axion condensates are gravitationally unstable. On the stable branch of masses $\kappa >.34$, the mass $M$ and radius\footnote{We use the common convention that $R_{99}$, the radius inside which $.99$ of the mass of the condensate is located, represents the ``size'' of the condensate.} $R_{99}$ of the condensate are fit by the functions \cite{ESVW}
\begin{equation} \label{MandR}
M(\kappa) \approx \frac{8.75}{\sqrt{\kappa}}\,\frac{M_P\,f}{m}, \qquad
R_{99}(\kappa) \approx 1.15\,\sqrt{\kappa}\,\frac{M_P}{f\,m}.
\end{equation}
We observe in Figure \ref{Mass_FDM} that at fixed $m$, the value of $f$ determines the position of the maximum mass, and thus the turnaround of the function $M(\Delta)$. A similar structure can be observed in Figure \ref{Radius_FDM} for the radius, where the position of the maximum mass corresponds to a slight dip in the otherwise straight line representing $R_{99}(\Delta)$. Trading $\kappa$ for $\Delta$ in eq. (\ref{MandR}), we see that the lines
\begin{equation}
M(\Delta) \approx 1.75\,\Delta\,\frac{M_P{}^2}{m} \qquad
R_{99}(\Delta) \approx \frac{5.75}{m\,\Delta}
\end{equation}
bound the full set of solutions from above.
For the specific choice of FDM parameters $m=10^{-22}$ eV and $f=5\times10^{16}$ GeV we find the maximum mass $M_c \approx 6\times10^{10} M_\odot$; this is lower than the value found in the non-interacting limit of $8\times10^{11} M_\odot$ \cite{Witten} by about an order of magnitude, due to the inclusion of attractive self-interactions. Our estimate of the maximum mass also agrees well with the recent analysis of \cite{MarshUL}, which also includes the leading attractive self-interaction. More generally, the mass and radius of FDM halo cores over a wide range of the scale parameter $\Delta$ and at different values of $f$ are illustrated in Figures \ref{Mass_FDM} and \ref{Radius_FDM}.
We can also analyze the relationship between $M$ and $R_{99}$, which were investigated for both attractive and repulsive self-interactions in \cite{ChavanisMR,ChavanisMR2}. In a recent paper \cite{Witten}, the authors present a bound on the product
\begin{equation} \label{MR_NI}
M\,R_{1/2} \geq 3.925\,\frac{M_P{}^2}{m^2} \qquad \text{(non-interacting bosons)},
\end{equation}
where $R_{1/2}$ is the radius inside which $.5$ of the mass of the condensate is contained; the inequality is saturated for stationary, ground state configurations, i.e. for the condensates considered here. In our calculation, on the stable branch of solutions (where $\kappa > .34$), we find the product
\begin{equation}
M\,R_{99} = 10.06\,\frac{M_P{}^2}{m^2},
\end{equation}
using eq. (\ref{MandR}). To find the relationship between $R_{99}$ and $R_{1/2}$, we calculated their ratio numerically and found that $R_{99}/R_{1/2} \approx 3.65$ holds within 1\%, in a range of $1/\Delta$ extending over many orders of magnitude. This implies
\begin{equation}
M\,R_{1/2} = 2.76\,\frac{M_P{}^2}{m^2} \qquad \text{(interacting axions)}.
\end{equation}
This product is below the lower bound (\ref{MR_NI}) presented in \cite{Witten} due to our inclusion of self-interactions.
It is worth noting also that in the limit $f \sim M_P$, the mass-radius relationship for axion condensates approaches that of a black hole. This is easy to see using eq. (\ref{MandR}):
\begin{equation}
\frac{G\,M}{R_{99}} = \frac{1}{M_P{}^2}\,
\frac{\frac{8.75}{\sqrt{\kappa}}\frac{M_P\,f}{m}}
{1.15\sqrt{\kappa}\frac{M_P}{f\,m}}
= \frac{7.6}{\kappa}\,\frac{f^2}{M_P{}^2}.
\end{equation}
Near $\kappa=\mathcal{O}(1)$ (the position of the maximum mass) and $f\sim M_P$, we find $G\,M/R_{99}\sim 1$, implying that $R_{99} \sim R_S$, the Schwarzschild radius.
We must also ensure that the weak-binding approximation, on which our analysis \cite{ESW} and the classical one of \cite{Witten} depends, is also valid.\footnote{A stability analysis for very strongly-bound condensates, with $\Delta=\mathcal{O}(1)$, is a task we plan to undertake in the near future.} We observe in Figure \ref{Radius_FDM} that cores of radius $R\lesssim 1$ pc have $\Delta \gtrsim .3$, and become relatively strongly bound. Such cores would not be well-described by our weak-binding analysis.
An estimate of the decay rate, obtained from the expression derived in \cite{ESW}, is given in eq. (\ref{lifetime}); it is a one-to-one function of the binding energy parameter $\Delta$ in the region of interest. Because the condensate mass $M$ is determined by the value of $\Delta$, it is easy to connect $\tau$ to $M$ as well. Following the analysis of \cite{ESW}, we find that axion condensates with $m\sim 10^{-22}$ eV which have $\Delta \gtrsim .1$ have lifetimes shorter than the age of the universe. This region is represented by the shaded regions in Figures \ref{Mass_FDM} and \ref{Radius_FDM}. For $f\lesssim 10^{18}$ GeV, the transition to decay instability occurs on the gravitationally unstable branch of solutions; however at $f\gtrsim 10^{18}$ GeV, the bounds from decay are as strong or stronger than those coming from gravitational stability. This can be an important constraint on bound structures originating in theories of Planck scale axions.
In Figure \ref{Radius_FDM}, it is particularly striking that almost regardless of the value of $f$, condensates with $R\lesssim 2$ pc lie in the unstable, shaded region. This implies a fundamental limiting radius of $R_{min}\sim 2$ pc for FDM cores composed of axions. Observe also that, in Figure \ref{Mass_FDM}, it is easy to read off the maximum mass of FDM condensates for each value of the decay constant $f$. For any axion theory with $f\ll M_P$, no stable condensate exists with a mass larger than about $M_{max} \approx 10^{12} M_\odot$.
\section{Conclusions} \label{ConclusionSec}
In a previous publication~\cite{ESVW} we established scaling relations for the mass and radius of weakly bound condensates of interacting axions, as functions of the mass of the axion, its decay constant, and the particle energy (or alternatively the central density). We also found the maximum mass and size of the bound states as functions of those parameters. In this paper we have applied those results to condensates of axions forming FDM, providing corrections to similar calculations which neglect the self-interaction of axions \cite{Sin,Hu,Goodman,Amendola,Schive,Marsh,Witten}.
In another publication~\cite{ESW} we developed methods to calculate the lifetime of axion condensates due to their self-interaction, through the four-axion interaction term in which three bound axions produce a single free relativistic axion. Here we have applied those results to estimate the lifetime of condensates formed from FDM. We have found that, provided the decay constant of FDM axions satisfies $f\lesssim 0.05 \,M_P$, all condensates having binding energy smaller than that of those of maximal mass have lifetimes greater than the age of the universe making them viable candidates for forming central regions of galactic halos. We have also explained in details the decay mechanism described in \cite{ESW} and further clarified the justification of its validity.
The methods we have described here, based on previous work in \cite{ESVW,ESW}, rely on a double expansion to leading order in $\delta$ and $\Delta$. This is appropriate for so-called dilute axion stars, which are weakly bound. However, it has been pointed out that an effective short-distance repulsive interaction in the axion potential also gives rise to dense axion stars \cite{Braaten,ELSW,ELSW2}, which are at least energetically stable. We plan to extend our methods to this regime to analyze the properties of these states in the near future.
Collapsing boson condensates have been investigated by a number of groups \cite{Stoof,Stoof2,Harko,ChavanisCollapse,ELSW,Tkachev2016,MarshCollapse,ELSW2}. Recently, we found that supercritical QCD axion condensates, having masses larger than the maximal allowed stable mass, collapse towards the global minimum of the effective axion potential~\cite{ELSW,ELSW2}. Similar arguments indicate that FDM axion condensates which exceed the maximum mass $M_c$ will also collapse in this way. Such supercritical condensates can form during galactic collisions, in a manner similar to the mechanism outlined in \cite{Collisions}; such events would lead to collapse, causing the condensate to emit a large number of relativistic particles. Consequences of such events will be studied in a future publication.
\section*{Acknowledgements}
We thank M. Amin, P. Argyres, E. Braaten, J. Brod, P. Fox, R. Harnik, A. Kagan, A. Mohapatra, K. Schutz, G. Semenoff, M. Takimoto, H. Zhang, and J. Zupan for fruitful discussions. The work of J.E. was supported by the U.S. Department of Energy, Office of Science, Office of Workforce Development for Teachers and Scientists, Office of Science Graduate Student Research (SCGSR) program. The SCGSR program is administered by the Oak Ridge Institute for Science and Education for the DOE under contract number DE-SC0014664. J.E. also thanks the Fermilab Theory Group and the Weizmann Institute Department of Physics for their hospitality. L.C.R.W. thanks Mainz Institute for Theoretical Physics for their hospitality, and the participants of the Quantum Vacuum and Gravitation program, especially M. Bartelmann, A. Mazumdar, and T. Prokopec, for discussions.
|
1,116,691,500,798 | arxiv | \section{Introduction}
This paper is a cleaned up and improved version of author's preprint \cite{LS}. It features more straightforward version of the main theorem and various improvements and clarifications.
\begin{subsection}{Main question} Let use denote by $D \subset \mathbb{C}$ the unit holomorphic disk and consider an analytic family $\pi: \tilde{X} \rightarrow D$ of complex surfaces over a disk, with smooth total space, with central fiber $X = \pi^{-1}(0)$ being a normal crossing surface. Denote by $X^{\varepsilon} = \pi^{-1}(\varepsilon)$ the nearby fiber, by $X_1, \cdots, X_n$ the irreducible components of $X$, and by $\hat{X}_1, \cdots, \hat{X}_n$ their normalizations.
Suppose $\hat{X}_i$ are rational surfaces, and normalizations of components of the singular locus are rational curves (we call such degeneration \textbf{geometrically maximal}). Then, the \textbf{dual complex} of the special fiber $X$ is intimately related to the topology of the general fiber - in particular, if the dual complex has trivial rational cohomology, then the general fiber has $h^{1,0}(X^{\varepsilon})=h^{2,0}(X^{\varepsilon})=0$. By the results of Kollar, Xu and de Fernex \cite{dFKX}, Theorem 4, if the general fiber was rational, then the dual complex needs to be contractible and even \textbf{collapsible}, which poses an interesting question of building (from smoothing techniques) examples of degenerations with dual complex being contractible but not collapsible. It would give (possibly new) surfaces with $h^{1,0} = h^{2,0} = 0$.
\end{subsection}
\begin{subsection}{Differences with earlier versions} As this text is an overhaul of the previous version of the paper, it might be worth stating the differences with the previous version. The main difference is the different proof of surjectivity of the map $O$ - instead of dealing with the asymptotics we just calculate it explicitly enough. We also give independent proofs of all the results we use for the smoothing of normal crossing spaces instead of referring to the literature, which, sadly, mostly written in the assumptions of simple normal crossing. Since the time we writing previous text, we have also managed to prove that our example is actually simply-connected, which we only suspected at that time.
\end{subsection}
\begin{subsection}{Purpose and organization of the paper}
This paper presents an example (first explicit, to our knowledge) of a geometrically maximal degeneration with non-collapsible contractible dual complex (topological duncehat), which answers affirmatively to the question of \cite{dFKX} on whether such degenerations exist. The generic fiber of this degeneration turns out to be some simply-connected surface of general type with $h^{1,1} = 9$ - same as Barlow's surface \cite{B}. We conjecture that it is in a deformation class of Barlow's surface. We would like to also point out that there are constructions which produce simply-connected surfaces with $p_g = 0$ by smoothing of isolated $\mathbb{Q}$-Gorenstein singularities \cite{LYP}. It would be extremely interesting to find out whether these constructions are related to the present work - in particular, what are the dual complexes of SNC-resolutions of these families.
Our initial motivation for this construction was an attempt to understand the extent to which the two-dimensional complexes play the role similar to those of graphs in the enumerative geometry of surfaces. This understanding is still yet to be achieved.
Another motivation is the fact that this construction could, in principle, provide us with the new examples of surfaces with $h^{1,0}=h^{2,0}=0$.
We prefer to work in the assumptions which are slightly weaker than most of the works in the field - i.e., in the literature the degeneration is commonly assumed to be simple normal crossing, while we prefer to work with just normal crossing. Due to this, and for readers and our own convenience we tried to keep the exposition self-contained when possible. We would like to point out that while in principle all constructions can be done in simple normal crossing case the combinatorics of the problem tend to become very messy while doing so.
\newline
The text consists of the following next parts:
~
\noindent \textbf{Notation} - we state the problem precisely, introduce our assumptions and fix the terminology and notation for triangulated spaces and dual complexes.
~
\noindent \textbf{Smoothing} - we present our view on the deformation theory of the normal crossings (which largely follows Friedman, but we explain the relation to the logarithmic deformation theory) and reduce the smoothing problem to the calculation of the fiber of a certain obstruction mapping. The main result of this part is the following observation, known in the slightly different form already to Friedman:
~
\noindent For the normal crossing surface $S$ which satisfies our geometric maximality assumption denote by $M_S$ the moduli space of locally trivial deformations of $S$, by $J_S$ the space of line bundles on the singular locus of $S$ (it can be naturally identified for all locally trivial deformations of $S$), by $O: M_S \longrightarrow J_S$ the map which sends the normal crossing surface to its first tangent cohomology $T^1(S)$, considered as a line bundle on its singular locus. Assume also that locally trivial deformations of $S$ are unobstructed.
\begin{nthm}[Corollary \ref{cor:moduli-smooth}]Suppose that the fiber $O^{-1}(\mathcal{O}_{\Sing(S)})$ is smooth as a subscheme of $M_S$ in a point corresponding to $S$. Then, $S$ is smoothable.
\end{nthm}
It is clear that if the fiber is not smooth, but reduced, we can find the locally trivial deformation which is smoothable. We also prove the slight improvement of the result above - the case when the fiber is not reduced but has an expected dimension (Th. \ref{thm:cool-smooth}) This improvement is not needed in what follows but might turn out to be useful later. \\~\\~
\textbf{Example} - we introduce our degeneration, the special fiber $X$.\\~\\~
\textbf{Obstruction mapping} - we calculate the obstruction explicitly, thus proving \begin{nthm}[Corollary \ref{cor:epicwin}] There exists an analytic family $\tilde{X}$ with a smooth total space, with geometrically maximal degeneration and dual complex homeomorphic to the duncehat. \end{nthm}
\noindent
\textbf{Properties of the smoothing} - we show that the generic fibers of $\tilde{X}$ are algebraic, of general type and minimal; we calculate the fundamental group and show that it vanishes;
\newline
\noindent \textbf{Questions} - we state and motivate few questions and conjectures based on our observations. The conjectures (one very simple and other maybe a bit too brave) are: \begin{ncnj}The generic fiber of our degeneration $X^{\varepsilon}$ is in a deformation class of Barlow's surface.\end{ncnj}
\begin{ncnj}Every simply-connected surface with $h^{1,0}=h^{2,0}=0$ admits a geometrically maximal degeneration with non-collapsible contractible dual complex.\end{ncnj}
\end{subsection}
\begin{subsection}{Acknowledgement}
I thank Konstantin Loginov and Vadim Vologodsky for their crucial impact on my understanding of the subject; Denis Teryoshkin, Dmitry Sustretov, Andrei Losev and Dmitry Korb for stimulating discussions; Mikhail Kapranov for his suggestion on the connection of this work with the possible combinatorial construction of the phantom categories.
\end{subsection}
\section{Notation}
\subsection{Normal crossing.} \hfill\\
We fix the terminology. Recall that the singular point is called \textbf{normal crossing} if it is locally (in etale or analytic topology) equivalent to a union of a collection of coordinate hyperplanes in $\mathbb{C}^n$.
\begin{definition}The variety with normal crossing singularities is called \textbf{normal crossing} variety.\end{definition}
This notion admits a few different versions, which we would like to distinguish between.
\begin{definition}
Suppose $X$ is a NC-variety such that all its irreducible components are smooth. Then, it is called \textbf{simple normal crossing} (SNC).
Suppose, in addition, that the intersection of any collection of its irreducible components is connected. Then, we will call it \textbf{strictly simple normal crossing} (SSNC).
\end{definition}
\subsection{Triangulated spaces.}\hfill\\
There are also few different but similar notions of a triangulated space.
\begin{definition}
The \textbf{semi-simplicial set} $S$ is a contravariant functor $S$ from the category of non-empty totally ordered finite sets with injective order-preserving morphisms $\Delta_+$ to the category of sets.
Explicitly, it yields a set $S_0 = S(\{0\})$ of vertices, set $S_1 = S(\{01\})$ of edges, set $S_2 = S(\{012\})$ of $2$-simplices e.t.c. together with natural face maps $\partial_k: S_n \rightarrow S_{n-1}, 0 \leq k \leq n$ corresponding to deletion of $k$-th vertex.
We denote the category of semi-simplicial sets as $\ssSet$.
\end{definition}
\begin{note} $1$-dimensional semi-simplicial set is the same as ordered graph (possibly with loops and double edges).
\end{note}
This notion, however, contains additional information which we do not actually need - every simplex in the semi-simplicial set has a chosen ordering of vertices. There is the following, slightly more general, notion which forgets this data.
\begin{definition}
The \textbf{triangulated set} $T$ is a contravariant functor $T$ from the category of finite sets with injective morphisms $\Fin$ to the category of sets with the following additional property: for any $F \in \Ob(\Fin)$ action of $\Aut(F)$ on $T(F)$ is free. We denote $T(n) = T_n$, as above.
The category of triangulated sets is denoted as $\tSet$.
\end{definition}
\begin{note}
$T_n$ should be thought of as the set of $n$-dimensional simplices of $T$ with chosen ordering of vertices.
\end{note}
\begin{example}
There is a functor $p: \ssSet \rightarrow \tSet$, defined as: $p(S)(X) = S(X) \times \Or(X)$, where $\Or(X)$ denotes the set of orderings of $X$, and the face maps are natural.
\end{example}
While every $1$-dimensional triangulated set comes from the semi-simplicial set, in dimension $2$ it is already false.
\begin{note}
Every semi-simplicial set (and every triangulated set) has a geometric realization functor. We will use the word ''triangulated space'' for the pair (semi-simplicial/triangulated set, its realization) and hope it won't cause any confusion.
\end{note}
The main triangulated space we want to use throughout our work, however, does come from a semi-simplicial set:
\begin{definition}\label{def:dunce}
The \textbf{topological duncehat} is a $2$-dimensional semi-simplicial set with one vertex, one edge and one triangle and obvious structure maps.
\end{definition}
\begin{note}
We can also define functor $q: \tSet \rightarrow \ssSet$ as follows.
For a triangulated set $T$ denote the set of facets without orientation $T_n^{\red} = T_n / \Aut(\{01...n\})$.
Now, $T^{\red}_n$ and $T^{\red}_{m}, m < n$ admit a correspondence, for $\alpha \in T^{\red}_{m}, \beta \in T^{\red}_n$ we say that $\alpha < \beta$ iff they have preimages in $\tilde{\beta} \in T_n$ and $\tilde{\alpha} \in T_{m}$ such that $\tilde{\alpha}$ is a face of $\tilde{\beta}$. Now, we define:
$q(T)_n = \{\text{ordered collections of the form }\alpha_0 < \alpha_1 < ... \alpha_n\text{ with }\alpha_i \in T^{\red}_{k_i}, k_0 < ... < k_n\}$. Such collections are called ''flags'', and composition of the functors $p$ and $q$ in either order corresponds to the barycentric subdivision of the corresponding triangulated spaces.
\end{note}
We also give two possible restrictions on the definition above.
\begin{definition}
The triangulated space will be called \textbf{simple} if any simplex has no coinciding faces (of any codimension). The triangulated space will be called \textbf{strictly simple} if any two simplices either have an empty intersection or have a maximal common face.
\end{definition}
\begin{note}
Strictly simple triangulated space with global ordering on the set of vertices is the same as simplicial complex.
\end{note}
\begin{note}
$1$-dimensional triangulated space is a graph. Being simple for it means that the corresponding graph has no loops. Strictly simple means that it also has no double edges.
\end{note}
\subsection{Dual complexes.}
\begin{definition} For every SNC-variety $X = \underset{i=1}{\overset{n}{\bigcup}} X_i$ define its dual complex $\Delta_X$ by the following semi-simplicial set: \[(\Delta_X)_k = \underset{1 \leq i_0 < ... < i_k \leq n}{\coprod} \pi_0(X_{i_0}\cap X_{i_1} \cap ... \cap X_{i_k})\] with face maps coming from natural inclusions of $X_{i_0}\cap X_{i_1} \cap ... \cap X_{i_k} \subset X_{i_0}\cap X_{i_1} \cap ... X_{i_{s-1}} \cap X_{i_{s+1}}... \cap X_{i_k}$
\end{definition}
\begin{note}
$\Delta_X$ is strictly simple iff $X$ is strictly SNC and simple iff $X$ is SNC.
\end{note}
We can also define the dual complex for some (but not all!) normal crossing varieties, however, it won't in general will be a semi-simplicial set, but a triangulated set.
\begin{definition}
Consider a normal crossing variety $X$ together with its natural stratification $Z_n \subset Z_{n-1} \subset ... \subset Z_{1} = \Sing(X) \subset Z_0 = X$. Denote by $\mathcal{F}_k$ the set of irreducible components of $Z_k$. Consider any $F \in \mathcal{F}_k$, $\nu_F: \hat{F} \rightarrow F$ - its normalization (it is smooth). For any point of $x \in \hat{F}$, denote by $B_x(\hat{F})$ the set of irreducible components of the analytic germs of $X$, containing $(\nu_F)_{*}(T_x(\hat{F}))$ (we will call it the set of ''branches'' in $F$). It forms a local system over $\hat{F}$.
We will say that $X$ satisfies \textbf{no branch switching} assumption if this set is a trivial local system for any $F \in \mathcal{F}_k$ for any $k$. In that case we will denote this (globally defined) set $B(\hat{F})$, emphasizing its independence on the point $x$.
\end{definition}
\begin{example}
The simplest example when this assumption fails is the following: consider $X$ to be some surface which contains an embedded elliptic curve $E$, denote by $\sigma: E \rightarrow E$ any free involution of $E$. Then, $X/(s \sim \sigma (s))$ does not satisfy the no branch switching assumption.
\end{example}
\begin{note}
No branch switching assumption holds automatically if every $F$ is simply connected. We are mostly interested in the case when $X$ is a surface and all components and all intersections are rational, so for our application it is indeed the case.
\end{note}
\begin{definition}
For a normal crossing variety $X$ satisfying no branch switching assumption the \textbf{dual complex} is a triangulated set $\Delta_X$ defined as:
\[(\Delta_X)_k = \{(F, \alpha)|F\in\mathcal{F}_k, \alpha \in \Or(B(\hat{F}))\}\]
and the face maps are defined as follows:
for any pair $F \in \mathcal{F}_k, F' \in \mathcal{F}_{k-1}$ such that $F \subset F'$ the ordering of the branches in $F$ induces the ordering of branches in $F'$ naturally (for the reason that any branch of $F$ but one can be analytically continued to the branch of $F'$). That defines the boundary operation: if $(F, \alpha)$ and $(F', \alpha')$ are such that $F \subset F'$ and the orderings are induced (and the element $k \in \alpha$ is the one which is not promoted to $\alpha'$) we say that $\partial_k (F, \alpha) = (F', \alpha')$
\end{definition}
\subsection{Maximal vs geometrically maximal degeneration}\hfill\\
Consider the family $\pi: \tilde{X} \longrightarrow D$ of complex Kahler varieties over a disk, with smooth total space and NC central fiber. In the theory of Calabi-Yau degenerations there is a maximality condition \cite{D}.
\begin{definition}[Deligne]
The degeneration of Calabi-Yau manifolds of dimension $n$ is called \textbf{maximal} if an operator of monodromy acting on the middle cohomology $H^n(X)$ has a Jordan block of length $n$.
\end{definition}
This condition is conjectured to give the dual complexes homeomorphic to a (homological) sphere. It is indeed true in the case of K3 surfaces, according to Kulikov \cite{Ku} and Freedman \cite{Fr}.
In the case of surfaces of general type with $h^{1,0}=h^{2,0}=0$ there is clearly no hope of characterizing the maximality of degeneration in terms of the limiting Hodge structure - there is no variation of Hodge structure anywhere at all, as the Hodge structure of the generic fiber is already of Hodge-Tate type. We, however, give a condition which coincides with the maximality in case of K3 surfaces. We do not know its precise relation to the notion of maximality given by Deligne in Calabi-Yau case.
\begin{definition}
The degeneration of the family of varieties is called \textbf{geometrically maximal} if the normalizations of all irreducible components of $Z_i$ are rational.
\end{definition}
\begin{note}If the dimension of the fiber is $> 2$, it is not clear whether we should maybe use ''rationally connected'' instead of ''rational'' in this definition. We are mostly concerned about the dimension $2$ case, so it doesn't really matter.\end{note}
We would also like to point out that this definition is a bit ad hoc - it is completely not obvious how to prove that some surface (even admitting a lot of deformations) has such a degeneration.
\subsection{Cohomology of the dual complex and limiting Hodge structure}
Consider, the analytic family of algebraic varieties $\pi: \tilde{X} \longrightarrow D$ over a disk, with smooth total space and normal crossing central fiber. We denote as $\pi^{-1}(\varepsilon) = X^{\varepsilon}$ the fiber over some generic nearby point $\varepsilon \neq 0$. Assume, in addition, that the degeneration is geometrically maximal. Consider $\Delta_X$ - the dual complex of the central fiber .
\begin{lemma}
$H^*(\Delta_X, \mathbb{C}) = 0$ implies $h^{*,0}(X^{\varepsilon}) = 0$
\end{lemma}
This lemma (for the SNC case) can be deduced from the results of Schmid and Steenbrink \cite{St} on the limiting mixed Hodge structure, and improved to NC case due to the invariance of the homotopy type of the dual complex under birational changes (see \cite{dFKX}). We, however, give a very simple and straightforward argument:
\begin{proof} Denote by $X = \underset{a \in \pi_0 (\hat{X})}{\coprod} X_a$ the decomposition of the central fiber into irreducible components. Denote their normalizations as $\nu_a: \hat{X}_a \rightarrow X_a$.
For the stratum $Z_k$ denote its irreducible components $X_{\mu}$, where $\mu \in \Delta_X$, their normalizations $\nu_{\mu}: \hat{X}_{\mu} \rightarrow X_{\mu}$.
Then, we have the following resolution of the sheaf $\mathcal{O}$ on $X^0$:
\[
0 \rightarrow \mathcal{O}(X) \rightarrow \bigoplus_{\mu \in (\Delta_X)_0} \pi_* \mathcal{O}(\hat{X}_{\mu}) \otimes L_{\mu} \rightarrow \] \begin{equation} \rightarrow
\bigoplus_{\mu \in (\Delta_X)_1} \pi_* \mathcal{O}(\hat{X}_{\mu}) \otimes L_{\mu} \rightarrow \bigoplus_{\mu \in (\Delta_X)_2} \pi_* \mathcal{O}(\hat{X}_{\mu}) \otimes L_{\mu} \rightarrow \cdots
\end{equation}
Here, $L_{\mu}$ is the $1$-dimensional orientation space of the simplex $\mu$. The maps are induced by the natural inclusions of the components $\hat{X}_{\mu} \rightarrow \hat{X}_{\mu'}$ and are taken with the sign depending on the chosen orientation of corresponding simplices.
Now, the components of this resolution are acyclic (being the direct images of $\mathcal{O}$ on some rational varieties via normalization morphism, which has no higher direct images), and, hence, it calculates the cohomology of $\mathcal{O}(X)$. Now, the complex
\[0 \rightarrow \bigoplus_{a \in (\Delta_X)_0} H^0(X, \pi_* \mathcal{O}(\hat{X}_{\mu})) \otimes(L_\mu) \rightarrow \bigoplus_{\mu \in (\Delta_X)_1} H^0(X, \pi_* \mathcal{O}(\hat{X}_{\mu})) \otimes(L_{\mu}) \rightarrow \] \[ \bigoplus_{(\mu) \in (\Delta_X)_2} H^0(X, \pi_* \mathcal{O}(\hat{X}_{\mu})) \otimes L_{\mu} \rightarrow \cdots\]
is clearly identified with the combinatorial cochain complex of $\Delta_X$ because all $H^0(X^0, *)$ factors are isomorphic to $\mathbb{C}$ canonically, which implies the desired result.
\end{proof}
\iffalse
We also outline without proofs the general picture of the limiting MHS, following the results of [?][?][?][?].
First of all, suppose one is given a family $\tau: X \rightarrow \Spec(\mathbb{C}\{t\})$ over a disk, with smooth total space and simple normal crossing central fiber, but possibly with multiplicities. Then, the semi-stable reduction theorem [?] states, that after the base change $t \rightarrow t^k$ and resolution of obtained toroidal singularities one obtains the family without multiple fibers. It also has the dual complex PL-homeomorphic to the original one.
For the normal crossing (without simplicity assumption) one, in principle, could resolve the family in such a way that the central fiber becomes the SNC divisor. Then, by the previous construction, one would obtain the SNC family with the dual complex homeomorphic to the original one: it is not hard to check that blow ups of the strata of the normal crossing do not change the homeomorphism type of the dual complex in the case where it can be defined, i.e. no branch switching. We shall not do it here, but see [dFKX] for the precise description on the effect of the blow-ups on the dual complex of the SNC, the argument for the NC without branch switching is completely analogous.
It allows one to restrict the attention to the SNC case in the search of complexes of the prescribed PL-homeomorphism type. The reduction process above, however, has the potential to drastically complicate the combinatorics of the degeneration, so author believes it is worth to actually consider non-SNC case.
Now, the results of [?] imply that
\begin{itemize}
\item Action of the monodromy operator $T$ on the cohomology of the generic fiber is quasi-unipotent.
\item Provided it is unipotent [BLABLABLA CLEMENS-SCHMID SEQUENCE IN PARTICULAR IN OUR CASE IT HAS 0 COHOMOLOGY]
\end{itemize}
\fi
\section{Smoothing}
In this part, we present our point of view on the smoothing theory of normal crossing surfaces. It largely follows Friedman's original work on K3 surfaces \cite{Fr}.
Suppose we are given a normal crossing surface $X$. Then, there is the following question: is it possible to construct a family over a disk
$\tilde{X} \rightarrow D)$ with smooth total space, no multiple fibers and the central fiber isomorphic to $X$? There is an obstruction to it:
Consider the sheaf of first tangent cohomology $T_{X}^1$, it might be thought of as the sheaf of local deformations of the scheme. For the normal crossing surface, it forms a line bundle over the singular locus $\Sing(X)$.
The obstruction is as follows: $T_X^1$ must be isomorphic to $\mathcal{O}_{\Sing(X)}$.
$T_X^1$ of any normal crossing variety admits the very explicit description, which we reproduce, following Kawamata and Namikawa \cite{KN}.
\begin{definition}Let us denote by $R_k$ the coordinate cross in $\mathbb{C}^n$ defined by the equation $x_1 \cdots x_k = 0$. The \textbf{log chart} $U$ on $X$ is an (analytic) open subset $U$ together with the holomorphic embedding $\phi: U \longrightarrow \mathbb{C}^n$, such that $\phi(U)$ is identified with an open neighborhood of the $0$ in $R_k$. For the chart $U$ we denote $u_i = \phi^{*}x_i$. We will refer to $u_i$ for $i \in \{k+1, ..., n\}$ as ''free coordinates'' \end{definition}
\begin{definition}
The \textbf{log atlas} is the collection of log charts $U^1, ..., U^s$ together for the following structure for the pair of $U^i$ and $U^j$:
Choose the permutation $\sigma$ such that any irreducible component of $U^i \cap U^j$ such that it has index $s$ on $U^i$ has an index $\sigma(s)$ on $U^j$ - i.e., vanishing loci of $u^i_s$ and $u^j_{\sigma(s)}$ coincide.
Then, one should have the collection of functions $z^{ij}_1, ..., z^{ij}_n$ such that $u^j_{\sigma(s)} = z^{ij}_s u^i_s$. This choice is unique on $U^i \cap U^j \cap \Sing(X)$ up to the permutation of free coordinates.
This functions should satisfy the following condition: $z^{ij}_1 \cdots z^{ij}_n = 1$
\end{definition}
\begin{definition}
Suppose now $X$ was a central fiber of the family $\pi: \tilde{X} \rightarrow D$, and suppose $z$ is a standard coordinate on $D$. Then, the collection of open subsets $U_i$ of $\tilde{X}$ endowed with the maps $\phi_i: U_i \rightarrow \mathbb{C}^n$ such that $\phi_i^*x_1 \cdots \phi_i^*x_n = \pi^*z$ is called the \textbf{extended log atlas}.
\end{definition}
\begin{note}
It is easy to see that the extended log atlas on $\tilde{X}$ always exists, moreover, it is clear that the restriction of the extended log atlas on $X$ gives a log atlas. Hence, the existence of the log atlas on $X$ is the necessary condition for smoothing.
\end{note}
\begin{definition}[Kawamata-Namikawa]\label{def:log-kawa-nami}
Two log atlases are called \textbf{equivalent} if their union is a log atlas. The class of equivalences of log atlases is called the \textbf{log structure} on $X$.
\end{definition}
\begin{note}
In modern terminology this structure is called the log structure of semistable type.
\end{note}
\begin{definition}\label{def:tx1-log}
Now, choose the collection of log charts, not necessary satisfying the log atlas condition. Then, we define the line bundle $T_X^1$ on $\Sing(X)$ by the gluing functions $\varphi^{ij} = (z^{ij}_1 \cdots z^{ij}_n)^{-1}$.
\end{definition}
The log structure is the same as the trivializing section of this bundle.
\begin{definition}
The variety with trivial $T_X^1$ is called $d$-semistable.
\end{definition}
Now, let us recall the relation of this definition of $T_X^1$ with the more standard ones. Exposition here closely follows Friedman \cite{Fr}.
Suppose $X$ is a normal crossing variety, locally (in analytic topology) embedded in a smooth variety $V$. Then, we have the following short exact conormal sequence, which is a locally free resolution of the sheaf of Kahler differentials on $X$: \begin{equation}\label{eqn:conormal-sequence} 0 \rightarrow I_X / I_X^2 \overset{e}{\rightarrow} \Omega^1_{V|X} \overset{r}{\rightarrow} \Omega^1_X \rightarrow 0\end{equation}
The dual sequence fails to be exact, and $T_X^1$ measures this failure: \begin{equation}\label{eqn:normal-sequence}0 \rightarrow T_X^0 \overset{r^*}{\rightarrow} T_{V|X} \overset{e^*}{\rightarrow} N_X \overset{\delta}{\rightarrow} \end{equation}
\[ \hspace*{-2cm} \rightarrow T_X^1 \rightarrow 0\]
The invariant version of this description is:\begin{equation}\label{eqn:tx1-invariant}T_X^1 = \text{\textit{Ext}}^1(\Omega_X, \mathcal{O}_X)\end{equation}
Now, the definition by the normal sequence from eq.\ref{eqn:normal-sequence} bears resemblance with our explicit description. Let us show it explicitly for the convenience.
\begin{lemma}\label{lem:tx1-equiv}The definition \ref{def:tx1-log} and definition by normal sequence (eq.\ref{eqn:normal-sequence}) are equivalent.\end{lemma}
\begin{proof}
Suppose we are given, as in the def. \ref{def:tx1-log}, the collection of local log charts. Consider, at first, one of the charts, $U$, together with its structure functions $u_1, ... u_n$, the structure embedding \[\phi = (u_1, ..., u_n): U \rightarrow \mathbb{C}^n = V\] Let its image be the open neighborhood of the zero in the coordinate cross $R_k$. The explicit generator of $I_{\phi(U)}/I_{\phi(U)}^2$ is the defining equation of $X_k$, the function $x_1 x_2 \cdots x_k$. As an element of $\Omega_{V|X}$ it is represented by the differential form \[\alpha = x_2 \cdots x_k dx_1 + ... + x_1 \cdots x_{k-1} dx_k\]
Now, we need to calculate explictly the map \[e^*: T_{V|X} \longrightarrow N_X\]
We will use an element $\alpha^* \in N_X$ as the trivializing section of the normal bundle. Then, for a tangent field $v \in T_{V|X}$, the following holds: \[e^*(v) = \alpha(v) \alpha^*\]
It is, thus, not too hard to calculate the image of this map: \[\im(e^*) = \langle x_2 \cdots x_k; x_1 x_3 \cdots x_k; ... ; x_1 \cdots x_{k-1}\rangle \alpha^*\]
That means that the cokernel of this map is supported on $\Sing(R_k)$ and is a line bundle on it, generated by the image of $\alpha^*$. It now remains to calculate the gluing map on the intersection of two charts.
To simplify the calculation, let us add to our collection of log charts all the intersections between them. The gluing map, so, w.l.o.g. can be calculated for the pair of charts $U' \subset U$. Note that for the specific choice of structure functions - structure functions on $U'$ being just the restrictions of the structure functions on $U$, the gluing is trivially $1$. So we can further assume w.l.o.g. that we are actually given one local chart $U$ with two different sets of structure functions, $(u_1, ..., u_n)$, $(u_1', ..., u_n')$, and we need to calculate the resulting relation of the elements $(\alpha^*)'/(\alpha^*)|_{\Sing(U)}$. Again, we can assume (because it is true up to the permutation) that zero locus of $u_i$ and $u_i'$ coincides for all $i$. That means that $z_i = u_i' / u_i$ is a well defined function on $\overline{U \setminus \{u(p)=0\}}$. Let us promote it (non-uniquely) to the holomorphic function $z_i$ on $U$, satisfying $u_i' = z_i u_i$.
As the next step, consider the structure map $\phi = (u_1, ..., u_n): U \rightarrow V$. Let us, further, find such invertible holomorphic functions $\tilde{z}_1, ..., \tilde{z}_n$ that $\phi^* (\tilde{z}_i) = z_i$. Then, define the new coordinate chart \[x_i' = \tilde{z}_i x_i\] Note that $\phi$ in this new coordinate chart is actually the structure map for the collection $(u_1', ..., u_n')$: \[\phi^*(x_i') = u_i'\]
That allows us to compare \[\alpha = d(x_1\cdots x_n))\] and \[\alpha = d(x_1'\cdots x_n') = \tilde{z}_1 \cdots \tilde{z}_n d(x_1 \cdots x_n) + x_1 \cdots x_n d(\tilde{z}_1 \cdots \tilde{z}_n)\]
The second term vanishes on $\im(\phi)$, which shows that \[\alpha' = z_1 \cdots z_n \alpha\]
which concludes the proof.
\end{proof}
Now, let us proceed with the general theory of smoothing of normal crossing varieties. The exposition, once again, closely follows Friedman \cite{Fr}, the general results allowing this description are by Tyurina and Palamodov \cite{P}.
\begin{definition}\label{def:tx-global}For any proper algebraic variety $X$ we have the tangent complex \[T_X = \text{\textit{RHom}}(\Omega_X, \mathcal{O}_X)\]\end{definition} its global first cohomology is the tangent space of local deformations, and the formal scheme of infinitesimal deformations of $X$ being the fiber of (non-linear formal) Kuranishi map: \begin{equation}K: H^1(T_X) \longrightarrow H^2(T_X)\end{equation}
In normal crossings case, $T_X$ has only 0-th and 1-st cohomology as the complex of sheaves. That gives the distinguished triangle \begin{equation} \label{eq:tx-dist-triangle}T_X^0 \rightarrow T_X \rightarrow T_X^1[1]\end{equation} leading to the following exact sequence:
\[0 \rightarrow H^0(T_X^0) \rightarrow H^0(T_X) \rightarrow 0 \rightarrow \]
\begin{equation}\label{eq:tx-exact-seq}\rightarrow H^1(T_X^0) \rightarrow H^1(T_X) \rightarrow H^0(T_X^1) \rightarrow\end{equation}
\[\rightarrow H^2(T_X^0) \rightarrow H^2(T_X^1) \rightarrow H^1(T_X) \rightarrow ...\]
Suppose now that the dimension of $X$ is $2$. Then, all further terms vanish automatically by dimension. We now restrict our interest to the smoothing formal deformations - i.e., such families \[\tilde{X} \rightarrow \Spec(\mathbb{C}[[t]])\] that the total space is smooth, and the central fiber is reduced. It implies the statement equivalent to Kulikov's triple point condition:
\begin{lemma}[reformulation of Kulikov's triple point condition] \label{lem:Kul-c1}
$ $ \newline $c_1(T_X^1)=0$ on every irreducible component of $\Sing(X)$.
\end{lemma}
\begin{proof}Actually, $T_X^1$ should be a trivial bundle since such deformation requires the existence of log structure (see def.\ref{def:log-kawa-nami})\end{proof}
Let us now describe the geometric significance of the sequence from eq.\ref{eq:tx-exact-seq}.
\begin{note}\label{note:geom-sense}
The $T_X^0$ is just the sheaf of derivations of $X$. So, the fact that $H^0(T_X^0)$ and $H(T_X)$ are identified is clear from the standpoint of the deformation theory - any infinitesimal automorphism of $X$ comes from the vector field on $X$, just as in the smooth case.
Now, the cohomologies of $T_X^0$ actually control \textbf{locally trivial} deformations - that is, deformations which preserve the structure of singularities. We will see it more explicitly later, or one could refer to \cite{Fr}. The locally trivial deformations embed into the tangent space of all deformations, $H^1(T_X)$. Any nontrivial deformation gives the section of $H^0(T_X^1)$. Now, assuming Kulikov's topological condition (see lemma \ref{lem:Kul-c1}) we get that either $h^0(T_X^1) = 1$ if $T_X^1$ is trivial, or $h^0(T_X^1) = 0$ if it is nontrivial, yet trivial topologically. In the latter case, any deformation is locally trivial (which improves already known fact that there are no smoothings in this case). In the former case, the mapping from $H^1(T_X)$ to $H^0(T_X^1)$ could still be zero. This, however, can be avoided in case $H^2(T_X^0) = 0$ (a natural assumption corresponding to the idea that $X$, as a gluing of components, is not ''overdetermined'', so it has non-obstructed locally trivial deformations). In that case, we will have $h^1(T_X) = h^1(T_X^0) + 1$, and the Kuranishi map $K$ is a formal map of the form \[K:H^1(T_X) \rightarrow H^1(T_X^1)\] Later, the very explicit description of this map will be given. Motivated by this, we from now on assume the following:
\end{note}
\begin{assumption}\label{as:local-unobstructed}$H^2(T_X^0) = 0$\end{assumption}
In particular, this assumption implies that the locally trivial deformations of $X$ are unobstructed.
Now, we would like to describe the Kuranishi mapping. Our main source of knowledge here is the discussion in the Chapter 4 in \cite{Fr}.
According to \cite{P} $T_X$ forms a sheaf of dg-algebras. The $[\cdot,\cdot]$ descends on cohomology, which gives the quadratic mapping
\[K_2: H^1(T_X) \rightarrow H^2(T_X)\]
\[K_2(\alpha) = [\alpha, \alpha]\]
By the general deformation theory it is the second term of the Kuranishi mapping $K = \frac{1}{2}K_2 + \frac{1}{6}K_3 + \frac{1}{24}K_4 + ...$
Now, in our case the cohomologies of $T_X$ have a filtration: \[0 \rightarrow H^*(T_X^0) \rightarrow H^*(T_X) \rightarrow H^*(T_X^1) \rightarrow 0 \]
This filtration is in agreement with the commutator - that is, $H^*(T_X^0 \oplus T_X^1[1])$ is an associated graded algebra of $H^*(T_X)$.
Moreover, the commutator in this subalgebra has the following explicit description:
for $a, b \in H^*(T_X^0):$ \[[a,b] \text{ comes from the commutator of vector fields}\]
for $a \in H^*(T_X^0), b \in H^*(T_X^1[1]):$ \[ [a,b] = L_a(b)\] comes from the action of vector fields on all natural objects.
for $a, b \in H^*(T_X^1[1])$: \[[a,b] = 0\] for obvious grading reasons.
Now, suppose $X$ had a log structure. Then, denote by $\xi \in H^0(T_X) = H^1(T_X^1[1])$ the corresponding trivializing section of the line bundle $T_X^1$ (this is the same as $\alpha^*$, but we use it in the cohomological calculations, and $\alpha^*$ in the calculations with local coordinates).
\begin{definition}\label{def:tx-log}
The sheaf $T_X(\log) \subset T_X$ is a subsheaf of all vector fields $v$ which preserve log structure: i.e. \[[v, \xi] = 0\]
It is seen by direct local inspection that this sheaf is locally free (and $T_X$ is not).
\end{definition}
It is readily seen (again, verified locally) that $T_X(\log)$ admits the following resolution:
\begin{equation}\label{eq:tx-log-resolved}0 \rightarrow T_X(\log) \overset{i}{\rightarrow} T_X^0 \overset{\ad_{\xi}}{\rightarrow} T_X^1 \rightarrow 0\end{equation}
\begin{lemma} $\ad_\xi$ is actually a morphism of \textbf{coherent} sheaves i.e. it is linear w.r.t. multiplication by functions from the ground ring. \end{lemma}
Statement of this lemma follows from proposition 4.3 of \cite{Fr}.
\begin{proof}
In the local chart $U$ with structure functions $u_1, ..., u_n$ we act by a vector field $v$. Its infinitesimal action is \[u_i \mapsto u_i + \epsilon \partial_v(u_i)\]
The change functions to this new set of structure functions, so, are \[z_i = 1 + \epsilon \frac{\partial_v(u_i)}{u_i}\] (here, note that $\partial_v(u_i)$ is ought to vanish on the same component $u_i$ does due to $v$ being a derivation, so $z_i$ are well defined on $\Sing(U)$). So, \[L_{v}(\alpha^*) = -(\underset{i=1}{\overset{n}{\sum}} \frac{\partial_v u_i}{u_i})\alpha^*\]
This map is clearly linear w.r.t. the multiplication of $v$ by functions, which means that $\ad_{\xi}$ is linear.
\end{proof}
Provided $X$ has a dimension $2$ and assumption \ref{as:local-unobstructed} holds, we have \newline $H^{*}(T_X(\log))$ (non-canonically) equal to the cohomology of $\ad_{\xi}$ acting on $H^*(T_X^0 \oplus T_X^1[1])$ (higher terms of spectral sequence vanish for dimensional reasons).
\begin{note} There are actually two approaches to the smoothing. Initial approach, due to Friedman, was only about dimension two, and assumption \ref{as:local-unobstructed} held trivially, so it was mainly concerned about the action of $\ad_{\xi}$ on $H^*(T_X^0 \oplus T_X^1[1])$. Latter approach, due to Kawamata and Namikawa worked in arbitrary dimension and was mostly concerned about logarithmic cohomology, but used Calabi-Yau condition, which allowed them to smooth out varieties of higher dimension. This approach was later generalized to (generalized) Fano varieties by Tziolas \cite{Tz}. We are neither in Fano, nor Calabi-Yau case - so we choose the former, earlier approach to analyze our problem.
\end{note}
From now on, we assume the dimension $2$ and assumption \ref{as:local-unobstructed}.
\begin{theorem} \label{thm:log-def} Suppose \[\ad_{\xi}: H^1(T_X^0) \rightarrow H^1(T_X^1) = H^2(T_X^1[1])\] is surjective (or, equivalently, $H^2(T_X(\log)) = 0$) Then, there are smoothing deformations of $X$. \end{theorem}
This is a Corollary 2.4. from \cite{KN}. We now give the geometric self-contained proof for the reader's convenience.
\begin{proof}Consider the quadratic map \[K_2: H^1(T_X) \rightarrow H^2(T_X)\] Choose the non-canonical splitting $H^1(T_X) = H^1(T_X^0) \oplus (\mathbb{C} = H^0(T_X^1))$. Denote the coordinates on $H^1(T_X^0)$ as $x_1, ..., x_d$ and the coordinate along $H^0(T_X^1)$ as $t$. Then, \[K_2(x_1, ..., x_n, t) = (\sum_i \ell_i x_i)t + r t^2, \ell_i, r \in H^1(T_X^1)\] - there are no quadratic terms in $x$ due to $[\cdot,\cdot]$ being in agreement with filtration. Moreover, the linear map $\sum_i \ell_i x_i: H^1(T_X^0) \rightarrow H^1(T_X^1)$ is precisely $\ad_{\xi}$. Now, note that one can force $r=0$ by the linear change of coordinates, due to our assumption that $\ad_{\xi}$ is surjective. Assume now w.l.o.g that it is the case.
Now, note that the whole series $K$ is actually divisble by $t$, either by noting that Massey products are also in agreement with the filtration or noting that the locus $t=0$ is a locus of locally-trivial deformations, which are unobstructed by assumption \ref{as:local-unobstructed} and so it lies in the formal scheme $K=0$.
Then, the series $K/t = \ad_{\xi} + ...$ start with the linear term which is a surjective linear mapping, so define a smooth formal subvariety by the formal implicit function theorem. This subvariety is also clearly transverse to the locus of locally trivial deformations $t=0$. This implies existence of the formal smoothing deformation, and to make it non-formal apply Artin's approximation theorem (see [KN] for details).
\end{proof}
Our main concern for the rest of this section is developing methods to calculate $H^*(T_X(\log))$. Our approach is geometric in nature - we interpret $T_X(\log)$ as a sheaf controlling deformation theory of a certain object, and then deem to consider the corresponding moduli space by other, geometric, means.
\begin{theorem}[Friedman, {\cite{Fr}}, Thm. 4.5] \label{thm:txlog}
$H^1(T_X(\log))$ is a tangent space to the locally trivial deformations which preserve $d$-semistability.
\end{theorem}
\begin{proof}
Consider the covering by log atlas $(U^1, ..., U^s)$, with corresponding structure functions, $\alpha$ - corresponding nonvanishing section of $(T_X^1)^*$, $v^{ij} \in T_X (U^i \cap U^j)$ - Cech cocycle. The Cech cocycle deforming $T_X^1$ is then $\varphi^{ij} = L_{v^{ij}} (\alpha) / \alpha$, which is trivial iff $[v, \xi] = 0$
\end{proof}
\begin{assumption}
From now on, $\Sing(X)$ only has rational irreducible components.
\end{assumption}
\begin{definition}
Denote as $M_X$ the space of locally trivial deformations of $X$. Denote as $J_X$ the space of line bundles on $\Sing(X)$ with $c_1 = 0$. Denote by $O: M_X \rightarrow J_X$ the map which sends $X$ to $T_X^1$.
\end{definition}
\begin{note}
While $\Sing(X)$ could indeed deform, $J_X$ is always $(\mathbb{C}^*)^{h^1(\Sing(X))}$, identified canonically. See Lemma \ref{lem:glue} for details.
\end{note}
\begin{corollary} \label{cor:moduli-smooth}
If the fiber $O^{-1}(\mathcal{O}_{\Sing(X)})$ is smooth in $X$, there exists a smoothing of $X$.
\end{corollary}
\begin{proof}
Inspecting the proof of the Theorem \ref{thm:txlog} we see that $\ad_{\xi}$ is a differential of the map $O$ in the point $X$. Fiber being smooth implies surjectivity of this differential, which, in turn, implies existence of smoothing by Theorem \ref{thm:log-def}.
\end{proof}
Our approach to calculation of $H^*(T_X(\log))$ is now the following: describe $M_X$ as explicitly as we can in terms of gluing of components and calculate $O$.
Summarising this section: results above follow easily from the results of Friedman and Kawamata-Namikawa, Corollary \ref{cor:moduli-smooth} is used (in some form) by Friedman. The references are for SNC case, however actual proofs do work for NC case just as well - at the cost of a bit more confusing notation. We tried, however, to present our own point of view, based largely on geometric consideration of the Kuranishi mapping. We conclude the section with additional theorem, an improvement of Corollary \ref{cor:moduli-smooth} which seems to be not known in the log deformation theory. This fact is not needed in the current work but might find its applications later.
\begin{theorem}\label{thm:cool-smooth}
Suppose $O^{-1}(\mathcal{O}_{\Sing(X)})$ has the dimension $\dim(M_X)-\dim(J_X) = h^1(T_X^0) - h^1(T_X^1)$, yet not necessary smooth or even reduced. There still exists the surface $X'$ with $\Delta_{X'}$ homeomorphic to $\Delta_X$, admitting the smoothing deformation.
\end{theorem}
Note, that in standard deformation theory, it is enough to prove that $h^1(T_X) > h^2(T_X)$ to guarantee the existence of deformations. In the logarithmic deformation theory, though, it is not the case. This is, basically, our version of obstructed deformation theory. Curiously, the proof uses almost no log deformations and is mainly about the geometry of the Kuranishi space.
\begin{proof} Assume the notation from the proof of Thm. \ref{thm:txlog}, i.e., choose the non-canonical splitting of $H^1(T_X^0) \rightarrow H^1(T_X) \rightarrow H^0(T_X^1) = \mathbb{C}$. Pick the coordinate $t$ to be the pullback of the standard coordinate on $H^0(T_X^1)$ dual to $\xi$, and $(x_1, ..., x_d)$ - the rest of coordinates, defining splitting. We now fix our attention to an open neighborhood of the zero in which Kuranishi mapping converges.
Denote the analytic subset $K=0$ as $Z$, and the hyperplane $t=0$, which is an image of $H^1(T_X^0)$ as $H$.
It is easy to see that $H \subset Z$ - that is true because we have demanded in Assumption \ref{as:local-unobstructed} that locally trivial deformations are unobstructed, and $H^1(T_X^0)$ is the space of local deformations.
That implies that $t|K$ - because $K$ is the defining equation of $Z$, and $t$ is prime. Denote as $Z'$ the analytic subset $K/t = 0$.
Now, the set $Z' \cap H$ is the set of locally trivial deformations that preserve $d$-semistability. We have done it in the smooth case by calculation in Theorem \ref{thm:txlog}, but it can be proved geometrically, too. $Z' \cap H$ is clearly the same as $\Sing(Z) \cap H$, i.e., it is the subscheme of such points $p \in H$ that have tangent space bigger than $T_pH$. However, being $d$-semistable exactly means having $H^1(T_X) \neq H^1(T_X^0)$.
\begin{center}
\includegraphics[clip, width=0.7\linewidth]{kuranishi.jpg}
\end{center}
Now, we proceed with dimension counting. Denote $h^1(T_X^0) = d$, $h^1(T_X) = d+1$, $h^2(T_X) = h^1(T_X^1) = r$.
$Z'$, being a fiber of the map $K/t: H^1(T_X) \rightarrow H^2(T_X)$, has $\dim(Z') \geq d+1-r$. By assumption of the theorem, $\dim(Z' \cap H) = d-r$. It implies that $Z' \setminus H$ is non-empty, moreover, we have an arc $\gamma: D \rightarrow Z'$ which is not contained in $H$ and $\gamma(0) = X$. It also implies $\dim(Z') = d+1-r$. Denote the coordinate on a disk $\tau$.
This $\gamma$ corresponds to some flat family $\tilde{X}$ with central fiber $X$, which, however, does not necessarily have the smooth total space.
Suppose $\gamma$ has tangency $k$ with $H$. Then, we would like to consider $k$-th Kodaira-Spencer differential $\partial_{\tau}^k (t \circ \gamma) (d\tau)^k = \lambda \xi (d\tau)^k$. Let's see that it is defined correctly.
Explicitly, consider the $(k-1)$-st order of deformation \[ \tilde{X}^{(k-1)} = \tilde{X} \underset{D}{\times} \Spec(\mathbb{C}[\tau]/\tau^k)\] It is still locally trivial at this stage, so one can promote $\alpha^*$ to it (it is a choice, we choose to do it in such a way that $\xi$ is dual to $dt$ in $H^1(T_X')$ for any locally trivial deformation $X'$, and our choice of $t$ was arbitrary). Then, possible liftings of $\tilde{X}^{(k-1)}$ to the family over $\mathbb{C}[\tau]/\tau^{k+1}$ are a torsor over $H^1(T_{\tilde{X}^{(k-1)}})$, and this torsor has a distinguished codimension $1$ subspace - the subspace of locally trivial liftings. The quotient by this subspace is, then, canonically $H^0(T_{\tilde{X}^(k-1)}^1) = \mathbb{C}$, which provides us with the desired mapping - so, as expected, $k$-th order Kodaira-Spencer derivative is not canonical, yet its projection which measures the deviation from the locally trivial deformation is.
Now, consider the same family in the neighborhood of a singular point on $X$, denote its germ as $\mathcal{X}$. The deformations of the germ are controlled by $\Gamma(T_{\mathcal{X}}^1)$, and indeed the deformation is locally trivial up to the $k$-th order, so it has $k$-th Kodaira-Spencer differential, which is equal to $\lambda \alpha^*(d\tau)^k$.
That means that the deformation locally has the form $x_1 ... x_s = \tau^k$, here $s = 2, 3$, which implies, first, that the generic fiber is indeed smooth, and, second, that the singularities of the family are toroidal. By the results of \cite{dFKX} we can resolve them and apply semistable reduction theorem without changing the homeomorphism type of the dual complex.
\end{proof}
\section{Example}
We present our example. Let us recall that we are interested in constructing smoothable geometrically maximally degenerate normal crossing surface $X$ with non-contractible, collapsible dual complex. We decided to use the simplest example of such complex, topological duncehat (see Def.\ref{def:dunce}).
That means that $X$ is some surface with one irreducible component, self-intersecting in one curve, and having one triple point. Consider the normalization $\nu:\hat{X}\rightarrow X$ and the gluing locus $\nu^{-1}(\Sing(X))$. Due to the absence of branch switching (which is due to $\Sing(X)$ having rational normalization) the gluing locus is a union of two distinct rational curves, which we denote $P, Q \subset \hat{X}$. Their normalizations are denoted $\hat{P}, \hat{Q} \overset{\nu_P, \nu_Q}{\longrightarrow} P, Q$ respectively. These curves only have nodal intersections and self intersections and are glued by some mapping $\varphi: \hat{P} \rightarrow \hat{Q}$. It is also clear that the gluing locus $P \cup Q$ should have three nodal points in total - as they glue by three into one triple point on $X$.
That leaves two possibilities for the combinatorial structure of $X$:
\begin{itemize}
\item \textbf{Wrong case} $P$ and $Q$ are two smooth curves intersecting each other in three nodes.
\item \textbf{Right case} $P$ and $Q$ both have one node and intersect each other once.
\end{itemize}
It turns out that the first case the dual complex is actually the different triangulated space, the triangle $[012]$ with edges $[01], [12]$ and $[20]$ glued together. It has a fundamental group $\mathbb{Z}/3\mathbb{Z}$, and is not too interesting to us (while our program can, indeed, be carried out for it).
The second case, however, is what we are interested in. Denote as $p_N, q_N$ the nodes of $P, Q$, as $n$ the intersection of $P$ and $Q$, as $p_1, p_2$ - preimages of $p_N$ on $\hat{P}$ (ordered arbitrarily), as $q_1, q_2$ - preimages of $q_N$ on $\hat{Q}$, as $p_3, q_3$ - preimages of $n$ on $\hat{P}, \hat{Q}$ - respectively,
Now, recall that $\hat{P}, \hat{Q}$ are genus $0$, so the map $\varphi$ can be defined by images of three points. We now define \[\varphi(p_1) = q_2\]
\begin{equation}\label{eq:phi-def} \varphi(p_2) = q_3\end{equation}\[\varphi(p_3) = q_1\]
Let us explain why this gluing indeed has $\Delta_X$ as its dual complex. Let us denote $b_{P;1}, b_{P;2}$ the branches of $P$ passing through $p_N$ which have $p_1, p_2$ respectively on their lift to $\hat{P}$, and do similarly for $b_{Q;1}, b_{Q;2}$. Denote as $b_{P; n}$, $b_{Q; n}$ the branches of $P$ and $Q$ passing through $n$.
\newpage
Then, it is readily seen that the neighborhoods of $p_N, q_N, n$ glue into the neighborhood of the coordinate cross in $\mathbb{C}^3$ - $b_{P;1}$ glues to $b_{Q;2}$ which passes through $b_{Q;1}$ which glues to $b_{P;n}$ which passes through $b_{Q;n}$ which glues to $b_{P;2}$ which passes through $b_{P;1}$ (see picture below).
\begin{center}
\includegraphics[clip, width=0.5\linewidth]{glue.jpg}
\end{center}
Now, the edges of the triangle in $\Delta_X$ correspond to the branches of $\Sing(X)$ passing through the triple point (namely, $b_{P;1} = b_{Q;2}, b_{Q;1} = b_{P;n}$, $b_{Q;n} = b_{P;2}$).
Vertices of the triangle correspond to the branches of $\hat{X}$ passing through the triple point, let us denote $[0]$ the branch passing through $p_N$, $[1]$ - passing through $q_N$, $[2]$ - branch passing through $n$.
The edge orientation space is the ordering of branches of $\hat{X}$ passing through the curve corresponding to the edge. Then, set that the edges are oriented from the branch passing through $P$ to the one passing through $Q$. This convention gives the following orientation of the edges: $[01], [21], [02]$. Up to the permutation, it gives the gluing of the duncehat complex. (see picture below)
\begin{center}
\includegraphics[clip, width=0.5\linewidth]{dual.jpg}
\end{center}
Now, we would like to find appropriate surface and curves on it with combinatorics described above. Another thing to look for is Kulikov's triple point formula, which states that
\begin{proposition}
For $\nu_C: C \rightarrow \Sing(X)$ any normalization of the irreducible component of $\Sing(X)$ we have:
\[c_1(\nu_C^*T_X^1) = c_1(N(C; B_1)) + c_1(N(C;B_2)) + \tau\]
where $B_1, B_2$ - branches of $X$ passing through $C$, $\tau$ - amount of triple points on $C$.
\end{proposition}
Kulikov's condition then states that aforementioned quantity is zero, and is a topological requirement for $d$-semistability (see Lemma \ref{lem:Kul-c1})
\begin{definition}
Consider a projective plane $\mathbb{P}^2$ with a pair of nodal cubics (denote them $P'$, $Q'$). Suppose these cubics do not pass through each other's nodes. Blow up $\mathbb{P}^2$ in $8$ of $9$ of their intersections (if $P', Q'$ are not tranverse, we blow up multiple intersections multiple times). Additionally, blow up any smooth point on the proper preimage of $P'$ except the ninth, remaining intersection. Denote the resulting blow up as $\hat{X}$, the proper preimages of $P'$ and $Q'$ as $P$ and $Q$. Make the choices of $p_1, p_2$ and $q_1, q_2$. The data above is called the \textbf{Main Construct}.
\end{definition}
The rest of proof is concerned with the space of moduli of these constructs and the mapping $O$.
\begin{proposition} Kulikov's triple point condition is satisfied for the main construct.
\end{proposition}
\begin{proof}
The (proper) self-intersection of the nodal cubic is $7$. Every blowup lowers both self-intersections by $1$, and one additional blowup makes the self intersection of $P$ to be $-2$, and self-intersection of $Q$ to be $-1$. Adding the amount of triple points on the normalization of $\Sing(X)$, which is $3$, we obtain $0$, as predicted by Kulikov's condition.
\end{proof}
\section{Obstruction mapping}
We now would like to understand the obstruction mapping. We need a few technical preparations.
\subsection{Setup}
At first, let us state precisely what are the spaces $M_X$ and $J_X$ are.
\begin{definition}
$M_X$ is the moduli space of constructs, i.e. moduli space of pairs of nodal cubics $P$, $Q$ on $\mathbb{P}^2$ (we have called them $P'$, $Q'$ in previous section, but drop the superscript from now on as it is unlikely to cause confusion).
These cubics are subject to following condiitons:
\begin{itemize}
\item $P$ and $Q$ are non-degenerate nodal cubics - so, they do not develop cusp or do not become reducible.
\item $P$ doesn't pass through the node of $Q$ and vice versa.
\end{itemize}
Also, these cubics are subject to following additional choices:
\begin{itemize}
\item $\nu_P^{-1}p_N$ and $\nu_Q^{-1}q_N$ are ordered, them in order are denoted $(p_1, p_2)$, $(q_1, q_2)$.
\item An intersection point $n \in P \cap Q$ is chosen.
\item An additional point $b \in P$, satisfying $b \neq n, p_N$, is chosen.
\end{itemize}
These objects are considered up to the projective plane transformations.
\end{definition}
\begin{proposition}
$\dim(M_X) = 9$
\end{proposition}
\begin{proof}
Each cubic has $8$ parameters. Projective transformations also subtract $8$, and one additional is for an additional point $b$.
\end{proof}
\begin{definition}
$J_X$ is the space of line bundles with $c_1 = 0$ on \newline $C = \mathbb{P}^1/(0 \sim 1 \sim \infty)$.
\end{definition}
\begin{proposition}
$\dim(J_X) = 2$. It is canonically equivalent to the two-dimensional torus $(\mathbb{C}^*)^3 / (\mathbb{C}^*)$ (quotient by the subgroup generated by $(1;1;1)$).
\end{proposition}
\begin{proof}
Consider the line bundle $L$ over $C$. $\nu_C^{*}L$ is a trivial bundle on $\mathbb{P}^1$, consider its trivializing section $\kappa$. Then, $(\kappa(0), \kappa(1), \kappa(\infty))$ is a defined up to the multiplication by constant triple of elements in the $1$-dimensional space $L(0 \sim 1 \sim \infty)$. It gives the desired element in $(\mathbb{C}^*)^3 / (\mathbb{C}^*)$.
The fact that the described mapping is an isomorphism is also fairly obvious (see lemma \ref{lem:glue} further for details).
\end{proof}
\subsection{Line bundles on multicomponent curves}
Suppose $R$ is a curve with triple points only, and with rational components only. Denote by $\mathcal{S}=(R_1, ..., R_k)$ the set of normalizations of components of $\nu_i: R_i \rightarrow R$, and by $\mathcal{T} = (t_1, ..., t_s)$ the set of triple points.
Then, there is a bipartite graph, $\mathcal{S} \overset{\Gamma}{\leftrightarrow} \mathcal{T}$, connecting each component with triple points on it. We allow multiple edges - in case a component passes through a triple point multiple times.
For each point $t_i \in \mathcal{T}$ form the group $G_i$ = $(\mathbb{C}^*)^3$, with factors of the product labeled by edges of $\Gamma$ coming from $t$, and denote by $\delta_i: \mathbb{C}^* \rightarrow G_i$ the embedding of the diagonal group (generated by $(1;1;1)$). Denote $G_i/\im(\delta_i) = \tilde{G}_i$
For each component $R_i$ form the group $H_i = \mathbb{C}^*$.
There is a natural map $q: \prod H_i \rightarrow \prod G_j$ which maps each $H_i$ to all $G_j$'s such that $(ij)$ is an edge of $\Gamma$, to the component(s) of $G_j$ corresponding to the edge(s) $(ij)$.
\begin{lemma}[gluing lemma]\label{lem:glue}
The space of line bundles with $c_1 = 0$ on $R$ is canonically $\prod \tilde{G}_i / \im(q)$. The group law is also in accord with this description. We will refer to the elements of this group as \textbf{gluing data}.
\end{lemma}
\begin{proof}
Suppose $L$ is a line bundle over $R$, and pick $\gamma_1, ..., \gamma_k$ - sections of $L_i = \nu^*_i(L)$. Then, in every fiber of $L$ over a triple point $t_j$ we obtain values of three sections - which gives an element in $\tilde{G}_j$. The choice of $\gamma_i$'s was up to a constant, which gives the quotient by $q$.
Moreover, it is clear that any line bundle can be obtained in this way - given an element in $\prod G_j$ we can glue the fibers of $L_i$ together using this data in an obvious way, the group law check is also straightforward.
\end{proof}
\subsection{Explicit description of $T_X^1$}
\begin{lemma} \label{lem:tx1-explicit}
Suppose $X$ is a normal crossing surface without branch switching, $\nu: C \rightarrow \Sing(X)$ - normalization of some irreducible component of the singular locus, $t_1, ..., t_k \in C$ - preimages of the triple points, $B_1, B_2$ - branches of $X$ passing through $C$.
Then, $\nu^*(T_X^1) = N_{C; B_1} \otimes N_{C; B_2} ([t_1] + ... + [t_k])$.
Suppose $t \in X$ - a triple point, $C_1, C_2, C_3$ - branches of $\Sing(X)$ passing through $t$, $B_{ij}$ - branches of $X$ in $t$ passing through $C_i, C_j$.
Then, the fiber of $N_{C_1; B_{12}} \otimes N_{C_1; B_{13}} ([t])$ in $t$ is canonically identified with $T_{t; C_1} \otimes T_{t; C_2} \otimes T_{t;C_3}$, and this identification gives the gluing of $T_X^1$ between the components.
\end{lemma}
\begin{proof}
It is easier to work with the dual bundle, $(T_X^1)^* = \Hom(T_X^1, \mathcal{O}_{\Sing(X)})$. For the log chart $(u_1, u_2, u_3)$ near the triple point $t$ we can form the object $\omega_1 = u_1 du_2 \otimes du_3$. Provided $C_1$ is a component on which $u_2 = u_3 = 0$, $\omega_1$ is a section of $N^*_{C_1; B_{12}} \otimes N^*_{C_1; B_{13}}([-t])$. It easy to check that equivalent log charts give the same $\omega_1$.
Now, in the same way log structure gives the sections $\omega_2$, $\omega_3$ on the branches $C_2$, $C_3$, which gives the gluing between fibers in $t$ - $N^*_{C_1; B_{12}} \simeq T_{C_2}^*$, $N^*_{C_1, B_{13}} \simeq T_{C_3}^*$, $\mathcal{O}_{C_1}(-[t]) \simeq T_{C_1}^*$. That identifies the fiber of $(T_X^1)^*$ in $t$ with $T^*_{C_1} \otimes T^*_{C_2} \otimes T^*_{C_3}$ canonically.
\end{proof}
\begin{corollary}\label{cor:tx1-explicit}
Given $X$ geometrically maximally degenerate, satisfying Kulikov's triple point condition, pick for every component $\nu_i: C^i \rightarrow \Sing(X)$ with branches $B^i_1$, $B^i_2$ the pair of conormal forms $\omega^i_1 \in N^*_{C^i, B^i_1}, \omega^j_2 \in N^*_{C^i, B^i_2}$ such that $\omega^i_1 \otimes \omega^i_2$ vanishes in all triple points on $C^i$.
Then, gluing data is obtained as differential of $\omega^i_1 \otimes \omega^i_2$ along $C^i$.
\end{corollary}
\subsection{Calculation in the example} Before the start of the computation, we must encourage the reader. The computation which will shortly follow looks very tedious on first sight, however most of the complexity comes from the fact that we need to somehow trivialize $1$-dimensional tangent spaces to the points $(p,q)_{(1,2,3)}$. There are different such trivializations, and different tensors look better in different trivializations. It leads to a lot of multiplicative factors depending on how these trivializations relate, and this dependence is contained in two parameters - the positions of flex points of the cubics in the projective coordinates given by points $p_1, p_2, p_3$. We will later promptly ignore such multiplicative factors and obtain a nice answer modulo them - and it will be enough to prove the desired result.
So, on the promise of the final answer being nice modulo factors we can ignore, we proceed with calculation of the gluing data of the main construct:
We assume the following convention: while calculations will take place on $\mathbb{P}^2$, the calculation of all divisors will take place on $\hat{X}$ directly.
Denote as $\Omega_P, \Omega_Q$ the sections of $\omega_{\mathbb{P}^2}$ such that $\frac{1}{\Omega_P}$ vanishes in $P$ and similarly for the other one. They are chosen uniquely up to a constant - which we will fix in a specific way.
Denote as $\alpha_P$, $\alpha_Q$ Poincare residues of these forms - these are $1$-forms on $\hat{P}$ and $\hat{Q}$, with residues in $p_1, p_2$ and $q_1, q_2$ respectively.
We choose $\Omega_P$ and $\Omega_Q$ in such a way that the $\Res_{p_1}(\alpha_P) = \Res_{q_1}(\alpha_Q) = 1$.
Denote as $\tau_p$ the projective invariant coordinate on $\hat{P}$ such that \[\tau_P(p_1) = 0\]\begin{equation} \tau_P(p_2) = \infty\end{equation} \[\tau_P(\psi_P) = 1\] where $\psi_P$ is a flex point of $P$. Do the same for $Q$.
Denote \begin{equation}n_P = \tau_P(p_3), n_Q = \tau_Q (q_3)\end{equation}
Pick some arbitrary vector fields $v_P, v_Q$ defined only in terms of these coordinates. Specifically, we use
\begin{equation}v_P = (\tau_P-1)^2 \frac{\partial}{\partial \tau_P}\end{equation} and the same for $Q$. The only requirement we have is that these fields are nonzero in $(p,q)_{1,2,3}$ provided position is generic enough.
Pick also some arbitrary affine chart, generic enough, and denote its infinity as $H$, and its volume form as $\Omega$. Define affine equations of $P$ and $Q$ as follows:
\begin{equation}
f_{P,Q} = \frac{\Omega}{\Omega_{P,Q}}
\end{equation}
We would also like to pick some functions which vanishes in our points of interest. We suggest following: \begin{equation}s = n_P^2\frac{\tau_P - n_P}{(\tau_P - 2n_P)^3}\end{equation}
This function is chosen in such a way that it has the same formula in the coordinate defined by $p_1, p_2, p_3$, independently on $n_P$.
Finally, we pick the sections of the conormal bundle to $P$ and $Q$:
\begin{equation}\label{eq:beta}
\beta_P = i_{v_P} \Omega, \beta_Q = i_{v_Q} \Omega
\end{equation}
From now on, we identify the curve $\hat{Q}$ with $\hat{P}$ via the gluing map $\varphi$. Finally, we consider the following meromorphic section of $T_X^1$ (pulling back objects from $Q$ via the gluing map): \begin{equation}\eta = s \beta_P \otimes \beta_Q \end{equation}
\begin{proposition}
$\eta$ vanishes once in $p_1$, $p_2$, $p_3$ for $P, Q$ in generic position.
\end{proposition}
\begin{proof}Direct check.\end{proof}
Now, denote \[\lambda_2 = \frac{v_P(p_1)}{\varphi^*(v_Q (q_2))}\]\begin{equation}
\lambda_3 = \frac{v_P(p_2)}{\varphi^*(v_Q (q_3))}
\end{equation}\[\lambda_1 = \frac{v_P(p_3)}{\varphi^*(v_Q (q_1))}\]
\begin{proposition}
The gluing data for the section $\eta$ is as follows:
(along the branch $p_1$)
\begin{equation} \label{eq:main1}
\frac{\partial s}{\partial v_P}(p_1) \times \langle \beta_P (p_1), v_P(p_2)\rangle \times \langle\beta_Q (q_2), \lambda_1 v_Q(q_1)\rangle =
\end{equation}
\[ \lambda_1 \frac{\partial s}{\partial v_P}(p_1) \times \Omega(v_P(p_1), v_P(p_2)) \times \Omega(v_Q(q_2), v_Q(q_1))
\]
(along the branch $p_2$)
\begin{equation}\label{eq:main2}
\frac{\partial s}{\partial v_P}(p_2) \times \langle \beta_P (p_2), v_P(p_1)\rangle \times \langle\beta_Q (q_3), v_P(p_3)\rangle =
\end{equation}
\[\frac{\partial s}{\partial v_P}(p_2) \times \Omega(v_P(p_2), v_P(p_1)) \times \Omega(v_Q(q_3), v_P(p_3))
\]
(along the branch $p_3$)
\begin{equation}\label{eq:main3}
\frac{\partial s}{\partial v_P}(p_3) \times \langle \beta_P(p_3), \lambda_3 v_Q(q_3) \rangle \times \langle \beta_Q(q_1), \lambda_2 v_Q(q_2) \rangle =
\end{equation}
\[\lambda_2 \lambda_3 \frac{\partial s}{\partial v_P}(p_3) \times \Omega(v_P(p_3), v_Q(q_3)) \times \Omega(v_Q(q_1), v_Q(q_2))
\]
\end{proposition}
\begin{proof}
We use the vector field $v_P$ to trivialize the tangent spaces $T_{p_1}, T_{p_2}, T_{p_3}$ (remark that $\lambda_{i+1} v_Q(q_{i+1})$ is the same vector as $v_P(p_i)$). Then, we just calculate the value of the section in this trivialization. Finally, we use the definition of $\beta$'s, see eq.\ref{eq:beta}.
\end{proof}
\begin{lemma} \label{lem:n_p-dependance}
$\lambda$'s and $\frac{\partial s}{\partial v_P}(p_i)$'s only depend on the values of $n_P$ and $n_Q$.
\end{lemma}
\begin{proof}
For $\frac{\partial s}{\partial v_P}(p_i)$ it is evident from the definitions. For $\lambda$'s, the reason is following - pick the coordinate $n_P^{-1} \tau_P$, i.e., such that $p_3 = 1$, $p_1 = 0$ $p_2 = \infty$. The positions of flex points of $P$ and $Q$ are functions of $n_P$ and $n_Q$, so the fields $v_P$ and $v_Q$ in this coordinate depend only on the aforementioned entities.
\end{proof}
Now, this gluing data is by no means final, because the section $\eta$ was actually only rational - it had zeroes and poles away from the set $p_1, p_2, p_3$. Now we need to find a function $r$ such that $r\eta$ has all these zeroes and poles cancelled - and multiply the gluing data obtained by $(r(p_1), r(p_2), r(p_3))$.
Split the divisor of the function $s$ into positive and negative parts:
\begin{equation}D(s) = D(s)_+ + D(s)_-\end{equation}
\[D(s)_+ = [p_1] + [p_2] + [p_3]\]
Perform the calculation of the divisor $D(\eta) - D(s)_+$:
\begin{equation}
D(\eta) - D(s)_+ = D(s)_- + D(\beta_P) + D(\varphi^*\beta_Q)
\end{equation}
\begin{equation}
D(\beta_P) = D(v_P) + D(\Omega|_P) =
\end{equation}
\[D(v_P) - 3[H|_P] + ([b] + [Q|_P] - [p_3])\]
Here, in brackets we put part of the divisor which occurs due to the change of $\Omega$ under blow-up. Similarly, for $Q$:
\begin{equation}
D(\beta_Q) = D(v_Q) + D(\Omega|_Q)) =
\end{equation}
\[D(v_Q) - 3[H|_Q] + ([P|_Q] - [q_3])\]
Denote \begin{equation}s_b =\frac{\tau_P - 1}{\tau_P - \tau_P(b)}\end{equation}
\begin{lemma} \label{lem:eta'-independent}
The divisor of the section \[\eta' = (f^{-1}_Q)|_P \cdot \varphi^*( (f^{-1}_P)|_Q) \cdot s_b \cdot \eta\]
is dependent only on $n_P, n_Q$
\end{lemma}
\begin{proof}
From the calculations above this divisor is \[D(\eta') = D(v_P) + \varphi^{-1}(D(v_Q)) - [p_3] - [p_2] + D(s) + [\psi]\]
$D(v_P)$ is immobile, $\varphi^{-1}(D(v_Q))$ depends only on relative position of flex points, i.e. on $n_P$ and $n_Q$, $D(s)$ is immobile.
\end{proof}
Pick a function $g = \frac{\tau_P - n_P}{(\tau_P - 1)^2}$. It is dependent only on $n_P$ and has a divisor \begin{equation}D(g) = [p_2] + [p_3] - 2[\psi_P]\end{equation}
\begin{corollary} \label{cor:geta'}
The section $g \eta'$ has a divisor dependent only on $n_P$, and vanishes exactly once in $p_{1,2,3}$
\end{corollary}
Now, finally, we can pick a function $h$ which will cancel this additional part - it will depend only on $n_P, n_Q$ and be defined up to a constant. In what follows, we will not care about the dependance of the gluing data on $n_P$ and $n_Q$, so there is no point in finding this function precisely.
To finalize the answer, we would like to calculate
\[g \cdot \frac{1}{f_Q|_{P}} \cdot \varphi^*( \frac{1}{f_P|_{Q}}) (p_1)
\]
\begin{equation}
g \cdot \frac{1}{f_Q|_{P}} \cdot \varphi^*( \frac{1}{f_P|_{Q}}) (p_2)
\end{equation}
\[g \cdot \frac{1}{f_Q|_{P}} \cdot \varphi^*( \frac{1}{f_P|_{Q}}) (p_3)
\]
Let us proceed. From now on, we will use the sign ''$\sim$'' to say that two values are related by something dependent only on $n_P$ and $n_Q$.
\begin{equation} \label{eq:ans1} g \cdot \frac{1}{f_Q|_{P}} \cdot \varphi^*( \frac{1}{f_P|_{Q}}) (p_1) = g(p_1) \cdot \frac{1}{f_Q(p_N) f_P(q_N)} \sim \frac{1}{f_Q(p_N) f_P(q_N)}
\end{equation}
It is harder for two other ones:
\begin{equation}\label{eq:ans2}
g \cdot \frac{1}{f_Q|_{P}} \cdot \varphi^*( \frac{1}{f_P|_{Q}}) (p_2) \sim \frac{\alpha_P}{i_{v_Q}\Omega}(n)\cdot \frac{1}{f_Q(p_N)}
\end{equation}
the reason for this is the fact that $\frac{g}{\varphi^{*}(f_P|_{Q})}$ in the point $p_2$ satisfies \[\frac{g}{\varphi^{*}(f_P|_{Q})} \sim 1/\frac{\partial}{\partial v_Q} f_P|_{Q}\] in the point $q_3$, due to $dg \sim v_Q (q_3)$. This chain continues:
\[1/\frac{\partial}{\partial v_Q} f_P|_{Q} = \frac{\alpha_P}{i_{v_Q}\Omega}\]
Similarly,
\begin{equation}\label{eq:ans3}
g \cdot \frac{1}{f_Q|_{P}} \cdot \varphi^*( \frac{1}{f_P|_{Q}}) (p_3) \sim \frac{\alpha_Q}{i_{v_P}\Omega} (n) \cdot \frac{1}{f_P(q_N)}
\end{equation}
Now, to cancel everything and put it into place, let use also use the following obvious formulas:
\begin{equation} \label{eq:subst-invariant-volumes}
f_Q(p_N) = \frac{\Omega(v_P(p_1), v_P(p_2))}{\Omega_Q (v_P(p_1), v_P(p_2))}
\end{equation}
\[ f_P(q_N) = \frac{\Omega(v_Q(q_1), v_Q(q_2))}{\Omega_P (v_Q(q_1), v_Q(q_2))}
\]
Using equations \ref{eq:main1}, \ref{eq:main2}, \ref{eq:main3}, taking into account corollary \ref{cor:geta'} and equations \ref{eq:ans1}, \ref{eq:ans2}, \ref{eq:ans3}, and using equation \ref{eq:subst-invariant-volumes} we obtain
\begin{theorem} The gluing data of $T_X^1$ is, up to some multiples dependent only on $n_P, n_Q$:
For $p_1$:
\begin{equation}s_b(p_1) \Omega_P(v_P(p_1), v_P(p_2)) \Omega_Q(v_Q(q_1), v_Q(q_2))\end{equation}
For $p_2$:
\[
s_b(p_2)\Omega_Q(v_P(p_1), v_P(p_2)) \Omega(v_Q(q_3), v_P(p_3))\frac{\alpha_P}{i_{v_Q}\Omega}(n) =
\]
\[= s_b(p_2) \Omega_Q(v_P(p_1), v_P(p_2)) (\alpha_P,v_P)(p_3)\]
And the quantity $(\alpha_P, v_P) (p_3)$ is actually $\sim 1$, so
\begin{equation}\sim s_b(p_2) \Omega_Q(v_P(p_1), v_P(p_2))\end{equation}
Similarly, for $p_3$:
\begin{equation}
s_b(p_3)\Omega_Q(v_Q(q_1), v_Q(q_2))
\end{equation}
\end{theorem}
Denote $1/\Omega_Q(v_Q(q_1), v_Q(q_2)) = a, 1/\Omega_P(v_P(p_1), v_P(p_2)) = b$.
Now, without loss of generality, we can normalize the gluing data in the following way:
\begin{equation}
\const \times (s_b(p_1), s_b(p_2), s_b(p_3)) \times (1; a; b)
\end{equation}
Here, $(s_b(p_1), s_b(p_2), s_b(p_3))$ is the gluing data of the bundle $\mathcal{O}([b]-[\psi_P])$, and $\const$ is some group element dependent only on $n_P, n_Q$ - so, geometrically, on the cross-ratio of $p_1, p_2, p_3, \psi_P$ and $q_1, q_2, q_3, \psi_Q$.
\begin{theorem} Consider the open subset of $M^u_X \subset M_X$ of such constructs that $p_N, q_N, n$ do not lie on the same line. Every fiber of the map $O: M^u_X \rightarrow J_X$ is smooth, and $O$ is surjective (restricted on this subset).
\end{theorem}
Consider a pair of nodal cubics $P$ and $Q$, satisfying the assumption above, pick a generic affine chart, and restrict ourselves with the following subspace of $M^u_X$:
\begin{definition}
$M_X^{0}$ is a space of such constructs $(P', b')$, $Q'$ that $(P, b)$ is affine equivalent to $(P', b')$ in our chosen chart, and transform conjugating them preserves the point $n$, and the same is true for $Q$ and $Q'$. Denote by $f_{P'}$, $f_{Q'}$ their defining equations
\end{definition}
The values which were dependent only on $n_P, n_Q$ are constant on such subspace.
\begin{proof}
Then, it is easy to see that on $M_X^0$ \[\Omega_Q(v_{p_1}, v_{p_2}) = \const f_Q(p_N)\]
and for $\Omega_P$ symmetrically.
We would like to show that for any pair $P, Q$ there exists a two-dimensional tangent subspace in $M_X^0$ such that the differential of the map $(f_P(q_N), f_Q(p_N))$ is surjective.
It is very clear, because one can pick the affine transform which preserves $n$ and $p_N$, and moves $q_N$ anywhere not in the line $(n, p_N)$ - in particular, moves it transversely to the level set of the function $f_P$. Moving $Q$ with such an affine transform does not change the value $f_Q(p_N)$, yet changes $f_P(q_N)$ in the first order. Similar argument, of course, works for $P$.
That proves the first part of the theorem - for any construct there exists a two-dimensional tangent subspace in $M_X^0$, and hence, in $M_X^u$, which is projects isomorphically to the tangent space of $J_X$. That implies that the fibers are all smooth.
It is not hard to also come up with the family of constructs which maps to $J_X = (\mathbb{C}^*)^2$ surjectively - one should start with any pair $P, Q$ as above, then uses the transforms described above to make the value $f_Q(p_N)$ arbitrary without changing $f_P(q_N)$, then does the same for $f_P(q_N)$.
\end{proof}
\begin{corollary}\label{cor:epicwin}
There exists a smoothable construct.
\end{corollary}
\section{Properties of the smoothing}
In this section, we discuss the topology of the generic fiber of our smoothing family. Let us pick the new convention - the specific family we have constructed in the previous section will be referred as $\tilde{Y}$ to avoid confusion in the general discussions about any family.
Suppose $\tilde{X}$ is any family with normal crossing central fiber $X$ and smooth total space. Denote as $Z_i^0$ the manifold $Z_i \setminus Z_{i+1}$, and consider the following the bundle $T_i \rightarrow Z_i^0$ by $i-1$-dimensional tori: the torus over a point of $Z_i^0$ is a product of spherisation of normal bundles to $Z_i^0$ inside branches of $Z_{i-1}$ passing through it.
\begin{lemma}
The generic fiber $X^\varepsilon$ is homeomorphic to some space $X^{\mathbb{R}}$ representable as the disjoint union of $T_i$'s.
\end{lemma}
\begin{proof}
This fact seems to be well known, however we did not manage to find the exact reference (there is a writeup by W. D. Gillam \cite{WDG} available on the net, but overall this construction seems to be the part of the folklore) so sketch the proof here. The construction is as follows: consider the real oriented blowup of the family $\tilde{X}$ in the central fiber $X$. It can be checked locally that the resulting space is a manifold with corners, admits canonical structure of the manifold with boundary, and is mapped into the real oriented blowup of $D$ in point $0$ with differential of the full rank. Then, by Ehresmann's lemma, the fibers are diffeomorphic, and the fiber over any preimage of $0$ will be of the form described above.
\end{proof}
\begin{corollary}[also well known]
\[\chi(X^{\varepsilon}) = \chi^c (X \setminus \Sing(X))\]
\end{corollary}
Here, $\chi^c$ is an Euler characteristic with compact support.
\begin{proof}
$\chi^c$ is additive on disjoint union and $0$ for all bundles with toral fibers.
\end{proof}
That alows the computation of cohomology of $Y^{\varepsilon}$:
\begin{corollary}
\[\chi(Y^{\varepsilon}) = 11\]
\[h^{1,1}(Y^\varepsilon) = 9\]
\end{corollary}
\begin{proof}
Let us calculate $\chi^c(Y \setminus \Sing(Y))$. Euler characteristic of $\mathbb{P}^2$ is $3$, we add $9$ for nine blow ups, subtract $4$ for removing two rational curves and add $3$ for three nodal points. $3 + 9 - 4 + 3 = 11$.
The second row just follows from $h^{1,0}=h^{2,0}=0$.
\end{proof}
Then, we would like to check that $Y^\varepsilon$ is a minimal surface of general type (the fact that it is of general type follows from it being not rational and geography of Chern numbers, but minimality does not). It also automatically implies that the fibers of the family are not only analytic, but algebraic projective surfaces.
\begin{lemma}\label{lem:general}
$Y^{\varepsilon}$ is a surface of general type.
\end{lemma}
\begin{proof}
By Noether's formula it has $c_1^2 + c_2 = 12\chi(\mathcal{O}_{Y^\varepsilon}) = 12$, and $c_2 = 11$. As we do not know it is minimal, we conclude that it could be a blow-up of some surface with $c_1^2 + c_2 = 12$ and $c_2 \leq 11$. Such surfaces are either rational or of general type by the classification of surfaces, and we know that $Y^{\varepsilon}$ is not rational, because $\Delta_Y$ is not collapsible.
\end{proof}
\begin{theorem}\label{thm:minimal}The generic fiber of $\tilde{Y}$ is minimal.
\end{theorem}
\begin{proof}
As we already know it is of general type, we can prove that there is no $K_{\tilde{Y}/D}$-negative contraction. Indeed, assume that the generic fiber admits some $K_{\tilde{Y}/D}$-negative curve. Then, there will be a limiting curve in $Y$, with one of its components still being $K_Y$-negative. However, note that $\nu_Y^*(K_Y) = K_{\hat{Y}}([P]+[Q])$. $\hat{Y}$ is a projective plane blown up in $9$ points, $8$ of which lie in $P cap Q$, and one lies on $P$. Denote as $\pi: \hat{Y} \rightarrow \mathbb{P}^2$ the projection on the projective plane, as $E_1, ... E_8$ - exceptional curves which do lie in the preimage of $P \cap Q$, $E_b$ - the last exceptional curve. Then, \[K_{\hat{Y}}([P]+[Q]) = \pi^*(\mathcal{O}(3)) - E_1 - ... - E_8\]
Such bundle is nonnegative on any curve - it is a pullback of $-K$ of the plane with $8$ points blown up, and as these points lie in, say $P$, any curve $C$ which passes through them $k$ times will have the degree at least $\lceil k/3 \rceil$.
\end{proof}
Now, we would like to calculate the fundamental group of $Y^{\varepsilon}$. We shall do it for our combinatorial model $Y^{\mathbb{R}}$.
\begin{lemma}\label{lem:pi1=0}
$\pi_1(Y \setminus \Sing(Y)) = 0$
\end{lemma}
\begin{proof}
By Severi's problem \cite{D2}, the fundamental group of the complement of the nodal curve $P \cup Q$ in $\mathbb{P}^2$ is abelian. The blowups can only add relations (because any loop can be perturbed in such a way that it doesn't go through the exceptional locus). Hence, $\pi_1(Y \setminus \Sing(Y)) = H_1(Y \setminus \Sing(Y))$.
Then, $H_1(\mathbb{P}^2 \setminus (P \cup Q))$ is generated by two generators - little loops going around $P$ and $Q$. The blow up in the point $b$ forces the loop around $P$ to become contractible. The blow up in any point of intersection of $P$ and $Q$ makes these loops homotopy equivalent (up to a sign).
\end{proof}
Next theorem we prefer to prove in a bit higher generality, it works for any family $X$ with normal crossing fiber.
\begin{theorem} \label{thm:dxcontractible-pi1}
Suppose $\pi_1(\Delta_X) = 0$, and every component of $X \setminus \Sing(X)$ also has trivial fundamental group. Then, $\pi_1(X^{\mathbb{R}}) = \pi_1(X^{\varepsilon}) = 0$.
\end{theorem}
We split the proof in a few parts. Let us start by perturbing a loop $\gamma$ in such a way that it is transverse to the stratification of $X^{\mathbb{R}}$. Then, the loop will intersect the real codimension $1$ stratum $T_1$ finite amount of times.
\begin{definition}
The \textbf{combinatorial image} of $\gamma$ is a edge-path $\gamma'$ in $\Delta_X$ which moves along the edges corresponding to the components $\gamma$ intersects.
\end{definition}
\begin{lemma}\label{lem:cancel}
In the assumptions of the Theorem \ref{thm:dxcontractible-pi1} every path $\tau$ with combinatorial image $e^{-1}e$ for some edge $e$ (i.e. path going through some codimension $1$ strata and then coming back through it) is homotopic as a path with fixed ends to the path with combinatorial image $1$ (not leaving the open part $X \setminus \Sing(X)$).
\end{lemma}
\begin{proof}
Note that the path $\tau$ enters and leaves through the same codimension $1$ stratum. As it is connected, the path $\tau$ is equivalent to a path of the form: \[\tau = \tau_2 m^{-1} \ell m\tau_1\]
where $\ell$ is a loop in the open part, $\tau_1, \tau_2$ - paths in the open part, $m$ is a short path going through the codimension $1$ component corresponding to $e$. Then, $\ell$ is contractible by the assumption of the theorem, and then $m^{-1}m$ is contracted.
\end{proof}
\begin{lemma}\label{lem:lift}
Any edge-path $\tau'$ in $\Delta_X$ admith a lift - a path $\tau$ in $X^{\mathbb{R}}$ which has the combinatorial image $\tau'$. We can choose the ends of $\tau$ as we wish - provided they lie in the components of $X \setminus \Sing(X)$ corresponding to the ends of $\tau'$
\end{lemma}
\begin{proof}
It is obvious from the fact that the components of $X\setminus\Sing(X)$ corresponding to vertices of $\Delta_X$ are connected - and this is true by the very construction of the dual complex.
\end{proof}
\begin{lemma}\label{lem:lasso}
Any edge-loop $\gamma'$ in $\Delta_X$ is a composition of the elementary loops of the form $(s'_i)^{-1}\sigma'_i(s'_i)$ where $s'_i$ are some paths and $\sigma'_i$ are loops going around exactly one triangle.
\end{lemma}
\begin{proof}
This is true for any contractible loop in any triangulated space and follows easily from the following well-known statement - any contractible edge-loop is contractible by the sequence of elementary operations either swapping an edge to the pair of edges moving around the other side of the triangle, or vice versa. This statement follows from the shellability of any triangulation of $2$-dimensional disk.
\end{proof}
\begin{lemma}\label{lem:global-lift}
Any (transverse to the stratification) loop $\gamma$ in $X^{\mathbb{R}}$ admits the analogous decomposition: \[\gamma = \gamma_n ... \gamma_1 \]
\[\gamma_i = s_i^{-1} \sigma_i s_i\]
with $s_i$ being some paths and $\sigma_i$ being loops with the combinatorial image going around the boundary of $1$ triangle.
\end{lemma}
\begin{proof}
Consider the decomposition of $\gamma'$ and lift $\sigma'_i$'s and $s'_i$'s in some arbitrary way (only thing required is that they are composable). Denote the composition of these lifts $\tilde{\gamma}$. The combinatorial image of $\tilde{\gamma}^{-1}\gamma$ is a path which is removable by operations from Lemma \ref{lem:cancel}, from which it follows that it is contractible.
\end{proof}
\begin{lemma}\label{lem:elementary-move}
For any triangle $m \in \Delta_X$ there exists a loop around the corresponding real codimension $2$ component in $X^{\mathbb{R}}$ which is contractible.
\end{lemma}
\begin{proof}
It is a local check, and the local model is a real oriented blowup of $\mathbb{C}^3$ in the coordinate cross $xyz = 0$. It is easy to see explicitly: the preimage of the point $0$ is a $2$-dimensional torus, and moves from the components $x=0$ to component $y=0$ are unique up to adding the generator $(1;0)$ of its first homology. Similarly, moves from $y =0$ to $z=0$ are unique up to adding another generator, and third move is unique up to adding $(-1;-1)$. That freedom allows to kill any element of the first homology, even in multiple ways.
\end{proof}
Now, we are ready to deduce the theorem.
\begin{proof}[Proof of the Theorem \ref{thm:dxcontractible-pi1}]
By Lemma \ref{lem:elementary-move} we can pick such $\sigma_i$'s in the of the proof of the Lemma \ref{lem:global-lift} that they are contractible.
\end{proof}
\begin{corollary}[of Thm.\ref{thm:dxcontractible-pi1} and Lem.\ref{lem:pi1=0}]
\[\pi_1(Y^{\varepsilon}) = 0\]
\end{corollary}
\section{Questions}
As we have already mentioned, these results are mostly proof of concept - they show that such beasts as families with non-collapsible, contractible dual complex exist, and probably abundant. Yet, the methods and our understanding of how to construct these is still lacking. Identifying example constructed in this way with some known surfaces also doesn't seem too easy.
However, we take the liberty to propose the following
\begin{conjecture} $Y^{\varepsilon}$ lies in the deformation class of Barlow's surface.
\end{conjecture}
The motivation for this is simply that this is the only known simply-connected surface of general type with such numerical invariants.
The two main roads of development are possible from the current point.
One could be an attempt to construct more examples - and figure out what are the possible constraints of the construction. On this path one will probably encounter some kind of ''tropical'' structure on the dual complex, analogous to the integral flat connections on the sphere arising in the mirror symmetry (see \cite{KS}). After the combinatorial part is settled, one will then need to understand the map $O$ much better than we currently do. This, however, will most likely be less of an issue compared to the first one.
The straightforward way of constructing some examples of interesting dual complexes would be taking a look on the constructions of Korean school \cite{LYP} and then resolving the families obtained by them. We didn't investigate in detail but are planning to.
The second road would be attempting to understand if the dual complex $\Delta_X$ of a geometrically maximal degeneration says anything useful about the deformation class of $X^{\varepsilon}$. Following this road, one would try to perform some elementary equivalences on $\Delta_X$ which would correspond to changes in topology while moving along the boundary of the moduli space of surfaces. Then, this ''simple type'' of the complex will be an invariant of the deformation class of $X^{\varepsilon}$.
The chances of this second program succeeding are rather slim - not only the boundary of the moduli space might not be connected, we can not currently prove the existence of even one geometrically maximal degeneration. We, however, would like to put it as a conjecture - not supported by much but a wishful thinking, but here it is:
\begin{conjecture}
Suppose we are given a \textbf{simply-connected} surface with $h^{1,0} = h^{2,0}$. Then, it admits a geometrically maximal degeneration, moreover, space of such degenerate surfaces is connected.
\end{conjecture}
\newpage
|
1,116,691,500,799 | arxiv | \section{Introduction}
Learning from heavy tailed or corrupted data is a long pursued challenge in statistics receiving considerable attention in literature~\cite{huber1964robust, hampel1971, huber2004robust, HeavyTails, diakonikolas2019robust, audibert2011robust} and gaining an additional degree of complexity in the high-dimensional setting~\cite{lugosi2019regularization,lecue2020robust, dalalyan2019outlier, balakrishnan2017computationally, liu2019high, liu2020high, fan2021shrinkage}.
Sparsity inducing penalization techniques~\cite{donoho2000high, buhlmann2011statistics} are the go-to approach for high-dimensional data and have found many applications in modern statistics~\cite{tibshirani1996regression, buhlmann2011statistics, donoho2000high, hastie2015statistical}.
A clear favourite is the Least Absolute Shrinkage and Selection Operator (LASSO)~\cite{tibshirani1996regression}. Theoretical studies have shown that under the so-called Restricted Eigenvalue (RE) condition, the latter achieves a nearly-optimal estimation rate~\cite{bickel2009simultaneous, buhlmann2011statistics, negahban2012unified}. Further, a rich literature extensively studies the oracle performances of LASSO in various contexts and conditions~\cite{bunea2007sparsity, lounici2008sup, zhang2008sparsity, zhang2009some, zhao2006model, zou2006adaptive, van2008high, lecue2018regularization, bellec2018slope}.
Other penalization techniques induce different sparsity patterns or lead to different statistical guarantees, such as, to cite but a few, SLOPE~\cite{bogdan2015slope,su2016slope} which is adaptive to the unknown sparsity and leads to the optimal estimation rate, OSCAR~\cite{bondell2008simultaneous} which induces feature grouping or group-$\ell_1$ penalization~\cite{yuan2006model, huang2010variable} which induces block-sparsity.
Other approaches include for instance Iterative Hard Thresholding (IHT)~\cite{blumensath2009iterative, blumensath2010normalized, jain2014iterative, shen2017tight, jain2016structured} whose properties are studied under the Restricted Isometry Property (RIP).
Another close problem is low-rank matrix recovery, involving the nuclear norm as a low-rank inducing convex penalization~\cite{koltchinskii2011nuclear, candes2009exact, candes2011tight, rohde2011estimation, negahban2011estimation, negahban2012restricted}.
The high-dimensional statistical inference methods cited above are, however, not robust: theoretical guarantees are derived under light-tails (generally sub-Gaussian) and the i.i.d assumption. Unfortunately, these assumptions fail to hold in general, for instance, it is known that financial and biological data often displays heavy-tailed behaviour~\cite{fan2021shrinkage} and outliers or corruption are not uncommon when handling massive amounts of data which are tedious to thoroughly clean. Moreover, the majority of the previous references focus on the oracle performance of estimators as opposed to providing guarantees for explicit algorithms to compute them.
A natural question is therefore : \emph{can one build alternatives to such high-dimensional estimators that are robust to heavy tails and outliers, that are computationally efficient and achieve rates similar to their non-robust counterparts ?}
Recent advances about robust mean estimation~\cite{catoni2012challenging, lugosi2021robust, diakonikolas2020outlier, Depersin2019RobustSE, lei2020fast} gave a strong impulse in the field of robust learning~\cite{lecue2020robust1, HeavyTails, pmlr-v97-holland19a, diakonikolas2019robust, cherapanamjeri2020optimal, bakshi2021robust}, including the high-dimensional setting~\cite{liu2019high,liu2020high, balakrishnan2017computationally} which led to significant progress towards a positive answer to this question.
However, to the best of our knowledge, the solutions proposed until now are all suboptimal in one way or another. The shortcomings either lie in the obtained statistical rate: which is sometimes significantly far from optimal, or in the robustness: most works consider heavy tailed and corrupted data separately and only very limited amounts of corruption, or in computational complexity: some corruption-filtering algorithms are too heavy and do not scale to real world applications.
In this paper, we propose explicit algorithms to solve multiple sparse estimation problems with high performances in all previous aspects. In particular, our algorithm for vanilla sparse estimation enjoys a nearly optimal statistical rate (up to a logarithmic factor), is simultaneously robust to heavy tails and strong corruption (when a fraction of the data is corrupted) and has a comparable computational complexity to a non robust method.
\subsection{Main contributions.}
This paper combines non-Euclidean optimization algorithms and robust mean estimators of the gradient into explicit algorithms in order to achieve the following main contributions.
\begin{itemize}
\item We propose a framework for robust high-dimensional linear learning using two linearly converging stage-wise algorithms for high-dimensional optimization based on Mirror Descent and Dual Averaging. These may be applied for smooth and non-smooth objectives respectively so that most common loss functions are covered. The previous algorithms may be plugged with an appropriate gradient estimator to obtain explicit robust algorithms for solving a variety of problems.
\item The central application of our framework is an algorithm for ``vanilla'' $s$-sparse estimation reaching the nearly optimal $s\log(d)/n$ statistical rate by combining stage-wise Mirror Descent (Section~\ref{sec:md}) with a simple trimmed mean estimator of the gradient. This algorithm is simultaneously robust to heavy-tailed distributions and $\eta$-corruption of the data\footnote{We say that data is $\eta$-corrupted for some number $0<\eta <1/2$ if an $\eta$ fraction of the samples is replaced by arbitrary (and potentially adversarial) outliers after data generation.}. This improves over previous literature which considered the two issues separately or required a very restricted value of the corruption rate $\eta$.
\item In addition to vanilla sparsity, we instantiate our procedures for group sparse estimation and low-rank matrix recovery, in which different metrics on the parameter space are induced and used to measure the statistical error on the gradient (Section~\ref{sec:applis}). For heavy-tailed data and $\eta$-corruption, the gradient estimator we propose for vanilla sparsity enjoys an optimal statistical rate with respect to the induced metric while the one proposed for group sparsity is nearly optimal up to a logarithmic factor. Moreover, for heavy tailed data and a limited number of outliers\footnote{For low-rank matrix recovery, our estimator is based on Median-Of-Means so that the number of tolerated outliers is up to $K/2$ where $K$ is the number of blocks used for estimation (see Section~\ref{sec:appliLowRank})}, our proposed gradient estimator for low-rank matrix recovery is nearly optimal. Thus, our solutions to each of these problems are the most robust yet in the literature.
\item Our algorithms offer a good compromise between robustness and computational efficiency with the only source of overhead coming from the robust gradient estimation component. In particular, for vanilla sparse estimation, this overhead is minimal so that the asymptotic complexity of our procedure is \emph{equivalent} to that of non-robust counterparts. This is in contrast with previous works requiring costly sub-procedures to filter out corruption.
\item We validate our results through numerical experiments using synthetic data for regression and real datasets for classification (Section~\ref{sec:exp}). Our experiments confirm our mathematical results and compare our algorithms to concurrent baselines from literature.
\item All algorithms introduced in this paper as well as the main baselines from literature we use for comparisons are implemented and easily accessible in a few lines of code through our \texttt{Python} library called \texttt{linlearn}, open-sourced under the BSD-3 License on \texttt{GitHub} and available here\footnote{\url{https://github.com/linlearn/linlearn}}.
\end{itemize}
\subsection{Related Works}
The early work of~\cite{agarwal2012stochastic} focuses on vanilla sparse recovery in the stochastic optimization setting and uses a multistage annealed LASSO algorithm where the penalty shrinks progressively. The method reaches the nearly optimal $s\log(d)/n$ rate, however, it is not robust since the data is assumed i.i.d sub-Gaussian. The subsequent work of~\cite{sedghi2014multi} extends this framework to other problems such as additive sparse and low-rank matrix decomposition by changing the optimization algorithm but the sub-Gaussian assumption remains necessary.
More recently,~\cite{Juditsky2020SparseRB} proposed a stochastic optimization mirror descent algorithm which computes multiple solutions on disjoint subsets of the data and aggregates them with a Median-Of-Means type procedure.
The final solution achieves the rate $s\log(d)/n$ under $s$-vanilla sparsity with sub-Gaussian deviation and an application to low-rank matrix recovery is also developed.
This aggregation method can handle some but not all heavy-tailed data.
For instance, if the data follows a Pareto($\alpha$) distribution then the analysis yields a statistical error with a factor of order $d^{1/\alpha}$ which is not acceptable in a high-dimensional setting.
Moreover, the presence of outliers is not considered so that the given bounds do not measure the impact of corruption.
Other works consider high-dimensional linear learning methods that are robust to corrupted data.
An outlier robust method for mean and covariance estimation in the sparse $\eta$-contaminated\footnote{$\eta$-contamination refers to the case where the data is sampled from a mixed distribution $(1-\eta)P + \eta Q$ where $P$ is the true data distribution and $Q$ is an arbitrary one.} high-dimensional setting is proposed by~\cite{balakrishnan2017computationally} along with theoretical guarantees.
By extension, these also apply to several problems of interest such as sparse linear estimation or sparse GLMs. The idea is to use an SDP relaxation of sparse PCA~\cite{d2004direct} in order to adapt the filtering approach from~\cite{diakonikolas2019robust}, which relies on the covariance matrix to detect outliers, to the high-dimensional setting. However, the need to solve SDP problems makes the algorithm computationally costly. In addition, the true data distribution is assumed Gaussian and the considered $\eta$-contamination framework is weaker than $\eta$-corruption which allows for adversarial outliers.
The previous ideas were picked up again in the later work of~\cite{liu2020high} who proposes a robust variant of IHT for sparse regression on $\eta$-corrupted data. Unfortunately, these results suffer from several shortcomings since data needs to be Gaussian with a known or sparse covariance matrix.
Moreover, the gradient estimation subroutine, which is reminiscent of~\cite{balakrishnan2017computationally}, is computationally heavy since it requires solving SDP problems as well and a number of samples scaling as $s^2$ instead of $s$ is required. This seems to come from the fact that sparse gradients need to be estimated, in which case the $s^2$ dependence is unavoidable based on an oracle lower bound for such an estimation~\cite{diakonikolas2017statistical}.
Recently,~\cite{dalalyan2019outlier} derived oracle bounds for a robust estimator in the linear model with Gaussian design and a number of adversarially contaminated labels.
Although optimal rates in terms of the corruption are achieved, this setting excludes corruption of the covariates and does not apply for heavy-tailed data distributions. In contrast, the very recent work of~\cite{sasai2022robust} considers sparse estimation with heavy-tailed and $\eta$-corrupted data and derives a nearly optimal estimation bound using an algorithm which filters the data before running an $\ell_1$-penalized robust Huber regression which corresponds to a similar approach to~\cite{pensia2020robust} where the non sparse case was treated. Although the $s\log(d)/n$ rate is achieved with optimal robustness, this claim only applies for regression under the linear model with some restrictive assumptions such as zero mean covariates. In addition, little attention is granted to the practical aspect and no experiments are carried out.
Finally,~\cite{liu2019high} proposes an IHT algorithm using robust coordinatewise gradient estimators.
These results cover the heavy-tailed and $\eta$-corrupted settings separately thanks to Median-Of-Means~\cite{alon1999space, JERRUM1986169, nemirovskij1983problem} and Trimmed mean~\cite{pmlr-v28-chen13h, yin2018byzantine} estimators respectively.
However, the corruption rate $\eta$ is restricted to be of order at most $O(1/(\sqrt{s}\log(nd)))$ and the question of elaborating an algorithm which is simultaneously robust to both corruption and heavy tails is left open.
We summarize the settings and results of the previously mentioned works, along with ours on vanilla sparse estimation, in Table~\ref{table:comparison} which focuses on robust papers with explicit algorithms.
\begin{table}[!ht
\centering
\begin{tabular}{l|ccccl}
\makecell{Method} & \makecell{Statistical \\rate} & \makecell{Iteration\\ Complexity} & \makecell{Data dist.\\and corruption} & \makecell{Loss} \\
\hline
\makecell{AMMD\\ This paper\\Section~\ref{sec:md}} & \makecell{$O\Big(\sqrt{s}\sqrt{\eta + \frac{\log(d/\delta)}{n}}\Big)$\\with $\mathbb{P} \geq 1-\delta$} & \makecell{$O(nd)$} & \makecell{$L_4$ covariates,\\$L_2$ labels,\\$\eta$-corruption} & \makecell{Lip. smooth,\\ QM} \\[5pt]
\makecell{AMDA\\ This paper\\Section~\ref{sec:da}} & \makecell{$O\Big(\sqrt{s}\sqrt{\eta + \frac{\log(d/\delta)}{n}}\Big)$\\with $\mathbb{P} \geq 1-\delta$} & \makecell{$O(nd)$} & \makecell{$L_4$ covariates,\\$L_2$ labels,\\$\eta$-corruption} & \makecell{Lipschitz, \\PLM} \\[5pt]
\makecell{Balakrishnan\\et al.~\cite{balakrishnan2017computationally}} & \makecell{$O\Big(\|\theta^\star\|_2 \big( \eta \log(1/\eta) +$\\$ s\sqrt{\log(d/\delta)/n} \big)\Big)$\\with $\mathbb{P} \geq 1-\delta$} & \makecell{$\Omega(nd^2 \!+\! d^3)$} & \makecell{Gaussian,\\ $\eta$-contamination} & \makecell{LSQ, GLMs,\\Logit} \\[5pt]
\makecell{Liu et al.,\\2018~\cite{liu2020high}} & \makecell{$O\big(\eta \vee s\sqrt{\log(d/\delta)/n}\big)$\\with $\mathbb{P} \geq 1-\delta$} & \makecell{$\Omega(nd^2 \!+\! d^3)$} & \makecell{Gaussian,\\$\eta$-corruption} & \makecell{LSQ} \\[5pt]
\makecell{Liu et al.,\\2019~\cite{liu2019high}\\(MOM)} & \makecell{$O\Big(\sqrt{s\log(d)/n}\Big)$\\with $\mathbb{P} \geq 1-d^{-2}$} & \makecell{$O(nd)$} & \makecell{$L_4$ covariates,\\linear/logit model} & \makecell{LSQ, Logit} \\[5pt]
\makecell{Liu et al.,\\2019~\cite{liu2019high}\\(TMean)} & \makecell{$O\Big(\eta \sqrt{s}\log(nd)$\\ $+ \sqrt{s\log(d)/n}\Big)$\\with $\mathbb{P} \geq 1-d^{-2}$} & \makecell{$O(nd \log(n))$} & \makecell{sub-Gaussian,\\ $\eta$-corruption} & \makecell{LSQ, Logit} \\[5pt]
\makecell{Juditsky\\et al.~\cite{Juditsky2020SparseRB}} & \makecell{$O\big(\!\sqrt{\!s\log(d)\!\log(1/\delta)/n}\big)$\\with $\mathbb{P} \geq 1-\delta$} & \makecell{$O(d)$\\(stochastic\\optim.)} & \makecell{$L_2$ gradient} & \makecell{Lip. smooth,\\ QM} \\[5pt]
\end{tabular}
\caption{Summary of the main hypotheses and results of our proposed algorithms and related works in the literature on vanilla sparse estimation. The statistical rate column gives the derived error bound on $\|\widehat \theta - \theta^\star\|_2$ between the estimated and true parameter and the associated confidence. In the ``Data distribution and corruption'' column, rows with no reference to corruption correspond to methods which do not consider it. In the ``Loss'' column, the following abbreviations are used : QM = quadratic minoration (Assumption~\ref{asm:quadgrowth}), PLM = pseudo-linear minoration (Assumption~\ref{asm:pseudolingrowth}), LSQ = least squares, GLM = generalized linear model, Logit = Logistic, Lip. smooth = Lipschitz smooth (gradient Lipschitz).}
\label{table:comparison}
\end{table}
\subsection{Agenda}
The remainder of this document is structured as follows : Section~\ref{sec:setting} lays out the setting including the definition of the objective and our assumptions on the data.
Sections~\ref{sec:md} and~\ref{sec:da} define optimization algorithms based on Mirror Descent and Dual Averaging addressing the cases of smooth and non-smooth losses respectively.
Both Sections state convergence results for their respective algorithms.
Section~\ref{sec:applis} considers instantiations of our general setting to vanilla sparse, group sparse and low-rank matrix estimation for a general loss.
In each case, the norm $\|\cdot\|$ and dual norm $\|\cdot\|_{*}$ are instantiated and a robust and efficient gradient estimator is proposed so that, combined with the results of Sections~\ref{sec:md} and~\ref{sec:da}, we obtain solutions with nearly optimal statistical rates (up to logarithmic terms) in each case.
Finally, Section~\ref{sec:exp} presents numerical experiments on synthetic and real datasets which demonstrate the performance of our proposed methods and compare them with baselines from recent literature.
\section{Setting, notation and assumptions}\label{sec:setting}
We consider supervised learning from a dataset $(X_i, Y_i)_{i=1}^n $ from which the majority is distributed as a random variable $(X, Y) \in \mathcal X \times \mathcal Y$ where the covariate space $\mathcal X$ is a high-dimensional Euclidean space and the label space $\mathcal Y$ is $\mathbb{R}$ or a finite set.
The remaining minority of the data are called outliers and may be completely arbitrary or even adversarial.
Given a loss function $\ell : \widehat \mathcal Y \times \mathcal Y \to \mathbb{R}$ where $\widehat \mathcal Y$ is the prediction space, our goal is to optimize the unobserved objective
\begin{equation}\label{eq:objective}
\mathcal{L} (\theta) = \mathbb{E} [\ell(\langle \theta, X \rangle , Y)]
\end{equation}
over a convex set of parameters $\Theta$,
where the expectation is taken w.r.t. the joint distribution of $(X, Y)$.
Given an optimum $\theta^\star = \argmin_{\theta \in \Theta} \mathcal{L}(\theta)$ (assumed unique), we are also interested in controlling the estimation error $\|\theta - \theta^\star\|$ where $\|\cdot\|$ is a norm on $\Theta$, which will be defined according to each specific problem (see Section~\ref{sec:applis} below).
Moreover, we assume that the optimal parameter is sparse according to an abstract sparsity measure $S : \Theta \to \mathbb{N}$.
\begin{assumption}\label{asm:sparse}
The optimal solution $\theta^\star$ is $s$-sparse for some integer $s$ significantly smaller than the problem dimension i.e. $S(\theta^\star) \leq s$. Additionally, for any $s$-sparse vector $\theta \in \Theta$ we have the inequality $\|\theta\| \leq \sqrt{s}\|\theta\|_2$ and an upper bound $\bar{s} \geq s$ on the sparsity is known.
\end{assumption}
The precise notion of sparsity will be determined later in Section~\ref{sec:applis} through the sparsity measure $m$ depending on the application at hand. The simplest case corresponds to the conventional notion of vector sparsity where $\mathcal X = \mathbb{R}^d$ for some large $d$, $\Theta \subset \mathbb{R}^d$ and the sparsity measure $S(\theta) = \sum_{j\in \setint{d}}\ind{\theta_j \neq 0}$ counts the number of non-zero coordinates. However, as we intend to also cover other forms of sparsity later, we do not fix this setting right away.
Since the objective~\eqref{eq:objective} is not observed due to the law of the data being unknown, statistical approximation will be necessary in order to recover an approximation of $\theta^\star$. Instead of estimating the objective itself (which is of limited use for optimization), we will rather compute estimates of the gradient
\begin{equation}
g(\theta) := \nabla_{\theta} \mathcal{L}(\theta)
\end{equation}
in order to run gradient based optimization procedures.
Note that, since we consider the high-dimensional setting, a standard gradient descent approach is excluded since it would incur an error strongly depending on the problem dimension. In order to avoid this, one must use a non Euclidean optimization method as is customary for high-dimensional problems~\cite{Juditsky2020SparseRB, agarwal2012stochastic}.
As commonly stated in the robust statistics literature \cite{catoni2012challenging, lugosi2021robust, lugosi2019mean}, estimating an expectation using a conventional empirical mean only yields values with far from optimal deviation properties in the general case. Several estimators have been proposed which enjoy sub-Gaussian deviations from the true mean and robustness to corruption. Notable examples in the univariate case are the median-of-means (MOM) estimator~\cite{alon1999space, JERRUM1986169, nemirovskij1983problem}, Catoni's estimator~\cite{catoni2012challenging} and the trimmed mean~\cite{lugosi2021robust}. However, in the multivariate case (estimating the mean of a random vector), the optimal sub-Gaussian estimation rate cannot be obtained by a straightforward extension of the previous methods and a line of works~\cite{lugosi2019sub, Hopkins2018MeanEW, pmlr-v99-cherapanamjeri19b, Depersin2019RobustSE, lugosi2021robust, lei2020fast} has pursued elaborating efficient algorithms to achieve it. Most recently, \cite{diakonikolas2020outlier} managed to show that stability based estimators enjoy sub-Gaussian deviations while being robust to corruption of a fraction of the data. However, it is important to remember that all the works we just mentioned measure the estimation error using the Euclidean norm while many other choices are possible which may require the estimation algorithm to be adapted in order to achieve optimal deviations with respect to the chosen norm. This aspect was studied in~\cite{lugosi2019near} who gave a norm-dependent formula for the optimal deviation and an algorithm to achieve it, although the latter has exponential complexity and does not consider the presence of outliers.
This is an important aspect to keep in mind in our high-dimensional setting since we will be measuring the statistical error on the gradient using the dual norm $\|\cdot\|_{*}$ of $\|\cdot\|$ which will never be the Euclidean one.
\begin{equation}
\|v\|_{*} = \sup_{\|x\| \leq 1} \langle v, x \rangle
\end{equation}
Of course, apart from the way it is measured, the quality of the estimations one can obtain also crucially depends on the assumptions made on the data. We formally state ours here. We denote $|A|$ as the cardinality of a finite set $A$ and use the notation $\setint k = \{ 1, \ldots, k\}$ for any integer $k \in \mathbb{N} \setminus \{ 0 \}$.
\begin{assumption}
\label{asm:data}
The indices of the training samples $\setint n$ can be divided into two disjoint subsets $\setint n = \mathcal{I} \cup \mathcal{O}$ of \emph{outliers} $\mathcal O$ and \emph{inliers} $\mathcal I$ for which
we assume the following\textup:~$(a)$ we have $|\mathcal I| > |\mathcal O|;$~$(b)$ the pairs $(X_i, Y_i)_{i \in \mathcal I}$ are i.i.d with distribution $P$ and the outliers $(X_i, Y_i)_{i \in \mathcal O}$ are arbitrary$;$~$(c)$ the distribution $P$ is such that :
\begin{equation}
\label{eq:XY-moments}
\mathbb{E}\big[ \|X\|_2^4 \big] < +\infty, \quad \mathbb{E}\big[\|Y X\|_2^2\big] < +\infty \quad
\text{and} \quad \mathbb{E}\big[|Y|^2\big] < +\infty.
\end{equation}
Moreover\textup, the loss function $\ell$ admits constants $C_{\ell, 1}, C_{\ell, 2}, C'_{\ell, 1}, C'_{\ell, 2}> 0$ such that for all $z, y \in \widehat{\mathcal Y}\times \mathcal Y$ \textup:
\begin{equation*}
|\ell(z, y)| \leq C_{\ell, 1} + C_{\ell, 2}|z-y|^2 \quad \text{and} \quad |\ell'(z, y)| \leq C_{\ell, 1}' + C_{\ell, 2}'|z-y|.
\end{equation*}
\end{assumption}
The above hypotheses are sufficient so that the objective function and its gradient exist for any parameter $\theta$ and the gradient admits a second moment. The distribution $P$ is allowed to be heavy-tailed and the conditions on $\ell$ do not go far beyond limiting it to a quadratic behavior and are satisfied by common loss functions for regression and classification\footnote{Including the square loss, the absolute loss, Huber's loss, the logistic loss and the Hinge loss.}. Note that depending on the loss function used and the moment requirements of gradient estimation, the previous moments assumption can be weakened as in~\cite[Assumption 2]{gaiffas2022robust} for instance. However, we stick to this version in this work for simplicity. Depending on the gradient estimator, the number of outliers $|\mathcal O|$ will be bounded in the subsequent statements either by a constant fraction $\eta n$ ($\eta$-corruption) of the sample for some $0 < \eta <1/2$, or by a constant number.
In the two following sections, we will assume that we have a gradient estimator $\widehat{g}$ at our disposal such that for all $\theta \in \Theta$ we have $\widehat g(\theta) = g(\theta) + \epsilon(\theta)$ where $\epsilon(\theta)$ represents the random estimation error of the gradient at $\theta$. Same as for the norm $\|\cdot\|$, this estimator will be precisely defined for each individual application in Section~\ref{sec:applis} in such a way that the error $\|\widehat{g}(\theta) - g(\theta)\|_* = \|\epsilon(\theta)\|_*$ is (nearly) optimally controlled.
\section{The smooth case with Mirror Descent}\label{sec:md}
In this section we will assume the loss $\ell$ is smooth, formally :
\begin{assumption}
\label{asm:lipsmoothloss}
For any $y \in \mathcal Y,$ the loss $z \mapsto \ell(z, y)$ is convex, differentiable and $\gamma$-smooth meaning that
\begin{equation}\label{eq:lipsmooth}
|\ell'(z, y) - \ell'(z', y)| \leq \gamma |z - z'|
\end{equation} for some $\gamma > 0$ and all $z, z' \in \widehat \mathcal Y$, where the derivative is taken w.r.t. the first argument.
\end{assumption}
The above assumption is stated in all generality for $z, z'$ belonging to the prediction space $\widehat \mathcal Y$. For regression or binary classification tasks, we will have $\widehat \mathcal Y = \mathbb{R}$ and $\ell '(\cdot, y) \in \mathbb{R}$ so that the absolute values are enough to interpret the required inequality. Nonetheless, it can also be extended for $K$-way multiclass classification where $\widehat \mathcal Y = \mathbb{R}^K$ and $\ell '(\cdot, y) \in \mathbb{R}^K$, in which case the absolute values on both sides of the above inequality should be interpreted as Euclidean norms.
We also make the following quadratic minoration assumption.
\begin{assumption}\label{asm:quadgrowth}
Let $\theta^\star \in \Theta$ be the optimum of the objective $\mathcal{L}$ and $\|\cdot\|_2$ the usual Euclidean norm. There exists a constant $\kappa > 0$ such that for all $\theta \in \Theta:$
\begin{equation}\label{eq:quadminoration}
\mathcal{L}(\theta) - \mathcal{L}(\theta^\star) \geq \kappa \|\theta - \theta^\star\|_2^2.
\end{equation}
\end{assumption}
Assumption~\ref{asm:quadgrowth} may be compared to strong convexity except that it is a weaker version thereof.
As a consequence of Assumptions~\ref{asm:data} and~\ref{asm:lipsmoothloss}, the objective gradient is $L$-Lipschitz continuous for some constant $L > 0$, meaning that we have :
\begin{equation}
\|g(\theta) - g(\theta')\|_* \leq L\|\theta - \theta'\| \quad \forall \theta, \theta' \in \Theta.
\end{equation}
This property is necessary to establish the convergence of the Mirror Descent algorithm~\cite{nemirovskij1983problem} proposed in this section.
Since we adopt a multistage mirror descent procedure as done in~\cite{Juditsky2020SparseRB}, our framework is also similar to theirs.
\begin{definition}\label{def:dgf}
A function $\omega : \Theta \to \mathbb{R}$ is a distance generating function if it is a real convex function over $\Theta$ which satisfies :
\begin{enumerate}
\item $\omega$ is continuously differentiable and strongly convex w.r.t. the norm $\|\cdot\|$ i.e.
\begin{equation*}
\langle \nabla \omega (\theta) - \nabla \omega (\theta'), \theta - \theta' \rangle \geq \|\theta - \theta'\|^2.
\end{equation*}
\item We have $\omega(\theta) \geq \omega(0) = 0$ for all $\theta \in\Theta$.
\item There exists a constant $\nu > 0$ called the quadratic growth constant such that we have \textup:
\begin{equation}\label{eq:dgf_quad_growth}
\omega(\theta) \leq \nu \|\theta\|^2 \quad \forall \theta \in \Theta.
\end{equation}
\end{enumerate}
\end{definition}
We shall see, for individual applications, that one needs to choose $\omega$ in such a way that it is strongly convex and the constant $\nu$ has only a light dependence on the dimension. For a reference point $\theta_0 \in \Theta$, we define $\omega_{\theta_0}(\theta) = \omega(\theta - \theta_0)$ and the associated Bregman divergence :
\begin{equation*}
V_{\theta_0}(\theta, \theta') = \omega_{\theta_0}(\theta) - \omega_{\theta_0}(\theta') - \langle \nabla \omega_{\theta_0}(\theta'), \theta - \theta' \rangle.
\end{equation*}
Given a step size $\beta > 0$ and a dual vector $u \in \Theta^*$, we define the following proximal mapping :
\begin{align*}
\prox_{\beta}(u, \theta ; \theta_0, \Theta) :=& \argmin_{\theta' \in \Theta} \{ \langle \beta u, \theta' \rangle + V_{\theta_0}(\theta', \theta)\} \\
=& \argmin_{\theta' \in \Theta} \{ \langle \beta u - \nabla \omega_{\theta_0} (\theta), \theta' \rangle + \omega_{\theta_0}(\theta')\}.
\end{align*}
The previous operator yields the next iterate of Mirror Descent for previous iterate $\theta$, gradient $u$ and step size $\beta$ with Bregman divergence defined according to the reference point $\theta_0$. Ideally, we would plug $g(\theta)$ as gradient $u$ but since the true gradient is not observed, we replace it with the estimator $\widehat{g}(\cdot)$. All in all, given an initial parameter $\theta_0$ we obtain the following iteration for Mirror Descent :
\begin{equation}\label{eq:iterMDuncorrected}
\theta_{t+1} = \prox_{\beta}(\widehat{g}(\theta_t), \theta_t; \theta_0, \Theta),
\end{equation}
with a step size $\beta$ to be defined later according to problem parameters. The previous proximal operator can be computed in closed form in each of the applications we consider in Section~\ref{sec:applis}, see Appendix~\ref{apd:closed-form-prox} for details. We state the convergence properties of the above iteration in the following proposition.
\begin{proposition}\label{prop:uncorrectedMD}
Grant Assumptions~\ref{asm:data} and \ref{asm:lipsmoothloss} so that the objective $\mathcal{L}$ is $L$-Lipschitz-smooth for some $L>0$. Let mirror descent be run with constant step size $\beta \leq 1/L$ starting from $\theta_0 \in \Theta$ with $\Theta = B_{\|\cdot \|}(\theta_0, R)$ for some radius $R>0$. Let $\theta_1, \dots, \theta_T$ denote the resulting iterates and $\widehat{\theta}_T = \sum_{t=1}^T \theta_t/T,$ then the following inequality holds \textup:
\begin{align*}
\mathcal{L}(\widehat{\theta}_T) - \mathcal{L}(\theta^\star) &\leq \frac{1}{T}\Big(\frac{1}{\beta }V_{\theta_0}(\theta^\star, \theta_0) + \sum_{t=0}^{T-1} \langle \epsilon_t, \theta^\star - \theta_{t+1} \rangle \Big) \\
&\leq \frac{\nu R^2}{\beta T} + 2\bar{\epsilon}R
\end{align*}
where $\epsilon_t = \widehat{g}(\theta_t) - g(\theta_t)$ and $\bar{\epsilon} = \max_{t=0\dots T-1}\|\epsilon_t\|_{*}.$
\end{proposition}
Proposition~\ref{prop:uncorrectedMD} is proven in Appendix~\ref{proof:uncorrectedMD} based on~\cite[Proposition 2.1]{Juditsky2020SparseRB} and quantifies the progress of mirror descent on the objective value while measuring the impact of the gradient errors.
The original version of~\cite{Juditsky2020SparseRB} considers a stochastic optimization problem in which a new sample arrives at each iteration providing an unbiased estimate of the gradient so that it is possible to obtain a bound with optimal quadratic dependence on the statistical error. Though the above result is suboptimal in this respect, we will show in the sequel that an optimal statistical rate can still be achieved using a multistage procedure.
Notice that the previous statement only provides guarantees for the average $\widehat{\theta}_T = \sum_{t=1}^T \theta_t/T$. While this is commonplace for online settings, we intuitively expect the last iterate $\theta_T$ to be the best estimate of $\theta^\star$ in our batch setting where all the data is available from the beginning.
In order to address this issue, we define a \emph{corrected} proximal operator given an upper bound $\bar{\epsilon}$ on the statistical error :
\begin{equation}\label{eq:iterMDcorrected}
\widehat{\prox}_{\beta}(u, \theta; \theta_0, \Theta) = \argmin_{\theta'\in\Theta} \{\langle \beta u, \theta' \rangle + \beta\bar{\epsilon}\|\theta' - \theta\| + V_{\theta_0}(\theta', \theta)\}.
\end{equation}
For this new operator, the following statement applies.
\begin{proposition}\label{prop:correctedMD}
In the setting of Proposition~\ref{prop:uncorrectedMD}, let mirror descent be similarly run with constant step size $\beta \leq 1/L$ starting from $\theta_0 \in \Theta$ with $\Theta = B_{\|\cdot \|}(\theta_0, R)$ for some radius $R>0$. Let $\theta_1, \dots, \theta_T$ denote the resulting iterates obtained through $\theta_{t+1} = \widehat{\prox}_{\beta}(\widehat{g}(\theta_t), \theta_t;\theta_0, \Theta)$ then the following inequality holds \textup:
\begin{align*}
\mathcal{L}(\theta_T) - \mathcal{L}(\theta^\star) &\leq \frac{1}{T}\Big(\frac{1}{\beta }V_{\theta_0}(\theta^\star, \theta_0) + 2\sum_{t=0}^{T-1} \langle \epsilon_t, \theta^\star - \theta_{t+1} \rangle \Big) \\
&\leq \frac{\nu R^2}{\beta T} + 4\bar{\epsilon}R,
\end{align*}
where $\bar{\epsilon} = \max_{t=0\dots T-1}\|\epsilon_t\|_{*}$.
\end{proposition}
The proof of Proposition~\ref{prop:correctedMD} is given in Appendix~\ref{proof:correctedMD} and mainly differs from that of Proposition~\ref{prop:uncorrectedMD} in that the introduced correction allows to show a monotonous decrease of the objective i.e. $\mathcal{L}(\theta_{t+1}) \leq \mathcal{L}(\theta_{t})$ letting us draw the conclusion on the last iterate. Nevertheless, we suspect that the correction is not really needed for this bound to hold on $\theta_T$ and consider it rather as an artifact of our proof.
Propositions~\ref{prop:uncorrectedMD} and~\ref{prop:correctedMD} only state a linear dependence of the final excess risk on the statistical error $\bar{\epsilon}$ which leads to a suboptimal statistical rate of $1/\sqrt{n}$. However, the optimal rate of $1/n$ can be achieved by leveraging the quadratic growth condition of Assumption~\ref{asm:quadgrowth} upon running \emph{multiple stages} of Mirror Descent~\cite{juditsky2011first, Juditsky2020SparseRB}. We now present the corresponding multistage mirror descent algorithm.
\paragraph{Algorithm : Approximate Multistage Mirror Descent (AMMD)}
\begin{itemize}
\item \textit{Initialization}: Initial parameter $\theta^{(0)}$ and $R>0$ such that $\theta^{\star} \in \Theta := B_{\|\cdot\|}(\theta_0, R)$.
Number of stages $K > 0$. Step size $\beta \leq 1/L$. Quadratic minoration constant $\kappa$.
High probability upperbound $\bar{\epsilon}$ on the error $\|\widehat{g}(\theta) - g(\theta)\|_*$.
Upperbound $\bar{s}$ on the sparsity $s$.
\item Set $R_0 = R$.
\item Loop over stages $k = 1\dots K$ :
\begin{itemize}
\item Set $\theta^{(k)}_0 = \theta^{(k-1)}$, $\Theta_k = B_{\|\cdot\|}(\theta^{(k)}_0, R_{k-1})$
\item Run iteration \begin{equation*}
\theta^{(k)}_{t+1} = \prox_{\beta}\big(\widehat{g}(\theta^{(k)}_t), \theta^{(k)}_t; \theta^{(k)}_0, \Theta_k \big),
\end{equation*} for $T_k$ steps with $T_k = \Big\lceil \frac{\nu R_{k-1} }{\beta \bar{\epsilon}}\Big\rceil$.
\item Set $\theta^{(k)} = \sparse_{\bar{s}} (\widetilde{\theta}^{(k)})$ where $\widetilde{\theta}^{(k)} = \theta^{(k)}_{T_k}$
\item Set $R_k = \frac{1}{2}(R_{k-1} + \frac{40\bar{s}\bar{\epsilon}}{\kappa})$.
\end{itemize}
\item \textit{Output}: The final stage estimate $\theta^{(K)}$.
\end{itemize}
The AMMD algorithm borrows ideas from~\cite{juditsky2011first, juditsky2014deterministic, Juditsky2020SparseRB} aiming to achieve linear convergence using mirror descent. The main trick lies in the fact that performing multiple stages of mirror descent allows to repeatedly restrict the parameter space into a ball of radius $R_k$ which shrinks geometrically with each stage. At the end of each stage, the last optimisation iterate is replaced by the closest $\bar{s}$-sparse parameter through $\sparse_{\bar{s}}(\cdot)$. In this work, we find that the radius $R_k$ evolves following a special contraction as a result of the statistical error being factored in. Additionally, we show that this procedure also allows to improve the result of Proposition~\ref{prop:correctedMD} to achieve a fast statistical rate.
The following statement expresses the theoretical properties of AMMD.
\begin{theorem}\label{thm:MDoptimalrate}
Grant Assumptions~\ref{asm:sparse},~\ref{asm:data},~\ref{asm:lipsmoothloss} and~\ref{asm:quadgrowth}. Let $L>0$ denote the Lipschitz smoothness constant for the objective $\mathcal{L}$. Assume approximate Mirror Descent is run with step size $\beta \leq 1/L$ starting from $\theta_0 \in\Theta$ such that $\theta^\star \in B_{\|\cdot\|}(\theta_0, R)$ for some $R > 0$ and using a gradient estimator $\widehat{g}$ with error upperbound $\bar{\epsilon}$ as in Proposition~\ref{eq:iterMDuncorrected}, then after $K$ stages we have the inequalities \textup:
\begin{equation*}
\big\|\theta^{(K)} - \theta^\star\big\| \leq \sqrt{2\bar{s}} \big\|\theta^{(K)} - \theta^\star\big\|_2 \leq 2\sqrt{2\bar{s}} \big\|\widetilde{\theta}^{(K)} - \theta^\star\big\|_2 \leq 2^{-(K-1)/2}R + \frac{40\bar{s}\bar{\epsilon}}{\kappa},
\end{equation*}
\begin{equation*}
\mathcal{L}(\widetilde{\theta}^{(K)}) - \mathcal{L}(\theta^\star) \leq 10\bar{\epsilon} \Big(2^{-K}R + \frac{40 \bar{s}\bar{\epsilon}}{\kappa}\Big).
\end{equation*}
Moreover\textup, the corresponding number of necessary iterations is bounded by \textup:
\begin{equation*}
T = \sum_{k=1}^K T_k \leq \frac{2R \nu}{\beta \bar{\epsilon}} + K\Big(1 + \frac{40 \nu \bar{s}}{\kappa \beta}\Big).
\end{equation*}
\end{theorem}
Theorem~\ref{thm:MDoptimalrate} is proven in Appendix~\ref{proof:thmMD} and may be compared to~\cite[Theorem 2.1]{Juditsky2020SparseRB} which uses similar ideas. The optimisation error vanishes exponentially with respect to the number of stages $K$, therefore, the same rate holds with respect to the iterations since successive stages contain a decreasing number of them.
As for the statistical error, one can see that Theorem~\ref{thm:MDoptimalrate} exhibits a dependence in $\bar{\epsilon}^2$ of the excess risk upperbound so that the suboptimal statistical rate in Propositions~\ref{prop:uncorrectedMD} and~\ref{prop:correctedMD} is improved into a fast rate as announced. This is accomplished by shrinking the size of the considered parameter set through the stages until it reaches the scale of the statistical error, yielding an optimal rate (see the proof for further details). We can now turn to the case of a non smooth loss function $\ell$.
\section{The non smooth case with Dual Averaging}\label{sec:da}
In the previous section, we saw how sparse estimation can be performed using the Mirror Descent algorithm to optimize a smooth objective with a non Euclidean metric on the parameter space. The smoothness property is necessary for these results to hold so that many loss functions not satisfying it are left uncovered. Therefore, we propose to use another algorithm for non smooth objectives. The alternative is the Dual Averaging algorithm~\cite{nesterov2009primal} which was already used for non smooth sparse estimation in~\cite{agarwal2012stochastic} for instance. Since the original algorithm requires to average the iterates to obtain a parameter with provable convergence properties, we instead use a variant~\cite{nesterov2015quasi} for which such properties apply for individual iterates.
The smoothness condition in Assumption~\ref{asm:lipsmoothloss} is no longer required but we still need to replace it with a Lipschitz property :
\begin{assumption}\label{asm:lipschitz}
There exists a positive constant $M > 0$ such that the objective $\mathcal{L}$ is $M$-Lipschitz w.r.t. the norm $\|\cdot\|$ i.e. for all $\theta, \theta' \in \Theta$ it holds that \textup:
\begin{equation*}
\mathcal{L}(\theta) - \mathcal{L}(\theta') \leq M \|\theta - \theta'\|.
\end{equation*}
\end{assumption}
We also replace Assumption~\ref{asm:quadgrowth} by the following weaker assumption :
\begin{assumption}\label{asm:pseudolingrowth}
There exist positive constants $\kappa, \lambda > 0$ such that the following inequality holds \textup:
\begin{equation*}
\mathcal{L} (\theta) - \mathcal{L}(\theta^\star) \geq \frac{\kappa\|\theta - \theta^\star\|_2^2}{\lambda + \|\theta - \theta^\star\|_2}.
\end{equation*}
\end{assumption}
We introduce this (to our knowledge) previously unknown assumption in the literature which we call the \emph{pseudo-linear} growth assumption in order to better suit the setting of this section. Indeed, few non-smooth loss functions, if any, result in quadratically growing objectives as Assumption~\ref{asm:quadgrowth} requires. Note that the lower bound of Assumption~\ref{asm:pseudolingrowth} is linear away from the optimum i.e. for big $\|\theta - \theta^\star\|_2$ and behaves quadratically around it. This assumption is also weaker than a linear lower bound proportional to $\|\theta - \theta^\star\|_2$ because of its quadratic behaviour around the optimum. We will show that, for $\kappa$ big enough, this minoration suffices to obtain linear convergence to a solution with fast statistical rate.
Analogously to Mirror Descent's distance generating function $\omega$, we let $\omega : \Theta \to \mathbb{R}^+$ be the \emph{prox-function}. We choose to denote it similarly since it plays an analogous role for Dual Averaging and has the same properties as those listed in Definition~\ref{def:dgf}.
Let $(a_t)_{t\geq0}$ be a sequence of step sizes and $(\gamma_t)_{t \geq 0}$ a non decreasing sequence of positive scaling coefficients. The DA procedure is defined, given an initial $\theta_0 \in \Theta$, by the following scheme :
\begin{align*}
s_t = \frac{1}{A_t} \sum_{i=0}^t a_i \widehat{g}_i \quad \text{with} \quad &\widehat{g}_i = \widehat{g}(\theta_i) \:\: \text{and}\:\: g_i = g(\theta_i) \quad \forall i =0, \dots , T. \\
A_t = \sum_{i=0}^t a_i \quad \text{and}\quad &\theta_t^+ = \argmin_{\theta\in \Theta}A_t \langle s_t, \theta \rangle + \gamma_t \omega(\theta). \\
\theta_{t+1} = (1 - \tau_t)\theta_t + &\tau_t \theta_t^+ \quad\quad \text{where} \quad \quad\tau_t = \frac{a_{t+1}}{A_{t+1}}.
\end{align*}
\begin{proposition}\label{prop:DA}
Grant Assumption~\ref{asm:lipschitz}, let Dual Averaging be run following the above scheme\textup, let $R>0$ such that $\Theta \subseteq B_{\|\cdot\|}(\theta_0, R)$ and denote $\bar{\epsilon} = \max_i \|\epsilon_i\|_*,$ we have the following inequality \textup:
\begin{equation*}
A_t(\mathcal{L}(\theta_t) - \mathcal{L}(\theta^\star)) + \frac{\gamma_t}{2}\|\theta_t^+ - \theta^\star\|^2 \leq \gamma_t \omega(\theta^\star) + \sum_{i=0}^t\frac{a_i^2}{2\gamma_{i-1}}\|g_i\|_*^2 + 4A_t R \bar{\epsilon}.
\end{equation*}
In particular\textup, by choosing $a_i = 1$ and $\gamma_i = \sqrt{i+1}$ for all $i$ we get \textup:
\begin{equation*}
\mathcal{L}(\theta_t) - \mathcal{L}(\theta^\star) \leq \frac{1}{\sqrt{t}}\big( \omega(\theta^\star) + M^2\big) + 4 R \bar{\epsilon}.
\end{equation*}
\end{proposition}
The proof of Proposition~\ref{prop:DA} is given in Appendix~\ref{proof:DA} and is inspired from~\cite[Theorem 3.1]{nesterov2015quasi}. In this result, we manage to obtain a statement in terms of the individual iterates $\theta_t$ thanks to the running average performed in the above scheme whereas the initial study of dual averaging focused on the \emph{average} of the iterates~\cite{nesterov2009primal}. Notice that the convergence speed degrades to $1/\sqrt{t}$ with the new assumptions as opposed to $1/t$ previously. This convergence speed is the fastest possible and cannot be improved with a different choice of $a_i$ and $\gamma_i$. Most importantly, Proposition~\ref{prop:DA} quantifies the impact of the errors on the gradients on the quality of the optimisation result and shows that it remains controlled in this case too.
As in the previous section, the statistical rate we initially obtain is suboptimal and a multistage procedure is needed to improve it. We explicit the algorithm in question below.
\paragraph{Algorithm : Approximate Multistage Dual Averaging (AMDA)}
\begin{itemize}
\item \textit{Initialization}: Initial parameter $\theta_0$ and $R>0$ such that $\theta^{\star} \in \Theta := B_{\|\cdot\|}(\theta_0, R)$.
Pseudolinear minoration constants $\kappa, \lambda$.
High probability upperbound $\bar{\epsilon}$ on the error $\|\widehat{g}(\theta) - g(\theta)\|_*$.
Upperbound $\bar{s}$ on the sparsity $s$.
\item Set $R_0 = R$ and $\tau = \frac{10\sqrt{8\bar{s}} \bar{\epsilon}}{\kappa}$ and $R^\star = \frac{80 \lambda \bar{s}\bar{\epsilon}}{\kappa}$.
\item Set $k=0$ and the per stage number of iterations $T' = \Big\lceil \Big(\frac{\nu + M^2}{\bar{\epsilon}}\Big)^2 \Big\rceil$
\item For $k = 1,\dots, K$ :
\begin{itemize}
\item Set $\theta^{(k)}_0 = \theta^{(k-1)}$, $\Theta_k = B_{\|\cdot\|}(\theta^{(k)}_0, R_{k-1}).$
\item Run Dual averaging with prox-function $\omega_{\widehat{\theta}^{(k-1)}}$ and steps $a_i = R_{k-1}$ for $T'$ iterations.
\item Set $\theta^{(k)} = \sparse_{\bar{s}} (\widetilde{\theta}^{(k)})$ where $\widetilde{\theta}^{(k)} := \theta_{T'}^{(k)}$.
\item Set $R_{k} =\max\big( \tau R_{k-1}, \frac{1}{2}(R_{k-1} + R^\star) \big)$.
\end{itemize}
\item \textit{Output}: The final stage estimate $\theta^{(K)}$.
\end{itemize}
The basic idea behind the AMDA algorithm is the same as for AMMD in Section~\ref{sec:md}. Namely, multiple optimisation stages are run through which the parameter space is increasingly restricted allowing to obtain similar benefits regarding convergence speed and statistical performance. However, it is worth noting that these improvements are obtained under much milder conditions here since the objective may not even be smooth and is only required to satisfy the mild growth condition of Assumption~\ref{asm:pseudolingrowth} whereas smoothness and quadratic minoration or strong convexity were indispensable in previous works~\cite{juditsky2011first, juditsky2014deterministic, Juditsky2020SparseRB}.
We state the convergence guarantees for this algorithm in the following Theorem.
\begin{theorem}\label{thm:DAoptimalrate}
Grant Assumptions~\ref{asm:sparse},~\ref{asm:data},~\ref{asm:lipschitz},~\ref{asm:pseudolingrowth} and assume that $\tau = \frac{10\sqrt{8\bar{s}} \bar{\epsilon}}{\kappa} < 1$. At the end of each stage $k \geq 1$, we have \textup:
\begin{equation}\label{eq:thmDAparamIneq}
\big\|\theta^{(k)} - \theta^{\star}\big\| \leq \sqrt{2\bar{s}}\big\|\theta^{(k)} - \theta^{\star}\big\|_2 \leq 2\sqrt{2\bar{s}}\big\|\widetilde{\theta}^{(k)} - \theta^{\star}\big\|_2 \leq R_k,
\end{equation}
\begin{equation}\label{eq:thmDAobjIneq}
\mathcal{L}(\widetilde{\theta}^{(k)}) - \mathcal{L}(\theta^\star) \leq 5\bar{\epsilon}R_{k-1}.
\end{equation}
Moreover, the total number of necessary iterations before $R_k \leq 2R^\star = \frac{160 \lambda \bar{s}\bar{\epsilon}}{\kappa}$ is at most
\begin{equation*}
\log(R_0 / R^\star) \Big(\frac{1}{\log(1/\tau)} + \frac{1}{\log(2)}\Big)\Big( \Big(\frac{\nu + M^2}{\bar{\epsilon}}\Big)^2 +1\Big).
\end{equation*}
\end{theorem}
The proof of Theorem~\ref{thm:DAoptimalrate} is given in Appendix~\ref{proof:thmDA} and shows that the optimization stages go through two phases: an initial \textit{linear} phase corresponding to the linear regime of the lower bound given by Assumption~\ref{asm:pseudolingrowth} and a later \textit{quadratic} phase during which the quadratic rate takes over. The success of the linear phase relies on the condition $\tau < 1$ which can be rewritten as $\bar{\epsilon} \leq O(\bar{s} \kappa)$ where the factor $\bar{s}$ is a byproduct of measuring the parameter error with the norm $\|\cdot\|$ while Assumption~\ref{asm:pseudolingrowth} is stated with the Euclidean one. In the linear regime, $\kappa$ acts as a lower bound for the gradient norm so that the condition ensures that the error is smaller than the actual gradient allowing the optimisation to make progress.
\section{Applications}\label{sec:applis}
We now consider a few problems which may be solved using the previous optimization procedures. As said earlier, we have omitted to quantify the gradient errors $\|\epsilon\|_*$ until now. This is because the definition of the dual norm $\|\cdot\|_*$ is problem dependent. In the next subsections, we consider a few instances and propose adapted gradient estimators for them. In each case, the existence of a second moment for the gradient random variable $G(\theta) := \ell'(\langle \theta, X \rangle, Y)X$ is required. This follows from the next Lemma proven in Appendix~\ref{proof:obj-grad-moment} based on Assumption~\ref{asm:data}.
\begin{lemma}
\label{lem:obj-grad-moment}
Under Assumption~\ref{asm:data} the objective $\mathcal{L}(\theta)$ is well defined for all $\theta \in \Theta$ and we have \textup:
\begin{equation*}
\mathbb{E}\big[ \big\| G(\theta) \big\|_*^2 \big] = \mathbb{E}\big[ \big\|\ell'(X^\top \theta, Y) X \big\|_*^2 \big] < +\infty.
\end{equation*}
\end{lemma}
In what follows, we will assume that, at each step of the optimization algorithm, the estimation of the gradient is performed with a new batch of data. For example, if the available dataset contains $n$ samples then it needs to be divided into $T$ disjoint splits in order to make $T$ optimization steps. This is necessary in order to guarantee that the gradient samples used for estimation at each step $t$ are independent from $\theta_t$, the (random) current parameter which depends on the data used before.
This trick was previously used for example in~\cite{HeavyTails} for the same reasons. A possible alternative is to use an $\epsilon$-net argument or Rademacher complexity in order to obtain uniform deviation bounds on gradient estimation over a compact parameter set $\Theta$. However, this entails extra dependence on the dimension in the resulting deviation bound which we cannot afford in the high-dimensional setting. For these reasons, we prefer to use data splitting in this work and regard it more like a proof artifact rather than a true practical constraint. Note that we do not implement it later in our experimental section.
\subsection{Vanilla sparse estimation}\label{sec:appliVanillaSparse}
In this section, we consider the problem of optimizing an objective $\mathcal{L} (\theta) = \mathbb{E} [\ell(\langle \theta, X \rangle , Y)]$ where the covariate space $\mathcal X$ is simply $\mathbb{R}^d$ and the labels are either real numbers $\mathcal Y = \mathbb{R}$ (regression) or binary labels (binary classification). In this case, the parameter space is a subset $\Theta \subset \mathbb{R}^d$, the sparsity of a parameter $\theta \in \Theta$ is measured as its number of nonzero entries $S(\theta) = \sum_{j\in\setint{d}}\ind{\theta_j\neq 0}$, and $\|\cdot\|$ is defined as the $\ell_1$ norm $\|\cdot\| = \|\cdot\|_1$ so that $\|\cdot\|_* = \|\cdot\|_{\infty}$. We define the distance generating function $\omega$ as :
\begin{equation*}
\omega(\theta) = \frac{1}{2}e\log(d)d^{(p-1)(2-p)/p} \|\theta\|_p^2 \quad \text{with }\quad p = 1 + \frac{1}{\log (d)}.
\end{equation*}
One can check that the above definition satisfies the requirements of Definition~\ref{def:dgf}. In particular, it is strongly convex w.r.t. $\|\cdot\|$ and quadratically growing with constant $\nu = \frac 1 2 e^2 \log(d)$ (see~\cite[Theorem 2.1]{nesterov2013first}).
We consider Assumption~\ref{asm:data} on the data with a constant fraction of outliers $|\mathcal O| \leq \eta n$ for some $\eta < 1/2$ ($\eta$-corruption) so that the gradient samples $g^i(\theta) := \ell'(\theta^\top X_i, Y_i) X_i$ may be both heavy-tailed and corrupted as well. We propose to compute $\widehat{g}(\theta)$ as the coordinatewise trimmed mean of the sample gradients i.e.
\begin{equation}\label{eq:coordTMestimator}
\widehat{g}_j(\theta) = \mathtt{TM}_{\alpha} \big(g_j^1(\theta), \dots, g_j^n(\theta)\big),
\end{equation}
where, assuming without loss of generality that $n$ is even, the trimmed mean estimator with parameter $\alpha$ for a sample $x_1,\dots, x_n \in \mathbb{R}$ is defined as follows
\begin{equation*}
\mathtt{TM}_{\alpha}(x_1, \dots, x_n) = \frac{2}{n} \sum_{i = n/2 +1}^n
q_\alpha \vee x_i \wedge q_{1 - \alpha},
\end{equation*}
where we denoted $a \wedge b:= \min(a, b)$ and $a \vee b:= \max(a, b)$ and used the quantiles $q_\alpha := x^{([\alpha n/2])}$ and $q_{1 - \alpha} = x^{([(1-\alpha) n/2])}$ with $x^{(1)} \leq \cdots \leq x^{(n/2)}$ the order statistics of $(x_i)_{i\in \setint{n/2}}$ and where $[\cdot]$ denotes the integer part.
The main hurdle to compute the trimmed mean estimator is to find the two previous quantiles. A naive approach for this task would be to sort all the values leading to an $O(n\log(n))$ complexity. However, this can be brought down to $O(n)$ using the median-of-medians algorithm (see for instance \cite[Chapter 9]{cormen2009introduction}) so that the whole procedure runs in linear time.
We now give the deviation bound satisfied by the estimator~(\ref{eq:coordTMestimator}).
We denote $x^{j}$ as the $j$-th coordinate of a vector $x$.
\begin{lemma}\label{lem:coordTM}
Grant Assumption~\ref{asm:data} with a fraction of outliers $|\mathcal O| \leq \eta n$ with $\eta < 1/8$. Fix $\theta \in \Theta$, let $\sigma_j^2 = \Var (\ell'(\theta^\top X, Y) X^j)$ for $j\in \setint{d}$ be the gradient coordinate variances and let $1> \delta > e^{-n/2}/4$ be a failure probability and consider the coordinatewise trimmed mean estimator~\eqref{eq:coordTMestimator} with parameter $\alpha = 8\eta + 12\frac{\log(4/\delta)}{n}$. Denoting $\sigma_{\max}^2 = \max_j \sigma_j^2$, we have with probability at least $1 - \delta$ :
\begin{equation}
\big\|\widehat{g}(\theta) - g(\theta)\big\|_{\infty} \leq 7 \sigma_{\max} \sqrt{4\eta + 6\frac{\log(4/\delta) + \log(d)}{n}}.
\end{equation}
\end{lemma}
\begin{proof}
This is an almost immediate application of~\cite[Lemma 9]{gaiffas2022robust} (see also~\cite[Theorem 1]{lugosi2021robust}). By an immediate application of the latter, we obtain for each $j\in\setint{d}$ that with probability at least $1 - \delta/d$ we have :
\begin{equation}
\big|\widehat{g}_j(\theta) - g_j(\theta)\big| \leq 7 \sigma_{j} \sqrt{4\eta + 6\frac{\log(4/\delta) + \log(d)}{n}}.
\end{equation}
Hence, the lemma follows by a simple union bound argument.
\end{proof}
For the sake of simplicity, this deviation bound is only stated for a square integrable gradient which yields a $\sqrt{\eta}$ dependence in the corruption rate. More generally, for a random variable admitting a finite moment of order $k$, one can derive a bound in terms of $\eta^{1 - 1/k}$ which reflects a milder dependence for greater $k$, see~\cite[Lemma 9]{gaiffas2022robust} for the bound in question.
In a way, the fact that the gradient error is measured with the infinity norm in this setting is a ``stroke of luck'' since the optimal dependence in the dimension for the statistical error becomes achievable using only a univariate estimator. This is in contrast with situations where multivariate robust estimators need to be used for which the combination of efficiency, sub-Gaussianity and robustness to $\eta$-corruption is hard to come by.
Based on Lemma~\ref{lem:coordTM} we obtain a gradient error of order $\bar{\epsilon} = O(\sqrt{\log(d)/n})$. Plugging this deviation rate into Theorem~\ref{thm:MDoptimalrate} yields the optimal $s\log(d)/n$ rate for vanilla sparse estimation. The same applies for Theorem~\ref{thm:DAoptimalrate} provided the condition $\tau < 1$ holds.
\begin{corollary}
In the context of Theorem~\ref{thm:MDoptimalrate} and Lemma~\ref{lem:coordTM}, let the AMMD algorithm be run starting from $\theta_0\in\Theta = B_{\|\cdot\|}(\theta_0, R)$ using the coordinatewise trimmed mean estimator with sample splitting i.e. at each iteration a different batch of size $\widetilde{n} = n/T$ is used for gradient estimation with confidence $\widetilde{\delta} = \delta/T$ where $T$ is the total number of iterations. Let $K$ be the number of stages and $\widehat{\theta}$ the obtained estimator. Denote $\sigma_{\max}^2 = \sup_{\theta \in \Theta} \max_{j\in\setint{d}}\Var (\ell'(\theta^\top X, Y) X^j)$, with probability at least $1 - \delta$, the latter satisfies :
\begin{equation*}
\big\|\widehat{\theta} - \theta^\star\big\|_2 \leq \frac{2^{-K/2}R}{\sqrt{\bar{s}}} + \frac{140\sqrt{2\bar{s}}\sigma_{\max}}{\kappa}\sqrt{4\eta + 6 \frac{\log(4/\widetilde{\delta}) + \log(d)}{\widetilde{n}}}.
\end{equation*}
\end{corollary}
\begin{proof}
The result is easily obtained by combining Theorem~\ref{thm:MDoptimalrate} and Lemma~\ref{lem:coordTM} with a union bound argument over all iterations $T$ in order to bound $\bar{\epsilon} = \max_{t=0,\dots, T-1}\|\epsilon_t\|_*$ as defined in Proposition~\ref{prop:correctedMD}.
\end{proof}
In the above upper bound, the optimisation error vanishes exponentially with the number of stages $K$ so that the final error can be attributed in large part to the second statistical error term. The latter achieves the nearly optimal $\sqrt{s\log(d)/n}$ rate and combines robustness to heavy tails and $\eta$-corruption. Moreover, this statement holds for a \emph{generic} loss function satisfying the assumptions of Section~\ref{sec:md}. These can be further weakened to those given in Section~\ref{sec:da} by using Dual Averaging while preserving the same statistical rate\footnote{Indeed, a combination of Lemma~\ref{lem:coordTM} with Theorem~\ref{thm:DAoptimalrate} is possible as well in order to obtain guarantees for non smooth objectives. However, we do not formulate another statement to avoid excessive repetition.}. To our knowledge, this is the first result with such properties for vanilla sparse estimation whereas previous results from the literature either focused on specific learning problems with a fixed loss function~\cite{dalalyan2019outlier, sasai2022robust} or isolated the issues of robustness by assuming the data to be either heavy tailed or Gaussian and corrupted~\cite{balakrishnan2017computationally, liu2019high, liu2020high, Juditsky2020SparseRB}. Furthermore, we stress that this error bound is achieved at a comparable computational cost to that of a standard non robust algorithm since, as mentioned earlier, the robust trimmed mean estimator can be computed in linear time.
A possible room for improvement is to try to remove the $\bar{s}$ factor multiplying the corruption rate $\eta$. The only works we are aware of achieving this are~\cite{liu2020high, sasai2022robust} but both involve costly data-filtering steps. We suspect this may be an inevitable price to pay for such an improvement.
\subsection{Group sparse estimation}
In the group sparse case, we again consider the covariate space $\mathcal X = \mathbb{R}^d$ where the coordinates $\setint{d}$ are arranged into groups $G_1, \dots, G_{N_G}$ which form a partition of the coordinates $\setint{d}$ and sparsity is measured in terms of these groups i.e. $S(\theta) = \sum_{j\in\setint{N_G}}\ind{\theta_{G_j} \neq 0}$ and we assume the optimal $\theta^\star = \argmin_{\theta}\mathcal{L}(\theta)$ satisfies $S(\theta^\star) \leq s$.
The norm $\|\cdot\|$ is set to be the $\ell_1/\ell_2$ norm : $\|\theta\|_{1, 2} = \sum_{j\in\setint{N_G}} \|\theta_{G_j}\|_2$ and the dual norm is the analogous $\ell_{\infty}/\ell_2$ norm $\|\theta\|_{\infty, 2} = \max_{j\in\setint{N_G}} \|\theta_{G_j}\|_2$. The label set $\mathcal Y$ may be equal to $\mathbb{R}$ (regression) or a finite set (binary or multiclass classification).
For simplicity, we assume that the groups are of equal size $m$ so that $d = m N_G$. This is for example the case when trying to solve a $d$-dimensional linear multiclass classification with $K$ classes by estimating a parameter $\theta\in \mathbb{R}^{d\times K}$ and predicting $\argmax_j (\theta^\top X)_j$ for a datapoint $X \in \mathbb{R}^d$. In this case, it makes sense to consider the rows $(\theta_{i,:})_{i\in\setint{d}}$ as groups which are collectively determined to be zero or not depending on the importance of feature $i$. For simplicity, we restrict ourselves to this setting until the end of this section with no loss of generality.
We assume the data corresponds to Assumption~\ref{asm:data} with $\eta$-corruption (i.e. $|\mathcal O| \leq \eta n$). Analogously to the vanilla case, we propose to estimate the gradient \emph{groupwise} i.e. one group of coordinates at a time. For this task, a multivariate, sub-Gaussian and corruption-resilient estimation algorithm is needed. We suggest to use the estimator advocated in \cite{diakonikolas2020outlier} for this purpose which remarkably combines these qualities. We refer to it as the DKP estimator and restate its deviation bound here for the sake of completeness.
\begin{proposition}[{\cite[Proposition 1.5]{diakonikolas2020outlier}}]\label{prop:dkk}
Let $T$ be an $\eta$-corrupted set of $n$ samples from a distribution $P$ in $\mathbb{R}^d$ with mean $\mu$ and covariance $\Sigma$.
Let $\eta' = \Theta(\log(1/\delta )/n + \eta) \leq c$ be given\textup, for a constant $c > 0$. Then any stability-based algorithm on input $T$ and $\eta'$\textup, efficiently computes $\widehat{\mu}$ such that with probability at least $1 - \delta$\textup, we have \textup:
\begin{equation}
\big\|\widehat{\mu} - \mu\big\|_2 = O\bigg(\sqrt{\frac{\Tr(\Sigma) \log\mathrm{r}(\Sigma)}{n}} + \sqrt{\|\Sigma\|_{\mathrm{op}} \eta} + \sqrt{\frac{\|\Sigma\|_{\mathrm{op}}\log(1/\delta)}{n}}\bigg),
\end{equation}
where $\mathrm{r}(\Sigma) = \Tr(\Sigma)/\|\Sigma\|_{\mathrm{op}}$ is the stable rank of $\Sigma$.
\end{proposition}
The above bound is almost optimal up to the $\sqrt{\log\mathrm{r}(\Sigma)}$ factor which is at most $\sqrt{\log(d)}$. Note that we are also aware that \cite[Proposition 1.6]{diakonikolas2020outlier} states that, by adding a Median-Of-Means preprocessing step, stability based algorithms can achieve the optimal deviation. Nevertheless, the number $k$ of block means required needs to be such that $k \geq 100\eta n$ so that the corruption rate $\eta$ is strongly restricted because necessarily $n \geq k$. Therefore, we prefer to stick with the result above.
An algorithm with the statistical performance stated in Proposition~\ref{prop:dkk} is given, for instance, in~\cite[Appendix A.2]{diakonikolas2020outlier}, we omit it here for brevity. Now, we can reuse the previous section's trick by estimating the gradients \emph{blockwise} this time to obtain the following lemma:
\begin{lemma}\label{lem:groupDKK}
Grant Assumption~\ref{asm:data} with a fraction of outliers $|\mathcal O| \leq \eta n$. Fix $\theta \in \Theta$ and denote $G_1(\theta), \dots, G_n(\theta)$ the gradient samples distributed according to $G(\theta) \in \mathbb{R}^{d\times K}$ (except for the outliers). Let $\Sigma_j = \Var (G(\theta)_{j,:}) \in \mathbb{R}^{K \times K}$ be the gradient block variances. Consider the groupwise estimator $\widehat{g}(\theta)$ defined such that $\widehat{g}(\theta)_{j,:}$ is the DKP estimator applied to $G(\theta)_{j,:}$. Then we have with probability at least $1 - \delta$ :
\begin{equation}
\big\|\widehat{g}(\theta) - g(\theta)\big\|_{\infty, 2} \leq O\bigg(\max_j \sqrt{\frac{\Tr(\Sigma_j) \log\mathrm{r}(\Sigma_j)}{n}} + \sqrt{\|\Sigma_j\|_{\mathrm{op}}}\bigg( \sqrt{\eta} + \sqrt{\frac{\log(1/\delta) + \log(d)}{n}} \bigg)\bigg) \label{eq:dkkDeviation}
\end{equation}
\end{lemma}
\begin{proof}
Inequality~\eqref{eq:dkkDeviation} is straightforward to obtain using Proposition~\ref{prop:dkk} and a union bound argument on $j\in\setint{d}$.
\end{proof}
One can easily see that, in the absence of corruption, the above deviation bound scales as $\widetilde{O}\Big(\sqrt{\frac{K}{n}} + \sqrt{\frac{\log (d)}{n}} \Big)$. Combined with the sparsity assumption given above, plugging this estimation, which applies for the gradient error $\|\epsilon_t\|_*$, into Theorems~\ref{thm:MDoptimalrate} or~\ref{thm:DAoptimalrate} yields near optimal (up to a logarithmic factor) estimation rates for the group-sparse estimation problem~\cite{negahban2012unified, lounici2009taking}. The following corollary formalizes this statement.
\begin{corollary}
In the context of Theorem~\ref{thm:MDoptimalrate} and Lemma~\ref{lem:groupDKK}, let the AMMD algorithm be run starting from $\theta_0\in\Theta = B_{\|\cdot\|}(\theta_0, R)$ and using the blockwise DKP estimator with sample splitting i.e. at each iteration a different batch of size $\widetilde{n} = n/T$ is used for gradient estimation with confidence $\widetilde{\delta} = \delta/T$ where $T$ is the total number of iterations. Let $N$ be the number of stages and $\widehat{\theta}$ the obtained estimator. With probability at least $1 - \delta$, the latter satisfies :
\begin{align*}
\big\|\widehat{\theta} - \theta^\star\big\|_2 &\leq \frac{2^{-N/2}R}{\sqrt{\bar{s}}} + \frac{\sqrt{\bar{s}}}{\kappa} O\bigg(\sup_{\theta\in\Theta}\max_{j\in\setint{d}} \sqrt{\frac{\Tr(\Sigma_{\theta, j}) \log\mathrm{r}(\Sigma_{\theta, j})}{\widetilde{n}}}\: + \\
&\quad \sqrt{\|\Sigma_{\theta, j}\|_{\mathrm{op}}}\bigg( \sqrt{\eta} + \sqrt{\frac{\log(1/\widetilde{\delta}) + \log(d)}{\widetilde{n}}} \bigg)\bigg) \\
&\leq \frac{2^{-N/2}R}{\sqrt{\bar{s}}} + \frac{\sqrt{\bar{s}}}{\kappa}\widetilde{O}\bigg(\sup_{\theta\in\Theta}\max_{j\in\setint{d}} \sqrt{\frac{K}{\widetilde{n}}} + \bigg( \sqrt{\eta} + \sqrt{\frac{\log(1/\widetilde{\delta}) + \log(d)}{\widetilde{n}}} \bigg)\bigg),
\end{align*}
where $\Sigma_{\theta, j} = \Var(G(\theta)_{j,:})$.
\end{corollary}
As before, the stated bound reflects a linearly converging optimisation and displays a statistical rate nearly matching the optimal rate for group-sparse estimation~\cite{negahban2012unified, lounici2009taking} up to logarithmic factors. In addition, robustness to heavy tails and $\eta$-corruption likely makes this result the first of its kind for group-sparse estimation since all robust works we are aware of focus on vanilla sparsity.
\subsection{Low-rank matrix recovery}\label{sec:appliLowRank}
We also consider the variant of the problem where the covariates belong to a matrix space $\mathcal X = \mathbb{R}^{p\times q}$ in which case the objective $\mathcal{L} (\theta) = \mathbb{E} [\ell(\langle \theta, X \rangle , Y)]$ needs to be optimized over $\Theta \subset \mathbb{R}^{p\times q}$. In this setting, $\langle \cdot, \cdot \rangle$ refers to the Frobenius scalar product between matrices
\begin{equation*}
\langle a, b \rangle = \Tr(a^\top b).
\end{equation*}
Without loss of generality, we assume that $p \geq q$ and sparsity is meant as the number of non zero singular values i.e. for a matrix $A\in \mathbb{R}^{p\times q}$, denoting $\sigma(A) = (\varsigma_j(A))_{j\in\setint{q}}$ the set of its singular values we define $S(A) = \sum_{j\in\setint{q}}\ind{\varsigma_j(A) \neq 0}$. We set $\|\cdot\|$ to be the nuclear norm $\|A\| = \|\sigma(A)\|_1$ and the associated dual norm is the operator norm $\|\cdot\|_* = \|\cdot\|_{\mathrm{op}}$.
On the optimization side, an appropriate distance generating function (resp. prox-function) needs to be defined for this setting before Mirror Descent (resp. Dual Averaging) can be run. Based on previous literature (see~\cite[Theorem 2.3]{nesterov2013first} and~\cite{Juditsky2020SparseRB}), we know that the following choice satisfies the requirements of Definition~\ref{def:dgf} :
\begin{equation*}
\omega(\theta) = 2e \log(2q) \Big( \sum_{j=1}^q \varsigma_j(\theta)^{1+r}\Big)^{2/(1+r)} \quad \text{with}\quad r = 1/(12 \log(2q)).
\end{equation*}
This yields a corresponding quadratic growth parameter $\nu = O(\log(q))$. In order to fully define our optimization algorithm for this problem, it remains to specify a robust estimator for the gradient. This turns out to be a challenging question since the estimated value is matricial and the operator norm $\|\cdot\|_* = \|\cdot\|_{\mathrm{op}}$ emerging in this case is a fairly exotic choice to measure statistical error.
In order to achieve a nearly optimal statistical rate we define a new estimator called ``CM-MOM'' (short for Catoni Minsker Median-Of-Means) which combines methods from~\cite{minsker2018sub} for sub-Gaussian matrix mean estimation and ideas from~\cite{minsker2015geometric, hsu2016loss} in order to apply a Median-Of-Means approach for multivariate estimation granting robustness to outliers provided these are limited in number. We now define this estimator in detail. Let $\psi$ be a function defined as
\begin{equation*}
\psi(x) = \log(1 + |x| + x^2/2)
\end{equation*}
We consider a restricted version of Assumption~\ref{asm:data} in which the number of outliers is limited as\footnote{In fact, one may allow up to $|\mathcal O|\leq K/2$ outliers at the price of worse constants in the resulting deviation bound. See the proof of Proposition~\ref{prop:spectralMOM}} $|\mathcal O| \leq K/12$ where $K$ is an integer such that $K < n$. Provided a sample of matrices $A_1, \dots, A_n \in \mathbb{R}^{p\times q}$ and a scale parameter $\chi > 0$, the CM-MOM estimator proceeds as follows :
\begin{itemize}
\item Split the sample into $K$ disjoint blocks $B_1, \dots, B_K$ of equal size $m = n/K$.
\item Compute the dilated block means $\xi^{(j)}$ for $j=1,\dots, K$ as
\begin{equation*}
\xi^{(j)} = \frac{1}{\chi m}\sum_{i\in B_j} \psi(\theta \widetilde{A}_i) \in \mathbb{R}^{(p+q)\times (p+q)},
\end{equation*}
where the dilation $\widetilde{A}$ of matrix $A\in \mathbb{R}^{p\times q}$ is defined as $\widetilde{A} = \begin{pmatrix}0 & A \\ A^\top & 0 \end{pmatrix} \in \mathbb{R}^{(p+q)\times (p+q)}$ which is symmetric and the function $\psi$ is applied to a symmetric matrix $S\in\mathbb{R}^{d\times d}$ by applying it to its eigenvalues i.e. let $S = UDU^\top$ be its eigendecomposition with $D=\mathrm{diag}((\lambda_j)_{j\in\setint{d}})$ then $\psi(S) = U\psi(D)U^\top = U\mathrm{diag}((\psi(\lambda_j))_{j\in\setint{d}})U^\top.$
\item Extract the block means $\widehat{\mu}_j \in \mathbb{R}^{p\times q}$ such that $\xi^{(j)} = \begin{pmatrix}\xi^{(j)}_{11} & \widehat{\mu}_j \\ \widehat{\mu}_j^\top & \xi^{(j)}_{22} \end{pmatrix}.$
\item Compute the pairwise distances $r_{jl} = \|\widehat{\mu}_j - \widehat{\mu}_l\|_{\mathrm{op}}$ for $j,l \in \setint{K}$.
\item Compute the vectors $r^{(j)} \in \mathbb{R}^K$ for $j\in \setint{K}$ where $r^{(j)}$ is the increasingly sorted version of $r_{j:}$.
\item return $\widehat{\mu}_{\widehat{i}}$ where $\widehat{i} \in \argmin_i r^{(i)}_{K/2}$.
\end{itemize}
One may guess that the choice of the scale parameter $\chi$ plays an important role to guarantee the quality of the estimate. This aspect is inherited from Catoni's original estimator for the mean of a heavy-tailed real random variable~\cite{catoni2012challenging} from which Minsker's estimator~\cite{minsker2018sub}, which we use to estimate the block means, is inspired. The following statement gives the optimal value for $\chi$ and the associated deviation bound satisfied by CM-MOM.
\begin{proposition}[CM-MOM]\label{prop:spectralMOM}
Let $A_1,\dots, A_n\in \mathbb{R}^{p\times q}$ be an i.i.d sample following a random variable $A$ with expectation $\mu = \mathbb{E} A$ such that a subset of indices $\mathcal O \subset \setint{n}$ are outliers and finite variance
\begin{equation*}
v(A) = \max \big( \big\|\mathbb{E} (A - \mu)(A - \mu)^\top \big\|_{\mathrm{op}}, \big\|\mathbb{E} (A - \mu)^\top(A - \mu)\big\|_{\mathrm{op}} \big)< \infty.
\end{equation*}
Let $\delta > 0$ be a failure probability and take $K = \lceil 18\log(1/\delta) \rceil < n$ blocks\textup, we assume $n = mK$. Let $\widehat{\mu}$ be the CM-MOM estimate as defined above with scale parameter
\begin{equation*}
\chi = \sqrt{\frac{2m\log(8(p+q))}{v(A)}}.
\end{equation*}
Assume we have $|\mathcal O| \leq K/12$ then with probability at least $1 - \delta$ we have \textup:
\begin{equation*}
\big\| \widehat{\mu} - \mu \big\|_{\mathrm{op}} \leq 18\sqrt{\frac{v(A)\log(8(p+q))\log(1/\delta)}{n}}
\end{equation*}
\end{proposition}
Proposition~\ref{prop:spectralMOM} is proven in Appendix~\ref{proof:spectralMOM} and enjoys a deviation rate which scales optimally, up to logarithmic factors, as $\sqrt{p+q}$ in the dimension~\cite{vershynin2018high}. This dependence is hidden by the factor $\sqrt{v(A)}$ which scales in that order (see for instance~\cite{tropp2015introduction}). Although the dependence of the optimal scale $\chi$ on the unknown value of $v(A)$ constitutes an obstacle, previous experience using Catoni-based estimators~\cite{catoni2012challenging, pmlr-v97-holland19a, gaiffas2022robust} has shown that the choice is lenient and good results are obtained as long as a value of the correct scale is used. Possible improvements for Proposition~\ref{prop:spectralMOM} are to derive a bound with an additive instead of multiplicative term $\log(1/\delta)$ or supporting $\eta$-corruption. However, we are not aware of a more robust solution for matrix mean estimation in the general case than the above result.
Now that we have an adapted gradient estimation procedure, we can proceed to combine its deviation bound with our optimization theorems in order to obtain guarantees on learning performance.
\begin{corollary}\label{cor:lowrankmatrix}
In the context of Theorem~\ref{thm:MDoptimalrate} and Proposition~\ref{prop:spectralMOM}, let the AMMD algorithm be run starting from $\theta_0\in\Theta = B_{\|\cdot\|}(\theta_0, R)$ and using the CM-MOM estimator with sample splitting i.e. at each iteration a different batch of size $\widetilde{n} = n/T$ is used for gradient estimation with confidence $\widetilde{\delta} = \delta/T$ where $T$ is the total number of iterations. Assume that each batch contains no more than $K/12$ outliers. Let $N$ be the number of stages and $\widehat{\theta}$ the obtained estimator. With probability at least $1 - \delta$, the latter satisfies :
\begin{align*}
\big\|\widehat{\theta} - \theta^\star\big\|_2 &\leq \frac{2^{-N/2}R}{\sqrt{\bar{s}}} + \sup_{\theta \in \Theta}\frac{360\sqrt{\bar{s}}}{\kappa}\sqrt{\frac{2v(G(\theta)) \log(8(p+q)) \log(1/\widetilde{\delta})}{\widetilde{n}}},
\end{align*}
where $\|\cdot\|_2$ denotes the Frobenius norm.
\end{corollary}
Corollary~\ref{cor:lowrankmatrix} matches the optimal performance bounds given in classical literature for low-rank matrix recovery~\cite{koltchinskii2011nuclear, rohde2011estimation, candes2011tight, negahban2011estimation} up to logarithmic factors. The previous statement is most similar to~\cite[Proposition 3.3]{Juditsky2020SparseRB} except that it applies for more general learning tasks and under much lighter data assumption.
\section{Implementation and numerical experiments}\label{sec:exp}
In this section, we demonstrate the performance of our proposed algorithms on synthetic and real data. Before we proceed, we prefer to point out that our implementation does not \emph{exactly} correspond to the previously given pseudo-codes. Indeed, as the reader may have noticed previously, certain instructions of AMMD and AMDA require the knowledge of quantities which are not available in practice (such as $R$, the distance to the optimum) and, even in a controlled setting, the estimation of some quantities (such as the maximum gradient error $\bar{\epsilon}$) may be overly conservative which generally impedes the proper convergence of the optimization. We list the main divergences of our implementation from the theoretically studied procedures of AMMD and AMDA given before :
\begin{itemize}
\item For AMMD, we only use the conventional $\prox$ operator and never the corrected $\widehat{\prox}$ operator defined in Section~\ref{sec:md}.
\item The radii $R_k$ are taken constant equal to a fixed arbitrary $R > 0$.
\item The stage lengths ($T_k$ for AMMD and $T'$ for AMDA) are constant and fixed arbitrarily.
\item The number of stages is determined through a maximum number of iterations but the algorithm stops after the last whole stage.
\item The within-stage step-sizes $a_i$ in AMDA are fixed to a small constant (smaller than $R$) for more stability.
\item For both AMMD and AMDA, the whole dataset is used at each step to compute a gradient estimate and no data-splitting is performed.
\end{itemize}
\subsection{Synthetic sparse linear regression}
We first test our algorithms on the classic problem of linear regression. We generate $n$ covariates $X_i \in \mathbb{R}^d$ following a non-isotropic distribution with covariance matrix $\Sigma$ and labels $Y_i = X_i^\top \theta^\star + \xi_i$ for a fixed $s$-sparse $\theta^\star \in \mathbb{R}^d$ and simulated noise entries $\xi_i$. The covariance matrix $\Sigma$ is diagonal with entries drawn uniformly at random in $[1, 10]$.
We use the least-squares loss in this experiment and the problem parameters are $n = 500 , d=5000, s=40$ and a sparsity upper bound $\bar{s}=50$ is given to the algorithms instead of the real value. The noise variables $\xi_i$ always follow a Pareto distribution with parameter $\alpha = 2.05$. Apart from that we consider three settings :
\begin{enumerate}[label=(\alph*)]
\item The gaussian setting : the covariates follow a gaussian distribution.
\item The heavytailed setting : the covariates are generated from a multivariate Student distribution with $\nu=4.1$ degrees of freedom.
\item The corrupted setting : the covariates follow the same Student distribution and $5\%$ of the data ($(X_i, Y_i)$ pairs) are corrupted.
\end{enumerate}
We run various algorithms:
\begin{itemize}
\item AMMD using the trimmed mean estimator (\texttt{AMMD}).
\item AMDA using the trimmed mean estimator (\texttt{AMDA}).
\item The iterative thresholding procedure defined in~\cite{liu2019high} using the MOM estimator (\texttt{LLC\_MOM}).
\item The iterative thresholding procedure defined in~\cite{liu2019high} using the trimmed-mean estimator (\texttt{LLC\_TM}).
\item Lasso with CGD solver and the trimmed mean estimator as implemented in~\cite{gaiffas2022robust} (\texttt{Lasso\_CGD\_TM}).
\item Lasso with CGD solver as implemented in Scikit Learn~\cite{scikit-learn} (\texttt{Lasso\_CGD}).
\end{itemize}
Another possible baseline is the algorithm proposed in~\cite{liu2020high}. Nevertheless, we do not include it here because it relies on the outlier removal algorithm inspired from~\cite{balakrishnan2017computationally}. The latter requires to run an SDP subroutine making it excessively slow as soon as the dimension is greater than a few hundreds.
Note that the ``trimmed mean'' estimator used in~\cite{liu2019high} is different from ours since they simply exclude the entries below and above a pair of empirical data quantiles. On the other hand, the estimator we define in Section~\ref{sec:appliVanillaSparse} simply replaces the extreme values by the exceeded threshold before computing an average. This is also called a ``Winsorized mean'' and enjoys better statistical properties.
The algorithms using Lasso~\cite{tibshirani1996regression} optimize an $\ell_1$ regularized objective. The regularization is weighted by a factor $2\sigma \sqrt{\frac{2\log(d)}{n}}$ where $\sigma^2 = \Var(\xi)$ is the noise variance. The previous regularization weight is known to ensure optimal statistical performance, see for instance~\cite{bickel2009simultaneous}.
\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{linreg_HD_all.pdf}
\caption{L2 error $\|\theta_t - \theta^\star\|_2$ ($y$-axis) against iterations ($x$-axis) for all the considered algorithms in the simulation settings.}
\label{fig:lin_reg}
\end{figure}
The experiment is repeated 30 times and the results are averaged. We do not display any confidence intervals for better readability. Figure~\ref{fig:lin_reg} displays the results. We observe that Lasso based methods quickly reach good optima in general and that the version using the robust trimmed mean estimator is sometimes superior in the presence of heavy tails and corruption in particular. \texttt{AMMD} reaches nearly equivalent optima, albeit significantly slower than Lasso methods as seen for settings (a) and (b). However, it is somehow more robust to corruption as seen on setting (c). Unfortunately, the \texttt{AMDA} algorithm struggles to closely approximate the original parameter. We mainly attribute this to slow convergence in settings (a) and (b). Nonetheless, \texttt{AMDA} is among the most robust algorithms to corruption as seen on setting (c). The remaining iterative thresholding based methods \texttt{LLC\_MOM} and \texttt{LLC\_TM} seem to generally stop at suboptimal optima. The Median-Of-Means variant \texttt{LLC\_MOM} is barely more robust than \texttt{Lasso\_CGD} in the corrupted setting (c). The \texttt{LLC\_TM} variant is better but still inferior to \texttt{AMMD}. This reflects the superiority of the (Winsorized) trimmed mean used by \texttt{AMMD} and \texttt{Lasso\_CGD\_TM} to the conventional trimmed mean in \texttt{LLC\_TM}.
\subsection{Sparse classification on real data}
We also carry out experiments on real high dimensional binary classification datasets. These are referred to as \texttt{gina} and \texttt{bioresponse} and were both downloaded from \texttt{openml.com}. We run \texttt{AMMD}, \texttt{AMDA}, \texttt{LLC\_MOM} and \texttt{LLC\_TM} with similar sparsity upperbounds and various levels of corruption and track the objective value (Logistic loss) for each of them. The results are displayed on Figure~\ref{fig:classif_iter} (average over 10 runs). In the non corrupted case, we see that all algorithms reach approximately equivalent optima whereas they display different levels of resilience when corruption is present. In particular, \texttt{LLC\_MOM} is unsurprisingly the most vulnerable since it is based on Median-Of-Means which is not robust to $\eta$-corruption. The rest of the algorithms cope better thanks to the use of trimmed mean estimators, although \texttt{LLC\_TM} seems to be a little less robust which is probably due to the previously mentioned difference in its gradient estimator. Finally, Figure~\ref{fig:classif_iter} also shows that \texttt{AMMD} and \texttt{AMDA} (respectively using Mirror Descent and Dual Averaging) tend to reach generally better final optima despite converging a bit slower than the other algorithms. They also prove to be more stable, even when high step sizes are used.
\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{classif_HD_iter_fig.pdf}
\caption{Log loss ($y$-axis) along training iterations ($x$-axis) on two datasets (rows) for $0\%$ corruption (first column), $10\%$ corruption (middle column) and $20\%$ corruption (last column).}
\label{fig:classif_iter}
\end{figure}
\section{Conclusion}
In this work, we address the problem of robust supervised linear learning in the high-dimensional setting. In order to cover both smooth and non-smooth loss functions, we propose two optimisation algorithms which enjoy linear convergence rates with only a mild dependence on the dimension. We combine these algorithms with various robust mean estimators, each of them tailored for a specific variant of the sparse estimation problem. We show that the said estimators are robust to heavy-tailed and corrupted data and allow to reach the optimal statistical rates for their respective instances of sparse estimation problems. Furthermore, their computation is efficient which favorably reflects on the computational cost of the overall procedure.
We also confirm our theoretical results through numerical experiments where we evaluate our algorithms in terms of speed, robustness and performance of the final estimates. Finally, we compare our performances with the most relevant concurrent works and discuss the main differences. Perspectives for future work include considering other types of sparsity, divising an algorithm capable of reaching the optimal $s\log(d/s)/n$ rate for vanilla sparsity or considering problems beyond recovery of a single parameter such as, for example, additive sparse and low-rank matrix decomposition.
\acks{This research is supported by the Agence Nationale de la Recherche as part of the ``Investissements d'avenir'' program (reference ANR-19-P3IA-0001; PRAIRIE 3IA Institute).
}
\newpage
|
1,116,691,500,800 | arxiv | \section{Introduction\footnote{A nonvisually accessible version of this paper is available at \url{https://github.com/elisakreiss/contextual-description-evaluation}}}
In the pursuit of ever more powerful image description systems, we need evaluation metrics that provide a clear window into model capabilities. At present, we are seeing a rise in \emph{referenceless} (or reference-free) metrics \cite{hessel2021clipscore,lee2021qace,lee2021umic,feinglass2021smurf}, building on prior work in domains such as machine translation \cite{lo2019yisi, zhao2020limitations} and summarization \cite{louis2013automatically, peyrard2018objective}. These metrics seek to estimate the quality of a text corresponding to an image without requiring ground truth labels, crowd worker judgments, or reference descriptions. In this work, we investigate the current value of such metrics for assessing the usefulness of image descriptions for blind and low vision (BLV) users.
Images have become central in all areas of digital communication, from scientific publishing to social media memes
\cite{hackett2003accessibility, bigham2006webinsight, buzzi2011web, morris2016most, voykinska2016how, gleason2019it}.
While images can be made nonvisually accessible with image descriptions, these are rare, with coverage as low as 0.1\% on English-language Twitter \cite{gleason2020twitter}. In light of this challenge, there are numerous efforts underway to investigate what makes descriptions useful and develop models to artificially generate such descriptions at scale.
\begin{figure}[t!]
\centering
\includegraphics[width=0.45\textwidth]{figures/contextual-relevance.pdf}
\caption{Whether an image description makes an image accessible depends on the context where the image appears. Referenceless metrics like CLIPScore can't capture such context-sensitivity. We provide experimental evidence with blind and low vision (BLV) participants that this makes current referenceless metrics insufficient for evaluating image description quality.}
\label{fig:contextual-relevance}
\end{figure}
Referenceless metrics offer the promise of quick and efficient evaluation of models that generate image descriptions, and they are even suggested to be more reliable than existing reference-based metrics \cite{kasai2021transparent,kasai2021bidimensional}. The question then arises of whether referenceless metrics can provide suitable guidance for meeting accessibility needs.
There are two main categories of referenceless metrics. \emph{Imaged-based} metrics assess a description's quality relative to its associated image. The most prominent example is CLIPScore \cite{hessel2021clipscore}, a metric based on CLIP -- a multi-modal model trained on a large image--text dataset \cite{radford2021learning}. CLIPScore provides a compatibility score for image--text pairs, leveraging the fact that CLIP was trained contrastively with positive and negative examples \citep{hessel2021clipscore, lee2021umic}. In contrast, \emph{text-based} metrics rely entirely on intrinsic properties of the description text. For instance, SPURTS rates a description based on its linguistic style by leveraging the information flow in DistilRoBERTa, a large language model \cite{feinglass2021smurf}.
Do referenceless metrics, of either type, align with what BLV users value in image descriptions for accessibility? Studies with BLV users highlight that the \emph{context} in which an image appears is important. For example, while the clothes a person is wearing are highly relevant when browsing through shopping websites, the identity of the person becomes important when reading the news \cite{stangl2021going, muehlbradt2022what,stangl2020person}.
Not only the domain but even the immediate context matters for selecting what is relevant. Consider the image in \figref{fig:contextual-relevance}, showing a park with a gazebo in the center and a sculpture on a pedestal in the foreground. This image could appear for instance in the Wikipedia article on sculptures or gazebos. However, an image description written for the image's occurrence in the article of gazebos (``A freestanding, open, hexagonal gazebo with a dome-like roof in an idyllic park area.'')\ becomes unhelpful for the occurrence of the image in the article on sculptures. Thus, context could play a central role in the assessment of description quality.
In this work, we report on studies with sighted and BLV participants that seek to provide rich, multidimensional information about what people value in accessibility image descriptions. Our central manipulation involves systematically varying the Wikipedia articles the images are presented as appearing in, and studying the effects this has on participants' judgments. For both sighted and BLV participants, we observe strong and consistent effects of context. However, by their very design, current referenceless metrics can't capture these effects, since they treat description evaluation as a context-less problem. This shortcoming goes undetected on most existing datasets and previously conducted human evaluations, which presume that image descriptions are context-independent.
Image accessibility is a prominent and socially important goal that image-based NLG systems are striving to reach \cite{gurari2020captioning}. Our results suggest that current referenceless metrics may not be reliable guides in these efforts.
\section{Background}
\begin{figure*}[t!]
\includegraphics[width=1\textwidth]{figures/expdesign.pdf}
\caption{Experimental design overview consisting of two main phases: (A) eliciting descriptions written for images occurring within varying contexts, (B) obtaining detailed evaluations of those descriptions from sighted and BLV participants. These evaluations give insights into the role that context needs to play for providing useful descriptions, and function as the gold standard that the results from referenceless metrics are then compared to.}
\label{fig:expdesign}
\end{figure*}
\subsection{Image Accessibility}
Screen readers provide auditory and braille access to Web content. To make images accessible in this way, screen readers use image descriptions embedded in HTML \texttt{alt} tags. However, such descriptions are rare. While frequently visited websites are estimated to have about 72\% coverage \cite{guinness2018caption}, this drops to less than 6\% on English-language Wikipedia \cite{kreiss2022concadia} and to 0.1\% on English-language Twitter \cite{gleason2019it}. This has severe implications especially for BLV users who have to rely on such descriptions to engage socially \cite{morris2016most, macleod2017understanding, buzzi2011web, voykinska2016how} and stay informed \cite{gleason2019it, morris2016most}.
Moreover, these coverage estimates are based on any description being available, without regard for whether the descriptions are useful. Precisely what constitutes a useful description is still an underexplored question. A central finding from work with BLV users is that one-size-fits-all image descriptions don't address image accessibility needs \cite{stangl2021going, muehlbradt2022what, stangl2020person}. \citet{stangl2021going} specifically tested the importance of the \emph{scenario} -- the source of the image and the informational goal of the user -- by placing each image within different source domains (e.g., news or shopping website) which were associated with specific goals (e.g., learning or browsing for a gift). They find that BLV users have certain description preferences that are stable across scenarios (e.g., people's identity and facial expressions, or the type of location depicted), whereas others are scenario-dependent (e.g., hair color).
We extend this previous work by keeping the scenario stable but varying the immediate context the image is embedded in.
Current referenceless metrics take the one-size-fits-all approach. We explicitly test whether this is sufficient to capture the ratings provided by BLV users when they have access to the broader context.
\subsection{Image-based Text Evaluation Metrics}
There are two evaluation strategies for automatically assessing the quality of a model's generated text from images: \emph{reference-based} and \emph{referenceless} (or reference-free) metrics.
Reference-based metrics rely on ground-truth texts associated with each image that were created by human annotators. The candidate text generated by the model is then compared with those ground-truth references, returning a similarity score. A wide variety of scoring techniques have been explored. Examples are BLEU \cite{papineni2002bleu}, CIDEr \cite{vedantam2015cider}, SPICE \cite{anderson2016spice}, ROUGE \cite{lin2004rouge}, and BERTscore \cite{zhang2019bertscore}. The more references are provided, the more reliable the scores, which requires datasets with multiple high-quality annotations for each image. Such datasets are expensive and difficult to obtain.
As discussed above, referenceless metrics dispense with the need for ground-truth reference texts. Instead, text quality is assessed based either on how the text relates to the image content \cite{hessel2021clipscore,lee2021umic,lee2021qace} or on text quality alone \cite{feinglass2021smurf}. As a result, these metrics can in principle be used anywhere without the need for an expensive annotation effort. How the score is computed varies between metrics. CLIPScore \cite{hessel2021clipscore} and UMIC \cite{lee2021umic} pose a classification problem where models are trained contrastively on compatible and incompatible image--text pairs. A higher score for a given image and text as input then corresponds to a high compatibility between them. QACE provides a high score if descriptions and images provide similar answers to the same questions \cite{lee2021qace}. SPURTS is a referenceless metric which judges text quality solely based on text-internal properties that can be conceptualized as maximizing unexpected content \cite{feinglass2021smurf}. SPURTS was originally proposed as part of the metric SMURF which additionally contains a reference-based component, specifically designed to capture the semantics of the description. However, \citeauthor{feinglass2021smurf} find that SPURTS alone already seems to approximate human judgments well, which makes it a relevant referenceless metric to consider. While varying in their approach, all current referenceless metrics share that they treat image-based text generation as a context-independent problem.
Reference-based metrics have the \emph{potential} to reflect context-dependence, assuming the reference texts are created in ways that engage with the context the image appears in. Referenceless methods are much more limited in this regard: if a single image--description pair should receive different scores in different contexts, but the metric operates only on image--description pairs, then the metric will be intrinsically unable to provide the desired scores.
\section{Experiment: The Effect of Context on Human Image Description Evaluation}
Efforts to obtain and evaluate image descriptions through crowdsourcing are mainly conducted out-of-context: images that might have originally been part of a tweet or news article are presented in isolation to obtain a description or evaluation thereof. Following recent insights on the importance of the domains an image appeared in \cite{stangl2021going, muehlbradt2022what, stangl2020person}, we seek to understand the role of context in shaping how people evaluate descriptions. \figref{fig:expdesign} provides an overview of the two main phases. Firstly, we obtained contextual descriptions by explicitly varying the context each image could occur in (\figref{fig:expdesign}A). We then explored how context affects sighted and BLV users' assessments of descriptions along a number of dimensions (\figref{fig:expdesign}B). Finally, in \secref{sec:metric-assess}, we compare these contextual evaluations with the results from the referenceless metrics CLIPScore \cite{hessel2021clipscore} and SPURTS \cite{feinglass2021smurf}.
\subsection{Data}
To investigate the effect of context on image descriptions, we designed a dataset where each image was paired with three distinct contexts, here Wikipedia articles. For instance, an image of a church was paired with the first paragraphs of the Wikipedia articles on \emph{Building material}, \emph{Roof}, and \emph{Christian cross}. Similarly, each article appeared with three distinct images. The images were made publicly available through Wikimedia Commons. Overall, we obtained 54 unique image--context pairs, consisting of 18 unique images and 17 unique articles. The dataset, experiments used for data collection, and analyses are made available.\footnote{\url{https://github.com/elisakreiss/contextual-description-evaluation}}
\subsection{Contextual Description Writing}
\begin{figure*}[t!]
\includegraphics[width=1\textwidth]{figures/blvsightedcorr.pdf}
\caption{Correlation of BLV and sighted participant ratings across questions. Sighted participants provided ratings twice -- before seeing the image (in green) and after (in blue). Each point denotes the average rating for a description. The Pearson correlations (R) are all statistically significant, as indicated by the asterisks. For all questions, higher ratings are associated with higher quality descriptions.}
\label{fig:corr-blv-sighted}
\end{figure*}
To obtain image descriptions, we recruited 74 participants on Amazon's Mechanical Turk for the task of providing image descriptions that could make the images accessible to users who can't see them.
\paragraph{Task} Each participant went through a brief introduction explaining the challenge and purpose of image descriptions and was then shown six distinct articles, each of them containing a different image they were asked to describe. To enable participants to judge their descriptions, the description then replaced the image and participants could choose to edit their response before continuing. The task did not contain any guidance on which information should or should not be included in the description. Consequently, any context-dependence is simply induced by presenting the images within contexts (Wikipedia articles) instead of in isolation.
\paragraph{Exclusions} We excluded six participants who indicated confusion about the task in the post-questionnaire and one participant for whom the experiment didn't display properly. Overall, each image--article pair received on average five descriptions.
\paragraph{Results} After exclusions, we obtained 272 descriptions that varied in length between 13 and 541 characters, with an average of 24.9 words. With the following human subject evaluation experiment, we evaluate to what extent the description content was affected by the image context.
\subsection{Contextual Description Evaluation}
After obtaining contextual image descriptions, we designed a description evaluation study which we conducted with BLV as well as sighted participants. Both groups can provide important insights. We consider the ratings of BLV participants as the primary window into accessibility needs. However, sighted participant judgments can complement these results, in particular by helping us determine whether a description is true for an image. Furthermore, the sighted participants' intuitions about what makes a good description are potentially informative since sighted users are usually the ones providing image descriptions.
\paragraph{Task} Sighted as well as BLV participants rated each image description as it occurred within the respective Wikipedia article. To get a better understanding of the kinds of content that might affect description quality, each description was evaluated according to five dimensions:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item \emph{Overall}: How good the description seemed overall.
\item \emph{Imaginability}: How well the description helped the participant imagine the image.
\item \emph{Relevance}: How well the description captured relevant information.
\item \emph{Irrelevance}: How much extra (irrelevant) information it added.
\item \emph{Fit}: How well the image seemed to fit within the article.
\end{enumerate}
The questions \emph{Imaginability} and \emph{Relevance} are designed to capture two central aspects of description content. While \emph{Imaginability} has no direct contextual component, \emph{Relevance} and \emph{Irrelevance} specifically ask about the contextually determined aspects of the description. These dimensions give us insights into the importance of context in the \emph{Overall} description quality ratings.
Responses were provided on 5-point Likert scales. In addition to 17 critical trials each participant completed, we further included two trials with descriptions carefully constructed to exhibit for instance low vs.~high context sensitivity. These trials allowed us to ensure that the questions and scales were interpreted as intended by the participants. Overall, each participant completed 19 trials, where each trial consisted of a different article and image. Trial order and question order were randomized between participants to avoid potential ordering biases.
\subsubsection{Sighted Participants}\label{sec:exp-eval-sighted}
\paragraph{Task} To ensure high data quality, sighted participants were asked a reading comprehension question before starting the experiment which also familiarized them with the overall topic of image accessibility. If they passed, they could choose to enter the main study, otherwise they exited the study and were only compensated for completing the comprehension task.
In each trial, participants first saw the Wikipedia article, followed by an image description. This \emph{no image} condition can be conceptualized as providing sighted participants with the same information a BLV user would be able to access. They then responded to the five questions and were asked to indicate if the description contained false statements or discriminatory language. After submitting the response, the image was revealed and participants responded again to four of the five questions. The \emph{Imaginability} question was omitted since it isn't clearly interpretable once the image is visible. Their previous rating for each question was made available to them so that they could reason about whether they wanted to keep or change their response.
\paragraph{Participants and Exclusions} 79 participants were recruited over Amazon's Mechanical Turk, 68 of whom continued past the reading comprehension question. We excluded eight participants since they spent less than 19 minutes on the task, and one participant whose logged data was incomplete. This resulted in 59 submissions for further analysis.
\subsubsection{BLV Participants}
The 68 most-rated descriptions across the 17 Wikipedia articles and 18 images were then selected to be further evaluated by BLV participants.
\paragraph{Task} To provide BLV participants with the same information as sighted participants, they similarly started with the reading comprehension question before continuing to the main trials. After reading the Wikipedia article and the image descriptions, participants first responded to the five evaluation dimensions. Afterwards, they provided answers to five open-ended questions about the description content. The main focus of the analysis presented here is on the Likert scale responses, but the open-ended explanations allow more detailed insights into description preferences. Each description was rated by exactly four participants.
\paragraph{Participants} 16 participants were recruited via targeted email lists for BLV users, and participants were unknowing about the purpose of the study. Participants self-described their level of vision as totally blind (7), nearly blind (3), light perception only (5), and low vision (1). 15 participants reported relying on screen readers (almost) always when browsing the Web, and one reported using them often.
We enrolled fewer blind participants than sighted participants, as they are a low-incidence population, requiring targeted and time-consuming recruitment. For example, crowd platforms that enable large sample recruitment are inaccessible to blind crowd workers \citep{Vashistha-etal:2018}.
\subsubsection{Evaluation Results}
The following analyses are based on the 68 descriptions, comprising 18 images and 17 Wikipedia articles. Each description is evaluated according to multiple dimensions by sighted as well as BLV participants for how well the description serves an accessibility goal.
\figref{fig:corr-blv-sighted} shows the correlation of BLV and sighted participant ratings across questions. We find that the judgments of the two groups are significantly correlated for all questions. The correlation is encouraging since it shows an alignment between the BLV participants' reported preferences and the sighted participants' intuitions. Whether sighted participants could see the image when responding didn't make a qualitative difference. The results further show that the dataset provides very poor to very good descriptions, covering the whole range of possible responses. This range is important for insights into whether a proposed evaluation metric can detect what makes a description useful.
\begin{figure}[t!]
\includegraphics[width=1\linewidth]{figures/human_length_corr.pdf}
\caption{Correlation of BLV and sighted participant judgments with description length (in characters). Human annotations are rescaled to the zero to one range.}
\label{fig:length-corr}
\end{figure}
We conducted a mixed effects linear regression analysis of the BLV participant judgments to investigate which responses are significant predictors of the overall ratings. We used normalized and centered fixed effects of the three content questions (\emph{Imaginability}, \emph{Relevance} and \emph{Irrelevance}), and random by-participant and by-description intercepts. If context doesn't affect the quality of a description, \emph{Imaginability} should be a sufficient predictor of the overall ratings. However, in addition to an effect of \emph{Imaginability} ($\beta = .42$, $\text{SE} = .06$, $p<.001$), we find a significant effect of \emph{Relevance} as well ($\beta = .44$, $\text{SE} = .05$, $p<.001$), suggesting that context plays an essential role in guiding what makes a description useful. This finding replicates with the sighted participant judgments.
A case where BLV and sighted participant ratings diverge is in the effect of description length (\figref{fig:length-corr}). While longer descriptions tend to be judged overall more highly by BLV participants, there is no such correlation for sighted participants. This finding contrasts with popular image description guidelines, which generally advocate for shorter descriptions.\footnote{E.g., \url{https://webaim.org/techniques/alttext/}} The lack of correlation between sighted participant ratings and description length might be linked to this potential misconception.
\section{Referenceless Metrics for Image Accessibility}\label{sec:metric-assess}
Referenceless metrics have been shown to correlate well with how sighted participants judge description quality when descriptions are written and presented out-of-context \cite{hessel2021clipscore,feinglass2021smurf,lee2021umic,kasai2021transparent}. While image accessibility is one of the main goals referenceless metrics are intended to facilitate \cite{kasai2021bidimensional,hessel2021clipscore,kasai2021transparent}, it remains unclear whether they can approximate the usefulness of a description for BLV users. Inspired by recent insights into what makes a description useful, we argue that the inherently decontextualized nature of current referenceless metrics makes them inadequate as a measure of image accessibility. We focus on two referenceless metrics to support these claims: CLIPScore \cite{hessel2021clipscore} and SPURTS \cite{feinglass2021smurf}.
CLIPScore is currently the most prominent referenceless metric, and it seems likely to inspire additional similar metrics, given the rising number of successful multimodal models trained under contrastive learning objectives. While CLIPScore has been tested on a variety of datasets and compared to crowd worker judgments, it has so far been disconnected from insights into what makes descriptions useful for accessibility.
\begin{figure}[t!]
\includegraphics[width=0.48\textwidth]{figures/compat_length.pdf}
\caption{Analyses of the capabilities of referenceless metrics. (A) CLIPScore can pick out whether a written description is compatible with the image. When shuffling image--description pairs, the average CLIPScore drops from 0.73 to 0.43. SPURTS can't make this distinction due to its image-independence. (B) Longer descriptions are associated with higher scores of SPURTS but not CLIPScore.}
\label{fig:clipscore-truthfulness}
\end{figure}
\begin{figure*}[t!]
\includegraphics[width=0.99\textwidth]{figures/human_clipscorespurts_corr.pdf}
\caption{Correlations of CLIPScore (top row) and SPURTS (bottom row) with human ratings provided by sighted and BLV participants. The Pearson correlations are computed over the human evaluators' average per-description rating. Sighted participants responded to the questions twice; once without (in green) and after seeing the image (in blue). There are no significant correlations between CLIPScore and human ratings. SPURTS correlates significantly with all responses provided by BLV participants but negatively with \emph{Irrelevance} and not at all with sighted participant ratings, indicating a fundamental mismatch.}
\label{fig:clipscore-blvsighted}
\end{figure*}
CLIPScore uses the similarity of CLIP's image and description embeddings as the predictor for description quality, as formulated in (\ref{eq:clipscore}). Denoting $\frac{x}{|x|}$ as $\overline{x}$, we can express CLIPScore as:
\begin{equation} \label{eq:clipscore}
\text{max}\left({\overline{\textnormal{image}}} \cdot {\overline{\textnormal{description}}}, 0\right)
\end{equation}
SPURTS is different from CLIPScore since it only considers the description itself, without taking image information into account. The main goal of SPURTS is to detect fluency and style, and it can be written as
\begin{equation} \label{eq:spurts}
\textnormal{median}_{\textnormal{layer}} \textnormal{max}_{\textnormal{head}} \ I_{\textnormal{flow}}(y_{w/o}, \theta),
\end{equation}
where $I_{\textnormal{flow}}$, which \citet{feinglass2021smurf} refer to as information flow, is normalized mutual information as defined in \citet{witten2005practical}. For an input text without stop words, $y_{w/o}$, and a transformer with parameters $\theta$, SPURTS computes the information flow for each transformer head at each layer, and then returns the layer-wise median of the head-wise maxima.
We turn now to assessing the extent to which CLIPScore and SPURTS approximate the ratings of the BLV and sighted users from our studies. Our central conclusion is that the context-independence of these metrics makes them inadequate for approximating description quality.
\subsection{Compatibility}
We first inspect the extent to which current referenceless metrics can capture whether a description is true for an image. SPURTS provides scores independent of the image and therefore inherently can't capture any notion of truthfulness.
In contrast, CLIPScore is trained to distinguish between fitting and non-fitting image--text pairs, returning a compatibility score. We test whether this generalizes to our experimental data by providing CLIPScore with the true descriptions written for each image and a shuffled variant where images and descriptions were randomly paired. As \figref{fig:clipscore-truthfulness}A demonstrates, CLIPScore rates the ordered pairs significantly higher compared to the shuffled counterparts ($\beta = 2.02$, $\text{SE} = .14$, $p<.001$),\footnote{Result from a linear effects analyses where the shuffled condition is coded as 0, and the ordered condition as 1.} suggesting that it captures image--text compatibility.
\subsection{Description Length Correlation}
Since the length of the description can already account for some of the variance of the BLV ratings, we further investigate whether description length is a general predictor for CLIPScore and SPURTS (see \figref{fig:clipscore-truthfulness}B). For CLIPScore, description length doesn't correlate with predicted quality of the description, which is likely a consequence of the contrastive learning objective, which only optimizes for compatibility but not quality. SPURTS scores, in contrast, significantly correlate with description length, which is aligned with the BLV ratings.
\subsection{Context Sensitivity}
Crucially, the descriptions were written and evaluated within contexts, i.e., their respective Wikipedia article, and previous work suggests that the availability of context should affect what constitutes a good and useful description.
Since current referenceless metrics can't integrate context, we expect that they shouldn't be able to capture the variation in the human description evaluations, and this is indeed what we find.
To investigate this hypothesis, we correlated sighted and BLV description evaluations with the CLIPScore and SPURTS ratings. As shown in \figref{fig:clipscore-blvsighted}, CLIPScore fails to capture any variation observed in the human judgments across questions. This suggests that, while CLIPScore can add a perspective on the compatibility of a text for an image, it can't get beyond that as an indication of how useful a description is if it's true for the image.
Like CLIPScore, SPURTS scores don't correlate with the sighted participant judgments (see \figref{fig:clipscore-blvsighted}, bottom). However, specifically with respect to the overall rating, SPURTS scores show a significant correlation with the BLV participant ratings. While this seems encouraging, further analysis revealed that this correlation is primarily driven by the fact that both BLV and SPURTS ratings correlate with description length. The explained variance of the BLV ratings from description length alone is $0.152$ and SPURTS score alone explains $0.08$ of the variance. In conjunction, however, they only explain $0.166$ of the variance, which means that most of the predictability of SPURTS is due to the length correlation. This is further supported by a mixed effects linear regression analysis in which we fail to find a significant effect of SPURTS ($\beta = .80$, $\text{SE} = .44$, $p > .05$) once we include length as a predictor ($\beta = .64$, $\text{SE} = .15$, $p < .001$).\footnote{We assume random intercepts by participant and description, and we rescaled description length to fall in the range between 0 and 1.}
\begin{table*}[tp]
\centering
\begin{tabular}{@{}r@{ \ }l rrrr@{}}\toprule
&& Overall & Imaginability & Relevance & Irrelevance \\ \midrule
CLIPScore:& BLV & 0.075 & 0.104 & 0.086 & 0.090 \\
With Context:& BLV & 0.201 & 0.182 & 0.202 & 0.142 \\ \hline
CLIPScore:& sighted, no img & $-$0.013 & 0.064 & 0.000 & $-$0.166 \\
With Context:& sighted, no img & 0.238 & 0.315 & 0.190 & $-$0.019 \\ \hline
CLIPScore:& sighted, w img & 0.139 & & 0.106 & $-$0.079 \\
With Context:& sighted, w img & 0.331 & & 0.240 & 0.052 \\ \bottomrule
\end{tabular}
\caption{Comparison of the human rating correlations with the original context-independent CLIPScore and the context-sensitive adaptation, using the same CLIP embeddings. Missing cells were not experimentally measured by design (\secref{sec:exp-eval-sighted}). Across questions and participant groups, correlations improve. The CLIPScore correlations are a replication of \figref{fig:clipscore-blvsighted}.}
\label{tab:clipscorewcontext}
\end{table*}
A further indication that SPURTS isn't capturing essential variance in BLV judgments is apparent from the negative correlation in the \emph{Irrelevance} question ($R=-0.25$). This suggests that SPURTS scores tend to be higher for descriptions that are judged to contain too much irrelevant information and low when participants assess the level of information to be appropriate. In the BLV responses, \emph{Irrelevance} is positively correlated with the \emph{Overall} ratings ($R=0.33$), posing a clear qualitative mismatch to SPURTS. Since what is considered extra information is dependent on the context, this is a concrete case where the metric's lack of context integration results in undesired behavior.
Finally, SPURTS' complete lack of correlation with sighted participant judgments further suggests that SPURTS is insufficient for picking up the semantic components of the descriptions. This aligns with the original conception of the metric, where a reference-based metric (SPARCS) is used to estimate semantic quality.
Overall, our results highlight that SPURTS captures the BLV participants' preferences for longer descriptions but falls short in capturing additional semantic preferences, and is inherently inadequate for judging the truthfulness of a description more generally. CLIPScore can't capture any of the variation in BLV or sighted participant ratings, uncovering clear limitations.
\subsection{Implications for Other Referenceless Metrics}
In the previous experiments, we established that the referenceless metrics CLIPScore and SPURTS can't get traction on what makes a good description when the images and descriptions are contextualized.
Other referenceless metrics such as UMIC \cite{lee2021umic} and QACE \cite{lee2021qace} face the same fundamental issue as CLIPScore and SPURTS due to their contextless nature. Like CLIPScore, UMIC is based on an image--text model (UNITER; \citet{chen2020uniter}) trained under a contrastive learning objective. Similarly, it produces an image--text compatibility score solely by receiving a decontextualized image and text as input.
QACE uses the candidate description to derive potential questions that should be answerable based on the image. The evaluation is therefore whether the description mentions aspects that are true of the image and not about which aspects of the image are relevant to describe. This again only provides insights into image--text compatibility but not contextual relevance.\footnote{Unfortunately, we are unable to provide quantitative results for these referenceless metrics since the authors haven't provided the code necessary (QACE), or the code relies on image features that can't be created for novel datasets with currently available hardware (UMIC, QACE).}
In summary, the current context-independence of all existing referenceless metrics is a major limitation for their usefulness. This is a challenge that needs to be addressed to make these metrics a useful tool for advancing image-based NLG systems.
\section{Discussion: Future Opportunities and Limitations}
So far, we have argued that CLIPScore and other referenceless metrics aren't a useful approximation for BLV (and sighted) user judgments of high-quality descriptions, primarily due to their contextless nature. Using the example of CLIPScore, we will now explore where future work on referenceless metrics in image-based NLG can progress and also discuss some underlying limitations.
\subsection{The Potential for Integrating Context into CLIPScore}
Can referenceless metrics like CLIPScore be made context sensitive? To begin exploring this question, as a proof of concept, we amend (\ref{eq:clipscore}) as follows:
\begin{multline}
\label{eq:clipwcontext}
\overline{\textnormal{description}} \cdot \textnormal{context} \ + \\
\textnormal{description} \cdot \left(\overline{\textnormal{image}} - \overline{\textnormal{context}}\right)
\end{multline}
Here, quality is a function of (a) the description's similarity to the context (first addend) and (b) whether the description captures the information that the image adds to the context (second addend). These two addends can be seen as capturing aspects of (ir)relevance and imaginability, respectively, though we anticipate many alternative ways to quantify these dimensions.
\tabref{tab:clipscorewcontext} reports correlations between this augmented version of CLIPScore and our sighted and BLV participant judgments. We find it encouraging that even this simple approach to incorporating context boosts correlations with human ratings for all the questions in our experiment.
For the \emph{Irrelevance} question, it even clearly captures the positive correlation with BLV ratings, which is negative for both CLIPScore and SPURTS, indicating a promising shift. We consider this an encouraging signal that large pretrained models such as CLIP might still constitute a resource for developing future referenceless metrics.
However, despite these promising signs, there are also reasons to believe that CLIP-based metrics have other restrictive limitations. Due to CLIP's training, images are cropped at the center region and texts need to be truncated at 77 tokens \cite{radford2021learning}. Specifically for the purpose of accessibility, the information this removes can be crucial for determining whether a description is useful or not. For instance, our experiments show that the length of a description is an important indicator for description quality -- information lost in CLIP-based metrics. Moreover, this disproportionately affects the ability to encode the context paragraphs, which are often longer than a typical description. These decisions are therefore likely reflected in any resulting metric and should therefore be reconsidered when devising a new metric.
\subsection{Referenceless Metrics for Image-Based NLG Beyond Accessibility}
While we have specifically focused on the usefulness of referenceless metrics for image accessibility, this isn't the only potential purpose an image-based text might address. \citet{kreiss2022concadia} distinguish \emph{descriptions}, i.e., image-based texts that are written to replace the image, and \emph{captions}, i.e., texts that are intended to appear alongside images, such as tweets or newspaper captions. This suggests that the same text can be very useful for contextualizing an image but fail at providing image accessibility, and vice versa.
To investigate this distinction, they asked participants to rate \texttt{alt} descriptions as well as image captions from Wikipedia according to (1) how much the text helped them imagine the image, and (2) how much they learned from the text that they couldn't have learned from the image. Descriptions were rated more useful for imagining the image, whereas captions were rated more useful for learning additional information. Captions used for contextualizing an image might therefore be another potential use domain for a referenceless metric such as CLIPScore.
\begin{figure}
\includegraphics[width=.48\textwidth]{figures/clipscore-purposecorr.pdf}
\caption{CLIPScore provides higher ratings for image-based texts that capture image content well. Whether the texts provide additional information to the image content doesn't affect the ratings.}
\label{fig:corr-clipscore-purpose}
\end{figure}
To see whether CLIPScore might be a promising resource for evaluating captions, we obtained CLIPScore ratings for the descriptions and captions in \citet{kreiss2022concadia}. CLIPScore ratings correlate with the reconstruction as opposed to the contextualization goal (see \figref{fig:corr-clipscore-purpose}), suggesting that CLIPScore is inherently less appropriate to be used for assessing caption datasets. This aligns with the original observation in \citet{hessel2021clipscore} that CLIPScore performs less well on the news caption dataset GoodNews \cite{biten2019gooda} compared to MSCOCO \cite{hessel2021clipscore}, a contextless description dataset.
Taken together, this suggests that the ``one-size-fits-all'' approach to referenceless image-based text evaluation is not sufficient for adequately assessing text quality for the contextualization or the accessibility domain.
\section{Conclusion}
The context an image appears in shapes the way high-quality accessibility descriptions are written. In this work, we reported on experiments in which we explicitly varied the contexts images were presented in and investigated the effects of this contextual evaluation on participant ratings. These experiments reveal strong contextual effects for sighted and BLV participants. We showed that this poses a serious obstacle for current referenceless metrics.
Our results have wide implications for research on automatic description generation. The central result is that context plays an important role in what makes a description useful. Not realizing that leads to datasets and evaluation methods that don't actually optimize for accessibility. Thus, we need to situate datasets and models within contexts, and evaluation methods need to be sensitive to context. In light of the benefits of referenceless metrics, we feel it is imperative to explore ways for them to incorporate context and to assess the resulting scores against judgments from BLV users.
\section*{Acknowledgements}
This work is supported in part by a grant from Google through the Stanford Institute for Human-Centered AI and by the NSF under project REU-1950223. We thank our experiment participants for their invaluable input.
\section{Introduction\footnote{A nonvisually accessible version of this paper is available at \url{https://github.com/elisakreiss/contextual-description-evaluation}}}
In the pursuit of ever more powerful image description systems, we need evaluation metrics that provide a clear window into model capabilities. At present, we are seeing a rise in \emph{referenceless} (or reference-free) metrics \cite{hessel2021clipscore,lee2021qace,lee2021umic,feinglass2021smurf}, building on prior work in domains such as machine translation \cite{lo2019yisi, zhao2020limitations} and summarization \cite{louis2013automatically, peyrard2018objective}. These metrics seek to estimate the quality of a text corresponding to an image without requiring ground truth labels, crowd worker judgments, or reference descriptions. In this work, we investigate the current value of such metrics for assessing the usefulness of image descriptions for blind and low vision (BLV) users.
Images have become central in all areas of digital communication, from scientific publishing to social media memes
\cite{hackett2003accessibility, bigham2006webinsight, buzzi2011web, morris2016most, voykinska2016how, gleason2019it}.
While images can be made nonvisually accessible with image descriptions, these are rare, with coverage as low as 0.1\% on English-language Twitter \cite{gleason2020twitter}. In light of this challenge, there are numerous efforts underway to investigate what makes descriptions useful and develop models to artificially generate such descriptions at scale.
\begin{figure}[t!]
\centering
\includegraphics[width=0.45\textwidth]{figures/contextual-relevance.pdf}
\caption{Whether an image description makes an image accessible depends on the context where the image appears. Referenceless metrics like CLIPScore can't capture such context-sensitivity. We provide experimental evidence with blind and low vision (BLV) participants that this makes current referenceless metrics insufficient for evaluating image description quality.}
\label{fig:contextual-relevance}
\end{figure}
Referenceless metrics offer the promise of quick and efficient evaluation of models that generate image descriptions, and they are even suggested to be more reliable than existing reference-based metrics \cite{kasai2021transparent,kasai2021bidimensional}. The question then arises of whether referenceless metrics can provide suitable guidance for meeting accessibility needs.
There are two main categories of referenceless metrics. \emph{Imaged-based} metrics assess a description's quality relative to its associated image. The most prominent example is CLIPScore \cite{hessel2021clipscore}, a metric based on CLIP -- a multi-modal model trained on a large image--text dataset \cite{radford2021learning}. CLIPScore provides a compatibility score for image--text pairs, leveraging the fact that CLIP was trained contrastively with positive and negative examples \citep{hessel2021clipscore, lee2021umic}. In contrast, \emph{text-based} metrics rely entirely on intrinsic properties of the description text. For instance, SPURTS rates a description based on its linguistic style by leveraging the information flow in DistilRoBERTa, a large language model \cite{feinglass2021smurf}.
Do referenceless metrics, of either type, align with what BLV users value in image descriptions for accessibility? Studies with BLV users highlight that the \emph{context} in which an image appears is important. For example, while the clothes a person is wearing are highly relevant when browsing through shopping websites, the identity of the person becomes important when reading the news \cite{stangl2021going, muehlbradt2022what,stangl2020person}.
Not only the domain but even the immediate context matters for selecting what is relevant. Consider the image in \figref{fig:contextual-relevance}, showing a park with a gazebo in the center and a sculpture on a pedestal in the foreground. This image could appear for instance in the Wikipedia article on sculptures or gazebos. However, an image description written for the image's occurrence in the article of gazebos (``A freestanding, open, hexagonal gazebo with a dome-like roof in an idyllic park area.'')\ becomes unhelpful for the occurrence of the image in the article on sculptures. Thus, context could play a central role in the assessment of description quality.
In this work, we report on studies with sighted and BLV participants that seek to provide rich, multidimensional information about what people value in accessibility image descriptions. Our central manipulation involves systematically varying the Wikipedia articles the images are presented as appearing in, and studying the effects this has on participants' judgments. For both sighted and BLV participants, we observe strong and consistent effects of context. However, by their very design, current referenceless metrics can't capture these effects, since they treat description evaluation as a context-less problem. This shortcoming goes undetected on most existing datasets and previously conducted human evaluations, which presume that image descriptions are context-independent.
Image accessibility is a prominent and socially important goal that image-based NLG systems are striving to reach \cite{gurari2020captioning}. Our results suggest that current referenceless metrics may not be reliable guides in these efforts.
\section{Background}
\begin{figure*}[t!]
\includegraphics[width=1\textwidth]{figures/expdesign.pdf}
\caption{Experimental design overview consisting of two main phases: (A) eliciting descriptions written for images occurring within varying contexts, (B) obtaining detailed evaluations of those descriptions from sighted and BLV participants. These evaluations give insights into the role that context needs to play for providing useful descriptions, and function as the gold standard that the results from referenceless metrics are then compared to.}
\label{fig:expdesign}
\end{figure*}
\subsection{Image Accessibility}
Screen readers provide auditory and braille access to Web content. To make images accessible in this way, screen readers use image descriptions embedded in HTML \texttt{alt} tags. However, such descriptions are rare. While frequently visited websites are estimated to have about 72\% coverage \cite{guinness2018caption}, this drops to less than 6\% on English-language Wikipedia \cite{kreiss2022concadia} and to 0.1\% on English-language Twitter \cite{gleason2019it}. This has severe implications especially for BLV users who have to rely on such descriptions to engage socially \cite{morris2016most, macleod2017understanding, buzzi2011web, voykinska2016how} and stay informed \cite{gleason2019it, morris2016most}.
Moreover, these coverage estimates are based on any description being available, without regard for whether the descriptions are useful. Precisely what constitutes a useful description is still an underexplored question. A central finding from work with BLV users is that one-size-fits-all image descriptions don't address image accessibility needs \cite{stangl2021going, muehlbradt2022what, stangl2020person}. \citet{stangl2021going} specifically tested the importance of the \emph{scenario} -- the source of the image and the informational goal of the user -- by placing each image within different source domains (e.g., news or shopping website) which were associated with specific goals (e.g., learning or browsing for a gift). They find that BLV users have certain description preferences that are stable across scenarios (e.g., people's identity and facial expressions, or the type of location depicted), whereas others are scenario-dependent (e.g., hair color).
We extend this previous work by keeping the scenario stable but varying the immediate context the image is embedded in.
Current referenceless metrics take the one-size-fits-all approach. We explicitly test whether this is sufficient to capture the ratings provided by BLV users when they have access to the broader context.
\subsection{Image-based Text Evaluation Metrics}
There are two evaluation strategies for automatically assessing the quality of a model's generated text from images: \emph{reference-based} and \emph{referenceless} (or reference-free) metrics.
Reference-based metrics rely on ground-truth texts associated with each image that were created by human annotators. The candidate text generated by the model is then compared with those ground-truth references, returning a similarity score. A wide variety of scoring techniques have been explored. Examples are BLEU \cite{papineni2002bleu}, CIDEr \cite{vedantam2015cider}, SPICE \cite{anderson2016spice}, ROUGE \cite{lin2004rouge}, and BERTscore \cite{zhang2019bertscore}. The more references are provided, the more reliable the scores, which requires datasets with multiple high-quality annotations for each image. Such datasets are expensive and difficult to obtain.
As discussed above, referenceless metrics dispense with the need for ground-truth reference texts. Instead, text quality is assessed based either on how the text relates to the image content \cite{hessel2021clipscore,lee2021umic,lee2021qace} or on text quality alone \cite{feinglass2021smurf}. As a result, these metrics can in principle be used anywhere without the need for an expensive annotation effort. How the score is computed varies between metrics. CLIPScore \cite{hessel2021clipscore} and UMIC \cite{lee2021umic} pose a classification problem where models are trained contrastively on compatible and incompatible image--text pairs. A higher score for a given image and text as input then corresponds to a high compatibility between them. QACE provides a high score if descriptions and images provide similar answers to the same questions \cite{lee2021qace}. SPURTS is a referenceless metric which judges text quality solely based on text-internal properties that can be conceptualized as maximizing unexpected content \cite{feinglass2021smurf}. SPURTS was originally proposed as part of the metric SMURF which additionally contains a reference-based component, specifically designed to capture the semantics of the description. However, \citeauthor{feinglass2021smurf} find that SPURTS alone already seems to approximate human judgments well, which makes it a relevant referenceless metric to consider. While varying in their approach, all current referenceless metrics share that they treat image-based text generation as a context-independent problem.
Reference-based metrics have the \emph{potential} to reflect context-dependence, assuming the reference texts are created in ways that engage with the context the image appears in. Referenceless methods are much more limited in this regard: if a single image--description pair should receive different scores in different contexts, but the metric operates only on image--description pairs, then the metric will be intrinsically unable to provide the desired scores.
\section{Experiment: The Effect of Context on Human Image Description Evaluation}
Efforts to obtain and evaluate image descriptions through crowdsourcing are mainly conducted out-of-context: images that might have originally been part of a tweet or news article are presented in isolation to obtain a description or evaluation thereof. Following recent insights on the importance of the domains an image appeared in \cite{stangl2021going, muehlbradt2022what, stangl2020person}, we seek to understand the role of context in shaping how people evaluate descriptions. \figref{fig:expdesign} provides an overview of the two main phases. Firstly, we obtained contextual descriptions by explicitly varying the context each image could occur in (\figref{fig:expdesign}A). We then explored how context affects sighted and BLV users' assessments of descriptions along a number of dimensions (\figref{fig:expdesign}B). Finally, in \secref{sec:metric-assess}, we compare these contextual evaluations with the results from the referenceless metrics CLIPScore \cite{hessel2021clipscore} and SPURTS \cite{feinglass2021smurf}.
\subsection{Data}
To investigate the effect of context on image descriptions, we designed a dataset where each image was paired with three distinct contexts, here Wikipedia articles. For instance, an image of a church was paired with the first paragraphs of the Wikipedia articles on \emph{Building material}, \emph{Roof}, and \emph{Christian cross}. Similarly, each article appeared with three distinct images. The images were made publicly available through Wikimedia Commons. Overall, we obtained 54 unique image--context pairs, consisting of 18 unique images and 17 unique articles. The dataset, experiments used for data collection, and analyses are made available.\footnote{\url{https://github.com/elisakreiss/contextual-description-evaluation}}
\subsection{Contextual Description Writing}
\begin{figure*}[t!]
\includegraphics[width=1\textwidth]{figures/blvsightedcorr.pdf}
\caption{Correlation of BLV and sighted participant ratings across questions. Sighted participants provided ratings twice -- before seeing the image (in green) and after (in blue). Each point denotes the average rating for a description. The Pearson correlations (R) are all statistically significant, as indicated by the asterisks. For all questions, higher ratings are associated with higher quality descriptions.}
\label{fig:corr-blv-sighted}
\end{figure*}
To obtain image descriptions, we recruited 74 participants on Amazon's Mechanical Turk for the task of providing image descriptions that could make the images accessible to users who can't see them.
\paragraph{Task} Each participant went through a brief introduction explaining the challenge and purpose of image descriptions and was then shown six distinct articles, each of them containing a different image they were asked to describe. To enable participants to judge their descriptions, the description then replaced the image and participants could choose to edit their response before continuing. The task did not contain any guidance on which information should or should not be included in the description. Consequently, any context-dependence is simply induced by presenting the images within contexts (Wikipedia articles) instead of in isolation.
\paragraph{Exclusions} We excluded six participants who indicated confusion about the task in the post-questionnaire and one participant for whom the experiment didn't display properly. Overall, each image--article pair received on average five descriptions.
\paragraph{Results} After exclusions, we obtained 272 descriptions that varied in length between 13 and 541 characters, with an average of 24.9 words. With the following human subject evaluation experiment, we evaluate to what extent the description content was affected by the image context.
\subsection{Contextual Description Evaluation}
After obtaining contextual image descriptions, we designed a description evaluation study which we conducted with BLV as well as sighted participants. Both groups can provide important insights. We consider the ratings of BLV participants as the primary window into accessibility needs. However, sighted participant judgments can complement these results, in particular by helping us determine whether a description is true for an image. Furthermore, the sighted participants' intuitions about what makes a good description are potentially informative since sighted users are usually the ones providing image descriptions.
\paragraph{Task} Sighted as well as BLV participants rated each image description as it occurred within the respective Wikipedia article. To get a better understanding of the kinds of content that might affect description quality, each description was evaluated according to five dimensions:
\begin{enumerate}\setlength{\itemsep}{0pt}
\item \emph{Overall}: How good the description seemed overall.
\item \emph{Imaginability}: How well the description helped the participant imagine the image.
\item \emph{Relevance}: How well the description captured relevant information.
\item \emph{Irrelevance}: How much extra (irrelevant) information it added.
\item \emph{Fit}: How well the image seemed to fit within the article.
\end{enumerate}
The questions \emph{Imaginability} and \emph{Relevance} are designed to capture two central aspects of description content. While \emph{Imaginability} has no direct contextual component, \emph{Relevance} and \emph{Irrelevance} specifically ask about the contextually determined aspects of the description. These dimensions give us insights into the importance of context in the \emph{Overall} description quality ratings.
Responses were provided on 5-point Likert scales. In addition to 17 critical trials each participant completed, we further included two trials with descriptions carefully constructed to exhibit for instance low vs.~high context sensitivity. These trials allowed us to ensure that the questions and scales were interpreted as intended by the participants. Overall, each participant completed 19 trials, where each trial consisted of a different article and image. Trial order and question order were randomized between participants to avoid potential ordering biases.
\subsubsection{Sighted Participants}\label{sec:exp-eval-sighted}
\paragraph{Task} To ensure high data quality, sighted participants were asked a reading comprehension question before starting the experiment which also familiarized them with the overall topic of image accessibility. If they passed, they could choose to enter the main study, otherwise they exited the study and were only compensated for completing the comprehension task.
In each trial, participants first saw the Wikipedia article, followed by an image description. This \emph{no image} condition can be conceptualized as providing sighted participants with the same information a BLV user would be able to access. They then responded to the five questions and were asked to indicate if the description contained false statements or discriminatory language. After submitting the response, the image was revealed and participants responded again to four of the five questions. The \emph{Imaginability} question was omitted since it isn't clearly interpretable once the image is visible. Their previous rating for each question was made available to them so that they could reason about whether they wanted to keep or change their response.
\paragraph{Participants and Exclusions} 79 participants were recruited over Amazon's Mechanical Turk, 68 of whom continued past the reading comprehension question. We excluded eight participants since they spent less than 19 minutes on the task, and one participant whose logged data was incomplete. This resulted in 59 submissions for further analysis.
\subsubsection{BLV Participants}
The 68 most-rated descriptions across the 17 Wikipedia articles and 18 images were then selected to be further evaluated by BLV participants.
\paragraph{Task} To provide BLV participants with the same information as sighted participants, they similarly started with the reading comprehension question before continuing to the main trials. After reading the Wikipedia article and the image descriptions, participants first responded to the five evaluation dimensions. Afterwards, they provided answers to five open-ended questions about the description content. The main focus of the analysis presented here is on the Likert scale responses, but the open-ended explanations allow more detailed insights into description preferences. Each description was rated by exactly four participants.
\paragraph{Participants} 16 participants were recruited via targeted email lists for BLV users, and participants were unknowing about the purpose of the study. Participants self-described their level of vision as totally blind (7), nearly blind (3), light perception only (5), and low vision (1). 15 participants reported relying on screen readers (almost) always when browsing the Web, and one reported using them often.
We enrolled fewer blind participants than sighted participants, as they are a low-incidence population, requiring targeted and time-consuming recruitment. For example, crowd platforms that enable large sample recruitment are inaccessible to blind crowd workers \citep{Vashistha-etal:2018}.
\subsubsection{Evaluation Results}
The following analyses are based on the 68 descriptions, comprising 18 images and 17 Wikipedia articles. Each description is evaluated according to multiple dimensions by sighted as well as BLV participants for how well the description serves an accessibility goal.
\figref{fig:corr-blv-sighted} shows the correlation of BLV and sighted participant ratings across questions. We find that the judgments of the two groups are significantly correlated for all questions. The correlation is encouraging since it shows an alignment between the BLV participants' reported preferences and the sighted participants' intuitions. Whether sighted participants could see the image when responding didn't make a qualitative difference. The results further show that the dataset provides very poor to very good descriptions, covering the whole range of possible responses. This range is important for insights into whether a proposed evaluation metric can detect what makes a description useful.
\begin{figure}[t!]
\includegraphics[width=1\linewidth]{figures/human_length_corr.pdf}
\caption{Correlation of BLV and sighted participant judgments with description length (in characters). Human annotations are rescaled to the zero to one range.}
\label{fig:length-corr}
\end{figure}
We conducted a mixed effects linear regression analysis of the BLV participant judgments to investigate which responses are significant predictors of the overall ratings. We used normalized and centered fixed effects of the three content questions (\emph{Imaginability}, \emph{Relevance} and \emph{Irrelevance}), and random by-participant and by-description intercepts. If context doesn't affect the quality of a description, \emph{Imaginability} should be a sufficient predictor of the overall ratings. However, in addition to an effect of \emph{Imaginability} ($\beta = .42$, $\text{SE} = .06$, $p<.001$), we find a significant effect of \emph{Relevance} as well ($\beta = .44$, $\text{SE} = .05$, $p<.001$), suggesting that context plays an essential role in guiding what makes a description useful. This finding replicates with the sighted participant judgments.
A case where BLV and sighted participant ratings diverge is in the effect of description length (\figref{fig:length-corr}). While longer descriptions tend to be judged overall more highly by BLV participants, there is no such correlation for sighted participants. This finding contrasts with popular image description guidelines, which generally advocate for shorter descriptions.\footnote{E.g., \url{https://webaim.org/techniques/alttext/}} The lack of correlation between sighted participant ratings and description length might be linked to this potential misconception.
\section{Referenceless Metrics for Image Accessibility}\label{sec:metric-assess}
Referenceless metrics have been shown to correlate well with how sighted participants judge description quality when descriptions are written and presented out-of-context \cite{hessel2021clipscore,feinglass2021smurf,lee2021umic,kasai2021transparent}. While image accessibility is one of the main goals referenceless metrics are intended to facilitate \cite{kasai2021bidimensional,hessel2021clipscore,kasai2021transparent}, it remains unclear whether they can approximate the usefulness of a description for BLV users. Inspired by recent insights into what makes a description useful, we argue that the inherently decontextualized nature of current referenceless metrics makes them inadequate as a measure of image accessibility. We focus on two referenceless metrics to support these claims: CLIPScore \cite{hessel2021clipscore} and SPURTS \cite{feinglass2021smurf}.
CLIPScore is currently the most prominent referenceless metric, and it seems likely to inspire additional similar metrics, given the rising number of successful multimodal models trained under contrastive learning objectives. While CLIPScore has been tested on a variety of datasets and compared to crowd worker judgments, it has so far been disconnected from insights into what makes descriptions useful for accessibility.
\begin{figure}[t!]
\includegraphics[width=0.48\textwidth]{figures/compat_length.pdf}
\caption{Analyses of the capabilities of referenceless metrics. (A) CLIPScore can pick out whether a written description is compatible with the image. When shuffling image--description pairs, the average CLIPScore drops from 0.73 to 0.43. SPURTS can't make this distinction due to its image-independence. (B) Longer descriptions are associated with higher scores of SPURTS but not CLIPScore.}
\label{fig:clipscore-truthfulness}
\end{figure}
\begin{figure*}[t!]
\includegraphics[width=0.99\textwidth]{figures/human_clipscorespurts_corr.pdf}
\caption{Correlations of CLIPScore (top row) and SPURTS (bottom row) with human ratings provided by sighted and BLV participants. The Pearson correlations are computed over the human evaluators' average per-description rating. Sighted participants responded to the questions twice; once without (in green) and after seeing the image (in blue). There are no significant correlations between CLIPScore and human ratings. SPURTS correlates significantly with all responses provided by BLV participants but negatively with \emph{Irrelevance} and not at all with sighted participant ratings, indicating a fundamental mismatch.}
\label{fig:clipscore-blvsighted}
\end{figure*}
CLIPScore uses the similarity of CLIP's image and description embeddings as the predictor for description quality, as formulated in (\ref{eq:clipscore}). Denoting $\frac{x}{|x|}$ as $\overline{x}$, we can express CLIPScore as:
\begin{equation} \label{eq:clipscore}
\text{max}\left({\overline{\textnormal{image}}} \cdot {\overline{\textnormal{description}}}, 0\right)
\end{equation}
SPURTS is different from CLIPScore since it only considers the description itself, without taking image information into account. The main goal of SPURTS is to detect fluency and style, and it can be written as
\begin{equation} \label{eq:spurts}
\textnormal{median}_{\textnormal{layer}} \textnormal{max}_{\textnormal{head}} \ I_{\textnormal{flow}}(y_{w/o}, \theta),
\end{equation}
where $I_{\textnormal{flow}}$, which \citet{feinglass2021smurf} refer to as information flow, is normalized mutual information as defined in \citet{witten2005practical}. For an input text without stop words, $y_{w/o}$, and a transformer with parameters $\theta$, SPURTS computes the information flow for each transformer head at each layer, and then returns the layer-wise median of the head-wise maxima.
We turn now to assessing the extent to which CLIPScore and SPURTS approximate the ratings of the BLV and sighted users from our studies. Our central conclusion is that the context-independence of these metrics makes them inadequate for approximating description quality.
\subsection{Compatibility}
We first inspect the extent to which current referenceless metrics can capture whether a description is true for an image. SPURTS provides scores independent of the image and therefore inherently can't capture any notion of truthfulness.
In contrast, CLIPScore is trained to distinguish between fitting and non-fitting image--text pairs, returning a compatibility score. We test whether this generalizes to our experimental data by providing CLIPScore with the true descriptions written for each image and a shuffled variant where images and descriptions were randomly paired. As \figref{fig:clipscore-truthfulness}A demonstrates, CLIPScore rates the ordered pairs significantly higher compared to the shuffled counterparts ($\beta = 2.02$, $\text{SE} = .14$, $p<.001$),\footnote{Result from a linear effects analyses where the shuffled condition is coded as 0, and the ordered condition as 1.} suggesting that it captures image--text compatibility.
\subsection{Description Length Correlation}
Since the length of the description can already account for some of the variance of the BLV ratings, we further investigate whether description length is a general predictor for CLIPScore and SPURTS (see \figref{fig:clipscore-truthfulness}B). For CLIPScore, description length doesn't correlate with predicted quality of the description, which is likely a consequence of the contrastive learning objective, which only optimizes for compatibility but not quality. SPURTS scores, in contrast, significantly correlate with description length, which is aligned with the BLV ratings.
\subsection{Context Sensitivity}
Crucially, the descriptions were written and evaluated within contexts, i.e., their respective Wikipedia article, and previous work suggests that the availability of context should affect what constitutes a good and useful description.
Since current referenceless metrics can't integrate context, we expect that they shouldn't be able to capture the variation in the human description evaluations, and this is indeed what we find.
To investigate this hypothesis, we correlated sighted and BLV description evaluations with the CLIPScore and SPURTS ratings. As shown in \figref{fig:clipscore-blvsighted}, CLIPScore fails to capture any variation observed in the human judgments across questions. This suggests that, while CLIPScore can add a perspective on the compatibility of a text for an image, it can't get beyond that as an indication of how useful a description is if it's true for the image.
Like CLIPScore, SPURTS scores don't correlate with the sighted participant judgments (see \figref{fig:clipscore-blvsighted}, bottom). However, specifically with respect to the overall rating, SPURTS scores show a significant correlation with the BLV participant ratings. While this seems encouraging, further analysis revealed that this correlation is primarily driven by the fact that both BLV and SPURTS ratings correlate with description length. The explained variance of the BLV ratings from description length alone is $0.152$ and SPURTS score alone explains $0.08$ of the variance. In conjunction, however, they only explain $0.166$ of the variance, which means that most of the predictability of SPURTS is due to the length correlation. This is further supported by a mixed effects linear regression analysis in which we fail to find a significant effect of SPURTS ($\beta = .80$, $\text{SE} = .44$, $p > .05$) once we include length as a predictor ($\beta = .64$, $\text{SE} = .15$, $p < .001$).\footnote{We assume random intercepts by participant and description, and we rescaled description length to fall in the range between 0 and 1.}
\begin{table*}[tp]
\centering
\begin{tabular}{@{}r@{ \ }l rrrr@{}}\toprule
&& Overall & Imaginability & Relevance & Irrelevance \\ \midrule
CLIPScore:& BLV & 0.075 & 0.104 & 0.086 & 0.090 \\
With Context:& BLV & 0.201 & 0.182 & 0.202 & 0.142 \\ \hline
CLIPScore:& sighted, no img & $-$0.013 & 0.064 & 0.000 & $-$0.166 \\
With Context:& sighted, no img & 0.238 & 0.315 & 0.190 & $-$0.019 \\ \hline
CLIPScore:& sighted, w img & 0.139 & & 0.106 & $-$0.079 \\
With Context:& sighted, w img & 0.331 & & 0.240 & 0.052 \\ \bottomrule
\end{tabular}
\caption{Comparison of the human rating correlations with the original context-independent CLIPScore and the context-sensitive adaptation, using the same CLIP embeddings. Missing cells were not experimentally measured by design (\secref{sec:exp-eval-sighted}). Across questions and participant groups, correlations improve. The CLIPScore correlations are a replication of \figref{fig:clipscore-blvsighted}.}
\label{tab:clipscorewcontext}
\end{table*}
A further indication that SPURTS isn't capturing essential variance in BLV judgments is apparent from the negative correlation in the \emph{Irrelevance} question ($R=-0.25$). This suggests that SPURTS scores tend to be higher for descriptions that are judged to contain too much irrelevant information and low when participants assess the level of information to be appropriate. In the BLV responses, \emph{Irrelevance} is positively correlated with the \emph{Overall} ratings ($R=0.33$), posing a clear qualitative mismatch to SPURTS. Since what is considered extra information is dependent on the context, this is a concrete case where the metric's lack of context integration results in undesired behavior.
Finally, SPURTS' complete lack of correlation with sighted participant judgments further suggests that SPURTS is insufficient for picking up the semantic components of the descriptions. This aligns with the original conception of the metric, where a reference-based metric (SPARCS) is used to estimate semantic quality.
Overall, our results highlight that SPURTS captures the BLV participants' preferences for longer descriptions but falls short in capturing additional semantic preferences, and is inherently inadequate for judging the truthfulness of a description more generally. CLIPScore can't capture any of the variation in BLV or sighted participant ratings, uncovering clear limitations.
\subsection{Implications for Other Referenceless Metrics}
In the previous experiments, we established that the referenceless metrics CLIPScore and SPURTS can't get traction on what makes a good description when the images and descriptions are contextualized.
Other referenceless metrics such as UMIC \cite{lee2021umic} and QACE \cite{lee2021qace} face the same fundamental issue as CLIPScore and SPURTS due to their contextless nature. Like CLIPScore, UMIC is based on an image--text model (UNITER; \citet{chen2020uniter}) trained under a contrastive learning objective. Similarly, it produces an image--text compatibility score solely by receiving a decontextualized image and text as input.
QACE uses the candidate description to derive potential questions that should be answerable based on the image. The evaluation is therefore whether the description mentions aspects that are true of the image and not about which aspects of the image are relevant to describe. This again only provides insights into image--text compatibility but not contextual relevance.\footnote{Unfortunately, we are unable to provide quantitative results for these referenceless metrics since the authors haven't provided the code necessary (QACE), or the code relies on image features that can't be created for novel datasets with currently available hardware (UMIC, QACE).}
In summary, the current context-independence of all existing referenceless metrics is a major limitation for their usefulness. This is a challenge that needs to be addressed to make these metrics a useful tool for advancing image-based NLG systems.
\section{Discussion: Future Opportunities and Limitations}
So far, we have argued that CLIPScore and other referenceless metrics aren't a useful approximation for BLV (and sighted) user judgments of high-quality descriptions, primarily due to their contextless nature. Using the example of CLIPScore, we will now explore where future work on referenceless metrics in image-based NLG can progress and also discuss some underlying limitations.
\subsection{The Potential for Integrating Context into CLIPScore}
Can referenceless metrics like CLIPScore be made context sensitive? To begin exploring this question, as a proof of concept, we amend (\ref{eq:clipscore}) as follows:
\begin{multline}
\label{eq:clipwcontext}
\overline{\textnormal{description}} \cdot \textnormal{context} \ + \\
\textnormal{description} \cdot \left(\overline{\textnormal{image}} - \overline{\textnormal{context}}\right)
\end{multline}
Here, quality is a function of (a) the description's similarity to the context (first addend) and (b) whether the description captures the information that the image adds to the context (second addend). These two addends can be seen as capturing aspects of (ir)relevance and imaginability, respectively, though we anticipate many alternative ways to quantify these dimensions.
\tabref{tab:clipscorewcontext} reports correlations between this augmented version of CLIPScore and our sighted and BLV participant judgments. We find it encouraging that even this simple approach to incorporating context boosts correlations with human ratings for all the questions in our experiment.
For the \emph{Irrelevance} question, it even clearly captures the positive correlation with BLV ratings, which is negative for both CLIPScore and SPURTS, indicating a promising shift. We consider this an encouraging signal that large pretrained models such as CLIP might still constitute a resource for developing future referenceless metrics.
However, despite these promising signs, there are also reasons to believe that CLIP-based metrics have other restrictive limitations. Due to CLIP's training, images are cropped at the center region and texts need to be truncated at 77 tokens \cite{radford2021learning}. Specifically for the purpose of accessibility, the information this removes can be crucial for determining whether a description is useful or not. For instance, our experiments show that the length of a description is an important indicator for description quality -- information lost in CLIP-based metrics. Moreover, this disproportionately affects the ability to encode the context paragraphs, which are often longer than a typical description. These decisions are therefore likely reflected in any resulting metric and should therefore be reconsidered when devising a new metric.
\subsection{Referenceless Metrics for Image-Based NLG Beyond Accessibility}
While we have specifically focused on the usefulness of referenceless metrics for image accessibility, this isn't the only potential purpose an image-based text might address. \citet{kreiss2022concadia} distinguish \emph{descriptions}, i.e., image-based texts that are written to replace the image, and \emph{captions}, i.e., texts that are intended to appear alongside images, such as tweets or newspaper captions. This suggests that the same text can be very useful for contextualizing an image but fail at providing image accessibility, and vice versa.
To investigate this distinction, they asked participants to rate \texttt{alt} descriptions as well as image captions from Wikipedia according to (1) how much the text helped them imagine the image, and (2) how much they learned from the text that they couldn't have learned from the image. Descriptions were rated more useful for imagining the image, whereas captions were rated more useful for learning additional information. Captions used for contextualizing an image might therefore be another potential use domain for a referenceless metric such as CLIPScore.
\begin{figure}
\includegraphics[width=.48\textwidth]{figures/clipscore-purposecorr.pdf}
\caption{CLIPScore provides higher ratings for image-based texts that capture image content well. Whether the texts provide additional information to the image content doesn't affect the ratings.}
\label{fig:corr-clipscore-purpose}
\end{figure}
To see whether CLIPScore might be a promising resource for evaluating captions, we obtained CLIPScore ratings for the descriptions and captions in \citet{kreiss2022concadia}. CLIPScore ratings correlate with the reconstruction as opposed to the contextualization goal (see \figref{fig:corr-clipscore-purpose}), suggesting that CLIPScore is inherently less appropriate to be used for assessing caption datasets. This aligns with the original observation in \citet{hessel2021clipscore} that CLIPScore performs less well on the news caption dataset GoodNews \cite{biten2019gooda} compared to MSCOCO \cite{hessel2021clipscore}, a contextless description dataset.
Taken together, this suggests that the ``one-size-fits-all'' approach to referenceless image-based text evaluation is not sufficient for adequately assessing text quality for the contextualization or the accessibility domain.
\section{Conclusion}
The context an image appears in shapes the way high-quality accessibility descriptions are written. In this work, we reported on experiments in which we explicitly varied the contexts images were presented in and investigated the effects of this contextual evaluation on participant ratings. These experiments reveal strong contextual effects for sighted and BLV participants. We showed that this poses a serious obstacle for current referenceless metrics.
Our results have wide implications for research on automatic description generation. The central result is that context plays an important role in what makes a description useful. Not realizing that leads to datasets and evaluation methods that don't actually optimize for accessibility. Thus, we need to situate datasets and models within contexts, and evaluation methods need to be sensitive to context. In light of the benefits of referenceless metrics, we feel it is imperative to explore ways for them to incorporate context and to assess the resulting scores against judgments from BLV users.
\section*{Acknowledgements}
This work is supported in part by a grant from Google through the Stanford Institute for Human-Centered AI and by the NSF under project REU-1950223. We thank our experiment participants for their invaluable input.
|
1,116,691,500,801 | arxiv | \section{Introduction}
\label{sec:intro}
With recent advances in machine learning and their efficient implementations, we are beginning to see an increasingly widespread use of deep neural networks (DNNs) on small or lightweight platforms such as mobile devices, edge computing systems, and embedded platforms~\cite{Chen2019,Lane2018,Tan_2019_CVPR}. For DNNs that are too large or complex to realize entirely within a mobile or edge device, a collaborative intelligence~\cite{eshratifar2019towards} approach can be used to split the DNN between the device and the cloud. The data output at the split location is signaled to the remainder of the DNN on the cloud. Determining the ideal place to split the network can depend upon both the application and the size of the activations or feature tensors that need to be signaled~\cite{Kang2017}. The feature tensors inside a DNN may be greatly expanded in terms of size and redundancy as compared to the data originally input to the network. Therefore, data reduction or compression methods are needed in order to reduce the bandwidth of the data to be transferred.
Existing approaches to reducing the data bandwidth between layers typically include quantization, data reorganization, pruning, and sometimes the addition of small neural networks to reduce the feature tensor dimensions. Quantization within DNNs is currently a very active area of research, not only for collaborative intelligence, but also for reducing overall complexity and memory requirements. In~\cite{Mishra2017_WRPN}, ResNet\nobreakdash-34 was modified to have more filters in each layer, and the weights and activations were quantized to 2 and 4 bits, respectively, while maintaining the accuracy of the unmodified network. In~\cite{Banner2018_8BitTraining}, weights, activations, and some of the gradient and back-propagation computations were quantized to 8 bits, along with a scaling modification made to batch normalization. In~\cite{Choi2019_2bit}, networks were trained with weights and activations quantized to two bits. Binary neural networks~\cite{Hubara2016_BNN,Rastegari2016_XNOR} use only one-bit weights and activations during inference. For collaborative intelligence applications, a convolutional layer can be inserted after the split layer to significantly reduce the dimensions of the feature tensor, followed by 8-bit quantization~\cite{eshratifar2019towards}. These methods, however, require end-to-end (re)training to obtain the final network weights and parameters. Given the complexity of the training process and availability of pre-trained floating-point weights for state of the art DNNs, methods for post-training quantization have been of great interest as well. Several post-training quantization techniques~\cite{Zhao2019_Quantization,banner2018_ACIQ,Banner2019_4bit,Krishnamoorthi2018_QuantizingDC} reduce the losses caused by quantization while quantizing weights and/or activations down to between 4 (or an average of 4) and 8 bits, and sometimes lower. Another method to reduce the data size is to compress tiled picture(s) of the activation tensors at the split layer using conventional image or video codecs~\cite{dfc_for_collab_object_detection, Choi2018NearLosslessDF, eshratifar2019bottlenet, eshratifar2019towards,Choi_BaF_2020}.
In this paper, we present a lightweight compression method that is well-suited for coding the output of a split DNN in edge-based devices. This method uses simple and very coarse scalar quantization along with clipping, binarization, and entropy coding to compress the activations. No retraining of network weights is needed. It is also capable of coding to a wide range of bit-rates. We also present comparisons to compression using the HEVC screen content coding extension~\cite{hevc_scc}, by tiling the activation channels into pictures as was done in~\cite{Choi2018NearLosslessDF}.
In Section~\ref{sec:lightweight}, we present the lightweight codec along with an examination of the effects of clipping. Section~\ref{sec:experiments} presents experimental results, followed by conclusions in Section~\ref{sec:conclusions}.
\section{Lightweight compression of tensors}
\label{sec:lightweight}
Fig.~\ref{fig:system} illustrates how the lightweight compression method can be used in a collaborative intelligence application. Layers in a DNN include operations such as convolutions, batch normalization operations, and activation functions. If the first several layers of a DNN are performed for example on a mobile or edge device, we would like to compress the outputs of the activation functions (feature tensors) for transmission to the platform that is performing the remaining layers of the DNN. Because the compression will be performed on an edge device, we would like to keep the complexity of the compression method relatively low, while not significantly compromising the accuracy of the DNN model. For the lightweight compression presented here, we rely on relatively simple operations such as clipping and very coarse scalar quantization to a few representative values. To further compress the data, we convert the quantized symbols to a binarized representation for passing to an entropy coder. The compressed bitstream is sent to the cloud or another computing platform, where it is decoded and converted to a reconstructed feature tensor, which is then processed by the remaining layers of the DNN. The net effect of this process on the DNN computations is that the output of one layer is clipped and quantized; namely the layer that will be transmitted.
\begin{figure}[tb]
\centering
\includegraphics[width=0.48\textwidth,viewport=2.391047 225.827993 553.571983 538.217984,clip]{figures/system.pdf}
\caption{Lightweight compression system overview
}
\label{fig:system}
\end{figure}
For post-training quantization of activations for applications such as object detection and classification, it is well known that uniformly quantizing
32-bit floating point activation values to 8 bits generates a small or negligible reduction in the network's accuracy~\cite{Vanhoucke2011}. This behavior also holds true for collaborative intelligence applications with quantization applied to where the network is split. The network's performance degrades significantly, however, when quantizing without refinements to 4 or fewer bits. For example, the Top\nobreakdash-1 classification accuracy of ResNet-50~\cite{He2015DeepRL} on the ImageNet ILSVRC2012~\cite{imagenet2015} validation data set is 75.8\%, but if we cut the network at the output of layer 21 and uniformly quantize each activation value to 8 levels (3 bits), then the accuracy becomes 59.7\%; a drop of 16.1\%.
By clipping the tensor elements prior to quantization, it is possible to eliminate this loss, without requiring any retraining of the DNN's weights. In this section, we first examine the effects of clipping, and then we present the lightweight compression technique which makes use of an entropy-constrained quantizer design process tailored for clipped activations.
\pagestyle{empty}
\subsection{Effects of clipping}
\label{subsec:clipping}
Fig.~\ref{fig:clip_range} shows the effects that clipping and coarse quantization have on the mean Average Precision (mAP) of the YOLOv3~\cite{Redmon2018_yolov3} object detection network when run on the COCO 2017~\cite{COCO} validation data set (IoU = 0.5). Here, the activations at the output of layer 12 are clipped (clamped) to be between $c_\mathrm{min} = 0.0$ and $c_\mathrm{max}$. Each clipped activation value, denoted as $x_\mathrm{clp}$, is processed by an $N$-level quantizer as follows:
\begin{equation}
\label{eq:clipquant}
Q(x_\mathrm{clp}) = \mathrm{round}\left(\frac{x_\mathrm{clp} - {c}_\mathrm{min}}{{c}_\mathrm{max} - {c}_\mathrm{min}}
\cdot (N-1)\right) \, ,
\end{equation}
where $\mathrm{round(\cdot)}$ rounds away from zero for halfway cases. Note that unlike related literature that focuses on reduced bit-depth architectures, our $N$ does not need to be a power of two, as the purpose of quantization is for compression and subsequent transmission or storage in a bit-stream. The mean-square quantization error (MSQE) computed between an unmodified activation $x$ and the inverse-quantized activation is also shown in Fig.~\ref{fig:clip_range}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.48\textwidth,viewport=7.740000 7.794000 701.243979 413.081987,clip]{figures/clip_range.pdf}
\caption{Effects of clipping layer 12 activations in YOLOv3}
\label{fig:clip_range}
\end{figure}
For 3-bit quantization, i.e. 8-level quantization, peak performance is achieved over a range of ${c}_\mathrm{max}$ values between roughly 3.0 and 5.0. The performance degrades when clipping extends further. As the number of quantization levels is decreased, the optimal ${c}_\mathrm{max}$ decreases, as does the range of ${c}_\mathrm{max}$ values that achieves peak performance. With 1-bit (2-level) quantization, the optimal range is quite narrow. When the quantization is not extremely coarse, e.g. 8-level (3-bit) or higher, the minimum MSQE generally coincides with the peak mAP performance. Earlier works, e.g.~\cite{banner2018_ACIQ,Banner2019_4bit} have leveraged this behavior to model the quantization error in order to select the optimal clipping range for all activations in a DNN. However, it is evident from these prior works that deviations from the models occur with extremely coarse quantization, e.g. 2-bit (4-level) and below, and when the distributions of the activations are non-Gaussian. We can see in Fig.~\ref{fig:clip_range} that the optimal ${c}_\mathrm{max}$ for 1-bit quantization is approximately 2.0, whereas the minimum MSQE occurs near
${c}_\mathrm{max}=1.75$.
Thus, choosing ${c}_\mathrm{max}$ based on minimizing MSQE can result in a potential loss in mAP of several percent when $N$ is small. For the experiments presented later in this paper, we empirically select the clipping ranges in order to determine the best achievable network accuracy when compression is used.
\subsection{Modified entropy-constrained quantization for clipped activations}
Although a uniform quantizer is simple, it is not optimal for signals that are not uniformly distributed. Moreover, because there is a trade-off between network accuracy and bit-rate when compressing activations, we would like be able to compress the tensors over a range of rates or file sizes. Entropy-constrained quantization~\cite{Chou1989} and rate-distortion optimization~\cite{Sullivan1998} are well-known methods for compressing data subject to minimizing a Lagrangian cost function $J = D + \lambda R$, where $D$ is a distortion metric, $R$ is a rate or size of the representation, and $\lambda$ is a scalar. Thus, we can easily obtain optimal quantizers in the mean-squared sense over a range of rates by using an entropy-constrained design process. However, we showed earlier that the accuracy of a DNN is quite sensitive to the clipping range of a layer's activation when quantized to a very low number of levels. The representative level for each bin of an $\ell^2$-norm optimized quantizer corresponds to the centroid of the data quantized to that bin. For example, suppose a one-bit quantizer divides $[0.0,2.0]$ into two bins, $[0.0,1.0]$ and $[1.0,2.0]$, and the representative values are 0.3 and 1.5, respectively. If a layer's activations are clipped to $[0.0,2.0]$, quantized, and transmitted, then the receiver would reconstruct data having only the values $\{0.3,1.5\}$, which span a range much smaller than the initial clipping range of $[0.0,2.0]$. To address this problem, we present a modified entropy-constrained quantizer design process to pin the reconstruction levels of the outermost bins to ${c}_\mathrm{min}$ and ${c}_\mathrm{max}$, to ensure that the reconstructed activations span the full clipping range. The reconstruction values for the interior bins and the threshold values between bins are not pinned and are free to vary under the design algorithm.
Using notation similar to that of~\cite{Girod_ecquant}, the conventional entropy-constrained quantizer design process for an $N$-level quantizer is summarized in Algorithm~\ref{alg:ec_conventional}. With this process, quantizers can be designed to cover a range of rate-distortion performance points, based upon the value of the Lagrange multiplier $\lambda$. To design quantizers that work well in a lightweight compression system for clipped activations, the modified quantizer design process is shown in Algorithm~\ref{alg:ec_mod}. The main modifications are related to pinning the representative levels for the boundary bins and using codeword lengths instead of probabilities for computing rate-related terms. The boundary pinning occurs in Step~\ref{stp:ec_mod_update}. Here, the first and last representative levels are always set to the minimum and maximum activation clipping values ${c}_\mathrm{min}$ and ${c}_\mathrm{max}$, respectively. The interior levels are computed as in Step~\ref{stp:ec_update} of Algorithm~\ref{alg:ec_conventional}.
For modifying the rate terms, we replace the probability-based estimate of the number of bits used to represent a bin, $\log_{2}(p_n)$, with the known length $b_n$ of the binarized codeword used to represent the bin. In the next subsection, we discuss the binarization and subsequent entropy coding.
\begin{algorithm}[!htb]
\small
\caption{Conventional entropy-constrained quantizer design process}
\label{alg:ec_conventional}
\hspace*{\algorithmicindent} \textbf{Input:} Training samples \\
\hspace*{2cm} $x
\in \{x_m; m = 0,1,\dotsc,M-1\}$ \\
\hspace*{1.6cm} Number of quantizer bins $N$ \\
\hspace*{1.6cm} Lagrange multiplier $\lambda$ \\
\hspace*{\algorithmicindent} \textbf{Output:} Quantizer representative levels \\
\hspace*{2.2cm} $\hat{x}_{n}, n \in \{0,1, \dotsc ,N-1\}$ \\
\hspace*{1.85cm} Quantizer decision thresholds \\
\hspace*{2.2cm} $t_{n}, n \in \{1, \dotsc ,N-1\}$
\begin{algorithmic}[1]
\State Initialize the representative levels $\hat{x}_n$ and probabilities $p_n$ for each bin, for $n \in \{0,1,\dotsc,N-1\}$ (e.g., uniform and equiprobable)
\State \label{stp:ec_assign}Assign each training sample $x_m$, $m \in \{0,1,\dotsc,M-1\}$ to quantizer bin $n$ having representative value $\hat{x}_n$ such that the Lagrangian rate-distortion cost is minimized:
\begin{equation*}
\argmin_{n} \left[ (x_m - \hat{x}_n)^{2} - \lambda \log_{2}p_n \right]
\end{equation*}
The subset of samples $x_m$ assigned to bin $n$ is denoted as $\bm{B}_n$.
\State \label{stp:ec_update}Update the probabilities $p_n$ based on the assignment from Step~\ref{stp:ec_assign}, and recompute the representative levels for each bin:
\begin{equation*}
\hat{x}_n = \frac{1}{|\bm{B}_n|}\sum_{x \in \bm{B}_n} x \, , \quad n \in \{0,1, \dotsc ,N-1\}
\end{equation*}
where $|\bm{B}_n|$ is the number of samples assigned to bin $n$.
\State Based on the recomputed representative levels, recompute the Lagrangian cost function, and repeat Steps~\ref{stp:ec_assign} and~\ref{stp:ec_update} until the reduction in the cost function is less than a threshold.
\State Compute $N-1$ quantizer decision thresholds:
\begin{equation*}
\begin{split}
t_{n} = \frac{\hat{x}_n + \hat{x}_{n-1}}{2}
+ \lambda \frac{\log_{2} p_n - \log_{2} p_{n-1}}{2(\hat{x}_n - \hat{x}_{n-1})} \, , \\
n \in \{1, \dotsc ,N-1\}
\end{split}
\end{equation*}
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[!htb]
\small
\caption{Modified entropy-constrained quantizer design process for clipped activations}
\label{alg:ec_mod}
\hspace*{\algorithmicindent} \textbf{Input:} Training samples \\
\hspace*{2cm} $x
\in \{x_m; m = 0,1,\dotsc,M-1\}$ \\
\hspace*{1.6cm} Number of quantizer bins $N$ \\
\hspace*{1.6cm} Codeword lengths $b_{n}, n \in \{1, \dotsc ,N-1\}$ \\
\hspace*{1.6cm} Lagrange multiplier $\lambda$ \\
\hspace*{1.6cm} Activation clipping range $\left[{c}_{\mathrm{min}}, {c}_{\mathrm{max}}\right]$ \\
\hspace*{\algorithmicindent} \textbf{Output:} Quantizer representative levels \\
\hspace*{2.2cm} $\hat{x}_{n}, n \in \{0,1, \dotsc ,N-1\}$ \\
\hspace*{1.85cm} Quantizer decision thresholds \\
\hspace*{2.2cm} $t_{n}, n \in \{1, \dotsc ,N-1\}$
\begin{algorithmic}[1]
\State Clip (clamp) the training samples $x$ to be within $\left[{c}_{\mathrm{min}}, {c}_{\mathrm{max}}\right]$, which is the clipping range applied to the activations
\State Initialize the representative levels $\hat{x}_n$ for each bin, for $n \in \{0,1,\dotsc,N-1\}$ (e.g., uniform)
\State \label{stp:ec_mod_assign}Assign each training sample $x_m$, $m \in \{0,1,\dotsc,M-1\}$ to quantizer bin $n$ having representative value $\hat{x}_n$ such that the Lagrangian rate-distortion cost is minimized:
\begin{equation*}
\argmin_{n} \left[ (x_m - \hat{x}_n)^{2} - \lambda b_n \right]
\end{equation*}
The subset of samples $x_m$ assigned to bin $n$ is denoted as $\bm{B}_n$.
\State \label{stp:ec_mod_update}Recompute the representative levels for each bin:
\begin{align*}
\hat{x}_0 &= {c}_{\mathrm{min}} \\
\hat{x}_{N-1} &= {c}_{\mathrm{max}} \\
\mathrm{if} N > 2:\\
\hat{x}_n &= \frac{1}{|\bm{B}_n|}\sum_{x \in \bm{B}_n} x \, , \quad n \in \{1, \dotsc ,N-2\}
\end{align*}
where $|\bm{B}_n|$ is the number of samples assigned to bin $n$.
\State Based on the recomputed representative levels, recompute the Lagrangian cost function, and repeat Steps~\ref{stp:ec_mod_assign} and~\ref{stp:ec_mod_update} until the reduction in the cost function is less than a threshold.
\State Compute $N-1$ quantizer decision thresholds:
\begin{equation*}
\begin{split}
t_{n} = \frac{\hat{x}_n + \hat{x}_{n-1}}{2}
+ \lambda \frac{b_{n} - b_{n-1}}{2(\hat{x}_n - \hat{x}_{n-1})} \, , \\
n \in \{1, \dotsc ,N-1\}
\end{split}
\end{equation*}
\end{algorithmic}
\end{algorithm}
\subsection{Binarization and entropy coding}
After quantizing an activation element, an index associated with the selected representative level is coded and signaled to a bit-stream. For the types of DNNs that we are splitting, the activation values tend to be skewed toward zero, as illustrated by the histogram in Fig.~\ref{fig:resnet50_hist} for the unclipped layer-21 activations of ResNet-50 when run over the ImageNet ILSVRC2012 validation data set.
Given that we will be able to achieve good performance when quantizing to very few bins, a truncated unary binarization scheme~\cite{Marpe2003_CABAC} is well suited for this purpose. For example, the bin indices $\{0,1,2,3\}$ of a 2-bit (4-level) quantizer are mapped to the binarized strings $\{0,10,110,111\}$ respectively.
\begin{figure}[tb]
\centering
\includegraphics[width=0.48\textwidth,viewport=6.840000 11.052000 703.107400 195.875432,clip]{figures/resnet50_hist.pdf}
\caption{Histogram of activation elements for layer 21 of ResNet-50}
\label{fig:resnet50_hist}
\end{figure}
The binarized strings must next be coded to a bit-stream. For this paper, we use Context-based Adaptive Binary Arithmetic Coding (CABAC)~\cite{Marpe2003_CABAC}, similar to that used in H.264/AVC and HEVC. One context is used for each bit position for the binarized string. For the 2-bit example given in the previous paragraph, three contexts are used.
\section{Experimental results}
\label{sec:experiments}
We applied our lightweight compression technique to activations output from the split layer of a DNN, for three different inference scenarios: YOLOv3 object detection at layer 12, VGG-16 classification at layer 6, and ResNet-50 classification at layer 21. The dimensions of the activations at these layers were 52$\times$52$\times$256, 56$\times$56$\times$128, and 32$\times$32$\times$512, respectively. Pre-trained network weights were obtained from~\cite{darknet_weights}. The software modified to run the experiments was the \textit{Darknet} version of~\cite{AlexeyAB_darknet}. For YOLOv3 with input size 416$\times$416, mAP (IoU~=~0.5) results were obtained using the COCO 2017 validation data set, which includes about 5k images. For VGG-16 and ResNet-50, classification accuracies were obtained using the ImageNet ILSVRC2012 validation data set, which has 50k images. For experiments using entropy-constrained quantization, the quantizer design algorithms were run on activations output when running the first part of the network on 100 images from the data set. After clipping, quantization, and coding to a bit-stream, the activations were decoded and inverse quantized and then passed to the remainder of the neural network. The bit-streams also included side information needed by the decoder, e.g. $c_{\mathrm{min}}$, $c_{\mathrm{max}}$, $N$, and some dimensional parameters for object detection, which together comprised 24 bytes for object detection and 12 bytes for classification networks. The clipping thresholds were empirically selected as described in Section~\ref{subsec:clipping}. The size of the compressed data is reported as bits per element, i.e. the size of the bit-stream divided by the number of elements in the activation's output feature tensor.
The performance when using a uniform quantizer is shown in Tables~\ref{tab:table_yolov3}, \ref{tab:table_vgg16}, and~\ref{tab:table_resnet50} for YOLOv3, VGG-16, and ResNet-50, respectively. For YOLOv3, no loss in performance occurred when quantizing all the way down to 16 levels (4-bit quantization). The drop in mAP was less than 1\% with a 4-level (2-bit) quantizer. For VGG-16, the clipping ranges were much larger than with the other networks, because VGG-16 does not use batch normalization. The Top-1 accuracy drop with an 8-level (3-bit) quantizer was 0.3\% and 1.2\% with 7-level quantization. For ResNet-50, the activations could be quantized to 8 levels with no loss, and at 4 levels (2 bits), the Top-1 loss was well below 1\%. One-bit quantization was feasible with YOLOv3 and ResNet-50, which yielded 4.8\% and 4.9\% losses, respectively, with compressed sizes of 0.39 and 0.41 bits per element. We can also see that as the number of quantization levels decreased, generally the optimal clipping range decreased as well, as discussed in Section~\ref{subsec:clipping}.
\begin{table}[tb]
\centering
\caption{Object detection performance for YOLOv3 with uniform quantization and lightweight compression at layer 12}
\label{tab:table_yolov3}
\resizebox{\columnwidth}{!}{
\begin{tabular}{cccccc}
&
\makecell{\textbf{quantized} \\ \textbf{bits}} &
\makecell{\textbf{quantizer} \\ \textbf{bins}} &
\makecell{\textbf{clip} \\ \textbf{min, max}} &
\makecell{\textbf{compressed} \\ \textbf{bits/element}} &
\makecell{\textbf{mAP (\%)}} \\
\midrule
\makecell{unmodified \\ (float32)} & 32 & --- & --- & --- & 67.4 \\
\arrayrulecolor{lightgray} \midrule \arrayrulecolor{defaultrulecolor}
\multirow{5}{*}{\makecell{with \\ lightweight \\ compression}}
& 4 & 16 & -0.8, 3.5 & 1.94 & 67.4 \\
& 3 & 8 & -1.0, 4.0 & 1.15 & 66.8 \\
& 2 & 4 & -0.1, 2.5 & 0.86 & 66.5 \\
& & 3 & -0.1, 2.7 & 0.60 & 66.2 \\
& 1 & 2 & \phantom{-}0.0, 2.0 & 0.39 & 62.6 \\
\end{tabular}
}
\end{table}
\begin{table}[tb]
\centering
\caption{Classifier performance for VGG-16 with uniform quantization and lightweight compression of layer 6 activations}
\label{tab:table_vgg16}
\resizebox{\columnwidth}{!}{
\begin{tabular}{c@{\hskip3pt}c@{\hskip3pt}c@{\hskip3pt}c@{\hskip3pt}c@{\hskip3pt}c@{\hskip3pt}c@{\hskip3pt}}
&
\makecell{\textbf{quantized} \\ \textbf{bits}} &
\makecell{\textbf{quantizer} \\ \textbf{bins}} &
\makecell{\textbf{clip} \\ \textbf{min, max}} &
\makecell{\textbf{compressed} \\ \textbf{bits/element}} &
\makecell{\textbf{Top-1} \\ \textbf{accuracy} \\ \textbf{(\%)}} &
\makecell{\textbf{Top-5} \\ \textbf{accuracy} \\ \textbf{(\%)}} \\
\midrule
\makecell{unmodified \\ (float32)} & 32 & --- & --- & --- & 70.4 & 89.8 \\
\arrayrulecolor{lightgray} \midrule \arrayrulecolor{defaultrulecolor}
\multirow{6}{*}{\makecell{with \\ lightweight \\ compression}}
& 4 & 16 & 0, 2600 & 2.33 & 70.2 & 89.7 \\
& 3 & 8 & 0, 2400 & 2.41 & 70.1 & 89.6 \\
& & 7 & 0, 2200 & 1.57 & 69.2 & 89.1 \\
& & 6 & 0, 2000 & 1.48 & 68.6 & 88.7 \\
& & 5 & 0, 1800 & 1.36 & 67.8 & 88.2 \\
& 2 & 4 & 0, 1600 & 1.20 & 66.1 & 87.2 \\
\end{tabular}
}
\end{table}
\begin{table}[tb]
\centering
\caption{Classifier performance for ResNet-50 with uniform quantization and lightweight compression at layer 21}
\label{tab:table_resnet50}
\resizebox{\columnwidth}{!}{
\begin{tabular}{c@{\hskip3pt}c@{\hskip3pt}c@{\hskip3pt}c@{\hskip3pt}c@{\hskip3pt}c@{\hskip3pt}c@{\hskip3pt}}
&
\makecell{\textbf{quantized} \\ \textbf{bits}} &
\makecell{\textbf{quantizer} \\ \textbf{bins}} &
\makecell{\textbf{clip} \\ \textbf{min, max}} &
\makecell{\textbf{compressed} \\ \textbf{bits/element}} &
\makecell{\textbf{Top-1} \\ \textbf{accuracy} \\ \textbf{(\%)}} &
\makecell{\textbf{Top-5} \\ \textbf{accuracy} \\ \textbf{(\%)}} \\
\midrule
\makecell{unmodified \\ (float32)} & 32 & --- & --- & --- & 75.8 & 92.9 \\
\arrayrulecolor{lightgray} \midrule \arrayrulecolor{defaultrulecolor}
\multirow{6}{*}{\makecell{with \\ lightweight \\ compression}}
& 3 & 8 & 0, 14 & 1.23 & 75.8 & 92.9 \\
& & 7 & 0, 12 & 1.22 & 75.7 & 92.9 \\
& & 6 & 0, 12 & 1.09 & 75.6 & 92.8 \\
& & 5 & 0, 12 & 0.94 & 75.5 & 92.8 \\
& 2 & 4 & 0, 10 & 0.86 & 75.2 & 92.7 \\
& 1 & 2 & 0, \phantom{1}7 & 0.41 & 70.9 & 90.4 \\
\end{tabular}
}
\end{table}
Object detection and classification performance when quantizing to between 1 and 2 bits is shown in Figs.~\ref{fig:yolov3_plot}--\ref{fig:resnet50_top5_plot}, for lightweight compression with both uniform and modified entropy-constrained quantization. Fig.~\ref{fig:yolov3_plot} also shows the YOLOv3 performance with a conventional entropy-constrained quantizer that does not pin the outermost reconstruction levels. Here, with 2-bit quantization, the modified method performed about 0.7\% better than with the conventional method, with additional gains at lower rates.
Fig.~\ref{fig:yolov3_plot} also shows that the entropy-constrained quantizer's ability to cover a range of rates enables us to improve the 1-bit performance by about 1\%. For the Top-1 and Top-5 accuracies of ResNet-50, Figs.~\ref{fig:resnet50_top1_plot} and~\ref{fig:resnet50_top5_plot} show that the modified quantizer design algorithm also allows us to obtain improved accuracies for 3- and 4-level (2-bit) quantizers. The relative behaviors among corresponding performance curves for Top-1 and Top-5 accuracies are consistent, with lower losses for Top-5. The range of achievable compressed bit-stream sizes for all these experiments with quantization to 2 and fewer bits was between about 0.3 to 1.0 bits per tensor element.
Figs.~\ref{fig:yolov3_plot}--\ref{fig:resnet50_top5_plot} also show the performance when coding the activations using the HM16.20~\cite{HM16.20} implementation of the HEVC screen content coding extension (HEVC-SCC). HEVC-SCC includes tools that help with the coding of non-camera-captured pictures. As shown in~\cite{Choi2018NearLosslessDF}, when activation channels are arranged to form a picture, they contain much high-frequency content. HEVC-SCC includes a transform skip (TS) mode that is available for all transform block sizes, so we show results when enabling TS for 4$\times$4 blocks only, and for TS enabled over all block sizes. Each set of activation channels were quantized to 8 bits and mosaicked into an 832$\times$832 picture for YOLOv3 and 1024$\times$512 for ResNet-50. Given the fineness of the quantizer, clipping was not necessary. The mosaicked activations for the validation set were coded by HEVC-SCC as an all-Intra sequence of monochrome (4:0:0) 8-bit pictures.
Even with the improved performance with TS on all block sizes, the lightweight compression outperformed HEVC-SCC by up to 1.3\%, depending upon rate. If we compare the operations needed by the lightweight codec to that reported for HEVC~\cite{Bossen1012_hevc_complexity}, we can see that the lightweight codec is well over 90\% less complex than HEVC, as we are only performing clipping, scalar quantization, binarization of only a few symbols, and entropy coding.
\begin{figure}[tb]
\centering
\includegraphics[width=0.46\textwidth,viewport=7.200000 7.218000 701.099979 421.829987,clip]{figures/yolov3_hevc_plot.pdf}
\caption{Object-detection mAP for YOLOv3 using lightweight compression at layer 12 with modified entropy-constrained quantization}
\label{fig:yolov3_plot}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[width=0.46\textwidth,viewport=7.200000 7.218000 701.063979 421.811987,clip]{figures/resnet50_hevc_top1_plot.pdf}
\caption{Top-1 accuracy for ResNet-50 using lightweight compression at layer 21 with modified entropy-constrained quantization}
\label{fig:resnet50_top1_plot}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[width=0.46\textwidth,viewport=7.200000 7.218000 701.081979 421.847987,clip]{figures/resnet50_hevc_top5_plot.pdf}
\caption{Top-5 accuracy for ResNet-50 using lightweight compression at layer 21 with modified entropy-constrained quantization}
\label{fig:resnet50_top5_plot}
\end{figure}
\section{Conclusions}
\label{sec:conclusions}
We presented an efficient post-training lightweight compression method for collaborative intelligence applications, suitable for when an edge device compresses the output of a split DNN and transmits it to the cloud, where the remainder of the DNN is processed.
The codec only requires clipping, coarse quantization, binarization, and entropy coding to compress feature tensors, so is well over 90\% less complex than existing image or video codecs such as HEVC that are typically used for picture compression. We also presented an entropy-constrained quantizer design process tailored specifically for clipped activations. With this lightweight lossy coding technique, we were able to quantize the 32-bit floating point activations at a split DNN layer to fewer than 2 bits per element and then compress them further to 0.6 to 0.8 bits per element, while keeping the network accuracy loss to less than 1\%. When coding the activations using HEVC, we showed that the lightweight codec yielded accuracies of up to 1.3\% higher than HEVC, depending upon the rate. The performance and simplicity of this lightweight compression technique makes it an attractive option for coding activations for edge/cloud DNN applications.
\bibliographystyle{IEEEbib-abbrev}
|
1,116,691,500,802 | arxiv | \section{Introduction}
Characterizing the general structure of star-forming regions is an issue which is subject of and worth of intensive study. The molecular clouds (MCs) associated with them are the original sites of star formation \citep[for reviews, see][]{MacLow_Klessen_04, McKee_Ostriker_07, Klessen_Glover_14}. Denser fragments of MCs, often labeled `cores' and/or `clumps', turn out to have mass distributions similar or identical to the initial stellar mass function \citep[][Table 1]{ALL_07, VDK_13}. This raises the problem whether there is a link between the general structure of a cloud and its star-forming properties. Some basic indicators of general cloud structure are, for instance: i) the existence of scaling relations of velocity dispersion and density \citep{Larson_81, Solomon_ea_87, Heyer_ea_09, KLN_13}; and, ii) the probability distribution of column density whose shape could be close to lognormal \citep{LAL_11, Brunt_15}, to a power-law function \citep{LAL_15} or a combination of both \citep{Kainulainen_ea_09}.
The
analysis of the indicator ii) is considered as a key to understanding the evolutionary status of the cloud and the dominant processes that govern its physics \citep[see][for discussion]{Schneider_ea_13, Schneider_ea_15a}. By use of the probability distribution function (pdf) one can calculate masses within chosen density thresholds and -- defining effective size in some way, -- study the intra-cloud mass-size relationship \citep{LAL_10, BP_ea_12}.
In this Paper we model general MC structure assuming power-law scaling relations of velocity dispersion and density and a lognormal density distribution at each scale. The scales are defined through iso-density contours within which an equipartition between gravitational, kinetic and thermal energy exists. The physical basis and the construction of the model are described in Sect. \ref{Model}. The predicted mass-size relationships and their comparison with observational data for several Galactic star-forming regions are presented in Sect. \ref{Model predictions}. Sect. \ref{Discussion} contains a discussion of the applicability of the model in terms of column-density range, cloud evolutionary stage and size of the star-forming region. A summary of this work is given in Sect. \ref{Summary}.
\section{Model of cloud structure}
\label{Model}
\subsection{Physical framework}
\label{Framework}
The cloud is considered to be at an early evolutionary stage, prior to formation of stars and/or stellar clusters in its densest parts. Its possible age is in the range $t_{1}\gtrsim5$~Myr, corresponding to fully developed supersonic turbulence, and $t_{2}\lesssim18-20$~Myr, corresponding to a global cloud contraction, as suggested by numerical simulations of cloud evolution \citep[e.g.][]{Baner_ea_09, VS_ea_07} or observations of nearby galaxies \citep{Fukui_ea_09, Meidt_ea_15}. Hence the general structure of the cloud is determined primarily by the interaction of supersonic turbulence and gravity. The fully developed turbulence shapes the density and velocity field at any spatial scale $L$ within the inertial range through a cascade possibly driven by the very process of cloud formation \citep{KH_10}. We set conservative limits of the inertial range: $0.1\lesssim L \lesssim 20$~pc. The lower limit is close to the transonic scale and to the typical size of dense (prestellar) cores. The upper limit of
$20$~pc is adopted to ensure that the gas is mainly molecular and isothermal (with temperature $T=10-20$~K). This estimate is plausible as well if one takes into account that the largest scale of the inertial range is about 3 times less than the injection scale and adopts for the latter $\sim50$~pc, which is above the typical size of giant MCs \citep{Kritsuk_ea_07, Padoan_ea_06}. An equipartition of gravitational vs. kinetic and thermal energy takes place within the mentioned evolutionary stage as gravity slowly takes over toward a global cloud contraction (Fig. 8 in \citealt{VS_ea_07}; see also \citealt{ZA_ea_12}).
\subsection{Basic assumptions}
\subsubsection{Scaling relations of velocity dispersion and mean density}
\label{Assumptions: scaling relations}
Power-law scaling relations of velocity dispersion $u_L$ and mean density $\langle n \rangle_L$ are assumed to hold within the adopted inertial range. Applied to MCs and cloud fragments, they were initially discovered by \citet{Larson_81} and therefore are often called ``Larson's first and second relations''. In our modelling, we use these relations in the form:
\begin{equation}
\label{eq_Larson_1}
u_L = u_0\,\Big(\frac{L}{1~{\rm pc}}\Big)^\beta~,
\end{equation}
\begin{equation}
\label{eq_Larson_2}
\langle n \rangle_L = n_0\,\Big(\frac{L}{3~{\rm pc}}\Big)^\alpha~.
\end{equation}
The suggested normalization was chosen in view of the scatter of original data \citep{Larson_81, Solomon_ea_87, Falgarone_McKee_15} and of the possible variations of the scaling index $0.33\lesssim\beta\lesssim0.50$ \citep{Larson_81, HB_04, Padoan_ea_06, Padoan_ea_09} where the classical value of \citet{K41} for incompressible turbulence is taken as a lower limit. The scaling indices $\alpha$ and $\beta$ are interdependent in the proposed model (see our next basic assumption \ref{Assumptions: equipartition}) and thus the variations of $\beta$ generate $-1.3\lesssim\alpha\lesssim-1$. To provide consistency of the mean-density scaling relation with such index values and within the inertial range a higher normalization factor of the scale was adopted in equation \ref{eq_Larson_2} (cf. Fig. \ref{fig_scaling_relations}, bottom).
Different estimates of the scaling coefficient $u_0$ can be found in the literature \citep{Heyer_ea_09, BP_ea_11a, Shetty_ea_12} while $n_0$ is less studied, in particular, due to the variety of ways to define discrete objects in MCs and their density. Reference values of $u_0$ and $n_0$ yielding scaling relations in agreement with observational data are given in Table \ref{table_var_scaling_coef} and two concrete scaling relations for fixed scaling coefficients are shown in Fig. \ref{fig_scaling_relations}. Note that $u_0= u (L=1~{\rm pc})\gtrsim1$~km/s is typical for dense cloud regions which are possible sites of star formation \citep[see][Fig. 3]{BP_ea_11a}.
\begin{figure}
\hspace{3.5em}
\begin{center}
\includegraphics[width=84mm]{Fig/fig_scaling_laws.eps}
\vspace{0.9cm}
\caption{Scaling relations of velocity (top) and mean density (bottom) when the scaling indices $\alpha$ and $\beta$ are varied and for a fixed set of coefficients $(u_0,n_0)$. Shaded areas show the data scatter. The `classical' case $\beta=0.50,~\alpha=-1$ (thick line) and that of shallow velocity scaling $\beta=0.33,~\alpha=-1.34$ (thin line) are plotted.}
\label{fig_scaling_relations}
\end{center}
\end{figure}
\begin{table}
\caption{Fiducial values of the scaling-relations coefficients $u_0$ and $n_0$ allowing variations of indices $\alpha$ and $\beta$ within the confidence ranges.
}
\label{table_var_scaling_coef}
\begin{center}
\begin{tabular}{cccc}
\hline
\hline
$u_0$ & $n_0$ & $-\alpha$ & $\beta$ \\
~[ km/s ] & [ \cc ] & ~ & ~ \\
\hline
1.0& $1.3\times10^{3}$ & 0.90-1.34& 0.33-0.55 \\
1.0& $4.4\times10^{3}$ & 0.90-1.34& 0.33-0.55 \\
1.4& $1.3\times10^{3}$ & 1.00-1.34& 0.33-0.50 \\
1.4& $4.4\times10^{3}$ & 0.90-1.34& 0.33-0.55 \\
\hline
\hline
\end{tabular}
\end{center}
\smallskip
\end{table}
\subsubsection{Equipartition between gravitational and kinetic energy, including thermal support}
\label{Assumptions: equipartition}
The equipartition relation is described by the equation:
\begin{equation}
\label{eq_equipartition}
|W|=f (E_{\rm kin} + E_{\rm th})~,
\end{equation}
where $W$, $E_{\rm kin}$ and $E_{\rm th}$ are gravitational, kinetic (turbulent) and thermal (internal) energy per unit volume $v$ and the coefficient $f$ is taken to vary from unity to 4, i.e. from weakly-gravitating to strongly gravitationally bound entities. As we consider cold molecular gas with $T=10-20$~K, the thermal energy term in the equation above is much less than the gravitational and the kinetic ones and thus contributes only for the fine energy balance.
\citet{BP_06} demonstrated (see Sect. 3.6 there) that in case of equipartition between gravitational and kinetic energy the scaling indices $\alpha$ and $\beta$ are interdependent:
\begin{equation}
\label{eq_alpha_beta_relation}
\beta=\frac{\alpha+2}{2}~~.
\end{equation}
Interestingly, \citet{BP_VS_95} found from numerical simulations that equipartitions of this type hold for regions of various size in turbulent interstellar medium, defined by some density threshold.
\subsubsection{Lognormal density distribution at each scale}
A lognormal volumetric distribution of density is found in numerous numerical simulations of supersonic turbulence \citep[e.g.][]{Klessen_00, LKM_03, Kritsuk_ea_07, Federrath_ea_10} and is described through a standard lognormal pdf:
\begin{equation}
\label{eq_v_pdf}
p_v(s)\,ds=\frac{1}{\sqrt{2\pi \sigma^2}}\,\exp{\Bigg[-\frac{1}{2}\bigg( \frac{s -s_{{\rm max,}\,v}}{\sigma}\bigg)^2 \Bigg]}\,ds~,
\end{equation}
where $s=\ln[n/\langle n \rangle_L]$ is the log density, $s_{\rm max}$ is the distribution peak and $\sigma$ is the standard deviation. The latter two parameters are interdependent (see \citealt{VS_94}) and are determined from the sonic Mach number ${\cal M}=u_L/c_{\rm s}$ ($c_{\rm s}$ is the sound speed) and turbulence forcing parameter $b$:
\begin{equation}
\label{eq_sigma_PDF}
\sigma^2={\rm ln}\,(1+b^2\,{\cal M}^2)~,~~~s_{{\rm max},\,v}=-\frac{\sigma^2}{2}
\end{equation}
In our model we also use the mass-weighted log-density pdf:
\begin{equation}
\label{eq_m_pdf}
p_m(s)\,ds=\frac{1}{\sqrt{2\pi \sigma^2}}\,\exp{\Bigg[-\frac{1}{2}\bigg( \frac{s -s_{{\rm max,}\,m}}{\sigma}\bigg)^2 \Bigg]}\,ds~,
\end{equation}
where $s_{{\rm max,}\,m}=-s_{{\rm max,}\,v}$ \citep[see][Sect. 3.3.1]{LKM_03}.
The turbulence forcing parameter $b$ is taken to span values between $0.33$, for purely solenoidal forcing, and $0.42$, for a natural mixture between solenoidal and compressive modes \citep{FKS_08, Federrath_ea_10, Konstandin_ea_15}.
\subsubsection{Introduction of physical scale}
\label{Physical scale}
Introducing a characteristic turbulent scale $L$ is straightforward. We define it as the linear size of a cube within which velocity dispersion and mean density are calculated according the assumed scaling relations (equations \ref{eq_Larson_1} and \ref{eq_Larson_2}). This quantity is essentially statistical since it is linked to statistical properties of fully developed turbulence. Our second assumption (Sect. \ref{Assumptions: equipartition}) requires another, deterministic definition of scale, through the total volume of regions wherein the balance of gravitational vs. kinetic and thermal energies is achieved. Inspired by the finding of \citet{BP_VS_95}, we define such {\it physical scale} $L_t\le L$ as the effective size of the sum of all regions delineated by log-density threshold level $t$ at which equation \ref{eq_equipartition} is satisfied.
We stress that the notion of physical scale is not to be confused in any way with a connected region or a clump (Fig. \ref{fig_physical_scale}). To create an intuitive reference to observable objects, we label the regions included in a physical scale `cloudlets'. The size of a single cloudlet can vary from a few pixels on a map (or, in the 3D case, numerical cube) up to a size of whole clouds.
\begin{figure}
\begin{center}
\includegraphics[width=50mm]{Fig/fig_physical_scale.eps}
\vspace{0.2cm}
\caption{The notion of physical scale $L_t$: the effective size of the sum of all regions (grey `cloudlets') delineated by given log-density threshold level $t$ (thick dark-grey line) and incorporated within turbulent scale $L$.}
\label{fig_physical_scale}
\end{center}
\end{figure}
\subsection{Parameters of the cloudlets in equipartition}
Let $t$ be the threshold level at which the equipartition of energies (equation \ref{eq_equipartition}) is achieved. Then the mean log density of set of cloudlets delineated by $t$ is:
\begin{eqnarray}
\label{eq_mean_log_density_t}
\overline{s}_t & = & \ln\,[\overline{n}_t/\langle n \rangle_L] =\int\limits_{t}^\infty s\,p_v(s)\,ds \nonumber \\
~ & = & \frac{1}{\sqrt{2\pi\sigma^2}} \int\limits_{t}^\infty s \exp{\Bigg[-\frac{1}{2}\bigg( \frac{s + \sigma^2/2}{\sigma}\bigg)^2 \Bigg]}\,ds~.
\end{eqnarray}
The total mass of these cloudlets is defined through the mass of the turbulent scale $M_L=\mu\langle n \rangle_L L^3$ using the mass-weighted log-density pdf (equation \ref{eq_m_pdf}):
\begin{eqnarray}
\label{eq_mass_t}
M_t & = & M_L \frac{1}{\sqrt{2\pi\sigma^2}} \int\limits_{t}^\infty \exp{\Bigg[-\frac{1}{2}\bigg( \frac{s - \sigma^2/2}{\sigma}\bigg)^2 \Bigg]}\,ds \nonumber \\
~ & = & M_L\, \frac{1}{2}{\rm erfc}(t_m)~,
\end{eqnarray}
where $t_m=(t-\sigma^2/2)/(\sigma \sqrt{2})$ and a mean particle mass $\mu=1.37 m_{\rm u}$ is adopted which accounts for Galactic abundances of atomic and molecular hydrogen and heavier elements \citep{Draine_11}.
The size of the physical scale is calculated straightforwardly from its volume $V_t$:
\begin{equation}
\label{eq_size_t}
L_t = V_t^{1/3}= L\,\bigg[\int\limits_{t}^\infty p_v(s)\,ds \bigg]^{1/3} = L\,\bigg[ \frac{1}{2}{\rm erfc}(t_v)\bigg]^{1/3}~,
\end{equation}
where $t_v=(t+\sigma^2/2)/(\sigma \sqrt{2})$.
\subsection{The equipartition equation}
By use of expressions (\ref{eq_mean_log_density_t})-(\ref{eq_size_t}) the gravitational, kinetic and thermal energies per unit volume read:
\begin{equation}
\label{eq_energy_expressions}
|W|=\frac{3z_c}{5}G\frac{M_t}{L_t}\mu \overline{n}_t~,~~E_{\rm kin}=\frac{1}{2}\mu \overline{n}_t u_t^2~,~~E_{\rm th}=\frac{3}{2}\overline{n}_t \Re T,
\end{equation}
where $\overline{n}_t=\langle n \rangle_L\,\exp(\overline{s}_t)$ and $u_t=u_L(L_t/L)^\beta$ are the mean density and velocity dispersion of cloudlets, respectively, and $\Re$ is the universal gas constant. The coefficient $1\le z_c\le 2$ accounts for the contribution of the mass outside the cloudlets to their total gravitational energy. In this work, we adopt $z_c=1.5$ like in \citet{DVK_11}.
Now the equipartition equation (\ref{eq_equipartition}) can be written in terms of the scaling indices $\alpha$ and $\beta$:
\begin{eqnarray}
\label{eq_equipartition_equation}
\noindent\frac{3z_c}{5} G \mu n_0 \bigg(\frac{L}{3~{\rm pc}}\bigg)^\alpha \bigg(\frac{L}{1~{\rm pc}}\bigg)^2 \bigg[0.5^{2/3}\frac{{\rm erfc}(t_m)}{{\rm erfc}(t_v)}\bigg] = \nonumber \\
~~~~~~ = \frac{f}{2}\bigg[u_0^2 \bigg(\frac{L}{1~{\rm pc}}\bigg)^{2\beta} \big[ 0.5\, {\rm erfc}(t_v)\big]^{2\beta/3} + 3 \Re T \bigg]~,
\end{eqnarray}
which becomes an equation for $\beta$ through equation (\ref{eq_alpha_beta_relation}). The other free parameters of the model are $f$, $u_0$, $n_0$ and the turbulence forcing parameter $b$ which is implicitly present in the error functions. Varying the threshold level $t$, one can find (if existing) a solution for fixed values of the scaling indices of velocity and mean density. The dynamic range of $b$ turns out to be constrained in the predominantly solenoidal regime -- no solutions were obtained for compressive forcing ($b\ge0.42$).
\section{Model predictions}
\label{Model predictions}
\subsection{Mass-size relationship}
\label{mass-size_relationship}
Mass-size diagrams are often used as a tool to study general structure of MCs and star-forming regions \citep{Lada_ea_08, LAL_10, Kauffmann_ea_10b, Beaumont_ea_12, Shetty_ea_12}. Basically a power-law mass-size relationship $M\propto L^{\gamma}$ has been found, where the index $\gamma$ is constant or changes slowly with the effective size $L$. However, the definition of $L$ introduced by various authors is different. The work of \citet{Kauffmann_ea_10b} makes use of the {\sc Dendrogram} clump-finding algorithm \citep{Rosolowsky_ea_08} and applies it to a set of MC maps, obtained from dust-continuum and dust-emission observations. These authors analyse the mass-size relationship of the extracted objects which build up a hierarchy of embedded connected regions with increasing mean density. \citet{LAL_10} study dust-extinction maps of nearby star-forming regions and delineate structures of given effective size varying stepwise the level of constant absorption. These objects are similar to the physical scales in
our model (Sect. \ref{Physical scale}) -- in both approaches a fixed density threshold defines a set of cloudlets to which a single effective size is ascribed. However, in contrast to the work of \citet{LAL_10}, the threshold value $t$ here is not arbitrary but is determined by the required equipartition of energies (equation \ref{eq_equipartition}).
\begin{figure}
\hspace{3.5em}
\begin{center}
\includegraphics[width=84mm]{Fig/fig_MR_var_beta.eps}
\vspace{0.2cm}
\caption{Mass-size diagrams of models with $f=2$, $n_0=1.3\times10^3$~\cc, $u_0=1.4$~km/s. The slopes at small scales ($\gamma\sim0.1$~pc), for $\beta=0.33$ and $\beta=0.50$, are indicated with arrows. The mass-size relationships in Polaris (derived from {\it Planck} data) and in Rosette ({\it Herschel} data), are plotted for comparison (see text).}
\label{fig_MR_var_beta}
\end{center}
\end{figure}
The mass-size relationship in our model is defined as a power-law relation $M_t\propto L_t^{\gamma}$ between the physical scale and the mass $M_t$ enclosed therein. From small to large scales within the inertial range $0.1~{\rm pc}\lesssim L_t \lesssim 20$~pc and for any combination of free model parameters, the scaling index $\gamma(0.1)\ge \gamma \ge \gamma(20)$ decreases monotonically -- see the plots $\gamma(L_t)$ in Appendix \ref{Appendix_gamma_variations}. Yet the variation of $\gamma$ in each considered model case does not exceed 0.3 dex (Table \ref{table_var_gamma}).
Fig. \ref{fig_MR_var_beta} illustrates the effect of varying the velocity scaling index and the turbulent forcing parameter. The latter evidently does not affect the predicted mass-size relationship for fixed $\beta$ (cf. top and bottom panel). On the other hand, gradual change of $\beta$ from $0.33$ (incompressible turbulence) to $0.50$ leads to corresponding general steepening of the slopes $\gamma$. When mass-size relationships are derived through imposing extinction or column density thresholds, the plausible slopes at small scales are $\le 2$, in view of the properties of the extinction/column density pdf \citep{BP_ea_12}. This is illustrated also by observational mass-size relationships for a region with (Rosette) and without star-forming activity (Polaris) in Fig. \ref{fig_MR_var_beta}. Such slopes can be reproduced by models with $\beta=0.33$ (or a bit larger; Table \ref{table_var_gamma}, Column 5 \& 6) which we take into further consideration.
\begin{table}
\caption{Variation of the slope $\gamma$ of the mass-size relationship from small ($L_t\sim0.1$~pc) to large scales ($L_t\sim20$~pc).}
\label{table_var_gamma}
\begin{center}
\begin{tabular}{cc@{~~~}c@{~~~}ccccc}
\hline
\hline
~ & ~ & ~ & ~ & \multicolumn{2}{c}{$\beta=0.33$} & \multicolumn{2}{c}{$\beta=0.50$} \\
$f$ & $n_0$ & $u_0$ & $b$ & $\gamma(0.1)$ & $\gamma(20)$ & $\gamma(0.1)$ & $\gamma(20)$\\
\hline
1 & 1.3 & 1.4 & 0.33 & 1.72 & 1.66 & 2.08 & 2.01\\
2 & 1.3 & 1.0 & 0.33 & 1.78 & 1.67 & 2.18 & 2.01\\
2 & 1.3 & 1.4 & 0.33 & 1.80 & 1.68 & 2.21 & 2.00\\
3 & 1.3 & 1.4 & 0.33 & 1.88 & 1.69 & 2.29 & 2.03\\
3 & 4.4 & 1.4 & 0.33 & 1.72 & 1.66 & 2.07 & 2.00\\
4 & 1.3 & 1.0 & 0.33 & 1.93 & 1.68 & 2.34 & 2.04\\
4 & 1.3 & 1.4 & 0.33 & 1.92 & 1.71 & 2.31 & 2.04\\
4 & 4.4 & 1.4 & 0.33 & 1.80 & 1.67 & 2.11 & ~2.01 \vspace{6pt}\\
1 & 1.3 & 1.4 & 0.40 & 1.70 & 1.66 & 2.04 & 2.00\\
2 & 1.3 & 1.0 & 0.40 & 1.73 & 1.66 & 2.11 & 2.01\\
2 & 1.3 & 1.4 & 0.40 & 1.76 & 1.67 & 2.15 & 2.01\\
3 & 1.3 & 1.4 & 0.40 & 1.81 & 1.68 & 2.20 & 2.03\\
3 & 4.4 & 1.4 & 0.40 & 1.69 & 1.66 & 2.02 & 2.00\\
4 & 1.3 & 1.0 & 0.40 & 1.86 & 1.69 & 2.27 & 2.02\\
4 & 1.3 & 1.4 & 0.40 & 1.85 & 1.69 & 2.29 & 2.02\\
4 & 4.4 & 1.4 & 0.40 & 1.74 & 1.66 & 2.09 & 2.01\\
\hline
\hline
\end{tabular}
\end{center}
\smallskip
\end{table}
\subsection{Comparison with recent observations}
\label{Observational test}
A more detailed observational test of the proposed model is made by use of publicly available {\it Planck} dust-opacity maps\footnote{Based on observations obtained with Planck (http://www.esa.int/Planck), an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada.} on the Galactic regions Polaris, Perseus, Pipe and Orion A and of {\it Herschel} data on Rosette \citep{Schneider_ea_12}. Those regions were selected to represent a wide variety of star-forming conditions: a diffuse medium with no signs of star formation (Polaris), a molecular cloud with a few identified young stellar objects (Pipe Nebula), a site of ongoing low- and intermediate-mass star formation (Perseus) and evolved giant MC complexes with star formation (Rosette, Orion A). The mass-size relationships (Fig. \ref{fig_MR_var_rho0_u0_panel}) were obtained from the column-density pdfs ($N$-pdfs) by imposing stepwise thresholds of decreasing column density, like in \citet{LAL_10}. The
uncertainties of the mass estimates reflect the uncertainties of distance or of distance gradient within the given region \citep[in the case of Perseus, see][]{Schlafly_ea_14}. More information on the selected regions is given in Appendix \ref{Appendix_SFRs} to which we refer the reader.
The ability of our model to describe the general structure of a given star-forming region is quantified through the mass scaling index at small scales $\gamma(0.1)$ and the upper scale $L_{\rm dev}$ of deviation of the model when the observational mass-size relationship is fitted at small scales. The values of $\gamma(0.1)$ (Table \ref{table_var_gamma}, Column 5) are basically consistent with the data in all studied regions. Thus a good agreement could be achieved through variation of the free model parameters (Fig. \ref{fig_MR_var_rho0_u0_panel}) and provided that $L_{\rm dev}$ is at least several pc, i.e. within the size range of typical MCs.
\begin{figure}
\hspace{3.5em}
\begin{center}
\includegraphics[width=83mm]{Fig/fig_MR_var_rho0_u0.eps}
\vspace{0.7cm}
\caption{Mass-size ($M_t$ -- $L_t$) relationships from models with $\beta=0.33$ compared with those of several Galactic star-forming regions, derived from {\it Planck} observations. Typical uncertainties of the mass estimates due to uncertainties or gradients of distance to/within given region are shown.}
\label{fig_MR_var_rho0_u0_panel}
\end{center}
\end{figure}
As mentioned in the previous section, the turbulence forcing parameter does not affect the mass-size relationships and therefore only models with purely solenoidal forcing ($b=0.33$) are shown. Variations of the velocity scaling coefficient $u_0$ toward the higher value $1.4$~km/s (cf. Table \ref{table_var_scaling_coef}) produce small increase of the model masses at a fixed $L_t$ (Fig. \ref{fig_MR_var_rho0_u0_panel}, cf. top and middle panels). Variations of the density scaling coefficient $n_0$ lead to a shift of the mass-size relationships by a factor of 2 or 3 toward higher masses (Fig. \ref{fig_MR_var_rho0_u0_panel}, cf. middle and bottom panels).
Variations of the equipartition coefficient $f$ evidently allow for modelling of three particular regions at scales $L\le L_{\rm dev}$ where $L_{\rm dev}$ is at least several pc. Models of weakly self-gravitating clouds ($f=1$) and velocity scaling coefficient $u_0=1$~km/s describe very well the structure of internal regions of Polaris Flare (Fig. \ref{fig_MR_var_rho0_u0_panel}, top). That should be expected in view of the sparsity of dense, possibly gravitating cloud cores in this region \citep{Andre_ea_10}. The Pipe Nebula could be described either with strongly gravitationally bound models of lower density scaling coefficient ($f=4$, $n_0=1.3\times10^3$~\cc) or with weakly self-gravitating, but denser models ($f=1$, $n_0=4.4\times10^3$~\cc; Fig. \ref{fig_MR_var_rho0_u0_panel}, middle and bottom panels). This follows from the equipartition equation (\ref{eq_equipartition_equation}) where the increase of $f$ is physically equivalent to the increase of the mean density through $n_0$. The structure of the
internal parts of the Rosette region could be approximated only by models where the medium is strongly gravitational bound and dense ($f=4,~n_0=4.4\times10^3$~\cc). The latter clearly cannot reproduce the structure of a huge star-forming complex like Orion A, although its index $\gamma(0.1)$ is in the range specified in Table \ref{table_var_gamma}. A possible explanation could be that the whole physical picture in this region is essentially different than the basic assumptions of the model. Orion A is an evolved complex with active but not recent star formation which has been propagating through it within the last dozen of Myr. Numerous dense prestellar cores are detected wherein most of the mass at small scales is concentrated \citep[see][for review, and Sect. 4.2]{Bally_08}. Nevertheless, small scales in this region, with sizes up to several pc, could be speculatively fitted through extreme increasing of the scaling coefficients $n_0$ and $u_0$ beyond the limits which yield scaling relations
consistent with observational data (see Fig. \ref{fig_scaling_relations}).
In obvious contrast with the structure of other selected regions, Perseus is characterized by a substantial change of the slope of the mass-size relationship from small to large scales. We return to this issue in Sect. \ref{gamma_variations}.
\section{Discussion on the model applicability}
\label{Discussion}
\subsection{The velocity-dispersion scaling index}
A traditional interpretation of the first Larson's relation (equation \ref{eq_Larson_1}) with scaling index $\beta\sim 0.33$ is that the interstellar medium is dominated by subsonic flows while values $\beta\sim0.50$ are held as indicative for highly compressible supersonic turbulence. However, such claims are justified largely by results from numerical simulations of isothermal {\it non self-gravitating} media \citep{Kritsuk_ea_07, Federrath_ea_10}. Also, the two Larson's relations should be considered interdependent even in a purely turbulent case -- through the scaling of the density-weighted velocity dispersion $\rho^{1/3}v$ which is sensitive to the driving mode \citep[solenoidal or compressive; see][for discussion]{Federrath_13}. In the model, proposed here, this interdependence might be additionally modified by gravity, through the assumed equipartitions at each scale in the inertial range. For instance, \citet{Stanchev_ea_15} found $\rho^{1/3}v \propto L^{-2/3}$ in Perseus region, under the
assumption of equipartition between gravitational and turbulent energy. This is nearly consistent with $\rho^{1/3}v \propto L^{-(2+\beta)/3}$ in this work, given that $\beta=0.33$.
In fact, some numerical works on magnetized clouds yield even shallower velocity power spectrum than predicted in the Kolmogorov theory. For instance, \citet{Collins_ea_12} measured $\beta= 0.23-0.29$ and \citet{Kritsuk_ea_09} obtained $\beta=0.25-0.31$, in consistence with \citet{Lemaster_Stone_09}.
In view of the abovementioned, models with velocity-scaling index $\beta=0.33$ could be considered appropriate to describe general structure of compressible turbulent molecular clouds with an essential role of self-gravity in the energy budget.
\subsection{Variety of cloud conditions}
The obtained values of $L_{\rm dev}$ from fitting of the mass-size relationship in the sampled regions (Table \ref{table_model_agreement}) are within the size range of typical MCs \citep{BT_07}. This suggests that the model is appropriate for description of the dense molecular phase in star-forming complexes. As expected in view of the physical framework of the model (Sect. \ref{Framework}), best fits are obtained for regions with sparse or with no star formation at all: Pipe and Polaris. Their $N$-pdfs have nearly lognormal shapes (Fig. \ref{fig_SF_regions_Npdfs}) with tiny powel-law (PL) tails of very steep slopes, which is typical for inactive complexes \citep{Kainulainen_ea_09, Schneider_ea_15b}. Large scales in Polaris with a mass-size relationship that cannot be fitted ($L>L_{\rm dev}$) correspond to column-density range $N\lesssim 1\times10^{21}$~\sqc~wherein the assumptions for purely molecular phase and, probably, isothermality might be not true \citep{VS_10, Hennebelle_ea_08}. Note, however, that
the
lognormal pdfs in our model are defined at abstract scales within the cloud and cannot easily be compared with the {\it single} pdf of the entire cloud which should be considered rather as a superposition of many scale pdfs.
Longer PL tails of the $N$-pdfs with shallower slopes ($\lesssim4$) are indication for gravitational contraction and other processes controlled by gravity which eventually lead to local events of star formation \citep{BP_ea_11b, Schneider_ea_15a}. In our sample, such regions are Rosette, Orion A and Perseus. The PL tail in Rosette is characterized by a slope of $\sim4$ and yields a mass-size relationship that can be modelled up to $L_{\rm dev}\sim4$~pc (Fig. \ref{fig_SF_regions_Npdfs} and Table \ref{table_model_agreement}). In contrary, the model fails to fit the general structure of Perseus and Orion A (Fig. \ref{fig_MR_var_rho0_u0_panel}). Their $N$-pdfs exhibit pronounced PL tails with slopes $2.1$ \citep{Stanchev_ea_15} and $2.7$, correspondingly. Numerical simulations show that such long tails with slopes $\lesssim 3$ characterize strongly self-gravitating media \citep{KNW_11, FK_13, Girichidis_ea_14}. Therefore we revisit the issue of the evolutionary status of the clouds whose structure the model aims
to represent.
\begin{table}
\caption{Upper scale $L_{\rm dev}$ of deviation of the model from observational mass-size relationships and its corresponding column density $N(L_{\rm dev})$. The lower limit $N_{\rm obs,\,PL}$ of the power-law tail of the observational $N$-pdf is given in Column 4. The slope of this tail is specified in Column 5.}
\label{table_model_agreement}
\begin{center}
\begin{tabular}{lcccc}
\hline
\hline
Region & $L_{\rm dev}$ & $N(L_{\rm dev})$ & $N_{\rm obs,\,PL}$ & $|$Slope$|$\\
~ & [ pc ] & [ 10$^{21}$~\sqc ] & [ 10$^{21}$~\sqc ] & \\
\hline
Polaris & $\sim15$ & $~1.2$ & ~3.7 & $>6$ \\
Pipe & $\sim~5$ & $~4.0$ & 13.5 & $>7$ \\
Rosette & $\sim~4$ & $14.0$ & 16.0 & $\sim 4$\\
Orion A & -- & -- & 33.0 & 2.7 \\
\hline
\hline
\end{tabular}
\end{center}
\smallskip
\end{table}
\begin{figure}
\hspace{3.5em}
\begin{center}
\includegraphics[width=83mm]{Fig/fig_model_consistency_pdfs.eps}
\vspace{0.7cm}
\caption{$N$-pdfs of the selected regions, extracted from {\it Planck} and {\it Herschel} (Rosette) data. Those parts that yield mass-size relationships our model is consistent with are shown with bullets.}
\label{fig_SF_regions_Npdfs}
\end{center}
\end{figure}
\subsection{Cloud evolutionary stage}
We chose numerical $N$-pdfs from two grid simulations that represent MC evolution at two different evolutionary stages. The simulation S10, analysed in the work of \citet{Shetty_ea_10}, provides a snapshot from the early cloud evolution: well-developed and driven turbulence with increasing contribution of gravity to the energy budget. The simulation S15 from \citet{Stanchev_ea_15} has been aimed to depict the late MC evolution, about and after the formation of first stars. Basic information about the used simulations is given in Table \ref{table_num_simulations}. Their set-ups are comparable in terms of resolution and magnetic field and essentially differ in regard to treatment of turbulence and geometry of clump/cloud formation. For instance, dense regions of gas in S10 form under combined influence of driven random Gaussian velocity field and gravity which is artificially switched on after several dynamical times. On the other hand, cloud formation in the simulation from S15 takes place through collision
of one-dimensional flows whereas turbulence at the considered late evolutionary stage is due to fluid motions. However, both different treatments of turbulence are realistic from the perspective of evolutionary time.
\begin{table}
\caption{Summary of the numerical simulations used to test the model. Notation of the reference: S10 - \citet{Shetty_ea_10}; S15 - \citet{Stanchev_ea_15}.}
\label{table_num_simulations}
\begin{center}
\begin{tabular}{lcc}
\hline
\hline
~ & S10 & S15 \\
\hline
Evolutionary time & $0.5$ free-fall times & $\gtrsim15$~Myr \\
Selected area & $10 \times 10$~pc & $40 \times 50$~pc \\
Turbulence & driven & decaying \\
Initial Mach number$^\ast$ & 9.0 & 0.4\\
Initial density & 200~\cc & 1~\cc\\
Magnetic field & 0.6 $\mu$G & 3 $\mu$G \\
Maximum resolution & $\lesssim 0.01$~pc & $0.03$~pc\\
Simulation code & {\sc ENZO} & {\sc FLASH} \\
\hline
\hline
\end{tabular}
\end{center}
\smallskip
$\ast$ The initial medium in S10 is isothermal with $T=10$~K while it is warm ($T=5000$~K) in S15.
\end{table}
In Fig. \ref{fig_MR_N-pdfs_simulations} we illustrate an analysis which is analogous to the one performed in the previous Section. As evident from the top panel, the mass-size relationship from S10 is located within the zone, covered by the set of models with $\beta=0.33$, $b=0.33$ and varying $f$, $n_0$ and $u_0$. Best fit is provided by a model with `virial-like' equipartition ($f=2$) as $L_{\rm dev}\sim 3$~pc is about the upper limit of the inertial range in the simulation. The discrepancy at small scales ($\lesssim 0.2$~pc) is probably caused by the end of the inertial range and/or resolution effects -- see the corresponding $N$-pdf tail in the bottom panel. The column-density range wherein the model is consistent with S10 falls entirely in the PL tail with {\it average} slope of about $2$ although the shape is close to part of a lognormal (cf. the $N$-pdf tails in Fig. \ref{fig_SF_regions_Npdfs}).
On the other hand, the model evidently cannot predict the mass-size relationship from S15 at time $\sim20$~Myr, i.e. after emergence of first stars, even when strong gravitational boundedness ($f=4$) is assumed. We attribute this to the physical conditions in the dense clumps which populate the small scales ($L<1$~pc) in the considered simulation box. Their mean volume densities are at least few times $10^4$~\cc ~(most often, $\sim10^5$~\cc) which hints at their prestellar nature. Typical linewidths of such objects are trans/subsonic and the analysis of their density profiles possibly suggests a lack of equilibrium \citep[for a discussion, see][]{BT_07}. Therefore their physics is inconsistent with the adopted assumptions for supersonic turbulence and energy equipartition. The applicability of our model is thus constrained to the early evolutionary stage of MCs -- with an upper age limit $t\lesssim15$~Myr, about the formation of first stars.
\begin{figure}
\hspace{3.5em}
\begin{center}
\includegraphics[width=83mm]{Fig/fig_MR_N-pdfs_simulations.eps}
\vspace{0.7cm}
\caption{Test of the model from simulations at early (S10) and late (S15) evolutionary stage (see Table \ref{table_num_simulations}). {\it Top:} Mass-size relationships, compared with the predictions of models with $\beta=0.33$, $b=0.33$, $1\le f \le 4$ (shaded area); {\it Bottom:} Numerical $N$-pdfs and the part where the model is consistent with the derived mass-size relationship (bullets). The regime of dense prestellar cores with $n\sim10^4 - 10^5$~\cc~from S15 (squares) is shown in both panels.}
\label{fig_MR_N-pdfs_simulations}
\end{center}
\end{figure}
\subsection{Variations of the intracloud mass scaling index}
\label{gamma_variations}
The variation of $\gamma$ in a given model with fixed $f$ is restricted (Table \ref{table_var_gamma}). Thus the discrepancy at $L>L_{\rm dev}$ is caused by a significant drop of the mass-scaling index at larger scales in real star-forming regions. The latter phenomenon could be explained with the characteristics of the cloud's $N$-pdf: lognormal shape with/without PL tail, width of the lognormal part, slope of the PL tail and typical density of transition between both regimes. From Fig. 11 in \citet{BP_ea_12} one could see that significant variations of $\gamma$ within a region are produced by $N$-pdfs which are combinations of a broad (lognormal) component and a shallow PL tail -- like in Rosette, Perseus and Orion A (Fig. \ref{fig_SF_regions_Npdfs}). The parameters of these two components reflect the balance between turbulence and gravity at different scales in the cloud.
It is physically consistent to expect that this balance is described at different scales by a different type of equipartition (if such is present at all). For example, whereas the densest cores evolve faster and local collapses take place, the global contraction of the cloud starts at a time when first stars have been already formed \citep{VS_ea_07, BP_ea_11b}. In view of this, small scales are to be described by an equipartition with stronger contribution of gravity ($f\ge2$) while the large ones, comparable to the size of entire cloud, should be characterized by $1\le f \le 2$ or less. A combination of models with different choice of $f$ for scales over $L_{\rm dev}$ could reproduce, in principle, the total observational mass-size relationships in some regions. That is evident from an eye inspection of Fig. \ref{fig_MR_var_rho0_u0_panel}. Fixing the other free model parameters, a decrease of $f$ produces less mass at a given scale. In case an observational mass-size relationship is well fitted through a
model with $f=f^\prime$ at scales $L\le L_{\rm dev}$, it can be successfully reproduced also at $L> L_{\rm dev}$ by a series of models with decreasing $f<f^\prime$.
\section{Summary}
\label{Summary}
We present a model of the general structure of molecular clouds (MCs) at their early evolutionary stage ($5 \lesssim t \lesssim 15$~Myr), characterized by developed supersonic isothermal turbulence and essential contribution of gravity to the energy balance at different spatial scales $L$. Here we consider the range $0.1~{\rm pc}\le L \le 20$~pc, adopting a turbulent injection scale above the typical size of giant MCs. Our model is very sensitive to the evolutionary stage of the cloud, as well as to the properties of its internal turbulence. In particular, it depends on the assumed power-law scaling relations of the velocity dispersion and the mean density, on the equipartition between gravitational and kinetic energy, including thermal support, i.e. $|W| \sim f(E_{\rm kin} + E_{\rm th})$ ($1\le f \le 4$), and on the validity of a lognormal probability density function (pdf) at each turbulent scale $L$. A physical scale $L_t\le L$ is defined as the effective size of the sum of all regions above a log-density
threshold level $t$ at which the equipartition equation is satisfied. Free parameters of the presented model are the velocity scaling index $\beta$, the coefficients in the scaling relations of velocity ($u_0$) and density ($n_0$), the coefficient of equipartition $f$ and the turbulence forcing parameter $b$. The predictive power of the model is put to test by comparison of the mass-size relationships $L_t - M_t$ with ones, derived from observational column-density pdfs in several Galactic regions of varying star-forming activity as well from two simulations of evolved MCs.
The results of this study are as follows:
\begin{itemize}
\item The model predictions of mass-size relationships are not sensitive to the value of $b$, given that turbulence forcing is predominantly compressive, while the variations of the parameters $u_0$ and $n_0$ lead to variations of model masses at a fixed scale $L_t$ within a factor of 3. A velocity scaling index $\beta$ which is significantly larger than the value in Kolmogorov theory ($0.33$) produces mass-size relationships that cannot fit the observational ones. However, it should be not considered as an indication for subsonic turbulence but is rather determined by interplay between gravity and highly compressible turbulence.
\item Variations of the equipartition coefficient $1\le f\le 4$ essentially shift the mass range and allow for modelling of some star-forming regions at scales within the size range of typical MCs ($\gtrsim4$~pc). Observed mass-size relationships at larger scales could be reproduced as well by a series of models with decreasing $f$ which is justified in view of the physical state of evolving MCs as revealed from numerical simulations.
\item The model is able to describe the general structure of regions with low or no star-forming activity, characterized by nearly lognormal $N$-pdf (Polaris, Pipe) as well the structure of some star-forming regions, given that their $N$-pdf is forming a short and steep power-law tail (Rosette).
\item Comparisons with two numerical simulations of cloud evolution at different stages show that the model is able to describe the general properties of a medium with driven turbulence and strong self-gravity in the energy balance but prior to eventual star formation -- which is consistent with the basic assumptions.
\end{itemize}
\vspace{12pt}
{\it Acknowledgement:} T.V. acknowledges support by the {\em Deutsche Forschungsgemeinschaft} (DFG) under grant KL 1358/20-1. We thank R. Shetty and B. K\"{o}rtgen for providing data from their simulations \citep[][respectively]{Shetty_ea_12, Stanchev_ea_15} and O. Stanchev for his support on software and technical issues. \\
|
1,116,691,500,803 | arxiv |
\section{Conclusion}
\label{sec:conclusion}
We have presented a framework to combine model learning and controller tuning with application to high-performance autonomous racing.
In particular, we proposed to encode the environmental condition via a residual dynamics model such that knowledge between different contexts can be shared to reduce the effort for controller tuning.
The benefits of the proposed approach have been demonstrated on a custom hardware platform with an extensive experimental evaluation.
Key for the method's success was the low-dimensional representation of the environment by means of the custom parametric model.
For more complex learning models such as neural networks, one could for example obtain a low-dimensional context via (variational) auto-encoders, which we leave for future research.
\section*{Acknowledgement}
The authors would like to thank all students that contributed to the racing platform.
Andrea Carron's research was supported by the Swiss National Centre of Competence in Research NCCR Digital Fabrication and the ETH Career Seed Grant 19-18-2.
\footnotesize
\bibliographystyle{IEEEtran}
\section{Experiments}
\label{sec:experiments}
\input{figures/control_learn_architecture/control_learn_architecture.tex}
We begin the experimental section by giving a short overview about the hardware platform and how the different components of the control architecture interact with each other.
Subsequently, we evaluate the non-linear feature functions from \cref{sec:model_learning} and demonstrate that both model learning as well as controller tuning are required for high-performance racing.
Further, we confirm the main hypothesis of this paper, i.e., different environmental conditions and cars require different controller parameters for optimal performance.
Last, we show that the regression coefficients of the residual model are capable of capturing different contexts and demonstrate that utilizing the contextual information for controller tuning leads to a significant speed up of finding the optimal controller parameters.
\subsection{The Racing Platform}
\label{sec:racing_platform}
A schematic overview of the components and their respective communication channels is depicted in \cref{fig:learning_architecture}.
At the heart of the control loop is the \gls{mpcc} that computes the new input $\action_\ti$ for a given state $\state_\ti$ at 35 Hz; the respective input is sent via remote control to the race cars.
The cars are equipped with reflective markers enabling high-accuracy position and orientation measurements $\observation_{\ti}$ via a motion capture system, from which the full state $\state_{\ti}$ is estimated via an \gls{ekf}.
Interacting with the \gls{mpcc} are the components for model learning and contextual \gls{bo} that provide the context information $\context$ and the controller parameters $\bovar$, respectively.
\subsection{Dynamics Learning}\label{sec:experiments_dynamics_learning}
\input{figures/model_prediction_error/model_prediction_error}
\input{figures/position_density/position_density.tex}
Although the linear model in \cref{eq:residual_model} that is used to learn the residual dynamics has only two parameters per dimension, it is able to approximate the residual dynamics to high accuracy.
\cref{fig:learning_error_signal} shows the predicted error (solid line) and the corresponding uncertainty estimates (shaded region) for the lateral velocity (top) and yaw rate (bottom), respectively.
The training data corresponds to two complete laps around the track (predicted values in green) and the testing data to one lap (predicted values in red).
Qualitatively, the model is capable of predicting the true error signal (gray) without overfitting.
For a more quantitative analysis, we compare our model with a \gls{gp} in terms of the RMSE in \cref{tab:prediction_error}.
Uncertainty estimates for the prediction error correspond to the standard deviation from 50 models trained on independently sampled training (2 laps) and testing data (1 lap).
We note that learning with either model helps to drastically improve the predictive capabilities over the nominal dynamics model.
Notably, \gls{blr} with $\bm{\phi}_{\text{Taylor}}$ even outperforms the \gls{gp} for the yaw rate indicating that our model strikes a good balance between complexity and expressiveness.
\subsection{Joint Optimization: Key to High-Performance Racing}
\label{sec:motivation}
Next, we want to demonstrate the importance of both model learning as well as controller tuning for high-performance racing.
To this end, we drive 20 laps with three different settings and record the respective lap times:
Setting \#1 serves as the benchmark, for which the controller weights are not tuned and we solely use the nominal model $f(\state, \action)$ in the \gls{mpcc}.
From \cref{fig:position_density} we can see that the car reliably makes it around the track albeit resulting in a clearly sub-optimal trajectory.
Especially after turns with high velocity (bottom left and right), the car repeatedly bumped into the track's boundaries.
For setting \#2, we use the same controller weights, but account for errors in the dynamics by learning a residual model $g(\state, \action)$.
The car's trajectory is slightly improved and does not bounce against the boundaries anymore.
Interestingly, while model learning leads to a change in the car's trajectory, it does not directly translate to a reduction of the lap time as depicted on the right in \cref{fig:position_density}.
In setting \#3, we changed the controller weights such that the driving behaviour is more aggressive.
Consequently, the car's trajectory is greatly improved and the lap time is reduced by well over a second.
Note that the combination of aggressive controller weights without model learning repeatedly lead to the car crashing into the track's boundary from which it could not recover.
We conclude that both accurate dynamics as well as a well-tuned controller are required for high-performance racing, justifying the joint optimization approach.
\subsection{Optimal Controllers Depend on Context}
\input{figures/laptime_response_surfaces/laptime_response_surfaces}
The main hypothesis of this paper is that the \gls{mpcc}'s parameters require adaptation under different environmental conditions.
In order to confirm this hypothesis, the goal of the next experiment is to approximately find the response surface of the following objective for different contexts
\begin{align}
\label{eq:regularized_objective}
J(\bovar) = T_{\mathrm{lap}} + \lambda \cdot \bar{\Delta}_{\mathrm{centerline}}
\end{align}
with $T_{\mathrm{lap}}$ denoting the lap time, $\bar{\Delta}_{\mathrm{centerline}}$ is the average distance of the car to the centerline in centimeters and \mbox{$\lambda$} is a parameter governing the trade-off between the two cost terms.
The regularizing term penalizes too aggressive behaviour of the controller resulting in the car crashing due to cutting the track's corners.
To emulate the different driving conditions, we use two different cars each with and without an additional mass of 40 grams (corresponding to about 30\% of the cars' total weight) attached to the chassis, leading to a total of four different contexts.
Note that the additional mass influences the inertia, traction and steering behaviour of the cars.
For each context, we record 200 laps, where after every second lap we change the controller's weights and evaluate the objective on the lap thereafter, thus accounting for transient effects due to the change of the weights.
For the first 70 iterations, the controller weights are sampled from a low-discrepancy sequence to sufficiently explore the full parameter space and the remaining 30 parameter vectors are chosen according to the \gls{ucb} acquisition function to exploit the objective around the vicinity of the estimated optimum.
We show the respective \gls{gp} mean estimates of the objective function as well as its minimum (red cross) in \cref{fig:laptime_response_surfaces}.
The response surfaces vary considerable across the contexts and so do the optimal controller weights.
However, there are also some common features, e.g., a low advancing parameter $Q_{\text{adv}}$ in \cref{eq:mpcc} generally leads to a cautious driving behaviour and therefore an increased lap time.
\input{figures/reg_coeff_distribution/reg_coeff_distribution}
The goal of this paper is to utilize the knowledge collected from previous experiments to accelerate the search for optimal controller parameters in new contexts.
As such, the residual model must be able to encode different environmental conditions via the regression coefficients $\C$ in \cref{eq:residual_model} as context and adapt accordingly if the conditions change.
To this end, we re-estimate the regression coefficients after every lap based on the telemetry data from the previous lap.
Due to the ever changing data, we expect the regression coefficients to fluctuate slightly, even for the same context.
We therefore investigate if a change in the car's true dynamics leads to distinguishably different context variables despite the stochasticity mentioned above.
Based on the data from the previous experiment (200 laps per context), we show the respective distributions of the four regression coefficients in \cref{fig:reg_coeff_distribution}.
While the respective distributions exhibit outliers, the different contexts lead to regression coefficients that are clearly distinguishable from each other and as such can serve as suitable descriptors for the respective changes in the dynamics.
We hypothesize that the outliers originate from situations where the cars shortly loose traction and start to drift, which is not well captured by either the nominal or residual linear model based on the custom features.
\subsection{Contextual Controller Tuning}
\input{figures/contextual_bayesopt/contextual_bayesopt}
In the final experiment, we show that utilizing the contextual information from previous runs accelerates the search for the optimal controller parameters.
We again consider the four different contexts from the previous experiments.
For each context, we first use standard \gls{bo} to minimize the regularized lap time \cref{eq:regularized_objective} by adapting the weights $Q_{\text{adv}}$ and $Q_{\text{cont}}$ in the \gls{mpcc} formulation.
The results for standard \gls{bo} (blue) are shown in \cref{fig:contextual_bayesopt}, where the solid lines represent the mean across three independent experiments and the shaded region corresponds to the standard deviation.
For each context, we see that \gls{bo} reliably finds the controller weights that lead to the objective's minimum.
However, note that each time the optimization is started from scratch such that the initial lap times are relatively high because the algorithm has to explore the full parameter space.
For contextual \gls{bo} (red), we now utilize the collected data from the other experiments to pre-train the objective's surrogate \gls{gp} model.
In particular, we accumulate the collected data in order, meaning that for context~2 the \gls{gp} is initialized with the 30 iterations from context~1.
For the last context, we utilize all 90 data points from the previous three contexts.
The results clearly show that by transferring the knowledge between the contexts, the optimization is accelerated drastically and even the first iterate already leads to a good behaviour.
\section{Introduction}
In recent years, learning-based methods have become increasingly popular to address challenges in autonomous racing due to advances in the field of machine learning and growing computational capabilities of modern hardware.
\Gls{mpc} is of particular interest for learning-based approaches as it can deal with state and input constraints and offers several ways to improve the closed-loop performance from data \cite{Hewing2020LearningMPC}.
A common method of using data to improve \gls{mpc} is by learning a predictive model of the open-loop plant from pre-recorded state transitions.
However, as environmental conditions can change, it is imperative to adapt the learned model during operation instead of keeping it fixed \cite{McKinnon2019learnFastForgetSlwo}.
In addition to the predictive model, the effective behaviour of \gls{mpc} is governed by several parameters.
While for simple control schemes such as PID controllers, there exist heuristics to tune the parameters \cite{Ziegler1942optimumSettings}, this is generally not the case for \gls{mpc}, such that practitioners oftentimes resort to hand-tuning, which is potentially inefficient and sub-optimal.
In contrast, an alternative approach to controller tuning is to treat it as black-box optimization problem and use zeroth order optimizers such as CMA-ES \cite{Hansen2001CMAES}, DIRECT \cite{Jones1993Direct} or \gls{bo} \cite{Frazier2018tutorialBO} to find the optimal parameters.
Due to its superior sample efficiency and ability to cope with noise-corrupted objective functions, \gls{bo} has become especially popular \cite{Chatzilygeroudis2019PolicySearchHandfulTrials} and therefore is also employed in our proposed approach.
In this paper, we argue that it is not sufficient to find the controller configuration leading to the best lap time only once and then keep it constant during operation -- akin to the need for adaptive model learning schemes.
Consider the following example for autonomous race cars:
Under good weather conditions, the tires exhibit sufficient traction such that the controller can maneuver the car aggressively.
If it starts raining, the traction reduces such that the driving behaviour needs to be adapted accordingly, rendering the previous controller sub-optimal.
There are several options to deal with the issue in the aforementioned example.
One choice for example is to assume that the optimal controller parameters change over time \cite{Bogunovic2016TimeVaryingBanditOptimization}.
This approach `forgets' old parameters over time and continue to explore the full parameter space over and over as time progresses.
However, simply forgetting old data is inefficient if previously observed scenarios re-occur.
Alternatively, one can try to find a `robust' controller parameterization that is suited for several different environmental conditions \cite{Tesch2011SnakeGaitBOChangingEnvironments,Froehlich2020NoisyInputES}.
However, the drawback of this approach is that the robust controller configuration performs well on average but is not necessarily optimal for any of the specific conditions.
In contrast, we aim at finding the optimal controller for each context to obtain the optimal lap time in all conditions.
\textbf{Contributions }
The contributions of this paper are fourfold:
First, we propose to jointly learn a dynamics model and optimize the controller parameters in order to fully leverage the available data and efficiently adapt to new conditions.
In particular, we encode the environmental conditions via the dynamics model as context variables, which in turn are leveraged via \emph{contextual} \gls{bo} for highly efficient controller tuning.
Second, we derive a novel parametric model that is tailored to account for model inaccuracies imposed by lateral tire forces, which are critical for high-performance autonomous racing.
Third, we demonstrate the effectiveness of our proposed framework with an extensive experimental evaluation, i.e., more than 2'000 driven laps, on a custom platform with 1:28 scale RC race cars.
To the best of our knowledge, this paper provides the first demonstration of contextual \gls{bo} for a robotic application on hardware.
Last, we open-source an implementation of (contextual) \gls{bo} for the \gls{ros} \cite{Quigley2009ros} framework at \linebreak\url{www.github.com/IntelligentControlSystems/bayesopt4ros} -- the first of its kind in the official \gls{ros} package distribution list.
\section{Proposed Learning Architecture}\label{sec:learning_architecture}
In the following, we describe the two components that form the core contribution of this paper: learning the dynamics model of the cars using custom feature functions in \cref{sec:model_learning} and the contextual controller tuning based on \gls{bo} in \cref{sec:contextual_controller_tuning}.
\subsection{Learning the Dynamics Model}\label{sec:model_learning}
The goal of learning the residual dynamics is to account for any effects that are not captured by the nominal model~$f(\state, \action)$ in order to improve the predictive performance of the full model~\cref{eq:discrete_model}.
Especially for controllers that rely on propagation of the model, such as \gls{mpc}, the predictive accuracy strongly impacts the control performance.
However, there is a trade-off to be struck between the learned model's accuracy and complexity as the full control loop has to run in real time.
To this end, we use a linear model with non-linear feature functions $\bm{\phi}(\state, \action) = [\phi_1(\state, \action), \dots, \phi_{n_\phi}(\state, \action)]^\top : \R^{n_x} \times \R^{n_u} \rightarrow \R^{n_\phi}$ to regress the residual prediction errors via
\begin{align*}
g(\state, \action) = \C \bm{\phi}(\state, \action): \R^{n_x} \times \R^{n_u} \rightarrow \R^{n_g},
\end{align*}
where the regression coefficients $\C = [\context_1, \dots, \context_{n_g}]^\top \in \R^{n_g \times n_\phi}$ encode the environmental condition as \emph{context}.
The two main benefits of the linear model are its computational efficiency as well as the explicit representation of the context.
While the expressiveness of a linear model is limited by its feature functions, as opposed to a non-parametric model such as a \gls{gp}, it does not require a complex management of data points to retain the real-time capability of the controller \cite{Kabzan2019learningMPC}.
To infer the regression coefficients $\C$, we opt for a probabilistic approach by means of \gls{blr} \cite[Ch.~3.3]{Bishop2006Book}.
Utilizing a Bayesian treatment instead of ordinary linear regression has two advantages.
On the one hand it leads to more robust estimates due to regularization via a prior belief over the coefficients and on the other hand it includes uncertainty quantification.
As such, it allows the controller to account for uncertainty in the model prediction akin to \cite{Hewing2019cautiousMPC} without considerable computational overhead.
For the considered application of autonomous racing, we only learn the residual dynamics of a subset of the states.
Notably, the change in position and orientation can readily be computed via integration of the respective time derivatives.
Further, we do not account for modeling errors of the longitudinal velocity $v_x$ as we found empirically that the nominal model was already sufficient for good predictive performance.
Consequently, the learned model accounts for the lateral velocity $v_y$ and the yaw rate $\omega$, such that
\begin{align}\label{eq:residual_model}
g(\state, \action) =
\begin{bmatrix}
g_{vy}(\state, \action) \\ g_{\omega}(\state, \action)
\end{bmatrix} =
\begin{bmatrix}
\context_{vy}^\top \\ \context_{\omega}^\top
\end{bmatrix}
\bm{\phi}(\state, \action).
\end{align}
There is a wide variety of general-purpose feature functions $\bm{\phi}$, such as polynomials, radial basis functions, or trigonometric functions.
In this paper, we construct feature functions that are specific to the dynamic bicycle model.
In particular, we assume that the discrepancy in the predictive model is due to misidentified physical parameters~$\parameter$ such as mass, inertia and the parameters of the Pacejka tire model, i.e., \mbox{$\parameter_{\mathrm{true}} = \parameter_{\mathrm{nom.}} + \Delta \parameter$}.
Computing the first order Taylor expansion of the model $f(\state, \action; \parameter)$ around the nominal parameters $\parameter_{\mathrm{nom.}}$ naturally gives rise to a set of feature functions $\bm{\phi}_{\mathrm{Taylor}} = \nabla_\parameter f$ such that $f(\state, \action; \parameter_\mathrm{true}) \approx f(\state, \action; \parameter_\mathrm{nom.}) + \bm{\phi}_{\mathrm{Taylor}}(\state, \action)^\top \Delta \parameter $.
Empirically, we found that the inertia $I_z$ and the parameters $D_{\mathrm{f}/\mathrm{r}}$ in the Pacejka tire model~\cref{eq:pacejka_model} have the largest influence on the model's prediction error.\footnote{We use the inverse inertia to retain linearity in the parameter.}
Consequently, we compute the Taylor expansion around those parameters such that the resulting feature functions used in \cref{eq:residual_model} are given by
\begin{align}
\label{eq:taylor_features}
\bm{\phi}_{\mathrm{Taylor}}(\state, \action) =
\begin{bmatrix}
\sin(C_{\mathrm{f}} \arctan(B_{\mathrm{f}} \alpha_{\mathrm{f}})) \cos(\delta) \\
\sin(C_{\mathrm{r}} \arctan(B_{\mathrm{r}} \alpha_{\mathrm{r}}))
\end{bmatrix}.
\end{align}
Note that by design the feature functions share close resemblance with the tire model \cref{eq:pacejka_model}.
Using the features \cref{eq:taylor_features} to characterize the residual model \cref{eq:residual_model} then leads to the context vector $\context = [\context_{vy}^\top, \context_{\omega}^\top]^\top \in \R^4$, which we obtain by concatenation of the respective regression coefficients.
\subsubsection*{Practical Considerations}
Given a data set with state and input trajectories $\data = \{ \state_\ti, \action_\ti \}_{\ti=0}^K$, the state prediction error is computed as $\error_\ti = \state_{\ti+1} - f(\state_\ti, \action_\ti)$.
Since we cannot measure the full state vector directly, we employ an \gls{ekf} to infer the state from position and orientation measurements.
However, the estimated state $\state_{\mathrm{EKF}}$ does not suffice as ground truth because the \gls{ekf} itself depends on the nominal model.
We therefore construct the prediction errors in a `semi-offline' manner.
In regular time intervals, such as after every lap, we take the raw position and orientation measurements $\observation$ and compute the respective time derivatives numerically.
Since this step is done in hindsight, we can use non-causal filtering techniques to get a reliable ground truth for the velocity states.
While this approach leads to accurate state estimates, the downside is that the data for model learning exhibits a time delay of a few seconds, which did not pose any problems in practice.
\subsection{Contextual Controller Tuning}\label{sec:contextual_controller_tuning}
\begin{algorithm}[t]
\begin{algorithmic}[1]
\State $\data^{\mathrm{BO}}_0 \gets$ Initialize BO (Sobol, random, etc.)
\State \textcolor{blue}{$\data^{\mathrm{tel}}_0 \gets$ Collect initial telemetry data}
\For{$n = 1 \leq N$}
\State \textcolor{blue}{$\context_n \gets$ Context from BLR using $\data^{\mathrm{tel}}_{n-1}$}
\State $\bovar_n \gets$ Optimize acquisition function $\alpha(\data^{\mathrm{BO}}_{0:n-1}, \textcolor{blue}{\context_{n}})$
\State \textcolor{blue}{Update residual dynamics model \cref{eq:residual_model} with $\context_n$}
\State Update \gls{mpcc} \cref{eq:mpcc} with parameters $\bovar_n$
\State Drive extra round to mitigate transient effects
\State $J_n, \textcolor{blue}{\data^{\mathrm{tel}}_{n-1}} \gets$ Observe cost \textcolor{blue}{and telemetry data}
\State $\data^{\mathrm{BO}}_{0:n} \gets \data^{\mathrm{BO}}_{0:n-1} \cup (\bovar_n, \textcolor{blue}{\context_n}, J_n)$ Extend BO data
\EndFor
\end{algorithmic}
\caption{Contextual \gls{bo} for controller tuning. Steps different from standard \gls{bo} are highlighted in blue.}
\label{alg:contextual_bo}
\end{algorithm}
In \cref{sec:controller_tuning}, we have discussed how standard \gls{bo} can be employed to address the controller tuning problem by posing it as a black-box optimization.
This section focuses on the extension of standard \gls{bo} to enable the sharing of knowledge between different environmental conditions, to which end we employ \emph{contextual} \gls{bo} \cite{Krause2011contextual}.
In order to account for the environmental conditions encoded in the context variable $\context \in \R^{\ncontext}$, we extend the parameter space of the objective function such that it additionally depends on the context, i.e., $J: \bodomain \times \R^{\ncontext} \rightarrow \R$.
Consequently, the \gls{gp} model approximating the objective $J$ accounts for the context variable as well.
We follow \cite{Krause2011contextual} and split the kernel for the joint parameter space into a product of two separate kernels such that
\begin{align*}
k([\bovar, \context], [\bovar', \context']) = k_{\bovar}(\bovar, \bovar') k_{\context}(\context, \context').
\end{align*}
In this way, we can leverage kernel-based methods by encoding prior knowledge in the choice of the respective kernels.
With this extended \gls{gp} model, we can now generate a joint model from all previous observations across different contexts $\data_n = \{ \bovar_i, J(\bovar_i; \context_i) + \epsilon_i \}_{i=1}^n$.
\cref{alg:contextual_bo} summarizes the required steps for contextual \gls{bo} with instructions that differ from the standard \gls{bo} setting highlighted in blue.
In Line~4, the context $\context_n$ is computed from the telemetry data collected during the previous lap as the regression coefficients of the residual model.
Since the context is governed by the environment itself we do not have influence over the respective values.
Accordingly, the acquisition function is only optimized with respect to $\bovar$ in Line~5, while $\context_n$ is kept constant.
That way, \gls{bo} continues to explore the parameter space when new contexts are observed due to the increased predictive uncertainty of the \gls{gp}.
In contrast, for contexts that are similar to the ones encountered in previous laps, \gls{bo} readily transfers the experience and thus accelerates optimization.
Continuously updating the residual dynamics model and utilizing it as context variable allows us to account for either slowly changing environmental conditions, e.g., an emptying fuel tank or tires wearing off, quickly varying conditions, e.g., different surface traction due to rain, or deliberate changes of the car's condition, such as new tires.
\section{Preliminaries}
\label{sec:preliminaries}
In this section, we describe the vehicle's dynamics model for which the tire forces are the prime source of modelling errors.
Subsequently, we review the \gls{mpc} scheme and discuss its relevant tuning parameters.
Lastly, we formalize the controller tuning problem as black-box optimization and how \gls{bo} can be employed to address it.
\subsection{Vehicle Model}
\label{sec:vehicle_model}
In this work, we model the race car's dynamics with the bicycle model~\cite{Rajamani2011Vehicle} (see \cref{fig:bicycle_model}).
The state of the model is given by $\state = \left[p_x, p_y, \psi, v_x, v_y, \omega \right]$, where $p_x$, $p_y$, $\psi$ denote the car's position and heading angle in the global coordinate frame, respectively.
The velocities and yaw rate in the vehicle's body frame are denoted by $v_x$, $v_y$, and $\omega$.
The car is controlled via the steering angle $\delta$ and the drive train command $\tau$, summarized as input $\action = [\delta, \tau]$.
The states evolve according to the discrete-time dynamics
\begin{align}\label{eq:discrete_model}
\state_{\ti+1} = f(\state_\ti, \action_\ti) + \mathbf{B} g(\state_\ti, \action_\ti),
\end{align}
where $f$ denotes the nominal model and $g$ the residual model accounting for un-modelled effects, which is learned from data.
The matrix $\mathbf{B}$ determines the state's subspace influenced by the residual dynamics.
The dynamic bicycle model is governed by the set of ordinary differential equations
\begin{align}\label{eq:continous_model}
\begin{bmatrix}
\dot{v}_x \\ \dot{v}_y \\ \dot{\omega}
\end{bmatrix} =
\begin{bmatrix}
\frac{1}{m} \left(F_x - F_{\mathrm{f}}\sin(\delta) + m v_y \omega \right) \\
\frac{1}{m} \left(F_{\mathrm{r}} + F_{\mathrm{f}}\cos(\delta) - m v_x \omega \right) \\
\frac{1}{I_z} \left(F_{\mathrm{f}} l_{\mathrm{f}}\cos(\delta) - F_{\mathrm{r}} l_{\mathrm{r}} \right)
\end{bmatrix}
\end{align}
with $m$ as the car's mass, $I_z$ being the moment of inertia, and $l_{\mathrm{f}/\mathrm{r}}$ define the distance between the center of gravity and the front and rear axles, respectively.
One of the most critical components in the bicycle model for the use in high-performance racing are the lateral forces acting on the tires.
To this end, we employ the simplified Pacejka model~\cite{Pacejka2002Tire}
\begin{align}\label{eq:pacejka_model}
F_{\mathrm{f}/\mathrm{r}} = D_{\mathrm{f}/\mathrm{r}} \sin(C_{\mathrm{f}/\mathrm{r}} \arctan(B_{\mathrm{f}/\mathrm{r}} \alpha_{\mathrm{f}/\mathrm{r}}))
\end{align}
for the front (f) and rear (r), respectively.
The parameters $B, C$ and $D$ are typically found via system identification and depend on the car itself as well as on the friction coefficient between the tires and road surface.
The so-called slip angles $\alpha_{\mathrm{f}/\mathrm{r}}$ are defined as
\begin{align*}
\alpha_\mathrm{f} = \arctan \left( \frac{v_y + l_{\mathrm{f}} \omega}{v_x} \right) - \delta, \hspace{0.5em} \alpha_\mathrm{r} = \arctan \left( \frac{v_y - l_{\mathrm{r}} \omega}{v_x} \right).
\end{align*}
For driving maneuvers at the edge of the car's handling limits, it is paramount to estimate the lateral tire forces as accurately as possible.
To this end, we derive a learnable parametric model tailored to account for inaccuracies of the tire model \cref{eq:pacejka_model} in \cref{sec:model_learning}.
\input{figures/dynamics_model}
\subsection{Model Predictive Contouring Control (MPCC)}\label{subsec:model_predictive_contouring_control}
In this section, we briefly review the \gls{mpcc} formulation for race cars introduced in \cite{Liniger2015RCCars}.
As opposed to tracking control, in which a pre-defined trajectory, e.g., the ideal racing line including the velocity profile, is tracked, the contouring control approach allows for simultaneous path planning and path tracking in real-time.
In particular, the goal in \gls{mpcc} is to maximize the progress along the track while satisfying the constraints imposed by the boundaries of the track as well as the car's dynamics.
The control problem is formalized by the following non-linear program
\begin{align}
\begin{split}
\label{eq:mpcc}
\action^* & = \argmin_{[\paction_0, \dots, \paction_{H}]} \sum_{i=0}^{H} \bm{\varepsilon}_i^\top \Q \bm{\varepsilon}_i - Q_{\mathrm{adv}} \gamma_{i} \\
\text{s.t. } & \pstate_0 = \state_\ti, \; \pstate_{i+1} = f(\pstate_i, \paction_i) + \B g(\pstate_i, \paction_i) \\
& \pstate_i \in \mathcal{X}, \; \paction_i \in \mathcal{U},\; \gamma_i > 0,
\end{split}
\end{align}
where $\bm{\varepsilon}$ denotes the lag and contouring error, respectively, and~$\gamma$ is the advancing parameter.
We solve an approximation to the above optimization problem using the state-of-the-art solver ForcesPRO~\cite{Domahidi2014ForcesPro,Zanelli2017ForcesNlp} in a receding horizon fashion such that the action $\action_\ti$ for a given state $\state_\ti$ at time step $\ti$ is given by the first element of the optimal action sequence~$\action^*$.
The parameters $\Q = \operatorname{diag}[Q_{\mathrm{lag}}, Q_{\mathrm{cont}}]$ and $Q_{\mathrm{adv}}$ weigh the relative contribution of the cost terms and determine the effective driving behaviour of the car.
\emph{Remark:}
Note that the objective in \cref{eq:mpcc} does not directly encode the closed-loop performance of interest, i.e., the lap time.
Notably, it is not known a-priori, what values of the cost parameters lead to the best performance and thus require careful tuning.
While a time-optimal formulation would not necessarily require a similar tuning effort, the optimization problem itself is significantly more complex and real-time feasibility becomes an issue.
\subsection{Controller Tuning via Bayesian Optimization}
\label{sec:controller_tuning}
To formalize the controller tuning problem, we assume that a controller is parameterized by a vector $\bovar \in \bodomain \subseteq \R^\nbovar$, which influences the controller's performance with respect to some metric \mbox{$J: \bodomain \rightarrow \R$}.
In the case of race cars, the metric of interest is typically the lap time.
As such, the controller tuning problem can be cast as
\begin{align}\label{eq:controller_tuning_problem}
\bovar^* = \argmin_{\bovar \in \bodomain} J(\bovar).
\end{align}
The optimization problem \cref{eq:controller_tuning_problem} has three caveats, which make it difficult to solve:
1) Generally, no analytical form of $J$ exists but we can only evaluate it pointwise.
2) Each observed function value is typically corrupted by noise and
3) each function evaluation corresponds to a full experiment that has to be run on hardware.
Therefore, we can only spend a limited amount of evaluations to find $\bovar^*$.
These challenges render standard numerical optimization methods impractical and we resort to \acrfull{bo}, a sample-efficient optimization algorithm designed to deal with stochastic black-box functions such as described above.
\gls{bo} works in an iterative manner and has two key ingredients: 1) a probabilistic surrogate model approximating the objective function and 2) a so-called acquisition function $\acqfunc(\bovar)$ to choose new evaluation points, trading off exploration and exploitation \cite{Frazier2018tutorialBO}.
A common choice for the surrogate model are \glspl{gp} \cite{Rasmussen2006Book}, which allow for closed-form inference of the posterior mean $\mu(\bovar)$ and variance $\sigma^2(\bovar)$ based on a set of noisy observed function values $\data_n = \{ \bovar_i, J(\bovar_i) + \epsilon_i\}_{i=1}^{n}$.
The next evaluation point $\bovar_{n+1}$ is then chosen by maximizing the acquisition function $\alpha(\bovar)$.
In this paper, we are employing the \gls{ucb} acquisition function given by $\alpha_{\mathrm{UCB}}(\bovar) = \mu(\bovar) + \beta \sigma(\bovar)$, with $\beta$ as so-called exploration parameter.
This process is iterated until the evaluation budget is exhausted or a sufficiently good solution has been found, see also \cref{alg:contextual_bo}.
\section{Related Work}
\label{sec:related_work}
\subsection{Combining Model Learning and Controller Tuning}
In recent years, \gls{bo} has found widespread applications in control, robotics and reinforcement learning, owing its success to superior sample efficiency \cite{Chatzilygeroudis2019PolicySearchHandfulTrials}.
In this section, we focus on approaches that combine model learning with \gls{bo}.
The algorithm proposed in \cite{Wilson2014TrajectoryBO} computes a prior mean function for the objective's surrogate model by simulating trajectories with a learned model.
They account for a potential bias introduced due to model inaccuracies by means of an additional parameter governing the relative importance of the simulated prior mean function, which can be inferred through evidence maximization.
The work in \cite{Bansal2017GoalDrivenBO} builds on the idea of `identification for control' \cite{Gevers2005IdentifcationForControl} and uses \gls{bo} to learn dynamics models not from observed state transitions but in order to directly maximize the closed-loop performance of \gls{mpc} controllers.
Alternatively, the work in \cite{Froehlich2019domainSelectionBO} utilizes a learned dynamics model to find an optimal subspace for linear feedback policies, enabling the application to high-dimensional controller parameterizations.
Recently, two general frameworks have been proposed to jointly learn a dynamics model and tune the controller parameters in an end-to-end fashion \cite{OKelly2020TunerCar, Edwards2021AutoMPC}.
Both frameworks have only been validated in simulation and neither considers varying environmental conditions to adapt the controller parameters.
\subsection{Contextual Bayesian Optimization for Robotics}
The foundation for the work on contextual \gls{bo} was first explored in the multi-armed bandit setting \cite{Krause2011contextual}.
Therein, a variant of the \gls{ucb} acquisition function was proposed and a multi-task \gls{gp} \cite{Bonilla2008MultiTaskGP} was employed to model the objective as a function of both the decision variables and the context.
The idea of using context information to share knowledge between similar tasks or environmental conditions has found widespread applications in the field of robotics.
The authors in \cite{Metzen2015ContextualPSwithBO} applied contextual \gls{bo} to control a robotic arm with the aim of throwing a ball to a desired target location.
In particular, they optimize the controller parameters with \gls{bo} and consider the target's coordinates as context variables.
Another promising application of contextual \gls{bo} is locomotion for robots wherein the context is typically describing the surface properties, such as inclination or terrain shape \cite{Yang2018ContextualBOLocomotionPrimitives,Seyde2019LocmotionContextBO}.
In \cite{Fiducioso2019SafeContextualBO}, a contextual variant of the SafeOpt algorithm \cite{Berkenkamp2016ControllerOptimization} is applied to tune a PID controller for temperature control where the outside temperature is considered as context variable.
Related to contextual \gls{bo} is also the `active' or `offline' contextual setting \cite{Fabisch2014activeContextualPolicySearch}, wherein the context can be actively selected, e.g., in simulations or controlled environments.
In this paper, we assume that the context information is given by the environment and either needs to be measured or inferred.
To the best of our knowledge, this paper is the first in the field of contextual \gls{bo} for control/robotics with an extensive study on hardware instead of pure simulation-based results.
\subsection{Model Predictive Control for Autonomous Racing}
\gls{mpc} is theoretically well understood and allows for incorporating state and input constraints to safely race on a track, making it a prime candidate approach for controlling autonomous cars.
Two common approaches to the racing problem are time-optimal control \cite{Metz1989TimeOptimalRacing} and contouring control \cite{Lam2010MPCC}, i.e., where the progress is maximized along the track.
Both approaches have been implemented successfully on hardware platforms akin to ours \cite{Verschueren2014TimeOptimalMPC,Liniger2015RCCars}.
In order to account for modeling errors, several learning approaches have been proposed in the context of race cars.
The authors in \cite{Hewing2019cautiousMPC} use \gls{gp} regression to learn a residual dynamics model and employ an adaptive selection scheme of data points to achieve real-time computation.
A similar approach has been applied to a full-scale race car \cite{Kabzan2019learningMPC}.
Alternatively, the algorithm proposed in \cite{Rosolia2020LearningRaceCar} uses time-varying locally linear models around previous trajectories without relying on a nominal model.
In this paper, we use the contouring \gls{mpc} approach based on \cite{Liniger2015RCCars} and learn a residual term based on a nominal dynamics model similar to \cite{Hewing2019cautiousMPC}.
\subsection{Other related work}
We want to additionally mention the following ideas that are related to this paper, but do not directly fit into any of the three previous categories.
The authors in \cite{Wischnewski2019SafeOptRacecarTuning} employ a safe variant of \gls{bo} \cite{Berkenkamp2016ControllerOptimization} to increase the controller's performance without violating pre-specified handling limits of the car.
The approach proposed in \cite{Jain2020racingLineBO} uses \gls{bo} to optimize the ideal racing line instead of relying on an optimal control formulation.
Lastly, the authors in \cite{McKinnon2020ContextAwareCostShaping} propose to learn the discrepancy between the predicted and observed \gls{mpc} cost in order to account for modeling errors of an autonomous vehicle in different contexts.
|
1,116,691,500,804 | arxiv | \section{Introduction}
Humans have the ability to process symbolic knowledge and maintain symbolic thought~\cite{coevoflangandbrain}. When reasoning, humans do not require combinatorial enumeration of examples but instead utilise invariant patterns where specific entities are replaced with placeholders. Symbolic cognitive models~\cite{cogmodelsymbolic} embrace this perspective with the human mind seen as an information processing system operating on formal symbols such as reading a stream of tokens in natural language. The language of thought hypothesis~\cite{languageofthought} frames human thought as a structural construct with varying sub-components such as ``X went to Y''. By recognising what varies across examples, humans are capable of lifting examples into invariant principles that account for other instances.
This symbolic thought with variables is learned at a young age through symbolic play~\cite{piagetsymthought}. For instance, a child learns that a sword can be substituted with a stick~\cite{sticksword} and engages in pretend play
\setlength{\intextsep}{0.5em}
\begin{wrapfigure}{r}{0.5\linewidth}
\centering
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{l}
\vard{bernhard} is a \vard{frog}\\
\vard{lily} is a \vard{frog}\\
\vard{lily} is \vard{green}\\
\hline
what colour is \vard{bernhard}\\
\hline
green
\end{tabular}
\caption{Invariant learned for bAbI task 16, basic induction, where \vard{bernhard} denotes a variable with default symbol bernhard. This single invariant accounts for all the training, validation and test examples of this task.}
\label{fig:rule_babi_task16}
\end{wrapfigure}
Although variables are inherent in symbolic formalisms and their models of computation, as in first-order logic~\cite{russell2016artificial}, they are pre-engineered and used to solve specific tasks by means of assigning values to them. However, when learning from data only, being able to recognise when and which symbols could act as variables and therefore take on different values is crucial for lifting examples into general principles that are invariant across multiple instances. Figure~\ref{fig:rule_babi_task16} shows an example invariant learned by our approach: \emph{if someone is the same thing as someone else then they have the same colour}. With this invariant, our approach solves \emph{all} training and test examples in task 16 of the bAbI dataset~\cite{babi}.
In this paper we address the question of whether a machine can learn and use the notion of a \emph{variable}, i.e. a symbol that can take on different values. For instance, given an example of the form ``bernhard is a frog'' the machine would learn that the token ``bernhard'' could be \emph{someone} else and the token ``frog'' could be \emph{something} else.
When the machine learns that a symbol is a variable, assigning a value can be reframed as attending to an appropriate symbol.
Attention models~\cite{bahdanauatt, nmtatt, attsurvey} allow neural networks to focus, attend to certain parts of the input often for the purpose of selecting a relevant portion. Since attention mechanisms are also differentiable they are often jointly learned within a task. This perspective motivates our idea of a unification mechanism for learning variables across examples that utilises attention and is fully differentiable. We refer to this approach as \emph{soft unification} that can jointly learn which symbols can act as variables and how to assign values to them.
Hence, we propose an end-to-end differentiable neural network approach for learning and utilising the notion of lifting examples into invariants, which can then be used by the network to solve given tasks. The main contribution of this paper is a novel architecture capable of learning and using variables by lifting a given example through soft unification. As a first step, we present the empirical results of our approach in a controlled environment using four synthetic datasets and then with respect to a real-world dataset along with the analysis of the learned invariants that capture the underlying patterns present in the tasks. Our implementation using Chainer~\cite{chainer} is publicly available at
{\footnotesize \url{https://github.com/nuric/softuni}} with the accompanying data.
\section{Unification Networks}\label{sec:uni_nets}
Reasoning with variables involves first of all identifying what variables are in a given context as well as defining the process by which they are assigned values. The intuition is that when the varying components, i.e. variables, of an example are identified, the example can be lifted into an invariant that captures its structure but with variables replacing its varying components. Such an invariant can account for multiple other instances of the same structure. We present our approach in Algorithm~\ref{algo:uninet} and detail the steps below (note that the algorithm is more compressed than the steps described). Refer to Table~\ref{tab:notation_table} in Appendix~\ref{apx:model_details} for a summary of symbols and notations used in the paper.
\begin{step}[Pick Invariant Example]\label{step:pick}
We start from an example data point to generalise from. If we assume that within a task there is \emph{one} common pattern then any example should be an instance of that pattern. Therefore, we can randomly pick any example within a task from the dataset as our invariant example $G$. In this paper, $G$ is one data point consisting of a context, query and answer (see Table~\ref{tab:samples}).
\end{step}
\begin{step}[Lift Invariant Example]\label{step:lift}
In order for the invariant example to predict other examples correctly, certain symbols might need to \emph{vary} such as \vard{bernard} in Figure~\ref{fig:rule_babi_task16}. We capture this degree of \emph{variableness} with a function $\psi:{\mathbb{S}} \rightarrow [0,1]$ for every symbol appearing in $G$ where ${\mathbb{S}}$ is the set of all symbols. When $\psi$ is a learnable function, the model learns to identify the variables and convert the data point into an invariant, i.e. learning invariants. For a certain threshold $\psi(s) \geq t$, we visualise them in bold with a \textbf{\textit{V}}~prefix. An invariant is thus a pair $I \triangleq (G, \psi)$, the example data point and the variableness of each symbol. Which symbols emerge as variables depend on whether they need to be assigned new values, but how do we know which values to assign? This brings us to unification.
\end{step}
\begin{step}[Compute Unifying Features]\label{step:ufeats}
Suppose now we are given a new data point $K$ that we would like to unify with our invariant $I$ from the previous step. $K$ might start with a question ``what colour is lily'' and our invariant ``what colour is \vard{bernard}''. We would like to match bernard with lily. However, if we were to just use $d$-dimensional representations of symbols $\phi: {\mathbb{S}} \rightarrow \mathbb{R}^d$, the representations of bernard and lily would need to be similar which might confound an upstream predictor network, e.g. when lily and bernard appear in the same story it would be difficult to distinguish them with similar representations. To resolve this issue, we learn \emph{unifying features} $\phi_U: {\mathbb{S}} \rightarrow \mathbb{R}^d$ that intuitively capture some common meaning of two otherwise distinct symbols.
For example, $\phi(\text{bernhard})$ and $\phi(\text{lily})$ could represent specific people whereas $\phi_U(\text{bernhard}) \approx \phi_U(\text{lily})$ the notion of \emph{someone}; similarly, in Figure~\ref{fig:diag_softuni}, $\phi_U(7) \approx \phi_U(3)$ the notion of the head of a sequence. Notice how we use the original symbol bernhard in computing the representation of \vard{bernhard}; this is intended to capture the variable's bound \textit{meaning} following the idea of referants~\cite{senseandreference}.
\end{step}
\setlength{\textfloatsep}{1em}
\begin{algorithm}[t]
\small
\DontPrintSemicolon
\SetKwFunction{ApplyMap}{map}
\SetKw{KwListIn}{in}
\SetKw{KwLet}{Let}
\SetKwComment{Comment}{$\triangleright$ }{}
\SetKwFunction{ExtractFeat}{extract}
\KwIn{Invariant $I$ consisting of example $G$ and variableness network $\psi$, example $K$, features network $\phi$, unifying features network $\phi_U$, upstream predictor network $f$}
\KwOut{Predicted label for example $K$}
\BlankLine
\Begin(\Comment*[h]{Unification Network}){
\Return{$f \circ g(I, K)$}\Comment*[f]{Predictions using Soft Unification $g$}
}
\Begin(\Comment*[h]{Soft Unification function $g$}){
\ForEach{symbol s \KwListIn G}{
${\bm{A}}_{s,:} \leftarrow \phi(s)$\Comment*[r]{Features of $G$, ${\bm{A}} \in \mathbb{R}^{|G|\times d}$}
${\bm{B}}_{s,:} \leftarrow \phi_U(s)$\Comment*[r]{Unifying features of $G$, ${\bm{B}} \in \mathbb{R}^{|G|\times d}$}
}
\ForEach{symbol s \KwListIn K}{
${\bm{C}}_{s,:} \leftarrow \phi(s)$\Comment*[r]{Features of $K$, ${\bm{C}} \in \mathbb{R}^{|K|\times d}$}
${\bm{D}}_{s,:} \leftarrow \phi_U(s)$\Comment*[r]{Unifying features of $K$, ${\bm{D}} \in \mathbb{R}^{|K|\times d}$}
}
\KwLet ${\bm{P}} = \operatorname{softmax}({\bm{B}}{\bm{D}}^T)$\Comment*[r]{Attention map over symbols, ${\bm{P}} \in \mathbb{R}^{|G| \times |K|}$}\label{algoline:uninet_uni_att}
\KwLet ${\bm{E}} = {\bm{P}}{\bm{C}}$\Comment*[r]{Attended representations of $G$, ${\bm{E}} \in \mathbb{R}^{|G|\times d}$}
\ForEach{symbol s \KwListIn G}{
${\bm{U}}_{s,:} \leftarrow \psi(s){\bm{E}}_{s,:} + (1-\psi(s)){\bm{A}}_{s,:}$\Comment*[f]{Unified representation of $I$, ${\bm{U}} \in \mathbb{R}^{|G| \times d}$}\label{algoline:uninet_inv_value}
}
\Return{${\bm{U}}$}
}
\caption{Unification Networks}
\label{algo:uninet}
\end{algorithm}
\begin{step}[Soft Unification]\label{step:unify}
Using learned unifying features, for every variable in $I$ we can now find a corresponding symbol in $K$. This process of unification, i.e. replacing variables with symbol values, is captured by the function $g$ that given an invariant and another example it updates the variables with appropriate values. To achieve this, we compute a soft attention for each symbol $s$ in the invariant using the unifying features $\phi_U(s)$ (line~\ref{algoline:uninet_uni_att} in Algorithm~\ref{algo:uninet}) and interpolate between its own and its variable value (line~\ref{algoline:uninet_inv_value} in Algorithm~\ref{algo:uninet}).
Since $g$ is a differentiable formulation and $\psi$, $\phi$ and $\phi_U$ can be learnable functions, we refer to this step as \emph{soft unification}. In Figure~\ref{fig:diag_softuni}, the variable \vard{7} is changed towards the symbol 3, having learnt that the unifiable feature is the head of the sequence.
\end{step}
\begin{step}[Predict]\label{step:predict}
So far we have constructed a unified data point of $I$ with the new example $K$ of which we would like to predict the answer. How do we predict? We use another, potentially upstream task specific, network $f$ that tries to predict the answer based on our unified input. Recall that our data points are triples of the form context, query and answer. In question answering, $f$ could be a memory network, or when working with grid like inputs, a CNN. By predicting on the output of our unification, we expect that $f \circ g(I,K) = f(K) = a$. If $f$ is differentiable, we can learn how to unify while solving the upstream task. We focus on $g$ and use standard networks for $f$ to understand which invariants are learned and the interaction of $f \circ g$ instead of the raw performance of $f$.
\end{step}
\section{Instances of Unification Networks}\label{sec:inst_uni_nets}
We present four architectures to model $f \circ g$ and demonstrate the flexibility of our approach towards different architectures and upstream tasks. Except in Unification RNN, the $d$-dimensional representation of symbols are learnable embeddings $\phi(s) = O[s]^T{\bm{E}}$ with ${\bm{E}} \in \mathbb{R}^{|{\mathbb{S}}| \times d}$ randomly initialised by $\mathcal{N}(0,1)$ and $O[s]$ the one-hot encoding of the symbol. The variableness of symbols are learnable weights $\psi(s) = \sigma(w_s)$ where ${\bm{w}} \in \mathbb{R}^{|{\mathbb{S}}|}$ and $\sigma$ is the sigmoid function. We consider every symbol independently as a variable irrespective of its surrounding context and leave further contextualised formulations as future work. However, unifying features $\phi_U$ can be context sensitive to disambiguate same symbol variables appearing in different contexts. Full details the of models, including hyper-parameters, are available in Appendix~\ref{apx:model_details}.
\textbf{Unification MLP (UMLP)} ($f$: MLP, $g$: RNN) We start with a sequence of symbols as input, e.g. a sequence of digits \texttt{4234}. Unifying features $\phi_U$, from Step~\ref{step:ufeats}, are obtained using the hidden states of a bi-directional GRU~\cite{gru} processing the embedded sequences. In Step~\ref{step:predict}, the upstream MLP predicts the answer based on the flattened representation of the unified sequence.
\textbf{Unification CNN (UCNN)} ($f$: CNN, $g$: CNN) To adapt our approach for a grid of symbols, we use \emph{separate} convolutional neural networks with the same architecture to compute unifying features $\phi_U$ as well as to predict the correct answer through $f$ in Step~\ref{step:predict}. We mask out padding in Step~\ref{step:unify} to avoid assigning null values to variables.
\textbf{Unification RNN (URNN)} ($f$: RNN, $g$: MLP) We start with a varying length sequence of words such as a movie review. We set ConceptNet word embeddings~\cite{numberbatch} as $\phi$ to compute $\phi_U(s) = \operatorname{MLP}(\phi(s))$ and $\psi(s) = \sigma(\operatorname{MLP}(\phi(s))$. Then, the final hidden state of $f$, an LSTM~\cite{lstm}, predicts the answer.
\begin{figure}[h]
\centering
\includegraphics[width=0.8\linewidth]{diag/unimemnn.pdf}
\caption{Graphical overview of soft unification within a memory network (UMN). Each sentence is processed by two bi-directional RNNs for memory and unification. At each iteration the context attention selects which sentences to unify and the invariant produces the same answer as the example.}
\label{fig:diag_unimemnn}
\end{figure}
\textbf{Unification Memory Networks (UMN)} ($f$: MemNN, $g$: RNN) Soft unification does not need to happen prior to $f$ in a $f \circ g$ fashion but can also be incorporated at any intermediate stage multiple times. To demonstrate this ability, we unify the symbols at different memory locations at each iteration of a Memory Network~\cite{memnn}. We take a list of lists as input such as a tokenised story, Figure~\ref{fig:diag_unimemnn}. The memory network $f$ uses the final hidden state of a bi-directional GRU (blue squares in Figure~\ref{fig:diag_unimemnn}) as the sentence representations to compute a context attention, i.e. select the next context sentence starting with the query. With the sentences attended to, we can unify the words of the sentences at each iteration following Steps~\ref{step:lift} to \ref{step:unify}. We use another bi-directional GRU (pink diamonds in Figure~\ref{fig:diag_unimemnn}) for unifying features $\phi_U$. Following line~\ref{algoline:uninet_inv_value} in Algorithm~\ref{algo:uninet}, the new unified representation of the memory slot (the sentence) is used by $f$ to perform the next iteration. The prediction is then based on the final hidden state of the invariant example. This setup, however, requires pre-training $f$ such that the context attentions match the correct pairs of sentences to unify which limits the performance of the combined network by how well $f$ performs.
Although we assume a single pattern in Step~\ref{step:pick}, a task might contain slightly different examples such as ``Where is X?'' and ``Why did X go to Y?''. To let the models potentially learn and benefit from different invariants, we can pick multiple examples to generalise from and aggregate the predictions from each invariant. One simple approach is to sum the predictions of the invariants $\sum_{I \in {\mathbb{I}}} f \circ g(I,K)$ used in UMLP, UCNN and URNN where ${\mathbb{I}}$ is the set of invariants. For UMN, at each iteration we weigh the hidden states from each invariant using a bilinear attention $\eta = \text{softmax}({\bm{h}}^0_I {\bm{W}} {\bm{h}}^{0^T}_K)$ where ${\bm{h}}^0_I$ and ${\bm{h}}^0_K$ are the representations of the query (at iteration 0).
\begin{table*}[h]
\small
\centering
\caption{Sample context, query and answer triples and their training sizes \emph{per task}. For the distribution of generated number of examples per task on Sequence and Grid data refer to Appendix~\ref{apx:dataset}.}
\label{tab:samples}
\begin{tabular}{@{}ccccc@{}}
\toprule
Dataset & Context & Query & Answer & Training Size\\
\midrule
Sequence & 8384 & duplicate & 8 & $\le$ 1k, $\le$ 50\\
Grid &
\setlength{\tabcolsep}{1mm}
\renewcommand{\arraystretch}{0.5}
\begin{tabular}{@{}ccc@{}}
0 & 0 & 3\\
0 & 1 & 6\\
8 & 5 & 7
\end{tabular} & corner & 7 & $\le$ 1k, $\le$ 50\\
bAbI &
\begin{tabular}{@{}l@{}}
Mary went to the kitchen.\\
Sandra journeyed to the garden.
\end{tabular} & Where is Mary? & kitchen & 1k, 50\\
Logic &
\begin{tabular}{@{}l@{}}
p(X) $\leftarrow$ q(X).\\
q(a).
\end{tabular} & p(a). & True & 2k, 100\\
Sentiment A. & easily one of the best films & Sentiment & Positive & 1k, 50\\
\bottomrule
\end{tabular}
\end{table*}
\section{Datasets}\label{sec:datasets}
We use five datasets consisting of context, query and an answer $(C, Q, a)$ (see Table~\ref{tab:samples} and Appendix~\ref{apx:dataset} for further details) with varying input structures: fixed or varying length sequences, grids and nested sequences (e.g. stories). In each case we use an appropriate model: UMLP for fixed length sequences, UCNN for grid, URNN for varying length sequences and UMN for iterative reasoning.
\textbf{Fixed Length Sequences} We generate sequences of length $l=4$ from 8 unique symbols represented as digits to predict (i) a constant, (ii) the head of the sequence, (iii) the tail and (iv) the duplicate symbol. We randomly generate 1000 triples and then only take the unique ones to ensure the test split contains unseen examples. The training is then performed over a 5-fold cross-validation. Figure~\ref{fig:diag_softuni} demonstrates how the invariant `\vard{7} 4' can predict the head of another example sequence `3 9'.
\begin{wrapfigure}{r}{0.50\linewidth}
\centering
\includegraphics[width=0.9\linewidth]{diag/softuni.pdf}
\caption{Graphical overview of predicting the head of a sequence using an invariant and soft unification where $g$ outputs the new sequence 3 4.}
\label{fig:diag_softuni}
\end{wrapfigure}
\textbf{Grid} To spatially organise symbols, we generate a grid of size $3 \times 3$ from 8 unique symbols. The grids contain one of (i) $2 \times 2$ box of identical symbol, (ii) a vertical, diagonal or horizontal sequence of length 3, (iii) a cross or a plus shape and (iv) a triangle. In each task we predict (i) the identical symbol, (ii) the head of the sequence, (iii) the centre of the cross or plus and (iv) the corner of the triangle respectively. We generate 1000 triples discarding any duplicates.
\textbf{bAbI} The bAbI dataset consists of 20 synthetically generated natural language reasoning tasks (refer to~\cite{babi} for task details). We take the 1k English set and use 0.1 of the training set as validation. Each token is lower cased and considered a unique symbol. Following previous works~\cite{qrn, memn2n}, we take multiple word answers also to be a unique symbol. To initially form the repository of invariants, we use the bag-of-words representation of the questions and find the most dissimilar ones based on their cosine similarity as a heuristic to obtain varied examples.
\textbf{Logical Reasoning} To distinguish our notion of a variable from that used in logic-based formalisms, we generate logical reasoning tasks in the form of logic programs using the procedure from~\cite{deeplogic}. The tasks involve learning $f(C,Q) = \text{True}$ if and only if $C \vdash Q$ over 12 classes of logic programs exhibiting varying paradigms of logical reasoning including negation by failure~\cite{nbf}. We generate 1k and 50 logic programs per task for training with 0.1 as validation and another 1k for testing. Each logic program has one positive and one negative prediction giving a total of 2k and 100 data points respectively. We use one random character from the English alphabet for predicates \emph{and} constants, e.g. $p(p,p)$ and an upper case character for logical variables, e.g. $p(\text{\texttt{X}},\text{\texttt{Y}})$. Further configurations such as restricting the arity of predicates to 1 are presented in Table~\ref{tab:deeplogic_results} Appendix~\ref{apx:further_results}.
\textbf{Sentiment Analysis} To evaluate on a noisy real-world dataset, we take the sentiment analysis task from~\cite{sochersentiment} and prune sentences to a maximum length of 20 words. We threshold the scores $\le 0.1$ and $\ge 0.9$ for negative and positive labels respectively to ensure unification cannot yield a neutral score, i.e. the model is forced to learn either a positive or a negative label. We then take 1000 or 50 training examples \emph{per label} and use the remaining $\approx 676$ data points as unseen test examples.
\section{Experiments}
We probe three aspects of soft unification: the impact of unification on performance over unseen data, the effect of multiple invariants and data efficiency. To that end, we train UMLP, UCNN and URNN with and without unification and UMN with pre-training using 1 or 3 invariants over either the entire training set or only 50 examples.
Every model is trained via back-propagation using Adam~\cite{adam} with learning rate 0.001 on an Intel Core i7-6700 CPU using the following objective function:
\begin{equation}\label{eq:objective_function}
J = \overbrace{\lambda_K \mathcal{L}_{\text{nll}}(f)}^\text{Original output} + \lambda_I [ \ \overbrace{\mathcal{L}_{\text{nll}}(f \circ g)}^\text{Unification output} + \overbrace{\tau \sum_{s \in {\mathbb{S}}} \psi(s) }^\text{Sparsity} \ ]
\end{equation}
where $\mathcal{L}_{\text{nll}}$ is the negative log-likelihood with sparsity regularisation over $\psi$ at $\tau = 0.1$ to discourage the models from utilising spurious number of variables.
We add the sparsity constraint over the variableness of symbols $\psi(s)$ to avoid the trivial solution in which every symbol is a variable and $G$ is completely replaced by $K$ still allowing $f$ to predict correctly. Hence, we would like the \emph{minimal} transformation of $G$ towards $K$ to expose the common underlying pattern.
For UMLP and UCNN, we set $\lambda_K = 0,\ \lambda_I = 1$ for training just the unified output and the converse for the non-unifying versions. For URNN, we set $\lambda_K = \lambda_I = 1$ to train the unified output and set $\lambda_I=0$ for non-unifying version. To pre-train the UMN, we start with $\lambda_K = 1,\ \lambda_I = 0$ for 40 epochs then set $\lambda_I = 1$ to jointly train the unified output. For UMN, we also add the mean squared error between hidden states of $I$ and $K$ at each iteration (see Appendix~\ref{apx:training_details}). In the strongly supervised cases, the negative log-likelihood of the context attentions (which sentences are selected at each iteration) are also added. Further training details including sample training curves are available in Appendix~\ref{apx:training_details}.
\begin{figure*}[t]
\centering
\includegraphics[width=1.0\textwidth]{plots/umlp_ucnn_acc_plot.pdf}
\caption{Test accuracy over iterations for UMLP, UCNN and URNN models with 1 invariant versus no unification. Soft unification aids with data efficiency of models against their plain counterparts.}
\label{fig:umlp_ucnn_acc}
\end{figure*}
Figure~\ref{fig:umlp_ucnn_acc} portrays how soft unification generalises better to unseen examples in test sets over plain models. Despite $f \circ g$ having more trainable parameters than $f$ alone, this data efficiency is visible across all models when trained with only $\le 50$ examples \emph{per task}. We believe soft unification architecturally biases the models towards learning unifying features that are common across examples, therefore, potentially also common to unseen examples. The data efficient nature is more emphasised with UMLP and UCNN on synthetic datasets in which there are unambiguous patterns in the tasks and they achieve higher accuracies in as few as 250 iterations (batch updates) against their plain counterparts. In the real-world dataset of sentiment analysis, we observe a less steady training curve for URNN and performance as good as if not better than its plain version. The fluctuations in accuracy around iterations 750 to 1000 in UCNN and iteration 700 in URNN are caused by penalising $\psi$ which forces the model to adjust the invariant to use less variables half way through training. Results with multiple invariants are identical and the models learn to ignore the extra invariants (Figure~\ref{fig:umlp_ucnn_invs} Appendix~\ref{apx:further_results}) due to the regularisation applied on $\psi$ zeroing out unnecessary invariants. Training with different learning rates overall paint a similar picture (Figure~\ref{fig:umlp_ucnn_learning_rates} Appendix~\ref{apx:training_details}).
\begin{table*}[h]
\footnotesize
\centering
\caption{Aggregate error rates (\%) on bAbI 1k for UMN and baselines N2N~\cite{memn2n}, GN2N~\cite{gn2n}, EntNet~\cite{entnet}, QRN~\cite{qrn} and MemNN~\cite{memnn} respectively. Full comparison is available in Appendix~\ref{apx:further_results}.}
\label{tab:babi_agg_results}
\begin{tabular}{@{}rccccc|ccccc@{}}
\toprule
Training Size & \multicolumn{4}{c|}{1k} & 50 & \multicolumn{5}{c}{1k}\\
Supervision & \multicolumn{2}{c|}{Weak} & \multicolumn{3}{c|}{Strong} & \multicolumn{4}{c|}{Weak} & Strong\\
\# Invs / Model & 1 & 3 & 1 & 3 & 3 & N2N & GN2N & EntNet & QRN & MemNN\\
\midrule
Mean & 18.8 & 19.0 & \textbf{5.1} & 6.6 & 28.7 & 13.9 & 12.7 & 29.6 & 11.3 & 6.7\\
\# $>5\%$ & 10 & 9 & 3 & 3 & 17 & 11 & 10 & 15 & 5 & 4\\
\bottomrule
\end{tabular}
\end{table*}
\begin{table*}[b]
\footnotesize
\centering
\caption{Aggregate task error rates (\%) on the logical reasoning dataset for UMN and baseline IMA~\cite{deeplogic}. Individual task results are available in Appendix~\ref{apx:further_results}.}
\label{tab:deeplogic_agg_results}
\begin{tabular}{@{}rccccc|cc@{}}
\toprule
Model & \multicolumn{5}{c|}{UMN} & \multicolumn{2}{c}{IMA}\\
Training Size & \multicolumn{4}{c|}{2k} & 100 & \multicolumn{2}{c}{2k}\\
Supervision & \multicolumn{2}{c|}{Weak} & \multicolumn{3}{c|}{Strong} & Weak & Strong\\
\# Invs & 1 & 3 & 1 & 3 & 3 & \multicolumn{2}{c}{-}\\
\midrule
Mean & 37.7 & 37.6 & \textbf{27.4} & 29.0 & 47.1 & 38.8 & 31.5\\
\# $>5\%$ & 10 & 10 & 10 & 11 & 12 & 11 & 11\\
\bottomrule
\end{tabular}
\end{table*}
For iterative reasoning tasks, Tables~\ref{tab:babi_agg_results} and \ref{tab:deeplogic_agg_results} aggregate the results for our approach against comparable baseline memory networks which are selected based on whether they are built on Memory Networks (MemNN)~\cite{memnn} and predict by iteratively updating a hidden state. For example, End-to-End Memory Networks (N2N)~\cite{memn2n} and Iterative Memory Attention (IMA)~\cite{deeplogic} networks update a hidden state vector after each iteration by attending to a single context sentence similar to our architecture. We observe that strong supervision and more data per task yield lower error rates which is consistent with previous work reflecting how $f \circ g$ can be bounded by the efficacy of $f$ modelled as a memory network. In a weak supervision setting, i.e. when sentence selection is not supervised, our model attempts to unify arbitrary sentences often failing to follow the iterative reasoning chain. As a result, only in the supervised case we observe a minor improvement over MemNN by 1.6 in Table~\ref{tab:babi_agg_results} and over IMA by 4.1 in Table~\ref{tab:deeplogic_agg_results}. Without reducing the performance of $f$, our approach is still able to learn invariants as shown in Figure~\ref{fig:invariants}. This dependency on $f$ also limits the ability of $f \circ g$ to learn from 50 and 100 examples per task failing 17/20 of bAbI and 12/12 of logical reasoning tasks respectively. The increase in error rate with 3 invariants in Table~\ref{tab:deeplogic_agg_results}, we speculate, stems from having more parameters and more pathways, rendering training more difficult and slower.
\begin{table}[h]
\small
\centering
\caption{Number of \emph{exact} matches of the learnt invariants with expected ones in synthetic datasets where the invariant is known. If the model does not get an exact match, it might use more or fewer variables than expected while still solving the task. For sequences and grid datasets, there are 5 folds each with 4 tasks giving 20 and the logic dataset has 12 tasks with 3 runs giving 36 invariants.}
\label{tab:correctness}
\begin{tabular}{@{}rcccccc@{}}
\toprule
Model & \multicolumn{2}{c}{UMLP} & \multicolumn{2}{c}{UCNN} & \multicolumn{2}{c}{UMN}\\
Dataset & \multicolumn{2}{c}{Sequences} & \multicolumn{2}{c}{Grid} & \multicolumn{2}{c}{Logic}\\
Train Size & $\leq1k$ & $\leq50$ & $\leq1k$ & $\leq50$ & \multicolumn{2}{c}{2k}\\
Supervision & \multicolumn{4}{c}{-} & Weak & Strong\\
\midrule
Correct / Total & 18/20 & 18/20 & 13/20 & 14/20 & 7/36 & 23/36\\
Accuracy (\%) & 90.0 & 90.0 & 65.0 & 70.0 & 19.4 & 63.9\\
\bottomrule
\end{tabular}
\end{table}
For synthetic sequences, grid and logic datasets in which we know exactly what the invariants \emph{can be}, Table~\ref{tab:correctness} shows how often our approach captures \emph{exactly} the expected invariant. We threshold $\psi$ as explained in Section~\ref{sec:analysis} and check for an exact match; for example for predicting the head of a sequence in UMLP, we compare the learnt invariant against the pattern ``\textbf{\textit{V}} \_ \_ \_''. Although with increasing dataset complexity the accuracy drops, it is important to note that just because the model does not capture the exact invariant it may still solve the task. In these cases, it may use extra or more interestingly fewer variables as further discussed in Section~\ref{sec:analysis}.
\section{Analysis}\label{sec:analysis}
\begin{wrapfigure}{r}{0.4\linewidth}
\centering
\includegraphics[width=1.0\linewidth]{plots/sentiment_inv.pdf}
\caption{The variableness $\psi(s)$ of an invariant for sentiment analysis with words such as `silly' emerging as variables.}
\label{fig:sentiment_inv}
\end{wrapfigure}
Figure~\ref{fig:sentiment_inv} shows an invariant for sentiment analysis in which words such as silly that contribute more to the sentiment have a higher $\psi$. Intuitively, if one replaces `silly' with the adjective `great', the sentiment will change. The replacement, however, is not a hard value assignment but an interpolation (line~\ref{algoline:uninet_inv_value} Algorithm~\ref{algo:uninet}) which may produce a new intermediate representation from $G$ towards $K$ different enough to allow $f$ to predict correctly. Since we penalise the magnitude of $\psi$ in equation~\ref{eq:objective_function}, we expect these values to be as low as possible. For synthetic datasets, we apply a threshold $\psi(s) > t$ to extract the learned invariants and set $t$ to be the mean of the variable symbols as a heuristic except for bAbI where we use $t=0.1$. The magnitude of $t$ depends on the amount of regularisation $\tau$, equation~\ref{eq:objective_function}, number of iterations and batch size. Sample invariants shown in Figure~\ref{fig:invariants} describe the patterns present in the tasks with parts that contribute towards the final answer becoming variables. Extra symbols such as `is' or `travelled' do not emerge as variables, as shown in Figure~\ref{fig:rule_babi_2}; we attribute this behaviour to the fact that changing the token `travelled' to `went' does not influence the prediction but changing the action, the value of \vard{left} to `picked' does. However, based on random initialisation, our approach can convert an arbitrary symbol into a variable and let $f$ compensate for the unifications it produces. For example, the invariant ``\vard{8} 5 2 2'' could predict the tail of another example by unifying the head with the tail using $\phi_U$ of those symbols in Step~\ref{step:ufeats}. Further examples are shown in Appendix~\ref{apx:further_results}. Pre-training $f$ as done in UMN seems to produce more robust and consistent invariants since, we speculate, a pre-trained $f$ encourages more $g(I,K) \approx K$.
\begin{figure*}[t]
\footnotesize
\centering
\begin{subfigure}[t]{0.48\textwidth}
\centering
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{l}
\vard{john} travelled to the \vard{office}\\
\vard{john} \vard{left} the \vard{football}\\
\hline
where is the \vard{football}\\
\hline
office
\end{tabular}
\caption{bAbI task 2, two supporting facts. The model also learns \vard{left} since people can also drop or pick up objects potentially affecting the answer.}
\label{fig:rule_babi_2}
\end{subfigure}
~
\begin{subfigure}[t]{0.49\textwidth}
\centering
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{l}
this \vard{morning} \vard{bill} went to the \vard{school}\\
yesterday \vard{bill} journeyed to the \vard{park}\\
\hline
where was \vard{bill} before the \vard{school}\\
\hline
park
\end{tabular}
\caption{bAbI task 14, time reasoning. \vard{bill} and \vard{school} are recognised as variables alongside \vard{morning} capturing \emph{when} someone went.}
\label{fig:rule_babi_14}
\end{subfigure}\\[1em]
\begin{subfigure}[t]{0.31\textwidth}
\centering
\setlength{\tabcolsep}{0.2em}
\begin{tabular}{rcr}
5 8 6 4 & const & 2\\
\vard{8} 3 3 1 & head & 8\\
8 3 1 \vard{5} & tail & 5\\
\vard{1} 4 3 \vard{1} & dup & 1
\end{tabular}
\caption{Successful invariants learned with UMLP using 50 training examples only shown as $(C,Q,a)$.}
\label{fig:rule_seq_tasks}
\end{subfigure}
~
\begin{subfigure}[t]{0.32\textwidth}
\centering
\begin{tabular}{ccc}
\setlength{\tabcolsep}{0.2em}
\begin{tabular}{@{}rrr@{}}
0 & \textbf{\textit{V}} & \textbf{\textit{V}}\\
0 & \textbf{\textit{V}} & \textbf{\textit{V}}\\
0 & 0 & 0
\end{tabular} &
\setlength{\tabcolsep}{0.2em}
\begin{tabular}{@{}rrr@{}}
0 & 1 & 0\\
6 & \textbf{\textit{V}} & 8\\
0 & 7 & 0
\end{tabular} &
\setlength{\tabcolsep}{0.2em}
\begin{tabular}{@{}rrr@{}}
0 & 0 & 1\\
0 & 5 & 4\\
7 & 8 & \textbf{\textit{V}}
\end{tabular}\\
box & centre & corner
\end{tabular}
\caption{Successful invariants learned with UCNN. Variable default symbols are omitted for clarity.}
\label{fig:rule_grid_tasks}
\end{subfigure}
~
\begin{subfigure}[t]{0.31\textwidth}
\centering
\begin{tabular}{l}
\vard{i} ( \texttt{T} ) $\leftarrow$ \vard{l} ( \texttt{T} ),\\
\vard{l} ( \texttt{U} ) $\leftarrow$ \vard{x} ( \texttt{U} ),\\
\vard{x} ( \texttt{K} ) $\leftarrow$ \vard{n} ( \texttt{K} ),\\
\vard{n} ( \vard{o} ) $\vdash$ \vard{i} ( \vard{o} )
\end{tabular}
\caption{Logical reasoning task 5 with arity 1. The model captures how \vard{n} could entail \vard{i} in a chain.}
\label{fig:rule_deeplogic_5}
\end{subfigure}
\caption{Invariants learned across the four datasets using the three architectures. For iterative reasoning datasets, bAbI and logical reasoning, they are taken from strongly supervised UMN.}
\label{fig:invariants}
\end{figure*}
\textbf{Interpretability versus Ability} A desired property of interpretable models is transparency~\cite{interpretability}. A novel outcome of the learned invariants in our approach is that they provide an approximation of the underlying general principle that may be present in the data. Figure~\ref{fig:rule_deeplogic_5} captures the structure of multi-hop reasoning in which a predicate \vard{n} can entail another \vard{i} with a matching constant \vard{o} if there is a chain of rules that connect the two predicates. However, certain aspects regarding the ability of the model such as how it performs temporal reasoning, are still hidden inside $f$. In Figure~\ref{fig:rule_babi_14}, although we observe \vard{morning} as a variable, the overall learned invariant captures nothing about how changing the value of \vard{morning} alters the behaviour of $f$, i.e. how $f$ uses the interpolated representations produced by $g$. The upstream model $f$ might look \emph{before} or \emph{after} a certain time point \vard{bill} went somewhere depending what \vard{morning} binds to. Without the regularising term on $\psi(s)$, we initially noticed the models using, what one might call extra, symbols as variables and binding them to the same value occasionally producing unifications such as ``bathroom bathroom to the bathroom'' and still $f$ predicting, unsurprisingly, the correct answer as bathroom. Hence, regularising $\psi$ with the correct amount $\tau$ in equation~\ref{eq:objective_function} to reduce the capacity of unification seems critical in extracting not just any invariant but one that represents the common structure.
\begin{figure}[h]
\centering
\begin{subfigure}[t]{0.30\textwidth}
\centering
\includegraphics[width=1.00\textwidth,height=8em,keepaspectratio]{vsym_plots/qa16_uni.pdf}
\caption{bAbI task 16. A one-to-one mapping is created between variables \vard{bernhard} with brian and \vard{frog} with lion.}
\label{fig:uni_babi_16}
\end{subfigure}
~
\begin{subfigure}[t]{0.30\textwidth}
\centering
\raisebox{-1em}{\includegraphics[width=1.00\textwidth,height=10em,keepaspectratio]{vsym_plots/qa06_single_uni.pdf}}
\vspace{-1em}
\caption{bAbI task 6. \vard{bathroom} is recognised as the only variable creating a one-to-many binding to capture the same information.}
\label{fig:uni_babi_06}
\end{subfigure}
~
\begin{subfigure}[t]{0.30\textwidth}
\centering
\includegraphics[width=0.63\textwidth,height=8em,keepaspectratio]{vsym_plots/dl01_ar2_to_ar1_uni.pdf}
\caption{Logical reasoning task 1. An arity 2 predicate is forced to bind with arity 1 creating a many-to-one binding.}
\label{fig:uni_dl_01_ar2_to_ar1}
\end{subfigure}
\caption{Variable bindings from line~\ref{algoline:uninet_uni_att} in Algorithm~\ref{algo:uninet}. Darker cells indicate higher attention values.}
\label{fig:uni}
\end{figure}
Attention maps from line~\ref{algoline:uninet_uni_att} in Algorithm~\ref{algo:uninet} reveal three main patterns: one-to-one, one-to-many or many-to-one bindings as shown in Figure~\ref{fig:uni} with more in Appendix~\ref{apx:further_results}. Figure~\ref{fig:uni_babi_16} captures what one might expect unification to look like where variables unify with their corresponding counterparts, e.g. \vard{bernhard} with brian and \vard{frog} with lion. However, occasionally the model can optimise to use less variables and \emph{squeeze} the required information into a single variable, for example by binding \vard{bathroom} to john and kitchen as shown in Figure~\ref{fig:uni_babi_06}. We believe this occurs due to the sparsity constraint on $\psi(s)$ encouraging the model to be as conservative as possible. Since the upstream network $f$ is also trained, it has the capacity to compensate for condensed or malformed unified representations; a possible option could be to freeze the upstream network while learning to unify. Finally, if there are more variables than needed as in Figure~\ref{fig:uni_dl_01_ar2_to_ar1}, we observe a many-to-one binding with \vard{w} and \vard{e} mapping to the same constant $q$. This behaviour begs the question how does the model differentiate between $p(q)$ and $p(q,q)$. We speculate the model uses the magnitude of $\psi(w) = 0.037$ and $\psi(e) = 0.042$ to encode the difference despite both variables unifying with the same constant.
\section{Related Work}
Learning an underlying general principle in the form of an invariant is often the means for arguing for generalisation in neural networks. For example, Neural Turing Machines~\cite{neuralturing} are tested on previously unseen sequences to support the view that the model might have captured the underlying pattern or algorithm. In fact, \cite{memnn} claim ``MemNNs can discover simple linguistic patterns based on verbal forms such as (X, dropped, Y), (X,
took, Y) or (X, journeyed to, Y) and can successfully generalise the meaning of their instantiations.'' However, this claim is based on the output of $f$ and unfortunately it is unknown whether the model has truly learned such a representation or indeed is utilising it. Our approach sheds light on this ambiguity and presents these linguistic patterns explicitly as invariants, ensuring their utility through $g$ without solely analysing the output of $f$ on previously unseen symbols. Although we associate these invariants with our existing understanding of the task to perhaps mistakenly anthropomorphise the machine, for example by thinking it has learned \vard{mary} as \emph{someone}, it is important to acknowledge that these are just symbolic patterns. They do not make our model, in particular $f$, more interpretable in terms of how these invariants are used or what they mean to the model. In these cases, our interpretations may not necessarily correspond to any understanding of the machine, relating to the Chinese room argument~\cite{chineseroom}.
Learning invariants by lifting ground examples is related to least common generalisation~\cite{leastcommongeneraliser} by which inductive inference is performed on facts~\cite{lcgfromfacts} such as generalising \textit{went(mary,kitchen)} and \textit{went(john,garden)} to \textit{went(X,Y)}. Unlike in a predicate logic setting, our approach allows for soft alignment and therefore generalisation between varying length sequences. Existing neuro-symbolic systems~\cite{neurosymbolic} focus on inducing rules that adhere to \emph{given} logical semantics of what variables and rules are. For example, $\delta ILP$~\cite{evans2018learning} constructs a network by rigidly following the given semantics of first-order logic. Similarly, Lifted Relational Neural Networks~\cite{liftedneuralnetworks} ground first-order logic rules into a neural network while Neural Theorem Provers~\cite{timntp} build neural networks using backward-chaining~\cite{russell2016artificial} on a given background knowledge base with templates.
However, the notion of a variable is pre-defined rather than learned with a focus on presenting a practical approach to solving certain problems, whereas our motivation stems from a cognitive perspective.
At first it may seem the learned invariants, Section~\ref{sec:analysis}, make the model more interpretable; however, this transparency is not of the model $f$ but of the data. The invariant captures patterns that potentially approximates the data generating distribution but we still do not know \emph{how} the model $f$ uses them upstream. Thus, from the perspective of explainable artificial intelligence (XAI)~\cite{xaisurvey}, learning invariants or interpreting them does not constitute an explanation of the reasoning model $f$ even though ``if \emph{someone} goes \emph{somewhere} then they are there'' might look like one. Instead, it can be perceived as causal attribution~\cite{xaisocialsci} in which someone being somewhere is attributed to them going there. This perspective also relates to gradient based model explanation methods such as Layer-Wise Relevance Propagation~\cite{layerwiserelevance} and Grad-CAM~\cite{gradcam,gradcamplus}. Consequently, a possible view on $\psi$, Section~\ref{sec:uni_nets}, is a gradient based usefulness measure such that a symbol utilised upstream by $f$ to determine the answer becomes a variable similar to how a group of pixels in an image contribute more to its classification. However, gradient based saliency methods have shown to be unreliable if based solely on visual assessment~\cite{sanitychecksforsaliencymaps}.
Finally, one can argue that our model maintains a form of counterfactual thinking~\cite{counterfactualthinking} in which soft unification $g$ creates counterfactuals on the invariant example to alter the output of $f$ towards the desired answer, Step~\ref{step:predict}. The question \emph{where Mary would have been if Mary had gone to the garden instead of the kitchen} is the process by which an invariant is learned through multiple examples during training. This view relates to methods of causal inference~\cite{pearlcausalinference,statisticscausalinference} in which counterfactuals are vital as demonstrated in structured models~\cite{pearlcausationcounterfactual}.
\section{Conclusion}
We presented a new approach for learning variables and lifting examples into invariants through the usage of soft unification. Application of our approach to five datasets demonstrates that Unification Networks perform comparatively if not better to existing architectures without soft unification while having the benefit of lifting examples into invariants that capture underlying patterns present in the tasks.
Since our approach is end-to-end differentiable, we plan to apply this technique to multi-modal tasks in order to yield multi-modal invariants for example in visual question answering.
\subsubsection*{Acknowledgements}
We would like to thank Murray Shanahan for his helpful comments, critical feedback and insights regarding this work. We also thank Anna Hadjitofi for proof-reading and improving clarity throughout the writing of the paper.
\section*{Broader Impact}
As it is with any machine learning model aimed at extracting patterns solely from data, learning invariants through soft unification is prone to being influenced by spurious correlations and biases that might be present in the data. There is no guarantee that even a clear, high accuracy invariant might correspond to a valid inference or casual relationship as discussed in Section~\ref{sec:analysis} with some mis-matching invariants presented in Appendix~\ref{apx:further_results}. As a result, if our approach succeeds in solving the task with an invariant, it does not mean that there is only pattern or in the case of failing to do so, a lack of patterns in the data. There has been recent work~\cite{invariantriskmin,invariantcausalpred} on tackling a different notion of invariance formed of features that are consistent (hence invariant) across different training dataset environments, to learn more robust predictors. Our method is instead targeted at research and researchers involved with combining cognitive aspects such as variable learning and assignment with neural networks under the umbrella of neuro-symbolic systems~\cite{neurosymbolic,neurosymsurvey}. A differentiable formulation of variables could accelerate the research of combining logic based symbolic systems with neural networks.
In summary, we regard this work as an experimental stepping stone towards better neuro-symbolic systems in the domain of artificial intelligence research.
\printbibliography
\clearpage
|
1,116,691,500,805 | arxiv | \section{Introduction}
There are many settings in random matrix theory for which the eigenvalues (assumed real) can be scaled in relation to the
matrix size in such a way that the limiting support is compact. This is referred to as a global scaling. As some concrete examples,
let $X$ be an $N \times N$ standard complex Gaussian matrix, and construct from this the Hermitian matrices
$H_1 = {1 \over 2} (X + X^\dagger)$ and $H_2 = X^\dagger X$. The set of matrices $H_1$ ($H_2$) are said to form the
Gaussian unitary ensemble (special case of the Laguerre unitary ensemble), and have joint eigenvalue probability
density function (PDF) proportional to
\begin{equation}\label{1.1}
\prod_{l=1}^N w(x_l) \prod_{1 \le j < k \le N} (x_k - x_j)^2, \qquad w(x) = \left \{ \begin{array}{cc} e^{-x^2}, & {\rm matrices} \: H_1 \\
e^{-x} \chi_{x > 0}, & {\rm matrices} \: H_2; \end{array} \right.
\end{equation}
see e.g.~\cite{Fo10,PS11}. Here $\chi_A = 1$ for $A$ true, $\chi_A = 0$ otherwise.
Scaling the eigenvalues $x_j \mapsto \sqrt{2N} x_j $ (matrices $H_1$) and $x_j \mapsto 4 N x_j $ (matrices $H_2$), it is a standard
result that as $N \to \infty$ the spectrum is supported on the intervals $(-1,1)$ and $(0,1)$ respectively. Among the endpoints of
the intervals of support, the point $x = 0$ for the global scaling of the matrices $H_2$ is special. Thus the region $x < 0$
to the other side of this endpoint has strictly zero eigenvalue density for all values of $N$, because $H_2$ is positive
definite. For this reason the endpoint $x = 0$ in this example is called a hard edge. The hard edge notion extends beyond the class of
matrix ensembles permitting a global scaling to include heavy tailed distributions --- an example of the latter is given in
Section \ref{S3.3} below. The essential point then is that the limiting eigenvalue density is nonzero for $x>0$, and strictly
zero for $x<0$.
In this paper our interest is in the approach to a limiting hard edge state for various ensembles of positive definite matrices.
A hard edge state refers to the statistical distribution formed when the eigenvalues are scaled to have nearest neighbour
spacing of order unity as $N \to \infty$. For the matrices $H_2$, or more generally the ensemble of matrices with
weight function
\begin{equation}\label{Lw}
w(x) = x^a e^{-x} \chi_{x > 0}
\end{equation}
(Laguerre weight, realised for $a = n - N \in \mathbb Z_{\ge 0}$ as the eigenvalue PDF of
matrices $X^\dagger X$ with $X$ an $n \times N$ complex standard Gaussian matrix) with parameter $a > - 1$, this takes place for the scaling of the eigenvalues
$x_j \mapsto x_j / 4N$, and gives rise to the hard edge state specified by the $k$-point correlations (see \cite[\S 7.2]{Fo10})
\begin{equation}\label{1.2}
\rho_{(k)}^{\rm hard}(x_1,\dots,x_k) = \det [ K^{\rm hard}(x_j, x_l;a) ]_{j,l=1}^k,
\end{equation}
where, with $J_a(u)$ denoting the Bessel function,
\begin{equation}\label{1.3}
K^{\rm hard}(x,y;a)
= {1 \over 4} \int_0^1 J_a(\sqrt{xt}) J_a(\sqrt{yt}) \, dt.
\end{equation}
For finite $N$ the $k$-point correlation function is defined in terms of the joint eigenvalue PDF, $P_N$ say,
according to
\begin{equation}\label{1.4}
\rho_{(k)}(x_1,\dots,x_k) = {N! \over (N - k)!} \int_{-\infty}^\infty dx_{k+1} \cdots \int_{-\infty}^\infty dx_N \, P_N(x_1,\dots, x_N).
\end{equation}
For eigenvalue PDFs of the form (\ref{1.1}), the correlation function (\ref{1.4}) admits the determinant evaluation (see e.g.~\cite[\S 5.1]{Fo10})
\begin{equation}\label{1.5a}
\rho_{(k)}(x_1,\dots,x_k) = \det [ K_N(x_j, x_l ) ]_{j,l=1}^k,
\end{equation}
where
\begin{align}\label{1.5b}
K_N(x,y) = \Big ( w(x) w(y) \Big )^{1/2} \sum_{n=0}^{N-1} {1 \over h_n} p_n(x) p_n(y) \nonumber \\
\end{align}
In (\ref{1.5b}) $\{p_n(x)\}$ refers to the set of orthogonal polynomials with respect to the weight function $w(x)$ ---
$p_n$ of degree $n$ and chosen to be monic for convenience --- with norm $h_n$,
\begin{equation}\label{1.5c}
\int_{-\infty}^\infty w(x) p_m(x) p_n(x) \, dx = h_n \delta_{m,n}.
\end{equation}
In the case of the Laguerre weight, the polynomials $p_n(x)$ are proportional to the Laguerre
polynomials $L_n^{(a)}(x)$.
Recently, attention has been given to the rate of convergence to the hard edge limiting kernel
(\ref{1.3}). One line of motivation came from a question posed by
Edelman, Guionnet and P\'ech\'e \cite{EGP16}. These authors, taking a viewpoint in numerical analysis, took up the problem of studying
finite $N$ effects in the hard edge scaling of the distribution of the smallest singular value of a (complex) standard Gaussian matrix.
With $E^{\rm LUE}(0;(0,s))$ denoting the probability that there are no eigenvalues in the interval $(0,s)$ of the LUE, it was conjectured
in \cite{EGP16} that
\begin{equation}\label{pk5}
E^{\rm LUE}(0;(0,s/(4N))) = E^{\rm hard}(0;(0,s)) + {a \over 2N} s {d \over ds} E^{\rm hard}(0;(0,s)) + O \Big ( {1 \over N^2} \Big ),
\end{equation}
where
$$
E^{\rm hard}(0;(0,s)) = \lim_{N \to \infty} E^{\rm LUE}(0;(0,s/(4N))),
$$
and thus \cite{Bo16,PS16},
\begin{equation}\label{pk7}
E^{\rm LUE}\bigg ( 0; \Big ( 0, {s \over 4 N + 2a} \Big ) \bigg ) = E^{\rm hard}(0;(0,s)) + O \Big ( {1 \over N^2} \Big ),
\end{equation}
which moreover is the optimal rate of convergence.
Subsequently Bornemann \cite{Bo16}
provided a proof of (\ref{pk5}) which involved extending the limit formula (\ref{1.2}) to the large $N$ expansion
\begin{align}\label{pk6}
{1 \over 4 N} K_N^{(L)} \Big ( {X \over 4 N} , {Y \over 4 N} \Big ) & = K^{\rm hard}(X,Y) + {1 \over N} {a \over 8} J_a(\sqrt{X}) J_a(\sqrt{Y}) + O \Big ( {1 \over N^2} \Big ) \nonumber \\
& = K^{\rm hard}(X,Y) + {1 \over N} {a \over 2} \bigg ( x {\partial \over \partial x} + y {\partial \over \partial y} + 1 \bigg ) K^{\rm hard}(X,Y) + O \Big ( {1 \over N^2} \Big ),
\end{align}
valid uniformly for $X,Y \in [0,s]$. In fact knowledge of (\ref{pk6}) is sufficient to establish (\ref{pk5}).
We remark too that analogous to (\ref{pk7}), it follows from (\ref{pk6}) that
\begin{equation}\label{pk8}
{1 \over 4 N + 2a } K_N^{(L)} \Big ( {X \over 4 N + 2a} , {Y \over 4 N + 2a} \Big ) = K^{\rm hard}(X,Y) + O \Big ( {1 \over N^2} \Big ),
\end{equation}
and this implies (\ref{pk7}).
Our aim in this work is to extend hard edge scaling results of the type (\ref{pk6}) to examples of a recently isolated structured class of random
matrices known as P\'olya ensembles \cite{KK16}. The definition of these ensembles, which include the Laguerre unitary ensemble, the Jacobi unitary ensemble,
products of these ensembles, and their Muttalib-Borodin generalisations, will be given in Section \ref{S2.1}. The benefit of the structures provided
by the P\'olya ensemble class is seen by our revision of the key formulas in Section \ref{S2.2}, where we also extend the theory by exhibiting differential
recurrences satisfied by the associated biothogonal pair, and a differential identity satisfied by the correlation kernel. In Section \ref{S2.3} we make note of some
asymptotic formulas relating to ratios of gamma functions which will be used in our subsequent large $N$ hard edge analysis. The latter is undertaken is Section \ref{S3},
starting with products of Laguerre ensembles, then the Laguerre Muttalib-Borodin ensemble, and finally products of Laguerre ensembles and their inverses,
with the latter including as a special case the Jacobi unitary ensemble.
The Jacobi unitary ensemble is specified by the eigenvalue PDF (\ref{1.1}) with weight
\begin{equation}\label{A.2}
x^a (1 - x)^b \chi_{0 < x < 1}.
\end{equation}
Our results of Section \ref{S3.3} imply that
\begin{align}\label{pk6a}
{1 \over 4 N^2} K_N^{(J)} \Big ( {X \over 4 N^2} , {Y \over 4 N^2} \Big ) & = K^{\rm hard}(X,Y) + { a+b \over 2 N} J_a(\sqrt{X}) J_a(\sqrt{Y}) + O \Big ( {1 \over N^2} \Big ) \nonumber \\
& = K^{\rm hard}(X,Y) + { a +b \over N} \bigg ( x {\partial \over \partial x} + y {\partial \over \partial y} + 1 \bigg ) K^{\rm hard}(X,Y) + O \Big ( {1 \over N^2} \Big ),
\end{align}
and thus
\begin{equation}\label{A.2a}
{1 \over 4 \tilde{N}^2} K_N^{(J)} \Big ( {X \over 4 \tilde{N}^2} , {Y \over 4 \tilde{N}^2} \Big ) \bigg |_{\tilde{N} = N + (a+b)/2} =
K^{\rm hard}(X,Y) + O \Big ( {1 \over N^2} \Big ).
\end{equation}
This gives an explanation for recent results in \cite{MMM19} relating to the large $N$ form of the distribution
of the smallest eigenvalue in the Jacobi unitary ensemble. In Appendix A large $N$ expansions of the latter
quantity are extended to all Jacobi $\beta$-ensembles with $\beta$ even.
\section{Preliminaries}
\subsection{P\'olya ensembles --- definitions}\label{S2.1}
The Vandermonde determinant identity tells us that
\begin{equation}\label{2.1}
\det [ x_k^{j-1} ]_{j,k=1}^N = \det [p_{j-1}(x_k) ]_{j,k=1}^{N} = \prod_{1 \le j < k \le N} (x_k - x_j),
\end{equation}
where $\{p_{l}(x)\}_{l=0}^{N-1}$ are arbitrary monic orthogonal polynomials, $p_l$ of degree $l$.
A generalisation of (\ref{1.1}) is therefore an eigenvalue PDF proportional to
\begin{equation}\label{2.2}
\det [p_{j-1}(x_k) ]_{j,k=1}^{N} \det [ w_{j-1}(x_k) ]_{j,k=1}^N
\end{equation}
for some polynomials $\{p_{l}(x)\}_{l=0}^{N-1}$ and functions
$\{ w_j(x) \}_{j=0}^{N-1}$ --- note though that in general there is no guarantee
(\ref{2.2}) will be positive. In \cite{KZ14} eigenvalue PDFs (\ref{2.2}) were given the name
polynomial ensembles.
In \cite{KK16,KK19} a further specialisation of (\ref{2.2}),
\begin{equation}\label{2.3}
\det [p_{j-1}(x_k) ]_{j,k=1}^{N} \det \bigg [ \Big ( - x_k {\partial \over \partial x_k} \Big )^{j-1} w(x_k) \bigg ]_{j,k=1}^N,
\end{equation}
was proposed. Assuming all the eigenvalues are positive, it was shown that this class of eigenvalue
PDF is closed under multiplicative convolution. At first PDFs of the form (\ref{2.3}) were referred to as
polynomial ensembles of derivative type, but subsequently with the requirement that they be non-negative,
it was pointed out in \cite{FKK17} that it is more apt to use the term P\'olya ensemble. The invariance of a
determinant under the elementary row operation of adding one multiple of a row to another shows
\begin{align}\label{2.4}
\det \bigg [ \Big ( - x_k {\partial \over \partial x_k} \Big )^{j-1} w(x_k) \bigg ]_{j,k=1}^N
& = \det \bigg [ \prod_{l=1}^{j-1} \Big ( - x_k {\partial \over \partial x_k} - l \Big ) w(x_k) \bigg ]_{j,k=1}^N \nonumber \\
& = \det \bigg [ {\partial^{j-1} \over \partial x_k^{j-1}} \Big ( (-x_k)^{j-1} w(x_k) \Big ) \bigg ]_{j,k=1}^N.
\end{align}
In relation to the second line, note that it is in fact an equality that
\begin{equation}\label{2.5}
\prod_{l=1}^{j-1} \Big ( - x {\partial \over \partial x} - l \Big ) w(x) = {d^{j-1} \over d x^{j-1}} \Big ( (-x)^{j-1} w(x) \Big ).
\end{equation}
The differential operator on the RHS of (\ref{2.5}) reveals that the Laguerre unitary ensemble fits the framework of
P\'olya ensembles. Thus choosing $w(x)$ to be given by (\ref{Lw}), the Rodrigues formula for the Laguerre polynomials
tells us that
\begin{equation}\label{2.6}
{d^{j-1} \over d x^{j-1}} \Big ( (-x)^{j-1} w(x) \Big ) = (-1)^{j-1} (j-1)! w(x) L_{j-1}^{(a)}(x),
\end{equation}
and so, up to proportionality, (\ref{2.3}) reduces to
\begin{equation}\label{2.7}
\prod_{l=1}^N x_l^a e^{-x_l} \det [ p_{j-1}(x_k) ]_{j,k=1}^N \det [ L_{j-1}^{(a)}(x_k) ]_{j,k=1}^N.
\end{equation}
In view of (\ref{2.1}), this corresponds to the eigenvalue PDF for the Laguerre unitary ensemble. The advantage in working
within the P\'olya ensemble framework is that it reveals a mechanism to obtain the asymptotic expansion of the
correlation kernel (\ref{1.5b}) at the hard edge, which applies at once to a much wider class of random matrix ensembles.
The reason for this are certain general structural formulas applicable to all P\'olya ensembles. These will be revised next.
\subsection{P\'olya ensembles --- biorthogonal system and correlation kernel}\label{S2.2}
It is standard in random matrix theory that the ensembles (\ref{2.2}) are determinantal, meaning that the $k$-point correlation
functions have the form (\ref{1.5a}). Moreover, if the polynomials $\{p_l(x) \}_{l=0}^N$ and the functions $\{q_j(x)\}_{j=0}^N$ --- the latter
chosen from ${\rm span} \, \{w_j(x) \}_{j=0}^{N}$ --- have the
biorthogonal property
\begin{equation}\label{2.8}
\int_{-\infty}^\infty p_m(x) q_n(x) \, dx = \delta_{m,n},
\end{equation}
then the correlation kernel has the simple form
\begin{equation}\label{2.9}
K_N(x,y) = \sum_{j=0}^{N-1} p_{j}(x) q_{j}(y);
\end{equation}
see e.g.~\cite[\S 5.8]{Fo10}.
While in general computation of the LU (lower/ upper triangular) decomposition of a certain inverse matrix
used to construct the biorthogonal
functions (see e.g.~\cite[Proof of Prop.~5.8.1]{Fo10}), this cannot be expected to result in
a tractable formula for (\ref{2.9}), permitting large $N$ analysis, without further structures.
It is at this stage that the utility of P\'olya ensembles shows itself: special functional forms for the
biorthogonal system hold true, and moreover there is a summed up form of the kernel
as an integral analogous to (\ref{1.3}), which together facilitate a large $N$ analysis.
The formulas, which are due to Kieburg and K\"osters \cite{KK16}, involve the Mellin transform
of the weight $w$ in (\ref{2.3}),
\begin{equation}\label{Me}
\mathcal M[w](s) := \int_0^\infty y^{s-1} w(y) \, dy.
\end{equation}
One has that the polynomials $\{p_l(x)\}_{l=0}^N$ in the biorthogonal pair $\{p_j, q_k\}$ are specified by
\begin{equation}\label{Me1}
p_n(x) = (-1)^n n! \mathcal M[w](n+1) \sum_{j=0}^n{ (-x)^{j} \over j! (n-j)! \mathcal M[w](j+1) } ,
\end{equation}
and that the functions $\{q_l(x)\}_{l=0}^N$ --- chosen from the span of the functions specifying the columns in (\ref{2.3}) ---
are specified by the Rodrigues type formula
\begin{equation}\label{Me2}
q_n(x) = {1 \over n! \mathcal M[w](n+1)} {d^n \over d x^n} \Big((-x)^n w(x)\Big).
\end{equation}
Moreover, the correlation kernel can be written in a form generalising the final expression in (\ref{1.3}),
\begin{equation}\label{Me3}
K_N(x,y) = - N { \mathcal M[w](N+1) \over \mathcal M[w](N)} \int_0^1 p_{N-1}(xt) q_N(yt) \, dt.
\end{equation}
In \cite{KK16} the integral form (\ref{Me3}) of the correlation kernel was derived by first converting
(\ref{Me1}) and (\ref{Me2}) to integral forms, which allow for the summation to be carried out
in closed form. The identification with the RHS of (\ref{Me3}) then follows after some manipulation.
In a special case this strategy was first given in \cite{KZ14}.
An alternative method of derivation is also possible, as we will now show, which involves first
identifying differential recurrences satisfied by each of the $p_n(x)$ and $q_n(x)$.
(We remark that other examples of differential recurrences can be found in a number of recent studies in random matrix
theory \cite{Ku19,FK19,FK20,Fo20a,Fo20b}.)
\begin{proposition}\label{PR1}
Let $p_n(x)$ and $q_n(x)$ be specified by (\ref{Me1}) and (\ref{Me2}).
These functions satisfy the differential recurrences
\begin{align}
x {d \over dx} p_n(x) & = n p_n(x) + n {\mathcal M [w](n+1) \over \mathcal M [w] (n) } p_{n-1}(x) \label{PQ1} \\
x {d \over dx} q_n(x) & = - {(n+1) \mathcal M[w](n+2) \over \mathcal M[w](n+1)} q_{n+1}(x) + (n+1) q_n(x).\label{PQ2}
\end{align}
A corollary of these recurrences is the differential identity
\begin{equation}\label{Me4}
\Big ( x{\partial \over \partial x} + y{\partial \over \partial y} + 1 \Big ) K_N(x,y) = - N { \mathcal M[w](N+1) \over \mathcal M[w](N)} p_{N-1}(x) q_N(y),
\end{equation}
which implies (\ref{Me3}).
\end{proposition}
\begin{proof}
From the formula (\ref{Me1}),
$$
x {d \over dx} p_n(x) = (-1)^n n! \mathcal M[w](n+1) \sum_{j=0}^n (-1)^{j} {j \over j! (n-j)! \mathcal M[w](j+1) } x^j.
$$
Rewrite the $j$ in the denominator of this expression as $n - (n - j)$, and use this to decompose the sum into two.
Upon some simple manipulation, the identity (\ref{PQ1}) results.
According to (\ref{2.5}), the formula (\ref{Me2}) can be rewritten
$$
q_n(x) = {1 \over n! \mathcal M[w](n+1)} \prod_{l=1}^{n} \Big ( - x {\partial \over \partial x} - l \Big ) w(x).
$$
Acting on both sides with $- x {d \over dx} - (n+1)$ shows
$$
\Big ( - x {d \over dx} - (n+1) \Big ) q_n(x) = {(n+1) \mathcal M[w](n+2) \over \mathcal M[w](n+1)} q_{n+1}(x).
$$
This gives (\ref{PQ2}).
With the differential recurrences (\ref{PQ1}) and (\ref{PQ2}) established, we can use them in the expression (\ref{2.9}) to give
\begin{multline}
\Big ( x{\partial \over \partial x} + y{\partial \over \partial y} \Big ) K_N(x,y) \\
= \sum_{n=0}^{N-1} \Big ( n p_n(x) + n {\mathcal M [w](n+1) \over \mathcal M [w] (n) } p_{n-1}(x) \Big )
\Big ( - {(n+1) \mathcal M[w](n+2) \over \mathcal M[w](n+1)} q_{n+1}(y) + (n+1) q_n(y) \Big ).
\end{multline}
Simple manipulation reduces this to (\ref{Me4}).
In (\ref{Me4}) scale $x$ and $y$ by writing as $xt$ and $yt$ respectively. The LHS of (\ref{Me4}) can then
be written
\begin{equation}\label{2.17a}
{d \over dt} t K_N(tx,ty) = - N { \mathcal M[w](N+1) \over \mathcal M[w](N)} p_{N-1}(tx) q_N(ty).
\end{equation}
Integrating both sides from $0$ to $1$, on the LHS noting $\lim_{t \to 0^+} t K_N(tx,ty) = 0$
as follows from (\ref{2.9}), reclaims (\ref{Me3}).
\end{proof}
\begin{remark}\label{R2}
We show in Appendix B how (\ref{2.17a}), combined with a recurrence formula of fixed depth of
$t p_{N-1}(t)$ known to hold for a number of the specific P\'olya ensembles considered
in Section \ref{S3}, provides a combinatorial based method to compute the leading
large $N$ form of the moments of the spectral density.
\end{remark}
\subsection{Asymptotics of ratios of gamma function}\label{S2.3}
The gamma function $\Gamma(z)$ is one of the most commonly occurring of special functions \cite{andrews99}, analytic in the
complex plane except for poles at 0 and the negative integers.
Since $\Gamma(z+1) = z \Gamma(z)$ and $\Gamma(1) = 1$, for $n$ a non-negative integer
\begin{equation}\label{0.1}
\Gamma(n+1) = n!,
\end{equation}
and so gives meaning to the factorial for
general complex $n$. Historically \cite{Pe24} Stirling's formula for the gamma function is the large $n$ approximation to the
factorial $n! \approx \sqrt{2 \pi } n^{n + 1/2} e^{-n}$, later extended to the asymptotic series \cite{WW65}
\begin{equation}
n! = \sqrt{2 \pi n} \Big ( {n \over e} \Big )^n \bigg ( 1 + {1 \over 12 n} + {1 \over 288 n^2} + O \Big ( {1 \over n^3} \Big ) \bigg ).
\end{equation}
Using (\ref{0.1}) and
truncating this asymptotic series at $O(1/n)$ leads to the large $|z|$ asymptotic expansion \cite{tricomi51}
\begin{align}\label{eq1}
\frac{\Gamma(z+a)}{\Gamma(z+b)}=z^{a-b}\left(
1+\frac{1}{2z}(a-b)(a+b-1)+O(z^{-2})
\right),\quad |z|\to\infty
\end{align}
valid for $|{\rm arg} \, z| < \pi$ and $a,b$ fixed. Furthermore,
specify $(u)_\alpha := \Gamma(u+\alpha)/\Gamma(u)$, which for $\alpha$ a positive integer corresponds to the
product $(u)_\alpha = (u) (u+1) \cdots (u+ \alpha - 1)$. From this definition, and under the assumption that
$\alpha$ is a positive integer, we see
\begin{align}\label{eq2}
(-N+k)_{\alpha}= (-1)^\alpha {\Gamma(N - k + 1) \over \Gamma(N - k + 1 - \alpha) } = (-N)^{\alpha}\left(
1-\frac{\alpha(2k+\alpha-1)}{2N}+O(N^{-2})
\right),\quad N\to\infty,
\end{align}
where the large $N$ form follows from (\ref{eq1}).
Our analysis of the rate of convergence for hard edge scalings will have use for both
(\ref{eq1}) and (\ref{eq2}).
\section{Hard edge scaling to $O(1/N)$ for some P\'olya ensembles}\label{S3}
\subsection{Products of Laguerre ensembles}\label{S3.1}
The realisation of the Laguerre unitary ensemble with $a = n - N$ noted below (\ref{Lw}) can equivalently be expressed
as being realised by the squared singular values of an $n \times N$ standard complex Gaussian matrix. A natural
generalisation, first considered in \cite{AKW13,AIK13}, is to consider the squared singular values of the product
of say $M$ rectangular standard complex Gaussian matrices (assumed to be of compatible sizes). Since each ensemble
in the product is individually a P\'olya ensemble, the closure property
of P\'olya ensembles under multiplicative convolution from \cite{KK16} tells us that the product ensemble can be formed by simply replacing $w(x)$
in (\ref{2.3}) by
\begin{equation}\label{W1}
w^{(M)}(x) := \int_0^\infty dx_1 \cdots dx_M \, \delta \Big ( x - \prod_{j=1}^M x_j \Big ) \prod_{l=1}^M w_l(x_l), \quad w_j(x) = {1 \over \Gamma(a_j + 1)} x^{a_j} e^{-x} .
\end{equation}
For the Mellin transform we have the factorised gamma function evaluation
\begin{equation}\label{W2}
\mathcal M[w^{(M)}](s) = \prod_{j=1}^M {\Gamma(a_j + s) \over \Gamma(a_j + 1)}.
\end{equation}
The formula for the inverse Mellin transform then gives
\begin{align}\label{W3}
w^{(M)}(x) & = \Big ( \prod_{j=1}^M {1 \over \Gamma(a_j + 1)} \Big ) {1 \over 2 \pi i } \int_{c - i \infty}^{c + i \infty} \prod_{j=1}^M \Gamma(a_j - s) \, x^{s} \, ds \nonumber \\
& = \prod_{j=1}^M {1 \over \Gamma(a_j + 1)} \,
\MeijerG{M}{0}{0}{M}{-}{a_1,\ldots,a_M}{x}.
\end{align}
Here $c$ is any positive real number, and $G^{0,M}_{M,0}$ denotes a particular Meijer G-function; see \cite{MSH09}.
Substituting (\ref{W2}) in (\ref{Me1}) and (\ref{W3}) in (\ref{Me2}) shows \cite{AIK13}
\begin{align}\label{W4}
p_n(x) & = (-1)^n n! \prod_{j=1}^M \Gamma(a_j + n + 1) \sum_{j=0}^n { (-x)^j \over j! (n - j)! \prod_{l=1}^M (a_l + 1)_j} \nonumber\\
& =
(-1)^n \prod_{j=1}^M {\Gamma(a_j + n + 1) \over \Gamma(a_j + 1)} \,
{}_1 F_M \bigg ( \begin{array}{cc} -n \\ a_1 + 1,\dots, a_M + 1 \end{array} \Big | x \bigg ),
\end{align}
with $ {}_1 F_M$ the notation for the particular hypergeometric series, and
\begin{align}\label{W5}
q_n(x) & = {(-1)^n \over n!} \prod_{j=1}^M {1 \over \Gamma(a_j + n + 1)} {1 \over 2 \pi i }
\int_{c - i \infty}^{c + i \infty} {\Gamma(n+s+1) \over \Gamma(s+1) } \prod_{j=1}^M \Gamma(a_j - s) \, x^{s} \, ds \nonumber \\
& = {(-1)^n \over n!} \prod_{j=1}^M {1 \over \Gamma(a_j +n+ 1)} \,
\MeijerG{M}{1}{1}{M+1}{-n}{a_1,\ldots,a_M,0}{x}.
\end{align}
According to (\ref{Me4}) and (\ref{Me3}), $K_N(x,y)$ is fully determined by $p_{N-1}(x)$ and $q_N(y)$. Since our aim is to
expand $K_N(x,y)$ for large $N$ with hard edge scaled variables, it suffices then to compute the hard edge expansion
of these particular biorthogonal functions.
\begin{proposition}\label{P3.1}
Denote
\begin{equation}\label{W6}
{}_0 F_M \bigg ( \begin{array}{cc} - \\ a_1 + 1,\dots, a_M + 1 \end{array} \Big | -x \bigg ) = \sum_{j=0}^\infty {(-x)^j \over j! \prod_{s=1}^M (a_s + 1)_j},
\end{equation}
as conforms with standard notation in the theory of hypergeometric functions. We have
\begin{multline}\label{W7}
{}_1 F_M \bigg ( \begin{array}{cc} -N + 1 \\ a_1 + 1,\dots, a_M + 1 \end{array} \Big | {x \over N} \bigg ) \\
= \bigg ( 1 - {1 \over 2 N} \Big ( x {d \over dx} +
\Big ( x {d \over dx} \Big )^2 \Big ) \bigg ) \, {}_0 F_M \bigg ( \begin{array}{cc} - \\ a_1 + 1,\dots, a_M + 1 \end{array} \Big | -x \bigg ) + O \Big ( {1 \over N^2} \Big ).
\end{multline}
Also
\begin{multline}\label{W8}
{1 \over N!} \MeijerG{M}{1}{1}{M+1}{-N}{a_1,\ldots,a_M,0}{{x \over N}} \\
= \bigg ( 1 + {1 \over 2 N} \Big ( x {d \over dx} +
\Big ( x {d \over dx} \Big )^2 \Big ) \bigg ) \MeijerG{M}{0}{1}{M+1}{-}{a_1,\ldots,a_M,0}{{x }} + O \Big ( {1 \over N^2} \Big ).
\end{multline}
In both (\ref{W7}) and (\ref{W8}) the bound on the remainder holds uniformly for $x \in [0,s]$, for any fixed $s \in \mathbb R_+$.
\end{proposition}
\begin{proof}
In the summation (\ref{W4}) defining the LHS of (\ref{W7}) the only $N$ dependence is the factor
$$
{(-N + 1)_j \over N^j} = (-1)^j \bigg ( 1 -\frac{j(j+1)}{2N} + O \Big ( {1 \over N^2} \Big )\bigg ),
$$
where the expansion follows from (\ref{eq2}). This result, valid for fixed $j$, can nonetheless be substituted in
the summation since the factor in the summand $(-N+1)_j/j! N^j$ is a rapidly decaying function of $j$. Doing this shows
\begin{multline*}
\sum_{j=0}^\infty {(-x)^j \over j! \prod_{s=1}^M (a_s + 1)_j} \bigg ( 1 - \frac{j(j+1)}{2N} + O \Big ( {1 \over N^2} \Big ) \bigg ) \\ =
\bigg ( 1 - {1 \over 2 N} \Big ( x {d \over dx} + \Big ( x {d \over dx} \Big )^2 \Big ) \bigg ) \, {}_0 F_M \bigg ( \begin{array}{cc} - \\ a_1 + 1,\dots, a_M + 1 \end{array} \Big | -x \bigg ) + O \Big ( {1 \over N^2} \Big ),
\end{multline*}
with the bound on the RHS uniform for $x \in [0,s]$.
In relation to (\ref{W8}), after multiplying through the prefactor $1/ N!$ inside the integrand of the integral (\ref{W5}) defining the LHS, we see
the only dependence on $N$ is the factor
$$
{\Gamma(N + s +1) \over N^s \Gamma(N+1)} = 1 + {s(s+1) \over 2N} + O \Big ( {1 \over N^2} \Big ),
$$
where the expansion follows from (\ref{eq2}). The result (\ref{W8}) now follows by noting
\begin{multline*}
{1 \over 2 \pi i } \int_{c - i \infty}^{c + i \infty} {1 \over \Gamma(s+1) } \prod_{j=1}^M \Gamma(a_j - s) \bigg ( 1 + {s(s+1) \over 2N} + O \Big ( {1 \over N^2} \Big ) \bigg ) \, x^{s} \, ds
\\ = \bigg ( 1 + {1 \over 2 N} \Big ( x {d \over dx} +
\Big ( x {d \over dx} \Big )^2 \Big ) \bigg ) \MeijerG{M}{0}{1}{M+1}{-}{a_1,\ldots,a_M,0}{x } + O \Big ( {1 \over N^2} \Big ),
\end{multline*}
and arguing in relation to the error term as above.
\end{proof}
Substituting the results of Proposition \ref{P3.1} in (\ref{W4}) with $n=N-1$ and in (\ref{W5}) with $n=N$, then substituting in (\ref{Me3}) shows
\begin{multline}\label{FG}
{1 \over N} K_N(x/N,y/N) \\
= \int_0^1 \bigg ( 1 - {1 \over 2 N} \Big ( x {d \over dx} +
\Big ( x {d \over dx} \Big )^2 \Big ) \bigg ) F(xt)
\bigg ( 1 + {1 \over 2 N} \Big ( y {d \over dy} +
\Big ( y {d \over dy} \Big )^2 \Big ) \bigg ) G(yt) \, dt + O \Big ( {1 \over N^2} \Big ),
\end{multline}
where $F$ denotes the function $ {}_0 F_M$ in (\ref{W7}) and $G$ denotes the function $G_{1,M+1}^{M,0}$ in (\ref{W8}).
Note that the error bound from asymptotic forms in Proposition \ref{P3.1} persist because the error bounds therein
are uniform with respect to $x,y$ when these variables are restricted to a compact set; see \cite{Bo16} on this
point in relation to (\ref{pk5}).
Independent of the details of these functions, the structure (\ref{FG}) permits simplification.
\begin{proposition}\label{P3.2}
The expression (\ref{FG}) has the simpler form
\begin{equation}\label{FG1}
{1 \over N} K_N(x/N,y/N)
= \int_0^1 F(xt) G(yt) \, dt - {1 \over 2N} \Big ( x {\partial \over \partial x} - y {\partial \over \partial y} \Big ) F(x) G(y) + O \Big ( {1 \over N^2} \Big ).
\end{equation}
\end{proposition}
\begin{proof}
At order $1/N$ the RHS of (\ref{FG}) reads
$$
- {1 \over 2 N} \int_0^1 G(yt) \Big ( x {d \over dx} + \Big ( x {d \over dx} \Big )^2 \Big ) F(xt) \, dt
+ {1 \over 2 N} \int_0^1 F(xt) \Big ( y {d \over dy} + \Big ( y {d \over dy} \Big )^2 \Big ) G(yt) \, dt.
$$
In this expression, both the derivatives with respect to $x$, and the derivatives with respect to $y$ can
be replaced by derivatives with respect to $t$. Performing one integration by parts for each of
the terms involving the second derivative, (\ref{FG1}) results.
\end{proof}
Recalling (\ref{1.5a}), we see from (\ref{FG1}) that in general for products of Laguerre unitary ensembles, the pointwise rate of
convergence to the hard edge limiting $k$-point correlation is $O(1/N)$. On the other hand, as noted in the text around
(\ref{pk6}), earlier works
\cite{EGP16,Bo16,PS16,HHN16,FT19} have demonstrated that for the Laguerre unitary ensemble itself (the case $M = 1$),
with the hard edge scaling variables as used in (\ref{FG1}), and with the Laguerre parameter $a=0$, the convergence
rate is actually $O(1/N^2)$. Moreover, these same references found that the $O(1/N^2)$ rate holds for general
Laguerre parameter $a>-1$ if each $N$ on the LHS of (\ref{FG1}) is replaced by $N+a/2$.
From the viewpoint of (\ref{FG1}), the special feature of the case $M=1$ is that then $F$ and $G$ are related
by
\begin{equation}\label{FGa}
G(x) = x^a F(x),
\end{equation}
as follows from the final paragraph of Section \ref{S2.1}. The term $O(1/N)$ in (\ref{FG1})
can therefore be written to involve only $F$,
\begin{equation}\label{FG2}
- {1 \over 2N} y^a \Big ( -a F(x) F(y) + \Big ( x {\partial \over \partial x} - y {\partial \over \partial y} \Big ) F(x) F(y) \Big ) \Big |_{M=1} .
\end{equation}
Substituting in (\ref{FG1}), then substituting the result in (\ref{1.5a}), we factor $x_l$ from each column to
effectively remove $y^a$ from (\ref{FG2}). The term involving partial derivatives in the latter is then antisymmetric,
and so does not contribute to an expansion of the determinant at order $1/N$, telling us that
\begin{align}\label{FG3}
&{1 \over N^k} \rho_{(k)}\Big ( {x_1 \over N} ,\dots, {x_k \over N} \Big ) \Big |_{M=1}
\nonumber \\
& \quad = \prod_{l=1}^k x_l^a \det \Big [ \Big ( \int_0^1 t^a F(x_jt) F(x_l t) \, dt + {a \over 2N}F(x_j) F(x_l) \Big ) \Big |_{M=1} \Big ]_{j,l=1}^k
+ O \Big ( {1 \over N^2} \Big ) \nonumber \\
& \quad = \det \Big [ \Big ( \int_0^1 \tilde{F}(x_jt) \tilde{F}(x_l t) \, dt + {a \over 2N}\tilde{F}(x_j) \tilde{F}(x_l) \Big ) \Big |_{M=1} \Big ]_{j,l=1}^k
+ O \Big ( {1 \over N^2} \Big ),
\end{align}
where $\tilde{F}(x) = x^{a/2} F(x)$, and the second equality follows from the first by multiplying each row $j$ by $x_j^{a/2}$ and each column
$k$ by $x_k^{a/2}$. In this latter form the kernel is symmetric. Comparison with (\ref{1.2}) and (\ref{1.3}) then
shows
$$
\tilde{F}(x) \Big |_{M=1} = J_a( \sqrt{4x} ), \qquad \int_0^1 \tilde{F}(xt) \tilde{F}(yt) \, dt \Big |_{M=1} = 4 K^{\rm hard}(4x,4y)
$$
(the reason for the factors of 4 comes from the choice of hard edge scaling $x \mapsto x/4N$ in (\ref{1.2}), (\ref{1.3})
rather than $x \mapsto x/N$ as in (\ref{FG3})). This is in agreement with the references cited above relating
to the hard edge expansion of the Laguerre unitary ensemble correlation kernel up to and including the
$O(1/N)$ term, and so has the property
that upon replacing $N$ by $N+a/2$ on the LHS, the convergence has the optimal rate of $O(1/N^2)$.
\subsection{Laguerre Muttalib-Borodin model}\label{S3.2}
The Laguerre Muttalib-Borodin model \cite{Mu95,Bo98,FW15,Zh15}, defined as the eigenvalue PDF
proportional to
\begin{equation}\label{MB1}
\prod_{l=1}^N x_l^a e^{-x_l} \prod_{1 \le j < k \le N} (x_j - x_k ) (x_j^\theta - x_k^\theta),
\end{equation}
with each $x_l$ positive is, with $\theta = M$ and upon the change of variables $x_l \mapsto x_l^{1/\theta}$,
known to be closely related to the product of $M$ matrices from the LUE. Specifically, there is a choice
of the Laguerre parameters $a_l$ for which the joint PDF of the latter reduces to this transformation of
(\ref{MB1}) \cite{KS14}. In particular, it follows that in the case $\theta = M$ at least, (\ref{MB1}) corresponds to a
P\'olya ensemble.
In fact it is known from \cite{KK16} that (\ref{MB1}) is an example of a P\'olya ensemble for general $\theta > 0$.
We can thus make use of the theory of Section \ref{S2.2} to study the hard edge expansion of the correlation
kernel.
The normalised weight function corresponding to (\ref{MB1}) after the stated change of variables is
\begin{equation}\label{MB2}
w^{({\rm MB}, L)}(x) = {1 \over \theta \Gamma(a+1)} x^{-1 + (a+1)/\theta} e^{- x^{1/\theta}},
\end{equation}
which has Mellin transform
\begin{equation}\label{MB3}
{\mathcal M} [ w^{({\rm MB}, L)} ](s) = {\Gamma(\theta(s-1) + a + 1) \over \Gamma (a + 1) }.
\end{equation}
Hence the polynomials $p_n(x)$ in (\ref{Me1}) read
\begin{equation}\label{MB4}
p_n^{({\rm MB}, L)}(x) = (-1)^n \Gamma(\theta n + a + 1) \sum_{j=1}^n { (-n)_j x^j \over j! \Gamma(\theta j + a + 1)},
\end{equation}
first identified in the work of Konhauser \cite{Ko67}.
Taking the inverse Mellin transform of (\ref{MB3}) gives the integral form of the weight,
$$
w^{({\rm MB}, L)}(x) = {1 \over \Gamma(a+1)} {1 \over 2 \pi i} \int_{c - i \theta}^{c + i \theta} \Gamma(-\theta (s + 1) + a + 1) x^s \, ds ,
$$
valid for $c > 0$. Using this in (\ref{Me2}) shows
\begin{equation}\label{MB5}
q_n^{({\rm MB}, L)}(x) = {(-1)^n \over n! \Gamma(\theta n + a + 1)}
{1 \over 2 \pi i } \int_{c - i \theta}^{c + i \theta} { \Gamma(s+n+1) \over \Gamma(s) } \Gamma(-\theta (s + 1) + a + 1) x^s \, ds.
\end{equation}
The dependence on $n$ in the summand of (\ref{MB4}) and integrand of (\ref{MB5}) is precisely the same as in
(\ref{W4}) and (\ref{W5}) respectively. Applying the working of Proposition \ref{P3.1} then gives hard
edge asymptotics that is structurally identical to $p_n(x)$ and $q_n(x)$ for products of Laguerre ensembles.
From this we conclude a formula structurally identical to (\ref{FG1}) for the hard edge asymptotics of the kernel.
\begin{proposition}
Define
$$
\tilde{p}_n^{({\rm MB}, L)}(x) = {(-1)^n \over \Gamma(\theta n + a + 1)} p_n^{({\rm MB}, L)}(x), \qquad
\tilde{q}_n^{({\rm MB}, L)}(x) = {(-1)^n \Gamma(\theta n + a + 1)} q_n^{({\rm MB}, L)}(x).
$$
Also define
$$
F^{({\rm MB},L)}(x) = \sum_{j=0}^\infty {x^j \over j! \Gamma(\theta j + a + 1)}, \qquad
G^{({\rm MB},L)}(x) = {1 \over 2 \pi i} \int_{c - i \theta}^{c + i \theta} { \Gamma(-\theta (s + 1) + a + 1) \over \Gamma (s) } x^s \, ds.
$$
We have
\begin{align*}
\tilde{p}_{N-1}^{({\rm MB}, L)}(x/N) & = \bigg ( 1 - {1 \over 2 N} \Big ( x {d \over dx} + \Big ( x {d \over dx} \Big )^2 \Big ) \bigg ) F^{({\rm MB},L)}(x) + O \Big ( {1 \over N^2} \Big ) \\
\tilde{q}_N^{({\rm MB}, L)}(x/N) & = \bigg ( 1 + {1 \over 2 N} \Big ( x {d \over dx} + \Big ( x {d \over dx} \Big )^2 \Big ) \bigg ) G^{({\rm MB},L)}(x) + O \Big ( {1 \over N^2} \Big ) ,
\end{align*}
and furthermore
\begin{multline*}
{1 \over N} K_N^{({\rm MB}, L)}(x/N,y/N) \\
= \int_0^1 F^{({\rm MB}, L)}(xt) G^{({\rm MB}, L)}(yt) \, dt - {1 \over 2N} \Big ( x {\partial \over \partial x} - y {\partial \over \partial y} \Big ) F^{({\rm MB}, L)}(x) G^{({\rm MB}, L)}(y) + O \Big ( {1 \over N^2} \Big ).
\end{multline*}
\end{proposition}
As in the discussion following Proposition \ref{P3.1}, this tells us that the rate of convergence to the hard edge scaled limit of the
$k$-point correlation is $O(1/N)$, with the case $\theta = 1$ (corresponding to the LUE) an exception, where by appropriate choice
of scaling variables, the rate is $O(1/N^2)$.
\subsection{Products of Laguerre ensembles and inverse Laguerre ensembles}\label{S3.3}
In the guise of the square singular values for the product of complex Gaussian matrices, times the inverse of a further
product of complex Gaussian matrices, the study of the eigenvalues of a product of Laguerre ensembles and inverses
was initiated in \cite{Fo14}. This was put in the context of P\'olya ensembles in \cite{KS14}. Moreover, in the case that there
are equal numbers of matrices and inverse matrices, such product ensembles can be related to a single weight function,
as we will now demonstrate. The essential point
is that the eigenvalues of $X_{b_1}^{-1} X_{a_1}$, where $X_{a_1}$, $X_{b_1}$ has eigenvalues from the Laguerre unitary ensemble
has eigenvalue PDF proportional to (see e.g.~\cite[Exercises 3.6 q.3]{Fo10})
\begin{equation}\label{Kq1}
\prod_{l=1}^N {x_l^{a_1} \over (1+ x_l)^{b_1 + a_1 + 2N}} \prod_{1 \le j < k \le N} ( x_k - x _j)^2
\end{equation}
and that this in turn is an example of a P\'olya ensemble (\ref{2.3}) with
\begin{equation}\label{Kq2}
w^{(\rm I)}(x) = {x^{a_1} \over (1 + x)^{b_1 + a_1+N+1}} \chi_{x > 0}
\end{equation}
(here the superscript (I) indicates `inverse').
Structurally, a key distinguishing feature relative to the weight (\ref{Lw}) is that (\ref{Kq2}) depends on $N$.
After normalising (\ref{Kq2}), proceeding as in the derivation of (\ref{W1}) shows the weight function
for the P\'olya ensemble of the corresponding product ensemble is
\begin{equation}\label{Kq3}
\mathcal M [ w^{({\rm I}, M)}](s) = \prod_{l=1}^M { \Gamma(a_l+s) \Gamma(b_l + N + 1- s) \over \Gamma(a_l+1) \Gamma(b_l + N) }.
\end{equation}
Use of (\ref{Kq3}) in (\ref{Me1}) shows
\begin{multline}\label{Kq4}
{(-1)^n \over \prod_{l=1}^M \Gamma(a_l+n+1) \Gamma(b_l + N - n ) }
p_n^{({\rm I}, M)}(x)
= \sum_{j=0}^n {(-n)_j \over j!} {x^j \over \prod_{l=1}^M \Gamma(a_l+j+1) \Gamma(b_l + N - j ) } .
\end{multline}
Further, using (\ref{Kq3}) to write $w^{(\rm I)}(x) $ as an inverse Mellin transform shows from (\ref{Me2}) that
\begin{multline}\label{Kq5}
{(-1)^n \over \prod_{l=1}^M \Gamma(a_l+n+1) \Gamma(b_l + N - n ) }q_n^{({\rm I}, M)}(x) \\ = {1 \over 2 \pi i} {1 \over n!} \int_{c - i \theta}^{c + i \theta}
{ \Gamma(s+n) \over \Gamma(s) } \Big ( \prod_{l=1}^M
\Gamma(a_l-s) \Gamma(b_l + N +1 + s) \Big ) x^s \, ds.
\end{multline}
Proceeding as in the derivation of Proposition \ref{P3.1}, and making use in particular of the asymptotic formula
(\ref{eq1}) for the ratio of two gamma functions, the large $N$ forms of (\ref{Kq4}) and (\ref{Kq5}) as relevant to (\ref{Me3})
can be deduced. This allows for the analogue of (\ref{FG}) to be deduced, which then proceeding as in the derivation of
Proposition \ref{P3.2} gives the analogue of (\ref{FG1}).
\begin{proposition}\label{P3.4}
Denote the LHS of (\ref{Kq4}) with $n = N -1$, and multiplied by $\prod_{l=1}^M \Gamma( N + b_l)$, by
$\tilde{p}_{N-1}^{({\rm I}, M) }(x)$, and let $F$ be specified as below (\ref{FG}). Also, denote the LHS of
(\ref{Kq5}) with $n = N $, and divided by $\prod_{l=1}^M \Gamma( N + b_l)$, by $\tilde{q}_{N}^{({\rm I}, M) }(x)$,
and let $G$ be as specified below (\ref{FG}). We have
\begin{equation}
p_{N-1}^{({\rm I}, M)}\Big ( {x \over N^{M+1}} \Big )
= \bigg ( 1 - {1 \over 2 N} \bigg (
\Big ( 1 + M - 2 \sum_{l=1}^M b_l \Big ) x {d \over dx} +
(1 + M) \Big ( x {d \over d x} \Big )^2 + O\Big ( {1 \over N^2} \Big ) \bigg ) \bigg ) F(x),
\end{equation}
\begin{multline}
{1 \over N^M} q_{N}^{({\rm I}, M)}\Big ( {x \over N^{M+1}} \Big ) \\
= \bigg ( 1 + {1 \over N} \sum_{l=1}^M b_l +{1 \over 2N} \bigg (
\Big ( 1 + M + 2 \sum_{l=1}^M b_l \Big ) x {d \over dx} +
(1 + M) \Big ( x {d \over d x} \Big )^2 + O\Big ( {1 \over N^2} \Big ) \bigg ) \bigg ) G(x)
\end{multline}
and
\begin{multline}\label{3.26}
{1 \over N^{M+1}} K_N \Big ( {x \over N^{M+1}}, {y \over N^{M+1}} \Big ) = \int_0^1 F(xt) G(yt) \, dt \\
- {1 \over 2 N} (1 + M) \Big ( G(y) x {d \over dx} F(x) - F(x) y {d \over d y} G(y) \Big )
+ {1 \over N} \Big ( \sum_{l=1}^M b_l \Big ) F(x) G(y) + O \Big ( {1 \over N^2} \Big ).
\end{multline}
\end{proposition}
The expansion (\ref{3.26}) shows that in general the leading correction to the
hard edge scaled limit of the $k$-point correlation
in the case of $M$ products of random matrices formed from the multiplication of a
Laguerre unitary ensemble and inverse Laguerre unitary ensemble is $O(1/N)$.
However, as for products studied in Section \ref{S3.1}, the case $M=1$ is special,
as then the relation (\ref{FGa}) between $F$ and $G$ holds. The $O(1/N)$ term in
(\ref{3.26}) the simplifies to read
\begin{equation}\label{3.27}
{1 \over N} y^a \bigg ( (a_1 + b_1) F(x) F(y) - \Big ( x {\partial \over \partial x} - y {\partial \over \partial y} \Big ) F(x) F(y) \bigg ) \bigg |_{M=1}
\end{equation}
Proceeding now as in the derivation of (\ref{FG3}), and with the same meaning of $\tilde{F}$ used
therein, we thus have
\begin{align}\label{FG3a}
&{1 \over N^{2k}} \rho_{(k)}\Big ( {x_1 \over N^2} ,\dots, {x_k \over N^2} \Big ) \Big |_{M=1}
\nonumber \\
& \quad = \prod_{l=1}^k x_l^{a_1} \det \Big [ \Big ( \int_0^1 t^{a_1} F(x_jt) F(x_l t) \, dt + {a_1 + b_1 \over N}F(x_j) F(x_l) \Big ) \Big |_{M=1} \Big ]_{j,l=1}^k
+ O \Big ( {1 \over N^2} \Big ) \nonumber \\
& \quad = \det \Big [ \Big ( \int_0^1 \tilde{F}(x_jt) \tilde{F}(x_l t) \, dt + {a_1 + b_1 \over N}\tilde{F}(x_j) \tilde{F}(x_l) \Big ) \Big |_{M=1} \Big ]_{j,l=1}^k
+ O \Big ( {1 \over N^2} \Big ).
\end{align}
As in the discussion below (\ref{FG3}), it follows that if on the LHS $N$ is replaced by $N + (a_1 + b_1)/2$, the convergence to
the hard edge limit has the optimal rate of $O(1/N^2)$.
\begin{remark}
1.~Changing variables $x_l = y_l/(1 - y_l)$, $0 < y_l < 1$ in (\ref{Kq1}) gives the functional form
\begin{equation}\label{Ja}
\prod_{l=1}^N y_l^{a_1} ( 1 - y_l)^{b_1} \prod_{1 \le j < k \le N} (y_k - y_j)^2,
\end{equation}
which up to proportionality is the eigenvalue PDF for the Jacobi unitary ensemble. In the recent work
\cite{MMM19} the corrections to the hard edge scaled limit of the distribution of the smallest
eigenvalue have been analysed, with results obtained consistent with (\ref{FGa}). In Appendix
A we present a large $N$ analysis of this distribution for the Jacobi $\beta$-ensemble (the Jacobi unitary ensemble
is the case $\beta = 2$) for general even $\beta$. \\
2.~The case $b_1 = 0$ of the Jacobi unitary ensemble is closely related to the Cauchy two-matrix
model \cite{BGS14}. The latter is determinantal, but since the PDF consists of two-components,
the determinant has a block structure. Nonetheless, each block can be expressed in terms of just
a single correlation kernel. The hard edge scaling of the latter has been undertaken in \cite{BGS14},
with a result analogous to (\ref{FG3a}) with $b_1 = 0$ obtained. Closely related to the Cauchy two-matrix
matrix model is the Bures ensemble, as first observed in \cite{BGS09}, and further developed
in \cite{FK16}, with a Muttalib-Borodin type extension given in \cite{FL19}. Since the elements of the
correlation kernel for the Bures ensemble (which is a Pfaffian point process) are given in terms of
the correlation kernel for the Cauchy two-matrix matrix model, it follows that by tuning the scaling
variables at the hard edge, an optimal convergence rate of $O(1/N^2)$ can be achieved.
\\
3.~A Muttalib-Borodin type generalisation of (\ref{Kq1}) is known \cite[Jacobi prime case]{FI18}. Working
analogous to that of Section \ref{S3.2} could be undertaken, although we refrain from doing that here.
It would similarly be possible to obtain the analogue of Proposition \ref{P3.4} for the singular values
of products of truncations of unitary ensembles \cite{KKS15}, which we know from \cite{KK16} can
be cast in a P\'olya ensemble framework as products of Jacobi unitary ensembles.
\end{remark}
\subsection*{Acknowledgements}
This research is part of the program of study supported
by the Australian Research Council Centre of Excellence ACEMS. We thank Mario Kieburg for
feedback on a draft of this work.
|
1,116,691,500,806 | arxiv | \section{Introduction}
Various astrophysical problems require to estimate stellar mass
from the photometry: for instance, to reconstruct the assembly
history of galaxies through cosmic times
\citep[e.g.][2004]{Bell_apjss_149} \nocite{Bell2004};
to define the stellar/baryonic mass Tully--Fisher relation,
which is more physically meaningful than
in any specific photometric band \citep{Bell_apj_550, McGaugh2005}; to
determine the inventory of baryons in the Universe (Fukugita,
Hogan \& Peebles 1998; Fukugita \& Peebles 2004; McGaugh et~al.\ 2010);
\nocite{Fukugita98, Fukugita2004, McGaugh2010} and to disentangle the
contribution of luminous versus dark matter in galactic dynamics
\citep[e.g.][and references therein]{PortSal2010}.
The key to it is the stellar mass--to--light ratio ($M_\star/L$).
For a Simple Stellar Population (ensemble of coeval
stars with the same chemical
composition, formed in the same burst of star formation), $M_\star/L$
depends on (a) the Initial Mass Function; (b) the age and (c) the metallicity.;
for a composite stellar population, (b) and (c) become, respectively,
age distribution (star formation history) and metallicity distribution.
Population synthesis techniques and chemo-photometric models of galaxies can
predict theoretical $M_\star/L$ ratios (including both living stars and remnants)
associated to the photometric properties; yet only in the past decade this
possibility
has been fully appreciated in the dynamical and extra-galactic community.
In particular, colour---$M_\star/L$ relations have become
a popular and handy tool to estimate stellar masses in external galaxies
(e.g.\ Kranz, Slyz \& Rix 2003; McGaugh 2004; Kassin, de Jong \& Weiner 2006;
Bakos, Trujillo \& Pohlen 2008; Treuthard, Salo \& Buta 2009; Torres-Flores
et~al.\ 2011).
\nocite{Kranz_apj_586, McGaugh2004, Kassin_apj_643, Bakos_apjl_683,
Treuthardt2009, Torres2011}
Early on, \citet{SarTin74} and \citet{LarTin78} reported tight relations
between colour and $M_\star/L$ for $B,V$ bands,
used by \citet{Tinsley81} to discuss the dark matter content in galaxies
of different Hubble type. Those papers presented the first linear equation
relating colour and log($M_\star/L$), and the insightful
remark that, adopting a different Initial Mass Function (IMF)
``All the models of a set could be arbitrarily moved up or down
in $\log M/L_B$ [...] The {\it slopes} of the relations would not be altered''.
After the pioneering papers of Tinsley and collaborators, the next analysis
of $M_\star/L$ ratios from galactic models was by \citet{JabAri1992},
who extended colour--$M_\star/L$ relations to near infrared (NIR) bands; it is of
historical interest that, in those early models, the $M_\star/L_H$ vs.\ $(B-H)$
relation had a {\it negative} slope, at odds with relations in optical
bands. Their models were used by \citet{Persic93} to discuss
the dark matter content of galaxies as a function of luminosity.
These early results on $M_\star/L$ from galactic models went otherwise mostly
unnoticed. The breakthrough introduction of ``colour--$M_\star/L$ relations''
to the wider dynamical and extra-galactic
community, was the extensive study of \citet{Bell_apj_550}: they
showed that a variety of disc galaxy models
(closed box models, open models with inflows or outflows, with different
formation age, with starbursts, etc.) all resulted in the same, robust
log--linear colour--$M_\star/L$ relations (CMLR). The tightest relations involve
optical bands, where the notorious age--metallicity degeneracy is, for
once, an advantage, as it concurs to keeping the relation tight.
Similar CMLR were found to hold {\it within} disc galaxy models, along
the radial colour profiles obtained in the inside--out scenario
\citep*{Portinari_mn_347}. Semi--empirical (as opposed to purely theoretical)
CMLR have been derived by \citet{Bell_apjss_149} from multi-band photometry
of galaxies in the SDSS+2MASS surveys.
A key feature of CMLR, also highlighted by \citet{Bell_apj_550}, is that the
zero--point of the relation is set by the stellar IMF, while the slope
of the relation is robust
versus this, and other, model assumptions. The zero--point in the optical
bands ($B,V,I$) seems to be well established \citep{Flynn2006}: models of the
local Galactic disc based on star counts yield a colour--$M_\star/L$ datapoint
for the ``solar cylinder'' that agrees with the normalization of typical
Solar Neighbourhood IMFs \citep{Kroupa:1998, Chabrier_apj_554, Chabrier2002}.
To estimate stellar mass, near infrared (NIR) photometry
is most often a favoured choice:
(i) NIR luminosity, less affected than optical bands by minor recent
star formation episodes (the ``frosting'' effect), is a better tracer of the
bulk of the stellar population. (ii) The $M_\star/L$ ratio in the NIR
varies, overall, less than in the optical \citep{Bell_apj_550} ---
though it is not constant and totally insensitive to the star
formation history, as often assumed in the past.
(iii) NIR luminosity is less
affected by dust extinction. (iv) Nowadays NIR studies benefit from
excellent large databases obtained from extensive surveys (2MASS, UKIDSS, etc.)
However, the integrated NIR light of a stellar population is heavily affected
by the contribution of its Asymptotic Giant Branch (AGB) stars
\citep{Maraston_mn_300, Maraston_mn_362, Girardi_mn_300, Mouhcine2002,
Mouhcine2003}.
At intermediate ages (0.3--3~Gyr) the AGB phase dominates the NIR emission,
lowering the K--band $M_\star/L$ ratio by a factor of 3--5 and inducing
a colour transition to the red, reaching $(V-K) \geq 3$. For high redshift
galaxies in the relevant age range ($z \sim 2$, ages 0.2--2~Gyr), population
synthesis models including the AGB phase yield about 1~mag brighter K--band
luminosities, 2~mag redder $(V-K)$ colours, and 60\% lower stellar masses and
ages than other models \citep{Maraston2006, Tonini2009}.
It is by now well established that, in spite of its short--lived nature,
the complex Thermally Pulsing (TP)-AGB phase has considerable impact
on NIR colours and $M_\star/L$, and needs to be accurately modelled.
Major advances in this respect
have been implemented in the recent release of the Padova isochrones.
In this paper, we explore their
consequences on theoretical CMLR.
The paper is organized as follows. In Section~2 we discuss CMLR for Simple
Stellar Populations, comparing the previous and recent release of the Padova
isochrones, differing only in the TP-AGB phase implementation. In Section~3
we derive CMLR for composite stellar populations resulting from extended
star formation histories.
In Section~4 we derive CMLR for more detailed disc galaxy models
with internal colour and metallicity gradients \citep{Portinari_mn_347}.
Dust can also influence CMLR relations, although to first order approximation,
it is believed to have little impact
on optical CMLR, as the dust extinction+reddening vector runs almost
parallel to the CMLR itself \citep{Bell_apj_550}.
The past decade saw major progress in modelling the effects
of interstellar dust on the integrated light of galaxies
\citep[e.g.][]{Silva98,Popescu_aa_362,Piovan2006}.
We therefore revisit the role of dust on CMLR by including the attenuation
effects derived from detailed radiative transfer models
\citep{Tuffs_aa_419}. Section~5 is thus dedicated
to the CMLR of (normal) dusty galaxies. In Section 6 we outline our summary
and conclusions.
\section{Colour--$M_\star/L$ relations for Simple Stellar Populations}
In this section we discuss CMLR for Simple Stellar Populations
\citep[SSPs;][]{Renzini_Buzzoni_1986}. We analyze the role of the TP-AGB phase
by comparing
SSPs based on the ``old'' \citep{Girardi_aas_141,Girardi_aa_391} and ``new''
(Marigo et~al.\ 2008; Girardi et~al.\ 2010)
\nocite{Marigo_aa_482, Girardi_apj_724}
isochrone dataset of the Padova
group\footnote{http://stev.oapd.inaf.it/cgi-bin/cmd}. The ``old'' isochrones
--- basically the same set used for the CMLR of
\citet{Portinari_mn_347} --- included the simplified TP-AGB prescriptions
of \citet{Girardi_mn_300}. These are now superseded by detailed, calibrated
evolutionary models of the TP-AGB phase
\citep{Marigo_aa_469} that have been implemented in the ``new'' isochrone set.
Notice that the ``old'' and the ``new'' datasets differ only in the
treatment of the TP-AGB phase, so that the onset of the AGB in the
SSPs is still at $t \sim 10^8$~yrs, corresponding to a turnoff mass of about
5~M$_\odot$. The Marigo TP-AGB models follow the
detailed evolution of stars through pulse cycles, core mass and luminosity
growth, III dredge--up, hot--bottom burning nucleosynthesis and overluminosity,
conversion from M--type to C--type star, mass loss and final superwind phase.
In their latest version \citep{Marigo_aa_469},
the models also consistently follow the variations of envelope opacities
with surface chemical composition, and of pulsation mode; and adjust the mass
loss rate accordingly. The main difference with respect to the ``old'' isochrone
set is the transition to C stars due to the III dredge--up, and their extended
red tail in the NIR HR diagram.
The models are calibrated to reproduce the luminosity function of thousands
of carbon stars in the Large and Small Magellanic Clouds (LMC and SMC),
as well as the M-- and C--type star
counts/lifetimes in Magellanic Cloud clusters; and successfully compare to
a variety of observables (periods and mass loss rates;
period--luminosity relations; initial--final mass relation). These models aim
at a comprehensive coverage of the complex TP-AGB
evolution; in contrast to other models, optimized for population synthesis
purposes, that simply calibrate the luminosity contribution of M-- and C--type
AGB stars as a function of SSP age
\citep{Maraston_mn_300, Maraston_mn_362}.
We retrieved isochrones with ages from $\log (t/\mathrm{yr})$ = 6.0 to 10.1,
in Johnson-Cousins \emph{UBVRIJHK}, Two Micron All Sky Survey (2MASS)
\emph{JHK$_s$} and Sloan Digital Sky Survey (SDSS) \emph{ugriz} filters.
Integrated SSP luminosities were computed populating the isochrones with
the \citet{Kroupa:1998} IMF, that is suitable to model the chemical
evolution of the Milky Way and provides the correct zero--point
for optical CMLR \citep{Boissier99, Flynn2006}.
The grid of isochrones and SSPs covers seven metallicities, with
corresponding colour coding in the figures: $Z$=0.0001 (cyan), 0.0004
(yellow), 0.001 (orange), 0.004 (blue), 0.008 (green), 0.019$=Z_{\odot}$ (red)
and 0.03 (black). Our SSPs are based on the \citet{Marigo_aa_482} isochrones
for $Z \geq 0.004$, where the AGB phase is accurately
calibrated on observations in the Milky Way and Magellanic Clouds;
for lower metallicities ($Z \leq 0.001$) we rely on the Girardi et~al.\ (2010)
release, calibrated on the resolved AGB population of old, metal poor dwarf
galaxies and on the white dwarf masses of globular clusters.
These very low metallicities are however of minor interest
for CMLR relevant to the general galaxy population: as noted by
\citet{Bell_apj_550}, the chemical enrichment caused by even modest amounts
of star formation raises the galaxy metallicity rapidly to at least $Z$=0.002
even in a closed box case; and since the ``G dwarf problem'' appears to be
ubiquitous both in disc galaxies like the Milky Way and in elliptical galaxies
\citep{Bressan94}, the low--$Z$ tail of the metallicity distribution function
is expected to be always little populated.
We express $M_\star/L$ ratios in solar units using the solar magnitudes
listed in Table~\ref{tab:solmag} obtained, consistently with the adopted
isochrones, by interpolating the bolometric corrections of the Padova
database for the corresponding solar model ($\log T_{eff}=3.762$,
$\log g=4.432$).
\begin{table}
\label{tab:solmag}
\caption{Adopted solar magnitudes in Johnson--Cousins--Glass bands,
SDSS and 2MASS bands.}
\begin{center}
\begin{tabular}{llll}
\hline
\hline
band & $\cal M_\odot$ & band & $\cal M_\odot$ \\
$B$ & 5.497 & $g$ & 5.144 \\
$V$ & 4.828 & $r$ & 4.676 \\
$R$ & 4.445 & $i$ & 4.569 \\
$I$ & 4.118 & $z$ & 4.553 \\
$J$ & 3.699 & $J^{2M}$ & 3.647 \\
$H$ & 3.356 & $H^{2M}$ & 3.334 \\
$K$ & 3.327 & $Ks^{2M}$ & 3.295 \\
\hline
\end{tabular}
\end{center}
\end{table}
The SSP mass $M_\star(t)$,
including both living stars and stellar remnants as a function of age,
is computed from the lifetimes and remnant masses of the Padova models
\citep*{Portinari_aa_334, Marigo2001}.
For a total initial SSP mass of 1~$M_\odot$, after a Hubble time the locked--up
fraction is typically 70\%
for the \citet{Kroupa:1998} or the Salpeter IMF. Fig.~\ref{fig:SSPmass}
shows the evolution of the SSP mass for these two IMFs. The Salpeter case
allows a direct comparison to the SSPs by \citet{Maraston_mn_362}
and \citet{Bruzual_mn_344}; considering that the three sets of models have
independent assumptions on stellar remnant masses and lifetimes, the agreement
is excellent, within few~\% : any significant difference in $M_\star/L$
between them is entirely due to the adopted luminosity and colour evolution,
not to the mass evolution.
(No direct comparison is presented for the Kroupa 1998 IMF, as the public
Kroupa SSPs by Maraston 2005 are rather based on the top--heavy IMF in Eq.~3
of Kroupa 2001). \nocite{Kroupa_mn_322}
Notice that, since the \citet{Kroupa:1998} IMF contains fewer massive stars,
and more long--lived stars, than the Salpeter IMF, a Kroupa SSP sheds its
mass more slowly, although the final locked--up fraction is about the same.
\begin{figure}
\begin{center}
\includegraphics[scale=0.25,angle=-90]{fig1}
\caption{Evolution of the SSP mass, including living stars and remnants,
for the Kroupa (1998) IMF adopted in this work, and for the Salpeter IMF.
Also shown are the masses of the SSPs from Maraston (2005) and
Bruzual \& Charlot (2003), with Salpeter IMF. The plot is for solar
metallicity, but we verified that, within in each model, the impact
of metallicity on $M_\star(t)$ is just few \%.}
\label{fig:SSPmass}
\end{center}
\end{figure}
\begin{figure*}
\includegraphics[scale=0.95]{fig2a}
\includegraphics[scale=0.95]{fig2b}
\caption[Bolometric and $K$-band luminosity ratios for the new
dataset]{Relative contributions of the TP-AGB phase to the total luminosity
of a SSP using the new isochrones. The ratios are plotted
as a function of SSP age, with metallicities listed in the legend.
\emph{Left panel}: bolometric ratios. \emph{Right panel}: $K$-band ratios. }
\label{fig:luminosity_ratios}
\end{figure*}
\subsection{The AGB phase contribution to the integrated light}
Updates in TP-AGB modelling are expected to affect CMLR involving NIR bands,
where TP-AGB stars dominate, by up to 80\%, the luminosity
of SSPs of intermediate ages 0.3--3~Gyr (Figure~\ref{fig:luminosity_ratios});
see \citet{Maraston_mn_300, Maraston_mn_362}. This is supported
by observations of LMC clusters \citep{Frogel_apj_352}, where AGB
stars contribute up to 40\% of the bolometric luminosity in said age range.
In Figure \ref{fig:sspdust}, we compare the $M_\star/L$ evolution
of the old and new SSPs in $K$ band. The onset
of the AGB phase, soon after 100~Myr, is smoother in the new models:
the ``AGB phase transition'' is not as sharp,
due to revised mass--loss prescriptions that reduce the lifetimes
of the most massive TP-AGB stars \citep{Marigo_aa_482}. Later on though,
the new models remain significantly brighter in the NIR (up to 0.5~mag),
predicting ``lighter'' $M_\star/L$ ratios for most of the SSP lifetime. In
bluer bands, the difference between old and new models is reduced, becoming
negligible in $UBV$.
As a minor detail, notice that the new $Z=0.004$ SSP presents a spike
of high $M_\star/L_K$ (and a corresponding blue spike in $(V-K)$, see
Fig.~\ref{fig:ssp_comparison}) around $\log t=8.4$.
As the NIR luminosity is very sensitive to the contribution of carbon stars,
this feature can be ascribed to the complex dependence of the carbon star phase
as a function of stellar mass and age for this particular metallicity
\citep[see Fig.~20 in][]{Marigo_aa_469}.
The spike is however smoothed away in the case of composite stellar populations.
For solar and LMC metallicity we also compare to the models of Maraston
(2005) --- her $Z=0.02$ and~0.01 SSPs respectively: the $M_\star/L_K$ evolution
is qualitatively similar to that of the new Padova models,
although the onset of the AGB contribution occurs at $t > 200$~Myr.
While this difference is relevant for star clusters within
that specific age range, it is less crucial for the integrated light
and CMLR of galaxies with extended star formation histories, where the two
models globally agree (see Fig.~5).
A detailed comparison to the Maraston models is beyond the scopes of this
paper; some comparison can be found in \citet{Marigo_aa_482} and
\citet{conroy2009, conroy2010} --- though notice that the latter authors adopt
different spectral libraries, and tailor their own version of the default
Padova models considered here.
\begin{figure}
\begin{center}
\includegraphics[scale=0.33]{fig3}
\caption[Circumstellar dust]{$K$ band $M_\star/L$ ratio as a function of age
for the ``old'' (dotted) and ``new'' (solid) SSPs. The effect of circumstellar
dust on the ``new'' SSPs is also shown (dashed lines).
\emph{Top panel:} $Z$=0.004; \emph{Mid panel:} $Z$=0.008;
\emph{bottom panel:} $Z=Z_\odot$=0.019. Notice that the luminosity peak
around $\log t=9.2$, most prominent for $Z=0.008$, is a non--spurious,
understood feature of SSPs \citep{Girardi_mn_300};
it is so short-lived, however, that it is smoothed away when considering
composite stellar populations with finite age range. For solar and LMC
metallicity (bottom panels) we also overplot the corresponding SSPs from
Maraston (2005)
rescaled, for the sake of this comparison, from Salpeter to Kroupa (1998) IMF
by accountig for the approriate evolutionary flux (Maraston 1998).}
\label{fig:sspdust}
\end{center}
\end{figure}
\subsection{Circumstellar dust} \label{sec:circumstellar_dust}
The new Padova isochrones can also include circumstellar dust around
AGB stars, following the recipes of \citet{Bressan_aa_332} or
\citet{Groenewegen_aa_448}. The circumstellar envelope reprocesses
a fraction of the stellar UV/optical light to the mid and far infrared.
The process is crucial to interpret the MIR HR diagram of the Magellanic
Clouds
\citep[Marigo et~al. 2008, 2010; see also][]{Boyer2009, Kelson2010, Barmby2012}
\nocite{Marigo2010} but NIR light is much less affected.
Figure~\ref{fig:sspdust} also compares the NIR $M_\star/L$ of new SSPs with
and without circumstellar dust; the difference is tiny compared to that
between the old and the new models.
Dust effects on NIR light are highlighted in two--colour
plots such as $(J-H)$ vs.\ $(H-K)$ \citep{Bressan_aa_332, Piovan2003}, but we
verified that, for the sake of CMLR, even for these colours the impact of dust
is smaller than the difference between the old and the new SSPs.
In bluer bands the difference between dusty and dust-free SSPs remains
negligible, as the optical luminosity of SSPs
is not so sensitive to the AGB phase
\citep{Maraston_mn_300, Maraston_mn_362};
so, albeit the optical emission of AGB stars does suffer from
dust reprocessing, this is of minor importance for the SSP as a whole.
In short, the impact of circumstellar dust on integrated SSP light
is far less crucial than the improved TP-AGB modelling,
and we shall neglect it in the remainder of this work.
\begin{figure*}
\includegraphics[scale=1]{fig4a}
\includegraphics[scale=1]{fig4b}
\caption{Comparison of SSP evolution and CMLR between the old (dashed)
and the new (solid) datasets. Three metallicities are shown: $Z$=0.0001
(cyan), 0.004 (blue) and 0.019 (red). }
\label{fig:ssp_comparison}
\end{figure*}
\subsection{Time evolution of colours and mass-to-light}
Figure \ref{fig:ssp_comparison} exemplifies the origin of CMLR for SSPs,
as the combined result of the time evolution of colours and $M_\star/L$.
We selected for illustration three metallicities: $Z$=0.0001
is the lowest in the database; $Z$=0.004 and 0.019,
corresponding to SMC and solar metallicity, cover the range
significant for the bulk of stellar populations in galaxies.
The top panels show a typical optical CMLR: both $M_\star/L$ and colour smoothly
increase with SSP age and the resulting CMLR does not significantly depend
on metallicity, at least in the range $Z=0.004-0.02$. The new implementation
of the TP-AGB phase has very mild effects in optical bands: the AGB
``phase transition'' is not apparent in optical colours, dominated by the
light of turn-off Main Sequence stars and core helium--burning stars
\citep{Renzini_Buzzoni_1986, Bressan94, Maraston_mn_300, Maraston_mn_362}.
Only at very low metallicities, where young SSPs are much bluer, we see an
AGB colour transition.
The bottom panels in Fig.~\ref{fig:ssp_comparison} illustrate a typical
optical--NIR CMLR, where the difference between old and new models is far more
evident.
The $(V-K)$ colour tends to saturate, becoming mainly a metallicity indicator,
after $t =0.3$~Gyr. Correspondingly, the optical---NIR CMLR for SSPs breaks
down at old ages (cf.\ the almost vertical lines in the middle panel).
With the new SSPs, the evolution of the $K$ band $M_\star/L$ ratio is very
similar for $0.004 \leq Z \leq 0.019$; this degeneracy with
respect to metallicity
was already remarked by \citet{Maraston_mn_362}.
\begin{table*}
\caption[$b$ and $\tau$ values]{Grid of $b$ parameter values and corresponding
e--folding timescales $\tau$ for our exponential models.}
\label{table_b_tau}
\begin{small}
\begin{tabular}{cccccccccccccccc}
\hline
\hline
\multirow{2}{*}{\normalsize $b$} & \vline & 0.01 & 0.05 & 0.10 & 0.15 & 0.20 & 0.30 & 0.40 & 0.50 & 0.60 & 0.70 & 0.80 & 0.90 & 1.00 \\
& \vline & 1.10 & 1.23 & 1.37 & 1.55 & 1.75 & 2.01 & 2.35 & 2.84 & 3.23 & 3.67 & 4.59 & 6.46 & 8.33 & 10.00\\
\hline
\multirow{2}{*}{\normalsize $\tau$} & \vline &1.55&2.20&2.80&3.25&3.80&4.90&6.20&8.00&10.50&15.00&23.00&50.00&$\infty$ \\
& \vline & -50.00 & -23.00 & -15.00 & -10.50 & -8.00 & -6.20 & -4.90 & -3.80 & -3.25 & -2.80 & -2.20 & -1.55 & -1.20 & -1.00 \\
\hline
\hline
\end{tabular}
\end{small}
\end{table*}
\section{Colour--$M_\star/L$ relations for exponential models}
\label{sec:simple_exp_models}
Composite stellar populations --- convolutions of SSPs of different age and
metallicity, according to a given star formation and chemical evolution history
--- are more relevant for practical applications of CMLR to real galaxies.
In this section we shall consider exponentially declining
(or increasing) Star Formation Rates (SFR), a common recipe
to mimic the photometric properties of the Hubble sequence.
The age of our models is $T=10$~Gyr \citep[the age estimate for the Milky Way
disc;][]{Carraro_2000}. For each metallicity, we compute
a grid of 27 exponential models with SFR $\Psi(t) \propto e^{-t/\tau}$:
declining SFR are modelled with e--folding timescales $\tau$ ranging from
1.55~Gyr to $\infty$ (constant star formation rate);
increasing SFR are modelled with negative values of $\tau$ ranging from
$-50.00$ to $-1.00$.
These star formation histories (SFHs) can be characterized by the
``birthrate parameter'' $b = \psi(T)/{\langle \psi \rangle}$:
the ratio between present--day and past average SFR.
This can be considered a tracer of morphological galaxy type,
with $b < 0.2$ for
Sa--Sab discs, $b \sim 0.4$ for Sb discs and $b \sim 1$ for Sc
discs \citep*{Kennicutt_apj_435,Sommer-Larsen_apj_596}.
Therefore, exponential models with
$b \leq 1$ schematically represent ``normal'' spiral galaxies;
models with $b > 1$ represent blue galaxies with prominent recent star
formation. Elliptical galaxies are
typically well represented by old SSPs ($b \longrightarrow 0$).
The range of adopted $\tau$ and $b$ values (from 0.01 to 10.0) are tabulated
in Table~\ref{table_b_tau}. Our grid of
exponential models is similar to that considered by \citet{Bell_apjss_149}.
\begin{figure*}
\begin{center}
\includegraphics[scale=0.35]{fig5a}
\includegraphics[scale=0.35]{fig5b}
\caption[$M/L$ versus $b$ parameter]{Mass-to-light ratio in $I$ and $K$ band
for exponential models of different metallicity, as a function of the $b$
parameter. Solid lines represent the new models and dashed lines the old
models. The right panels show the full range of model SFH with $b=0.1-10$; the
left panels zoom on the more ``normal'' $b \leq 1$. The black dot--dashed line
shows the K--band $M_\star/L$ from the Maraston SSPs (Z=0.01).}
\label{fig:ML_vs_b}
\end{center}
\end{figure*}
\begin{figure}
\begin{center}
\includegraphics[scale=0.9]{fig6}
\caption{$K$ band mass-to-light vs.\ $B-R$ (upper panel) and $V-K$ (lower
panel). Solid/dashed lines for new/old models, as in Figure \ref{fig:ML_vs_b}.
The thick part of the curves marks the case $b \leq 1$, representative of the
SFH of ``normal'' galaxies. }
\label{fig:ML_vs_col}
\end{center}
\end{figure}
\subsection{Effects of updated TP-AGB models}
We now compare the CMLR of exponential SFHs, derived from the old and the new
SSPs, mostly for $Z \geq 0.004$ as the relevant metallicity range for
integrated galaxy light.
In Figure \ref{fig:ML_vs_b} we plot the $M_\star/L$ ratio of exponential models
versus $b$ parameter. As already shown for SSPs, the optical luminosity is only
marginally affected by the update: the new models are brighter
just by $\leq$0.03~dex in $I$ band.
In NIR bands, the new TP-AGB implementation renders
the models brighter up to 0.1~dex (or 25\%); the difference
is largest at solar metallicity. With the new models, the dependence of
$M_\star/L_K$ on metallicity is largely reduced (Fig.~\ref{fig:ssp_comparison}
and Maraston 2005): NIR $M_\star/L$ ratios depend on the SFH of the system
but not much on the underlying chemical enrichment history and metallicity
distribution. This is convenient when deriving stellar masses from NIR
luminosity; in comparison, for $M_\star/L_I$ the metallicity dependence
is as strong as the SFH dependence
(at least for $b \leq 1$).
In Fig.~\ref{fig:ML_vs_col} we plot CMLR for $M_\star/L_K$ versus
{\mbox{$(B-R)$}} and $(V-K)$ colours, a popular choice for estimating
stellar masses \citep{Kranz_apj_586,Kassin_apj_643}.
The effect of the new SSPs on the $M_\star/L_K - (B-R)$ relation is an overall
brightening by 0.1--0.15~dex at any fixed $(B-R)$, as the latter
is quite insensitive to TP-AGB modelling (see Fig.~\ref{fig:ssp_comparison}).
This is in general the case for CMLR based on optical colours.
In optical--NIR colours instead, the new models are up to 0.3~mag redder than
the old models. As a combined
result of brightening and reddening, the $M_\star/L_K$ ratio {\it at a given
colour} is up to 0.3~dex, or 2 times, lighter. Factor--of--2
lighter masses were indeed derived by \citet{Maraston2006} from multi-band
photometry of galaxies at $z \sim 2$, thanks to the AGB contribution in their
models.
Also, the $M_\star/L_K - (V-K)$ relation depends more strongly on metallicity
for the new models, as $(V-K)$ is more of a metallicity tracer than
a $M_\star/L$ tracer (see Fig.~\ref{fig:ssp_comparison}).
Similar comments hold for other optical--NIR colours.
In Figure~\ref{fig:exp_BRIK_full} we show the new CMLR for $B$ and $K$
band $M_\star/L$ versus $(B-R)$ and $(I-K)$, for all the metallicities
(coloured solid lines) and a representative sampling of the $b$--parameter
range 0.01--10 (dashed lines).
The CMLR for a 10 Gyr old SSP is added as a dot--dashed line, smoothly extending
the CMLR of the exponential models with the lowest $b$ values.
We confirm the tight correlation between $B$ band $M_\star/L$ ratio
and the optical $(B-R)$ colour, very little affected by SFH and metallicity
--- other than possibly for the lowest $Z=0.0001$ case. Such a degree of
age and metallicity degeneracy ensures robust CMLR in the optical.
CMLR remain quite robust even for $K$ band $M_\star/L$ versus $(B-R)$:
the age--metallicity degeneracy is quite tight for $Z \geq 0.004$,
breaking down only for lower metallicities that are seldom relevant in
integrated galaxy light.
In $(I-K)$, the age--metallicity degeneracy breaks down
\citep{Bell_apj_550} for this is mainly a metallicity indicator, like $(V-K)$.
(For the lowest metallicities, the dependence of $(I-K)$ on $Z$ gets
``inverted'' for $b>1$ --- i.e.\ when the system is dominated
by young stellar populations. We ascribe this behaviour to the fact that
the extended C star phase of the more massive AGB stars causes
a greater colour transition toward the red, the lower the metallicity;
see Fig.~20 in Marigo \& Girardi 2007).
CMLR in $(I-K)$ are not very meaningful. The same holds
for redder (purely NIR, e.g.\ $J-K$) colours and for $(V-K)$; while
$(B-K)$ yields reasonably tight CMLR for $Z \geq 0.008$, the LMC metallicity.
We provide in Table~\ref{tab:fit_bmodels} log--linear CMLR for exponential
models, for those colours that reasonably trace the $M_\star/L$ ratio,
at least in the metallicity range relevant for galaxies ($Z \geq 0.004$).
These CMLR fit the detailed model results
within $\pm$0.1~dex (25\% accuracy in $M_\star/L$); asterisks indicate
less tight CMLR, accurate within $\pm$0.13~dex (35\% error).
As a rule, the fits are based on models with $0.004 \leq Z \leq 0.03$, but for
optical--optical colours the same fits are good also down to much lower
metallicities. In few cases, meaningful fits are limited to $Z \geq 0.008$.
The metallicity range where the fit is valid is indicated in the table.
Among optical colours, $(R-I)$ and $(r-i)$ are the least reliable $M_\star/L$
indicators; their use is further discouraged by their short baseline, as the
error on the colour will have a significant impact on the practical estimate
of $M_\star/L$ \citep[see also][]{Gallazzi_Bell2009}.
In Table~\ref{tab:fit_bmodels} we also provide CMLR for SDSS colours;
they are steeper --- lighter at the blue end --- than those
of \citet{Bell_apjss_149}.
This is partly due to the fact that their semi--empirical relations
are intrinsically flatter than the theoretical ones (BdJ or PST04), partly
due to the fact that our new models result in somewhat steeper
CMLR even for optical colours (see Section~\ref{sect:PST04}).
\begin{table*}
\caption{Fitting coefficients of CMLR of the form :
$\log (M_\star/L) = s \times {\rm colour} + z$ from exponential SFH models
with Kroupa IMF; the colour range covered by the models is indicated at
the bottom of each sub-table.
The rightmost column indicates the metallicity range $Z \geq Z_{min}$ where
the fitted CMLR apply (see text). Asterisks indicate less tight CMLR (accuracy
of $\pm$0.13~dex rather than $\pm$0.1~dex).}
\label{tab:fit_bmodels}
\begin{scriptsize}
\begin{tabular}{lcccl}
\hline
\hline
colour & $\log (M_\star/L)$ & $s$ & $z$ & \multicolumn{1}{c}{$Z_{min}$} \\
\hline
$(B-V)$ & $B$ & 1.866 & -1.075 & 0.0004 \\
$(B-V)$ & $V$ & 1.466 & -0.807 & 0.0004 \\
$(B-V)$ & $R$ & 1.277 & -0.720 & 0.0004 \\
$(B-V)$ & $I$ & 1.147 & -0.704 & 0.0004 \\
$(B-V)$ & $J$ & 1.047 & -0.850 & 0.004 \\
$(B-V)$ & $H$ & 1.030 & -0.992 & 0.004 \\
$(B-V)$ & $K$ & 1.064 & -1.066 & 0.004 \\
$(B-V)$ & $J^{2M}$ & 1.031 & -0.835 & 0.004 \\
$(B-V)$ & $H^{2M}$ & 1.024 & -0.991 & 0.004 \\
$(B-V)$ & $Ks^{2M}$ & 1.055 & -1.066 & 0.004 \\
0.2--1.0 & & & & \\
\hline
$(B-R)$ & $B$ & 1.272 & -1.287 & 0.0004 \\
$(B-R)$ & $V$ & 1.000 & -0.975 & 0.0004 \\
$(B-R)$ & $R$ & 0.872 & -0.866 & 0.0004 \\
$(B-R)$ & $I$ & 0.783 & -0.836 & 0.001 \\
$(B-R)$ & $J$ & 0.714 & -0.969 & 0.004 \\
$(B-R)$ & $H$ & 0.701 & -1.109 & 0.004 \\
$(B-R)$ & $K$ & 0.724 & -1.186 & 0.004 \\
$(B-R)$ & $J^{2M}$ & 0.702 & -0.952 & 0.004 \\
$(B-R)$ & $H^{2M}$ & 0.697 & -1.107 & 0.004 \\
$(B-R)$ & $Ks^{2M}$ & 0.718 & -1.185 & 0.004 \\
0.5--1.6 & & & & \\
\hline
$(B-I)$ & $B$ & 1.041 & -1.549 & 0.001 \\
$(B-I)$ & $V$ & 0.819 & -1.182 & 0.001 \\
$(B-I)$ & $R$ & 0.714 & -1.047 & 0.001 \\
$(B-I)$ & $I$ & 0.641 & -0.997 & 0.004 \\
$(B-I)$ & $J$ & 0.582 & -1.112 & 0.004 \\
$(B-I)$ & $H$ & 0.571 & -1.249 & 0.004 \\
$(B-I)$ & $K$ & 0.589 & -1.330 & 0.004 \\
$(B-I)$ & $J^{2M}$ & 0.572 & -1.094 & 0.004 \\
$(B-I)$ & $H^{2M}$ & 0.567 & -1.245 & 0.004 \\
$(B-I)$ & $Ks^{2M}$ & 0.583 & -1.327 & 0.004 \\
0.7--2.2 & & & & \\
\hline
$(B-K)$ & $B$ & 0.898 & -3.009 & 0.008 \\
$(B-K)$ & $V$ & 0.710 & -2.335 & 0.008 \\
$(B-K)$ & $R$ & 0.620 & -2.054 & 0.008 \\
$(B-K)$ & $I$ & 0.556 & -1.901 & 0.008 \\
$(B-K)$* & $J$ & 0.498 & -1.927 & 0.008 \\
$(B-K)$* & $H$ & 0.487 & -2.047 & 0.008 \\
$(B-K)$* & $K$ & 0.498 & -2.141 & 0.008 \\
2.6--4.2 & & & & \\
\hline
$(B-Ks)$ & $B$ & 0.894 & -3.047 & 0.008 \\
$(B-Ks)$ & $V$ & 0.707 & -2.367 & 0.008 \\
$(B-Ks)$ & $R$ & 0.618 & -2.082 & 0.008 \\
$(B-Ks)$ & $I$ & 0.555 & -1.928 & 0.008 \\
$(B-Ks)$* & $J^{2M}$ & 0.489 & -1.922 & 0.008 \\
$(B-Ks)$* & $H^{2M}$ & 0.484 & -2.069 & 0.008 \\
$(B-Ks)$* & $Ks^{2M}$ & 0.494 & -2.166 & 0.008 \\
2.6--4.2 & & & & \\
\hline
$(V-R)$ & $B$ & 3.963 & -1.728 & 0.004 \\
$(V-R)$ & $V$ & 3.124 & -1.325 & 0.004 \\
$(V-R)$ & $R$ & 2.724 & -1.172 & 0.004 \\
$(V-R)$ & $I$ & 2.443 & -1.109 & 0.004 \\
$(V-R)$ & $J$ & 2.220 & -1.215 & 0.004 \\
$(V-R)$ & $H$ & 2.178 & -1.349 & 0.004 \\
$(V-R)$ & $K$ & 2.248 & -1.434 & 0.004 \\
$(V-R)$ & $J^{2M}$ & 2.186 & -1.195 & 0.004 \\
$(V-R)$ & $H^{2M}$ & 2.166 & -1.346 & 0.004 \\
$(V-R)$ & $Ks^{2M}$ & 2.228 & -1.431 & 0.004 \\
0.25--0.65 & & & & \\
\hline
& & & & \\
& & & & \\
& & & & \\
\end{tabular}
\begin{tabular}{lccccc}
\hline
\hline
colour & $\log (M_\star/L)$ & $s$ & $z$ & \multicolumn{1}{c}{$Z_{min}$} \\
\hline
$(V-I)$ & $B$ & 2.312 & -2.111 & 0.004 \\
$(V-I)$ & $V$ & 1.826 & -1.629 & 0.004 \\
$(V-I)$ & $R$ & 1.593 & -1.438 & 0.004 \\
$(V-I)$ & $I$ & 1.426 & -1.346 & 0.004 \\
$(V-I)$ & $J$ & 1.285 & -1.420 & 0.004 \\
$(V-I)$* & $H$ & 1.257 & -1.547 & 0.004 \\
$(V-I)$* & $K$ & 1.296 & -1.637 & 0.004 \\
$(V-I)$ & $J^{2M}$ & 1.265 & -1.397 & 0.004 \\
$(V-I)$* & $H^{2M}$ & 1.249 & -1.541 & 0.004 \\
$(V-I)$* & $Ks^{2M}$ & 1.282 & -1.630 & 0.004 \\
0.6--1.2 & & & & \\
\hline
$(R-I)$* & $B$ & 5.436 & -2.594 & 0.004 \\
$(R-I)$* & $V$ & 4.304 & -2.016 & 0.004 \\
$(R-I)$* & $R$ & 3.756 & -1.776 & 0.004 \\
$(R-I)$* & $I$ & 3.357 & -1.645 & 0.004 \\
$(R-I)$* & $J$ & 2.955 & -1.674 & 0.008 \\
$(R-I)$* & $H$ & 2.898 & -1.804 & 0.008 \\
$(R-I)$* & $K$ & 2.965 & -1.891 & 0.008 \\
$(R-I)$* & $J^{2M}$ & 2.934 & -1.661 & 0.008 \\
$(R-I)$* & $H^{2M}$ & 2.907 & -1.813 & 0.008 \\
$(R-I)$* & $Ks^{2M}$ & 2.975 & -1.906 & 0.008 \\
0.35--0.60 & & & & \\
\hline
$(g-r)$ & $g$ & 1.774 & -0.783 & 0.0004 \\
$(g-r)$ & $r$ & 1.373 & -0.596 & 0.0004 \\
$(g-r)$ & $i$ & 1.227 & -0.576 & 0.001 \\
$(g-r)$ & $z$ & 1.158 & -0.619 & 0.001 \\
$(g-r)$ & $J^{2M}$ & 1.068 & -0.728 & 0.004 \\
$(g-r)$ & $H^{2M}$ & 1.060 & -0.884 & 0.004 \\
$(g-r)$ & $Ks^{2M}$ & 1.091 & -0.956 & 0.004 \\
0.1--0.85 & & & & \\
\hline
$(g-i)$ & $g$ & 1.297 & -0.855 & 0.001 \\
$(g-i)$ & $r$ & 1.005 & -0.652 & 0.001 \\
$(g-i)$ & $i$ & 0.897 & -0.625 & 0.001 \\
$(g-i)$ & $z$ & 0.845 & -0.665 & 0.004 \\
$(g-i)$ & $J^{2M}$ & 0.779 & -0.769 & 0.004 \\
$(g-i)$ & $H^{2M}$ & 0.772 & -0.924 & 0.004 \\
$(g-i)$ & $Ks^{2M}$ & 0.794 & -0.997 & 0.004 \\
0.1--1.2 & & & & \\
\hline
$(g-z)$ & $g$ & 1.152 & -0.991 & 0.004 \\
$(g-z)$ & $r$ & 0.896 & -0.759 & 0.004 \\
$(g-z)$ & $i$ & 0.800 & -0.721 & 0.004 \\
$(g-z)$ & $z$ & 0.752 & -0.754 & 0.004 \\
$(g-z)$ & $J^{2M}$ & 0.683 & -0.844 & 0.004 \\
$(g-z)$* & $H^{2M}$ & 0.673 & -0.995 & 0.004 \\
$(g-z)$* & $Ks^{2M}$ & 0.691 & -1.069 & 0.004 \\
0.3--1.5 & & & & \\
\hline
$(r-i)$* & $g$ & 4.707 & -1.025 & 0.004 \\
$(r-i)$* & $r$ & 3.650 & -0.784 & 0.004 \\
$(r-i)$* & $i$ & 3.251 & -0.741 & 0.004 \\
$(r-i)$* & $z$ & 3.054 & -0.773 & 0.004 \\
$(r-i)$* & $J^{2M}$ & 2.806 & -0.867 & 0.004 \\
$(r-i)$* & $H^{2M}$ & 2.768 & -1.018 & 0.004 \\
$(r-i)$* & $Ks^{2M}$ & 2.843 & -1.093 & 0.004 \\
0.07--0.34 & & & & \\
\hline
$(r-z)$ & $g$ & 3.101 & -1.323 & 0.008 \\
$(r-z)$ & $r$ & 2.395 & -1.007 & 0.008 \\
$(r-z)$ & $i$ & 2.128 & -0.937 & 0.008 \\
$(r-z)$ & $z$ & 1.995 & -0.958 & 0.008 \\
$(r-z)$ & $J^{2M}$ & 1.864 & -1.067 & 0.008 \\
$(r-z)$ & $H^{2M}$ & 1.843 & -1.223 & 0.008 \\
$(r-z)$ & $Ks^{2M}$ & 1.885 & -1.302 & 0.008 \\
0.2--0.62 & & & & \\
\hline
\end{tabular}
\end{scriptsize}
\end{table*}
\begin{figure*}
\begin{center}
\includegraphics[scale=0.9]{fig7}
\caption[CMLR for exponential SFH]{CMLR combining $B$ and $K$ band $M_\star/L$
versus $(B-R)$ and $(I-K)$ colours for the full range of $b$ parameters and
metallicities. The solid lines connect models with the same
metallicity (colour coding as in Fig.~\protect{\ref{fig:luminosity_ratios}});
the dashed lines connect models with the same $b$ parameter:
from top to bottom, $b=$ 0.01, 0.15, 0.40, 1.00, 1.75, 3.67, 10.00.
The dash-dotted line represents a pure SSP of age 10 Gyr.}
\label{fig:exp_BRIK_full}
\end{center}
\end{figure*}
\subsection{Comparison to old literature and to the no--TPAGB case}
To highlight the importance of the TP-AGB phase for CMLR, we compare
to the previous theoretical relations of \citet[][hereafter BdJ]{Bell_apj_550},
based on an early release of the GALAXEV population synthesis package by
\citet{Bruzual_mn_344}. Our Fig.~\ref{fig:exp_BRIK} mimics Fig.~2 of BdJ
by limiting to exponentially decreasing SFHs ($b \leq 1$, representative
of ``normal'' galaxies) and mostly plotting the same metallicities (albeit
their plot extends up to $Z=0.05$ and ours down to $Z=0.0001$). The two figures
are directly comparable, safe for a systematic offset in $M_\star/L$ zero-point
of about 0.2~dex, due to the different IMF (Salpeter vs.\ Kroupa) and
older age (12 vs.\ 10~Gyr) in their models.
The bottom panel of Fig.~\ref{fig:exp_BRIK} excludes the contribution of
the TP-AGB phase (we computed no--TPAGB SSPs from the Padova
isochrones, halting integration at the mass point where the Mhec flag marks
the extension to the TP-AGB phase).
The outcome is remarkably similar to Fig.~2 of BdJ, reasserting that early
versions of GALAXEV effectively neglected the TP-AGB contribution
\citep{Maraston_mn_362},
although this has improved in later versions \citep{Bruzual_aspc_374}.
With respect to the no--TPAGB case, the optical CMLR becomes marginally
less tight in the range $Z=0.004-0.03$, while $(I-K)$ becomes an even
more neat metallicity indicator, independent of SFH.
Most interesting is the $M_\star/L_K$--$(B-R)$ CMLR:
the ``wedge'' pattern in the no--TPAGB case, also seen in BdJ, significantly
changes when the TP-AGB phase is included: this CMLR becomes lighter,
steeper, and tighter in the metallicity range $Z=0.004-0.03$.
The same applies to NIR $M_\star/L$ versus optical colours in general.
\begin{figure}
\begin{center}
\includegraphics[scale=0.56]{fig8a}
\vspace{0.5truecm}
\includegraphics[scale=0.56]{fig8b}
\caption[CMLR with limited $b$ parameter values]{Same as
Fig.~\ref{fig:exp_BRIK_full}, but limited to $b \leq 1$.
In the lower panel, the TP-AGB phase has been excluded.}
\label{fig:exp_BRIK}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.4]{fig9}
\caption{Reproduction of Fig.~20 by \citet{Bell_apjss_149}: location of
galaxies with multi-band SDSS+2MASS photometry in the $M/L_K$ vs.\ $(B-R)$
plane; the solid line is the ``semi--empirical'' CMLR derived by Bell et~al.\
the dashed line is the theoretical relation of BdJ.
Overlayed is a grid of exponential models obtained by BdJ with population
synthesis models available at the time.}
\label{fig:bell03}
\end{center}
\end{figure}
\begin{figure*}
\begin{center}
\includegraphics[scale=0.62,angle=270]{fig10}
\caption{CMLR for the chemo--photometric models of disc galaxies of
PST04 with Salpeter and Kroupa IMF, updated with the new SSP set.
Symbols as in Fig.~B1 of PST04. Open symbols: one--zone models for individual
disc annuli; filled symbols: multi--zone models (integrated photometry
of bulge region, disc region, or global galaxy); asterisks, ``optical disc''
region; see PST04 for more details. The solid lines represent the linear fits
for the different IMFs; the dashed line is the superseded CMLR of PST04 for the
Kroupa IMF.}
\label{fig:CMLR_PST04}
\end{center}
\end{figure*}
The ``wedge'' pattern in the $M_\star/L_K$--$(B-R)$ relation, typical of
population synthesis models of the early 2000's, directly reflected in the
multi--band analysis of the galaxy mass function of
\citet{Bell_apjss_149}. In Fig.~\ref{fig:bell03} we reproduce their Fig.~20,
where each galaxy was assigned a location in the CMLR plane by
$\chi^2$--optimization within the underlying grid of exponential SFH models;
to this we overlay the theoretical grid
from Fig.~2 of BdJ, vertically shifted to adjust it to the IMF normalization
adopted by Bell et~al.\footnote{Although the model grid of
\citet{Bell_apjss_149}
was based on the PEGASE package rather than on GALAXEV, \citet{Maraston_mn_362}
showed that the PEGASE and GALAXEV synthesis models shared the same limits
as to TP-AGB phase implementation.}
Clearly the colour--$M_\star/L$ pattern predicted at the time beared on
the estimated $M_\star/L$ ratios and on the resulting
``semi--empirical'' CMLR of \citet{Bell_apjss_149}, that is much flatter
than the original theoretical relation of BdJ and has been widely used
thereafter. Considering the rather different CMLR predicted by modern models,
and the crucial role of NIR
light in determining stellar mass, those results on the stellar mass function
and semi--empirical CMLR are worth a revision.
Also, we argue against the blind application of
semi-empirical relations established for the galaxy population as a whole,
when interpreting photometric properties within individual galaxies
\citep[e.g.][]{Kassin_apj_643}. The very flat slope of the semi--empirical
CMLR of \citet{Bell_apjss_149} is evidently driven by a population of
objects with very blue optical colours but high NIR $M_\star/L$ ratios,
that stands aside of the rest of the trend as very metal--poor (dwarf?)
galaxies with old stellar populations.
Their chemical and
photometric properties are then quite different from those found within normal
disc galaxies, so that CMLR derived including these objects do not apply
to colour profiles in spiral galaxies.
\begin{figure*}
\begin{center}
\includegraphics[scale=1]{fig11}
\caption{CMLR for $M_\star/L_B$ vs.\ $(B-V)$ for the exponential models
with $b \leq 2$ (top panels) or $b \leq 1$ (bottom panels). {\it Left}:
CMLR from the ``new'' SSP dataset; {\it middle}: excluding the contribution
of the TP-AGB phase; {\it right}: CMLR from the ``old'' dataset.}
\label{fig:CMLR_break}
\end{center}
\end{figure*}
\section{Colour--$M_\star/L$ relations for disc galaxy models}
\label{sect:PST04}
In this section we recompute the CMLR for the disc galaxy models of
\citet[][hereinafter PST04]{Portinari_mn_347} and
discuss the updated CMLR relevant {\it within} individual galaxies.
Fig.~\ref{fig:CMLR_PST04} reproduces Fig.~B1 of PST04 with the new
photometric models --- showing only the Salpeter and Kroupa IMF case for
clarity. The old CMLR of PST04 for the Kroupa IMF case are shown as dashed
lines for comparison.
For purely optical CMLR (top panels) the new and old
CMLR differ at the blue end: the old CMLR were linear down to a certain break
point in colour (the blue limit of the dashed lines; see also
Fig.~B1 in PST04), below which $M_\star/L$ quickly dropped.
In the new CMLR, the relation steepens toward the blue (say, below 0.6 in
$B-V$, 0.9 in $B-R$) with no abrupt break; this smoother trend reflects
that of the new SSPs in Fig.~\ref{fig:ssp_comparison} (top mid panel).
Still, a single linear fit is adequate over a wide colour range, with a slope
somewhat steeper than in the CMLR of PST04. The fitting coefficients are listed
in Table~\ref{tab:fit_PST04} and Table~\ref{tab:fit_PST04_sdss}.
(Notice that disc models cover a smaller colour range than the exponential
models, as SFHs and metallicities do not get as extreme as considered
in the previous section.)
The break/steepening of the optical CMLR at the blue end seems supported
by observations \citep{McGaugh2005} so it is worth commenting in detail.
In the inside--out scenario, disc galaxies are characterized by radial
gradients in metallicity and SFH: blue colours in the disc outskirts result
from a combination of low metallicities {\it and} slow SFHs ($b \simeq 1$).
To illustrate the consequence of this folding,
Fig.~\ref{fig:CMLR_break} displays the $M_\star/L_B - (B-V)$ relation
from exponential models. For SFHs typical of ``normal'' galaxies ($b \leq 1$,
bottom panels) around $(B-V)=0.5$ there's a shift from $Z \geq 0.004$ to
$Z < 0.004$: bluer colours can only be obtained for lower metallicities,
which for a given SFH (e.g., $b=1$) have systematically lower optical
$M_\star/L$. Indeed, the old CMLR of PST04 had a break in $M_\star/L$ at
$(B-V) \simeq 0.5$. The effect is stronger when the
TP-AGB phase is properly included (left panels vs.\ mid panels)
and was even stronger with the old TP-AGB
prescriptions of \citet{Girardi_mn_300} (right panels), that led to the
abrupt break in the CMLR of PST04 --- while no such effect was seen
in the BdJ models for global galaxies.
This shows that, albeit linear $\log M_\star/L$---colour relations are
a reasonable and handy approximation over a very large range in colours,
for detailed studies it is worth to consider realistic chemical evolution
models and the role of the mass--metallicity relation (for galaxies in general)
or of metallicity gradients (within individual objects).
As the drop in $M_\star/L$ at the blue end is favoured by the data
\citep{McGaugh2005}, it would be interesting to apply the new, smoother
but steeper CMLR.
In any case, in the outskirts of disc galaxies the $M_\star/L$ ratio should
be lower than predicted by the BdJ recipe
\citep[and by the even flatter semi--empirical relations
of][]{Bell_apjss_149}.
When considering red or NIR bands (bottom panels in
Fig.~\ref{fig:CMLR_PST04}), the CMLR relation flattens out at the blue end,
with a large scatter. This is also consequence of the metallicity gradients:
blue colours correspond to the metal--poor outskirts of discs, and
at low $Z$ red--NIR $M_\star/L$ ratios tend to be larger and do not follow
tight CMLR (Fig.~\ref{fig:exp_BRIK}, left panels). In Table~\ref{tab:fit_PST04}
and~\ref{tab:fit_PST04_sdss} we indicate the blue limit for the log--linear
CMLR, below which flattening occurs and scatter in $M_\star/L$ at given
colour becomes significant, exceeding $\pm 0.1$~dex.
The bottom right panel of Fig.~\ref{fig:CMLR_PST04} shows $M_\star/L_K$
vs.\ $(V-K)$, representative of optical--NIR CMLR in general.
As $(V-K)$ is mostly a metallicity tracer, rather than a $M_\star/L$ tracer,
the new relation is very steep at the red end
(central regions of disc galaxies with high metallicities) and flattens out
below $(V-K) = 2.6$, presenting everywhere a large scatter.
Though we still provide CMLR based on optical--NIR
colours in Table~\ref{tab:fit_PST04}, we remark with asterisks that these
colours are not good tracers of $M_\star/L$.
Optical colours should always be preferred --- even when estimating NIR
$M_\star/L$ (Fig.~\ref{fig:CMLR_BR_MLK}). The new
optical colour---NIR $M_\star/L$ relations are steeper,
with lower $M_\star/L$ at a given colour; and
are nicely tight down to some blue limit ($\sim$0.9 for $B-R$)
below which they display the typical flattening and large scatter of the low
metallicity regimes.
We remark again that, when studying the profiles of disc galaxies (for the sake
of decomposing rotation curves, for instance) this sort of steep CMLR should be
adopted, as they correspond to self--consistent folding of SF and metal
enrichment histories, with low metallicities always associated to slow
SFH and young stellar populations in the outer regions.
``Global'' CMLR for the
galaxy population as a whole \citep{Bell_apjss_149} are much flatter due to
the contribution of galaxies with metal--poor old populations
(Fig.~\ref{fig:bell03}), that have no counterpart in disc galaxies.
Depending on the type of problem at hand, the suitable set of CMLR (``local''
or ``global'', theoretical or semi--empirical) should be adopted.
\begin{figure}
\begin{center}
\includegraphics[scale=0.26,angle=270]{fig12}
\caption{$M_\star/L_K$ vs.\ $(B-R)$ relation for the disc galaxy models of PST04.
Lines and symbols as in Fig.~\protect{\ref{fig:CMLR_PST04}}.}
\label{fig:CMLR_BR_MLK}
\end{center}
\end{figure}
\begin{table*}
\caption{Fitting coefficients of CMLR of the form :
$\log (M_\star/L) = s \times {\rm colour} + z$ from disc galaxy models.
Zero--points are given for the Kroupa and the Salpeter IMF normalizations.
Asterisks warn against those colours that are not optimal $M_\star/L$ tracers.
For NIR $M_\star/L$ ratios, we also indicate the blue limit below which the
relation flattens out (see text). The colour range effectively covered by the
disc galaxy models is indicated at the bottom of each sub-table.}
\label{tab:fit_PST04}
\begin{scriptsize}
\begin{tabular}{lccccc}
\hline
\hline
colour & $M_\star/L$ & $s$ & $z_{Kr}$ & $z_{Salp}$ & blue lim \\
\hline
$(B-V)$ & $B$ & 2.027 & -1.168 & -0.933 & \\
$(B-V)$ & $V$ & 1.627 & -0.900 & -0.665 & \\
$(B-V)$ & $R$ & 1.438 & -0.811 & -0.580 & \\
$(B-V)$ & $I$ & 1.294 & -0.782 & -0.559 & \\
$(B-V)$ & $J$ & 1.074 & -0.832 & -0.634 & $>$0.5 \\
$(B-V)$ & $H$ & 1.042 & -0.958 & -0.770 & $>$0.5 \\
$(B-V)$ & $K$ & 1.070 & -1.023 & -0.836 & $>$0.5 \\
$(B-V)$ & $J^{2M}$ & 1.050 & -0.812 & -0.618 & $>$0.5 \\
$(B-V)$ & $H^{2M}$ & 1.032 & -0.953 & -0.767 & $>$0.5 \\
$(B-V)$ & $Ks^{2M}$ & 1.054 & -1.020 & -0.837 & $>$0.5 \\
0.35--0.9 & & & & & \\
\hline
$(B-R)$ & $B$ & 1.374 & -1.384 & -1.164 & \\
$(B-R)$ & $V$ & 1.106 & -1.077 & -0.855 & \\
$(B-R)$ & $R$ & 0.974 & -0.964 & -0.744 & \\
$(B-R)$ & $I$ & 0.870 & -0.912 & -0.699 & \\
$(B-R)$ & $J$ & 0.753 & -0.976 & -0.785 & $>$0.9 \\
$(B-R)$ & $H$ & 0.731 & -1.098 & -0.916 & $>$0.9 \\
$(B-R)$ & $K$ & 0.750 & -1.168 & -0.987 & $>$0.9 \\
$(B-R)$ & $J^{2M}$ & 0.735 & -0.953 & -0.765 & $>$0.9 \\
$(B-R)$ & $H^{2M}$ & 0.723 & -1.093 & -0.913 & $>$0.9 \\
$(B-R)$ & $Ks^{2M}$ & 0.739 & -1.162 & -0.985 & $>$0.9 \\
0.7--1.45 & & & & & \\
\hline
$(B-I)$ & $B$ & 1.141 & -1.688 & -1.480 & $>$1.25 \\
$(B-I)$ & $V$ & 0.916 & -1.317 & -1.103 & $>$1.25 \\
$(B-I)$ & $R$ & 0.814 & -1.187 & -0.974 & $>$1.25 \\
$(B-I)$ & $I$ & 0.741 & -1.136 & -0.928 & $>$1.25 \\
$(B-I)$ & $J$ & 0.638 & -1.166 & -0.977 & $>$1.40 \\
$(B-I)$ & $H$ & 0.627 & -1.297 & -1.115 & $>$1.40 \\
$(B-I)$ & $K$ & 0.641 & -1.367 & -1.187 & $>$1.40 \\
$(B-I)$ & $J^{2M}$ & 0.627 & -1.145 & -0.958 & $>$1.40 \\
$(B-I)$ & $H^{2M}$ & 0.625 & -1.297 & -1.116 & $>$1.40 \\
$(B-I)$ & $Ks^{2M}$ & 0.638 & -1.370 & -1.192 & $>$1.40 \\
1.0--2.0 & & & & & \\
\hline
$(B-K)$ & $B$ & 0.892 & -2.860 & -2.741 & $>$3.0 \\
$(B-K)$ & $V$ & 0.719 & -2.269 & -2.125 & $>$3.0 \\
$(B-K)$ & $R$ & 0.643 & -2.046 & -1.895 & $>$3.0 \\
$(B-K)$ & $I$ & 0.592 & -1.942 & -1.789 & $>$3.0 \\
$(B-K)$* & $J$ & 0.508 & -1.855 & -1.721 & $>$3.1 \\
$(B-K)$* & $H$ & 0.495 & -1.960 & -1.833 & $>$3.1 \\
$(B-K)$* & $K$ & 0.504 & -2.037 & -1.914 & $>$3.1 \\
2.4--3.9 & & & & & \\
\hline
$(B-Ks)$ & $B$ & 0.903 & -2.950 & -2.831 & $>$3.1 \\
$(B-Ks)$ & $V$ & 0.724 & -2.330 & -2.185 & $>$3.1 \\
$(B-Ks)$ & $R$ & 0.646 & -2.096 & -1.944 & $>$3.1 \\
$(B-Ks)$ & $I$ & 0.595 & -1.988 & -1.834 & $>$3.1 \\
$(B-Ks)$* & $J^{2M}$ & 0.509 & -1.886 & -1.754 & $>$3.2 \\
$(B-Ks)$* & $H^{2M}$ & 0.507 & -1.034 & -0.908 & $>$3.2 \\
$(B-Ks)$* & $Ks^{2M}$ & 0.516 & -2.116 & -1.995 & $>$3.2 \\
2.5--3.9 & & & & & \\
\hline
$(V-R)$ & $B$ & 4.321 & -1.867 & -1.682 & $>$0.38 \\
$(V-R)$ & $V$ & 3.478 & -1.465 & -1.270 & $>$0.38 \\
$(V-R)$ & $R$ & 3.078 & -1.313 & -1.118 & $>$0.38 \\
$(V-R)$ & $I$ & 2.773 & -1.235 & -1.044 & $>$0.38 \\
$(V-R)$ & $J$ & 2.430 & -1.270 & -1.101 & $>$0.40 \\
$(V-R)$ & $H$ & 2.311 & -1.359 & -1.201 & $>$0.40 \\
$(V-R)$ & $K$ & 2.386 & -1.442 & -1.285 & $>$0.40 \\
$(V-R)$ & $J^{2M}$ & 2.353 & -1.230 & -1.064 & $>$0.40 \\
$(V-R)$ & $H^{2M}$ & 2.267 & -1.341 & -1.185 & $>$0.40 \\
$(V-R)$ & $Ks^{2M}$ & 2.315 & -1.415 & -1.263 & $>$0.40 \\
0.32--0.58 & & & & & \\
\hline
$(V-I)$ & $B$ & 2.677 & -2.423 & -2.253 & $>$0.85 \\
$(V-I)$ & $V$ & 2.156 & -1.916 & -1.732 & $>$0.85 \\
$(V-I)$ & $R$ & 1.922 & -1.724 & -1.538 & $>$0.85 \\
$(V-I)$ & $I$ & 1.756 & -1.632 & -1.448 & $>$0.85 \\
$(V-I)$ & $J$ & 1.549 & -1.629 & -1.464 & $>$0.88 \\
$(V-I)$ & $H$ & 1.508 & -1.738 & -1.580 & $>$0.88 \\
$(V-I)$ & $K$ & 1.546 & -1.822 & -1.667 & $>$0.88 \\
$(V-I)$ & $J^{2M}$ & 1.513 & -1.592 & -1.429 & $>$0.88 \\
$(V-I)$ & $H^{2M}$ & 1.495 & -1.729 & -1.573 & $>$0.88 \\
$(V-I)$ & $Ks^{2M}$ & 1.528 & -1.812 & -1.659 & $>$0.88 \\
0.70--1.14 & & & & & \\
\hline
\end{tabular}
\begin{tabular}{lccccc}
\hline
\hline
colour & $M_\star/L$ & $s$ & $z_{Kr}$ & $z_{Salp}$ & blue lim \\
\hline
$(V-J)$* & $B$ & 1.648 & -2.924 & -2.862 & $>$1.7 \\
$(V-J)$* & $V$ & 1.348 & -2.358 & -2.259 & $>$1.7 \\
$(V-J)$* & $R$ & 1.213 & -2.141 & -2.030 & $>$1.7 \\
$(V-J)$* & $I$ & 1.126 & -2.047 & -1.930 & $>$1.7 \\
$(V-J)$* & $J$ & 0.992 & -1.991 & -1.890 & $>$1.8 \\
$(V-J)$* & $H$ & 0.985 & -2.127 & -2.031 & $>$1.8 \\
$(V-J)$* & $K$ & 1.009 & -2.220 & -2.128 & $>$1.8 \\
1.35--2.15 & & & & & \\
\hline
$(V-J^{2M})$* & $B$ & 1.478 & -2.669 & -2.611 & $>$1.75 \\
$(V-J^{2M})$* & $V$ & 1.197 & -2.126 & -2.031 & $>$1.75 \\
$(V-J^{2M})$* & $R$ & 1.072 & -1.923 & -1.815 & $>$1.75 \\
$(V-J^{2M})$* & $I$ & 0.988 & -1.832 & -1.718 & $>$1.75 \\
$(V-J^{2M})$* & $J^{2M}$ & 0.947 & -1.951 & -1.850 & $>$1.85 \\
$(V-J^{2M})$* & $H^{2M}$ & 0.959 & -2.129 & -2.033 & $>$1.85 \\
$(V-J^{2M})$* & $Ks^{2M}$ & 0.982 & -2.226 & -2.135 & $>$1.85 \\
1.4--2.2 & & & & & \\
\hline
$(V-H)$* & $B$ & 1.610 & -4.011 & -3.969 & $>$2.45 \\
$(V-H)$* & $V$ & 1.336 & -3.297 & -3.214 & $>$2.45 \\
$(V-H)$* & $R$ & 1.180 & -2.927 & -2.830 & $>$2.45 \\
$(V-H)$* & $I$ & 1.094 & -2.773 & -2.670 & $>$2.45 \\
$(V-H)$* & $J$ & 0.981 & -2.679 & -2.589 & $>$2.50 \\
$(V-H)$* & $H$ & 0.979 & -2.823 & -2.738 & $>$2.50 \\
$(V-H)$* & $K$ & 1.000 & -2.925 & -2.844 & $>$2.50 \\
1.9--2.9 & & & & & \\
\hline
$(V-H^{2M})$* & $B$ & 1.571 & -3.956 & -3.917 & $>$2.45 \\
$(V-H^{2M})$* & $V$ & 1.282 & -3.194 & -3.114 & $>$2.45 \\
$(V-H^{2M})$* & $R$ & 1.153 & -2.892 & -2.797 & $>$2.45 \\
$(V-H^{2M})$* & $I$ & 1.070 & -2.744 & -2.643 & $>$2.45 \\
$(V-H^{2M})$* & $J^{2M}$ & 0.941 & -2.599 & -2.512 & $>$2.50 \\
$(V-H^{2M})$* & $H^{2M}$ & 0.954 & -2.789 & -2.707 & $>$2.50 \\
$(V-H^{2M})$* & $Ks^{2M}$ & 0.976 & -2.899 & -2.822 & $>$2.50 \\
1.95--2.90 & & & & & \\
\hline
$(V-K)$* & $B$ & 1.689 & -4.465 & -4.443 & $>$2.60 \\
$(V-K)$* & $V$ & 1.373 & -3.595 & -3.528 & $>$2.60 \\
$(V-K)$* & $R$ & 1.231 & -3.242 & -3.159 & $>$2.60 \\
$(V-K)$* & $I$ & 1.141 & -3.064 & -2.973 & $>$2.60 \\
$(V-K)$* & $J$ & 1.034 & -2.970 & -2.892 & $>$2.65 \\
$(V-K)$* & $H$ & 1.035 & -3.123 & -3.050 & $>$2.65 \\
$(V-K)$* & $K$ & 1.054 & -3.222 & -3.153 & $>$2.65 \\
2.05--3.00 & & & & & \\
\hline
$(V-Ks^{2M})$* & $B$ & 1.628 & -4.382 & -4.363 & $>$2.65 \\
$(V-Ks^{2M})$* & $V$ & 1.331 & -3.549 & -3.485 & $>$2.65 \\
$(V-Ks^{2M})$* & $R$ & 1.198 & -3.214 & -3.134 & $>$2.65 \\
$(V-Ks^{2M})$* & $I$ & 1.113 & -3.046 & -2.958 & $>$2.65 \\
$(V-Ks^{2M})$* & $J^{2M}$ & 0.979 & -2.867 & -2.791 & $>$2.72 \\
$(V-Ks^{2M})$* & $H^{2M}$ & 0.991 & -3.056 & -2.985 & $>$2.72 \\
$(V-Ks^{2M})$* & $Ks^{2M}$ & 1.015 & -3.176 & -3.110 & $>$2.72 \\
2.1--3.1 & & & & & \\
\hline
$(R-I)$ & $B$ & 7.003 & -3.315 & -3.172 & $>$0.45 \\
$(R-I)$ & $V$ & 5.614 & -2.620 & -2.457 & $>$0.45 \\
$(R-I)$ & $R$ & 5.005 & -2.354 & -2.186 & $>$0.45 \\
$(R-I)$ & $I$ & 4.605 & -2.223 & -2.054 & $>$0.45 \\
$(R-I)$ & $J$ & 3.964 & -2.099 & -1.955 & $>$0.47 \\
$(R-I)$ & $H$ & 3.862 & -2.196 & -2.059 & $>$0.47 \\
$(R-I)$ & $K$ & 3.955 & -2.290 & -2.156 & $>$0.47 \\
$(R-I)$ & $J^{2M}$ & 3.874 & -2.052 & -1.910 & $>$0.47 \\
$(R-I)$ & $H^{2M}$ & 3.834 & -2.186 & -2.050 & $>$0.47 \\
$(R-I)$ & $Ks^{2M}$ & 3.915 & -2.278 & -2.146 & $>$0.47 \\
0.37--0.57 & & & & & \\
\hline
& & & & & \\
& & & & & \\
& & & & & \\
& & & & & \\
& & & & & \\
& & & & & \\
& & & & & \\
& & & & & \\
& & & & & \\
& & & & & \\
& & & & & \\
& & & & & \\
\end{tabular}
\end{scriptsize}
\end{table*}
\begin{table}
\caption{Same as Table~\protect{\ref{tab:fit_PST04}} but for SDSS colours.}
\label{tab:fit_PST04_sdss}
\begin{footnotesize}
\begin{tabular}{lccccc}
\hline
\hline
colour & $\log (M_\star/L)$ & $s$ & $z_{Kr}$ & $z_{Salp}$ & blue lim \\
\hline
$(g-r)$ & $g$ & 1.930 & -0.851 & -0.634 & \\
$(g-r)$ & $r$ & 1.530 & -0.663 & -0.445 & \\
$(g-r)$ & $i$ & 1.370 & -0.633 & -0.420 & \\
$(g-r)$ & $z$ & 1.292 & -0.665 & -0.462 & $>$0.3 \\
$(g-r)$ & $J^{2M}$ & 1.139 & -0.732 & -0.544 & $>$0.4 \\
$(g-r)$ & $H^{2M}$ & 1.128 & -0.880 & -0.699 & $>$0.4 \\
$(g-r)$ & $Ks^{2M}$ & 1.153 & -0.945 & -0.767 & $>$0.4 \\
0.25--0.75 & & & & & \\
\hline
$(g-i)$ & $g$ & 1.385 & -0.899 & -0.698 & $>$0.5 \\
$(g-i)$ & $r$ & 1.098 & -0.702 & -0.498 & $>$0.5 \\
$(g-i)$ & $i$ & 0.985 & -0.669 & -0.468 & $>$0.5 \\
$(g-i)$ & $z$ & 0.898 & -0.675 & -0.484 & $>$0.5 \\
$(g-i)$ & $J^{2M}$ & 0.868 & -0.804 & -0.621 & $>$0.6 \\
$(g-i)$ & $H^{2M}$ & 0.861 & -0.952 & -0.776 & $>$0.6 \\
$(g-i)$ & $Ks^{2M}$ & 0.879 & -1.019 & -0.847 & $>$0.6 \\
0.35--1.05 & & & & & \\
\hline
$(g-z)$ & $g$ & 1.294 & -1.097 & -0.908 & $>$0.7 \\
$(g-z)$ & $r$ & 1.033 & -0.867 & -0.669 & $>$0.7 \\
$(g-z)$ & $i$ & 0.946 & -0.838 & -0.642 & $>$0.7 \\
$(g-z)$ & $z$ & 0.894 & -0.861 & -0.672 & $>$0.7 \\
$(g-z)$ & $J^{2M}$ & 0.789 & -0.906 & -0.734 & $>$0.8 \\
$(g-z)$ & $H^{2M}$ & 0.787 & -1.060 & -0.893 & $>$0.8 \\
$(g-z)$ & $Ks^{2M}$ & 0.803 & -1.128 & -0.965 & $>$0.8 \\
0.7--1.3 & & & & & \\
\hline
$(r-i)$ & $g$ & 5.926 & -1.293 & -1.116 & $>$0.20 \\
$(r-i)$ & $r$ & 4.738 & -1.025 & -0.838 & $>$0.20 \\
$(r-i)$ & $i$ & 4.338 & -0.982 & -0.795 & $>$0.20 \\
$(r-i)$ & $z$ & 4.093 & -0.996 & -0.816 & $>$0.20 \\
$(r-i)$ & $J^{2M}$ & 3.684 & -1.043 & -0.878 & $>$0.21 \\
$(r-i)$ & $H^{2M}$ & 3.659 & -1.192 & -1.033 & $>$0.21 \\
$(r-i)$ & $Ks^{2M}$ & 3.740 & -1.264 & -1.109 & $>$0.21 \\
0.12--0.32 & & & & & \\
\hline
$(r-z)$ & $g$ & 3.902 & -1.592 & -1.466 & $>$0.38 \\
$(r-z)$ & $r$ & 3.116 & -1.263 & -1.116 & $>$0.38 \\
$(r-z)$ & $i$ & 2.863 & -1.205 & -1.054 & $>$0.38 \\
$(r-z)$ & $z$ & 2.716 & -1.214 & -1.067 & $>$0.38 \\
$(r-z)$ & $J^{2M}$ & 2.355 & -1.196 & -1.067 & $>$0.40 \\
$(r-z)$ & $H^{2M}$ & 2.344 & -1.346 & -1.223 & $>$0.40 \\
$(r-z)$ & $Ks^{2M}$ & 2.384 & -1.416 & -1.297 & $>$0.40 \\
0.22--0.56 & & & & & \\
\hline
\end{tabular}
\end{footnotesize}
\end{table}
\section{Attenuation by interstellar dust}\label{sec:attenuation}
We have discussed in Section~\ref{sec:circumstellar_dust} that circumstellar
dust around AGB stars has a negligible effect on optical and NIR CMLR, so that
we can disregard it. In this section we address the effect of interstellar
dust, that both reddens and dims stellar luminosity.
\citet{Bell_apj_550} argue that, to first order, in optical CMLR the two
effects compensate each other, as the dust vector runs almost parallel to the
dust--free CMLR (age--metallicity--dust degeneracy).
We revisit the effects of interstellar dust on optical and NIR
CMLR over galactic scales, taking advantage of recent recipes based on
detailed radiative transfer models
\citep{Tuffs_aa_419}. Our goal is to provide CMLR that are statistically
applicable to large galaxy samples that include a range of morphologies,
intrinsic colours and random inclinations.
\subsection{The attenuated spiral galaxy models}
We construct simple models of spiral galaxies consisting of bulge+disc,
and then apply the dust corrections of
\citet{Tuffs_aa_419}.
The models include a bulge component, represented by a
10~Gyr old SSP, and a disc component with exponential SFH with $b \leq 2$
(Section~\ref{sec:simple_exp_models}):
$b \leq 1$ values represent ``normal'' disc galaxy morphologies,
and we extend to somewhat bluer objects with recent intensive SF.
For simplicity, the dust--free models were calculated only for solar
metallicity: we shall see that dust effects on galactic scales
are statistically more relevant than metallicity effects.
The models have a random distribution in $b=0.01 \div 2$ and in
bulge--to--total ratio B/T=$0 \div 0.6$, typical of disc galaxies
\citep{Allen_mn_371}; the corresponding luminosity ratios (B/T)$_\lambda$
entering Eq.~\ref{eq:optical_attenuation} below, are computed
self--consistently from the SFHs for bulge and disc.
Dust attenuation is then added for random inclinations from face-on to
edge--on, $0 \leq \cos(i) \leq 1$, for a total of 20000 models.
Dust attenuation in the various bands is computed
following the prescriptions of \citet{Tuffs_aa_419}, to which the reader
is referred for all details. Their dust models have been
successfully tested both on multi--band emission of individual galaxies
\citep{Popescu_aa_362} and on large galaxy surveys \citep{Driver_mn_379}.
The attenuation prescriptions of Tuffs et~al. depend only on the
properties of interstellar dust, not on the incident stellar radiation field,
and are presented as applicable to real disc galaxies irrespectively of
their detailed stellar SED; likewise, they are applicable to model disc
galaxies irrespectively of the specific population synthesis model used
to generate them.
\begin{figure*}
\includegraphics[scale=0.25,angle=-90]{fig13a}
\includegraphics[scale=0.25,angle=-90]{fig13b}
\includegraphics[scale=0.25,angle=-90]{fig13c}
\includegraphics[scale=0.25,angle=-90]{fig13d}
\caption{CMLR for dust--free model galaxies
(black dots) and attenuated models (red dots). The solid line indicates
the linear fit to the attenuated models.}
\label{fig:dust}
\end{figure*}
In short, the dust model includes a clumpy dust component associated
with the star-forming regions in a thin disc \citep[which affects only
UV light, and can be neglected in the
optical/NIR;][]{Popescu_aa_362,Tuffs_aa_419} and a diffuse dust component
residing both in the young thin disc and in the older disc. The bulge
is intrinsically dust--free, but is heavily affected by the dust in the disc,
to the extent that it turns out to be the most attenuated galaxy component;
a counter--intuitive result confirmed by observational trends
\citep{Driver_mn_379}.
The equation for the composite attenuation of the galaxy at wavelength
$\lambda$ is:
\begin{equation} \label{eq:optical_attenuation}
\Delta m_\lambda = -2.5 \log \Bigg{(}
r^\mathrm{0,disc}_\lambda 10^{-0.4\Delta m^\mathrm{disc}_\lambda} +
r^\mathrm{0,bulge}_\lambda 10^{-0.4\Delta m^\mathrm{bulge}_\lambda } \Bigg{)}
\end{equation}
where $r^0_\lambda$ is the fraction of the bulge or disc to the total intrinsic
luminosity in the band $\lambda$: (B/T)$_\lambda$ and
(1-B/T)$_\lambda$, respectively. (We recast the equation in terms of the
intrinsic B/T ratio of the galaxy models,
while the original Eq.~18 in Tuffs et~al.\ was expressed in terms of
the apparent, dust--attenuated B/T ratio, to be applied to observed galaxies.)
$\Delta m^\mathrm{disc,bulge}_\lambda$ is the attenuation value in magnitudes
for the disc or bulge, provided by Tuffs et~al.\ in the
form of polynomial equations in $(1-\cos i)$. The polynomial coefficients
are tabulated for various optical and NIR photometric bands, and optical depth
values $\tau_B^f$.
We use the fixed value $\tau^f_B=4.0$, consistent with the
empirical mean opacity $\tau_B^f = 3.8 \,\pm \, 0.7$ found by
\citet{Driver_mn_379} based on 10095 galaxies with bulge-disc decomposition.
We notice in passing that
Eq.~\ref{eq:optical_attenuation} becomes inconsistent for high B/T values,
as it predicts an ever--increasing attenuation for increasing bulge fraction,
while in the limit {\mbox{B/T $\longrightarrow$ 1}} attenuation should
vanish together
with the disc component. We suspect that this is a consequence of extending
the disc+dust to the very centre, rather than modelling a hole in the dusty
disc in correspondence to the bulge component. However, here we only consider
galaxies with an important disc component (B/T $\leq$ 0.6) where the dust
prescriptions of Tuffs et al.
are well supported by observational evidence \citep{Driver_mn_379}.
\subsection{Colour--$M_\star/L$ relations for dusty galaxies}
Fig.~\ref{fig:dust} shows the effects of dust on various CMLR.
The dust--free galaxies (black dots) are intrinsically redder with decreasing
disc $b$--parameter and increasing B/T ratio. They follow, within the quoted
uncertainty of 0.1~dex, the CMLR predicted by the exponential models of
Section~\ref{sec:simple_exp_models}: the
superposition of two components, disc+bulge (exponential SFH + old SSP)
still closely follow the one--component CMLR ---
but there is hardly any CMLR in $(I-K)$, as discussed in
Section~\ref{sec:simple_exp_models}.
When dust attenuation is taken into account (red dots), the models spread on
a much wider area in the plot, the brighter (fainter) bound
corresponding to face--on (edge--on) inclinations.
The top panels show optical CMLR. The spread in $\log M_\star/L$ at a fixed
colour is about 0.5~dex (a factor--of--3), with dusty galaxies being
tipically less luminous than the dust--free case. Notice that the red end
of the dust--free
distribution is scattered much further to the red by attenuation, than the
blue end; this is another counter--intuitive result of the bulge component
suffering more extinction than the disc.
The largest departures from the dust--free case correspond to the
highest inclinations, that are statistically less frequent; altogether the
{\it statistical} CMLR of attenuated galaxies (solid line) is not too far
from the dust--free exponential case of Section~\ref{sec:simple_exp_models}.
In the $\log M_\star/L$ vs.\ $(B-R)$ plane the dusty CMLR
is mildly offset (0.1~dex) from the dust--free one,
toward heavier $M_\star/L_B$ at fixed colour.
The (statistical) effect of dust is smallest for $M_\star/L_V$ vs.\ $(V-I)$:
the attenuated CMLR just shows a bit steeper slope than the
dust--free one, and the difference is even less than 0.1~dex. This is
interesting as
\citet{Gallazzi_Bell2009} have recently selected $(g-i)$ as the best
suited colour for stellar mass estimates in the dust--free case. As this colour
is quite close to $(V-I)$, we argue that dust corrections are small
for CMLR involving {\mbox{$(g-i)$}, so that this is a robust optical
$M_\star/L$ indicator also with respect to dust effects.
(We cannot address
the role of dust attenuation directly in SDSS bands, as the Tuffs et~al.\
prescriptions are only provided for Johnson bands.)
\begin{table*}
\caption{Fitting coefficients of CMLR of the form :
$\log (M_\star/L) = s \times {\rm colour} + z$ for dusty galaxy models.}
\label{tab:fit_dustymodels}
\begin{tabular}{lccc}
\hline
\hline
colour & $\log (M_\star/L)$ & $s$ & $z$ \\
\hline
$(B-V)$ & $B$ & 1.836 & -0.905 \\
$(B-V)$ & $V$ & 1.493 & -0.681 \\
$(B-V)$ & $R$ & 1.337 & -0.627 \\
$(B-V)$ & $I$ & 1.228 & -0.646 \\
$(B-V)$ & $J$ & 1.048 & -0.803 \\
$(B-V)$ & $H$ & 0.941 & -0.909 \\
$(B-V)$ & $K$ & 0.866 & -0.926 \\
0.5--1.1 & & & \\
\hline
$(B-R)$ & $B$ & 1.272 & -1.173 \\
$(B-R)$ & $V$ & 1.040 & -0.905 \\
$(B-R)$ & $R$ & 0.934 & -0.832 \\
$(B-R)$ & $I$ & 0.860 & -0.837 \\
$(B-R)$ & $J$ & 0.730 & -0.960 \\
$(B-R)$ & $H$ & 0.652 & -1.047 \\
$(B-R)$ & $K$ & 0.599 & -1.051 \\
0.9--1.8 & & & \\
\hline
$(B-I)$ & $B$ & 1.049 & -1.492 \\
$(B-I)$ & $V$ & 0.859 & -1.167 \\
$(B-I)$ & $R$ & 0.772 & -1.069 \\
$(B-I)$ & $I$ & 0.711 & -1.057 \\
$(B-I)$ & $J$ & 0.602 & -1.142 \\
$(B-I)$ & $H$ & 0.537 & -1.209 \\
$(B-I)$ & $K$ & 0.493 & -1.200 \\
1.4--2.5 & & & \\
\hline
$(B-J)$ & $B$ & 0.886 & -2.203 \\
$(B-J)$ & $V$ & 0.724 & -1.746 \\
$(B-J)$ & $R$ & 0.649 & -1.585 \\
$(B-J)$ & $I$ & 0.597 & -1.527 \\
$(B-J)$ & $J$ & 0.505 & -1.541 \\
$(B-J)$ & $H$ & 0.452 & -1.567 \\
$(B-J)$ & $K$ & 0.416 & -1.532 \\
2.4--3.9 & & & \\
\hline
$(B-H)$ & $B$ & 0.803 & -2.604 \\
$(B-H)$ & $V$ & 0.656 & -2.073 \\
$(B-H)$ & $R$ & 0.588 & -1.876 \\
$(B-H)$ & $I$ & 0.540 & -1.793 \\
$(B-H)$ & $J$ & 0.457 & -1.767 \\
$(B-H)$ & $H$ & 0.409 & -1.771 \\
$(B-H)$ & $K$ & 0.377 & -1.720 \\
3.2--4.8 & & & \\
\hline
$(B-K)$ & $B$ & 0.734 & -2.496 \\
$(B-K)$ & $V$ & 0.600 & -1.984 \\
$(B-K)$ & $R$ & 0.537 & -1.797 \\
$(B-K)$ & $I$ & 0.493 & -1.719 \\
$(B-K)$ & $J$ & 0.418 & -1.705 \\
$(B-K)$ & $H$ & 0.374 & -1.716 \\
$(B-K)$ & $K$ & 0.345 & -1.670 \\
3.4--5.2 & & & \\
\hline
\end{tabular}
\begin{tabular}{lccc}
\hline
\hline
colour & $\log (M_\star/L)$ & $s$ & $z$ \\
\hline
$(V-R)$ & $B$ & 3.966 & -1.697 \\
$(V-R)$ & $V$ & 3.308 & -1.356 \\
$(V-R)$ & $R$ & 2.953 & -1.227 \\
$(V-R)$ & $I$ & 2.700 & -1.191 \\
$(V-R)$ & $J$ & 2.301 & -1.266 \\
$(V-R)$ & $H$ & 2.071 & -1.328 \\
$(V-R)$ & $K$ & 1.911 & -1.314 \\
0.4--0.7 & & & \\
\hline
$(V-I)$ & $B$ & 2.404 & -2.223 \\
$(V-I)$ & $V$ & 1.959 & -1.756 \\
$(V-I)$ & $R$ & 1.752 & -1.589 \\
$(V-I)$ & $I$ & 1.606 & -1.526 \\
$(V-I)$ & $J$ & 1.365 & -1.547 \\
$(V-I)$ & $H$ & 1.225 & -1.577 \\
$(V-I)$ & $K$ & 1.129 & -1.542 \\
0.9--1.4 & & & \\
\hline
$(V-J)$ & $B$ & 1.587 & -3.129 \\
$(V-J)$ & $V$ & 1.296 & -2.501 \\
$(V-J)$ & $R$ & 1.161 & -2.259 \\
$(V-J)$ & $I$ & 1.066 & -2.143 \\
$(V-J)$ & $J$ & 0.902 & -2.065 \\
$(V-J)$ & $H$ & 0.808 & -2.038 \\
$(V-J)$ & $K$ & 0.744 & -1.966 \\
1.9--2.7 & & & \\
\hline
$(V-H)$ & $B$ & 1.266 & -3.428 \\
$(V-H)$ & $V$ & 1.034 & -2.746 \\
$(V-H)$ & $R$ & 0.927 & -2.479 \\
$(V-H)$ & $I$ & 0.851 & -2.345 \\
$(V-H)$ & $J$ & 0.729 & -2.235 \\
$(V-H)$ & $H$ & 0.645 & -2.191 \\
$(V-H)$ & $K$ & 0.595 & -2.110 \\
2.7--3.8 & & & \\
\hline
$(V-K)$ & $B$ & 1.048 & -2.979 \\
$(V-K)$ & $V$ & 0.856 & -2.379 \\
$(V-K)$ & $R$ & 0.767 & -2.150 \\
$(V-K)$ & $I$ & 0.704 & -2.044 \\
$(V-K)$ & $J$ & 0.597 & -1.981 \\
$(V-K)$ & $H$ & 0.535 & -1.966 \\
$(V-K)$ & $K$ & 0.414 & -1.640 \\
2.9--4.2 & & & \\
\hline
$(I-K)$ & $B$ & 0.970 & -1.632 \\
$(I-K)$ & $V$ & 0.885 & -1.480 \\
$(I-K)$ & $R$ & 0.831 & -1.428 \\
$(I-K)$ & $I$ & 0.786 & -1.430 \\
$(I-K)$ & $J$ & 0.627 & -1.376 \\
$(I-K)$ & $H$ & 0.505 & -1.298 \\
$(I-K)$ & $K$ & 0.386 & -1.114 \\
2.0--2.7 & & & \\
\hline
\end{tabular}
\end{table*}
The role of dust is remarkable in optical--NIR CMLR
(bottom panels): inclination strongly influences the colours, and the strong
reddening renders the dusty optical--NIR CMLR {\it lighter}, at a given colour,
than the dust--free case.
Since the dust--free optical--NIR CMLR depend on metallicity,
we checked that changing the metallicity
of the galaxy models (within a factor of 2--3 from solar, as relevant for
global galaxy metallicities) has very little impact on the resulting dusty
CMLR: the effect of dust is dominant over metallicity effects.
In some optical--NIR CMLR, attenuation and reddening combine so as to maintain
tight CMLR --- albeit different from the dust--free case. The best
example is $(B-K)$, with a $M_\star/L$ scatter of at most $\pm$0.15~dex in all
bands (bottom left panel). Also $(V-H)$ is a good $M_\star/L$ tracer for
dusty galaxies, with a scatter of $\pm$0.15~dex in $M_\star/L_V$ or $M_\star/L_H$
at given $(V-H)$.
Even for the $(I-K) - M_\star/L_I$ relation, the scatter ($\pm$0.2~dex) is no
worse than that for optical dusty CMLR (bottom right).
In summary, when dust attenuated galaxies are considered, optical--NIR CMLR
are no worse (in terms of scatter) than optical--optical CMLR; in some cases
--- notably $(B-K)$ --- they are even favoured. The best option in dusty
optical--NIR CMLR is to derive $M_\star/L$ in one of the bands involved
in the base colour; a very common case in practice.
\section{Summary and conclusions}
In this paper we rediscuss theoretical colour--stellar mass-to-light relations
(CMLR) in the light of modern populations synthesis models including an accurate
implementation of the TP-AGB phase, and of the effects of interstellar dust
as predicted from radiative transfer models of disc galaxies.
The importance of the AGB phase for the integrated NIR luminosity of
stellar populations has been extensively discussed in recent years
\citep[][and other papers of the same series]{Maraston_mn_362, Maraston2006,
Tonini2009}.
In intermediate--age stellar populations (0.3--2 Gyr), TP-AGB stars dominate
the NIR luminosity, lowering the $M_\star/L$ ratio by up to a factor
of 2 and driving very red optical--NIR colours ($V-K \geq 3$);
NIR $M_\star/L$ ratios are quite independent of metallicity. Optical
luminosities and colours, on the other hand, are not severely affected.
We update theoretical CMLR for composite stellar populations by means
of the latest Padova isochrones (Marigo et~al.\ 2008; Girardi et ~al.\ 2010),
which include a far more refined treatment of the TP-AGB phase than
the previous dataset \citep{Girardi_mn_300, Girardi_aa_391}.
For most of their evolution after the onset of the AGB at $\sim$100~Myr,
the updated Single Stellar Populations are significanly brighter in the NIR
(up to 0.5~mag), with correspondingly ``lighter'' $M_\star/L$.
The effect of circumstellar dust on the integrated optical and NIR luminosity
is negligible.
Considering both Simple and composite Stellar Populations (the latter with
exponentially decreasing/increasing star formation histories mimicking the
Hubble sequence, as well as from full chemo--photometric galaxy models)
we highlight the following characteristics of updated CMLR.
\begin{itemize}
\item
Optical CMLR
are little affected by the upgrade in the AGB models, and remain tight thanks
to a strong age--metallicity degeneracy
\citep[cf.][]{Bell_apj_550}.
\item
The integrated NIR luminosity is increased with minor effects in the optical:
the resulting NIR $M_\star/L$ is about 0.1~dex lighter, at a given optical
colour.
\item
The integrated NIR luminosity is also less sensitive to metallicity --- at
least for $Z \geq 0.004$ which is representative of the bulk of the stellar
populations in galaxies. This favours a more robust estimate of stellar mass
from NIR light, with tighter NIR $M_\star/L$---optical colour relations
(like $M_\star/L_K$ vs.\ $B-R$) compared to previous predictions.
\item
In optical--NIR colours, such as $(V-K)$, the new models are both lighter
and redder, so that the new CMLR are much lighter (up to 0.3~dex, a factor
of 2) at a given colour.
\item
As noticed by \citet{Bell_apj_550}, optical--NIR colours like $(V-K)$ or
$(I-K)$ are mostly metallicity indicators, while being very poor $M_\star/L$
tracers; this is even more true with the new models.
\item
The new models suggests a revision of results obtained from multi--band
analysis of the galaxy population spanning from optical to NIR, including
semi--empirical CMLR \citep{Bell_apjss_149}.
\item
We argue against the use of semi--empirical CMLR established for the
general galaxy population, when studying individual galaxies: the CMLR
resulting from a coherent star formation and chemical evolution history
(and their radial gradients) within a single galaxy is potentially
different from the CMLR obtained as a statistical average of galaxies.
This is evident for instance in the ``break'' of the CMLR
at blue optical colours, seen in chemo--photometric models of disc galaxies
\citep{Portinari_mn_347} and not when considering a wider range of
(uncorrelated) SFH and metallicities describing the galaxy population
in general.
\end{itemize}
We finally warn that recent
observational results suggest that the newest population synthesis
models may actually overestimate the luminosity contribution of AGB stars
\citep[Kriek et~al.\ 2010;][]{Zibetti2012}, \nocite{Kriek2010}
partly due to an excess of rare, luminous AGB stars in the models
(Melbourne et~al.\ 2010) \nocite{Melbourne2012} and partly due to the effects
of circumstellar dust (Meidt et~al.\ 2012) \nocite{Meidt2012}.
Future, better calibrated AGB models may converge on CMLR
intermediate between the ``classic'' ones of the early 2000's and those
presented here.
All of the above refers to dust--free CMLR. It is usually
assumed that, at least for optical CMLR, dust is a second--order effect thanks
to the age--metallicity--dust degeneracy \citep{Bell_apj_550}. We revisited
this issue considering a more realistic implementation of dust effects on
galactic scales, and found that dust has a non--negligible role.
\begin{itemize}
\item
The combined effect of reddening and attenuation introduces an enormous scatter
even in optical CMLR: highly inclined disc
galaxies can be 0.5~dex ``heavier'' at a given colour, than predicted by
dust--free CMLR. So, for individual galaxies, the CMLR does not
apply unless good inclination information and dust corrections are available.
\item
Nontheless, we can still define ``dusty'' CMLR that {\it statistically}
apply to large
galaxy samples where we lack detailed morphological and inclination information
for individual objects (but are at least able to distinguish a disc--like
galaxy from a pure spheroid with no dust). In the optical, statistical dusty
CMLR are somewhat heavier (about 0.1~dex) at a given colour, than the dust-free
case.
\item
The smallest change with respect to the dust--free case is found
for $M_\star/L_V$ vs.\ $(V-I)$.
This suggests that the $(g-i)$ colour, recently selected as optimal
stellar mass tracer in the dust--free case \citep{Gallazzi_Bell2009},
remains a good tracer also when considering (or neglecting!) the effects
of dust.
\item
Dust reddening strongly alters optical--NIR CMLR, making them lighter,
at given colour, than dust--free CMLR.
\item
For some optical--NIR CMLR, extinction and reddening combine to yield rather
tight dusty CMLR --- albeit very different from the dust--free ones.
$(B-K)$ is an excellent stellar mass tracer for dusty galaxies,
with a scatter in $M_\star/L$ within $\pm$0.15~dex in all bands. $M_\star/L_{V,H}$
vs.\ $(V-H)$ CMLR are also similarly tight. For dusty galaxies, these
CMLR are better mass tracers than optical--optical CMLR.
\end{itemize}
In Tables~\ref{tab:fit_bmodels} through~\ref{tab:fit_dustymodels} we give
our updated CMLR for the various cases: exponential models (for those colours
where defining a CMLR is meaningful, at least for $Z \geq 0.004$);
chemo--photometric
disc galaxy models, where CMLR result from a consistent convolution of
SF and chemical evolution history, to be best applied within individual
(dust--free or dust--corrected) galaxies; and ``dusty'' CMLR that can be
statistically applied to large galaxy samples.
CMLR are a robust, handy tool to estimate
stellar masses; we can optimize their use by choosing the colour and
relation most suitable for each specific problem.
\section*{Acknowledgments}
We thank L\'eo Girardi and Paola Marigo for useful discussions on the
Padova stellar models.
This study was financed by the Academy of Finland (grants nr.~130951 and 218317)
and by the Magnus Ehrnrooth foundation.
\bibliographystyle{mn2e}
|
1,116,691,500,807 | arxiv | \section{Introduction}
\label{section:introduction}
In recent years, 3D vehicle localization is becoming an essential component in CVIS and ITS. Vehicle behaviors \cite{2019behaviour, 2020behaviour} and traffic flow statistics \cite{2020vehiclecount} can be analyzed and used for traffic state estimation \cite{2020Vehicle} and traffic management \cite{2020trafficcontrol}, which is of great research significance and practical value. In practical applications, obtaining vehicle localization and dimension in 3D space is more important than vehicle information in 2D images. Different from sensors in autonomous vehicles, roadside sensors are usually installed on higher to obtain full-road vehicle information easily. In CVIS, by estimating 3D vehicle localization, vehicle locations can be obtained and sent to each vehicle for accurate path planning to avoid collisions.
The most common sensors currently used for 3D vehicle localization are lidar \cite{2019pointrcnn, 2020pvrcnn, 2019pointpillars, 2018voxelnet} and stereo vision systems represented by binocular and RGB-D depth cameras \cite{2017stereo, 2018rgbd, 2019stereorcnn}. We can obtain accurate 3D location provided by point cloud data directly. However, they are usually expensive and environmentally demanding. In contrast, monocular RGB cameras \cite{2019monogrnet, 2019mono, 2020SMOKE, 2020d4lcn, 2021MonoFlex, 2021fcos3d} are cost-effective due to widespread deployment and fast data processing speed, which is more suitable for large-scale traffic scenes. Currently, monocular 3D vehicle localization is achieved by monocular 3D vehicle detection methods, which are focused on the onboard view. Differently, the roadside view is fixed with more geometric priors. At the same time, the roadside view is higher and wider than the onboard, which is more suitable for large-area perception. However, due to missing depth information, challenges like occlusion and congestion still exist in monocular roadside surveillance scenes.
Monocular 3D vehicle detection methods can be divided into the following two categories: (1) geometric constraint-based methods \cite{2017deep3dbox, 2017deepmanta, 2019cubeslam, 2019gs3d}. (2) 3D feature estimation based methods \cite{20193dbox, 2020keypoint, 2020rtm3d, 2020perspective, 2021litefpn, 2021km3d}. Methods of the first category are implemented based on mature 2D detectors, which use 3D bounding boxes or CAD models to represent vehicles. Then, 3D vehicle models are solved by establishing constraints of 3D bounding boxes fitting closely to 2D boxes. However, this geometric constraint is not enough for generating unambiguous results. Sub-networks will also reduce the processing efficiency. Compared with the first category, 3D vehicle information like keypoint, orientation, and dimension is directly extracted without 2D detectors. Due to missing depth information in monocular images, additional geometric inference modules are needed, which will also increase processing time to some extent.
To solve the above problems, we propose a monocular 3D vehicle localization method for surveillance cameras in traffic scenes. Firstly, the transformation matrix between 2D image space and 3D world space is solved by camera calibration, which is further used for 3D vehicle localization. Secondly, a one-stage 3D vehicle localization network CenterLoc3D is proposed, which contains three modules: backbone, multi-scale feature fusion, and multi-task detection head. In multi-scale feature fusion, we propose a weighted-fusion module to fuse five feature maps containing multi-scale information with different weights for multi-scale vehicle detection. In multi-task detection head, the multi-scale feature map is used as the input to obtain outputs. Outputs contain four branches: vehicle type, centroid, eight vertexes, and the dimension of 3D bounding boxes. To improve the precision of 3D vehicle localization without sacrificing efficiency, a loss with spatial constraints embedding is proposed, including reprojection constraint of 2D-3D transformation obtained by camera calibration and IoU constraint of 3D box projection. Finally, we also propose a benchmark including a dataset, an annotation tool, and evaluation metrics for 3D vehicle localization in roadside monocular traffic scenes. Through experimental validation, it can be proved that our method is efficient and robust.
The main contributions of this paper are summarized as follows:
\begin{itemize}
\item A monocular 3D vehicle localization network CenterLoc3D for roadside surveillance cameras in traffic scenes is proposed, which directly predicts accurate 3D vehicle projection vertexes and dimensions.
\item A weighted-fusion module is proposed in multi-scale feature fusion, which further enhances feature extraction capability.
\item A loss with spatial constraints embedding is proposed, which can effectively improve the accuracy of 3D vehicle localization.
\item A benchmark including a dataset, an annotation tool, and evaluation metrics is proposed for experimental validation, which is helpful for the development of monocular 3D vehicle localization in roadside monocular traffic scenes.
\end{itemize}
\section{Related Work}
\label{section:related work}
Deep learning-based monocular 3D vehicle detection methods can be divided into the following two categories: geometric constraint-based and 3D feature estimation-based.
\textbf{Geometric Constraint-Based Methods.}
Due to the continuous development of convolutional neural networks (CNNs), many excellent 2D object detection methods \cite{2017fasterrcnn, 2016ssd, 2020yolov4, 2017retinanet, 2019centernet} have emerged. Some methods \cite{2016mono3ddet, 2019deepfit} directly apply the regional proposal in 2D object detection to 3D object detection. Mono3D \cite{2016mono3ddet} introduces sliding windows for 3D vehicle detection. Firstly, a series of candidate bounding boxes in 3D space are generated and projected to the image plane by camera calibration. Secondly, priori information like vehicle segmentation contours is combined to further obtain vehicle areas and 3D bounding boxes. The best results are finally selected by non-maximum suppression (NMS). However, the search range in 3D space is much larger than that in 2D space while obtaining priori information is time-consuming, which greatly reduces the efficiency. Currently, 2D object detection methods are still used in most 3D vehicle detection methods \cite{2019monogrnet, 2019mono, 2017deep3dbox, 2017deepmanta, 2019cubeslam, 2019gs3d} to obtain vehicle location, dimension and orientation. To improve efficiency, generating candidate boxes in 3D space is replaced by using geometric constraints of 3D bounding boxes fitting closely to 2D boxes. Deep3DBox \cite{2017deep3dbox} obtains 3D vehicle bounding boxes and orientation by constructing 2D-3D box constraints and multi-bin loss function. Deep MANTA \cite{2017deepmanta}, a multi-task network, uses 3D CAD models and region proposal network (RPN) to obtain vehicle type, 2D bounding boxes, location, visibility, and similarity to CAD model, which is used to select the best 3D CAD model. CubeSLAM \cite{2019cubeslam} obtains three orthogonal vanishing points by extracting straight line segments of vehicles and constructs 3D vehicle bounding boxes by geometric constraints between vanishing points and 2D bounding boxes. GS3D \cite{2019gs3d} is able to correct 3D bounding boxes, which contains two sub-networks: 2D+O and 3D sub-network. The 2D+O sub-network adds a vehicle orientation regression branch to Faster R-CNN \cite{2017fasterrcnn}, which is used to obtain 2D vehicle bounding boxes and orientation simultaneously. With camera calibration and the above information, a coarse 3D bounding box called guidance can be obtained. The 3D sub-network is used to extract visible 3D features of the guidance and complete the correction to obtain more accurate detection results. The above methods based on geometric constraints usually require 2D bounding boxes and time-consuming priori information extraction. The constraints that 2D bounding boxes can provide are limited, which leads to ambiguous results. At the same time, sub-networks will additionally increase the processing time.
\textbf{3D Feature Estimation-Based Methods.}
To avoid ambiguous results from geometric constraints and further improve accuracy and efficiency, many new methods \cite{20193dbox, 2020keypoint, 2020rtm3d, 2020perspective, 2020SMOKE, 2021MonoFlex, 2021fcos3d, 2021litefpn, 2021km3d} focus on direct 3D information extraction in images which can be used for 3D bounding box regression by CNNs. Mono3DBox \cite{20193dbox} proposes an end-to-end detector that can directly predict the center point of the vehicle bottom in the image. Then, the point is converted into 3D space by a look-up table to realize 3D vehicle detection and localization. MonoGRK \cite{2020keypoint} proposes an end-to-end keypoint-based 3D vehicle detection and localization network, using ResNet-101 \cite{2016resnet} as the backbone. The detection head contains three sub-networks: 2D vehicle detection, 2D keypoint regression, and 3D vehicle dimension regression. Besides, 3D CAD models are also used like DeepMANTA \cite{2017deepmanta}, but only 14 keypoints are labelled for each model, which describes the CAD model coarsely. 2D-3D space is linked in the geometric inference module, which enables the model end-to-end training. Transformer3D \cite{2020perspective} proposes a roadside 3D vehicle detection method using CNNs. Automatic camera calibration \cite{2017autocalibimprove} is firstly used to obtain three orthogonal vanishing points and scale factors for perspective transformation. Then, the transformed image is fed into RetinaNet \cite{2017retinanet} to obtain 2D bounding boxes and conversion parameters for 3D boxes. Finally, 3D bounding boxes are recovered by camera calibration results and the conversion parameters for speed measurement. The pipeline is simple, but 3D bounding boxes are not directly evaluated in the experimental results. RTM3D \cite{2020rtm3d} views 3D vehicle detection as keypoint detection without leveraging 2D detection results. The network is constructed based on ResNet-18 \cite{2016resnet} and DLA-34 \cite{2018dla}, which directly predicts center points and eight vertexes of 3D bounding boxes in the image. Then, vehicle location, dimension, and orientation can be solved by minimizing the reprojection error of 2D-3D bounding boxes. Lite-FPN \cite{2021litefpn} proposes a light-weight keypoint-based feature pyramid network, which contains three parts: backbone, detection head and post-processing. Top-k operation is used in the detection head to connect keypoints and regression sub-network. The attention module is added in keypoint detection loss to effectively solve the problem of mismatch between keypoint confidence and location. KM3D \cite{2021km3d} proposes a 3D vehicle detection network that contains keypoint detection and geometric inference modules. Outputs include center point, eight vertexes, orientation, dimension, and confidence. In the network, keypoints are used to further guide vehicle orientation regression and 3D IoU strategy is used to help confidence branch training. The above method based on 3D feature estimation can improve the accuracy of 3D vehicle detection by designing a geometric inference module in the network. However, this module will increase model complexity and inference time.
Among the reviewed literature, geometric constraint-based methods leverage 2D detectors and additional geometric constraint modules to obtain 3D vehicle information, which reduces the efficiency. 3D feature estimation-based methods embed geometric inference modules into networks without 2D detectors, which increases the complexity. Moreover, almost all the reviewed research uses datasets of onboard scenes, instead of roadside perspective, which is limited in large-scale 3D perception.
In this paper, we focus on an efficient 3D vehicle detection structure for roadside surveillance cameras to directly obtain 3D vehicle information without 2D detectors and geometry inference modules, to tackle the above-mentioned research gaps and weaknesses.
\section{CenterLoc3D for 3D vehicle localization}
\label{section:methods}
\subsection{Framework}
\label{subsec:framework}
The overall framework of the proposed method is shown in Figure \ref{fig:fig_framework}, which consists of two components: camera calibration and 3D vehicle localization. Firstly, the transformation matrix between 2D image space and 3D world space is solved by camera calibration. Secondly, vehicle type, centroid, eight vertexes, and dimensions of 3D bounding boxes are obtained by CenterLoc3D. Finally, 3D vehicle centroids can be obtained by camera calibration for 3D vehicle localization.
\begin{figure}[t]
\centering
\includegraphics[width=0.7\linewidth]{Figure1_framework}
\caption{\leftskip=0pt \rightskip=0pt plus 0cm The overall framework of the proposed method. The method consists of two parts: camera calibration and 3D vehicle localization. A single image is the input of CenterLoc3D to obtain vehicle type, centroid, eight vertexes, and dimensions of 3D bounding boxes. Combined with camera calibration, 3D vehicle localization results can be further obtained.}
\label{fig:fig_framework}
\end{figure}
\subsection{Camera Calibration}
\label{subsec:camera calibration}
To complete 3D vehicle localization in traffic scenes, the transformation matrix between 2D image space and 3D world space must be derived through camera calibration. We refer to the previous work \cite{2010Taxonomy,2019Automatic} to define coordinate systems, establish the camera calibration model, and choose the single vanishing point-based calibration method VWL (One Vanishing Point, Known Width and Length) \cite{2010Taxonomy} to complete camera calibration.
\begin{figure}[t]
\centering
\includegraphics[width=0.7\linewidth]{Figure2_calibration}
\caption{\leftskip=0pt \rightskip=0pt plus 0cm Schematic diagram of coordinate systems and camera calibration model. The world coordinate system is defined as ${O_w} - x,{\rm{ }}y,{\rm{ }}z$, camera coordinate system ${O_c} - {x_c},{\rm{ }}{y_c},{\rm{ }}{z_c}$, image coordinate system ${O_i} - u,{\rm{ }}v$. Parameters of calibration model include camera focal length $f$, camera height from the ground $h$, camera tilt angle $\phi$ and camera pan angle $\theta$.}
\label{fig:camera_model}
\end{figure}
Schematic diagram of coordinate systems and camera calibration model is shown in Figure \ref{fig:camera_model}. Three coordinate systems are defined, all of which are right-handed. The world coordinate system is defined by $x,{\rm{ }}y,{\rm{ }}z$ axis. The origin ${O_w}$ is located at the projection point of the camera on the road plane, whereas $z$ is perpendicular to the road plane upwards. The camera coordinate system is defined by ${x_{\rm{c}}},{\rm{ }}{y_{\rm{c}}},{\rm{ }}{z_{\rm{c}}}$ axis. The origin ${O_c}$ is located at the camera optical center. ${x_{\rm{c}}}$ is parallel to $x$. ${z_{\rm{c}}}$ points to the ground along the camera optical axis. ${y_{\rm{c}}}$ is perpendicular to the plane ${x_c}{O_c}{z_c}$. The image coordinate system is defined by $u,{\rm{ }}v$ axis. The origin ${O_i}$ is located at image center. In the image coordinate system, $u$ is horizontal right and $v$ is vertical downward. ${z_{\rm{c}}}$ intersects the road plane at $r = ({c_x},{c_y})$ in the image coordinate system, which is called the principal point and the default location is at image center. ${c_x}$ and ${c_y}$ represent half of the image width and height, respectively.
In addition to the above parameters, another parameter is the roll angle, which can be expressed by a simple image rotation and has no effect on calibration results. Therefore, it is not considered. In this paper, VWL and road marks \cite{2019Automatic} are used to solve and optimize calibration parameters $f,h,\phi ,\theta $. The vanishing point $VP = ({u_0},{v_0})$ along the direction of traffic flow is extracted by road edge lines. To illustrate the road space in a straightforward way, ${O_i}$ and $y$ axis are adjusted. Firstly, ${O_i}$ is moved to the upper left corner of the image, corresponding to the intrinsic parameter matrix $K$:
\begin{equation}
\label{equation_K}
K = \left[ {\begin{array}{*{20}{c}}
f&0&{{c_x}}\\
0&f&{{c_y}}\\
0&0&1
\end{array}} \right]
\end{equation}
Then, $y$ axis is adjusted to the direction along traffic flow. The rotation matrix $R$ contains a rotation of $\phi + {\pi \mathord{\left/
{\vphantom {\pi 2}} \right.
\kern-\nulldelimiterspace} 2}$ about $x$ axis and $\theta $ about $z$ axis, which can be defined as:
\begin{equation}
\label{equation_R}
\begin{aligned}
R &= {R_x}(\phi + {\pi \mathord{\left/
{\vphantom {\pi 2}} \right.
\kern-\nulldelimiterspace} 2}){R_z}(\theta )\\
&= \left[ {\begin{array}{*{20}{r}}
{\cos \theta }&{ - \sin \theta }&0\\
{ - \sin \phi \sin \theta }&{ - \sin \phi \cos \theta }&{ - \cos \phi }\\
{\cos \phi \sin \theta }&{\cos \phi \cos \theta }&{ - \sin \phi }
\end{array}} \right]
\end{aligned}
\end{equation}
The translation matrix $T$ is:
\begin{equation}
\label{equation_T}
T = \left[ {\begin{array}{*{20}{c}}
1&0&0&0\\
0&1&0&0\\
0&0&1&{ - h}
\end{array}} \right]
\end{equation}
The adjusted transformation formula from world point $(x,y,z)$ to image point $(u,v)$ is:
\begin{equation}
\label{equation_H}
s\left[ {\begin{array}{*{20}{c}}
u\\
v\\
1
\end{array}} \right] = KRT\left[ {\begin{array}{*{20}{c}}
x\\
y\\
z\\
1
\end{array}} \right] = H\left[ {\begin{array}{*{20}{c}}
x\\
y\\
z\\
1
\end{array}} \right]
\end{equation}
where $H = \left[ {{h_{ij}}} \right],i = 1,2,3;j = 1,2,3,4$ is the $3 \times 4$ projection matrix from world to image coordinate. $s$ is the scale factor.
Finally, according to the above derivation, the adjusted transformation between world and image can be described as follows:
\begin{equation}
\label{equa_xyz2uv}
\left\{ \begin{array}{l}
u = \frac{{{h_{11}}x + {h_{12}}y + {h_{13}}z + {h_{14}}}}{{{h_{31}}x + {h_{32}}y + {h_{33}}z + {h_{34}}}}\\
v = \frac{{{h_{21}}x + {h_{22}}y + {h_{23}}z + {h_{24}}}}{{{h_{31}}x + {h_{32}}y + {h_{33}}z + {h_{34}}}}
\end{array} \right.
\end{equation}
\begin{equation}
\label{equa_uv2xyz}
\left\{ \begin{array}{l}
x = \frac{{{b_1}({h_{22}} - {h_{32}}v) - {b_2}({h_{12}} - {h_{32}}u)}}{{({h_{11}} - {h_{31}}u)({h_{22}} - {h_{32}}v) - ({h_{12}} - {h_{32}}u)({h_{21}} - {h_{31}}v)}}\\
y = \frac{{ - {b_1}({h_{21}} - {h_{31}}v) + {b_2}({h_{11}} - {h_{31}}u)}}{{({h_{11}} - {h_{31}}u)({h_{22}} - {h_{32}}v) - ({h_{12}} - {h_{32}}u)({h_{21}} - {h_{31}}v)}}
\end{array} \right.
\end{equation}
where $\left\{ \begin{array}{l}
{b_1} = u({h_{33}}z + {h_{34}}) - ({h_{13}}z + {h_{14}})\\
{b_2} = v({h_{33}}z + {h_{34}}) - ({h_{23}}z + {h_{24}})
\end{array} \right.$.
\subsection{CenterLoc3D}
\label{subsec:centerloc3d}
In the roadside camera view, 3D vehicle localization can be described in terms of 3D vehicle detection. We propose a 3D vehicle localization network CenterLoc3D for roadside cameras, which uses a single RGB image as the input and the outputs include vehicle type, centroid, vertexes, and dimensions of 3D bounding boxes. Combined with camera calibration in Section \ref{subsec:camera calibration}, 3D vehicle localization results can be calculated.
\subsubsection{Network Architecture}
\label{subsubsec:network arch}
The overall architecture of the network is shown in Figure \ref{fig:arch_centerloc}, which contains three parts: backbone, multi-scale feature fusion and multi-task detection head. Firstly, the RGB image is scaled to $I \in { \mathbb{R} ^{H \times W \times 3}}$ as input, where $H = W = 512$. Then, the image is downsampled with a factor of $S = 4$.
\begin{figure*}[htbp]
\centering
\includegraphics[width=1.0\linewidth]{Figure3_network}
\caption{\leftskip=0pt \rightskip=0pt plus 0cm The architecture of CenterLoc3D. The network mainly consists of three parts: backbone, multi-scale feature fusion, and multi-task detection head. Multi-scale weighted-fusion feature map is obtained in the first two parts. 3D bounding box regression includes vehicle type, centroid, eight vertexes, and dimensions, which are decoded to final outputs.}
\label{fig:arch_centerloc}
\end{figure*}
\textbf{Backbone.} To make a trade-off between accuracy and efficiency, we use ResNet-50 \cite{2016resnet} as our backbone, containing residual and inverted bottleneck structures, which have good feature extraction capability. We extract the last three features with channels of 512, 1024, and 2048 for feature fusion.
\textbf{Multi-Scale Feature Fusion.} The series of feature maps obtained by the backbone are hierarchical. The high-resolution feature map retains more accurate local features and is suitable for small object detection. The low-resolution feature map contains higher-level semantic information and is suitable for large object detection. In monocular roadside traffic scenes, vehicles are distributed in different locations of different sizes. To make the network adaptive to vehicles of different sizes, a weighted-fusion module is proposed in multi-scale feature fusion based on RetinaNet \cite{2017retinanet}. Figure \ref{fig:fig_weighted-fusion} shows the schematic diagram of weighted-fusion module. The feature maps ${C_3},{C_4},{C_5}$ of size $64 \times 64$, $32 \times 32$, and $16 \times 16$ are extracted by backbone and used to construct feature pyramid ${F_p} = \left\{ {{P_3},{P_4},{P_5},{P_6},{P_7}} \right\}$. Different from YOLOv4 \cite{2020yolov4}, which directly retains the feature pyramid, we use deconvolution and unsample \cite{2017fpn} to unify the five feature maps of different sizes in ${F_p}$ to the same size $\overline {{F_p}} = \left\{ {\overline {{P_3}} ,\overline {{P_4}} ,\overline {{P_5}} ,\overline {{P_6}} ,\overline {{P_7}} } \right\}$. Then, weighted-fusion module is added by the fusion strategy shown in Equation \ref{equa_feature_fusion}. Finally, a fused feature map $F \in {\mathbb{R}^{\frac{H}{S} \times \frac{W}{S} \times 64}}$ containing multi-scale information of vehicles is obtained. This feature fusion module not only generates multi-scale feature maps, but also reduces computational effort in the prediction process and improve the overall efficiency.
\begin{equation}
\label{equa_feature_fusion}
F = \sum\limits_{i = 3}^7 {{w_i} \times \overline {{P_i}} }
\end{equation}
where ${w_i}$ denotes the weight of the feature map with $\sum\limits_{i = 3}^7 {{w_i}} = 1$, which can be set according to the importance degree. We set 0.5, 0.2, 0.1, 0.1 and 0.1 respectively in the experiment.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{Figure4_weighted-fusion}
\caption{Schematic diagram of weighted-fusion module.}
\label{fig:fig_weighted-fusion}
\end{figure}
\textbf{Multi-Task Detection Head.} Based on the multi-scale fusion feature map, a multi-task detection head is designed based on the actual demand for 3D vehicle localization in roadside scenes. The detection head is comprised of four branches: vehicle type, centroid, eight vertexes, and dimensions of 3D bounding boxes. To further improve the ability to distinguish different types of vehicles, the type branch is implemented by fully convolutional layers and attention module \cite{2018senet}. The remaining branches are implemented by fully convolutional layers only. Inspired by CenterNet \cite{2019centernet}, vehicle type is defined as centroid heatmap ${M_c} \in {[0,1]^{\frac{H}{S} \times \frac{W}{S} \times C}}$, where $C$ denotes the number of vehicle types. The remaining branches are defined as centroid offset ${M_{co}} \in {[0,1]^{\frac{H}{S} \times \frac{W}{S} \times 2}}$, vertexes ${M_v} \in {\mathbb{R}^{\frac{H}{S} \times \frac{W}{S} \times 16}}$ and dimension ${M_s} \in {\mathbb{R}^{\frac{H}{S} \times \frac{W}{S} \times 3}}$. To ensure the stability during training, we normalize centroids and vertexes to the fusion feature map with size ${H \mathord{\left/
{\vphantom {H S}} \right.
\kern-\nulldelimiterspace} S},{W \mathord{\left/
{\vphantom {W S}} \right.
\kern-\nulldelimiterspace} S}$.
\subsubsection{Loss Function}
\label{subsubsec:loss func}
In the training process, the loss function contains six components: vehicle classification, centroid offset, vertexes, dimension, and a loss with spatial constraints embedding for improving the precision of 3D vehicle localization. The spatial constraints include reprojection constraint of 2D-3D transformation obtained by camera calibration and IoU constraint of 3D box projection.
\textbf{Vehicle Classification Loss.} To solve the problem of imbalanced positive and negative samples, we use focal loss \cite{2017retinanet} as vehicle classification loss:
\begin{equation}
\label{equa_focal_loss}
{L_c} = - \frac{1}{N}\sum\limits_{k = 1}^C {\sum\limits_{i = 1}^{{W \mathord{\left/
{\vphantom {W S}} \right.
\kern-\nulldelimiterspace} S}} {\sum\limits_{j = 1}^{{H \mathord{\left/
{\vphantom {H S}} \right.
\kern-\nulldelimiterspace} S}} {\left\{ {\begin{array}{*{20}{c}}
{{{(1 - {{\hat p}_{cij}})}^\alpha }\log ({{\hat p}_{cij}})}&{if{\rm{ }}{p_{cij}} = 1}\\
{\hat p_{cij}^\alpha {{(1 - {p_{cij}})}^\beta }\log (1 - {{\hat p}_{cij}})}&{if{\rm{ }}{p_{cij}} < 1}
\end{array}} \right.} } }
\end{equation}
\noindent where $N$ is the number of positive samples, $\alpha $ and $\beta $ are hyper-parameters used to adjust loss weights of positive and negative samples, which are usually set to 2 and 4. ${p_{cij}}$ is the response value of each ground truth vehicle in the heatmap described by the Gaussian kernel function ${e^{ - \frac{{{{(x - {p_{cijx}})}^2} + {{(y - {p_{cijy}})}^2}}}{{2{\sigma ^2}}}}}$. $\sigma $ is the standard deviation calculated from the ground truth vehicle dimension in image space \cite{2019centernet}.
\textbf{3D Vehicle Information Regression Loss.} We use L1 regression loss for vehicle centroid offset, vertex, and dimension loss:
\begin{equation}
\label{equa_lco}
{L_{co}} = \frac{1}{N}\sum\limits_{i = 1}^{{W \mathord{\left/
{\vphantom {W S}} \right.
\kern-\nulldelimiterspace} S}} {\sum\limits_{j = 1}^{{H \mathord{\left/
{\vphantom {H S}} \right.
\kern-\nulldelimiterspace} S}} {\mathbbm{1}_{ij}^{obj}\lvert {M_{co}^{ij} - ({{{p_{center}}} \mathord{\left/
{\vphantom {{{p_{center}}} S}} \right.
\kern-\nulldelimiterspace} S} - {{\tilde p}_{center}})} \rvert} }
\end{equation}
\begin{equation}
\label{equa_lv}
{L_v} = \frac{1}{N}\sum\limits_{i = 1}^{{W \mathord{\left/
{\vphantom {W S}} \right.
\kern-\nulldelimiterspace} S}} {\sum\limits_{j = 1}^{{H \mathord{\left/
{\vphantom {H S}} \right.
\kern-\nulldelimiterspace} S}} {\mathbbm{1}_{ij}^{obj}\lvert {M_v^{ij} - {{{p_{vertex}}} \mathord{\left/
{\vphantom {{{p_{vertex}}} S}} \right.
\kern-\nulldelimiterspace} S}} \rvert} }
\end{equation}
\begin{equation}
\label{equa_ls}
{L_s} = \frac{1}{N}\sum\limits_{i = 1}^{{W \mathord{\left/
{\vphantom {W S}} \right.
\kern-\nulldelimiterspace} S}} {\sum\limits_{j = 1}^{{H \mathord{\left/
{\vphantom {H S}} \right.
\kern-\nulldelimiterspace} S}} {\mathbbm{1}_{ij}^{obj}\lvert {M_s^{ij} - \tilde M_s^{ij}} \rvert} }
\end{equation}
where $\mathbbm{1}_{ij}^{obj}$ denotes whether centroid appears at $i,j$. ${p_{center}}$ and ${p_{vertex}}$ denote the ground truth centroids and vertexes of 3D bounding boxes in input image with size $H,W$. $\tilde p_{center}^{}$ denotes the ground truth centroids in the fusion feature map with size ${H \mathord{\left/
{\vphantom {H S}} \right.
\kern-\nulldelimiterspace} S},{W \mathord{\left/
{\vphantom {W S}} \right.
\kern-\nulldelimiterspace} S}$. ${\tilde M_s}$ denotes the ground truth feature map of 3D vehicle dimension.
\textbf{Loss with spatial constraints embedding.} To further improve the precision of 3D vehicle localization, loss functions with spatial constraints embedding are designed, including reprojection constraints using camera calibration and vehicle IoU constraints.
\begin{figure}[htbp]
\centering
\subcaptionbox{\centering Reprojection constraint\label{subfig:projection_constraint}}
\includegraphics[height=3.0cm]{Figure5_a
}
\subcaptionbox{\centering IoU constraint\label{subfig:iou_constraint}}
\includegraphics[height=3.0cm]{Figure5_b
}
\caption{Schematic diagram of spatial constraints.}
\label{fig:fig_spatial_constraint}
\end{figure}
The schematic diagram of reprojection constraint and vehicle IoU constraint is shown in Figure \ref{fig:fig_spatial_constraint}. We define the predicted and ground truth vertexes in image as $p_i^{pred} = (u_i^{pred},v_i^{pred})$ and $p_i^{gt} = (u_i^{gt},v_i^{gt})$. The corresponding vertexes in world are $P_i^{pred} = (x_i^{pred},y_i^{pred},z_i^{pred})$ and $P_i^{gt} = (x_i^{gt},y_i^{gt},z_i^{gt})$. The projection of predicted vertexes in world and image are defined as $P_i^{proj} = (x_i^{proj},y_i^{proj},z_i^{proj})$ and $p_i^{proj} = (u_i^{proj},v_i^{proj})$, $i = 1,2, \cdots ,8$. The predicted dimension is $D_v^{pred} = (l_v^{pred},w_v^{pred},h_v^{pred})$. Based on $p_i^{pred}$ from CenterLoc3D, $P_i^{proj}$ can be obtained by Equation \ref{equa_uv2xyz} and Table \ref{tab:table_3d_proj_vertexes}. Then, $p_i^{proj}$ can be obtained by Equation \ref{equa_xyz2uv}. As shown in Figure \ref{subfig:projection_constraint}, $p_i^{proj}$ and $p_i^{pred}$ constitute the projection constraints, where predicted, ground truth, and projection boxes are represented in blue, red, and green, respectively.
The minimum external rectangles of predicted and ground truth vertexes are $v_{rec}^{pred}$ and $v_{rec}^{gt}$. As shown in Figure \ref{subfig:iou_constraint}, blue indicates the predicted box while red is the ground truth. The green area indicates the overlap between the two boxes.
\begin{table}[htbp]
\centering
\caption{Calculation of 3D bounding box projection vertexes in world space.}
\label{tab:table_3d_proj_vertexes}
\begin{tabular}{cc}
\toprule
Vertex & World coordinate \\
\midrule
$P_1^{proj}$& $(x_2^{gt} + w_v^{pred},y_2^{gt},z_2^{gt})$ \\
$P_2^{proj}$& $(x_2^{gt},y_2^{gt},z_2^{gt})$ \\
$P_3^{proj}$& $(x_2^{gt},y_2^{gt} + l_v^{pred},z_2^{gt})$ \\
$P_4^{proj}$& $(x_2^{gt} + w_v^{pred},y_2^{gt} + l_v^{pred},z_2^{gt})$ \\
$P_5^{proj}$& $(x_2^{gt} + w_v^{pred},y_2^{gt},z_2^{gt} + h_v^{pred})$ \\
$P_6^{proj}$& $(x_2^{gt},y_2^{gt},z_2^{gt} + h_v^{pred})$ \\
$P_7^{proj}$& $(x_2^{gt},y_2^{gt} + l_v^{pred},z_2^{gt} + h_v^{pred})$ \\
$P_8^{proj}$& $(x_2^{gt} + w_v^{pred},y_2^{gt} + l_v^{pred},z_2^{gt} + h_v^{pred})$ \\
\bottomrule
\end{tabular}
\end{table}
The loss comprised of two constraints shown in Figure \ref{fig:fig_spatial_constraint} is defined as follows:
\begin{equation}
\label{equa_lproj}
{L_{proj}} = \frac{1}{N}\sum\limits_{i = 1}^{{W \mathord{\left/
{\vphantom {W S}} \right.
\kern-\nulldelimiterspace} S}} {\sum\limits_{j = 1}^{{H \mathord{\left/
{\vphantom {H S}} \right.
\kern-\nulldelimiterspace} S}} {\mathbbm{1}_{ij}^{obj}\lvert {M_{proj}^{ij}(H,M_s^{ij}) - {\bar M_v^{ij}} } \rvert} }
\end{equation}
\begin{equation}
\label{equa_liou}
{L_{iou}} = \frac{1}{N}\sum\limits_{i = 1}^{{W \mathord{\left/
{\vphantom {W S}} \right.
\kern-\nulldelimiterspace} S}} {\sum\limits_{j = 1}^{{H \mathord{\left/
{\vphantom {H S}} \right.
\kern-\nulldelimiterspace} S}} {\mathbbm{1}_{ij}^{obj} \cdot IoU(R_{bbox}^{ij} - \tilde R_{bbox}^{ij})} }
\end{equation}
where $M_{proj}^{}(H,M_s^{ij})$ denotes the feature map of projection vertexes calculated by camera calibration matrix $H$ in Section \ref{subsec:camera calibration} and vehicle dimension feature map ${M_s}$ predicted by the network. $\bar {M_v^{}} $ denotes predicted 3D bounding boxes vertexes in the original image. $R_{bbox}^{ij}$ and $\tilde R_{bbox}^{ij}$ denote the predicted and ground truth minimum external rectangles. $IoU$ denotes the IoU loss strategy and we use CIoU loss \cite{2020ciou} in the experiment.
With the above six loss functions, the multi-task loss function can be defined as follows:
\begin{equation}
\label{equa_lossfunc}
L = {\lambda _c}{L_c} + {\lambda _{co}}{L_{co}} + {\lambda _v}{L_v} + {\lambda _s}{L_s} + {\lambda _{proj}}{L_{proj}} + {\lambda _{iou}}{L_{iou}}
\end{equation}
where $\lambda $ is the weight to balance the loss of each component. We set ${\lambda _c} = 1$, ${\lambda _{co}} = 1$ , ${\lambda _v} = 0.1$ , ${\lambda _s} = 0.1$ , ${\lambda _{proj}} = 0.1$ and ${\lambda _{iou}} = 1$ in the experiment.
\section{Dataset of 3D Vehicle Localization}
\label{sec:datasets}
Most currently available 3D vehicle localization datasets are based on onboard views \cite{2012kitti, 2020waymo, 2020nuscenes} instead of roadside, which is difficult for large-scale 3D perception and validation of roadside vehicle perception methods. Therefore, we propose a 3D vehicle localization dataset (SVLD-3D) for roadside surveillance cameras and an annotation tool (LabelImg-3D) for experimental validation.
\subsection{Dataset Composition}
\begin{figure}[htbp]
\centering
\subcaptionbox{\centering Scene A\label{subfig:dataset_samples_a}}
\includegraphics[width=0.3\linewidth]{Fig_6a}\quad
\includegraphics[width=0.3\linewidth]{Fig_6b}\quad
\includegraphics[width=0.3\linewidth]{Fig_6c}
}
\subcaptionbox{\centering Scene B\label{subfig:dataset_samples_b}}
\includegraphics[width=0.3\linewidth]{Fig_6d}\quad
\includegraphics[width=0.3\linewidth]{Fig_6e}\quad
\includegraphics[width=0.3\linewidth]{Fig_6f}
}
\subcaptionbox{\centering Scene C\label{subfig:dataset_samples_c}}
\includegraphics[width=0.3\linewidth]{Fig_6g}\quad
\includegraphics[width=0.3\linewidth]{Fig_6h}\quad
\includegraphics[width=0.3\linewidth]{Fig_6i}
}
\subcaptionbox{\centering Scene
D\label{subfig:dataset_samples_d}}
\includegraphics[width=0.3\linewidth]{Fig_6j}\quad
\includegraphics[width=0.3\linewidth]{Fig_6k}\quad
\includegraphics[width=0.3\linewidth]{Fig_6l}
}
\subcaptionbox{\centering Scene
E\label{subfig:dataset_samples_e}}
\includegraphics[width=0.3\linewidth]{Fig_6m}\quad
\includegraphics[width=0.3\linewidth]{Fig_6n}\quad
\includegraphics[width=0.3\linewidth]{Fig_6o}
}
\caption{\leftskip=0pt \rightskip=0pt plus 0cm Samples in SVLD-3D. Five different scenes are listed from the first to the fifth row.\label{fig:dataset_samples}}
\end{figure}
Scenes in SVLD-3D dataset are from BrnoCompSpeed \cite{2019brnocompspeed} and self-collected urban scenes with resolution of ${\rm{1920}} \times {\rm{1080}}$ and ${\rm{1080}} \times {\rm{720}}$, respectively. BrnoCompSpeed is public and provided by Brno University of Technology, which contains six highway scenes with three views (left, center, and right). Each frame contains ground truth road marks and vanishing points (used for camera calibration (section \ref{subsec:camera calibration})). Each vehicle contains ground truth location and speed collected by Lidar and GPS (used for vehicle dimension annotation (section \ref{subsec:label proc}) and training loss function (section \ref{subsubsec:loss func})). However, only highway scenes with low traffic volumes and single-vehicle types exist in BrnoCompSpeed. To further expand dataset diversity, urban scenes with more vehicle types, congestion, and occlusion are also included in SVLD-3D.
SVLD-3D contains five typical scenes with three vehicle types (car, truck, and bus), with a total of 14593 images in the training and validation dataset and 2273 images in the test dataset. Some samples in SVLD-3D are shown in Figure \ref{fig:dataset_samples}. Table \ref{tab:table_calib_results} shows detailed information of different scenes in the dataset, where the effective field of view ${D_r} = ({D_{ry}},{D_{rx}})$ is the maximum distance that the roadside camera can perceive along and perpendicular to the road direction.
\begin{table}[htbp]
\centering
\caption{Details of different scenes in SVLD-3D.}
\label{tab:table_calib_results}
\begin{tabular}{ccccccc}
\toprule
\multirow{2}{*}{Scene} & \multirow{2}{*}{${D_{ry}}$} & \multirow{2}{*}{${D_{rx}}$} & \multicolumn{4}{c}{Camera Calibration Parameters} \\ \cline{4-7}
& & & $f$ & $\phi /rad$ & $\theta /rad$ & $h/mm$ \\ \midrule
A & 120 & 25 & 2878.13 & 0.17874 & 0.26604 & 10119.08 \\
B & 120 & 25 & 3994.17 & 0.15717 & 0.35346 & 8071.00 \\
C & 60 & 15 & 3384.25 & 0.26295 & -0.24869 & 8126.49 \\
D & 80 & 10 & 3743.78 & 0.11225 & -0.07516 & 7353.40 \\
E & 60 & 10 & 1142.26 & 0.33372 & 0.14387 & 7166.44 \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Label Process}
\label{subsec:label proc}
Since the original data in SVLD-3D only contains images and 3D vehicle location provided by Lidar, we develop an annotation tool LabelImg-3D and relabel vehicles to obtain ground truth vehicle type, centroid, vertexes, and dimensions of 3D bounding boxes. The ground truth 3D and 2D vehicle centroid, and dimension are denoted as $P_{cen}^{gt} = (x_{cen}^{gt},y_{cen}^{gt},z_{cen}^{gt})$, $p_{cen}^{gt} = (u_{cen}^{gt},v_{cen}^{gt})$ and $D_v^{gt} = (l_v^{gt},w_v^{gt},h_v^{gt})$.
Compared with variable views of autonomous vehicles, the roadside camera usually has certain installation standards and the pan angle is relatively fixed. Therefore, only the scenes with typical pan angles are selected. To reduce label effort, previous work \cite{2021Spatial} and 2D bounding boxes provided by YOLOv4 \cite{2020yolov4} are used as guidance. The specific label steps are as follows:
\begin{enumerate}[(1)]
\item Vehicle type. If the vehicle type obtained by YOLOv4 is consistent with the ground truth, label this type as ground truth. Otherwise, adjust to the correct type in LabelImg-3D.
\item Vehicle dimension. Firstly, YOLOv4 is used to obtain 2D vehicle bounding boxes as guidance. Secondly, the geometric constraint of 3D boxes fitting closely to 2D is used to adjust vehicle dimension by camera calibration. At the same time, vehicle dimensions can also be obtained by observing vehicle models and referring to relevant documents during the labeling process. These two sub-steps can be helpful to obtain more accurate vehicle dimensions $D_v^{gt}$.
\item Vehicle centroid. $p_{cen}^{gt}$ can be obtained by Equation \ref{equa_xyz2uv} when $P_{cen}^{gt}$ is known by lidar.
\item Vehicle vertexes. Based on $P_{cen}^{gt}$ and $D_v^{gt}$ in step (2), $p_i^{gt}$ can be obtained by Table \ref{tab:table_3d_gt_vertexes} and Equation \ref{equa_xyz2uv}.
\end{enumerate}
\begin{table}[htbp]
\centering
\caption{Calculation of 3D bounding box ground truth vertexes in world space.}
\label{tab:table_3d_gt_vertexes}
\begin{tabular}{cc}
\toprule
Vertex & World coordinate \\
\midrule
$P_1^{gt}$& $(x_{cen}^{gt} + {{w_v^{gt}} \mathord{\left/
{\vphantom {{w_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2},y_{cen}^{gt} - {{l_v^{gt}} \mathord{\left/
{\vphantom {{l_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2},z_{cen}^{gt} - {{h_v^{gt}} \mathord{\left/
{\vphantom {{h_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2})$ \\
$P_2^{gt}$& $(x_{cen}^{gt} - {{w_v^{gt}} \mathord{\left/
{\vphantom {{w_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2},y_{cen}^{gt} - {{l_v^{gt}} \mathord{\left/
{\vphantom {{l_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2},z_{cen}^{gt} - {{h_v^{gt}} \mathord{\left/
{\vphantom {{h_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2})$ \\
$P_3^{gt}$& $(x_{cen}^{gt} - {{w_v^{gt}} \mathord{\left/
{\vphantom {{w_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2},y_{cen}^{gt} + {{l_v^{gt}} \mathord{\left/
{\vphantom {{l_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2},z_{cen}^{gt} - {{h_v^{gt}} \mathord{\left/
{\vphantom {{h_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2})$ \\
$P_4^{gt}$& $(x_{cen}^{gt} + {{w_v^{gt}} \mathord{\left/
{\vphantom {{w_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2},y_{cen}^{gt} + {{l_v^{gt}} \mathord{\left/
{\vphantom {{l_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2},z_{cen}^{gt} - {{h_v^{gt}} \mathord{\left/
{\vphantom {{h_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2})$ \\
$P_5^{gt}$& $(x_{cen}^{gt} + {{w_v^{gt}} \mathord{\left/
{\vphantom {{w_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2},y_{cen}^{gt} - {{l_v^{gt}} \mathord{\left/
{\vphantom {{l_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2},z_{cen}^{gt} + {{h_v^{gt}} \mathord{\left/
{\vphantom {{h_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2})$ \\
$P_6^{gt}$& $(x_{cen}^{gt} - {{w_v^{gt}} \mathord{\left/
{\vphantom {{w_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2},y_{cen}^{gt} - {{l_v^{gt}} \mathord{\left/
{\vphantom {{l_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2},z_{cen}^{gt} + {{h_v^{gt}} \mathord{\left/
{\vphantom {{h_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2})$ \\
$P_7^{gt}$& $(x_{cen}^{gt} - {{w_v^{gt}} \mathord{\left/
{\vphantom {{w_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2},y_{cen}^{gt} + {{l_v^{gt}} \mathord{\left/
{\vphantom {{l_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2},z_{cen}^{gt} + {{h_v^{gt}} \mathord{\left/
{\vphantom {{h_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2})$ \\
$P_8^{gt}$& $(x_{cen}^{gt} + {{w_v^{gt}} \mathord{\left/
{\vphantom {{w_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2},y_{cen}^{gt} + {{l_v^{gt}} \mathord{\left/
{\vphantom {{l_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2},z_{cen}^{gt} + {{h_v^{gt}} \mathord{\left/
{\vphantom {{h_v^{gt}} 2}} \right.
\kern-\nulldelimiterspace} 2})$ \\
\bottomrule
\end{tabular}
\end{table}
Finally, vehicle type, centroid, vertexes, and dimensions are recorded in the annotation file. Figure \ref{fig:fig_draw_bbox_result} shows the flowchart of label process and visualization of annotations.
\begin{figure}[htbp]
\centering
\subcaptionbox{\centering Flowchart of label process\label{subfig:flowchart_label_process}}
\includegraphics[width=0.5\linewidth]{Figure7_a
}
\subcaptionbox{\centering Visualization of annotations\label{subfig:vis_annotation}}
\includegraphics[width=0.38\linewidth]{Figure7_b_label
}
\caption{Label process and visualization.}
\label{fig:fig_draw_bbox_result}
\end{figure}
\section{Experimental Protocols}
\label{section:experimental protocols}
In this section, we introduce implementation details and evaluation metrics for our experiments.
\subsection{Implementation Details}
\label{subsec:implem datails}
We implement our network using PyTorch platform with Core i7-8700 CPU and one GTX 1080Ti GPU. The original image is scaled to $512 \times 512$ for training and testing. We split the dataset into the training set and validation set with a ratio of 9:1. We use Adam optimizer with a base learning rate of 0.001 for 100 epochs. The learning rate reduces by a factor of 10 when validation loss no longer decreases for three continuous epochs. The pretrained model on ImageNet is used for fine-tuning. We train our network using weights freezing in the first 60 epochs with a batch size of 16. The batch size drops $2 \times$ in the rest epochs. When validation loss no longer decreases for 7 continuous epochs, the training process will be stopped by early stopping.
We use random color jitter, horizontal flip, and perspective transformation as image augmentation. As shown in Figure \ref{fig:data_aug}, perspective transformation is used as the simulation of roadside view change according to the camera imaging principle. After data augmentation, the training and validation set can be expanded to 58372 images, which quadruples the original volume.
\begin{figure}[htbp]
\centering
\subfloat[\centering ]{\includegraphics[width=0.22\linewidth]{Fig_8a
\label{fig:data_aug_a}}
\hfil
\subfloat[\centering ]{\includegraphics[width=0.22\linewidth]{Fig_8b
\label{fig:data_aug_b}}
\hfil
\subfloat[\centering ]{\includegraphics[width=0.22\linewidth]{Fig_8c
\label{fig:data_aug_c}}
\hfil
\subfloat[\centering ]{\includegraphics[width=0.22\linewidth]{Fig_8d
\label{fig:data_aug_d}}
\newline
\subfloat[\centering ]{\includegraphics[width=0.22\linewidth]{Fig_8e
\label{fig:data_aug_e}}
\hfil
\subfloat[\centering ]{\includegraphics[width=0.22\linewidth]{Fig_8f
\label{fig:data_aug_f}}
\hfil
\subfloat[\centering ]{\includegraphics[width=0.22\linewidth]{Fig_8g
\label{fig:data_aug_g}}
\hfil
\subfloat[\centering ]{\includegraphics[width=0.22\linewidth]{Fig_8h
\label{fig:data_aug_h}}
\newline
\caption{\leftskip=0pt \rightskip=0pt plus 0cm Schematic diagram of data augmentation. (a) No augmentation. (b) Color jitter (CJ). (c) Horizontal flip (HF). (d) Perspective transformation (PT). (e) CJ + HF. (f) CJ + PT. (g) HF + PT. (h) CJ + HF + PT.}
\label{fig:data_aug}
\end{figure}
\subsection{Evaluation Metrics}
\label{subsec:eval metrics}
Evaluation metrics include average precision (AP), frame per second (FPS), precision and error of 3D vehicle localization and dimension prediction.
\subsubsection{Average Precision and Speed}
\label{subsubsec:ap and fps}
Referring to evaluation metrics of existing 3D vehicle detection datasets, we use $A{P_{3D}}$ \cite{2016mono3ddet} to evaluate 3D average precision and FPS to evaluate speed.
$A{P_{3D}}$ is similar to $A{P_{2D}}$ in 2D detection provided by VOC dataset \cite{2010pascalvoc}. Differently, 3D IoU is used instead of 2D in calculating $A{P_{3D}}$. The equation of $AP$ can be expressed as:
\begin{equation}
\label{equa_ap}
\begin{array}{l}
AP = \frac{1}{{11}}\sum\limits_{r \in \{ 0,0.1, \ldots ,1\} } {{p_{{\rm{interp}}}}(r)} \\
{p_{{\rm{interp}}}}(r) = \mathop {\max }\limits_{\tilde r:\tilde r \ge r} p(\tilde r)
\end{array}
\end{equation}
The precision at each recall level $r$ is interpolated by taking the maximum precision measured for a method for which the corresponding recall exceeds $r$, $p(\tilde r)$ is the measured precision at recall $\tilde r$.
The equation of FPS is defined as:
\begin{equation}
\label{equa_fps}
FPS = {1 \mathord{\left/
{\vphantom {1 {{t_{proc}}}}} \right.
\kern-\nulldelimiterspace} {{t_{proc}}}}
\end{equation}
where $t_{proc}$ represents the processing time (measured in second) of the network for a single frame.
\subsubsection{3D Vehicle Localization Precision and Error}
\label{subsubsec:3dlocpe}
Combined with camera calibration in Section \ref{subsec:camera calibration}, 3D vehicle centroids can be further obtained by Equation \ref{equa_uv2xyz} and used for 3D vehicle localization.
The predicted and ground-truth 3D vehicle centroid are denoted as $P_{cen}^{pred} = (x_{cen}^{pred},y_{cen}^{pred},z_{cen}^{pred})$ and $P_{cen}^{gt} = (x_{cen}^{gt},y_{cen}^{gt},z_{cen}^{gt})$.
3D vehicle localization precision and error can be defined as follows:
\begin{equation}
\label{equa_ploc}
{P_{loc}} = (1 - \sum\limits_{k \in \{ x,y\} } {\frac{{\lvert {k_{cen}^{pred} - k_{cen}^{gt}} \rvert}}{{{{{D_{rk}}} \mathord{\left/
{\vphantom {{{D_{rk}}} 2}} \right.
\kern-\nulldelimiterspace} 2}}}} ) \times 100\%
\end{equation}
\begin{equation}
\label{equa_eloc}
{E_{loc}} = \sum\limits_{k \in \{ x,y\} } {\lvert {k_{cen}^{pred} - k_{cen}^{gt}} \rvert}
\end{equation}
where ${D_r}$ can be found in Table \ref{tab:table_calib_results}.
\subsubsection{3D Vehicle Dimension Precision and Error}
\label{subsubsec:3dsizepe}
We design a 3D vehicle dimension regression branch in the network, which can be used for 3D vehicle dimension prediction, including length, width, and height in meters.
The predicted and ground-truth 3D dimension are denoted as $D_v^{pred} = (l_v^{pred},w_v^{pred},h_v^{pred})$ and $D_v^{gt} = (l_v^{gt},w_v^{gt},h_v^{gt})$.
3D vehicle dimension precision and error can be defined as follows:
\begin{equation}
\label{equa_pdim}
{P_{dim}} = (1 - \sum\limits_{k \in \{ {l_v},{w_v},{h_w}\} } {\frac{{\lvert {{k^{pred}} - {k^{gt}}} \rvert}}{{{k^{gt}}}}} ) \times 100\%
\end{equation}
\begin{equation}
\label{equa_edim}
{E_{dim}} = \sum\limits_{k \in \{ {l_v},{w_v},{h_w}\} } {\lvert {{k^{pred}} - {k^{gt}}} \rvert}
\end{equation}
\section{Results and Discussions}
\label{section:results and discussions}
In this section, we provide experimental results, an ablation study and discussions to demonstrate the effectiveness of CenterLoc3D.
\subsection{Average Precision and Speed of CenterLoc3D}
\label{subsec:ap and fps of centerloc3d}
$A{P_{3D}}$ and FPS of different monocular 3D vehicle detection methods on validation and test set are compared in Table \ref{tab:table_compare_kitti}, which are calculated by Equation \ref{equa_ap} and Equation \ref{equa_fps}. In addition, FLOPS and parameters are also important metrics, with 28.61GFlops and 34.95M for our network. KITTI validation and test set are used in onboard scenes at easy, moderate, and hard settings, which are determined by vehicle size in image space. We use the proposed SVLD-3D dataset for experimental validation. The IoU thresholds are 0.5 and 0.7 respectively. SVLD-3D dataset has only one validation set without difficulty settings.
\begin{table*}[htbp]
\centering
\caption{Comparison of $A{P_{3D}}$ and FPS of different monocular 3D vehicle detection methods.}
\label{tab:table_compare_kitti}
\resizebox{\textwidth}{!}
\begin{tabular}{ccccccccccc}
\toprule
\multirow{2}{*}{Method} & \multirow{2}{*}{Scene} & \multirow{2}{*}{Backbone} & \multirow{2}{*}{GPU} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}$A{P_{3D}}(IOU > 0.5)$ \\ $[va{l_1}/va{l_2}]$ \end{tabular}} & \multicolumn{3}{c}{\begin{tabular}[c]{@{}c@{}}$A{P_{3D}}(IOU > 0.7)$\\ $[va{l_1}/va{l_2}/test]$\end{tabular}} & \multirow{2}{*}{FPS} \\ \cline{5-10}
& & & & Easy & Moderate & Hard & Easy & Moderate & Hard & \\
\midrule
MonoGRNet \cite{2019monogrnet} & onboard & VGG-16 & GTX Titan X & 50.51/54.21 & 36.97/39.69 & 30.82/33.06 & 13.88 / 24.97 / - & 10.19 / 19.44 / - & 7.62 / 16.30 / - & 16.7 \\
Deep3DBox \cite{2017deep3dbox} & onboard & VGG-16 & - & 27.04 / - & 20.55 / - & 15.88 / - & 5.85 / - / - & 4.10 / - / - & 3.84 / - / - & - \\
GS3D \cite{2019gs3d} & onboard & VGG-16 & - & 32.15/30.60 & 29.89/26.40 & 26.19/22.89 & 13.46/11.63/7.69 & 10.97/10.51/6.29 & 10.38/10.51/6.16 & 0.4 \\
\multirow{2}{*}{RTM3D \cite{2020rtm3d}} & \multirow{2}{*}{onboard} & ResNet-18 & \multirow{2}{*}{GTX 1080Ti×2} & 47.43/46.52 & 33.86/32.61 & 31.04/30.95 & 18.13 / 18.38 / - & 14.14 / 14.66 / - & 13.33 / 12.35 / - & 28.6 \\
& & DLA-34 & & 54.36/52.59 & 41.90/40.96 & 35.84/34.95 & 20.77/19.47/13.61 & 16.86/16.29/10.09 & 16.63/15.57/8.18 & 18.2 \\
SMOKE \cite{2020SMOKE} & onboard & DLA-34 & GTX TITAN X×4 & \multicolumn{3}{c}{-} & 14.76/19.99/14.03 & 12.85/15.61/9.76 & 11.50/15.28/7.84 & 33.3 \\
\multirow{2}{*}{KM3D \cite{2021km3d}} & \multirow{2}{*}{onboard} & ResNet-18 & \multirow{2}{*}{GTX 1080Ti} & 47.23/47.13 & 34.12/33.31 & 31.51/25.84 & 19.48/18.34/12.65 & 15.32/14.91/8.39 & 13.88/12.58/7.12 & 47.6 \\
& & DLA-34 & & 56.02/54.09 & 43.13/43.07 & 36.77/37.56 & 22.50/22.71/16.73 & 19.60/17.71/11.45 & 17.12/16.15/9.92 & 25.0 \\
\multirow{3}{*}{Lite-FPN \cite{2021litefpn}} & \multirow{3}{*}{onboard} & ResNet-18 & \multirow{3}{*}{GTX 2080Ti} & \multicolumn{3}{c}{\multirow{3}{*}{-}} & 17.04 / - / - & 14.02 / - / - & 12.23 / - / - & 88.57 \\
& & ResNet-34 & & \multicolumn{3}{c}{} & 18.01 / - / 15.32 & 15.29 / - / 10.64 & 14.28 / - / 8.59 & 71.32 \\
& & DLA-34 & & \multicolumn{3}{c}{} & 19.31 / - / - & 16.19 / - / - & 15.47 / - / - & 42.37 \\
Ours & roadside & ResNet-50 & GTX 1080Ti & \multicolumn{3}{c}{91.34 / -} & \multicolumn{3}{c}{79.36 / - / 51.30} & 41.18 \\
\bottomrule
\end{tabular
}
\end{table*}
Figure \ref{fig:vis_results_all} illustrates visualization results on SVLD-3D test set. Different views and types of vehicles in SVLD-3D test set are tested, with occlusion of environment and other vehicles. Vehicles are widely distributed in the scene. From Table \ref{tab:table_compare_kitti} and Figure \ref{fig:vis_results_all}, detected vehicles all response with red circle in the heatmaps. It can be seen that our network achieves real-time 3D vehicle detection and is adaptive to occlusion and small vehicles.
\begin{figure*}[!h]
\centering
\subcaptionbox{\centering Scene A\label{subfig:vis_results_all_a}}
\includegraphics[height=2.1cm]{Fig_9a
\includegraphics[height=2.1cm]{Fig_9a_heatmap}\quad
\includegraphics[height=2.1cm]{Fig_9b
\includegraphics[height=2.1cm]{Fig_9b_heatmap
}
\subcaptionbox{\centering Scene B\label{subfig:vis_results_all_b}}
\includegraphics[height=2.1cm]{Fig_9c
\includegraphics[height=2.1cm]{Fig_9c_heatmap}\quad
\includegraphics[height=2.1cm]{Fig_9d
\includegraphics[height=2.1cm]{Fig_9d_heatmap
}
\subcaptionbox{\centering Scene C\label{subfig:vis_results_all_c}}
\includegraphics[height=2.1cm]{Fig_9e
\includegraphics[height=2.1cm]{Fig_9e_heatmap}\quad
\includegraphics[height=2.1cm]{Fig_9f
\includegraphics[height=2.1cm]{Fig_9f_heatmap
}
\subcaptionbox{\centering Scene D\label{subfig:vis_results_all_d}}
\includegraphics[height=2.1cm]{Fig_9g
\includegraphics[height=2.1cm]{Fig_9g_heatmap}\quad
\includegraphics[height=2.1cm]{Fig_9h
\includegraphics[height=2.1cm]{Fig_9h_heatmap
}
\subcaptionbox{\centering Scene E\label{subfig:vis_results_all_e}}
\includegraphics[height=2.1cm]{Fig_9i
\includegraphics[height=2.1cm]{Fig_9i_heatmap}\quad
\includegraphics[height=2.1cm]{Fig_9j
\includegraphics[height=2.1cm]{Fig_9j_heatmap
}
\caption{\leftskip=0pt \rightskip=0pt plus 0cm Visualization results on SVLD-3D test set. 3D vehicle detection results (left) are represented by color bounding boxes. Centroids of heatmaps (right) are represented in red circles. \label{fig:vis_results_all}}
\end{figure*}
3D vehicle information which can be obtained by different methods is compared in Table \ref{tab:table_compare_func}. It can be seen that our method obtains not only 3D bounding boxes but also 3D centroids, dimensions, and 3D locations, with real-time performance.
\begin{table}[!h]
\centering
\caption{Comparison of 3D vehicle information obtained by different methods.}
\label{tab:table_compare_func}
\begin{tabular}{ccccc}
\toprule
Method & BBox & Centroid & Dimension & Location \\
\midrule
MonoGRNet \cite{2019monogrnet} & \checkmark & & \checkmark & \checkmark \\
Deep3DBox \cite{2017deep3dbox} & \checkmark & & & \\
GS3D \cite{2019gs3d} & \checkmark & & \checkmark & \\
RTM3D \cite{2020rtm3d} & \checkmark & \checkmark & \checkmark & \\
SMOKE \cite{2020SMOKE} & \checkmark & \checkmark & \checkmark & \\
KM3D \cite{2021km3d} & \checkmark & & \checkmark & \\
Lite-FPN \cite{2021litefpn} & \checkmark & & \checkmark & \\
Ours & \checkmark & \checkmark & \checkmark & \checkmark \\
\bottomrule
\end{tabular}
\end{table}
\subsection{3D Vehicle Localization Precision and Error of CenterLoc3D}
\label{subsec:3dlocpe of centerloc3d}
Table \ref{tab:table_loc_results} and Figure \ref{fig:vis_loc_results} show 3D vehicle localization results and precision of different scenes and types in SVLD-3D test set, which is calculated by Equation \ref{equa_ploc}. It can be seen that our network can also achieve good results in 3D vehicle localization, with an average precision of 98\%.
Top views of 3D vehicle localization of different frames in SVLD-3D test set are shown in Figure \ref{fig:vis_loc_results_top}, which also shows that our network has high precision in 3D vehicle localization. Each sub-figure in Figure \ref{fig:vis_loc_results_top} represents different frames in different scenes. At the same time, vehicles far from the roadside camera can also be detected and localized.
\begin{table}[htbp]
\centering
\caption{3D vehicle localization results and precision on SVLD-3D test set.}
\label{tab:table_loc_results}
\resizebox{0.7\textwidth}{!}
\begin{tabular}{ccccc}
\toprule
Vehicle & Type & ${P_{centroid}}$ & ${\widetilde P_{centorid}}$ & Precision \\
\midrule
1 & Car & 24.927, 88.430, 0.673 & 24.952, 88.519, 0.760 & 0.996 \\
2 & Car & 8.611, 68.119, 0.717 & 8.585, 67.729, 0.770 & 0.991 \\
3 & Car & -0.015, 82.595, 0.755 & -0.015, 82.595, 0.755 & 0.999 \\
4 & Car & 8.188, 105.823, 0.780 & 8.217, 105.714, 0.825 & 0.996 \\
5 & Car & 11.508, 72.225, 0.731 & 11.434, 71.608, 0.785 & 0.984 \\
6 & Car & 18.829, 62.322, 0.727 & 18.791, 61.883, 0.790 & 0.990 \\
7 & Truck & 21.315, 43.322, 0.958 & 21.219, 43.156, 0.875 & 0.990 \\
8 & Car & 8.538, 38.425, 0.735 & 8.451, 38.100, 0.730 & 0.988 \\
9 & Car & 18.144, 67.893, 0.674 & 18.165, 68.067, 0.700 & 0.995 \\
10 & Car & 0.336, 43.784, 0.730 & 0.382, 43.959, 0.700 & 0.993 \\
11 & Car & 3.812, 46.123, 0.697 & 3.794, 46.043, 0.710 & 0.997 \\
12 & Car & 11.035, 65.584, 0.730 & 11.261, 65.935, 0.740 & 0.976 \\
13 & Car & 0.142, 81.101, 0.721 & 0.100, 80.344, 0.770 & 0.984 \\
14 & Car & -14.298, 59.759, 0.662 & -14.297, 59.758, 0.680 & 0.999 \\
15 & Car & -6.232, 39.097, 0.672 & -6.193, 39.138, 0.750 & 0.993 \\
16 & Car & -9.671, 38.371, 0.705 & -9.754, 38.703, 0.665 & 0.978 \\
17 & Car & -6.249, 56.957, 0.681 & -6.275, 56.747, 0.690 & 0.989 \\
18 & Car & -10.033, 64.324, 0.740 & -10.032, 64.321, 0.740 & 0.999 \\
19 & Car & -1.770, 53.300, 0.756 & -1.820, 52.683, 0.860 & 0.975 \\
20 & Bus & -5.174, 71.789, 1.452 & -5.341, 73.016, 1.410 & 0.936 \\
21 & Car & -7.645, 57.673, 0.800 & -7.593, 57.247, 0.800 & 0.975 \\
22 & Car & 0.356, 22.313, 0.739 & 0.340, 22.388, 0.670 & 0.994 \\
23 & Car & 0.862, 37.990, 0.735 & 0.804, 37.053, 0.765 & 0.957 \\
24 & Car & 1.376, 40.059, 0.825 & 1.412, 40.053, 0.860 & 0.993 \\
\bottomrule
\end{tabular}
}
\end{table}
\begin{figure}[htbp]
\centering
\subfloat[\centering Scene A-4093]{\includegraphics[width=0.15\linewidth]{Fig_10a
\label{fig:vis_loc_results_top_a}}
\hfil
\subfloat[\centering Scene A-4172]{\includegraphics[width=0.15\linewidth]{Fig_10b
\label{fig:vis_loc_results_top_b}}
\hfil
\subfloat[\centering Scene A-4391]{\includegraphics[width=0.15\linewidth]{Fig_10c
\label{fig:vis_loc_results_top_c}}
\newline
\subfloat[\centering Scene B-6647]{\includegraphics[width=0.15\linewidth]{Fig_10d
\label{fig:vis_loc_results_top_d}}
\hfil
\subfloat[\centering Scene B-6751]{\includegraphics[width=0.15\linewidth]{Fig_10e
\label{fig:vis_loc_results_top_e}}
\hfil
\subfloat[\centering Scene B-8001]{\includegraphics[width=0.15\linewidth]{Fig_10f
\label{fig:vis_loc_results_top_f}}
\newline
\subfloat[\centering Scene C-8241]{\includegraphics[width=0.15\linewidth]{Fig_10g
\label{fig:vis_loc_results_top_g}}
\hfil
\subfloat[\centering Scene C-8287]{\includegraphics[width=0.15\linewidth]{Fig_10h
\label{fig:vis_loc_results_top_h}}
\hfil
\subfloat[\centering Scene C-8351]{\includegraphics[width=0.15\linewidth]{Fig_10i
\label{fig:vis_loc_results_top_i}}
\newline
\subfloat[\centering Scene D-4752]{\includegraphics[width=0.15\linewidth]{Fig_10j
\label{fig:vis_loc_results_top_j}}
\hfil
\subfloat[\centering Scene D-5743]{\includegraphics[width=0.15\linewidth]{Fig_10k
\label{fig:vis_loc_results_top_k}}
\hfil
\subfloat[\centering Scene D-7930]{\includegraphics[width=0.15\linewidth]{Fig_10l
\label{fig:vis_loc_results_top_l}}
\newline
\subfloat[\centering Scene E-4060]{\includegraphics[width=0.15\linewidth]{Fig_10m
\label{fig:vis_loc_results_top_m}}
\hfil
\subfloat[\centering Scene E-5116]{\includegraphics[width=0.15\linewidth]{Fig_10n
\label{fig:vis_loc_results_top_n}}
\hfil
\subfloat[\centering Scene E-6712]{\includegraphics[width=0.15\linewidth]{Fig_10o
\label{fig:vis_loc_results_top_o}}
\hfil
\newline
\caption{\leftskip=0pt \rightskip=0pt plus 0cm Top views of 3D vehicle localization of different frames in SVLD-3D test set. The red star indicates ground truth 3D vehicle locations. The blue square indicates predicted 3D vehicle locations. The number below each sub-figure indicates the frame number.}
\label{fig:vis_loc_results_top}
\end{figure}
\begin{figure*}[htbp]
\centering
\subfloat[\centering ]{\includegraphics[width=0.28\linewidth]{Fig_11a
\label{fig:vis_loc_results_a}}
\hfil
\subfloat[\centering ]{\includegraphics[width=0.28\linewidth]{Fig_11b
\label{fig:vis_loc_results_b}}
\hfil
\subfloat[\centering ]{\includegraphics[width=0.28\linewidth]{Fig_11c
\label{fig:vis_loc_results_c}}
\newline
\subfloat[\centering ]{\includegraphics[width=0.28\linewidth]{Fig_11d
\label{fig:vis_loc_results_d}}
\hfil
\subfloat[\centering ]{\includegraphics[width=0.28\linewidth]{Fig_11e
\label{fig:vis_loc_results_e}}
\hfil
\subfloat[\centering ]{\includegraphics[width=0.28\linewidth]{Fig_11f
\label{fig:vis_loc_results_f}}
\newline
\subfloat[\centering ]{\includegraphics[width=0.28\linewidth]{Fig_11g
\label{fig:vis_loc_results_g}}
\hfil
\subfloat[\centering ]{\includegraphics[width=0.28\linewidth]{Fig_11h
\label{fig:vis_loc_results_h}}
\hfil
\subfloat[\centering ]{\includegraphics[width=0.28\linewidth]{Fig_11i
\label{fig:vis_loc_results_i}}
\newline
\caption{\leftskip=0pt \rightskip=0pt plus 0cm Visualization results of 3D vehicle localization on SVLD-3D test set. Color boxes indicate the predicted results. Pink boxes indicate the ground truth results. Numbers on the arrow pointing to vehicles are consistent with numbers in Table \ref{tab:table_loc_results}.}
\label{fig:vis_loc_results}
\end{figure*}
\begin{figure*}[htbp]
\centering
\subfloat[\centering Scene A]{\includegraphics[width=0.32\linewidth]{Fig_12a
\label{fig:error_loc_curves_a}}
\subfloat[\centering Scene B]{\includegraphics[width=0.32\linewidth]{Fig_12b
\label{fig:error_loc_curves_b}}
\subfloat[\centering Scene C]{\includegraphics[width=0.32\linewidth]{Fig_12c
\label{fig:error_loc_curves_c}}
\newline
\begin{center}
\subfloat[\centering Scene D]{\includegraphics[width=0.32\linewidth]{Fig_12d
\label{fig:error_loc_curves_d}}
\subfloat[\centering Scene E]{\includegraphics[width=0.32\linewidth]{Fig_12e
\label{fig:error_loc_curves_e}}
\end{center}
\caption{\leftskip=0pt \rightskip=0pt plus 0cm 3D vehicle localization error of X- (green), Y- (red), Z- (blue) axis, and total error (orange) according to the distances between vehicles and roadside cameras in different scenes of SVLD-3D test set.}
\label{fig:error_loc_curves}
\end{figure*}
3D vehicle localization error is calculated by Equation \ref{equa_eloc}. In Figure \ref{fig:error_loc_curves}, we can see that 3D vehicle localization error increases as the distance grows between vehicles and roadside cameras of different scenes in SVLD-3D test set. The Y-axis error is the largest among the X-, Y-, and Z-axis. The largest error in scene B is larger than that in other scenes. This is due to the fact that the camera pan angle in scene B is closer to 0° than in the other scenes, which leads to incomplete vehicle feature learning along the vehicle length direction.
\subsection{3D Vehicle Dimension Precision and Error of CenterLoc3D}
\label{subsec:3dsizepe of centerloc3d}
Table \ref{tab:table_size_results} and Figure \ref{fig:vis_size_results} show 3D vehicle dimension prediction results and precision of different scenes and types in SVLD-3D test set, which is calculated by Equation \ref{equa_pdim}. It can be seen that our network can also achieve good results in 3D vehicle dimension prediction, with an average precision of 85\%.
\begin{table}[htbp]
\centering
\caption{3D vehicle dimension prediction results and precision on SVLD-3D test set.}
\label{tab:table_size_results}
\resizebox{0.7\textwidth}{!}
\begin{tabular}{ccccc}
\toprule
Vehicle & Type & ${D_v}$ & ${\widetilde D_v}$ & Precision \\
\midrule
1 & Car & 3.60, 1.71, 1.37 & 3.79, 1.70, 1.27 & 0.860 \\
2 & Car & 3.26, 1.67, 1.31 & 3.18, 1.61, 1.25 & 0.890 \\
3 & Car & 4.05, 1.76, 1.40 & 3.92, 1.80, 1.40 & 0.942 \\
4 & Car & 4.51, 1.81, 1.47 & 4.42, 1.88, 1.46 & 0.935 \\
5 & Car & 4.43, 1.78, 1.37 & 4.40, 1.78, 1.48 & 0.915 \\
6 & Car & 4.74, 1.80, 1.46 & 4.33, 1.77, 1.40 & 0.843 \\
7 & Car & 4.58, 1.82, 1.45 & 4.96, 1.86, 1.54 & 0.845 \\
8 & Car & 4.50, 1.79, 1.40 & 4.50, 1.70, 1.36 & 0.912 \\
9 & Car & 3.74, 1.64, 1.27 & 4.07, 1.68, 1.30 & 0.872 \\
10 & Car & 4.55, 1.80, 1.42 & 4.58, 1.68, 1.40 & 0.910 \\
11 & Car & 3.57, 1.80, 1.35 & 4.11, 1.80, 1.38 & 0.844 \\
12 & Car & 3.71, 1.76, 1.36 & 3.90, 1.80, 1.33 & 0.912 \\
13 & Car & 3.34, 1.77, 1.32 & 3.70, 1.76, 1.25 & 0.838 \\
14 & Bus & 12.83, 2.71, 2.75 & 12.00, 2.76, 2.82 & 0.886 \\
15 & Car & 4.74, 1.87, 1.48 & 4.77, 1.83, 1.53 & 0.939 \\
16 & Car & 5.00, 1.89, 1.48 & 4.75, 1.86, 1.56 & 0.880 \\
17 & Car & 4.69, 1.84, 1.44 & 4.60, 1.81, 1.37 & 0.914 \\
18 & Car & 4.68, 1.85, 1.43 & 4.56, 1.81, 1.34 & 0.885 \\
19 & Car & 4.64, 1.84, 1.45 & 4.68, 1.82, 1.50 & 0.947 \\
20 & Bus & 12.74, 2.68, 2.62 & 12.00, 2.76, 2.82 & 0.838 \\
\bottomrule
\end{tabular}
}
\end{table}
\begin{figure*}[htbp]
\centering
\subfloat[\centering ]{\includegraphics[width=0.28\linewidth]{Fig_13a
\label{fig:vis_size_results_a}}
\hfil
\subfloat[\centering ]{\includegraphics[width=0.28\linewidth]{Fig_13b
\label{fig:vis_size_results_b}}
\hfil
\subfloat[\centering ]{\includegraphics[width=0.28\linewidth]{Fig_13c
\label{fig:vis_size_results_c}}
\newline
\subfloat[\centering ]{\includegraphics[width=0.28\linewidth]{Fig_13d
\label{fig:vis_size_results_d}}
\hfil
\subfloat[\centering ]{\includegraphics[width=0.28\linewidth]{Fig_13e
\label{fig:vis_size_results_e}}
\hfil
\subfloat[\centering ]{\includegraphics[width=0.28\linewidth]{Fig_13f
\label{fig:vis_size_results_f}}
\newline
\subfloat[\centering ]{\includegraphics[width=0.28\linewidth]{Fig_13g
\label{fig:vis_size_results_g}}
\hfil
\subfloat[\centering ]{\includegraphics[width=0.28\linewidth]{Fig_13h
\label{fig:vis_size_results_h}}
\hfil
\subfloat[\centering ]{\includegraphics[width=0.28\linewidth]{Fig_13i
\label{fig:vis_size_results_i}}
\newline
\caption{\leftskip=0pt \rightskip=0pt plus 0cm Visualization results of 3D vehicle dimension prediction on SVLD-3D test set. Color boxes indicate the predicted results. Pink boxes indicate the ground truth results. Numbers on the arrow pointing to vehicles are consistent with numbers in Table \ref{tab:table_size_results}.}
\label{fig:vis_size_results}
\end{figure*}
\begin{figure*}[htbp]
\centering
\subfloat[\centering Scene A]{\includegraphics[width=0.32\linewidth]{Fig_14a
\label{fig:error_size_curves_a}}
\subfloat[\centering Scene B]{\includegraphics[width=0.32\linewidth]{Fig_14b
\label{fig:error_size_curves_b}}
\subfloat[\centering Scene C]{\includegraphics[width=0.32\linewidth]{Fig_14c
\label{fig:error_size_curves_c}}
\newline
\begin{center}
\subfloat[\centering Scene D]{\includegraphics[width=0.32\linewidth]{Fig_14d
\label{fig:error_size_curves_d}}
\subfloat[\centering Scene E]{\includegraphics[width=0.32\linewidth]{Fig_14e
\label{fig:error_size_curves_e}}
\end{center}
\caption{\leftskip=0pt \rightskip=0pt plus 0cm 3D vehicle dimension prediction error of length (green), width (red), height (blue), and total error (orange) according to the distances between vehicles and roadside cameras in three different scenes of SVLD-3D test set.}
\label{fig:error_size_curves}
\end{figure*}
In Figure \ref{fig:error_size_curves}, we can see that 3D vehicle dimension error increases continuously as the distance grows between vehicles and roadside cameras of different scenes in SVLD-3D test set. The vehicle length error is the largest among length, width, and height. Since the driving directions of vehicles are usually parallel to the road direction, partial feature occlusion exists along the vehicle length direction. Therefore, the vehicle dimension prediction error in the length direction is larger than width and height.
3D vehicle dimension prediction error is calculated by Equation \ref{equa_edim}. The errors of different methods are shown in Table \ref{tab:compare_dimensions}, from which it can be seen that our network has certain advantages in 3D vehicle dimension prediction.
\begin{table}[htbp]
\centering
\caption{Comparison of 3D vehicle dimension prediction errors of different methods.}
\label{tab:compare_dimensions}
\begin{tabular}{cccc}
\toprule
Method & Length/m & Width/m & Height/m \\
\midrule
3DOP \cite{2017stereo} & 0.504 & 0.094 & 0.107 \\
Mono3D \cite{2016mono3ddet} & 0.582 & 0.103 & 0.172 \\
MonoGRNet \cite{2019monogrnet} & 0.412 & 0.084 & 0.084 \\
MonoGRK \cite{2020keypoint} & 0.403 & 0.091 & 0.101 \\
Ours & 0.137 & 0.031 & 0.030 \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Ablation Study of CenterLoc3D}
\label{subsec:ablation study}
To further validate the effect of the weighted-fusion module and loss with spatial constraint embedding in our network, ablation experiments are conducted on SVLD-3D validation and test set. CenterNet \cite{2019centernet} with ResNet-50 \cite{2016resnet} is used as our baseline. We added the proposed modules one by one for validation. Ablation results are shown in Table \ref{tab:ablation_study}. For the improvement column, only $A{P_{3D}}$ of IoU threshold 0.7 on the test set is listed for comparison. In Table \ref{tab:ablation_study}, we can see that the results of adding three modules one by one outperform the former model by 6.64\%, 2.48\%, and 5.98\% in $A{P_{3D}}$ respectively. Therefore, conclusions can be summarized as follows: 1) Weighted-fusion module not only enables the network adaptive to different vehicle sizes but also increases network generalization capability. 2) Spatial constraints of camera calibration and vehicle IoU in loss help accurate 3D bounding box learning.
\begin{table*}[htbp]
\centering
\caption{Ablation study with different modules in CenterLoc3D on SVLD-3D validation and test set.}
\label{tab:ablation_study}
\resizebox{0.9\textwidth}{!}
\begin{tabular}{ccccccc}
\toprule
\multirow{2}{*}{Model} & \multicolumn{3}{c}{Modules} & \multirow{2}{*}{$A{P_{3D}}(IOU > 0.7)$} & \multirow{2}{*}{FPS} & \multirow{2}{*}{Improvement of $A{P_{3D}}$} \\ \cline{2-4}
& Weighted-Fusion & Reprojection & IoU & & & \\
\midrule
${M_{base}}$ & & & & 52.52 / 36.20 & 46.73 & - \\
${M_1}$ & \checkmark & & & 57.07 / 42.84 & 43.23 & 6.64 \\
${M_2}$ & \checkmark & \checkmark & & 68.38 / 45.32 & 41.31 & 2.48 \\
${M_3}$ & \checkmark & \checkmark & \checkmark & 79.36 / 51.30 & 41.18 & 5.98 \\
\bottomrule
\end{tabular}
}
\end{table*}
Figure \ref{fig:ablation_study_loc_error} shows 3D vehicle localization error of ablation study on SVLD-3D test set. Only three views (left, middle and right) of BrnoCompSpeed scenes in SVLD-3D test set are included in this study. In this figure, it can be seen that 3D vehicle localization error from ${M_{base}}$ to ${M_3}$ in the same scene decreases gradually, which indicates that the designed modules can effectively reduce 3D vehicle localization error.
\begin{figure*}[htbp]
\centering
\subfloat[\centering ${M_{base}}$-Scene A-0.34236]{\includegraphics[width=0.32\linewidth]{Fig_15_Mbase_a
\label{fig:ablation_study_loc_error_a}}
\hfil
\subfloat[\centering ${M_{base}}$-Scene B-0.35048]{\includegraphics[width=0.32\linewidth]{Fig_15_Mbase_b
\label{fig:ablation_study_loc_error_b}}
\hfil
\subfloat[\centering ${M_{base}}$-Scene C-0.29607]{\includegraphics[width=0.32\linewidth]{Fig_15_Mbase_c
\label{fig:ablation_study_loc_error_c}}
\newline
\subfloat[\centering ${M_1}$-Scene A-0.26307]{\includegraphics[width=0.32\linewidth]{Fig_15_M1a
\label{fig:ablation_study_loc_error_d}}
\hfil
\subfloat[\centering ${M_1}$-Scene B-0.26794]{\includegraphics[width=0.32\linewidth]{Fig_15_M1b
\label{fig:ablation_study_loc_error_e}}
\hfil
\subfloat[\centering ${M_1}$-Scene C-0.25254]{\includegraphics[width=0.32\linewidth]{Fig_15_M1c
\label{fig:ablation_study_loc_error_f}}
\newline
\subfloat[\centering ${M_2}$-Scene A-0.29456]{\includegraphics[width=0.32\linewidth]{Fig_15_M2a
\label{fig:ablation_study_loc_error_g}}
\hfil
\subfloat[\centering ${M_2}$-Scene B-0.30740]{\includegraphics[width=0.32\linewidth]{Fig_15_M2b
\label{fig:ablation_study_loc_error_h}}
\hfil
\subfloat[\centering ${M_2}$-Scene C-0.24080]{\includegraphics[width=0.32\linewidth]{Fig_15_M2c
\label{fig:ablation_study_loc_error_i}}
\newline
\subfloat[\centering ${M_3}$-Scene A-0.29415]{\includegraphics[width=0.32\linewidth]{Fig_15_M3a
\label{fig:ablation_study_loc_error_j}}
\hfil
\subfloat[\centering ${M_3}$-Scene B-0.29399]{\includegraphics[width=0.32\linewidth]{Fig_15_M3b
\label{fig:ablation_study_loc_error_k}}
\hfil
\subfloat[\centering ${M_3}$-Scene C-0.22648]{\includegraphics[width=0.32\linewidth]{Fig_15_M3c
\label{fig:ablation_study_loc_error_l}}
\newline
\caption{\leftskip=0pt \rightskip=0pt plus 0cm 3D vehicle localization error of ablation study on SVLD-3D test set. The results of ${M_{base}}$, ${M_1}$, ${M_2}$ and ${M_3}$ of different scenes are shown from the first to fourth row. 3D vehicle localization error of X-, Y-, Z-axis, and total error according to the distances between vehicles and roadside cameras on SVLD-3D test set are represented by green, red, blue, and orange lines. The number below each sub-figure indicates the average 3D vehicle localization error in meters of the corresponding scene.}
\label{fig:ablation_study_loc_error}
\end{figure*}
\section{Conclusion}
\label{section:conclusion}
Through experimental validation, CenterLoc3D achieves good performance on 3D vehicle detection, localization, and dimension prediction for roadside surveillance cameras, with $A{P_{3D}}$ of 51.30\%, average 3D localization precision of 98\%, average 3D dimension precision of 85\% and real-time performance with FPS of 41.18. Our contributions are as follows: 1) A 3D vehicle localization network CenterLoc3D for roadside monocular cameras is proposed, which can directly obtain 3D bounding boxes and 3D dimensions without leveraging 2D detectors. 2) A weighted-fusion strategy is proposed, which can effectively enhance feature extraction and improve generalization. 3) Loss function with constraints of camera calibration and vehicle IoU are embedded in CenterLoc3D, which reduces 3D vehicle localization error. In addition, we also propose a benchmark including a dataset, an annotation tool, and evaluation metrics, which provides a data basis for experimental validation.
However, CenterLoc3D is still needed to be improved for practical and advanced applications. When the camera pan angle is close to 0°, the features along the vehicle length direction are incomplete, leading to an increase in 3D vehicle localization error. In future work, the dataset needs to be further expanded to contain scenes with more views and more types of vehicles. In the meanwhile, more effective feature extraction modules and loss functions are needed to be designed to further improve 3D vehicle localization precision in roadside monocular traffic scenes.
|
1,116,691,500,808 | arxiv | \section{Introduction}
By using and generalizing the extended lattice operations due to Gowda, Sznajder and Tao
\cite{GowdaSznajderTao2004}, in \cite{NemethNemeth2012a} and \cite{NemethNemeth2012b}
it has been shown that the projection onto a closed convex set is
isotone with respect to the order defined by a cone if and only if the set is invariant with
respect to the extended lattice operations defined by the cone. We shall call such a set simply
invariant with respect to the cone, or if there is no ambiguity, lattice-like, or shortly l-l. We also showed that
the a closed convex set with interior points is l-l if and only if all of its tangent hyperplanes are l-l.
These results were motivated by iterative methods for variational
inequalities similar to the ones for complementarity problems in
\cite{IsacNemeth1990b,IsacNemeth1990c,Nemeth2009a,AbbasNemeth2011}. More specifically, a
variational inequality defined by a closed convex set $C$ and
a function $f$ can be equivalently written as the fixed point problem $\m x=P_C(\m x-f(\m x))$, where
$P_C$ is the projection onto the closed convex set $C$. If the Picard iteration
$\m x_{k+1}=P_C(\m x_k-f(\m x_k))$ is convergent and $f$ continuous, then the limit of $\m x_k$ is a
solution of the variational inequality defined by $f$ and $C$. Therefore, it is important to
give conditions under which the Picard iteration is convergent. This idea has been exploited
in several papers, such as \cite{Auslender1976,Bertsekas1989,Iusem1997,Khobotov1987,Korpelevich1976,Marcotte1991,Nagurney1993,Sibony1970,Solodov1999,Solodov1996,Sun1996}. However, none of these
papers used the monotonicity of the sequence $\m x_k$. If one can show that
$\m x_k$ is
monotone increasing (decreasing) and bounded from above (below) with respect to an order
defined by a regular cone (that is, a cone for which all such sequences are convergent), then
it is convergent and its limit is a solution of the variational inequality defined by $f$ and
$C$. In \cite{IsacNemeth1990b,IsacNemeth1990c,Nemeth2009a,AbbasNemeth2011} the convergence of the
sequence $\m x_k$ was proved by using its monotonicity. Although they use
non-iterative methods, we also mention the paper of H. Nishimura and E. A. Ok
\cite{NishimuraOk2012}, where the isotonicity of the projection onto a closed convex set is used
for studying the solvability of variational inequalities and related equilibrium problems.
To further accentuate the importance of ordered vector structures let us also mention that
recently they are getting more and more ground in studying various fixed point and related
equilibrium problems (see the book \cite{CarlHeikilla2011} of S. Carl and S Heikkil\"a and the
references therein). The case of a self-dual
cone is of special importance because of the elegant examples for invariant sets with respect
to the nonnegative orthant and the Lorentz cone \cite{NemethNemeth2012a}.
Moreover, properties of self-dual cones are becoming increasingly important because of conic
optimization and applications of the analysis on symmetric cones. Especially important
self-dual cones in applications are the nonnegative orthant, the Lorentz cone and the positive
semidefinite cone, however the class of self-dual cones is much larger \cite{BarkerForan1976}.
The results of \cite{NemethNemeth2012a} and \cite{NemethNemeth2012b} extend the results
of \cite{Isac1996} and \cite{NishimuraOk2012}. G. Isac showed in \cite{Isac1996} that the
projection onto a closed convex sublattice of the Euclidean space ordered by the nonnegative
orthant is isotone. H. Nishimura and E. A. Ok proved an extension of this
result and its converse to Hilbert spaces in \cite{NishimuraOk2012}. The study of invariant sets with respect to
the nonnegative orthant goes back to the results of D. M. Topkis \cite{Topkis1976} and A. F.
Veinott Jr. \cite{Veinott1981}, but it wasn't until quite recently when all such invariant
sets have been determined by M. Queyranne and F. Tardella \cite{QueyranneTardella2006}. The
same results have been obtained in \cite{NemethNemeth2012a} in a more geometric way. Although
\cite{NemethNemeth2012a} also determined the invariant sets with respect to the Lorentz cone,
it left open the question of finding the invariant sets with respect to the cone $\smp$ of
$n\times n$ positive semidefinite matrices, called the positive semidefinite cone.
As a particular case we show that if
$n\geq 3$, then there is no proper closed convex l-l set with nonempty interior in
the space $(\sm,\smp)$ (the space $\sm$ of $n\times n$ symmetric matrices ordered by the
cone $\smp$ of symmetric positive semidefinite matrices). For this it is
enough to show that there are no invariant hyperplanes because the closed convex invariant sets
with nonempty interior are the ones which have all tangent hyperplanes invariant.
All these problems can be handled in the unifying context of the Euclidean Jordan algebras.
This way we can augment this field to an approach, where the order induced
by the cone of squares (the basic notion of the Jordan algebra) becomes emphasized.
To shorten our exposition, we assume the knowledge of basic facts and results on
Euclidean Jordan algebras. We strive to be in accordance with the terminology
in \cite{FarautKoranyi1994}. A concise introduction of the used
basic notions and facts in the field can be found in \cite{GowdaSznajderTao2004}.
\section{Preliminaries}
Denote by $\R^m$ the $m$-dimensional Euclidean space endowed with the scalar
product $\lng\cdot,\cdot\rng:\R^m\times\R^m\to\R,$ and the Euclidean norm $\|\cdot\|$ and topology
this scalar product defines.
Throughout this note we shall use some standard terms and results from convex geometry
(see e.g. \cite{Rockafellar1970} and \cite{Zarantonello1971}).
Let $K$ be a \emph{convex cone} in $\R^m$, i.e., a nonempty set with
(i) $K+K\subset K$ and (ii) $tK\subset K,\;\forall \;t\in \R_+ =[0,+\infty)$.
The convex cone $K$ is called \emph{pointed}, if $K\cap(-K)=\{\m 0\}.$
The convex cone $K$ is {\it generating} if $K-K=\R^m$.
For any $\m x,\m y\in \R^m$, by the equivalence $\m x\leq_K\m y\Leftrightarrow\m y-\m x\in K$, the
convex cone $K$
induces an {\it order relation} $\leq_K$ in $\R^m$, that is, a binary relation, which is
reflexive and transitive. This order relation is {\it translation invariant}
in the sense that $\m x\leq_K\m y$ implies $\m x+\m z\leq_K\m y+\m z$ for all $\m z\in \R^m$, and
{\it scale invariant} in the sense that $\m x\leq_K\m y$ implies $t\m x\leq_K t\m y$ for any $t\in \R_+$.
If $\leq$ is a translation invariant and scale invariant order relation on $\R^m$, then
$\leq=\leq_K$, where $K=\{\m x\in\R^m:\m 0\leq\m x\}$ is a convex cone. If $K$ is pointed, then $\leq_K$ is
\emph{antisymmetric} too, that is $\m x\leq_K\m y$ and $\m y\leq_K\m x$ imply that $\m x=\m y.$
The elements $\m x$ and $\m y$ are called \emph{comparable} if $\m x\leq_K\m y$
or $\m y\leq_K\m x.$
We say that $\leq_K$ is a \emph{latticial order} if for each pair
of elements $\m x,\m y\in \R^m$ there exist the least upper bound
$\sup\{\m x,\m y\}$ and the greatest lower bound $\inf\{\m x,\m y\}$ of
the set $\{\m x,\m y\}$ with respect to the order relation $\leq_K$.
In this case $K$ is said a \emph{latticial or simplicial cone},
and $\R^m$ equipped with a latticial order is called an
\emph{Euclidean vector lattice}.
The \emph{dual} of the convex cone $K$ is the set
$$K^*:=\{\m y\in \R^m:\;\lng\m x,\m y\rng \geq 0,\;\forall \;\m x\in K\},$$
with $\lng\cdot,\cdot\rng $ the standard scalar product in $\R^m$.
The convex cone $K$ is called \emph{self-dual}, if $K=K^*.$ If $K$
is self-dual, then it is a generating pointed closed convex cone.
In all that follows we shall suppose that $\R^m$ is endowed with a
Cartesian reference system with a basis $\m e_1,\dots,\m e_m$. If $\m x\in \R^m$, then
$$\m x=x_1 \m e_1+...+x_m \m e_m$$ can be characterized by the ordered $m$-tuple of real numbers
$x_1,...,x_m$, called
\emph{the coordinates of} $x$ with respect to the given reference system, and we shall write
$\m x=(x_1,...,x_m).$ With this notation we have $\m e_i=(0,...,0,1,0,...,0),$
with $1$ in the $i$-th position and $0$ elsewhere. Let
$\m x,\m y\in \R^m$, $\m x=(x_1,...,x_m)$, $\m y=(y_1,...,y_m)$, where $x_i$, $y_i$ are the coordinates of
$\m x$ and $\m y$, respectively with respect to the reference system. Then, the scalar product of $\m x$
and $\m y$ is the sum
$\lng\m x,\m y\rng =\sum_{i=1}^m x_iy_i.$
It is easy to see that $\m e_1,\dots,\m e_m$ is an orthonormal system of vectors with respect to
this scalar product, in the sense
that $\lng \m e_i,\m e_j\rng =\delta_i^j$, where $\delta_i^j$ is the Kronecker symbol.
The set
\[\R^m_+=\{\m x=(x_1,...,x_m)\in \R^m:\; x_i\geq 0,\;i=1,...,m\}\]
is called the \emph{nonnegative orthant} of the above introduced Cartesian
reference system. A direct verification shows that $\R^m_+$ is a
self-dual cone.
The set
\begin{equation}\label{lorentzcone}
\mathcal L^{m+1}_+=\{(\m x,x_{m+1})\in\R^{m}\otimes\R=\R^{m+1}:\;\|\m x\|\leq x_{m+1}\},
\end{equation}
is a self-dual cone called the
\emph{$m+1$-dimensional second order cone}, or the
\emph{$m+1$-dimensional Lorentz cone}, or the \emph{$m+1$-dimensional ice-cream cone}
\cite{GowdaSznajderTao2004}.
The nonnegative orthant $\R^m_+$ and the Lorentz cone $L$ defined above are the
most important and commonly used self-dual cones in the Euclidean space.
But the family of self-dual cones is rather rich
\cite{BarkerForan1976}.
\section{Generalized lattice operations}\label{Generalized lattice operations}
A \emph{hyperplane through the origin}, is a set of form
\begin{equation}\label{hypersubspace}
H(\m 0,\m a)=\{\m x\in \R^m:\;\lng\m a,\m x\rng =0\},\;\;\m a\not=\m 0.
\end{equation}
For simplicity the hyperplanes through $\m 0$ will also be denoted by $H$.
The nonzero vector $\m a$ in the above formula is called \emph{the normal}
of the hyperplane.
A \emph{hyperplane through $\m u\in\R^m$ with the normal $\m a$} is the set of the form
\begin{equation}\label{hyperplane}
H(\m u,\m a)=\{\m x\in \R^m:\;\lng\m a,\m x\rng =\lng\m a,\m u\rng, \;\m a\not=\m 0\}.
\end{equation}
A hyperplane $H(\m u,\m a)$ determines two \emph{closed halfspaces} $H_-(\m u,\m a)$ and
$H_+(\m u,\m a)$ of $\R^m$, defined by
\[H_-(\m u,\m a)=\{\m x\in \R^m:\; \lng\m a,\m x\rng \leq \lng\m a,\m u\rng\},\]
and
\[H_+(\m u,\m a)=\{\m x\in \R^m:\; \lng\m a,\m x\rng \geq \lng\m a,\m u\rng\}.\]
Taking a Cartesian reference system in $\R^m$ and using the above introduced
notations,
the
\emph{coordinate-wise order} $\leq$ in $\R^m$ is defined by
\[\m x=(x_1,...,x_m)\leq\m y=(y_1,...,y_m)\;\Leftrightarrow\;x_i\leq y_i,\;i=1,...,m.\]
By using the notion of the order relation induced by a cone, defined in the preceding
section, we see that $\leq =\leq_{\R^m_+}$.
With the above representation of $\m x$ and $\m y$, we define
$$\m x\wedge\m y=(\min \{x_1,y_1\},...,\min \{x_m,y_m\}),\;\;\textrm{and}\;\;\m x\vee\m y=(\max \{x_1,y_1\},...,\max \{x_m,y_m\}).$$
Then, $\m x\wedge\m y$ is the greatest lower bound and $\m x\vee\m y$ is the least upper bound of
the set $\{\m x,\m y\}$ with respect to the coordinate-wise order. Thus, $\leq$ is a lattice order in $\R^m.$
The operations $\wedge$ and $\vee$ are called \emph{lattice operations}.
A subset $M\subset \R^m$ is called a \emph{sublattice of
the coordinate-wise ordered Euclidean space} $\R^m$, if from
$\m x,\m y\in M$ it follows that $\m x\wedge\m y,\;\m x\vee\m y\in M.$
Denote by $P_D$
the projection mapping onto a nonempty closed convex set $D\subset \R^m,$
that is the mapping which associates
to $\m x\in \R^m$ the unique nearest point of $x$ in $D$ (\cite{Zarantonello1971}):
\[ P_D\m x\in D,\;\; \textrm{and}\;\; \|\m x-P_D\m x\|= \inf \{\|\m x-\m y\|: \;\m y\in D\}. \]
The nearest point $P_D\m x$ can be characterized by
\begin{equation}\label{charac}
P_D\m x\in D,\;\;\textrm{and}\;\;\lng P_D\m x -\m x,P_D\m x-\m y\rng \leq 0 ,\;\forall\m y\in D.
\end{equation}
From the definition of the projection and the characterization (\ref{charac}) there follow immediately
the relations:
\begin{equation}\label{en}
P_D(-\m x)=-P_{-D}\m x ,
\end{equation}
\begin{equation}\label{et}
P_{\m x+D}\m y=\m x+P_D(\m y-\m x)
\end{equation}
for any $\m x,\m y\in\R^m$,
For a closed convex cone $K$ we define the following operations in $\R^m$:
\[\m x\sa_K\m y=P_{\m x-K}\m y,\;\,\textrm{and}\;\;\m x\su_K\m y=P_{\m x+K}\m y\]
(see \cite{GowdaSznajderTao2004}). Assume the operations $\su_K$ and $\sa_K$ have precedence over the addition of vectors and
multiplication of vectors by scalars.
A direct checking yields that if $K=\R^m_+$, then $\sa_K =\wedge$, and $\su_K =\vee$.
That is $\sa_K$ and $\su_K$ are some \emph{generalized lattice operations}.
Moreover: $\sa_K$ and $\su_K$ \emph{are lattice operations if and only if
the self-dual cone used in their definitions is a nonnegative orthant of
some Cartesian reference system.}
This suggest to call the operations $\sa_K$ and $\su_K$ \emph{lattice-like operations},
while a subset $M\subset \R^m$ which is \emph{invariant with respect to
$\sa_K$ and $\su_K$} (i.e. if for any $\m x,\m y\in M$ we have $\m x\sa_K\m y,\;\m x\su_K\m y\in M$),
a \emph{lattice-like} or simply an \emph{l-l} subset of $(\R^m,K)$.
The following assertions are direct consequences of the definition
of lattice-like operations:
\begin{lemma}\label{l0}
The following relations hold for any $\m x,\m y\in (\R^m,K)$:
$$\m x\sa_K\m y=\m x-P_K(\m x-\m y),$$
$$\m x\su_K\m y=\m x+P_K(\m y-\m x).$$
\end{lemma}
If $K$ is a nonzero closed convex cone, then the closed convex set $C\subset \R^m$ is called a
\emph{$K$-isotone projection set} or simply \emph{$K$-isotone}
if $\m x\leq_K\m y$ implies $P_C\m x\leq_K\m P_C\m y$. In this case we use equivalently
the term \emph{$P_C$ is $K$-isotone}.
We shall refer next often to the following theorems:
\begin{theorem}\cite{NemethNemeth2012b}\label{ISOINV}
Let $K\subset \R^m$ be a closed convex cone. Then, $C$ is a lattice-like set, if and only if $P_C$ is
$K$-isotone.
\end{theorem}
\begin{theorem}\cite{NemethNemeth2012a}\label{FOOO}
The closed convex set $C$ with nonempty interior in $(\R^m,K)$ is lattice-like,
if and only if it is of form
\begin{equation*}
C=\bigcap_{i\in \n} H_-(\m u_i,\m a_i),
\end{equation*}
where each hyperplane $H(\m u_i,\m a_i)$ through $\m u_i$ with the normal $\m a_i$ is tangent to $C$ and is lattice-like.
\end{theorem}
\section{Characterization of the lattice-like subspaces of $(\R^m,K)$}
Denote by $K$ a closed convex cone in $\R^m$ and by $(\R^m,K)$ the
resulting ordered vector space.
The notation $G\Subset H$ will mean
\emph{$H$ and $G$ are subspaces of $\R^m$ and $G$ is a subspace of $H$}.
Let $H\Subset \R^m$ and $L\subset H$ a closed convex cone.
The notation $G\sqsubset_L H$ will mean \emph{$G$ is an l-l subspace of $(H,L)$}.
We gather some results from Theorem 1 \cite{NemethNemeth2012b} and Lemma 6 \cite{NemethNemeth2012a} and particularize them for subspaces:
\begin{corollary}\label{fooalt}
Let $H$ a subspace in $(\R^m,K)$.
the following assertions are equivalent:
\begin{enumerate}
\item $H\sqsubset_K \R^m$,
\item $P_{K} H\subset H$,
\item $P_{H} K \subset K$.
\end{enumerate}
\end{corollary}
\begin{proof}
The corollary is in fact a reformulation of Theorem \ref{ISOINV} for
the case of $D=H$ a subspace. Indeed, condition 2 is nothing else as
the l-l property of of $H$ since if $\m x,\m y\in H$, then by
Lemma \ref{l0}, one has
$$\m x\sa_K\m y=\m x-P_K(\m x-\m y)\in H,$$
since $\m x,\,\m x-\m y,\,P_K(\m x-\m y)\in H.$
Similarly, $\m x\su_K\m y\in H.$
Condition 3 expresses, by the linearity of $P_H$ its $K$-isotonicity.
\end{proof}
\begin{corollary}\label{fooaltkov}
Let $G\Subset H$ and $H\sqsubset_K \R^m$. Then, $G\sqsubset_{K\cap H} H \Leftrightarrow G\sqsubset_K \R^m$.
\end{corollary}
\begin{proof}
In our proof we shall use without further comments the equivalences in Corollary \ref{fooalt}.
Let $G\Subset H$ and $H\sqsubset_K \R^m$.
First suppose that $$G\sqsubset_K \R^m,$$ which is equivalent to $$P_GK\subset K.$$ Hence,
$$P_G(H\cap K)\subset P_GK\subset H\cap K,$$ since $P_G(K)\subset G\subset H.$ Thus, $G\sqsubset_{K\cap H}H$.
Conversely, assume that $G\sqsubset_{H\cap K} H$. Take $\m x,\,\m y\in \R^m$ with $\m x\leq_K\m y.$ Then, from $H\sqsubset_K \R^m$ we have
$$P_H\m x\leq_K P_H\m y,\;\textrm{that is},\;P_H\m y-P_H\m x\in H\cap K,$$
hence
$$P_H\m x\leq_{H\cap K}P_H\m y.$$
From $G\sqsubset_{H\cap K} H$ it follows that
$$ P_G P_H\m x\leq_{H\cap K} P_G P_H\m y.$$
From the property of orthogonal projections one has
$$P_G=P_G P_H.$$
Thus, the above relation writes as
$$P_G\m x\leq_{H\cap K} P_G\m y,$$
or $P_G\m y -P_G\m x\in H\cap K \subset K.$
That is,
$$P_G\m x\leq _K P_G\m y,$$
which shows that $G\sqsubset_K \R^m.$
\end{proof}
The following lemma is a direct consequence of
Lemma 3 and Lemma 8 in \cite{NemethNemeth2012b}:
\begin{lemma}\label{ti}
Suppose that $K$ is a closed convex cone in $\R^m$.
Let $H(\m 0,\m a)\subset\R^m$ be a hyperplane
through the origin with unit normal vector $\m a\in\R^m$. Then, the following
assertions are equivalent:
(i) $P_{H(\m 0,\m a)}$ is $K$-isotone;
(ii) $P_{H(\m b,\m a)}$ is $K$-isotone for any $\m b\in \R^m$;
(iii) \[\lng\m x,\m y\rng\ge\lng\m a,\m x\rng\lng\m a,\m y\rng,\] for any $\m x,\m y\in K$.
\end{lemma}
\section{Lattice-like subspaces of the Euclidean Jordan algebra}
In the particular case of a self-dual cone $K\subset \R^m$, J. Moreau's theorem (\cite{Moreau1962}) reduces to
the following lemma:
\begin{lemma}\label{lm}
Let $K\subset \R^m$ be a self-dual cone. Then, for any $\m x\in \R^m$
the following two conditions are equivalent:
(i) $\m x=\m u-\m v,\;\m u,\m v\in K, \lng\m u,\m v\rng =0,$
(ii) $\m u=P_K\m x,\;\m v=P_K(-\m x).$
\end{lemma}
In all what follows we will consider that the ordered Euclidean space is
$(V,Q)$, the Euclidean Jordan
algebra $V$ of unit $\m e$ ordered by the cone $Q$ of squares in $V$. All the terms concerning $V$
will be equally used for $(V,Q)$.
Since the hyperplanes in Theorem \ref{FOOO} play an important role,
and since the l-l property is invariant with respect to translations
(Lemma 3, \cite{NemethNemeth2012a}), it is natural to study the l-l subspaces in $V$ which are naturally connected with
the algebraic structure of this space.
\begin{theorem}\label{mell}
Any Jordan subalgebra of $(V,Q)$ is a lattice-like subspace.
\end{theorem}
\begin{proof}
Take a Jordan subalgebra $L$ in $V$ and denote by
$Q_0$ its cone of squares. We have
\begin{equation}\label{omeg}
Q_0=\{\m x^2:\,x\in L\}\subset \{\m x^2:\,x\in V\}=Q.
\end{equation}
We shall prove that
\begin{equation}\label{omega=}
\m x\in L \Rightarrow P_{Q}\m x=P_{Q_0}\m x \in L.
\end{equation}
Indeed, we have, by Lemma \ref{lm} applied in the ordered
vector space $(L, Q_0)$, that
\begin{equation}\label{e8b}
\m x=P_{Q_0}\m x-P_{Q_0}(-\m x),\;\; \lng P_{Q_0}\m x,P_{Q_0}(-\m x)\rng =0,
\end{equation}
By (\ref{omeg})
$$P_{Q_0}\m x,\;\; P_{Q_0}(-\m x)\in Q_0 \subset Q,$$
which, by equations \eqref{e8b} and Lemma \ref{lm}, yield $P_{Q_0}\m x=P_{Q}\m x$, or equivalently
(\ref{omega=}).
Accordingly $P_QL\subset L,$ which by Corollary \ref{fooalt},
translates into $L\sqsubset_Q V.$
\end{proof}
\section{The Pierce decomposition of the Euclidean Jordan algebra and its lattice-like subspaces}
\label{pierce}
Let $r$ be the rank of $V$ and
$\{\m c_1,\dots,\m c_r\}$ be an arbitrary Jordan frame in $V$, that is, $\m c_k$ are primitive idempotents such that
$$\m c_i\m c_j=0,\;\; \textrm{if}\;\; i\not=j,\;\; \m c_i^2=\m c_i,$$
$$\m c_1+\dots +\m c_r=\m e.$$
With the notation
$$V_{ii}=V(\m c_i,1)=\R \m c_i,$$
$$V_{ij}= V\lf(\m c_i,\frac{1}{2}\rg)\cap V\lf(\m c_j,\frac{1}{2}\rg),$$
(where for $\lambda \in \R$, $V(\m c_i, \lambda)=\{\m x\in V:\,\m c_i\m x=\lambda\m x\}$),
we have by Theorem IV.2.1. \cite{FarautKoranyi1994} the following orthogonal
decomposition (the so-called \emph{Pierce decomposition}) of $V$:
\begin{equation}\label{Oplus}
V=\bigoplus_{i\leq j} V_{ij},
\end{equation}
where
\begin{equation}\label{Oplus1}
V_{ij}V_{ij}\subset V_{ii}+V_{jj};\;
V_{ij}V_{jk}\subset V_{ik},\;\;\textrm{if}\;\;i\not= k;\;
V_{ij}V_{kl}=\{0\}, \;\;\textrm{if}\;\; \{i,j\}\cap \{k,l\}=\emptyset.
\end{equation}
Taking for $1\leq k< r$
\begin{equation}\label{Oplus2}
V^{(k)}= \bigoplus_{i\leq j\leq k} V_{ij}
\end{equation}
is a Jordan algebra
with the unit
$$\m e_k=\m c_1+\dots +\m c_k.$$
Indeed, relations (\ref{Oplus1}) imply the invariance of $V^{(k)}$
with respect to the Jordan product. The same relations and the definitions imply
$\m e_k\m x_{ii}=\m c_i\m x_{ii}=\m x_{ii}$, for any $\m x_{ii}\in V_{ii}$ and $i\le k$;
$\m c_lV_{ij}=\{0\}\textrm{ if }l\notin \{i,j\}$;
$\m e_k \m x_{ij}=(\m c_i+\m c_j)\m x_{ij}=\m x_{ij}$, for any $\m x_{ij}\in V_{ij}$ and $i,j\le k$, $i\ne j$.
Hence $\m e_k$ is the unity of $V^{(k)}$. These relations also imply that
\begin{equation}\label{vkszubalg}
V^{(k)}=V(\m e_k,1)=\{\m x\in V:\,\m e_k\m x=\m x\}.
\end{equation}
Thus, $V(\m e_k,1)$ is a subalgebra (this follows also by Proposition IV.1.1 in \cite{FarautKoranyi1994} since
$\m e_k$ is idempotent). Hence
by Theorem \ref{mell}, $V(\m e_k,1)$ is an l-l subspace in $(V, Q)$.
A Jordan algebra is said \emph{simple} if it contains no nontrivial ideal.
A consequence of the above cited theorem and
the content of paragraph IV.2. of \cite{FarautKoranyi1994} is that
$V$ is simple if and only if $V_{ij}\not= \{0\}$
for any $V_{ij}$ in (\ref{Oplus}). By the same conclusion $V^{(k)}$ given by (\ref{Oplus2}) is simple too, and
by Corollary IV.2.6. in \cite{FarautKoranyi1994} the spaces $V_{ij},\;i\not= j$ have
the common dimension $d$, hence by (\ref{Oplus2})
$$\dme V^{(k)} =k+\frac{d}{2}k(k-1).$$
The subcone $F\subset Q$ is called a \emph{face of $Q$} if whenever $0\leq_Q\m x\leq_Q\m y$ and $\m y\in F$
it follows that $\m x\in F$.
It is well known that for an arbitrary face $F$ of $Q$ one has $P_{\spa F}Q\subset Q$
(see e.g. Proposition II.1.3 in \cite{Iochum1984}). Hence by Corollary \ref{fooalt} it follows the assertion:
\begin{corollary}\label{facealg}
Each subspace generated by some face of $Q$ is a lattice-like subspace in $(V,Q)$.
\end{corollary}
We give an independent proof of this.
\begin{proof}
Let $\{\m c_1,...,\m c_r\}$ be a Jordan frame in $V$, $k\leq r$. If
$$\m e_k=\m c_1+\dots+\m c_k, \quad 0\leq k\leq r,$$
then by Theorem 3.1 in \cite{GowdaSznajder2006}
$$F=V(\m e_k,1)\cap Q=\{x\in Q:\,\m e_k\m x=\m x\}$$
is a face of $Q$ and each face of $Q$ can be represented in this form for some Jordan frame.
The cone $F=V(\m e_k,1)\cap Q$ is the cone of squares in the subalgebra $V(\m e_k,1)$,
hence its relative interior is non-empty, accordingly
$$V(\m e_k,1)=\spa F=F-F.$$
Since $V(\m e_k,1)$ is a subalgebra, by Theorem \ref{mell} it is an l-l subspace.
\end{proof}
\section{The subalgebras and the lattice-like subspaces of
the space spanned by a Jordan frame}\label{orthant}
Suppose that the dimension of the Euclidean Jordan algebra $V$ is at least $2$.
Let $\{\m c_1,...,\m c_r\}$ be a Jordan frame in $V$.
Then,
$$V_{r}:=\spa\{\m c_1,...,\m c_r\}$$
is a Jordan subalgebra of $V$. Obviously, $V_r=V_{11}\oplus\dots\oplus V_{rr}$. If
$\m x,\m y\in V_r$, then $$\m x\m y=(x_1y_1,...,x_ry_r),$$
where $x_i$ and $y_i$ are the coordinates of $\m x$, respectively $\m y$
with respect to the above Jordan frame.
By using the notations of the above section, denote $Q_r=Q\cap V_r$ and let us show that
$$Q_r=\cone \{\m c_1,...,\m c_r\}:=\lf\{\sum_{i=1}^r\lambda_i\m c_i:\lambda_i\ge0,\textrm{ }\forall 1\le i\le r\rg\}.$$
The inclusion $\cone\{\m c_1,...,\m c_r\}\subset Q_r$ is obvious. Next, we show that $Q_r\subset\cone\{\m c_1,...,\m c_r\}$. Suppose to the contrary, that
there exists $\m x\in Q_r\setminus\cone\{\m c_1,...,\m c_r\}$. It follows that $\lng \m c_k,\m x\rng <0$ for some $k\in\{1,\dots,r\}$. Since $Q$ is selfdual,
this implies $\m x\notin Q$, which is a contradiction.
The ordered vector space $(V_r,Q_r)$ can be considered
an $r$-dimensional Euclidean vector space ordered
with the positive orthant $Q_r$ engendered by the Jordan frame.
Let $H_{r-1}$ be an l-l hyperplane in $(V_r,Q_r)$, with the unit normal
$\m a\in V_r$. Thus, the results in \cite{NemethNemeth2012a} and \cite{NemethNemeth2012b} applies,
hence if
\begin{equation}\label{a}
\m a=(a_1,...,a_r),
\end{equation}
then we must have
\begin{equation}\label{aa}
a_ia_j\leq 0,\;\;\textrm{if}\;\; i\not=j.
\end{equation}
Then, there are two possibilities:
\begin{description}
\item {\bf Case 1}. There exists an $i$ such that $a_i=1$ and $a_j=0$ for $j\not=i$.
\item {\bf Case 2}. There are only two nonzero coordinates, say $a_k$ and $a_l$ with $a_ka_l<0$.
\end{description}
\begin{description}
\item {\bf Ad 1}. In the {\bf Case 1} $$H_{r-1}=\spa\{\m c_1,...,\m c_{i-1},\m c_{i+1},...,\m c_r\}$$ and $H_{r-1}$ is obviously a Jordan algebra.
\item {\bf Ad 2}. In the {\bf Case 2} $$H_{r-1} = \{\m x\in V_r: a_kx_k+a_lx_l=0\}.$$ We know from the above cited result, that $H_{r-1}$ is an l-l
subspace in $(V_r,Q_r)$ and since $V_r$ is a subalgebra of $V$, by Theorem \ref{mell}, $V_r\sqsubset_Q V$. By using Corollary
\ref{fooaltkov} we have, for the l-l subspace $H_{r-1}\sqsubset_{Q_r} V_r$, that
$$H_{r-1}\sqsubset_Q V.$$
\end{description}
In the case {\bf Ad 1} the l-l hyperplane $H_{r-1}$ is also a Jordan algebra.
Suppose that {\bf Ad 2} holds. We would like to see under which condition
the l-l hyperplane $H_{r-1}$ is a Jordan algebra.
Let us suppose that $H_{r-1}$ is a Jordan algebra, and take
$\m x\in H_{r-1}$, $\m x=(x_1,...,x_r)$. Then, $\m x^2 =(x_1^2,...,x_r^2)\in H_{r-1}.$
Take $\m x$ with $x_l=a_k$ and $x_k=-a_l$. Then, $\m x\in H_{r-1}$ and we
must have $\m x^2\in H_{r-1}$. Hence
$$a_ka_l^2+a_la_k^2=a_ka_l(a_l+a_k)=0,$$
and since $a_ka_l\not=0,$ we must have
$$ a_k=\frac{\sqrt 2}{2}\qquad a_l=-\frac{\sqrt 2}{2},$$
or conversely.
In this case
\begin{equation}\label{k=l}
H_{r-1}= \{\m x:x_k=x_l\}
\end{equation}
is obviously a subalgebra.
\begin{remark}\label{nemalgebra}
For any hyperplane $H_{r-1}$ in $V_r$ with the unit normal $\m a$ having only
two nonzero components with opposite signs and different absolute values $H_{r-1}$
is an l-l subspace, but not a Jordan subalgebra.
\end{remark}
If $\m a\in V$ is arbitrary, then there exists a Jordan frame $\{\m c_1,...,\m c_r\}$ such that
$\m a$ can be represented in the form (\ref{a}) (Theorem III.1.2 in \cite{FarautKoranyi1994}). We will call such a Jordan frame as
being \emph{attached to} $\m a$.
\begin{corollary}\label{aaa}
Let $H$ be a lattice-like hyperplane in $(V,Q)$ with the normal
$\m a$ and $\{\m c_1,...,\m c_r\}$ be a Jordan frame attached to it.
If $\m a$ is represented by (\ref{a}), then the coordinates $a_i,\,i=1,...,r$
of $\m a$ satisfy the relations (\ref{aa}).
\end{corollary}
\begin{proof}
If $V_r=\spa\{\m c_1,...,\m c_r\}$, then $H_{r-1}=H\cap V_r$ is
an l-l hyperplane in $(V_r,Q_r)$ with the normal $\m a$ because it is the
intersection of two l-l sets: $V_r$ (a subalgebra) and $H$. Thus, we
can apply the characterization of l-l hyperplanes in $(V_r,Q_r)$ described
above in this section.
\end{proof}
Denote by $\mc F (Q)$ the family of faces of $Q$, by $\mc A$ the family of subalgebras of $V$
and by $\mc L$ the family of the l-l subspaces in $V$. Then, by the above reasonings, we conclude the
\begin{corollary}
We have the following strict inclusions:
$$\{\spa F:\,F\in \mc F (Q)\} \subset \mc A \subset \mc L.$$
\end{corollary}
\begin{proof}
The second strict inclusion follows from Remark \ref{nemalgebra}.
The first inclusion is strict since for instance
the subspaces in (\ref{k=l}) are subalgebras which
are not generated by faces of $Q$. Indeed, take in $V_r$ the reference system
engendered by $\m c_1,...,\m c_r$ and let
$$H_{r-1}=\{(t,t,x_{r+3},...,x_r)\in V_r:t,x_j\in \R\}.$$
Take $\m y=(1,1,0,...,0)\textrm{ and }\m x=(1,0,0,...,0)\textrm{ in }Q_r=Q\cap V_r.$
Since in $V_r,\;\leq_Q=\leq_{Q_r}$ and the latter is coordinate-vise ordering,
$$0\leq_Q\m x\leq_Q\m y,$$
and we have $\m y\in H_{r-1}\cap Q$, but $\m x\notin H_{r-1}\cap Q$, which shows that
$H_{r-1}\cap Q$ is not a face.
\end{proof}
\section{ The inexistence
of lattice-like hyperplanes in\\ simple Euclidean Jordan algebras of rank $r\geq 3$.}
\begin{theorem}\label{nincs}
Suppose that $V$ is a simple Euclidean Jordan algebra of rank $r\geq 3$.
Then, $V$ does not contain lattice-like hyperplanes.
\end{theorem}
\begin{proof}
Assume the contrary: $H$ is an l-l hyperplane through $0$ in $V$ with the
unit normal $\m a$.
Consider a Jordan frame $\{\m c_1,...,\m c_r\}$ attached to $\m a$.
The set
$$H_{r-1}=H\cap V_r$$
is obviously a hyperplane through $0$ in $V_r$.
Since by hypothesis $H\sqsubset_Q V$, by Corollary \ref{fooaltkov}, $H_{r-1}\sqsubset_{Q_r} V_r,$
where $Q_r=Q\cap V_r$.
If $\m a=(a_1,...,a_r)$ is the representation of $\m a$
in the reference system engendered by the Jordan frame,
then using Corollary \ref{aaa},
the l-l property of $H_{r-1}$ in $(V_r,Q_r)$ implies
that one of the following cases must hold:
\begin{description}
\item {\bf Case 1}. For some $i$ $a_i=1$ and $a_j=0$ for $j\not=i$.
\item {\bf Case 2}. There are only two nonzero coordinates, say $a_i$ and $a_j$ with $a_ia_j<0$.
\end{description}
Suppose that $i=1,\,j=2$.
Since $V$ is simple, $V_{12}\not= \{0\}$ (by Proposition IV.2.3 \cite{FarautKoranyi1994}), hence we
can take $\m x\in V_{12}$
with $\|\m x\|^2=2$. Then, by Exercise IV. 7 in \cite{FarautKoranyi1994},
we have that
\begin{equation}\label{xn}
\m u=\frac{1}{2}\m c_1+\frac{1}{2}\m c_2+\frac{1}{2}\m x,\qquad \textrm{and} \qquad \m v=\frac{1}{2}\m c_1+\frac{1}{2}\m c_2-\frac{1}{2}\m x
\end{equation}
are idempotent elements, hence $\m u,\m v\in Q.$
We further have
$$\m u\m v= \lf(\frac{1}{2}\m c_1+\frac{1}{2}\m c_2\rg)^2-\frac{1}{4}\m x^2,$$
whereby, by using Proposition IV.1.4 in \cite{FarautKoranyi1994}, we have \[\m x^2=\frac12\|\m x\|^2(\m c_1+\m c_2)=\m c_1+\m c_2,\]
and after raising to the second power and substitution
$$\m u\m v= \frac{1}{4}\m c_1+\frac{1}{4}\m c_2-\frac{1}{4}(\m c_1+\m c_2)=0.$$
Hence
$$\lng\m u,\m v\rng=0.$$
Since $H_{r-1}$ is l-l, we have by Lemma \ref{ti},
$$0= \lng\m u,\m v\rng \geq \lng \m a,\m u\rng \lng \m a,\m v\rng.$$
If $a_1=1, $ and $a_j=0$ for $j\not= 1$, the above relation becomes $0\geq \frac{1}{4}\|\m c_1\|^4,$ which is impossible.
Assume $a_1a_2<0$ and $a_j=0$ for $j>2$.
Take now
\begin{equation}\label{yn}
\begin{array}{l}
\m w=\ds\frac{1}{2}\m c_1+\frac{1}{2}\m c_3+\frac{1}{2}\m y\\\\
\m z=\ds\frac{1}{2}\m c_1+\frac{1}{2}\m c_3-\frac{1}{2}\m y
\end{array}
\end{equation}
with $\m y\in V_{13},\;\|\m y\|^2=2.$
Then, $\m w,\m z\in Q$ (similarly to $\m u,\m v\in Q$) and, by using the mutual orthogonality of the elements
$\m c_1,\,\m c_2,\,\m c_3,\,\m y$ and Lemma \ref{ti}, it follows that
$$0=\lng\m w,\m z\rng\geq\lng\m a,\m w\rng\lng\m a,\m z\rng =$$
$$\lng a_1\m c_1+a_2\m c_2,\frac{1}{2}\m c_1+\frac{1}{2}\m c_3+\frac{1}{2}\m y\rng\lng a_1\m c_1+a_2\m c_2,\frac{1}{2}\m c_1+\frac{1}{2}\m c_3-\frac{1}{2}\m y\rng
=\frac{1}{4}a_1^2\|\m c_1\|^2,$$
which is a contradiction.
\end{proof}
This theorem conferred with Theorem \ref{FOOO} and Lemma 3 in \cite{NemethNemeth2012a} yields the
\begin{corollary}
In the ordered Euclidean Jordan algebra $(V,Q)$ of rank at least $3$
there are no proper closed convex lattice-like set with nonempty interior.
In particular, for $n\geq 3$ the ordered space $(\sm,\smp)$
contains no proper, closed, convex lattice-like set with nonempty interior.
\end{corollary}
\section{The case of the simple Euclidean Jordan algebras of rank 2}
A simple Euclidean Jordan algebra of rank 2 is isomorphic to an
algebra associated with a positive definite bilinear form
(Corollary IV.1.5 \cite{FarautKoranyi1994}). This
is in fact a Jordan algebra associated with the Lorentz cone.
Hence the problem of the existence of l-l hyperplanes
in this case is answered positively in \cite{NemethNemeth2012a}
and \cite{NemethNemeth2012b}. In this section we use
the formalism developed in the preceding sections to this case too.
\begin{lemma}\label{ketdim}
Suppose that $\m a$ is the unit normal to a lattice-like hyperplane $H$
through $0$ in the simple Euclidean Jordan algebra $V$ of
rank 2. Let $\{\m c_1,\m c_2\}$ be the Jordan frame attached to $\m a$ and $\m a=a_1\m c_1+a_2\m c_2$. Then, supposing $a_1>0$, we obtain
\begin{equation}\label{keta2}
\m a= \frac{\sqrt 2}{2} \m c_1 -\frac{\sqrt 2}{2} \m c_2.
\end{equation}
\end{lemma}
\begin{proof}
Take $\m u$ and $\m v$ as in the formula (\ref{xn}). Then, $\m u,\m v\in Q$
and using Lemma \ref{ti} we obtain
\begin{gather*}
0=\lng\m u,\m v\rng\geq\lng\m a,\m u\rng\lng\m a,\m v\rng\\
=\lng a_1\m c_1+a_2\m c_2,\frac{1}{2}\m c_1+\frac{1}{2}\m c_2+\frac{1}{2}\m x\rng
\lng a_1\m c_1+a_2\m c_2,\frac{1}{2}\m c_1+\frac{1}{2}\m c_2-\frac{1}{2}\m x\rng
=\frac{1}{4}(a_1+a_2)^2,
\end{gather*}
whereby our assumption follows.
\end{proof}
\begin{theorem}\label{ketdim1}
Let $V$ be a simple Euclidean Jordan algebra of rank $2$ and $H$ be a hyperplane through $0$ with unit normal $\m a$ in $V$. Then, $H$
is lattice-like if and only if
$\m a=\sqrt 2/2 \m c_1 -\sqrt 2/2 \m c_2$ in its Jordan frame representation.
In this case $H$ is a subalgebra.
\end{theorem}
\begin{proof}
Suppose that $H=\ker\m a,\;\|\m a\|=1$ is l-l,
and that the Jordan frame attached to $\m a$ is $\{\m c_1,\m c_2\}.$
Then, by Lemma \ref{ketdim}, it follows that $\m a$ is
of form (\ref{keta2}).
Suppose that the Jordan frame representation of $\m a$ is of form (\ref{keta2}).
Then, equations \eqref{Oplus} and \eqref{Oplus1} imply that
$$ \ker\m a=\{t(\m c_1+\m c_2)+\m x=t\m e+\m x:\,t\in \R, \,\m x\in V_{12}\}.$$
Then, for two arbitrary
elements $\m u,\m v\in \ker\m a$, we have the representations:
$$\m u =t_1\m e+\m x;\textrm{ }\m v =t_2\m e+\m y;\textrm{ }\m x,\m y\in V_{12};\textrm{ }t_i\in \R,\textrm{ }i=1,2.$$
Then,
$$\m u\m v=t_1t_2\m e+t_1\m y+t_2\m x+ \m x\m y.$$
Since $\m x\m y=(1/4)((\m x+\m y)^2-(\m x-\m y)^2)$,
by using Proposition IV.1.4 in \cite{FarautKoranyi1994}, we conclude
that $\m x\m y=q(\m c_1+\m c_2)=q\m e$
with $q\in \R.$
Hence
$$\m u\m v=(t_1t_2+q)\m e+t_1\m y+t_2\m x\in \ker\m a.$$
This shows that $H=\ker\m a$ is a subalgebra, and hence an l-l set.
\end{proof}
\begin{remark}\label{ketketket}
With the notations in the above proof we have that $\spa \{\m c_1,\,\m c_2\}$ is a subalgebra of dimension $2$ in $V$.
Similarly to Remark \ref{nemalgebra}, it follows that there exist
l-l subspaces of dimension 1 in $\spa \{\m c_1,\,\m c_2\}$ which are not subalgebras.
\end{remark}
Collating Theorem \ref{ketdim1} and Theorem \ref{FOOO} it follows the result:
\begin{corollary}\label{konvlor}
The closed convex set with nonempty
interior $M\subset V$ is a lattice-like set if and only if it is of the form:
\begin{equation*}
M=\bigcap_{i\in \n} H_-(\m u_i,\m a_i),
\end{equation*}
with the $\m a_i$ normal unit vectors represented in their Jordan
frame $\m c^i_1, \,\m c^i_2 $ by
\begin{equation}\label{keta}
\m a_i=\varepsilon_i\left(\frac{\sqrt 2}{2}\m c^i_1-\frac{\sqrt 2}{2}\m c^i_2\right),\textrm{ }
\varepsilon_i=1\textrm{ or }-1.
\end{equation}
\end{corollary}
\begin{example}\label{lorentz}
Write the elements of $\R^{m+1}$ in the form
$(\m x,x_{m+1})$ with $\m x\in \R^m$ and $x_{m+1}\in \R.$
The Jordan product in $\R^{m+1}$ is defined by
$$(\m x,x_{m+1})\circ (\m y,y_{m+1})=(y_{m+1}\m x+x_{m+1}\m y, \lng\m x,\m y\rng+x_{m+1}y_{m+1}),$$
where $\lng \m x,\m y\rng$ is the usual scalar product in $\R^m.$
The space $\R^{m+1}$ equipped with the usual scalar product and the
operation $\circ$ just defined becomes an Euclidean Jordan algebra of rank 2, denoted by
$\mathcal L^{m+1}$ \cite{GowdaSznajderTao2004}, with the cone of squares $Q=\mathcal L^{m+1}_+$, the Lorentz cone
defined by (\ref{lorentzcone}).
The unit element in $\mathcal L^{m+1}$ is $(\m 0,1)$, where $\m 0$ is the zero
vector in $\R^m$.
The Jordan frame attached to $(\m x,x_{m+1})\in \mathcal L^{m+1}$ with $\m x\not=0$ is
$$\m c_1=\frac{1}{2}\left(\frac{\m x}{\|\m x\|},1\right),\quad \m c_2=
\frac{1}{2}\left(-\frac{\m x}{\|\m x\|},1\right).$$
The unit normal $\m a$ from Lemma \ref{ketdim} will be then parallel with
$(\m b,0)$ with some $\m b\in \R^m,\;\m b\not=\m 0.$
This means, that the hyperplanes $H(\m u_i,\m a_i)$ in Corollary \ref{konvlor}
are parallel with the $m+1$-th axis, and the closed convex set in
the corollary is in fact of form
$$M=C\times \R,$$
with $C$ closed convex set with nonempty interior in $\R^m.$
This is exactly the result in Example 1 of \cite{NemethNemeth2012b}.
\end{example}
\section{The general case}
For a general Euclidean Jordan algebra $V$, gathering the results
of Proposition III.4.4, Proposition III.4.5, and Theorem V.3.7,
of \cite{FarautKoranyi1994}, in Theorem 5 of \cite{GowdaSznajderTao2004} the following result is
stated:
\begin{theorem}\label{split}
Any Euclidean Jordan algebra $V$ is, in unique way, a direct sum
\begin{equation}\label{dirossz}
V=\oplus_{i=1}^k V_i
\end{equation}
of simple Euclidean Jordan algebras $V_i,\,i=1,...,k$.
Moreover the cone of squares $Q$ in $V$ is, in a unique way,
a direct sum
\begin{equation}\label{kupdirossz}
Q=\oplus_{i=1}^k Q_k
\end{equation}
of the cones of squares $Q_i$ in $V_i,\,i=1,...,k$.
\end{theorem}
(Here the direct sum, (by a difference to that in the
Pierce decomposition), means Jordan-algebraic
and hence also orthogonal direct sum.)
Let $C\subset V$ a closed convex set.
From the results in Theorem \ref{split} it follows easily (using the notations in the
theorem), that
\begin{equation}\label{projossz}
P_C = \sum_{i=1}^k P_{C_i},
\end{equation}
with $C_i=C\cap V_i,\;i=1,...,k$.
Collating these results with Corollary \ref{fooaltkov}, we have the following:
\begin{corollary}\label{ossz}
With the notations in Theorem \ref{split},
for the subspace $M\Subset V$ we have the
equivalence:
\begin{equation}\label{llossz}
M\sqsubset_Q V \Leftrightarrow M\cap V_i\sqsubset_{Q_i} V_i,\;i=1,...,k.
\end{equation}
For the closed convex set $C$ the projection $P_C$
is $Q$-isotone if and only if $P_{C\cap V_i}$ is $Q_i$-isotone in $(V_i,Q_i),\;i=1,...k.$
\end{corollary}
\begin{corollary}\label{osszhyp}
If $H$ is a lattice-like hyperplane in $V$ represented as (\ref{dirossz}) in Theorem \ref{split},
then $V_i\Subset H$ for each simple subalgebra in (\ref{dirossz}) of rank at least $3$.
\end{corollary}
\begin{proof}
Assume the contrary. Then, $H\cap V_i$ is an l-l hyperplane in $V_i$, which
contradicts Theorem \ref{nincs}.
\end{proof}
Gathering the results in Theorem \ref{FOOO}, Section 7, Corollary \ref{konvlor} and Corollary \ref{osszhyp}
we have
\begin{theorem}
Suppose that $V$ is an Euclidean Jordan algebra of form
(\ref{dirossz}) with $V_i$ simple subalgebras. Let us write this sum as
\begin{equation}\label{partition}
V=W_1\oplus W_2\oplus W_3
\end{equation}
where
\begin{equation}\label{partition1}
W_1=\oplus_{i\in I_1}V_i ,\;\;W_2=\oplus_{i\in I_2} V_i,\;\; W_3=\oplus_{i\in I_3} V_i,
\end{equation}
such that $V_i$ for $i\in I_1$ are the subalgebras of rank $1$, for $i\in I_2$,
the subalgebras of rank $2$, and for $i\in I_3$ the subalgebras of rank at least $3$.
Then, $C\subset V$ is a closed convex lattice-like subset with nonempty interior
if and only if the following conditions hold:
\begin{equation}\label{alap}
C=\bigcap_{i\in \n} H_-(\m u_i,\m a_i),
\end{equation}
where each hyperplane $H(\m u_i,\m a_i)$ through $\m u_i$ and with the unit normal $\m a_i$ is tangent to $C$ and is lattice-like.
Let $\{\m c^i_1,\dots,\m c^i_r\}$ be a Jordan frame attached to $\m a_i$.
The last conditions hold if and only if
\begin{equation}\label{aire}
\m a_i= a^i_1\m c^i_1+...+a^i_{r_1}\m c^i_{r_1} +a^i_{r_1+1}\m c^i_{r_1+1}+...+a^i_{r_2}\m c^i_{r_2}+a^i_{r_2+1}\m c^i_{r_2+1}+...+a^i_{r}\m c^i_{r}
\end{equation}
with $\m c^i_1,...,\m c^i_{r_1}\in W_1;\;\;\m c^i_{r_1+1},...,\m c^i_{r_2}\in W_2$, and
$\m c^i_{r_2+1},...,\m c^i_r\in W_3$,
and exactly one of the following two cases hold:
\begin{enumerate}
\item[(i)] There exists a $k\in\{1,\dots,r_1\}$ with $a^i_k\not= 0$ and exactly one of the
following two statements is true
\begin{enumerate}
\item[(i)'] The equality $a^i_j=0$ holds for $j\ne k$,
\item[(i)''] There exists an $l\in\{1,\dots,r_1\}$ such that
$a^i_la^i_k<0$ and $a^i_j=0$, for $j\notin\{k,l\}$,
\end{enumerate}
\item[(ii)] There exists $k,l\in\{r_1+1,\dots,r_2\}$ and $p\in I_2$ such that $\m c^i_k,\m c^i_l\in V_p$, $a^i_k=\sqrt{2}/2$,
$a^i_l=-\sqrt{2}/2$ and $a^i_j=0$, for $j\notin\{k,l\}$.
\end{enumerate}
\end{theorem}
\begin{proof}
Observe first that using the representation (\ref{aire}) of $a_i$ and
the partition (\ref{partition1}) of $V$, we have the following relations:
$$r_1=|I_1|,\;r_2-r_1=2|I_2|,\;r-r_2\geq 3|I_3|.$$
The representation (\ref{alap}) follows from Theorem \ref{FOOO}.
Let us see first that the alternative (i), respectively (ii) is
sufficient for $H(\m u_i,\m a_i)$ to be an l-l set.
If (i) holds, then the hyperplane $H_{r_1-1}$ through $0$ with the normal
$\m a^i_{W_1}=a^i_1\m c^i_1+...+a^i_{r_1}\m c^i_{r_1}$ is by Section 7 an l-l set in $\spa \{\m c^i_1,...,\m c^i_{r_1}\}$ ordered
by the orthant engendered by $\m c_1,...,\m c_{r_1}$.
Hence $H(\m u_i,\m a_i)=(H_{r_1-1}+\m u_i)\bigoplus W_2\bigoplus W_3$ is l-l in $V$ (by Theorem \ref{ISOINV}
and Lemma \ref{ti}).
If (ii) holds, then the hyperplane $H'$ through $0$ with the normal $\m a^i_0=a^i_k\m c^i_k+a^i_l\m c^i_l=
(\sqrt 2/2)(\m c^i_k-\m c^i_l)$ in $V_p$ is l-l (by Theorem \ref{ketdim1}, Theorem \ref{ISOINV} and Lemma
\ref{ti}), hence
$H(\m u_i,\m a_i)=(H'+\m u_i)\bigoplus (\bigoplus_{j\not= p}V_j)$ is l-l in $V$.
To complete the proof we have to show the necessity of the alternatives (i) and (ii).
Observe first that if $H(\m u_i,\m a_i)$ is l-l, then in the representation (\ref{aire})
of $\m a_i$, by Corollary \ref{osszhyp}, we must have $a^i_j=0$
whenever $j>r_2$. Thus, if $a^i_j\not =0$, then $j\leq r_2.$
Suppose that $a^i_k\not= 0$ for for some $\m c^i_k\in W_2$.
Then, there exists an $a^i_l\not= 0$ and $\m c^i_k,\,\m c^i_l\in V_p$, for some $V_p$ in the representation of $W_2$.
Indeed, in the case $a^i_k\not= 0$ it follows that $a^i_k\m c^i_k\in V_p\setminus \{0\}$
for some $V_p\subset W_2$, hence $H(\m u_i,\m a_i)\cap V_p$
is a hyperplane in $V_p$ and our assertion follows from Lemma \ref{ketdim} (and in particular one of
$a^i_k$ and $a^i_l$ is $\sqrt 2/2$ the other is $-\sqrt 2/2$).
From Corollary \ref{aaa} it follows then that
$a^i_j=0$ for $j\notin \{k,l\}$. Thus, the alternative (ii) must hold.
Suppose now that $a^i_j\not= 0$ for some $j\leq r_1$.
Then from the reasoning of the above paragraph and Corollary \ref{aaa}, we must have $a^i_k=0$ if $k>r_1$.
In this case two situations are possible: (i)' $a^i_j=1$ and $a^i_l=0$ for $l\not=j$,
and (i)'' there exists $a^i_l\not= 0,\,(l\leq r_1)$ with $a^i_ja^i_l<0$ and $a^i_k=0$ for $k\notin \{j,l\}.$
Thus, the alternative (i) must hold.
\end{proof}
\begin{example}\label{simplesplit}
Let $V$ be a simple Euclidean Jordan algebra with
the Pierce decomposition given by (\ref{Oplus}) and (\ref{Oplus1}), and $d$ the common dimension of $V_{ij}$, $i\ne j$ (see Corollary IV.2.6 \cite{FarautKoranyi1994}).
Denote
$$W_{k,l}=\bigoplus_{k\leq i\leq j\leq l}V_{ij}.$$
Then, $W_{k,l}$ is a subalgebra, hence an l-l subspace.
The sum
$$W_{1,k}\bigoplus W_{k+1,r},\quad k<r$$
is a subalgebra too, and hence an l-l subspace.
Suppose that $r\geq 4$ and $2\leq k \leq r-2$. Let $H_0$
be an l-l hyperplane in $W_{k+1,r}$ which is not its subalgebra.
Then,
$$W_{1,k}+H_0$$
is an l-l subspace in $V$ of dimension $k+(d/2)k(k-1)+ r-k-1$
which is not an algebra.
\emph{Question}: Is every l-l subspace of $V$ which is not a subalgebra of this type?
\end{example}
\section*{Acknowledgement} The authors express their gratitude to Roman Sznajder for his
helpful comments and information on Euclidean Jordan algebras.
\bibliographystyle{hieeetr}
|
1,116,691,500,809 | arxiv | \section{Introduction}
Let ${\mathbb X}$ be a Riemannian symmetric space of non-compact type. Then ${\mathbb X}$ is isomorphic to $G/K$, where $G$ is a connected real
semisimple Lie group, and $K$ a maximal compact subgroup. Consider further the Oshima compactification \cite{oshima78} $\widetilde {\mathbb X}$ of ${\mathbb X}$,
a simply connected closed real-analytic manifold on which $G$ acts analytically. The orbital decomposition
of $\widetilde {\mathbb X}$ is of normal crossing type, and the open orbits are isomorphic to $G/K$, the number of them being equal
to $2^l$, where $l$ denotes the rank of $G/K$. In this paper, we shall study the invariant integral operators
\begin{equation}
\label{eq:1}
\pi(f)= \int _G f(g) \pi(g) d_G (g),
\end{equation}
where $\pi$ is the regular representation of $G$ on the Banach space $\mathrm{C}(\widetilde {\mathbb X})$ of continuous
functions on $\widetilde {\mathbb X}$, $f$ a smooth, rapidly decreasing function on $G$, and $d_G$ a Haar measure on $G$.
These operators play an important role in representation theory, and our interest will be directed towards the
elucidation of their microlocal structure within the theory of pseudodifferential operators. Since the underlying
group action on $\widetilde {\mathbb X}$ is not transitive, the operators $\pi(f)$ are not smooth, and the orbit structure
of $\widetilde {\mathbb X}$ is reflected in the singular behavior of their Schwartz kernels.
As it turns out, the operators in question can be characterized as pseudodifferential operators belonging to a particular class
which was first introduced in \cite{melrose} in connection with boundary problems. In fact, if $\widetilde {\mathbb X}_\Delta$ denotes a component
in $\widetilde {\mathbb X}$ isomorphic to $G/K$, we prove that the restrictions
\begin{equation*}
\pi(f)_{|\overline{\widetilde {\mathbb X}_{\Delta}}}:{\rm C^{\infty}_c}(\overline{\widetilde {\mathbb X}_{\Delta}}) \longrightarrow
{\rm C^{\infty}}(\overline{\widetilde {\mathbb X}_{\Delta}})
\end{equation*}
of the operators $\pi(f)$ to the manifold
with corners $\overline{\widetilde {\mathbb X}_{\Delta}}$
are totally characteristic pseudodifferential operators of class $\L^{-\infty}_b$. A similar
description of invariant integral operators on prehomogeneous vector spaces was obtained by the second author in \cite{ramacher06}.
We then consider the holomorphic semigroup generated by a strongly elliptic operator $\Omega$ associated to the regular
representation $(\pi, \mathrm{C}(\widetilde {\mathbb X}))$ of $G$, as well as its resolvent. Since both the holomorphic semigroup and the
resolvent can be characterized as operators of the form \eqref{eq:1}, they can be studied with the previous methods, and relying on the
theory of elliptic operators on Lie groups \cite{robinson} we obtain a description of the asymptotic behavior of the semigroup and resolvent kernels
on $\widetilde {\mathbb X}_\Delta\simeq {\mathbb X}$ at infinity. In the particular case of the Laplace-Beltrami operator on ${\mathbb X}$,
these questions have been intensively studied before.
While for the classical heat kernel on ${\mathbb X}$ precise upper and lower bounds were previously obtained in \cite{anker-ji99}
using spherical analysis, a detailed description of the analytic properties of the resolvent of the Laplace-Beltrami
operator on ${\mathbb X}$ was given in \cite{mazzeo-melrose87}, \cite{mazzeo-vasy05}.
The paper is organized as follows. In Section \ref{Sec:2} we briefly recall those parts of the structure theory of real semisimple
Lie groups that are relevant to our purposes. We then describe the $G$-action on the homogeneous spaces $G/P_{\Theta}(K)$,
where $P_{\Theta}(K)$ is a closed subgroup of $G$ associated naturally to a subset $\Theta$ of the set of simple roots, and
the corresponding fundamental vector fields. This leads to the definition of the Oshima compactification $\widetilde{{\mathbb X}}$ of the
symmetric space ${\mathbb X}\simeq G/K$, together with a description of the orbital decomposition of $\widetilde{{\mathbb X}}$. Since this decomposition
is of normal crossing type, it is well-suited for our analytic purposes. A thorough and unified description of the various
compactifications of a symmetric space is given in \cite{borel-ji}. Section \ref{sec:PDO} contains a summary with some of the basic
facts in the theory pseudodifferential operators needed in the sequel. In particular, the class of totally characteristic
pseudodifferential operators on a manifold with corners is introduced. Section \ref{Sec:4} is the central part
of this paper. By analyzing the orbit structure of the $G$-action on $\widetilde{{\mathbb X}}$, we
are able to elucidate the microlocal structure of the convolution operators $\pi(f)$, and characterize them as
totally characteristic pseudodifferential operators on the manifold with corners $\overline{\widetilde{{\mathbb X}}_\Delta}$. This
leads to a description of the asymptotic behavior of their Schwartz kernels on $\widetilde {\mathbb X}_\Delta\simeq {\mathbb X}$ at infinity.
In Section \ref{Sec:5}, we consider the holomorphic semigroup $S_\tau$ generated by the closure $\overline \Omega$ of a strongly
elliptic differential operator $\Omega$ associated to the representation $\pi$. Since $S_\tau=\pi(K_\tau)$, where $K_\tau(g)$
is a smooth and rapidly decreasing function on $G$, we can apply our previous results to describe
the Schwartz kernel of $S_\tau$. The Schwartz kernel of the resolvent $(\lambda { \bf 1}+ \overline{\Omega})^{-\alpha}$,
where $\alpha > 0$, and $\Re \lambda$ is sufficiently large, can be treated similarly, but is more subtle
due to the singularity of the corresponding group kernel $R_{\alpha,\lambda}(g)$ at the identity.
\newpage
\section{The Oshima compactification of a Riemannian symmetric space}
\label{Sec:2}
Let $G$ be a connected real semisimple Lie group with finite centre and Lie algebra ${\bf \mathfrak g}$, and denote by $\langle X,Y\rangle = \tr \, (\mathrm{ad}\, X\circ \mathrm{ad}\, Y)$ the \emph{Cartan-Killing form} on ${\bf \mathfrak g}$. Let $\theta$ be the Cartan involution of ${\bf \mathfrak g}$, and
$${\bf \mathfrak g} = \k\oplus{\bf \mathfrak p}$$
the Cartan decomposition of ${\bf \mathfrak g}$ into the eigenspaces of $\theta$, corresponding to the eigenvalues $+1$ and $-1$ , respectively, and put $\langle X,Y\rangle _\theta:=-\langle X,\theta Y\rangle $. Note that the Cartan decomposition is orthogonal with respect to $\langle,\rangle_{\theta}$. Consider further a maximal Abelian subspace $\a$ of ${\bf \mathfrak p}$. Then $\mathrm{ad}\,(\a)$ is a commuting family of self-adjoint operators on ${\bf \mathfrak g}$. Indeed, for $X,Y,Z\in{\bf \mathfrak g}$ one computes
\begin{align*}
\langle \mathrm{ad}\, X(Z),Y\rangle _\theta&=-\langle [X,Z],\theta Y\rangle =-\langle Z,[\theta Y,X]\rangle =-\langle Z,\theta [Y,\theta X]\rangle =\langle Z,[Y,\theta X]\rangle _\theta\\&=\langle Z,-[\theta X,Y]\rangle _\theta=\langle Z,-\mathrm{ad}\,\theta X(Y)\rangle _\theta.
\end{align*}
Therefore $-\mathrm{ad}\,\theta X$ is the adjoint of $\mathrm{ad}\, X$ with respect to $\langle ,\rangle _\theta$. So, if we take $X\in{\bf \mathfrak p}$, the -1 eigenspace of $\theta$, $\mathrm{ad}\, X$ is self-adjoint with respect to $\langle ,\rangle _\theta$. The dimension $l$ of $\a$ is called the \emph{real rank of $G$} and \emph{the rank of the
symmetric space $G/K$}. Next, one defines for each $\alpha\in\a^*$, the dual of $\a$, the simultaneous eigenspaces ${\bf \mathfrak g}^\alpha=\{X\in{\bf \mathfrak g}:[H,X]=\alpha(H)X \, \text{for all } H\in\a\}$ of $\mathrm{ad}\,(\a)$. A functional $0 \not =\alpha\in\a^\ast $ is called a \emph{(restricted) root} of $({\bf \mathfrak g},\a)$ if ${\bf \mathfrak g}^\alpha\neq\{0\}$, and setting $\Sigma=\{\alpha\in\a^*:\alpha\neq0,{\bf \mathfrak g}^\alpha\neq\{0\}\}$, we obtain the decomposition
\begin{equation*}
{\bf \mathfrak g}={\bf\mathfrak m}\oplus \a\oplus \bigoplus_{\alpha\in \Sigma}{\bf \mathfrak g}^\alpha,
\end{equation*}
where ${\bf\mathfrak m}$ is the centralizer of $\a$ in $\k$. Note that this decomposition is orthogonal with respect to $ \langle \cdot ,\cdot\rangle _\theta$. With respect to an ordering of $\a^\ast$, let $\Sigma^{+}=\{\alpha\in\Sigma:\alpha> 0\}$ denote the \emph{set of positive roots}, and $\Delta=\{\alpha_1,\dots ,\alpha_l\}$ the \emph{set of simple roots}. Let $\rho=\frac{1}{2}\Sigma_{\alpha\in\Sigma^{+}}\alpha$, and put $m(\alpha)=$ dim ${\bf \mathfrak g}^\alpha$ which is, in general, greater than $1$.
Define ${\bf\mathfrak n}^+=\bigoplus_{\alpha\in\Sigma^{+}}{\bf \mathfrak g}^\alpha, \, {\bf\mathfrak n}^-=\theta({\bf\mathfrak n}^+)$, and write $K, A, N^+$ and $N^- $ for the analytic subgroups of $G$ corresponding to $\k,\, \a,\, {\bf\mathfrak n}^+$, and $ {\bf\mathfrak n}^-$, respectively. The \emph{Iwasawa decomposition} of $G$ is then given by
\begin{equation*}
G=KAN^{\pm}.
\end{equation*}
Next, let $M=\{k\in K:\mathrm{Ad}\,(k)H=H \text{ for all } H\in\a\}$ be the centralizer of $\a$ in $K$ and $M^*=\{k\in K:\mathrm{Ad}\,(k)\a\subset\a\}$ the normalizer of $\a$ in $K$. The quotient $W={M^*}/{M}$ is the \emph{Weyl group} corresponding to $({\bf \mathfrak g},\a)$, and acts on $\a$ as a group of linear transformations via the adjoint action. Alternatively, $W$ can be characterized as follows. For each $\alpha_i \in \Delta $, define a reflection in $\a^*$ with respect to the Cartan-Killing form $\langle \cdot , \cdot \rangle $ by
\begin{equation*}
w_{\alpha_i}:\lambda\mapsto \lambda -2\alpha_{i}\langle \lambda,\alpha_{i}\rangle /\langle \alpha_i,\alpha_i\rangle,
\end{equation*}
where $\langle \lambda,\alpha_{}\rangle=\langle H_\lambda, H_\alpha \rangle.$ Here $H_\lambda$ is the unique element in $\a$ corresponding to a given $\lambda\in\a^\ast$, and determined by the non-degeneracy of the Cartan-Killing form. One can then identify the Weyl group $W$ with the group generated by the reflections $\{w_{\alpha_i}:\alpha_i\in\Delta\}$. For a subset $\Theta$ of $\Delta$, let now $W_{\Theta}$ denote the subgroup of W generated by reflections corresponding to elements in $\Theta$, and define
\begin{equation*}
P_{\Theta}=\bigcup_{w\in W_{\Theta}}Pm_{w}P,
\end{equation*}
where $m_w$ denotes a representative of $w$ in $M^\ast$, and $P=MAN^+$ is a minimal parabolic subgroup. It is then a classical result in the theory of parabolic subgroups \cite{warner72} that, as $\Theta$ ranges over the subsets of $\Delta$, one obtains all the parabolic subgroups of $G$ containing $P$. In particular, if $\Theta = \emptyset$, $P_\Theta=P$. Let us now introduce for $\Theta\subset\Delta$ the subalgebras
\begin{equation*}
\a_{\Theta}=\{H\in\a:\alpha(H)=0 \, \text{ for all } \alpha\in\Theta\}, \qquad \a(\Theta)=\{H\in\a:\langle H,X\rangle _{\theta}=0 \text{ for all } X\in\a_{\Theta}\}.
\end{equation*}
Note that, when restricted to the $+1$ or the $-1$ eigenspace of $\theta$, the orthogonal complement of a subspace with respect to $\langle \cdot ,\cdot \rangle $ is the same as its orthogonal complement with respect to $\langle \cdot ,\cdot \rangle _{\theta}$. We further define
\begin{align*}
{\bf\mathfrak n}_{\Theta}^{+}&=\sum_{\alpha\in\Sigma^{+}\setminus\langle \Theta\rangle ^{+}}{\bf \mathfrak g}^{\alpha},
\hspace{4cm} {\bf\mathfrak n}_{\Theta}^{-}=\theta({\bf\mathfrak n}_{\Theta}^{+}),\\
{\bf\mathfrak n}^+(\Theta)&=\sum_{\alpha\in\langle \Theta\rangle ^{+}}{\bf \mathfrak g}^{\alpha}, \hspace{4cm} {\bf\mathfrak n}^-(\Theta)=\theta({\bf\mathfrak n}^+(\Theta)), \\
{\bf\mathfrak m}_{\Theta}&={\bf\mathfrak m}+{\bf\mathfrak n}^+(\Theta)+{\bf\mathfrak n}^-(\Theta)+\a(\Theta),\hspace{1.1cm} {\bf\mathfrak m}_{\Theta}(K)={\bf\mathfrak m}_{\Theta}\cap\k,
\end{align*}
where $\left\langle \Theta\right\rangle ^{+}=\Sigma^{+}\cap\sum_{\alpha_{i}\in\Theta}{\mathbb R}\alpha_{i}$, and denote by $A_{\Theta},A(\Theta), \,N_{\Theta}^{\pm},N^{\pm}(\Theta),M_{\Theta,0}$, and $M_{\Theta}(K)_{0}$ the corresponding connected analytic subgroups of $G$, obtaining the decompositions $A=A_\Theta A(\Theta)$ and $N^\pm =N_\Theta^\pm N(\Theta)^\pm$, the second being a semi-direct product. Let next $M_{\Theta}=MM_{\Theta,0}, \, M_{\Theta}(K)=MM_{\Theta}(K)_{0}$. One has the \emph{Iwasawa decompositions}
\begin{equation*}
M_{\Theta}=M_{\Theta}(K)A(\Theta)N^{\pm}(\Theta),
\end{equation*}
and the \emph{Langlands decompositions}
\begin{align*}
P_{\Theta}&=M_{\Theta}A_{\Theta}N_{\Theta}^{+} =M_{\Theta}(K)AN^{+}.
\end{align*}
In particular, $P_\Delta= M_\Delta =G$, since $m_\Delta = {\bf\mathfrak m} \oplus\a \oplus \bigoplus_{\alpha \in \Sigma} {\bf \mathfrak g}^\alpha$, and $\a_\Delta, {\bf\mathfrak n}_\Delta ^+$ are trivial.
One then defines
$$P_{\Theta}(K)=M_{\Theta}(K)A_{\Theta}N_{\Theta}^{+}.$$
$P_\Theta(K)$ is a closed subgroup, and $G$ is a union of the open and dense submanifold $N^- A(\Theta) P_\Theta(K)=N^-_\Theta P_\Theta$, and submanifolds of lower dimension, see \cite{oshima78}, Lemma 1.
For $\Delta=\{ \alpha_{1},\dots ,\alpha_{l}\}$, let next $\{H_{1},\dots ,H_{l}\}$ be the basis of $\a$, dual to $\Delta$, i.e. $\alpha_{i}(H_{j})=\delta_{ij}$. Fix a basis $\{X_{\lambda,{i}}:1\leq i \leq m(\lambda)\}$ of ${\bf \mathfrak g}^{\lambda}$ for each $\lambda\in\Sigma^{+}$. Clearly,
\begin{equation*}
[H, - \theta X_{\lambda,i}]=-\theta [\theta H, X_{\lambda,i}]=-\lambda(H) (-\theta X_{\lambda,i}), \qquad H \in \a,
\end{equation*}
so that setting $X_{-\lambda,i}=-\theta(X_{\lambda,{i}})$ one obtains a basis $\{X_{-\lambda,{i}}:1\leq i \leq m(\lambda)\}$ of ${\bf \mathfrak g}^{-\lambda} \subset {\bf\mathfrak n}^-$. One now has the following lemma due to Oshima.
\begin{lemma}
\label{lemma:fundvec}
Fix an element $g\in G$, and identify $N^{-}\times A(\Theta)$ with an open dense submanifold of the homogeneous space $G/P_{\Theta}(K)$ by the map $(n,a)\mapsto gna P_{\Theta}(K)$. For $Y\in{\bf \mathfrak g}$, let $Y_{|G/P_{\Theta}(K)}$ be the fundamental vector field corresponding to the action of the one-parameter group $\exp (sY),s\in{\mathbb R},$ on $G/P_{\Theta}(K)$. Then, at any point $p=(n,a)\in N^{-}\times A(\Theta)$, we have
\begin{align*}
(Y_{|G/P_{\Theta}(K)})_{p}&=\sum_{\lambda\in \Sigma^+}\sum_{i=1}^{m(\lambda)} c_{-\lambda,i}(g,n)(X_{-\lambda,i})_p+ \sum_{\lambda\in \langle \Theta\rangle ^+} \sum_{i=1}^{m(\lambda)}c_{\lambda,i}(g,n)e^{-2\lambda( \log a)}(X_{-\lambda,i})_p \\ &+ \sum_{\alpha_i\in\Theta} c_{i}(g,n)(H_i)_p
\end{align*}
with the identification $T_{n}N^{-}\bigoplus T_{a}(A(\Theta))\simeq T_{p}(N^{-}\times A(\Theta))\simeq T_{gnaP_{\Theta}(K)}G/P_{\Theta}(K)$. The coefficient functions $c_{\lambda,{i}}(g,n),c_{-\lambda,i}(g,n),c_{i}(g,n)$ are real-analytic, and are determined by the equation
\begin{equation}
\label{eq:2211}
\mathrm{Ad}\,^{-1}(gn)Y=\sum_{\lambda\in \Sigma^+}\sum_{i=1}^{m(\lambda)}(c_{\lambda,i}(g,n)X_{\lambda,i}+c_{-\lambda,i}(g,n)X_{-\lambda,i})+\sum_{i=1}^{l}c_{i}(g,n)H_{i} \mod {\bf\mathfrak m}.
\end{equation}
\end{lemma}
\begin{proof}
Due to its importance, and for the convenience of the reader, we shall give a detailed proof of the lemma, following the original proof given in \cite{oshima78}, Lemma 3. Let $s\in {\mathbb R}$, and assume that $|s|$ is small. According to the direct sum decomposition ${\bf \mathfrak g}={\bf\mathfrak n}^-\oplus \a\oplus{\bf\mathfrak n}^+\oplus{\bf\mathfrak m}$ one has for an arbitrary $Y\in {\bf \mathfrak g}$
\begin{equation}
\label{eq:2}
(gn)^{-1}\exp(sY)gn=\exp N_1^-(s)\exp A_1(s)\exp N_1^+(s)\exp M_1(s) ,
\end{equation}
where $N_1^-(s)\in {\bf\mathfrak n}^-$, $A_1(s)\in \a$, $N_1^+(s)\in {\bf\mathfrak n}^+$, and $ M_1(s)\in {\bf\mathfrak m}$. The action of $\exp(sY)$ on the homogeneous space $G/P_{\Theta}(K)$ is therefore given by
\begin{align*}
\exp(sY)gnaP_{\Theta}(K)&=gn\exp N_1^-(s)\exp A_1(s)\exp N_1^+(s)\exp M_1(s)aP_{\Theta}(K)\\ & = gn\exp N_1^-(s)\exp A_1(s)\exp N_1^+(s)a\exp M_1(s)P_{\Theta}(K) \\ &=gn\exp N_1^-(s)\exp A_1(s)\exp N_1^+(s)aP_{\Theta}(K),
\end{align*}
since $M$ is the centralizer of $A$ in $K$, and $\exp M_1(s) \in M M_\Theta(K)_0 \subset P_\Theta(K)$. The Lie algebra of $P_{\Theta}(K)$ is ${\bf\mathfrak m}_{\Theta}(K)\oplus \a_{\Theta}\oplus {\bf\mathfrak n}^+_{\Theta}$, which we shall henceforth denote by ${\bf \mathfrak p}_\Theta(K)$. Using the decomposition ${\bf \mathfrak g}={\bf\mathfrak n}^-\oplus\a(\Theta)\oplus {\bf \mathfrak p}_\Theta(K)$ we see that
\begin{equation}
\label{eq:3}
a^{-1}\exp N_1^+(s)a=\exp N_2^-(s)\exp A_2(s)\exp P_2(s),
\end{equation}
where $N_2^-(s)\in {\bf\mathfrak n}^-$, $A_2(s)\in\a(\Theta)$, and $P_2(s)\in{\bf \mathfrak p}_\Theta(K)$. From this we obtain that
\begin{align*}
gn\exp N_1^-(s)&\exp A_1(s)\exp N_1^+(s)aP_{\Theta}(K)\\& =gn\left(\exp N_1^-(s)\exp A_1(s)a\exp N_2^-(s)\right)\exp A_2(s)\exp P_2(s)P_{\Theta}(K)\\&= gn\left(\exp N_1^-(s)\exp A_1(s)a\exp N_2^-(s)a^{-1}\right)a\exp A_2(s)P_{\Theta}(K).
\end{align*}
Noting that $[\a,{\bf\mathfrak n}^-]\subset{\bf\mathfrak n}^-$ one deduces the equality $\exp N_1^-(s)\exp A_1(s)a\exp N_2^-(s)a^{-1}\exp A_1(s)^{-1}=\exp N_3^-(s)\in N^-$, and consequently
\begin{equation}
\label{eq:4}
\exp N_1^-(s)\exp A_1(s)a\exp N_2^-(s)a^{-1}=\exp N_3^-(s)\exp A_1(s),
\end{equation}
which in turn yields
\begin{align*}
gn\exp N_1^-(s)\exp A_1(s)\exp N_1^+(s)aP_{\Theta}(K)&=gn\exp N_3^-(s)\exp A_1(s)a\exp A_2(s)P_{\Theta}(K)\\&=gn\exp N_3^-(s)a\exp (A_1(s)+A_2(s))P_{\Theta}(K).
\end{align*}
The action of ${\bf \mathfrak g}$ on $G/ P_{\Theta}(K)$ can therefore be characterized as
\begin{equation}
\label{eq:action}
\exp(sY)gnaP_{\Theta}(K)=gn\exp N_3^-(s)a\exp (A_1(s)+A_2(s))P_{\Theta}(K).
\end{equation}
Set ${dN_i^-(s)}/{ds}|_{s=0}= N_i^-$, ${dN_1^+(s)}/{ds}|_{s=0}= N_1^+$, ${dA_i(s)}/{ds}|_{s=0}= A_i$, and ${dP_2(s)}/{ds}|_{s=0}= P_2$, where $i=1,2$, or $3$. By differentiating equations \eqref{eq:2}-\eqref{eq:4} at $s=0$ one computes
\begin{align}
\label{eq:Ad}
\mathrm{Ad}\,^{-1}(gn)Y&=N_1^-+A_1+N_1^+ \qquad \text{mod} \quad {\bf\mathfrak m}, \\
\label{eq:Ad2}
\mathrm{Ad}\,^{-1}(a)N_1^+&=N_2^-+A_2 +P_2,\\
N_1^-+\mathrm{Ad}\,(a)N_2^-&=N_3^-.
\end{align}
In what follows, we express $N_1^\pm \in {\bf\mathfrak n}^\pm$ in terms of the basis of ${\bf\mathfrak n}^\pm$, and $A_1$ in terms of the one of $\a$, as
\begin{align*}
N_1^\pm&=\sum_{\lambda \in \Sigma^+}\sum_{i=1}^{m(\lambda)}c_{\pm\lambda,i}(g,n)X_{\pm\lambda,i},\\
A_1&=\sum_{i=1}^lc_i(g,n)H_i=\sum_{\alpha_i \in \Theta}c_i(g,n)H_i\quad \text{ mod} \, \a_\Theta.
\end{align*}
For a fixed $X_{\lambda,i}$ one has $[H,X_{\lambda,i}]=\lambda(H)X_{\lambda,i}$ for all $H\in \a$. Setting $H=-\log a$, $a \in A$, we get $\mathrm{ad}\,(-\log a)X_{\lambda,i} =-\lambda(\log a)X_{\lambda,i}$. By exponentiating we obtain $e^{\mathrm{ad}\,(-\log a)}X_{\lambda,i}=e^{-\lambda(\log a)}X_{\lambda,i}$, which together with the relation $e^{\mathrm{ad}\,(-\log a)}=\mathrm{Ad}\,(\exp(-\log a))$ yields
$$\mathrm{Ad}\,^{-1}(a)X_{\lambda,i}=e^{-\lambda(\log a)}X_{\lambda,i}.$$
Analogously, one has $[H,X_{-\lambda,i}]=\theta[ \theta H, -X_{\lambda,i}]= - \lambda(H)X_{-\lambda,i}$ for all $H\in \a$, so that
\begin{equation}
\label{eq:7}
\mathrm{Ad}\,^{-1}(a)X_{-\lambda,i}=e^{\lambda(\log a)}X_{-\lambda,i}.
\end{equation}
We therefore arrive at
\begin{align*}\mathrm{Ad}\,^{-1}(a)X_{\lambda,i}&=e^{-\lambda(\log a)}(X_{\lambda,i}-X_{-\lambda,i})+e^{-\lambda(\log a)}X_{-\lambda,i}\\ &=e^{-\lambda(\log a)}(X_{\lambda,i}-X_{-\lambda,i})+e^{-2\lambda(\log a)}\mathrm{Ad}\,^{-1}(a)X_{-\lambda,i}.
\end{align*}
Now, since $\theta(X_{\lambda,i}-X_{-\lambda,i})=\theta(X_{\lambda,i})-\theta(X_{-\lambda,i})=-X_{-\lambda,i}-(-X_{\lambda,i})=X_{\lambda,i}-X_{-\lambda,i},$ we see that $X_{\lambda,i}-X_{-\lambda,i}\in \k$. Consequently, if $\lambda$ is in $\left\langle \Theta\right\rangle ^{+}$, one deduces that
$X_{\lambda,i}-X_{-\lambda,i} \in \left({\bf\mathfrak m}+{\bf\mathfrak n}^+(\Theta)+{\bf\mathfrak n}^-(\Theta)+\a(\Theta)\right)\cap\k={\bf\mathfrak m}_{\Theta}(K).$
On the other hand, if $\lambda$ is in $\Sigma^+-\left\langle \Theta\right\rangle ^{+}$, then $\mathrm{Ad}\,^{-1}(a)X_{\lambda,i}=e^{-\lambda(\log a)}X_{\lambda,i}$ belongs to ${\bf\mathfrak n}_\Theta^+$.
Collecting everything we obtain
\begin{align*}
\mathrm{Ad}\,^{-1}(a)N_1^+&=\sum_{\lambda \in \Sigma^+}\sum_{i=1}^{m(\lambda)}c_{\lambda,i}(g,n)\mathrm{Ad}\,^{-1}(a)X_{\lambda,i}
\\&=\sum_{\lambda \in \left\langle \Theta\right\rangle ^{+}}\sum_{i=1}^{m(\lambda)}c_{\lambda,i}(g,n)\mathrm{Ad}\,^{-1}(a)X_{\lambda,i}+\sum_{\lambda \in \Sigma^+-\left\langle \Theta\right\rangle ^{+}}\sum_{i=1}^{m(\lambda)}c_{\lambda,i}(g,n)\mathrm{Ad}\,^{-1}(a)X_{\lambda,i}
\\&=\sum_{\lambda \in \left\langle \Theta\right\rangle ^{+}}\sum_{i=1}^{m(\lambda)}c_{\lambda,i}(g,n)\left(e^{-2\lambda(\log a)}\mathrm{Ad}\,^{-1}(a)X_{-\lambda,i} +e^{-\lambda(\log a)}(X_{\lambda,i}-X_{-\lambda,i})\right)\\ & +\sum_{\lambda \in \Sigma^+-\left\langle \Theta\right\rangle ^{+}}\sum_{i=1}^{m(\lambda)}c_{\lambda,i}(g,n)e^{-\lambda( \log a)}X_{\lambda,i} \\&=\sum_{\lambda \in \left\langle \Theta\right\rangle ^{+}}\sum_{i=1}^{m(\lambda)}c_{\lambda,i}(g,n)e^{-2\lambda(\log a)}\mathrm{Ad}\,^{-1}(a)X_{-\lambda,i} \\&+\sum_{\lambda \in \left\langle \Theta\right\rangle ^{+}}\sum_{i=1}^{m(\lambda)}c_{\lambda,i}(g,n)e^{-\lambda(\log a)}(X_{\lambda,i}-X_{-\lambda,i})+\sum_{\lambda \in \Sigma^+-\left\langle \Theta\right\rangle ^{+}}\sum_{i=1}^{m(\lambda)}c_{\lambda,i}(g,n)e^{-\lambda( \log a)}X_{\lambda,i}.
\end{align*}
Comparing this with the expression \eqref{eq:Ad2} we had obtained earlier for $\mathrm{Ad}\,^{-1}(a)N_1^+$, we obtain that
\begin{equation}
\label{eq:A2}
A_2=0 ,
\end{equation}
and $N_2^-=\sum_{\lambda \in \left\langle \Theta\right\rangle ^{+}}\sum_{i=1}^{m(\lambda)}c_{\lambda,i}(g,n)e^{-2\lambda(\log a)}\mathrm{Ad}\,^{-1}(a)X_{-\lambda,i}$, since ${\bf \mathfrak g}=\k\oplus \a \oplus {\bf\mathfrak n}^-$, and $ {\bf \mathfrak p}_\Theta(K)\cap \a(\Theta)=\{0\}$. Therefore
\begin{align}
\label{eq:8}
\begin{split}
N_3^-&=N_1^-+\mathrm{Ad}\,(a)N_2^-\\=\sum_{\lambda \in \Sigma^+}\sum_{i=1}^{m(\lambda)}c_{-\lambda,i}(g,n)X_{-\lambda,i} &+ \sum_{\lambda \in \left\langle \Theta\right\rangle ^{+}}\sum_{i=1}^{m(\lambda)}c_{\lambda,i}(g,n)e^{-2\lambda(\log a)}X_{-\lambda,i},\\
A_1+A_2& =\sum_{\alpha_i \in \Theta}c_i(g,n)H_i\quad \text{ mod} \, \a_\Theta.
\end{split}
\end{align}
As $N^-\times A(\Theta)$ can be identified with an open dense submanifold of the homogeneous space $G/P_\Theta(K)$, we have the isomorphisms $T_{gnaP_{\Theta}(K)}G/P_{\Theta}(K)\simeq T_{p}(N^{-}\times A(\Theta)) \simeq T_{n}N^{-}\bigoplus T_{a}(A(\Theta))$, where $p=(n,a)\in N^- \times A(\Theta)$. Therefore, by equation \eqref{eq:action} and the expressions for $N_3^- $ and $A_1+A_2$, we finally deduce that the fundamental vector field $Y_{|G/P_{\Theta}(K)}$ at a point $p$ corresponding to the action of $\exp(sY)$ on $G/P_\Theta(K)$ is given by
\begin{align*}
(Y_{|G/P_{\Theta}(K)})_{p}&=\sum_{\lambda\in \Sigma^+}\sum_{i=1}^{m(\lambda)} c_{-\lambda,i}(g,n)(X_{-\lambda,i})_p+ \sum_{\lambda\in \langle \Theta\rangle ^+} \sum_{i=1}^{m(\lambda)}c_{\lambda,i}(g,n)e^{-2\lambda \log a}(X_{-\lambda,i})_p \\ &+ \sum_{\alpha_i\in\Theta} c_{i}(g,n)(H_i)_p,
\end{align*}
where $Y \in {\bf \mathfrak g}$, and the coefficients are given by \eqref{eq:2211}.
\end{proof}
Let us next state the following
\begin{lemma}
\label{lemma:8}
Let $Y\in {\bf\mathfrak n}^{-}\oplus \a$ be given by $Y=\sum_{\lambda\in \Sigma^+}\sum_{i=1}^{m(\lambda)}c_{-\lambda,i} X_{-\lambda,i} +\sum_{j=1}^{l}c_{j} H_j$,
and introduce the notation $t^\lambda=t_1^{\lambda(H_1)}\cdots t_l^{\lambda(H_l)}$. Then, via the identification of $N^-\times{\mathbb R}^l_+$ with $N^-A$ by $(n,t)\mapsto n\cdot exp(-\sum_{j=1}^{l}H_j\log t_j)$, the left invariant vector field on the Lie group $N^-A$ corresponding to $Y$ is expressed as
\begin{equation*}
\tilde Y_{|N^-\times {\mathbb R}^l_+}=\sum_{\lambda\in \Sigma^+}\sum_{i=1}^{m(\lambda)}c_{-\lambda,i}t^{\lambda}X_{-\lambda,i}-\sum_{j=1}^{l}c_{j}t_j\frac{\partial}{\partial t_j},
\end{equation*}
and can analytically be extended to a vector field on $N^-\times{\mathbb R}^l$.
\end{lemma}
\begin{proof}
The lemma is stated in Oshima, \cite{oshima78}, Lemma 8, but for greater clarity, we include a proof of it here. Let $X_{-\lambda,i}$ be a fixed basis element of ${\bf\mathfrak n}^-$. The corresponding left-invariant vector field on the Lie group $N^-A$ at the point $na $ is given by
\begin{align*}
\frac{d}{ds}f(na\exp(sX_{-\lambda,i}))_{|s=0}&=\frac{d}{ds}f(n(a\exp(sX_{-\lambda,i})a^{-1})a)_{|s=0}= \frac{d}{ds}f(n\e{sAd(a)X_{-\lambda,i}}a)_{|s=0},
\end{align*}
where $f$ is a smooth function on $N^-A$. Regarded as a left invariant vector field on $ N^ - \times {\mathbb R}^l_+$, it is therefore given by
\begin{equation*}
\tilde X_{-\lambda,i|N^-\times {\mathbb R}^l_+}= \mathrm{Ad}\,(a)X_{-\lambda,i} =e^{-\lambda( \log a)}X_{-\lambda,i}=t^{\lambda}X_{-\lambda,i},
\end{equation*}
compare \eqref{eq:7}. Similarly, for a basis element $H_i$ of $\a$ the corresponding left invariant vector field on $N^-A$ reads
\begin{gather*}
\frac{d}{ds}f(na\exp(sH_i))_{|s=0}=\frac{d}{ds}f(n\exp(-\sum_{j=1}^{l}\log t_jH_j)\exp(sH_i))_{|s=0}\\=\frac{d}{ds}f\Big (n\exp(-\sum_{j=1}^{l}\log t_jH_j+sH_i)\Big )_{|s=0}=\frac{d}{ds}f\Big (n\exp(-\sum_{j\neq i}\log t_jH_j-\log (t_ie^{-s})H_i)\Big )_{|s=0},
\end{gather*}
and with the identification $N^- A \simeq N^ - \times {\mathbb R}^l_+$ we obtain
$$\tilde H_{i|N^-\times {\mathbb R}^l_+}=-t_i\frac{\partial}{\partial t_i}.$$
As there are no negative powers of $t$, $\tilde Y_{N^- \times {\mathbb R}^l_+}$ can be extended analytically to $N^-\times{\mathbb R}^l$, and the lemma follows.
\end{proof}
Similarly, by the identification $G/K \simeq N^-\times A\simeq N^- \times {\mathbb R}^l_+$ via the mappings $(n,t)\mapsto n\cdot exp(-\sum_{i=1}^{l}H_i\log t_i)\cdot a \mapsto gnaK$ one sees that
the action on $G/K$ of the fundamental vector field corresponding to
$\exp (sY)$ , $Y \in {\bf \mathfrak g}$, is given by
\begin{equation}
\label{eq:2.13}
Y_{|N^-\times {\mathbb R}^l_+}=\sum_{\lambda\in \Sigma^+}\sum_{i=1}^{m(\lambda)}(c_{\lambda,i}(g,n)t^{2\lambda}+ c_{-\lambda,i}(g,n))X_{-\lambda,i}-\sum_{i=1}^{l}c_{i}(g,n)t_i\frac{\partial}{\partial t_i},
\end{equation}
where the coefficients are given by \eqref{eq:2211}.
Again, the vector field \eqref{eq:2.13} can be extended analytically to $N^-\times{\mathbb R}^l$, but in contrast to the left invariant vector field $\tilde Y_{N^-\times {\mathbb R}^l}$, $Y_{N^-\times {\mathbb R}^l}$ does not necessarily vanish if $t_1=\dots t_l=0$.
We come now to the description of the Oshima compactification of the Riemannian symmetric space $G/K$. For this, let $\hat{{\mathbb X}}$ be the product manifold $G\times N^-\times {\mathbb R}^l$. Take $\hat{x}=(g,n,t)\in\hat{{\mathbb X}}$, where $g\in G,\,n\in N^-,\,t=(t_{1},\dots ,t_{l})\in{\mathbb R}^l$, and define an action of $G$ on $\hat{{\mathbb X}}$ by $g'\cdot (g,n,t):=(g'g,n,t),\, g'\in G.$ For $s \in {\mathbb R}$, let
\begin{equation*}
\sgn s = \left \{\begin{array}{cl} s/ |s|, & s \not=0, \\ 0, & s=0,
\end{array} \right.
\end{equation*}
and put $\sgn \hat{x}=(\sgn t_1,\dots ,\sgn t_l)\in\{-1,0,1\}^l$. We then define the subsets $ \Theta_{\hat{x}}=\{\alpha_i\in\Delta: t_i\neq 0\}$. Similarly, let $ a(\hat{x})=\exp (-\sum_{t_{i}\neq 0} H_i\log|t_i|) \in A(\Theta_{\hat{x}})$. On $\hat{{\mathbb X}}$, define now an equivalence relation by setting
\begin{equation*}
\hat{x}=(g,n,t)\sim \hat{x}'=(g',n,'t') \quad \Longleftrightarrow \quad \left \{
\begin{array}{l}
a) \, \sgn \hat{x}=\sgn \hat{x}', \\
b) \, g\, n\, a(\hat{x})\, P_{\Theta_{\hat{x}}}(K)=g'\, n'\, a(\hat{x}')\, P_{\Theta_{\hat{x}'}}(K).
\end{array} \right.
\end{equation*}
Note that the condition $\sgn \hat{x}=\sgn \hat{x}'$ implies that $\hat{x},\hat{x}' $ determine the same subset $\Theta_{\hat{x}}$ of $\Delta$, and consequently the same group $P_{\Theta_{\hat{x}}}(K)$, as well as the same homogeneous space $G/P_{\Theta_{\hat{x}}}(K)$, so that condition $b)$ makes sense. It says that $gna(\hat{x}), \, g'n'a(\hat{x}') $ are in the same $P_{\Theta_{\hat{x}}}(K)$ orbit on $G$, corresponding to the right action by $P_{\Theta_{\hat{x}}}(K)$ on $G$. We now define
\begin{equation*}
\widetilde{{\mathbb X}}:=\hat{{\mathbb X}}/\sim,
\end{equation*}
endowing it with the quotient topology, and denote by $\pi:\hat{{\mathbb X}}\rightarrow \widetilde{{\mathbb X}}$ the canonical projection. The action of $G $ on $\hat{{\mathbb X}}$ is compatible with the equivalence relation $\sim$, yielding a $G$-action
$g'\cdot \pi (g,n,t):=\pi (g'g,n,t)$ on $\widetilde {\mathbb X}$. For each $g\in G$, one can show that the maps
\begin{equation}
\label{eq:coord}
\phi_g:N^-\times{\mathbb R}^l\rightarrow \widetilde{U}_g: (n,t)\mapsto \pi(g,n,t), \qquad \widetilde{U}_g=\pi (\{g\} \times N^- \times {\mathbb R}^l),
\end{equation}
are bijections. One has then the following
\begin{theorem}
\label{Thm.1}
\begin{enumerate}
\item $\widetilde{{\mathbb X}}$ is a simply connected, compact, real-analytic manifold without boundary.
\item $\widetilde{{\mathbb X}}=\cup_{w \in W} \widetilde{U}_{m_w}=\cup_{g\in G}\widetilde{U}_g $. For $ g\in G$, $ \widetilde{U}_g $ is an open submanifold of $\widetilde{{\mathbb X}}$ topologized in such a way that the coordinate map $\phi_g$ defined above is a real-analytic diffeomorphism. Furthermore, $\widetilde{{\mathbb X}}\setminus \widetilde{U}_g$ is the union of a finite number of submanifolds of $\widetilde{{\mathbb X}}$ whose codimensions in $\widetilde{{\mathbb X}}$ are not lower than $2$.
\item The action of $G$ on $\widetilde{{\mathbb X}}$ is real-analytic. For a point $\hat{x}\in\hat{{\mathbb X}} $, the $G$-orbit of $\pi(\hat{x})$ is isomorphic to the homogeneous space $G/P_{\Theta_{\hat{x}}}(K)$, and for $\hat{x}, \hat{x}' \in \hat{{\mathbb X}}$ the $G$-orbits of $\pi(\hat{x})$ and $\pi(\hat{x}')$ coincide if and only if $\sgn \hat{x}=\sgn \hat{x}'$. Hence the orbital decomposition of $\widetilde{{\mathbb X}}$ with respect to the action of $G$ is of the form
\begin{equation}
\label{eq:decomp}
\widetilde{{\mathbb X}}\simeq \bigsqcup_{\Theta\subset\Delta} 2^{\#\Theta}(G/P_\Theta(K)) \quad \text{(disjoint union)},
\end{equation}
where $\#\Theta$ is the number of elements of $\Theta$ and $2^{\#\Theta}(G/P_\Theta(K))$ is the disjoint union of $2^{\#\Theta}$ copies of $G/P_\Theta(K)$.
\end{enumerate}
\end{theorem}
\begin{proof}
See Oshima, \cite{oshima78}, Theorem 5.
\end{proof}
Next, for $\hat{x}=(g,n,t)$ define the set $B_{\hat{x}}=\{ (t_1'\dots t_l')\in {\mathbb R}^l:\sgn t_i=\sgn t_i' ,1\leq i\leq l\}$. By analytic continuation, one can restrict the vector field \eqref{eq:2.13} to $N^-\times B_{\hat{x}}$, and with the identifications $G/P_{\Theta_{\hat{x}}}(K) \simeq N^-\times A(\Theta_{\hat{x}})\simeq N^-\times B_{\hat{x}}$ via the maps
$$ gnaP_{\Theta_{\hat{x}}}\leftarrow(n,a)\mapsto(n,\sgn t_1e^{-\alpha_1(\log a)},\dots, \sgn t_le^{-\alpha_l (\log a)}),$$
one actually sees that this restriction coincides with the vector field in Lemma \ref{lemma:fundvec}. The action of the fundamental vector field on $\widetilde {{\mathbb X}}$ corresponding to $\exp{sY}, Y\in {\bf \mathfrak g}$, is therefore given by the extension of \eqref{eq:2.13} to $N^-\times {\mathbb R}^l$.
Note that for a simply connected nilpotent Lie group $N$ with Lie algebra ${\bf\mathfrak n}$, the exponential $\exp:{\bf\mathfrak n}\rightarrow N$ is a diffeomorphism. So, in our setting, we can identify $N^-$ with ${\mathbb R}^k$.
Thus, for every point in $\widetilde{{\mathbb X}}$, there exists a local coordinate system $(n_1,\dots , n_k,t_1,\dots , t_l)$ in a neighbourhood of that point such that two points $(n_1,\dots ,n_k,t_1,\dots ,t_l)$ and $(n'_1,\dots ,n'_k,t'_1,\dots ,t'_l)$ belong to the same $G$-orbit if, and only if, $\sgn t_j=\sgn t'_j$, for $j=1,\dots, l$. This means that the orbital decomposition of $\widetilde{{\mathbb X}}$ is of \emph{normal crossing type}. In what follows, we shall identify the open $G$-orbit $\pi(\{\hat{x}=(e,n,t) \in\hat {\mathbb X}:\sgn \hat{x}=(1,\dots ,1)\})$ with the Riemannian
symmetric space $G/K$, and the orbit $\pi(\{\hat{x}\in\hat{{\mathbb X}}:\sgn \hat{x}=(0,\dots ,0)\}$ of lowest dimension with its Martin boundary $G/P$.
\section{Review of pseudodifferential operators}
\label{sec:PDO}
{\bf{Generalities.}} This section is devoted to an exposition of some basic facts about pseudodifferential operators needed to formulate our main results in the sequel. For a detailed introduction to the field, the reader is referred to \cite{hoermanderIII} and \cite{shubin}. Consider first an open set $U$ in ${\mathbb R}^n$, and let $x_1,\dots ,x_n$ be the standard coordinates. For any real number $l$, we denote by ${\rm S}^l(U\times {\mathbb R}^n)$ the class of all functions $a(x,\xi)\in {\rm C^{\infty}}(U\times {\mathbb R}^n)$ such that, for any multi-indices $\alpha,\beta$, and any compact set $\mathcal{K}\subset U$, there exist constants $C_{\alpha,\beta,\mathcal{K}}$ for which
\begin{equation}
\label{H}
|(\gd ^\alpha_\xi\gd ^\beta_x a)(x,\xi)| \leq C_{\alpha,\beta,\mathcal{K}} \eklm{\xi}^{l-|\alpha|}, \qquad x \in \mathcal{K}, \quad \xi \in {\mathbb R}^n,
\end{equation}
where $\eklm{\xi}$ stands for $(1+|\xi|^2)^{1/2}$, and $|\alpha|=\alpha_1+\dots +\alpha_n$. We further put ${\rm S}^{-\infty}(U\times {\mathbb R}^n)=\bigcap _{l \in {\mathbb R}} {\rm S}^l(U\times {\mathbb R}^n)$. Note that, in general, the constants $C_{\alpha,\beta,K}$ also depend on $a(x,\xi)$. For any such $a(x,\xi)$ one then defines the continuous linear operator
\begin{displaymath}
A:{\rm C^{\infty}_c}(U) \longrightarrow {\rm C^{\infty}}(U)
\end{displaymath}\large\Large\normalsize
by the formula
\begin{equation}
\label{I}
Au(x)=\int e^{ix \cdot \xi} a(x,\xi) \hat u(\xi) {\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi,
\end{equation}
where $\hat u$ denotes the Fourier transform of $u$, and ${\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi=(2\pi)^{-n} \d \xi$. \footnote{Here and in what follows we use the convention that, if not specified otherwise, integration is to be performed
over Euclidean space.} An operator $A$ of this form is called a \emph{pseudodifferential operator of order l}, and we denote the class of all such operators for which $a(x,\xi) \in {\rm S}^l(U\times {\mathbb R}^n)$ by $\L^l(U)$. The set $\L^{-\infty}(U)=\bigcap_{l\in {\mathbb R}} \L^l (U)$ consists of all operators with smooth kernel. They are called \emph{smooth operators}. By inserting in \eqref{I} the definition of $\hat u$, we obtain for $Au$ the expression
\begin{equation}
\label{II}
Au(x)=\int \int e^{i(x-y) \cdot \xi} a(x,\xi) u(y) \d y \,{\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi,
\end{equation}
which has a suitable regularization as an oscillatory integral. The Schwartz kernel of $A$ is a distribution $K_A\in {\mathcal D} '(U\times U)$ which is given the oscillatory integral
\begin{equation}
\label{III}
K_A(x,y)=\int e^{i(x-y)\cdot \xi} a(x,\xi) \,{\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi.
\end{equation}
It is a smooth function off the diagonal in $U\times U$.
Consider next a $n$-dimensional paracompact ${\rm C^{\infty}}$ manifold ${\bf X}$, and let $\{(\kappa_\gamma, \widetilde U^\gamma)\}$ be an atlas for ${\bf X}$. Then a linear operator
\begin{equation}
\label{IIIa}
A:{\rm C^{\infty}_c}({\bf X}) \longrightarrow {\rm C^{\infty}}({\bf X})
\end{equation}
is called a \emph{pseudodifferential operator on ${\bf X}$ of order $l$} if for each chart diffeomorphism $\kappa_\gamma:\widetilde U^\gamma \rightarrow U^\gamma= \kappa_\gamma(\widetilde U^\gamma)$, the operator $A^{\gamma} u = [A_{|\widetilde U^\gamma} ( u\circ \kappa_{\gamma})] \circ \kappa_{\gamma}^{-1}$ given by the diagram
\begin{displaymath}
\begin{CD}
{\rm C^{\infty}_c}(\widetilde U^{\gamma}) @>{A_{|\widetilde U^\gamma}}>> {\rm C^{\infty}}(\widetilde U^\gamma) \\
@A {\kappa_{\gamma}^\ast}AA @AA {\kappa_{\gamma}^\ast}A\\
{\rm C^{\infty}_c}( U^{\gamma}) @> {A^{\gamma}}>> {\rm C^{\infty}}( U^\gamma)
\end{CD}
\end{displaymath}
is a pseudodifferential operator on $U^\gamma$ of order $l$, and its kernel $K_A$ is smooth off the diagonal. In this case we write $A \in \L^l({\bf X})$. Note that, since the $\widetilde U^\gamma$ are not necessarily connected, we can choose them in such a way that ${\bf X}\times {\bf X}$ is covered by the open sets $\widetilde U^\gamma \times \widetilde U^\gamma$. Therefore the condition that $K_A$ is smooth off the diagonal can be dropped.
Now, in general, if ${\bf X}$ and ${\bf Y}$ are two smooth manifolds, and
\begin{equation*}
A: {\rm C^{\infty}_c}({\bf X}) \longrightarrow {\rm C^{\infty}}({\bf Y}) \subset {\mathcal D}'({\bf Y})
\end{equation*}
is a continuous linear operator, where ${\mathcal D}'({\bf Y})=({\rm C^{\infty}_c}({\bf Y},\Omega))'$ and $\Omega=|\Lambda^n({\bf Y})|$ is the density bundle on ${\bf Y}$, its Schwartz kernel is given by the distribution section $K_A \in {\mathcal D}'({\bf Y} \times {\bf X}, {\bf 1} \boxtimes \Omega_{{\bf X}})$, where ${\mathcal D}'({\bf Y}\times {\bf X} ,1 \boxtimes \Omega_{{\bf X}}) = ({\rm C^{\infty}_c}({\bf Y} \times {\bf X}, ({{\bf 1}} \boxtimes \Omega_{{\bf X}})^\ast \otimes \Omega_{{\bf Y}\times {\bf X}}))'$. Observe that ${\rm C^{\infty}_c} ({\bf Y}, \Omega_{{\bf Y}}) \otimes {\rm C^{\infty}} ({\bf X}) \simeq {\rm C^{\infty}} ( {\bf Y} \times {\bf X}, ({\bf 1} \boxtimes \Omega_{{\bf X}})^\ast \otimes\Omega_{{\bf Y} \times {\bf X}})$.
In case that ${\bf X}={\bf Y}$ and $A\in \L^l({\bf X})$, $A$ is given locally by the operators $A^{\gamma}$, which can be written in the form
\begin{equation*}
A^{\gamma}u(x) = \int \int e^{i(x-y) \cdot \xi} a ^{\gamma}(x,\xi) u(y) \d y {\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi,
\end{equation*}
where $u \in {\rm C^{\infty}_c}(U^{\gamma})$, $x \in U^\gamma$, and $a^{\gamma}(x,\xi) \in {\rm{S}}^l(U^\gamma, {\mathbb R}^n)$. The kernel of $A$ is then determined by the kernels $K_{A^{\gamma}} \in {\mathcal D}'(U^\gamma \times U ^{\gamma})$. For $l < -\dim {\bf X}$, they are continuous, and given by absolutely convergent integrals.
In this case, their restrictions to the respective diagonals in $U^\gamma \times U^\gamma$ define continuous functions
\begin{equation*}
k^\gamma(m)= K_{A^{\gamma}} (\kappa_\gamma (m),\kappa _\gamma (m)), \qquad m \in \widetilde U^\gamma,
\end{equation*}
which, for $m \in \widetilde U^{\gamma_1} \cap \widetilde U^{\gamma_2}$, satisfy the relations $ k^{\gamma_2}(m) =| \det (\kappa_{\gamma_1} \circ \kappa ^{-1}_{\gamma_2})' | \circ \kappa_{\gamma_2}(m) k^{\gamma_1}(m)$, and therefore define a density $k \in C ({\bf X},\Omega)$ on $\Delta_{{\bf X}}\times {\bf X} \simeq {\bf X}$.
If ${\bf X}$ is compact, this density can be integrated, yielding the trace of the operator $A$,
\begin{equation}
\label{eq:trace}
\tr A=\int _{{\bf X}} k=\sum_\gamma \int_{U^\gamma} (\alpha_\gamma \circ \kappa_\gamma^{-1}) (x) \, K_{A^\gamma}(x,x) \d x,
\end{equation}
where $\{\alpha_\gamma\}$ denotes a partition of unity subordinated to the atlas $\{(\kappa_\gamma, \widetilde U^\gamma)\}$, and $dx$ is Lebesgue measure in ${\mathbb R}^n$.
{\bf{Totally characteristic pseudodifferential operators.}} We introduce now a special class of pseudodifferential operators associated in a natural way to a ${\rm C^{\infty}}$ manifold ${\bf X}$ with boundary $\gd {\bf X}$. Our main reference
will be \cite{melrose} in this case. Let ${\rm C^{\infty}}({\bf X})$ be the space of functions on ${\bf X}$ which are ${\rm C^{\infty}}$ up to the boundary, and ${\rm \dot C^{\infty}}({\bf X})$ the subspace of functions vanishing to all orders on $\gd {\bf X}$. The standard spaces of distributions over ${\bf X}$ are
\begin{equation*}
{\mathcal D}'({\bf X})= ({\rm \dot C^{\infty}_c}({\bf X},\Omega))', \qquad \dot {\mathcal D}({\bf X})' =({\rm C^{\infty}_c}({\bf X},\Omega))',
\end{equation*}
the first being the space of \emph{extendible distributions}, whereas the second is the space of \emph{distributions supported by ${\bf X}$}. Consider now the translated partial Fourier transform of a symbol $a(x,\xi) \in {\rm S}^l({\mathbb R}^n\times {\mathbb R}^n)$,
\begin{equation*}
Ma(x,\xi';t)=\int e^{i(1-t)\xi_1} a(x,\xi_1,\xi') d\xi_1,
\end{equation*}
where we wrote $\xi=(\xi_1,\xi')$. $Ma(x,\xi';t)$ is ${\rm C^{\infty}}$ away from $t=1$, and one says that $a(x,\xi)$ is \emph{lacunary} if it satisfies the condition\begin{equation}
\label{V}
Ma(x,\xi';t)=0 \qquad \text{ for } t<0.
\end{equation}
The subspace of lacunary symbols will be denoted by ${\rm S}^l_{la}({\mathbb R}^n\times {\mathbb R}^n)$. Let $Z=\overline{{\mathbb R}^+} \times {\mathbb R}^{n-1}$ be the standard manifold with boundary with the natural coordinates $x=(x_1,x')$. In order to define on $Z$ operators of the form \eqref{II},
where now $a (x,\xi)=\widetilde a(x_1,x',x_1\xi_1, \xi')$ is a more general amplitude and $\widetilde a(x,\xi)$ is lacunary, one rewrites the formal adjoint of $A$ by making a singular coordinate change. Thus, for $u \in {\rm C^{\infty}_c}(Z)$, one considers
\begin{equation*}
A^\ast u(y) =\int\int e^{i(y-x)\cdot \xi} \overline{a}(x,\xi) u(x) \, \d x {\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi.
\end{equation*}
By putting $\lambda=x_1\xi_1$, $s=x_1/y_1$, this can be rewritten as
\begin{equation}
\label{VII}
A^\ast u(y)=(2\pi)^{-n}\int \int \int \int e^{i(1/s-1,y'-x')\cdot (\lambda,\xi')}\overline {\widetilde a}(y_1 s, x',\lambda,\xi') u(y_1 s,x')d\lambda \frac{ds}s dx' d\xi'.
\end{equation}
According to \cite{melrose}, Propositions 3.6 and 3.9, for every $\widetilde a \in {\rm S}_{la}^{-\infty}(Z\times {\mathbb R}^n) $, the successive integrals in \eqref{VII} converge absolutely and uniformly, thus defining a continuous bilinear form
\begin{equation*}
{\rm S}^{-\infty}_{la}(Z\times {\mathbb R}^n) \times {\rm C^{\infty}_c}(Z) \longrightarrow {\rm C^{\infty}}(Z),
\end{equation*}
which extends to a separately continuous form
\begin{equation*}
{\rm S}^\infty_{la}(Z\times {\mathbb R}^n) \times {\rm C^{\infty}_c}(Z) \longrightarrow {\rm C^{\infty}}(Z).
\end{equation*}
If $\widetilde a \in {\rm S}^\infty_{la}(Z \times {\mathbb R}^n)$ and $a(x,\xi)=\widetilde a(x_1,x',x_1\xi_1,\xi')$, one then defines the operator
\begin{equation}
\label{VIII}
A:\dot {\mathcal E}'(Z) \longrightarrow \dot {\mathcal D}'(Z),
\end{equation}
written formally as \eqref{II}, as the adjoint of $A^\ast$. In this way, the oscillatory integral \eqref{II} is identified with a separately continuous bilinear mapping
\begin{equation*}
{\rm S}^\infty_{la}(Z\times {\mathbb R}^n) \times \dot {\mathcal E}'(Z) \longrightarrow \dot {\mathcal D}'(Z).
\end{equation*}
The space $\L^l_b(Z)$ of \emph{totally characteristic pseudodifferential operators on $Z$ of order $l$} consists of those continuous linear maps \eqref{VIII} such that for any $u,v \in {\rm C^{\infty}_c}(Z)$, $v Au$ is of the form \eqref{II} with $a(x,\xi)=\widetilde{a}(x_1,x',x_1\xi_1,\xi ')$ and $\widetilde a(x,\xi)\in {\rm S}^l_{la}(Z\times {\mathbb R}^n)$. Similarly,
a continuous linear map \eqref{IIIa} on a smooth manifold $\bf X$ with boundary $\gd \bf X$ is said to be an element of the space $\L^l_{b}({\bf X})$ of \emph{totally characteristic pseudodifferential operators on ${\bf X} $ of order $l$}, if for a given atlas $(\kappa_\gamma,\widetilde U^\gamma)$ the operators $A^\gamma u=[A_{|\widetilde U^\gamma} (u \circ \kappa_\gamma)] \circ \kappa^{-1}_\gamma$ are elements of $\L^l_b(Z)$,
where the $\widetilde U^\gamma$ are coordinate patches isomorphic to subsets in $Z$.
In an analogous way, it is possible to introduce the concept of a totally characteristic pseudodifferential operator on a manifold with corners. As the standard manifold with corners, consider
\begin{equation*}
{\mathbb R}^{n,k}=[0,\infty)^k \times {\mathbb R}^{n-k}, \qquad 0 \leq k \leq n,
\end{equation*}
with coordinates $x=(x_1,\dots, x_k, x')$. A \emph{totally characteristic pseudodifferential operator on ${\mathbb R}^{n,k}$ of order $l$} is locally given by an oscillatory integral \eqref{II} with $a(x,\xi) =\widetilde a (x, x_1 \xi_1,\dots, x_k \xi_k, \xi')$, where now $\widetilde a(x,\xi)$ is a symbol of order $l$ that satisfies the lacunary condition for each of the coordinates $x_1,\dots, x_k$, i.e.
\begin{equation*}
\int e^{i(1-t)\xi_j} a(x, \xi) \d \xi_j =0 \qquad \text{for } t<0 \text{ and } 1\leq j \leq k.
\end{equation*}
In this case we write $\widetilde a(x,\xi) \in {\rm S}^l_{la}({\mathbb R}^{n,k}\times {\mathbb R}^n)$.
A continuous linear map \eqref{IIIa} on a smooth manifold $\bf X$ with corners is then said to be an element of the space $\L^l_{b}({\bf X})$ of \emph{totally characteristic pseudodifferential operators on ${\bf X} $ of order $l$}, if for a given atlas $(\kappa_\gamma,\widetilde U^\gamma)$ the operators $A^\gamma u=[A_{|\widetilde U^\gamma} (u \circ \kappa_\gamma)] \circ \kappa^{-1}_\gamma$ are totally characteristic pseudodifferential operator on ${\mathbb R}^{n,k}$ of order $l$, where the $\widetilde U^\gamma$ are coordinate patches isomorphic to subsets in ${\mathbb R}^{n,k}$. For an extensive treatment, we refer the reader to \cite{loya98}.
\section{Invariant integral operators}
\label{Sec:4}
Let $\widetilde {\mathbb X}$ be the Oshima compactification of a Riemannian symmetric space ${\mathbb X}\simeq G/K$ of non-compact type. As was already explained, $G$ acts analytically on $\widetilde {\mathbb X}$, and the orbital decomposition is of normal crossing type. Consider the Banach space $\mathrm{C}(\widetilde {\mathbb X})$ of continuous, complex valued functions on $\widetilde {\mathbb X}$, equipped with the supremum norm, and let $(\pi,\mathrm{C}(\widetilde {\mathbb X}))$ be the corresponding continuous regular representation of $G$ given by
\begin{equation*}
\pi(g) \phi(\tilde x) =\phi(g^{-1} \cdot \tilde x), \qquad \phi \in \mathrm{C}(\widetilde {\mathbb X}).
\end{equation*}
The representation of the universal enveloping algebra ${\mathfrak U}$ of the complexification ${\bf \mathfrak g}_{\mathbb C}$ of ${\bf \mathfrak g}$ on the space of differentiable vectors $\mathrm{C}(\widetilde {\mathbb X})_\infty$ will be denoted by $d\pi$. We will also consider the regular representation of $G$ on ${\rm C^{\infty}}(\widetilde {\mathbb X})$ which, equipped with the topology of uniform convergence on compact subsets, becomes a Fr\'{e}chet space. This representation will be denoted by $\pi$ as well. Let $(L,{\rm C^{\infty}} (G))$ be the left regular representation of $G$. With respect to the left-invariant metric on $G$ given by $\langle,\rangle_{\theta}$, we define $d(g,h)$ as the distance between two points $g,h \in G$, and set $|g|=d(g,e)$, where $e$ is the identity element of $G$. A function $f$ on $G$ is \emph{at most of exponential growth}, if there exists a $\kappa>0$ such that $|f(g)| \leq C e^{\kappa|g|}$ for some constant $C>0$.
As before, denote a Haar measure on $G$ by $d_{G}$. Consider next the space $\S(G)$ of rapidly decreasing functions on $G$ introduced in \cite{ramacher06}.
\begin{definition}
The space of rapidly decreasing functions on $G$, denoted by $\S(G)$, is given by all functions $f \in {\rm C^{\infty}}(G)$ satisfying the following conditions:
\begin{itemize}
\item[i)] For every $\kappa \geq 0$, and $X \in {\mathfrak U}$, there exists a constant C such that
$$|dL(X)f(g)| \leq C e^{-\kappa |g|} ;$$
\item[ii)] for every $\kappa \geq 0$, and $X \in {\mathfrak U}$, one has $dL(X)f \in \L^1(G,e^{\kappa|g|}d_G)$.
\end{itemize}
\end{definition}
For later purposes, let us recall the following integration formulas.
\begin{proposition}
\label{prop:A}
Let $f_1\in\S(G)$, and assume that $f_2 \in {\rm C^{\infty}}(G)$, together with all its derivatives, is at most of exponential growth. Let $X_1, \dots, X_d$ be a basis of ${\bf \mathfrak g}$, and for $X^\gamma=X^{\gamma_1}_{i_1}\dots X^{\gamma_r}_{i_r}$ write $X^{\tilde \gamma}=X^{\gamma_r}_{i_r}\dots X^{\gamma_1}_{i_1}$, where $\gamma$ is an arbitrary multi-index. Then
\begin{equation*}
\int_{G}f_1(g) dL(X^\gamma) f_2(g) d_{G}(g)=(-1)^{|\gamma|} \int _{G}
dL(X^{\tilde \gamma}) f_1(g) f_2(g) d_{G}(g).
\end{equation*}
\end{proposition}
\begin{proof}
See \cite{ramacher06}, Proposition 1.
\end{proof}
Next, we associate to every $f\in \S(G)$ and $\phi \in \mathrm{C}(\widetilde {\mathbb X})$ the element $\int _{G} f(g)
\pi(g) \phi \d_{G}(g)\in \mathrm{C}(\widetilde {\mathbb X})$. It is defined as a Bochner integral, and the continuous linear operator on $\mathrm{C}(\widetilde {\mathbb X} )$ obtained this way is denoted by \eqref{eq:1}. Its restriction to ${\rm C^{\infty}}(\widetilde {\mathbb X})$ induces a continuous linear operator
\begin{equation*}
\pi(f):{\rm C^{\infty}}(\widetilde {\mathbb X}) \longrightarrow {\rm C^{\infty}}(\widetilde {\mathbb X}) \subset {\mathcal D}'(\widetilde {\mathbb X}),
\end{equation*}
with Schwartz kernel given by the distribution section $\mathcal{K}_f \in {\mathcal D}'(\widetilde {\mathbb X} \times \widetilde {\mathbb X}, {{\bf 1}} \boxtimes \Omega_{\widetilde {\mathbb X}})$. The properties of the Schwartz kernel $\mathcal{K}_f$ will depend on the analytic properties of $f$, as well as the orbit structure of the underlying $G$-action, and our main effort will be directed towards the elucidation of the structure of $\mathcal{K}_f$. For this, let us consider the orbital decomposition \eqref{eq:decomp} of $\widetilde {\mathbb X}$, and remark that the restriction of $\pi(f) \phi$ to any of the connected components isomorphic to $G/P_\Theta(K)$ depends only on the restriction of $\phi\in \mathrm{C}(\widetilde {\mathbb X})$ to that component, so that we obtain the continuous linear operators
\begin{equation*}
\pi(f)_{|\widetilde {\mathbb X}_\Theta}:{\rm C^{\infty}_c}(\widetilde {\mathbb X}_\Theta) \longrightarrow {\rm C^{\infty}}(\widetilde {\mathbb X}_\Theta),
\end{equation*}
where $\widetilde {\mathbb X}_\Theta$ denotes a component in $\widetilde {\mathbb X}$ isomorphic to $G/P_\Theta(K)$. Let us now assume that $\Theta=\Delta$, so that $P_\Theta(K)=K$. Since $G$ acts transitively on $\widetilde {\mathbb X}_{\Delta}$ one deduces that $\pi(f)_{|\widetilde {\mathbb X}_\Delta} \in \L^{-\infty}(\widetilde {\mathbb X}_{\Delta})$, c.p. \cite{ramacher06}, Section 4. The main goal of this section is to prove that the restrictions of the operators $\pi(f)$ to the manifolds with corners $\overline{\widetilde {\mathbb X}_{\Delta}}$ are totally characteristic pseudodifferential operators of class $\L^{-\infty}_b$.
Let $\mklm{(\widetilde U_{m_w}, \phi_{m_w}^{-1})}_{w\in W}$ be the finite atlas on the Oshima compactification $\widetilde {\mathbb X}$ defined earlier. For each $\tilde x \in \widetilde {\mathbb X}$, let $\widetilde W_{\tilde x}$ be an open neighborhood of $\tilde x$ contained in some $\widetilde U_{m_w}$ such that $\mklm{h \in G: h \widetilde W_{\tilde x} \subset \widetilde {U}_{m_w}}$
acts transitively on the $G$-orbits of $\widetilde{W}_{\tilde x}$, c.p. \cite{ramacher06}, Section 6. We obtain a finite atlas $\mklm{(\widetilde{W}_\gamma, \phi^{-1}_{m_{w_\gamma}})}_{\gamma \in I}$ of $\widetilde {\mathbb X}$ satisfying the following properties:
\begin{itemize}
\item[i)] For each $\widetilde{W}_\gamma$, there exist open sets $V_\gamma \subset V_\gamma^1 \subset G$, stable under inverse, that act transitively on the $G$-orbits of $\widetilde{W}_\gamma$;
\item[ii)] For all $\gamma \in I$ one has $V_\gamma^1 \cdot \widetilde{W}_\gamma \subset \widetilde{U}_{m_{w_\gamma}}$ for some $m_{w_\gamma}\in M^*$.
\end{itemize}
To simplify notation, we shall write $\phi_\gamma$ instead of $ \phi_{m_{w_\gamma}}$.
Consider now the localization of the operators $\pi(f)$ with respect to the finite atlas $\mklm{(\widetilde{W}_\gamma, \phi^{-1}_\gamma)}_{\gamma \in I}$ given by
\begin{equation*}
A_{f}^{\gamma} u=[\pi(f)_{|\widetilde W_{\gamma}}(u\circ \phi_{\gamma}^{-1})]\circ \phi_{\gamma}, \qquad u\in{\rm C^{\infty}_c}(W_{{\gamma}}), \, W_{\gamma}=\phi^{-1}_{\gamma}(\widetilde W_\gamma),
\end{equation*}
see Section \ref{sec:PDO}. Writing $\phi_{\gamma}^{g}= \phi_{\gamma}^{-1}\circ g^{-1}\circ \phi_{\gamma}$ and $x=(n,t)\in {W}_\gamma$ we obtain
\begin{equation*}
A_{f}^{\gamma} u(x)=\int_{G}f(g)\pi(g)(u\circ \phi_{\gamma}^{-1})(\phi_{\gamma}(x))dg=\int_{G}f(g)(u\circ \phi_{\gamma}^{g})(x)dg.
\end{equation*}
Since we can restrict the domain of integration to $V_\gamma$, the latter integral can be rewritten as
\begin{equation*}
A_{f}^{\gamma} u(x)=\int_{G}c_{\gamma}(g) f(g)(u\circ \phi_{\gamma}^{g})(x)dg,
\end{equation*}
where $c_{\gamma}$ is a smooth bounded function on $G$ with support in $V_\gamma^1$ such that $c_{\gamma}\equiv 1$ on $V_\gamma$. Define next
\begin{equation}
\hat{f}_\gamma(x,\xi)=\int_{G}e^{i\phi_{\gamma}^{g}(x)\cdot\xi} c_{\gamma}(g)f(g) dg, \qquad a_{f}^{\gamma}(x,\xi)=e^{-ix\cdot \xi}\hat{f}_\gamma(x,\xi).
\end{equation}
Differentiating under the integral we see that $\hat{f}_\gamma(x,\xi),a_{f}^{\gamma}(x,\xi) \in{\rm C^{\infty}}(W_{{\gamma}} \times {\mathbb R}^{k+l})$.
Let us next state the following
\begin{lemma}
\label{lemma:expansion}
For any $\tilde x=\phi_{\gamma} (n,t) \in \widetilde{W}_\gamma$ and $g\in V_\gamma^1$ we have the power series expansion
\begin{equation}
\label{eq:expansion}
t_j(g\cdot \tilde x)=\displaystyle\sum_{\substack{\alpha,\beta \\ \beta_{j} \neq 0}} c^{j}_{\alpha,\beta}(g)n^{\alpha}(\tilde x)t^{\beta}(\tilde x), \qquad j=1,\dots, l,
\end{equation}
where the coefficients $c^{j}_{\alpha,\beta}(g)$ depend real-analytically on $g$, and $\alpha, \beta$ are multi-indices.
\end{lemma}
\begin{proof}
By Theorem \ref{Thm.1}, a $G$-orbit in $\widetilde {\mathbb X}$ is locally determined by the signature of any of its elements. In particular, for $\tilde x \in \widetilde W_\gamma$, $ g\in V_\gamma^1$ we have $\sgn{t_j(g\cdot \tilde x)}=\sgn{t_j(\tilde x)}$ for all $j=1,\dots, l$. Hence, $t_j(g\cdot \tilde x)=0$ if and only if $t_j(\tilde x)=0$. Now, due to the analyticity of the coordinates $(\phi_{\gamma}, \widetilde W_\gamma)$, there is a power series expansion
\begin{equation*}
t_j(g\cdot \tilde x)=\displaystyle\sum_{\alpha,\beta } c^{j}_{\alpha,\beta}(g)n^{\alpha}(\tilde x)t^{\beta}(\tilde x), \qquad \tilde x\in \widetilde{W}_\gamma, \, g\in V_\gamma^1,
\end{equation*}
for every $j=1,\dots,l$, which can be rewritten as
\begin{equation}\label{eqn.10}
t_j(g\cdot \tilde x)=\displaystyle\sum_{\substack{\alpha,\beta \\ \beta_{j} \neq 0}} c^{j}_{\alpha,\beta}(g)n^{\alpha}(\tilde x)t^{\beta}(\tilde x) + \displaystyle\sum_{\substack{\alpha,\beta \\ \beta_{j}= 0}} c^{j}_{\alpha,\beta}(g)n^{\alpha}(\tilde x)t^{\beta}(\tilde x).
\end{equation}
Suppose $t_j(\tilde x)=0$. Then the first summand of the last equation must vanish, as in each term of the summation a non-zero power of $t_j(\tilde x)$ occurs. Also, $t_j(g\cdot \tilde x) =0$. Therefore \eqref{eqn.10} implies that the second summand must vanish, too. But the latter is independent of $t_j$. So we conclude
\begin{equation*}
\displaystyle\sum_{\substack{\alpha,\beta \\ \beta_{j}= 0}} c^{j}_{\alpha,\beta}(g)n^{\alpha}(\tilde x)t^{\beta}(\tilde x)\equiv{0}
\end{equation*}
for all $\tilde x \in \widetilde{W}_\gamma$, $ g\in V_\gamma^1$, and the assertion follows.
\end{proof}
From Lemma \ref{lemma:expansion} we deduce that
\begin{equation}
\label{eq:powers of t}
t_j(g\cdot \tilde x)=t_j^{q_j}(\tilde x) \chi_j(g,\tilde x), \qquad \tilde x \in \widetilde{W}_\gamma, \, g \in V_\gamma^1,
\end{equation}
where $\chi_j(g,\tilde x)$ is a function that is real-analytic in $g$ and in $\tilde x$, and $q_j$ is the lowest power of $t_j$ that occurs in the expansion \eqref{eq:expansion}, so that
\begin{equation}
\label{eq:notzero}
\chi_j(g,\tilde x) \not= 0 \qquad \forall \, \tilde x \in \widetilde W_\gamma, \, g \in V_\gamma^1.
\end{equation}
Indeed, $\chi_j(g,\tilde x)$ can only vanish if $t_j(\tilde x)=0$. But if this were the case, $q_j$ would not be the lowest power, and we obtain \eqref{eq:powers of t}. Furthermore, since $t_j(g \cdot \tilde x) = t_j(\tilde x)$ for $g=e$, one has $q_1=\dots= q_l$.
Thus, for $\tilde x=\phi_{\gamma} (x) \in \widetilde{W}_\gamma$, $x=(n,t)$, $g \in V_\gamma^1$, we have
\begin{equation*}
\phi^{g}_{\gamma} (x)=(n_1(g\cdot \tilde x),\dots,n_k(g\cdot \tilde x),t_1(\tilde x)\chi_{{}_1}(g,\tilde x),\dots,t_l(\tilde x)\chi_{{}_l}(g,\tilde x)).
\end{equation*}
Note that similar formulas hold for $\tilde x \in \widetilde U_{m_w}$ and $g$ sufficiently close to the identity. The following lemma describes the $G$-action on $\widetilde {\mathbb X}$ as far as the $t$-coordinates are concerned.
\begin{lemma}
\label{lemma:char}
Let $X_{-\lambda,i}$ and $H_j$ the basis elements for ${\bf\mathfrak n}^-$ and $\a$ introduced in Section \ref{Sec:2}, $w \in W$, and $\tilde x \in \widetilde U_{m_{w}}$. Then, for small $s\in {\mathbb R}$,
\begin{equation*}
\chi_{{}_{j}}(\e{sH_i},\tilde x)=e^{-c_{ij}(m_{w})s},
\end{equation*}
where the $c_{ij}(m_{w})$ represent the matrix coefficients of the adjoint representation of $M^\ast$ on $\a$, and are given by $\mathrm{Ad}\,(m_{w}^{-1})H_i=\sum_{j=1}^lc_{ij}(m_{w})H_j$.
Furthermore, when $\tilde x =\pi(e,n,t)$,
$$\chi_{{}_j}(\e{sX_{-\lambda,i}},\tilde x)\equiv 1 .$$
\end{lemma}
\begin{proof}
Let $Y \in {\bf \mathfrak g}$. As we saw in the proof of Lemma \ref{lemma:fundvec}, the action of the one-parameter group $\exp(sY)$ on the homogeneous space $G/P_{\Theta}(K)$ is given by equation
\eqref{eq:action}, where $N_3^-(s) \in{\bf\mathfrak n}^-,A_1(s)\in\a, A_2(s) \in\a(\Theta)$. Denote the derivatives of $N_3^-(s)$, $A_1(s)$, and $A_2(s)$ at $s=0$ by $N_3^-$, $ A_1$, and $A_2$ respectively. The analyticity of the $G$-action implies that $N_3^-(s),A_1(s),A_2(s)$ are real-analytic functions in $s$. Furthermore, from \eqref{eq:action} it is clear that $N_3^-(0)=0$, $A_1(0)+A_2(0)=0$, so that for small $s$ we have
\begin{align*}
A_1(s)+A_2(s)&=( A_1+A_2)\, s+ \frac{1}{2}\frac{d^2}{d s^{2}}(A_1(s)+A_2(s))|_{s=0} \, s^2+\dots \\
N_3^-(s)&=N_3^- \, s+\frac{1}{2}\frac{d^2}{d s^{2}}N_3^-(s)|_{s=0} \, s^2+\dots.
\end{align*}
Next, fix $m_{w} \in M^*$ and let $\Theta = \Delta$. The action of the one-parameter group corresponding to $H_i$ at $\tilde x =\pi(m_{w},n,t) \in \widetilde U_{m_{w}} \cap \widetilde {\mathbb X}_\Delta $ is given by
\begin{align*}
\exp(sH_i)m_{w}naK&=
m_{w}\left(m_{w}^{-1}\exp(sH_i)m_{w}\right)naK= m_{w} \exp(s\mathrm{Ad}\,(m_{w}^{-1})H_i)naK.
\end{align*}
As $m_{w}$ lies in $M^*$, $\exp(s\mathrm{Ad}\,(m_{w}^{-1})H_i)$ lies in $A$. Since $A$ normalizes $N^-$, we conclude that $\exp(s\mathrm{Ad}\,(m_{w}^{-1})H_i)n\exp(-s\mathrm{Ad}\,(m_{w}^{-1})H_i)$ belongs to $N^-$. Writing $$n^{-1}\exp(s\mathrm{Ad}\,(m_{w}^{-1})H_i)n\exp(-s\mathrm{Ad}\,(m_{w}^{-1})H_i)=\exp N_3^-(s)$$
we get
$$\exp(sH_i)m_{w}naK=m_{w}n\exp N_3^-(s)a\exp(s\mathrm{Ad}\,(m_{w}^{-1})H_i)K.$$
In the notation of \eqref{eq:action} we therefore obtain $A_1(s)+A_2(s)=s\mathrm{Ad}\,(m_{w}^{-1})H_i$, and by writing $\mathrm{Ad}\,(m_{w}^{-1})H_i=\sum_{j=1}^lc_{ij}(m_{w})H_j$ we arrive at
\begin{align*}
a\exp(A_1(s)+A_2(s))&=\exp\Big (\sum_{j=1}^l(c_{ij}(m_{w})s-\log t_j)H_j\Big ).
\end{align*}
In terms of the coordinates this shows that $t_j(\exp(sH_i)\cdot \tilde x)=t_j(\tilde x)e ^{-c_{ij}(m_{w})s}$ for $\tilde x \in \widetilde U_{m_w} \cap \widetilde {\mathbb X}_\Delta$, and by analyticity we obtain that $\chi_{{}_{j}}(\e{sH_i},\tilde x)=e^{-c_{ij}(m_{w})s}$ for arbitrary $\tilde x \in \widetilde U_{m_w}$.
On the other hand, let $Y=X_{-\lambda,i}$, and $\tilde x =\phi_e(n,t)\in \widetilde U_e\cap \widetilde {\mathbb X}_\Delta$. Then the action corresponding to $X_{-\lambda,i}$ at $\tilde x $ is given by
\begin{align*}
\exp(sX_{-\lambda,i})naK=n\exp N_3^-(s)aK,
\end{align*}
where we wrote $\exp N_3^-(s)=s \mathrm{Ad}\,(n^{-1})\exp X_{-\lambda,i}$. In terms of the coordinates this implies that $t_j(\exp(sX_{-\lambda,i})\cdot \tilde x)=t_j(\tilde x)$ showing that $\chi_{{}_j}(\e{sX_{-\lambda,i}},\tilde x)\equiv 1$ for $\tilde x \in \widetilde U_e \cap \widetilde {\mathbb X}_\Delta$, and, by analyticity, for general $\tilde x \in \widetilde U_e$, finishing the proof of the lemma.
\end{proof}
Let now $x=(n,t)\in W_\gamma$, and define the matrix
\begin{equation}
T_x = \begin{pmatrix} t_1& & 0\\& \ddots &\{\rm 0}& & t_l\end{pmatrix},
\end{equation}
so that for $\tilde x=\phi_{\gamma} (x) \in \widetilde{W}_\gamma$, $\, g \in V_\gamma^1$,
\begin{equation*}
({\bf 1}_k\otimes T^{-1}_{x})(\phi^{g}_{\gamma} (x))=(x_1(g\cdot \tilde x),\dots,x_k(g\cdot \tilde x),\chi_{{}_1}(g,\tilde x),\dots,\chi_{{}_l}(g,\tilde x)),
\end{equation*}
and set
\begin{equation*}
\psi^\gamma_{\xi,x}(g)=e^{i({\bf 1}_k\otimes T^{-1}_{x})(\phi^{g}_{\gamma} (x))\cdot\xi},
\end{equation*}
where $\xi=(\xi_1,\dots,\xi_{k+l})\in{\mathbb R}^{k+l}$. Also, introduce the auxiliary symbol
\begin{equation}
\label{eq:auxsym}
\tilde{a}_f^{\gamma}(x,\xi)=a_{f}^{\gamma}(x,({\bf 1}_k\otimes T^{-1}_{x})\xi)=e^{-i(x_1,\dots,x_k,1,\dots,1).\xi}\int_{G}\psi^\gamma_{\xi,x}(g) c_{\gamma}(g) f(g) dg.
\end{equation}
Clearly, $\tilde{a}_f^{\gamma}(x,\xi)\in{\rm C^{\infty}}(W_{{\gamma}} \times {\mathbb R}^{k+l})$. Our next goal is to show that $\tilde{a}_f^{\gamma}(x,\xi)$ is a lacunary symbol. To do so, we shall need the following
\begin{proposition}
\label{prop:1}
Let $(L,{\rm C^{\infty}} (G))$ be the left regular representation of $G$. Let $X_{-\lambda,i},H_j $ be the basis elements of ${\bf\mathfrak n}^-$ and $\a$ introduced in Section \ref{Sec:2}, and $(\widetilde W_\gamma, \phi_{\gamma})$ an arbitrary chart. With $x=(n,t)\in W_\gamma$, $\tilde x=\phi_{\gamma} (x) \in \widetilde{W}_\gamma$, $\, g \in V_\gamma^1$ one has
\begin{equation}
\label{eq:23}
\begin{pmatrix} dL(X_{-\lambda,1})\psi^\gamma_{\xi,x}(g)\\ \vdots\\ dL(H_l)\psi^\gamma_{\xi,x}(g)\end{pmatrix} =i\psi^\gamma_{\xi,x}(g)\Gamma(x,g)\xi,
\end{equation}
with
\begin{equation}
\label{eq:Gamma}
\Gamma(x,g)=
\left(\begin{array}{cc}
\Gamma_1 & \Gamma_2 \\
\Gamma_3 &\Gamma_4\\
\end{array}\right)
=
\left(\begin{array}{cccc}
dL(X_{-\lambda,i})n_{j,\tilde x}(g)& \multicolumn{1}{c|}{} & &dL(X_{-\lambda,i})\chi_j(g,\tilde x) \\
& \multicolumn{1}{c|}{} & &\\
\cline{1-4} & \multicolumn{1}{c|}{} & &\\
dL(H_i)n_{j,\tilde x}(g) & \multicolumn{1}{c|}{} & &dL(H_i)\chi_j(g,\tilde x)\\
\end{array}\right)
\end{equation}
belonging to $\mathrm{GL}(l+k,{\mathbb R})$, where $n_{j,\tilde x}(g)=n_j(g\cdot \tilde x)$.
\end{proposition}
\begin{proof}
Fix a chart $(\widetilde W_\gamma, \phi_{\gamma})$, and let $x$, $\tilde x$, $ g$ be as above. For $X\in{\bf \mathfrak g}$, one computes that
\begin{align*}
dL(X)\psi^\gamma_{\xi,x}(g)&=\frac{d}{ds}e^{i({\bf 1}_k\otimes T^{-1}_{t})\phi^{\e{-sX}g}_{\gamma} (x)\cdot\xi}|_{s=0}=i\psi^\gamma_{\xi,x}(g)\Big [\sum_{{i}=1}^{k}\xi_{i}dL(X)n_{i,\tilde x}(g)\\ &+\sum_{{j}=k+1}^{l+k}\xi_{j}dL(X)\chi_{j}(g,\tilde x)\Big ],
\end{align*}
showing the first equality. To see the invertibility of the matrix $\Gamma(x,g)$, note that for small $s$
\begin{equation*}
\chi_j(\e{-sX}g, \tilde x) = \chi_j(g,\tilde x) \chi_j (\e{-sX},g\cdot \tilde x).
\end{equation*}
Lemma \ref{lemma:char} then yields
\begin{align*}
dL(H_i) \chi_j(g,\tilde x)&= \chi_j(g,\tilde x) \frac d {ds} \Big ( e^{c_{ij}(m_{w_\gamma})s} \Big )_{|s=0} =\chi_j(g,\tilde x)c_{ij}(m_{w_\gamma}).
\end{align*}
This means that $\Gamma_4$ is the product of the matrix $\left (c_{ij}(m_{w_\gamma})\right )_{i,j}$ with the diagonal matrix whose $j$-th diagonal entry is $\chi_j(g,\tilde x)$. Since $\left (c_{ij}(m_{w_\gamma})\right )_{i,j}$ is just the matrix representation of $\mathrm{Ad}\,(m_{w_\gamma}^{-1}) $ relative to the basis $\{H_1,\dots, H_l\}$ of $\a$, it is invertible. On the other hand, $\chi_j(g,\tilde x)$ is non-zero for all $j \in \{1,\dots,l\}$ and arbitrary $g $ and $ \tilde x$. Therefore $\Gamma_4$, being the product of two invertible matrices, is invertible. Next, let us show that the matrix $\Gamma_1$ is non-singular. Its $(ij)^{th}$ entry reads
\begin{equation*}
dL(X_{-\lambda,i})n_{j,\tilde x}(g)=\frac d{ds} n_{j,\tilde x}(\e{-sX_{-\lambda,i}}\cdot g )_{|s=0}=(-X_{-\lambda,{i}|\widetilde{{\mathbb X}}})_{g\cdot \tilde x} (n_{j}).
\end{equation*}
For $\Theta\subset\Delta$, $q\in {\mathbb R}^l$, we define the $k$-dimensional submanifolds
\begin{equation*}
\mathfrak{L}_{\Theta}(q)=\{ \tilde x=\phi_{\gamma}(n,q)\in \widetilde W_{\gamma}: q_i\neq 0 \Leftrightarrow \alpha_i \in \Theta \},
\end{equation*}
and consider the decomposition $T_{g\cdot \tilde x}\widetilde{{\mathbb X}}_{\Theta}=T_{g\cdot \tilde x}\mathfrak{L}_{\Theta}(q)\oplus N_{g\cdot \tilde x}\mathfrak{L}_{\Theta}(q)$ of $T_{g\cdot \tilde x}\widetilde{{\mathbb X}}_{\Theta}$ into the tangent and normal space to $\mathfrak{L}_{\Theta}(q)$ at the point $g\cdot \tilde x \in \widetilde{{\mathbb X}}_{\Theta}$.
Since $\widetilde{{\mathbb X}}_{\Theta}$ is a $G$-orbit, the group $G$ acts transitively on it. Now, as $g$ varies over $G$ in Lemma \ref{lemma:fundvec}, one deduces that $N^{-}\times A(\Theta)$ acts locally transitively on $\widetilde{{\mathbb X}}_{\Theta}$. In addition, by the definition of $\mathfrak{L}_{\Theta}(q)$, $N_{g\cdot \tilde x}\mathfrak{L}_{\Theta}(q)$ is spanned by the vector fields $\{ -t_i\frac{\partial}{\partial t_i}\}_{ \alpha_i \in\Theta}$. Consequently, $T_{g\cdot \tilde x}\mathfrak{L}_{\Theta}(q)$ must be equal to the span of the vector fields $\{X_{-\lambda,{i}|\widetilde{{\mathbb X}}}\}$, which means that $N^-$ acts locally transitively on $\mathfrak{L}_{\Theta}(q)$ for arbitrary $\Theta$. Since the latter is parametrized by the coordinates $(n_1,\dots,n_k)$, one concludes that the matrix $((X_{-\lambda,{i}|\widetilde{{\mathbb X}}})_{g \cdot \tilde x}(n_j))_{ij}$ has full rank. Thus, $\Gamma_1$ is non-singular. On the other hand, if $\tilde x =\pi(e,n,t)\in \tilde U_e$, Lemma \ref{lemma:char} implies
\begin{equation*}
dL(X_{-\lambda,i}) \chi_j(g,\tilde x)= \chi_j(g,\tilde x) \frac {d} {ds}\Big ( \chi_j(e^{-s X_{-\lambda,i}}, g \cdot \tilde x) \Big )_{|s=0} =0, \\
\end{equation*}
showing that $\Gamma_2$ is identically zero, while $\Gamma_4$ is a non-singular diagonal matrix in this case. Geometrically, this amounts to the fact that the fundamental vector field corresponding to $H_j$ is transversal to the hypersurface defined by $t_j=q\in {\mathbb R}\setminus \{0\}$, while the vector fields corresponding to the Lie algebra elements $X_{-\lambda,i}, H_i, \, i\neq j$, are tangential. We therefore conclude that $\Gamma(x,g)$ is non-singular if $\tilde x \in \widetilde U_e$. But since the different copies $\widetilde {\mathbb X}_{\Theta_{(e,n,t)}}$ of $G/P_{\Theta_{(e,n,t)}} (K)\simeq N^- \times B_{(e,n,t)}\subset N^- \times {\mathbb R}^l \simeq \widetilde U_e$ in $\widetilde {\mathbb X}$ are isomorphic to each other, the same must hold if $\tilde x$ lies in one of the remaining charts $\widetilde U_{m_{w_\gamma}}$, and the assertion of the lemma follows.
\end{proof}
We can now state the main result of this paper. In what follows, $\{(\widetilde W_\gamma, \phi_{\gamma})\}_{\gamma \in I}$ will always denote the atlas of $\widetilde {\mathbb X}$ constructed above.
\begin{theorem}
\label{thm:3}
Let $\widetilde {\mathbb X}$ be the Oshima compactification of a Riemannian symmetric space ${\mathbb X}\simeq G/K$ of non-compact type, and $f\in \S({G})$ a rapidly decaying function on $G$. Let further $\mklm{( \widetilde W_\gamma, \phi_\gamma^{-1})}_{\gamma \in I}$ be the atlas of $\widetilde {\mathbb X}$ construced above. Then the operators $\pi(f)$ are locally of the form
\begin{gather}
\label{20}
A^\gamma_fu(x)= \int e ^{i x \cdot \xi} a_f^\gamma(x,\xi)\hat u(\xi) {\,\raisebox{-.1ex}{\={}}\!\!\!\!d}\xi, \qquad u \in {\rm C^{\infty}_c}(W_\gamma),
\end{gather}
where $a_f^\gamma(x,\xi)=\tilde a_f^\gamma(x,\xi_1, \dots, \xi_k, x_{k+1} \xi_{k+1}, \dots, \xi_{k+l} x_{k+l})$, and $\tilde a_f^\gamma(x,\xi) \in {\rm S^{-\infty}_{la}}(W_\gamma \times {\mathbb R}^{k+l}_\xi)$ is given by \eqref{eq:auxsym}. In particular, the kernel of the operator $A^\gamma_f$ is determined by its restrictions to $W_\gamma^\ast \times W_\gamma^\ast$, where $W_\gamma^\ast=\{ x=(n,t) \in W_\gamma: t_1 \cdots t_l \not=0\}$, and given by the oscillatory integral
\begin{equation}
\label{20b}
K_{A_f^\gamma} (x,y)=\int e^{i(x-y) \cdot \xi} a^\gamma_f(x,\xi) {\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi.
\end{equation}
\end{theorem}
As a consequence, we obtain the following
\begin{corollary}
\label{corollary}
Let $\widetilde {\mathbb X}_\Delta$ be an open $G$-orbit in $\widetilde {\mathbb X}$ isomorphic to $G/K$. Then the continuous linear operators
\begin{equation*}
\pi(f)_{|\overline{\widetilde {\mathbb X}_\Delta}}:{\rm C^{\infty}_c}(\overline{\widetilde {\mathbb X}_\Delta}) \longrightarrow {\rm C^{\infty}}(\overline{\widetilde {\mathbb X}_\Delta}),
\end{equation*}
are totally characteristic pseudodifferential operators of class $\L^{-\infty}_b$ on the manifolds with corners $\overline{\widetilde {\mathbb X}_\Delta}$.
\end{corollary}
\qed
\begin{proof}[Proof of Theorem \ref{thm:3}]
Our considerations will essentially follow the proof of Theorem 4 in \cite{ramacher06}.
Let $\Gamma(x,g)$ be the matrix defined in \eqref{eq:Gamma}, and consider its extension as an endomorphism in ${\mathbb C}^1[{\mathbb R}^{k+l}_\xi]$ to the symmetric algebra ${\rm{S}}({\mathbb C}^1[{\mathbb R}^{k+l}_\xi])\simeq {\mathbb C}[{\mathbb R}^{k+l}_\xi]$. Since for $x \in W_\gamma$, $g \in V^1_\gamma$, $\Gamma(x,g)$ is invertible, its extension to $ {\rm{S}}^N({\mathbb C}^1[{\mathbb R}^{k+l}_\xi])$ is also an automorphism for any $N\in{\mathbb N}$. Regarding the polynomials $\xi_1,\dots,\xi_{k+l}$ as a basis in ${\mathbb C}^1[{\mathbb R}^{k+l}_\xi]$, let us denote the image of the basis vector $\xi_j$ under the endomorphism $\Gamma(x,g)$ by $\Gamma \xi_j$, so that by \eqref{eq:23}
\begin{align*}
\Gamma \xi_j&= -i \psi^\gamma_{-\xi,x}(g) dL(X_{-\lambda,j})\psi^\gamma_{\xi,x}(g), \qquad 1\leq j \leq k, \\
\Gamma \xi_j&= -i \psi^\gamma_{-\xi,x}(g) dL(H_j) \psi^\gamma_{\xi,x}(g), \qquad k+1\leq j \leq k+l.
\end{align*}
Every polynomial $\xi_{j_1} \otimes \dots \otimes \xi_{j_N}\equiv \xi_{j_1} \dots \xi_{j_N}$ can then be written as a linear combination
\begin{equation}
\label{24}
\xi^\alpha =\sum _\beta \Lambda^\alpha_\beta (x,g) \Gamma \xi_{\beta_1} \cdots \Gamma \xi_{\beta_{|\alpha|}},
\end{equation}
where the $\Lambda^\alpha_\beta(x,g)$ are real-analytic functions on $W_\gamma \times V^1_\gamma$. We need now the following
\begin{lemma}
\label{lem:3}
For arbitrary indices $\beta_1,\dots, \beta_r$, one has
\begin{align}
\label{25}
\begin{split}
i^r \psi^\gamma_{\xi,x}(g) \Gamma\xi_{\beta_1} \cdots \Gamma\xi_{\beta_r}&= dL(X_{\beta_1} \cdots X_{\beta_r}) \psi^\gamma_{\xi,x}(g)\\&+ \sum_{s=1}^{r-1} \sum _{\alpha_1,\dots, \alpha_s} d ^{\beta_1,\dots, \beta_r}_{\alpha_1,\dots, \alpha_s} (x ,g) dL(X_{\alpha_1} \cdots X_{\alpha_s}) \psi^\gamma_{\xi,x}(g),
\end{split}
\end{align}
where the coefficients $ d ^{\beta_1,\dots, \beta_r}_{\alpha_1,\dots, \alpha_s} (x ,g) \in {\rm C^{\infty}}(\tilde W_\gamma \times \supp c_\gamma)$ are at most of exponential growth in $g$, and independent of $\xi$.
\end{lemma}
\begin{proof}
The lemma is proved by induction. For $r=1$ one has $i \psi^\gamma_{\xi,x}(g) \Gamma \xi_p =d L(X_p) \psi^\gamma_{\xi,x}(g)$, where $1 \leq p \leq d$. Differentiating the latter equation with respect to $X_j$, and writing $\Gamma \xi_p = \sum _{s=1}^{k+l} \Gamma_{ps} (x,g) \, \xi_s$, we obtain with \eqref{24} the equality
\begin{gather*}
-\psi^\gamma_{\xi,x}(g) \Gamma \xi_j \Gamma \xi_p = dL(X_jX_p) \psi^\gamma_{\xi,x}(g)-\sum_{s,r=1}^{k+l} (dL (X_j) \Gamma_{ps}) (x,g) \Lambda ^s_r(x,g) dL (X_r) \psi^\gamma_{\xi,x}(g).
\end{gather*}
Hence, the assertion of the lemma is correct for $r=1,2$. Now, assume that it holds for $r\leq N$. Setting $r=N$ in equation \eqref{25}, and differentiating with respect to $X_p$, yields for the left hand side
\begin{gather*}
i^{N+1} \psi^\gamma_{\xi,x}(g) \Gamma \xi_p \Gamma\xi_{\beta_1} \cdots \Gamma \xi_{\beta_N} \\+ i^N \psi^\gamma_{\xi,x}(g) \Big ( \sum_{s,q=1}^{k+l} (dL(X_p) \Gamma_{\beta_1s})(x,g) \Lambda^s_{q} (x,g) \Gamma \xi_q \Big ) \Gamma \xi _{\beta_2} \cdots \Gamma \xi_{\beta_N} + \dots.
\end{gather*}
By assumption, we can apply \eqref{25} to the products $\Gamma \xi_q \Gamma\xi_{\beta_2} \cdots\Gamma \xi_{\beta_N}, \dots$ of at most $N$ factors. Since the functions $n_{i,m}(g)$ and $ \chi_j(g,m)$, and consequently the coefficients of $\Gamma(x,g)$, are at most of exponential growth in $g$, the assertion of the lemma follows.
\end{proof}
\noindent
\emph{End of proof of Theorem \ref{thm:3}}.
Let us next show that $\tilde a_f^\gamma(x,\xi) \in {\rm S^{-\infty}}(W_\gamma \times {\mathbb R}^{k+l}_\xi)$. As already noted, $\tilde a_f^\gamma(x,\xi) \in {\rm C^{\infty}}(W_\gamma \times {\mathbb R}^{k+l}_\xi)$. While differentiation with respect to $\xi$ does not alter the growth properties of $\tilde a^\gamma_f(x,\xi)$, differentiation with respect to $x$ yields additional powers in $\xi$.
Now, as an immediate consequence of equations \eqref{24} and \eqref{25}, one computes for arbitrary $N \in {\mathbb N}$
\begin{equation}
\label{26}
\psi^\gamma_{\xi,x}(g)(1+\xi^2)^N= \sum_{r=0}^{2N} \sum_{|\alpha| =r} b^N_\alpha(x,g) d L(X^\alpha) \psi^\gamma_{\xi,x}(g),
\end{equation}
where the coefficients $b^N_\alpha(x,g) \in {\rm C^{\infty}}(W_\gamma \times V^1_\gamma)$ are at most of exponential growth in $g$. Now, $(\gd^\alpha_\xi \gd^\beta_x \tilde a ^\gamma_f)(x,\xi)$ is a finite sum of terms of the form
\begin{equation*}
\xi^\delta e^{-i(x_1,\dots x_k,1, \dots, 1) \cdot \xi} \int_{G} f(g) d_{\delta \beta}(x,g) \psi^\gamma_{\xi,x}(g)c_\gamma(g) dg,
\end{equation*}
the functions $d_{\delta\beta}(x,g) \in {\rm C^{\infty}} (W_\gamma \times V^1_\gamma)$ being at most of exponential growth in $g$. Making use of equation \eqref{26}, and integrating according to Proposition \ref{prop:A}, we finally obtain for arbitrary $\alpha, \beta$ the estimate
\begin{equation*}
|(\gd ^\alpha_\xi \gd ^\beta _x \tilde a_f^\gamma) (x,\xi) | \leq \frac 1 {(1+\xi^2)^N} C_{\alpha,\beta,\mathcal{K}} \qquad x \in \mathcal{K},
\end{equation*}
where $\mathcal{K}$ denotes an arbitrary compact set in $W_\gamma$, and $N\in {\mathbb N}$. This proves that $\tilde a^\gamma_f(x,\xi) \in {\rm S^{-\infty}}(W_\gamma \times {\mathbb R}^{k+l}_\xi)$. Since equation \eqref{20} is an immediate consequence of Fourier inversion formula, it remains to show that $\tilde a_f^\gamma(x,\xi)$ satisfies the lacunary condition \eqref{V} for each of the coordinates $t_i$. Now, it is clear that $a^\gamma_f \in \mathrm{S}^{-\infty} ( W_\gamma ^\ast \times {\mathbb R}^{k+l}_\xi$), since $G$ acts transitively on each $\widetilde {\mathbb X}_{\Delta}$. As a consequence, the Schwartz kernel of the restriction of the operator $A^\gamma_f:{\rm C^{\infty}_c} (W_\gamma) \rightarrow {\rm C^{\infty}}(W_\gamma)$ to $W^\ast_\gamma$ is given by the absolutely convergent integral
\begin{equation*}
\int e^{i(x-y) \cdot \xi} a^\gamma_f(x,\xi) {\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi \in {\rm C^{\infty}}( W_\gamma^\ast \times W_\gamma^\ast).
\end{equation*}
Next, let us write $W_\gamma=\bigcup _{\Theta \subset \Delta} W_\gamma^\Theta$, where $W_\gamma^\Theta=\mklm{x=(n,t): t_i \not=0 \Leftrightarrow \alpha_i \in \Theta}$. Since on $W_\gamma^\Theta$ the function $A_f^\gamma u$ depends only on the restriction of $u \in {\rm C^{\infty}_c}(W_\gamma)$ to $W_\gamma^\Theta$, one deduces that
\begin{equation}
\label{eq:33}
\supp K_{A_f^\gamma} \subset \bigcup _{\Theta \subset \Delta} \overline{W_\gamma^\Theta} \times \overline{W_\gamma^\Theta}.
\end{equation}
Therefore, each of the integrals
\begin{equation*}
\int e ^{i(x_j-y_j) \xi_j} \tilde a^\gamma _f (x,({\bf{1}}_k \otimes T_x)\xi) \d \xi_j,\qquad j=k+1, \dots, k+l,
\end{equation*}
which are smooth functions on $W_\gamma^\ast \times W_\gamma^\ast$, must vanish if $x_j$ and $y_j$ do not have the same sign. With the substitution $r_j=y_j/x_j -1$, $\xi_j x_j =\xi_j'$ one finally arrives at the conditions
\begin{equation*}
\int e^{-ir_j \xi_j} \tilde a^\gamma_f(x,\xi) \d \xi_j =0 \qquad \mbox{ for } r_j < -1, \, x \in W_\gamma^\ast.
\end{equation*}
But since $\tilde a^\gamma_f$ is rapidly decreasing in $\xi$, the Lebesgue bounded convergence theorem implies that these conditions must also hold for $x \in W_\gamma$. Thus, the lacunarity of the symbol $\tilde a_f ^\gamma$ follows. The fact that the kernel $K_{A^\gamma_f}$ must be determined by its restriction to $W_\gamma^\ast \times W_\gamma^\ast$, and hence by the oscillatory integral \eqref{20b}, is now a consequence of \cite{melrose}, Lemma 4.1, completing the proof of Theorem \ref{thm:3}.
\end{proof}
As a consequence of Theorem \ref{thm:3}, we can locally write the kernel of $\pi(f)$ in the form
\begin{align}
\label{27}
\begin{split}
K_{A_f^\gamma}(x,y) &= \int e ^{i(x-y) \cdot \xi} a^\gamma _f (x,\xi) {\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi=\int e^{i(x-y) \cdot ({\bf{1}}_k \otimes T_x^{-1}) \xi} \tilde a_f^\gamma(x,\xi) |\det ({\bf{1}}_k \otimes T_x^{-1})'(\xi)| {\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi\\
&=\frac 1 {|x_{k+1}\cdots x_{k+l}|} \tilde A_f^\gamma(x,x_1-y_1, \dots, 1 - \frac{y_{k+1}}{x_{k+1}}, \dots), \qquad x_{k+1}\cdots x_{k+l} \not=0,
\end{split}
\end{align}
where $\tilde A_f^\gamma(x,y)$ denotes the inverse Fourier transform of $\tilde a_f^\gamma(x,\xi)$,
\begin{equation}\label{eq:34}
\tilde A_f^\gamma(x,y)= \int e ^{i y\cdot \xi} \tilde a_f ^\gamma(x,\xi) \, {\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi.
\end{equation}
Since for $x \in W^\gamma$ the amplitude $\tilde a_f^\gamma(x,\xi)$ is rapidly falling in $\xi$, it follows that $\tilde A_f^\gamma(x,y) \in \S({\mathbb R}^n_y)$, the Fourier transform being an isomorphism on the Schwartz space. Therefore $K_{A^\gamma_f}(x,y)$ is rapidly decreasing as $| x_{j}| \to 0 $ if $x_j\not=y_j$ and $k+1\leq j\leq k+l$. Furthermore, by the lacunarity of $\tilde a ^\gamma_f$, $ K_{A_f^\gamma}(x,y)$ is also rapidly decaying as $| y_{j}| \to 0 $ if $x_j\not=y_j$ and $k+1\leq j\leq k+l$.
\section{Holomorphic semigroup and resolvent kernels}
\label{Sec:5}
In this section, we shall study the holomorphic semigroup generated by a strongly elliptic operator $\Omega$ associated to the regular representation $(\pi, \mathrm{C}(\widetilde {\mathbb X}))$ of $G$, as well as its resolvent. Both the holomorphic semigroup and the resolvent can be characterized as convolution operators of the type considered before, so that we can study them by the methods developed in the previous section. In particular, this will allow us to obtain a description of the asymptotic behavior of the semigroup and resolvent kernels on $\widetilde {\mathbb X}_\Delta\simeq {\mathbb X}$ at infinity.
Let us begin by recalling some basic facts about elliptic operators and parabolic evolution equations on Lie groups, our main reference being \cite{robinson}. Let ${\mathcal G}$ be a Lie group, and $\pi$ a continuous representation of ${\mathcal G}$ on a Banach space ${\mathcal B}$. Let further $X_1,\dots, X_d$ be a basis of the Lie algebra $\mathrm{Lie}({\mathcal G})$ of ${\mathcal G}$, and
\begin{equation*}
\Omega= \sum_{|\alpha| \leq q} c_\alpha \d \pi (X^\alpha)
\end{equation*}
a \emph{strongly elliptic differential operator of order $q$} associated with $\pi$, meaning that for all $\xi \in {\mathbb R}^d$ one has the inequality $\Re (-1)^{q/2} \sum_{|\alpha| =q} c_{\alpha} \xi^\alpha \geq \kappa |\xi|^q$ for some $\kappa >0$. By the general theory of strongly continuous semigroups, its closure generates a strongly continuous holomorphic semigroup of bounded operators given by
\begin{equation*}
S_\tau=\frac 1 {2\pi i} \int _\Gamma e^{\lambda \tau} ( \lambda { \bf 1}+\overline{\Omega})^{-1} d\lambda,
\end{equation*}
where $\Gamma$ is a appropriate path in ${\mathbb C}$ coming from infinity and going to infinity such that $\lambda \notin \sigma(\overline{\Omega})$ for $\lambda \in \Gamma$. Here $|\arg \tau|< \alpha$ for an appropriate $\alpha \in (0,\pi/2]$, and the integral converges uniformly with respect to the operator norm. Furthermore, the subgroup $S_\tau$ can be characterized by a convolution semigroup of complex measures $\mu_\tau$ on ${\mathcal G}$ according to
\begin{equation*}
S_\tau=\int_{\mathcal G} \pi(g) d\mu_\tau(g),
\end{equation*}
$\pi$ being measurable with respect to the measures $\mu_\tau$. The measures $\mu_\tau$ are absolutely continuous with respect to Haar measure $d_{\mathcal G}$ on ${\mathcal G}$, and denoting by $K_\tau (g)\in L^1({\mathcal G},d_{\mathcal G})$ the corresponding Radon-Nikodym derivative, one has
\begin{equation*}
S_\tau=\pi(K_\tau)=\int_{\mathcal G} K_\tau(g) \pi(g) d_{\mathcal G}(g).
\end{equation*}
The function $K_\tau(g)\in \L^1({\mathcal G},d_{\mathcal G})$ is analytic in $\tau$ and $g$,
and universal for all Banach representations. It satisfies the parabolic differential equation
\begin{equation*}
\frac{\gd K_\tau}{\gd \tau} (g) + \sum_{|\alpha| \leq q } c_\alpha \, dL(X^\alpha) K_\tau(g)=0, \qquad \lim_{\tau \to 0 } K_\tau(g) = \delta(g),
\end{equation*}
where $(L, {\rm C^{\infty}}({\mathcal G}))$ denotes the left regular representation of ${\mathcal G}$. As a consequence, $K_\tau$ must be supported on the identity component ${\mathcal G}_0$ of ${\mathcal G}$. It is called the \emph{Langlands kernel} of the holomorphic semigroup $S_\tau$, and satisfies the following $\L^1$- and $\L^\infty$-bounds.
\begin{theorem}
\label{thm:4}
For each $\kappa \geq 0$, there exist constants $a,b,c>0$, and $\omega\geq 0$ such that
\begin{equation}
\label{eq:35}
\int_{{\mathcal G}_0} |dL(X^\alpha)\gd^\beta_\tau K_\tau(g)| e^{\kappa |g|}\d_{{\mathcal G}_0}(g) \leq a b^{|\alpha|}
c^\beta {|\alpha|}!\, \beta!(1+\tau^{-\beta-{|\alpha|}/q })e^{\omega \tau},
\end{equation}
for all $\tau>0$, $\beta=0,1,2,\dots$ and multi-indices $\alpha$. Furthermore,
\begin{equation}
\label{eq:36}
|dL(X^\alpha)\gd^\beta_\tau K_\tau(g)|\leq a b^{|\alpha|}
c^\beta {|\alpha|}!\, \beta!(1+\tau^{-\beta-({|\alpha|}+d+1)/q })e^{\omega \tau}e^{-\kappa |g|},
\end{equation}
for all $g \in {\mathcal G}_0$, where $d=\dim {\mathcal G}_0$, and $q$ denotes the order of $\Omega$.
\end{theorem}
A detailed exposition of these facts can be found in \cite{robinson}, pages 30, 152, 166, and 167. Let now ${\mathcal G}=G$, and $(\pi,{\mathcal B})$ be the regular representation of $G$ on $C(\widetilde {\mathbb X})$. Theorem \ref{thm:4} implies that the Langlands kernel $K_\tau$ belongs to the space $\S(G)$ of rapidly falling functions on $G$. As a consequence of the previous considerations we obtain
\begin{theorem}
\label{thm:heatoperator}
Let $\Omega$ be a strongly elliptic differential operator of order $q$ associated with the regular representation $(\pi,C(\widetilde {\mathbb X}))$, and $S_\tau=\pi(K_\tau)$ the holomorphic semigroup of bounded operators generated by $\overline \Omega$. Then the operators $S_\tau$ are locally of the form \eqref{20} with $f$ being replaced by $K_\tau$, and totally characteristic pseudodifferential operators of class $\L^{-\infty}_b$ on the manifolds with corners $\overline{\widetilde{\mathbb X}_\Delta}$. Furthermore, on $W_\gamma \times W_\gamma$, the kernel of $S_\tau$ is given by
\begin{align*}
\begin{split}
S^\gamma_\tau(x,y)=K_{A_{K_\tau}^\gamma}(x,y) &= \int e ^{i(x-y) \cdot \xi} a^\gamma _{K_\tau} (x,\xi) {\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi =\frac 1 {|x_{k+1}\cdots x_{k+l}|} \tilde A_{K_\tau}^\gamma(x,({\bf{1}}_k \otimes T_x^{-1})(x-y)),
\end{split}
\end{align*}
where $x_{k+1}\cdots x_{k+l} \not=0$, and $\tilde A_{K_\tau}^\gamma(x,y)$ was defined in \eqref{eq:34}.
In particular, $S^\gamma_\tau(x,y)$ is rapidly falling at infinity as $| x_{j}| \to 0 $, or $| y_{j}| \to 0 $, as long as $x_j\not=y_j$, where $k+1\leq j\leq k+l$. In addition,
\begin{equation}
\label{eq:38z}
|\tilde A_{K_\tau}^\gamma(x,y)| \leq \begin{cases}
c_1 (1+\tau^{-(l+k+1)/q}), & 0 < \tau \leq 1, \\
c_2 e^{\omega \tau}, & 1< \tau,
\end{cases}
\end{equation}
uniformly on compact subsets of $ W_\gamma\times W_\gamma$ for some constants $c_i>0$.
\end{theorem}
\begin{proof}
The first assertions are immediate consequences of Theorem \ref{thm:3}, and its corollary. In order to prove \eqref{eq:38z}, note that for large $N\in {\mathbb N}$ one computes with \eqref{eq:auxsym}, \eqref{26}, and \eqref{eq:34}
\begin{align*}
|\tilde A_{K_\tau}^\gamma(x,y)|& \leq \int_{{\mathbb R}^{k+l}} |\tilde a_{K_\tau} ^\gamma(x,\xi)| \, {\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi =\int _{{\mathbb R}^{k+l}}\Big |\int_{G}\psi^\gamma_{\xi,x}(g) c_{\gamma}(g) K_\tau(g) d_G(g) \Big | {\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi \\
&= \int_{{\mathbb R}^{k+l}} (1+|\xi|^2)^{-N} \Big | \int_G c_{\gamma}(g) K_\tau(g)\sum_{r=0}^{2N} \sum_{|\alpha| =r} b^N_\alpha(x,g) d L(X^\alpha) \psi^\gamma_{\xi,x}(g)
d_G(g) \Big | {\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi.
\end{align*}
If we now apply Proposition \ref{prop:A}, and take into account the estimate \eqref{eq:35} we obtain
\begin{align*}
|\tilde A_{K_\tau}^\gamma(x,y)|& \leq \int (1+|\xi|^2)^{-N} \Big | \int_G \psi^\gamma_{\xi,x}(g) \sum_{r=0}^{2N} \sum_{|\alpha| =r} d L(X^{\tilde \alpha}) [ b^N_\alpha(x,g)c_{\gamma}(g) K_\tau(g)]
d_G(g) \Big | {\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi\\
&\leq \begin{cases}
c_1 (1+\tau^{-2N/q}), & 0 < \tau \leq 1, \\
c_2 e^{\omega \tau}, & 1< \tau,
\end{cases}
\end{align*}
for certain constants $c_i>0$. Expressing $\xi^{k+l+1}_j \psi^\gamma_{\xi,x}(g)$ on $\mklm{\xi \in \mathbb{R}^n: |\xi_i| \leq |\xi_j| \, \text{for all } \, i}$ as left derivatives of $\psi^\gamma_{\xi,x}(g)$ according to \eqref{24} and \eqref{25}, and estimating the maximum norm on $\mathbb{R}^n$ by the usual norm,
a similar argument shows that the last estimate is also valid for $N=(k+l +1)/2$, compare \eqref{eq:49}. The proof is now complete.
\end{proof}
Let us now turn to the resolvent of the closure of the strongly elliptic operator $\Omega$. By \eqref{eq:35} one has the bound $\norm{S_\tau}\leq c e^{\omega \tau}$ for some constants $c \geq 1, \omega \geq 0$. For $\lambda \in {\mathbb C}$ with $\Re \lambda > \omega$, the resolvent of $\overline \Omega$ can then be expressed by means of the Laplace transform according to
\begin{equation*}
(\lambda { \bf 1} + \overline{\Omega})^{-1} = {\Gamma(1)}^{-1} \int_0^\infty e^{-\lambda \tau} S_\tau \d \tau,
\end{equation*}
where $\Gamma$ is the $\Gamma$-function.
More generally, one can consider for arbitrary $\alpha >0$ the integral transforms
\begin{equation*}
(\lambda { \bf 1} + \overline{\Omega})^{-\alpha} = {\Gamma(\alpha)}^{-1} \int_0^\infty e^{-\lambda \tau} \tau^{\alpha-1} S_\tau \d \tau.
\end{equation*}
As it turns out, the functions
\begin{equation*}
R_{\alpha, \lambda}(g) = \Gamma(\alpha)^{-1} \int_0 ^\infty e^{-\lambda \tau} \tau^{\alpha-1} K_\tau(g) \d \tau
\end{equation*}
are in $\L^1(G,e^{\kappa |g|}d_G)$, where $\kappa \geq 0$ is such that $\norm{\pi(g)} \leq c e^{\kappa|g|}$ for some $c\geq 1$. This implies that the resolvent of $\overline \Omega$ can be expressed as the convolution operator
\begin{equation*}
(\lambda { \bf 1}+ \overline \Omega)^{-\alpha} =\pi(R_{\alpha, \lambda})= \int_G R_{\alpha, \lambda}(g) \pi(g) \d_G(g).
\end{equation*}
The resolvent kernels $R_{\alpha,\lambda}$ decrease exponentially as $|g| \to \infty$, but they are singular at the identity if $d \geq q \alpha$. More precisely, one has the following
\begin{theorem}
\label{thm:res.est}
There exist constants $b,c, \lambda_0>0$, and $a_{\alpha,\lambda}>0$, such that
\begin{equation*}
|dL(X^\delta) R_{\alpha, \lambda} (g) | \leq \begin{cases} a_{\alpha, \lambda} |g|^{-(d+|\delta|-q\alpha)}e^{-(b (\Re \lambda)^{1/q}-c)|g|}, & d>q\alpha, \\a_{\alpha, \lambda} (1+ |\log |g|| ) e^{-(b (\Re \lambda)^{1/q}-c)|g|},& d=q\alpha, \\ a_{\alpha, \lambda} e^{-(b (\Re \lambda)^{1/q}-c)|g|}, & d < q \alpha
\end{cases}
\end{equation*}
for each $\lambda \in {\mathbb C}$ with $\Re \lambda > \lambda_0$.
\end{theorem}
A proof of these estimates is given in \cite{robinson}, pages 238 and 245. Our next aim is to understand the microlocal structure of the operators $\pi(R_{\alpha, \lambda})$ on the Oshima compactification $\widetilde {\mathbb X}$
of ${\mathbb X}\simeq G/K$. Consider again the atlas $\mklm{(\widetilde{W}_\gamma, \phi^{-1}_{\gamma})}_{\gamma \in I}$ of $\widetilde {\mathbb X}$ introduced in Section \ref{Sec:4}, and the local operators
\begin{equation}
\label{eq:38a}
A_{R_{\alpha,\lambda}}^{\gamma} u=[\pi(R_{\alpha,\lambda})_{|\widetilde W_{\gamma}}(u\circ \phi_{\gamma}^{-1})]\circ \phi_{\gamma},
\end{equation}
where $u\in{\rm C^{\infty}_c}(W_{{\gamma}})$ and $W_{\gamma}=\phi^{-1}_{\gamma}(\widetilde W_\gamma)$. By the Fourier inversion formula, $A_{R_{\alpha,\lambda}}^\gamma$ is given by the absolutely convergent integral
\begin{equation}
\label{eq:38b}
A_{R_{\alpha,\lambda}}^\gamma u (x)=
\int_{\mathbb{R}^n} e^{i x \cdot \xi} a^\gamma_{R_{\alpha,\lambda}} (x,\xi) \hat u(\xi) {\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi,
\end{equation}
where
\begin{align*}
a^\gamma_{R_{\alpha,\lambda}}(x,\xi)&= \int_{G}e^{i(\phi_\gamma^{g}(x)-x) \cdot\xi} c_\gamma(g)R_{\alpha, \lambda}(g) d_G(g), \\
\tilde a^\gamma_{R_{\alpha,\lambda}}(x,\xi)&= \int_{G}e^{i[({ \bf 1}_k \otimes T_x^{-1}) (\phi_\gamma^{g}(x)-x)]\cdot \xi} c_\gamma(g)R_{\alpha, \lambda}(g) d_G(g)
\end{align*}
are smooth functions on $W_\gamma \times {\mathbb R}^{k+l}$,
since $R_{\alpha,\lambda} \in \L^1(G,e^{\kappa |g|}d_G)$, the notation being the same as in Section \ref{Sec:4}. Moreover, in view of the $\L^1$-bound \eqref{eq:35}, the functions $e^{-\lambda \tau} \tau^{\alpha-1} \tilde a ^\gamma_{K_\tau}(x,\xi)$ and $e^{-\lambda \tau} \tau^{\alpha-1} a ^\gamma_{K_\tau}(x,\xi)$ are integrable in $\tau$ over $(0, \infty)$, and by Fubini we obtain the equalities
\begin{align*}
a^\gamma_{R_{\alpha,\lambda}}(x,\xi)&=\Gamma(\alpha)^{-1} \int _0 ^\infty e^{-\lambda \tau} \tau^{\alpha -1} a ^\gamma_{K_\tau}(x,\xi) d\tau, \\
\tilde a^\gamma_{R_{\alpha,\lambda}}(x,\xi)&= \Gamma(\alpha)^{-1} \int _0 ^\infty e^{-\lambda \tau} \tau^{\alpha -1} \tilde a ^\gamma_{K_\tau}(x,\xi) d\tau.
\end{align*}
In what follows, we shall describe the microlocal structure of the resolvent $( \lambda { \bf 1} + \overline \Omega)^{-\alpha}$ on $\widetilde {\mathbb X}$, and in particular, its kernel.
\begin{proposition}
\label{prop:3}
Let $Q$ be the largest integer such that $Q < q\alpha$. Then $ \tilde a^\gamma_{R_{\alpha,\lambda}}(x,\xi)\in {\rm S}^{-Q}_{la}(W_\gamma \times {\mathbb R}^{k+l})$. That is, for any compactum $\mathcal{K} \subset W_\gamma$, and arbitrary multi-indices $\beta,\epsilon$ there exist constants $C_{\mathcal{K},\beta,\epsilon}>0$ such that
\begin{equation}
\label{eq:ressym}
|(\gd^{\epsilon}_x \gd_\xi ^{ \beta} \tilde a^\gamma_{R_{\alpha,\lambda}})(x,\xi) | \leq C_{\mathcal{K},\beta,\epsilon} (1+ |\xi|^2)^{(-Q-|\beta|)/2}, \qquad x \in \mathcal{K}, \, \xi \in {\mathbb R}^{k+l},
\end{equation}
and $ \tilde a^\gamma_{R_{\alpha,\lambda}}$ satisfies the lacunary condition \eqref{V} for each of the coordinates $x_j$, $k+1 \leq j \leq k+l$.
\end{proposition}
\begin{proof}
For a fixed a chart chart $(\widetilde W_\gamma, \phi_{\gamma})$ of $\widetilde {\mathbb X}$ we write $x=(n,t)\in W_\gamma$, $\tilde x=\phi_{\gamma} (x) \in \widetilde{W}_\gamma$ as usual. As a consequence of Proposition \ref{prop:1} and Lemma \ref{lem:3} one computes with \eqref{26} for arbitrary $N \in {\mathbb N}$
\begin{gather*}
(\gd_\xi ^{2 \beta} \tilde a^\gamma_{R_{\alpha,\lambda}})(x,\xi)= \int_G e^{i[({ \bf 1}_k \otimes T_x^{-1}) (\phi_\gamma^{g}(x)-x)] \cdot \xi} [i({ \bf 1}_k \otimes T_x^{-1}) (\phi_\gamma^{g}(x)-x)]^{2\beta} c_\gamma(g)R_{\alpha, \lambda}(g) d_G(g)\\
=(1 + |\xi|^2) ^{-N} e^{-i(x_1, \dots, x_k,1, \dots, 1) \cdot \xi} \sum_{r=0}^{2 N} \sum_{|\delta| =r}\int_G b^{N}_\delta(x,g) d L(X^\delta) \psi^\gamma_{\xi,x}(g)\\
\cdot [i({ \bf 1}_k \otimes T_x^{-1}) (\phi_\gamma^{g}(x)-x)]^{2\beta} c_\gamma(g)R_{\alpha, \lambda}(g) d_G(g).
\end{gather*}
Now, $n_r(g\cdot \tilde x)\to n_r(\tilde x)$ and $ \chi_r(g,\tilde x)\to 1$ as $g \to e $, so that due to the analyticity of the $G$-action on $\widetilde {\mathbb X}$ one deduces
\begin{equation}
\label{eq:37}
|({ \bf 1}_k \otimes T_x^{-1}) (\phi_\gamma^{g}(x)-x)|=|(n_1(g\tilde x)- n_1, \dots, \chi_1(g\tilde x)-1, \dots ) | =C_\mathcal{K} |g|, \qquad x \in \mathcal{K}.
\end{equation}
Indeed, let
\begin{equation*}
(\zeta_1, \dots, \zeta_d) \mapsto e^{\zeta_1 X_1 + \dots +\zeta_d X_d} =g
\end{equation*}
be canonical coordinates of the first type near the identity $e \in G$. We then have the power expansions
\begin{equation}
\label{eq:38}
\chi_r(g,\tilde x) -1 = \sum_{\alpha,\beta,\gamma} c^r_{\alpha,\beta, \gamma} n^\alpha t^\beta \zeta^\gamma, \qquad n_r(g\cdot \tilde x)-n_r(\tilde x) = \sum_{\alpha,\beta,\gamma} d^r_{\alpha,\beta, \gamma} n^\alpha t^\beta \zeta^\gamma,
\end{equation}
where the constant term vanishes, that is, $c^r_{\alpha,\beta,\gamma},\, d^r_{\alpha,\beta,\gamma}=0$ if $|\gamma|=0$. Hence,
$$|n_r(g\cdot \tilde x)-n_r(\tilde x)|, \, |\chi_r(g,\tilde x) -1|\leq C_1 |\zeta| \leq C_2 |g|,$$
compare \cite{robinson}, pages 12-13, and we obtain \eqref{eq:37}. With Theorem \ref{thm:res.est}, we therefore have the pointwise estimates
\begin{equation*}
| [({ \bf 1}_k \otimes T_x^{-1}) (\phi_\gamma^{g}(x)-x)]^{\beta'} dL(X^{\delta'}) R_{\alpha, \lambda} (g) | \leq C_{\mathcal{K}, \alpha, \lambda} |g|^{-(d+|\delta'|-q\alpha-|\beta'| )}e^{-(b (\Re \lambda)^{1/q}-c)|g|}
\end{equation*}
for some constant $C_{\mathcal{K},\alpha, \lambda}>0$ uniformly on $\mathcal{K}\times V_\gamma^1$. Now, let $ 2\tilde Q$ be the largest even number strictly smaller than $q\alpha$. Applying the same reasoning as in the proof of Proposition \ref{prop:A}, one obtains for $N= \tilde Q+|\beta|$
\begin{gather*}
(\gd_\xi ^{2 \beta} \tilde a^\gamma_{R_{\alpha,\lambda}})(x,\xi)
=(1 + |\xi|^2) ^{-\tilde Q-|\beta|} \sum_{r=0}^{2\tilde Q+2|\beta|} \sum_{|\delta| =r}(-1)^{|\delta|} \int_G e^{i[({ \bf 1}_k \otimes T_x^{-1}) (\phi_\gamma^{g}(x)-x)] \cdot \xi} \\ \cdot d L(X^{\tilde \delta})\big [ b^{\tilde Q +|\beta|}_\delta(x,g) [i({ \bf 1}_k \otimes T_x^{-1}) (\phi_\gamma^{g}(x)-x)]^{2\beta} c_\gamma(g)R_{\alpha, \lambda}(g) \big ] d_G(g),
\end{gather*}
since all the occuring combinations $ [({ \bf 1}_k \otimes T_x^{-1}) (\phi_\gamma^{g}(x)-x)]^{\beta'} dL(X^{\delta'}) R_{\alpha, \lambda} (g)$ on the right hand side are such that $q\alpha +|\beta'| -|\delta'|>0$, implying that the corresponding integrals over $G$ converge. Equality then follows by the left-invariance of $d_G(g)$, and Lebesgue's Theorem on Dominated Convergence. To show the estimate \eqref{eq:ressym} in general for $\epsilon=0$, let $x \in \mathcal{K}$, and $\xi \in {\mathbb R}^{k+l}$ be such that $|\xi| \geq 1$, and $|\xi|_{\mathrm{max}} = \max \mklm{|\xi_r|: 1 \leq r \leq k+l}= |\xi_j|$.
Using \eqref{24} and \eqref{25} we can express $\xi^{Q +|\beta|}_j \psi^\gamma_{\xi,x}(g)$ as left derivatives of $\psi^\gamma_{\xi,x}(g)$, and repeating the previous argument we obtain the estimate
\begin{align}
\begin{split}
\label{eq:49}
|(\gd_\xi ^{ \beta} \tilde a^\gamma_{R_{\alpha,\lambda}})(x,\xi)| &= |\xi_j| ^{-Q-|\beta|} \Big | \sum_{r=0}^{Q + |\beta|} \sum_{|\delta| =r}\int_G b^{j}_\delta(x,g) d L(X^\delta) \psi^\gamma_{\xi,x}(g)\\
\cdot [i({ \bf 1}_k \otimes T_x^{-1}) (\phi_\gamma^{g}(x)-x&)]^{\beta} c_\gamma(g)R_{\alpha, \lambda}(g) d_G(g) \Big |\leq \tilde C_{\mathcal{K},\beta} \frac 1 {|\xi|_{\mathrm{max}}^{Q+ |\beta|}} \leq C_{\mathcal{K}, \beta} \frac 1 {|\xi|^{Q+ |\beta|}},
\end{split}
\end{align}
where the coefficients $b^{j}_\delta(x,g)$ are at most of exponential growth in $g$. But since $\tilde a^\gamma_{R_{\alpha,\lambda}}(x,\xi)\in {\rm C^{\infty}}(W_\gamma \times {\mathbb R}^{k+l})$, we obtain \eqref{eq:ressym} for $\epsilon =0$.
Let us now turn to the $x$-derivatives. We have to show that the powers in $\xi$ that arise when differentiating $(\gd^{ \beta} _\xi \tilde a^\gamma_{R_{\alpha,\lambda}})(x,\xi) $ with respect to $x$ can be compensated by an argument similar to the previous considerations. Now, \eqref{eq:38} clearly implies
\begin{equation*}
{\gd_x^\epsilon} \, (\chi_r(g,\tilde x) -1)=O(|g|), \qquad
{\gd_x^\epsilon}\, (n_r(g\cdot \tilde x)-n_r(\tilde x) )=O(|g|).
\end{equation*}
Thus, each time we differentiate the exponential $e^{i[({ \bf 1}_k \otimes T_x^{-1}) (\phi_\gamma^{g}(x)-x)] \cdot \xi}$ with respect to $x$, the result is of order $O(|\xi||g|)$. Therefore, expressing the ocurring powers $\xi^{\epsilon'} \psi^\gamma_{\xi,x}(g)$ as left derivatives of $\psi^\gamma_{\xi,x}(g)$, we can repeat the preceding argument to absorb the powers in $\xi$, and \eqref{eq:ressym} follows.
Note next that the previous argument also implies $ a^\gamma_{R_{\alpha,\lambda}}(x,\xi) \in \mathrm{S}^{-Q} ( W_\gamma ^\ast \times {\mathbb R}^{k+l}_\xi$), where $W_\gamma^\ast=\mklm{x=(n,t) \in W_\gamma: t_1\cdots t_l\not=0}$, the $G$-action being transitive on each $\widetilde {\mathbb X}_\Delta$. The Schwartz kernel $K_{A_{R_{\alpha,\lambda}}^\gamma}$ of the restriction of the operator \eqref{eq:38a} to $W^\ast_\gamma$ is therefore given by the oscillatory integral
\begin{equation*}
\int e^{i(x-y) \cdot \xi} a^\gamma_{R_{\alpha,\lambda}}(x,\xi) {\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi \in {\mathcal D}'( W_\gamma^\ast \times W_\gamma^\ast),
\end{equation*}
which is ${\rm C^{\infty}}$ off the diagonal. As in \eqref{eq:33} we have
$
\supp K_{A_{R_{\alpha,\lambda}}^\gamma} \subset \bigcup _{\Theta \subset \Delta} \overline{W_\gamma^\Theta} \times \overline{W_\gamma^\Theta}
$,
so that each of the integrals
\begin{equation*}
\int e ^{i(x_j-y_j) \xi_j} \tilde a^\gamma_{R_{\alpha,\lambda}} (x,({\bf{1}}_k \otimes T_x)\xi) \d \xi_j,\qquad j=k+1, \dots, k+l,
\end{equation*}
must vanish if $x_j$ and $y_j$ do not have the same sign. Hence,
\begin{equation*}
\int e^{-ir_j \xi_j} \tilde a^\gamma_{R_{\alpha,\lambda}}(x,\xi) \d \xi_j =0 \qquad \mbox{ for } r_j < -1, \, x \in W_\gamma^\ast.
\end{equation*}
Since $\tilde a^\gamma_{R_{\alpha,\lambda}}(x,\xi) \in \mathrm{S}^{-Q} ( W_\gamma \times {\mathbb R}^{k+l}_\xi)$, these integrals are absolutely convergent for $r_j\not=0$. Lebesgue's Theorem on Bounded Convergence theorem then implies that these conditions must also hold for $x \in W_\gamma$. The proof of the proposition is now complete.
\end{proof}
\begin{remark}
One would actually expect that $ \tilde a^\gamma_{R_{\alpha,\lambda}}(x,\xi)\in {\rm S}_{la}^{-q\alpha}(W_\gamma \times {\mathbb R}^{k+l})$, being the local symbol of the resolvent $(\lambda { \bf 1} +\overline \Omega)^{-\alpha}$. Nevertheless, the general estimates of Theorem \ref{thm:res.est} for the resolvent kernels $R_{\alpha,\lambda}$, which correctly reflect the singular behavior at the identity, are not sufficient to show this, and more information about them is required. Indeed, $dL(X^\beta) R_{\alpha,\lambda} \in L_1(G,d_G(g))$ only holds if $0< q\alpha - |\beta|$.
\end{remark}
We are now able to describe the microlocal structure of the resolvent $(\lambda{ \bf 1}+ \overline \Omega)^{-\alpha}$.
\begin{theorem}
\label{thm:resolvent}
Let $\Omega$ be a strongly elliptic differential operator of order $q$ associated with the representation $(\pi,C(\widetilde {\mathbb X}))$ of $G$. Let $\omega\geq 0$ be given by Theorem \ref{thm:4}, and $\lambda \in {\mathbb C}$ be such that $\Re \lambda > \omega$. Let further $\alpha>0$, and denote by $Q $ the largest integer such that $Q < q\alpha$. Then $(\lambda{ \bf 1}+ \overline \Omega)^{-\alpha}=\pi(R_{\alpha,\lambda})$ is locally of the form \eqref{eq:38b}, where $ a^\gamma_{R_{\alpha,\lambda}} (x,\xi)= \tilde a^\gamma_{R_{\alpha,\lambda}} (x,({ \bf 1}_k \otimes T_x) \xi)$, and $ \tilde a^\gamma_{R_{\alpha,\lambda}}(x,\xi)\in {\rm S}_{la}^{-Q}(W_\gamma \times {\mathbb R}^{k+l})$. In particular, $(\lambda{ \bf 1}+ \overline \Omega)^{-\alpha}$ is a totally characteristic pseudodifferential operators of class $\L^{-Q}_b$ on the manifolds with corners $\overline{\widetilde{\mathbb X}_\Delta}$. Furthermore, its kernel is locally given by the oscillatory integral
\begin{align*}
\begin{split}
R^\gamma_{\alpha,\lambda}(x,y)&= \int e ^{i(x-y) \xi} a^\gamma _{R_{\alpha,\lambda}} (x,\xi) {\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi =\frac 1 {|x_{k+1}\cdots x_{k+l}|} \int e ^{i({\bf{1}}_k \otimes T_x^{-1})(x-y) \cdot\xi} a^\gamma _{R_{\alpha,\lambda}} (x,\xi) {\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi,
\end{split}
\end{align*}
where $x_{k+1}\cdots x_{k+l} \not=0, \, x , y \in W_\gamma$. $ R^\gamma_{\alpha,\lambda}(x,y)$ is smooth off the diagonal, and rapidly falling at infinity as $| x_{j}| \to 0 $, or $| y_{j}| \to 0 $, as long as $x_j\not=y_j$, where $k+1\leq j\leq k+l$.
\end{theorem}
\begin{proof}
The assertions of the theorem are direct consequences of our previous considerations, except for the behavior of $R^\gamma_{\alpha,\lambda}(x,y)$ at infinity. Let $k+1\leq j\leq k+l$. While the behavior as $|y_j| \to 0$ is a direct consequence of the lacunarity of $\tilde a^\gamma_{R_{\alpha,\gamma}}$, the behavior as $|x_j| \to 0$ is a direct consequence of the fact that, as oscillatory integrals,
\begin{equation*}
\int e ^{i(x-y) \cdot \xi} a^\gamma _{R_{\alpha,\lambda}} (x,\xi) {\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi= \frac 1 {|x-y|^{2N}} \int e ^{i(x-y) \cdot \xi} ( \gd_{\xi_1}^2 + \dots +\gd_{\xi_{k+l}}^2 )^N a^\gamma _{R_{\alpha,\lambda}} (x,\xi) {\,\raisebox{-.1ex}{\={}}\!\!\!\!d} \xi,
\end{equation*}
where $x\not = y$, and $N$ is arbitrarily large.
\end{proof}
\begin{remark}
The singular behavior of $R_{\alpha,\lambda}(g)$ at the identity corresponds to the fact that, as a pseudodifferential operator of class $L^{-Q}_b$, $(\lambda { \bf 1} + \overline \Omega)^{-\alpha}$ has a kernel which is singular at the diagonal.
\end{remark}
To conclude, let us say some words about the classical heat kernel on a Riemannian symmetric space of non-compact type.
Consider thus the regular representation $(\sigma,\mathrm{C}(\widetilde {\mathbb X}))$ of the solvable Lie group $S=AN^-\simeq {\mathbb X}\simeq G/K$ on the Oshima compactification $\widetilde {\mathbb X}$ of ${\mathbb X}$,
and associate to every $f\in \S(S)$ the corresponding convolution operator
\begin{equation*}
\int _{S} f(g) \sigma(g) \d_{S}(g).
\end{equation*}
Its restriction to ${\rm C^{\infty}}(\widetilde {\mathbb X})$ induces again a continuous linear operator
\begin{equation*}
\sigma(f):{\rm C^{\infty}}(\widetilde {\mathbb X}) \longrightarrow {\rm C^{\infty}}(\widetilde {\mathbb X}) \subset {\mathcal D}'(\widetilde {\mathbb X}),
\end{equation*}
and an examination of the arguments in Section \ref{Sec:4} shows that an analogous analysis applies to the operators $\sigma(f)$. In particular, Theorem \ref{thm:3} holds for them, too.
Let $\rho$ be the half sum of all positiv roots, and
\begin{equation*}
{C}=\sum_j H_j^2 - \sum_{j} Z_j^2-\sum_j [ X_j \theta(X_j) + \theta(X_j) X_j]\equiv \sum_j H_j^2 -2\rho +2 \sum X_j^2 \mod {\mathfrak U}({\bf \mathfrak g}) \k
\end{equation*}
be the Casimir operator in ${\mathfrak U}({\bf \mathfrak g})$, where $\mklm{H_j}$, $\mklm{Z_j}$, and $\mklm{X_j}$ are orthonormal basis of $\a$, ${\bf\mathfrak m}$, and ${\bf\mathfrak n}^-$, respectively, and put $C'=\sum_j H_j^2 -2\rho +2 \sum X_j^2$. Though $-d\pi(C')$ is not a strongly elliptic operator in the sense defined above, $\Omega=-d\sigma(C')$ certainly is. Consequently, if $K'_\tau(g) \in \S(S)$ denotes the corresponding Langlands kernel, Theorems \ref{thm:heatoperator} and \ref{thm:resolvent} yield descriptions of the Schwartz kernels of $\sigma(K'_\tau)$ and $(\lambda { \bf 1} + \overline \Omega)^{-\alpha}$ on $\widetilde {\mathbb X}$. On the other hand, denote by $\Delta$ the Laplace-Beltrami operator on ${\mathbb X}$. Then
\begin{equation*}
\Delta \phi(gK)=\phi(g:C)= \phi(g:C'), \qquad \phi \in {\rm C^{\infty}}({\mathbb X}),
\end{equation*}
and the associated heat kernel $h_\tau(g)$ on ${\mathbb X}$ coincides with the heat kernel on $S$ associated to $C'$. But the latter is essentially given by the Langlands kernel $K'_\tau(g)$, being the solution of the parabolic equation
\begin{equation*}
\frac{\gd K'_\tau}{\gd \tau} (g) - dL(C') K'_\tau(g)=0, \qquad \lim_{\tau \to 0 } K'_\tau(g) = \delta(g)
\end{equation*}
on $S$. In this particular case, optimal upper and lower bounds for $h_\tau$ and the Bessel-Green-Riesz kernels were
given in \cite{anker-ji99} using spherical analysis under certain restrictions coming from the lack of control in the
Trombi-Varadarajan expansion for spherical functions along the walls. Our asymptotics for the kernels of
$\sigma(K'_\tau)$ and $(\lambda { \bf 1} + \overline \Omega)^{-\alpha}$ on $\widetilde {\mathbb X}_\Delta\simeq {\mathbb X}$ are free
of restrictions, and in concordance with those of \cite{anker-ji99}, though, of course, less explicit.
A detailed description of the resolvent of $\Delta$ on ${\mathbb X}$ was given in \cite{mazzeo-melrose87}, \cite{mazzeo-vasy05}.
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1,116,691,500,810 | arxiv | \section{Introduction}
Consider some arbitrary classical (i.e., non-quantum) experimental procedure,
wherein experiments are performed and measurements are taken to quantify some
characteristic of some component. Suppose this component is to be included as
sub-component of a larger system and an engineer is charged with performing an
analysis (likely via computer simulation) of how variability of the sub-components
affects the overall system performance. Such studies are commonly denoted
sensitivity analysis, risk analysis, or uncertainty quantification
\cite{iman1988investigation,saltelli2000sensitivity} and such techniques are of
practical concern in a variety of contexts.
Suppose the engineer tracks down the experimental results for a given component
(e.g., length tolerances for some packaging) and is able to obtain the sample
data generated by the experimentalist, or more likely, the reported mean $\mu$
and variance $\sigma^2$. In either case, it is likely that the engineer will
model the distribution of the characteristic using the normal distribution
$\mathcal{N}(\mu,\sigma^2)$, and indeed, there is mathematical reasoning to do
so \cite{jaynes1957information}, beyond any computational reasons for doing so.
The engineer might use samples from this distribution as part of a Monte Carlo
analysis, or perhaps the functional form of the probability distribution
function can be used as part of a closed-form analysis.
A similar situation exists in quantum information, but it is often unremarked.
On one hand, the evolution of an open quantum system can often be treated as a
noisy evolution, through the stochastic Liouville equation
\cite{kubo1963stochastic}. This type of description applies to some of the
most common types of qubits and their predominant decoherence mechanisms (see,
e.g. Refs. \cite{Kubo1957, Schulten1978, ernst1987principles, Schneider1998,
Grigorescu1998, abergel2003, Cheng2004, Wilhelm2007}). On the other hand,
especially in the context of quantum gates or circuits, we speak of ``the
superoperator'' or ``the error channel'' for a given quantum operation, and
not a stochastic process. Unlike the classical case, it is impossible to
recover the quantum operation associated with an individual quantum trajectory,
only measurements collected at the end of an experiment, which individually
contain only partial information about the average quantum channel that
generated the collections of measurements. Thus, techniques for converting
sets of quantum measurements into estimates of quantum states or channels,
called tomography, are essentially computing estimates of the average
quantities of interest, in much the same way that descriptive statistics such
as the mean and variance are used in classical contexts. This suggests that a
rigorous statistical approach must be developed in order to develop accurate
assessments of the impacts of noise and imperfections on quantum systems, when
only sample averages of an underlying probability distribution are available.
Here, we develop such a statistical approach applicable to quantum information by
showing the relationships between a number of probability distributions common
in the field of directional or orientation statistics
\cite{mardia2009directional,chikuse2012statistics}, but somewhat obscure
outside of this field, and a number of relevant structures in quantum
information. In particular, we relate the parameters of these
distributions to certain forms of quantum operations. The hope is that these
distributions can be used both in closed-form and Monte Carlo analysis in the
simulation of quantum circuits, as well a providing additional
foundation for different inference problems concerning quantum systems.
In the discussion below, we first cover a number of relevant forms of quantum
operations that we will later draw connections to from the field of orientation
statistics. Next, we introduce the concept of an exponential family, which is
a geometric concept that generalizes many of the familiar properties of normal
random variables, such as the \textit{sufficiency} of the sample mean and
variance fully describing the sampled data, as well as the maximum entropy property
that the normal distribution exhibits among all random variables with a fixed
variance. The preliminary discussion is concluded by a brief overview of
Stiefel manifolds, which will be the sample space (instead of Euclidean space)
on which the relevant probability distributions will be defined.
\subsection{CPTP Maps}
In quantum information, a quantum state is represented by a density operator
$\rho$, where $\rho\in\mathbb{C}^{N\times N}$ is a positive semi-definite,
Hermitian matrix with $\Tr(\rho) =1$ \cite{nielsen2010quantum}. Quantum
operations are then completely positive, trace-preserving (CPTP) maps
\cite{nielsen2010quantum}. Here, we will make the additional assumption that
the quantum maps of interest map to density operators of the same dimension as
the input dimension, but this can be generalized. Below we will summarize some
relevant properties and equivalent representations of CPTP maps that will help to
draw the connection to the field of orientation statistics. More information
about various forms of CPTP maps can be found in e.g.
\cite{fujiwara1999one,bruzda2009random,wood2014tensor}.
Choi's theorem on completely positive (CP) maps \cite{choi1975completely}
states that any CP map $\Phi(\rho)$ can be written in the Kraus form
\begin{equation}
%
\Phi(\rho)=\sum_{i=1}^m A_i\rho A_i^\dagger
%
\end{equation}
where $A_i\in\mathbb{C}^{N\times N}$ and $m$ is said to be the Kraus rank of
the map $\Phi$ and $m\leq N^2$. The added characteristic of trace preserving (TP)
is equivalent to $\sum_iA_i^\dagger A_i \triangleq \mathbbm{1}_N$. Note that the Kraus
form is not unique, that is, different sets of Kraus operators $A_i$ can create
equivalent maps.
Let $|\cdot\rangle\rangle$ denote the vectorization operator, and
$\langle\langle\cdot|=|\cdot\rangle\rangle^\dagger$. From the Kraus form, two
additional representations of CPTP maps can be defined, the Choi matrix form
\begin{equation}\label{eq:choi}
%
\Lambda = \sum_{i=1}^m|A_i\rangle\rangle\langle\langle A_i|\,,
%
\end{equation}
and the Liouvillian superoperator (or dynamical matrix form)
\begin{equation}
%
\mathcal{L} = \sum_{i=1}^m A_i^*\otimes A_i\,,
%
\end{equation}
where $*$ denotes entry-wise conjugation, not conjugate transposition
($\dagger$).
These forms are of course equivalent and are related by a reshuffling
involution \cite{bruzda2009random,wood2014tensor}.
Let $\Tr_{2}$ denote the partial trace over the second Hilbert space in a
composite system, so that $\Tr_2(A\otimes B) =A$. The TP property implies that
$\Tr_2(\Lambda) = \mathbbm{1}_N$ (and thus $\Tr(\Lambda)=N$) and it is obvious
from the form in (\ref{eq:choi}) that $\Lambda$ is a positive semi-definite
Hermitian matrix. As (\ref{eq:choi}) has the same functional form as a density
operator on $\mathbb{C}^{N^2\times N^2}$, there is an isomorphism between
$\Lambda/N$ and density operators on the higher dimensional space $N^2$, called
the \textit{Jamio\l{}kowski} isomorphism \cite{jamiolkowski1972linear}. Since
$\Lambda$ is a Hermitian positive semi-definite matrix, we can diagonalize it
as $\Lambda = KDK^\dagger$ where $K$ is a unitary matrix (in particular, its
columns are orthonormal) and $D$ is a diagonal matrix whose entries
$D_{ii}\in[0,N]$ and $\sum D_{ii} = N$. Letting $|K_i\rangle\rangle$ denote
the $i$th column of the above decomposition, $\Lambda$ can be used to define
``canonical'' Kraus operators $\sqrt{D_{ii}}K_i$.
While $\Lambda$ enjoys a number of useful structural properties, the
Liouvillian superoperator is convenient for the propagation of state, as
$\mathcal{L}(\rho) = \mathcal{L}|\rho\rangle\rangle$. In this work, the
primary use of the superoperator will be as it relates to the Pauli transfer
matrix or affine form \cite{fujiwara1999one}. For a single qubit system ($N=2$),
the Pauli transfer matrix is the Liouvillian superoperator expressed in the
basis spanned by the vectorized Pauli matrices $|\sigma_I\rangle\rangle$, $|\sigma_X\rangle\rangle$,
$|\sigma_Y\rangle\rangle$, $|\sigma_Z\rangle\rangle$. Higher dimensional analogues exist
\cite{fujiwara1999one}, but the key concept is that states $\rho$ (in say, the
computational basis) are mapped to Bloch vectors $\varphi\in\mathbb{R}^{N^2-1}$
with $||\varphi||_2\leq1$ and unitary quantum operations are represented as
rotations (elements of $SO(N^2-1)$) of $\varphi$ along a hypersphere. The
affine form of general CPTP maps can include two rotation components, a
contraction component, and linear shift, but the focus here will be on affine
representations of unitary operations.
\subsection{Sufficient Statistics and Exponential Families}
A probability distribution $p(x;\theta)$ parameterized by $\theta$ is said to be an
exponential family if $p(x;\theta)$ can be expressed as
\begin{equation}
%
p(x;\theta) = \exp(\langle\theta,T(x)\rangle - \psi(\theta) + \kappa(x))\,,
%
\end{equation}
where
\begin{itemize}
\item $\theta$ are the natural parameters
\item $T(x)$ are the sufficient statistics
\cite{koopman1936distributions,pitman1936sufficient}, typically linearly independent
\item $\psi(\theta)$ is the log-normalizer which causes $p(x;\theta)$ to integrate to one
\item $k(x)$ is the carrier measure that determines the support of the
distribution, for example on the positive reals or some manifold
embedded in Euclidean space
\end{itemize}
Exponential families play a prominent role in statistics
\cite{amari2007methods,barndorff2014information}, particularly in the context
of maximum-likelihood estimation. Many common distributions such as Gaussian,
exponential, Bernoulli, etc., are exponential families. Two basic facts of
exponential families motivate their study here. The first is that the
so-called expectation parameters $\eta$ of a probability distribution are
defined by $\eta = E_{p(x;\theta)}[T(x)]$ and are in one-to-one correspondence with
the natural parameters $\theta$, meaning that any estimates $\hat{\theta}$
of $\theta$ are functions of sample averages of $T(x)$
\cite{amari2007methods,barndorff2014information}. This is the key principle
that allows for samples from a normal distribution to be completely summarized
by the mean and variance of the data, and is essentially a unique property to
exponential families.
The second motivating fact is that among all possible exponential families with
sufficient statistics $T'$ that contain $T$, the exponential family whose
sufficient statistics are just $T$ maximizes the entropy $E[-\log(p(x))]$
\cite{amari2007methods}. This property is one reason why it is reasonable to
assume normal distributions on some random parameter when only a mean and
variance are reported without any additional information, such as a Bayesian
prior or valid range on the parameter.
Note that any two sets of statistics
that span the same set in function space define the same distribution and will
enjoy the same maximum entropy property. For example, this is why mean $E[x]$
and variance $E[(x-E[x])^2]$ or mean and second moment $E[x^2]$ will both lead
to the normal distribution (when the domain is the entire real line).
In quantum information, we often speak of \textit{the} superoperator (or other
equivalent representation) associated with a quantum channel, when in physical
reality various external sources (generally grouped into semiclassical noise or bath) result
in a (seemingly) random quantum operation even when the input control pulses
are meant to be identical. Furthermore, given the mathematics of quantum
measurement and tomography, it is not possible to measure individual quantum
trajectories and produce a sample of superoperators to apply statistical
techniques to. Instead, the tomographic process produces an estimate of an
average superoperator $\hat{\mathcal{L}}$ or an equivalent representation \cite{Chuang1997, Merkel2013, blume2016certifying}. The remainder of this paper discusses probability
distributions for random quantum states and CPTP maps for whom the statistic
$\hat{\mathcal{L}}$ (or an equivalent representation) is sufficient in the
sense described above.
\subsection{Stiefel Manifolds}\label{sec:stiefel}
Suppose $X\in\mathbb{F}^{n\times k}$
($\mathbb{F}=\mathbb{R}$,$\mathbb{C}$,$\mathbb{H}$) with $X^\dagger X=\mathbbm{1}_k$,
then $X$ is called a $k$-frame of orthogonal vectors in $\mathbb{F}^n$. The set of
all such $X$ forms the Stiefel manifold $V_k(\mathbb{F}^n)$. In particular:
\begin{itemize}
\item $V_n(\mathbb{R}^n)\cong O(n)$, $V_n(\mathbb{C}^n)\cong U(n)$
\item $V_1(\mathbb{R}^n)=S^{n-1}$, $V_1(\mathbb{C}^n)=S^{2n-1}$ ~ ~ (i.e., unit
spheres)
\item $V_{n-1}(\mathbb{R}^n)\cong SO(n)$, $V_{n-1}(\mathbb{C}^n) \cong
SU(n)$
\end{itemize}
Thus, for random quantum operations that are unitary, $V_{n-1}(\mathbb{C}^n)$ is
of interest. Furthermore, $V_{n-1}(\mathbb{R}^n)$ is relevant for unitary
rotations both from the Bloch representation of the operation (i.e., the PTM),
and also from decomposing $U$ into its real and imaginary parts, constructing a
real-valued matrix. Fortunately, Stiefel manifolds are the natural sample
space for generalizations of directional statistics (often called orientation
statistics) \cite{mardia2009directional,chikuse2012statistics} so a number of
exponential families have been defined on Stiefel manifolds, along with
inference \cite{mardia2009directional,chikuse2012statistics,sei2013properties}
and methods for generating random samples
\cite{mezzadri2006generate,hoff2009simulation,kent2013new}.
Since all Stiefel manifolds are compact, they admit a Haar measure and can thus
be sampled uniformly. This fact will be exploited in the discussion below, and
here for completeness we describe a known method for uniform sampling of real
and complex Stiefel manifolds \cite{chikuse2012statistics}. Let
$X\in\mathbb{R}^{m\times n}$, $m\leq n$ be matrix whose entries are independent
samples from a real Gaussian distribution with zero mean and unit variance,
then $X$ is a sample of the real-valued \textit{Ginibre} ensemble
\cite{mehta2004random}. By taking the $QR$-decomposition of $X$, the polar
matrix $Q$ is an element of $V_m(\mathbb{C}^n)$ (see
\cite{mezzadri2006generate} for some technical implementation issues).
Similarly, if $X$ and $Y$ are independent samples from the Ginibre ensemble,
then the $QR$ decomposition of $Z=X+iY$ yields uniform random elements in
$V_m(\mathbb{C}^n)$.
In addition to the spaces $U(N)$, $SU(N)$, and $SO(N^2-1)$ which naturally arise in
the context of quantum operations, consider a set of $k$ matrices
$\{A_j\}\in\mathbb{C}^{N\times N}$. Let $\mathcal{S}$ denote the matrix formed by
stacking the $A_j$, so that
\begin{equation}
%
S=\begin{bmatrix} A_1\\A_2\\\vdots\\A_k\end{bmatrix}\,.
%
\end{equation}
Since $\mathcal{S}^\dagger \mathcal{S} = \sum_j A_j^\dagger A_j$, we have that
if $\mathcal{S}\in
V_{N}(\mathbb{C}^{kN})$, then the matrices $A_j$ are valid Kraus operators for a
CPTP map. Thus we have that there is a correspondence between Kraus operators
and columns of unitary matrices, as noted in \cite{bruzda2009random}. We will
consider CPTP maps defined by $N^2$ Kraus operators (which is always possible)
and call the associated matrix $\mathcal{S}\in V_{N}(\mathbb{C}^{N^3})$ the
\textit{Stiefel} form of a CPTP map. Note that since we can rearrange the
index of the Kraus operators, this representation is not unique.
\section{Random Quantum States}
Quantum states are represented by density operators, which are Hermitian trace
one matrices. This space is compact and can be sampled uniformly, for example
by uniformly sampling the eigendecomposition $\rho=KDK^\dagger$ of the density
operator $\rho$. The columns of the matrix $K$ are complex and orthonormal,
and are thus unitary matrices which can be sampled uniformly according to the
method described in Section~\ref{sec:stiefel}. The matrix $D$ is diagonal and
must have $\Tr D =1$, which amounts to uniformly sampling from the simplex,
which can be accomplished (for example), by sampling $N+1$ numbers uniformly in
$[0,1]$, sorting them, and taking their difference as the sample elements on
the diagonal of $D$ \cite{Devroye1986nonuniform}. With probability one,
density operators generated in this fashion will have rank $N$, but by setting
some diagonal elements of $D$ to zero, we can achieve an arbitrary purity level
(although for pure states it is more efficient to sample uniformly from from
$V_1(\mathbb{C}^N)$.
Suppose, however, we have performed state tomography on a quantum system and
have produced an (average) estimate for a state, $\hat{\rho}$. Unless
$\hat{\rho}$ is close to $\mathbbm{1}/N$, then a uniform model is unlikely to be
representative. Certainly, we would hope that state preparation is very close
to the targeted pure state. Below, we discuss two classes of distribution that
are capable of producing non-uniform and concentrated distributions of random
states.
\subsection{Random Pure States}
First, consider a random \textit{pure} state with (random) density operator
$\rho$. Since the space of density operators is convex, we have that the
average $\hat{\rho}$ of random pure states is also a valid density operator,
but it is not necessarily pure (and will only be so if the random variable is
constant). In this case, a natural representation of the pure state $\rho$ is
a unit vector $\varphi$ called a (generalized) Bloch vector
\cite{nielsen2010quantum} which is also the state representation used in an
affine form of a CPTP map \cite{fujiwara1999one}. The unit vector $\varphi$ is
an element of $V_1(\mathbb{R}^{N^2-1})$ by definition, and the average
Bloch vector $\hat{\varphi}$ is the sufficient statistic for the vector Von
Mises-Fisher distribution on $V_1(\mathbb{R}^{N^2-1})$. The vector Von
Mises-Fisher distribution is specified by two natural parameters, the
\textit{mean direction} $\mu$ (a unit vector), and the concentration parameter
$\kappa\in(0,\infty)$. The probability density function of the vector Von
Mises-Fisher distribution is
\begin{equation}
%
p_{VF}(X; \mu,\kappa) = c_{VF}(\mu,\kappa)\exp(\kappa\mu^\top X)
%
\end{equation}
where $c_{VF}$ is a normalization involving $\Gamma$ and Bessel
functions \cite{mardia2009directional}. Methods for computing the natural
parameters from an average $\hat{\varphi}$ can be found in
\cite{mardia2009directional}.
\subsection{Random Mixed States}
Consider an element $X\in V_k(\mathbb{C}^N)\subset\mathbb{C}^{N\times k}$, and
denote the (orthonormal, by assumption) columns of $X$ by $\vec{x}_i$,
$i=1,\dots,k$, then
\begin{equation}
%
\begin{aligned}
%
\Tr(XX^\dagger) &= \sum_{\ell=1}^k\sum_{i=1}^N\sum_{j=1}^N x_{\ell
i}x_{\ell j}^*\\
%
& = \sum_{i,j=1}^k\langle\vec{x}_i,\vec{x}_j\rangle\\
%
& = k\,.\\
%
\end{aligned}
%
\end{equation}
Noting $XX^\dagger$ is Hermitian and positive semi-definite (consider
(\ref{eq:choi})), when properly normalized by $1/k$ it is a density density
operator. Thus, elements $X\in V_k(\mathbb{C}^N)$ can be identified with a
density operator density operators of dimension $N$ with
purity $k$.
Consider the family of probability distributions defined on
$V_k(\mathbb{C}^{N})$ with the following form:
\begin{equation}
%
p_{MB}(X;A) = c_{MB}(A)\exp\left(\Tr\left(-X^\dagger A X\right)\right)
%
\end{equation}
where $c_{MB}(A)$ is a normalizer. These distributions are known as the
(complex) matrix-Bingham distributions
\cite{mardia2009directional,chikuse2012statistics} and are often used to model
axial data $x$ in which $\pm x$ are indistinguishable, as opposed to
directional data $\theta$, i.e., angles. These distributions have also found
applications in the analysis of shape \cite{kent1994complex}, where the scale
of the data is normalized and there is a desire for rotation invariance.
Given a Stiefel manifold on $\mathbb{R}^N$ or $\mathbb{C}^N$, the
matrix-Bingham distribution is defined by a natural parameter matrix $A$ whose
dual is the expectation parameter $E[XX^\dagger]$. This distribution is known
to maximize the Shannon entropy relative to the Haar measure on a given Stiefel
manifold among all distributions with the same moment criterion specified by
$E[X]=0$ and $E[XX^\dagger]$. Note that the moment constraint $E[X]=0$ is
consistent with the phase ambiguity in quantum mechanics. Thus, given an
average density operator, we can associate with it a probability distribution
that has sufficient statistics specified by that average, and is in some sense
the least informative distribution on that Stiefel manifold.
Note that the literature is predominately focused on the real-valued matrix
Bingham distribution. However, given $A$ as the natural parameter (i.e.,
concentration matrix) for a complex matrix-Bingham distribution on
$V_k(\mathbb{C}^N)$, let
\begin{equation}\label{eq:complexreal}
%
A_R=\begin{bmatrix}\Re A & -\Im A\\\Im A &\Re A\end{bmatrix}
%
\end{equation}
and let $Y\sim\exp(\Tr Y A_R Y^\top)$ be distributed as a real-valued
matrix-Bingham distribution on $V_k(\mathbb{R}^{2N})$. Then, let $Y_R$ denote
the first $N$ rows of $Y$ and $Y_I$ denote the remaining $N$ rows. The random
variable $Z = Y_R+iY_I$ will be distributed as a complex matrix-Bingham
variable with natural parameter $A$ on $V_k(\mathbb{C}^N)$ \cite{kent2013new}.
\section{Random CPTP Maps}
Sampling uniformly from the space of CPTP maps was discussed in
\cite{bruzda2009random} (note their notation is in some sense dual to the
notation here), by first selecting the rank of a Choi matrix and using the
complex Ginibre ensemble with an appropriate normalization. A statistically
identical approach can be achieved by generating $\mathcal{S}$ matrices using
the Ginibre ensemble and the QR decomposition. In either case, by assuming a
Kraus rank of 1, one can sample uniformly from the space of unitary operations.
Thus, there are known mechanisms for sampling uniformly from various spaces of
CPTP maps.
The focus on fault tolerant quantum computation, however, has emphasized the
generation of gates that are strongly concentrated (in some sense) about the
ideal, so much so that the ensemble average of the random quantum operations
is nearly identical to the ideal gate. For the purposes of inference (e.g.,
tomography), circuit-level simulation, and system modeling this indicates that we need
distributions on appropriate spaces that are capable of producing strongly
concentrated distributions, and can be defined in terms of the average
CPTP map (i.e., the output of some process tomographic algorithm).
\subsection{Random Unitary Operations and the matrix Fisher
Distribution}\label{sec:cptp_unit}
Let $X$ be an element of $V_N(\mathbb{C}^N)$, as noted above, $X$ is a unitary
matrix, and since representations such as the Choi matrix and Liouvillian
superoperator are inherently quadratic in terms of the entries of $X$, we might
tempted to use the matrix-Bingham distribution with its sufficient statistic
$XX^\dagger$, and look for correspondences to these matrix representations.
However, since $X$ is unitary, $XX^\dagger=\mathbbm{1}_N$ for all $X$, meaning the only
valid matrix-Bingham distribution on this space is equivalent to the uniform
distribution.
The next logical step is to consider $V_{N-1}(\mathbb{C}^N)$ to consider random
special unitary operations. If we decompose $X\in V_{N-1}(\mathbb{C}^N)$ into
its columns by $X=[x_1,x_2,\dots,x_{N-1}]$ and let $\tilde{x}$ denote the
unique vector such that the matrix $[x_1, \dots, x_{N-1}, \tilde{x}]$ is an
element of $SU(N)$. In this case, we have have that the sufficient statistic
of the matrix-Bingham distribution $XX^\dagger$, can be written as
$\sum_{j=1}^{N-1}x_jx_j^\dagger$. From the this column-representation, we have
that the expected Choi matrix of the element $SU(N)$ associated with random $X$
is
\begin{equation}\label{eq:su_bing}
%
\begin{aligned}
%
\Lambda & = E\left[|[x_1,x_2, \dots,x_{N-1},\tilde{x}]\rangle\rangle
\langle\langle [x_1,x_2,\dots x_{N-1},\tilde{x}]|\right]\\
&=E\left[\begin{bmatrix}x_1x_1^\dagger & x_1x_2^\dagger &\cdots
&x_1\tilde{x}^\dagger\\ x_2x_1^\dagger & x_2x_2^\dagger& \dots &
x_2\tilde{x}^\dagger\\ \vdots &\vdots &\ddots &\vdots\\
\tilde{x}x_1^\dagger
&\tilde{x}x_2^\dagger&\cdots&\tilde{x}\tilde{x}^\dagger
\end{bmatrix}\right]\,.
%
\end{aligned}
%
\end{equation}
From (\ref{eq:su_bing}) we have that the sufficient statistic for a
matrix-Bingham distribution on $V_{N-1}(\mathbb{C}^N)$ can be associated with a
Choi-matrix by summing the first $N-1$ terms on the block-diagonal in
(\ref{eq:su_bing}). The quantity $E[\tilde{x}\tilde{x}^\dagger]$ is determined
by $\sum_{j=1}^{N-1}x_jx_j^\dagger$ from the partial trace constraints on
$\Lambda$, but the off diagonal terms are essentially being ignored. In this
sense, the matrix-Bingham distribution is a statistical sub-model of
distributions on $SU(N)$ that have Choi matrices as sufficient statistics, and
information contained in these off-diagonal terms are being discarded. This is
notionally similar to the relationship between a Guassian random vector with
non-diagonal covariance matrix and its diagonalization.
In order to capture this missing information in a sufficient statistic, we turn
to the Pauli transfer matrix or affine form of a quantum map. In this form, a
given quantum operation maps Bloch (or Bloch-like) vectors $\varphi$ to
$A\varphi+\tau$. In the case of a random unitary map, the channel will be
unital and thus $\tau=0$, meaning the average map is defined only by $A$, and
special unitary operations of the density operator will correspond to special
orthogonal operations on $\varphi$. Thus, we should look for distributions on
$SO(N^2-1)$ whose sufficient statistics correspond to average elements of
$SO(N^2-1)$. One such candidate is a generalization of the Von Mises-Fisher
distribution called the \textit{matrix-Fisher} distribution.
A random matrix $X$ is said to have the matrix-Fisher distribution with
parameter $F$ if its probability density function (relative to an appropriate
Haar measure) is of the form
\begin{equation}\label{eq:fisher}
%
p_{MF}(X;F) = c_{MF}(F)\exp\left(\Tr\left(F^\top X)\right)\right)
%
\end{equation}
where $c_{MF}(F)$ is a normalizer.
In a similar manner to the way the matrix-Bingham distribution generalizes a
single vector to multiple axes, the matrix-Fisher distribution generalizes
distributions of random angles into random orientations. Unlike the
matrix-Bingham distribution, the matrix-Fisher distribution distinguishes
between $\pm x$. The matrix-Fisher distribution is determined by a natural
parameter matrix $F$ whose dual is the expectation parameter $E[X]$, the
additive mean. Like the matrix-Bingham distribution, the matrix-Fisher
distribution is also a maximum entropy distribution on its associated manifold
with the moment criterion $E[X]$.
In the literature, there are two subtly distinct families of matrix-Fisher
distributions that generate random elements from $SO(N^2-1)$. These have the
same general functional form as (\ref{eq:fisher}), but draw from different
sample spaces, namely the Stiefel manifold $V_{n-1}(\mathbb{R}^n)$ and the
rotation group $SO(n)$. Since $V_{n-1}(\mathbb{R}^n)$ can be uniquely
identified with an element of $SO(n)$, one might think these two distributions
are identical, but they are in fact different. The parameter space for the
matrix-Fisher distribution on $V_{n-1}(\mathbb{R}^n)$ is a $n\times(n-1)$
matrix, whereas the parameter space for the matrix-Fisher distribution on
$SO(n)$ is a $n\times n$ matrix. The two families are related, however,
suppose $F_1$ is the natural parameter for the matrix-Fisher distribution on
$V_{n-1}(\mathbb{R}^n)$ and let $F_2=[F_1;\vec{0}]$ be $F_1$ with column of
zeros appended. The matrix $F_2$ is a valid parameter matrix for the
matrix-Fisher distribution on $SO(n)$, and the two distributions specify the
same distribution of random elements on $SO(n)$. Thus, the matrix-Fisher
distribution on $V_{n-1}(\mathbb{R}^{n})$ is a statistical sub-model of the
matrix-Fisher distribution on $SO(n)$ (in fact a strict sub-model for $n>2$).
Additional details of the difference between these two classes of distributions
can be found in \cite{sei2013properties}. Unfortunately, the version of the
matrix-Fisher distribution on $V_{n-1}(\mathbb{R}^{n})$ appears to be the more
well studied version in statistics, but less relevant for quantum applications,
where it seems unlikely that rank-deficient average rotation will be used
often. That said, using a sign-preserving singular value decomposition, it is
possible to sample from the matrix-Fisher distribution on $SO(n)$ using similar
techniques to the Stiefel manifold version \cite{kent2013new,sei2013properties}.
\subsection{General CPTP Maps}\label{sec:cptp_gen}
As is the case with unitary maps, the matrix-Bingham distribution initially
appears to be an attractive choice for the generation of non-uniform CPTP maps.
Noting that a Choi matrix $\Lambda$ is hermitian, positive semi-definite and
has trace $N$, one might be inclined to use it as the sufficient statistic for a
matrix Bingham distribution, in an identical fashion as the random quantum
state case. However, even though $\Lambda$ is CPTP, the random output map
$XX^\dagger$ is in general not TP (the use of matrix-Bingham distributions
would be ideal for non-uniform CP maps, however). One might be tempted to
accept this and hope that the output maps are nearly TP when $\Lambda$
indicates a strongly concentrated matrix-Bingham distribution, and that the TP
normalization process would not affect the distribution too much. We
attempted to use this technique, and in general it produces CPTP maps that are
not concentrated near the target map.
Another avenue for a statistical model for non-uniform CPTP maps is to note
that the Stiefel representation $\mathcal{S}$ could be generated by a (complex)
matrix von Mises-Fisher distribution. The complex matrix von-Mises Fisher
distribution can be derived from the real valued distribution via a similar
stacking trick as the Bingham distribution. This distribution initially shows
more promise than the matrix-Bingham distribution, but it too presents problems
for more subtle reasons. While a given Choi matrix can be used to generate
Stiefel representation (ignoring for a moment the issue of a non-unique
ordering of the Kraus operators), we have that the average of random Choi
matrix is again a Choi matrix (by convexity), whereas the average of the
equivalent Stiefel representations will lie ``inside'' the Stiefel manifold,
and thus not be the Stiefel representation of a CPTP map. Geometrically, we are
sacrificing convexity for a simple manifold structure. Thus, using an average
Choi matrix to map to an average Stiefel representation $\mathcal{S}$ will
always result in a degenerate (impulsive) distribution that has all probability
mass at $\mathcal{S}$. That said, it is conceptually possible to ``scale`` a
given $\mathcal{S}$ by $(1-\varepsilon)$, $\varepsilon\in(0,1)$ that can be
used to define a distribution that generates random Stiefel representations
with concentration about $\mathcal{S}$ parameterized by $\varepsilon$.
Converting the random Stiefel representations to Choi matrices and taking the
average CPTP map does not empirically appear to be the desired average map, but
it can be made quite close as $\varepsilon\to0$. For some use cases this
approximation may be sufficient, and an example application of this approach is
shown in Section~\ref{sec:example:amp}. It may be possible to compute the
average of the resulting CPTP maps in such a way that the process could be
inverted to find distributions whose average is exactly the desired, but would
still have the issue of Choi matrices being identified with many $\mathcal{S}$.
Having shown that the previously discussed statistical models cannot generate exact
non-uniform samples of arbitrary CPTP maps, we will use the theory of
exponential families to introduce a class of probability distributions
defined on Stiefel manifolds that uses a Choi matrix as its sufficient
statistic.
Given a set of Kraus operators $A_k$, $k=1,\cdots,N^2$, the entries of the
equivalent Choi matrix $\Lambda$ are
\begin{equation}
%
\Lambda_{ij} = \sum_{k}|A_k\rangle\rangle_i|A_k\rangle\rangle^*_j
%
\end{equation}
where $|A_k\rangle\rangle_i$ denotes the $i$th entry of the vector
$|A\rangle\rangle$. Next, consider a reshuffling of the rows of the Stiefel
representation $\mathcal{S}$, defined by the same $A_k$ where
\begin{equation}
%
\xi = \begin{bmatrix}
|A_1\rangle\rangle_1&|A_1\rangle\rangle_{N+1}&\hdots&|A_1\rangle\rangle_{N(N-1)+1}\\
|A_2\rangle\rangle_1&|A_2\rangle\rangle_{N+1}&\hdots&|A_2\rangle\rangle_{N(N-1)+1}\\
\vdots&\vdots&\ddots&\vdots\\
|A_N\rangle\rangle_1&|A_N\rangle\rangle_{N+1}&\hdots&|A_N\rangle\rangle_{N(N-1)+2}\\
|A_1\rangle\rangle_2&|A_1\rangle\rangle_{N+2}&\hdots&|A_1\rangle\rangle_{N(N-1)+2}\\
\vdots&\vdots&\ddots&\vdots\\
|A_N\rangle\rangle_N&|A_N\rangle\rangle_{2N}&\hdots&|A_N\rangle\rangle_{N^2}\\\end{bmatrix}\,,
%
\end{equation}
and decompose this into a block form
\begin{equation}
\xi=\begin{bmatrix} \xi_1&\xi_{N+1}&\hdots&\xi_{N(N-1)+1}\\
\xi_2&\xi_{N+2}&\hdots&\xi_{N(N-1)+2}\\ \vdots &\vdots&\ddots&\vdots\\
\xi_N&\xi_{2N}&\hdots&\xi_{N^2}\\ \end{bmatrix}\,.
\end{equation}
Then, we have that $\Lambda_{ij}=\xi^\dagger_j\xi_i=\langle
\xi_j,\xi_i\rangle$. Since we only re-shuffled rows, $\xi$ is still an element
of the same Stiefel manifold as $\mathcal{S}$. Treating an average Choi matrix
$\Lambda$ as a sufficient statistic for $\xi$ is in effect specifying average
inner products between the components in the block structure of $\xi$ in a way
that is not captured by the matrix-Fisher or matrix-Bingham distributions.
Such a distribution would have exponential form
\begin{equation}\label{eq:schultz_dist}
%
p(\xi;\Theta)=C(\Theta)\exp\left(\sum_{i,j=0}^{N-1}
%
\begin{bmatrix}\xi_{Ni+1}\\\xi_{Ni+1}\\\vdots\\\xi_{Ni+N}\end{bmatrix}^\dagger
%
\mathcal{A}_{i,j}
%
\begin{bmatrix}\xi_{Nj+1}\\\xi_{Nj+1}\\\vdots\\\xi_{Nj+N}\end{bmatrix}
%
%
\right)
%
\end{equation}
where each $\mathcal{A}_{i,j}$ denotes the matrix
\begin{equation}
\mathcal{A}_{i,j}=\begin{bmatrix}\Theta_{Ni+1,Nj+1}I_N &
\Theta_{Ni+1,Nj+2}I_N &\cdots &\Theta_{Ni+1, Nj+N}I_N\\
%
\Theta_{Ni+2,Nj+1}I_N & \Theta_{Ni+2,Nj+2}I_N &\cdots &\Theta_{Ni+2,
Nj+N}I_N\\
%
\vdots & \vdots&\ddots &\vdots\\
\Theta_{Ni+N,Nj+1}I_N & \Theta_{Ni+N,Nj+2}I_N &\cdots &\Theta_{Ni+N,
Nj+N}I_N\\\end{bmatrix}
%
\end{equation}
and $\Theta$ is the natural form of the parameter (i.e., the dual coordinate
system to the expectation parameter $\Lambda/N$ \cite{amari2007methods}). For
positive semi-definite $\mathcal{A}_{i,j}$, the distribution in
(\ref{eq:schultz_dist}) is the complex version of a generalized frame-Bingham
distribution \cite{kume2013saddlepoint,arnold2013statistics}. The real-valued
frame-Bingham distribution can be Gibbs sampled via the techniques of
\cite{hoff2009simulation} as per the discussion in
\cite{kume2013saddlepoint,arnold2013statistics}, and using standard tricks for
converting complex vector operations to real ones (see e.g.,
(\ref{eq:complexreal})) the complex variant can be generated from an
appropriate real-valued distribution. As far as inference procedures for the
frame-Bingham distribution, \cite{kume2013saddlepoint} introduces a procedure
for approximating the normalizer $C(\Theta)$, but we conjecture that given the
additional structure imposed by $E[\xi\xi^\dagger]=\Lambda\otimes\mathbbm{1}_N$
the estimation process is replicated using the traditional Bingham
distribution. Showing this explicitly is an area of future research.
\subsection{Other Relevant Distributions}
There are other non-uniform distributions on that appear in the literature that
may be relevant in the context of quantum information. The matrix-Fisher and
matrix-Bingham distribution are actually subfamilies of the generalized matrix
Bingham-von Mises-Fisher distribution, which has the form
\begin{equation}
%
p_{BMF}(X;A,B,C) = c_{BF}(A,B,C) \exp\left(\Tr \left(C^\dagger X+BX^\dagger
AX\right)\right)\,.
%
\end{equation}
For spherical data (that is data on $V_1(\mathbb{R}^n)$), there are notions of
bivariate extensions of the Fisher distribution (see \cite[\textsection
11.4]{mardia2009directional} for a brief discussion and references).
Additionally, the frame-Bingham distribution described above is itself a
sub-model of products of the generalized matrix-von Mises-Fisher distribution
\cite{kume2013saddlepoint}. Such bivariate and multivariate extensions to the
general Stiefel manifolds will likely be needed for analysis of correlated gate
sequences.
\subsubsection{Non-Exponential Families}
All of the families of probability distributions discussed above are
exponential families, with the exception of the uniform distributions (which
are generally limits of exponential families), and as such enjoy the
sufficiency and maximum entropy properties, among a host of other geometrically
motivated properties \cite{amari2007methods}. There are a number of other
statistical models in the literature that are not exponential families, but
nevertheless have other attractive qualities, and we
will briefly touch on them here. In general, we are primarily exploiting
isomorphisms in the geometry of a given Stiefel manifold and linking them to
classes of quantum objects. Thus, any probability distribution defined on a
relevant Stiefel manifold will define random quantum objects of that class, but
may not have as clear of a link in terms of average states or maps.
The matrix angular central Gaussian (MACG) distribution
\cite{chikuse1990matrix} is another model for axial data that is easily
generated from a matrix normal random variate
\cite{gupta1999matrix,chikuse2012statistics}. Like the matrix Bingham
distribution, it is antipodally symmetric, and is specified by a concentration
matrix $A$ that determines the qualitative shape of distribution on the Stiefel
manifold. Furthermore, MACG and matrix Bingham distributions specified by the
same $A$ will have similar shapes. In fact, it is used in a rejection sampling
scheme for Bingham random variables \cite{kent2013new}. In \cite{kent2013new}
the relationship between the matrix Bingham distribution and MACG distribution
is described as analogous to the one between the normal and Cauchy
distributions.
Another broad class of distributions that are worth mentioning are so-called
wrapped distributions \cite{mardia2009directional}. For the angular case, this
corresponds to taking some random variable $x$ on the real line, and
considering the angular random variable $\theta=x \text{ mod } 2\pi$. Common
examples include the wrapped normal and Cauchy distributions. This concept can
be extended to more general manifolds by considering multi-variate
distributions on the tangent space of the manifold
\cite{chikuse2012statistics}. Due to the wrapping, it can be difficult to
estimate parameters of the underlying distribution but in the case of the
wrapped normal, the von-Mises distribution and its generalizations has been
shown to be close in shape to the wrapped normal and in fact asymptotically
approach it as the distributions become more concentrated.
Due to particular properties of the Fourier transform of the normal
distribution's probability density function, it satisfies the property that the
sum of two wrapped normals is also a wrapped normal
\cite{jammalamadaka2001topics}, a composition property that the von-Mises
distribution does not satisfy. Since angular addition is equivalent to the
composition of rotation operations (in this dimension), further exploration of
wrapped distributions in the more exotic spaces considered here would be useful
if we desire a family probability distributions that model random quantum
operations that are closed under composition.
\section{Examples}
\subsection{Dephasing Noise}
Consider a single qubit system with a Hamiltonian of the form
\begin{equation}\label{eq:dephasing}
%
H_z(t) = \zeta(t)\sigma_Z
%
\end{equation}
where $\zeta(t)$ is a stochastic process. Then, the quantity $U(t)$ defined by
the solution to the differential equation
\begin{equation}
%
i\frac{d}{dt}U(t) = H_z(t)U(t)\,,\,\,\,\,U(0)=\mathbbm{1}_N
%
\end{equation}
is a random unitary operation. Fix $t=T$ and let $U\triangleq U(T)$. Since
all of the $H_z$ are proportional to $\sigma_Z$, $U$ will be of the form
\begin{equation}
%
U =
\begin{pmatrix}\alpha&0\\0&\alpha^*\end{pmatrix}
%
\end{equation}
for some $\alpha\in\mathbb{C}$ with $|\alpha|=1$. Alternatively, since
$|\alpha|=1$, $\alpha=\exp{i\theta}$, where $\theta =
\int_0^T\zeta(t)\,dt$ for a particular trajectory of $\zeta$.
The corresponding Liouvillian to $U$ will be of the form
\begin{equation}\label{eq:liou}
%
\mathcal{L}=U^*\otimes U = \begin{bmatrix}1&0&0&0\\ 0&\alpha^{*^2}&0&0\\
0&0&\alpha^2&0\\ 0&0&0&1\\\end{bmatrix}\,.
%
\end{equation}
Let
$T=[|\sigma_I\rangle,|\sigma_X\rangle,|\sigma_Y\rangle,|\sigma_Z\rangle]^\dagger$,
then the Pauli Transfer matrix is
\begin{equation}\label{eq:ptm}
%
\mathcal{R} = T^{-1}\mathcal{L}T = \begin{bmatrix}1&0&0&0\\ 0&\Re
\alpha^2&\Im\alpha^2&0\\ 0&-\Im\alpha^2&\Re\alpha^2&0\\ 0&0&0&1\end{bmatrix}\,,
%
\end{equation}
which has corresponding affine form
\begin{equation}
%
\varphi\to\underbrace{\begin{bmatrix}\Re
\alpha^2&\Im\alpha^2&0\\ -\Im\alpha^2&\Re\alpha^2&0\\
0&0&1\end{bmatrix}}_{A}\varphi+\underbrace{\begin{bmatrix}0\\0\\0\end{bmatrix}}_{\tau}
%
\end{equation}
Let $A$ denote the $3\times3$ sub-matrix in the lower-right corner of
$\mathcal{R}$. Since $U$ is special unitary, $A\in SO(3)$, and thus $A$ is a random
element of $SO(3)$. Suppose we are given an \textit{average} dephasing Pauli
transfer matrix $\bar{A}$ generated by the dynamics in (\ref{eq:dephasing}),
then it is
\begin{equation}
%
\bar{A} = \begin{bmatrix}\Re E[\alpha^2]&\Im E[\alpha^2]&0\\ -\Im
E[\alpha^2]&\Re E[\alpha^2]&0\\ 0&0&1\end{bmatrix}\,.
%
\end{equation}
To sample random matrices from $SO(3)$ using the matrix-Fisher distribution
with this average $\bar{A}$ is problematic, since it is degenerate, as the
third column must always be $[0,0,1]^\top$. Instead, the matrix-Fisher
distribution on $SO(2)$ should be used and used to generate the upper-left
$2\times2$ portion of the random $SO(3)$ element. Furthermore, the
matrix-Fisher distribution on $SO(2)$ is equivalent to the matrix-Fisher
distribution on $V_1(\mathbb{R}^2)$ \cite{sei2013properties} which is in turn
equivalent to the Von Mises distribution on the circle
\cite{mardia2009directional}, although the latter is typically expressed in
terms of an angle $\theta$ rather than an element of $\mathbb{R}^2$.
Figure~\ref{fig:dephasing} shows random dephasing operations generated using
the above scheme, with average dephasing strength $E[\alpha^2] = .9$.
\begin{figure}[h!]
\centering
\begin{tabular}{cc}
\subfloat[][]{\label{fig:dephasing:a}
\includegraphics[width=.4\columnwidth]{DephasingSphere}
}
&
\subfloat[][]{\label{fig:dephasing:b}
\includegraphics[width=.5\columnwidth]{DephasingHist}
}
\end{tabular}
\caption{\protect\subref{fig:dephasing:a} Randomly sampled dephasing operations
drawn from the Von Mises distribution as described above and applied to the
Bloch vector $[0,1,0]^\top$. \protect\subref{fig:dephasing:b} Histogram of
random samples of dephasing angle $\theta$ sampled according to the Von
Mises distribution corresponding to the data in \protect\subref{fig:dephasing:a}.}
\label{fig:dephasing}
\end{figure}
\subsection{Depolarizing noise}
The Choi matrix for Pauli channel depolarizing noise is
$(1-p_x-p_y-p_z)|\sigma_I\rangle\rangle\langle\langle \sigma_I| +
p_x|\sigma_X\rangle\rangle\langle\langle
\sigma_X|+p_y|\sigma_Y\rangle\rangle\langle\langle
\sigma_Y|+p_z|\sigma_Z\rangle\rangle\langle\langle \sigma_Z|$, with
$p_i\in[0,1]$, $\sum p_i\leq 1$. Converting this to a Pauli transfer matrix
yields
\begin{equation}
%
\begin{aligned}
%
\mathcal{R} &= (1-p_x-p_y-p_z)I_4\\
%
&+p_x\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}
+p_y\begin{bmatrix}1&0&0&0\\0&-1&0&0\\0&0&1&0\\0&0&0&-1\end{bmatrix}
+p_z\begin{bmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&1\end{bmatrix}
%
\end{aligned}
%
\end{equation}
so the corresponding affine form is
\begin{equation}\label{eq:depol_affine}
%
\varphi\to \underbrace{\begin{bmatrix} 1 -2(p_y+p_z)&0&0\\0&1-2(p_x+p_z)&0\\
0&0&1-2(p_x+p_y)\end{bmatrix}}_{A_p}\varphi+\begin{bmatrix}0\\0\\0\end{bmatrix}\,.
%
\end{equation}
Note that this is consistent with (\ref{eq:ptm}) under the assumption that
$\zeta(t)$ in (\ref{eq:dephasing}) is zero mean so that $E[\Im\alpha^2]=0$. If
we treat the contraction matrix $A_p$ in (\ref{eq:depol_affine}) as an average of random
elements in $SO(3)$, we can apply the matrix-Fisher distribution on
$SO(3)$. Figure~\ref{fig:depol1} shows random samples of depolarizing
channels drawn from the matrix-Fisher distribution on $SO(3)$ whose average
corresponds to $p_x=0.001$, $p_y=0.01$ and $p_z=0.1$.
\begin{figure}[h!]
\centering
\begin{tabular}{c}
\includegraphics[width=.4\columnwidth]{DepolarizingSphere}
\end{tabular}
\caption{Randomly generated unitary maps whose average is the depolarizing
channel with $p_x=0.001$, $p_y=0.01$, and $p_z=0.1$ sampled according to the
Von Mises-Fisher distribution on $SO(3)$ applied to Bloch vector
$[\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2},0]^\top$.
}
\label{fig:depol1}
\end{figure}
Since the depolarizing channel is expressed as a mixture over unitary maps, it
is natural to treat a depolarizing channel as the average of random unitary
operations as in Figure~\ref{fig:depol1}. However, the same depolarizing
channel in Choi form could serve as the sufficient statistic for a
frame-Bingham distribution as described in Section~\ref{sec:cptp_gen}. In
this case, the random CPTP maps generated would be non-unitary with
probability 1. Figure~\ref{fig:depol2:a} shows random samples generated
from the frame-Bingham distribution applied to the same input state as
Figure~\ref{fig:depol1}. Note that the shapes of the output distributions
are are distinctly different. Furthermore, although it cannot be seen from
the figures themselves, since the operations in Figure~\ref{fig:depol1} are
unitary, the output states are all on the surface of the Bloch sphere.
However, in the case of the frame-Bingham distribution, the states are
pulled in to the Bloch sphere, resulting in a distribution of norms of
Bloch vectors as in Figure~\ref{fig:depol2:b}.
\begin{figure}[h!]
\centering
\begin{tabular}{cc}
\subfloat[][]{\label{fig:depol2:a}
\includegraphics[width=.4\columnwidth]{DepolarizingSphereFrameBingham}
}
&
\subfloat[][]{\label{fig:depol2:b}
\includegraphics[width=.5\columnwidth]{DepolarizingSphereBlochAmplitudeFrameBingham}
}
\end{tabular}
\caption{\protect\subref{fig:depol2:a} Randomly generated CPTP maps whose
average is the depolarizing channel with $p_x=0.001$, $p_y=0.01$, and
$p_z=0.1$ sampled according to the frame-Bingham distribution applied to
Bloch vector $[\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2},0]^\top$.
%
\protect\subref{fig:depol2:b} Histogram of norms of output Bloch vectors
corresponding to the data in \protect\subref{fig:depol2:a}.}
\label{fig:depol2}
\end{figure}
\subsection{Amplitude Damping}{\label{sec:example:amp}
An example amplitude damping channel \cite{nielsen2010quantum}, parameterized
by $\gamma$ is defined by Kraus operators
\begin{equation}
%
\begin{aligned}
%
A_1 &= \begin{bmatrix} 1&0\\0&\sqrt{1-\gamma}\end{bmatrix}\,,&&&
%
A_2 &= \begin{bmatrix} 0&\sqrt{\gamma}\\0&0\end{bmatrix}\,.
%
\end{aligned}
%
\end{equation}
The corresponding PTM is
\begin{equation}
%
\mathcal{R} =
\begin{bmatrix}1&0&0&0\\0&\sqrt{1-\gamma}&0&0\\0&0&\sqrt{1-\gamma}&0\\
\gamma&0&0&1-\gamma\end{bmatrix}
%
\end{equation}
which the first column indicates is non-unital, meaning the techniques in
Section~\ref{sec:cptp_unit} do not apply. Furthermore, amplitude damping
belongs to a class of CPTP maps that cannot be generated as the average of a
random (i.e., non-constant) CPTP map, as discussed in
Section~\ref{sec:representable}.
That said, we can generate distributions whose average is nearly the amplitude
damping channel by using the Stiefel representation
\begin{equation}
%
\mathcal{S} = \begin{bmatrix} 1&0\\0&\sqrt{1-\gamma}\\
%
0&\sqrt{\gamma}\\0&0\\ 0&0\\ 0&0\\ 0&0\\ 0&0\\ \end{bmatrix}\,,
%
\end{equation}
and set $(1-\varepsilon)\mathcal{S}$ (for small $\varepsilon$) as the average
value for complex matrix-Fisher distribution as described in
Section~\ref{sec:cptp_gen}. Figure~\ref{fig:ampdamp1} shows the action of
randomly sampled approximate amplitude damping channels ($\gamma=0.01$) on the
initial state $|0\rangle$, for decreasing $\varepsilon$. For reference, we
find empirically that for $\varepsilon=0.001$ the diamond norm error between
the average approximate amplitude damping channel and the actual is around
$0.05$.
\begin{figure}[h!]
\centering
\begin{tabular}{c}
\includegraphics[width=.4\columnwidth]{AppxAmpDamping}
\end{tabular}
\caption{Attempts to approximate an amplitude
damping channel ($\gamma=0.01)$ using the matrix-Fisher distribution on the Stiefel
representation for varying $\varepsilon$. Random samples are applied to
the state $|0\rangle$ for presentation.}
\label{fig:ampdamp1}
\end{figure}
In addition to the matrix-Fisher approach, we can also use the frame-Bingham
distribution to attempt to approximate the amplitude damping channel using the
corresponding Choi matrix
\begin{equation}
%
\Lambda_\gamma = \begin{bmatrix}1&0&0&\sqrt{1-\gamma}\\
0&0&0&0\\
0&0&\gamma&0\\
\sqrt{1-\gamma}&0&0&1-\gamma\end{bmatrix}\,\,.
%
\end{equation}
The
first caveat to note is that the Choi matrix for an amplitude damping channel
is in some sense singular, since it indicates that $\xi_2$ in the the
alternative stiefel representation $\xi$ is always 0. This singularity is
easily handled by a slight modification to the projection step in
\cite{hoff2009simulation} for the first column of $\xi$. Even with this
additional step, the frame-Bingham distribution is still not capable of
averaging to an amplitude damping channel. Instead, estimating the parameters
in an attempt match the diagonal terms of the Choi matrix with a frame-Bingham
distribution, while trying to concentrate the distribution as strongly as
possible is shown in Figure~\ref{fig:ampdamp2:a}.
\begin{figure}[h!]
\centering
\begin{tabular}{cc}
\subfloat[][]{\label{fig:ampdamp2:a}
\includegraphics[width=.4\columnwidth]{AmplitudeDampingCPTP}
}
&
\subfloat[][]{\label{fig:ampdamp2:b}
\includegraphics[width=.4\columnwidth]{AmplitudeDampingMixture}
}
\end{tabular}
\caption{\protect\subref{fig:depol2:a} Randomly generated CPTP maps whose
average is the depolarizing channel with $p_x=0.001$, $p_y=0.01$, and
$p_z=0.1$ sampled according to the frame-Bingham distribution applied to
Bloch vector $[\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2},0]^\top$.
%
\protect\subref{fig:depol2:b} Histogram of norms of output Bloch vectors
corresponding to the data in \protect\subref{fig:depol2:a}.}
\label{fig:depol2}
\end{figure}
An alternative approximation that also uses the frame-Bingham distribution is
to further restrict the distribution generated in the projection step to be a
random amplitude damping channel. Again, the average of random amplitude
damping channels will not be an amplitude damping channel, but we attempt to
approximate by matching the diagonal terms of the Choi matrix. This approach
is shown in Figure~\ref{fig:ampdamp2:b}. In this case, since every operation
produced is an amplitude damping channel for some random $\gamma$, the map of
the state $|0\rangle$ lies on the axis, unlike the other approximation cases.
\subsection{A Non-Unital Example}
Consider the amplitude damping channel above followed by the depolarizing
channel. In PTM form, this channel is
\begin{equation}\label{eq:nonunitalexample}
\mathcal{R} = \begin{bmatrix}1&0&0&0\\
0 & (1-2(p_y+p_z))\sqrt(1-\gamma) & 0 & 0\\
0 & 0 & (1-2(p_x+p_z))\sqrt(1-\gamma) & 0\\
(1-2(p_x+p_y))\gamma &0 &0 & (1-2(p_x+p_y))(1-\gamma)\\\end{bmatrix}
\end{equation}
which is non-unital, but unlike the pure amplitude damping channel, it can be
represented by a frame-Bingham distribution as shown in Figure~\ref{fig:nonunital}.
\begin{figure}[h!]
\centering
\begin{tabular}{cc}
\subfloat[][]{\label{fig:nonunital:a}
\includegraphics[width=.4\columnwidth]{NonUnitalFrameBingham}
}
&
\subfloat[][]{\label{fig:nonunital:b}
\includegraphics[width=.5\columnwidth]{NonUnitalAmplitudeFrameBingham}
}
\end{tabular}
\caption{\protect\subref{fig:depol2:a} Randomly generated CPTP maps whose
average is the nonunital channel in (\ref{eq:nonunitalexample}) with
$\gamma=0.01$, $p_x=0.001$, $p_y=0.01$, and
$p_z=0.1$ sampled according to the frame-Bingham distribution applied to
Bloch vector $[\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2},0]^\top$.
%
\protect\subref{fig:depol2:b} Histogram of norms of output Bloch vectors
corresponding to the data in \protect\subref{fig:nonunital:a}.}
\label{fig:nonunital}
\end{figure}
\section{CPTP Maps Representable as Random CPTP Maps}\label{sec:representable}
In this section we will give a geometric characterization the set of CPTP maps
(of a given size) which can be represented by the average of a random CPTP map
of the same size. Note that this applies to any method of generating random
CPTP maps beyond what is presented here, including Pauli-channel error models
and stochastic master equation approaches.
For a convex set $\mathcal{C}$, a point $x\in\mathcal{C}$ is said to be an
\textit{extreme} point if $\mathcal{C}\setminus\{x\}$ is still a convex set. This
definition implies that an extreme point cannot be represented as the convex
combination of other points in $\mathcal{C}$, and as such an extreme point $x$
cannot be the average of a non-trivial random variable taking elements in
$\mathcal{C}$. Note that there is a distinction between extreme points and
\textit{boundary} points of $\mathcal{C}$, all extreme points are boundary
points, but not all boundary points are extreme (consider the edges of a
square, for example). Points that are not extreme points must be equal to some
convex combination of extreme points, and thus can be trivially represented as
the average over these points when the extreme points are sampled according to
their weight in the convex combination. Thus, the question of representability
in this manner is an exercise in classifying the extreme points of the space of
CPTP maps.
For general $N$, this appears to be an open question, however for the case where
$N=2$, the convex geometry of CPTP maps has been completely
characterized in \cite{ruskai2002analysis}. In particular they show
\begin{theorem}[Theorem 13 in \cite{ruskai2002analysis}]
%
If a CPTP map
written in Kraus form $\{A_k\}$ requires exactly two Kraus operators, then
it is either unital or an extreme point.
\end{theorem}
Since the amplitude damping channel requires two Kraus operators and is
non-unital, it must be an extreme point. Thus, it cannot serve as the average
for any non-trivial random CPTP map operating on $N\times N$ density
operators, and as such we were unable to exactly reproduce the amplitude
damping channel in Section~\ref{sec:example:amp}. Furthermore, any random CP
map on $N\times N$ density operators that has an amplitude damping channel as
its average, must take values that are not TP with non-zero probability (c.f.
\cite{burgarth2016can}).
\section{Conclusion}
In this manuscript, we have presented a number of connections between
distributions studied in directional and orientation statistics to various
representations of quantum states and quantum operations. Furthermore, by
connecting the notion of an average quantum state or operation to a sufficient
statistic of an exponential family we are able to define the unique probability
distribution that maximizes entropy while still satisfying the target average.
From a modeling and simulation perspective, we foresee a number of applications
of this work to the study of quantum systems. The generation of random quantum
states and operations with a specified average that can be sampled in an
absolutely continuous manner (as compared to the standard Pauli error channel)
can have a drastic effect on simulation results for e.g., error correction and
threshold computations \cite{BarnesPaper}. Additionally, this sort of
statistical foundation allows for the exploration of correlated quantum errors
which will be present in a non-Markovian environment.
From an inference and analytic perspective, we expect that the statistical
models presented here will be useful for developing statistical tests and
analysis. For example, tests for correlation from orientation statistics could
be adapted to provide tests and measures for non-Markovianity in quantum
channels. One can also envision adapting concepts such as graphical models
from applied statistics and machine learning to complex inference problems of
relevance to quantum information, such as tomography.
The work here also touches on some interesting questions on the geometry of
quantum operations, for example, further investigation is warranted of the
exact connection between the vastly different geometries of the Stiefel
representation (a differential geometry on a surface) and the convex geometry
of Choi matrices. Such a viewpoint may be useful in characterizing extreme
points in the convex geometry of quantum operations. Additionally, there may be
interesting connections between the information geometry of the exponential
families presented here and traditional concepts from quantum information. For
example, does the Kullback-Leibler divergence between elements of the
exponential family relate to any known distance measure between quantum
operations or are there alternative information geometries which maximize
expected diamond norm?
\acknowledgments{This project was supported by the Intelligence Advanced
Research Projects Activity via Department of Interior National Business Center
contract number 2012-12050800010. The U.S. Government is authorized to
reproduce and distribute reprints for Governmental purposes notwithstanding any
copyright annotation thereon. The views and conclusions contained herein are
those of the authors and should not be interpreted as necessarily representing
the official policies or endorsements, either expressed or implied, of IARPA,
DoI/NBC, or the U.S. Government.}
\bibliographystyle{apsrev4-1}
|
1,116,691,500,811 | arxiv | \section{Introduction}
\label{s:introduction}
\subsection{Background and main results}
\label{ss:1.1}
Let $n\geq 2$ be a positive integer and $\Delta$ be the unit disc in $\mathbb C$. Let $p: (\mathcal{X}, \mathcal L)\rightarrow \Delta$ be a flat polarized degenerating family of $n$-dimensional Calabi-Yau varieties. More precisely, we assume that $\mathcal X$ is normal with $X_t\equiv p^{-1}(t)$ smooth for $t\neq 0$ and $X_0$ singular, the relatively canonical line bundle $K_{\mathcal X/\Delta}$ is trivial, and $\mathcal L$ is relatively ample. Yau's proof of the Calabi conjecture \cite{Yau} yields for each $t\neq 0$ a unique smooth Ricci-flat K\"ahler metric (\emph{the Calabi-Yau metric}) $\omega_{CY,t}$ on $X_t$ in the cohomology class $2\pi c_1(\mathcal L|_{X_t})$. The following is a folklore problem:
\begin{problem}
\label{pb:1-1} What is the limiting geometric behavior of $(X_t,\omega_{CY,t})$ in the Gromov-Hausdorff sense as $|t|\to0$, and how to describe its connection with the algebraic geometry associated to this degeneration?
\end{problem}
A particularly intriguing and challenging situation is when \emph{collapsing} occurs, or equivalently, when the diameters of $(X_t,\omega_{CY, t})$ tend to infinity. In this case if we rescale the diameters to be fixed then the Gromov-Hausdorff limit is a lower dimensional space. In the special case of large complex structure limits, the above problem is related to the limiting version of the SYZ Conjecture, as proposed by Gross-Wilson \cite{GW} and Kontsevich-Soibelman \cite{KonSo}. Based on the main results in this paper (Theorem \ref{t:main-theorem}), we will propose a more general conjecture (Conjecture \ref{cj:generalized-SYZ}) for general degenerations.
The goal in this paper is to give an answer to Problem \ref{pb:1-1} for a
special class of complex structure degenerations. More precisely, we give a complete description of the collapsing geometry of the family of Calabi-Yau metrics {along the degeneration of complex structures}. We mainly focus on the special class of examples below, but the crucial techniques involved apply to more abstract situations, and the strategy can be extended to deal with more general classes of collapsing (see the discussions in Section \ref{s:discussions}).
Let $f_1, f_2, f$ be homogeneous polynomials in $n+2$ variables of degree $d_1$, $d_2$ and
$d_1+d_2=n+2$ respectively. Let $\mathcal X\subset \mathbb C\mathbb{P}^{n+1}\times \Delta$ be the family of Calabi-Yau hypersurfaces $X_t$ in $\mathbb C\mathbb{P}^{n+1}$ defined by the equation $F_t(x)=0$ (see Figure \ref{f: the degeneration}), where
\begin{equation} \label{eqn1.1}
F_t(x)\equiv f_1(x)f_2(x)+tf(x),\quad x\in\mathbb{C}\mathbb{P}^{n+1},
\end{equation}
and $t$ is a complex parameter on the unit disc $\Delta\subset \mathbb C$. The projection map $p: \mathcal X\rightarrow \Delta$ is the natural one. The relative ample line bundle $\mathcal L$ comes from restriction of the natural $\mathcal{O}(1)$ bundle over $\mathbb C\mathbb{P}^{n+1}$.
We further assume that $f_1, f_2, f$ are sufficiently general so that the following hold:
\begin{enumerate}[(i)]
\item $Y_1=\{f_1(x)=0\}$ and $Y_2=\{f_2(x)=0\}$ are smooth hypersurfaces in $\mathbb C\mathbb{P}^{n+1}$;
\item $X_t$ is smooth for $t\neq 0$ and $|t|\ll1$;
\item $D=\{f_1(x)=f_2(x)=0\}$ is a smooth complete intersection in $\mathbb C\mathbb{P}^{n+1}$;
\item $H=\{f_1(x)=f_2(x)=f(x)=0\}$ is a smooth complete intersection in $\mathbb C\mathbb{P}^{n+1}$.
\end{enumerate}
Then $Y_1 \cap Y_2 = D$ and $H \subset D$. Naturally we may view $Y_1, Y_2$ and $D$ as subvarieties in $X_0\subset \mathcal X$, and $X_0=Y_1\cup Y_2$, as illustrated in Figure \ref{fig1.1}. By adjunction formula, $D$ is also Calabi-Yau, and sits as an anti-canonical divisor in both $Y_1$ and $Y_2$. Moreover, for $i\in\{1,2\}$, the normal bundle $L_i$ of $D$ in $Y_i$ is given by $\mathcal{O}(d_{3-i})|_D$.
Notice the total space $\mathcal X$ has singularities along $H\times \{0\}$, and transverse to $H\times \{0\}$ the singularities are locally modeled on a three dimensional ordinary double point. The dual intersection complex of the singular fiber $X_0$ is a one dimensional interval.
For $i\in\{1, 2\}$, it has been shown by Tian-Yau \cite{TY} that $Y_i\setminus D$ admits a complete Ricci-flat K\"ahler metric $\omega_{TY, i}$ with interesting asymptotics governed by the \emph{Calabi model space} (c.f. Section \ref{ss:Calabi model space}) , and the latter in turn depends on the Calabi-Yau metric on $D$ in the cohomology class $2\pi \mathcal{O}(1)|_D$. Notice that the construction of Tian-Yau can be viewed as a generalization of Yau's proof of the Calabi conjecture to the non-compact case, but it remains an interesting question that in what sense the metrics $\omega_{TY, i}$ are uniquely or canonically associated to the pair $(Y_i, D)$.
\begin{figure}
\label{fig1.1}
\begin{tikzpicture}[scale=0.7]
\draw (-5, 3) to [out=-75, in=75] (-5, -3);
\draw (-5, 3) to (-3.5, 4);
\draw (-5, -3) to (-3.5, -2);
\draw (-3.5, 4) to [out=-75, in=75] (-3.5, -2);
\draw (-3, 3) to [out=-75, in=75] (-3, -3);
\draw (-3, 3) to (-1.5, 4);
\draw (-3, -3) to (-1.5, -2);
\draw (-1.5, 4) to [out=-75, in=75] (-1.5, -2);
\draw (3.0, 3) to [out=-75, in=75] (3.0, -3);
\draw (3.0, 3) to (4.5, 4);
\draw (3, -3) to (4.5, -2);
\draw (4.5, 4) to [out=-75, in=75] (4.5, -2);
\draw (1.0, 3) to [out=-75, in=75] (1.0, -3);
\draw (1.0, 3) to (2.5, 4);
\draw (1.0, -3) to (2.5, -2);
\draw (2.5, 4) to [out=-75, in=75] (2.5, -2);
\draw (-1, 3) to (0.5, 4);
\draw (-1, 3) to [out=-65, in=80] (-0.7, -0.5);
\draw (-0.7, -0.5) to [out=-70, in=65] (-1, -3);
\draw (-1, -3) to (0.5, -2);
\draw (0.5, 4) to [out=-65, in=80] (0.8, 0.5);
\draw (0.8, 0.5) to [out=-70, in=65] (0.5, -2);
\draw[blue, very thick] (-0.7, -0.5) to (0.8, 0.5);
\node[red] at (0.05, 0) {$\bullet$};
\node[red] at (-1.75, 0.1) {$\bullet$};
\node[red] at (-3.75, 0.4) {$\bullet$};
\node[red] at (2.25, 0.2) {$\bullet$};
\node[red] at (4.25, 0.6) {$\bullet$};
\node at (-4, 2.5) {$X_t$};
\node at (0, 2.5) {$Y_1$};
\node at (0, -1.5) {$Y_2$};
\node[blue] at (0.3, 0.7) {$D$};
\node[red] at (-6, 1.4) {$H\times\{t\}$};
\draw[->] (-5.2, 1.3) to (-3.9, 0.5);
\node at (0, -4.5) {$X_0=Y_1\cup_D Y_2$};
\draw[->] (0, -4) to (0, -2.5);
\draw[thick, red, dashed] plot[smooth] coordinates {(-3.75, 0.4) (-1.75, 0.1) (0.05, 0) (2.25, 0.2) (4.25, 0.6)};
\end{tikzpicture}
\caption{The algebraic family $\mathcal X$}
\label{f: the degeneration}
\end{figure}
The main result of this paper provides an answer to Problem \ref{pb:1-1} for the above family. Roughly speaking, we prove that for $|t|$ sufficiently small, the Calabi-Yau metrics $\omega_{CY, t}$ on $X_t$ can be constructed by gluing the Tian-Yau metrics on $Y_1\setminus D$ and $Y_2\setminus D$, together with an approximate Calabi-Yau metric on a neck region. Let us denote the {\it renormalized metric} and the {\it renormalized measure} as follows,
\begin{align}
\tilde\omega_{CY,t} \equiv (\diam_{\omega_{CY, t}}(X_t))^{-2}\cdot \omega_{CY, t}
,\quad d\underline{\nu}_t\equiv (\Vol_{\omega_{CY, t}}(X_t))^{-1}\cdot\dvol_{\omega_{CY, t}},
\end{align}
so that $\tilde\omega_{CY, t}$ has unit diameter and $d\underline{\nu}_t$ is a probability measure.
The more precise version of the main result is as follows (see Figure \ref{f:glued-manifold} for a geometric description).
\begin{theorem}
\label{t:main-theorem}
The following statements hold:
\begin{enumerate}
\item (Diameter estimate) There exists a constant $C>0$ such that for $0<|t|\ll1$ we have
\begin{equation}
C^{-1}\cdot (\log|t|^{-1})^{\frac{1}{2}}\leq \diam_{\omega_{CY, t}}(X_t)\leq C \cdot (\log|t|^{-1})^{\frac{1}{2}}.
\end{equation}
\item (Convergence) As $|t|\rightarrow 0$, the spaces $(X_{t}, \tilde\omega_{CY,t}, d\underline{\nu}_t)$ converge, in the measured Gromov-Hausdorff sense, to $(\mathbb{I}, dx^2, d\underline{\nu}_{\mathbb{I}})$, where $\mathbb{I}= [0,1]\subset\mathbb{R}$ is a unit interval endowed with the standard metric $dx^2$ and the singular measure has an explicit expression
\begin{align}
d\underline{\nu}_{\mathbb{I}} &=C_{n,d_1,d_2}\cdot \mathscr{V}_{\mathbb{I}}(x)dx,\\ \mathscr{V}_{\mathbb{I}}(x)&=
\begin{cases}
(\frac{x}{d_1})^{\frac{n-1}{n+1}}, & x\in[0,\frac{d_1}{d_1+d_2}],
\\
(\frac{1-x}{d_2})^{\frac{n-1}{n+1}}, & x \in[\frac{d_1}{d_1+d_2}, 1],
\end{cases}\end{align}
for some constant $C_{n,d_1,d_2}>0$ depending only on $n$, $d_1$ and $d_2$.
\item (Singular fibration) For $0<|t|\ll 1$, there is a continuous surjective map $\mathcal{F}_t: X_t\rightarrow\mathbb{I}$ with the following properties:
\begin{enumerate}[(a)]
\item (Almost distance preserving) For all $p,q\in X_t$,
\begin{equation}\Big||\mathcal{F}_t(p)-\mathcal{F}_t(q)|-d_{\tilde\omega_{CY,t}}(p,q)\Big|<\tau(t),\quad \lim\limits_{t\to0}\tau(t)=0.\end{equation} \item (Regular fiber) For each $x\in (0,1)\setminus\{\frac{d_1}{d_1 + d_2}\}$, $\mathcal{F}_t^{-1}(x)$ is an $S^1$-bundle over $D$ with the first Chern class \begin{align}
c_1(\mathcal{F}_t^{-1}(x))
=
\begin{cases}
c_1(\mathcal{O}(d_2)|_D), & x\in (0,\frac{d_1}{d_1+d_2}),
\\
c_1(\mathcal{O}(-d_1)|_D), & x\in (\frac{d_1}{d_1+d_2},1).
\end{cases}
\end{align}
\item (Singular fiber and deepest bubble) The fiber $\mathcal{F}_t^{-1}(\frac{d_1}{d_1 + d_2})$ is a singular $S^1$-fibration over $D$ with vanishing circles along $H\subset D$. Suitable rescalings around the vanishing circles on $\mathcal{F}_t^{-1}(\frac{d_1}{d_1 + d_2})$ converge to the Riemannian product $\mathbb C_{TN}^2\times\mathbb C^{n-2}_{flat}$, where $\mathbb{C}_{TN}^2$
is the Taub-NUT space and $\mathbb C^{n-2}_{flat}$ is the Euclidean space.
\item (End bubble) Suitable rescalings around the ends $x=0$ and $x=1$ converge to the complete Tian-Yau metrics $\omega_{TY, 1}$ and $\omega_{TY, 2}$ on $Y_1\setminus D$ and $Y_2\setminus D$ respectively.
\end{enumerate}
\end{enumerate}
\end{theorem}
\begin{figure}
\begin{tikzpicture}[scale=0.8]
\draw (0,0) to [out = 90, in = 180] (1,.5) to (2,.5);
\draw (2,.5) to [out = 0, in = 235] (4,2);
\draw (4,2) to [out = 55, in = 180] (6,3);
\draw (6,3) to [out = 0, in = 125] (8,2);
\draw (8,2) to [out = -55, in = 180] (10,.5);
\draw (10,.5) to (11,.5);
\draw (11,.5) to [out = 0, in = 90] (12,0);
\draw (0,0) to [out = 270, in = 180] (1,-.5) to (2,-.5);
\draw (2,-.5) to [out = 0, in = 125] (4,-2);
\draw (4,-2) to [out = -55, in = 180] (6,-3);
\draw (6,-3) to [out = 0, in = -125] (8,-2);
\draw (8,-2) to [out = 55, in = 180] (10,-.5);
\draw (10,-.5) to (11,-.5);
\draw (11,-.5) to [out = 0, in = -90] (12,0);
\draw[blue](2,0) ellipse (.2 and .5);
\draw[red](2,0) ellipse (.2 and .05);
\draw[blue] (3,0) ellipse (.1 and .82);
\draw[red](3,0) ellipse (.1 and .05);
\draw[blue](4,0) ellipse (.05 and 2);
\draw[red ](4,0) ellipse (.05 and .02);
\draw[blue](10,0) ellipse (.2 and .5);
\draw[red](10,0) ellipse (.2 and .05);
\draw[blue](9,0) ellipse (.1 and .82);
\draw[red](9,0) ellipse (.1 and .05);
\draw[blue](8,0) ellipse (.05 and 2);
\draw[red ](8,0) ellipse (.05 and .02);
\draw[red](5,0) ellipse (.03 and .02);
\draw[blue](5,0) ellipse (.03 and 2.82);
\draw[red](7,0) ellipse (.03 and .02);
\draw[blue](7,0) ellipse (.03 and 2.82);
\draw (1.2,-.05) arc [ radius = .5, start angle = 45, end angle = 135];
\draw (11.5,-.05) arc [ radius = .5, start angle = -45, end angle = -135];
\draw[very thick] (6,-1.5) -- (6, 1.5);
\node at (6.25, 0) {$H$};
\draw (0,-5) -- (12,-5);
\draw (0,-5.1) -- (0,-4.9);
\draw (12,-5.1) -- (12,-4.9);
\draw (6,-5.1) -- (6,-4.9);
\draw[->, very thin] (0,-.3) -- (0,-4.5);
\draw[->, very thin] (1.25,-.6) -- (.2,-4.5);
\draw[->, very thin] (3,-.9) -- (3,-4.5);
\draw[->, very thin] (6,-3.1) -- (6,-4.5);
\draw[->, very thin] (5,-2.9) -- (5.8,-4.5);
\draw[->, very thin] (12,-.3) -- (12,-4.5);
\draw[->, very thin] (10.75,-.6) -- (11.8,-4.5);
\draw[->, very thin] (9,-.9) -- (9,-4.5);
\draw[->, very thin] (6,-3.1) -- (6,-4.5);
\draw[->, very thin] (7,-2.9) -- (6.2,-4.5);
\draw[->, thick] (-2,1.5) -- (-2,-4.5);
\node at (-1.5,-1.5) {$\mathcal{F}_t$};
\node at (1,.8) {$Y_1\setminus D$};
\node at (11,.8) {$Y_2\setminus D$};
\node at (6,3.3) {$\mathcal M$};
\node at (0,-5.5) {$x=0$};
\node at (6,-5.5) {$x=\frac{d_1}{d_1+d_2}$};
\node at (-2,2.0) {$X_t$};
\node at (-2,-5.0) {$\mathbb{I}$};
\node at (12,-5.5) {$x=1$};
\end{tikzpicture}
\caption{The collapsing Calabi-Yau metric $\omega_{CY,t}$ on $X_t$}
\label{f:glued-manifold}
\end{figure}
\begin{remark}
A consequence of the theorem is that the Tian-Yau metrics $\omega_{TY, i}$ on $Y_i\setminus D$, though not a priori canonical by construction, is indeed canonically associated to the above degeneration family of compact Calabi-Yau manifolds.
\end{remark}
\begin{remark}
Using the gluing construction in this paper, we obtain a fairly precise description of the multi-scale collapsing of the Calabi-Yau metrics $(X_t, \tilde\omega_{CY, t})$ as $t\rightarrow 0$. For each $x\in (0, 1)\subset \mathbb{I}$, not only does $\mathcal{F}^{-1}_{t}(x)$ collapse as $t\rightarrow0$, each $S^1$-fiber of $\mathcal{F}^{-1}_{t}(x)$ collapses also, but at a faster rate.
This iterated collapsing in effect gives rise to various bubbles of different geometric properties.
See Section \ref{ss:regularity-scales} for detailed studies on the rescaled limit geometries. \end{remark}
\begin{remark}
Transverse to the divisor $H\subset D$, the singular fibration $\mathcal{F}_t$ is topologically modeled on the composition of the Hopf fibration
\begin{equation}\mathbb C^2\rightarrow \mathbb C\oplus \mathbb R\simeq \mathbb R^3, \quad (z_1, z_2)\mapsto \Big(z_1z_2, \frac{1}{2}(|z_1|^2-|z_2|^2)\Big), \end{equation}
and the projection map
\begin{equation}\mathbb C\oplus \mathbb R\rightarrow \mathbb R,\quad (y, z)\mapsto z.\end{equation}
\end{remark}
\begin{remark}
In Theorem \ref{t:main-theorem},
the singularities of $d\underline{\nu}_{\mathbb{I}}$ coincide with the singular loci of the map $\mathcal F_t$. The explicit formula of $d\underline\nu_{\mathbb{I}}$ has applications in understanding the properties of general collapsing limits with lower Ricci curvature bounds. See Section \ref{ss:renormalized-measure} for more discussions.
\end{remark}
It is worth emphasizing that, in the volume collapsing case, Theorem \ref{t:main-theorem} provides the first {\it explicit and precise} descriptions on the metric limiting behavior and singularity formation along an algebraically degenerating family, beyond the easy case of abelian varieties.
Prior to our complete answer to Problem \ref{pb:1-1} for the special family obtained in this paper, an initial progress was established in \cite{HSVZ} which focused on a gluing construction of hyperk\"ahler metrics for $n=2$. However, \cite{HSVZ} only described the construction at the level of {\it Riemannian metrics} without studying the relationships with the complex structure degenerations. So both the results in Theorem \ref{t:main-theorem} and the ideas in the proof are essentially new even in the case of complex dimension two.
In this paper, the formulation and the proof of Theorem \ref{t:main-theorem} naturally require us to directly work on a fixed algebraic degeneration family and perform the metric construction at the level of K\"ahler potentials. This framework puts us into a much more rigid situation and leads to a primary challenge, which demands a number of new ideas and techniques.
A more challenging part of this paper, compared to the 2 dimensional case studied in \cite{HSVZ}, lies in the construction of the neck region which are significantly harder. This involves studying the reduction of the Calabi-Yau equation with an $S^1$-symmetry, and in our setting we need to allow the $S^1$-action to have fixed points, which means we need to study the solutions of the reduced equation with singularities which generically are of real codimension 3.
When $n=2$, this reduction gives rise to a \emph{linear} PDE (Gibbons-Hawksing ansatz), and the singularities are \emph{isolated}. When $n>2$, the reduced equation is a PDE, however this time \emph{non-linear}, an equation we call the \emph{non-linear Gibbons-Hawking ansatz}. Moreover, we need to study the solutions with non-isolated singularities, which involves new technical ingredients (see Sections \ref{s:Greens-currents}-\ref{s:neck}).
A fundamental strategy is to exploit the adiabatic feature of the problem, i.e., the smallness of the $S^1$-orbits, to construct \emph{approximate} solutions to the nonlinear Gibbons-Hawking ansatz via \emph{linearization}.
\subsection{Outline of the proof of the main theorem}
The proof of Theorem \ref{t:main-theorem} consists of four parts.
The first part involves an algebraic modification of the family $\mathcal X$ (see Section \ref{ss:algebraic-geometry}). The overarching strategy in the proof of Theorem \ref{t:main-theorem} is to start with the Tian-Yau metrics on $X_0\setminus D=(Y_1\setminus D)\cup (Y_2\setminus D)$, and graft them to nearby fibers $X_t$ for $|t|$ small to get K\"ahler metrics which are \emph{approximately} Calabi-Yau. However, the existence of singularities of the total space $\mathcal X$ along $H$ imposes difficulties in performing a reasonable construction. To overcome this issue, we first modify the family $\mathcal X\rightarrow \Delta$ to obtain another family $\widehat{\mathcal X}\rightarrow\Delta$ by using base change and birational modifications (see Figure \ref{f: the modified family}). The new family $\widehat{\mathcal X}$ essentially agrees with $\mathcal X$ away from $X_0$, and the new central fiber $\widehat{X}_0$ comprises of a chain of \emph{three} components, with the two end components isomorphic to $Y_1, Y_2$ respectively, and the middle component $\mathcal N$ is given by a conic bundle over $D$, and is realized as a natural hypersurface in the projective bundle $\mathbb{P}(L_1\oplus L_2\oplus \mathbb C)$ cut out by the equation $s_1\otimes s_2+s_3f(x)=0$ (viewing $f$ as a section of $L_1\otimes L_2$). The conic fibers degenerate precisely along the divisor $H$ in $D$. The component $\mathcal N$ intersects transversally with $Y_1, Y_2$ along $D_1, D_2$, which are both naturally isomorphic to $D$. Notice that $\widehat{\mathcal X}$ is not necessarily smooth. Instead it has singularities along $D_1\cup D_2$ which is of codimension $2$. However, it turns out that working with $\widehat{\mathcal X}$ is the correct thing to do.
The second part involves the \emph{construction of the neck region}. We aim at constructing Calabi-Yau metrics on the smooth locus of the central fiber of $\widehat{\mathcal X}$. For the two end components, these are provided by the complete Tian-Yau metrics. For the middle component, with a moment's thought, one realizes that it is not possible to construct a complete Calabi-Yau metric on $\mathcal N^0=\mathcal N\setminus (D_1\cup D_2)$. The reason is that if such a metric existed, it would have two ends, and hence Cheeger-Gromoll splitting theorem would imply that the complete Ricci-flat manifold $\mathcal N^0$ must isometrically split off a line, which is not compatible with the complex geometry of $\mathcal N^0$. Instead we will try to construct a family of incomplete Calabi-Yau metrics defined on larger and larger open subsets in $\mathcal N^0$. The fact that $\mathcal N$ has a natural holomorphic $\mathbb C^*$-action suggests us to look for Calabi-Yau metrics with an $S^1$-symmetry.
In complex dimension 2, the hyperk\"ahler structure of the neck region was achieved in \cite{HSVZ} using the classical Gibbons-Hawking ansatz and studying the reduced linear equations, where the underlying complex manifold of the neck was not identified. In higher dimensions, the technologies employed in the construction are much more involved due to the appearance of the much wilder singularity behavior, and the framework is substantially
different from hyperk\"ahler triple method in \cite{HSVZ}. This is based on the study of higher dimensional generalizations of the Gibbons-Hawking ansatz, see Section \ref{s:torus-symmetries}. An essential difference from the case in complex dimension $2$ is that the corresponding reduced equations in higher dimensions are still nonlinear. Via a linearization we are led to study certain solutions to a linear elliptic PDE with singularities along a submanifold. The existence and local regularity of such solutions, namely \emph{Green's currents}, are studied in detail in Section \ref{s:Greens-currents}, which will be used to construct a family of incomplete K\"ahler metrics on open subsets of $\mathcal N^0$. The fact that the singularities of the Green's currents are non-isolated causes essential difficulties in understanding the regularity of the K\"ahler metrics. In actuality we only prove the metrics are $C^{2,\alpha}$ and this suffices for our purpose. Another difference in higher dimensions is that these metrics are only \emph{approximately} Calabi-Yau. Since we work on the fixed algebraic family $\widehat{\mathcal X}$, this requires us to obtain a precise formula and establish some uniform estimates for the K\"ahler potentials of these K\"ahler metrics, which is crucial for our gluing construction. In Section \ref{ss:glued-metrics}, we graft the incomplete approximately Calabi-Yau metrics constructed in Section \ref{s:neck} and the complete Tian-Yau metrics on $Y_i\setminus D$ to $C^{1,\alpha}$-K\"ahler metrics on $X_t$ for $|t|\ll1$, which are approximately Calabi-Yau.
The third part involves using \emph{weighted analysis} and the implicit function theorem to perturb the above approximately Calabi-Yau metrics on $X_t$ to genuine Calabi-Yau metrics. A crucial point is to prove some uniform weighted estimates, see Proposition \ref{p:estimate-L-neck} and Proposition \ref{p:global-injectivity-estimate} for both the incomplete and complete versions. To do this, including the analysis on the regularity scales in Section \ref{s:neck}, another crucial analytic input we need is a Liouville theorem on the Tian-Yau spaces in higher dimensions, which is proved in \cite{SZ-Liouville}.
One can directly see from the implicit function theorem that the Gromov-Hausdorff collapsing behavior of the Calabi-Yau metrics.
Finally, to prove Theorem \ref{t:main-theorem}, we start with the family of approximately Calabi-Yau metrics constructed above and the Tian-Yau metrics on the two end components of $\widehat X_0$. We then perturb them to approximately Calabi-Yau metrics on the nearby fibers $\widehat X_t$ for $|t|\ll1$. In this step we need to work at the level of K\"ahler potentials and use suitable cut-off functions.
Finally we perturb them to genuine Calabi-Yau metrics on $\widehat X_t$ in the K\"ahler class $2\pi c_1(\mathcal O(1)|_{\widehat X_t})$, so by well-known uniqueness of Calabi-Yau metrics we know the latter must agree with $\omega_{CY, t}$. This means that we have obtained an approximate geometric description of the metrics $\omega_{CY, t}$ for $|t|\ll1$. Then conclusion of Theorem \ref{t:main-theorem} follows.
\subsection{Organization of the paper}
We briefly overview the remainder of the paper.
In Section \ref{s:torus-symmetries}, we study the Calabi-Yau equation with torus symmetry and particularly we will focus on the reduced equations which yield certain singular behavior. We also formulate the linearization of such singular equations in Section \ref{ss:linearized equation}. This serves as model objects in understanding the degenerations.
In Section \ref{s:Greens-currents}, we discuss both the existence and local regularity theory for the linearized equations with singularities along a submanifold as formulated in Section \ref{ss:linearized equation}. To do this, this section will provide detailed analysis on the existence and local regularity theory of Green's currents.
In Section \ref{s:neck}, we construct a family of degenerating family of K\"ahler metrics which are approximately Calabi-Yau on the neck region. In Sections \ref{ss:kaehler-structures}-\ref{ss:complex-geometry}, we will identify the underlying complex manifold and carry out detailed analysis on the K\"ahler potentials of the metrics.
To precisely understand the metric degenerations, we study in Section \ref{ss:regularity-scales} the regularity of the metrics in a quantitative fashion and classify the various rescaled limiting geometries for this family of metrics. In Section \ref{ss:neck-weighted-analysis}, we set up the weighted spaces and prove several fundamental weighted estimates.
Section \ref{ss:perturbation of complex structures}
focuses on the estimates for the perturbation of the complex structures. This is necessary since our actual degenerating family of complex structures is only a perturbation of the neck region.
In Section \ref{s:neck-perturbation}, we apply the implicit function theorem and weighted estimates to show that the family of approximately Calabi-Yau metrics on the neck can be perturbed to genuine Calabi-Yau metrics, all of which will be done in Sections \ref{ss:perturbation-framework}-\ref{ss:incomplete weighted analysis}.
Here an important subtlety is that, to perturb those incomplete K\"ahler metrics, we will invoke Neumann boundary conditions instead of Dirichlet boundary conditions, which more naturally fits in with the formulation of Proposition \ref{p:neck-uniform-injectivity}.
In Section \ref{ss:renormalized-measure}, we also discuss the renormalized limit measures and their singular behavior, which may be of its own interest.
In Section \ref{s:gluing}, we will complete the proof of
Theorem \ref{t:main-theorem}.
There are three main points in the this section. The algebraic modification as mentioned above is formulated in Section \ref{ss:algebraic-geometry}.
In Section \ref{ss:glued-metrics}, we will construct the glued metrics on $X_t$ which are approximately Calabi-Yau. Finally, we will perform the perturbation in Section \ref{ss:global analysis}. It is worth mentioning that in the proof of Theorem \ref{t:main-theorem} we do not use genuine Calabi-Yau metrics on the neck region as in Theorem \ref{t:neck-CY-metric}. But the details of the final perturbation step is essentially the same as the proof of Theorem \ref{t:neck-CY-metric}. With the detailed perturbation argument on the neck region, we can safely omit the details in the final perturbation step.
In Section \ref{s:discussions}, we will give some possible extensions of our results and we will propose a conjecture (Conjecture \ref{cj:generalized-SYZ}) based on Theorem \ref{t:main-theorem}.
\subsection{Acknowledgements}
We would like to thank Lorenzo Foscolo, Mark Haskins, and Shouhei Honda for helpful discussions. We thank Ronan Conlon for comments which improved the presentation.
We are also grateful to Hans-Joachim Hein and Jeff Viaclovsky for stimulating discussions on the study of collapsing hyperk\"ahler metrics on K3 surfaces which led to an earlier joint paper \cite{HSVZ}. We thank Yang Li for communications regarding the draft of his preprint \cite{Li} and the first draft of the current paper in January 2019.
Substantial parts of this paper were written when the second author was visiting Princeton University, Academia Sinica, and ShanghaiTech University in the academic year 2018-2019. He would like to thank those institutions for their hospitality and support.
\section{Calabi-Yau metrics with torus symmetry}
\label{s:torus-symmetries}
In this section, we discuss the Calabi-Yau metrics which are preserved by a compact torus action, and the symmetry reduction of the Calabi-Yau equation. We will explain why we expect these metrics to provide local models for the collapsing of Calabi-Yau metrics when the complex structure degenerates. The ideas of this section will be used in Section \ref{s:neck} to construct the approximately Calabi-Yau neck region.
In Section \ref{ss:dimension reduction} we explain the motivation and study the dimension reduction of the Calabi-Yau equation for the $S^1$-action. In Sections \ref{ss:Calabi model space}-\ref{ss:2d standard model}, we will give some exact solutions to the reduced equation, which will serve as important local models in later sections. In Section \ref{ss:linearized equation} we consider the linearized equation and explain a natural class of singular solutions given by Green's currents. Section \ref{ss:higher rank torus} is to discuss the Calabi-Yau metrics admitting a torus action of higher rank.
\subsection{Motivation and dimension reduction of the Calabi-Yau equation}
\label{ss:dimension reduction}
We begin by recalling the familiar theory in complex dimension two. In this case, Calabi-Yau metrics are locally hyperk\"ahler and such metrics with free $S^1$-action are locally given by the classical \emph{Gibbons-Hawking ansatz}, in terms of a positive harmonic function on a domain in $\mathbb R^3$. Notice that in the classical Gibbons-Hawking ansatz, the hyperk\"ahler metrics admit an $S^2$-family of parallel compatible complex structures. A priori there is no preferred choice. If we do make a choice of complex structure, then the base $\mathbb R^3$ also has a natural splitting into $\mathbb C\oplus \mathbb R$, and we refer to Section \ref{ss:2d standard model} for further discussions. When the $S^1$-action is not free, the fixed points correspond to simple poles, i.e., Dirac type singularities of the harmonic function, locally given by $\frac{1}{2r}$ plus a smooth function, where $r$ is the Euclidean distance to the origin $0^3\in\mathbb{R}^3$. The local topological model for the $S^1$-fibration near a singularity is the standard Hopf fibration $\pi: \mathbb R^4\rightarrow \mathbb R^3$. Applying the Gibbons-Hawking ansatz to the entire $\mathbb R^3$ with a positive Green's function $G_{\lambda}\equiv\frac{1}{2r}+\lambda$ for $\lambda>0$, one obtains a homothetic scaling family of the \emph{Taub-NUT} metrics $g_{\lambda}$ on $\mathbb R^4$, so that $g_{\lambda}$ converges to the Euclidean metric on $\mathbb R^4$ when $\lambda\rightarrow 0$, and to the Euclidean metric on $\mathbb R^3$ when $\lambda\rightarrow\infty$.
We can also apply the Gibbons-Hawking ansatz to three dimensional flat manifolds with slower than cubic volume growth, but then there will not exist any non-trivial global positive harmonic function with only simple poles.
Nevertheless, the Gibbons-Hawking construction still yields various interesting families of \emph{incomplete} hyperk\"ahler metrics. Important examples are given by the Green's functions on $S^1\times \mathbb R^2$ (the Ooguri-Vafa metric, c.f. \cite{GW}) and $\mathbb{T}^2\times \mathbb R$ (c.f. \cite{HSVZ}). These metrics are important in understanding the collapsing behaviors of hyperk\"ahler metrics on K3 surfaces \cite{GW, HSVZ}.
In higher dimensions, the algebro-geometric consideration concerning complex structure degenerations also suggests the significance of Calabi-Yau metrics with torus symmetry. We will explain the motivation in a local model situation. Let $p:\mathcal{N}\to\Delta\subset \mathbb{C}$ be a degenerating family of smooth complex algebraic varieties $\mathcal{N}_t\equiv p^{-1}(t)$ such that as $t\to0$, $\mathcal{N}_t$ degenerates into $\mathcal N_0$ which is a union of irreducible components. Our primary observation is that, in the generic situation, near a point on $\mathcal N_0$ where the $(k+1)$-components intersect transversally, the degenerating family is locally modeled by the equation
\begin{equation}z_0\cdot \ldots \cdot z_{k}=t\cdot(f(z_{k+1}, \cdots, z_{n})+g),\end{equation}
where $g$ is contained in the analytic ideal generated by $z_0 , \ldots , z_k$. Near a point with $z_0=\ldots=z_{k}=0$, this can be further approximated by omitting the term $g$, which results in a $(\mathbb C^*)^k$-fibration
\begin{equation}z_0\cdot \ldots \cdot z_k=t\cdot f(z_{k+1}, \ldots, z_n)\end{equation}
over an $(n-k)$-dimensional base. The fibers are orbits of the $(\mathbb C^*)^k$-action, where $(\mathbb C^*)^k$ is naturally a subgroup in
$(\mathbb C^*)^{k+1}=\{(\lambda_0, \cdots, \lambda_k)|\lambda_i\in \mathbb C^*\}$
defined by the relation $\lambda_0\cdots\lambda_k=1$.
More generally, one can consider a complex manifold $D$ and $(k+1)$ holomorphic line bundles $L_0, \ldots, L_k$ over $D$. Let us denote $E\equiv\bigoplus\limits_{j=0}^k L_j$ and fix a holomorphic section $f$ of the tensor product $L_0\otimes \cdots\otimes L_k\cong \det (E)$. Then we can consider the hypersurface $\mathcal N$ in $E\times \mathbb C$ cut-out by the equation
\begin{equation}s_0\otimes\ldots\otimes s_k=t\cdot f(x),\end{equation}
where $(x, [s_0, \cdots, s_k])$ is a point in $E$ and $t\in \mathbb C$. We can view $\mathcal N$ as a family of hypersurfaces in $E$ parametrized by $t\in \mathbb C$. There is a natural $\mathbb C^*$ action on $\mathcal N$ given by
\begin{equation}
\lambda(\zeta)\cdot \Big(x, [s_0, \cdots, s_k], t\Big) \equiv \Big(x, [\zeta s_0, \cdots, \zeta s_k], \zeta^{k+1} t\Big).
\end{equation}
It induces isomorphisms between $\mathcal N_t$ and $\mathcal N_1$ for all $t\neq 0$, and it preserves $\mathcal N_0$.
For simplicity we only consider the generic case when the zeroes of $f$ form smooth hypersurface, then for $t\neq 0$, $\mathcal N_t$ is smooth but the projection map $\pi_t: \mathcal N_t\rightarrow D$ is still singular precisely along the union of $\Pi_{ij}\equiv \{(x, [s_0, \cdots, s_k]\in \mathcal N|s_i=s_j=0, f(x)=0\}$ for all pairs $(i, j)$ with $i\neq j$. Notice this union is also the singular set of the total space $\mathcal N$. When $t=0$, $\mathcal N_0$ is simply the union of the zero sections of $L_j$.
Suppose that the base $D$ has a holomorphic volume form $\Omega_D$ and a Calabi-Yau metric $\omega_D$, satisfying the equation $\omega_D^n=C\Omega_D\wedge\bar\Omega_D$. Then one can easily write down a $(\mathbb C^*)^k$-invariant holomorphic volume form $\Omega_t$ on $\mathcal N_t$ for $t\neq 0$, which is given by
\begin{equation}
\Omega_t=\sum_{j=0}^k (-1)^j \frac{ds_0}{s_0}\wedge\cdots \widehat{\frac{ds_j}{s_j}}\wedge\cdots\wedge \frac{ds_k}{s_k}\wedge \pi_t^*\Omega_D.
\end{equation}
Here the notation $\frac{ds_j}{s_j}(j=0, \cdots k)$ should be understood after choosing a local holomorphic section of $L_j$, but it is easy to see that this definition of $\Omega_t$ does not depend on the particular choice. Also a priori $\Omega_t$ is defined away from the singular fibers of the projection map $\pi_t$, and it is not difficult to see that $\Omega_t$ extends to a nowhere vanishing holomorphic volume form on $\mathcal N_t$.
Let $T^k=(S^1)^k\subset (\mathbb C^*)^k$ be the obvious maximal compact subgroup. Naturally one would ask for $T^k$-invariant Calabi-Yau metrics $\omega_t$ on (part of) $\mathcal N_t$, satisfying the equation $\omega_t^n=C\Omega_t\wedge \bar\Omega_t$, and we are then led to study dimension reduction of the Calabi-Yau equation under the $T^k$ action. This has been written down by Matessi \cite{Matessi} and we now explain the details in the case $k=1$, and we briefly discuss the case of general $k$ in Section \ref{ss:higher rank torus}.
\
Let $(X, \omega, J)$ be an $n$-dimensional K\"ahler manifold admitting an $S^1$-action which is holomorphic and Hamiltonian with a moment map function $z$, i.e.,
\begin{equation}\label{momentmap}
dz=\xi\lrcorner\omega,
\end{equation}
where $\xi$ is the vector field generating the $S^1$-action.
We now assume in addition that the $S^1$-action is free. Locally in a neighborhood of an $S^1$-orbit we can complexify the $S^1$-action and obtain a complex quotient $D$ which is an $(n-1)$-dimensional complex manifold. Then the local $S^1$-quotient $Q$ can be identified as a differentiable manifold with $D\times I$, where $I$ is an interval with coordinate function $z$.
Denote by $\{w_1, \cdots, w_{n-1}\}$ the local holomorphic coordinates on $D$. Then they can be viewed as local holomorphic functions on $X$. Let $t$ be an arbitrary local function with $\xi(t)=1$, Then $\{z, t, w_1, \cdots, w_{n-1}\}$ gives a local coordinate system on $X$, and we have $\xi=\partial_t$. Let us write $w_i=x_i+\sqrt{-1} y_i$. Then we can express the complex structure $J$ on $X$ in terms of the local coordinates as
\begin{equation}
Jdx_i=dy_i, \quad Jdy_i=-dx_i, \quad Jdz=h^{-1}\Theta,
\end{equation}
where $h>0$ is a local function and $\Theta$ is a local 1-form which can be written as
\begin{equation}
\Theta=-dt+\theta,
\end{equation}
such that $\theta$ does not have $dt$ component. The above negative sign appears due to the fact that
\begin{equation}
(Jdz)(\partial_t)=-dz(J\partial_t)=-\omega(\xi, J\xi)<0.
\end{equation}
This also gives an intrinsic geometric interpretation for $h^{-1}$, as the squared norm of the Killing field $\xi$. In particular, $h$ is $S^1$-invariant and hence it descends to a function on $Q$.
By the $S^1$-invariance
\begin{equation}\mathcal L_{\xi} (Jdz)=0,\quad \mathcal L_{\xi}(dt)=0,
\end{equation}
we obtain
$\mathcal L_{\xi}\theta=0$.
Therefore, $\theta$ can be also viewed as a 1-form on $Q$.
We can write the K\"ahler form $\omega$ on $X$ as
\begin{equation}
\omega=dz\wedge (-dt+\theta)+\tilde\omega,
\end{equation}
where $\tilde\omega$ is a $(1,1)$-form without $dz$ or $dt$ component. This is due to \eqref{momentmap} and the fact that $\omega$ is of type $(1,1)$.
Since $\mathcal L_{\xi}\omega=0$ we also have $\mathcal L_{\xi}\tilde\omega=0$, so the coefficients of $\tilde\omega$ also descend to $Q$. In particular, we may view $\tilde\omega=\tilde\omega(z)$ as a family of $(1,1)$-forms on $D$. The condition $d\omega=0$ is equivalent to
\begin{equation}
\label{omegaequation}
\begin{cases}
d_D\tilde\omega(z)=0\\
\partial_z\tilde\omega(z)=d_D\theta,
\end{cases}
\end{equation}
where $d_D$ denotes the differential along $D$.
Now we consider the integrability condition of the complex structure $J$. It is direct to check that
\begin{equation}J\partial_t=h^{-1}\theta_z\partial_t+h^{-1}\partial_z, \end{equation}
so the holomorphic vector field generating the $\mathbb C^*$-action is given by
\begin{equation}\xi^{1,0}=\frac{1}{2}(\partial_t-\sqrt{-1} J\partial_t)=\frac{1}{2}(1-\sqrt{-1} h^{-1}\theta_z)\partial_t-\frac{1}{2}\sqrt{-1} h^{-1}\partial_z.
\end{equation}
The dual holomorphic $(1,0)$-form is
\begin{equation}\kappa=\sqrt{-1}(hdz+\sqrt{-1}\Theta+\kappa'),\end{equation}
where $\kappa'$ only involves $dx_i, dy_i$. The integrability condition for $J$ can be expressed as
\begin{equation}
d\kappa\wedge \kappa\wedge dw_1\wedge\cdots\wedge dw_{n-1}=0,\end{equation}
which is equivalent to
\begin{equation}\label{thetaequation}
\begin{cases}
d_D\theta\wedge dw_1\wedge \cdots\wedge dw_{n-1}=0\\
\partial_z\theta=-d_D^ch ,
\end{cases}
\end{equation}
where $d_D^c\equiv J_D \circ d_D$.
The first equation follows from the second equation in \eqref{omegaequation} which implies $d_D\Theta$ is of type $(1,1)$ on $D$. Notice that \eqref{omegaequation} and \eqref{thetaequation} together can be re-organized as a system
\begin{equation}\label{omegahequation}
\begin{cases}\partial_z^2\tilde\omega+d_Dd_D^ch=0\\
d\Theta=\partial_z\tilde\omega-dz\wedge d_D^ch.
\end{cases}
\end{equation}
It is not difficult to globalize the above discussion and the upshot is that a K\"ahler form on $X$ with a free $S^1$-action gives rise to a family of K\"ahler forms $\tilde\omega(z)$ on a complex manifold $D$, together with a positive function $h$ on $D\times I$, satisfying \eqref{omegahequation}. This is the familiar procedure in \emph{K\"ahler reduction}. The 1-form $-\sqrt{-1}\Theta$ can be viewed as a family of connection 1-forms on the natural $S^1$-bundle over $D$, so as a consequence $\partial_z\tilde\omega=d_D\Theta$ determines an integral cohomology class in $2\pi H^2(D; \mathbb Z)$.
Conversely, suppose that we are given $(\tilde\omega(z),h)$ that satisfies \eqref{omegahequation}, and suppose that the deRham class $[\partial_z\tilde\omega_z]$ lies in $2\pi H^2(D; \mathbb Z)$. Then by general theory we can choose a connection 1-form $-\sqrt{-1}\Theta$ on an $S^1$-bundle $X$ over $D\times I$ satisfying \eqref{omegahequation}, and this gives rise to a K\"ahler structure on $X$ with $S^1$-action.
The isomorphism class of the resulting K\"ahler structures on $X$ depends only on the gauge equivalent classes of the connection 1-form, so is unique if the first Betti number of $D$ vanishes.
Now we specialize to study Calabi-Yau metrics with $S^1$-symmetry, so we additionally assume that $X$ has a holomorphic volume form $\Omega$.
Denote the holomorphic $(n-1)$-form on $X$
\begin{equation}
\tilde\Omega=\xi^{1,0}\lrcorner \Omega.
\end{equation}
The fact that $\Omega$ is $S^1$-invariant and holomorphic implies that $\tilde\Omega$ descends to a holomorphic $(n-1, 0)$-form $\Omega_D$ on $D$, and we also have
$\Omega=\kappa\wedge\Omega_D$.
By definition,
\begin{align}\omega^{n}&=-n\cdot dz\wedge dt\wedge \tilde\omega^{n-1},
\\
\Omega\wedge\bar\Omega&=2\sqrt{-1} (-1)^{n-1} h\cdot dz\wedge dt\wedge \Omega_D\wedge\bar\Omega_D.
\end{align}
So the Calabi-Yau equation on $X$
\begin{equation}\frac{\omega^{n}}{n!}=\frac{(\sqrt{-1})^{n^2}}{2^n}\cdot\Omega\wedge\bar\Omega
\end{equation}
becomes
\begin{equation}\label{volumeformequation}
\frac{\tilde\omega^{n-1}}{(n-1)!}=\frac{(\sqrt{-1})^{(n-1)^2}}{2^{n-1}}\cdot h\cdot \Omega_D\wedge\bar\Omega_D.
\end{equation}
Combining \eqref{omegahequation} and \eqref{volumeformequation}, we obtain that
\begin{equation} \label{nonlinearGH}
\partial_z^2\tilde\omega+d_Dd_D^c\frac{2^{n-1}\tilde\omega^{n-1}}{(\sqrt{-1})^{(n-1)^2}\Omega_D\wedge\bar\Omega_D}=0.
\end{equation}
Again it is easy to see this discussion can be globalized so we get a complex Calabi-Yau manifold $(D, \Omega_D)$ together with a family of K\"ahler forms $\tilde\omega(z)$ satisfying \eqref{nonlinearGH}.
Conversely, the study of $n$-dimensional Calabi-Yau manifolds $(X, \omega, \Omega)$ with a free $S^1$-action is reduced to the study of the equation \eqref{nonlinearGH}.
Now we make a few observations. First, when $n=2$, equation \eqref{nonlinearGH} is reduced to a linear equation. Notice that $\frac{\sqrt{-1}}{2}\Omega_D\wedge\bar\Omega_D$ is a flat K\"ahler form when $n=2$. Then we can write
\begin{equation}
\tilde\omega=\frac{\sqrt{-1}}{2}\cdot V \cdot \Omega_D\wedge\bar\Omega_D,
\end{equation}
for a real-valued function $V$ on $Q=D\times I$. Then equation \eqref{volumeformequation} is equivalent to
\begin{equation} \label{3dGH}
\partial_z^2V-\Delta_D V=0,
\end{equation}
where $\Delta_D=d_D^*d_D$ is the Hodge Laplace operator with respect to the above flat metric on $D$. Equation \eqref{3dGH} is now exactly the Laplace equation on $Q$, and the above discussion is reduced to the classical Gibbons-Hawking ansatz which produces hyperk\"ahler 4-manifolds. The minor difference is that here we have made a distinguished choice of the complex structure so that $Q$ naturally splits as $D\times I$.
When $n>2$, \eqref{nonlinearGH} is a non-linear equation, which is much harder to deal with. We call \eqref{nonlinearGH} the \emph{non-linear Gibbons-Hawking ansatz} for the Calabi-Yau metrics with $S^1$-symmetry This equation was first written down by Matessi \cite{Matessi}, in a slightly different form.
\subsection{Calabi model spaces}
\label{ss:Calabi model space}
In general it is not easy to directly solve equation \eqref{nonlinearGH}, but we can easily see some special solutions, which will be important for us.
Suppose that $(D, \Omega_D)$ is an $(n-1)$-dimensional compact Calabi-Yau manifold, and $\omega_D$ is a Calabi-Yau metric on $D$ with $[\omega_D]\in 2\pi H^2(D;\mathbb Z)$, satisfying
\begin{equation}
\frac{\omega_D^{n-1}}{(n-1)!}=\frac{(\sqrt{-1})^{(n-1)^2}}{2^{n-1}} \Omega_D\wedge\bar\Omega_D.
\end{equation}
If we set
\begin{equation}
\label{e:Calabi model solution}
\begin{cases}
\tilde\omega(z)=z\cdot \omega_D \\
h=z^{n-1},
\end{cases}
\end{equation}
then as long as $z>0$, $(\tilde\omega, h)$ clearly satisfies \eqref{nonlinearGH} and the integrality condition $[\partial_z\tilde\omega(z)]\in 2\pi H^2(D;\mathbb Z)$ is also achieved. So the above gives incomplete Calabi-Yau metrics in dimension $n$.
This metric has already appeared in K\"ahler geometry, which is usually expressed in terms of a K\"ahler potential. To explain this,
we fix a holomorphic line bundle $L_D$ with first Chern class $\frac{1}{2\pi} [\omega_D]$, and also fix a hermitian metric on $L_D$ whose curvature form is $-\sqrt{-1}\omega_D$. Then we consider the subset $\mathcal{C}^n$ of the total space of $L_D$ consisting of all elements $\xi$ with $0<|\xi|< 1$. It is endowed with a holomorphic volume form $\Omega_{\mathcal{C}^n}$ and a Ricci-flat K\"ahler metric $\omega_{\mathcal{C}^n}$ which is incomplete as $|\xi| \to 1$ and complete as $|\xi| \to 0$. The holomorphic volume form $\Omega_{\mathcal{C}^n}$ is given by (as in Section \ref{ss:complex-geometry})
\begin{equation}
\Omega_{\mathcal{C}^n}=\sqrt{-1}\cdot\frac{d\xi}{\xi}\wedge \Omega_D.
\end{equation}
The metric $\omega_{\mathcal{C}^n}$ is given by the \emph{Calabi ansatz}
\begin{equation}\label{calabiansatz}\omega_{\mathcal{C}^n}=\frac{n}{n+1} \sqrt{-1}\partial\bar{\partial} (-{\log |\xi|^2})^{\frac{n+1}{n}}. \end{equation}
It is straightforward to check that
\begin{equation}\omega_{\mathcal{C}^n}^n=\frac{1}{n\cdot 2^{n-1}}(\sqrt{-1})^{n^2} \Omega_{\mathcal{C}^n}\wedge\overline\Omega_{\mathcal{C}^n}.\end{equation}
Clearly the Calabi-Yau structure $(\omega_{\mathcal{C}^n}, \Omega_{\mathcal{C}^n})$ is invariant under the natural $S^1$-action on $L_D$. Applying the $S^1$-reduction as in Section \ref{ss:dimension reduction}, we get that the moment map is given by
\begin{equation}
z=(-{\log |\xi|^2})^{1/n},
\end{equation}
and the reduced family of K\"ahler metrics on $D$ is given by
\begin{equation}
\tilde\omega=z\cdot\omega_D.
\end{equation}
The function $h$ is
\begin{equation}
h=\frac{2}{n}\cdot z^n.
\end{equation}
So we see this gives rise to the above solution to \eqref{e:Calabi model solution} (up to a multiplicative constant on $h$), We call the space $(\mathcal{C}^n, \omega_{\mathcal{C}^n}, \Omega_{\mathcal{C}^n})$ a \emph{Calabi model space}. In Remark \ref{r:Calabi model potential} we will see the formula \eqref{calabiansatz} can also be recovered from \eqref{e:Calabi model solution}, and this works in a more general situation.
Now in the above Calabi ansatz the connection 1-form $-\sqrt{-1}\Theta$ is given by the Chern connection 1-form on $L_D$. We claim that by varying the holomorphic structures on $L_D$ we obtain all possible gauge equivalence classes of $\Theta$. This follows from the fact that there is a natural isomorphism between the group $\mathcal S_h$ of the isomorphism classes of holomorphic line bundles with $c_1=0\in H^2(D;\mathbb R)$ and the group $\mathcal S_f$ of gauge equivalence classes of flat $U(1)$ connections on $D$. Abstractly, we know the first group fits into an exact sequence
\begin{equation}
0\rightarrow \frac{H^1(D;\mathcal{O})}{H^1(D;\mathbb Z)}\rightarrow \mathcal S_h\rightarrow H^2_{\text{tor}}\rightarrow 0
\end{equation}
obtained from the exponential sequence on $D$,
where $H^2_{tor}$ denotes the torsion subgroup in $H^2(D;\mathbb Z)$. The second group is given by $\text{Hom}(\pi_1(D), S^1)=\text{Hom}(H_1(D; \mathbb Z), S^1)$, and it
fits into a short exact sequence
\begin{equation}
0\rightarrow \frac{H^1(D; \mathbb R)}{H^1(D;\mathbb Z)}\rightarrow \mathcal S_f\rightarrow \text{Hom}(H_{1, \text{tor}}, S^1)\rightarrow 0 ,
\end{equation}
where $H_{1, tor}$ is the torsion subgroup in $H_1(D; \mathbb Z)$. The isomorphism between $\mathcal S_h$ and $\mathcal S_f$ induces an isomorphism on the torsion quotients, which coincides with the isomorphism\begin{equation}
H^2_{\text{tor}}\simeq \text{Ext}(H_1(D;\mathbb Z), \mathbb Z)\simeq \text{Hom}(H_{1, tor}, S^1)
\end{equation}
given by the universal coefficient theorem.
We mentioned above that gauge equivalent choices of the connection 1-form $-\sqrt{-1}\Theta$ yield isomorphic Calabi-Yau structures on $\mathcal{C}^n$. Now we observe that for different choices of gauge equivalence classes which differ only by an element in the identity component of $\mathcal S_f$, the resulting Calabi-Yau structures are also isomorphic, via a diffeomorphism that covers a holomorphic isometry on $D$.
To see this, let us fix a choice of $\Theta$. Then given any vector field $Z$ on $D$, let $\hat{Z}$ be the horizontal lift of $Z$ to the $S^1$-bundle with respect to the connection form $-\sqrt{-1}\Theta$. The infinitesimal variation of $\Theta$ along the flow of $\hat{Z}$ is $\mathcal L_{\hat{Z}} \Theta=\hat{Z}\lrcorner d\Theta$, which is the pull-back of the form $Z\lrcorner \omega_D$ on $D$.
For a different choice $\Theta'$, modulo gauge equivalence we may assume $\Theta'-\Theta$ is a harmonic 1-form on $D$, so it must be parallel by Bochner's theorem. Let $Z$ be the vector field on $D$ that satisfies $Z\lrcorner \omega_D=\Theta'-\Theta$. Then $Z$ is also parallel, so $Z$ is holomorphic and Killing. In particular, $\mathcal{L}_{Z}\omega_D=0$. Then it follows that $\Theta'$ and $\Theta$ are related by the flow of the lifted vector field $\hat{Z}$.
\subsection{Two dimensional standard model spaces}
\label{ss:2d standard model}
In the Gibbons-Hawking ansatz, to get non-trivial topology one often needs to allow the $S^1$-action to have fixed points. This corresponds to the harmonic function $V$ having Dirac type singularities. For the convenience of later discussions, let us briefly recall the relevant formulae in this model situation, using our description with a preferred complex structure.
Let $\mathbb C^2$ be equipped with the standard holomorphic coordinates $(u_1, u_2)$ and the flat K\"ahler metric
\begin{equation}
\begin{cases}
\omega_{\mathbb C^2}=\frac{\sqrt{-1}}{2}(du_1\wedge d\bar u_1+du_2\wedge d\bar u_2)\\
\Omega_{\mathbb C^2}=du_1\wedge du_2.
\end{cases}
\end{equation}
Consider the $S^1$-action on $\mathbb C^2$,
\begin{equation}e^{\sqrt{-1} t}\cdot (u_1, u_2)\equiv(e^{-\sqrt{-1} t}u_1, e^{\sqrt{-1} t}u_2)\label{e:2D-model-action}
\end{equation}
with an infinitesimal generator
\begin{equation}\partial_t=-\sqrt{-1}(u_1\partial_{u_1}-u_2\partial_{u_2})+\sqrt{-1}(\bar u_1\partial_{\bar u_1}-\bar u_2\partial_{\bar u_2}).
\end{equation}
Then we have a moment map $z$ for the $S^1$-action with respect to $\omega_{\mathbb C^2}$ and a complex moment map
$y$ for the complexified $\mathbb C^*$-action with respect to $\Omega_{\mathbb C^2}$. So we obtain the standard Hopf map \begin{align}\pi: \mathbb C^2\rightarrow Q_0\equiv \mathbb C\times \mathbb R,\quad (u_1,u_2)\mapsto (y, z),\end{align} where
\begin{equation}
\begin{cases}
z=\frac{1}{2}(|u_1|^2-|u_2|^2)\\
y=u_1u_2.
\end{cases}
\end{equation}
Then the holomorphic quotient is $D_0=\mathbb C$ with holomorphic coordinate $y=y_1+\sqrt{-1} y_2$. Let us define \begin{equation} \label{modelquantities}
\begin{cases}
\tilde\omega_0\equiv\frac{\sqrt{-1}}{4r}dy\wedge d\bar y\\
\Omega_0\equiv dy\\
h_0\equiv\frac{1}{2r}\\
V_0\equiv\frac{1}{2r},
\end{cases}
\end{equation}
where $r=\sqrt{y_1^2+y_2^2+z^2}$ is the standard radial function on $Q_0$, and we have the relation
\begin{equation}
r=\frac{1}{2}(|u_1|^2+|u_2|^2).
\end{equation}
The connection 1-form $-\sqrt{-1}\Theta_0$ on $\mathbb C^2$ can also be written down explicitly as
\begin{equation}
\Theta_0=h_0 \cdot J_0(dz)=\sqrt{-1}\cdot \frac{u_1d\bar u_1-\bar u_1 du_1+\bar u_2du_2-u_2d\bar u_2}{2(|u_1|^2+|u_2|^2)}.\label{e:model-connection}
\end{equation}
Let us define the curvature 2-form on $Q_0$,
\begin{equation}
\Upsilon_0\equiv \partial_z\tilde\omega_0-dz\wedge d_{\mathbb C}^ch_0.
\end{equation}
So we have that
\begin{equation}\label{modelcurvature}
\Upsilon_0=-\frac{\sqrt{-1}}{4r^3} (zdy\wedge d\bar y+yd\bar y\wedge dz-\bar y dy\wedge dz).
\end{equation}
From our above discussion we have the following holds
\begin{equation}
\label{e:d Theta0}
\begin{cases}d\Theta_0=\Upsilon_0\\
\tilde \omega_0+dz\wedge \Theta_0=\omega_{\mathbb C^2},
\end{cases}
\end{equation}
where we have implicitly viewed a form on $Q_0$ as a form on $\mathbb C^2$ using the pull-back $\pi^*$. In other words, the flat metric on $\mathbb C^2$ together with the above $S^1$-action can be recovered via the Gibbons-Hawking ansatz applied to the function $V_0=\frac{1}{2r}$ on $Q_0=\mathbb C\times \mathbb R$.
The above flat metric admits a one-parameter non-flat perturbation, which corresponds to replacing $V_0$ in the Gibbons-Hawking ansatz by $V_0+\lambda$ for a positive constant $\lambda$ . We define
\begin{equation}
\begin{cases}
\tilde\omega_{0, \lambda}\equiv (\frac{1}{2r}+\lambda)\frac{\sqrt{-1}}{2}dy\wedge d\bar y\\
h_{0, \lambda}\equiv\frac{1}{2r}+\lambda,\\
\end{cases}
\end{equation}
which yields a family of \emph{Taub-NUT metrics} $(\omega_{TN, \lambda}, \Omega_{TN, \lambda})$ on $\mathbb R^4$ with
\begin{equation}
\label{Taub-NUT}
\begin{cases}\omega_{TN, \lambda}\equiv (\frac{1}{2r}+\lambda)\frac{\sqrt{-1}}{2}dy\wedge d\bar y+dz\wedge \Theta_0\\
\Omega_{TN, \lambda}\equiv \sqrt{-1} ((\frac{1}{2r}+\lambda)dz+\Theta_0)\wedge dy.
\end{cases}
\end{equation}
We can still view these metrics as defined on $\mathbb R^4$ with coordinates $u_1, u_2, \bar u_1, \bar u_2$ via the above Hopf map, but the coordinate functions $u_1, u_2$ are no longer holomorphic. Indeed one can write down explicitly the holomorphic volume form
\begin{equation}
\Omega_{TN, \lambda}=du_1\wedge du_2+\frac{\lambda}{2}dz\wedge dy.
\end{equation}
We also have
\begin{equation}\label{e: modelholomorphic1form}
h_{0, \lambda}\cdot dz+\sqrt{-1} \Theta_{0}=\lambda\cdot dz+\frac{1}{2}(\frac{du_1}{u_1}-\frac{du_2}{u_2}).
\end{equation}
LeBrun \cite{LeBrun} showed that if we make a (non-holomorphic) coordinate change on $\mathbb C^2$,
\begin{equation}
\begin{cases}
\eta_+=u_1e^{\frac{\lambda}{2}(|u_1|^2-|u_2|^2)}\\
\eta_-=u_2e^{\frac{\lambda}{2}(|u_2|^2-|u_1|^2)},
\end{cases}
\end{equation}
then we have
$\Omega_{TN, \lambda}=d\eta_+\wedge d\eta_-$.
So the underlying complex manifold is still bi-holomorphic to $\mathbb C^2$ with holomorphic coordinates $\eta_{+}$ and $\eta_-$, and one can write down a global K\"ahler potential
\begin{equation}
\label{e:TaubNUT potential}
\begin{cases}
\omega_{TN, \lambda}=\sqrt{-1} \partial\bar{\partial} \varphi_{\lambda}
\\
\varphi_{\lambda}=\frac{1}{2}(|u_1|^2+|u_2|^2)+\frac{T}{4}(|u_1|^4+|u_2|^4).
\end{cases}
\end{equation}
Again in Remark \ref{r:TaubNUT potential} we will see this follows from a more general fact.
\subsection{Linearized equation and singularities}
\label{ss:linearized equation}
Now we return to the higher dimensional situation. One interesting local model is given by the product of a flat space $\mathbb C^{n-2}$ with trivial $S^1$-action and the above 2-dimensional model $(\mathbb C^2, \omega_{\mathbb C^2}, \Omega_{\mathbb C^2})$. Let $Q_0\equiv \mathbb{C}\times\mathbb{R}$ and $Q\equiv\mathbb C^{n-2}\times Q_0$. Then the singular set of $\tilde\omega$ and $h$ is a real codimension-3 subspace $P=\mathbb C^{n-2}\oplus\{0\}\subset Q$, and they both have transversal Dirac type singularities along $P$. In our applications, we need to consider the non-linear situation. So $D$ is an $n-1$ dimensional complex manifold and $H\subset D$ is a smooth complex hypersurface, and we want our solution $(\tilde\omega, h)$ to the equation \eqref{nonlinearGH} to satisfy a distributional equation on $Q$ of the form
\begin{equation} \label{distributionequation}
\Big(\partial_z^2\tilde\omega+d_Dd_D^c\frac{2^{n-1}\tilde\omega^{n-1}}{(\sqrt{-1})^{(n-1)^2}\Omega_D\wedge\bar\Omega_D}\Big)\wedge dz=2\pi \cdot \delta_{P},
\end{equation}
where $P\equiv H\times\{0\}$ and $\delta_P$ is a degree $3$ current given by integration along $P$. This equation has appeared in the literature \cite{Zharkov} in a slightly different form. A solution to this equation, with suitable regularity, will give rise to a Calabi-Yau metric with an $S^1$-action whose fixed point locus is a complex codimension-2 submanifold and transverse to which the action is given the model \eqref{e:2D-model-action}. This is exactly what we are motivated to search for from the algebro-geometric discussion at the beginning of this section.
Unfortunately, solving the equation \eqref{distributionequation} in general seems very difficult.
Motivated by recent results in the study of adiabatic limits of $G_2$-manifolds \cite{Don, FHN}, we attempt to study the equation when the $S^1$-orbit is very small. Again suppose $(D, \omega_D, \Omega_D)$ is $(n-1)$-dimensional Calabi-Yau, then for $T$ large we know there are trivial constant solutions to \eqref{nonlinearGH}with $\tilde\omega=T\omega_D$ and $h=T^{n-1}$. Now we look for a perturbation $\tilde\omega=T\omega_D+\psi$ for $T$ large. To the first order we know $\psi$ must satisfy the linearized equation at $T\omega_D$, hence
\begin{equation}
\partial_z^2\psi+T^{n-2}d_Dd_D^c \Big(\Tr_{\omega_D}\psi\Big)=0,
\end{equation}
which by K\"ahler identities is equivalent to
\begin{equation}
\partial_z^2\psi-T^{n-2}d_Dd_D^*\psi=0.
\end{equation}
Up to a scaling of the $z$ variable this is equivalent to the equation
\begin{equation}
\partial_z^2\psi-d_Dd_D^*\psi=0.
\end{equation}
If $\psi$ also satisfies $d_D\psi=0$, then $\psi\wedge dz$ is a harmonic $3$-form on the product $Q=D\times \mathbb R_z$. Again, the interesting case is when $\psi$ has singularities, and we want to study the case when the singular set is of the form $H\times \{0\}\subset Q$ for $H$ a smooth hypersurface in $D$, and correspondingly $\psi\wedge dz$ satisfies
\begin{equation}
\Delta_Q (\psi \wedge dz) = 2\pi\cdot \delta_P.
\end{equation}
This is a generalization of Green's functions to 3-forms and we call $\psi \wedge dz$ a \emph{Green's current}, which is our main object of study in Section \ref{s:Greens-currents}.
When $n=2$, the above Green's current is simply a Green's function which has been used in \cite{HSVZ} to obtain exact solutions to a family of incomplete Calabi-Yau metrics by Gibbons-Hawking construction. In higher dimensions, using Green's currents, we can apply \eqref{omegahequation} to define a family of approximately Calabi-Yau metrics. This is our main object in Section \ref{s:neck}.
\subsection{Higher rank torus symmetry}\label{ss:higher rank torus}
Now we assume that an $n$-dimensional K\"ahler manifold $(X, \omega, J)$ admits an $T^k$-action which is holomorphic and Hamiltonian. We first assume the action is free. Let $(z_1, \cdots, z_k)$ be the moment map. Then similar discussion to that in Section \ref{ss:dimension reduction} yields locally a family of K\"ahler forms $\tilde\omega$ on the complex quotient, parametrized by $(z_1, \ldots z_k)\in \mathbb R^k$, a family of connection $1$-forms $-\sqrt{-1}\Theta_j(j=1, \cdots, k)$ and a positive definite $k\times k$ real symmetric matrix $W=(W_{ij})$ with the inverse matrix
$W^{ij}=\langle \partial_{t_i}, \partial_{t_j}\rangle$ such that the following system of equations hold
\begin{equation}
\begin{cases}\partial_{z_j}\tilde\omega = d_D\Theta_j
\\
\partial_{z_j}\Theta_i=-d^c_D W_{ij}
\\
\partial_{z_l}W_{ij}=\partial_{z_j}W_{il}.
\end{cases}\end{equation}
As before the first two equations combine to give an equation on $(\tilde\omega, W_{ij})$
\begin{equation}
\label{e:higher Tk}
\partial_{z_i}\partial_{z_j}\tilde\omega+d_Dd_D^cW_{ij}=0.
\end{equation}
Now suppose the complex quotient $D$ is Calabi-Yau with a holomorphic volume form $\Omega_D$, then the Calabi-Yau equation on $X$ becomes
\begin{equation}
\frac{\tilde{\omega}^{n-k}}{(n-k)!}=\frac{(\sqrt{-1})^{(n-k)^2}}{2^{n-k}} \cdot \det(W_{ij}) \cdot \Omega_D\wedge\bar\Omega_D.
\end{equation}
This equation has been derived by Matessi and Zharkov (see \cite{Matessi}, \cite{Zharkov}). Again when the $T^k$-action is not free one should replace \eqref{e:higher Tk} by a distributional equation. We will discuss a simple example in Section \ref{ss:general-situations}. In the most extreme case when $k=n$ is the complex dimension of $X$, this becomes the real Monge-Amp\`ere equation
$\det(W_{ij})=C$.
\section{Green's currents}
\label{s:Greens-currents}
In this section, we study in detail some existence and regularity theory of {\it Green's currents}. Our main motivation for studying these arises from Section \ref{s:torus-symmetries}, where we see that the Green's currents appear as Dirac type singular solutions to the linearization of dimension reduced Calabi-Yau equation by the $S^1$-symmetry. It is possible that this study will also have applications to other geometric problems, especially to those concerning adiabatic limits.
This section is organized as follows. In Section \ref{ss:geodesic-coordinates} we recall the generalized geodesic normal coordinates for an embedded submanifold. In Section \ref{ss:real-case} we discuss the definition, local existence and regularity properties of Green's currents in the general Riemannian setting. In Section \ref{ss:complex-greens-currents}, we refine these results in the special case related to K\"ahler geometry. In Section \ref{ss:global-existence}, we prove a global existence result which will be immediately used in Section \ref{s:neck}.
\subsection{Normal coordinates for an embedded submanifold}\label{ss:geodesic-coordinates}
We start our discussion by introducing the basic notions of the {\it normal exponential map} and {\it normal coordinates} with respect to an embedded submanifold.
This part seems to be standard in Riemannian geometry, in the language of Fermi coordinates (see \cite{Gray} for example). Since we cannot find a reference for the precise formulae we will need, we include detailed
discussions and proofs in this section.
Let $(Q,g)$ be an oriented Riemannian manifold of dimension $m$ and let $P\subset Q$ be a closed embedded oriented submanifold of codimension $k_0$ in $Q$.
In our later applications, we only need the case $k_0=3$.
Denote by ${N}$ the normal bundle of $P$ in $Q$, equipped with the induced fiberwise Riemannian inner products. For any $p\in P$, denote by $N(p)$ the fiber of $N$ over $p$.
The {\it normal exponential map} of $P$ in $Q$ is defined as
\begin{equation}\Exp_P: {N}\rightarrow Q,\ (p, v)\mapsto \Exp_p(v),\ v\in N(p),\end{equation}
where $\Exp_p:T_pQ\to Q$ is the usual exponential map at $p\in Q$.
By implicit function theorem, we know that $\Exp_P: {N}\to Q$ is a diffeomorphism from some neighborhood of the zero section in ${N}$ to some tubular neighborhood of $P$ in $Q$.
Now we define the {\it normal coordinates} for $P$.
Fix $p\in P$. First we choose local coordinates $\{x_1', \ldots, x_{m-k_0}'\}$ in a small neighborhood $U\subset P$ of $p$ such that at $p$
\begin{equation}\langle \partial_{x_i'}, \partial_{x_j'}\rangle=\delta_{ij},\ 1\leq i,j\leq m-k_0.\end{equation} We also assume that $\partial_{x_1'}\wedge\cdots\wedge \partial_{x_{m-k_0}'}$ is compatible with the orientation on $U$.
Next, we pick local orthonormal sections $\{e_1, \ldots, e_{k_0}\}$ of the normal bundle $N$
such that
\begin{equation}
\langle e_{\alpha}, e_{\beta}\rangle=\delta_{\alpha\beta}, \ 1\leq \alpha,\beta\leq k_0,
\end{equation}
on $U$. Again we assume that $e_1\wedge\cdots\wedge e_{k_0}$ is compatible with the orientation on $N$, i.e., $e_1\wedge \cdots \wedge e_{k_0}\wedge \partial_{x_1'}\wedge\cdots \partial_{x_{m-k_0}'}$ is compatible with the orientation on $Q$. Then we can find $\epsilon>0$ such that $\Exp_p$ restricts to a diffeomorphism from
\begin{equation}\mathfrak{S}_0=\{(q, v)\in N|q\in U,\ |v|<\epsilon\}\end{equation}
to a neighborhood $\mathcal U$ of $p$. In particular, $U=\mathcal U\cap P$.
\begin{definition}[Normal coordinates] For any $(q,v)\in \mathfrak{S}_0$ with $v=\sum\limits_{\alpha=1}^{k_0}v_{\alpha}e_{\alpha}$, the local normal coordinates are defined as follows
\begin{align}
\begin{cases}
x_j(\Exp_P(q,v))\equiv x_j'(q), & 1\leq j\leq m-k_0,
\\
y_{\beta}(\Exp_P(q,v))\equiv v_{\beta}, & 1\leq\beta\leq k_0.
\end{cases}
\end{align}
\end{definition}
By definition for each fixed point $(y_1,\ldots, y_{k_0},x_1,\ldots, x_{m-k_0})$ in $\mathcal{U}$, the curve \begin{equation}\vartheta(t)\equiv (ty_1,\ldots, ty_{k_0},x_1,\ldots, x_{m-k_0}),\label{e:normal-geodesic}\end{equation} is a {\it normal geodesic} which is orthogonal to $U\subset P$.
Let
\begin{equation}
\partial_{y_1},\ldots, \partial_{y_{k_0}}, \partial_{x_1},\ldots, \partial_{x_{m-k_0}}
\end{equation}
be the induced coordinate vector fields. Then $\partial_{y_\alpha}|_{y=0}=e_\alpha$ when both are viewed as sections of $N$ over $U$.
For $1\leq i,j\leq m-k_0$
and $1\leq\alpha\leq k_0$, we denote
\begin{equation}g_{ij}\equiv\langle\partial_{x_i}, \partial_{x_j}\rangle,\ g_{i\alpha}\equiv\langle\partial_{x_i}, \partial_{y_\alpha}\rangle, \ g_{\alpha\beta}\equiv\langle\partial_{y_\alpha}, \partial_{y_\beta}\rangle.\end{equation}
Obviously we have
\begin{equation}g_{ij}(0, 0)=\delta_{ij},\ g_{\alpha\beta}(x, 0)=\delta_{\alpha\beta},\ g_{i\alpha}(x,0)=0,\end{equation}
for all $x\in U\subset P$.
The second fundamental form of the embedding $U\hookrightarrow\mathcal U$ can be written as $\IIs=\IIs_{ij}^{\alpha} \partial_{y_\alpha}\otimes (dx_i\otimes dx_j)$, where
\begin{equation}\IIs^\alpha_{ij}\equiv\langle\nabla_{\partial_{x_i}}e_\alpha, \partial_{x_j}\rangle\end{equation}
Denote by $\overrightarrow{H}=H^{\alpha}e_{\alpha}$ the mean curvature vector. Then we have
$H^\alpha\equiv g^{ij}\IIs^\alpha_{ij}$.
Let us define the function $r$
in the normal coordinates as follows, \begin{equation}r(x,y)\equiv|y|=\Big(\sum\limits_{\alpha=1}^{k_0} y_{\alpha}^2\Big)^{\frac{1}{2}}.\label{e:normal-distance-function}\end{equation}
The following lemma gives an extension of the usual Gauss Lemma.
\begin{lemma}[Generalized Gauss Lemma] \label{l:generalized-Gauss} In the above notations, for any $p\in P$, let $\{x_i, y_{\alpha}\}$ be a normal coordinate system defined in a neighborhood $\mathcal{U}$ of $p$ in $Q$ with $U=\mathcal{U}\cap P$.
Let $r$ be the function defined by \eqref{e:normal-distance-function}. Then $r$ satisfies the following properties in $\mathcal{U}$:
\begin{enumerate}
\item $\nabla r = \partial_r$ holds in $\mathcal{U}\setminus U$. In particular, $r$ is the normal distance function
in $\mathcal{U}$, i.e., $r(q)=d(q, P)$ for all $q\in \mathcal{U}$.
\item $\partial_r$ is orthogonal to $\partial_{x_i}$'s in $\mathcal{U}\subset Q$, and hence \begin{equation}\label{Gausslemma}
\langle\partial_{x_i},r\partial_r\rangle=\langle \partial_{x_i}, \sum_{\alpha=1}^{k_0} y_\alpha \partial_{y_\alpha}\rangle=\sum_{\alpha=1}^{k_0} y_\alpha g_{i\alpha}=0, \ 1\leq i\leq m-k_0.
\end{equation}
\end{enumerate}
\end{lemma}
The proof is by straightforward computations, so we omit the details.
For the convenience of later discussions, we introduce several notations concerning the {\it normal regularity order} near the submanifold $P$. It will be frequently used throughout the paper.
\begin{definition}[Normal regularity order] \label{d:normal-regularity}
Let $T(x,y)$ be a tensor field locally defined on $\mathcal{U}$ which is $C^{\infty}$ on $\mathcal U\setminus U$. Then for every $k\in\mathbb{N}$, we define the following notion as $r\rightarrow 0$:
\begin{enumerate}
\item
$T(x,y)=O'(r^k)$ if for each $\epsilon>0$
\begin{align}
\Big|\partial_t^I\partial_{n}^{J}T(x,y)\Big|=\begin{cases}
O(1), & |J|\leq k-1,\\
O(r^{k-|J|-\epsilon}), & |J|\geq k,
\end{cases}\end{align}
for all multi-indices $I$ and $J$. Here we work on the normal coordinate system and take the usual derivatives on the coefficients of $T$. The tangential derivative $\partial_t$ denotes one of the $\partial_{x_j}$'s, and the normal derivative $\partial_n$ denotes one of the $\partial_{y_\alpha}$'s.
\item $T(x,y)=r^kO'(1)$
if $r^{-k}T(x,y)=O'(1)$.
\item
$T(x,y) = \widetilde{O}(r^k)$ if $T(x,y)\in C^{\infty}(\mathcal{U})$ and
$T(x,y)=r^kO'(1)$. In other words, $T(x,y)$ is smooth on $\mathcal{U}$
and has vanishing normal derivatives along $U$ up to order $k-1$.
\end{enumerate}
\end{definition}
Notice that that defining condition does not depend on the choice of the local coordinates $\{y_\alpha, x_i\}$, as long as they satisfy that $y_\alpha=0$ along $P$ for $1\leq\alpha\leq k_0$. We also give the following instructive example.
\begin{example} In the above notations, we have the following:
\begin{enumerate}
\item If $T=O'(1)$, then $T \in L^q(\mathcal{U})$ for any $q>1$.
\item If $T\in C^{\infty}(\mathcal{U})$, then $T=O'(r^k)$ for all $k\in\mathbb{Z}_+$.
\item If $T=r^k O'(1)$ for some $k\in\mathbb{Z}_+$, then $T=O'(r^k)$ and $T$ vanishes along $U$.
\item For any $k\in\mathbb{Z}_+$ and $\alpha\in(0,1)$, if $T=O'(r^k)$, then $T$ is $C^{k-1, \alpha}(\mathcal{U})$.
\end{enumerate}
\end{example}
\begin{remark}
The purpose of introducing this notation is because we will frequently meet the terms such as $r^{-j}\cdot y_{\alpha_1}\cdots y_{\alpha_l}$ which naturally lie in $O'(r^{l-j})$. The reason to allow the defect of $\epsilon$ is because such tensors are well-behaved under the $W^{2,q}$-elliptic estimate in Lemma \ref{l:higher-regularity}.
\end{remark}
The following metric tensor expansion will be used in our regularity analysis.
\begin{lemma} \label{l:metric-expansion}
In the above normal coordinates, we have the following expansions of the metric tensor $g$ of $Q$ along the normal directions,
\begin{align}
g_{\alpha\beta}|_{(x,y)}&=\delta_{\alpha\beta}-\frac{1}{3}\Rm_{\alpha\gamma\xi\beta}\Big|_{(x,0)}y_{\gamma} y_{\xi} +\widetilde{O}(r^3),
\\
g_{ij}|_{(x, y)}&=g^P_{ij}(x)+2\IIs^\alpha_{ij}\Big|_{(x,0)} y_{\alpha} -(\Rm_{i\gamma \xi j} - \langle \nabla_{\partial_{x_i}} \partial_{y_\gamma}, \nabla_{\partial_{x_j}} \partial_{y_\xi}\rangle)\Big|_{(x,0)}y_\gamma y_\xi+\widetilde{O}(r^3),
\\
g_{i\alpha}|_{(x,y)}&=\langle \nabla_{\partial_{x_i}}\partial_{y_\gamma}, \partial_{y_\alpha} \rangle\Big|_{(x,0)} y_\gamma -\frac{2}{3}\Rm_{i\gamma\xi\alpha}\Big|_{(x,0)}y_{\gamma}y_{\xi} + \widetilde{O}(r^3),
\end{align}
where $g^P=(g^P_{ij})$ is the restriction of the metric $g$ to $U\subset P$, $\Rm$ is the Riemann curvature tensor of $g$.
\end{lemma}
\begin{proof}
This can be proved using the Jacobi fields. For any fixed point $q=(0^{k_0},x_1,\ldots, x_{m-k_0})\in U\subset P$, we choose a unit vector $v=\sum\limits_{\alpha=1}^{m-k_0}v_{\alpha}\partial_{y_{\alpha}}\in N(q)\cong \mathbb R^{k_0}$ with $|v|=1$. Let $\vartheta$ be the radial geodesic in $\mathcal{U}$
\begin{equation}\vartheta(t)\equiv \Exp_P(q,tv) = \Exp_q(tv).\end{equation}
In particular $\vartheta(0)=q$, and $\vartheta'(0)=v$. In the normal coordinates, the geodesic $\vartheta$ can represented as
$\vartheta(t)=(tv_1,\ldots, tv_{k_0}, x_1,\ldots,x_{m-k_0})$.
For each $1\leq \alpha\leq k_0$ and $1\leq i\leq m - k_0$, we define the geodesic variations
\begin{align}
\sigma_{\alpha}(t,s) &\equiv ((tv_1,\ldots, t(v_{\alpha}+s), \ldots, tv_{k_0}, x_1,\ldots,x_{m-k_0}),
\\
\sigma_i(t,s) &\equiv (tv_1,\ldots, tv_{k_0}, x_1, \ldots, x_i + s, \ldots x_{m-k_0}).
\end{align}
Then the variation fields of $\sigma_{\alpha}(t,s)$ and $\sigma_i(t,s)$ at $s=0$ give
the following Jacobi fields along the radial geodesic $\vartheta(t)$ respectively:
\begin{align}
\begin{cases}
J_{\alpha}(t)=t\cdot \partial_{y_{\alpha}}, & 1\leq \alpha\leq k_0
\\
J_{i}(t)= \partial_{x_i}, & 1\leq i \leq m- k_0.
\end{cases}
\end{align}
By definition,
\begin{equation}
J_{\alpha}(0)=0,\ J_i(0)=\partial_{x_i}.
\end{equation}
Taking covariant derivatives at $t=0$ we get
\begin{equation}
J_{\alpha}'(0) = \partial_{y_{\alpha}},\ J_i'(0)=v_{\alpha}\nabla_{\partial_{x_i}}\partial_{y_{\alpha}}.
\end{equation}
Then applying the Jacobi equation along the geodesic $\vartheta$
\begin{align}
\begin{cases}
J_{\alpha}''+\Rm(J_{\alpha},\vartheta')\vartheta'=0,\\
J_{i}''+\Rm(J_{i},\vartheta')\vartheta'=0,\\
\end{cases}
\end{align}
where $\Rm(X,Y)Z\equiv \nabla_X\nabla_YZ - \nabla_Y\nabla_X Z - \nabla_{[X,Y]}Z$ is the Riemann curvature tensor of $g$, we obtain
\begin{align}
J''_\alpha(0)=0, \ J'''_\alpha(0)=-v_\gamma v_\xi \Rm(\partial_{y_\alpha},\partial_{y_\gamma})\partial_{y_\xi},\\
J''_i(0)=-v_\gamma v_\xi \Rm(\partial_{x_i},\partial_{y_\gamma})\partial_{y_\xi}.
\end{align}
Therefore,
\begin{align}g_{\alpha\beta}&=t^{-2}g(J_\alpha, J_\beta)=\delta_{\alpha\beta}-\frac{1}{3}\Rm_{\alpha\gamma\xi\beta}v_\gamma v_\xi t^2+\widetilde{O}(t^3),
\\
g_{ij}&=g(J_i, J_j)=g^P_{ij}+2 v_\alpha\IIs_{ij}^\alpha t-\Big(\Rm_{i\gamma\xi j} - \langle \nabla_{\partial_{x_i}} \partial_{y_\gamma}, \nabla_{\partial_{x_j}} \partial_{y_\xi}\rangle\Big)v_\gamma v_\xi t^2+\widetilde{O}(t^3),
\\
g_{i\alpha}&=t^{-1}g(J_i, J_\alpha)= \langle \nabla_{\partial_{x_i}}\partial_{y_\gamma}, \partial_{y_\alpha}\rangle v_\gamma t-\frac{2}{3}\Rm_{i\gamma\xi\alpha}v_{\gamma}v_{\xi}t^2+\widetilde{O}(t^3).\end{align}
Notice $y_{\alpha} = tv_{\alpha}$, and the conclusion follows.
\end{proof}
As a digression we briefly discuss the intrinsic meaning of the above expansion. The point is that locally the Riemannian metric $g$ is approximated by a Riemannian metric $g_N$ on the normal bundle $N$ up to the first order. Notice we have the natural projection $\pi: N\rightarrow P$ and $N$ is a Riemannian vector bundle together with an induced ``normal" connection, given by the normal component of the Levi-Civita connection of $g$. The latter hence gives rise to a distribution of horizontal subspaces at each point of $N$, which in our coordinates is spanned by $\partial_{x_i}-A_{i\alpha\beta}y_\beta \partial_{y_\alpha}$, where
\begin{equation} \label{eqn2-10}
A_{i\alpha\beta}(x)\equiv\langle \nabla_{\partial_{x_i}}\partial_{y_\beta}, \partial_{y_\alpha}\rangle|_{y=0}
\end{equation}
is a smooth function on $U$.
We define $g_N$ so that at each point of $N$, the vertical and horizontal subspaces are orthogonal and on the vertical part is given by the bundle metric on $N$, and on the horizontal part is given by the perturbation of the base metric $g_P$ using the second fundamental form. In this way we get a coordinate free description of the above expansion up to the first order.
We also define
\begin{equation}A_{ij\alpha\beta}\equiv \frac{1}{2}(\partial_{x_i} A_{j\alpha\beta}-\partial_{x_j}A_{i\alpha\beta}).\end{equation}
Then the curvature of the normal connection is given by
\begin{equation}\Omega_{ij\alpha\beta}\equiv A_{ij\alpha\beta}+\frac{1}{2}(A_{i\alpha\gamma}A_{j\gamma\beta}-A_{i\beta\gamma}A_{j\gamma\alpha}).\end{equation}
\subsection{Green's currents for Riemannian submanifolds} \label{ss:real-case}
First we recall and introduce the basic terminology. Let $(Q, g)$ be an oriented Riemannian manifold of dimension $m$. Denote by $\Omega^l_0(Q)$ the space of differential $l$-forms with compact support in $Q$.
\begin{definition}
[$k$-current] A $k$-current on $Q$ is a linear functional \begin{equation}
T: \Omega^{m-k}_0(Q)\to \mathbb{R}, \quad \chi\mapsto (T, \chi),
\end{equation}
which is continuous in the sense of distributions, i.e., suppose that $\chi_j\in \Omega_0^{m-k}(Q)$ is a sequence of differential forms with all derivatives uniformly converging to $0$ as $j\to\infty$, then
$\lim\limits_{j\to\infty}(T, \chi_j)= 0$.
\end{definition}
The notion of currents unifies the notion of differential forms and submanifolds. In particular, a locally integrable $k$-form $\beta$ can be naturally viewed as a $k$-current via the pairing
\begin{equation}
(\beta, \chi)\equiv \int_Q\beta\wedge\chi, \ \ \ \ \chi\in \Omega^{m-k}_0(Q),
\end{equation}
and an oriented submanifold $P$ of \emph{codimension}-$k$ also defines a $k$-current $\delta_P$ via
\begin{equation}
(\delta_P, \chi)\equiv \int_P \chi, \ \ \ \ \chi\in \Omega^{m-k}_0(Q).
\end{equation}
The usual exterior differential operator $d$ and the Hodge star operator $*$ acting on differential forms naturally extend to currents. Given a $k$-current $T$ and $\chi\in\Omega_0^{m-k}(Q)$, then we define
\begin{align}
(dT, \chi) & \equiv (-1)^{k+1}(T, d\chi),
\\(*T, \chi) & \equiv (-1)^{k(m-k)}(T, *\chi).\end{align}
Let $d^*$ be the codifferential operator and denote by $\Delta \equiv dd^*+d^*d$ the Hodge Laplacian, then it follows that for every $k$-current $T$ and $\chi\in \Omega_0^{m-k}(Q)$,
\begin{align}
(d^*T, \chi)& =(-1)^k(T, d^*\chi),
\\
(\Delta T, \chi)&=(T, \Delta \chi).\end{align}
A $k$-current $T$ is called {\it harmonic} if $\Delta T=0$. It follows from the standard elliptic regularity theory that a harmonic $k$-current can be represented by a smooth harmonic $k$-form.
Now let $P\subset Q$ be a (not necessarily closed) embedded oriented submanifold. Although the following discussion applies to more general setting, for our purpose in this paper we will only consider the case when $P$ is of codimension-$3$ in $Q$. The importance of the codimension-$3$ case in our setting is related to the fact that there is a Hopf fibration $\mathbb R^4\rightarrow \mathbb R^3$ which is a singular $S^1$-fibration with a smooth total space and a codimension-3 discriminant locus on the base. The codimension-3 condition also appears in other geometric settings, for example, Hitchin's theory of Gerbes \cite{Hitchin}.
\begin{definition}[Green's current] Suppose that $P$ is of codimension-3 in $Q$.
A Green's current $G_P$ for $P$ in $Q$ is a locally integrable $3$-form which solves the following current equation on $Q$,
\begin{equation}\label{Greencurrent}
\Delta G_P=2\pi \cdot \delta_P.
\end{equation}
\end{definition}
\begin{example}The above normalization constant is chosen such that in the case $Q\equiv \mathbb{R}^3$ and $P\equiv 0^3\in\mathbb{R}^3$, then
\begin{equation}G_P=\frac{1}{2|y|}dy_1\wedge dy_2\wedge dy_3\end{equation}
solves the current equation $\Delta_0 G_P=2\pi\cdot \delta_P$ for the standard Hodge Laplacian $\Delta_0$ on $\mathbb{R}^3$. \end{example}
In particular, $G_P$ is harmonic and hence smooth outside $P$. Notice that a Green's current $G_P$ for $P$ is unique up to the addition of a harmonic $3$-form. So the singular behavior near $P$ does not depend on the particular choice of $G_P$. Also it is clear that if $Q'\subset Q$ is an open submanifold, then the restriction of $G_P$ to $Q'$ is a Green's current for $P'=P\cap Q'$ in $Q'$, so that we can study the regularity problem in a local fashion. Our goal of this subsection is to understand the local existence and regularity of $G_P$ via the approximation by the standard model, i.e., the product Euclidean space $\mathbb R^3\times \mathbb R^{n-3}$.
To begin with, we have the following simple regularity result for $d(G_P)$.
\begin{proposition}\label{regularityd}
Given a Green's current $G_P$ for $P$ in $Q$, its differential $d(G_P)$ extends to a smooth $4$-form across $P$.
\end{proposition}
\begin{proof}
This is a local result so we can work with the geodesic ball $B_r(p)$ for any $p\in P$ such that $\overline{B_r(p)}\subset \subset Q$ and $\overline{B_r(p)}\cap P \subset \subset P$. Given any test form $\chi\in \Omega^{m-4}_0(B_r(p))$ we have
\begin{equation}(\Delta (d(G_P)), \chi)=(d\Delta G_P, \chi)=(\Delta G_P, d\chi)=(2\pi \delta_P, d\chi)=2\pi\int_P d\chi=2\pi\int_{\partial B_r(p)\cap P} \chi=0. \end{equation}
Therefore, $d(G_P)$ is a harmonic $4$-current in $B_r(p)$ and hence it is smooth in $B_r(p)$.
\end{proof}
From now on, let us fix a point $p\in P$ and let $\{x_j, y_\alpha\}$ be a normal coordinate system for $P$ in a neighborhood $\mathcal U$ of $p$ in $Q$ with $U=\mathcal U\cap P$. We also need the following notation.
\begin{notation}
\label{n:differential-form-notation} Given $k\in\mathbb{Z}_+$ and $0\leq q\leq 3$, the notations $\Pi_q^{(k)}$ always represents a general $q$-form for, which is defined in a neighborhood of $p$ in $Q$, and can be expressed in normal coordinates as
\begin{equation}\Pi_q^{(k)}\equiv A_{i,\alpha_1,\ldots, \alpha_{q-1}}^{(k)}(x,y)(dy_{\alpha_1}\wedge\ldots \wedge dy_{\alpha_{q-1}})\wedge dx_i + B_{\alpha_1,\ldots, \alpha_q}^{(k)}(x,y)dy_{\alpha_1}\wedge\ldots\wedge dy_{\alpha_q},\label{e:3-form-polynomial-coe}\end{equation}
where $A_{i,\alpha_1,\ldots, \alpha_{q-1}}^{(k)}(x,y)$ and $B_{\alpha_1,\ldots, \alpha_q}^{(k)}(x,y)$ are homogeneous polynomial functions in $y$ of degree $k$ with coefficient functions smooth on $U\subset P$.\end{notation}
\begin{remark}
Notice that this expression depends on the choice of local coordinates, but under a change of coordinates, a $p$-form $\Pi_p^{(k)}$ still yields such an expression, modulo a term of order $\widetilde{O}(r^{k+1})$.
\end{remark}
Now we are ready to state the first main theorem of this section, which gives a local existence for Green's current, and describes its leading singular behavior.
\begin{theorem} \label{t:Green-expansion} Given the above oriented codimension-3 submanifold $P$ in $Q$. For any $p\in P$, shrinking $\mathcal U$ if necessary. Then there exists a Green's current $G_{U}$ for $U=\mathcal U\cap P$ which satisfies
\begin{align}
\Delta G_U = 2\pi\cdot \delta_U \quad \text{in} \ \mathcal{U},
\end{align}
and in normal coordinates we have the local expansion
\begin{align} \label{e:Green expansion}
G_U
=&\frac{1}{2r}(1-\frac{H^\alpha y_\alpha}{2}) dy_1\wedge dy_2\wedge dy_3+\frac{1}{2r}y_{\beta}A_{i\alpha\beta}dx_i \wedge dy_{\widehat{\alpha}}
-\frac{1}{4} A_{ij\alpha\beta}r\cdot dy_{\widehat{\alpha\beta}}\wedge dx_i\wedge dx_j\nonumber
\\+& \frac{3}{16}(A_{i\alpha, \alpha+1}A_{j\alpha, \alpha+2}-A_{i\alpha, \alpha+2}A_{j\alpha, \alpha+1})d(ry_\alpha)\wedge dx_i\wedge dx_j
+r^{-3}\Pi_3^{(4)}+O'(r^2),\end{align}
where $\overrightarrow{H}=H^{\alpha}e_{\alpha}$ is the mean curvature of $P\subset Q$ and the $3$-form $\Pi_3^{(4)}$ is defined in \eqref{e:3-form-polynomial-coe}.
\end{theorem}
Here and in the following we use the notation that for $\alpha\in \{1, 2, 3\}$, $\widehat{\alpha}=(\alpha+1, \alpha+2)$ (with the convention $3+1=1$), $dy_{\alpha\beta}\equiv dy_\alpha\wedge dy_\beta$, and that
\begin{align}
dy_{\widehat{\alpha\beta}}\equiv\begin{cases}
dy_{\alpha+2}, & \beta=\alpha+1,\\
-dy_{\alpha+1}, & \beta=\alpha+2,\\
0, & \beta=\alpha.
\end{cases}
\end{align}
\begin{remark}
Any Green's current $G_P$ for $P$ yields the same singular expansion \eqref{e:Green expansion} near $p\in P$.
\end{remark}
\begin{remark}
Theorem \ref{t:Green-expansion} is certainly not optimal. Our choice of order of expansion is dictated by our applications, and the above exactly suits our purposes. See Remark \ref{r:C2alpharegularity} and Remark \ref{r:C1alphavsC2alpha}. In general one would expect a ``poly-homogeneous" expansion.
Similarly
in the proof we will not keep track of the explicit form of $\Pi_3^{(4)}$ because it is not needed in our applications. However, it is possible to obtain the precise expression with more work. Given the above expansion, there are also some constraint for $\Pi_3^{(4)}$ following from the fact that $d(G_{U})$ is smooth by Proposition \ref{regularityd}.
\end{remark}
Before starting the proof of Theorem \ref{t:Green-expansion}, we need some preliminary computations. For the convenience of our calculations, we introduce three $1$-forms
\begin{equation}\eta_\alpha \equiv dy_\alpha+p_{i\alpha}dx_i \label{e:definition-of-eta-alpha}\end{equation} such that
\begin{equation}\label{eqn3-29}
\langle\eta_\alpha, dx_j\rangle=0\end{equation}
for all $j$ and $\alpha$ at all points of $\mathcal U$. Then the linear span of the $\eta_\alpha$'s is orthogonal to the linear span of the $dx_i$'s.
The following elementary lemma is crucial in the proof of Theorem \ref{t:Green-expansion}.
\begin{lemma} \label{l:coe-crossing} The coefficient $p_{i\alpha}$ in \eqref{e:definition-of-eta-alpha} has the expansion
\begin{equation}p_{k\alpha}=g_{k\alpha}+\widetilde{O}(r^3), \quad 1\leq k\leq m-3,\ 1\leq \alpha\leq 3.
\end{equation}
\end{lemma}
\begin{proof}
We write the full matrix expression of the metric $g$ as
\begin{equation}
g= \left[ {\begin{array}{cc}
g_{\alpha\beta}& 0 \\
0 & g_{ij} \\
\end{array} } \right]+
\left[ {\begin{array}{cc}
0 & S \\
S^t & 0 \\
\end{array} } \right],
\end{equation}
where $S=(g_{\alpha i})=\widetilde{O}(r)$. We denote by $(h_{\alpha\beta})$ and $(h_{ij})$ the inverse matrix of $(g_{\alpha\beta})$ and $(g_{ij})$ respectively.
Then by elementary consideration
\begin{eqnarray}
g^{-1} &=& \left[ {\begin{array}{cc}
h_{\alpha\beta}& 0 \\
0 & h_{ij} \\
\end{array} } \right]-\left[ {\begin{array}{cc}
h_{\alpha\beta}& 0 \\
0 & h_{ ij} \\
\end{array} } \right]
\left[ {\begin{array}{cc}
0 & S \\
S^t & 0 \\
\end{array} } \right]\left[ {\begin{array}{cc}
h_{\alpha\beta}& 0 \\
0 & h_{ij} \\
\end{array} } \right]
\nonumber\\
&+& \left[ {\begin{array}{cc}
h_{\alpha\beta}& 0 \\
0 & h_{ij} \\
\end{array} } \right] \left[ {\begin{array}{cc}
0 & S \\
S^t & 0 \\
\end{array} } \right]\left[ {\begin{array}{cc}
h_{\alpha\beta}& 0 \\
0 & h_{ ij} \\
\end{array} } \right]
\left[ {\begin{array}{cc}
0 & S \\
S^t & 0 \\
\end{array} } \right]\left[ {\begin{array}{cc}
h_{\alpha\beta}& 0 \\
0 & h_{ ij} \\
\end{array} } \right]
+\widetilde{O}(r^3).
\end{eqnarray}
Notice that the third term does not have off-diagonal contributions, so the matrix $g^{-1}=(g^{IJ})$ satisfies
\begin{align}g^{i\alpha} &=-h_{ij}g_{j\beta}h_{\beta\alpha}+\widetilde{O}(r^3)
\label{e:inverse-crossing}
\\
g^{ij} &= h_{ij}+\widetilde{O}(r^2),\label{eqn3-35}
\\
g^{\alpha\beta} &= h_{\alpha\beta}+\widetilde{O}(r^2)=\delta_{\alpha\beta}+\widetilde{O}(r^2),\label{e:inverse normal}
\end{align}
where we used Lemma \ref{l:metric-expansion}.
By \eqref{eqn3-29} we have
$p_{i\alpha}g^{ij}+g^{j\alpha}=0$.
Let $(\hat{g}_{ij})$ be the inverse of the matrix $(g^{ij})$
with $1\leq i,j\leq m-3$, i.e., $g^{ij}\hat{g}_{jk}=\delta_{ik}$. Then
\begin{equation}
p_{k\alpha} = -g^{j\alpha}\hat{g}_{jk}.\label{e:coe-coor}
\end{equation}
We claim that for any $1\leq i,j \leq m-3$,
\begin{equation}
\hat{g}_{ij} - g_{ij} =\widetilde{O}(r^2).
\end{equation}
In fact, since $
g^{ij}g_{jk} + g^{i\alpha}g_{\alpha k} =\delta_{ik}$,
we get
\begin{equation}
\hat{g}_{li}(g^{ij}g_{jk} + g^{i\alpha}g_{\alpha k})=\hat{g}_{lk}.\end{equation}
So this implies that
\begin{equation}
\hat{g}_{lk} - g_{lk} = \hat{g}_{li} g^{i\alpha} g_{\alpha k} = \widetilde{O}(r^2), \label{e:db-inverse-order}
\end{equation}
which proves the claim.
Therefore, combining \eqref{e:inverse-crossing},\eqref{e:inverse normal}, \eqref{e:coe-coor} and \eqref{e:db-inverse-order}, we obtain
\begin{eqnarray}
p_{k\alpha}&=&
-(g_{kj}+\widetilde{O}(r^2))g^{j\alpha}
\nonumber\\
&=& -g_{kj}g^{j\alpha} + \widetilde{O}(r^3)
\nonumber\\
&=&h_{\alpha\beta}g_{k\beta}+\widetilde{O}(r^3)
\nonumber\\
&=&g_{k\alpha}+\widetilde{O}(r^3).\end{eqnarray}
\end{proof}
The following symmetry property of $p_{i\alpha}$ will also frequently used in our calculations below.
By Lemma \ref{l:metric-expansion} and Lemma \ref{l:coe-crossing}, we may write
\begin{equation} \label{e:symmetry-of-coefficient-p}
p_{i\alpha}=A_{i\alpha\beta}y_\beta+\frac{1}{2}B_{i\alpha\beta\gamma} y_\beta y_\gamma+\widetilde{O}(r^3).
\end{equation}
Here the connection term $A_{i\alpha\beta}\equiv \langle\nabla_{\partial_{x_i}}\partial_{y_{\beta}}, y_{\alpha}\rangle|_{(x,0)}$
is skew-symmetric in $\alpha$, $\beta$, and the curvature term $B_{i\alpha\beta\gamma}\equiv -\frac{2}{3}(R_{i\beta\gamma\alpha}+R_{i\gamma\beta\alpha})$ is symmetric in $\beta$, $\gamma$.
\begin{lemma}
For every $1\leq \alpha,\beta\leq 3$,
\begin{equation} \label{etaorthogonal}
\langle \eta_\alpha, \eta_\beta\rangle=\delta_{\alpha\beta}+\widetilde{O}(r^2).
\end{equation}
\end{lemma}
\begin{proof}
By definition
$\langle \eta_\alpha, \eta_\beta\rangle=\langle dy_\alpha+p_{i\alpha} dx_i, dy_\beta+p_{j\beta}dx_j\rangle$. By \eqref{e:inverse normal} we get
\begin{equation}\langle dy_\alpha, dy_\beta\rangle=\delta_{\alpha\beta}+\widetilde{O}(r^2).\end{equation}
Also we have $p_{i\alpha}=\widetilde{O}(r)$ and $\langle dx_i, dy_\beta\rangle=\widetilde{O}(r)$ for all $i$ and $\alpha$. The conclusion then follows.
\end{proof}
Using the above differential $1$-forms $\eta_{\alpha}$'s, we can decompose the volume form $\dvol_g$ in the horizontal and vertical directions, which will substantially simplify the computations regarding the Hodge Laplacian.
The volume form of $g$ is given by
\begin{equation}\dvol_g=\sqrt{\det(g)}\cdot dy_1\wedge dy_2\wedge dy_3\wedge dx_1\wedge \cdots\wedge dx_{m-3},\end{equation}
where we have used the orientation fixed above.
We define the normal and tangential volume forms by
\begin{eqnarray}\begin{cases}
{\dvol}_N \equiv \eta_1\wedge \eta_2\wedge \eta_3\\
{\dvol}_T \equiv \sqrt{\det(g_{ij}^P)}\cdot dx_1\wedge \cdots\wedge dx_{m-3}.
\end{cases}
\end{eqnarray}
By the expansion formula \eqref{e:symmetry-of-coefficient-p}, the normal volume form $\dvol_N$ has the following expansion,
\begin{eqnarray} \label{e:dvol-N-expansion}
{\dvol}_N=dy_1\wedge dy_2\wedge dy_3+\Big(A_{i\alpha\beta} y_{\beta}+\frac{1}{2}B_{i\alpha\beta\gamma}y_\beta y_\gamma \Big) dx_i\wedge dy_{\widehat\alpha}\nonumber \\
+A_{i,\alpha+1,\beta}A_{j, \alpha+2, \gamma} y_\beta y_\gamma dy_{\alpha}\wedge dx_i\wedge dx_j+
\widetilde{O}(r^3).
\end{eqnarray}
In addition, by the definition of $\eta_{\alpha}$'s, it holds that for each $1\leq \alpha \leq 3$,
\begin{equation}
\eta_{\alpha} \wedge \dvol_T = dy_{\alpha} \wedge \dvol_T,
\end{equation}
and hence
\begin{equation}
\dvol_N\wedge \dvol_T = dy_1\wedge dy_2\wedge dy_3 \wedge \dvol_T = \frac{\sqrt{\det(g_{ij}^P)}}{\sqrt{\det(g)}}\cdot \dvol_g.
\end{equation}
\begin{lemma}\label{l:v-T-N}Denote by $*:\Omega^k(Q)\to\Omega^{m-k}(Q)$ the Hodge $*$-operator. Then we have the following:
\begin{enumerate}
\item The tangential volume form $\dvol_T$ satisfies
\begin{equation}
*\dvol_T=(-1)^{m+1}\dvol_N\cdot (1-H^{\alpha}y_{\alpha}+\widetilde{O}(r^2)).
\end{equation}
\item For any $\alpha\in\{1,2,3\}$,
\begin{equation}
*(\eta_{\alpha}\wedge \dvol_T) = \lambda_{1}\eta_{\widehat{\alpha}} + \lambda_{2}\eta_{\widehat{\alpha+1}} + \lambda_{3}\eta_{\widehat{\alpha+2}},
\end{equation}
where $\lambda_1=1-H^{\alpha}y_{\alpha}+\widetilde{O}(r^2)$, $\lambda_2=\widetilde{O}(r^2)$, $\lambda_3=\widetilde{O}(r^2)$.
\end{enumerate}
\end{lemma}
\begin{proof}
First, we prove item (1). By \eqref{eqn3-29} we have
\begin{equation}(-1)^{m+1}*\dvol_T=\lambda\cdot\dvol_N\end{equation}
for a function $\lambda>0$. The function $\lambda$ is given by
\begin{equation}\lambda=\frac{|\dvol_T|^2\sqrt{\det(g)}}{\sqrt{\det(g_{ij}^P)}}=\sqrt{\det(g)}\det(g^{ij})\sqrt{\det(g_{ij}^P)}.\label{e:coe-def}\end{equation}
Now we compute the expansion of $\lambda$. Applying the expansions of $g_{ij}$, $g_{\alpha\beta}$ and $g_{i\alpha}$ in Lemma \ref{l:metric-expansion}, one can directly obtain the following,
\begin{align}\det(g)&=\det(g_{\alpha\beta})\cdot\det(g_{ij})+\widetilde{O}(r^2),\label{e:v}
\\\det(g_{\alpha\beta})&=1+\widetilde{O}(r^2),\label{e:vertical-v}
\\\det(g_{ij})&=\det(g_{ij}^P)\cdot(1+2H^\alpha y_\alpha)+\widetilde{O}(r^2).\label{e:horizontal-v}\end{align}
Plugging \eqref{e:vertical-v} and \eqref{e:horizontal-v} into \eqref{e:v},
\begin{equation}
\det(g) = \det(g_{ij}^P)\cdot (1 + 2H^{\alpha} y_{\alpha}) + \widetilde{O}(r^2).\label{e:g-expansion}
\end{equation}
Let $(h_{ij})$ be the inverse of the matrix $(g_{ij})$. Since $g^{ij}=h_{ij} + \widetilde{O}(r^2)$ by (\ref{eqn3-35}), so it follows that
\begin{equation}\det(g^{ij})=\det(h_{ij})+\widetilde{O}(r^2)=(\det(g_{ij}))^{-1}+\widetilde{O}(r^2).
\end{equation}
Plugging \eqref{e:horizontal-v} into the above,
\begin{equation}\det(g^{ij})=\det(g_{ij}^P)^{-1}\cdot (1-2H^{\alpha}y_{\alpha})+\widetilde{O}(r^2).\label{e:inv-expansion}\end{equation}
Therefore, substituting \eqref{e:g-expansion} and \eqref{e:inv-expansion} into \eqref{e:coe-def},
\begin{equation}
\lambda = 1 - H^{\alpha} y_{\alpha} + \widetilde{O}(r^2),
\end{equation}
which completes the proof of item (1).
Now we prove item (2).
For each $\alpha\in\{1,2,3\}$, we can write
\begin{equation}
*(\eta_\alpha\wedge \dvol_T)=\lambda_1 \cdot \eta_{\widehat{\alpha}}+\lambda_2\cdot \eta_{\widehat{\alpha+1}}+\lambda_3 \cdot \eta_{\widehat{\alpha+2}}.\end{equation}
Taking point-wise wedge product with $\eta_\alpha\wedge \dvol_T$, and noticing $\eta_{\widehat{\alpha+1}}\wedge \eta_{\alpha}$, $\eta_{\widehat{\alpha+2}}\wedge \eta_{\alpha}$ are both zero, we obtain
\begin{equation}
\lambda_1 \cdot \eta_{\widehat{\alpha}} \wedge \eta_\alpha\wedge \dvol_T = (\eta_\alpha\wedge \dvol_T) \wedge *(\eta_\alpha\wedge \dvol_T).
\end{equation}
Therefore, by \eqref{eqn3-29} and \eqref{etaorthogonal} we get
\begin{equation}
\lambda_1 =\frac{|\eta_{\alpha}\wedge\dvol_T|^2\sqrt{\det(g)}}{\sqrt{\det(g_{ij}^P)}}=1-H^{\alpha}y_{\alpha}+\widetilde{O}(r^2),
\end{equation}
Similarly taking wedge product with $\eta_{\alpha+1}\wedge \dvol_T$ and $\eta_{\alpha+2}\wedge \dvol_T$ respectively, and again by \eqref{etaorthogonal} we obtain
that
\begin{equation}
\lambda_2=\widetilde{O}(r^2), \lambda_3=\widetilde{O}(r^2).
\end{equation}
These imply that
\begin{equation}
*(\eta_\alpha\wedge \dvol_T) =\lambda_1 \cdot \eta_{\widehat{\alpha}} +\lambda_2 \cdot \eta_{\widehat{\alpha+1}} + \lambda_3 \cdot \eta_{\widehat{\alpha+2}},
\end{equation}
where $\lambda_1 = 1+H^{\alpha}y_{\alpha}+\widetilde{O}(r^2)$, $\lambda_2 = \widetilde{O}(r^2)$ and $\lambda_3 = \widetilde{O}(r^2)$.
\end{proof}
Now we proceed to prove Theorem \ref{t:Green-expansion}. This will be done in several steps.
\
\noindent {\bf Step 1:}
We start
by defining a 3-form
\begin{equation}
\phi_1\equiv(-1)^{m+1}\frac{1}{2r} *\dvol_T. \label{e:def-phi-1}\end{equation}
By item (1) of Lemma \ref{l:v-T-N}, immediately we have
\begin{align}
\phi_1=\frac{\dvol_N}{2r}\cdot (1-H^{\alpha}y_{\alpha}+\widetilde{O}(r^2)).\label{e:phi-1-expansion}
\end{align}
Then applying the expansion of $\dvol_N$ in \eqref{e:dvol-N-expansion}, $\phi_1$ has a further expansion,
\begin{align}\phi_1&=\frac{1-H^\alpha y_\alpha}{2r}dy_1\wedge dy_2 \wedge dy_3+\frac{1}{2r}A_{i\alpha\beta} y_\beta dx_i\wedge dy_{\widehat{\alpha}}\nonumber\\
&+\frac{1}{2r}A_{i,\alpha+1,\beta}A_{j, \alpha+2, \gamma} y_\beta y_\gamma dy_{\alpha} \wedge dx_i\wedge dx_j+r^{-1}\Pi_3^{(2)}+O'(r^2)\nonumber\\
&=\frac{1-H^\alpha y_\alpha}{2r}dy_1\wedge dy_2 \wedge dy_3+\frac{1}{2r}A_{i\alpha\beta} y_\beta dx_i\wedge dy_{\hat\alpha}\nonumber\\
&+\frac{1}{4}(A_{i\alpha, \alpha+1}A_{j\alpha, \alpha+2}-A_{i\alpha, \alpha+2}A_{j\alpha, \alpha+1})y_\alpha \cdot dr\wedge dx_i\wedge dx_j+r^{-1}\Pi_3^{(2)}+O'(r^2),
\label{e:phi1expansion-2}
\end{align}
where $\Pi_3^{(2)}$ is the $3$-form introduced in Notation \ref{n:differential-form-notation} and
the last step follows from the following lemma:
\begin{lemma}[Re-arrangement Lemma]
\label{l:rearrange}
\begin{equation}
A_{i,\alpha+1,\beta}A_{j, \alpha+2, \gamma} y_\beta y_\gamma dy_{\alpha} \wedge dx_i\wedge dx_j=\frac{1}{2}(A_{i\alpha, \alpha+1}A_{j\alpha, \alpha+2}-A_{i\alpha, \alpha+2}A_{j\alpha, \alpha+1})y_\alpha \cdot rdr\wedge dx_i\wedge dx_j.
\end{equation}
\end{lemma}
\begin{proof}
First by writing out the terms and re-arranging the subscripts and using the skew symmetry of $A_{i\alpha\beta}$ we get
\begin{eqnarray}
&&A_{i,\alpha+1,\beta}A_{j, \alpha+2, \gamma} y_\beta y_\gamma dy_{\alpha}\nonumber \\
&=&\Big(A_{i, \alpha+1, \alpha}A_{j, \alpha+2, \alpha} y_\alpha^2+A_{i, \alpha+1, \alpha}A_{j, \alpha+2, \alpha+1} y_\alpha y_{\alpha+1}\nonumber\\
&&+A_{i, \alpha+1, \alpha+2}A_{j, \alpha+2, \alpha} y_\alpha y_{\alpha+2} +A_{i, \alpha+1, \alpha+2}A_{j, \alpha+2, \alpha+1} y_{\alpha+1}y_{\alpha+2}\Big)dy_\alpha\nonumber\\
&=&A_{i, \alpha, \alpha+1}A_{j, \alpha, \alpha+2} y_\alpha^2dy_\alpha-A_{i\alpha, \alpha+2} A_{j\alpha, \alpha+1} y_\alpha y_{\alpha+2}dy_{\alpha+2}\nonumber\\
&&-A_{i, \alpha, \alpha+2}A_{j, \alpha, \alpha+1}y_\alpha y_{\alpha+1}dy_{\alpha+1}-A_{i, \alpha, \alpha+1}A_{j, \alpha, \alpha+1}y_\alpha y_{\alpha+1}dy_{\alpha+2}.
\label{eqn3-98}\end{eqnarray}
So we have
\begin{equation} A_{i,\alpha+1,\beta}A_{j, \alpha+2, \gamma} y_\beta y_\gamma dy_{\alpha} \wedge dx_i\wedge dx_j=\frac{1}{2} (A_{i,\alpha+1,\beta}A_{j, \alpha+2, \gamma}-A_{j,\alpha+1,\beta}A_{i, \alpha+2, \gamma}) y_\beta y_\gamma dy_{\alpha} \wedge dx_i\wedge dx_j.\end{equation}
Correspondingly by skew-symmetrizing each term of (\ref{eqn3-98}) with respect to $i$ and $j$, we get
\begin{eqnarray}
&&\frac{1}{2} (A_{i,\alpha+1,\beta}A_{j, \alpha+2, \gamma}-A_{j,\alpha+1,\beta}A_{i, \alpha+2, \gamma})y_\beta y_\gamma dy_\alpha\nonumber\\
&=&\frac{1}{2}(A_{i, \alpha, \alpha+1}A_{j, \alpha, \alpha+2}- A_{i\alpha, \alpha+2}A_{j\alpha, \alpha+1}) y_\alpha (y_\alpha dy_\alpha+y_{\alpha+1}dy_{\alpha+1}+y_{\alpha+2}dy_{\alpha+2})\nonumber\\
&=& \frac{1}{2} (A_{i\alpha, \alpha+1}A_{j\alpha, \alpha+2}-A_{i\alpha, \alpha+2}A_{j\alpha, \alpha+1})y_\alpha \cdot rdr.
\end{eqnarray}
\end{proof}
\
\noindent {\bf Step 2:} In this step will explicitly compute the singular terms in the expansion of $\Delta\phi_1$. Mainly, we will prove the following proposition.
\begin{proposition}
\label{p:Delta-phi-1-expansion}
Let $\phi_1$ be the $3$-form defined in \eqref{e:def-phi-1}, then we have
\begin{align}
\Delta\phi_1=& -\frac{H^\alpha y_\alpha}{2r^3}dy_1\wedge dy_2\wedge dy_3-\Omega_{ij\alpha\beta} (\frac{1}{4r} dy_{\widehat{\alpha\beta}} +\frac{1}{4r^2} y_{\widehat{\alpha\beta}}dr )\wedge dx_i\wedge dx_j
\nonumber\\
&+A_{ij\alpha\beta}(\frac{1}{4r^2}y_{\widehat{\alpha\beta}}dr-\frac{1}{4r}dy_{\widehat{\alpha\beta}})\wedge dx_i\wedge dx_j +r^{-5} \Pi_3^{(4)}+O'(1).
\label{e:Delta-phi-1}
\end{align}
\end{proposition}
\begin{proof}
The proof consists of two steps.
The first step focuses on the computation for $d^*d\phi_1$. Starting with the expansion of $\phi_1$ in \eqref{e:phi-1-expansion}, we have \begin{equation}
d\phi_1 = \frac{-1+H_{\alpha}y_{\alpha}+\widetilde{O}(r^2)}{2r^3} \cdot rdr\wedge \dvol_N + \frac{1}{2r} d\Big((1-H^{\alpha}y_{\alpha}+{\widetilde{O}}(r^2)) \dvol_N\Big).\label{e:d-phi-1}
\end{equation}
To deal with the first term, we use Lemma \ref{l:coe-crossing} and \eqref{Gausslemma} in Lemma \ref{l:generalized-Gauss}, then \begin{eqnarray} \label{eqn2-14}
y_\alpha \eta_\alpha&=&y_{\alpha}dy_{\alpha}-y_{\alpha}p_{i\alpha}dx_i\nonumber\\
&=&rdr-y_\alpha g_{i\alpha}dx_i+\widetilde{O}(r^4)
\nonumber\\
&=&rdr+\widetilde{O}(r^4),\label{e:gauss-cancel}
\end{eqnarray}
which yields
\begin{equation}
rdr\wedge \dvol_N=(y_\alpha\eta_\alpha)\wedge \dvol_N+\widetilde{O}(r^4)=\widetilde{O}(r^4).\label{e:normal-times-vertical-v}
\end{equation}
So it follows that
\begin{equation}
d\phi_1 = \frac{1}{2r} d\Big((1-H^{\alpha}y_{\alpha}+\widetilde{O}(r^2)) \dvol_N\Big) + O'(r).
\end{equation}
It is easy to see that
\begin{equation}d(\widetilde{O}(r^2)\dvol_N)=\widetilde{O}(r^2).\end{equation}
So we obtain
\begin{equation}
d\phi_1= \frac{1}{2r} d\Big((1-H^{\alpha}y_{\alpha}) \dvol_N\Big) + O'(r).\label{e:d-phi-1-dv-N}
\end{equation}
Next, let us compute the expansion for $d(\dvol_N)$. By definition, \begin{equation}d(\dvol_N)=d(\eta_1\wedge\eta_2\wedge\eta_3)=d\eta_\alpha\wedge\eta_{ \widehat\alpha}.\end{equation}
By \eqref{e:symmetry-of-coefficient-p},
\begin{eqnarray}
\label{detaalpha}
d\eta_\alpha&=&
d(p_{i\alpha})\wedge dx_i
\nonumber\\
&=&A_{i\alpha\beta}dy_{\beta} \wedge dx_i+A_{ji\alpha\beta} y_{\beta} dx_j\wedge dx_i+B_{i\alpha\beta\gamma}y_{\beta}dy_{\gamma}\wedge dx_i+\widetilde{O}(r^2).\end{eqnarray}
So we have
\begin{equation}d(\dvol_N)=A_{i\alpha\beta}dy_{\beta} \wedge dx_i \wedge \eta_{\widehat{\alpha}}+ y_\beta (A_{ji\alpha\beta} dx_j\wedge dx_i+B_{i\alpha\beta\gamma} dy_\gamma \wedge dx_i )\wedge dy_{\widehat\alpha}
+\widetilde{O}(r^2).
\label{e:dv-N-splitting}\end{equation}
Now we need to rearrange the above expansion. Since $A_{i\alpha\beta}$ is skew symmetric in $\alpha$ and $\beta$, we have for $\alpha\in \{1, 2, 3\}$,
\begin{equation}A_{i\alpha\alpha}=0,\end{equation} so the leading order in the first term vanishes, hence
\begin{eqnarray}A_{i\alpha\beta}dy_\beta\wedge dx_i \wedge \eta_{\widehat{\alpha}}
&=&A_{i\alpha\beta}A_{j\mu\gamma}y_{\gamma}dy_\beta \wedge dx_i\wedge dx_j\wedge dy_{\widehat{\alpha\mu}}+\widetilde{O}(r^2)
\nonumber\\
&=&A_{i\alpha\beta}A_{j\beta\gamma}y_\gamma dy_{\widehat\alpha}\wedge dx_i\wedge dx_j+ \widetilde{O}(r^2)
\nonumber\\
&=&\frac{1}{2}(A_{i\alpha\beta}A_{j\beta\gamma}-A_{i\gamma\beta}A_{j\beta\alpha})y_\gamma dy_{\widehat\alpha}\wedge dx_i\wedge dx_j+ \widetilde{O}(r^2).
\end{eqnarray}
Therefore,
\begin{equation}
d(\dvol_N) =\Omega_{ij\alpha\beta}\cdot y_\beta \cdot dy_{\widehat\alpha}\wedge dx_i\wedge dx_j+B_{i\alpha\beta\alpha}\cdot y_\beta \cdot dy_1\wedge dy_2\wedge dy_3\wedge dx_i+\widetilde{O}(r^2).\label{e:1-d-N}
\end{equation}
By \eqref{e:dvol-N-expansion} we have
\begin{equation} \label{e:H-d-N}
d(H^{\alpha}y_{\alpha})\wedge \dvol_N=(H^\alpha A_{i\alpha\beta}-\partial_i(H^\beta))\cdot y_{\beta}\cdot dy_1\wedge dy_2\wedge dy_3\wedge dx_i+\widetilde{O}(r^2).
\end{equation}
Now substituting \eqref{e:1-d-N} and \eqref{e:H-d-N} into \eqref{e:d-phi-1-dv-N},
\begin{align} \label{e:d-phi-1-exp}
d\phi_1
&= \frac{1}{2r}\Omega_{ij\alpha\beta}\cdot y_\beta \cdot dy_{\widehat\alpha}\wedge dx_i\wedge dx_j\nonumber\\
&+\frac{1}{2r}\Big(B_{i\alpha\beta\alpha}-(H^\alpha A_{i\alpha\beta}-\partial_i(H^\beta)\Big)\cdot y_\beta \cdot dy_1\wedge dy_2\wedge dy_3\wedge dx_i+O'(r).
\end{align}
Now we need to take $d^*$ of this. Notice that the leading order of $d^*d\phi_1$ can be obtained using the flat Euclidean model, so we obtain
\begin{eqnarray} \label{e:d*dphi1-curvature-to-be-simplified}
d^*d\phi_1&=& \Omega_{ij\alpha\beta} (\frac{1}{2r} dy_{\widehat{\beta\alpha}} -\frac{1}{2r^3} y_\mu y_\beta dy_{\widehat{\mu\alpha}})dx_i\wedge dx_j+r^{-3}\Pi_3^{(2)}+O'(1).
\end{eqnarray}
In our next step, let us compute $dd^*\phi_1$. First,
\begin{equation}*\phi_1=\frac{1}{2r}\dvol_T.\end{equation}
Notice that $d(\dvol_T)=0$, so
\begin{equation}d*\phi_1=-\frac{1}{2r^3} \cdot rdr\wedge \dvol_T.\end{equation}
By \eqref{e:gauss-cancel}, $rdr= y_{\alpha}\eta_{\alpha}+{\widetilde{O}}(r^4)$, then
\begin{equation}
d*\phi_1=-\frac{y_{\alpha}}{2r^3}{}\eta_{\alpha}\wedge \dvol_T +O'(r).\end{equation}
Applying item (2) of Lemma \ref{l:v-T-N},
\begin{equation}*d*\phi_1=-\frac{y_{\alpha}}{2r^3}{}(1-H^{\beta}y_{\beta} + \widetilde{O}(r^2))\eta_{\widehat{\alpha}} + O'(r).\end{equation}
So it follows that
\begin{eqnarray} \label{eqn2-19}
d^*\phi_1
&=&- *d*\phi_1= \frac{y_{\alpha}}{2r^3}{}(1-H^{\beta}y_{\beta})\eta_{\widehat{\alpha}}{} + \frac{\widetilde{O}(r^2)}{r^3}y_\alpha dy_{\widehat\alpha} + O'(r).
\end{eqnarray}
Taking $d$ and applying Lemma \ref{l:generalized-Gauss},
\begin{align}
dd^*\phi_1=&(1-H^{\beta}y_{\beta})\Big(-\frac{3y_{\alpha}}{2r^5}{} rdr\wedge \eta_{\widehat{\alpha}}+\frac{1}{2r^3} d(y_\alpha \eta_{\widehat{\alpha}})\Big){}
-\frac{H^{\beta}y_{\alpha}}{2r^3}\eta_{\widehat{\alpha}}\wedge dy_{\beta}+r^{-5}\Pi_3^{(4)}+O'(1)
\nonumber\\
=&(1-H^{\beta}y_{\beta})\Big(-\frac{3y_{\alpha}}{2r^5}{} rdr\wedge \eta_{\widehat{\alpha}}+\frac{1}{2r^3} d(y_\alpha \eta_{\widehat{\alpha}})\Big)
-\frac{H^{\alpha}y_{\alpha}}{2r^3}dy_1\wedge dy_2\wedge dy_3
\nonumber\\
+&r^{-5}\Pi_3^{(4)}+O'(1).\label{e:dd^*}\end{align}
Now we simplify this expression. By \eqref{eqn2-14},
\begin{equation}
-\frac{3y_{\alpha}}{2r^5}{} rdr\wedge \eta_{\widehat{\alpha}} =- \frac{3y_\alpha y_\beta}{2r^5} {}\eta_\beta \wedge \eta_{\widehat{\alpha}}+O'(1)=-\frac{3}{2r^3}{}\dvol_N+{O}'(1).\label{e:first-term}\end{equation}
Also
\begin{eqnarray}
\frac{1}{2r^3} d(y_\alpha \eta_{\widehat{\alpha}}) &=& \frac{1}{2r^3}dy_{\alpha}\wedge \eta_{\widehat{\alpha}} +\frac{1}{2r^3}y_{\alpha}d\eta_{\widehat{\alpha}}
\nonumber\\
&=&\frac{1}{2r^3}(\eta_{{\alpha}} - p_{i\alpha} dx_i)\wedge \eta_{\widehat{\alpha}} +\frac{1}{2r^3}y_\alpha (d{\eta_{\alpha+1}}\wedge {\eta_{\alpha+2}}-{\eta_{\alpha+1}}\wedge d{\eta_{\alpha+2}})
\nonumber\\
&=& \frac{3}{2r^3}\dvol_N -\frac{1}{2r^3}(A_{i\alpha\beta}y_\beta+\frac{1}{2}B_{i\alpha\beta\gamma}y_\beta y_\gamma)dx_i\wedge\eta_{\widehat{\alpha}}
\nonumber\\
&+&\frac{1}{2r^3}y_\alpha (d{\eta_{\alpha+1}}\wedge {\eta_{\alpha+2}}-{\eta_{\alpha+1}}\wedge d{\eta_{\alpha+2}})+{O'}(1).
\end{eqnarray}
So it follows that
\begin{align}
dd^*\phi_1 =& -\frac{1}{2r^3}\Big(A_{i\alpha\beta}\cdot y_{\beta}\cdot dx_i\wedge\eta_{\widehat{\alpha}} -y_\alpha (d{\eta_{\alpha+1}}\wedge {\eta_{\alpha+2}}-{\eta_{\alpha+1}}\wedge d{\eta_{\alpha+2}})\Big)
\nonumber\\
-&\frac{H^{\alpha}y_{\alpha}}{2r^3}dy_1\wedge dy_2\wedge dy_3+r^{-5}\Pi_3^{(4)}+O'(1).\label{e:dd*phi1-before-key-cancellation}
\end{align}
Next, we will show a crucial cancellation for the first term of the above $dd^*\phi_1$, which gives a further order improvement.
\begin{lemma}[Cancellation Lemma]\label{l:key-cancellation} The following holds:
\begin{align}
& A_{i\alpha\beta}\cdot y_{\beta}\cdot dx_i\wedge \eta_{\widehat{\alpha}}
-y_{\alpha} (d{\eta_{\alpha+1}}\wedge{\eta_{\alpha+2}}-{\eta_{\alpha+1}}\wedge d{\eta_{\alpha+2}})
\nonumber\\
=& -A_{ij\alpha\beta} y_\beta y_\mu dy_{\widehat{\mu\alpha}}\wedge dx_i\wedge dx_j
+\Pi_3^{(2)}+{\widetilde{O}}(r^3).\end{align}
\end{lemma}
\begin{proof}
Directly applying the definition of $\eta_{\alpha}$, we have
\begin{align} \label{e:cancellation-first-term}
& A_{i\alpha\beta}\cdot y_{\beta}\cdot dx_i\wedge \eta_{\widehat{\alpha}}\nonumber\\
=& A_{i\alpha\beta} y_\beta dx_i\wedge dy_{\widehat{\alpha}}+ A_{i\alpha\beta} y_\beta y_\gamma (A_{j, \alpha+1, \gamma} dy_{\alpha+2}-A_{j, \alpha+2, \gamma} dy_{\alpha+1})\wedge dx_i\wedge dx_j+\widetilde{O}(r^3).
\end{align}
By \eqref{detaalpha}, we get
\begin{align} \label{e:eta-d-eta}
&y_\alpha (d{\eta_{\alpha+1}}\wedge{\eta_{\alpha+2}}-{\eta_{\alpha+1}}\wedge d{\eta_{\alpha+2}})\nonumber\\
=& y_\alpha (A_{i, \alpha+1, \beta}dy_\beta \wedge dx_i \wedge dy_{\alpha+2}-A_{i, \alpha+2, \beta}dy_\beta \wedge dx_i \wedge dy_{\alpha+1})\nonumber\\
+&y_\alpha y_\gamma (A_{i, \alpha+1, \beta} A_{j, \alpha+2, \gamma}-A_{i, \alpha+2, \beta} A_{j, \alpha+1, \gamma} )dy_\beta\wedge dx_i\wedge dx_j\nonumber\\
+&y_\alpha y_\beta (A_{ij, \alpha+1, \beta} dy_{\alpha+2}-A_{ij, \alpha+2, \beta} dy_{\alpha+1})\wedge dx_i \wedge dx_j\nonumber\\
+&\Pi_3^{(2)}+\widetilde{O}(r^3).
\end{align}
Rearranging the subscripts of the first groups of terms in \eqref{e:eta-d-eta},
\begin{eqnarray}
&&y_\alpha (A_{i, \alpha+1, \beta}dy_\beta \wedge dx_i \wedge dy_{\alpha+2}-A_{i, \alpha+2, \beta}dy_\beta \wedge dx_i \wedge dy_{\alpha+1})\\
&=& y_\alpha (A_{i, \alpha+1, \alpha}dy_\alpha \wedge dx_i \wedge dy_{\alpha+2}-A_{i, \alpha+2, \alpha}dy_\alpha \wedge dx_i \wedge dy_{\alpha+1})\\
&=& y_{\alpha+2} A_{i, \alpha, \alpha+2}dy_{\alpha+2} \wedge dx_i \wedge dy_{\alpha+1}-y_{\alpha+1}A_{i, \alpha, \alpha+1}dy_{\alpha+1} \wedge dx_i \wedge dy_{\alpha+2}\\
&=& A_{i\alpha\beta} y_\beta dx_i \wedge dy_{\widehat{\alpha}},
\end{eqnarray}
which matches the first term of \eqref{e:cancellation-first-term}. As in the proof of Lemma \ref{l:rearrange}, one can see that the second groups of terms in \eqref{e:cancellation-first-term} and \eqref{e:eta-d-eta} are both equal to
\begin{equation}
(A_{i\alpha, \alpha+1}A_{j\alpha, \alpha+2}-A_{i\alpha, \alpha+2}A_{j\alpha, \alpha+1})y_\alpha \cdot rdr\wedge dx_i\wedge dx_j.
\end{equation}
Next, the third group of terms in \eqref{e:eta-d-eta} can be rewritten as follows,
\begin{eqnarray}
&&y_\alpha y_\beta (A_{ij, \alpha+1, \beta} dy_{\alpha+2}-A_{ij, \alpha+2, \beta} dy_{\alpha+1})\wedge dx_i \wedge dx_j
\nonumber\\
&=&
A_{ij\alpha\beta} y_\beta (y_{\alpha+2} dy_{\alpha+1}-y_{\alpha+1}dy_{\alpha+2})\wedge dx_i\wedge dx_j
\nonumber\\
&=&A_{ij\alpha\beta} y_\beta y_\mu dy_{\widehat{\mu\alpha}}\wedge dx_i\wedge dx_j.
\end{eqnarray}
The conclusion just follows.
\end{proof}
Let us return to the expansion of
$dd^*\phi_1$ given by
\eqref{e:dd*phi1-before-key-cancellation}. Applying Lemma \ref{l:key-cancellation}, we have
\begin{align}
dd^*\phi_1 &= -\frac{H^\alpha y_\alpha}{2r^3}dy_1\wedge dy_2\wedge dy_3+\frac{1}{2r^3} A_{ij\alpha\beta} y_\beta y_\mu dy_{\widehat{\mu\alpha}}\wedge dx_i\wedge dx_j\nonumber\\
&+r^{-5} \Pi_3^{(4)}+O'(1). \label{e:dd*phi1-A-to-be-simplified}
\end{align}
To further simplify, we need the following lemma.
\begin{lemma}[Re-arrangement Lemma] The following identities hold:
\begin{align}
\Omega_{ij\alpha\beta} (\frac{1}{2r} dy_{\widehat{\beta\alpha}} -\frac{1}{2r^3} y_\mu y_\beta dy_{\widehat{\mu\alpha}}) &=-\Omega_{ij\alpha\beta} (\frac{1}{4r} dy_{\widehat{\alpha\beta}} +\frac{1}{4r^2} y_{\widehat{\alpha\beta}}dr ), \label{e:curvature-simplification}
\\
\frac{1}{2r^3} A_{ij\alpha\beta} y_\beta y_\mu dy_{\widehat{\mu\alpha}}&=A_{ij\alpha\beta}(\frac{1}{4r^2}y_{\widehat{\alpha\beta}}dr-\frac{1}{4r}dy_{\widehat{\alpha\beta}}).\label{e:connection-simplification}
\end{align}
\end{lemma}
\begin{proof}We only prove \eqref{e:curvature-simplification} because the other equality follows from the same computations.
Using the fact that $\Omega_{ij\alpha\beta}=-\Omega_{ij\beta\alpha}$, we can write out the left hand side as
\begin{align}
&\Omega_{ij\alpha\beta} (\frac{1}{2r} dy_{\widehat{\beta\alpha}} -\frac{1}{2r^3} y_\mu y_\beta dy_{\widehat{\mu\alpha}})\nonumber\\
=&
\Omega_{ij\alpha, \alpha+1}\Big(-\frac{1}{r} dy_{\alpha+2} -\frac{1}{2r^3} (y_{\alpha+2} y_{\alpha+1} dy_{y_{\alpha+1}}-y_{\alpha+1}^2dy_{\alpha+2} )+\frac{1}{2r^3} (y_\alpha^2 dy_{\alpha+2}-y_\alpha y_{\alpha+2} dy_{\alpha})\Big)\nonumber\\
=&\Omega_{ij\alpha, \alpha+1}\Big(-\frac{1}{2r}dy_{\alpha+2}-\frac{1}{2r^2} y_{\alpha+2}dr\Big)\nonumber\\
=& -\frac{1}{2}\Omega_{ij\alpha, \beta}\Big(\frac{1}{2r}dy_{\widehat{\alpha\beta}}+\frac{1}{2r^2} y_{\widehat{\alpha\beta}}dr\Big).
\end{align}
\end{proof}
Applying the above lemma, \eqref{e:d*dphi1-curvature-to-be-simplified} and \eqref{e:dd*phi1-A-to-be-simplified} can
be simplified as follows,
\begin{align}
d^*d\phi_1 &= -\Omega_{ij\alpha\beta}(\frac{1}{4r}dy_{\widehat{\alpha\beta}}+\frac{1}{4r^2}y_{\widehat{\alpha\beta}}dr)\wedge dx_i\wedge dx_j + r^{-3}\Pi_3^{-2} + O'(1),
\\
dd^*\phi_1 &= -\frac{H^\alpha y_\alpha}{2r^3}dy_1\wedge dy_2\wedge dy_3+ A_{ij\alpha\beta} (\frac{1}{4r^2}y_{\widehat{\alpha\beta}}dr-\frac{1}{4r}dy_{\widehat{\alpha\beta}})\wedge dx_i\wedge dx_j\nonumber\\
&+r^{-5} \Pi_3^{(4)}+O'(1).
\end{align}
Therefore,
\begin{align}
\Delta \phi_1 = & (d^*d+dd^*)\phi_1
\nonumber\\
= & -\frac{H^\alpha y_\alpha}{2r^3}dy_1\wedge dy_2\wedge dy_3-\Omega_{ij\alpha\beta} (\frac{1}{4r} dy_{\widehat{\alpha\beta}} +\frac{1}{4r^2} y_{\widehat{\alpha\beta}}dr )\wedge dx_i\wedge dx_j
\nonumber\\
&+A_{ij\alpha\beta}(\frac{1}{4r^2}y_{\widehat{\alpha\beta}}dr-\frac{1}{4r}dy_{\widehat{\alpha\beta}})\wedge dx_i\wedge dx_j +r^{-5} \Pi_3^{(4)}+O'(1).
\end{align}
The proof is done.
\end{proof}
\noindent{\bf Step 3:} In this step we modify $\phi_1$ to kill the unbounded terms on the right hand side of \eqref{e:Delta-phi-1}.
We first we recall some elementary computations involving the standard Euclidean Hodge Laplacian.
\begin{lemma}\label{l:euclidean-laplacian}
Let $\Delta_0$ be the standard Hodge Laplacian on the Euclidean space $\mathbb R^3$, then the following holds:
\begin{enumerate}
\item Let $\{y_1,y_2,y_3\}$ be the Cartesian coordinates of $\mathbb R^3$, then
\begin{align}
\begin{cases}
\Delta_0r=-\frac{2}{r},\\
\Delta_0(\frac{y_\alpha}{r})=\frac{2y_\alpha}{r^3},\\
\Delta_0(\frac{y_\alpha y_\beta}{r})=\frac{4y_\alpha y_\beta}{r^3}, & \alpha\neq \beta,\\
\Delta_0((\frac{y_\alpha^2}{r}-r))=\frac{4y_\alpha^2}{r^3}.
\end{cases}
\end{align}
\item Denote by $\mathcal P_4$ the space of all homogeneous degree 4 polynomials on $\mathbb R^3$, then the operator
\begin{equation}\square: \mathcal P_4\rightarrow \mathcal P_4; f\mapsto r^{5}\Delta_0(r^{-3}f)\end{equation}
is an isomorphism.
\end{enumerate}
\end{lemma}
\begin{proof}
Item (1) is a direct calculation. An convenient way to see this is to use the following two facts:
\begin{enumerate}
\item A homogeneous polynomial degree $k$ polynomial restricts to an eigenfunction of the Hodge-Laplacian $\Delta_{S^2}$ on the unit sphere $S^2$, with eigenvalue $k(k+1)$.
\item Given an eigenfunction $h$ of $\Delta_{S^2}$ on the unit sphere with eigenvalue $k$, for any $l$, we can extend $h$ to a homogeneous function $h_l$ on $\mathbb R^3\setminus\{0\}$ of degree $l$, and
\begin{equation}\Delta_0 h_l=r^{-2}(k-l(l+1))h_l.\end{equation}
\end{enumerate}
For the second item it is possible to write down an explicit inverse to $\Delta_0$. Here we provide a quick abstract proof. First we notice $\square:\mathcal{P}_4\to\mathcal{P}_4$ is a well-defined linear map. This follows from the standard computations
\begin{eqnarray*}
r^5\Delta_0(r^{-3}f)&=&r^5\Delta_0(r^{-3})\cdot f-2 r^5\nabla (r^{-3})\cdot \nabla f+r^{2}\Delta_0 f\\
&=& -6 f-3\nabla (r^2)\cdot \nabla f + r^2\Delta_0f.
\end{eqnarray*}
Since each term in the above formula is a polynomial in $\mathcal{P}_4$, so $\square f\in \mathcal{P}_4$.
Now to prove $\square$ is an isomorphism it suffices to prove it has a trivial kernel. Let $u\equiv r^{-3}f$, then $u=O(r)$ for both $r\to0$ and $r\to\infty$. If $\Delta_0(u)=0$, then $u$ is harmonic on $\mathbb R^3\setminus\{0\}$. The removable singularity theorem implies that $u$ extends smoothly on $\mathbb R^3$. Since $u=O(r)$ as $r\to\infty$, by Liouville theorem, $u$ is a linear function. Since $f\in\mathcal{P}_4$, we conclude $f\equiv 0$. The proof is done.
\end{proof}
Next, we want to find a bounded correction $3$-form
$\mathfrak{B}_0=O'(1)$ such that
\begin{equation}\Delta(\phi_1+\mathfrak{B}_0)=O'(1)\ \text{on}\ \mathcal{U}\setminus P.\end{equation}
Now the main part is to eliminate the unbounded terms in $\Delta\phi_1$ which relies on the following explicit calculations for $\Delta\mathfrak{B}_0$.
In fact, the leading terms of $\Delta\mathfrak{B}_0$ are exactly given by the Euclidean Laplacian $\Delta_0$ acting on the normal components
such that the explicit computations in Lemma \ref{l:euclidean-laplacian} can be effectively used in our context.
Precisely, we have the following lemma.
\begin{lemma}
\label{l:laplacian-error} Let $\Delta_0$ be the Hodge Laplacian on $\mathbb{R}^3$, then the following holds:
\begin{enumerate}
\item
Denote by $\nu_y $ one of the following differential forms $dy_{\alpha}$, $dy_{\alpha}\wedge dy_{\beta}$ or $dy_1\wedge dy_2\wedge dy_3$. Similarly, let $\tau_x$ be a tangential $p$-form given by $\tau_x \equiv dx_1\wedge\ldots \wedge dx_{\alpha_p}$ with $0\leq p\leq m-3$.
Let $\omega \equiv f(x)h(y) \nu_y \wedge \tau_x$, where $f(x)$ is a smooth function defined on $U$ and $h(y)=O'(|y|^k)$ for some $k\in\mathbb{Z}_+$. Then
\begin{equation}
\Delta \omega - f(x)\cdot \Delta_0(h(y))\cdot \nu_y\wedge \tau_x = O'(|y|^{k-1}).
\end{equation}
\item Let $f$ be a smooth function defined on $U\subset P$
and let
$\omega \equiv f(x)\cdot \frac{y_{\alpha}}{r}dy_1\wedge dy_2\wedge dy_3$. Then
\begin{align}
\Delta \omega = 2r^{-2}\omega+ r^{-5}\Gamma_3^{(4)}+O'(1),
\end{align}
where the $3$-form $\Gamma_3^{(4)}$ has the form of \eqref{e:3-form-polynomial-coe} in Notation \ref{n:differential-form-notation}.
\end{enumerate}
\end{lemma}
\begin{proof}
First, we prove item (1).
By definition, $\Delta=dd^* + d^*d$.
We only prove the case $\nu_y= dy_{\alpha}$ for $1\leq \alpha\leq 3$ and $1\leq p\leq m-3$. The proof of the remaining cases is identical.
First, we compute $d^*d\omega$.
\begin{align}
d\omega &= f(x) \cdot \frac{\partial h(y)}{\partial y_{\beta}} dy_{\beta}\wedge dy_{\alpha}\wedge \tau_x + \frac{\partial f(x)}{\partial x_j} \cdot h(y)\cdot dx_j\wedge \nu_y\wedge \tau_x,\end{align}
which implies that
\begin{align}
*d\omega &=(-1)^p f(x)\cdot\frac{\partial h(y)}{\partial y_{\beta}} dy_{\widehat{\beta\alpha}}\wedge *_T(\tau_x) + O'(|y|^k)
\nonumber\\
&=(-1)^p f(x)\Big(\frac{\partial h(y)}{\partial y_{\alpha-1}}dy_{\alpha+1} -\frac{\partial h(y)}{\partial y_{\alpha+1}}dy_{\alpha-1}\Big)\wedge *_T(\tau_x)+ O'(|y|^k).
\end{align}
Differentiating the above equality,
\begin{align}
d*d\omega & =(-1)^p f\cdot\Big(\frac{\partial^2h}{\partial y_{\alpha}\partial y_{\alpha-1}}dy_{\alpha}\wedge dy_{\alpha+1}
- \frac{\partial^2h}{\partial y_{\alpha}\partial y_{\alpha+1}}dy_{\alpha}\wedge dy_{\alpha-1}
\nonumber\\
&+ \Big(\frac{\partial^2h}{\partial y_{\alpha-1}^2}+\frac{\partial^2h}{\partial y_{\alpha+1}^2}\Big)dy_{\alpha-1}\wedge dy_{\alpha+1}\Big)\wedge *_T(\tau_x)+ O'(|y|^{k-1}).
\end{align}
Then it follows that
\begin{align}
d^*d\omega & = (-1)^{mp+m+1}*d*d\omega
\nonumber\\
&=f\cdot\Big(\frac{\partial^2h}{\partial y_{\alpha}\partial y_{\alpha-1}}dy_{\alpha-1}+\frac{\partial^2h}{\partial y_{\alpha}\partial y_{\alpha+1}}dy_{\alpha+1}-\Big(\frac{\partial^2h}{\partial y_{\alpha-1}^2}+\frac{\partial^2h}{\partial y_{\alpha+1}^2}\Big)dy_{\alpha}\Big)\wedge \tau_x
\nonumber\\
& + O'(|y|^{k-1}).\label{e:d*d-omega}
\end{align}
On the other hand,
\begin{align}
*\omega = f(x)h(y)dy_{\widehat{\alpha}}\wedge *_T(\tau_x),
\end{align}
which implies
\begin{equation}
d*\omega = f(x)\frac{\partial h}{\partial y_{\alpha}}\dvol_N\wedge *_T(\tau_x) + O'(|y|^k).
\end{equation}
So it follows that
\begin{equation}
d^*\omega = (-1)^{mp+1}*d*\omega= -f\cdot\frac{\partial h}{\partial y_{\alpha}}\cdot\tau_x + O'(|y|^k),
\end{equation}
and hence
\begin{equation}
dd^*\omega = - f\cdot \Big( \frac{\partial^2h}{\partial y_{\alpha-1}\partial y_{\alpha}} dy_{\alpha-1} + \frac{\partial^2h}{\partial y_{\alpha}^2}dy_{\alpha}+\frac{\partial^2h}{\partial y_{\alpha+1}\partial y_{\alpha}}dy_{\alpha+1}\Big)\cdot dy_{\alpha} \wedge \tau_x + O'(|y|^{k-1}).\label{e:dd*-omega}
\end{equation}
Therefore, combining \eqref{e:d*d-omega} and \eqref{e:dd*-omega},
\begin{equation}
\Delta\omega= (d^*d+dd^*)\omega = - f\cdot (\Delta_0 h(y) ) dy_{\alpha}\wedge \tau_x + O'(|y|^{k-1}),
\end{equation}
where $\Delta_0(h(y))=-\frac{\partial^2 h}{\partial y_1^2}-\frac{\partial^2 h}{\partial y_2^2}-\frac{\partial^2 h}{\partial y_3^2}$.
The proof of (1) is done.
Now we prove item (2). Let $\omega = f\cdot \frac{y_{\alpha}}{r}dy_1\wedge dy_2\wedge dy_3$ and the first step is to compute the term $d^*d\omega$.
By Lemma \ref{l:generalized-Gauss},
$dr=\frac{y_{\gamma}dy_{\gamma}}{r}$, then
\begin{equation}d(\frac{y_{\alpha}}{r}) \wedge dy_1\wedge dy_2\wedge dy_3=0.\end{equation}
This implies that
\begin{equation}
d\omega = \frac{\partial f}{\partial x_i} \cdot \frac{y_{\alpha}}{r} dx_i\wedge dy_1\wedge dy_2\wedge dy_3,\end{equation}
and hence
\begin{equation}
*d\omega = - \frac{\partial f}{\partial x_i} \cdot \frac{y_{\alpha}}{r} *_{T}(dx_i) + O'(r).
\end{equation}
Differentiating the above equality and applying Lemma \ref{l:generalized-Gauss} again,
\begin{align}
d*d\omega = -\frac{\partial f}{\partial x_i}\Big(\frac{dy_{\alpha}}{r} - \frac{ y_{\alpha}y_{\beta}\cdot dy_{\beta}}{r^3}
\Big)*(dx_i)
+O'(1).
\end{align}
It follows that
\begin{align}
*d*d\omega = (-1)^{m+1}\cdot \frac{\partial f}{\partial x_i}\cdot\frac{dy_{\widehat{\alpha}}\wedge dx_i}{r} + (-1)^m\frac{\partial f}{\partial x_i}\cdot\frac{ y_{\alpha}y_{\beta}}{r^3}\cdot dy_{\widehat{\beta}} \wedge dx_i + O'(1).
\end{align}
Therefore,
\begin{align}
d^*d\omega &= (-1)^{m+1}*d*d\omega
\nonumber\\
&= \frac{\partial f}{\partial x_i}\cdot\frac{dy_{\widehat{\alpha}}\wedge dx_i}{r} - \frac{\partial f}{\partial x_i}\cdot\frac{ y_{\alpha}y_{\beta}}{r^3}\cdot dy_{\widehat{\beta}} \wedge dx_i + O'(1). \label{e:d*d-w}
\end{align}
Now we compute $dd^*\omega$.
By Lemma \ref{l:v-T-N} and the expansion of $\dvol_N$ in \eqref{e:dvol-N-expansion},
\begin{align}
*(dy_1\wedge dy_2\wedge dy_3)
&= * (\dvol_N + A_{i\gamma\beta} y_{\gamma}dy_{\widehat{\gamma}}\wedge dx_i) + \widetilde{O}(r^2)
\nonumber\\
&= (1+H^{\beta}y_{\beta})\dvol_T+A_{i\gamma\beta}y_{\gamma}dy_{\gamma}\wedge *_T(dx_i) + \widetilde{O}(r^2),
\end{align}
so we have
\begin{align}
*\omega &= f\cdot \frac{y_{\alpha}}{r}\cdot \dvol_T + f\cdot H^{\beta}\cdot \frac{y_{\alpha}y_{\beta}}{r}\cdot \dvol_T+ f\cdot A_{i\gamma\beta}\frac{y_{\alpha}y_{\gamma}}{r}dy_{\gamma}\wedge *_T(dx_i) + O'(r^2)
\nonumber\\
&\equiv \FT_1 + \FT_2 + \FT_3 + O'(r^2).
\end{align}
By collecting the leading terms, it is easy to compute the leading term in the above equality,
\begin{align}
d*d(\FT_1)&=\frac{\partial f}{\partial x_i}\Big( \frac{dx_i\wedge dy_{\widehat{\alpha}}}{r} - \frac{y_{\alpha}y_{\beta}dx_i\wedge dy_{\widehat{\beta}}}{r^3}\Big)-2\omega + O'(1)
\\
d*d(\FT_2)&= r^{-5}\Pi_3^{(4)},\ d*d(\FT_3)= r^{-5}\Pi_3^{(4)}.
\end{align}
Therefore,
\begin{equation}
dd^*\omega = -\frac{\partial f}{\partial x_i}\cdot \frac{dx_i\wedge dy_{\widehat{\alpha}}}{r} + \frac{\partial f}{\partial x_i}\cdot \frac{y_{\alpha}y_{\beta}}{r^3} \cdot dx_i\wedge dy_{\widehat{\beta}} + 2\omega + r^{-5} \Pi_3^{(4)} + O'(1).\label{e:dd*-w}
\end{equation}
By \eqref{e:d*d-w} and \eqref{e:dd*-w} we obtain the expansion
\begin{align}
\Delta\omega = (d^*d + dd^*)\omega = 2 \omega + r^{-5} \Pi_3^{(4)} + O'(1).
\end{align}
So the proof is done.
\end{proof}
Now we finish Step 2 by proving the following
\begin{proposition}
There is some $3$-form $\Lambda_3^{(4)}$ (given in the form in Notation \ref{n:differential-form-notation}) such that
if we choose \begin{align}
\mathfrak{B}_0 & \equiv \frac{H^\alpha y_\alpha}{4r}dy_1\wedge dy_2\wedge dy_3
\nonumber\\
&+ \Big(\Omega_{ij\alpha\beta}(\frac{1}{16}y_{\widehat{\alpha\beta}}dr-\frac{3}{16} rdy_{\widehat{\alpha\beta}})-\frac{1}{16}A_{ij\alpha\beta}(y_{\widehat{\alpha\beta}} dr+rdy_{\widehat{\alpha\beta}})\Big)\wedge dx_i\wedge dx_j + r^{-3}\Lambda_3^{(4)},
\end{align}
then the corrected $3$-form \begin{align}
\phi_2 \equiv & \phi_1 + \mathfrak{B}_0
\label{e:phi_2-correction}
\end{align} satisfies
$\Delta \phi_2 = O'(1)$
in $\mathcal{U}\setminus P$
and has the expansion
\begin{align}
\phi_2 = & \frac{1}{2r}(1-\frac{H^\alpha y_\alpha}{2}) dy_1\wedge dy_2\wedge dy_3+\frac{1}{2r}y_{\beta}A_{i\alpha\beta}dx_i \wedge dy_{\widehat{\alpha}}
-\frac{1}{4} A_{ij\alpha\beta}r\cdot dy_{\widehat{\alpha\beta}}\wedge dx_i\wedge dx_j\nonumber
\\&+ \frac{3}{16}(A_{i\alpha, \alpha+1}A_{j\alpha, \alpha+2}-A_{i\alpha, \alpha+2}A_{j\alpha, \alpha+1})d(ry_\alpha)\wedge dx_i\wedge dx_j
+r^{-3}\Pi_3^{(4)}+O'(r^2).
\end{align}
\end{proposition}
\begin{proof}
Let $\mathfrak{b}_0 \equiv \frac{H^\alpha y_\alpha}{4r}dy_1\wedge dy_2\wedge dy_3$. Then item (2) of Lemma \ref{l:laplacian-error} tells us that
\begin{equation}
\Delta\mathfrak{b}_0 = \frac{H^\alpha y_\alpha}{2r^3}dy_1\wedge dy_2\wedge dy_3 + r^{-5}\Gamma_3^{(4)} + O'(1).
\end{equation}
Let $\Pi^{(4)}$ be the $3$-form in the expansion of $\Delta\phi_1$ given by \eqref{e:Delta-phi-1} in Proposition \ref{p:Delta-phi-1-expansion}.
Next, Lemma \ref{l:euclidean-laplacian} and Lemma \ref{l:laplacian-error} tell us that
there are $3$-forms $\widehat{\Gamma}_3^{(4)}$
and $\widehat{\Pi}_3^{(4)}$ which are also of the form as in \eqref{e:3-form-polynomial-coe} such that
\begin{align}
\Delta(r^{-3}\widehat{\Gamma}_3^{(4)}) = - r^{-5}\Gamma_3^{(4)} + O'(1),
\\
\Delta (r^{-3}\widehat{\Pi}_3^{(4)}) = -r^{-5}\Pi_3^{(4)} + O'(1).
\end{align}
Now let
$\mathfrak{b}_1 \equiv r^{-3}\widehat{\Gamma}_3^{(4)} + r^{-3}\widehat{\Pi}_3^{(4)}$. Then the correction term $\mathfrak{b}_0 + \mathfrak{b}_1$ is chosen as the above such that $\Delta(\mathfrak{b}_0 + \mathfrak{b}_1)$ in fact eliminates the $O'(r^{-2})$-term and implicit
$O'(r^{-1})$-terms in the expansion of $\Delta\phi_1$ (see Proposition \ref{p:Delta-phi-1-expansion}).
In the following, we will make a further correction such that those explicit $O'(r^{-1})$-terms will be cancelled out as well. In fact,
we define
\begin{equation}
\mathfrak{b}_2 \equiv \Big(\Omega_{ij\alpha\beta}(\frac{1}{16}y_{\widehat{\alpha\beta}}dr-\frac{3}{16} rdy_{\widehat{\alpha\beta}})-\frac{1}{16}A_{ij\alpha\beta}(y_{\widehat{\alpha\beta}} dr+rdy_{\widehat{\alpha\beta}})\Big)\wedge dx_i\wedge dx_j,
\end{equation}
applying Lemma \ref{l:euclidean-laplacian} and Lemma \ref{l:laplacian-error} again, then
\begin{equation}
\Delta\mathfrak{b}_2 =\Omega_{ij\alpha\beta} \Big(\frac{1}{4r} dy_{\widehat{\beta\alpha}} +\frac{y_{\widehat{\alpha\beta}}}{4r^2} dr\Big)\wedge dx_i\wedge dx_j
-A_{ij\alpha\beta}\Big(\frac{y_{\widehat{\alpha\beta}}}{4r^2}dr-\frac{1}{4r}dy_{\widehat{\alpha\beta}}\Big)\wedge dx_i\wedge dx_j,
\end{equation}
and hence
$\Delta(\phi_1+\mathfrak{b}_0 + \mathfrak{b}_1 + \mathfrak{b}_2) = O'(1)$. Therefore,
it suffices to choose the correction term
\begin{equation}
\mathfrak{B}_0 \equiv \mathfrak{b}_0 + \mathfrak{b}_1 + \mathfrak{b}_2,
\end{equation}
which gives
$\Delta(\phi_1+\mathfrak{B}_0) =
O'(1)$.
Notice that, $\mathfrak{b}_2$ has a further cancellation,
\begin{align}
\mathfrak{b}_2 = &\Big(\Omega_{ij\alpha\beta}(\frac{1}{16}y_{\widehat{\alpha\beta}}dr-\frac{3}{16} rdy_{\widehat{\alpha\beta}})-\frac{1}{16}A_{ij\alpha\beta}(y_{\widehat{\alpha\beta}} dr+rdy_{\widehat{\alpha\beta}})\Big)\wedge dx_i\wedge dx_j,
\nonumber\\
=& -\frac{1}{4}A_{ij\alpha\beta}rdy_{\widehat{\alpha\beta}}\wedge dx_i\wedge dx_j - \frac{1}{16}(A_{i,\alpha,\alpha+1}A_{j,\alpha,\alpha+2}-A_{i,\alpha,\alpha+2}A_{j,\alpha,\alpha+1})y_{\alpha}dr\wedge dx_i\wedge dx_j
\nonumber\\
& + \frac{3}{16}(A_{i,\alpha,\alpha+1}A_{j,\alpha,\alpha+2}-A_{i,\alpha,\alpha+2}A_{j,\alpha,\alpha+1})rdy_{\alpha} \wedge dx_i\wedge dx_j.
\end{align}
Therefore,
\begin{align}
\phi_2 = &
\phi_1 + \mathfrak{B}_0
\nonumber\\
=& \frac{1}{2r}(1-\frac{H^\alpha y_\alpha}{2}) dy_1\wedge dy_2\wedge dy_3+\frac{1}{2r}y_{\beta}A_{i\alpha\beta}dx_i \wedge dy_{\widehat{\alpha}}
-\frac{1}{4} A_{ij\alpha\beta}r\cdot dy_{\widehat{\alpha\beta}}\wedge dx_i\wedge dx_j\nonumber
\\&+ \frac{3}{16}(A_{i\alpha, \alpha+1}A_{j\alpha, \alpha+2}-A_{i\alpha, \alpha+2}A_{j\alpha, \alpha+1})d(ry_\alpha)\wedge dx_i\wedge dx_j
+r^{-3}\Pi_3^{(4)}+O'(r^2),
\end{align}
and
$\Delta \phi_2 = O'(1)$ on $\mathcal{U}\setminus P$.
\end{proof}
\noindent{\bf Step 4:} In this step we compute $\Delta\phi_2$ as a current on $\mathcal U$.
\begin{lemma}\label{l:almost-Green's-current}
For any $p\in P$, let $\mathcal{U}$ be a neighborhood of $p$ in $Q$ and let $U = \mathcal{U}\cap P$, then we have
\begin{equation}
\Delta\phi_2=2\pi\delta_U+O'(1).
\end{equation}
\end{lemma}
\begin{proof}
Let $\chi\in \Omega_0^{m-3}(\mathcal{U})$ be a compactly supported test form. Then applying integration by parts once,
\begin{equation}(\phi_2, \Delta\chi)=\int_{\mathcal{U}} \phi_2\wedge (dd^*+d^*d)\chi=\int_{\mathcal{U}} d\phi_2\wedge d^*\chi-\int_{\mathcal{U}} d^*\phi_2\wedge d\chi.\end{equation}
Notice that the integration by parts works here because $\phi_2=O(r^{-1})$. Also notice that \eqref{e:d-phi-1-exp} and \eqref{e:phi_2-correction} imply $d\phi_2=O'(1)$, so
\begin{equation}\int_{\mathcal{U}} d\phi_2\wedge d^*\chi=\int_{\mathcal{U}} d^*d\phi_2\wedge \chi. \end{equation}
On the other hand, by (\ref{eqn2-19}),
\begin{equation}d^*\phi_2=-\frac{y_\alpha}{2r^{3}} {} dy_{\hat\alpha}+\zeta,\end{equation}
where $\zeta$ is a $2$-form satisfying $\zeta=O'(r^{-1})$.
Denote by $S_{\epsilon}^2$ the normal geodesic sphere bundle $\{r=\epsilon\}$, then we get that
\begin{eqnarray}-\int_{\mathcal U} d^*\phi_2\wedge d\chi
&=&\int_{\mathcal U} dd^* \phi_2 \wedge \chi + \int_{S_{\epsilon}^2} d^*\phi_2\wedge \chi
\nonumber\\
&=&\int_{\mathcal U} dd^* \phi_2 \wedge \chi + \lim_{\epsilon\rightarrow 0} \frac{1}{2\epsilon^3}{}\int_{S_\epsilon} (y_\alpha dy_{\hat\alpha} +\epsilon^2 \zeta)\wedge \chi.\end{eqnarray}
By direct calculation of the last term on the right hand side we obtain
\begin{equation}
-\int_{\mathcal U} d^*\phi_2\wedge d\chi=\int_{\mathcal U} dd^* \phi_2 \wedge \chi+2\pi{}\int_{P}\chi.
\end{equation}
This concludes the proof.
\end{proof}
\noindent{\bf Step 5:} Now we study the regularity of the solution of $\Delta T=v$ with $v=O'(1)$.
\begin{lemma}\label{l:higher-regularity}
Let $p\in P$ and let $\mathcal{U}$ be a neighborhood of of $p$ in $Q$. Assume that $v$ is a $3$-form on $\mathcal{U}$ which satisfies $v=O'(1)$ and $v\in C^{\infty}(\mathcal{U}\setminus U)$, where $U = \mathcal{U}\cap P$. Then the equation \begin{equation}\Delta T= v, \label{e:current-poisson} \end{equation}
has a weak solution $T \in W^{2,p}(\mathcal{V})$ for any $p>1$ which satisfies $T=O'(r^2)$ and $T\in C^{\infty}(\mathcal{V}\setminus U)$. Here $\mathcal{V} \subset\subset \mathcal{U}$ is a smaller neighborhood of $p$ in $Q$, and $r$ is the distance to $P$.
\end{lemma}
\begin{proof} Since the assumption implies $v\in L^q(\mathcal{U})$ for any $q>1$, by the standard elliptic theory, equation \eqref{e:current-poisson} has a solution $T$ which is $W^{2,q}$ for any $q>1$, and $T$ is $C^{\infty}$ away from $U$. Applying the $W^{2,q}$-regularity to $\Delta T=v$ and using the Sobolev embedding, we have that \begin{equation}T\in W^{2,q}(\mathcal{U}_1)\cap C^{1, \alpha}(\mathcal{U}_1)\label{e:sobolev-weaker}\end{equation} for any $q>1$ and $0<\alpha<1$, where $\mathcal{U}_1\subset \subset \mathcal{U}$ a smaller neighborhood of $p$ in $Q$.
Therefore, $|T|=O(1)$ and $|\partial T|=O(1)$.
It remains to show $T=O'(r^2)$.
The proof consists of two primary steps.
\begin{flushleft}{\bf Step 1:}\end{flushleft}
{\it Let us fix any fixed point $p\in P$. In the normal coordinate system around $p$, we will show that}
\begin{equation}
|\partial^k T|=O(r^{2-(k+\epsilon)}), \label{e:general-nabla}
\end{equation}
{\it where $k\geq 2$ and $\epsilon>0$ are arbitrary, and $\partial^k$ denotes the mixed derivatives on the coefficient functions of $T$ of order $k$.}
The regularity \eqref{e:general-nabla} will be proved by the induction on the derivative order $k\geq 2$.
In the base step, we compute $\Delta\Phi$, where $\Phi\equiv \partial^2 T$.
Differentiating the equation $\Delta T = v$ by $\partial^2$, it schematically yields
\begin{equation}\Delta\Phi + Q_1* \partial\Phi +Q_2*\Phi+Q_3*\partial T + Q_4* T=\partial^2 v = O'(r^{-2}),\label{e:initial-eq}\end{equation}
where $Q_i$'s involve the derivatives of the metric $g$, and the shorthand $A*B$ denotes a linear combination of $A_I\cdot B_J$ with coefficient functions $A_I$, $B_J$ respectively. Notice that $T \in C^{1,\alpha}(\mathcal{U}_{1})$, so equation \eqref{e:initial-eq} can be rewritten as
\begin{equation}
\Delta\Phi + Q_1 * \partial\Phi + Q_2 * \Phi = \partial^2 v-Q_3*\partial T - Q_4*T.\label{e:Del-Phi-original}
\end{equation}
Since $\partial^2v-Q_3*\partial T - Q_4* T= O(r^{-2-\epsilon})$ which is not $L^q$-integrable around $U$ for $q$ sufficiently large, the desired regularity of $\Phi$ does not directly follow from the standard $W^{2,q}$-estimate.
To obtain the refined estimate for $\Phi$, we will need the following rescaling argument. One can choose a smaller neighborhood $\mathcal U_2\subset\subset\mathcal{U}_1$ such that $B_{r_x/2}(x)\subset \mathcal{U}_1$ for every $x\in \mathcal{U}_2$, where $r_x\equiv r(x)\in(0,1)$. For every fixed $x\in\mathcal{U}_2\setminus U$, let us take the rescaled metric
$\tilde{g} = (r_x)^{-2} \cdot g$ and the dilation $\widetilde{\Phi}(y)\equiv \Phi(r_x\cdot y)$ with $y\in B_{1/2}^{\tilde{g}}(x)$. Then equation \eqref{e:Del-Phi-original} in $B_{r_x/2}(x)\subset \mathcal{U}_1$ becomes
\begin{equation}\widetilde{\Delta} \widetilde{\Phi}+r_x\cdot\widetilde{Q}_1*\widetilde{\partial}\widetilde{\Phi}+(r_x)^2\cdot\widetilde{Q}_2*\widetilde{\Phi}= \tilde{\eta} \quad \text{in} \ B_{1/2}^{\tilde{g}}(x),\label{e:rescaled-Phi-eq}\end{equation}
where $|\tilde{\eta}|_{L^{\infty}(B_{\frac{1}{2}^{\tilde{g}}(x)})}\leq C_{\epsilon}\cdot r^{-\epsilon}$ for each $\epsilon>0$.
Notice that the $L^{\infty}$-norms of the coefficients in \eqref{e:rescaled-Phi-eq} are uniformly bounded (independent of $x$), and \eqref{e:sobolev-weaker} implies $|\Phi|_{L^q(\mathcal{U}_1)}=|\partial^2T|_{L^q(\mathcal{U}_1)}\leq C_q$ for any $q>1$. Using simple rescaling, we have that for every $q>1$,
$|\widetilde{\Phi}|_{L^q(B_1^{\tilde{g}}(x))}\leq C_q\cdot (r_x)^{-\frac{n}{q}}$, where $C_q>0$ is a uniform constant independent of $x\in\mathcal{U}_2\setminus U$.
Applying the $W^{2,q}$-estimate to \eqref{e:rescaled-Phi-eq} and applying the Sobolev embedding, we have that for each $q>1$\begin{align}|\widetilde{\Phi}|_{W^{2,q}(B_{1/2}^{\tilde{g}}(x))} \leq C_q \cdot (r_x)^{-\frac{n}{q}}.\label{e:Phi-(2,q)}\end{align}
For any fixed $\epsilon>0$, letting $q>1$ be sufficiently large and applying the Sobolev embedding, we obtain that
\begin{align}
|\widetilde{\Phi}|_{C^{1,\alpha}(B_{1/4}^{\tilde{g}}(x))}\leq C_{\alpha,\epsilon}\cdot (r_x)^{-\epsilon}.
\end{align}
Rescaling back to the original metric $g$, we have that \begin{equation}|\Phi|_{L^{\infty}(B_{r_x/4}(x))}+|r_x\cdot \partial \Phi|_{L^{\infty}(B_{r_x/4}(x))}\leq C_{\epsilon} \cdot (r_x)^{-\epsilon},\label{e:Phi-base-step}\end{equation}
where $C_{\epsilon}>0$ is independent of the base point $x\in \mathcal{U}_{2}\setminus U$.
This completes the base step.
Based on the initial step $k=2$, we are now ready to finish the induction step. For any fixed integer $k\geq 3$, suppose that the estimate in item (1) holds for all $2\leq j \leq k-1$. That is, for every $\epsilon>0$, $2\leq j \leq k-1$ and $q>1$, there is a smaller neighborhood $\mathcal{U}_{j}\subset\subset \mathcal{U}$ on which the following estimates hold:\begin{align}
&|\partial^{j-2} \Phi|=O(r^{2-j-\epsilon}),\label{e:hypothesis-1}
\end{align}
where $C_{k,q,\epsilon}>0$ is independent of the choice of the base point $x\in \mathcal{U}_{j}\setminus U$. Then we will show that for all $\epsilon>0$, $|\partial^k T|=O(r^{2-(k+\epsilon)})$.
Differentiating equation \eqref{e:initial-eq} by $\partial^{k-3}$, we have that for every $\epsilon>0$,
\begin{equation}
\Delta(\partial^{k-3}\Phi) + \sum\limits_{j=0}^{k-2}Q_j * \partial^j \Phi = O(r^{1-k-\epsilon}), \label{e:Delta-nabla-(k-2)}
\end{equation}
where $Q_j$'s arise from the derivatives of the metric coefficients.
As before, to obtain the higher regularity of $\Phi$, we will study \eqref{e:Delta-nabla-(k-2)} under the rescaling $\tilde{g} = (r_x)^{-2}\cdot g$ and the dilation $\widetilde{\Phi}(y) = \Phi(r_x\cdot y)$ for every fixed $x\in \mathcal{U}_k\setminus U$, where $\mathcal{U}_k$ is some smaller neighborhood.
Applying the induction hypothesis \eqref{e:hypothesis-1} and absorbing the lower order terms of the \eqref{e:Delta-nabla-(k-2)} in the rescaled form, we have that for every $\epsilon>0$,
\begin{equation}
\widetilde{\Delta}(\widetilde{\partial}^{k-3}\widetilde{\Phi}) + r_x \cdot \widetilde{Q}_{j-1} * \widetilde{\partial}(\widetilde{\partial}^{k-3} \widetilde{\Phi}) + \widetilde{Q}_{j-3} * (\widetilde{\partial}^{k-3}\widetilde{\Phi}) = O(r^{-\epsilon}).\end{equation}
Then the elliptic regularity implies that for each $\epsilon>0$ and $q>1$,
\begin{equation}
|\widetilde{\partial}^{k-3}\widetilde{\Phi}|_{W^{2,q}(B_{1/4}^{\tilde{g}}(x))} \leq C_{k,q,\epsilon}\cdot (r_x)^{-\epsilon},
\end{equation}
where $C_{k,q,\epsilon}>0$ is independent of the base point $x\in\mathcal{U}_k\setminus U$. Applying the Sobolev embedding and scaling back to the original metric $g$, we have that for each $\epsilon>0$,
\begin{equation}|\partial^k T|=|\partial^{k-2}\Phi|_{L^{\infty}(B_{r_x/4}(x))}\leq C_{k,\epsilon}\cdot (r_x)^{2-k-\epsilon}\end{equation} where $C_{k,\epsilon}>0$ is independent of the base point $x\in\mathcal{U}_k\setminus U$.
The proof of \eqref{e:general-nabla} is done.
\begin{flushleft}{\bf Step 2:}\end{flushleft}
{\it We prove that for each $\epsilon>0$, $k\geq 2$, $\ell \in \mathbb{Z}_+$, }
\begin{equation} |\partial_t^{\ell}T|=O(1),\quad |\partial_t^{\ell}\partial T|=O(1),\quad
|\partial_t^{\ell} \partial^k T|=O(r^{2-k-\epsilon}), \label{e:mixed-nabla}
\end{equation}
{\it where $\partial_t^{\ell}$ denotes the derivatives in the tangential directions of order $\ell$. This will be proved by induction on the both $\ell$ and $k$.}
Before going through the proof, we first outline the induction scheme and several sub-steps that we will establish:
\begin{itemize}
\item Step 2.0: For every $\ell\in\mathbb{Z}_+$, the following tangential regularity holds for $T$ and $\partial T$, \begin{align}|\partial_t^{\ell}T|=O(1),
\quad |\partial_t^{\ell}\partial T|=O(1). \label{e:tangential-T-pT}\end{align}
\item Step 2.1: For every $\epsilon>0$ and $k\geq 2$, the following holds
\begin{align}
|\partial_t\partial^k T| = O(r^{2-k-\epsilon}).\label{e:first-order-tangential}
\end{align}
\item Step 2.2: Under the induction hypothesis that the higher regularity
\begin{align}
|\partial_t^i\partial^k T| = O(r^{2-k-\epsilon})
\label{e:main-induction-hypothesis}\end{align}
holds for every $\epsilon>0$, $k\geq 2$, $1\leq i \leq \ell -1$, we will prove that for every $\epsilon>0$ and $k\geq 2$,
\begin{align}
|\partial_t^{\ell}\partial^k T| = O(r^{2-k-\epsilon}).\label{e:mixed-regularity-induction-step}
\end{align}
\end{itemize}
{\bf Step 2.0:} {\it To prove \eqref{e:tangential-T-pT}, it suffices to show $\partial_t^{\ell} T\in W^{2,q}$ for any $q>1$.}
To this end, differentiating $\Delta T=v$ by the tangential derivative $\partial_t$, we have that
\begin{equation}
\Delta(\partial_t T) + Q_1 * \partial^2 T + Q_2 * \partial T = \partial_t v = O'(1),\label{e:tangential-T}
\end{equation}
where $Q_1$ and $Q_2$ involve the derivatives of the metric.
Applying the elliptic regularity to \eqref{e:tangential-T}, in some smaller neighborhood $\mathcal{U}'\subset \subset \mathcal{U}$, we have that for any $q>1$
\begin{align}\partial_t T\in W^{2,q}(\mathcal{U}').\label{e:tangent-T-(2,q)}\end{align} Furthermore, given any $\ell\in\mathbb{Z}_+$, computing $\partial_t^{\ell}(\Delta T) =\partial_t^{\ell} v$, we have
\begin{align}
\Delta(\partial_t^{\ell} T) + \sum\limits_{j=0}^{\ell-1} \Big(E_j*\partial^2 \partial_t^j T + F_j*\partial\p_t^j T + H_j*\partial_t^j T\Big)=\partial_t^{\ell} v= O'(1).
\end{align}
Based on \eqref{e:tangent-T-(2,q)} and simple induction argument, we obtain
$\partial_t^{\ell} T \in W^{2,q}(\mathcal{U}'')$
for all $q>1$ and $\ell\in\mathbb{Z}_+$ in a smaller neighborhood $\mathcal{U}''\subset\subset \mathcal{U}'$.
{\bf Step 2.1:} {\it This step is to prove the estimate \eqref{e:first-order-tangential} by induction on $k\geq 2$.}
Let $\Phi = \partial^2 T$ and the base step here is to prove \eqref{e:first-order-tangential} for $k=2$.
Using \eqref{e:general-nabla}, it is straightforward to check that
\begin{align}
\Delta(\partial_t \Phi) + Q_1 * \partial(\partial_t\Phi) + Q_2 *(\partial_t\Phi)=\sum\limits_{j=0}^4Q_j'*\partial^j T + \partial_t\partial^2v = O(r^{-2-\epsilon}) ,\label{e:Delta-p_t-Phi}\end{align} for all $\epsilon>0$. Since by \eqref{e:tangent-T-(2,q)}, $\partial_t\Phi=\partial_t\partial^2 T\in L^q$ for any $q>1$, applying the previous rescaling argument, we have that \begin{align}|\partial_t\Phi|_{L^{\infty}(B_{r_x/2}(x))}+r_x\cdot |\partial_t\partial\Phi|_{L^{\infty}(B_{r_x/2}(x))}\leq C_{\epsilon}\cdot (r_x)^{-\epsilon}\label{e:first-tangential-Phi}\end{align} in terms of the original metric $g$,
where $C_{\epsilon}>0$ is independent of the choice
of the base point $x$. Next, computing $\Delta(\partial_t\partial^{k-3}\Phi)$ for any $k\geq 3$, it follows that
\begin{align}
\Delta(\partial_t\partial^{k-3}\Phi)
=\partial^{k-3}\Delta(\partial_t\Phi) + \sum\limits_{j=1}^{k-2}Q_j*\partial_t\partial^j\Phi.\label{e:Delta-1-(k-1)}
\end{align}
Applying the induction hypothesis $|\partial_t\partial^j\Phi|=O(r^{-j-\epsilon})$ for $0\leq j\leq k-3$, equation \eqref{e:Delta-1-(k-1)} can be rewritten as the following for any
\begin{align}
\Delta(\partial_t\partial^{k-3}\Phi) + Q_1 * \partial(\partial_t\partial^{k-3}\Phi) + Q_2*(\partial_t\partial^{k-3}\Phi) = O(r^{1-k-\epsilon}).
\end{align}
The same rescaling argument implies that
$|\partial_t\partial^kT|=|\partial_t\partial^{k-2}\Phi|=O(r^{2-k-\epsilon})$ for every $\epsilon>0$. This completes Step 2.1.
{\bf Step 2.2:} {\it We will prove \eqref{e:mixed-regularity-induction-step} under the induction hypothesis \eqref{e:main-induction-hypothesis}.
}
The proof is by induction on $k\geq 2$ and the arguments consist of two parts.
To begin with, we will show that the following regularity\begin{align}
|\partial_t^{\ell}\Phi | =O(r^{-\epsilon})\label{e:base-case-of-(2.0)}
\end{align}
holds for any $\ell\in\mathbb{Z}_+$ and $\epsilon>0$. Recall that the case $\ell=1$
has been proved in \eqref{e:first-tangential-Phi}.
Now let us compute $\Delta(\partial_t^{\ell}\Phi)$ for $\ell>2$. By straightforward computations,
\begin{align}
\Delta(\partial_t^{\ell}\Phi) = \partial_t^{\ell}\Delta\Phi + \sum\limits_{\ell'=1}^{\ell-1}(E_j*\partial^2\partial_t^{\ell'}\Phi + F_j*\partial\p_t^{\ell'}\Phi).\label{e:Del-tangential-ell-Phi}
\end{align}
Notice that by \eqref{e:initial-eq} and the lower regularity \eqref{e:tangential-T-pT}
\begin{align}
\partial_t^{\ell}\Delta\Phi = Q_1*\partial(\partial_t^{\ell}\Phi)+Q_2*(\partial_t^{\ell}\Phi) + O(r^{-2-\epsilon}).\label{e:tangential-Delta-Phi}
\end{align}
Plugging \eqref{e:tangential-Delta-Phi} into \eqref{e:Del-tangential-ell-Phi}, and applying
the induction hypothesis \eqref{e:main-induction-hypothesis} to the above lower order terms, we eventually have that
\begin{align}
\Delta(\partial_t^{\ell}\Phi) + Q_1*\partial(\partial_t^{\ell}\Phi)+Q_2*(\partial_t^{\ell}\Phi) =O(r^{-2-\epsilon})\end{align}
for every $\epsilon>0$.
Since we have proved in Step 2.0 that $|\partial_t^{\ell}\Phi|\in L^q$ for any $q>1$, the previous rescaling argument implies that for every $\epsilon>0$,
\begin{align}
|\partial_t^{\ell}\Phi|=O(r^{-\epsilon}), \quad |\partial_t^{\ell}\partial\Phi|=O(r^{-1-\epsilon}).
\end{align}
This completes the proof of \eqref{e:base-case-of-(2.0)}.
Let $k\geq 3$ be any fixed positive integer.
Based on \eqref{e:base-case-of-(2.0)} and the hypothesis $|\partial_t^{\ell}\partial^j \Phi| = O(r^{-j-\epsilon})$
for every $\epsilon>0$, $\ell\in\mathbb{Z}_+$ and $0\leq j\leq k-3$, we proceed to prove the induction step. That is,
we will show that \begin{align}
|\partial_t^{\ell}\partial^{k-2}\Phi| = O(r^{2-k-\epsilon}).
\end{align}
The proof is the same as before. Computing the Laplcian of $\partial_t^{\ell}\partial^{k-3}\Phi$, we have that
\begin{align}
\Delta(\partial_t^{\ell}\partial^{k-3}\Phi) & = \partial^{k-3}\Delta(\partial_t^{\ell}\Phi) + Q_1 * \partial (\partial_t^{\ell}\partial^{k-3}\Phi) + Q_2 * (\partial_t^{\ell}\partial^{k-3}\Phi) +\sum\limits_{j=1}^{k-4} F_j*\partial_t^{\ell}\partial_j\Phi.
\end{align}
Using the induction hypothesis, one can eventually obtain
\begin{align}
\Delta(\partial_t^{\ell}\partial^{k-3}\Phi) + Q_1 * \partial (\partial_t^{\ell}\partial^{k-3}\Phi) + Q_2 * (\partial_t^{\ell}\partial^{k-3}\Phi) = O(r^{1-k-\epsilon})\end{align}
for every $\epsilon>0$. The same rescaling argument implies that
\begin{align}
|\partial_t^{\ell}\partial^{k-3}\Phi| = O(r^{3-k-\epsilon}), \quad |\partial_t^{\ell}\partial^{k-2}\Phi| = O(r^{2-k-\epsilon}).
\end{align}
which proves \eqref{e:mixed-regularity-induction-step}. Therefore, the proof is done.
\end{proof}
For our purpose later, we also need the following lemma.
\begin{lemma}Let $\Delta$ denote the Hodge Laplacian on $Q$, then
\begin{equation}\Delta(r^2)=-6+\widetilde{O}(r^2).\end{equation}
\end{lemma}
\begin{proof}
This follows from similar, and simpler arguments as above. First,
\begin{equation}dr^2=2rdr=2y_\alpha \eta_\alpha+\widetilde{O}(r^4),\end{equation}
so it follows that
\begin{align}*dr^2&=2y_\alpha \eta_{\widehat\alpha}\wedge \dvol_T+\widetilde{O}(r^3)\\
d*dr^2&=6\dvol_N\wedge \dvol_T+\widetilde{O}(r^2)=6\dvol_g+\widetilde{O}(r^2).\end{align}
Hence $\Delta(r^2)=d^*dr^2=-*d*dr^2=-6+\widetilde{O}(r^2)$.\end{proof}
\subsection{Green's currents on a cylinder} \label{ss:complex-greens-currents}
In this subsection we assume $Q\equiv D\times\mathbb{R}$ is a Riemannian product of a K\"ahler manifold $(D, \omega_D, J_D)$ of complex dimension $n-1$, and the real line $\mathbb R$ with coordinate $z$.
Given a smooth divisor $H\subset D$, let $P\equiv H\times \{0\}\subset D\times\{0\}$. In our discussion sometimes we also naturally identify $H$ with $P$.
The results of this subsection will be purely local so $D$ and $H$ are not necessarily compact. Our goal is to prove a few local expansion results, which will be used in Section \ref{s:neck}.
The splitting of a line $\mathbb{R}$ allows us to study the normal exponential map in $Q$ in terms of the normal exponential map in $D$. Notice that the normal bundle of $P$ in $Q$ is naturally a Riemannian direct sum
\begin{equation}{N}=N_0\oplus \mathbb R_z,\end{equation}
where $N_0$ is the normal bundle of $H$ in $D$ given as the orthogonal complement $(TH)^{\perp}$ of $TH$ in $TD|_H$ (with respect to $\omega_D$). So $N_0$ is naturally a hermitian line bundle. We also naturally identify $N_0$ with the holomorphic normal bundle $(TD|_H)/TH$, as complex line bundles. Therefore, $N_0$ can be viewed as a holomorphic hermitian line bundle. The {\it normal exponential map} of $H$ in $D$ is defined by
\begin{equation}
\Exp_H: N_0\to D,\ (p,v)\mapsto\Exp_p(v),\end{equation}
which gives a local diffeomorphism from a neighborhood of the zero section in $N_0$ to a tubular neighborhood of $H$ in $D$. Immediately, \begin{equation}
\label{eqn3-1}
d\text{Exp}_H: TN_0|_{H} \longrightarrow TD|_{H}\end{equation}
is the identity map under the natural isomorphisms $TN_0|_{H}\cong N_0\oplus TH$ and $TD|_{H}\cong N_0\oplus TH
$.
Given any point $p\in H$, we may choose local holomorphic coordinates $\{w_i\}_{i=1}^{n-1}$ on $D$, centered at $p$, such that $H$ is locally defined by $w_1=0$, and the coordinate vector fields $\partial_{w_i}$ are orthonormal at $p$.
Then $dw_1, w_2, \ldots, w_{n-1}$ induces local holomorphic coordinates on $N_0$, which we denote by $\{\zeta, w_2', \ldots, w_{n-1}'\}$.
Given any $(p,v)\in N_0$, its coordinates are by definition given as
\begin{align}
\begin{cases}
\zeta= (dw_1)_p(v), \\
w_j'=w_j(p), & j\geq 2.
\end{cases}\end{align}
Under the normal exponential map $\Exp_H$, these coordinates can also be viewed as local (non-holomorphic) coordinates on $D$, and when restricted to $H$ we have $w_j'=w_j(j\geq 2)$ and $d\zeta=dw_1$. In particular, $\{w_2', \cdots, w_{n-1}'\}$ still give holomorphic coordinates on $H$.
Similarly using $\Exp_H$, the coordinate vector field $\partial_\zeta$, originally defined on the normal bundle $N_0$, can also be viewed as a local (non-holomorphic) vector field on $D$. When restricted to $H$, the vector field $\partial_\zeta$ can hence be identified with the local section $\sigma$ of $(TH)^{\perp}\subset TD|_H$ given by the orthogonal projection of the holomorphic vector field $\partial_{w_1}$.
Then we obtain a local unitary frame $e$ of $(TH)^{\perp}$ given by
\begin{equation}e\equiv\sigma/|\sigma|.
\end{equation}
These generate fiber coordinates $y, \bar y$ on $N_0$ such that
\begin{equation}y =|\sigma|\cdot \zeta.\end{equation}
In this way we obtain local coordinates $\{y,\bar{y}, y_3=z, w_2', \bar w_2', \ldots, w_{n-1}', \bar w_{n-1}'\}$ in a neighborhood of $p$ in $Q$. To match with the notation in the previous subsection, with respect to the local orthonormal basis $\{\sigma+\bar\sigma, \sqrt{-1}(\sigma-\bar\sigma), \partial_z\}$, the normal geodesic coordinates are given by $\{y_1=Re(y), y_2=Im(y), y_3=z\}$, and
$r^2=|y|^2+z^2.$
Also, the convention for the orientation is given such that
\begin{equation}2^{-(n-1)} \sqrt{-1} dy\wedge d\bar y\wedge dz \wedge \prod\limits_{j=2}^{n-1}(\sqrt{-1}dw_j' \wedge d\bar w_j')\end{equation} defines a positive volume form. In the following, we will also use $\langle\cdot, \cdot\rangle$ to denote the hermitian inner product on $(1,0)$-type vectors. The relation with the Riemannian inner product is seen as
\begin{equation}\label{unitary}\langle \xi, \xi\rangle=2\langle Re(\xi), Re(\xi)\rangle.\end{equation}
What the notation means will be clear in the context.
By making $P$ and $Q$ smaller we get local existence of Green's current $G_P$ for $P$ in $Q$, by
Theorem \ref{t:Green-expansion}, with the expansion given there. In our case the formula can be written in terms of the above complex coordinates as
\begin{proposition}\label{p:complex-Green-expansion}
Let $G_P$ be a Green's current for $P\equiv H\times\{0\}$ in $Q$, then locally
\begin{equation}G_P=\psi\wedge dz+\mathcal{R},\end{equation}
where $\mathcal{R}\in C^{\infty}$ and $\psi$ is family of real-valued $(1,1)$-forms on $D$ parametrized by $z$, satisfying
$\psi(-z)-\psi(z)\in C^\infty.$ Moreover, in terms of the above local coordinates we can write
\begin{equation}
\psi=\frac{\sqrt{-1}}{4r} dy\wedge d\bar y+\frac{1}{2r}(yd\bar y+\bar yd y)\wedge\Gamma +r\cdot d\Gamma+r^{-3}\Pi_2^{(4)}+ O'(r^2),\label{e:singular-2-form-expansion}
\end{equation}
where $\Gamma$ is a smooth real-valued $1$-form locally defined on $H$ given by
\begin{equation}\Gamma(v)\equiv-\frac{\sqrt{-1}}{2} \langle\nabla_{v}\partial_y, \partial_y\rangle\Big|_H, \ v\in T_pH,\end{equation}
and $\Pi_2^{(4)}$ is the $2$-form given by Notation \ref{n:differential-form-notation} and each term in it contains at least one of the $dy$ or $d\bar{y}$.
\end{proposition}
\begin{proof}
This essentially follows from the fact that $P$ is located on the slice $\{z=0\}$ and $H$ is a complex submanifold of $D$. Indeed, we can decompose
$G_P=\psi_1\wedge dz+\mathcal{R}_1,$
where $\mathcal{R}_1$ does not involve $dz$.
Given any compactly supported test form $\chi\in\Omega_0^{2n-4}(Q)$, we can write
$\chi=\beta\wedge dz+\gamma,$
where $\gamma$ does not involve $dz$. Immediately,
\begin{align}
(\mathcal R_1, \Delta\gamma)&=\int_Q\mathcal R_1\wedge \Delta\gamma=0,
\\
(\psi_1\wedge dz, \Delta(\beta\wedge dz)) &= \int_Q\psi_1\wedge dz\wedge \Delta(\beta\wedge dz)=0.
\end{align}
So it follows that \begin{equation}(\mathcal R_1, \Delta\chi)=(\mathcal R_1, \Delta(\beta\wedge dz))=(G_P, \Delta(\beta\wedge dz))=2\pi\int_P(\beta\wedge dz)=0.\end{equation}
This implies that $\Delta\mathcal R_1=0$ in the distributional sense. By the elliptic regularity, we have $\mathcal R_1\in C^{\infty}$.
Now we write $\psi_1=\psi+\psi_2,$
where $\psi$ is $J_D$-invariant, i.e., of type $(1,1)$ in $D$, and $\psi_2$ is anti-$J_D$-invariant. Since $H$ is a complex submanifold of $D$, the Dirac current $\delta_P$ is $J_D$-invariant, and hence $J_D(G_P)$ is also a Green's current for $P$. So we see that $\psi_2=\frac{1}{2}(G_P-J(G_P))$ is smooth. Then we have that
\begin{equation}G_P=\psi\wedge dz+\mathcal{R} ,\end{equation}
where $\mathcal R=\mathcal R_1+\psi_2\wedge dz$ is smooth.
Similarly, since $\delta_P$ is invariant under $z\mapsto -z$, the difference $\psi(z)-\psi(-z)$ is smooth.
To see the expansion of $\psi$, notice that $H$ is K\"ahler, and hence it is a minimal submanifold of $D$. So the mean curvature of $H$ in $D$ vanishes. Also notice that $\partial_z$ is parallel on $Q$ so $A_{i\alpha\beta}=0$ if either $\alpha=3$ or $\beta=3$. This implies that
\begin{equation}
\psi=\frac{\sqrt{-1}}{4r} dy\wedge d\bar y+\frac{1}{2r}(yd\bar y+\bar yd y)\wedge\Gamma +r\cdot \mathbb A+r^{-3}\Pi_2^{(4)}+ \widetilde{O}(r^2),
\end{equation}
where
\begin{align}
\Gamma =-\frac{1}{2}A_{i12}dx_i\quad \text{and}\quad \mathbb{A} =-\frac{1}{4}A_{ij\alpha\beta}dx_i\wedge dx_j.\end{align}
In particular, $\mathbb A=d\Gamma$. Re-writing
$A_{i12}=\langle \nabla_{\partial_{x_i}}\partial_{y_2}, \partial_{y_1}\rangle$
in terms of the coordinates $y, \bar y$ and keeping in mind \eqref{unitary}, we obtain the desired formula for $\Gamma$.
\end{proof}
Proposition \ref{p:complex-Green-expansion} has a quick corollary which will be used in our later calculations.
\begin{corollary}\label{c:psipower}
For any positive integer $k\geq 2$, we have
$
\psi^k = O'(r^{k-1}).
$
\end{corollary}
\begin{proof}
By Proposition \ref{p:complex-Green-expansion}, we may write $\psi = \FT_1 + \FT_2 + \FT_3$, where
\begin{align}
\FT_1 & \equiv \frac{\sqrt{-1}}{4r} dy\wedge d\bar y,
\\
\FT_2 & \equiv \frac{1}{2r}(yd\bar y+\bar yd y)\wedge\Gamma=O'(1),
\\
\FT_3 & \equiv r\cdot d\Gamma+r^{-3}\Pi_2^{(4)}+ O'(r^2)=O'(r).
\end{align}
Immediately we have that
$(\FT_1)^2 = (\FT_2)^2=\FT_1\wedge \FT_2=0$ and for all $k\geq 1$, $\FT_2\wedge (\FT_3)^{k}=(\FT_3)^k=O'(r^k)$.
Moreover,
\begin{equation}
\FT_1\wedge \FT_3 = \frac{\sqrt{-1}}{4}dy\wedge d\bar{y}\wedge d\Gamma +\frac{\sqrt{-1}}{4r^4}dy\wedge d\bar{y}\wedge \Pi_2^{(4)} + O'(r).
\end{equation}
Notice that $\frac{\sqrt{-1}}{4}dy\wedge d\bar{y}\wedge d\Gamma$ is a smooth term and by definition $dy\wedge d\bar{y}\wedge\Pi_2^{(4)}=0$. Then
$
\FT_1\wedge \FT_3 = O'(r).
$
So for all $k\geq 1$,
$
\FT_1\wedge (\FT_3)^k=O'(r^k).
$
Now by direct calculation,
\begin{eqnarray}
\psi^{k}=k\Big((\FT_1)\wedge(\FT_3)^{k-1} + (\FT_2) \wedge (\FT_3)^{k-1}\Big) + (\FT_3)^k.
\end{eqnarray}
The conclusion then follows.
\end{proof}
Notice that the above local coordinates $\{y, \bar y\}$ are not canonical, and depend on the initial choice of the local coordinates $\{w_i\}_{i=1}^{n-1}$ on $D$. However, a different choice of local holomorphic coordinates on $D$ will induce the coordinates $\tilde y, \bar{\tilde y}$ on fibers of $N_0$ such that
\begin{equation}y=e^{\sqrt{-1}\phi}\cdot\tilde y \label{e:y-coordinate-change}\end{equation}
for some real function $\phi$ on $H$. In particular, we have the transformations
\begin{align}
dy\wedge d\bar y&=d\tilde y \wedge d\bar\tilde y-\sqrt{-1} d|y|^2\wedge d\phi,
\\
\Gamma&=\widetilde{\Gamma}-\frac{1}{2}d\phi.\label{e:Gamma-transform}
\end{align}
This suggests viewing $\Gamma$ as a connection 1-form on the normal bundle. Indeed this is exactly the case.
\begin{lemma} \label{l:lemmagamma}
$2\sqrt{-1}\cdot\Gamma$ is the Chern connection 1-form of the normal bundle $N_0$ with respect to the above hermitian holomorphic structure, in the local holomorphic frame $\sigma$. In other words,
\begin{equation}\Gamma=\frac{1}{2}d^c_H \log |\sigma|.\end{equation}
\end{lemma}
\begin{proof}
By definition,
$\sigma=f\partial_y=\partial_{w_1}-\sum_{j\geq 2}\mu_j \partial_{w_j},$
where $f=|\sigma|>0$ is local real valued function on $H$, and $\mu_2, \cdots, \mu_{n-1}$ are local complex valued function on $H$. The key property we will use is that along $H$, $\nabla_{\partial_{\bar w_k}} \sigma$ is tangential to $H$ for $k\geq 2$. In fact, the K\"ahler condition implies $\nabla_{\bar{\partial}_{w_k}}\partial_{w_j}=0$ for all $j$, and hence
\begin{equation}\nabla_{\partial_{\bar w_k}}\sigma=\nabla_{\partial_{\bar w_k}}\Big(\partial_{w_1}-\sum_{j=2}^{n-1}\mu_j\partial_{w_j}\Big)=-\sum_{j= 2}^{n-1}\partial_{\bar w_k}(\mu_j) \partial_{w_j}. \end{equation}
Therefore,
\begin{equation}\partial_{w_k}f=\partial_{w_k} \langle \partial_y, \sigma \rangle= \langle \nabla_{ \partial_{w_k} }\partial_y, f\partial_y\rangle+\langle \partial_y, \nabla_{\partial_{\bar w_k}} \sigma\rangle=f\langle \nabla_{\partial_{w_k}}\partial_y, \partial_y\rangle,\end{equation}
and hence
\begin{equation}\langle \nabla_{\partial_{w_k}}\partial_y, \partial_y\rangle=f^{-1}\partial_{w_k}f=\partial_{w_k}(\log f).\end{equation}
Differentiating $|\partial_y|^2=1$, we get
\begin{equation}\langle \nabla_{\partial_{w_k}}\partial_y, \partial_y\rangle+\langle \partial_y, \nabla_{\partial_{\bar w_k}}\partial_y\rangle=0,\end{equation}
which implies
\begin{equation}\langle\nabla_{\partial_{\bar w_k}}\partial_y, \partial_y\rangle=-\partial_{\bar w_k} \log f.\end{equation}
Therefore,
\begin{align}\Gamma
&=-\frac{\sqrt{-1}}{2} \Big(\sum_{k\geq 2}\langle \nabla_{\partial_{w_k}}\partial_y, \partial_y\rangle dw_k+\sum_{k\geq 2}\langle \nabla_{\partial_{\bar w_k}}\partial_y, \partial_y\rangle d\bar w_k\Big)
\nonumber\\
&=-\frac{\sqrt{-1}}{2} \Big(\sum_{k\geq 2}\partial_{w_k}(\log f) dw_k - \sum_{k\geq 2}\partial_{\bar w_k}(\log f) d\bar w_k\Big)
\nonumber\\
&=-\frac{\sqrt{-1}}{2}(\partial_H \log f-\bar{\partial}_H \log f)
\nonumber\\
&=\frac{1}{2}d^c_H\log f.\end{align}
\end{proof}
For later applications, we need a few more local expansion results. We will also use the notation $O'$ and $\widetilde{O}$ in Definition \ref{d:normal-regularity} when we discuss the expansion in a neighborhood of $H$ in $D$, and the distance function is locally given by $|y|$. Notice the following expansions are given in the local (non-holomorphic) coordinates $\{y, \bar y, w_2', \bar w_2', \cdots, w_{n-1}, \bar w_{n-1}'\}$, and by definition we have $w_j'|_H=w_j|_H$ for $j\geq 2$.
\begin{proposition} \label{p:d^c|y|^2}
Locally near the point $p\in H$ we have
\begin{equation} \label{eqn3-4}
d^c_D|y|^2=\sqrt{-1}(yd\bar y-\bar ydy)+4|y|^2\cdot\Gamma+\widetilde{O}(|y|^3).
\end{equation}
\end{proposition}
The proof relies on
the following expansions
of the holomorphic coordinate functions $w_j$.
\begin{lemma}\label{l:w_1-expansion}
We have the expansion
\begin{align}\begin{cases}w_1=a_1y+a_2y^2+\widetilde{O}(|y|^3)
\\
w_j=w_j'+c_jy+d_jy^2+ \widetilde{O}(|y|^3), & j\geq 2,
\end{cases}
\end{align}
where $a_1=|\sigma|^{-1}>0$, $a_2$, $c_j$, $d_j$ are local smooth functions on $H$.
\end{lemma}
\begin{proof}
By definition, $\sigma$ is the orthogonal projection of $\partial_{w_1}$ onto $(TH)^{\perp}$, so we have that
\begin{equation}\label{eqn3-188}a_1^{-1}\partial_y|_H=\partial_{w_1}+\sum_{j= 2}^{n-1} b_j \partial_{w_j},\end{equation}
where $a_1=|\sigma|^{-1}>0$ and $b_j$ are smooth functions on $H$. Now write
\begin{equation}\label{eqn2-22}
\partial_y=\sum_{j=1}^{n-1} \frac{\partial w_j}{\partial y} \partial_{w_j}+\sum_{j= 1}^{n-1} \frac{\partial \bar w_j}{\partial { y}} \partial_{\bar w_j}.
\end{equation}
Then we get
\begin{align}\label{eqn2-34}
\frac{\partial \bar w_j}{\partial y}\Big|_H=0, \ j\geq 1,
\end{align}
which in particular implies
\begin{equation}
\frac{\partial w_j}{\partial \bar{y}}\Big|_H=\overline{\frac{\partial \bar w_j}{\partial y}}\Big|_H=0,
\ j\geq 1.
\end{equation}
Now by the definition of the normal exponential map, we have that at $p$,
\begin{equation}\nabla_{\partial_y}\partial_y=\nabla_{\partial_{\bar y}}\partial_y=\nabla_{\partial_{\bar y}}\partial_{\bar y}=0. \end{equation}
Using the K\"ahler condition we have
\begin{equation}\nabla_{\partial_{w_j}}{\partial_{\bar w_k}}=\nabla_{\partial_{\bar w_j}}\partial_{w_k}=0, \ \ j, k\geq 1.\end{equation}
Then by \eqref{eqn2-22} we get
\begin{equation}
\frac{\partial^2 w_j}{\partial y\partial \bar y}\Big|_H=\frac{\partial ^2 w_j}{\partial\bar y^2}\Big|_H=0, \ \ \ j\geq 1.
\end{equation}
Thus the conclusion follows.
\end{proof}
\begin{proof}[Proof of Proposition \ref{p:d^c|y|^2}]
Given the above lemma we first obtain that
\begin{align}
d\bar w_1&=a_1 d\bar y+\bar y (da_1+2\bar a_2d\bar y)+\widetilde{O}(|y|^2),
\\
w_1d\bar w_1&=a_1^2yd\bar y+|y|^2a_1da_1+a_1y(2\bar a_2\bar y+a_2 y)d\bar y+\widetilde{O}(|y|^3).
\end{align}
Hence
\begin{equation} \label{eqn1001}
d_D^c |w_1|^2=\sqrt{-1} a_1^2(yd\bar y-\bar ydy)+\sqrt{-1} a_1\bar a_2 \bar y (2yd\bar y-\bar ydy)-\sqrt{-1} a_1 a_2 y(2\bar y dy-yd\bar y)+\widetilde{O}(|y|^3).
\end{equation}
On the other hand, we have
\begin{equation}
|w_1|^2=a_1^2|y|^2+a_1(a_2y+\bar a_2 \bar y)|y|^2+\widetilde{O}(|y|^4).
\end{equation}
So
\begin{equation}\label{eqn1000}
d_D^c |w_1|^2=a_1^2 d_D^c|y|^2+|y|^2 d_D^c a_1^2+d_D^c (a_1(a_2y+\bar a_2 \bar y)|y|^2)+\widetilde{O}(|y|^3).
\end{equation}
Now by Lemma \ref{l:w_1-expansion},
\begin{equation}d_D^c (a_1y)=d_D^c w_1+\widetilde{O}(|y|)=-\sqrt{-1}dw_1+\widetilde{O}(|y|),\end{equation}
so
\begin{equation}
d_D^cy=-\sqrt{-1} dy+\widetilde{O}(|y|).
\end{equation}
Similarly, $d_D^c\bar{y}=\sqrt{-1} d\bar{y}+\widetilde{O}(|y|)$.
Plugging these into \eqref{eqn1000}, and compare with \eqref{eqn1001} we obtain
\begin{align}d_D^c |y|^2=\sqrt{-1}(yd\bar y-\bar ydy)-2|y|^2d_D^c \log a_1+\widetilde{O}(|y|^3).
\end{align}
Thanks to Lemma \ref{l:w_1-expansion}, $a_1=|\sigma|^{-1}$ which is a smooth function on $H$, so
\begin{align}
d_D^c |y|^2=\sqrt{-1}(yd\bar y-\bar ydy)+2|y|^2d_H^c \log |\sigma|+\widetilde{O}(|y|^3).
\end{align}
By Lemma \ref{l:lemmagamma}, $\Gamma=\frac{1}{2}d_H^c\log |\sigma|$, so we conclude
\begin{equation}d_D^c |y|^2=\sqrt{-1}(yd\bar y-\bar ydy)+4|y|^2\Gamma +\widetilde{O}(|y|^3).\end{equation}
\end{proof}
Now we prove an expansion result for the trace of $\psi$.
\begin{proposition} \label{p:trace-expansion} Let $\psi$ be the $2$-form on $Q$ given as in \eqref{e:singular-2-form-expansion}, then we have the following expansion in a neighborhood of $p$ in $Q$
\begin{equation}
\label{e:trace expansion equation}\Tr_{\omega_D}\psi=\frac{1}{2r}+O'(r).\end{equation}
\end{proposition}
Using \eqref{e:singular-2-form-expansion} it is easy to see $\Tr_{\omega_D}\psi$ admits an expansion of the form
\begin{equation}\Tr_{\omega_D}\psi=\frac{A_0}{r}+\frac{A_1y+\bar A_1\bar y}{r}+O'(r).\label{e:tr-exp}\end{equation}
for local functions $A_0, A_1$ defined on $H$. It suffices to show $A_0\equiv 1$ and $A_1\equiv 0 $. Since the left hand side is independent of the choice of local holomorphic coordinates, it suffices to work on the slice $z=0$ with special local holomorphic coordinates in a neighborhood of $p\in H$, and it suffices to understand the Taylor expansion along the fiber $N_0(p)$ of $N_0$ over the fixed point $p$.
\begin{lemma}\label{l:holo-coor-at-a-point}
We may choose the above holomorphic coordinates $\{w_i\}_{i=1}^{n-1}$ centered at $p$, so that $H$ is locally given by $w_1=0$ and
\begin{equation}
\omega_D=\frac{\sqrt{-1}}{2}g_{i\bar j}dw_i\wedge d\bar w_j,
\end{equation}
where
\begin{align}
\begin{cases}
g_{i\bar j}(0)=\delta_{ij}, & 1\leq i, j\leq n-1,\\
\partial_{w_1}g_{i\bar j}(0)=0, & 1\leq i, j\leq n-1,
\\
\partial_{w_k}g_{1\bar 1}(0)=\partial_{w_k}g_{i\bar j}(0)=0, & 2\leq i,j, k\leq n-1.
\end{cases}\label{e:at-a-point}
\end{align}
\end{lemma}
\begin{remark}
In fact, the only non-trivial Christoffel symbols at $p$ are
\begin{equation}\Gamma_{ij}^1 (0)=\partial_{i}g_{j\bar 1}(0), \ \ \Gamma_{\bar i\bar j}^{\bar 1}=\partial_{\bar i}g_{1\bar j}(0)
\end{equation}
for $i, j\geq 2$. This is due to the constraint that the equation $w_1=0$ defines $H$, which prevents us from using substitutions like \begin{equation}w_1=z_1+\sum\limits_{i, j=2}^{n-1} C_{1ij}z_i z_j.\end{equation} Intrinsically, $\{\Gamma^1_{ij}\}_{i, j\geq 2}$ captures the second fundamental form of the complex hypersurface $H$ at $p$.
\end{remark}
\begin{proof}[Proof of Lemma \ref{l:holo-coor-at-a-point}]
This follows from elementary manipulation. First, the holomorphic coordinates $\{w_i\}_{i=1}^{n-1}$ can be chosen such that $g_{i\bar j}(0)=\delta_{ij}$ for all $1\leq i, j\leq n-1$. By the substitution of the form
\begin{equation}\begin{cases}
w_i=z_i+\frac{1}{2}\sum\limits_{j, k= 2}^{n-1} C_{ijk}z_j z_k+\sum\limits_{j= 2}^{n-1} D_{ij} z_1z_j+E_{i}z_1^2, \ \ 2\leq i\leq n-1,\\
w_1=z_1+\sum\limits_{j= 1 }^{n-1} F_j z_1z_j,
\end{cases}\label{e:co-change}
\end{equation}
with suitable choices of coefficients, where $C_{ijk}=C_{ikj}$ for $2\leq i,j,k\leq n-1$.
One can plug both the Taylor expansion of $g_{i\bar{j}}$ along $z_k$'s and \eqref{e:co-change} into $\omega_D$. Comparing the coefficients, then it follows that,
\begin{align}
\begin{cases}
C_{ijk} = - \partial_{w_k}g_{j\bar{i}}(0),
\\
D_{ij} = -\partial_{w_j} g_{1\bar{i}}(0),
\\
E_i = -\frac{1}{2}\partial_{w_1}g_{1\bar{j}}(0),
\\
F_i = - \partial_{w_j}g_{1\bar{1}}(0),
\\
F_1 = -\frac{1}{2}\partial_{w_1}g_{1\bar{1}}(0),
\end{cases}
\end{align}
where $2\leq i,j,k\leq n-1$. Then we can achieve \eqref{e:at-a-point} with $\{w_i\}_{i=1}^{n-1}$ replaced by $\{z_i\}_{i=1}^{n-1}$.
\end{proof}
Now we prove Proposition \ref{p:trace-expansion}.
\begin{proof}[Proof of Proposition \ref{p:trace-expansion}] The goal is to show $A_0=1$ and $A_1=0$ in the expansion \eqref{e:tr-exp}.
We work in the above special holomorphic coordinates given by Lemma \ref{l:holo-coor-at-a-point}.
The first step is to show that the $O'(1)$-term in the expansion of $\psi$
given by Proposition \ref{p:complex-Green-expansion}
in fact vanishes along $N_0(p)$.
To this end, notice that $\partial_y=\sigma=\partial_{w_1}$ at $p\in H$ and hence by Lemma \ref{l:w_1-expansion},
\begin{equation}w_1 = y + a_2 y^2 + \widetilde{O}(|y|^3).\label{e:unit-leading-coe}
\end{equation}
Since the only non-trivial Christofell symsbols at $p$ are $\Gamma_{ij}^1$ and $\Gamma_{\bar i\bar j}^{\bar 1}$ for $i, j\geq 2$, it easily follows that
\begin{equation}d|\sigma|(p)=0.\label{e:d-sigma=0}\end{equation}
Combining \eqref{e:d-sigma=0} and Lemma \ref{l:lemmagamma}, \begin{equation}\Gamma(p)=\frac{1}{2}(d_H^c\log|\sigma|)(p)=0,\end{equation} for each $p\in H$. Therefore, along the fiber $N_0(p)$ of the normal bundle $N_0(p)$,
the expansion of $\psi$ in Proposition \ref{p:complex-Green-expansion} becomes\begin{equation}\psi=\frac{\sqrt{-1}}{4|y|} dy\wedge d\bar y+O(|y|).\end{equation}
The next is to compute $A_0(p)$ and $A_1(p)$ in \eqref{e:tr-exp}.
As in the proof of Lemma \ref{l:w_1-expansion}, we obtain that
\begin{align}\frac{\partial w_j}{\partial y}(p)&=\frac{\partial w_j}{\partial \bar y}=0, \ \ j\geq 2,
\\
\frac{\partial^2 w_j}{\partial y^2}(p)&=\frac{\partial^2 w_j}{\partial y\partial\bar y}(p)=\frac{\partial^2 w_j}{\partial \bar y^2}(p)=0, \ \ j\geq 1. \end{align}
This particularly implies that $a_2(p)=0$ and
along the fiber $N_0(p)$,
\begin{equation}
\label{eqn3.291}w_j=O(|y|^3), \ \ j\geq 2.\end{equation}
By Lemma \ref{l:holo-coor-at-a-point}, $\partial_{w_1}g_{i\bar{j}}(p)=0$ for all $1\leq i,j\leq n-1$, then the expansion of $\omega_D$ along the fiber $N_0(p)$ is at least quadratic in the $w_1$-direction, i.e.,
\begin{eqnarray}
\omega_D&=&\frac{\sqrt{-1}}{2} \Big(dw_1\wedge d\bar w_1+\sum_{j=2}^{n-1} dw_j\wedge d\bar w_j\Big)+O\Big(\sqrt{\sum_{j=2}^{n-1}|w_j|^2}\Big)+O(|w_1|^2)\nonumber\\
&=&\frac{\sqrt{-1}}{2} \Big(dw_1\wedge d\bar w_1+\sum_{j=2}^{n-1} dw_j\wedge d\bar w_j\Big)+O(|y|^2).
\end{eqnarray}
By \eqref{e:unit-leading-coe} and \eqref{eqn3.291}, along the fiber $N_0(p)$, we have
\begin{equation}dw_1=dy+O(|y|^2),\quad dw_j=dw_j'+O(|y|^2), \ \ j\geq 2.
\end{equation}
So we get
\begin{equation}
\omega_D=\frac{\sqrt{-1}}{2}\Big(dy\wedge d\bar y+\sum_{j=2}^{n-1} dw_j'\wedge d\bar w_j'\Big)+O(|y|^2)
\end{equation}
Since by definition,
\begin{equation}\Big(\Tr_{\omega_D}\psi\Big)\cdot \frac{\omega_D^{n-1}}{(n-1)!}=\psi\wedge \frac{ \omega_D^{n-2}}{(n-2)!}.\end{equation}
by elementary manipulations we get that $A_0(p)=1$ and $A_1(p)=0$.
\end{proof}
We close this subsection by proving an expansion of a local holomorphic volume form on $D$. Given the choice of local holomorphic coordinates on $D$ as before, let $\Omega_D$ be a local holomorphic volume form in a neighborhood of $p$, then we can always write
\begin{equation}\Omega_D=f\cdot dw_1\wedge dw_2\cdots \wedge dw_{n-1},\label{e:Omega-D-in-w-coordinates}\end{equation}
for a local nowhere vanishing holomorphic function $f$.
Denote the local holomorphic volume form on $H$
\begin{equation}
\Omega_H\equiv dw_2'\wedge\cdots dw_{n-1}'=(dw_2\wedge\cdots \wedge dw_{n-1})|_H.
\end{equation}
Then $\Omega_H$ can be naturally viewed as a complex $(n-2)$-form in some neighborhood of $p$ in $D$, in the coordinate system given by $\{y, \bar y, w_2', \bar w_2', \cdots, w_{n-1}', \bar w_{n-1}'\}$.
\begin{proposition}\label{lem3-6} We have the following expansion
\begin{equation}\Omega_D=F(dy+2\sqrt{-1} y \Gamma)\wedge \Omega_H+\widetilde{O}(|y|)dy+\widetilde{O}(|y|^2) \label{e:Omega-D-expansion}\end{equation}
for some smooth function $F$ locally defined on $H$.
\end{proposition}
\begin{proof}
We need to calculate the expansion for $dw_1\wedge \ldots\wedge dw_{n-1}$ in \eqref{e:Omega-D-in-w-coordinates}.
First, by Lemma \ref{l:w_1-expansion},\begin{equation}
dw_1=a_1(dy+yd_H\log a_1)+\widetilde{O}(|y|) dy+\widetilde{O}(|y|^2),
\end{equation}
where $a_1=|\sigma|^{-1}$.
Notice that
\begin{equation}
d_H\log a_1 =2\partial_H\log a_1 - \sqrt{-1}d_H^c \log a_1.
\end{equation}
Applying Lemma \ref{l:lemmagamma},
\begin{equation}
d_H\log a_1=2(\partial_H\log a_1+\sqrt{-1}\Gamma)
.\end{equation}
Next, applying Lemma \ref{l:w_1-expansion} to $w_j$'s for $j\geq 2$,
\begin{equation}
dw_j=dw_j'+c_jdy+ydc_j+\widetilde{O}(|y|)dy+\widetilde{O}(|y|^2).
\end{equation}
Since it holds that
\begin{equation}
\partial_H \log a_1\wedge\Omega_H\equiv 0,
\end{equation}
then taking the wedge product,\begin{equation}
dw_1\wedge\cdots \wedge dw_{n-1}=a_1(dy+2\sqrt{-1} y\Gamma)\wedge \Omega_H+\widetilde{O}(|y|)dy+\widetilde{O}(|y|^2).
\end{equation}
On the other hand, we have the expansion of $f$ in \eqref{e:Omega-D-in-w-coordinates},
\begin{equation}
f=f|_H+\frac{\partial f}{\partial y}\Big|_H\cdot y+\frac{\partial f}{\partial\bar y}\Big|_H\cdot \bar y+\widetilde{O}(|y|^2).
\end{equation}
Therefore,
\begin{equation}
\Omega_D=f|_H \cdot a_1\cdot (dy+2\sqrt{-1} y\Gamma)\wedge \Omega_H+\widetilde{O}(|y|)dy+\widetilde{O}(|y|^2).
\end{equation}
So we obtain the conclusion by taking $F\equiv f|_H\cdot a_1$.
\end{proof}
\subsection{A global existence result}\label{ss:global-existence}
We assume the same set-up as in Section \ref{ss:complex-greens-currents}.
We further assume that $D$ is compact K\"ahler, and $H$ is a smooth divisor in $D$ which is Poincar\'e dual to $\frac{k}{2\pi}[\omega_D]$ for some positive integer $k$.
The following proposition establishes a global existence for Green's current in this setting. We thank Lorenzo Foscolo for discussions concerning the constructive proof.
\begin{proposition} \label{p:existence-Greens-current}
Given any constants $k_-, k_+\in \mathbb R$ with $
k_--k_+=k,$
there exists a unique global Green's current $G_P$ for $P$ in $Q$ such that the following properties hold:
\begin{enumerate}
\item $G_P$ is of the form \begin{equation}G_P=\psi(z)\wedge dz, \end{equation}
where $\psi(z)$ is a family of real-valued closed $(1,1)$-forms on $D$ parametrized by $z$ and satisfies the expansion \eqref{e:singular-2-form-expansion} near $P$.
\item
For any $k\in\mathbb{N}$ and $\delta\in(0,10^{-2})$, we have
\begin{align}
\label{e:greens-current-exp-asymp}
\begin{cases}
|\nabla^k(\psi(z)-(k_-z)\cdot \omega_D)|=O(e^{(1-\delta)\sqrt{\lambda_1}z}),& z\rightarrow -\infty,\\
|\nabla^k(\psi(z)-(k_+z)\cdot \omega_D)|=O(e^{-(1-\delta)\sqrt{\lambda_1} z}), & z\rightarrow \infty,
\end{cases}
\end{align}
where $\lambda_1>0$ is the first eigenvalue of the Hodge Laplacian acting on real-valued closed $(1,1)$-forms on $D$, and
\eqref{e:greens-current-exp-asymp} are with respect to the fixed product metric on $Q$.
\end{enumerate}
\end{proposition}
\begin{proof}[Proof of Proposition \ref{p:existence-Greens-current}]
We first prove the existence part. Let $\{\phi_j\}_{j=0}^{\infty}$ be a complete $L^2$-orthonormal basis of eigenvectors for the Hodge Laplacian $\Delta_D$ acting on real-valued closed $(1,1)$-forms on $D$. We suppose $\Delta_D \phi_j=\lambda_j\phi_j$, and $\lambda_j\geq 0$ is increasing in $j$. Our basic strategy is to first construct a formal series and then prove the convergence.
To begin with, the Dirac $3$-current $\delta_P$ of $P\subset Q$ has a formal expansion along the $D$ direction,
\begin{equation}2\pi \delta_P=\sum_{j=0}^{\infty} f_j(z)\phi_j\wedge dz. \end{equation}
Here $f_j(z)$ is a $0$-current on $\mathbb R$ given by
\begin{equation}f_j(z) \equiv 2\pi\Big( \int_H *_D\phi_j \Big)\delta_{0}(z),
\end{equation}
where $\delta_0(z)$ is the standard Dirac $0$-current acting on functions on $\mathbb R$, supported at $\{z=0\}$.
Furthermore, if $\Delta_D\phi_j=0$, then $*_D\phi_j$ is closed and
\begin{equation}\int_H*_D\phi_j=\int_D \frac{k}{2\pi} \omega_D\wedge *_D\phi_j=\frac{k}{2\pi}\langle \omega_D, \phi_j\rangle_{L^2(D)}. \end{equation}
It follows that there is exactly one $j$, which we may assume to be $0$, such that $\lambda_j=0$ and $f_j$ is non-zero. The corresponding eigenform $\phi_0$ is a multiple of $\omega_D$. Now suppose $G_P$ is given as a formal series
\begin{equation}G_P=\sum\limits_{j=0}^{\infty} h_j(z) \phi_j \wedge dz,\end{equation}
then we need $h_j$ to satisfy \begin{equation}\label{eqn3333}-h_j''(z)+\lambda_j\cdot h_j(z)=f_j(z).\end{equation}
If $\lambda_j>0$, we can write down a solution
\begin{align}h_j(z)&=\frac{1}{-2\sqrt{\lambda_j}}\Big(e^{-\sqrt{\lambda_j}\cdot z}\int_{-\infty}^z e^{\sqrt{\lambda_j}\cdot u}f_j(u)du+e^{\sqrt{\lambda_j}\cdot z} \int_z^\infty e^{-\sqrt{\lambda_j}\cdot u}f_j(u)du\Big)
\nonumber\\
&=\begin{cases}\frac{\pi}{-\sqrt{\lambda_j}}\cdot e^{-\sqrt{\lambda_j}\cdot z}\cdot \int_H *_D\phi_j, & z>0,
\\
\frac{\pi}{-\sqrt{\lambda_j}}\cdot e^{\sqrt{\lambda_j}\cdot z}\cdot \int_H *_D\phi_j, & z\leq 0.
\end{cases}
\label{e:formal-solution-h-j}\end{align}
If $j>0$ and $\lambda_j=0$ we simply set $h_j(z)=0$.
If $j=0$, we can write down a solution
\begin{align}h_0(z)=\begin{cases}
k_+z \langle \omega_D, \phi_0\rangle_{L^2(D)}, & z\geq 0,\\
k_-z \langle \omega_D, \phi_0\rangle_{L^2(D)}, & z\leq0.
\end{cases}
\end{align}
With this choice of $h_j$, we can define $G_P$ as the formal series given above.
Next we claim that $G_P$ is well-defined as a $3$-current on $Q$. For any test form $\chi\in\Omega_0^{m-3}(Q)$, we need to show the sum $\sum\limits_{j=0}^N\Big(h_j(z) \phi_j \wedge dz, \chi\Big)$ converges as $N\rightarrow\infty$.
So it suffices to show that
\begin{equation}
\sum\limits_{j=0}^\infty\Big|\Big(h_j(z)\phi_j \wedge dz, \chi\Big)\Big|<\infty.\label{e:bounded-sum-current}\end{equation}
To see this, for each $j$, we write
\begin{eqnarray}\Big(h_j(z)\phi_j \wedge dz, \chi\Big)&=&\int_Q h_j(z) \phi_j\wedge dz \wedge \chi
\nonumber\\
&=&\int_{\mathbb{R}} \Big( h_j(z) \cdot \int_D\langle \chi, *_D\phi_j\rangle\dvol_{\omega_D}\Big) dz.\label{e:fubini}\end{eqnarray}
We first derive a uniform bound on the integral $\int_D\langle \chi, *_D\phi_j\rangle\dvol_{\omega_D}$. If $\lambda_j>0$, then for all $\ell\geq 1$,
\begin{eqnarray}\int_D\langle \chi(z), *_D\phi_j\rangle \dvol_{\omega_D} &=&
\frac{1}{(\lambda_j)^{\ell}}\int_D\langle \chi(z), *_D(\Delta_D)^{\ell}(\phi_j)\rangle \dvol_{\omega_D}
\nonumber\\
&=&\frac{1}{(\lambda_j)^{\ell}}\int_D\langle (\Delta_D)^{\ell}\chi(z), *_D\phi_j\rangle\dvol_{\omega_D}.\end{eqnarray}
By standard elliptic regularity we have \begin{equation}
\|\phi_j\|_{C^0(D)} \leq C \cdot (\lambda_j)^{\frac{n-1}{2}},
\end{equation}
where $C$ depends only on $n$ and the metric $\omega_D$.
So it follows that
\begin{equation}
\Big|\int_D\langle \chi(z), *_D\phi_j\rangle \dvol_{\omega_D}\Big| \leq C\cdot \|\chi\|_{C^{2\ell}(Q)}\cdot \frac{1}{(\lambda_j)^{\ell-\frac{n-1}{2}}}.\end{equation}
Notice this estimate is independent of $z$.
Next, we have
\begin{eqnarray}
\int_{\mathbb{R}}|h_{j}(z)|dz &\leq& \int_{-\infty}^{-1}|h_j(z)|dz + \int_{-1}^{1}|h_j(z)|dz + \int_{1}^{+\infty} |h_j(z)|dz
\nonumber\\
&\leq & C(1+\lambda_j^{\frac{n}{2}}).
\end{eqnarray}
Combining the above estimates, we have
\begin{equation}
|(h_j(z) \phi_j \wedge dz, \chi)|=\Big|\int_{\mathbb{R}} \Big( h_j(z) \cdot \int_D\langle \chi, *_D\phi_j\rangle\dvol_{\omega_D}\Big) dz\Big| \leq C\cdot \|\chi\|_{C^{2\ell}(Q)}\cdot \frac{C(1+\lambda_j^{\frac{n}{2}})}{(\lambda_j)^{\ell-\frac{n}{2}}}.
\end{equation}
Applying Weyl's law, we see that if we fix $\ell$ sufficiently large, then \eqref{e:bounded-sum-current} holds, and this completes the proof of the claim.
Now we show $G_P$ as defined above satisfies the current equation
\begin{equation}\label{eqn3347}
\Delta G_P =2\pi\delta_P .
\end{equation}
Given a test form $\chi\in\Omega_0^{m-3}(Q)$,
applying the definition of $f_j$, $h_j$ and integration by parts, it is straightforward to see that for each $j$,
\begin{equation}(h_j(z) \phi_j \wedge dz, \Delta\chi)=(f_j(z) \phi_j \wedge dz, \chi).\end{equation}
So
\begin{equation}
(G_P, \Delta\chi)=\sum_j (f_j(z) \phi_j\wedge dz, \chi)=\sum_j 2\pi (\int_H *_D\phi_j) \int_D \phi_j\wedge \chi|_{z=0}.
\end{equation}
By Hodge decomposition we can write the $(n-2,n-2)$ component of $\chi|_{z=0}$ as $ d_D\alpha+\beta$ with $d_D^*\beta=0$. Since $\phi_j$ is a closed $(1,1)$ form, we have
\begin{equation}
\int_{D} \phi_j\wedge \chi|_{z=0}=\int_{D}\phi_j\wedge \beta=\langle \beta, *_D\phi_j\rangle_{L^2(D)}.
\end{equation}
Notice $\beta=\sum_{j} \langle \beta, *_D\phi_j\rangle_{L^2(D)}*_D\phi_j$, we see $(G_P, \Delta\chi)=2\pi \int_{P}\chi, $ which proves \eqref{eqn3347}. In particular, we know $G_P$ is smooth away from $P$. Clearly $G_P$ satisfies (1) with $\psi=\sum_j h_j(z) \phi_j$.
Finally we study the asymptotics of $G_P$ as $z\rightarrow\pm\infty$. If $\lambda_j>0$, by standard elliptic regularity we have for all $\ell\geq 0$
\begin{equation}
\| \nabla^{\ell}\phi_j \|_{C^0(D)}\leq C\cdot (\lambda_j)^{\frac{n-1}{2}+\frac{\ell+1}{2}}.
\end{equation}
This implies that for any $z>10^{n^2+\ell^2}$, we have \begin{equation}
\|\nabla^{\ell}_Q(h_j(z)\cdot \phi_j)\|_{C^0(D\times\{z\})} \leq C(\lambda_j)^{\frac{n+\ell}{2}}e^{-\sqrt{\lambda_j} z}.
\end{equation}
By elementary computations, for each $\delta\in(0,10^{-2})$ and for $z$ sufficiently large, we have
\begin{equation}
\sum\limits_{\lambda_j>0} \|\nabla^{\ell}_Q(h_j(z)\cdot \phi_j)\|_{C^0(D\times\{z\})} \leq C e^{-(1-\delta)\sqrt{\lambda_1}z}\cdot \sum\limits_{j=0}^{\infty}(\lambda_j)^{-6n}.
\end{equation}
By Weyl's law, the above series converges, and hence for each $\ell\in\mathbb{N}$,
$\nabla_Q^{\ell}(G_P-h_0(z)\phi_0\wedge dz)$ is exponentially decaying as $z\rightarrow +\infty$. The argument is identical for $z<0$. Then $G_P$ satisfies the \eqref{e:greens-current-exp-asymp}.
To see the uniqueness, suppose there is another Green's current $\widetilde G_P=\tilde \psi(z)\wedge dz$ also satisfying (1) and (2), then $\widetilde G_P-G_P=(\tilde \psi(z)-\psi(z))\wedge dz$ is a global harmonic 3-form with exponential decay at infinity. Applying Fourier expansion to $\tilde\psi(z)-\psi(z)$ along the $D$ direction, similar to what is done in the above, it is easy to see that $\tilde\psi(z)=\psi(z)$.
\end{proof}
The constants $k_-$ and $k_+$ determine some information of the above $\psi$, which will be used later.
\begin{proposition}\label{p:cohomology-constant}
Let $\psi$ be the $(1,1)$-current in Proposition \ref{p:existence-Greens-current}, then the following holds:
\begin{enumerate}\item The cohomology class $[\psi(z)]\in H^2(D; \mathbb R)$ is given by $k_-z[\omega_D]$ and $k_+z[\omega_D]$ for $z<0$ and $z>0$ respectively.
\item We have
\begin{equation}\label{eqn2-43}
\partial_z\psi|_{z=0}=\frac{1}{2}(k_-+k_+)\omega_D.
\end{equation}
In particular, $\partial_z\psi|_{z=0}$ extends smoothly across $P$.
\end{enumerate}
\end{proposition}
\begin{proof}
Since $Q$ is a Riemannian product, we have for $z\neq 0$, \begin{equation}\frac{d^2}{dz^2}\psi(z)=\Delta_D \psi(z)=d_Dd_D^*\psi(z)\end{equation}
is exact, which implies that the cohomology class $[\partial_z\psi(z)]\in H^2(D; \mathbb R)$ is locally constant for $z\in \mathbb{R}\setminus\{0\}$. On the other hand, by the exponential decay property in \eqref{e:greens-current-exp-asymp} we see that
\begin{equation}\lim_{z\rightarrow\pm \infty}(\psi(z)-k_\pm z[\omega_D]) =0.\end{equation}
So (1) follows.
For item (2), denote
\begin{equation}\tilde\psi(z)\equiv\psi(-z)+(k_-+k_+)z\omega_D. \end{equation}
Then $\tilde\psi\wedge dz$ is also a Green current for $P$ and it is also asymptotic to $(k_{\pm}z)\cdot \omega_D$ as $z\rightarrow \pm\infty$. Therefore by uniqueness, $\tilde\psi(z)=\psi(z)$.
Taking the $z$-derivative at $z=0$ we get the conclusion.
\end{proof}
\section{The approximately Calabi-Yau neck region}
\label{s:neck}
In this section, we build the {\it the neck region}. It is one of the key geometric ingredients in this paper.
For all dimensions, we will construct a family of incomplete K\"ahler metrics with $S^1$-symmetry, on certain singular $S^1$-fibrations over a cylindrical base. These will serve to interpolate between the geometries at the ends of two Tian-Yau metrics.
Our construction is motivated by the non-linear Gibbons-Hawking ansatz in Section \ref{s:torus-symmetries}. However, as explained in Section \ref{s:torus-symmetries}, it does not seem easy to solve the non-linear reduced equation directly. Instead, we will use a singular solution to the \emph{linearized} ansatz, namely, the Green's current constructed in Section \ref{s:Greens-currents}, to obtain a family of K\"ahler metrics with $S^1$-symmetry, parametrized by a large parameter $T\gg 1$. Here are two technical points to note:
\begin{itemize}
\item These metrics will not be shown to be smooth along the fixed loci of the $S^1$-action. Indeed, we only prove that they are $C^{2, \alpha}$ for all $\alpha\in (0, 1)$ (Proposition \ref{p:C2alpha}). For our gluing construction, we need a further perturbation which lowers the regularity to be $C^{1,\alpha}$. This turns out to be sufficient for our analysis.
\item These metrics are only \emph{approximately} Calabi-Yau, in an appropriate weighted sense (Proposition \ref{p:CY-error-small}). One can also perturb them to genuine incomplete Calabi-Yau metrics (see Section \ref{s:neck-perturbation}).
\end{itemize}
Let us first set up some notations. Throughout this section we fix integers $n\geq 2$ and $k>0$.
Let $(D, \omega_D, \Omega_D)$ be a closed Calabi-Yau manifold of complex dimension $n-1$. Here $\omega_D$ is a K\"ahler form in the class $2\pi c_1(L)$ for some ample holomorphic line bundle $L$, $\Omega_D$ is a holomorphic volume form on $D$, and we assume the following normalized Calabi-Yau equation holds \begin{equation}
\label{e:CY equation on D}
\frac{1}{(n-1)!}\omega_D^{n-1}=\frac{(\sqrt{-1})^{(n-1)^2}}{2^{n-1}}\Omega_D\wedge\bar\Omega_D.
\end{equation}
We fix a hermitian metric on $L$ whose curvature form is $-\sqrt{-1} \omega_D$. This naturally induces a hermitian metric on all tensor powers of $L$.
We also fix a smooth divisor $H$ in the linear system $L^{\otimes k}$ and a defining section $S_H$ of $H$. Fix $r_D>0$ such that for any $p\in H$, the local coordinate system $\{y, \bar y, w_2', \bar w_2', \cdots, w_{n-1}', \bar w_{n-1}'\}$ introduced in Section \ref{ss:complex-greens-currents} exists in the ball $B_{10 r_D}(p)$.
Let $Q\equiv D\times \mathbb R$ be the Riemannian product of $D$ and the real line $\mathbb R$ parametrized by the
coordinate $z\in (-\infty, \infty)$. We denote
$P\equiv H\times \{0\}$ the codimension-$3$ submanifold of $Q$.
Using the normal exponential map on $D$ (resp. $Q$), we may always implicitly identify a tubular neighborhood of $P$ in $D$ (resp. $Q$) with a neighborhood of the zero section in the normal bundle $N_0$ (resp. $N=N_0\oplus \mathbb R$). Here we adopt the notation in Section \ref{ss:complex-greens-currents}, so $N_0$ is a hermitian line bundle and $N$ is a Riemannian vector bundle.
Let us fix $k_-, k_+\in \mathbb Z$ with $k_->0$, $k_+<0$ and $k_--k_+=k$. Applying Proposition \ref{p:existence-Greens-current}, there is a unique Green's current $G_P=\psi\wedge dz$ for $P$ in $Q$ such that the asymptotics \eqref{e:greens-current-exp-asymp} holds.
We also make the following notational conventions for this section:
\begin{itemize}
\item $\epsilon_T$ denotes a family of smooth functions on $D$, parametrized by $T\geq 1$, such that for each $k\geq 0$, its $k$-th derivative with respect to $\omega_D$ is of the form $O(e^{-\delta_k T})$ as $T\rightarrow\infty$, for some $\delta_k>0$ (independent of $T$).
\item $\underline \epsilon_T$ denotes a function of $T$ which is $O(e^{-\delta T})$ as $T\rightarrow\infty$, for some $\delta>0$
\item $\epsilon(z)$ denotes a smooth function defined on a subdomain in $Q$ for $|z|\gg1$ such that its all derivatives decay exponentially fast as $|z|\rightarrow\infty$.
\item $B_T$ denotes a family of smooth functions on $D$, parametrized by $T\gg1$, such that for each $k\geq 0$, its $k$-th derivatives with respect to $\omega_D$ is bounded independent of $T$.
\item $\underline B_T$ denotes a function of $T$ which is uniformly bounded as $T\rightarrow\infty$.
\item $B(z)$ denotes a smooth function of $z$, such that all its derivatives are uniformly bounded.
\end{itemize}
The organization of this section is as follows. In Section \ref{ss:kaehler-structures} we construct a family of $S^1$-invariant K\"ahler structures. Special attention is paid to understand the singularity structure near the fixed loci of the $S^1$-action. We will first construct a smooth compactification and write an explicit local model, and then study the regularity of the K\"ahler structures. In Section \ref{ss:complex-geometry} we show the underlying complex manifold is an open subset in an explicit $\mathbb C^*$ fibration over $D$, and derive a formula for the K\"ahler potential of our family of K\"ahler metrics. In Section \ref{ss:regularity-scales} we study and classify the limiting geometry of our family of metrics at regularity scales, which forms a foundation for our weighted analysis. In Section \ref{ss:neck-weighted-analysis} we define the relevant weighted H\"older spaces and prove a local weighted Schauder estimate. We also show our family of K\"ahler metrics are approximately Calabi-Yau by providing an estimate of the error in a weighted H\"older space. In Section \ref{ss:perturbation of complex structures} we deal with a perturbation of the complex structures of the underlying complex manifold, and estimate the error in a weighted H\"older space.
\subsection{Construction of a family of $C^{2,\alpha}$-K\"ahler structures}
\label{ss:kaehler-structures}
In this subsection we use \eqref{omegahequation} to construct a family of $C^{2, \alpha}$-K\"ahler structures on certain singular $S^1$-fibrations over an increasing family of domains in $Q$. In Section \ref{sss:incomplete} we construct smooth K\"ahler metrics on principal $S^1$-bundles over $Q\setminus P$. In Section \ref{sss:smooth compactification} we construct a smooth compactification by adding fixed points of the $S^1$-action which lie over points in $P$. In Section \ref{sss:metric compactification} we study the regularity of the K\"ahler metric near the fixed loci. Most of the quantities defined in this subsection will depend on the parameter $T$, but for simplicity of notation, we will not always keep track of this if the meaning is clear from the context.
\subsubsection{K\"ahler metrics on $S^1$-bundles away from $P$}
\label{sss:incomplete}
For $T\gg1$, we define
\begin{equation}
\tilde\omega=T\omega_D+\psi.
\end{equation}
It can be viewed as a family of closed $(1,1)$-forms $\tilde\omega(z)$ on $D$ parametrized by $z\in \mathbb R$.
Using the K\"ahler identities, we obtain
\begin{equation} \label{compatibility}\partial_z^2\tilde\omega=\Delta_{D} \tilde\omega=-d_Dd_D^c \Tr_{\omega_D}\tilde\omega, \end{equation}
If we define
\begin{equation}
h\equiv \Tr_{\omega_D}\tilde\omega+q(z)
\end{equation}
for any smooth function $q(z)$, then the pair $(\tilde\omega, h)$ satisfies the first equation in \eqref{omegahequation}:
\begin{equation} \label{eqn4445}
\partial_z^2\tilde\omega+d_Dd_D^ch=0.
\end{equation}
For our purpose we need to make a special choice of the function $q(z)$. First we choose a function $q_0(z)$ that satisfies
the following cohomological condition
\begin{equation} \label{eqn5-3}
q_0(z)\int_D \omega_D^{n-1}+(n-1)\int_D\tilde\omega(z)\wedge \omega_D^{n-2}=T^{2-n}\int_D \tilde\omega(z)^{n-1}, \ \ \forall z\in\mathbb{R}.
\end{equation}
By Proposition \ref{p:cohomology-constant}, the cohomology class $[\psi(z)]\in H^2(D; \mathbb R)$ is piecewise linear in $z\in \mathbb R$, so
\begin{align}\label{q0definition}
q_0(z)=
\begin{cases}
T^{2-n}(T+k_+ z)^{n-1}-(n-1)(T+k_+ z), & z>0,
\\
T^{2-n}(T+k_- z)^{n-1}-(n-1)(T+k_- z), & z<0.
\end{cases}
\end{align}
It follows that $q_0(z)$ is identically zero if $n=2$. But if $n>2$ then $q_0(z)$ is only $C^{1,1}$ at $z=0$ and we need to smooth it. We fix throughout this section a smooth function $L_0: \mathbb R\rightarrow \mathbb R$ satisfying
\begin{align} \label{e:l(z)}
L_0(z)\equiv
\begin{cases}
k_+z, & z>1,
\\
0 , & z=0,
\\
k_-z, & z<-1.
\end{cases}
\end{align}
and let
\begin{equation} \label{e:LT z}
L_T(z)\equiv T+L_0(z).
\end{equation}
Let us we define
$q(z)\equiv T^{2-n}L_T(z)^{n-1}-(n-1)L_T(z)$.
Then
\begin{align}
q(z)
=
\begin{cases}
q_0(z), & |z|\geq 1,
\\
q_0(z) + T^{-1}B(z), & |z|<1.
\end{cases}
\end{align}
It is also easy to see from \eqref{eqn5-3} that \begin{align} \label{haverage}
\int_D h\omega_D^{n-1}=
\begin{cases}T^{2-n}\int_D \tilde\omega(z)^{n-1}, & |z|\geq 1, \\
T^{2-n}\int_D \tilde\omega(z)^{n-1}+T^{-1}B(z), & z\in [-1, 1].
\end{cases}
\end{align}
We refer to Remark \ref{r:error-function} for an explanation of this particular choice of $q(z)$.
To apply the construction in Section \ref{s:torus-symmetries}, we need to restrict to the region in $Q$ where $\tilde\omega(z)$ is a positive $(1,1)$-form and $h$ is a positive function.
For $T$ large we define $T_+>0$ and $T_-<0$ by
\begin{equation}\label{e:define-T-plus-minus}
\begin{cases}
L_T(T_+)=T+k_+T_+=T^{\frac{n-2}{n}}\\
L_T(T_-)=T+k_-T_-=T^{\frac{n-2}{n}}.
\end{cases}
\end{equation}
and denote by $Q_T\subset Q$ the closed subset defined by $z\in [T_-, T_+]$. The reason for this choice of $T_\pm$ is also explained in Remark \ref{r:error-function}.
\begin{lemma}
For $T$ large, over $Q_T\setminus P$, both $\tilde\omega$ and $h$ are positive. Moreover, we have
\begin{align}
h &=T^{2-n}(T+k_\pm z)^{n-1}+\epsilon(z), \quad |z|\geq 1,\label{e:h-asymptotics}
\\h & =T+\frac{1}{2r}+O'(r) + T^{-1}B(z), \quad |z|\leq 1,\label{e:h-bounded-z}
\end{align}
where $O'(r)$ is independent of $T$, and its singular behavior near $P$ is given by Definition \ref{d:normal-regularity}.
\end{lemma}
\begin{proof}
We first consider $\tilde\omega$.
As $z\rightarrow\pm\infty$ the behavior of $\tilde\omega$ is governed by \eqref{e:greens-current-exp-asymp}, so for $T\gg 1$ we know $\tilde\omega$ is positive over the region where $z\in [-k_-^{-1}(T-1), -k_+^{-1}(T-1)]\setminus [-C, C]$ for some number $C>0$ independent of $T$. By the expansion of $\psi$ in a neighborhood of $P$ given in Proposition \ref{p:complex-Green-expansion}, for $T$ sufficiently large, $\tilde\omega$ is also positive when $z\in [-C, C]$. Hence $\tilde\omega$ is positive over the region where $z\in [-k_-^{-1}(T-1), -k_+^{-1}(T-1)].$ Since this contains $Q_T$ we see in particular $\tilde\omega$ is positive over $Q_T\setminus P$.
To deal with $h$ we need to analyze $q(z)$. When $|z|\geq 1$, we have
\begin{equation}\label{qexpansion}
q(z)=q_0(z)=T^{2-n}(T+k_\pm z)^{n-1}-(n-1)(T+k_\pm z),
\end{equation}
where the choice of $+$ or $-$ depends on whether $z>0$ or $z<0$.
By \eqref{e:greens-current-exp-asymp} we then get
\begin{equation}
h=T^{2-n}(T+k_\pm z)^{n-1}+\epsilon(z).
\end{equation}
So we can find $C>0$ such that $h$ is positive when $z\in [T_-, T_+]\setminus [-C, C]$. On the other hand, on $[-C, C]$ we know by definition
\begin{equation}\label{e:q-bounded-distance}
q(z)=(2-n)T+T^{-1}B(z).
\end{equation}
Hence by the expansion in Proposition \ref{p:trace-expansion} we obtain \eqref{e:h-bounded-z}. This implies that for $T\gg1$, $h$ is also positive when $z\in [-C, C]$.
\end{proof}
Now on $Q\setminus P$ we define the 2-form
\begin{equation}
\Upsilon\equiv \partial_z\tilde\omega-dz\wedge d_D^ch.
\end{equation}
Then \eqref{compatibility} implies that $\Upsilon$ is closed on $Q\setminus P$ and hence $[\Upsilon]\in H^2(Q\setminus P, \mathbb R)$.
Moreover, we have
\begin{lemma}\label{l:integrality}
The cohomology class $\frac{1}{2\pi}[\Upsilon]\in H^2(Q\setminus P; \mathbb R)$ is integral.
\end{lemma}
\begin{proof}
As mentioned in the beginning of this section, we identify a tubular neighborhood of $P$ in $Q$ with a neighborhood of the zero section in its normal bundle $N=N_0\oplus \mathbb R$. For simplicity we may assume this neighborhood is given by $\mathcal B_\epsilon$, the 2-ball bundle over $P$ consisting of the set of all elements in $N_0\oplus \mathbb R$ with norm at most $\epsilon$, and we denote by $\mathcal S_\epsilon$ the boundary of $\mathcal B_\epsilon$.
Fix $z_0>0$, then the composition of the natural maps
\begin{equation}
D\simeq D\times \{z_0\}\hookrightarrow Q\setminus P \hookrightarrow Q\rightarrow D
\end{equation}
is the identity map, which implies that for all $k$, the map $H_k(Q\setminus P; \mathbb Z)\rightarrow H_k(Q;\mathbb Z)$ is surjective and we have a natural splitting
\begin{equation}
H_2(Q\setminus P; \mathbb Z)=H_2(D; \mathbb Z)\oplus K
\end{equation}
for some $K$.
By assumption for $z>0$,
\begin{equation}[\partial_z\tilde\omega(z)]=[\partial_z\psi(z)]=k_+[\omega_D]=2\pi k_+c_1(L),
\end{equation}
so $\frac{1}{2\pi}[\Upsilon]|_{D\times \{z_0\}}=k_+c_1(L)$ is integral. Hence it suffices to show the integral of $\frac{1}{2\pi}\Upsilon$ over any element in $K$ is also an integer.
By the Mayer-Vietoris sequence applied to $Q=(Q\setminus P)\cup \mathcal B_\epsilon$, we get
\begin{equation}
0\rightarrow H_2(\mathcal S_\epsilon; \mathbb Z)\rightarrow H_2(Q\setminus P; \mathbb Z)\oplus H_2(\mathcal B_\epsilon; \mathbb Z)\rightarrow H_2(Q;\mathbb Z)\simeq H_2(D;\mathbb Z)\rightarrow 0.
\end{equation}
So we obtain the exact sequence
\begin{equation} \label{e: 4-23}
0\rightarrow K\rightarrow H_2(\mathcal S_\epsilon; \mathbb Z)\rightarrow H_2(\mathcal B_\epsilon; \mathbb Z)\simeq H_2(P; \mathbb Z).
\end{equation}
On the other hand,
by the Gysin sequence applied to the 2-sphere bundle $p:\mathcal S_\epsilon\rightarrow P$ we get
\begin{equation}
0\rightarrow H^2(P; \mathbb Z) \xrightarrow{p^*} H^2(\mathcal S_\epsilon; \mathbb Z)\xrightarrow{\int} H^0(P; \mathbb Z)\xrightarrow{\wedge e} H^3(P; \mathbb Z)\rightarrow\cdots
\end{equation}
where $\int$ denotes integration over the 2-sphere fibers, and $\wedge e$ denotes the wedge product with Euler class of $\mathcal S_\epsilon$.
Since the Euler class $e$ of $N_0\oplus \mathbb R$ vanishes, the above becomes
\begin{equation}
\label{e: 4-25}
0\rightarrow H^2(P; \mathbb Z)\xrightarrow{p^*} H^2(\mathcal S_\epsilon; \mathbb Z)\xrightarrow{\int} H^0(P; \mathbb Z)\simeq \mathbb Z\rightarrow 0.
\end{equation}
\eqref{e: 4-23} and \eqref{e: 4-25} together imply that modulo torsion, $K$ is generated by the homology class of a 2-sphere fiber of $p$. So we just need to show $\int \frac{1}{2\pi}[\Upsilon]|_{\mathcal S_\epsilon}$ is an integer.
By the expansion of $\psi$ and $h$ in Proposition \ref{p:complex-Green-expansion} and Proposition \ref{p:trace-expansion}, it is easy to check that by restricting to the fiber of $N$ over $p$, we have
\begin{equation}
\Upsilon|_{N(p)}=-\frac{\sqrt{-1}}{4r^3}(zdyd\bar y+(yd\bar y-\bar ydy)dz)+O(1).
\end{equation}
Further restricting to the $2$-sphere with radius $\epsilon$, we get
\begin{equation}
\Upsilon|_{\mathcal S_\epsilon(p)}=-\frac{1}{2\epsilon^2}\dvol_{S^2_\epsilon}+O(1),
\end{equation}
where $\dvol_{S^2_\epsilon}$ is the area form of the standard $\epsilon$-sphere in $\mathbb R^3$. Taking the integral and letting $\epsilon\rightarrow 0$,\begin{equation}
\int_{\mathcal S_\epsilon(p)} \Upsilon=-2\pi.
\end{equation}
\end{proof}
By Lemma \ref{l:integrality}, standard theory yields a $U(1)$ connection $1$-form $-\sqrt{-1}\Theta$ on a principal $S^1$-bundle
\begin{equation}\pi: \mathcal M^* \rightarrow Q_T\setminus P\end{equation}
with curvature form $-\sqrt{-1}\Upsilon$. Moreover, $\mathcal M^*$ restricts to the standard Hopf bundle on each normal $S^2$ to $P$ (it has degree $-1$ if we use the natural orientation). Then we have the second equation in \eqref{omegahequation}:
\begin{equation} \label{d Theta equation}
d\Theta=\partial_z\tilde\omega-dz\wedge d_D^c h.
\end{equation}
On ${\mathcal M^*}$ we define a real-valued 2-form and a complex-valued $n$-form
as follows
\begin{align}
\omega &\equiv T^{\frac{2-n}{n}}(\pi^*\tilde\omega+dz\wedge \Theta),\label{e:def-omega}
\\
\Omega &\equiv\sqrt{-1}(hdz+\sqrt{-1}\Theta)\wedge \pi^*\Omega_D.\label{e:def-Omega}
\end{align}
One can directly check that both $\omega$ and $\Omega$ are closed. By the discussion in Section \ref{s:torus-symmetries}, we know $(\omega, \Omega)$ defines a smooth K\"ahler metric on ${\mathcal M^*}$, so that $\Omega$ is the holomorphic volume form and $\omega$ is the K\"ahler form. Also $h^{-1}$ has an intrinsic geometric meaning as the norm squared of the Killing field generating the $S^1$-action.
By \eqref{e:CY equation on D} and straightforward calculations, we have
\begin{equation}
\frac{(\sqrt{-1})^{n^2}2^{-n}\Omega\wedge\bar\Omega}{\omega^{n}/n!}=\frac{T^{-1}h\omega_D^{n-1}} {(\omega_D+T^{-1}\psi)^{n-1}}.\label{e:error}
\end{equation}
\begin{definition}
\label{d:error-function} Given the above constructed K\"ahler metric $\omega$, the error function is defined by
\begin{equation}
\mathrm{Err}_{CY}\equiv \frac{T^{-1}h\omega_D^{n-1}} {(\omega_D+T^{-1}\psi)^{n-1}}-1.
\end{equation}
\end{definition}
In particular, $\omega$ is a Calabi-Yau metric if $\mathrm{Err}_{CY}=0$.
\begin{remark} \label{r:error-function}
We now explain the reason for the various of choices of constants in the definition above.
First, the constants $T_{\pm}$ are chosen so that on the boundary $\{z=T_\pm\}$, the size of the base $D\times \{z\}$ and the size of the $S^1$-circles are comparable, both of order $O(1)$. This can be seen from \eqref{e:h-asymptotics} and \eqref{e:greens-current-exp-asymp}.
Secondly, the function $q(z)$ and the rescaling factor $T^{\frac{n-2}{n}}$ in the above definition of $\omega$ are chosen to make the K\"ahler metric $(\omega, \Omega)$ approximately Calabi-Yau in the following sense:
\begin{enumerate}
\item Applying \eqref{e:greens-current-exp-asymp} and \eqref{e:h-asymptotics}, we have $\mathrm{Err}_{CY}=T^{-2}\epsilon(z(\bm{x}))$ for $\bm{x}\in \mathcal M^*$ satisfying $|z(\bm{x})|\geq C$.
\item Applying \eqref{e:trace expansion equation} and \eqref{e:h-bounded-z}, we have $\mathrm{Err}_{CY}=O(T^{-2})$ for
$\bm{x}\in\mathcal M^*$ satisfying $|z(\bm{x})|\leq C$ and $d_Q(\bm{x}, P)\geq d_0>0$, where $d_0>0$ is some definite constant.
\end{enumerate}
Later we need a more precise weighted estimate on $\mathrm{Err}_{CY}$, which will be shown in Proposition \ref{p:CY-error-small}.
\end{remark}
\begin{remark} \label{remark4-2-2}
As explained in Section \ref{s:torus-symmetries}, a priori these structures depend on the choice of $\Theta$. Given two such $\Theta$ and $\Theta'$, the difference $\Theta'-\Theta$ is a closed 1-form on $Q_T\setminus P$. Since $P$ has codimension $3$ in $Q$, it follows that $H^1(Q_T\setminus P; \mathbb R)\simeq H^1(Q; \mathbb R)\simeq H^1(D; \mathbb R)$. Then
$
\Theta'-\Theta=df+\beta
$
for a function $f$ on $Q_T\setminus P$ and a harmonic 1-form $\beta$ on $D$. If $b_1(D)=0$ then $\beta=0$, and the isomorphism class of the K\"ahler structure $(\omega, \Omega)$ is independent of the choice of $\Theta$. If $b_1(D)>0$, up to gauge equivalence, $\Theta-\Theta'$ is the pull-back of a flat connection on $D$. In Section \ref{sss:complex manifold}, we will fix this choice of flat connection, by complex-geometric considerations, so then we have a unique choice of $\Theta$ up to gauge equivalence.
\end{remark}
\begin{remark}\label{r:bundle fixed}
Later we will study the behavior of these metrics as $T\rightarrow\infty$. It is convenient to notice that the above $U(1)$ bundle and the connection 1-form $-\sqrt{-1}\Theta$ can be defined over the entire $Q\setminus P$. Hence as $T$ varies we may view $(\omega, \Omega)$ as a family of pairs of forms on the fixed $U(1)$ bundle, but they only define a K\"ahler structure over $Q_T\setminus P$.
\end{remark}
\subsubsection{Smooth compactification}
\label{sss:smooth compactification}
Next we move on to the study the compactified geometry of ${\mathcal M^*}$ near $P$. We first construct a smooth model for the compactification and then study the regularity of the K\"ahler structure $(\omega, \Omega)$ on this model.
As before we always identify a neighborhood $\mathcal U$ of $P$ in $Q$ with a tubular neighborhood of the zero section in $N_0\oplus \mathbb R$ over $H$.
Denote by $\mathbb L_1$ and $\mathbb L_2$ the complex line bundles over $H$ given by the restriction
\begin{equation}
\mathbb L_1\equiv L^{\otimes -k_+}|_H; \ \ \mathbb L_2\equiv L^{\otimes k_-}|_H.
\end{equation}
Then as complex line bundles $N_0$ is isomorphic to $L^{\otimes k}|_H\simeq \mathbb L_1\otimes \mathbb L_2$, and we fix such an isomorphism now. Notice $N_0$ is equipped with a natural hermitian metric induced from the K\"ahler metric $\omega_D$ on $D$ (c.f. Section \ref{ss:complex-greens-currents}). This then determines a hermitian metric on $L$ hence on $\mathbb L_1$ and $\mathbb L_2$. Define
\begin{equation}\mathbb L\equiv \mathbb L_1\oplus \mathbb L_2,\end{equation}
and consider the map
\begin{equation}
\tau: \mathbb L \rightarrow N_0\oplus \mathbb R; (s_1, s_2)\mapsto (s_1\otimes s_2, \frac{|s_1|^2-|s_2|^2}{2}).
\end{equation}
Away from the zero section in $\mathbb L$, $\tau$ is a principal $S^1$-bundle, with the $S^1$-action given by
\begin{equation}\label{S1action equation}
e^{\sqrt{-1}\mathfrak{t}}\cdot (s_1, s_2)=(e^{-\sqrt{-1} t}s_1, e^{\sqrt{-1} t}s_2).
\end{equation}
As in Section \ref{ss:complex-greens-currents}, we choose local holomorphic coordinates $\{w_1, \cdots, w_{n-1}\}$ on $D$ centered at $p\in H$. These give rise to local coordinates $\{y, \bar y, w_2', \bar w_2', \cdots, w_{n-1}', \bar w_{n-1}'\}$ on $N_0$, and also a local unitary section of $N_0$ in the form $$\bm{e}\equiv|\sigma|^{-1}\cdot \sigma.$$
Then we choose a local section $\bm{e}_L$ of $L|_H$ with $\bm{e}_L^{\otimes k}=\bm{e}$. Correspondingly we get local unitary sections $$\bm{e}_1\equiv \bm{e}_L^{\otimes -k_+}, \ \ \ \ \bm{e}_2\equiv \bm{e}_L ^{\otimes k_-}$$ of $\mathbb L_1, \mathbb L_2$ respectively. Then we obtain local fiber coordinates $u_1, u_2, y$ on $\mathbb L_2, \mathbb L_1, N_0$ respectively by writing
\begin{equation}s_1=u_1\bm{e}_1, \ \ \ s_2=u_2\bm{e}_2, \ \ \ s= y\bm{e}. \end{equation}
Then the map $\tau$ can be represented in coordinates as
\begin{equation} \label{e:y definition equation}
\begin{cases}
y=u_1u_2\\
z=\frac{1}{2}(|u_1|^2-|u_2|^2)
\end{cases}\end{equation}
Hence $\tau$ is the standard Hopf fibration $\mathbb C^2\rightarrow\mathbb R^3$ when restricting to each fiber.
\begin{lemma}\label{lem4-3}
Over \ $\mathcal U\setminus P$, the principal $S^1$-bundles ${\mathcal M^*}$ and $\mathbb L$ are isomorphic.
\end{lemma}
\begin{proof}
Notice a principal $S^1$-bundle is topologically determined by its first Chern class. It suffices to compare the first Chern classes of ${\mathcal M^*}$ and $\mathbb L$ over the sphere bundle $\mathcal S_\epsilon$ for a small $\epsilon$. As in the proof of Lemma \ref{l:integrality} the Gysin sequence gives
\begin{equation}0\rightarrow H^2(P; \mathbb Z)\xrightarrow{p^*} H^2(\mathcal S_\epsilon; \mathbb Z)\xrightarrow{\int} H^0(P; \mathbb Z)\simeq \mathbb Z\rightarrow 0.\end{equation}
From the proof of Lemma \ref{l:integrality} we know
\begin{equation}
\int c_1({\mathcal M^*})=\int_{\mathcal S_\epsilon(p)} \frac{1}{2\pi}\Upsilon=-1.
\end{equation}
Also by \eqref{modelcurvature} we have
\begin{equation}
\int c_1(\mathbb L)=\int_{S^2\subset\mathbb R^3}\frac{1}{2\pi}\Upsilon_0=-1.
\end{equation}
So we have that
${\mathcal M^*}=\mathbb L\otimes p^* L'$
for some $U(1)$ bundle $L'$ over $P$. Now we restrict both ${\mathcal M^*}$ and $\mathbb L$ to the subset $H_0\subset \mathcal U$ where $y=0$ and $z=z_0$ for a fixed $z_0<0$. We can identify $H_0$ with $H$ by the projection map. Now we claim both restrictions have first Chern class equal to $k_- c_1(\mathbb L_2)$. For ${\mathcal M^*}$ this follows from construction and for $\mathbb L$ we notice that $z=z_0<0$ implies that $s_2\neq 0$ and $s_1=0$, so the projection map $(s_1, s_2)\mapsto |2z_0|^{1/2}\cdot s_2$ gives an isomorphism between the restriction of $\mathbb L$ and the unit circle bundle in $\mathbb L_2$. This also explains the choice of the weight of the $S^1$-action in \eqref{S1action equation}.
It follows from the claim that $L'$ is indeed a trivial principal $S^1$-bundle, and this finishes the proof.
\end{proof}
By Lemma \ref{lem4-3} we may glue ${\mathcal M^*}$ and $\mathbb L$ together to obtain a differentiable compactfication $\mathcal M$ of ${\mathcal M^*}$. The projection map $\pi$ naturally extends to a map
\begin{equation}\pi: \mathcal M\rightarrow Q_T\end{equation}
which is a singular $S^1$-fibration, with discriminant locus given by $P$. We identify
\begin{equation}
\mathcal P\equiv\pi^{-1}(P)
\end{equation}
with the zero section in $\mathbb L$, and identify a neighborhood of $\mathcal P$ with a neighborhood of the zero section in $\mathbb L$ and the projection map $\pi$ with the above $\tau$.
\subsubsection{Regularity of the K\"ahler structures}
\label{sss:metric compactification}
To study the regularity of the K\"ahler structure $(\omega, \Omega)$ on the compactification $\mathcal M$, we will make a special choice of the connection 1-form $-\sqrt{-1}\Theta$ on a neighborhood $\mathcal V$ of $\mathcal P$ in $\mathbb L$, with curvature form $-\sqrt{-1}\Upsilon$, which has explicit regularity behavior across $\mathcal{P}$. To do this, we need to proceed in a few steps. First, we notice that $\{u_1, \bar u_1, u_2, \bar u_2, w_2', \bar w_2', \cdots, w_{n-1}', \bar w_{n-1}'\}$ provides local coordinates on $\mathbb L$, and we can define a local model connection 1-form on $\mathbb L$ by simply taking the model formula \eqref{e:model-connection}:
\begin{equation}
\Theta_0=-\sqrt{-1}\cdot\frac{\bar u_1 du_1-u_1d\bar u_1-\bar u_2du_2+u_2d\bar u_2}{2(|u_1|^2+|u_2|^2)}.
\end{equation}
Just as in the discussion in Section 2, we see $\Theta_0(\partial_t)=-1$, where $\partial_t$ is the vector field generating the $S^1$-action. It is clear that the definition of $\Theta_0$ depends only on the choice of $\sigma$ and does not depend on the choice of $\bm{e}_1$ and $\bm{e}_2$ (which has the freedom of multiplying by a constant root of unity).
To make a globally defined connection 1-form, we need to add a correction term, and we define
\begin{equation}
\Theta_1=\Theta_0+\frac{z}{r}\Gamma-\frac{k_-+k_+}{k_--k_+}\Gamma,
\end{equation}
where $\Gamma$ is the local 1-form given in Section \ref{ss:complex-greens-currents}, and we have implicitly viewed forms on $H$ as forms on $\mathbb L$ using the pull-back $\pi^*$.
\begin{proposition}\label{p: prop4-4}
$-\sqrt{-1}\Theta_1$ is a globally-defined connection 1-form on the $S^1$-bundle $\tau: \mathbb L\setminus \mathcal P\rightarrow N\setminus H$, and we have
\begin{equation}
d\Theta_1-\tau^*\Upsilon=O'(s)
\end{equation}
where
\begin{equation}
s^2\equiv |u_1|^2+|u_2|^2=2r,
\end{equation}
and we have adopted the $O'$ notation in Section \ref{ss:geodesic-coordinates} for the submanifold $\mathcal P\subset \mathbb L$. In particular, $d\Theta_1-\tau^*\Upsilon$ extends continuously across $\mathcal P$. Moreover, it is cohomologically trivial in a neighborhood of $\mathcal P$. \end{proposition}
\begin{proof}
To see $\Theta_1$ is a well-defined, we consider the change of unitary frame $\bm{e}$ on $N_0$ to $\tilde \bm{e}=e^{\sqrt{-1} k\phi}\bm{e}$, then we have
\begin{equation}
\tilde y=e^{-(k_--k_+)\sqrt{-1}\phi}y; \ \ \tilde u_1=e^{k_+\sqrt{-1}\phi}u_1;\ \ \tilde u_2=e^{-k_-\sqrt{-1}\phi}u_2.
\end{equation}
for some local real-valued function $\phi$ on $H$. Then we get
\begin{align}
\bar u_1 du_1-u_1d\bar u_1 &=\bar {\tilde u}_1d\tilde u_1-\tilde u_1d\bar {\tilde u}_1-2k_+\sqrt{-1} |u_1|^2d\phi,
\\
\bar u_2 du_2-u_2d\bar u_2 &=\bar {\tilde u}_2d\tilde u_2-\tilde u_2d\bar {\tilde u}_2+2k_-\sqrt{-1} |u_2|^2d\phi,
\\
\Gamma&=\widetilde\Gamma-\frac{k_--k_+}{2}d\phi.
\end{align}
Then it is a straightforward to compute that $\widetilde{\Theta}_1=\Theta_1$, which shows that $\Theta_1$ is globally defined.
Now we consider the local expansion of $\Upsilon$. First, differentiating the expansion of $\psi$ in Proposition \ref{p:complex-Green-expansion},\begin{equation}
\partial_z\tilde\omega=-\sqrt{-1}\frac{z}{2r^3}dy\wedge
d\bar y-\frac{z}{2r^3} (yd\bar y+\bar ydy)\wedge\Gamma+\frac{z}{r} d\Gamma+O'(1)dy+O'(1)d\bar y+O'(r).
\end{equation}
Next, applying Proposition \ref{p:trace-expansion} and Proposition \ref{p:d^c|y|^2}, we obtain
\begin{equation}
d_D^ch=d_D^c(\frac{1}{2r}+O'(r))=-\frac{1}{4r^3} d_D^c |y|^2+O'(1)=-\frac{\sqrt{-1}(yd\bar y-\bar y dy)+4|y|^2\Gamma}{4r^3}+O'(1).
\end{equation}
Putting together these, and noting that $d\Theta_0$ is given as in \eqref{e:d Theta0}, we obtain
\begin{equation}
d\Theta_1-\tau^*\Upsilon=O'(1)dy+O'(1)d\bar y+O'(r)+O'(1)dz.
\end{equation}
Now translating into the coordinates $u_1, u_2$ on $\mathbb L$, and noticing that $\tau^*dx, \tau^*dy, \tau^*dz$ are all in $\tilde O(s)$ and $2r=s^2$, we obtain that $d\Theta_1-\tau^*\Upsilon=O'(s)$.
To see the last statement,
we can restrict to the slice with $z>0$ and $y=0$, then by Proposition \ref{p:cohomology-constant} we know $\Upsilon$ is cohomologous to $k_+\omega_D$. On the other hand, by definition $d\Theta_1$ on this slice is given by $(1-\frac{k_++k_-}{k_--k_+})d\Gamma$, which is also cohomologous to $k_+\omega_D$, using Lemma \ref{l:lemmagamma} and the fact that $2\pi\cdot c_1(N_0)=2\pi c_1(L)=(k_--k_+)[\omega_D]$.
\end{proof}
The next Lemma allows us to correct the $O'(s)$ term on the right hand side. We fix an arbitrary $S^1$-invariant Riemannian metric on $\mathbb L$.
\begin{lemma}\label{l:theta-regularity}
There exists a local 1-form $\theta$ on a neighborhood of $\mathcal P$ in $\mathbb L$ with the following properties:\begin{enumerate}
\item $\theta=O'(s^2)$,
\item $\theta$ is smooth away from $\pi^{-1}(P)$,
\item $\mathcal L_{\partial_t}\theta=0$,
\item $\partial_t\lrcorner \theta=0$,
\item $d(\Theta_1+\theta)=\tau^*\Upsilon$.
\end{enumerate}
\end{lemma}
\begin{proof}
From Proposition \ref{p: prop4-4}, the above proposition we know that in a neighborhood of $\mathcal P$, $d\Theta_1-\tau^*\Upsilon$ is in $C^{\alpha}$ for all $\alpha\in (0, 1)$ and is cohomologous to zero. The existence of a solution $\theta$ to $d(\Theta_1+\theta)=\tau^*\Upsilon$ is obtained by adding the gauge fixing condition $d^*\theta=0$, and solving the elliptic system with Neumann boundary condition
\begin{equation}
\begin{cases}
d\theta=\tau^*\Upsilon-d\Theta_1, \\
d^*\theta=0, \\
\theta(\nu)=0 \ \ \text{on} \ \ \partial\mathcal V,
\end{cases}
\end{equation}
on a tubular neighborhood $\mathcal V$ of $\mathcal P$ in $\mathbb L$.
Standard elliptic regularity guarantees a solution $\theta\in C^{1, \alpha}$ and is smooth away from $\mathcal P$. Since both $\tau^*\Upsilon$ and $\Theta_1$ are $S^1$-invariant, by averaging we may assume $\theta$ is $S^1$-invariant too, hence $\mathcal L_{\partial_t}\theta=0$ on the smooth part. Also since $\tau^*\Upsilon$ and $d\Theta_1$ are pulled-back from the base $Q_T\setminus P$, we have
$
\partial_t \lrcorner \tau^*\Upsilon=\partial_t\lrcorner d\Theta_1=0.
$
So we get
\begin{equation}
d(\partial_t\lrcorner \theta)=\mathcal L_{\partial_t}\theta-\partial_t\lrcorner(d\theta)=0.
\end{equation}
This implies $\partial_t\lrcorner \theta$ is a constant. Now as we approach $\mathcal P$, the norm of $\partial_t$, with respect to the fixed metric on $\mathbb L$, must go to zero, hence we see
$
\partial_t\lrcorner \theta=0.
$
The higher regularity of $\theta$ follows just as in the proof of Lemma \ref{l:higher-regularity}, since $\Delta\theta=d^*d\theta=d^*(\tau^*\Upsilon-d\Theta_1)=O'(1)$.
\end{proof}
Now we define a fixed 1-form on $\mathbb L$
\begin{equation}
\Theta_{m} \equiv \Theta_1 + \theta,
\end{equation}
Therefore, in a neighborhood of $\mathcal P\subset\mathbb L$ minus $\mathcal P$, the original choice of $\Theta$ can be written as
\begin{equation}
\Theta=\Theta_m+\theta_f,\label{e:fixed-connection-form}
\end{equation}
where $\theta_f$ is a flat connection, and hence it is gauge equivalent to the pull-back of a flat connection on $D$. Without loss of generality, we can then assume $\theta_f$ is smooth.
\begin{proposition}\label{p:C2alpha}With respect to the choice of the connection form $\Theta$ given in \eqref{e:fixed-connection-form},
$(\omega, \Omega)$ defined by \eqref{e:def-omega} and \eqref{e:def-Omega} gives a $C^{2, \alpha}$ (for all $\alpha\in (0, 1)$) K\"ahler structure on $\mathbb L$ which is invariant under the natural $S^1$-action and is smooth outside $\mathcal P$.
\end{proposition}
\begin{proof}
We first analyze the regularity of $\omega$, defined in \eqref{e:def-omega}.
To start with, let us compute the lifting $\pi^*\psi$. By \eqref{e:singular-2-form-expansion},
\begin{equation}\pi^*\psi=\pi^*\tilde\omega_0+\frac{1}{2r}(yd\bar y+\bar ydy)\wedge \Gamma+rd\Gamma+\pi^*(O'(r)dy+O'(r)d\bar y)+\pi^*O'(r^2),\end{equation}
where
$\tilde\omega_0=\frac{\sqrt{-1}}{4r} dy\wedge d\bar y$
is the standard form in the model setting \eqref{modelquantities}.
We also notice that
\begin{align}
\pi^*(O'(r)dy+O'(r)d\bar y)&=sO'(s^2),
\\
\pi^*O'(r^2)&=O'(s^4).
\end{align}
Now by definition
\begin{equation}
\Theta =\Theta_0+\frac{z}{r}\Gamma+\frac{k_-+k_+}{k_--k_+}\Gamma + \theta+\theta_f =\Theta_0+\frac{z}{r}\Gamma + O'(s^2).\label{e:Theta-near-p}
\end{equation}
Moreover, according to the discussions in Section \ref{s:torus-symmetries}, we have
$\pi^*\tilde\omega_0+dz\wedge \Theta_0=\omega_{\mathbb C^2},
$
where $\omega_{\mathbb C^2}=\frac{\sqrt{-1}}{2} (du_1\wedge d\bar u_1+du_2\wedge d\bar u_2)$ is the standard K\"ahler form of $\mathbb C^2$.
Therefore,
\begin{align}
\pi^*\psi + dz\wedge \Theta =& \omega_{\mathbb C^2} + rd\Gamma+
\frac{1}{2r}(yd\bar y+\bar ydy)\wedge \Gamma+dz\wedge (\frac{z}{r}\Gamma) + O'(s^3).
\end{align}
Using the relation $r^2=|y|^2+z^2$ and the simple computation
\begin{equation}
d(r\Gamma)= rd\Gamma+ dr\wedge \Gamma = rd\Gamma+\frac{1}{2r}(yd\bar y+\bar ydy)\wedge \Gamma+dz\wedge (\frac{z}{r}\Gamma),
\end{equation}
we have
\begin{align}
\pi^*\psi + dz\wedge \Theta =\omega_{\mathbb C^2}+ d(r \Gamma) + O'(s^3)
\nonumber=\omega_{\mathbb C^2}+ O'(s^3),
\end{align}
where we use the fact that $r=\frac{1}{2}s^2$ and hence $r\Gamma=s^2\Gamma$ is smooth on $\mathbb L$. Then it follows that
\begin{equation}T^{\frac{n-2}{n}}\omega =T\pi^*\omega_D+\omega_{\mathbb C^2}+O'(s^3).\end{equation}
Hence we see the $(1,1)$-form $\omega$ locally extends to a $C^{2, \alpha}$-form across the subset $\{u_1=u_2=0\}$.
Now we analyze the regularity of the holomorphic volume form $\Omega$, as defined by \eqref{e:def-Omega}.
By Lemma \ref{lem3-6}, locally we have
\begin{equation}\pi^*\Omega_D=F(u_1du_2+u_2du_1+2\sqrt{-1} u_1u_2\Gamma)\wedge \pi^*\Omega_H+\widetilde{O}(s^2)(u_1du_2+u_2du_1)+\widetilde{O} (s^3).
\end{equation}
Also
\begin{equation}
hdz+\sqrt{-1}\Theta=q(z)dz+\frac{1}{|u_1|^2+|u_2|^2}(-\bar u_2 du_2+\bar u_1 du_1+\sqrt{-1}(|u_1|^2-|u_2|^2)\Gamma)+O'(s^2).
\end{equation}
Therefore,
\begin{equation}\Omega=Fdu_1\wedge du_2\wedge \Omega_H +\sqrt{-1} F(u_2du_1-u_1du_2)\wedge \Gamma\wedge \Omega_H+\widetilde{O}(s^2)+sO'(s^2).\end{equation}
This implies that $\Omega$ also extends to a $C^{2, \alpha}$ form across $\{u_1=u_2=0\}$. This is equivalent to saying that the almost complex structure $J$ determined by $\Omega$ extends to a $C^{2, \alpha}$ almost complex structure on $\mathcal M$.
\end{proof}
\begin{remark}
\label{r:C2alpharegularity}
The reason we need the $C^{2, \alpha}$ regularity will be explained in Remark \ref{r:C1alphavsC2alpha}. If we only need weaker regularity then we only need to work out fewer terms in the expansion in Theorem \ref{t:Green-expansion}; similarly, with more work one can probably improve the regularity here, which is not needed for our purpose. But we point out that in general there is no reason to expect the above defined K\"ahler structures to be smooth. Only after we perturb these metrics to exactly Calabi-Yau metrics (see Section \ref{s:neck-perturbation} and \ref{s:gluing}) we can gain smoothness via elliptic regularity. This is in sharp contrast to the 2 dimensional case, where the Gibbons-Hawking ansatz yields an exact solution of the Calabi-Yau equation and simultaneously the metric completion near the (isolated) $S^1$-fixed points is smooth.
\end{remark}
Using the Newlander-Nirenberg theorem , we may locally find $C^{3, \alpha}$ holomorphic coordinates, making the complex structure locally standard while still keeping the K\"ahler form with $C^{2,\alpha}$ regularity .
By construction the K\"ahler structure $(\omega, \Omega)$ is preserved by the natural $S^1$-action. The corresponding Killing field is given by \begin{equation}\partial_t\equiv-\sqrt{-1}(u_1\partial_{u_1}-u_2\partial_{u_2})+\sqrt{-1}(\bar u_1\partial_{\bar u_1}-\bar u_2\partial_{\bar u_2}).
\end{equation}
The zero set $\mathcal P$
is a complex submanifold of $\mathcal M$ which bi-holomorphic to $H\subset D$.
We denote the corresponding holomorphic vector field by \begin{equation} \label{xi 10}
\xi^{1,0}\equiv\frac{1}{2}(\partial_t-\sqrt{-1} J\partial_t).
\end{equation}
We also have a smooth holomorphic projection $\pi: \mathcal M\rightarrow D\setminus H$ whose fibers are holomorphic cylinders (isomorphic to annuli in $\mathbb C$).
In the next subsection we will understand the underlying complex manifold and the K\"ahler potentials on $\mathcal M$.
\subsection{K\"ahler geometry}
\label{ss:complex-geometry}
A key feature in the analysis on K\"ahler manifolds is that we can describe the geometry locally in terms of a single \emph{potential function}. This has led to a vast simplification of formulae in K\"ahler geometry as compared to more general Riemannian geometric setting, and it also has allowed various techniques from PDE and several complex variables etc. to be exploited.
The goal of this subsection is to derive a formula for the relative K\"ahler potential for our K\"ahler manifold $(\mathcal M, \omega, \Omega)$. This is one of the most crucial observations in this paper.
In Section \ref{sss:complex manifold} we identify the underlying complex manifold of the family of K\"ahler metrics constructed in Section \ref{ss:kaehler-structures} as a family of open subsets of a fixed complex manifold. In Section \ref{sss:kahler potential} we derive a formula for the relative K\"ahler potential.
\subsubsection{The underlying complex manifold}
\label{sss:complex manifold}
We define the following holomorphic line bundles on $D$
\begin{equation}
\begin{cases}
L_+\equiv L^{-\otimes k_+}, \\
L_-\equiv L^{\otimes k_-},
\end{cases}
\end{equation}
and we denote by $J_\pm$ the complex structure on the total space of $L_\pm$.
Denote by $\mathcal N^0$ the hypersurface in the total space of $L_+\oplus L_-$ defined by the equation
\begin{equation}\label{e: SH definition}\zeta_+\otimes \zeta_-=S_H(x),\end{equation}
where $\zeta_\pm$ denotes points on the fibers of $L_\pm$ over $x\in D$, and $S_H$ is the section we fixed at the beginning of this section. Since $H$ is smooth, $\mathcal N^0$ is also smooth and the submanifold
\begin{equation}
\mathcal H\equiv\{\zeta_+=\zeta_-=0\}
\end{equation}
of $\mathcal N^0$
is naturally isomorphic to $H$.
The fixed hermitian metric on $L$ then induces hermitian metrics on $L_\pm$, and these yield norm functions on $L_\pm$:
\begin{equation}
r_\pm(\zeta_\pm)\equiv\|\zeta_\pm\|.
\end{equation}
Then by pulling back to $\mathcal N^0$ through the projection maps to $L_\pm$ we may also view $r_\pm$ as functions on $\mathcal N^0$.
There is a natural holomorphic volume form on $\mathcal N^0$ given by
\begin{equation}\label{e:Omega N0 definition}
\Omega_{\mathcal N^0}\equiv\frac{\sqrt{-1}}{2} (\frac{d\zeta_+}{\zeta_+}-\frac{d\zeta_-}{\zeta_-})\wedge \Omega_D,
\end{equation}
where $\Omega_D$ means the pull-back of $\Omega_D$ to $\mathcal N^0$ and for simplicity of notation we omit the pull-back notation when the meaning is clear from the context. The expression on the right hand side of \eqref{e:Omega N0 definition} should be understood after choosing a local holomorphic frame $\sigma$ of $L$, so that $\zeta_\pm$ becomes local holomorphic functions on $L_\pm$. One can check the definition does not depend on the choice of $\sigma$, and $\Omega_{\mathcal N^0}$ is a well-defined holomorphic volume form on $\mathcal N^0$.
There is a natural $\mathbb C^*$ action on $\mathcal N^0$ given by
\begin{equation}
\lambda\cdot (\zeta_+, \zeta_-)\equiv(\lambda^{-1}\zeta_+, \lambda \zeta_-), \ \ \lambda\in \mathbb C^*.
\end{equation}
and we denote by
\begin{equation}
\xi_{\mathcal N^0}\equiv\sqrt{-1}(-\zeta_+\partial_{\zeta_+}+\zeta_-\partial_{\zeta_-})
\end{equation}
the corresponding holomorphic vector field (the choice of coefficients is made so that the real part of $\xi_{\mathcal N^0}$ is twice the real vector field generated by the induced $S^1$-action, as given in \eqref{xi 10}).
One checks that $
\xi_{\mathcal N^0}\lrcorner\ \Omega_{\mathcal N^0}=\Omega_D
$ and $\mathcal L_{\xi_{\mathcal N^0}}\Omega_{\mathcal N^0}=0$.
Our main goal is to holomorphically embed $(\mathcal M, \Omega, \xi^{1,0})$ into $(\mathcal N^0, \Omega_{\mathcal N^0}, \xi_{\mathcal N^0})$, for appropriate choice of $\Theta$ (which is necessary in the case $b_1(D)>0$, see Remark \ref{remark4-2-2}). This will be summarized in Proposition \ref{p:complex geometry}. Notice for our later purpose the quantitative choice of various constants in the arguments below will be important.
First let us fix an arbitrary choice of $\Theta$. Denote
\begin{equation}
\begin{cases}
\mathcal M_-\equiv {\mathcal M^*}\setminus \pi^{-1}(H\times [0, \infty))\\
\mathcal M_+\equiv {\mathcal M^*}\setminus \pi^{-1}(H\times (\infty, 0]).
\end{cases}
\end{equation}
On $\mathcal M_-$ we can trivialize the $U(1)$ connection $-\sqrt{-1}\Theta$ along the $z$ direction so that the $z$ component $\Theta_z$ vanishes identically. Denote by $\Theta|_z$ the restriction of $\Theta$ to the slice $D\times \{z\}$ for $z<0$ and to $(D\setminus H)\times \{z\}$ for $z\geq 0$. From \eqref{d Theta equation} we see that that curvature form of $-\sqrt{-1}\Theta|_z$ is given by $-\sqrt{-1}\partial_z\tilde\omega$.
By definition $\mathcal M|_{z=T_-}$ is a principal $S^1$-bundle over $D$ endowed with a unitary connection with curvature $-\sqrt{-1}\partial_{z}\tilde\omega|_{z=T_-}$. So we may assume $\mathcal M|_{z=T_-}$ embeds into a holomorphic line bundle $\tilde L_-$ over $D$, as the unit circle bundle defined by a hermitian metric $\|\cdot\|_{\sim}^2$, and the connection 1-form $-\sqrt{-1} \Theta|_{T_-}$ agrees with the restriction of the Chern connection form. Denote by $\tilde r_-$ the norm function on $\tilde L_-$ corresponding to the hermitian metric. By Proposition \ref{p:cohomology-constant} we have $[\partial_z\tilde\omega]|_{z=T_-}=k_-[\omega_D]\in H^2(D; \mathbb R).
$
If $b_1(D)=0$, then $\tilde L_-$ is isomorphic to $L_-$ as holomorphic line bundles. In general $\tilde L_-$ is isomorphic to $L_-\otimes \mathcal F_-$ for a flat holomorphic line bundle $\mathcal F_-$.
Furthermore, we may extend $-\sqrt{-1} \Theta|_{T_-}$ naturally to the complement of the zero section ${\bf 0}_{\tilde L_-}$ in $\tilde L_-$, via the fiberwise projection, and the resulting 1-form coincides with the Chern connection form
$\sqrt{-1}\tilde {r}_-^{-1}\tilde J_-d\tilde r_-$, where $\tilde J_-$ denotes the complex structure on $\tilde L_-$.
Now we define a map $\Phi_-: {\mathcal M}_-\rightarrow \tilde L_-\setminus {\bf 0}_{\tilde L_-}$, where ${\bf 0}_{\tilde L_-}$ denotes the zero section in $\tilde L_-$. First at $z=T_-$ we define $\Phi_-$ to be the natural inclusion map as above, multiplied by $e^{A_-}$ for some constant $A_-$ to be determined later. Then using the trivialization of the $S^1$-bundle ${\mathcal M}_-$ along the $z$ direction and the natural scaling map on $\tilde L_-$, we extend the map to the whole ${\mathcal M}_-$ by setting
\begin{equation}
\label{e:tilde r}\tilde r_-=e^{A_--\int_{T_-}^z h(u) du}.
\end{equation}
Notice both $\mathcal M_-$ and $\tilde L_-$ have natural projection maps to $D$, and $\Phi_-$ clearly commutes with the projection maps to $D$, so $\Phi_-^*\alpha=\alpha$ for any $1$-form $\alpha$ which is a pull-back from $D$ (again we omit the pull-back notation here). Since
\begin{equation}
\partial_z\Theta|_z=d_D^ch=-J_D d_Dh
\end{equation}
we have
\begin{equation}\tilde r_-^{-1}\Phi_-^*d\tilde{r}_-=-hdz-\int_{T_-}^z du\wedge d_Dh=-hdz-J_D(\Theta|_{z}-\Theta|_{T_-}),
\end{equation}
noticing that $\Theta|_z-\Theta|_{T_-}$ is a 1-form which is a pull-back from $D$.
So
\begin{equation}\tilde{r}_-^{-1}\Phi_-^*(d\tilde{r}_-+\sqrt{-1} \tilde J_-d\tilde{r}_-)=-hdz-\sqrt{-1} \Theta|_z-\sqrt{-1}(\Theta|_{T_-}-\Theta|_z)-J_D(\Theta|_{z}-\Theta_{T_-})
\end{equation}
is a $(1, 0)$ form on ${\mathcal M}_-$. It then follows that $\Phi_-$ is holomorphic. It is also clear that $\Phi_-$ is $S^1$-equivariant with respect to the natural $S^1$-action on $\mathcal M_-$ and $\tilde L_-$.
Since $h$ is positive we see that the image of $\Phi_-$ is bounded in $L_-$. Since $\mathcal M\setminus \mathcal M_-$ is of complex codimension one, by the removable singularity theorem for bounded holomorphic functions, $\Phi_-$ extends to a holomorphic map on the entire $\mathcal M$.
Similarly we get a holomorphic embedding
$
\Phi_+: \mathcal M_+\rightarrow \tilde L_+
$
with
\begin{equation}
\label{tilde r+}
\tilde r_+=e^{A_+-\int_{z}^{T_+}h(u)du}
\end{equation}
for a constant $A_+$ to be determined. Here $\tilde L_+$ is the hermitian holomorphic line bundle determined by $\sqrt{-1}\Theta|_{T_+}$, and we have $\tilde L_+=L_+\otimes \mathcal F_+$ for a flat holomorphic line bundle $\mathcal F_+$. Notice there is a sign difference here due to the fact that $L_+=L^{-\otimes k_+}$. Again $\Phi_+$ is equivariant with respect to the natural $S^1$-action on $\mathcal M_+$ and the inverse of the natural $S^1$-action on $\tilde L_+$ (due to the sign difference above), and it extends to a holomorphic map on $\mathcal M$.
Together we obtain
\begin{equation}\Phi\equiv(\Phi_+, \Phi_-): \mathcal M\rightarrow \tilde L_+\oplus \tilde L_-,
\end{equation}
which is an embedding on $\mathcal M\setminus \mathcal P$. It commutes with projections maps to $D$.
Now we claim $\tilde L_+\otimes \tilde L_-$ is isomorphic to $L^{\otimes k}$ as holomorphic line bundles. First notice that by $S^1$-equivariancy, we know the map
\begin{equation}
\det\Phi\equiv \Phi_+\otimes \Phi_-: \mathcal M\setminus (H\times (-\infty, \infty)) \rightarrow \tilde L_+\otimes \tilde L_-
\end{equation}
has image lying on a holomorphic section, say $\tilde S$, of $\tilde L_+\otimes \tilde L_-$ over $D\setminus H$. Since $h$ is smooth away from $P=H\times \{0\}$, $\tilde S$ is nowhere zero on $D\setminus H$.
Again since $h$ is positive we know the image of $\Phi$ is bounded in $\tilde L_+\oplus \tilde L_-$, with respect to the norm defined by $\tilde r_\pm$, so $\tilde S$ is a bounded section with respect to the norm $\tilde r\equiv \tilde r_+\otimes \tilde r_-$. Thus again by removable singularity theorem for bounded holomorphic functions it extends to a holomorphic section on the entire $D$.
By our assumption that $[H]$ is Poincar\'e dual to $c_1(L)=c_1(\tilde L_+\otimes \tilde L_-)$, we see $H$ is exactly the zero locus of $\tilde S$ with multiplicity 1. So $\tilde L_+\otimes \tilde L_-$ is isomorphic to the holomorphic line bundle defined by the divisor $H$, which is exactly $L^{\otimes k}$. This proves the claim.
Now modify the choice of $\Theta$ so that $\tilde L_{\pm}$ is isomorphic to $L_{\pm}$ as holomorphic line bundles. By the correspondence between gauge equivalence classes of flat $S^1$-connections on $D$ and isomorphism classes of flat holomorphic line bundles on $D$ mentioned in Section \ref{ss:Calabi model space}, we can add a flat $S^1$-connection to $\Theta$, and make $\tilde L_-$ isomorphic to $L_-$ as holomorphic line bundle. The above claim then implies $\tilde L_+$ is also isomorphic to $L_+$ as holomorphic line bundles. So from now on we will simply identify $\tilde L_\pm$ with $L_\pm$.
By \eqref{e:greens-current-exp-asymp}, we have
$\partial_z\tilde\omega|_{z=T_\pm}=k_\pm\omega_D+\epsilon_T,
$
so we may assume that under this identification $
\log \tilde r_\pm=\log r_\pm+\epsilon_T$. From the above we also know there is a nonzero constant $C$ such that
\begin{equation}
\tilde S=C\cdot S_H,
\end{equation}
where $S_H$ is the section of $L^{\otimes k}$ we fixed at the beginning of this section.
Multiplying $\Phi_-$ by an element in $S^1$-we may assume $C$ is a positive real number.
Notice by definition locally once we choose a holomorphic trivialization $\sigma$ of $L_-$, we can write
$
\tilde r_-^2=|\zeta_-|^2 \cdot \|\sigma\|_{\sim}^2.
$
So
\begin{equation}
\frac{d\zeta_-}{\zeta_-}=\frac{d\tilde r_-}{\tilde r_-}+\sqrt{-1} J_-\frac{d\tilde r_-}{\tilde r_-}+\partial_D \log |\sigma|^2.
\end{equation}
Therefore we obtain
$\Phi_-^*\Omega_{L_-}=\Omega,
$
where
\begin{equation}\Omega_{L_-}\equiv -\sqrt{-1}\frac{d\zeta_-}{\zeta_-}\wedge \Omega_D
\end{equation}
is a natural holomorphic volume form on $L_-\setminus {\bf 0}_{L_-}$. In particular $\Phi_-$ is a holomorphic embedding. Also, we have
\begin{equation}
d\Phi_-(\xi^{1,0})=\sqrt{-1} \zeta_-\partial_{\zeta_-}
\end{equation}
is the natural holomorphic vector field on $L_-$.
So similarly we obtain that
\begin{equation} \label{e:d Phi xi}
d\Phi(\xi^{1,0})=\sqrt{-1}(\zeta_-\partial_{\zeta_-}-\zeta_+\partial_{\zeta_+}).
\end{equation}
Now we show that with appropriate choice of $A_\pm$, $\Phi$ maps $\mathcal M$ into $\mathcal N^0$.
Notice
\begin{equation}\log C=\frac{1}{\int_D \omega_D^{n-1}} \int_D \log \|{\tilde S}\| \omega_D^{n-1}-\frac{1}{\int_D \omega_D^{n-1}} \int_D \log \|S_H\|\omega_D^{n-1}, \end{equation}
where $\|\cdot\|$ denotes the norm on $L^k$ determined by the fixed metric on $L$.
The second term is a constant independent of $T$. For the first term, by definition we have
\begin{equation}-\log \|\tilde S\|=\int_{T_-}^{T_+}hdz-(A_-+A_+)+\epsilon_T.
\end{equation}
By \eqref{haverage} and Proposition \ref{p:cohomology-constant},
\begin{eqnarray}\int_{T_-}^{T_+} \int_D h \omega_D^{n-1}&=&\int_{T_-}^{T_+} T^{2-n} \int_D (T\omega_D+\psi)^{n-1}dz+T^{-1}\underline B_T\\ &=&\frac{1}{n}\int_D \omega_D^{n-1}(\frac{1}{k_-}-\frac{1}{k_+}) (T^2-1)+T^{-1}\underline B_T
\end{eqnarray}
So we get that
\begin{equation}
-\log C=\frac{1}{n}(\frac{1}{k_-}-\frac{1}{k_+})(T^2-1)-(A_-+A_+)+\frac{1}{\int_D \omega_D^{n-1}} \int_D \log \|S_H\|+T^{-1}\underline B_T
\end{equation}
Setting $C=1$ gives one condition on $A_-$ and $A_+$, but this does not determine $A_-$ and $A_+$. This corresponds to the fact that there is a $\mathbb C^*$ action on $\mathcal N^0$. For our later purposes (c.f. Remark \ref{r:Aplusminus}), we need an additional balancing condition
\begin{equation}\label{balancing condition}
k_-A_-=k_+A_+.
\end{equation}
Together these determine $A_-$ and $A_+$ as
\begin{align}
A_- &\equiv \frac{1}{nk_-}(T^2-1)-\frac{-k_+}{2(k_--k_+)}\frac{1}{\int_D \omega_D^{n-1}} \int_D \log \|S_H\|+T^{-1}\underline B_T,
\\
A_+ & \equiv \frac{1}{-nk_+}(T^2-1)-\frac{k_-}{2(k_--k_+)}\frac{1}{\int_D \omega_D^{n-1}} \int_D \log \|S_H\|+T^{-1}\underline B_T.
\end{align}
So we obtain the following
\begin{proposition}\label{p:complex geometry}
With the above choice of $\Theta$, we have a holomorphic embedding $\Phi: (\mathcal M, \Omega)\rightarrow \mathcal N^0$ as a relatively compact open subset containing $\mathcal H$, such that the following holds
\begin{enumerate}
\item $\Phi$ commutes with the projection maps to $D$;
\item $\Phi^*\Omega_{\mathcal N^0}=\Omega$;
\item $d\Phi(\xi^{1,0})=\xi_{\mathcal N^0}$. In particular, $\Phi$ maps $\mathcal P$ bi-holomorphically onto $\mathcal H$.
\end{enumerate}
\end{proposition}
For our purpose later, we list a few more results here. First we compare the function $z$ with the norm $r_-$ and $r_+$ near each end.
Given $C>0$ fixed, then by \eqref{e:h-asymptotics} we have
\begin{equation}\label{e: compare r and z}
\begin{cases}
-\log r_-=\frac{1}{nk_-}T^{2-n}(T+k_-z)^{n}-A_-+\epsilon_T+\epsilon(z), \ \ z\leq -C; \\
-\log r_+=-\frac{1}{nk_+}T^{2-n}(T+k_+ z)^{n}-A_++\epsilon_T+\epsilon(z), \ \ z\geq C,
\end{cases}
\end{equation}
by noticing that for example
\begin{equation}
\int_{T_-}^z\epsilon(z)dz=\epsilon_T+\epsilon(z), z\leq -C.
\end{equation}
So we have
\begin{equation} \label{e: compare z and r}
\begin{cases}
(T+k_-z)^n=T^{n-2}nk_-(A_--\log r_-+\epsilon_T+\epsilon(z)), \ \ z\leq -C;\\
(T+k_+z)^n=-T^{n-2}nk_+(A_+-\log r_-+\epsilon_T+\epsilon(z)), \ \ z\geq C.
\end{cases}
\end{equation}
Next we give a description of the behavior of the metric $\omega$ when we restrict to the region $|z|\geq 1$. From the asymptotics of $\tilde\omega$ and $h$ we know the metric is asymptotic to the ends of the Calabi model space in Section \ref{ss:Calabi model space}. Locally on $D$ we fix holomorphic coordinates $\{w_1, \cdots, w_{n-1}\}$ and choose a holomorphic trivialization of $L$ as before, then we obtain fiber holomorphic coordinates $\zeta_\pm$ on $L_\pm$. Denote
\begin{equation}
\omega_{\pm, cyl}\equiv \sum_{i\geq 1} \sqrt{-1} dw_i\wedge d\bar w_i+\frac{\sqrt{-1} d\zeta_\pm\wedge d\bar\zeta_\pm}{|\zeta_\pm|^2}
\end{equation}
the local cylindrical type metrics on $L_\pm$ respectively. Then we have
\begin{lemma}\label{l:neck cylindrical compare}
On $|z|\geq 1$, we have
\begin{equation}
C^{-1}T^{\frac{(n-2)(1-n)}{n}}(T+k_\pm z)^{1-n} \omega_{\pm, cyl}\leq \omega\leq C T^{\frac{2-n}{n}}(T+k_\pm z)\cdot \omega_{\pm, cyl}.
\end{equation}
Furthermore, for all $k\geq 1$, there exists $m_k, C_k$ such that
\begin{equation}
|\nabla^k_{\omega_{\pm, cyl}}\omega|_{\omega_{\pm, cyl}}\leq C_k (T^{\frac{2-n}{n}}(T+k_\pm z))^{m_k}.
\end{equation}
\end{lemma}
\begin{proof}
We only consider the case $z\leq -1$. Since
\begin{equation}
\frac{d\zeta_-}{\zeta_-}=\frac{dr_-}{r_-}+\sqrt{-1} J\frac{dr_-}{r_-}=-hdz+\epsilon_T-\sqrt{-1} J hdz
\end{equation}
Then we can estimate the coefficient of the metric $\omega$ in the frame given by $dw_1, \cdots dw_{n-1}$ and $\frac{d\zeta_-}{\zeta_-}$, using the asymptotics of $h$ \eqref{e:h-asymptotics} and $\tilde\omega$
\eqref{e:greens-current-exp-asymp}. From this the conclusion follows.
\end{proof}
We also need to understand the the level set $r_\pm=C$ under the projection to $D\times \mathbb R$, for a fixed $C>0$ and for $T$ large. First we have
\begin{proposition} \label{l:zeta formula}
We have
\begin{equation}
A_--\int_{T_-}^0 h(u)du=\frac{1}{2}\log {\|S_H\|}+B_T,
\label{e:asympotics-A-}
\end{equation}
\begin{equation}
A_+-\int_{0}^{T_+} h(u)du=\frac{1}{2}\log {\|S_H\|}+B_T.\label{e:asympotics-A+}
\end{equation}
\end{proposition}
\begin{proof}
We denote
\begin{equation}
\hat h_-=A_--\int_{T_-}^0 h(u)du-\frac{1}{2}\log \|S_H\|.
\end{equation}
By the Poincar\'e-Lelong equation we have
\begin{equation}
d_Dd_D^c \log {\|S_H\|}^2=4\pi \delta_H-(k_--k_+)\omega_D,
\end{equation}
where $\delta_H$ denotes the current of integration along $H$.
By directly taking derivatives and use \eqref{eqn4445} we obtain that outside $H$,
\begin{equation}
d_Dd_D^c (\int_{T_-}^{0}h(z)dz)=\int_{T_-}^{0}d_Dd_D^c h(z)dz=-\int_{T_-}^{0} \partial_z^2\tilde\omega(z)dz=\partial_z\tilde\omega|_{z=T_-}-\partial_z\tilde\omega|_{z=0}
\end{equation}
By \eqref{e:greens-current-exp-asymp} and \eqref{eqn2-43}, the right hand side is given by $\frac{1}{2}(k_--k_+)\omega_D+\epsilon_T$. Now using the asymptotics of $h$ near $P$ in \eqref{e:h-bounded-z}, one sees that $\hat h_-$ is bounded near $H$. So globally as currents on $D$, we have $
d_Dd_D^c\hat h_-=\epsilon_T.
$ Now
\begin{equation}
\int_{D} \hat h_- \omega_D^{n-1}=A_-\int_D \omega_D^{n-1}-\int_{D}\int_{T_-}^{0}h\omega_D^{n-1} dz=B_T
\end{equation}
By standard elliptic regularity we get the conclusion for $\hat h_-$. The proof of \eqref{e:asympotics-A+} is the same.
\end{proof}
\begin{remark}\label{r:Aplusminus}
This proposition explains the reason for choosing the constants $A_\pm$ to satisfy the balancing condition \eqref{balancing condition}: this makes the image under $\Phi$ of the slice $\{z=0\}$ lie in a bounded region in $\mathcal N^0$, which is not distorted as $T\rightarrow\infty$.
\end{remark}
For $|z|\leq 1$, since $h(z)=T+\frac{1}{2r}+O'(r)+O(T^{-1})$, we easily see that in a fixed distance (with respect to $\omega_D$) away from $H$, $r_\pm \leq C$ is equivalent to $ B_T\cdot T^{-1}\mp z\geq 0$. We need a refinement of this. Fix a normal coordinate chart $(y, \bar y, w_2', \cdots, \bar w_{n-1}')$ on $D$ as given in Section \ref{ss:complex-greens-currents}. Recall we have locally $r^2=|y|^2+z^2$.
\begin{proposition}
\label{p:z r- relation}
If $|z|\leq 1$, then we have
\begin{align}
\log r_-&=-Tz+\frac{1}{2}\log (r-z)+ B_T,
\\
\log r_+&=Tz+\frac{1}{2}\log (r+z)+ B_T.
\end{align}
\end{proposition}
\begin{proof}
By the previous proposition,
\begin{equation}
A_--\int_{T_-}^zh(u)=\frac{1}{2}\log {\|S_H\|}+B_T+\int_{z}^0 h(u)du.
\end{equation}
When $|z|\leq 1$ we have by \eqref{e:h-bounded-z} that
\begin{equation}
\int_{0}^z h(u)du=B_T+Tz+\frac{1}{2}(\log(r+z)-\log |y|).
\end{equation}
Comparing \eqref{e:y definition equation} and \eqref{e: SH definition} we see $\log {\|S_H\|}=\log |y|+B_T$ , it follows that
\begin{equation}
\log r_-=B_T-Tz+\frac{1}{2}\log (r-z).
\end{equation}
Similarly we get the estimate for $
\log r_+$.
\end{proof}
The following corollary will be used frequently later.
\begin{corollary}\label{c:r zeta relation}
The following hold:
\begin{enumerate}
\item Let $C>0$ be fixed. Then for $T$ large, $r_\pm\leq C$ implies $\frac{3}{4}T^{-1}\log T\mp z\geq0$.
\item Let $c\in (0, 1/2)$ be fixed. Then for $T$ large $r \leq cT^{-1}\log T$ implies
$\log r_\pm\leq -\frac{1}{2}(\frac{1}{2}-c) \log T.$
\item Let $C\geq 1$ be fixed. Then for $T$ large, $C\pm z\geq 0$ implies $\log r_\mp\leq (C+1)T$.
\end{enumerate}
\end{corollary}
\begin{proof}
The first two items are easy consequences of the previous proposition. For the last item to see the bound on $r_-$ we simply notice that for $C\geq 1$,
\begin{equation}
\int_{-C}^{-1} h(u)du=\int_{-C}^{-1} (T^{2-n}(T+k_-u)^{n-1}+\epsilon(u))du\leq CT.
\end{equation}
The bound for $r_+$ can be proved similarly.
\end{proof}
\subsubsection{K\"ahler potentials}\label{sss:kahler potential}We look for an $S^1$-invariant function $\phi$ on $\mathcal M$ satisfying the equation
\begin{equation} \label{eqn7-1}
T\pi^*\omega_{D}+dd^c\phi=T^{\frac{n-2}{n}}\omega
\end{equation}
We write
$d\phi=d_D\phi+\phi_zdz$
where as before $d_D\phi$ is the differential along $D$ direction and $\phi_z=\partial_z\phi$ is the derivative along $z$ direction. Then
$d^c\phi=d^c_D\phi+\phi_zh^{-1} \Theta, $
and
\begin{equation}dd^c\phi=d_Dd_D^c\phi+dz\wedge (d^c_D\phi_z)+d(\phi_z h^{-1})\wedge\Theta+\phi_z h^{-1} (\partial_z\tilde\omega-dz\wedge d_D^ch)\end{equation}
Since
$T^{\frac{n-2}{n}}\omega=\pi^* \tilde\omega+dz\wedge \Theta,
$
we see (\ref{eqn7-1}) is equivalent to the system of equations
\begin{equation}\label{eqn7-9}
\begin{cases}
\tilde\omega=T\omega_D+d_Dd^c_D\phi+\phi_zh^{-1} \partial_z\tilde\omega\\
d_D^c\phi_z-\phi_zh^{-1} d_D^ch=0\\
d(\phi_z h^{-1})=dz.
\end{cases}
\end{equation}
To solve these equations, we first notice that the last equation in (\ref{eqn7-9}) is equivalent to
\begin{equation}\phi_zh^{-1}=z+C\end{equation}
for a constant $C$.
So we obtain \footnote{In the case when $n=2$ for the classical Gibbons-Hawking ansatz this formula was derived by the authors together with Hans-Joachim Hein in the office of the first author at Stony Brook in the Fall of 2017. }
\begin{equation}\label{eqn7-2}
\phi(z)=\int_{z_0}^z (u+C)h du+\phi(z_0)
\end{equation}
for a function $\phi(z_0)$ on $D$.
The second equation of (\ref{eqn7-9}) then holds automatically, and if we take $\partial_z$ on the first equation then it also holds. So in order for $\phi$ defined in (\ref{eqn7-2}) to satisfy (\ref{eqn7-9}), it suffices that at a fixed $z=T_+$ the following holds
\begin{equation}T\omega_D+d_Dd^c_D \phi= \tilde\omega-(T_++C)\partial_z\tilde\omega.\end{equation}
Comparing the cohomology classes of both sides yields that $C$ must be zero.
Then we can solve $\phi(T_+)$ uniquely up to addition of a constant. After fixing a choice of $\phi(T_+)$ we may define $\phi$ by
\begin{equation} \label{eqn7-10}
\phi(z)=\int_{T_+}^z uh du+\phi(T_+),
\end{equation}
and we can view it as either a function on $Q_T$ or an $S^1$-invariant function on $\mathcal M$.
\begin{proposition}
The function $\phi$ is smooth on ${\mathcal M^*}$, and $C^{3, \alpha}$ on $\mathcal M$ with respect to the smooth topology as defined in Section \ref{ss:kaehler-structures}, and satisfies (\ref{eqn7-1}).
\end{proposition}
\begin{proof}
Since $h$ is smooth on $Q_T\setminus H\times (-\infty, 0]$, so is $\phi$. Using \eqref{e:h-bounded-z} it is easy to see that $\phi$ extends continuously on the whole $Q_T$. Hence for all fixed $z$, the following equation holds as currents on $D$
\begin{equation}
\label{eqn4149}T\omega_D+d_Dd_D^c\phi(z)=\tilde\omega(z)-z\partial_z\tilde\omega(z).
\end{equation}
Elliptic regularity implies that $\phi$ is smooth on each slice $\{z\}\times D$ for $z\neq 0$. Now for $z\leq 0$ we can write
\begin{equation}\phi(z)=\int_{T_-}^{z} uh du+\phi(T_-).
\end{equation}
We then see that $\phi$ is indeed smooth on $ Q_T\setminus P$. Over the $S^1$-fibration $\mathcal M$, we know $\phi$ is globally continuous, and it is smooth and satisfies the equation (\ref{eqn7-1}) on ${\mathcal M^*}$. Since $\mathcal P=\mathcal M\setminus \mathcal M^*$ is a complex submanifold, and $\phi$ is continuous across $\mathcal P$, by standard theory on extension of pluri-subharmonic functions we conclude the current equation \eqref{eqn4149} holds globally on $\mathcal M$. Since $\omega$ is $C^{2, \alpha}$ in local holomorphic coordinates on $\mathcal M$, elliptic regularity gives that $\phi$ is $C^{4, \alpha}$ in local holomorphic coordinates. This implies that $\phi$ is $C^{3, \alpha}$ in the smooth topology we defined, since the holomorphic coordinate functions are $C^{3,\alpha}$.
\end{proof}
\begin{remark}
\label{r:Calabi model potential}As a by-product we can also recover the formula of the Calabi model metric in terms of K\"ahler potentials as mentioned in Section \ref{ss:Calabi model space}. In this case as in \eqref{e:Calabi model solution} we take $\tilde\omega= z\omega_D$ and $h=z^{n-1}$. Then we can write
$\tilde\omega=dd^c\phi$
with
\begin{equation}\phi=\int_0^z u^{n} du=\frac{1}{n+1}z^{n+1}\end{equation}
To match with the formula for Calabi ansatz in \eqref{calabiansatz}, we notice that $z^{n+1}=(-\log |\xi|)^2$, and there is a factor of $\frac{n}{2}$ due to the normalization of the Calabi-Yau equation and that $dd^c=2\sqrt{-1} \partial\bar{\partial}$.
\end{remark}
\begin{remark}
\label{r:TaubNUT potential}
Notice the argument above does not essentially require the compactness of $D$, except to solve the equation \eqref{eqn4149} on one slice. Using similar idea one can get the expression of the Taub-NUT metric on $\mathbb C^2$ in terms of K\"ahler potentials, as mentioned in Section \ref{ss:2d standard model}. Here we take $D$ to be $\mathbb C$ with the standard flat structure, and
$
\tilde\omega(z)=\frac{\sqrt{-1}}{2} Vdy\wedge d\bar y$, $h=V$,
with $
V=\frac{1}{2r}+T.$
Suppose we want to find $\phi$ with
$
\omega=dd^c\phi,
$
then by \eqref{eqn7-2} we have
$
\phi(z)-\phi(0)=\frac{1}{2}r-\frac{1}{2}|y|+\frac{T}{2}z^2.
$
The equation \eqref{eqn4149} for $z=0$ becomes
$
4\partial_y\partial_{\bar y}\phi(0)=\tilde\omega(0)=\frac{1}{2|y|}+T,
$
and a solution is given by
$
\phi(0)=\frac{1}{2}|y|+\frac{T}{4}|y|^2.
$
So we get
\begin{equation}
\phi=\frac{1}{2}r+\frac{T}{2}z^2+\frac{T}{4}|y|^2=\frac{1}{4}(|u_1|^2+|u_2|^2)+\frac{T}{8}(|u_1|^4+|u_2|^4).
\end{equation}
This agrees with formula \eqref{e:TaubNUT potential} up to a constant $2$, again caused by the fact that $dd^c=2\sqrt{-1} \partial\bar{\partial}$.
\end{remark}
For our purpose later we need to express $\omega$ as $dd^c$ of an explicit function on the two ends $z\rightarrow\pm\infty$.
\begin{proposition}
When $z\leq -C$, we have
\begin{equation}
T^{\frac{n-2}{n}}\omega=T^{\frac{n-2}{n}}dd^c\phi_-,
\end{equation}
with
\begin{equation} \label{neck potential negative side}
\phi_-\equiv \frac{1}{n+1}n^\frac{n+1}{n}k_-^{-\frac{n-1}{n}} (A_-+\epsilon_T+\epsilon(z)-\log r_-)^{\frac{n+1}{n}}-T^{\frac{2}{n}}k_-^{-1} A_-+\epsilon_T+T^{\frac{2}{n}}\epsilon(z).
\end{equation}
Similarly for $z\geq C$, we have
\begin{equation}
T^{\frac{n-2}{n}}\omega=T^{\frac{n-2}{n}}dd^c\phi_+,\end{equation}
with
\begin{equation} \label{neck potential positive side}
\phi_+\equiv \frac{1}{n+1}n^\frac{n+1}{n}(-k_+)^{-\frac{n-1}{n}} (A_++\epsilon_T+\epsilon(z)-\log r_+)^{\frac{n+1}{n}}-T^{\frac{2}{n}}k_+^{-1} A_++\epsilon_T+T^{\frac{2}{n}}\epsilon(z).
\end{equation}
\end{proposition}
The goal of the rest of this subsubsection is to prove this proposition.
First notice from the above discussion we know for each fixed $z$, $\phi(z)$ is uniquely determined up to a constant on $D$ by the equation
\begin{equation}\label{eqn7-11}
T\omega_D+d_Dd^c_D \phi(z)=\tilde\omega(z)-z\partial_z \tilde\omega(z),
\end{equation}
and the integration formula (\ref{eqn7-10}) exactly gives a coherent way of fixing all the constants for each $z$, so the overall freedom in only up to a global constant.
Notice by \eqref{e:greens-current-exp-asymp} we have for $z\gg 1$,
\begin{equation}\tilde\omega(z)-z\partial_z\tilde\omega(z)-T\omega_D=\psi(z)-z\partial_z\psi(z)=\epsilon(z).\end{equation}
Standard elliptic estimate allows us to fix a solution $\phi(T_+)$ which is $\epsilon_T$. By \eqref{e:h-asymptotics} we obtain that for $z\geq C$
\begin{equation} \label{eqn8-7}
\phi(z)=C_++T^{2-n}k_{+}^{-2} (\frac{(k_+z+T)^{n+1}}{n+1}-\frac{T(k_+z+T)^{n}}{n}),
\end{equation}
where
\begin{equation}C_{+}=\epsilon_T+\epsilon(z)-T^{2-n}k_+^{-2}(\frac{1}{n+1}T^{\frac{(n+1)(n-2)}{n}}-\frac{1}{n}T^{n-2}).\end{equation}
For the other end $z\leq -C$, similarly we have
\begin{equation} \label{eqn8-8}
\phi(z)-\phi(T_-)=C_-+T^{2-n}k_{-}^{-2} (\frac{(k_-z+T)^{n+1}}{n+1}-\frac{T(k_-z+T)^{n}}{n}),
\end{equation}
where
\begin{equation}C_-=\epsilon_T+\epsilon(z)+T^{2-n}k_-^{-2}(\frac{1}{n+1}T^{\frac{(n+1)(n-2)}{n}}-\frac{1}{n}T^{n-2}).\end{equation}
To understand $\phi(T_-)$ we need the following
\begin{lemma}
We have
\begin{equation}\label{eqn7-6}
\phi(T_-)=\epsilon_T+T^{-1}\underline B_T.
\end{equation}
\end{lemma}
\begin{proof}
We have $\phi(T_-)=\phi(T_+)-\Psi,$ where
$\Psi=\int_{T_-}^{T_+}zhdz.$
Away from $H$ we have
\begin{equation}d_Dd_D^c\Psi=\int_{T_-}^{T_+} zd_Dd_D^c h dz=-\int_{T_-}^{T_+} z\partial_z^2\tilde\omega dz.\end{equation}
Integration by parts we get
\begin{equation}\label{e:small-ddc}
d_Dd_D^c\Psi=(-z \partial_z\tilde\omega+\tilde\omega)|^{T_+}_{T_-}=\epsilon_T.
\end{equation}
Notice since there is a factor $z$ in the integrand we do not get residue term at $z=0$. Notice $\Psi$ is continuous on $D$, and the right hand side is smooth on $D$, so elliptic regularity implies that $\Psi$ is indeed smooth on $D$, and the equation holds globally on $D$.
On the other hand, we have
\begin{equation}\int_D \Psi\omega_D^{n-1}=\int_{T_-}^{T_+}z \int _D h\omega_D^{n-1} dz.\end{equation}
Using \eqref{haverage}, we see
\begin{align}\label{e:quotient-int}
\frac{\int_D \Psi\omega_D^{n-1}}{\int_D \omega_D^{n-1}}= & T^{2-n}k_{+}^{-2} (\frac{(k_+z+T)^{n+1}}{n+1}-\frac{T(k_+z+T)^{n}}{n})
\nonumber\\&-T^{2-n}k_{-}^{-2} (\frac{(k_-z+T)^{n+1}}{n+1}-\frac{T(k_-z+T)^{n}}{n})+T^{-1}\underline B_T,
\end{align}
where we used the definition of $T_-$ and $T_+$.
(\ref{e:small-ddc}) and \eqref{e:quotient-int} together yield the conclusion.
\end{proof}
Now notice that by \eqref{e: compare r and z}
\begin{equation}-T^{3-n}k_{-}^{-2} \frac{(k_-z+T)^{n}}{n}= \frac{T}{k_-} (\log r_-- A_-+\epsilon_T+\epsilon(z))+\epsilon_T.\end{equation}
We may also write by definition
$\omega_D=-\frac{1}{k_-}dd^c\log r_-$. Then \eqref{neck potential negative side} and \eqref{neck potential positive side} follow from
\eqref{eqn7-1} and simple computation.
\
\subsection{Geometries at regularity scales}
\label{ss:regularity-scales}
In this subsection, we take a closer look at the Riemannian geometric behavior of the family of incomplete K\"ahler metrics $(\mathcal M_T, \omega_T)$ constructed in Section \ref{ss:kaehler-structures} as $T\rightarrow \infty$. For clarity we now re-install the parameter $T$ throughout the rest of this section. The K\"ahler metric $\omega_T$ is given by
\begin{equation}
\omega_T\equiv T^{\frac{2-n}{n}}\cdot\Big(\pi^*(T\omega_D+\psi)+dz\wedge \Theta\Big).\end{equation}
Denote the corresponding Riemannian metric by $g_T$, which has the form
\begin{equation}g_T=T^{\frac{2-n}{n}}\cdot\Big(\pi^*(Tg_0+g_1+h_Tdz^2)+h_T^{-1}\Theta^2\Big),\label{e:g_T-submersion}
\end{equation}
where $g_0$ is the Riemannian metric corresponding to $\omega_D$, and $g_1$ is the symmetric 2-tensor corresponding to $\psi$.
Notice that by Remark \ref{r:bundle fixed} we may fix the $U(1)$-connection $\Theta$ over the entire $Q\setminus P$. Then $\omega_T$ and $g_T$ can be viewed as families of tensors on a fixed space. Moreover, they are positive definite when restricted to $T_-\leq z\leq T_+$ and $T_{\pm}$ are defined in \eqref{e:define-T-plus-minus}.
It is easy to see that as the parameter $T\to+\infty$, the curvatures are unbounded around the singular set
$\mathcal{P}\subset\mathcal M_T$ such that the standard uniform elliptic estimates fail.
Instead, we will define some appropriate weighted H\"older spaces
and establish uniformly weighted a priori estimates, which will be done in Section \ref{ss:neck-weighted-analysis}.
Geometrically, the weighted elliptic estimate that we pursue is intimately connected with
the {\it effective regularity at definite scales} of the points in $(\mathcal M_T,\omega_T)$. More rigorously, we need the following notion.
\begin{definition}[Local regularity]
\label{d:local-regularity} Let $(M^n,g,p)$ be a Riemannian manifold and $p\in M^n$. Given $r>0$, $\epsilon>0$, $k\in\mathbb{N}$, $\alpha\in(0,1)$, we say $(M^n,g,p)$ is $(r,k+\alpha,\epsilon)$-regular at $p$ if the metric $g$ is at least $C^{k+\alpha}$ in $B_{2r}(p)$ and satisfies the following property: Let $(\widetilde{B_{2r}(p)},\tilde{p})$ be the Riemannian universal cover of $B_{2r}(p)$. Then $B_r(\tilde{p})$ is diffeomorphic to a disc $\mathbb{D}^n$ or a half disc $\mathbb{D}_+^n$ in the Euclidean space $\mathbb{R}^n$ such that $g$ in coordinates satisfies
\begin{equation}
|g_{ij}-\delta_{ij}|_{C^0(B_r(\tilde{p}))}+\sum\limits_{m=1}^k r^m\cdot|\nabla^m g_{ij}|_{C^0(B_r(\tilde{p}))} + r^{k+\alpha}[g_{ij}]_{C^{k,\alpha}(B_r(\tilde{p}))} < \epsilon.
\end{equation}
\end{definition}
\begin{remark}
The case that $B_r(\tilde{p})$ is diffeomorphic to a half Euclidean disc $\mathbb D_+^n$ will be used to discuss the regularity of a manifold with boundary.
\end{remark}
\begin{definition}
[$C^{k,\alpha}$-regularity scale] Let $(M^n,g)$ be a Riemannian manifold with a $C^{k,\alpha}$-Riemannian metric $g$. The $C^{k,\alpha}$-regularity scale at $p$, denoted by $r_{k,\alpha}(p)$, is defined as
the supremum of all $r>0$ such that $M^n$ is $(r,k+\alpha,10^{-6})$-regular at $p$.
\end{definition}
Intuitively, the $C^{k,\alpha}$-regularity scale
is the maximal zooming-in scale at which
the {\it nontrivial} $C^{k,\alpha}$-geometry is uniformly bounded on the local universal cover. Clearly, if we work in a scale smaller than the regularity scale, then the corresponding $C^{k,\alpha}$ geometry is also uniformly bounded. So in the following we are mostly interested in a lower bound of the regularity scale.
\begin{example}
If $g$ is a $C^{k,\alpha}$-metric on $M^n$, then for any $p\in M^n$, we have $r_{k,\alpha}(p)>0$. Here the size of $r_{k,\alpha}(p)$ depends on $p$.
\end{example}
\begin{example}
Let $(M^n,g)$ satisfy $|\Rm_g|\leq 1$ in $B_2(p)$. Then the following holds:
\begin{enumerate} \item there exists a dimensional constant $r_0(n)>0$ such that $r_{1,\alpha}(x)\geq r_0(n)>0$ for all $x\in B_1(p)$ and $\alpha\in(0,1)$. Moreover,
$r_{1,\alpha}(p) \geq r_0(n)\cdot r_{|\Rm|}(p) >0$, where
\begin{equation}r_{|\Rm|}(p)\equiv \sup\Big\{r>0\Big| |\Rm|_{C^0(B_r(p))}\leq r^{-2}\Big\}\end{equation} denotes the curvature scale at $p$.
\item In particular, if $\Rm_g\equiv 0$ on a complete manifold $M^n$, then $r_{k,\alpha}(x)=+\infty$ for all $x\in M^n$, $k\in\mathbb{Z}_+$ and $\alpha\in(0,1)$.
\end{enumerate}
\end{example}
Notice that the construction of the K\"ahler manifolds $(\mathcal M_T, \omega_T)$ in Section \ref{ss:kaehler-structures} are fairly explicit. In this subsection, we estimate a lower bound of the $C^{k, \alpha}$-regularity scale on $\mathcal M_T$ for $T$ large.
Before the technical discussion, it is helpful to present the scenario of geometric transformations on $\mathcal M_T$ from the singular set $\mathcal{P}$ to the boundary $\partial \mathcal M_T$.
First, as $T\to+\infty$, curvatures blow up if the reference point $\bm{x}$ is located around $\mathcal{P}$, and suitably rescaling the metric $\omega_T$
gives rise to a product bubble limit $\mathbb{C}_{TN, \lambda}^2\times\mathbb{C}^{n-2}$, where $\mathbb{C}_{TN, \lambda}^2$ is the Taub-NUT space (c.f. Section \ref{ss:2d standard model}) for some $\lambda>0$. This is a {\it deepest bubble (rescaling limit)} in our context.
When the distance from $\bm{x}$ to $\mathcal{P}$ is increasing,
the length of $S^1$-fiber at the infinity
of the Taub-NUT space $\mathbb{C}_{TN, \lambda}^2\times\mathbb{C}^{n-2}$ is decreasing which corresponds to $\lambda$ is increasing. The next level of bubble corresponds to $\lambda\rightarrow\infty$, and this amounts to getting the tangent cone at infinity of the product $\mathbb{C}_{TN}^2\times\mathbb{C}^{n-2}$, which is $\mathbb{R}^{2n-1}\equiv \mathbb{R}^3\times \mathbb{C}^{n-2}$. This is of codimension-$1$ collapse, with locally uniformly bounded curvature away from $\{0^{3}\}\times \mathbb{C}^{n-2}$. When $\bm{x}$ is getting further away from $\mathcal{P}$, the size of $D$ will be shrinking and the next level of bubble is $D \times \mathbb{R}$. This is again a codimension-$1$ collapse, with locally uniformly bounded curvature away $P=H\times \{0\}\subset D\times \mathbb{R}$. As we move further away from $\mathcal P$, we may still see the bubble $D\times \mathbb R$, but this time the codimension-1 collapse is with locally bounded curvature and $P$ gets pushed to infinity.
Finally, as $\bm{x}$ moves close to the boundary $\partial\mathcal M_T$, the metrics will converge to the incomplete Calabi model metrics $\mathcal C^n_-$ and $\mathcal C^n_+$, which corresponds to applying the construction in Section \ref{ss:Calabi model space} to the line bundle $L^{k_-}$ and $L^{k_+}$ over $D$.
\vspace{0.5cm}
Now we make a subdivision for $\mathcal M_T$ (see Figure \ref{f:neck-subdivision}).
Given $\bm{x}\in \mathcal M_T$. Denote by $r(\bm{x})$ the distance from $\pi(\bm{x})\in Q$
to $P$ with respect to the product metric $g_Q = g_D + dz^2$ on the base $Q$.
{\bf Region $\bf{I}_1$:}
This region consists of the points $\bm{x}$ which satisfy
$r(\bm{x})\leq T^{-1}$.
{\bf Region $\bf{I}_2$:}
This region consists of the points $\bm{x}$ which satisfy
$\frac{ T^{-1} }{2}\leq r(\bm{x}) \leq 1$.
{\bf Region $\bf{I}_3$:}
This region consists of the points
$\bm{x}$ which satisfy
$r(\bm{x}) \geq \frac{1}{2}$ and
$T_- \leq z(\bm{x}) \leq T_+$.
\vspace{0.5cm}
Notice that the above regions completely cover the neck $\mathcal M_T$, and
overlapping regions have the same geometric behavior.
So we will just ignore these overlaps in the following discussions.
\begin{figure}
\begin{tikzpicture}
\draw (-6, 2) to (6, 2);
\draw (-6, -2) to (6, -2);
\draw (0,0) circle (3 mm);
\draw (0, 0) circle (10mm);
\draw (-6, 2) to (-6, -2);
\draw (6, 2) to (6, -2);
\draw[blue] (-2, 2) to (-2, -2);
\node[blue] at (-2.2, 0) {$D$};
\node[red] at (0, 0) {$\bullet$};
\node at (-6, -2.5) {$z=T_-$};
\node at (6, -2.5) {$z=T_+$};
\node at (0, -2.5) {$z=0$};
\node at (0, 1.5) {$\bf{I}_1$};
\draw[dashed, ->] (0, 1.2) to (0.2,0);
\node[red] at (-1, -1.5) {$\mathcal{P}$};
\draw[ ->, red] (-0.8, -1.2) to (-0.1, -0.1);
\node at (0.7, 0) {$\bf{I}_2$};
\node at (3, 0) {$\bf{I}_3$};
\end{tikzpicture}
\caption{Subdivision of $\mathcal M_T$ into various regions}
\label{f:neck-subdivision}
\end{figure}
For convenience, we define a continuous function $\mathfrak{r}$ which is uniformly equivalent to $r(\bm{x})$: \begin{align}\label{e:definition-fr}
\mathfrak{r}(\bm{x}) \equiv \begin{cases}
T^{-1}, & r(\bm{x}) \leq T^{-1}, \\
r(\bm{x}), & 2T^{-1}\leq r(\bm{x}) \leq \frac{1}{4},\\
1, & r(\bm{x})\geq \frac{1}{2}.
\end{cases}
\end{align}
The following proposition gives an explicit lower bound estimate of the $C^{k,\alpha}$-regularity scale on $\mathcal{M}_T$.
\begin{proposition}
\label{p:regularity-scale}
Let us define \begin{align}\label{e:explicit-scales}
\mathfrak{s}(\bm{x}) \equiv \Big(\frac{L_T(\bm{x})}{T}\Big)^{\frac{1}{2}}\cdot \mathfrak{r}(\bm{x}) \cdot T^{\frac{1}{n}}, \quad \bm{x}\in \mathcal M_T,
\end{align}
where $L_T(\bm{x})$ is defined in \eqref{e:LT z}.
Then the following properties hold:
\begin{enumerate}
\item Given $k\in\mathbb{Z}_+$ and $\alpha\in(0,1)$,
there is a uniform constant $\underline{v}_0$ (depending on $k$ and $\alpha$) such that for all sufficiently large $T$ and $\bm{x}\in\mathcal{M}_T$, it holds that
\begin{equation} \label{e: regularity scale bound}
r_{k,\alpha}(\bm{x}) \geq \underline v_0\cdot \mathfrak{s}(\bm{x}),\end{equation}
where $k\leq 2$ if $\bm{x}$ is in Region $\bf{I}_1$, and for all $k\in\mathbb{Z}_+$ if $\bm{x}$ is in Region $\bf{I}_2$ and $\bf{I}_3$.
\item
There are uniform constants $\underline{v}_0>0$ and $\overline{v}_0>0$ independent of $T\gg1$ such that for each $\bm{x}\in \mathcal M_T$, we have
\begin{equation}
\underline{v}_0 \leq \frac{\mathfrak{s}(\bm{y})}{\mathfrak{s}(\bm{x})} \leq \overline{v}_0 \quad \text{for all} \quad \bm{y}\in B_{\mathfrak{s}(\bm{x})/4}(\bm{x}).
\end{equation}
\item Let $T_j$ be a sequence tending to infinity. Then for a sequence of points $\bm{x}_j\in\mathcal M_{T_j}$, the rescaled spaces $(\mathcal M_{T_j}, \mathfrak{s}(\bm{x}_j)^{-2} \cdot g_{T_j},\bm{x}_j)$ converge in the pointed Gromov-Hausdorff sense to one of the following as $T_j\to+\infty$:
\begin{itemize}
\item the Riemannian product $\mathbb{C}_{TN}^2\times \mathbb{C}^{n-2}$ where $\mathbb{C}_{TN}^2$ is the Taub-NUT space;
\item the product Euclidean space $\mathbb{R}^3\times \mathbb{C}^{n-2}$;
\item the cylinder $D\times\mathbb{R}$;
\item the Calabi model spaces $(\mathcal C_{\pm}^n, g_{\mathcal C_{\pm}^n})$.
\end{itemize}
\end{enumerate}
\end{proposition}
\begin{remark}
The Calabi model space $(\mathcal C_{-}^n, \omega_{\mathcal C_{-}^n}, \Omega_{\mathcal{C}_-^n})$ (reps. $(\mathcal C_{+}^n, \omega_{\mathcal C_{+}^n}, \Omega_{\mathcal C_{+}^n})$ ) is defined as the $n$ dimensional Calabi-Yau manifold with boundary, obtained by applying the construction in Section \ref{ss:Calabi model space} with $\omega_D$ replaced by $k_-\omega_D$ (resp. $-k_+\omega_D$), and with the interval $z\in[1, \infty)$. When $b_1(D)>0$, we also make the choice of the corresponding connection 1-form similar to the discussion in Section \ref{sss:complex manifold}, so that the underlying complex manifold is naturally embedded into the holomorphic line bundle given by $L_-$ (resp. $L_+$).
\end{remark}
\begin{remark}
In \eqref{e:explicit-scales}, $\frac{L_T(\bm{x})}{T} = 1 + O(T^{-1})$ if $|z(\bm{x})|$ is bounded.
\end{remark}
\begin{proof}
[Proof of Proposition \ref{p:regularity-scale}.
]
We will prove \eqref{e: regularity scale bound}
by contradiction.
Suppose that no such uniform constant $\underline{v}_0$ exists with respect to fixed $k\in\mathbb{Z}_+$ and $\alpha\in(0,1)$. That is, there are a sequence $T_j\to+\infty$ and a sequence of points $\bm{x}_j\in \mathcal M_{T_j}$ such that
\begin{align}
\frac{r_{k,\alpha}(\bm{x}_j)}{\mathfrak{s}(\bm{x}_j)}\to 0.\label{e:rescaled-r-limiting-to-zero}
\end{align}
Let us consider the rescaled sequence $(\mathcal M_{T_j},\tilde{g}_{T_j},\bm{x}_j)$ with $\tilde{g}_{T_j}\equiv \mathfrak{s}(\bm{x}_j)^{-2} \cdot g_{T_j}$ as $T_j\to+\infty$. In the proof, we will show that $C^{k,\alpha}$-regularity scale at $\bm{x}_j$ with respect to $\tilde{g}_{T_j}$ is uniformly bounded from below as $T_j\to +\infty$ which contradicts \eqref{e:rescaled-r-limiting-to-zero}.
We will produce a contradiction in each of the following cases depending upon the location of $\bm{x}_j$ in $\mathcal M_{T_j}$. We will also identify the rescaled limit in each case.
\vspace{0.5cm}
\begin{flushleft}
{\bf Case (1):} There is a constant $\sigma_0\geq 0$ such that $r(\bm{x}_j)\cdot T_j\to \sigma_0$ as $j\rightarrow\infty$.
\end{flushleft}
In this case, by definition $\mathfrak{s}(\bm{x}_j) = (1+O(T_j^{-\frac{1}{2}}))\cdot \mathfrak{r}(\bm{x}_j) \cdot T_j^{\frac{1}{n}}$. It suffices to show that $r_{k,\alpha}(\bm{x}_j)$ with respect to the rescaled metric $\mathfrak{r}(\bm{x}_j)^{-2}\cdot T_j^{-\frac{2}{n}}\cdot g_{T_j}$ (still denoted by $\tilde{g}_{T_j}$) is uniformly bounded from below as $T_j\to+\infty$.
First, we consider the case $\sigma_0\leq 1$.
Then by definition \eqref{e:definition-fr}, we have $\mathfrak{r}(\bm{x}_j)= T_j^{-1}$.
Denote the rescaled K\"ahler form and the rescaled holomorphic volume form by
\begin{equation}
\tilde{\omega}_{T_j} \equiv T_j^{\frac{2n-2}{n}}\cdot\omega_{T_j},\quad \widetilde{\Omega}_{T_j} \equiv T_j^{n-1}\cdot\Omega_{T_j}.
\end{equation}
As in \eqref{Taub-NUT}, for a parameter $\lambda>0$,
let us denote by
$(\omega_{TN,\lambda} ,\Omega_{TN,\lambda})$ the K\"ahler form and the holomorphic volume form of the Taub-NUT space
$\mathbb{C}_{TN, \lambda}^2$.
Clearly $r(\bm{x}_j)\rightarrow0$. Passing to a subsequence we may assume $\pi(\bm{x}_j)$ converges to $p\in P$.
Denote $\bm{p}\equiv \pi^{-1}(p)\in\mathcal P$.
In the following we will prove that as $T_j\rightarrow\infty$, \begin{equation} \label{I1convergence}(\mathcal M_{T_j}, \tilde{\omega}_{T_j}, \widetilde{\Omega}_{T_j}, \bm{x}_j)\xrightarrow{C^{2,\alpha}}(\mathbb{C}_{TN,1}^2\times \mathbb{C}^{n-2}, \omega_{TN,1}\oplus \omega_{\mathbb{C}^{n-2}}, \Omega_{TN, 1}\wedge \Omega_{\mathbb C^{n-2}}, \bm{0}^*)\end{equation} in the pointed $C^{2,\alpha}$-topology
and the fixed point $\bm{p}$ converges to $\bm{0}^*\equiv (\bm{0}^2, \bm{0}^{n-2})$, where $\bm{0}^2$ is the origin of $\mathbb{C}_{TN,1}^2$ and $(\omega_{\mathbb{C}^{n-2}}, \Omega_{\mathbb C^{n-2}})$ is the flat K\"ahler structure on $\mathbb C^{n-2}$ with the origin $\bm{0}^{n-2}\in\mathbb{C}^{n-2}$.
As in Section \ref{sss:metric compactification}, we work with the local coordinate system $\{y, \bar y, z, w_2', \bar w_2', \cdots, w_{n-1}', \bar w_{n-1}'\}$ in a neighborhood of $p$ in $Q$, which gives a local coordinate system in a neighborhood of $\bm{p}$ in $\mathcal M_T$, denoted by $\{u_1, \bar u_1, u_2, \bar u_2, w_2', \bar w_2', \ldots, w_{n-1}', \bar w_{n-1}'\}$. From the computation in Section \ref{sss:metric compactification},
\begin{equation}
T_j^{\frac{2n-2}{n}}\cdot\omega_{T_j} =\Big(T_j\cdot\omega_{TN, T_j}\oplus T_j^2\cdot\omega_{\mathbb C^{n-2}}\Big)+T_j^2\cdot\pi^*(\omega_D-\omega_{\mathbb C^{n-1}})+T_j\cdot O'(s^3)+T_j\cdot d(s^2\Gamma),
\end{equation}
where $\omega_{TN, T_j}$ is the Taub-NUT metric on $\mathbb C^2_{u_1, u_2}$ with a parameter $T_j$, and
\begin{equation}\omega_{\mathbb C^{n-2}}\equiv \frac{\sqrt{-1}}{2}\sum_{j= 2}^{n-1} dw_j'\wedge d\bar w_j', \ \ \omega_{\mathbb C^{n-1}}\equiv \frac{\sqrt{-1}}{2}\cdot dy\wedge d\bar y+\frac{\sqrt{-1}}{2}\sum_{j= 2}^{n-1} dw_j'\wedge d\bar w_j'.
\end{equation}
Notice that we have already used the relations \begin{equation}\pi^*(O'(r^p))=O'(s^{2p}) \ \text{for}\ p\geq 1,\quad \pi^*(dy)=\widetilde{O} (s), \quad \pi^*(dz)=\widetilde{O} (s).\end{equation}
We perform a change of coordinates
\begin{equation}
z=T_j^{-1}\cdot\underline{z}, \quad y=T_j^{-1}\cdot\underline{y}, \quad
w_j'=T_j^{-1}\cdot\underline{w_j}',\quad
u_k=T_j^{-1/2}\cdot\underline{u_k},
\end{equation}
and denote
${\bm w}\equiv (\underline w_2', \cdots, \underline w_{n-1}')$, ${\bm u}\equiv (\underline u_1, \underline u_2)$, $\underline{s}=|{\bm u}|$.
In these rescaled coordinates, the rescaled K\"ahler structure $(T_j\cdot \omega_{TN, T_j},T_j\cdot\Omega_{TN,T_j}) $ can be identified with $(\omega_{TN, 1},\Omega_{TN, 1})$. Moreover, we have that
\begin{align}
T_j^2\cdot \pi^*(\omega_D-\omega_{\mathbb C^{n-1}}) &= O((|{\bm w} |+|{\bm u}|^2)T_j^{-1}),\\
T_j\cdot O'(s^3) &= O(T_j^{-3/2}\underline s^3),\\
T_j \cdot d(s^2\Gamma) &= O(T_j^{-3/2}\underline s).
\end{align}
The above computations imply that on the region $|\bm w|+|\bm u|\leq C$ for a fixed $C>0$. Therefore, \begin{equation}|T_j^{\frac{2n-2}{n}}\omega_T-(\omega_{TN, 1}\oplus\omega_{\mathbb C^{n-2}})|_{C^{2, \alpha}}=O(T_j^{-1}),
\end{equation}
where the norm is measured with respect to the product metric
$\omega_{TN, 1}\oplus\omega_{\mathbb C^{n-2}}$.
Similarly, one can also obtain the expansion for the rescaled holomorphic form $\widetilde{\Omega}_{T_j}$,
\begin{equation}
\widetilde{\Omega}_{T_j} = T_j^{n-1}\cdot\Omega_{T_j}=\Omega_{TN, 1}\wedge d\underline w_2'\wedge \cdots\wedge d\underline w_{n-1}'+O((|{\bm w}|+|{\bm u}|^2)T_j^{-1}).
\end{equation}
This implies that the convergence \eqref{I1convergence} holds, and hence
there is $\underline{v}_0>0$ is independent of $T_j\gg1$ such that under the rescaled metric $\tilde{g}_{T_j}$,
\begin{equation}
r_{2,\alpha}(\bm{x}_j)\geq \underline{v}_0,
\end{equation}
which contradicts \eqref{e:rescaled-r-limiting-to-zero}. This completes the proof in the case $\sigma_0\leq 1$.
In the case $\sigma_0>1$, the proof is the same. We only need to notice that the K\"ahler structure on the limiting Taub-NUT space is given by
\begin{align}\label{e:parameter-TN}
\begin{cases}
\omega_{TN,\sigma_0^2} \equiv (\frac{1}{2r}+\sigma_0^2)\cdot \frac{\sqrt{-1}}{2}\cdot dy\wedge d\bar{y}
+dz\wedge \Theta_0
\\
\Omega_{TN,\sigma_0^2} \equiv \sqrt{-1} ((\frac{1}{2r}+\sigma_0^2)\cdot dz + \Theta_0)\wedge dy.
\end{cases}
\end{align}
The rest of the computations are the same.
So the proof in Case (1) is done.
\vspace{0.5cm}
\begin{flushleft}
{\bf Case (2):} $r(\bm{x}_j)\cdot T_j\rightarrow\infty$ and $r(\bm{x}_j)\rightarrow 0$ as $j\rightarrow\infty$.\end{flushleft}
In this case, by definition $\mathfrak{s}(\bm{x}_j) =(1+O(T_j^{-\frac{1}{2}}))\cdot r(\bm{x}_j)\cdot T_j^{\frac{1}{n}}$. It suffices to show that with respect to the rescaled metric
$r_j^{-2}\cdot T_j^{-\frac{2}{n}}\cdot g_j$ (still denoted by $\tilde{g}_{T_j}$)
with $r_j\equiv r(\bm{x}_j)$, the regularity scale
$r_{k,\alpha}(\bm{x}_j)$ is uniformly bounded from below, which contradicts \eqref{e:rescaled-r-limiting-to-zero}.
In the following computations, it is more convenient to consider the Riemannian metric $g_T$ corresponding to $\omega_{T_j}$, which is given by \begin{equation}g_{T_j}=T_j^{\frac{2-n}{n}}\cdot\Big(\pi^*(T_j\cdot g_0+g_1+h_{T_j}dz^2)+h_{T_j}^{-1}\cdot \Theta^2\Big).\label{e:g_T-submersion}
\end{equation}
Again we emphasize that $\Theta$ is independent of $T_j$.
Since $r_j\rightarrow 0$, we may assume that $\pi(\bm{x}_j)$ converges to a fixed point $p\in P$. In a neighborhood of $P$ in $Q$, we have local coordinates $\{y, \bar y, z, w_2', \bar w_2', \cdots, w_{n-1}', \bar w_{n-1}'\}$ as in Section \ref{ss:complex-greens-currents} and we rescale them by
\begin{equation}
z= r_j\cdot \underline z, \ y= r_j\cdot\underline y, \
w_p'= r_j\cdot\underline w_p',\ \ p=2, \ldots, n-1.
\label{e:rescaled-coordinates}
\end{equation}
Notice that $\pi(\bm{x}_j)$ has a definite distance away from the subspace $\{y=z=0\}\subset P$ under $\tilde{g}_{T_j}$.
We consider the pointed Gromov-Hausdorff limit of the metrics $(\mathcal M_T, \tilde g_j, \bm{x}_j)$. Notice that
\begin{equation}\tilde{g}_{T_j}=r_j^{-2}\cdot T_j^{-1}\cdot\Big(\pi^*(T_jg_0+g_1+h_{T_j}dz^2)+h_{T_j}^{-1}\Theta^2\Big).
\end{equation}
As before, we can see that in the rescaled coordinates, the first term converges smoothly to $g_{\mathbb C^{n-2}}\oplus dz^2$ away from the subspace $\{y=z=0\}$, where $g_{\mathbb C^{n-2}}$ denotes the flat metric on $\mathbb C^{n-2}_{w_2', \cdots, w_{n-1}'}$. On the other hand, since $h_{T_j}$ has uniformly positive lower bound in this region (by \eqref{e:h-bounded-z}), and $T_j\cdot r_j\rightarrow +\infty$, it is easy to see that the length of the $S^1$-fibers tends to zero uniformly.
Analyzing more closely using the behaviors of $\psi, h_{T_j}$ and $\Theta$ near $p$ (see \eqref{e:singular-2-form-expansion}, \eqref{e:h-bounded-z} and \eqref{e:Theta-near-p}), one can see that the metrics $\tilde{g}_j$ are collapsing with uniformly bounded curvature away from the subspace $\{y=z=0\}\subset P$. Moreover, around $\bm{x}_j$, if we pass to the local universal cover, we have $C^{k,\alpha}$-bounded geometry in a ball of definite size. This already shows that there is a constant $\underline{v}_0>0$ independent of $T_j$ such that $r_{k,\alpha}(\bm{x}_j)\geq \underline{v}_0$ with respect to $\tilde{g}_{T_j}$, which contradicts \eqref{e:rescaled-r-limiting-to-zero}.
With more analysis one can actually show that the rescaled metrics $(\mathcal M_{T_j}, \tilde g_{T_j}, \bm{x}_j)$ converges in the pointed Gromov-Hausdorff sense to the product Euclidean space $\mathbb R^3\times \mathbb C^{n-2}$. The details follow from explicit but lengthy tensor computations, so we omit them. Notice that $\mathbb R^3\times \mathbb C^{n-2}$ is
the tangent cone at infinity of the product space $\mathbb C^2_{TN,1}\times \mathbb C^{n-2}$ which appears as the rescaled limit in Case (1).
\vspace{0.5cm}
\begin{flushleft}
{\bf Case (3):} There is a constant $\underline{T}_0>0 $ such that $r(\bm{x}_j)\geq \underline{T}_0$ and $L_{T_j}(\bm{x}_j)^{-1}\cdot T_j^{\frac{n-2}{n}}\rightarrow 0$ as $j\rightarrow\infty$.
\end{flushleft}
In this case, $\mathfrak{s}(\bm{x}_j) = \Big(\frac{L_{T_j}(\bm{x}_j)}{T_j}\Big)^{\frac{1}{2}}\cdot r(\bm{x}_j) \cdot T_j^{\frac{1}{n}}$. Passing to a subsequence there are two sub-cases.
First, we consider the case $z(\bm{x}_j)\to z_0$ and $r(\bm{x}_j)\to r_0>0$. Then $\mathfrak{s}(\bm{x}_j) = (1+O(T_j^{-\frac{1}{2}}))\cdot r_0 \cdot T_j^{\frac{1}{n}}$. So it suffices to work with the rescaled metric
$r_0^{-2}\cdot T_j^{-\frac{2}{n}}\cdot g_{T_j}$ (again denoted by $\tilde{g}_{T_j}$) and show that the regularity scale
$r_{k,\alpha}(\bm{x}_j)$ is uniformly bounded from below. The rescaled metric $\tilde{g}_{T_j}$ has the explicit form
\begin{equation}
\tilde{g}_{T_j}=r_0^{-2}\cdot T_j^{-1}\cdot\Big(\pi^*(T_j\cdot g_0+g_1+h_{T_j}\cdot dz^2)+h_{T_j}^{-1}\cdot\Theta^2\Big).\end{equation}
Now using the asymptotics of $\psi$ and $h_{T_j}$ in \eqref{e:singular-2-form-expansion} and \eqref{e:h-bounded-z}, we see that $\tilde{g}_{T_j}$ converges smoothly as tensors to $\pi^*(g_0+dz^2)$ away from $P$. Since the $S^1$-fibers are collapsing, and we have uniformly bounded $C^{k,\alpha}$-geometry on a definite size ball on the local universal cover around $\bm{x}_j$. This implies that there is a constant $\underline{v}_0>0$ such that $r_{k,\alpha}(\bm{x}_j)\geq \underline{v}_0$ under the metric $\tilde{g}_{T_j}$, which contradicts to \eqref{e:rescaled-r-limiting-to-zero}.
Again with a little more work one can see that the rescaled metrics $(\mathcal M_{T_j}, \tilde g_j, \bm{x}_j)$ converges in the pointed Gromov-Hausdorff topology to the product space $D\times \mathbb{R}$, and $S^1$-fibers collapse. The collapsing is with uniformly bounded curvature away from $P$. We also omit the details. Notice that this limit is also related to the rescaled limit in Case (2): Any point $p\in D\times\mathbb{R}$ has a tangent cone $\mathbb R^3\times \mathbb C^{n-2}$.
Next, let us consider the case $|z_j|\to+\infty$ and $L_{T_j}(\bm{x}_j)^{-1}\cdot T_j^{\frac{n-2}{n}}\rightarrow 0$, where $z_j\equiv z(\bm{x}_j)$.
Without loss of generality, we may assume $z_j<0$. First, we have $\mathfrak{s}(\bm{x}_j)= \Big(\frac{L_j}{T_j}\Big)^{\frac{1}{2}} \cdot T_j^{\frac{1}{n}}$, where $L_j\equiv L_{T_j}(\bm{x}_j)$.
Then the rescaled metric $\tilde{g}_{T_j}\equiv \mathfrak{s}(\bm{x}_j)^{-2}\cdot g_{T_j}$ is given by \begin{align}
\tilde{g}_j=L_j^{-1}\cdot\Big(\pi^*((T_j\cdot g_0+g_1)+h_{T_j}\cdot dz^2)+h_{T_j}^{-1}\cdot\Theta^2\Big).\label{e:cylinder-rescaled-metric}\end{align}
We perform a change of coordinate
$z = z_j + (\frac{T_j}{L_j})^{\frac{n-2}{2}}\cdot w$.
Then using the asymptotics of $\psi$ and $h_{T_j}$ in \eqref{e:singular-2-form-expansion} and \eqref{e:h-asymptotics}, we can see that the $S^1$-fibers are collapsing. The pointed Gromov-Hausdorff limit is again the cylinder $D\times\mathbb{R}$ and the collapsing sequence has uniformly bounded curvature. It is then easy to obtain a contradiction.
Geometrically, the above two situations are related as follows. The rescaled limit $D\times\mathbb{R}_z$ in the second situation can be viewed as translating $D\times\mathbb{R}_z$ along $z$ towards $\pm\infty$, so that the singular set $P$ disappears.
\vspace{0.5cm}
\begin{flushleft}
{\bf Case (4):} There is a constant $c_0>0$ such that $L_{T_j}(\bm{x}_j)^{-1}\cdot T_j^{\frac{n-2}{n}}\to c_0$.
\end{flushleft}
In this case, by definition $\mathfrak{s}(\bm{x}_j) = (c_0)^{-1}\cdot (1+o(1))$.
One can see that $d_{g_j}(\bm{x}_j,\partial\mathcal M_{T_j})$ is uniformly bounded. Without loss of generality we may assume $z(\bm{x}_j)<0$. We rescale the $z$-coordinate by
\begin{equation}
w=T_j^{\frac{2-n}{n}}(T_j+k_-\cdot z).
\end{equation}
Then using the asymptotics of $h_{T_j}, \psi$, one can obtain convergence of the metric tensor $g_{T_j}$ in the $w$ coordinate. From this one easily see that $(\mathcal M_{T_j}, \tilde{g}_{T_j}, \bm{x}_j)$ converges smoothly to the Calabi model space $(\mathcal{C}_-^n, g_{\mathcal{C}_-^n}, \bm{x}_\infty)$ in the pointed Gromov-Hausdorff sense. Clearly we obtain a contradiction to \eqref{e:rescaled-r-limiting-to-zero}.
Notice that the Calabi model space has a boundary. To relate to the limit in Case (3), one can think of the space $D\times \mathbb R$ as a pointed Gromov-Hausdorff limit of certain scale down of the Calabi model space at infinity. We choose a sequence of points towards infinity, and scale down so that the base $D$ has fixed size. Then $\partial\mathcal{C}_-^n$ gets pushed to infinity and also the $S^1$-fibers are collapsing.
\end{proof}
\subsection{Fundamental estimates in the weighted H\"older spaces}
\label{ss:neck-weighted-analysis}
Based on the above detailed studies of the regularity scales,
we are now ready to define the weighted H\"older space on the neck.
To start with,
let us recall the notation,
\begin{align}
\mathcal M_T &\equiv \Big\{\bm{x}\in \mathcal{M}\Big|T_- \leq z(\bm{x})\leq T_+\Big\},
\\
\mathring{\mathcal{M}}_T &\equiv \Big\{\bm{x}\in \mathcal{M}\Big|T_- \leq z(\bm{x}) \leq T_+,\ d_{\omega_T}(\bm{x}, \partial\mathcal M_T)\geq 1\Big\}.
\end{align}
\begin{definition}
[Weight function] \label{d:weight-function}
Given fixed real parameters $n\geq 2$, $T\gg1$, $\delta>0$, $\nu,\mu\in\mathbb{R}$. For each $k\in\mathbb{N}$, $\alpha\in(0,1)$,
the weight function $\rho_{\delta,\nu,\mu}^{(k+\alpha)}$
is defined as follows,
\begin{align}
\label{e:definition of weights}
\rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})=e^{\delta\cdot U_T(\bm{x})}\cdot \mathfrak{s}(\bm{x})^{\nu+k+\alpha}\cdot T^{\mu},
\end{align}
where $\mathfrak{s}(\bm{x})$ is the regularity scale at $\bm{x}$ given by Proposition \ref{p:regularity-scale} and
\begin{align}
U_T(\bm{x}) \equiv T\Big(1-(\frac{L_T(\bm{x})}{T})^{\frac{n}{2}}\Big),
\quad
L_T(\bm{x}) \equiv L_T(z(\bm{x})) = T + L_0(z(\bm{x})),
\label{d:def-U_T}\end{align}
where the functions $L_T$ and $L_0$ are defined in \eqref{e:LT z}.
\end{definition}
To better understand the weight function \eqref{e:definition of weights}, we give several remarks.
\begin{remark}
In the case that $D$ is not flat, by the proof of Proposition \ref{p:regularity-scale}, one can see that $\mathfrak{s}(\bm{x})$ is uniformly equivalent to the $C^{k,\alpha}$-regularity scale $r_{k,\alpha}(\bm{x})$ at $\bm{x}$.
\end{remark}
\begin{remark}
The function $e^{\delta\cdot U_T(\bm{x})}$
is the dominating
factor at large scales on $\mathcal M_T$ which
behaves like an exponential function. The term $U_T(\bm{x})$ is used for unifying the weighted analysis for different ``large scales'' on $\mathcal M_T$, which will be seen in the proof of Proposition \ref{p:neck-uniform-injectivity} in Section \ref{s:neck-perturbation}. For intuition, there are two cases in which $U_T$ has simple expressions:
\begin{align}
\begin{cases}
U_T(\bm{x}) = - L_0(z) , & n=2,
\\
U_T(\bm{x}) \approx - \frac{n}{2}\cdot L_0(z) , & n> 2,\ |z(\bm{x})|\ll T.
\end{cases}
\end{align}
\end{remark}
\begin{remark}
The constant factor $T^\mu$ in the definition of the weight function is needed to deal with the non-linear term in the application of the implicit function theorem (see Proposition \ref{p:nonlinear-neck}). When $n=2$ the non-linear term is quadratic and this constant term is unnecessary, but when $n>2$ we need to choose appropriate $\mu$ (see \eqref{e:fix-mu-neck}) so that the weight function has a uniform lower bound independent of $T$.
\end{remark}
\begin{lemma}[Lower bound estimate for the weight function]\label{l:weight-function-lower-bound-estimate}
For fixed constants $\delta>0$, $\mu,\nu\in\mathbb{R}$, $\alpha\in(0,1)$ and $k\in\mathbb{N}$, then for all $T\gg1$ and $\bm{x}\in\mathcal M_T$,
\begin{align}
\rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})\geq \begin{cases}
T^{(\frac{1}{n}-1)(\nu+k+\alpha)+\mu}, & \nu+k+\alpha \geq 0,
\\
T^{\frac{\nu+k+\alpha}{n}+\mu}, & \nu+k+\alpha < 0.
\end{cases}
\end{align}
\end{lemma}
\begin{proof}
This lower bound estimate can be obtained by analyzing the regularity scale $\mathfrak{s}(\bm{x})$.
Denote by $w=w(\bm{x}) \equiv \frac{L_T(\bm{x})}{T}$ and recall that the two end points $T_-,T_+$ satisfy
\begin{align}
\begin{cases}
L_T(T_-) = T^{\frac{n-2}{n}}
\\
L_T(T_+) = T^{\frac{n-2}{n}}.
\end{cases}
\end{align}
Then we have
$w\in[T^{-\frac{2}{n}},1]$. So it follows that
\begin{equation}
\rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{x}) = F(w)\cdot \mathfrak{r}(\bm{x})^{\nu+k+\alpha}\cdot T^{\frac{\nu+k+\alpha}{n}+\mu},
\end{equation}
where $F(w) \equiv e^{\delta \cdot T(1-w^{\frac{n}{2}})}\cdot w^{\frac{\nu+k+\alpha}{2}}$.
By the definition of $\mathfrak{r}(\bm{x})$, immediately we have \begin{align}T^{-1}\leq\mathfrak{r}(\bm{x})\leq 1\end{align} for all $\bm{x}\in\mathcal M_T$, so it follows that
\begin{align}
\rho_{\delta,\nu,\mu}^{(k+\alpha)} (\bm{x})
\geq
\begin{cases}
F(w)\cdot T^{(\frac{1}{n}-1)(\nu+k+\alpha)+\mu}, & \nu+k+\alpha\geq 0,
\\
F(w)\cdot T^{\frac{\nu+k+\alpha}{n}+\mu}, & \nu+k+\alpha < 0,
\end{cases}
\end{align}
Now it suffices to
compute the lower bound of $F(w)$. To this end,
there are two cases to analyze depending on the sign of $\nu+k+\alpha$. First, let $\nu+k+\alpha\leq 0$, then obviously $F(w)\geq F(1) = 1$
and hence
\begin{equation}
\rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})\geq T^{(\frac{1}{n}-1)(\nu+k+\alpha)+\mu}.
\end{equation}
Next, we consider the case $\nu+k+\alpha>0$. Simple calculus shows that
$F(w)$ achieves its minimum in $[T^{-\frac{2}{n}},1]$ either at $w= 1$ or at $w=T^{-\frac{2}{n}}$. Notice that
$F(T^{-\frac{2}{n}})\gg F(1)$ as $T\gg1$.
This tells us that
\begin{equation}
\rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})\geq T^{\frac{\nu+k+\alpha}{n}+\mu}.\end{equation}
The proof is done.
\end{proof}
Using the above weight function, we define weighted H\"older spaces as follows.
\begin{definition}
[Weighted H\"older norm] \label{d:weighted-space} Let $\mathcal{K}\subset\mathcal M_T$ be a compact subset. Then the weighted H\"older norm of a tensor field $\chi\in T^{r,s}(\mathcal{K})$ of type $(r,s)$ is defined as follows:
\begin{enumerate}
\item The weighted $C^{k,\alpha}$-seminorm of $\chi$ is defined by
\begin{align}
[\chi]_{C_{\delta,\nu,\mu}^{k,\alpha}}(\bm{x}) & \equiv \sup\Big\{\rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{x}) \cdot\frac{|\nabla^k\tilde{\chi}(\tilde{\bm{x}})- \nabla^k\tilde{\chi}(\tilde{\bm{y}})|}{(d_{\tilde{g}_T}(\tilde{\bm{x}},\tilde{\bm{y}}))^{\alpha}} \ \Big| \ \tilde{\bm{y}}\in B_{r_{k,\alpha}(\bm{x})}(\tilde{\bm{x}})\Big\}\ \text{for}\ \bm{x}\in\mathcal{K},
\\
[\chi]_{C_{\delta,\nu,\mu}^{k,\alpha}(\mathcal{K})} & \equiv \sup\Big\{[\chi]_{C_{\delta,\nu,\mu}^{k,\alpha}}(\bm{x})\Big|\bm{x} \in \mathcal{K}\Big\},
\end{align}
where $r_{k,\alpha}(\bm{x})$ is the $C^{k,\alpha}$-regularity scale at $\bm{x}$, $\tilde{\bm{x}}$ denotes a lift of $\bm{x}$ to the universal cover of $B_{2r_{k,\alpha}(\bm{x})}(\bm{x})$,
the difference of the two covariant derivatives is defined in terms of parallel translation in $B_{r_{k,\alpha}(\bm{x})}(\tilde{\bm{x}})$,
and $\tilde{\chi}$, $\tilde{g}_T$ are the lifts of $\chi$, $g_T$ respectively.
\item The weighted $C^{k,\alpha}$-norm of $\chi$ is defined by
\begin{align}
\|\chi\|_{C_{\delta,\nu, \mu}^{k,\alpha}(\mathcal{K})}
\equiv \sum\limits_{m=0}^k\Big\|\rho_{\delta,\nu,\mu}^{(m)} \cdot\nabla^m \chi\Big\|_{C^0(\mathcal{K})} + [\chi]_{C_{\delta,\nu,\mu}^{k,\alpha}(\mathcal{K})},
\end{align}
\end{enumerate}
\end{definition}
\begin{remark}
By definition, it is direct to check that
\begin{equation}
\|\chi\|_{C_{\delta,\nu,\mu}^k(\mathcal{K})} = \sum\limits_{m=0}^k \|\nabla^m \chi\|_{C_{\delta,\nu+m,\mu}^0(\mathcal{K})}.
\end{equation}
\end{remark}
With the above definition of the weighted H\"older space, we are ready to give a local uniform weighted Schauder estimate with respect to the Laplacian on the neck $(\mathcal M_T, \omega_T)$.
\begin{proposition}
[Weighted Schauder estimate, the local version] \label{p:local-weighted-schauder}
For every sufficiently large parameter $T\gg1$, let
$\mathcal M_T$ be the neck region with
an $S^1$-invariant K\"ahler metric $\omega_T$
constructed in Section \ref{ss:kaehler-structures}.
Then the following estimates hold:
\begin{enumerate}
\item (Interior estimate)
Given $k\in\{0,1\}$ and $\alpha\in(0,1)$, there is some uniform constant $C_{k,\alpha}>0$
such that for any
$\bm{x}\in\mathring{\mathcal{M}}(T_-,T_+)$, $r\in(0,1/8]$, $u\in C^{k+2,\alpha}(B_{2 r\cdot \mathfrak{s}(\bm{x})}(\bm{x}))$,
\begin{align}
&r^{k+2+\alpha}\cdot \|u\|_{C_{\delta,\nu,\mu}^{k+2,\alpha}(B_{r\cdot \mathfrak{s}(\bm{x})}(\bm{x}))}
\nonumber\\
\leq & C_{k,\alpha} \Big(\|\Delta u\|_{C_{\delta,\nu+2,\mu}^{k,\alpha}(B_{2r\cdot \mathfrak{s}(\bm{x})}(\bm{x}))} + \| u\|_{C_{\delta,\nu,\mu}^0(B_{2r\cdot \mathfrak{s}(\bm{x})}(\bm{x}))}\Big),
\label{e:local-schauder}
\end{align}
where $\mathfrak{s}(\bm{x})$ is the function defined in \eqref{e:explicit-scales}.
\item (Higher order estimate away from $\mathcal{P}$) If $\bm{x}\in\mathring{\mathcal M}_T$ satisfies $r(\bm{x})\geq 2T^{-1}$, then the uniform Schauder estimate \eqref{e:local-schauder} holds for all $k\in\mathbb{Z}_+$, $\alpha\in(0,1)$, $r\in(0,r_0)$ with $r_0$ independent of $T$.
\item (Boundary estimate)
For any $k\in\mathbb{Z}_+$ and $\alpha\in(0,1)$, there exists some uniform constant $C_{k,\alpha}>0$ such that for all $\bm{x}\in\partial\mathcal{M}_T$, $r\in(0,1/8]$ and $u\in C^{k+2,\alpha}(B_{2r\cdot \mathfrak{s}(\bm{x})}^+(\bm{x}))$,
\begin{align}
& r^{k+2+\alpha}\cdot \|u\|_{C_{\delta,\nu,\mu}^{k+2,\alpha}(B_{r\cdot \mathfrak{s}(\bm{x})}^+(\bm{x}))}\nonumber
\\ \leq & C_{k,\alpha} \cdot \Big(\|\Delta u\|_{C_{\delta,\nu+2,\mu}^{k,\alpha}(B_{2r\cdot \mathfrak{s}(\bm{x})}^+(\bm{x}))} + \Big\| \frac{\partial u}{\partial n}\Big\|_{C_{\delta,\nu+1,\mu}^{k+1,\alpha}(B_{2r\cdot \mathfrak{s}(\bm{x})}^+(\bm{x})\cap \partial M_T)} \nonumber\\
& + \| u\|_{C_{\delta,\nu,\mu}^0(B_{2r\cdot \mathfrak{s}(\bm{x})}^+(\bm{x}))}\Big),
\label{e:boundary-local-estimate}
\end{align}
where $B_s^+(\bm{x})\equiv B_s(\bm{x})\cap \mathcal{M}_T$ and $\frac{\partial}{\partial n}$ is the exterior normal vector field on $\partial M_T$.
\end{enumerate}
\end{proposition}
\begin{proof}
We first describe the proof of \eqref{e:local-schauder}.
Without loss of generality, we only prove the estimate by assuming the scale parameter $r=1/8$ and $k=0$. The estimate in the general case $r\in(0,1)$ can be achieved by simple rescaling.
Since we have shown in item (1) of Proposition \ref{p:regularity-scale} that, for any $\bm{x}\in\mathring{\mathcal{M}}_T(T_-,T_+)$, under the rescaled metric
$\tilde{g}_T = \mathfrak{s}(\bm{x})^{-2}\cdot g_T$, the $C^{2,\alpha}$-regularity scale of
$B_{1/2}^{\tilde{g}_T}(\bm{x})$ is uniformly bounded from below (independent of $T$) for each $\alpha\in(0,1)$. So there is a uniform constant $C>0$ (independent of $T$) such that the standard Schauder estimate holds for $k\in\{0,1\}$ and for every $u\in C^{2,\alpha}(B_{2 r\cdot \mathfrak{s}(\bm{x})}(\bm{x}))$ and $\bm{x}\in \mathring{\mathcal{M}}(T_-,T_+)$, \begin{equation}
\|u\|_{C^{2,\alpha}(B_{1/8}^{\tilde{g}_T}(\bm{x}))} \leq C\Big(\| \Delta_{\tilde{g}}u \|_{C^{\alpha}(B_{1/4}^{\tilde{g}_T}(\bm{x}))} +\|u\|_{C^{0}(B_{1/4}^{\tilde{g}_T}(\bm{x}))}\Big).\label{e:ball-standard-schauder}
\end{equation}
Then rescaling back to $g_T$ and using the definition of the weighted functions, we have that
\begin{align}
\begin{split}
&\sum\limits_{m=0}^2\|\rho_{\delta,\nu,\mu}^{(m)}(\bm{x})\cdot\nabla^m u\|_{C^{0}(B_{\mathfrak{s}(\bm{x})/8}(\bm{x}))}
+
[\rho_{\delta,\nu,\mu}^{(2+\alpha)}(\bm{x})\cdot\nabla^2 u]_{C^{0,\alpha}(B_{\mathfrak{s}(\bm{x})/8}(\bm{x}))} \\
\leq & C\cdot\Big(\|\rho_{\delta,\nu+2,\mu}^{(0)}(\bm{x})\cdot \Delta u\|_{C^{0}(B_{\mathfrak{s}(\bm{x})/4}(\bm{x}))}
+[\rho_{\delta,\nu+2,\mu}^{(\alpha)}(\bm{x})
\cdot \Delta u]_{C^{0,\alpha}(B_{\mathfrak{s}(\bm{x})/4}(\bm{x}))}
\\
&+\|\rho_{\delta,\nu,\mu}^{(0)}(\bm{x}) \cdot u\|_{C^0(B_{\mathfrak{s}(\bm{x})/4}(\bm{x}))}\Big).
\end{split}
\end{align}
By the definition of the weighted norms, the next is to verify that for every $\bm{x}\in \mathring{\mathcal{M}}(T_-,T_+)$,
the weight function $\rho_{\delta,\nu,\mu}^{(k+\alpha)}$ is roughly a constant in the ball $B_{\mathfrak{s}(\bm{x})/4}(\bm{x})$ in the sense that there is a uniform constant $C=C_{k,\alpha}>0$ (independent of $T$) such that for any $\bm{y}\in B_{\mathfrak{s}(\bm{x})/4}(\bm{x})$,
\begin{equation}
C^{-1}\cdot \rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})\leq \rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{y}) \leq C\cdot \rho_{\delta,\nu,\mu}^{(k+\alpha)}(\bm{x}).\label{e:weight-function-control}
\end{equation}
The verification of \eqref{e:weight-function-control} follows from the comparison on $\mathfrak{s}(\bm{x})$ given in item (2) of Proposition \ref{p:regularity-scale}. So we have that
\begin{align}
\begin{split}
&\sum\limits_{m=0}^2\|\rho_{\delta,\nu,\mu}^{(m)}\cdot\nabla^m u\|_{C^{0}(B_{\mathfrak{s}(\bm{x})/8}(\bm{x}))}
+
[\rho_{\delta,\nu,\mu}^{(2+\alpha)}\cdot\nabla^2 u]_{C^{\alpha}(B_{\mathfrak{s}(\bm{x})/8}(\bm{x}))}\\
\leq &
C\Big(\|\rho_{\delta,\nu+2,\mu}^{(0)}\cdot \Delta u\|_{C^{0}(B_{\mathfrak{s}(\bm{x})/4}(\bm{x}))}+[\Delta u]_{C_{\delta,\nu+2,\mu}^{0,\alpha}(B_{\mathfrak{s}(\bm{x})/4}(\bm{x}))}+\|\rho_{\delta,\nu,\mu}^{(0)} \cdot u\|_{C^0(B_{\mathfrak{s}(\bm{x})/4}(\bm{x}))}\Big).
\end{split}
\end{align}
Then we obtain the weighted Schauder estimate \eqref{e:local-schauder}.
The proof of item (2) is the same as item (1). We just notice that higher order estimate holds in a geodesic ball without touching the singular locus $\mathcal{P}$.
We now prove
item (3), which
follows from the Schauder estimate for Neumann boundary problem.
As before, we only consider $r=1$ and prove the estimate for $\bm{x}\in\{T=T_-\}\subset \partial M_T$.
Let us choose the rescaled metric $\tilde{g} = \mathfrak{s}(\bm{x})^{-2} \cdot g$ with
\begin{equation}
\mathfrak{s}(\bm{x}) = (L_T(T_-))^{\frac{1}{2}}\cdot T^{\frac{2-n}{2n}}=1.\end{equation}
By the proof of Proposition \ref{p:regularity-scale}, for $T\gg1$ sufficiently large,
$(\mathcal M_T, \tilde{g}, \bm{x})$ is $C^k$-close to a fixed incomplete Calabi space $(\mathcal{C}_-^n, g_{\mathcal{C}_-^n},\bm{x}_{\infty})$ for any $k\in\mathbb{Z}_+$.
Moreover, the geodesic ball $B_{1/4}^+(\bm{x})$ with respect to the rescaled metric $\tilde{g}_T$ has uniformly bounded $C^{k,\alpha}$-geometry (independent of $T$) for any $k\in\mathbb{Z}_+$. So the standard Schauder estimate for the Neumann boundary problem reads as follows (see Section 6.7 of \cite{GT} for instance):
\begin{align}
\|u\|_{C^{k+2,\alpha}(B_{1/8}^+(\bm{x}))}
\leq C_{k,\alpha} \Big(\|\Delta u\|_{C^{k,\alpha}(B_{1/4}^+(\bm{x}))} + \Big\| \frac{\partial u}{\partial n}\Big\|_{C^{k+1,\alpha}(B_{1/4}^+(\bm{x})\cap \partial\mathcal M_T)}+ \| u\|_{C^0(B_{1/4}^+(\bm{x}))}\Big),
\end{align}
where $\frac{\partial}{\partial n}$ is the exterior normal vector field on $\partial\mathcal M_T$. So we obtain the desired weighted estimate.
\end{proof}
We finish this subsection with a weighted error estimate for the Calabi-Yau equation.
\begin{proposition}[Weighted error estimate] \label{p:CY-error-small}
Let $\mathrm{Err}_{CY}$ be the error function given by Definition \ref{d:error-function}.
For fixed parameters $\delta>0$, $\mu,\nu\in\mathbb{R}$ and $\alpha\in(0,1)$ which satisfy
\begin{align}
0<\delta < \delta_{e} & \equiv \frac{\sqrt{\lambda_1}}{n(|k_-|+|k_+|)},
\\
\nu + \alpha & >0,
\end{align}
where the constants $\lambda_1>0$, $k_->0$ and $k_+<0$ are given in Proposition \ref{p:existence-Greens-current}.
Then the weighted $C^{0,\alpha}$-estimate holds, \begin{equation}\|\mathrm{Err}_{CY}\|_{C^{0,\alpha}_{\delta, \nu, \mu}(\mathcal M_T)}=O(T^{-2+\frac{\nu+\alpha}{n}+\mu}).
\end{equation}
\end{proposition}
\begin{proof}
We again divide into different regions and estimate separately.
For $|z(\bm{x})|\leq 1$, applying Corollary \ref{c:psipower}, we have
\begin{equation}
(\omega_D+T^{-1}\psi)^{n-1}=\omega_D^{n-1}(1+T^{-1}\Tr_{\omega_D}\psi+\sum_{k\geq 2}T^{-k}\Phi_k),
\end{equation}
where $\Phi_k=O'(r^{k-1})$ is independent of $T$.
By \eqref{e:q-bounded-distance} we have
\begin{equation}
T^{-1}h=1+T^{-1}\Tr_{\omega_D}\psi+T^{-2}\underline B(z).
\end{equation}
Using \eqref{e:trace expansion equation}, it is easy to see that
$\|\mathrm{Err}_{CY}\|_{C^0(\{|z(\bm{x})|\leq 1\})}=O(T^{-2})$.
Immediately, by the definition of the weighted $C^0$-norm, we have
\begin{equation}
\|\mathrm{Err}_{CY}\|_{C_{\delta,\nu,\mu}^0(\{|z(\bm{x})|\leq 1\})}=O(T^{-2+\frac{\nu}{n}+\mu}),
\end{equation}
Next we consider the region $|z(\bm{x})|\geq 1$. Then by \eqref{e:greens-current-exp-asymp} we may write
\begin{align}
\psi=
\begin{cases}
(k_- z)\cdot \omega_D+\xi, & z\leq -1,
\\
(k_+ z)\cdot \omega_D+\xi, & z\geq 1,
\end{cases}
\end{align}
where $\xi=\epsilon(z)$.
So it follows that
\begin{align}
&(\omega_D+T^{-1}\psi)^{n-1}\nonumber\\
=&\Big((1+T^{-1}k_{\pm} z)\omega_D+T^{-1}\xi\Big)^{n-1}
\nonumber\\
=&\omega_D^{n-1}\Big((1+T^{-1}k_\pm z)^{n-1}+(1+T^{-1}k_\pm z)^{n-2}T^{-1}\Tr_{\omega_D}\xi+O(T^{-2})\Big).
\end{align}
By \eqref{e:h-asymptotics}, we have
$T^{-1}\cdot h=(1+T^{-1}k_\pm z)^{n-1}+T^{-1}\cdot \Tr_{\omega_D}\xi$.
So we obtain
\begin{equation}
\mathrm{Err}_{CY}=\Big((1+T^{-1}k_\pm z)^{-1}-(1+T^{-1}k_\pm z)^{-n+1}\Big)T^{-1}\Tr_{\omega_D}\xi+O(T^{-2}).
\end{equation}
Since for $z\in [T_-, T_+]$,
\begin{equation}
U_T(z)=T-T^{-\frac{n-2}{2}}(T+k_\pm z)^{\frac{n}{2}}=T(1-(1+T^{-1}k_{\pm}z)^{\frac{n}{2}})\leq -\frac{n}{2}\cdot k_{\pm}z.\label{e:U-upper-bound}
\end{equation}
Here we have used the following inequality: $(1-x)^p\geq 1-px$ for any $p\geq 1$ and $x\in(0,1)$.
By Proposition \ref{p:existence-Greens-current}, the asymptotics $\xi=\epsilon(z)$ has an explicit exponential decaying rate
$\epsilon(z)=O(e^{-(1-\tau)\sqrt{\lambda_1}z})$ for any $\tau\in(0,1)$. Applying \eqref{e:U-upper-bound} and the
the assumption
\begin{equation}0<\delta<\delta_e \equiv \frac{\sqrt{\lambda_1}}{n(|k_-|+|k_+|)},\end{equation} we conclude that, as $|z(\bm{x})|\to+\infty$,
the growth rate of $e^{\delta\cdot U_T(z(\bm{x}))}$
is slower than the decaying rate of $\epsilon(z)$.
Therefore, we have that \begin{align}
\|\mathrm{Err}_{CY}\|_{C^0(\{\|z(\bm{x})\|\geq 1\})} &= O(T^{-2}),
\\
\|\mathrm{Err}_{CY}\|_{C_{\delta,\nu,\mu}^0(\{\|z(\bm{x})\|\geq 1\})} &= O(T^{-2+\frac{\nu}{n}+\mu}).
\end{align}
The weighted $C^{0,\alpha}$-estimate can be obtained in a similar way.
It suffices to analyze the H\"older regularity around the singular set $\mathcal{P}$.
Notice that a fixed function in $O'(r)$ has bounded $C^{0,\alpha}$ norm, so
the weighted $C^{0,\alpha}$-estimate yields\begin{equation}
\|\mathrm{Err}_{CY}\|_{C_{\delta,\nu,\mu}^{\alpha}(\mathcal M_T)}=O(T^{-2+\frac{\nu+\alpha}{n}+\mu}).
\end{equation}
The proof is done.
\end{proof}
\subsection{Perturbation of complex structures}
\label{ss:perturbation of complex structures}
In Section \ref{ss:complex-geometry} we have identified the underlying complex manifold of our family of $C^{2, \alpha}$ K\"ahler metrics $(\mathcal M_T, \omega_T)$. In our gluing argument in Section \ref{ss:glued-metrics} we need to perturb the complex structure and accordingly perturb the K\"ahler forms. This section is devoted to the estimate of error caused by such a perturbation.
Under the holomorphic embedding of $\mathcal M_T$ into $\mathcal N^0$ defined in Section 4.2, $\Omega_T$ is identified with the standard holomorphic volume form $\Omega_{\mathcal N^0}$.
Fix $C_0>0$, and let $\mathcal V$ be the open neighborhood of $\mathcal P$ in $\mathcal N^0$ defined by
$\{r_+<C_0, r_-<C_0\}$.
Fix a smooth K\"ahler metric $\omega_{\mathcal N^0}$ on $\mathcal N^0$.
Assume that there are a family of complex structures $J_T'$ on $\mathcal V$ with holomorphic volume forms $\Omega_T'$ satisfying for all $k\geq 0$,
\begin{equation}
\sup_{\bm{x}\in \mathcal V}|\nabla^k_{\omega_{\mathcal N^0}}(\Omega_T'-\Omega_{\mathcal N^0})(\bm{x})|_{\omega_{\mathcal N^0}}\leq \underline\epsilon_{T^2}.
\end{equation}
We also assume there is a deformation of the form $\pi^*\omega_D$ over $\mathcal V$ to $\omega_{D, T}$, which is a closed $(1,1)$ form with respect $J_T'$, and satisfies that for all $k\geq 0$
\begin{equation}
\sup_{\bm{x}\in \mathcal V}|\nabla^k_{\omega_{\mathcal N^0}}(\omega_{D, T}-\pi^*\omega_D)(\bm{x})|_{\omega_{\mathcal N^0}}\leq \underline\epsilon_{T^2}.
\end{equation}
These assumptions will be met in our applications.
Let $\phi$ be the K\"ahler potential defined in \eqref{eqn7-10}. Then we define the new family of closed forms on $\mathcal V$ by
\begin{equation}
T^{\frac{n-2}{n}}\omega_T'\equiv T\omega_{D, T}+dJ_T'd\phi.
\end{equation}
\begin{proposition} \label{p: perturbation of complex structures}
For all sufficiently large $T$, the above $(\omega_T', \Omega_T')$ defines a family of $C^{1, \alpha}$-K\"ahler structures on $\mathcal V$ which satisfies that for all fixed $\alpha\in(0,1)$, $\delta,\mu, \nu\in \mathbb{R}$, we have
\begin{align}\label{e:Omega perturbation} \|\Omega_T'-\Omega_T\|_{C^{2, \alpha}_{\delta, \mu, \nu}(\mathcal V)}&=\underline \epsilon_{T^2},
\\
\label{e:omega perturbation}
\|\omega_T'-\omega_T\|_{C^{1, \alpha}_{\delta, \mu, \nu}(\mathcal V)}&=\underline \epsilon_{T^2}.\end{align}
\end{proposition}
\begin{remark} \label{r:C1alphavsC2alpha}
Notice that we lose one derivative control on $\omega_T'$. This is due to the fact that in the above definition of $\omega_{T}'$ we also have a derivative on the complex structure $J_T'$. Later we need at least $C^{1,\alpha}$ regularity on the metrics to obtain $C^{2, \alpha}$ weighted Schauder estimates for the Laplacian operator.
This technical issue is the reason why we need to show $(\omega_T, \Omega_T)$ defined in Section \ref{ss:kaehler-structures} is $C^{2,\alpha}$, and it determines the order we need in the expansion of the Green's current in Theorem \ref{t:Green-expansion}.
\end{remark}
\begin{proof}[Proof of Propsoition \ref{p: perturbation of complex structures}]
We first reduce the estimate to a local one. To do this, let us
choose finite holomorphic charts $\{(U_\beta, w_1, \ldots, w_{n-1})\}$ of $D$ and each chart $U_{\beta}$ is given by $\{|w_i|<1\}_{i=1}^{n-1}$, such that the following holds:
\begin{enumerate}
\item The smaller charts $V_{\beta}\subset U_{\beta}$ given by $\{|w_i|<1/2\}_{i=1}^{n-1}$ also cover $D$.
\item If $U_{\beta}\cap H\neq \emptyset$, then $U_{\beta}$ satisfies that for some $p\in H$, $w_i(p)=0$ for every $1\leq i\leq n-1$, and $H\cap U_{\beta}$ is given by $\{w_1=0\}$.
\item The line bundle $L$ restricted to every $U_\beta$ has a holomorphic trivialization $\sigma_{\beta}$ under which $\zeta_{\pm}$ can be viewed as local holomorphic functions on $\mathcal{N}^0$, and $\mathcal V\cap \pi^{-1}(U_\beta)$ is locally defined by $|\zeta_\pm|<C\cdot|\sigma_\beta|^{-|k_\pm|}$.
\end{enumerate}
The above gives an open cover of $\mathcal V$ by $\mathcal V\cap \pi^{-1}(V_\beta)$, and it suffices to prove the estimates in each of those open sets.
We only work with the charts that satisfy $U_\beta\cap H\neq \emptyset$, and the other case can be proved in a similar manner. For such charts, in $\pi^{-1}(U_\beta)$, by the definition of $\mathcal N^0$, we have that
\begin{equation}
\zeta_+\cdot \zeta_-=w_1 \cdot F(w_1, \ldots, w_{n-1})
\end{equation}
for a non-zero holomorphic function $F$. Without loss of generality, we may assume that $\{\zeta_+,\zeta_-, w_2, \ldots, w_{n-1}\}$ are holomorphic coordinates on $\pi^{-1}(U_\beta)$.
We first prove \eqref{e:Omega perturbation}.
The hypothesis implies that
\begin{equation} \label{e: holomorphic volume form difference}
\Omega_T'-\Omega_{T}=G_{i_1\ldots i_n} \cdot e_{i_1}\wedge\ldots\wedge e_{i_n},
\end{equation}
where each $e_j$ is one of $d\zeta_{\pm}, d\bar\zeta_{\pm}, dw_j, d\bar w_j$ for $2\leq j\leq n-1$, and $G_{i_1\ldots i_n}$ is a smooth function in $\zeta_{\pm}, \bar{\zeta}_{\pm}, w_j, \bar{w}_j $ for $2\leq j\leq n-1$ and its $k$-th derivative over $\mathcal V$ with respect to the fixed metric $\omega_{\mathcal N^0}$ is bounded by $\underline \epsilon_{T^2}$ for all $k$. Since a holomorphic function is automatically harmonic with respect to any K\"ahler metric, we have
\begin{equation}
\Delta_{\omega_T}\zeta_{\pm}=\Delta_{\omega_T}w_j=0,\quad 2\leq j\leq n-1.
\end{equation}
By item (1) of Corollary \ref{c:r zeta relation}, we have that $\mathcal V$ is contained in the region $|z|\leq 1$. Also notice by the discussion in Section \ref{ss:regularity-scales}, there is a constant $C>1$ such that for each $\bm{x} \in \mathcal V\cap \pi^{-1}(V_\beta)$, the ball $B_{C^{-1}\mathfrak{s}(\bm{x})}(\bm{x})$ is contained in $\pi^{-1}(U_\beta)\cap \{|z|\leq 2\}$. In this proof we denote by $C>0$ a constant which is independent of $T$ but may vary from line to line. Again by Corollary \ref{c:r zeta relation}, item (3) on $\pi^{-1}(U_\beta)\cap \{|z|\leq 2\}$, we have
\begin{equation}
|\zeta_\pm|\leq Cr_\pm \leq Ce^{3T}.
\end{equation}
Now applying Proposition \ref{p:local-weighted-schauder} to every $\bm{x}\in \mathcal V\cap \pi^{-1}(V_\beta)$ we obtain
\begin{equation}\label{e:eta pm estimate}
|\zeta_\pm|_{C^{3, \alpha}_{\delta, \mu, \nu}(\mathcal V\cap \pi^{-1}(V_\beta))}\leq C|\zeta_\pm|_{C^{0}_{\delta, \mu, \nu}(\pi^{-1}(U_\beta)\cap \{|z|\leq 1\})}\leq Ce^{3T}.
\end{equation}
Similarly, since on $U_\beta$ we have $|w_j|<1$, we get for $2\leq j\leq n-1$,
\begin{equation}
|w_j|_{C^{3, \alpha}_{\delta, \mu, \nu}(\mathcal V\cap \pi^{-1}(V_\beta))}\leq |w_j|_{C^{0}_{\delta, \mu, \nu}(\pi^{-1}(U_\beta)\cap \{|z|\leq 1\})}\leq C.
\end{equation}
Then using the chain rule and induction we get that
\begin{equation}
|G_{i_1\cdots i_n}|_{C^{3, \alpha}_{\delta, \nu,\mu}(\mathcal V\cap \pi^{-1}(V_\beta))}=\underline\epsilon_{T^2}\cdot Ce^{3T}=\underline \epsilon_{T^2}
\end{equation}
and hence
$|\Omega_T'-\Omega_T|_{C^{2, \alpha}_{\delta, \mu, \nu}(\mathcal V\cap \pi^{-1}(V_\beta))}=\underline \epsilon_{T^2}$.
Notice that the complex structure $J_T'$ is pointwise determined by the holomorphic volume form $\Omega_T'$ algebraically, so we have \begin{equation}
|J_T'-J_T|_{C^{2, \alpha}_{\delta, \nu,\mu}(\mathcal V\cap \pi^{-1}(V_\beta))}=\underline\epsilon_{T^2}.
\end{equation}
To prove \eqref{e:omega perturbation}, we write
\begin{equation}
\omega_T'-\omega_T = T(\omega_{D, T}-\pi^*\omega_D)+d((J_T'-J_T)d\phi).
\end{equation}
By assumption, and the above discussion, using \eqref{e:eta pm estimate} we get
\begin{equation}
|\omega_{D, T}-\pi^*\omega_D|_{C^{2, \alpha}_{\delta, \nu,\mu}(\pi^{-1}(V_\beta)\cap \{|z|\leq 1\})}=\underline\epsilon_{T^2}
\end{equation}
It is also easy to see that for two functions $V_1, V_2$, we have
\begin{equation}
|V_1\cdot V_2|_{C^{2, \alpha}_{\delta, \nu,\mu}(\mathcal V\cap \pi^{-1}(V_\beta))}\leq C\cdot T^m\cdot |V_1|_{C^{2, \alpha}_{\delta, \nu,\mu}(\mathcal V\cap \pi^{-1}(V_\beta))}\cdot|V_2|_{C^{2, \alpha}_{\delta, \nu,\mu}(\mathcal V\cap \pi^{-1}(V_\beta))},
\end{equation}
for some $m>0$ independent of $V_1$, $V_2$.
So the proof of \eqref{e:omega perturbation} is reduced to the following claim.
\vspace{0.5cm}
{\bf Claim:} There is a constant $C>0$ such that $|\phi|_{C^{3, \alpha}_{\delta, \nu,\mu}(\mathcal V\cap \pi^{-1}(V_\beta))}\leq e^{C\cdot T}$
holds for all $T\gg1$.
\vspace{0.5cm}
Let us prove the claim. Since by construction
$T^{\frac{n-2}{n}}\omega=T\pi^*\omega_D+dd^c\phi$,
we have that
\begin{equation}
\Delta_{T^{\frac{2n-2}{n}}\omega}\phi=n-T\cdot \Tr_{T^{\frac{2n-2}{n}}\omega}(\pi^*\omega_D).
\end{equation}
Since $\pi^*\omega_D$ is smooth on $\mathcal V$ and $\omega$ is parallel, again the above discussion gives that
\begin{equation}
|\Tr_{T^{\frac{2n-2}{n}}\omega}(\pi^*\omega_D)|_{C^{1, \alpha}_{\delta, \nu, \mu}(\mathcal V\cap\pi^{-1}(V_{\beta}))}\leq e^{CT}.
\end{equation}
Then Proposition \ref{p:local-weighted-schauder} implies that
\begin{equation}
|\phi|_{C^{3, \alpha}_{\delta, \nu, \mu}(\mathcal V\cap \pi^{-1}(V_\beta))}\leq e^{CT}+C|\phi|_{C^0_{\delta, \nu, \mu}(\pi^{-1}(U_\beta)\cap \{|z|\leq 1\})}.
\end{equation}
To bound the right hand side we use the formula
\begin{equation}
\phi=\int_{T_+}^z uh(u)du+\phi(T_+)=\int_{T_+}^0 uh(u)du+\phi(T_+)+\int_0^z uh(u)du.
\end{equation}
Hence by \eqref{e:h-asymptotics} and \eqref{e:h-bounded-z}, \begin{equation}
\phi=\frac{T}{2}z^2+\frac{1}{2}r+B_T+O(1),
\end{equation}
which gives
$|\phi|_{C^0_{\delta, \nu,\mu}(\pi^{-1}(V_\beta)\cap \{|z|\leq 1\})}\leq O(T^m)$
for some $m>0$. The conclusion then follows.
\end{proof}
\begin{remark}
In principle, it is possible to obtain more refined estimates on the higher order weighted norms of $\zeta_\pm$ and $\phi$ by more direct calculation. The above argument using weighted Schauder estimates avoids the lengthy computations, and it suffices for our purpose since in our setting the error caused by complex structure perturbation is at the scale $e^{-CT^2}$ while the weighted analysis in the region $\{|z|\leq 1\}$ only introduces at most $e^{CT}$ error.
\end{remark}
\section{Perturbation to Calabi-Yau metrics on the neck}
\label{s:neck-perturbation}
In Section \ref{ss:kaehler-structures}, we constructed a family of $C^{2,\alpha}$-K\"ahler structures $(\omega_T,\Omega_T)$ on $\mathcal M_T$ which are very close to Calabi-Yau metrics with weighted error estimates in Proposition \ref{p:CY-error-small}. The main result of this section is Theorem \ref{t:neck-CY-metric}.
Our goal in this section is to perturb $\omega_T$ to a genuine Calabi-Yau metric for any sufficiently large $T$.
Technically, this amounts to proving the uniform estimates demanded by the implicit function theorem (Lemma \ref{l:implicit-function}). Those estiamtes will be proved in Sections \ref{ss:perturbation-framework}-\ref{ss:incomplete weighted analysis}. In Section \ref{ss:renormalized-measure}, we will discuss some geometric information naturally arising in the {\it measured Gromov-Hausdorff convergence} of those Calabi-Yau metrics obtained in Theorem \ref{t:neck-CY-metric} on appropriate scales. The results in this section may have independent interest.
As mentioned in the introduction of this paper, it is the proof, but not the statement of Theorem \ref{t:neck-CY-metric} itself, that will be used in the proof of Theorem \ref{t:main-theorem}.
\subsection{Framework of perturbation}
\label{ss:perturbation-framework}
The studies and applications of the implicit function theorem have been well developed
in various contexts. We refer the readers
to the book \cite{Krantz} for seeing the comprehensive discussions and the history of the whole methodology.
For our practical and specific applications, we need the following quantitative version of implicit function theorem (Lemma \ref{l:implicit-function}), which is based on Banach contraction mapping principle.
To avoid confusions, we clarify several notations as follows:
\begin{itemize}
\item Let $\mathscr{L}: \mathfrak{S}_1 \to \mathfrak{S}_2$ be a bounded linear operator between normed linear spaces $\mathfrak{S}_1$ and $\mathfrak{S}_2$. Then the operator norm of $\mathscr{L}$ is defined by
\begin{equation}
\|\mathscr{L}\|_{op} \equiv \inf\Big\{ M_0\in \mathbb{R}_+ \Big| \ \|\mathscr{L}(\bm{v})\|_{\mathfrak{S}_2}\leq M_0\cdot \|\bm{v}\|_{\mathfrak{S}_1} , \ \forall \bm{v}\in \mathfrak{S}_1 \Big\}.
\end{equation}
\item We use the common notation $\bm{0}$ for the zero vector in every normed linear space.
\end{itemize}
\begin{lemma}
[Implicit function theorem]
\label{l:implicit-function}
Let $\mathscr{F}:\mathfrak{S}_1 \to \mathfrak{S}_2$ be a map between two Banach spaces such that for all $\bm{v}\in \mathfrak{S}_1$,
\begin{equation}\mathscr{F}(\bm{v})-\mathscr{F}(\bm{0})=\mathscr{L}(\bm{v})+\mathscr{N}(\bm{v}),\label{e:functional-expansion}\end{equation} where the operator $\mathscr{L}:\mathfrak{S}_1\to\mathfrak{S}_2$ is linear and the operator $\mathscr{N}:\mathfrak{S}_1\to \mathfrak{S}_2$ satisfies $\mathscr{N}(\bm{0})=\bm{0}$. Additionally we assume the following properties:
\begin{enumerate}
\item (Bounded inverse) $\mathscr{L}:\mathfrak{S}_1\to\mathfrak{S}_2$ is an isomorphism and there is some constant $C_L>0$ such that \begin{equation}\| \mathscr{L}^{-1} \|_{op} \leq C_L,\label{e:bounded-inverse}\end{equation} where $\mathscr{L}^{-1}$ is the inverse of $\mathscr{L}$.
\item There are constants $C_N>0$ and $r_0\in(0,\frac{1}{2C_L C_N})$ which satisfy the following:
\begin{enumerate}\item (Controlled nonlinear error) for all $\bm{v}_1,\bm{v}_2\in \overline{B_{r_0}(\bm{0})}\subset \mathfrak{S}_1$,
\begin{equation}\| \mathscr{N}(\bm{v}_1) - \mathscr{N}(\bm{v}_2) \|_{\mathfrak{S}_2} \leq C_N\cdot r_0 \cdot \| \bm{v}_1 - \bm{v}_2 \|_{\mathfrak{S}_1}.\label{e:nonlinear-term-control}\end{equation}
\item (Controlled initial error) $\mathscr{F}(\bm{0})$
is effectively controlled as follows,
\begin{equation}\| \mathscr{F}(\bm{0}) \|_{\mathfrak{S}_2}\leq \frac{r_0}{4C_L}.\label{e:very-small-initial-error}\end{equation}
\end{enumerate}
\end{enumerate}
Then the equation $\mathscr{F}(\bm{x})=\bm{0}$ has a unique solution $\bm{x}\in B_{r_0}(\bm{0})$ with the estimate
\begin{equation}
\|\bm{x}\|_{\mathfrak{S}_1} \leq 2C_L \cdot \|\mathscr{F}(\bm{0})\|_{\mathfrak{S}_2}.\label{e:apriori-estimate-of-the-fixed-point}
\end{equation}
\end{lemma}
\begin{remark}
In our applications, the constants $C_L>0$, $C_N>0$ and $r_0>0$ will be fixed as uniform constants (independent of $T\gg1$), which will be stated and proved in the next subsections. With the specified weight parameters $\delta,\mu,\nu$,
the error estimate in Proposition \ref{p:CY-error-small} in fact guarantees
$\|\mathrm{Err}_{CY}\|_{\mathfrak{S}_2} \to 0 $ as $T\to\infty$, which particularly implies $\| \mathscr{F}(\bm{0}) \|_{\mathfrak{S}_2}\to 0$ so that $\mathscr{F}$ satisfies item (2b) in the above lemma.
\end{remark}
To set up the perturbation problem in our setting,
we define the Banach spaces
\begin{align}
\begin{split}
\mathfrak{S}_1 & \equiv \Big\{\sqrt{-1}\partial\bar{\partial}\phi\in\Omega^{1,1}(\mathcal M_T)\Big| \phi\in C^{2, \alpha}(\mathcal M_T)\ \text{is}\ S^1\text{-invariant and satisfies} \ \frac{\partial \phi}{\partial n}\Big|_{\partial \mathcal M_T}=0\Big\},
\\
\mathfrak{S}_2 & \equiv \Big\{f\in C^{0,\alpha}(\mathcal M_T)\Big|f\ \text{is}\ S^1\text{-invariant and} \ \int_{\mathcal M_T} f \cdot \omega_T^n= 0\Big\},
\end{split}
\end{align}
endowed with the weighted H\"older norms
\begin{align}
\|\sqrt{-1}\partial\bar{\partial}\phi \|_{\mathfrak{S}_1} &\equiv \|\sqrt{-1}\partial\bar{\partial}\phi\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(X_t)}, \label{e:norm-of-S1-space}
\\
\|f \|_{\mathfrak{S}_2} &\equiv \|f\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(X_t)}.
\label{e:norm-of-S2-space}
\end{align}
Notice that an $S^1$-invariant function $\phi$ on $\mathcal M_T$ can be identified with a function on the quotient $Q_T$, and the Neumann condition $\frac{\partial\phi}{\partial n}|_{\partial \mathcal M_T}=0$ amounts to the condition $\partial_z\phi=0$ on $\partial Q_T$.
In this section, the parameters in the weighted norms are specified as follows:
\begin{enumerate}
\item[(NP1)] (Fix $\nu$) The parameters $\nu\in\mathbb{R}$ is chosen such that
\begin{align}
\nu\in(-1,0). \label{e:fix-nu-neck}
\end{align}
In our context, Lemma \ref{l:harmonic-removable-singularity} requires $\nu\in(-1,1)$. Furthermore, we need $\nu\in(-1,0)$ for effectively applying Proposition \ref{p:CY-error-small} to Lemma \ref{l:implicit-function}.
\item[(NP2)] (Fix $\alpha$) The H\"older exponent $\alpha\in(0,1)$ is chosen sufficiently small such that
\begin{equation}
\nu+\alpha<0.\label{e:fix-alpha-neck}
\end{equation}
\item[(NP3)] (Fix $\delta$) $\delta>0$ is chosen such that \begin{equation}
0<\delta < \delta_N \equiv \frac{1}{n\cdot (|k_-| + |k_+|)^n}\cdot \min\{\delta_b, \delta_e,\sqrt{\lambda_D}\},\label{e:fix-delta-neck}
\end{equation}
where
$\sqrt{\lambda_D}$ is in Lemma \ref{l:liouville-cylinder} (Liouville theorem for harmonic functions on the cylinder $Q$),
$\delta_e>0$ is in the error estimate Proposition \ref{p:CY-error-small}, $\delta_b\equiv \min \{\delta_{\mathcal{C}_-^n}, \delta_{\mathcal{C}_+^n}\}$ with the constants $\delta_{\mathcal{C}_{\pm}^n}$ given in Lemma \ref{l:Liouville-Calabi-space-SZ} which in turn depend on the two Calabi model spaces $(\mathcal C^n_{\pm}, g_{\mathcal C^n_{\pm}})$ respectively.
\item[(NP4)] (Fix $\mu$) The parameter $\mu$ is fixed by
\begin{equation}
\mu= \Big(1-\frac{1}{n}\Big)(\nu+2+\alpha).\label{e:fix-mu-neck}
\end{equation}
This condition guarantees that the weight function $\rho_{\delta,\nu,\mu}^{(\alpha)}$ with parameters specified as the above is uniformly bounded from below, which will be used in proving Proposition \ref{p:nonlinear-neck}.
\end{enumerate}
Now we explicitly write down the nonlinear functional $\mathscr{F}$ in our context. For $T\gg1$, starting with the $C^{2,\alpha}$-K\"ahler structure $(\omega_T,\Omega_T)$, we will solve the Calabi-Yau equation
\begin{equation}
\frac{1}{n!} (\omega_T+\sqrt{-1}\partial\bar{\partial}\phi)^n = \frac{(\sqrt{-1})^{n^2}}{2^n}\cdot \Omega_T\wedge \overline{\Omega}_T.\label{e:CY-eq-neck-T-large}
\end{equation}
To begin with, we appropriately normalize the holomorphic volume form $\Omega_T$.
Notice that the right hand side of \eqref{e:CY-eq-neck-T-large} satisfies
\begin{align}\int_{\mathcal M_T} \frac{(\sqrt{-1})^{n^2}}{2^n}\Omega_T\wedge\overline{\Omega}_T &= \int_{\mathcal M_T} h\cdot \frac{\omega_D^{n-1}}{(n-1)!} dz\wedge \Theta=\int_{T_-}^{T_+}\Big(\int_D h\cdot\frac{\omega_D^{n-1}}{(n-1)!}\Big)dz.\label{e:RHS-CY-eq}
\end{align}
By Proposition \ref{p:cohomology-constant}, we have
\begin{align}
[\tilde{\omega}(z)]
=
\begin{cases}
(T + k_-\cdot z)[\omega_D], & z<0,
\\
(T + k_+\cdot z)[\omega_D],& z>0,
\end{cases}
\end{align}
as a cohomology class in $H^2(D;\mathbb{R})$, which implies \begin{align}
\int_{\mathcal M_T} \frac{\omega_T^n}{n!} = T^{2-n}\int_{\mathcal M_T} \frac{\tilde\omega(z)^{n-1}}{(n-1)!}dz\wedge \Theta =\Big(\frac{k_+-k_-}{n\cdot k_-\cdot k_+}\Big)(T^2-1).\label{e:LHS-CY-eq}\end{align}
Comparing \eqref{e:RHS-CY-eq} and \eqref{e:LHS-CY-eq} and using \eqref{haverage}, we have that
\begin{equation}
\int_{\mathcal M_T} \frac{(\sqrt{-1})^{n^2}}{2^n}\Omega_T\wedge\overline{\Omega}_T=(1+O(T^{-2}))\int_{\mathcal M_T}\frac{\omega_T^n}{n!}.
\end{equation}
Recall that the error function $\mathrm{Err}_{CY}$ given in Definition \ref{d:error-function} satisfies \begin{equation}
\frac{(\sqrt{-1})^{n^2}}{2^n}\Omega_T\wedge\overline{\Omega}_T=(1+\mathrm{Err}_{CY}) \frac{\omega_T^n}{n!}.
\end{equation}
Therefore, multiplying $\Omega_T$ by a $T$-dependent constant $c_T$ of the form $1+O(T^{-2})$, we may assume
\begin{align}
\int_{\mathcal M_T}\mathrm{Err}_{CY}\cdot\omega_T^n=0.\label{e:integral-error-zero}\end{align}
Applying Proposition \ref{p:CY-error-small} and \eqref{e:fix-mu-neck},
\begin{equation}
\label{e:new error estimate}\|\mathrm{Err}_{CY}\|_{C^{0,\alpha}_{\delta, \nu+2, \mu}(\mathcal M_T)}=O(T^{\nu+\alpha}).
\end{equation}
Now we define
\begin{equation}
\mathscr{F}: \mathfrak{S}_1\rightarrow C^{0, \alpha}(\mathcal M_T);\quad \sqrt{-1}\partial\bar{\partial}\phi \mapsto \frac{(\omega_T+\sqrt{-1}\partial\bar{\partial}\phi)^n}{\omega_T^n}-(1-\mathrm{Err}_{CY}).
\end{equation}
\begin{lemma}
$\mathscr{F}$ maps $\mathfrak{S}_1$ into $\mathfrak{S}_2$, and $ \|\mathscr{F}(\bm{0})\|_{\mathfrak{S}_2}=O(T^{\nu+\alpha}).
$
\end{lemma}
\begin{proof}The lemma amounts to proving that
$\int_{\mathcal M_T} \mathscr{F}(\sqrt{-1}\partial\bar{\partial} \phi)\omega_T^n=0$.
Since \eqref{e:integral-error-zero} holds for the normalized $\Omega_T$,
by Stokes' theorem, we have that
\begin{equation}\int_{\mathcal M_T} (\omega_T +\sqrt{-1}\partial\bar{\partial}\phi)^n-\int_{\mathcal M_T}\omega_T^n=\int_{\partial\mathcal M_T} \gamma,\end{equation}
where $\gamma$ is the sum of terms which contain a factor $d^c\phi$ and other factors either $dd^c\phi$ or $\omega_T$. We claim that $\gamma$ identically vanishes on $\partial\mathcal M_T$. It suffices to show $\partial_t\lrcorner\gamma=0$. Since by assumption $\phi$ is $S^1$-invariant, so we have $\partial_t \phi=0$. By the Neumann boundary condition, we also have $d^c\phi(\partial_t)=0$ on $\partial\mathcal M_T$. This follows from the observation that $J\partial_t=\nabla z$.
Now
\begin{align}\partial_t\lrcorner \omega_T|_{\partial\mathcal M_T}&=dz|_{\partial\mathcal M_T}=0,
\\
\partial_t\lrcorner dd^c\phi&=\mathcal L_{\partial_t} (d^c\phi)-d(\partial_t\lrcorner d^c\phi)=-d(\partial_t\lrcorner d^c\phi).\end{align}
The last term vanishes on $\partial \mathcal M_T$ since $d^c\phi(\partial_t)=0$ pointwise on $\partial\mathcal M_T$.
\end{proof}
Now we are ready to state the main result in this section.
\begin{theorem}[Existence of $S^1$-invariant Calabi-Yau metrics] \label{t:neck-CY-metric}
For each sufficiently large $T$, there exists an $S^1$-invariant Calabi-Yau metric of the form $\omega_{T, CY}=\omega_T + \sqrt{-1}\partial\bar{\partial}\phi(T)
$, where $\phi(T)\in \mathfrak{S}_1$ and
\begin{equation} \label{e:eqn524}
\|\sqrt{-1}\partial\bar{\partial}\phi(T)\|_{\mathfrak{S}_1}\leq C_0\cdot T^{\nu+\alpha},
\end{equation}
where $C_0>0$ is a uniform constant independent of $T\gg1$ and
the weighted H\"older norm of $\mathfrak{S}_1$ is defined in \eqref{e:norm-of-S1-space} for parameters $\nu$, $\alpha$, $\delta$ and $\mu$ satisfying \eqref{e:fix-nu-neck},
\eqref{e:fix-alpha-neck}, \eqref{e:fix-delta-neck} and \eqref{e:fix-mu-neck}.
\end{theorem}
To prove Theorem \ref{t:neck-CY-metric}, we write
\begin{equation}
\mathscr{F}(\sqrt{-1}\partial\bar{\partial}\phi) - \mathscr{F}(\bm{0}) = \mathscr{L}(\sqrt{-1}\partial\bar{\partial}\phi) + \mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi),
\end{equation}
for any $\sqrt{-1}\partial\bar{\partial}\phi \in \mathfrak{S}_1$, where
\begin{align}
\mathscr{L}(\sqrt{-1}\partial\bar{\partial}\phi) & \equiv \Delta \phi,\label{e:linear-MT}
\\
\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi)\cdot \omega_T^n & \equiv (\omega_T+\sqrt{-1}\partial\bar{\partial}\phi)^n-\omega_T^n - n \omega_T^{n-1}\wedge \sqrt{-1}\partial\bar{\partial}\phi.\label{e:nonlinear-MT}
\end{align}
By definition of the weight function and Lemma \ref{l:weight-function-lower-bound-estimate}, we have the following nonlinear error estimate.
\begin{lemma}
[Nonlinear error estimate]\label{p:nonlinear-neck} There exists a constant $C_N>0$ independent of
$T\gg1$ such that for all
$\varrho\in (0,\frac{1}{2})
$
and
\begin{equation}\sqrt{-1}\partial\bar{\partial}\phi_1\in \overline{B_{\varrho}(\bm{0})} \subset \mathfrak{S}_1, \quad \sqrt{-1}\partial\bar{\partial}\phi_2\in \overline{B_{\varrho}(\bm{0})}\subset\mathfrak{S}_1,\end{equation} we have
\begin{align} \|\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_1)-\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_2)\|_{\mathfrak{S}_2}
\leq C_N \cdot \varrho \cdot \|\sqrt{-1}\partial\bar{\partial}(\phi_1-\phi_2)\|_{\mathfrak{S}_1}.
\end{align}
\end{lemma}
\begin{proof}
By definition,
\begin{align}
& \Big(\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_1)-\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_2)\Big)\cdot \omega_T^n
\nonumber\\
= & \sum\limits_{k=2}^n\begin{pmatrix}
n \\
k
\end{pmatrix} \cdot \omega_T^{n-k}\wedge \Big((\sqrt{-1}\partial\bar{\partial}\phi_1)^k - (\sqrt{-1}\partial\bar{\partial}\phi_2)^k \Big).
\end{align}
By the definition of the norm on $\mathfrak{S}_1$,
we have
\begin{equation}
\|\sqrt{-1}\partial\bar{\partial}\phi_1\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal M_T)} \leq \varrho , \quad
\|\sqrt{-1}\partial\bar{\partial}\phi_2\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal M_T)} \leq \varrho .
\end{equation}
With $\mu$ specified by \eqref{e:fix-mu-neck}, Lemma \ref{l:weight-function-lower-bound-estimate} shows that the weight function $\rho_{\delta,\nu+2,\mu}^{(\alpha)}:\mathcal M_T\to \mathbb{R}_+$ has a uniform lower bound $\rho_{\delta,\nu+2,\mu}^{(\alpha)}(\bm{x}) \geq 1$ for any $\bm{x}\in \mathcal M_T$,
So we have the pointwise estimates on $\mathcal M_T$,
\begin{equation}
|\sqrt{-1}\partial\bar{\partial}\phi_1 | \leq C_0\cdot \varrho \quad \text{and} \quad
|\sqrt{-1}\partial\bar{\partial}\phi_2 | \leq C_0\cdot \varrho,
\end{equation}
where $C_0>0$ is a uniform constant independent of $T\gg1$. This implies the pointwise estimate
\begin{equation}
|\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_1)-\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_2)
| \leq C_N \cdot \varrho \cdot |\sqrt{-1}\partial\bar{\partial}(\phi_1-\phi_2)|,
\end{equation}
where
$C_N>0$ is a uniform constant independent of $T$.
Write the above in terms of the weighted norms, we have
\begin{equation}
\|\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_1)-\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_2)
\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal M_T)} \leq C_N \cdot \varrho \cdot \|\sqrt{-1}\partial\bar{\partial}(\phi_1-\phi_2)\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal M_T)}.
\end{equation}
The proof is done.
\end{proof}
To apply the implicit function theorem,
we still need to prove the weighted linear estimate, which will be completed in the following subsections.
\subsection{Some Liouville type theorems and removable singularity theorems}
\label{ss:some liouville theorems}
In this subsection, we introduce some removable singularity and Liouville type theorems, which will be needed in the proof of Proposition \ref{p:global-injectivity-estimate}.
For the convenience of discussions, we give precise statement here.
\begin{lemma}[Removable singularity]
\label{l:harmonic-removable-singularity}Let $(M^m,g)$ be a Riemannian manifold with $m\geq 3$ such that $B_R(p)$
has a compact closure in $B_{2R}(p)$. Let $K\subset M^m$ be a smooth submanifold with $\dim(K)=k_0\leq m-3$. If $u$ is harmonic in $B_R(p)\setminus K$ and there is some
$\epsilon\in(0,1)$ such that
\begin{equation}
|u(x)|\leq \frac{C}{d_g(x,K)^{(m-2-k_0)-\epsilon}},
\end{equation}
then $u$ is harmonic in $B_R(p)$.
\end{lemma}
\begin{proof}
The point is to apply integration by parts to show that $u$ is a weak solution to $\Delta u =0$ on $B_R(p)$. The computations are routine and standard in the literature, so we just skip it.
\end{proof}
\begin{lemma}
[Liouville theorem on $\mathbb{R}^{m+n}$]\label{l:liouville-product} Given $m,l\in\mathbb{Z}_+$ with $m+l\geq 3$, Let $\mu_p\in(-1,1)\setminus\{0\}$ and let $u\in C^{\infty}(\mathbb{R}^{m+l})$ be a harmonic function on the Euclidean space $(\mathbb{R}^{m+l}, g_{\mathbb{R}^m}\oplus g_{\mathbb{R}^l})$. If $u$ satsifies
\begin{align}
|u(x,y)| \leq \frac{C}{|x|^{\mu_p}} \quad \text{for every}\ (x,y)\in (\mathbb{R}^m\setminus \{0^m\})\times\mathbb{R}^l,\label{e:partial-control}
\end{align}
then $u\equiv 0$ on $\mathbb{R}^{m+l}$.
\end{lemma}
\begin{proof}
The proof is done by using separation of variables.
For the simplicity of notations, we denote
\begin{equation}d\equiv m+l \geq 3.\end{equation}
Let $(r,\Theta)\in\mathbb{R}^{d}$ be the polar coordinate system in $\mathbb{R}^{d}$, so the Laplacian of $u$ can be written as
\begin{equation}
\Delta_{\mathbb{R}^{d}} u = \frac{\partial^2u}{\partial r^2} + \frac{d-1}{r}\cdot \frac{\partial u}{\partial r} + \frac{1}{r^2}\cdot\Delta_{\mathbb{S}^{d-1}}u.
\end{equation}
We make separation of variables on the punctured Euclidean space $\mathbb{R}^{d}\setminus\{0^{d}\}$. Let
\begin{equation}\lambda_j\equiv j(j+d-2),\ j\in\mathbb{N},\end{equation} be the spectrum of the unit round sphere $\mathbb{S}^{d-1}$. Correspondingly, let $\varphi_j\in C^{\infty}(\mathbb{S}^{d-1})$ satisfy
\begin{equation}
-\Delta_{\mathbb{S}^{d-1}}\varphi_j(\Theta) = \lambda_j \varphi_j(\Theta).
\end{equation}
Then the function $u(r,\Theta)$ has the expansion along the fiber $\mathbb{S}^{d-1}$,
\begin{equation}
u(r,\Theta) = \sum\limits_{j=0}^{\infty} u_j(r)\cdot\varphi_j(\Theta).\label{e:L2-exp-polar}
\end{equation}
Immediately, for each $j\in\mathbb{N}$, the coefficient function $u_j(r)$ solves the Euler-Cauchy equation,
\begin{equation}
u_j''(r) + \frac{d-1}{r}\cdot u_j'(r) - \frac{1}{r^2} \cdot\lambda_j \cdot u_j(r) =0,
\end{equation}
which has a general solution
\begin{equation}
u_j(r)=C_j\cdot r^{p_j} + C_j^*\cdot r^{q_j},\label{e:polar-general-solution}
\end{equation}
where $p_j=\frac{2-d+\sqrt{(d-2)^2+4\lambda_j}}{2}\geq 0$
and $q_j=\frac{2-d-\sqrt{(d-2)^2+4\lambda_j}}{2}<0$ solve the quadratic equation
\begin{equation}
w^2 + (d-2)w - \lambda_j = 0.
\end{equation}
So it is obvious
\begin{align}
p_0 & = 0 , \quad q_0=2-d\leq -1,\nonumber\\
p_j &\geq p_1 =1,\nonumber \\
q_j & \leq q_1 = 1-d \leq -2, \quad j\in\mathbb{Z}_+.\label{e:radial-gap}
\end{align}
In the following, we will show that, given the growth condition \eqref{e:partial-control} for $\mu_p\in(-1,1)\setminus\{0\}$, then for each $j\in\mathbb{N}$ and for each $r>0$, the coefficient $u_j(r)$ satisfies
\begin{equation}
|u_j(r)| \leq \frac{Q_j}{r^{\mu_p}},
\end{equation}
where $Q_j\in\mathbb{R}$.
In fact, so it follows from the expansion \eqref{e:L2-exp-polar} that for each $j\in\mathbb{N}$,
\begin{equation}
u_j(r) = \int_{\mathbb{S}^{d-1}} u(r,\Theta) \cdot \varphi_j \dvol_{\mathbb{S}^{d-1}},
\end{equation}
which implies
\begin{equation}
|u_j(r)| \leq |\varphi_j|_{L^{\infty}(\mathbb{S}^{d-1})}\cdot \int_{\mathbb{S}^{d-1}}\frac{1}{|x|^{\mu_p}} \dvol_{\mathbb{S}^{d-1}}.
\end{equation}
Next, we will write the above integral in the polar coordinates \begin{align}\Theta\equiv(\theta_1,\ldots,\theta_{d-1}),\quad \theta_1,\ldots,\theta_{d-2}\in[0,\pi],\quad \theta_{d-1}\in[0,2\pi].\end{align} Denote by $d\Theta\equiv d\theta_1\wedge d\theta_2\wedge \ldots \wedge d\theta_{d-1}$, then it is by elementary calculations that,
$|x| = r^m\cdot \prod\limits_{k=1}^{d-m}|\sin\theta_{k}|$ and
$\dvol_{\mathbb{S}^{d-1}} = \prod\limits_{k=1}^{d-2}(\sin^{d-k-1}\theta_k)\cdot d\Theta$.
Therefore,
\begin{equation}
\int_{\mathbb{S}^{d-1}}\frac{1}{|x|^{\mu_p}} \dvol_{\mathbb{S}^{d-1}}=\frac{1}{r^{\mu_p}}\int_{\mathcal{D}_{\Theta}} \frac{\prod\limits_{k=1}^{d-2}(\sin^{d-k-1}\theta_k)}{\prod\limits_{k=1}^{d-m}|\sin\theta_{k}|^{\mu_p}}\cdot d\Theta,
\end{equation}
where $\mathcal{D}_{\Theta}\equiv\{0\leq \theta_1,\ldots, \theta_{d-2}\leq \pi, \ 0\leq \theta_{d-1}\leq 2\pi\}$.
By assumption, $\mu_p\in(-1,1)\setminus\{0\}$, then the following is integrable
\begin{equation}
\mathcal{I}_0\equiv\int_{\mathcal{D}_{\Theta}}\prod\limits_{k=1}^{d-2}(\sin^{d-k-1}\theta_k)(\prod\limits_{k=1}^{d-m}|\sin\theta_{k}|^{\mu_p})^{-1}\cdot d\Theta.
\end{equation}
Therefore, for each
$j\in\mathbb{N}$, it holds that for all $r>0$,
\begin{equation}
|u_j(r)| \leq \frac{\mathcal{I}_0\cdot|\varphi_j|_{L^{\infty}(\mathbb{S}^{d-1})}}{r^{\mu_p}} \equiv \frac{Q_j}{r^{\mu_p}}
\end{equation}
Now we go back to the representation of $u_j(r)$ in \eqref{e:polar-general-solution} and we analyze the growth behavior of function as $r\ll1$ and $r\gg1$. Applying the assumption $\mu_p\in(-1,1)\setminus\{0\}$ and the gap obtained in \eqref{e:radial-gap}, we have that, for each $j\in\mathbb{N}$, $C_j=C_j^*=0$. Therefore, $u\equiv 0$.
\end{proof}
We will also need the following lemma which holds on a cylindrical space $D\times\mathbb{R}$.
The proof follows from the rather standard separation of variables. We omit the details.
\begin{lemma}
[Liouville theorem on a cylinder]
\label{l:liouville-cylinder}
Let $(Q,g_Q)\equiv(D\times\mathbb{R},g_Q)$ be a cylinder with a product Riemannian metric $g_Q = g_D\oplus dz^2$, where $(D,g_D)$ is a closed Riemannian manifold. Denote by $\lambda_D>0$ the lowest eigenvalue of the Laplace-Beltrami operator of $(D,g_D)$ acting on functions. If $u$ is a harmonic function on $Q$ satisfying the growth control
\begin{equation}|u| = O(e^{\lambda_c\cdot z})\end{equation} for some $\lambda_c\in(0,\sqrt{\lambda_D})$, then
$u\equiv 0$.
\end{lemma}
Finally we quote a Liouville theorem on the Calabi model space, which is proved in \cite{SZ-Liouville} (see Corollary 5.3.1 in this paper).
\begin{lemma}[Liouville theorem on Calabi model space,
\cite{SZ-Liouville}] \label{l:Liouville-Calabi-space-SZ} Let $(\mathcal{C}^n, g_{\mathcal{C}^n})$ be a Calabi model space. There exists a constant $\delta_{\mathcal{C}^n}>0$ such that the following holds:
Let $u$ be a harmonic function on $\mathcal{C}^n$ which satisfies the Neumann condition $\frac{\partial u}{\partial n}|_{\partial\mathcal{C}^n}=0$. If $u=O(e^{\delta_{\mathcal{C}^n}\cdot z^{\frac{n}{2}}})$ as $z\rightarrow\infty$, then $u$ is a constant on $\mathcal{C}^n$.
\end{lemma}
\subsection{Weighted analysis and existence of incomplete Calabi-Yau metrics}
\label{ss:incomplete weighted analysis}
The main focus of this subsection is to prove the following proposition.
\begin{proposition}[Uniform injectivity estimate on the neck]\label{p:estimate-L-neck}
For any sufficiently large parameter $T\gg1$, the linearized operator defined in \eqref{e:linear-MT}
\begin{equation}\mathscr{L}: \mathfrak{S}_1\rightarrow \mathfrak{S}_2, \quad \sqrt{-1}\partial\bar{\partial}\phi\mapsto \Delta\phi\end{equation}
is an isomorphism and satisfies the uniform injectivity estimate,
\begin{equation}
\|\sqrt{-1}\partial\bar{\partial}\phi\|_{\mathfrak{S}_1}\leq C_L \cdot \|\Delta\phi\|_{\mathfrak{S}_2}.
\end{equation}
Here the constant $C_L>0$ is independent of the parameter $T\gg1$.
\end{proposition}
A preliminary ingredient in proving Proposition \ref{p:estimate-L-neck} is the following weighted Schauder estimate.
\begin{proposition}[Weighted Schauder estimate on the neck]\label{p:neck-weighted-schauder}
There exists a constant $C>0$ such that for $T\gg1$, and all $u\in C^{2, \alpha}(\mathcal M_T)$, the following holds
\begin{equation}
\|u \|_{C_{\delta,\nu,\mu}^{2,\alpha}(\mathcal M_T)} \leq C \Big(\|\Delta u\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal M_T)} + \Big\|\frac{\partial u}{\partial n}\Big\|_{C_{\delta,\nu+1,\mu}^{1,\alpha}(\partial\mathcal M_T)}+\|u\|_{C_{\delta,\nu,\mu}^0(\mathcal M_T)}\Big).
\end{equation}
\end{proposition}
\begin{proof}
The proof follows directly from
Proposition \ref{p:local-weighted-schauder} and standard covering argument. \end{proof}
Next, the key part of the injectivity estimate in Proposition \ref{p:estimate-L-neck} is the following weighted
estimate for higher derivatives with respect to the Neumann boundary value problem.
\begin{proposition}[Uniform derivative estimates on the neck] \label{p:neck-uniform-injectivity}
Let the parameters $\mu$, $\nu$, $\alpha$, $\delta$ satisfy the conditions in \eqref{e:fix-nu-neck}, \eqref{e:fix-alpha-neck}, \eqref{e:fix-delta-neck} and \eqref{e:fix-mu-neck}. Then
there exists a uniform constant $C>0$ such that for $T\gg1$, and for all
$u\in C^{2,\alpha}(\mathcal M_T)$ satisfying the boundary condition $\frac{\partial u}{\partial n}|_{\partial\mathcal M_T}=0$, we have
\begin{align}
\|\nabla u\|_{C_{\delta,\nu+1,\mu}^{0}(\mathcal M_T)} + \|\nabla^2 u\|_{C_{\delta,\nu+2,\mu}^{0}(\mathcal M_T)} + [u]_{C_{\delta,\nu,\mu}^{2,\alpha}(\mathcal M_T)} \leq C \cdot \|\Delta u\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal M_T)}. \label{e:hoelder-neck}
\end{align}
\end{proposition}
\begin{proof}
The proof of \eqref{e:hoelder-neck} consists of two primary steps:
In the first step, we will prove the weighted $C^1$ and $C^2$ estimates,
\begin{equation}
\|\nabla u\|_{C_{\delta,\nu+1,\mu}^{0}(\mathcal M_T)} + \|\nabla^2 u\|_{C_{\delta,\nu+2,\mu}^{0}(\mathcal M_T)} \leq C \cdot \|\Delta u\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal M_T)},\label{e:neck-C2-estimate}\end{equation}
where $C>0$ is independent of $T$. Once \eqref{e:neck-C2-estimate} is accomplished, we will prove the weighted $C^{2,\alpha}$-estimate for some uniform constant $C>0$ independent of $T$,
\begin{equation}
[u]_{C_{\delta,\nu,\mu}^{2,\alpha}(\mathcal M_T)} \leq C \cdot \|\Delta u\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal M_T)}.\label{e:neck-top-semi-norm}\end{equation}
\begin{flushleft}
{\bf Step 1} (weighted $C^1$ and $C^2$ estimates):
\end{flushleft}
We will prove \eqref{e:neck-C2-estimate} by contradiction.
Suppose no such uniform constant $C>0$ exists. That is, for fixed parameters $(\nu, \alpha, \delta, \mu)$,
there are the following sequences:
\begin{enumerate}
\item
A sequence $T_j\to+\infty$.
\item A sequence of $C^{2,\alpha}$-functions $u_j$ on $\mathcal M_j\equiv \mathcal M_{T_j}$ which satisfy
\begin{align}
\begin{split}
\frac{\partial u_j}{\partial n}\Big|_{\partial\mathcal M_j} & = 0, \\
\|\nabla u_j\|_{C_{\delta,\nu+1,\mu}^{0}(\mathcal M_j)} + \|\nabla^2 u_j\|_{C_{\delta,\nu+2,\mu}^{0}(\mathcal M_j)} & = 1,
\\
\|\Delta u_j\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal M_j)} &\to 0, \quad j\to+\infty.
\end{split}
\end{align}
\end{enumerate}
So it follows that either
$\|\nabla u_j\|_{C_{\delta,\nu+1,\mu}^{0}(\mathcal M_j)}\geq \frac{1}{2}$ or
$\|\nabla^2 u_j\|_{C_{\delta,\nu+2,\mu}^{0}(\mathcal M_j)}\geq \frac{1}{2}$.
Without loss of generality, we only consider the first case and let $\bm{x}_j\in\mathcal M_j$ satisfy
\begin{equation}
|\rho_{\delta,\nu+1,\mu}^{(0)}(\bm{x}_j)\cdot\nabla u_j(\bm{x}_j)|=\|\nabla u_j\|_{C_{\delta,\nu+1,\mu}^{0}(\mathcal M_j)}\geq\frac{1}{2}.\end{equation}
Let us
renormalize $u_j$ by taking
$ v_j(\bm{x}) \equiv u_j(\bm{x}) - u_j(\bm{x}_j)$.
Immediately, $v_j(\bm{x}_j) = 0$, $\frac{\partial v_j}{\partial n}|_{\partial\mathcal M_j}=0$ and
\begin{align}\label{e:v_j-conditions}
\begin{split}
|\rho_{\delta,\nu+1,\mu}^{(0)}(\bm{x}_j)\cdot\nabla v_j(\bm{x}_j)|=\|\nabla v_j\|_{C_{\delta,\nu+1,\mu}^{0}(\mathcal M_j)}&\geq\frac{1}{2},
\\
\|\nabla v_j\|_{C_{\delta,\nu+1,\mu}^{0}(\mathcal M_j)} + \|\nabla^2 v_j\|_{C_{\delta,\nu+2,\mu}^{0}(\mathcal M_j)} & = 1,
\\
\|\Delta v_j\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal M_j)} &\to 0,
\\
\|v_j\|_{C_{\delta,\nu,\mu}^{0}(\mathcal M_j)} &\leq C_0,
\end{split}
\end{align}
where $C_0>0$ is independent of $T_j$.
Applying the weighted Schauder estimate in Proposition \ref{p:neck-weighted-schauder}, we have that
$ \|v_j\|_{C_{\delta,\nu,\mu}^{2,\alpha}(\mathcal M_j)} \leq C_0$ for some uniform constant $C_0>0$ independent of $T_j$.
We will rescale $(\mathcal M_j, g_{T_j})$ around the above reference points $\bm{x}_j$
such that the desired contradiction will arise on the limiting space.
Denote the rescaling factors as follows:
\begin{enumerate}
\item {\bf Rescaling of the metrics:}
Let $\tilde{g}_{T_j}\equiv \mathfrak{s}(\bm{x}_j)^{-2}\cdot g_{T_j}$, then with respect to the fixed reference point $\bm{x}_j\in\mathcal M_j$ picked as the above and passing to a subsequence, we have the convergence
$(\mathcal M_j, \tilde{g}_{T_j}, \bm{x}_j) \xrightarrow{GH} (X_{\infty}, \tilde{d}_{\infty}, \bm{x}_{\infty})$ with the rescaled limit $X_{\infty}$ identified in Proposition \ref{p:regularity-scale}.
\item {\bf Rescaling of the solutions:}
Let $\kappa_j>0$ be a sequence of rescaling factors which will be determined later, such that $\tilde{v}_j \equiv \kappa_j \cdot v_j$.
\item {\bf Rescaling of the weight functions:}
Denote by $\tilde{\rho}_{j,\delta,\nu,\mu}^{(k+\alpha)}$ be the weight functions on the rescaled spaces $(\mathcal M_j, \tilde{g}_{T_j},\bm{x}_j)$ which are given by by
$\tilde{\rho}_{j,\delta,\nu,\mu}^{(k+\alpha)} = \tau_j \cdot \rho_{j,\delta,\nu,\mu}^{(k+\alpha)}$, so that $\tilde{\rho}_{j,\delta,\nu,\mu}^{(k+\alpha)}$ converges to $\tilde{\rho}_{\infty,\delta,\nu,\mu}^{(k+\alpha)}$ on $(X_{\infty}, \tilde{g}_{\infty},\bm{x}_{\infty})$. Notice that the rescaling factor $\tau_j$ depends on $k$ and $\alpha$.
\end{enumerate}
In the following, we study the convergence of the renormalized functions $\tilde{v}_j$ with respect to the pointed convergence of the rescaled spaces $(\mathcal M_j,\tilde{g}_{T_j},\bm{x}_j)$ for $\bm{x}_j$ in every case in the proof of Proposition \ref{p:regularity-scale}.
The main goal is to show that $\tilde{v}_{\infty}\equiv 0$ on each rescaled limit $X_{\infty}$ which gives the desired contradiction.
To begin with, we fix the rescaling factors. Notice that as computing the weighted norms of $\tilde{v}_j$ in terms of $\tilde{g}_{T_j}$ and $\tilde{\rho}_{j,\delta,\nu,\mu}^{(k+\alpha)}$, the property
$\|\tilde{v}_j\|_{C_{\delta,\nu,\mu}^{k,\alpha}(\mathcal M_j)}=\|v_j\|_{C_{\delta,\nu,\mu}^{k,\alpha}(\mathcal M_j)}$ holds if we require the rescaling factors to satisfy
\begin{align}\frac{\tau_j \cdot \kappa_j}{(\mathfrak{s}(\bm{x}_j)^{-1})^{k+\alpha}}= 1.\end{align} So the rescaling factors $\tau_j$ and $\kappa_j$ are fixed as follows such that \eqref{e:v_j-conditions} retains for $\tilde{v}_j$ and $\bm{x}_j$:
\begin{itemize}
\item
If $|z(\bm{x}_j)|$ is uniformly bounded as $j\to+\infty$, \begin{align}
\begin{cases}
\tau_j = (\mathfrak{s}_j^{-1})^{\nu+k+\alpha}\cdot T_j^{-\mu}
\\
\kappa_j = (\mathfrak{s}_j^{-1})^{-\nu} \cdot T_j^{\mu}.
\end{cases}
\end{align}
\item If $|z(\bm{x}_j)|\to +\infty$,
\begin{align}
\begin{cases}
\tau_j = (\mathfrak{s}_j^{-1})^{\nu+k+\alpha} \cdot e^{-T_j}\cdot T_j^{-\mu}
\\
\kappa_j = (\mathfrak{s}_j^{-1})^{-\nu} \cdot e^{T_j} \cdot T_j^{\mu}.
\end{cases}
\end{align}
\end{itemize}
As in the proof of Proposition \ref{p:regularity-scale}, we will produce a contradiction in each of the following four cases depending upon the location of $\bm{x}_j\in\mathcal M_j$.
\vspace{0.5cm}
\begin{flushleft}
{\it Case (1): There is a constant $\sigma_0\geq 0$ such that $r(\bm{x}_j)\cdot T_j\to \sigma_0$}.
\end{flushleft}
\vspace{0.5cm}
Recall the proof of Proposition \ref{p:regularity-scale} that the pointed $C^{2,\gamma}$-convergence
\begin{equation}
(\mathcal M_j , \tilde{g}_{T_j}, \bm{x}_j) \xrightarrow{C^{2,\gamma}} (\mathbb{C}_{TN}^2 \times \mathbb{C}^{n-2}, \tilde{g}_{\infty}, \bm{x}_{\infty}) ,
\end{equation}
holds for any $\gamma\in(0,1)$,
where $\tilde{g}_{\infty} \equiv g_{TN} \oplus g_{\mathbb{C}^{n-2}}$ is the product metric of the Taub-NUT metric $g_{TN}\equiv g_{TN,\sigma_0^2}$ given in \eqref{e:parameter-TN} and the Euclidean metric $g_{\mathbb{C}^{n-2}}$.
Moreover, the rescaled weight function $\tilde{\rho}_{j,\delta,\nu,\mu}^{(k+\alpha)}$ converges to
\begin{align}
\tilde{\rho}_{\infty,\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})
=
\begin{cases}
1, & \bm{x}\in T_1(\Sigma_{0}),\\
(d_{\tilde{g}_{\infty}}(\bm{x}, \Sigma_0))^{\nu+k+\alpha}, & \bm{x}\in(\mathbb{C}_{TN}^2 \times \mathbb{C}^{n-2})\setminus T_1(\Sigma_{0}),
\end{cases}
\end{align}
where $\Sigma_0 \equiv \{p_{\infty}\} \times \mathbb{C}^{n-2}\subset\mathbb{C}_{TN}^2\times \mathbb{C}^{n-2}$ for some $p_{\infty}\in \mathbb{C}_{TN}^2$, is the Gromov-Hausdorff limit of the lifted divisor $\mathcal{P}\equiv \pi^{-1}(P)\subset \mathcal M_j$ with respect to the rescaled metrics $\tilde{g}_j$ such that
and
\begin{equation}T_1(\Sigma_0) \equiv \{\bm{x}\in \mathbb{C}_{TN}^2\times \mathbb{C}^{n-2} | d_{\tilde{g}_{\infty}}(\bm{x}, \Sigma_0) \leq 1\}.\end{equation}
It is straightforward that, the rescaled functions $\tilde{v}_j$ converge to $\tilde{v}_{\infty}$ in the $C_{loc}^{2,\alpha'}$-norm for each $\alpha'\in(0,\alpha)$ such that the following properties hold,
\begin{enumerate}
\item $\|\nabla_{\tilde{g}_{\infty}}\tilde{v}_{\infty}\|_{C_{\delta,\nu+1,\mu}^0(\mathbb{C}_{TN}^2\times \mathbb{C}^{n-2},\tilde{g}_{\infty})} +\|\nabla_{\tilde{g}_{\infty}}^2\tilde{v}_{\infty}\|_{C_{\delta,\nu+2,\mu}^0(\mathbb{C}_{TN}^2\times \mathbb{C}^{n-2},\tilde{g}_{\infty})} = 1$,
\item $\tilde{v}_{\infty}(\bm{x}_{\infty})=0$,
\item $\Delta_{\tilde{g}_{\infty}}\tilde{v}_{\infty} \equiv 0$ on $\mathbb{C}_{TN}^2\times \mathbb{C}^{n-2}$.
\end{enumerate}
We will prove that $\tilde{v}_{\infty} \equiv 0$ on $\mathbb{C}_{TN}^2\times \mathbb{C}^{n-2}$.
To start with, we will show that $\tilde{v}_{\infty}$ is constant on the Euclidean factor $\mathbb{C}^{n-2}$. Indeed,
we write $\bm{x}\equiv (\bm{x}', \bm{x}'') \in \mathbb{C}_{TN}^2\times\mathbb{C}^{n-2}$, so it suffices to prove that for every $1\leq k\leq 2n-4$, we have
\begin{equation}
|\nabla_{k} \tilde{v}_{\infty}| \equiv 0\ \text{on}\ \mathbb{C}^{n-2},\label{e:gradient-vanishing}
\end{equation}
where the partial derivative $\nabla_{k} \tilde{v}_{\infty}(\bm{x}) \equiv \frac{\partial \tilde{v}_{\infty}}{\partial x_k''}(\bm{x}',\bm{x}'')$ is taken in the directions of $\mathbb{C}^{n-2}$.
Now for every $1\leq k \leq 2n-4$,
\begin{equation}
\Delta_{\tilde{g}_{\infty}} (\nabla_{k} \tilde{v}_{\infty}) = \Delta_{\mathbb{C}_{TN}^2}(\nabla_{k} \tilde{v}_{\infty}) + \Delta_{\mathbb{C}^{n-2}} (\nabla_{k} \tilde{v}_{\infty}).
\end{equation}
Notice that $\tilde{g}_{\infty} = g_{TN} \oplus g_{\mathbb{C}^{n-2}}$ is a product metric and $\nabla_k$ in effect acts on the Euclidean factor $\mathbb{C}^{n-2}$, so $\nabla_k$ commutes with both $\Delta_{\mathbb{C}_{TN}^2}$ and $\Delta_{\mathbb{C}^{n-2}}$. Therefore,
\begin{eqnarray}
\Delta_{\tilde{g}_{\infty}} (\nabla_{k} \tilde{v}_{\infty}) = \nabla_k (\Delta_{\mathbb{C}_{TN}^2} \tilde{v}_{\infty} + \Delta_{\mathbb{C}^{n-2}}\tilde{v}_{\infty}) = 0.
\end{eqnarray}
The weighted bound implies the estimates
\begin{align}
\begin{cases}
|\nabla_k\tilde{v}_{\infty}(\bm{x})|\leq 1, & d_{\tilde{g}_{\infty}}(\bm{x},\Sigma_0)\leq 1,
\\
|\nabla_k\tilde{v}_{\infty}(\bm{x})|\leq d_{\tilde{g}_{\infty}}(\bm{x},\Sigma_0)^{-(\nu+1)}, & d_{\tilde{g}_{\infty}}(\bm{x},\Sigma_0)\geq 1.\label{e:D_k-decay}
\end{cases}
\end{align}
Since we have assumed $\nu\in(-1,0)$, so
it is straightforward
$-(\nu+1)\in (-1,0)$.
The above implies that $|\nabla_k\tilde{v}_{\infty}| \leq 1$ on $\mathbb{C}_{TN}^2\times\mathbb{C}^{n-2}$.
Applying Cheng-Yau's gradient estimate to the harmonic function $\nabla_k\tilde{v}_{\infty}$ on the Ricci-flat manifold $\mathbb{C}_{TN}^2\times\mathbb{C}^{n-2}$, we conclude that
$\nabla_k \tilde{v}_{\infty}$ is constant on $\mathbb{C}_{TN}^2\times \mathbb{C}^{n-2}$.
By \eqref{e:D_k-decay}, $\nabla_k\tilde{v}_{\infty}\equiv0$ for every $1\leq k\leq 2n-4$.
Therefore, $\tilde{v}_{\infty}$ is constant
on the Euclidean factor $\mathbb{C}^{n-2}$.
By the above argument, the limiting function $\tilde{v}_{\infty}$ can be viewed as a harmonic function on the Taub-NUT space $(\mathbb{C}_{TN}^2,g_{TN})$.
Now applying Bochner's formula,
\begin{equation}
\frac{1}{2}\Delta_{g_{TN}}|\nabla_{g_{TN}}\tilde{v}_{\infty}|^2 = |\nabla_{g_{TN}}^2 \tilde{v}_{\infty}|^2 \geq 0.
\end{equation}
Since $\tilde{v}_{\infty}$
satisfies the weighted bound
\begin{equation}
\|\nabla_{g_{TN}}\tilde{v}_{\infty}\|_{C_{\delta,\nu+1,\mu}^{0}(\mathbb{C}_{TN}^2)} + \|\nabla_{g_{TN}}^2\tilde{v}_{\infty}\|_{C_{\delta,\nu+2,\mu}^{0}(\mathbb{C}_{TN}^2)} =1 ,
\end{equation}
so we have for any $\bm{x}\in \mathbb{C}_{TN}^2\setminus B_1(\bm{x}_{\infty})$,
\begin{equation}
|\nabla_{g_{TN}}\tilde{v}_{\infty}(\bm{x})| \leq d_{g_{TN}}(\bm{x},\bm{x}_{\infty})^{-(\nu+1)}.
\end{equation}
By assumption $\nu\in(-1,0)$, then
$|\nabla_{g_{TN}}\tilde{v}_{\infty}| \equiv 0 $ on $\mathbb{C}_{TN}^2$ and hence $\tilde{v}_{\infty}$ is constant on $\mathbb{C}_{TN}^2$. Notice that
$\tilde{v}_{\infty}(\bm{x}_{\infty})=0$, so we conclude that
$\tilde{v}_{\infty}(\bm{x}_{\infty})\equiv 0$.
\vspace{0.5cm}
\begin{flushleft}
{\it Case (2): $r(\bm{x}_j)\cdot T_j\to \infty$ and $r(\bm{x}_j)\to 0$ as $j\rightarrow\infty$.} \end{flushleft}
\vspace{0.5cm}
As in the proof of Proposition \ref{p:regularity-scale}, we have the pointed Gromov-Hausdorff convergence,
\begin{equation}
(\mathcal M_j , \tilde{g}_{T_j} , \bm{x}_j) \xrightarrow{GH} (\mathbb{R}^3\times \mathbb{C}^{n-2}, g_0, \bm{x}_{\infty}),
\end{equation}
where the metric $g_0$ is the product Euclidean metric on $\mathbb{R}^3\times\mathbb{C}^{n-2}$. In this rescaled limit, the limiting reference point $\bm{x}_{\infty}$ satisfies $d_{g_0}(\bm{x}_{\infty},\Sigma_{0^3})=1$ and $\Sigma_{0^3}\equiv \{0^3\}\times\mathbb{C}^{n-2}\subset \mathbb{R}^3\times \mathbb{C}^{n-2}$ is the singular slice. Moreover, the convergence keeps curvature uniformly bounded away from the singular slice $\Sigma_{0^3}$. By passing to the local universal covers, in fact one can see that away from $\Sigma_{0^3}\subset \mathbb{R}^3\times \mathbb{C}^{n-2}$,
the functions $\tilde{v}_j$ converge to $\tilde{v}_{\infty}$ in the $C_{loc}^{2,\alpha'}$-norm for each $\alpha'\in(0,\alpha)$, such that the following properties hold,
\begin{enumerate}
\item $\|\nabla_{g_0}\tilde{v}_{\infty}\|_{C_{\delta,\nu+1,\mu}^0(\mathbb{R}^3\times\mathbb{C}^{n-2})} +\|\nabla_{g_0}^2\tilde{v}_{\infty}\|_{C_{\delta,\nu+2,\mu}^0(\mathbb{R}^3\times\mathbb{C}^{n-2})} = 1$,
\item $\tilde{v}_{\infty}(\bm{x}_{\infty})=0$,
\item $\Delta_{g_0}\tilde{v}_{\infty} \equiv 0$ in $(\mathbb{R}^3\times\mathbb{C}^{n-2})\setminus\Sigma_{0^3}$,\end{enumerate}
where the limiting weight function is
\begin{equation}
\rho_{\infty, \delta, \nu, \mu}^{(k+\alpha)} (\bm{x}) = (d_{g_0} (\bm{x}, \Sigma_{0^3}))^{\nu+k+\alpha},\ \bm{x}\in\mathbb{R}^3\times\mathbb{C}^{n-2}.
\end{equation}
Our goal is to show that $\tilde{v}_{\infty}\equiv 0$ on $\mathbb{R}^3\times\mathbb{C}^{n-2}$, which
consists of the following ingredients:
First, we will prove that $\tilde{v}_{\infty}$ in fact globally harmonic in $\mathbb{R}^3\times\mathbb{C}^{n-2}$. To show the singular slice $\Sigma_{0^3}$ is removable, for each $q\in \Sigma_{0^3}$, we take a unit ball $B_1(q)\subset\Sigma_{0^3}$, and for any $r\in(0,1)$, we choose the tubular neighborhood $T_r(B_1(q))\subset\mathbb{R}^3\times \mathbb{C}^{n-2}$.
Notice that $\nabla_{g_0}\tilde{v}_{\infty}$ satisfies the uniform estimate
\begin{equation}\|\nabla_{g_0}\tilde{v}_{\infty}\|_{C_{\delta,\nu+1,\mu}^{0}(\mathbb{R}^3\times\mathbb{C}^{n-2})}\leq 1,
\end{equation}
integrating the above weighted bound, then for any $\bm{x}\in T_r(B_1(q))\setminus B_1(q)$,
\begin{equation}
|\tilde{v}_{\infty}(\bm{x})| \leq C\cdot d(\bm{x},B_1(q))^{-(\nu)}.
\end{equation}
By Lemma \ref{l:harmonic-removable-singularity}, $B_1(q)$ is a removable singular set in $T_r(B_1(q))$ and hence $\tilde{v}_{\infty}$ is harmonic in $T_r(B_1(q))$.
Next, we will show that $\tilde{v}_{\infty}$ is constant in $\mathbb{C}^{n-2}$. It is straightforward that for each $1\leq k\leq 2n-4$, the partial derivative $\nabla_k\tilde{v}_{\infty}\equiv \frac{\partial}{\partial x_k''}\tilde{v}_{\infty}(\bm{x}',\bm{x}'')$ satisfies
\begin{equation}
\Delta_{g_0}(\nabla_k\tilde{v}_{\infty})=0 \ \text{in}\ \mathbb{R}^3\times\mathbb{C}^{n-2}.
\end{equation}
The weighted condition implies that $\nabla_k\tilde{v}_{\infty}$ satisfies the uniform estimate,
\begin{equation}
|\nabla_k\tilde{v}_{\infty}| \leq d(\bm{x},\Sigma_{0^3})^{-(\nu+1)} \quad \text{for every}\ \bm{x}\in \mathbb{R}^3\times\mathbb{C}^{n-2}.
\end{equation}
Since
we have assumed $\nu\in(-1,0)$,
Lemma \ref{l:liouville-product} implies that
$|\nabla_k\tilde{v}_{\infty}| \equiv 0$ on $\mathbb{R}^3\times\mathbb{C}^{n-2}$ and hence $\tilde{v}_{\infty}$ is constant in $\mathbb{C}^{n-2}$.
Therefore, $\tilde{v}_{\infty}$ can be viewed as a harmonic function in the Euclidean space $(\mathbb{R}^3,g_{\mathbb{R}^3})$.
By assumption, $\tilde{v}_{\infty}$ satisfies
\begin{equation}
|\tilde{v}_{\infty}(\bm{x})| \leq d_{g_{\mathbb{R}^3}}(\bm{x}, 0^3)^{-\nu}.
\end{equation}
Since $\nu\in(-1,0)$,
applying the standard Liouville theorem for sublinear growth harmonic functions on a Euclidean space,
we conclude that $\tilde{v}_{\infty}$
is a constant. The last step is to use the renormalization $\tilde{v}_{\infty}(\bm{x}_{\infty})=0$, which gives $\tilde{v}_{\infty}\equiv 0$. So the proof of Case (2) is done.
\vspace{0.5cm}
\begin{flushleft}
{\it Case (3): There is a constant $\underline{T}_0>0$ such that $r(\bm{x}_j)\geq \underline{T}_0$ and $L_{T_j}(\bm{x}_j)^{-1}\cdot T_j^{\frac{n-2}{n}}\to 0$.} \end{flushleft}
\vspace{0.5cm}
There are two situations to consider.
First, we consider the case $z(\bm{x}_j)\to z_0$. By Proposition \ref{p:regularity-scale},
the limit of $(\mathcal M_j,\tilde{g}_{T_j},\bm{x}_j)$ is the cylinder
$(Q,g_c,\bm{x}_{\infty})\equiv (D\times\mathbb{R}, g_{D}\oplus dz^2, \bm{x}_{\infty})$,
where $(D, g_{D})$
is a closed Calabi-Yau manifold.
The limiting function $\tilde{v}_{\infty}$ of $\tilde{v}_j$ satisfies
\begin{enumerate}
\item $\|\nabla_{g_c}\tilde{v}_{\infty}\|_{C_{\delta,\nu+1,\mu}^0(Q)} +\|\nabla_{g_c}^2\tilde{v}_{\infty}\|_{C_{\delta,\nu+2,\mu}^0(Q)} = 1$,
\item $\tilde{v}_{\infty}(\bm{x}_{\infty})=0$,
\item $\Delta_{g_c}\tilde{v}_{\infty} \equiv 0$ in $Q\setminus P$,
\end{enumerate}
where the limiting weight function is given by
$ \tilde{\rho}_{\infty,\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})= e^{-\frac{n}{2}\cdot \delta \cdot L_0(z)}\cdot \mathfrak{r}(\bm{x})^{\nu+k+\alpha}$ and
\begin{align}
L_0(z)=
\begin{cases}
k_+z , & z>1,
\\
k_-z, & z<-1,
\end{cases}
\end{align}
as defined in \eqref{e:l(z)}.
Similar to Case (2), first we need to extend the limiting function $\tilde{v}_{\infty}$ across
the singular set $P$.
Integrating $\nabla\tilde{v}_{\infty}$ around $P$, we have that $\tilde{v}_{\infty}$
satisfies the growth estimate
\begin{equation}
|\tilde{v}_{\infty}(\bm{x})| \leq C\cdot d_{g_c}(\bm{x},P)^{-\nu}.
\end{equation}
Since we have assumed $\nu\in(-1,0)$,
so Lemma \ref{l:harmonic-removable-singularity} implies that the singular set $P$ is removable. Now we have obtained that
$\tilde{v}_{\infty}$ is harmonic on $Q$ and satisfies
\begin{equation}
|\tilde{v}_{\infty}(\bm{x})| \leq C e^{-\delta\cdot \frac{n}{2}\cdot(|k_+|+|k_-|)\cdot |z(\bm{x})|},
\end{equation}
for $|z(\bm{x})|\gg1$.
By the choice of $\delta$, we conclude that
$\tilde{v}_{\infty}\equiv 0$ on $Q$.
The next situation is that the sequence $\bm{x}_j$ satisfies $|z(\bm{x}_j)|\to \infty$ and $L_{T_j}(\bm{x}_j)^{-1}\cdot T_j^{\frac{n-2}{n}}\to 0$.
We need to perform the coordinate change centered at the reference point $\bm{x}_j$,
\begin{equation}
z(\bm{x}) = z_j + \Big(\frac{T_j}{L_{T_j}(z_j)}\Big)^{\frac{n-2}{2}}w(\bm{x}), \quad z_j\equiv z(\bm{x}_j).
\end{equation}
Without loss of generality, we only consider the case $z_j \ll 0$.
Recall the proof of Proposition \ref{p:regularity-scale}, the rescaled limit is $Q=D\times\mathbb{R}$ with a product metric
$g_Q = g_D + dw^2$.
Moreover, the limiting weight function is
$\tilde{\rho}_{\infty,\delta,\nu,\mu}^{(k+\alpha)}(\bm{x})
= e^{-\frac{ \delta \cdot n\cdot k_-}{2} \cdot w(\bm{x})}$.
Now the growth condition implies that the limiting function $\tilde{v}_{\infty}$ satisfies
\begin{align}
\begin{cases}
\Delta_Q \tilde{v}_{\infty}(\bm{x}) = 0 , & \forall \bm{x} \in Q,
\\
|\tilde{v}_{\infty}(\bm{x})| \leq e^{\frac{\delta\cdot n\cdot k_-}{2} \cdot w(\bm{x})}, & w\in\mathbb{R}.
\end{cases}
\end{align}
By \eqref{e:fix-delta-neck}, we have
$\frac{\delta\cdot n\cdot k_-}{2} < \frac{\sqrt{\lambda_D}}{2}$.
Applying Lemma \ref{l:liouville-cylinder}, $ \tilde{v}_{\infty}(\bm{x}) \equiv 0$ on $Q$.
Combining the above situations, the proof of Case (3) is complete.
\vspace{0.5cm}
\begin{flushleft}
{\it Case (4): There is a constant $c_0>0$ such that $r(\bm{x}_j)\geq \underline{T}_0$ and $L_{T_j}(\bm{x}_j)^{-1}\cdot T_j^{\frac{n-2}{n}}\to c_0$.} \end{flushleft}
\vspace{0.5cm}
Without loss of generality we may assume $z(\bm{x}_j)<0$. Then the reference points $\bm{x}_j$ are close to the boundary $\{z=T_-\}\subset\partial\mathcal M_j$ such that we can exploit the Neumann boundary condition. Without loss of generality, we just assume $c_0=1$. We have shown in Section \ref{ss:regularity-scales} that the limit of the rescaled spaces $(\mathcal M_j, \tilde{g}_{T_j}, \bm{x}_j)$
is the Calabi model space $(\mathcal{C}^n_- , g_{\mathcal{C}^n_-}, \bm{x}_{\infty})$.
Let us perform the coordinate change as follows \begin{equation}
k_-\cdot (z(\bm{x}) - T_-) = T_j^{\frac{n-2}{n}}\cdot(w(\bm{x})-1).
\end{equation}
Moreover, the limiting weight function is
$\tilde{\rho}_{\infty,\delta,\nu,\mu}^{(k+\alpha)}(\bm{x}) = e^{-\delta\cdot w^{\frac{n}{2}}}\cdot w^{\frac{\nu+k+\alpha}{2}}$.
So $\tilde{v}_{\infty}$ satisfies
\begin{align}
\begin{cases}
\Delta_{g_{\mathcal{C}_-^n}} \tilde{v}_{\infty}(\bm{x}) = 0, & \bm{x}\in\mathcal{C}_-^n,
\\
|\tilde{v}_{\infty}(\bm{x})| \leq e^{\delta\cdot w^{\frac{n}{2}}}\cdot w^{-\frac{\nu}{2}} , & w(\bm{x})\gg1,
\\
\frac{\partial \tilde{v}_{\infty}(\bm{x})}{\partial w} = 0, & w(\bm{x})=1.
\end{cases}
\end{align}
Applying the renormalization condition $\tilde{v}_{\infty}(\bm{x}_{\infty})=0$ and Lemma \ref{l:Liouville-Calabi-space-SZ}, we have $\tilde{v}_{\infty}\equiv 0 $ on $\mathcal{C}_-^n$.
Combining all the above, we have completed the proof of the weighted estimates in Step 1.
\begin{flushleft}
{\bf Step 2} (weighted $C^{2,\alpha}$-estimate):
\end{flushleft}
It remains to prove the uniform weighted $C^{2,\alpha}$-estimate
\begin{equation}
[u]_{C_{\delta,\nu,\mu}^{2,\alpha}(\mathcal M_T)} \leq C \cdot \|\Delta u\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal M_T)}\label{e:weighted-higher-order}\end{equation}
for some uniform constant $C>0$ independent of $T$. This will again be proved by contradiction. As before, suppose that there are have a sequence of parameters $T_j\to+\infty$ and a sequence of functions $u_j$
on $(\mathcal M_j, g_{T_j})$
which satisfy
\begin{align}\label{e:u_j-contradicting-higher-order}
\begin{split}
\frac{\partial u}{\partial n}\Big|_{\partial M_j} & = 0,
\\
[u_j]_{C_{\delta,\nu,\mu}^{2,\alpha}}(\bm{x}_j)=[u_j]_{C_{\delta,\nu,\mu}^{2,\alpha}(\mathcal M_j)} &= 1,
\\
\|\Delta_{g_{T_j}} u_j\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal M_j)} &\to 0,\quad j\to+\infty.
\end{split}
\end{align}
We normalize the functions $u_j$ by
$v_j(\bm{x}) \equiv u_j(\bm{x}) - u_j(\bm{x}_j)$. Then \eqref{e:u_j-contradicting-higher-order} retains for $v_j$ and $\bm{x}_j$. Applying \eqref{e:neck-C2-estimate} to $v_j$, we have
\begin{align}
\|\nabla_{g_{T_j}} v_j\|_{C_{\delta,\nu+1,\mu}^0(\mathcal M_j)} + \|\nabla_{g_{T_j}}^2 v_j\|_{C_{\delta,\nu+2,\mu}^0(\mathcal M_j)} \leq C \cdot \|\Delta_{g_{T_j}} v_j\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\mathcal M_j)},
\end{align}
where $C>0$ is independent of $T_j$. So it follows that
\begin{align}
\|\nabla_{g_{T_j}} v_j\|_{C_{\delta,\nu+1,\mu}^0(\mathcal M_j)} + \|\nabla_{g_{T_j}}^2 v_j\|_{C_{\delta,\nu+2,\mu}^0(\mathcal M_j)} \to 0 \label{e:weighted-C2-limit-to-0}
\end{align}
as $j\to +\infty$. Denote $\mathfrak{s}_j\equiv \mathfrak{s}(\bm{x}_j)$, by straightforward computations, there is some uniform constant $C_0>0$ such that
$\|v_j\|_{C_{\delta,\nu,\mu}^0(B_{\mathfrak{s}_j}(\bm{x}_j))} \leq C_0 \cdot \|\nabla v_j\|_{C_{\delta,\nu+1,\mu}^0(B_{\mathfrak{s}_j}(\bm{x}_j))}$ and hence
\begin{align}
\|v_j\|_{C_{\delta,\nu,\mu}^0(B_{\mathfrak{s}_j}(\bm{x}_j))} \to 0 \ \text{as} \ j\to+\infty.\label{e:weighted-C0-limit-to-0}
\end{align}
By Proposition \ref{p:local-weighted-schauder},
$\|v_j\|_{C_{\delta,\nu,\mu}^{2,\alpha}(B_{\mathfrak{s}_j/10}(\bm{x}_j))} \to 0$, which contradicts \eqref{e:u_j-contradicting-higher-order}.
The proof of \eqref{e:weighted-higher-order}
is done.
Since we have proved \eqref{e:neck-C2-estimate} and \eqref{e:weighted-higher-order}, the proof of the Proposition is complete.
\end{proof}
Combining all the above estimates, we are ready to complete the proof of Theorem \ref{t:neck-CY-metric}.
\begin{proof}
[Proof of Theorem \ref{t:neck-CY-metric}]
It suffices to verify that $\mathscr{F}$ satisfies all the conditions in Lemma \ref{l:implicit-function}. Proposition \ref{p:estimate-L-neck} and Proposition \ref{p:nonlinear-neck} show that $\mathscr{F}$ satisfies
item (1) and item (2a). In our context, $C_L>0$
and $C_N>0$ are uniform constants. $r_0>0$
can be chosen as any fixed constant in $(0,\frac{1}{2C_LC_N})$. To verify item (2b)
in Lemma \ref{l:implicit-function}, we just need to use \eqref{e:new error estimate}. In fact, we have assumed $\nu+\alpha<0$, then
\begin{equation}
\|\mathscr{F}(\bm{0})\|_{\mathfrak{S}_2} \leq C\cdot T^{\nu+\alpha} \ll \frac{r_0}{4C_L},
\end{equation}
as $T$ is sufficiently large. This completes the proof.
\end{proof}
\subsection{Geometric singularity and normalized limit measure}
\label{ss:renormalized-measure}
The goal of this subsection is to understand the measured Gromov-Hausdorff limits of the sequence of incomplete Calabi-Yau metrics $(\mathcal M_T, \omega_{T,CY})$ (scaled to fixed diameter) constructed in Theorem \ref{t:neck-CY-metric}. As can be easily seen, the results are parallel to the statements in Theorem \ref{t:main-theorem}. So we will not repeat the arguments in Section \ref{s:gluing}.
To begin with, we recall the notion of \emph{measured Gromov-Hausdorff convergence}. We refer the readers to \cite{ChC1} for the general theory about this.
\begin{definition}[Measured Gromov-Hausdorff convergence] Let $(M_j^m, g_j, p_j)$ be a sequence of Riemannian manifolds with $\Ric_{g_j}\geq -(m-1)$ such that
\begin{equation}
(M_j^m, g_j, p_j) \xrightarrow{GH} (X_{\infty},d_{\infty},p_{\infty})
\end{equation}
for some metric space $(X_{\infty},d_{\infty},p_{\infty})$. Then by passing to a subsequence, the renormalized measures
\begin{equation}
d\underline{\nu}_j \equiv \frac{\dvol_{g_j}}{\Vol_{g_j}(B_1(p_j))}\label{e:renormalized measure}
\end{equation}
converge to a Radon measure $d\underline{\nu}_{\infty}$ on $X_\infty$ which is called the {\it renormalized limit measure}. The Gromov-Hausdorff convergence together with the convergence of the renormalzied measures is called the measured Gromov-Hausdorff convergence.
\end{definition}
In the general context of collapsed sequences with Ricci curvature bounded from below, $d\underline{\nu}_{\infty}$ behaves quite differently from the Hausdorff measures on $X_{\infty}$ induced by the limiting metric $d_{\infty}$.
In our specific context, $d\underline{\nu}_{\infty}$ has an explicit form and it effectively reveals the geometric singularity information in the collapsing spaces.
In our context, we are interested in the measured Gromov-Hausdorff limits of $(\mathcal M_T, \omega_{T, CY},d\underline{\nu}_{T, CY})$, where $d\underline{\nu}_{T, CY}$ is the renormalized volume measure of the metric $\omega_{T, CY}$. By the weighted estimate on the solution $\phi(T)$ given in \eqref{e:eqn524} we see that on the regularity scale the metrics $\omega_{T, CY}$ and $\omega_{T}$ are very close in $C^{ \alpha}$-topology for $T$ large. Lemma \ref{l:weight-function-lower-bound-estimate} and our choice of parameters ensure that the weight function has a positive lower bound independent of $T$, so we also know that for $T\gg1$,
\begin{equation}
(1+O(T^{\nu+\alpha}))\omega_{T}\leq \omega_{T, CY}\leq (1+O(T^{\nu+\alpha}))\omega_T.
\end{equation}
So on large scales the metrics are also close in $L^\infty$ topology. This allows us to effectively replace $\omega_{T, CY}$ by $\omega_T$ in the computation of limit geometries. Then we are led to do the following explicit calculations.
\begin{flushleft}{\bf Gromov-Hausdorff limit:}
\end{flushleft}
By construction and direct calculation, one can see that $g_T$ in large scale is approximated by the $1$ dimensional metric tensor $T^{\frac{(2-n)(n+1)}{n}} L_T(z)^{n-1} dz^2$. In particular, the diameter is of order $T^{\frac{n+1}{n}}$. This suggests rescaling the metric $g_T$ by $T^{-\frac{2(n+1)}{n}}$ in order to obtain bounded diameter. Indeed, upon the change of variable $z=T\cdot \xi$, we see $T^{-\frac{2(n+1)}{n}}g_T$ converges to the one dimensional metric $(1+k_{\mp}\xi)^{n-1}d\xi^2$, $\xi\in [-k_-^{-1}, -k_+^{-1}]$ in the Gromov-Hausdorff sense.
The above limit can be transformed into the standard metric on the unit interval $(\mathbb{I}, dv^2)$ via a constant rescaling and the following coordinate change
\begin{equation}
\begin{cases}
1+k_+(\frac{1}{k_-}-\frac{1}{k_+})v=(1+k_+\xi)^{\frac{n+1}{2}}, & \xi>0, \\
1+k_-(\frac{1}{k_-}-\frac{1}{k_+})v=(1+k_-\xi)^{\frac{n+1}{2}}, & \xi<0.
\end{cases}
\end{equation}
\begin{flushleft}
{\bf Renormalized limit measure:}
\end{flushleft}
Again we first calculate by definition
\begin{equation}
\Vol_{\omega_T}(\mathcal M_{a\leq z\leq b})=\int_a^b \Big( \frac{T^{2-n}}{(n-1)!} \int_D \tilde\omega(z)^{n-1}\Big)dz.
\end{equation}
Using the change of variable $z=T\cdot \xi$, we have that
\begin{equation}
\Vol_{\omega_T}(\mathcal M_{a\leq \xi\leq b})=T^2 \int_a^b\Big(\int_D \frac{1}{(n-1)!}(1+k_\pm \xi)^{n-1}\omega_D^{n-1}\Big)d\xi.
\end{equation}
So up to constant, the renormalized limit measure has density function given by $(1+k_{\pm}\cdot \xi)^{n-1}d\xi$. Changing to the $v$-variable this becomes (again up to constant multiplication)
\begin{equation}
d\underline\nu_\infty=
\begin{cases}
(\frac{v}{-k_+}+\frac{1}{k_--k_+})^{\frac{n-1}{n+1}}dv, & v\in [\frac{k_+}{k_--k_+}, 0],\\
(\frac{v}{-k_-}+\frac{1}{k_--k_+})^{\frac{n-1}{n+1}}dv, & v\in [0, \frac{k_-}{k_--k_+}].
\end{cases}
\end{equation}
\begin{flushleft}
{\bf Diameter and volume:}
\end{flushleft}
By straightforward computations, if the diameter of $\mathcal M_T$ is rescaled to be $1$, then the volume of $\mathcal M_T$ is collapsing at rate $T^{-2n}$. On the other hand, if the volume of $\mathcal M_T$ is rescaled to be $1$, then the diameter of $\mathcal M_T$ is of order $T$.
\begin{flushleft}
{\bf Fibration structure:}
\end{flushleft}
There is an obvious fibration of $\mathcal M_T$ over $[T_-, T_+]$ using the coordinate function $z$. Composing with above coordinate changes, we obtain a fibration
\begin{equation}
\mathcal F_T: \mathcal M_T\rightarrow \mathbb{I},\quad \bm{x}\mapsto v(\bm{x}).
\end{equation}
Then for any $v\neq 0$, $\mathcal F_T^{-1}(v)$ is an $S^1$-bundle over $D$, whose first Chern class is given by $c_1(L_\pm)$ depending on the sign of $v$, and $\mathcal F_T^{-1}(0)$ is an singular $S^1$-fibration over $D$ with vanishing circles along $H$.
\begin{flushleft}
\textbf{Bubble classification:}
\end{flushleft}
By Proposition \ref{p:regularity-scale}, it is clear that suitable rescalings around the vanishing circles in $\mathcal{F}_T^{-1}(0)$ converge to $\mathbb{C}_{TN}^2\times \mathbb C^{n-2}$. Also suitable rescalings around the ends $z=T_\pm$ gives the Calabi model spaces.
\
We close this section by giving the following remarks regarding the renormalized limit measure.
\begin{remark}
It can be seen from the above formulae that the limiting density function $\mathscr{V}_{\infty}=\frac{d\underline\nu_\infty}{dv}$ is a Lipschitz function on $\mathbb{I}$ and it is smooth everywhere in the interior of $\mathbb{I}$ except at $v=0$. On the other hand, the singular fiber of $\mathcal{F}$ precisely appears at $t=0$.
So in our context, the singularity of the renormalized limit measure $d\underline{\nu}_{\infty}$ effectively characterizes the singularity behavior of the collapsing geometry.
\end{remark}
\begin{remark}
By Cheeger-Colding (see \cite{ChC3}, theorem 4.6), in the regular set $\mathcal{R}$ of a general Ricci-limit space, the density function $\mathscr{V}_{\infty}$ of the renormalized limit measure always exists and is H\"older continuous. Our example tells us that, in general, one cannot expect the regularity of $\mathscr{V}_{\infty}$ to be differentiable in $\mathcal{R}$ (even though $\mathcal{R}$ is a smooth Riemannian manifold). We thank Shouhei Honda for pointing this out.
\end{remark}
\begin{remark}
If we use rescale the metrics further around the point $v=0$
such that the sequence of spaces collapse to the complete real line $\mathbb{R}$,
then $d\underline{\nu}_{\infty}$ coincides with the standard Lebesgue measure on $\mathbb{R}$. This fact can be quickly seen by scaling-up the coordinate $v$.
This is compatible with the general theory of Ricci-limit spaces. That is, due to Cheeger-Colding, the renormalized limit measure always splits off the Lebesgue measure of $\mathbb{R}$ if the limit space isometrically splits off $\mathbb{R}$ (see proposition 1.35 in \cite{ChC1}).
\end{remark}
\section{Proof of the main theorem}
\label{s:gluing}
The goal of this section is to prove Theorem \ref{t:main-theorem}. We will work with the special family of Calabi-Yau varieties $\mathcal X$ defined in the Introduction. In Section \ref{ss:algebraic-geometry} we show how to modify the family $\mathcal X$ to a new family $\widehat{\mathcal X}$ such that the new central fiber consists of a chain of three components, with the middle component given by the compactification of the space $\mathcal N^0$ defined in Section \ref{ss:complex-geometry}. Notice in Section \ref{s:neck} a family of neck metrics are constructed on an exhausting family of domains in $\mathcal N^0$. In Section \ref{ss:tian-yau} we review general facts about the Tian-Yau metrics on the complement of a smooth anti-canonical divisor in a Fano manifold. These give Ricci-flat K\"ahler metrics on the other two components of the central fiber in $\widehat{\mathcal X}$. In Section \ref{ss:glued-metrics} we explain how to graft the above neck metrics and Tian-Yau metrics on the central fiber of $\widehat{\mathcal X}$ to the nearby smooth fibers, and obtain approximately Calabi-Yau metrics in a suitable sense. In Section \ref{ss:global analysis} we finish the proof of Theorem \ref{t:main-theorem}. The arguments are very similar to those in Section \ref{ss:incomplete weighted analysis} and \ref{ss:renormalized-measure}, so we will not provide full details for the final perturbation arguments.
\subsection{Algebro-geometric aspect}
\label{ss:algebraic-geometry}
\subsubsection{Poincar\'e residue}
We first recall some general facts about \emph{Poincar\'e residues}. Given a smooth divisor $Z$ in a complex manifold $M$ of dimension $m$, the Poincar\'e residue map
\begin{equation}\Res: H^0(M, K_M\otimes [Z])\rightarrow H^0(Z, K_Z)
\end{equation}
can be defined as follows. Given a holomorphic $m$ form $\Omega$ on $M$ with a simple pole along $Z$, locally if we choose a defining function $h$ of $Z$, then $h\Omega$ is a holomorphic $m$ form, and we can write
$
h\Omega=dh\wedge \tilde\Omega
$
for some locally defined holomorphic $m-1$ form $\tilde\Omega$.
The Poincar\'e residue of $\Omega$ along $Z$ is given by
$\Res(\Omega)\equiv \tilde\Omega|_Z.$
It is straightforward to check that this definition does not depend on the choice of $h$ and $\tilde\Omega$, and gives rise to a well-defined holomorphic $m-1$ form $\Omega_Z$ globally on $Z$.
If we choose local holomorphic coordinates $z_1, \cdots, z_m$ on $M$, then we may write
\begin{equation}\Omega=\frac{g}{h} dz_1\wedge \cdots dz_m, \end{equation}
where $g$ is holomorphic.
At a point on $Z$ where $\frac{\partial h}{\partial z_1}\neq 0$, we have then by definition
\begin{equation}\Res(\Omega)=\frac{g}{\frac{\partial h}{\partial z_1}}dz_2\wedge \cdots \wedge dz_m.
\end{equation}
From the local expression one can see that if $Z$ is an anti-canonical divisor in $M$, and we pick a holomorphic volume form $\Omega_M$ on $M\setminus Z$ with a simple pole along $Z$, and then $\Res(\Omega_M)$ gives a holomorphic volume form $\Omega_Z$ on $Z$.
A special case arises when we have a globally defined holomorphic function $h: M\rightarrow \mathbb C$, and we are given a holomorphic volume form $\Omega$ on $M$, then for each $w\in \mathbb C$, we can apply the above construction to the meromorphic form $(h-w)^{-1}\Omega$. In this way we obtain a nowhere vanishing section $\Omega'$ of the relative canonical bundle $K_M\otimes (h^*K_{\mathbb C})^{-1}$ on the set where $h$ is a submersion, and it satisfies the equation
$dh\wedge \Omega'=\Omega. $
We may also view $\Omega'$ as a holomorphic varying family of holomorphic volume forms on the fibers of $h$.
\subsubsection{A model partial resolution of singularities}
Let $\mathcal S$ be a two dimensional $A_{k-1}(k\geq 2)$ singularity, which is a hypersurface in $\mathbb C^3$ with defining equation
$
z_1z_2+z_3^k=0.
$
Given two positive integers $a_1\geq a_2$ with $a_1+a_2=k$, we can define a \emph{partial} resolution of $\mathcal S$ as follows. Let $\overline{\mathcal S}$ be the subvariety in the product space $\mathbb C^3\times \mathbb C\mathbb{P}^2$ cut out by the following system of equations
\begin{equation}
\begin{cases}
z_3^{a_1}u_1=z_1u_3; \\
z_3^{a_2}u_2=z_2u_3; \\
u_1u_2+u_3^2=0;\\
z_3^{a_1-a_2}u_1z_2=u_2z_1; \\
z_3^{a_2}u_3+u_1z_2=0.
\end{cases}
\end{equation}
where $[u_1:u_2: u_3]$ denotes homogeneous coordinates on $\mathbb C\mathbb{P}^2$. Alternatively, $\overline{\mathcal S}$ can also be described as the closure in $\mathbb C^3\times\mathbb C\mathbb{P}^2$ of the graph of the rational map \begin{align}\mathcal S\rightarrow \mathbb C\mathbb{P}^2; (z_1, z_2, z_3)\mapsto \left[\frac{z_1}{z_3^{a_1}}: \frac{z_2}{z_3^{a_2}}:1\right].\end{align} On the affine chart $\{u_i\neq 0\}$ we denote by $v_j=u_j/u_i (j\neq i)$ the affine coordinates.
\begin{lemma}
$\overline{\mathcal S}$ has at most two singularities, which are of type $A_{a_1-1}$ and $A_{a_2-1}$ respectively, and the projection map $\overline{\mathcal S}\rightarrow \mathcal S$ is a partial resolution, with exceptional divisor isomorphic to $\mathbb C\mathbb{P}^1$.
\end{lemma}
\begin{proof}
We first show that the system of equations implies $z_1z_2+z_3^k=0$, so that $\overline{\mathcal S}$ does project to $\mathcal S$. To see this, we notice the first three equations imply
\begin{equation}
u_3^2(z_1z_2+z_3^k)=0.
\end{equation}
If $u_3\neq 0$, then we get $z_1z_2+z_3^k=0$. If $u_3=0$, then by the third equation we get that either $u_1\neq 0, u_2=0$ or $u_1=0, u_2\neq 0$. In the first case using the remaining equations we get $z_3=z_2=0$. In the second case we get $z_3=z_1=0$. In both cases the equation $z_1z_2+z_3^k=0$ is indeed satisfied.
Now we study singularities of $\overline{\mathcal S}$. In the affine chart $\{u_1\neq 0\}$, we get
\begin{equation}
\begin{cases}
v_2+v_3^2=0; \\
z_2+z_3^{a_2}v_3=0,
\end{cases}
\end{equation}
so we reduce the defining equations to a single equation in the $z_1, z_3, v_3$ variable given by
\begin{equation}
z_3^{a_1}=z_1v_3.
\end{equation}
This has exactly one $A_{a_1-1}$ singularity at $\{z_1=z_3=v_3=0\}$. Notice by convention an $A_0$ singularity is a smooth point. Similarly, on the affine chart $\{u_2\neq 0\}$ we reduce the equations to
\begin{equation}
z_3^{a_2}=z_2v_3.
\end{equation}
This has exactly one $A_{a_2-1}$ singularity at $\{z_2=z_3=v_3=0\}$.
On the affine chart $\{u_3\neq 0\}$, we reduce the equations to
$v_1v_2+1=0$ which is smooth.
It is then easy to verify that the projection map $\overline{\mathcal S}\rightarrow \mathcal S$ is an isomorphism outside the point $\{z_1=z_2=z_3=0\}$, and if $z_1=z_2=z_3=0$, we get the equation
$u_1u_2+u_3^2=0$,
which gives a conic in $\mathbb C\mathbb{P}^2$.
\end{proof}
From another point of view, we can think of both $\mathcal S$ and $\overline{\mathcal S}$ as families of algebraic curves by projecting to the $z_3$ variable. For $\mathcal S$ this is simply the standard nodal degeneration of conics in $\mathbb C^2$, modified by a base change. The family corresponding to $\overline{\mathcal S}$ is isomorphic to $\mathcal S$ over any general fiber $\{z_3\neq 0\}$, and the special fiber of $\overline{\mathcal S}$ is now given by a chain consisting of three components, two of which are given by the proper transforms of the two lines $\{z_1=0\}$ and $\{z_2=0\}$ in $\mathbb C^2$, and the middle component is the conic $\{u_1u_2+u_3^2=0\}$ in $\mathbb C\mathbb{P}^2$. In the special case when $a_1=a_2=1$, $\overline{\mathcal S}$ is smooth and the projection map is precisely the minimal resolution of singularity.
It is well-known that $\mathcal S$ has a canonical singularity, meaning that the canonical line bundle $K_{\mathcal S}$ is trivial. An explicit holomorphic volume form $\Omega_{\mathcal S}$ can be written by applying the Poincar\'e residue to the standard meromorphic form $\frac{1}{z_1z_2+z_3^k}dz_1\wedge dz_2\wedge dz_3$ on $\mathbb C^3$. In the chart $\{z_1\neq 0\}$, it is given by
\begin{equation}
\Omega_{\mathcal S}=\frac{dz_2\wedge dz_3}{z_2}.
\end{equation}
Notice $\mathcal S$ is isomorphic to the quotient $\mathbb C^2/\mathbb Z_k$, and $\Omega_{\mathcal S}$ pulls-back to a multiple of the standard holomorphic volume form on $\mathbb C^2$.
Viewing $\mathcal S$ as fibered over $z_3\in\mathbb C$, we further get a relative holomorphic volume form
\begin{equation}
\Omega'=-\frac{dz_2}{z_2}=\frac{dz_1}{z_1}.
\end{equation}
One can see $\Omega'$ is smooth away from the singularity $\{z_1=z_2=z_3=0\}$, and on each component of the singular fiber it is a meromorphic 1-form with a simple pole along the singularity.
The partial resolution $\overline{\mathcal S}$ is a \emph{crepant} resolution, i.e., the canonical line bundle $K_{\overline{\mathcal S}}$ is also trivial. Indeed the pull-back $\Omega_{\overline{\mathcal S}}$ of $\Omega_{\mathcal S}$ is nowhere vanishing on $\overline{\mathcal S}$, and by applying the Poincar\'e residue to the function $z_3$, we then get a meromorphic 1-form on each component of the special fiber. On the conic $\{u_1u_2+u_3^2=0\}$ the meromorphic 1-form is given by $v_1^{-1}dv_1=-v_2^{-1}dv_2$. The upshot is that we still get a meromorphic section of the relative canonical bundle, which is smooth away from the two singularities $\{u_1=u_3=z_1=z_2=z_3=0\}$ and $\{u_2=u_3=z_1=z_2=z_3=0\}$ of $\overline S$.
\subsubsection{A modification of the degenerating family}
We now recall the setup in the introduction. We adopt the notations there, and let $p:\mathcal \mathcal X \rightarrow\Delta$ be the family defined by \eqref{eqn1.1}. We also assume the genericity assumptions (i)-(iv) in the Section \ref{ss:1.1} hold. The total space $\mathcal X$ is singular along $H\times\{0\}$ and transverse to $H\times \{0\}$ the singularities are locally modeled on a two dimensional ordinary double point. For our purpose we need to perform certain birational transformations to $\mathcal X$. We first do a base change $t\mapsto t^{n+2}$, and work on the new family, which we still denote by $\mathcal X$. Then
$\mathcal X$ now has singularities along $D\times \{0\}$, transversal to which generically it is a two dimensional $A_{d-1}$ singularity, which becomes worse along $H\times \{0\}$. This is usually referred to as a compounded Du Val (cDV) singularity .
Now we apply the family version of the above model partial resolution to $\mathcal X$. Let $\widehat{\mathcal X}$ be the subvariety in the projective bundle $\mathbb{P}(\mathcal{O}(d_2)\oplus \mathcal{O}(d_1)\oplus \mathbb C)$ over $\mathbb C\mathbb{P}^{n+1}\times \Delta$ cut out by the equations
\begin{equation}
\begin{cases}
t^{d_1}s_1=s_3f_2(x);\\
t^{d_2}s_2=s_3f_1(x); \\
s_1\otimes s_2+s_3^2 f(x)=0;\\
t^{d_1-d_2}s_1\otimes f_1(x)=f_2(x)\otimes s_2; \\
t^{d_2}s_3f(x)+s_1\otimes f_1(x)=0.
\end{cases}
\end{equation}
where naturally we view $f_i\in H^0(\mathbb C\mathbb{P}^{n+1}, \mathcal{O}(d_i))$, $f\in H^0(\mathbb C\mathbb{P}^{n+1}, \mathcal{O}(n+2))$, and $[s_1: s_2: s_3]$ denotes a point in the fiber of the projective bundle over the point $(x, t)\in \mathbb C\mathbb{P}^{n+1}\times \Delta$.
For our discussion in the rest of this section, we will always take $[x_0:x_1:\cdots: x_{n+1}]$ to be the homogeneous coordinates of a point $x$ on $\mathbb C\mathbb{P}^{n+1}$. On the affine chart $\{x_i\neq 0\}$ of $\mathbb C\mathbb{P}^{n+1}$ we denote by $u=\{u_j=x_j/x_i, j\neq i\}$ the affine coordinates, and we view $x_i$ as a local trivialization of $\mathcal{O}(1)$. Then on this chart we can view holomorphic sections of powers of $\mathcal{O}(1)$ as local holomorphic functions. In particular, for a homogeneous function $R(x)$, we denote by $R(u)$ the corresponding inhomogeneous function. On the
affine trivialization of the projective bundle $\{s_i\neq 0\}$, we denote by $\{\zeta_j=s_j/s_i, j\neq i\}$ the affine coordinates on the fibers.
We also define
\begin{align}
D_1 & \equiv \{f_1(x)=f_2(x)=t=0, s_2=s_3=0\},
\\
D_2 & \equiv \{f_1(x)=f_2(x)=t=0, s_1=s_3=0\}.
\end{align}
\begin{lemma}
$\widehat{\mathcal X}$ is smooth away from the union $D_1\cup D_2$, and transverse to each $D_i$ the singularity is a two dimensional $A_{d_i-1}$ singularity.
\end{lemma}
\begin{proof}
We know $\widehat{\mathcal X}$ is isomorphic to $\mathcal X$ away from $D\times \{0\}$, so it suffices to consider around a point $(x, 0)$ where $f_1(x)=f_2(x)=0$. Locally in an affine chart $\{s_1\neq 0\}$, $\widehat{\mathcal X}$ is then cut out by the equations
\begin{equation}\label{e: s1 nonzero region all equations}\begin{cases}
f_2(u)\zeta_3=t^{d_1};\\
f_1(u)\zeta_3=t^{d_2}\zeta_2;\\
\zeta_2+\zeta_3^2f(u)=0;\\
f_2(u)\zeta_2=t^{d_1-d_2}f_1(u);\\
t^{d_2}\zeta_3 f(u)+f_1(u)=0.
\end{cases}\end{equation}
These can be reduced to two equations on the coordinates $u$, $t$ and $\zeta_3$, given by
\begin{equation} \label{e: s1 nonzero region}
\begin{cases}
f_2(u)\zeta_3-t^{d_1}=0\\
t^{d_2}\zeta_3 f(u)+f_1(u)=0.
\end{cases}
\end{equation}
By our assumption (iii) locally we may use $v_1=f_1(u)$ and $v_2=f_2(u)$ to replace $u_1, u_2$ (for example) as local holomorphic coordinates on a neighborhood of $x$ in $\mathbb C\mathbb{P}^{n+1}$. Then it is easy to see the corresponding subvariety is smooth if $\zeta_3\neq 0$, and has transversal $A_{d_1-1}$ singularities along $D_1$.
So this gives the local description of $\widehat{\mathcal X}$ in a neighborhood of $D_1$. Similarly on $\{s_2\neq 0\}$ we also know the space is smooth except with transversal $A_{d_2-1}$ singularities along $D_2$.
On $\{s_3\neq 0\}$, we use $u, t, \zeta_1, \zeta_2$ as coordinates, and we get the constraint equations
\begin{equation}\label{e: s3 nonzero region}
\begin{cases}
\zeta_1\zeta_2+f(u)=0,\\
f_2(u)-t^{d_1}\zeta_1=0,\\
f_1(u)-t^{d_2}\zeta_2=0.\end{cases}
\end{equation}
We only need to consider the points where $\zeta_1=\zeta_2=t=0$, so in particular we also have $f(u)=0$. At such a point, the differentials of these three equations are $(\nabla f(u), \nabla f_2(u), \nabla f_1(u))$. This is non-zero by our assumption (iv).
\end{proof}
One can see that the new central fiber $\hat{X}_0$ consists of a chain of three smooth components intersecting transversally, given by the proper transforms $\hat Y_1, \hat Y_2$ of $Y_1, Y_2$ respectively and the submanifold $\mathcal N$ in the projective bundle $\mathbb{P}(L_1\oplus L_2\oplus \mathbb C)$ over $D$ cut out by the equation $s_1s_2=s_3^2 f(x)$ (so that $\mathcal N$ is a quadric bundle over $D$, and singular fibers are over $H$). Notice $\mathcal N$ itself is a smooth manifold.
\begin{figure}
\begin{tikzpicture}
\draw (-5, 3) to [out=-75, in=75] (-5, -3);
\draw (-5, 3) to (-3.5, 4);
\draw (-5, -3) to (-3.5, -2);
\draw (-3.5, 4) to [out=-75, in=75] (-3.5, -2);
\draw (-3, 3) to [out=-75, in=75] (-3, -3);
\draw (-3, 3) to (-1.5, 4);
\draw (-3, -3) to (-1.5, -2);
\draw (-1.5, 4) to [out=-75, in=75] (-1.5, -2);
\draw (3.0, 3) to [out=-75, in=75] (3.0, -3);
\draw (3.0, 3) to (4.5, 4);
\draw (3, -3) to (4.5, -2);
\draw (4.5, 4) to [out=-75, in=75] (4.5, -2);
\draw (1.0, 3) to [out=-75, in=75] (1.0, -3);
\draw (1.0, 3) to (2.5, 4);
\draw (1.0, -3) to (2.5, -2);
\draw (2.5, 4) to [out=-75, in=75] (2.5, -2);
\draw (-1, 3) to (0.5, 4);
\draw (-1, 3) to [out=-65, in=60] (-0.7, 0.5);
\draw[red, very thick] (-0.7, 0.5) to (0.8, 1.5);
\draw (-0.7, -1.5) to [out=-60, in=65] (-1, -3);
\draw[red, very thick] (-0.7, -1.5) to (0.8, -0.5);
\draw (-1, -3) to (0.5, -2);
\draw (0.5, 4) to [out=-65, in=60] (0.8, 1.5);
\draw (0.8, -0.5) to [out=-60, in=65] (0.5, -2);
\draw[thick] (-0.7, 0.5) to [out=-65, in=65] (-0.7, -1.5);
\draw[thick] (0.8, 1.5) to [out=-65, in=65] (0.8, -0.5);
\draw[thick] (-0.5, 0.66) to [out=-65, in=65] (-0.5, -1.34);
\draw[thick] (0.4, 1.26) to [out=-65, in=65] (0.4, -0.74);
\draw[very thick] (-0.2, 0.8) to (0.05, 0);
\draw[very thick] (0.05, 0) to (-0.2, -1.19);
\node[blue] at (0.05, 0) {$\bullet$};
\node[blue] at (-1.75, 0.1) {$\bullet$};
\node[blue] at (-3.75, 0.4) {$\bullet$};
\node[blue] at (2.25, 0.2) {$\bullet$};
\node[blue] at (4.25, 0.6) {$\bullet$};
\node at (-4, 2.5) {$\widehat{X}_t$};
\node at (0, 2.5) {$\hat Y_1$};
\node at (0, -1.85) {$\hat Y_2$};
\node[red] at (0.3, 1.6) {$D_1$};
\node[red] at (0.3, -1.2) {$D_2$};
\node at (0.3, 0.5) {$\mathcal N$};
\node[blue] at (-6, 1.4) {$H\times\{t\}$};
\draw[->] (-5.2, 1.3) to (-3.9, 0.5);
\node at (0, -4.5) {$\widehat{X}_0=\hat Y_1\cup_{D_1}\mathcal N\cup_{D_2} \hat Y_2$};
\draw[->] (0, -4) to (0, -2.5);
\draw[thick, blue, dashed] plot[smooth] coordinates {(-3.75, 0.4) (-1.75, 0.1) (0.05, 0) (2.25, 0.2) (4.25, 0.6)};
\end{tikzpicture}
\caption{The modified family $\widehat{\mathcal X}$}
\label{f: the modified family}
\end{figure}
Then we have
\begin{equation}D_1=\hat Y_1\cap \mathcal N, \ \ D_2=\hat Y_2\cap \mathcal N.\end{equation}
It is straightforward to see that the normal bundle of $D_i$ in $\mathcal N$ is $L_i^{-1}$.
Next we consider holomorphic volume forms.
Viewing $\mathcal X$ as an anti-canonical divisor in $\mathbb C\mathbb{P}^{n+1}\times \Delta$, then away from $D\times \{0\}$, $\mathcal X$ is smooth and we then obtain a holomorphic volume form $\Gamma$. In the affine chart $\{x_0\neq 0\}\times \Delta\subset \mathbb C\mathbb{P}^{n+1}\times \Delta$, the meromorphic volume form is given by
\begin{equation}\frac{1}{F_t(u)} dt\wedge du_1\wedge \cdots \wedge du_{n+1}.\end{equation}
So the Poincar\'e residue on $\mathcal X$ is
\begin{equation}\Gamma=-\frac{1}{(n+2)t^{n+1}f(u)} du_1\wedge \cdots du_{n+1}. \end{equation}
Using the equation and the genericity assumptions, $\Gamma$ is indeed holomorphic on $\mathcal X\setminus D\times \{0\}$.
Now applying the above discussion to the global function $t$ on $\mathcal X$, then we get a holomorphic family of holomorphic volume forms $\Gamma_t$ on each $\widehat{X}_t$.
Differentiating the equation $F_t(u)=f_1(u)f_2(u)+t^{n+2}f(u)=0$, we get
$(n+2)t^{n+1}f(u)dt+d_uF_t=0.$
In the above affine chart, on the set where $\frac{\partial F_t}{\partial u_1}\neq 0$, we have
\begin{equation}
\Gamma_t=\frac{1}{\frac{\partial F_t(u)}{\partial u_1}} du_2\wedge \cdots du_{n+1}.
\end{equation}
This is indeed well-defined on $\widehat{X}_t$ for $t\neq 0$ and also on $X_0\setminus D$. On each component $Y_i$ of $X_0$, it has a simple pole along $D$. Notice $\Gamma_t$ is also the natural holomorphic volume form on $\widehat{X}_t$ when we apply the Poincar\'e residue to the divisor $\widehat{X}_t$ in $\mathbb C\mathbb{P}^{n+1}$.
Now we pass to the resolution $\widehat{\mathcal X}$. Abusing notation we still denote by $\Gamma$ its pull-back.
\begin{lemma}
$\Gamma$ extends to a global holomorphic volume form on $\widehat{\mathcal X}\setminus (D_1\cup D_2)$.
\end{lemma}
\begin{proof}
We only need to consider around a point $(x, t, s)$ on the exceptional set $\mathcal N$, so $(x, t)\in D\times \{0\}$. Without loss of generality may assume $x_0\neq 0$. Since $D$ is a complete intersection by assumption (iii), we may use $v_1=f_1(u)$ and $v_2=f_2(u)$ to replace $u_1, u_2$ (say) as local holomorphic coordinates on a neighborhood of $x$ in $\mathbb C\mathbb{P}^{n+1}$. So we can write
\begin{equation}\Gamma=-\frac{J^{-1}}{(n+2)t^{n+1}f(u)}dv_1\wedge dv_2\wedge du_3\cdots\wedge du_{n+1}, \end{equation}
where $J=\frac{\partial f_1}{\partial u_1} \frac{\partial f_2}{\partial u_2} -\frac{\partial f_1}{\partial u_2}\frac{\partial f_2}{\partial u_1}.$ is the Jacobian.
Suppose first we work on the affine chart $\{s_1\neq 0\}$. Then we get the local equations for $\widehat{\mathcal X}$ given by \eqref{e: s1 nonzero region}. Since we are away from $D_1$, we must have $\zeta_3\neq 0$. Then we can use $\zeta_3, t, u_3, \cdots, u_{n+1}$ as local holomorphic coordinates on $\widehat{\mathcal X}$. We have
\begin{align}dv_1 &=-t^{d_2}fd\zeta_3-d_2t^{d_2-1}\zeta_3 f dt-t^{n+2}\zeta_3 df,
\\
dv_2&=d_1t^{d_1-1} \zeta_3^{-1}dt-\zeta_3^{-2} t^{d_1}d\zeta_3,
\\
df&=\frac{\partial f}{\partial v_1} dv_1+\frac{\partial f}{\partial v_2} dv_2+\sum_{j\geq 3} \frac{\partial f}{\partial u_j} du_j. \end{align}
So we get
\begin{align}&(1+t^{d_2} \zeta_3\frac{\partial f}{\partial v_1}) dv_1
\nonumber\\
=&(-t^{d_2}f+t^{n+2} \zeta_3^{-1} \frac{\partial f}{\partial v_2})d\zeta_3-(d_2t^{d_2-1}\zeta_3 f+d_1t^{n+1}\frac{\partial f}{\partial v_2})dt & \mod (du_3, \cdots ,du_{n+1}).\end{align}
Hence we get
\begin{equation} \label{eqn8-3}
\Gamma=\frac{\zeta_3^{-1}}{(1+t^{d_2}\zeta_3\frac{\partial f}{\partial v_1})} J^{-1}d\zeta_3\wedge dt\wedge du_3\wedge\cdots\wedge du_{n+1}.\end{equation}
Near $t=0$ we see $\Gamma$ is smooth around such a point. Similarly we can deal with the chart $\{s_2\neq 0\}$.
Now on $\{s_3\neq 0\}$, we only need to consider a point on $D$ where $f=0$, then by our assumption (iv) we may use $v_3=f$ as a local holomorphic coordinate to replace $u_3$ for instance. Then we can write
\begin{equation}\Gamma=-\frac{1}{(n+2)t^{n+1}f} K^{-1} dv_1\wedge dv_2\wedge dv_3\wedge du_4\cdots \wedge du_{n+1},\end{equation}
where $K$ is the Jacobian for the change of coordinates. We have
\begin{equation}dv_3=-(\zeta_1d\zeta_2+\zeta_2d\zeta_1),\end{equation}
\begin{equation}dv_1=t^{d_2}d\zeta_2+d_2\zeta_2 t^{d_2-1} dt,\end{equation}
\begin{equation}dv_2=t^{d_1}d\zeta_1+d_1\zeta_1 t^{d_1-1}dt.\end{equation}
Then we get
\begin{equation} \label{eqn8-4}
\Gamma=K^{-1}dt\wedge d\zeta_1\wedge d\zeta_2\wedge du_4\cdots \wedge du_{n+1}
,\end{equation}
which is smooth.
\end{proof}
We can apply the previous Poincar\'e residue to the function $t$ on $\widehat{\mathcal X}$. Since the exceptional set of the resolution lies over $D\times \{0\}$, we still get $\Gamma_t$ for $t\neq 0$. On the central fiber $\hat X_0$, we still get $\Gamma_0$ on $\hat Y_1\setminus D_1$ and $\hat Y_2\setminus D_2$. Over $\mathcal N\setminus (D_1\cup D_2)$, using (\ref{eqn8-3}) and (\ref{eqn8-4}) we get the corresponding Poincar\'e residue
\begin{equation}\Gamma_{\mathcal N}=J^{-1}\frac{d\zeta_1}{\zeta_1}\wedge du_3\wedge \cdots\wedge du_{n+1}=-J^{-1}\frac{d\zeta_2}{\zeta_2}\wedge du_3\wedge \cdots\wedge du_{n+1}.
\end{equation}
Notice by applying Poincar\'e residue twice to the complete intersection $D=\{f_1=f_2=0\}$, we obtain a holomorphic volume form $\Omega_D$ on $D$, which in the above local coordinates can be written as
\begin{equation}\Omega_D=J^{-1}du_3\wedge \cdots\wedge du_{n+1}.\end{equation}
So we have that
$\Gamma_{\mathcal N}=\frac{d\zeta_1}{\zeta_1} \wedge \Omega_D$. This means that up to multiplying by $-\sqrt{-1}$, $\Gamma_{\mathcal N}$ agrees with the natural holomorphic volume form $\Omega_0$ on $\mathcal N_0$ defined in Section \ref{ss:complex-geometry}, under the identification $k_-=d_2, k_+=-d_1$.
\begin{remark}
A priori there could be various different choices of birational transformations extracting an extra component as above. For example, one can directly perform a blow-up on the original family $\mathcal X$ (without doing the base change $t\mapsto t^{n+2}$). The reason for our particular choice of birational transform is related to a matching condition required when we glue the Calabi-Yau metrics on the three components on the central fiber $\hat{X}_0$ to the nearby smooth fiber $X_t$. See \eqref{eqn6-70} and \eqref{eqn6-71}.
\end{remark}
\subsection{Tian-Yau metrics}
\label{ss:tian-yau}
In this subsection we briefly review the complete Ricci-flat K\"ahler metrics, constructed in \cite{TY} on the complement of a smooth anti-canonical divisor in a Fano manifold. We will state some facts on the asymptotics of these metrics. Interested readers are referred to \cite{HSVZ} (Section 3) for the proof and more details.
Let $Y$ be an $n$ dimensional Fano manifold, $D$ a smooth anti-canonical divisor in $Y$, and denote $Z=Y\setminus D$. By adjunction formula $D$ itself is Calabi-Yau, and we can find a Ricci-flat K\"ahler metric $\omega_D\in 2\pi c_1(L_D)$, where $L_D$ is the restriction of $K_Y^{-1}$ to $D$.
Fixing a defining section $S$ of $D$, we can view $S^{-1}$ as a holomorphic $n$-form $\Omega_{Z}$ on $Z$ with a simple pole along $D$. Rescaling suitably we may assume that the Poincar\'e residue of $\Omega_{Z}$ gives a holomorphic volume form $\Omega_D$ on $D$ which satisfies the normalization condition \eqref{e:CY equation on D}.
As before we can fix the hermitian metric $|\cdot|$ on $L_D$ whose curvature form is $-\sqrt{-1}\omega_D$ and we also fix a smooth extension to $Y$ with strictly positive curvature.
Then
\begin{equation}\omega_{Z} \equiv\frac{n}{n+1}\sqrt{-1} \partial\bar{\partial} (-{\log |S|^2})^{\frac{n+1}{n}} \end{equation}
defines a K\"ahler form on a neighborhood of infinity in $Z$. The Tian-Yau metric $\omega_{TY}$ on $Z$ is then obtained by solving a Monge-Amp\`ere equation with reference metric $\omega_{Z}$. Let $\mathcal{C}^n$ be the Calabi model space constructed using $(D, L_D, \omega_D)$, as in Section \ref{ss:Calabi model space}.
\begin{proposition}[\cite{TY}, see also \cite{HSVZ}] \label{t:hein}
There is a smooth function $\phi$ on $Z$ such that $\omega_{TY}\equiv \omega_{Z}+\sqrt{-1}\partial\bar{\partial}\phi$ is a complete Ricci-flat K\"ahler metric on $Z$ solving the Monge-Amp\`ere equation
\begin{equation}\omega_{TY}^n=\frac{1}{n\cdot 2^{n-1}}(\sqrt{-1})^{n^2}\Omega_{Z}\wedge\overline\Omega_{Z}. \end{equation}
Moreover, there is a diffeomorphism $\Phi: \mathcal{C}^n\setminus K'\rightarrow Y\setminus K$, where $K \subset Z$ is compact and $K' = \{|\xi| \geq \frac{1}{2}\}$ and constant $\delta_{Z}>0$, such that the following holds uniformly for all $z$ large:
\begin{enumerate}
\item The K\"ahler potential $\phi$ satisfies the asymptotics
\begin{equation}\label{lalilu}|\nabla_{g_Z}^k \phi|_{g_Z} = O(e^{-\delta_Z \cdot (-\log |S|^2)^{\frac{1}{2}}}) \ \text{for all} \ k\in\mathbb{N}.
\end{equation}
\item We have the complex structure asymptotics
\begin{equation}
|\nabla_{g_{\mathcal{C}^n}}^k(\Phi^*J_{Z}-J_{\mathcal{C}^n})|_{g_\mathcal{C}^n}=O(e^{-(\frac{1}{2}-\epsilon)z^n})\ \text{for all} \ k \in\mathbb{N}, \epsilon > 0.
\end{equation}
\item We have the holomorphic $n$-form asymptotics
\begin{equation}\label{e:n-form-asympt}
|\nabla_{g_{\mathcal{C}^n}}^k(\Phi^*\Omega_Z-\Omega_{\mathcal{C}^n})|_{g_{\mathcal C}^n}=O(e^{-(\frac{1}{2}-\epsilon)z^n})\ \text{for all} \ k \in\mathbb{N}, \epsilon > 0.
\end{equation}
\item We have the K\"ahler form asymptotics \begin{equation}|\nabla_{g_{\mathcal{C}^n}}^k(\Phi^*\omega_{TY}-\omega_{\mathcal{C}^n})|_{g_{\mathcal{C}^n}}=O(e^{-{\delta_Z} z^{n/2}}).\end{equation}
\item There is a constant $C>0$ such that
\begin{equation}
C^{-1}\cdot z\leq \Phi^*((-\log |S|^2)^{\frac{1}{n}})\leq C\cdot z.
\end{equation}
\end{enumerate}
\end{proposition}
In particular, the space $(Z, \omega_{TY})$ is $\delta_Z$-asymptotically Calabi in the sense of \cite{SZ-Liouville}. For later purposes we also need a simple observation regarding the asymptotics of $\omega_{TY}$. Fix a local holomorphic chart $\{U, w_1, \cdots, w_n\}$ centered at a point $p\in D$, i.e., $w_i(p)=0$ for all $i$, and such that $S$ is locally defined by $w_1=0$. Define a cylindrical type K\"ahler metric as follows
\begin{equation}
\omega_{cyl}\equiv \sum_{j=2}^n \sqrt{-1} dw_j\wedge d\bar w_j+\sqrt{-1} |w_1|^{-2}dw_1\wedge d\bar w_1.
\end{equation}
By a straightforward computation, we have the following.
\begin{lemma} \label{l: TY cylindrical compare}
On $U\setminus D$, there is a constant $C>0$ such that
\begin{equation}
C^{-1}(-\log|S|^2)^{\frac{1}{n}-1}\omega_{cyl}\leq \omega_{TY}\leq C(-\log |S|^2)^{\frac{1}{n}}\omega_{cyl},
\end{equation}
and for all $k\in\mathbb{Z}_+$, there are constants $C_k, m_k>0$ such that
\begin{equation}
|\nabla^k_{\omega_{cyl}}\omega_{TY}|_{\omega_{cyl}}\leq C_k(-\log |S|^2)^{m_k}.
\end{equation}
\end{lemma}
Using this lemma, later when we do estimates for quantities using the Tian-Yau metric, we can do computations using the cylindrical metric which becomes much simpler, and in the end we only get an error which is of polynomial order in $-\log |S|^2$.
Finally we need a crucial Liouville theorem on the Tian-Yau spaces, which is proved in \cite{SZ-Liouville}.
\begin{theorem}[Theorem 1.2, \cite{SZ-Liouville}]
\label{t:Liouville on Tian-Yau}
There exists a constant $\epsilon_{Z}>0$ such that if $u$ is a harmonic function on the above Tian-Yau space $(Z, \omega_{TY})$ and $|u|\leq e^{\epsilon_Z\cdot z^{\frac{n}{2}}}$ as $z\rightarrow\infty$,
then $u$ is a constant.
\end{theorem}
\subsection{Construction of approximately Calabi-Yau metrics}
\label{ss:glued-metrics}
We will work on the setup of Section \ref{ss:algebraic-geometry}. Let us recall some notation from previous discussion.
The algebro-geometric setup is
\begin{itemize}
\item We have the family of Calabi-Yau varieties $p:\widehat{\mathcal X} \rightarrow \Delta$ in $\mathbb C\mathbb{P}^{n+1}\times \Delta$. Let us denote by $\widehat{X}_t$ the fiber $p^{-1}(t)$. By construction, for $t\neq 0$ we know $\widehat{X}_t$ can be identified with $X_{t^{n+2}}$ in the original family.
\item The central fiber $\widehat{X}_0$ is given by the union of three smooth components: $\hat Y_1$, $\hat Y_2$ and $\mathcal N$, with $\hat Y_j\cap \mathcal N=D_j$ both canonically isomorphic to $D$.
\item Under the identification $k_-=d_2$, $k_+=-d_1$, and $L=\mathcal O(1)|_D$, $\mathcal N\setminus (D_1\cup D_2)$ is naturally identified with the space $\mathcal N^0$ defined in Section \ref{ss:complex-geometry}.
\item The normal bundle of $D_j$ in $\hat Y_j$ is $L_j=\mathcal{O}(d_{3-j})|_{D}$ and in $\mathcal N$ is $L_j^{-1}$.
\item There is a relative holomorphic volume form $\Gamma_t(t\in\Delta)$ defined on $\widehat{\mathcal X}\setminus \{D_1\cup D_2\}$. We denote
\begin{equation}
\begin{cases}
\Gamma_{0, 1}\equiv \Gamma_0|_{Z_1}\\
\Gamma_{0,2}\equiv \Gamma_0|_{Z_2},
\end{cases}
\end{equation}
where $Z_j\equiv \hat Y_j\setminus D_j$, and we know
\begin{equation}
\Gamma_0|_{\mathcal N^0}=-\sqrt{-1} \Omega_0,
\end{equation}
where $\Omega_0$ is the holomorphic volume form on $\mathcal N^0$ defined in \eqref{e:Omega N0 definition}.
\end{itemize}
The corresponding metric ingredients are
\begin{itemize}
\item We have the Calabi-Yau metric $\omega_D\in 2\pi c_1(L)$ on $D$, where $L=\mathcal{O}(1)|_D$. We fix a hermitian metric on $L$ with curvature $-\sqrt{-1}\omega_D$. We also extend this hermitian metric to the whole $\mathbb C\mathbb{P}^{n+1}$ such that its curvature form defines a smooth K\"ahler metric $\omega_{\mathbb C\mathbb{P}^{n+1}}$. This then induces hermitian metrics on $\mathcal{O}(l)$ for all $l$, and also on the pull-back of $\mathcal{O}(l)$ to the projective bundle $\mathbb{P}(\mathcal{O}(d_2)\oplus \mathcal{O}(d_1)\oplus \mathbb C)$. Later when $s$ is a holomorphic section of some $\mathcal{O}(l)$, $|s|$ will always mean the norm of $s$ with respect to this fixed hermitian metric.
\item Applying the construction in Section \ref{ss:tian-yau} to the line bundle $L_j\rightarrow D_j$, we have the Tian-Yau metrics $\omega_{TY, j}$ on $Z_j$ for $j\in\{1, 2\}$, and the Calabi-Yau metrics $\omega_{D_j}=d_{3-j}\cdot \omega_D$. So $\omega_{TY, j}$ is asymptotic to
\begin{equation}
\omega_{Z_j}=\frac{n}{n+1}\sqrt{-1} \partial\bar{\partial} (-\log |f_{3-j}|^2)^{\frac{n+1}{n}},
\end{equation}
and
\begin{equation}
\omega_{TY, j}^n=\frac{(\sqrt{-1})^{n^2}}{n\cdot 2^{n-1}}\cdot d_{3-j}^{n-1} \cdot \Gamma_{0, j}\wedge \bar\Gamma_{0, j},
\end{equation}
where the coefficient $d_{3-j}^{n-1}$ arises from the fact we are using $\omega_{D_j}$ instead of $\omega_D$ in the construction.
\item The family of incomplete $C^{2, \alpha}$ approximately Calabi-Yau metrics $(\omega_T, \Omega_T)$ on $\mathcal M_T$, and $(\mathcal M_T, \Omega_T)$ is embedded in $(\mathcal N^0, \Omega_0)$ as in Section \ref{ss:complex-geometry}, with $k_-=d_2$ and $k_+=-d_1$.
\end{itemize}
Our goal in this subsection is to construct for each $t$ small a $C^{1,\alpha}$ K\"ahler metric $\omega(t)$ on $\widehat{X}_t$ which is approximately Calabi-Yau in a suitable weighted sense.
\subsubsection{Matching between the parameters $t$ and $T$}
The relationship between the parameters $t$ and $T$ can be determined by studying the matching between the Tian-Yau ends and the neck region.
In our setting, we need to normalize the Tian-Yau metrics $\omega_{TY, i}$ on $Z_i$ (as in Section \ref{ss:tian-yau}) by defining \begin{equation}
\tilde\omega_{TY, j}\equiv 2^{\frac{-1}{n}}n^{\frac{1}{n}}d_{3-j}^{-\frac{n-1}{n}}\omega_{TY, j}.
\end{equation}
Then we have
\begin{equation}
\tilde\omega_{TY, j}=\frac{(\sqrt{-1})^{n^2}}{2^n} \Gamma_{0, j}\wedge \bar\Gamma_{0,j}.
\end{equation}
By definition we can write
\begin{equation}
\tilde\omega_{TY, j}= dd^c\phi_j=2\sqrt{-1} \partial\bar{\partial}\phi_j,\end{equation}
where
$\phi_j=\eta_j+\psi_j$
such that
\begin{align}
\eta_j=\frac{1}{n+1}\cdot k_{3-j}^{\frac{1-n}{n}}\cdot n^{\frac{n+1}{n}}\cdot (-\log |f_{3-j}|)^{\frac{n+1}{n}}\quad \text{and} \quad |\nabla^k \psi_1|=O(e^{-\delta_0 (-\log |f_{3-j}|^2)^{1/2}}),\end{align}
for all $k\in \mathbb{N}$, and where the derivatives and norms are taken with respect to the Tian-Yau metric itself (which is equivalent to taking with respect to the metric $\omega_{Z_j}$).
\
Now on the neck $\mathcal M_T$ we have the asymptotics of the K\"ahler potential given in Section \ref{ss:complex-geometry}. By the discussion there we identify $\mathcal M_T$ with an open set in $\mathcal N^0$, and the latter is naturally an open set in $\mathcal N$. Moreover, we can write
\begin{equation}
\label{e:asymptotics of phi-}
T^{\frac{n-2}{n}}\omega_T=dd^c\phi_T,
\end{equation}
with
\begin{equation}
\phi_T=
\begin{cases}
\phi_-\equiv\varphi_-+\psi_-, \ \ \ \ z< 0;\\
\phi_+\equiv\varphi_++\psi_+, \ \ \ \ z> 0,
\end{cases}
\end{equation}
where
\begin{equation}
\label{e:definition of phi-}
\begin{cases}
\varphi_-=\frac{1}{n+1}n^{\frac{n+1}{n}} k_-^{-\frac{n-1}{n}}(A_--\log |s_1/s_3|)^{\frac{n+1}{n}};\\
\varphi_+=\frac{1}{n+1}n^{\frac{n+1}{n}} (-k_+)^{-\frac{n-1}{n}}(A_+-\log |s_2/s_3|)^{\frac{n+1}{n}}.\end{cases}
\end{equation}
Notice $\omega_T$ is defined by $z\in [T_-, T_+]$, so by \eqref{e: compare r and z} $\varphi_-$ is well-defined for $A_--\log |s_1/s_3|\gg1$ and
$\varphi_2$ is well-defined for $A_+-\log |s_2/s_3|\gg1$. Moreover
for $|z|\geq 1$, we have
\begin{equation}
|\psi_\pm|=\epsilon(z)+\epsilon_T.
\end{equation}
Now on $\mathcal M_T$ for $|t|>0$ small we have \begin{equation}
t^{d_1}s_1=s_3f_2(x)
\end{equation}
which gives
\begin{equation}-d_1\log |t|-\log \frac{|s_1|}{|s_3|}=-\log |f_2|.\end{equation}
So if we want to graft the metrics on the three components of $\widehat{X}_0$ to nearby $\widehat{X}_t$ so that they match with small errors, then we need
\begin{equation} \label{eqn6-70}
d_1\log |t|=-A_-.
\end{equation}
Similarly at the positive end we need
\begin{equation}\label{eqn6-71}
d_2\log |t|=-A_+.
\end{equation}
This suggests that we should choose
\begin{equation}
\label{e:t T relation}
|t|=e^{-\frac{1}{d_1}A_-}=e^{-\frac{1}{d_2}A_+}.
\end{equation}
Given $|t|$ small we can find $T$ big so that \eqref{e:t T relation} holds. It is not necessary that $T$ is uniquely determined by $t$, but we always fix a particular choice for each $t$ throughout this section so that \eqref{e:t T relation} holds. With this choice it is easy to see that
\begin{equation} \label{eqn6667}
C^{-1}e^{-\frac{1}{d_1d_2n} T^2}\leq |t|\leq Ce^{-\frac{1}{d_1d_2n} T^2}.
\end{equation}
\subsubsection{Fixing the constants in the definition of weighted spaces}
\label{sss:parameters-fixed}
From now on, we fix weight parameters in the definition of weight spaces, which allows us to prove the uniform injectivity estimate in Proposition \ref{p:global-injectivity-estimate} and apply the implicit function theorem to complete the proof of the main theorem in Section \ref{ss:global analysis}. The parameters $\delta$, $\mu$, $\nu$ are fixed as follows (similar to the specification of the parameters in Section \ref{ss:perturbation-framework}):
\begin{enumerate}
\item[(GP1)] (Fix $\nu$) The parameter $\nu\in\mathbb{R}$ is chosen such that
$\nu\in(-1,0)$.
\item[(GP2)] (Fix $\alpha$) The H\"older constant $\alpha\in(0,1)$ is chosen such that
$\nu+\alpha<0$.
\item[(GP3)] (Fix $\delta$) The constant $\delta>0$ is chosen such that \begin{equation}
0<\delta < \delta_G \equiv \frac{1}{n\cdot (|k_-| + |k_+|)}\cdot \min\{\delta_e, \delta_{Z_1}, \delta_{Z_2}, \epsilon_{Z_1}, \epsilon_{Z_2}, \sqrt{\lambda_D}\},\label{e:global-delta}
\end{equation}
where $\sqrt{\lambda_D}$ is in Lemma \ref{l:liouville-cylinder},
$\delta_e>0$ is in Proposition \ref{p:CY-error-small}, $\delta_{Z_1}, \delta_{Z_2}$ are the constants in Proposition \ref{t:hein} applied to $Z_1, Z_2$, and $\epsilon_{Z_1},\epsilon_{Z_2}$ are the constants in Theorem \ref{t:Liouville on Tian-Yau} applied to $Z_1, Z_2$.
\item[(GP4)] (Fix $\mu$) The parameter $\mu>0$ is chosen as
$\mu=(1-\frac{1}{n})(\nu+2+\alpha)$.
\end{enumerate}
\subsubsection{Construction of $\omega(t)$}
We will divide a neighborhood of $\widehat{X}_0$ into various regions (c.f. Figure \ref{f: the division of region}):
\begin{itemize}
\item Region $\bf{I}$ is given by $2|s_3|\geq \max(|s_1|, |s_2|)$;
\item Region $\bf{II}_-$ is given by $s_1\neq 0$, and $|s_3|\leq 2|s_1|, -\log |f_2|\geq -\frac{d_1}{2}\log |t|$;
\item Region $\bf{III}_-$ is given by $s_1\neq 0$, and $|f_2|\leq 1/2$, $-\log |f_2|\leq -\frac{d_1}{2}\log |t|+1$;
\item Region $\bf{IV}_-$ is
given by $s_1\neq 0$ and $|f_2|\geq 1/4$;
\item Region $\bf{II}_+$ is given by $s_2\neq 0$, and $|s_3|\leq 2|s_2|, -\log |f_1|\geq -\frac{d_2}{2} \log |t|$;
\item Region $\bf{III}_+$ is
given by $s_2\neq 0$ and $|f_1|\leq 1/2$, $-\log |f_1|\geq -\frac{d_2}{2} \log |t|+1$;
\item Region $\bf{IV}_+$ is
given by $s_2\neq 0$ and $|f_1|\geq 1/4$.
\end{itemize}
\begin{figure}
\begin{tikzpicture}
\draw[thick] (0, 2) to (3, 5);
\draw[thick] (0, 2) to (0, -2);
\draw[thick] (0, -2) to (3, -5);
\node at (0, 2) {$\bullet$};
\node at (0, -2) {$\bullet$};
\node at (1, 4.5) {$Y_1$};
\draw[dashed, ->] (1.2, 4.2) to (1.6, 3.8);
\node at (1, -4.5) {$Y_2$};
\draw[dashed, ->] (1.2, -4.2) to (1.6, -3.8);
\node[red] at (3.8, 5) {$\widehat{X}_t$};
\node at (-0.3, 1.7) {$D_1$};
\node at (-0.3, -1.7) {$D_2$};
\node at (-2, -0) {$\mathcal N$};
\draw[dashed, ->] (-1.8, 0) to (-0.1, 0);
\node at (6, 4) {Region $\bf{IV}_-$};
\node at (6, 2) {Region $\bf{III}_-$};
\node at (6, 1) {Region $\bf{II}_-$};
\node at (6, 0) {Region $\bf{I}$};
\node at (6, -4) {Region $\bf{IV}_+$};
\node at (6, -2) {Region $\bf{III}_+$};
\node at (6, -1) {Region $\bf{II}_+$};
\draw[blue,->, dashed] (4.8, 4) to (3.1, 4);
\draw[blue,->, dashed] (4.8, 2) to (2.3, 2);
\draw[blue,->, dashed] (4.8, 1) to (1.3, 1);
\draw[blue,->, dashed] (4.8, 0) to (0.8, 0);
\draw[blue,->, dashed] (4.8, -4) to (3.1, -4);
\draw[blue,->, dashed] (4.8, -2) to (2.3, -2);
\draw[blue,->, dashed] (4.8, -1) to (1.3, -1);
\draw[red, thick] plot[smooth] coordinates {(3.5, 5) (0.5, 2) (0.5, -2) (3.5, -5)};
\draw[blue] (-0.5, 1) to (0.5, 1);
\draw[blue] (-0.5, 1) to (-0.5, -1);
\draw[blue] (-0.5, -1) to (0.5, -1);
\draw[blue] (0.5, 1) to (0.5, -1);
\draw[blue] (-1, 0.7) to (1, 0.7);
\draw[blue] (-1, 2) to (1, 2);
\draw[blue] (-1, 0.7) to (-1,2);
\draw[blue] (1, 0.7) to (1, 2);
\draw[blue] (-0.8, 2.4) to (1.2, 1.4);
\draw[blue] (0.2, 3.4) to (2.2, 2.4);
\draw[blue] (-0.8, 2.4) to (0.2, 3.4);
\draw[blue] (2.2, 2.4) to (1.2, 1.4);
\draw[blue] (0.4, 3) to (1.2, 2.4);
\draw[blue] (2.8, 5.4) to (0.4, 3);
\draw[blue] (1.2, 2.4) to (3.6, 4.8);
\draw[blue] (-1, -0.7) to (1, -0.7);
\draw[blue] (-1, -2) to (1, -2);
\draw[blue] (-1, -0.7) to (-1,-2);
\draw[blue] (1, -0.7) to (1, -2);
\draw[blue] (-0.8, -2.4) to (1.2, -1.4);
\draw[blue] (0.2, -3.4) to (2.2, -2.4);
\draw[blue] (-0.8, -2.4) to (0.2, -3.4);
\draw[blue] (2.2, -2.4) to (1.2, -1.4);
\draw[blue] (0.4, -3) to (1.2, -2.4);
\draw[blue] (2.8, -5.4) to (0.4, -3);
\draw[blue] (1.2, -2.4) to (3.6, -4.8);
\end{tikzpicture}
\caption{Division of a neighborhood of $\widehat{X}_0$}
\label{f: the division of region}
\end{figure}
For all $|t|$ sufficiently small, then we also get a division of $\widehat{X}_t$ into 7 regions.
Notice that we have non-empty intersections between these regions so that we need a cut-off on the overlap.
For the convenience of later analysis, we now fix a finite cover $\mathcal U=\{U_\beta^1, U_\gamma^2, U_{\mathcal N}, U_{-}, U_{+}\}$ of a neighborhood of $\widehat{X}_0$ in $\widehat{\mathcal X}$ obtained as follows.
We first cover a neighborhood of $D_1$.
Given any point in $(x, t, [s_1:s_2:s_3])\in D_1$, we have $t=f_1(x)=f_2(x)=s_2=s_3=0, s_1\neq0$. On the open subset $\{x_j\neq 0\}$ in $\mathbb C\mathbb{P}^{n+1}$, we can view $\sigma=x_j$ as a trivialization of $\mathcal{O}(1)$. Without loss of generality we may assume $j=0$. Then we get affine coordinates $\{u_i=x_i/x_0(i=1,\cdots, n+1)\}$, and we can $v_1=f_1(u)$ and $v_2=f_2(u)$ as local holomorphic functions on $\mathbb C\mathbb{P}^{n+1}$. Further without loss of generality we can assume $\{v_1, v_2, u_i=x_i/x_0 (i=3, \cdots)\}$ yield local holomorphic coordinates in a neighborhood of $x$ in $\mathbb C\mathbb{P}^{n+1}$. Correspondingly we can pull back these to be local holomorphic functions on the projective bundle $\mathbb{P}(\mathcal{O}(d_2)\oplus \mathcal{O}(d_1)\oplus \mathbb C)$. As before we also introduce local holomorphic functions $\zeta_3=s_3/s_1, \zeta_2=s_2/s_1$ on the projective bundle, and the space $\widehat{\mathcal X}$ is then defined by the equations as in \eqref{e: s1 nonzero region all equations}, which essentially reduces to one relation $v_2\zeta_3=t^{d_1}$ in the three variables $v_2, \zeta_3, t$. We denote by $U_\beta^1$ an open subset in $\widehat{\mathcal X}$ defined by the inequalities $|\zeta_3|< 3 |\sigma|^{d_2}$, $|v_2|<3|\sigma|^{d_2}$, and $|u_i|< C(i=3, \cdots)$ for some fixed $C>0$. Denote the trivializing section $\sigma$ by $\sigma_\beta^1$. For $|t|$ small, $U^1_{\beta, t}\equiv U^1_\beta\cap \widehat{X}_t$ is then defined by the equation $v_2\zeta_3=t^{d_1}$.
We have the natural projection maps
\begin{equation}\pi_\beta^1: U^1_{\beta, t}\rightarrow U^{1}_{\beta, 0}\cap Y_1; (x, t, v_2, \zeta_3)\mapsto (v_2, 0),\end{equation}
\begin{equation}\pi_\beta^{ \mathcal N}: U^1_{\beta, t}\rightarrow U^{1}_{\beta, 0}\cap \mathcal N; (x, t, v_2, \zeta_3)\mapsto (0, \zeta_3),\end{equation}
\begin{equation}\pi^{1, D}_{\beta}: U^1_{\beta, t}\rightarrow D; (x, t, v_2, \zeta_3)\rightarrow x.\end{equation}
Then the union of images $\pi^{1, D}_\beta(U^1_{\beta, t})$ form an open cover of $D$. By compactness we can choose and then fix finitely many of them which also cover $D$, and we put these $U^1_\beta$'s in $\mathcal U$. Then we obtain also a cover of a neighborhood of $D_1$ in $Y_1$ by $\{U^{1}_{\beta, 0}\cap Y_1\}$ and a cover of a neighborhood of $D_1$ in $\mathcal N$ by $\{U^{1}_{\beta, 0}\cap \mathcal N\}$ so that on each element in the cover we have holomorphic coordinates. Without loss of generality we may assume these cover the neighborhood defined by $|f_2|\leq 3$ and $|s_3/s_1|\leq 3$. So in particular they contain Regions $\bf{II}_-$ and $\bf{III}_-$.
We can do the same with $D_2$, and add the corresponding elements $U_\gamma^2$ to $\mathcal U$. Now away from $D_1\cup D_2$ we may find a trivialization of the fibration $\widehat{\mathcal X}\rightarrow\Delta$. So we can obtain three open subsets of $\widehat{\mathcal X}$, each of which has a differentiable trivialization over $\Delta$. Call these $U_{\mathcal N}$, $U_-$, $U_+$. Adding these to $\mathcal U$ we then obtain an open cover of a neighborhood of $\widehat{X}_0$. Over each of the three subsets we also have the projection map $\pi_{-}, \pi_{+}$, and $\pi_{{\mathcal N}}$ from them into $\widehat{X}_0\setminus (D_1\cup D_2)$. We may assume that Region $\bf{I}$ is contained in $U_{\mathcal N}$, Region $\bf{IV}_\pm$ is contained in $U_\pm$.
Let us fix a partition of unity $\chi_\beta^1$ of $D$ subordinate to the cover $\pi_\beta^{1, D}(U^1_{\beta, t})$, and ${\chi_\gamma^2}$ of $D$ subordinate to the cover $\pi_\gamma^{1, D}(U^2_{\gamma, t})$. We view these naturally as functions on the corresponding $U^1_{\beta, t}$ and $U^2_{\gamma, t}$, though not compactly supported (along the fiber direction).
\
Below we define the approximately Calabi-Yau metric $\omega(t)$ on $(\widehat{X}_t, \Gamma_t)$ for each region above, and we also define the weight function $\rho_t(\bm{x})$ simultaneously and measure the following error of the Calabi-Yau equation in the weighted sense:
\begin{equation}
\mathrm{Err}_{t}\equiv \frac{(\sqrt{-1})^{n^2}2^{-n}\Gamma_t\wedge\bar\Gamma_t}{\omega(t)^n/n!}-1.
\end{equation}
Obviously $\mathrm{Err}_{t}=0$ if and only if $\omega(t)$ is Calabi-Yau.
In the course we also discuss the gluing in the intersection of neighboring regions.
\
{\bf Region $\bf{I}$}. In this region we define
\begin{equation}\omega(t)=T^{\frac{2-n}{n}}(T\omega_{\mathbb C\mathbb{P}^{n+1}}|_{\widehat{X}_t}+dd^c\phi_{t, \mathcal N}),
\end{equation}
where
$\phi_{t,\mathcal N}=\pi_{\mathcal N}^*\phi_T$.
Using the fixed diffeomorphism $\pi_{\mathcal N}$ we may view the $C^{1,\alpha}$ K\"ahler structures $(\omega(t), \Omega(t)=\Gamma_t)$ on $\widehat{X}_t$ as a perturbation of the K\"ahler structure $(\omega_T, \Omega_0)$ on $\mathcal N^0$.
Notice by Corollary \ref{c:r zeta relation} it is not difficult to see that $\bf{I}\cap \mathcal N$ is contained in the union $\bf{I}_1\cup \bf{I}_2$ (as defined in Section \ref{ss:neck-weighted-analysis}). So we can define
\begin{equation}
z(\bm{x})\equiv z(\pi_{\mathcal N}(\bm{x})), \ \ L_t(\bm{x})\equiv L_t(\pi_{\mathcal N}(\bm{x})), \ \ \mathfrak{r}(\bm{x})\equiv \mathfrak{r}(\pi_{\mathcal N}(\bm{x}))
\end{equation}
and then use \eqref{e:definition of weights} to define the weight function $\rho_t(\bm{x})$. Applying Proposition \ref{p: perturbation of complex structures}, we conclude that
\begin{equation}
\begin{cases}|\Omega(t)-\Omega_0|_{C^{1, \alpha}_{\delta, \nu,\mu}}=\underline\epsilon_{T^2}, \\
|\omega(t)-\omega_t|_{C^{1, \alpha}_{\delta, \nu,\mu}}=\underline\epsilon_{T^2}.
\end{cases}
\end{equation}
Then by Proposition \ref{p:CY-error-small} we get an error estimate
\begin{equation}
\label{e:error region I}
\|\mathrm{Err}_{t}\|_{C^{0, \alpha}_{\delta,\nu+2, \mu}({\bf{I}}\cap \widehat{X}_t)}=O(T^{\nu+\alpha}).
\end{equation}
At the two ends of ${\bf{I}}\cap {\widehat{X}_t}$, we can write down the metric $\omega(t)$ in potential form. In the negative end we have $f_2\neq 0$, so we can write
\begin{equation}
\omega_{\mathbb C\mathbb{P}^{n+1}}|_{\widehat{X}_t}=-\frac{1}{d_2}dd^c \log |f_2|,
\end{equation}
and
\begin{equation}
\omega(t)=T^{-\frac{n-2}{n}}dd^c \phi_{t, \bf{I}_-},
\end{equation}
where
\begin{equation}
\label{e:Kahler potential I-}
\phi_{t, \bf{I}_-}=-\frac{T}{d_2}\log |f_2|+\phi_{t,\mathcal N}.
\end{equation}
Similarly at the positive end we have
\begin{equation}
\omega(t)=T^{-\frac{n-2}{n}}dd^c\phi_{t, \bf{I}_+},
\end{equation}
where
\begin{equation}
\phi_{t, \bf{I}_+}=-\frac{T}{d_1}\log |f_1|+\phi_{t, \mathcal N}.
\end{equation}
\
\
{\bf Region $\bf{IV}_\pm$.} We only consider the region $\bf{IV}_-$, and $\bf{IV}_+$ is similar. We define
\begin{equation}
\omega(t)=dd^c(\phi_1\circ \pi_-).
\end{equation}
Then for $|t|$ small we can view $(\widehat{X}_t\cap {\bf{IV}_-}, \omega(t))$ as a perturbation of the Tian-Yau metric $\tilde\omega_{TY, 1}$. It is easy to see that in the intersection $\widehat{X}_t\cap \bf{IV}_-$, for all $k\geq 0$ we have
\begin{equation}
\begin{cases}
|\nabla_{\tilde\omega_{TY, 1}}^k(\omega(t)-\tilde\omega_{TY, 1})|_{\tilde\omega_{TY, 1}}=\underline\epsilon_{T^2};\\
|\nabla_{\tilde\omega_{TY, 1}}^k((\Gamma(t)-\Gamma_{0, 1}))|_{\tilde\omega_{TY, 1}}=\underline\epsilon_{T^2}.
\end{cases}
\end{equation}
To define the weight we let
\begin{equation}
L_t(\bm{x})\equiv T^{\frac{n-2}{n}}(nk_-)^{\frac{1}{n}} (-\log 4)^{\frac{1}{n}},
\end{equation}
\begin{equation}
U_t(\bm{x})=T-T^{1-\frac{n}{2}}L_t(\bm{x})^{\frac{n}{2}},
\end{equation}
and then define $\rho_t(\bm{x})$ as in \eqref{e:definition of weights}.Then we obtain that
\begin{equation}
\label{e:error region IV-}
\|\mathrm{Err}_{t}\|_{C^{0, \alpha}_{\delta, \nu+2, \mu}({\bf{IV}_-}\cap \widehat{X}_t)}=\underline\epsilon_{T^2}.
\end{equation}
We also have by assumption the asymptotics at the end
\begin{equation}
\label{e:Kahler potential region IV-}
\phi_1\circ\pi_-=\eta_1\circ\pi_-+\psi_1\circ\pi_-.
\end{equation}
\
\
{\bf Region $\bf{II}_\pm$.} Again we only consider the region $\bf{II}_-$.
We define
\begin{equation}
\omega(t)=T^{\frac{2-n}{n}}dd^c \phi_{t, \bf{II}_-},
\end{equation}
where
\begin{equation}
\phi_{t, \bf{II}_-}=\sum {\chi_\beta^1} \cdot \phi_-\circ \pi_{\beta}^{\mathcal N}.
\end{equation}
By definition of $\phi_-$ we know this is well-defined in region $\bf{II}_-$ for $|t|>0$ small.
We need the following Lemma.
\begin{lemma}
\label{l:transition function expansion}
We have the following
\begin{enumerate}
\item On $\pi^1_{\beta}(U^1_{\beta, t}\cap U^1_{\beta',t})$, we write $\pi^{1}_{\beta'}\circ( \pi^1_\beta)^{-1}(\bm{x})=\bm{x}'$. Suppose $\bm{x}$ and $\bm{x}'$ have coordinates given by $(v_2, 0, u_i)$ and $(v_2', 0, u_i')$ in the chart $U^1_{\beta, 0}\cap Y_1$. Then we have
\begin{equation}
\begin{cases}
v_2'=v_2 \cdot (1+v_1 F_2)
\\
u_i'=u_i+v_1 G_i,
\end{cases}
\end{equation}
where $F_2$ and $G_i$ are smooth functions in $v_1, v_2, u_i$, and $v_1$ is implicitly determined by $v_2, u_i$ and $t$ by the equation \eqref{e: s1 nonzero region}
\item On $\pi^{\mathcal N}_{\beta}(U^1_{\beta, t}\cap U^1_{\beta',t})$, we write $\pi^{\mathcal N}_{\beta'}\circ( \pi^{\mathcal N}_\beta)^{-1}(\bm{x})=\bm{x}'$. Suppose $\bm{x}$ and $\bm{x}'$ have coordinates given by $(0, \zeta_3, u_i)$ and $(0, \zeta_3', u_i')$ in the chart $U^1_{\beta, 0}\cap\mathcal N$. Then we have
\begin{equation}
\begin{cases}
\zeta_3'=\zeta_3 \cdot (1+v_2 \tilde F_3)
\\
u_i'=u_i+v_2 \tilde G_i,
\end{cases}
\end{equation}
where $\tilde F_2$ and $\tilde G_i$ are smooth functions in $v_1, v_2, u_i$, and $v_1$ is implicitly determined by $\zeta_3, u_i$ and $t$ by the equation \eqref{e: s1 nonzero region}.
\end{enumerate}
\end{lemma}
\begin{proof}
This involves only local discussion. By construction we get overlapping local holomorphic charts on $\mathbb C\mathbb{P}^{n+1}$ given by $\{v_1, v_2, u_i(i\geq 3)\}$ and $\{v_1', v_2', u_i' (i\geq 3)\}$. Given a point in this overlap with coordinates $(v_1, v_2, u_i)$ and $(v_1', v_2', u_i')$ in these two coordinate charts respectively, then we have
\begin{equation}
\begin{cases}
v_1'=v_1\cdot Q_1(v_1, v_2, u_i)\\
v_2'=v_2\cdot Q_2(v_1, v_2, u_i)\\
u_i'=R_i'(v_1, v_2, u_i).
\end{cases}
\end{equation}
where $Q_1, Q_2$ are smooth and non-vanishing along $D$. More precisely, we have
\begin{equation}
Q_i=(\sigma_{\beta'}^1/\sigma_\beta^1)^{d_i}.
\end{equation}
Correspondingly we obtain the transition maps on $U^1_\beta\cap U^1_{\beta'}$ given by
\begin{equation}
\begin{cases}
v_2'=v_2\cdot Q_2(v_1, v_2, u_i); \\
\zeta_3'=\zeta_3\cdot Q_2^{-1}(v_1, v_2, u_i); \\
u_i'=R_i'(v_1, v_2, u_i);
\end{cases}
\end{equation}
where using \eqref{e: s1 nonzero region} we can write $v_1$ implicitly as a function of $v_2, \zeta_3'$ and $u_i$. In particular, we obtain the transition function on $Y_1\cap U_\beta^1\cap U_{\beta'}^1$ given by
\begin{equation}
\begin{cases}
v_2'=v_2\cdot Q_2(0, v_2, u_i); \\
u_i'=R_i'(0, v_2, u_i),
\end{cases}
\end{equation}
and on $\mathcal N\cap U_\beta^1\cap U_{\beta'}^1$ given by
\begin{equation}
\begin{cases}
\zeta_3'=\zeta_3\cdot Q_2^{-1}(v_1, 0, u_i); \\
u_i'=R_i'(v_1, 0, u_i).
\end{cases}
\end{equation}
Then the conclusion follows by a direct calculation.
\end{proof}
\begin{proposition}
\label{p:region II- error on overlapping region}
In the Region ${\bf{II}_-}\cap U_{\beta, t}^1$, we have for all $k\geq 0$
\begin{equation}
|\nabla^k(\phi_{t, \bf{II}_-}\circ (\pi_{\beta}^{\mathcal N})^{-1}-\phi_-)|=\underline\epsilon_{T^2},
\end{equation}
where the derivative and norm are taken with respect to the metric $\omega_T$.
\end{proposition}
\begin{proof}
We may write
\begin{equation}
\phi_{t, \bf{II}_-}\circ (\pi_{\beta}^{\mathcal N})^{-1}(\bm{x})-\phi_-(\bm{x})=\sum_{\beta': q\in U_{1, \beta'}} \chi_{\beta'}^1(\bm{x}) (\phi_-\circ \pi_{\beta'}^{\mathcal N}\circ(\pi_{\beta}^{\mathcal N})^{-1}(\bm{x})-\phi_-(\bm{x})).
\end{equation}
Write
\begin{equation}
\pi_{\beta'}^{\mathcal N}\circ (\pi_{\beta}^{\mathcal N})^{-1}(\bm{x})=\bm{x}'=(\zeta_3', u_i').
\end{equation}
Then we write
\begin{equation} \label{eqn-6109}
\phi_-(\bm{x}')-\phi_-(\bm{x})=\int_0^{1}\langle\nabla_{\omega_T}\phi_-(t\bm{x}'+(1-t)\bm{x}), \bm{x}'-\bm{x}\rangle_{\omega_T} dt.
\end{equation}
\textbf{Claim:} For any $k\geq 0$, there is a $C_k>0$ such at for all $\bm{x}\in \bf{II}_-$,
\begin{equation}
|\nabla^k_{\omega_T} \phi_-(\bm{x})|\leq e^{C_kT}.
\end{equation}
To see this we notice by definition $\phi_-$ satisfies the equation
\begin{equation}
\Delta_{\omega_T}\phi_-=\Tr_{\omega_T}\omega_T=n.
\end{equation}
Notice by Corollary \ref{c:r zeta relation} item (2), given $c\in (0, 1/2)$ we have for all $\bm{x}\in \bf{II}_-$,
\begin{equation}
r(\bm{x})\geq cT^{-1}\log T.
\end{equation}
Hence by Proposition \ref{p:regularity-scale}, for every point $\bm{y}$ in the regularity ball $B_{\mathfrak s(x)}(\bm{x})$, we have
\begin{equation}
r(\bm{y})\geq \frac{c}{2}T^{-1}\log T.
\end{equation}
So we can apply the item (2) in Proposition \ref{p:local-weighted-schauder}, and it suffices to show a bound on the $C^0$ norm of $\phi_-$. By \eqref{e:definition of phi-} it suffices to bound $-\log |f_2|$. By our definition for $\bm{x}\in \bf{II}_-$ we have
\begin{equation}
-\log |f_2|=-d_1\log |t|-\log \frac{|s_1|}{|s_3|}\leq -\frac{d_1}{2}\log |t|-\log 2\leq CT^2.\end{equation}
This then proves the Claim.
Now it suffices to bound the norm of the vector field $\bm{x}'-\bm{x}$ and its convariant derivatives. To this end we divide into two cases.
\textbf{Case 1}: $z\leq -1$.
Notice by Lemma \ref{l:neck cylindrical compare} comparing with the cylindrical metric, we obtain the norm of the tangent vectors $|\bm{x}'-\bm{x}|\leq |v_2|T^{m}$ for some $m>0$. On the other hand we have $|v_2|\leq C|f_2|\leq \underline\epsilon_{T^2}$. So we obtain
\begin{equation}
|\phi_-(\bm{x}')-\phi_-(\bm{x})|=\underline\epsilon_{T^2}.
\end{equation}
The higher order derivatives follows similarly by differentiating \eqref{eqn-6109} and Lemma \ref{l:neck cylindrical compare}, using the fact that all derivatives of the vector field $\bm{x}'-\bm{x}$ in the cylindrical metric is bounded by $C|v_2|$.
\
\textbf{Case 2}. $z\geq -1$. Then we instead compare the metric $\omega_T$ with the standard metric
\begin{equation}\omega_{std}\equiv \sum_{j=1}^{n-1}\sqrt{-1} dw_j\wedge d\bar w_j+\sqrt{-1} d\zeta_3\wedge d\bar \zeta_3.\end{equation}
As in the proof of Proposition \ref{p: perturbation of complex structures} we first notice
\begin{equation}\Delta_{\omega_T}w_j=\Delta_{\omega_T}\zeta_3=\Delta_{\omega_T}\zeta_3^{-1}=0.\end{equation}
By assumption we have $|\zeta_3|\leq C$ in this case, and also by Corollary \ref{c:r zeta relation}, item (3) we get $|\zeta_3^{-1}|\leq Ce^{CT}$.
Then we again apply Schauder estimates Proposition \ref{p:local-weighted-schauder}, item (2), to get \begin{equation}
|\nabla^k w_j|\leq Ce^{C_kT}, |\nabla^k \zeta_3|\leq Ce^{C_kT}.
\end{equation}
Hence we get for all $k\geq 0$.
\begin{equation}|\nabla^k_{\omega_T} \omega_{std}|_{\omega_T}\leq Ce^{C_kT}.\end{equation}
Now we use
\begin{equation}
\omega_T^{n}\leq C\Omega_T\wedge\bar\Omega_T\leq C|\zeta_3|^{-2} \omega_{std}^n.
\end{equation}
to get that
\begin{equation}
\omega_{std}\geq Ce^{-CT}\omega_T.
\end{equation}
Now we again can first estimate the norm of $\bm{x}'-\bm{x}$ and its derivatives using the standard metric, and use the above information to conclude.
\end{proof}
Now we define the weight function $\rho_t$. We first define
\begin{equation}
L_t(\bm{x})=\sum_\beta \chi_\beta(\bm{x})\cdot L(\pi_{\beta}^{\mathcal N}(\bm{x})), \ \ \ \ \mathfrak{r}(\bm{x})\equiv e^{\sum_\beta \chi_\beta(\bm{x})\cdot \log \mathfrak{r}(\pi_\beta^{\mathcal N}(\bm{x}))}.
\end{equation}
Then we define the weight function $\rho_t(\bm{x})$ as \eqref{e:definition of weights}.
Notice we have that on ${\bf{II}_-}\cap \widehat{X}_t\cap U_\beta^1$,
\begin{equation}
\label{e:l function on II-}
L_t(\bm{x})=T^{\frac{n}{2}-1}(nk_-)^{\frac{1}{n}}(A_--\log |r_-(\pi_\beta^{\mathcal N}(\bm{x}))|+\epsilon(z))^{\frac{1}{n}}\end{equation}
From this we get that
\begin{equation}|\omega(t)-(\pi_{\beta}^{\mathcal N})^*\omega_T|_{C^{1, \alpha}_{\delta, \nu, \mu}({\bf{II}_-}\cap \widehat{X}_t)}=\underline\epsilon_{T^2}. \end{equation}
Now we understand the holomorphic volume form. Using \eqref{eqn8-3} we get that
\begin{equation}
\Gamma_t= (1+H)(\pi_{\beta}^{\mathcal N})^*\Omega_0,
\end{equation}
where $H$ is a holomorphic function in $\zeta_3, v_2, w_2, \cdots, w_{n-1}$, and its derivatives is of order $\underline\epsilon_{T^2}$ in these coordinates. Then we again apply weighted Schauder estimates to get
that
\begin{equation}
|H|_{C^{0,\alpha}_{\delta, \nu, \mu}({\bf{II}_-}\cap \widehat{X}_t) }=\underline\epsilon_{T^2}.
\end{equation}
So by Proposition \ref{p:CY-error-small} we obtain
\begin{equation}
\label{e:error in region II-}
\|\mathrm{Err}_{t}\|_{C^{0, \alpha}_{\delta, \nu+2, \mu}({\bf{II}_-}\cap \widehat{X}_t)}=O(T^{\nu+\alpha}).
\end{equation}
\
Notice ${\bf{II}_-}\cap \widehat{X}_t$ has two ends. Along one end it is close to the negative end of Region $\bf{I}$.
\begin{proposition}
\label{p:potential difference II- I}
On the intersection ${\bf{II}_-}\cap {\bf{I}}\cap \widehat{X}_t$ we have for all $k\geq 0$
\begin{equation}|\nabla^k(\phi_{t, \bf{II}_-}-\phi_{t, \bf{I}_-}-d_1\log t)|=\underline\epsilon_{T^2},
\end{equation}
where the derivative and norm are taken with respect to $\omega(t)$.
\end{proposition}
\begin{proof}
We work in $U_\beta^1$ for a fixed $\beta$. We have
\begin{equation}
\phi_{t, \bf{I}_-}(\bm{x})=\frac{T}{d_2}\log |f_2(\bm{x})|+\pi_{\mathcal N}^* \phi_t(\bm{x})
\end{equation}
and
\begin{equation}
(\pi_{\beta}^{\mathcal N})^* \phi_-(\bm{x})=\phi_t(\bm{y})-\frac{T}{d_2}\log r_-(\bm{y})
\end{equation}
where $\bm{y}=\pi_\beta^{\mathcal N}(\bm{x})$. By definition it is easy to see that $\bm{y}-\bm{x}$ is of order $\underline\epsilon_{T^2}$ in the coordinates in $v_2, \zeta_3, w_2, \cdots, w_{n-1}$.
By our choice of $T$ in terms of $t$ we have
\begin{equation}-\log |r_-(\bm{y})|=d_1\log |t|-\log |f_2(\bm{y})|.\end{equation}
Then by Lemma \ref{l:transition function expansion}, and use weighed Schauder estimates as above we get the conclusion.
\end{proof}
By Proposition \ref{p:potential difference II- I}, we can easily glue the the potentials in Region $\bf{I}_-$ and $\bf{II}_-$, using a simple cut-off function of the form
\begin{equation}
\phi(t)\equiv\chi(r_-(\bm{x}))\cdot \phi_{t, \bf{I}_-}+(1-\chi(r_-(\bm{x})))\cdot \phi_{t, \bf{II}_-}
\end{equation}
where $\chi$ is a cut-off function in $s$ satisfying
\begin{equation}
\chi(s)=
\begin{cases}
1, s\leq 3/4\\
0, s\geq 5/4.
\end{cases}
\end{equation}
Along the other end, Region $\bf{II}_-$ is close to the region $\bf{III}_-$.
\begin{proposition}
\label{p:potential difference II- III-}
On the intersection ${\bf{II}_-}\cap {\bf{III}_-}\cap \widehat{X}_t$, we have for all $k\geq 0$
\begin{equation}
|\nabla^k(\phi_{t, \bf{II}_-}-\eta_1)|=O(e^{-\delta_e T}),
\end{equation}
where the derivative and norm are taken with respect to $\omega(t)$, and $\delta_e$ is defined as in Proposition \ref{p:CY-error-small}.
\end{proposition}
\begin{proof}
The proof is similar to the previous Proposition. One works in a fixed $U_\beta^1$, and then we use the asymptotics of $\phi_-$ (c.f. \eqref{e:asymptotics of phi-}) and the relation between $t$ and $T$ (c.f. \eqref{e:t T relation}). We omit the details.
\end{proof}
\
{\bf Region $\bf{III}_\pm$.} Again we only consider the Region $\bf{III}_-$. The discussion here is very similar to the case of Region $\bf{II}_-$ so the computations are sketchy. We define
\begin{equation}
\omega(t)=dd^c \phi_{t, \bf{III}_-},
\end{equation}
where
\begin{equation}
\phi_{t, \bf{III}_-}(\bm{x})=\sum\chi_\beta^1(\bm{x})\cdot \phi_{1}\circ \pi_\beta^1(\bm{x}).
\end{equation}
\begin{proposition}
In the intersection ${\bf{III}_-}\cap {U_{\beta, t}^1}$, we have for all $k\geq 0$
\begin{equation}
|\nabla^k(\phi_{t, \bf{III}_-}\circ (\pi_{\beta}^{1})^{-1}-\phi_1)|=\underline\epsilon_{T^2},
\end{equation}
where derivative is taken with respect to the metric $\omega_{TY, 1}$.
\end{proposition}
The proof is very similar to that of Proposition \ref{p:region II- error on overlapping region}, except that one compares with the cylindrical metric and uses Lemma \ref{l: TY cylindrical compare}. We omit the details.
To define the weight, we also define the function $L_t$ by setting
\begin{equation}
L_t(\bm{x})=T^{\frac{n}{2}-1}(nk_-)^{\frac{1}{n}}(-\log |f_2(\bm{x})|)^{\frac{1}{n}}
\end{equation}
and correspondingly the weight $\rho_t$ using \eqref{e:definition of weights}.
Similar to the case of Region $\bf{II}_-$, we have the holomorphic volume form
\begin{equation}
\Gamma_t=(1+H)(\pi_\beta^1)^*\Omega_0
\end{equation}
where $H$ is a holomorphic function in $v_2, \zeta_3, w_2, \cdots, w_{n-1}$ and is of order $\underline\epsilon_{T^2}$. Therefore, we obtain
\begin{equation}
\label{e:error in region III-}
\|\mathrm{Err}_{t}\|_{C^{0, \alpha}_{\delta, \nu+2, \mu}({\bf{III}_-}\cap \widehat{X}_t)}=\underline \epsilon_{T^2}.
\end{equation}
Region $\bf{III}_-$ has two ends. One end intersects Region $\bf{IV}-$.
\begin{proposition}
\label{p:potential difference III- IV-}
On $\bf{III}_-\cap \bf{IV}_-$, we have for all $k\geq 0$
\begin{equation}
|\nabla^k(\phi_{t, \bf{III}_-}-\phi_1)|=\underline\epsilon_{T^2},
\end{equation}
where the derivative and norm are taken with respect to $\omega(t)$.
\end{proposition}
This is fairly easy to see, by working in a fixed $U^1_\beta$.
\
The other end is close to the Region $\bf{II}_-$.
\begin{proposition}
\label{p:potential difference III- II-}
On $\bf{III}_-\cap \bf{II}_-$ we have for all $k\geq 0$
\begin{equation}
|\nabla^k(\phi_{t, \bf{III}_-}-\eta_1)|=O(e^{-\delta_{Z_1} T}),
\end{equation}
where the derivative and norm are taken with respect to $\omega(t)$, and $\delta_{Z_1}$ is the constant in Proposition \ref{t:hein} applied to $Z_1$.
\end{proposition}
To see this we only need to work in a fixed $U_\beta^1$ and use the asymptotics of the Tian-Yau metric $\tilde\omega_{TY, 1}$.
Now by Proposition \ref{p:potential difference II- III-} and \ref{p:potential difference III- II-}, we can choose a cut-off function to glue together $\phi_{t, \bf{II}_-}$ and $\phi_{t, \bf{III}_-}$. Similarly we may also glue the corresponding weight function $\rho_t(\bm{x})$. Here we need to use \eqref{e:l function on II-}, the fact that
\begin{equation}
-\log |f_2|=-\log \frac{|s_1|}{|s_3|}-d_1\log |t|,
\end{equation}
and the relation between $|t|$ and $T$ \eqref{e:t T relation}.
We choose a cut-off function to glue $\phi_{t, \bf{III}_-}$ and $\phi_{t, \bf{IV}_-}$, and also glue the corresponding weight function $\rho_t(\bm{x})$.
Similarly, we can define the metrics $\omega(t)$ on $\bf{II}_+, \bf{III}_+, \bf{IV}_+$ and glue them together in the overlapping regions, and we also glue the weight functions.
\
To sum up, we have constructed a family of $C^{1, \alpha}$ K\"ahler metrics $\omega(t)$ on $\widehat{X}_t$ for $|t|$ small such that in the above defined weighted norm
$\|\mathrm{Err}_{t}\|_{C^{0, \alpha}_{\delta, \nu+2, \mu}(\widehat{X}_t)}=O(T^{\nu+\alpha})$.
\begin{remark}
By the above gluing construction, $\omega(t)$ lies in the cohomology class $T^{\frac{2}{n}}\cdot 2\pi c_1(\mathcal O(1)|_{\widehat{X}_t})$. Hence we get the volume
\begin{equation}
\int_{\widehat{X}_t}\frac{\omega(t)^n}{n!}=C\cdot T^2\sim (-\log |t|)^{-1}.
\end{equation}
The above error estimate in particular gives
\begin{equation}
\int_{\widehat{X}_t}\Gamma_t\wedge\bar\Gamma_t\sim T^2\sim (-\log |t|)^{-1}.
\end{equation}
\end{remark}
For our analysis in the next subsection we define the normalized holomorphic volume form as
\begin{equation}
\Omega(t)\equiv (\frac{2^n\int_{\widehat{X}_t}\omega(t)^n}{(\sqrt{-1})^{n^2} \int_{\widehat{X}_t}\Gamma_t\wedge\bar\Gamma_t})^{\frac{1}{2}} \cdot \Gamma_t.
\end{equation}
Abusing notation we define $\mathrm{Err}_{t}$ by
\begin{equation}
\frac{(\sqrt{-1})^{n^2}}{2^n}\Omega(t)\wedge\bar\Omega(t)=(1+\mathrm{Err}_{t})\frac{\omega(t)^n}{n!},
\end{equation}
where
$
\int_{\widehat{X}_t}\mathrm{Err}_{t}\cdot\omega(t)^n=0
$
and
\begin{equation}
\|\mathrm{Err}_{t}\|_{C^{0, \alpha}_{\delta, \nu+2, \mu}(\widehat{X}_t)}=O(T^{\nu+\alpha}).\label{e:global-error-estimate}
\end{equation}
\subsection{Global weighted analysis on $\widehat{X}_t$ and the proof of the main theorem}
\label{ss:global analysis}
We are now in a position to set up the global weighted analysis on the glued manifold.
To begin with,
let $(\omega(t),\Omega(t))$
be the $C^{1,\alpha}$-K\"ahler structure on $\widehat{X}_t$ constructed in Section \ref{ss:glued-metrics}. So we define the Banach
\begin{align}
\mathfrak{S}_1 & \equiv \Big\{\sqrt{-1}\partial\bar{\partial}\phi\in\Omega^{1,1}(\widehat{X}_t)\Big| \phi\in C^{2, \alpha}(\widehat{X}_t)\Big\},
\nonumber\\
\mathfrak{S}_2 & \equiv \Big\{f\in C^{0,\alpha}(\widehat{X}_t)\Big| \int_{\widehat{X}_t} f\cdot \frac{\omega(t)^n}{n!} = 0\Big\},
\end{align}
which are equipped with the weighted norms defined similar to \eqref{e:norm-of-S1-space} and \eqref{e:norm-of-S2-space}, with parameters $\delta, \nu, \mu, \alpha$ given in
Section \ref{sss:parameters-fixed}
For $|t|\ll1$, we want to solve
\begin{equation}
\label{e:CY-eq}\frac{1}{n!} (\omega(t)+\sqrt{-1}\partial\bar{\partial}\phi)^n = (\sqrt{-1})^{n^2}2^{-n}\cdot \Omega(t)\wedge\overline{\Omega}(t).\end{equation}
Let $\mathscr{F}: \mathfrak{S}_1\rightarrow \mathfrak{S}_2$ be defined by
\begin{equation}
\mathscr{F}(\sqrt{-1}\partial\bar{\partial}\phi)\cdot\omega(t)^n\equiv (\omega(t)+\sqrt{-1}\partial\bar{\partial}\phi)^n-\omega(t)^n(1-\mathrm{Err}_{t}).
\end{equation}
Then \eqref{e:CY-eq} is equivalent to $
\mathscr{F}(\sqrt{-1}\partial\bar{\partial}\phi) = 0.\label{e:converted-CY-eq}
$
Now we write
\begin{equation}
\mathscr{F}(v) - \mathscr{F}(0) = \mathscr{L}(v) + \mathscr{N}(v),
\end{equation}
for any $v\in \mathfrak{S}_1$, where
$\mathscr{L}(\sqrt{-1}\partial\bar{\partial}\phi) = \Delta \phi$ is the linearization of $\mathscr{F}$ and
\begin{align}
\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi)\cdot \omega(t)^n & =(\omega(t)+\sqrt{-1}\partial\bar{\partial}\phi)^n-\omega(t)^n - n \omega(t)^{n-1}\wedge \sqrt{-1}\partial\bar{\partial}\phi.\label{e:nonlinear-term}
\end{align}
The proof of the following is identical to Proposition \ref{p:nonlinear-neck}.
\begin{proposition}
[Nonlinear error estimate]\label{l:nonlinear-error-estimate}
There exists a constant $C_N>0$ independent of
$0<|t|\ll 1$ such that for all
$\varrho\in (0,\frac{1}{2})$
and
\begin{equation}\sqrt{-1}\partial\bar{\partial}\phi_2\in \overline{B_{\varrho}(\bm{0})} \subset \mathfrak{S}_1, \quad \sqrt{-1}\partial\bar{\partial}\phi_2\in \overline{B_{\varrho}(\bm{0})}\subset\mathfrak{S}_1,\end{equation} we have the pointwise estimate
\begin{align} \|\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_1)-\mathscr{N}(\sqrt{-1}\partial\bar{\partial}\phi_2)\|_{\mathfrak{S}_2}
\leq C_N \cdot \varrho \cdot \|\sqrt{-1}\partial\bar{\partial}(\phi_1-\phi_2)\|_{\mathfrak{S}_1}.
\end{align}
\end{proposition}
The following is an analogue of Proposition \ref{p:neck-weighted-schauder}.
\begin{proposition}[Weighted Schauder estimate, the global version] \label{p:global-weighted-schauder}
There exists a uniform constant $C>0$ (independent of $0<|t|\ll1 $) such that for every
$u\in C^{2,\alpha}(\widehat{X}_t, \omega(t))$
\begin{align}
\| u\|_{C_{\delta,\nu,\mu}^{2,\alpha}(\widehat{X}_t)} \leq C \Big( \|\Delta u\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\widehat{X}_t)} + \| u\|_{C_{\delta,\nu,\mu}^0(\widehat{X}_t)} \Big).
\end{align}
\end{proposition}
The local version of Proposition
\ref{p:global-weighted-schauder} is given in Proposition \ref{p:local-weighted-schauder} and they share very similar proof. From the construction of the metric $\omega(t)$ in Section \ref{ss:glued-metrics} we see that the rescaled limit geometries as $|t|\to 0$ are the same as those of the neck $\mathcal M_T$ as $T\to\infty$ studied in Section \ref{ss:regularity-scales}, except that the two incomplete Calabi model space limits are replaced by the two complete Tian-Yau metrics on the ends. Another difference to note is that due to the perturbation of the neck the metric $\omega(t)$ now has only $C^{1,\alpha}$ regularity so that we can only obtain the $C^{2,\alpha}$ estimate here. We omit the details of the proof.
\begin{proposition}[Global injectivity estimates] \label{p:global-injectivity-estimate}
There exists a uniform constant $C>0$ (independent of $0<|t|\ll1$) such that for every
$u\in C^{2,\alpha}(\widehat{X}_t, \omega(t))$,
\begin{align}
\|\nabla u\|_{C_{\delta,\nu+1,\mu}^{0}(\widehat{X}_t)} + \|\nabla^2 u\|_{C_{\delta,\nu+2,\mu}^{0}(\widehat{X}_t)} +[u]_{C_{\delta,\nu,\mu}^{2,\alpha}(\widehat{X}_t)}\leq C \cdot \|\Delta u\|_{C_{\delta,\nu+2,\mu}^{0,\alpha}(\widehat{X}_t)}. \label{e:global-injectivity-estimate}
\end{align}
\end{proposition}
The proof is very similar to the proof of Proposition \ref{p:neck-uniform-injectivity}, using a contradiction argument and applying various Liouville theorems. We omit the details and only mention two points. The first point is that from our construction of $\omega(t)$ on $\widehat{X}_t$, if we rescale around points in Region $\bf{IV}_\pm$, then we obtain the Tian-Yau spaces as limits, instead of the incomplete Calabi model spaces, and we need to use Theorem \ref{t:Liouville on Tian-Yau}. The second point is that the other rescaled limits will be exactly the same as considered in the proof of Proposition \ref{p:neck-uniform-injectivity}, and this follows from the fact that by construction our metric $\omega(t)$ away from the region $\bf{IV}_\pm$ is essentially a small perturbation of the neck region $(\mathcal M_T, \omega_T)$.
As $|t|>0$ is sufficiently small, the existence of solution of the Calabi-Yau equation \eqref{e:CY-eq} is the same as the proof of Theorem \ref{t:neck-CY-metric}. In fact, combining Proposition \ref{p:global-injectivity-estimate} with the error estimate \eqref{e:global-error-estimate} and nonlinear estimate in Proposition \ref{l:nonlinear-error-estimate}, one can apply the implicit function theorem (Lemma \ref{l:implicit-function}), which gives a solution $\phi(t)\in \mathfrak{S}_1$ of \eqref{e:CY-eq}. Since $\omega(t)$ lies in the cohomology class $T^{\frac{2}{n}}\cdot 2\pi c_1(\mathcal O(1)|_{\widehat{X}_t})$, by the well-known uniqueness of the solution to the Calabi-Yau equation, the rescaled metric $T^{-\frac{2}{n}}\cdot(\omega(t)+\sqrt{-1} \partial\bar{\partial} \phi(t))$ must agree with the Calabi-Yau metric $\omega_{CY, t^{n+2}}$ on $X_{t^{n+2}}\simeq \widehat X_{t}$ in the Introduction. The geometric statements in Theorem \ref{t:main-theorem} then follow from similar arguments as in Section \ref{ss:renormalized-measure}. Notice that $\diam_{\omega(t)}(\widehat{X}_t)$ is of order $T^{\frac{n+1}{n}}$, and the relation between $T$ and $|t|$ is given by \eqref{eqn6667}. Then we have that
\begin{align}
C^{-1}(\log|t|^{-1})^{\frac{1}{2}} \leq \diam_{\omega_{TY,t^{n+2}}}(X_{t^{n+2}})\leq C\cdot(\log|t|^{-1})^{\frac{1}{2}},
\end{align}
where $C>0$ is independent of $T$. We omit other details.
\section{Extensions and conjectures}
\label{s:discussions}
In this section we discuss some possible extensions and questions related to our results in this paper.
\subsection{More general situation}
\label{ss:general-situations}
As discussed in Section \ref{ss:dimension reduction}, our motivation is to more general degenerations of Calabi-Yau manifolds. During the preparation of this paper in the Fall of 2018, we also made some preliminary progress towards understanding the case of maximal degenerations (which is related to the SYZ Conjecture in mirror symmetry), based on similar ideas to that of Section \ref{s:torus-symmetries} and \ref{s:neck}, and partly motivated by \cite{Morr}. We hoped in a future paper to work out the details of constructing local models generalizing the Ooguri-Vafa metric to higher dimensions. In January 2019, we received a preprint from Yang Li (\cite{Li}) who, partly motivated by \cite{HSVZ}, had essentially achieved most of what we were planning to do (in complex dimension three). Thus we decided not to expand in this direction beyond what we have written at the time we learned about \cite{Li}. On the other hand, we still present the original brief discussions here (so the arguments are rather sketchy and there will be NO theorems). We hope this may still be of some interest to the readers, since it seems to shed a slightly different light from \cite{Li}.
We start with a lemma on Green's function on certain non-compact spaces.
\begin{lemma}
\label{l:green-function-asymp} Let $(X^{m+n},g)\equiv (\mathbb{R}^m\times K^n, g_{\mathbb{R}^m}\oplus h)$ be a Riemannian product of a Euclidean space $(\mathbb{R}^m, g_{\mathbb{R}^m})$ and a compact Riemannian manifold $(K^n, h)$. For any point $p=(p_1, p_2)\in X^{m+n}$, there exists a Green's function $G_p$ on $X$ such that
\begin{enumerate}
\item $-\Delta_g G_p = 2\pi\delta_p$.
\item There are constants $\epsilon>0$, $R>0$ and $C>0$, independent of $p$, such that
\begin{equation}
|G_p(x) - \Phi_{m, p_1}(x_1)| \leq C \cdot e^{-\epsilon \cdot |x_1-p_1|}
\end{equation}
for any $x=(x_1, x_2)\in X^{m+n}\setminus B_R(p)$, where $\Phi_m:\mathbb{R}^m\setminus\{0^m\}\to\mathbb{R}$ is the standard Green's function given by
\begin{align}
\Phi_{m,p_1}(x_1)
\equiv
\begin{cases}
\frac{2\pi}{(m-2)\cdot \Area(\partial B_1(0^m))}\cdot |x_1 - p_1|^{2-m},& m \geq 3
\\
-\log |x_1 - p_1|, & m = 2,
\\
-2\pi |x_1-p_1|, & m=1.
\end{cases}
\end{align}
\end{enumerate}
\end{lemma}
\begin{proof}
The proof is by separation of variables, and is similar to Proposition \ref{p:existence-Greens-current}. So we will not provide all the details, except pointing out one key point. For simplicity of notation we may assume $p_1=0$. After separation of variables we need to solve a PDE of the form on $\mathbb{R}^m$
\begin{equation}
- \Delta_{\mathbb{R}^m} u_{\lambda} + \lambda \cdot u_{\lambda} =2\pi\cdot \delta_{0^m}, \label{e:delta-Rm}
\end{equation}
where $\lambda$ is non-negative. When $\lambda=0$, a solution is given by the Green's function of $-\Delta_{\mathbb{R}^m}$, so we only deal with the case $\lambda>0$. When $m=1$, this is the equation \eqref{eqn3333}. When $m\geq 2$, we look for a radial solution $u_{\lambda}=u_{\lambda}(r)$, then \eqref{e:delta-Rm} reduces to an ODE
\begin{equation}
-u_{\lambda}''(r) - \frac{m-1}{r}\cdot u_{\lambda}'(r) + \lambda \cdot u_{\lambda}(r) = 0, r\in (0, \infty)\end{equation}
We make the transformation
$f(r) \equiv u_{\lambda}(r)\cdot r^{-\alpha}$,
where $\alpha$ is to be determined.
Then it follows that
\begin{equation}
r^2f''(r) + (2\alpha+m-1)\cdot r\cdot f'(r) + \Big(\alpha(\alpha+m-2)-\lambda \cdot r^2\Big)\cdot f(r) = 0.
\end{equation}
Now let $2\alpha+m-1=1$, i.e., $\alpha=\frac{2-m}{2}$, and let $\sqrt{\lambda}\cdot r =s$, then we get the modified Bessel equation \begin{equation}
s^2f''(s) + s f'(s) - (\alpha^2 + s^2) f(s) = 0.
\end{equation}
Then we get a solution $u(r)=K_\alpha(\sqrt{\lambda} r)\cdot r^{\alpha}$, where $K_{\alpha}$ is the modified Bessel function.
So it follows that \begin{align}
u(r) \sim
\begin{cases}
\varphi_{\lambda}(p)\cdot r^{2-m}, & m\geq 3,
\\
\varphi_{\lambda}(p)\cdot \log r , & m=2,
\end{cases}
\end{align}
as $r\to0$.
In particular $u$ satisfies the distribution equation \eqref{e:delta-Rm}. Then we can define $G_p$ using a formal expansion, and the convergence and the asymptotic behavior follow from the uniform estimates on $K_\alpha(\sqrt{\lambda} r)$ for $r\geq 1$ (see Proposition 3.3 in \cite{SZ-Liouville}). We omit the detailed proof. \end{proof}
\begin{remark}
Applying this to $\mathbb R\times \mathbb{T}^2$ and $\mathbb R^2\times S^1$, we obtain an alternative treatment to the constructions in \cite{HSVZ} (Theorem 2.6) and \cite{GW} (Lemma 3.1).
\end{remark}
We are interested in studying the Green's currents in the situation of Section \ref{ss:global-existence} with $D$ replaced by the non-compact Calabi-Yau manifold $(\mathbb C^*)^n$, and with $H$ replaced by a smooth algebraic hypersurface in $D$ defined by a Laurent polynomial $F$.
Here $D$ is endowed with the standard flat K\"ahler metric
\begin{equation}\omega_D=\sum_{j=1}^n \frac{1}{2}\sqrt{-1}\partial\log w_j \wedge \bar{\partial} \log w_j,\end{equation}
where $\{w_1, \cdots, w_n\}$ are standard holomorphic coordinates on $(\mathbb C^*)^n$.
Denote $\xi_j=-\log w_j=u_j+\sqrt{-1} v_j$, which gives an identification $(\mathbb C^*)^n$ with $\mathbb R^n\times (S^1)^n$ equipped with the standard flat product metric.
Let $\pi: (\mathbb C^*)^n\rightarrow \mathbb R^n$ be the projection map. The \emph{amoeba} $\mathcal A(F)$ of $F$ is by definition the image $\pi(H)$.
We want to solve
$\Delta G_P=2\pi\cdot \delta_P$.
In terms of the coordinates $\{\xi_j\}$, we can view $\delta_P$ as a matrix of distributions by the decomposition
\begin{equation}\delta_P=\sum_{\alpha, \beta} f_{\alpha\beta}\widehat{\delta}_P\frac{\sqrt{-1}}{2}d\xi_\alpha \wedge d\bar\xi_\beta \wedge dz, \end{equation}
where $\widehat{\delta}_P$ is a $2n$-current given by setting
$(\widehat\delta_P, \phi)= \int_P \phi\dvol_P.$
Then by definition it is not difficult to see that at every point on $P$,
\begin{equation}
\label{eqn812}
f_{\alpha\beta} d\xi_\alpha \wedge d\xi_\beta=\frac{\partial F\wedge\bar{\partial} F}{|\partial F|^2}.
\end{equation}
If we decompose
\begin{equation}G_P=\sum h_{\alpha\beta} \frac{\sqrt{-1}}{2}d\xi_\alpha \wedge d\xi_\beta \wedge dz.\end{equation}
Then we need to solve a matrix of distributional equations
$\Delta h_{\alpha\beta}=f_{\alpha\beta}\widehat\delta_P.$
Let us write
\begin{equation}
\widehat\delta_P=\int_{P} \delta_{y} \dvol_P(y).
\end{equation}
Then one can write down a solution in the form
\begin{equation}h_{\alpha\beta}(x) \equiv \int_P(G_y(x)-\mathcal G(y))f_{\alpha\beta}(y)\dvol_P(y),
\end{equation}
where $G_y(x)$ is the Green's function on $Q$ constructed in Lemma \ref{l:green-function-asymp}, and $\mathcal G(y)$ is a renormalization function to make the integral converge. For example, we can take
\begin{equation}
\mathcal G(y)=\frac{c_n}{|\pi(y)|^{n-2}+1}.
\end{equation}
Now we consider an illustrating example when $n=2$, and
\begin{equation}
F(w_1, w_2)\equiv w_1+w_2+1.
\end{equation}
The amoeba $\mathcal A(F)$ is a well-known shape on $\mathbb R^2$ with three branches at infinity. Moreover, it is not difficult to show by direct calculation that $\mathcal A(F)$ converges exponentially fast (in the Hausdorff sense) to its \emph{tropicalization}, $T(F)$ which is given by the union of three half lines $P_1, P_2, P_3$ emanating from $0$ in $\mathbb R^2$, along the directions of $e_1, e_2, -e_1-e_2$.
In this case, one also expects that the Green's current $G_P$, viewed as a matrix $(h_{\alpha\beta})$, is asymptotic to the matrix of Green's functions defined using $T(F)$ on $\mathbb R^2$. This asymptotics should hold in suitable regions away from $T(F)$.
The point is that we should remember more information on $T(F)$ than simply a subspace in $\mathbb R^2$. Notice each $P_i$ is a straight half line and it has a unit normal $n_i$ in $\mathbb R^2$ (well-defined up to sign), and $n_i$ naturally arises if one notices \eqref{eqn812}. Then there is a well-defined matrix valued distribution
$\delta_{T(F)}\equiv \sum\limits_{i=1}^3 \widehat\delta_{P_i} \cdot n_i\otimes n_i$ on $\mathbb{R}^3=\mathbb{R}^2\times \mathbb{R}$,
where we view $P_i\subset \mathbb{R}^3$ as $P_i\times \{0\}$.
Then we can solve for a matrix value Green's function $G_{T(F)}$ for $T(F)$ in $\mathbb{R}^3$ such that
$-\Delta_{\mathbb R^3} G_{T(F)}=2\pi \delta_{T(F)}$.
For this purpose we first solve the Green's function for $P_1$ in $\mathbb{R}^3$. Again this is easy to write down explicitly as
\begin{equation}G_{P_1}(x)=\int_{0}^\infty \Big(\frac{1}{\sqrt{(x_1-t)^2+x_2^2+x_3^2}}-\frac{1}{t+1}\Big)dt=-\log (\sqrt{x_1^2+x_2^2+x_3^2}-x_1)+\log 2.\end{equation}
This has interesting asymptotics. Let us write $r^2=x_1^2+x_2^2+x_3^2$ and $u=(x_2, x_3)$. Then
\begin{align}G_{P_1}(x)
\sim \begin{cases} -\log r+\log 2 +\frac{x_1}{r}-\frac{x_1^2}{r^2}+\cdots, & |x_1|\leq C |u|,
\\
-2\log |u|+\log |x_1|+O(|u|^2x_1^{-2}), & |x_1|\gg|u|.
\end{cases}
\end{align}
Now the Green's function for $T(F)$ can be written down as a matrix
\begin{equation}G_{T(F)}= \left[ {\begin{array}{cc}
G_{P_2}+\frac{1}{2}G_{P_3} & -\frac{1}{2}G_{P_3} \\
-\frac{1}{2}G_{P_3} & G_{P_1}+\frac{1}{2}G_{P_3} \\
\end{array} } \right].
\end{equation}
Away from the three direction, the asymptotics as $r\rightarrow\infty$ is given by
\begin{equation}
-\left[ {\begin{array}{cc} \frac{3}{2}\log r & -\frac{1}{2}\log r \\
-\frac{1}{2}\log r & \frac{3}{2}\log r \\
\end{array} } \right].
\end{equation}
Now using the Green's current $G_P$ and its asymptotics at infinity as describe above, one can construct an $S^1$-invariant incomplete three dimensional K\"ahler metrics as in Section \ref{ss:kaehler-structures}. Notice as in \ref{ss:kaehler-structures} there are various parameters. First one can change the flat metric on $\mathbb{R}^2$. Also in the equation
\begin{equation}
\partial_z^2\tilde\omega+d_Dd_D^ch=0,
\end{equation}
one is free to add a function of $z$ to $h$, and add a closed $(1,1)$-form on $D=(\mathbb C^*)^2$ to $\tilde\omega$. For appropriate choices of parameters one can make this K\"ahler metric approximately Calabi-Yau, and then the goal is to use weighted analysis to perturb to a family of genuine (incomplete) Calabi-Yau metrics. In appropriate scales, these metrics should collapse to a limit which is given as a domain in $\mathbb{R}^3$. One unsatisfactory point from our point of view is that comparing with the general expectation in SYZ metric collapsing conjecture, these incomplete metrics live on a too small region, since here the collapsing limit is flat whereas in general we should get a limit which is singular along the union of $P_i$'s. In other words, what one constructs here is only an \emph{infinitesimal} model for the collapsing.
In complex dimension three, one can also consider $\mathbb{T}^2$-invariant Calabi-Yau metrics. As discussed in Section \ref{ss:higher rank torus}, the corresponding dimension reduced equation has slightly different form and the linearized equation in the case when there are stabilizers also motivates us study certain Green's currents.
Again we consider the model case $Q=\mathbb R^2\times \mathbb C^*$ is the quotient space and over $P=P_1\cup P_2\cup P_3\subset \mathbb R^2\times\{1\}$ we have stabilizers.
In this case we are interested in a matrix valued Dirac current
\begin{equation}\delta_P=\sum\limits_{i=1}^3 \widehat\delta_{P_i}\cdot n_i\otimes n_i,\end{equation}
where $P_i$ is naturally viewed as a submanifold in $\mathbb R^2\times \mathbb C^*$,
and the corresponding matrix valued Green's function $G_P$ which satisfies
$\Delta G_P=2\pi\delta_P$.
In large scale this is modeled by the corresponding current in $\mathbb R^3=\mathbb R^2\times \mathbb R$, and this has been discussed in the above. Near the vertex of $P$ one can consider the model $\mathbb R^2\times \mathbb C$, and find the corresponding Green's function for $P\subset \mathbb R^2\times 0$. This is similar to the calculation above. For example, one gets
\begin{equation}G_{P_1}(x)=\int_{0}^\infty \frac{1}{(x_1-t)^2+x_2^2+x_3^2+x_4^2}dt=\frac{1}{v}(\frac{\pi}{2}+\tan^{-1}\frac{x_1}{v}),\end{equation}
where $u=x_3+\sqrt{-1} x_4$ is the coordinate on $\mathbb C$, and
$v^2=|u|^2+x_2^2$.
We define
\begin{align}(W_{ij}) \equiv \left[ {\begin{array}{cc}
G_{P_2}+\frac12G_{P_3} & -\frac12G_{P_3} \\
-\frac12G_{P_3} & G_{P_1}+\frac12G_{P_3} \\
\end{array} } \right] \quad \text{and}
\quad
\tilde\omega \equiv \frac{\sqrt{-1}}{2}\Tr(W_{ij}) dw\wedge d\bar w.
\end{align}
Then one can check the equation \eqref{e:higher Tk} is satisfied, and one obtains away from the singular locus a $\mathbb{T}^2$-invariant K\"ahler metric.
Naively one expects to compactify this metric along singular locus. We compare this with the standard local holomorphic model, which is the standard flat holomorphic structure $(\omega_{\mathbb C^3}, \Omega_{\mathbb C^3})$ on $\mathbb C^3$ under the natural $\mathbb{T}^2$-action
\begin{equation}
(e^{\sqrt{-1} \theta_1}, e^{\sqrt{-1} \theta_2})\cdot (z_1, z_2, z_3)=(e^{\sqrt{-1}(\theta_1+\theta_2)}z_1, e^{-\sqrt{-1} \theta_1}z_2, e^{-\sqrt{-1}\theta_2}z_3).
\end{equation}
The corresponding quotient map is given by
\begin{equation}\mathcal Q: \mathbb C^3\rightarrow \mathbb R^2\oplus \mathbb C,\quad (z_1, z_2, z_3)\mapsto (\frac{1}{2}(|z_2|^2-|z_1|^2), \frac{1}{2}(|z_3|^2-|z_1|^2), z_1z_2z_3).\end{equation}
Also one can compute
\begin{align}W_{ij}&=
\frac{1}{|z_1|^2|z_2|^2+|z_3|^2|z_1|^2+|z_2|^2|z_3|^2} \left[ {\begin{array}{cc}
|z_1|^2+|z_3|^2 & -|z_1|^2 \\
-|z_1|^2 & |z_1|^2+|z_2|^2 \\
\end{array} } \right]
\\
\tilde\omega &=\frac{1}{|z_1|^2|z_2|^2+|z_3|^2|z_1|^2+|z_2|^2|z_3|^2}\cdot \frac{\sqrt{-1}}{2} {dw}\wedge d\bar w.\end{align}
So comparing with the previous formula they do not naturally match. This suggests that we might need to do something different near the vertex.
Now if we take the above formula of Green's current, but work instead on $\mathbb R^2\times \mathbb C$, then one can see the above matrix actually has strictly positive lower bound at infinity. This makes us suspect the existence of a complete Calabi-Yau metric on $\mathbb C^3$ which is approximately the above ansatz at infinity. One approach is by using this ansatz as background metric at infinity and solve the Calabi-Yau equation as in \cite{TY}. This should be similar to the result of Yang Li constructing a complete Calabi-Yau metric $\mathbb C^3$ with infinity tangent cone $\mathbb C^2/\mathbb Z_2\times \mathbb C$. If such a metric can be constructed, then it should have a $\mathbb{T}^2$-symmetry and at infinity has $r^4$ volume growth and the tangent cone at infinity is $\mathbb R^2\oplus\mathbb C$ with locus of the singular fibration given by the $Y$-vertex. The situation is analogous to the Taub-NUT space fibered over $\mathbb R\oplus \mathbb C$. The difference is that here we need to have discriminant locus essentially due to topological reasons.
The existence of such a complete Calabi-Yau metric on $\mathbb C^3$ also resolves the above concern regarding the bad singularity behavior of the ansatz metric near the vertex.
\subsection{Further remarks}
\label{ss:a few remarks}
We list several further remarks.
\
{\bf (1)} From the proof of Theorem \ref{t:main-theorem}, it follows that similar results hold in the following more general situation. We leave it for the readers to check the details.
\begin{itemize}
\item $p:\mathcal X\rightarrow \Delta$ is a proper holomorphic map from an $(n+1)$-dimensional normal complex analytic variety onto a disc $\Delta$ in $\mathbb C$.
\item For $t\neq 0$, $X_t\equiv p^{-1}(t)$ is a smooth $n$ dimensional compact complex manifold.
\item $X_0\equiv p^{-1}(0)$ is a union of two smooth $n$ dimensional Fano manifolds $Y_1$ and $Y_2$, and $Y_1\cap Y_2$ is a smooth $(n-1)$-dimensional Calabi-Yau manifold $D$.
\item $\mathcal L$ is a relatively ample holomorphic line bundle on $\mathcal X$.
\item Two positive integers $d_1, d_2$, and we denote $k=d_1+d_2$.
\item Holomorphic sections $f_1$, $f_2$, $f$ of $\mathcal L^{d_1}, \mathcal L^{d_2}, \mathcal L^{d_1+d_2}$ respectively satisfying
$
f_1f_2+tf=0.
$
\item $Y_1\cap Y_2$ is a smooth $(n-1)$-dimensional Calabi-Yau manifold $D$.
\item $\mathcal X$ is singular along a smooth divisor $H\subset D$ given by $\{f_1=f_2=f=t=0\}$ and transverse to $H$ the singularity is modeled on $\{x_1x_2+tx_3=0\}$.
\item There is a holomorphic volume form on the smooth locus of $\mathcal X$.
\end{itemize}
More generally, one may also try to understand the situation when the central fiber is given by a chain of irreducible components. This would require constructing a neck region using a Green's current on $D\times \mathbb R$ with singularities along the union of $H_j\times \{z_j\}$ for finitely many $z_j$.
\
{\bf (2)} In connection with algebro-geometric study of degenerations of Calabi-Yau manifolds, Theorem \ref{t:main-theorem} shows that the normalized Gromov-Hausdorff limit in our setting is \emph{topologically} the same as the \emph{essential skeleton} of the degeneration $\mathcal X$. In the other extremal case, namely, the case of large complex structure limit of Calabi-Yau manifolds, it is a folklore conjecture (see Gross-Wilson \cite{GW} and Kontsevich-Soibelman \cite{KonSo, KS-affine}) that the normalized Gromov-Hausdorff limit is \emph{topologically} the same as the essential skeleton of the degeneration. Motivated by our work in this paper, it is then natural to expect the following more general conjecture:
\begin{conjecture}\label{cj:generalized-SYZ}
Given a normal flat polarized degenerating family of Calabi-Yau manifolds $(\mathcal X, \mathcal L)\rightarrow \Delta$. Let $\omega_{CY, t}$ be the Calabi-Yau metric on $X_t$ in the class $2\pi c_1(\mathcal L|_{X_t})$. Let $\Delta(\mathcal X)$ be the essential skeleton of the degeneration with $\dim(\Delta(\mathcal X))=d\in\mathbb{Z}_+$. Let us denote \begin{align}
\tilde\omega_{CY,t} \equiv (\diam_{\omega_{CY, t}}(X_t))^{-2}\cdot \omega_{CY, t},\quad d\underline{\nu}_t\equiv (\Vol_{\omega_{CY, t}}(X_t))^{-1}\cdot\dvol_{\omega_{CY, t}}.
\end{align}
Then $(X_t, \tilde\omega_{CY, t}, d\underline{\nu}_t)$ converges in the measured Gromov-Hausdorff sense, to a compact measured metric space $(B,d_{B},d\underline{\nu}_0)$ such that $B$ is homeomorphic to $\Delta(\mathcal X)$, and the dimension of $B$ (in the sense of Colding-Naber \cite{Colding-Naber}) is equal to $d$.
\end{conjecture}
We also expect more refined relationship between the metric collapsing and algebraic geometry:
\begin{itemize}
\item It is interesting to understand the algebro-geometric meaning of the normalized limit measure in Theorem \ref{t:main-theorem}, see \cite{BJ} for related algebro-geometric work. Furthermore, it seems plausible to view the limit metric measure space as a solution to certain non-Archimedean Monge-Amp\`ere equation. In the case $n=2$, there is also a plausible connection with the compactification of moduli space of hyperk\"ahler metrics on K3 manifolds (see \cite{OO}).
\item It is interesting to understand the algebro-geometric meaning of the rescaled limits. In this paper, the Tian-Yau spaces naturally fit into some components $Y_1, Y_2$ of an algebraic degeneration, and the collapsing limit $D\times \mathbb R$ naturally maps to the intersecting divisor $Y_1\cap Y_2$. But the Taub-NUT limit spaces are not easily directly in the degeneration. This picture resembles the familiar relationship between the nodal degeneration of higher genus algebraic curves and degeneration of hyperbolic metrics. Certainly one expects similar phenomenon in more general settings.
\end{itemize}
\
{\bf (3)} As is mentioned in the Introduction, it remains an interesting question to directly glue two Tian-Yau metrics with the same divisor $D$, without a priori assuming the existence of the complex family $\mathcal X$. As mentioned in the Introduction, in the case $n=2$ this was done in \cite{HSVZ} using $SU(2)$-structures, and in the case $n=3$ it is possible to use deformations of $SU(3)$-structures. This would require certain analysis (in particular Liouville theorem) on forms instead of functions. We leave this for future study.
\
{\bf (4)}
There is a different class of Tian-Yau spaces, constructed on the complement of a smooth anti-canonical divisor in a projective manifold with trivial normal bundle. In particular the ambient manifold can not be Fano. These spaces have different asymptotics at infinity from the ones we considered in this paper. Namely, they are \emph{asymptotically cylindrical}. Given a smooth Fano manifold $Y$ and a pencil of anti-canonical divisors with smooth base locus $B$, let $Y'$ be the blown-up of $Y$ along $B$, and let $D'$ be the proper transform of a smooth element $D$ in the pencil. Then there is such an asymptotically cylindrical Calabi-Yau metric on $Y\setminus B$ (in every K\"ahler class). Asymptotically cylindrical Tian-Yau spaces have been important ingredients in the \emph{twisted connected sum} construction of examples of compact $G_2$ holonomy manifolds. It is interesting to see whether the ideas of this paper can be used to construct new examples of $G_2$ holonomy manifolds by gluing together a suitably twisted circle fibration over various pieces.
\
{\bf (5)}
Our main result approximately reduces the understanding on the geometry of part of the Calabi-Yau manifolds $(X_t, \omega_{CY, t})$ (the neck region) for $|t|\ll1$ to the geometry of the Calabi-Yau metric on the one lower dimensional space $D$. One expects this can possibly lead to an inductive way to study the geometry of Calabi-Yau metrics in higher dimensions through iterated degenerations. Correspondingly, it is also interesting to relate the submanifold geometry of the neck to that of $D$. For example, suppose we have a special Lagrangian fibration on a region in $D$, can we construct special Lagrangian fibrations on the neck which are $S^1$-invariant? At the two ends of the neck it is easy to see the pre-image of a special Lagrangian fibration under the projection map is approximately special Lagrangian. Near the singular fibers of the $S^1$-fibration the situation is more complicated and one expects certain singular perturbation techniques are needed. There are also similar discussions in \cite{Li} in the setting of Section \ref{ss:general-situations}.
\
{\bf (6)}
Theorem \ref{t:main-theorem} can be viewed as understanding the first order expansion of the family of Calabi-Yau metrics on $\widehat{\mathcal X}$ near $t=0$. One may ask whether it is possible to obtain a refined \emph{asymptotic expansion}. In spirit, it is similar to the case of a family of hyperbolic metrics on nodal degeneration of Riemann surfaces (see \cite{MZ} by Melrose-Zhu), and it is very likely similar techniques will be useful here. We thank Dominic Joyce and Xuwen Zhu for conversations on this.
\bibliographystyle{amsalpha}
|
1,116,691,500,812 | arxiv | \section{Introduction}
\label{intro}
In this paper we consider two types of self-similarity for tree graphs related to Horton-Strahler \cite{Horton45, Strahler} and Tokunaga \cite{Tok78} indexing schemes for the tree branches.
These schemes and self-similarity definitions were introduced in hydrology in the mid-20th
century to describe the dendritic structure of river networks and have penetrated other
areas of sciences since then.
Horton-Strahler indexing assigns positive orders to the tree branches
according to their relative importance in the hierarchy.
{\it Hotron self-similarity} refers to geometric decay of the number
$N_k$ of branches of order $k$.
{\it Tokunaga self-similarity} is a stronger constraint that addresses {\it side branching}
-- merging of branches of distinct orders.
A Tokunaga index $T_{i(i+k)}$ is the average number of branches of order $i\ge 1$
that merged a branch of order $(i+k)$, $k>0$.
Having Tokunaga self-similarity implies that different
levels of a hierarchical system have the same statistical structure,
in the sense $T_{i(i+k)}\equiv T_k = ac^{k-1}$ for a positive pair $(a,c)$
of Tokunaga parameters.
A classical model that exhibits Horton and Tokunaga self-similarity
is a critical binary Galton-Watson branching \cite{BWW00,Pec95,Pitman},
also known in hydrology as Shreve's random topology model for river
networks \cite{Shreve66}.
Recently, Horton and Tokunaga self-similarity were established
for the level-set tree representation of a homogeneous
symmetric Markov chain and a regular Brownian motion \cite{ZK12}.
This study is a first step towards exploring Horton and Tokunaga self-similarity of
trees generated by coalescent processes.
We focus on the tree generated by Kingman's coalescent
and its finite version, Kingman's $N$-coalescent process with
a constant collision kernel.
The main result is a particular form of Horton self-similarity,
called here {\it root-Horton law} for Kingman's coalescent.
We also establish a very close relation between the combinatorial
trees of Kingman's $N$-coalescent and a sequence of i.i.d. random variables
(referred to as {\it discrete white noise}),
which allows us to extend Horton self-similarity to the level set
tree of an infinite i.i.d. sequence.
These findings add two important classes of processes to the realm of
Horton self-similar systems.
Finally, we perform numerical experiments that suggest that Kingman's
coalescent, and hence the level-set tree of a white noise, are Horton
self-similar in a regular stronger sense as well as asymptotically Tokunaga
self-similar.
The paper is organized as follows.
Section~\ref{self} describes Horton-Strahler ordering of tree branches
and the related concept of Horton self-similarity.
Kingman's coalescent processes are defined
in Sect.~\ref{coalescent}.
The main results are summarized in Sect.~\ref{results}.
Section~\ref{SHSODE} introduces the Smoluchowski-Horton system of equations
that describes the dynamics of Horton-Strahler branches in Kingman's coalescent.
A proof of the existence of
root-Horton law for Kingman's coalescent is presented in Section~\ref{existence}.
Section~\ref{white} establishes the connection between
the combinatorial tree representation of Kingman's $N$-coalescent process and
combinatorial level-set tree of a discrete white noise.
This section also introduces an infinite Kingman's tree as a limit of
Kingman's $N$-coalescent trees viewed from a leaf.
Section~\ref{numerics} defines Tokunaga self-similarity and collects
numerical results and conjectures.
Smoluchowski-Horton system for a general coalescent process with
collision kernel is written in Sect.~\ref{general}.
Section~\ref{discussion} concludes.
\section{Self-similar trees}
\label{self}
This section describes {\it Horton self-similarity} for rooted
binary trees.
It is based on Horton-Strahler orders of the tree vertices.
\subsection{Rooted trees}
A {\it graph} $\mathcal{G}=(V,E)$ is a collection of vertices
$V=\{v_i\}$, $1\le i \le N_V$ and edges
$E=\{e_k\}$, $1\le k \le N_E$.
In a {\it simple} undirected graph each edge is defined as an unordered
pair of distinct vertices:
$\forall\, 1\le k \le N_E, \exists! \, 1\le i,j \le N_V, i\ne j$
such that $e_k=(v_i,v_j)$ and
we say that the edge $k$ {\it connects} vertices $v_i$ and $v_j$.
Furthermore, each pair of vertices in a simple graph may have
at most one connecting edge.
A {\it tree} is a connected simple graph $T=(V,E)$ without cycles.
In a {\it rooted} tree, one node is designated as a root; this
imposes a natural {\it direction} of edges as well as the
parent-child relationship between the vertices.
Specifically, of the two connected vertices the one closest to
the root is called {\it parent}, and the other -- {\it child}.
Sometimes we consider trees embedded in a plane ({\it planar trees}),
where the children of the same parent are ordered.
A {\it time-oriented tree} $T=(V,E,S)$ assigns time marks $S=\{s_i\}$,
$1\le i \le N_V$ to the tree vertices in such a way that
the parent mark is always larger (smaller) than that of its children.
A {\it combinatorial tree} $\textsc{shape}(T)\equiv(V,E)$ discards
the time marks of a time-oriented tree $T$, as well as possible
planar embedding, and only preserves its graph-theoretic structure.
We often work with a space ${\mathcal{T}}_N$ of combinatorial (not labeled,
not embedded) rooted binary trees with $N$ leaves, and space ${\mathcal{T}}$
of all (finite or infinite) rooted binary trees.
\subsection{Horton-Strahler orders}
\label{hst}
Horton-Strahler ordering of the vertices of a finite
rooted binary tree is performed in a hierarchical fashion,
from leaves to the root \cite{Pec95,NTG97,BWW00}:
(i) each leaf has order $r({\rm leaf})=1$;
(ii) when both children, $c_1, c_2$, of a parent vertex $p$ have
the same order $r$, the vertex $p$ is assigned order $r(p)=r+1$;
(iii) when two children of vertex $p$ have different orders, the vertex $p$
is assigned the higher order of the two.
Figure~\ref{fig_HST}a illustrates this definition.
A {\it branch} is defined as a union of connected vertices with
the same order.
The order $\Omega(T)$ of a finite tree $T$ is the order
of its root.
By $N_r$ we denote the total number of branches of order $r$.
\subsection{Horton self-similarity}
\label{sst}
Let ${\mathcal{Q}}_N$ be a probability measure on ${\mathcal{T}}_N$ and
$N_k^{( {\mathcal{Q}}_N)}$ be the number of nodes of Horton-Strahler
order $k$ in a tree generated according to the measure ${\mathcal{Q}}_N$.
\begin{Def}
We say that a sequence of probability laws $\{ {\mathcal{Q}}_N\}_{N \in \mathbb{N}}$ has {\it well-defined asymptotic Horton ratios} if for each $k \in \mathbb{N}$, random variables
$\left(N_k^{( {\mathcal{Q}}_N)}/N\right)$ converge in probability, as $N \rightarrow \infty$, to a constant value ${\mathcal{N}}_k$, called the {\it asymptotic ratio} of the branches of order $k$.
\end{Def}
{\it Horton self-similarity} implies that the sequence ${\mathcal{N}}_k$ decreases in a
geometric fashion as $k$ goes to infinity.
\begin{Def}
A sequence $\{ {\mathcal{Q}}_N\}_{N \in \mathbb{N}}$ of probability laws on ${\mathcal{T}}$ with well-defined asymptotic Horton ratios is said to obey a {\it root-Horton self-similarity law} if and only if
the following limit exists and is finite and positive:
$\lim\limits_{k \rightarrow \infty}
\Big( {\mathcal{N}}_k \Big)^{-{1 \over k}}=R>0.$
The constant $R$ is called the {\it Horton exponent}.
\end{Def}
\section{Coalescent processes, trees}
\label{coalescent}
This section reviews finite and infinite Kingman's coalescent
processes and introduces a tree representation of Kingman's
$N$-coalescent process.
\subsection{Kingman's $N$-coalescent process}
We start by considering a general finite coalescent process defined by a collision kernel \cite{Bertoin,Pitman,Berestycki}.
The process begins with $N$ particles (clusters) of mass one.
The cluster formation is governed by a collision rate
kernel $K(i,j)=K(j,i)>0$.
Namely, a pair of clusters with masses $i$ and $j$ coalesces at the
rate $K(i,j)$, independently of the other pairs, to
form a new cluster of mass $i+j$.
The process continues until there is a single cluster of mass $N$.
\medskip
\noindent
Formally, for a given $N$ consider the space ${\mathcal{P}}_{[N]}$ of
partitions of $[N]=\{1,2,\hdots,N\}$.
Let $\Pi^{(N)}_0$ be the initial partition in singletons,
and $\Pi^{(N)}_t ~~(t \geq 0)$ be a strong Markov process such that
$\Pi^{(N)}_t$ transitions from partition $\pi \in {\mathcal{P}}_{[N]}$ to
$\pi' \in {\mathcal{P}}_{[N]}$ with rate $K(i,j)$ provided that partition
$\pi'$ is obtained from partition $\pi$ by merging two clusters
of $\pi$ of masses $i$ and $j$.
If $K(i,j) \equiv 1$ for all positive integer masses $i$ and $j$,
the process $\Pi^{(N)}_t$ is known as Kingman's $N$-coalescent process,
or $N$-coalescent.
\subsection{Kingman's coalescent}
The infinite Kingman's coalescent $\Pi^{(\infty)}_t$ is a coalescent process defined over the space ${\mathcal{P}}_{\mathbb{N}}$ of all partitions of $\mathbb{N}=\{1,2,\hdots,\}$ such that
\cite{Bertoin,Pitman,Berestycki}
\begin{itemize}
\item $\Pi^{(\infty)}_0$ is the initial partition of $\mathbb{N}$ into singletons;
\item {\it (consistency)} a Markov process obtained by restricting $\Pi^{(\infty)}_t$ to $[N]$ is equivalent in distribution law to the $N$-coalescent $\Pi^{(N)}_t$.
\end{itemize}
In addition to the existence and uniqueness of such process $\Pi^{(\infty)}_t$, which follows
from consistency of finite restrictions and Kolmogorov's extension theorem,
Kingman \cite{Kingman82b} also provides an explicit probabilistic construction of the process.
In this construction, each cluster at a given time is indexed by the lowest element of $\mathbb{N}$
contained in the cluster.
Thus the index of the cluster a particle belongs to is nonincreasing.
For each particle, the process tracks the index of the cluster it belongs to throughout time.
\subsection{Coalescent tree}
A merger history of Kingman's $N$-coalescent process can
be naturally described by a time-oriented binary tree $T^{(N)}_{\rm K}$ constructed as follows.
Start with $N$ leaves that represent the initial $N$ particles and have time mark $t=0$.
When two clusters coalesce (a transition occurs), merge the corresponding vertices
to form an internal vertex with a time mark of the coalescent.
The final coalescence forms the tree root.
The resulting time-oriented tree represents the history of the process.
It is readily seen that there is one-to-one map from the trajectories
of an $N$-coalescence process onto the time-oriented trees
with $N$ leaves.
Finally, observe that the combinatorial version of the above coalescent tree
is invariant under time scaling $t_{\rm new}=C\,t_{\rm old}$, $C>0$.
Thus without loss of generality we let $K(i,j) \equiv 1/N$ in Kingman's
$N$-coalescent process.
Slowing the process's evolution $N$ times is natural in Smoluchowski coagulation
equations that describe the dynamics of the fraction of clusters of different
masses.
\section{Statement of results}
\label{results}
The main result of this paper is Horton self-similarity for the
combinatorial tree $\textsc{shape}\left(T^{(N)}_{\rm K}\right)$ of the
Kingman's $N$-coalescent process, as $N$ goes to infinity.
Specifically, let $N_k$ denote the number of branches of Horton-Strahler order $k$ in
the tree $T^{(N)}_{\rm K}$ that describes the $N$-coalescent.
We show in Sect.~\ref{SHSODE}, Lemma~\ref{lem1} that for each $k$, $N_k/N$ converges in probability to the asymptotic Horton ratio
\[{\mathcal{N}}_k=\lim_{N\to\infty} N_k/N.\]
Moreover, these ${\mathcal{N}}_k$ are finite, and can be expressed as
\[{\mathcal{N}}_k = \frac{1}{2}\int_0^\infty g_k^2(x)\,dx,\]
where the sequence $g_k(x)$ solves the following system of
ordinary differential equations:
\[g'_{k+1}(x)-{g^2_k(x) \over 2}+g_k(x) g_{k+1}(x)=0,\quad x\ge 0\]
with $g_1(x)=2/(x+2)$, $g_k(0)=0$ for $k \geq 2$.
The root-law Horton self-similarity is proven in Section \ref{existence}
in the following statement.
\begin{Mthm}
The asymptotic Horton ratios ${\mathcal{N}}_k$ exist and finite and satisfy
the convergence
$\lim\limits_{k \rightarrow \infty}\left({\mathcal{N}}_k \right)^{-{1 \over k}}=R$
with $2 \le R \le 4$.
\end{Mthm}
Section~\ref{level} introduces a {\it level set tree} $\textsc{level}(X)$ that describes
the structure of the level sets of a finite time series $X_k$ and provides a
one-to-one map between rooted planar time-oriented trees and sequences
of the local extrema of a time series.
Furthermore, let $X=(X_k)$ be a time series with $N$ local maxima separated by
$N-1$ internal local minima that are independent and identically
distributed with a common continuous distribution $F$;
we call $X$ an {\it extended discrete white noise}.
Section~\ref{finite} establishes the following equivalence.
\begin{Mthm2}
The combinatorial level set tree of an extended discrete white noise $X$
with $N$ local maxima has the same distribution
on ${\mathcal{T}}_N$ as the combinatorial tree generated by Kingman's $N$-coalescent and a constant collision kernel.
\end{Mthm2}
In Sect.~\ref{infinite} we construct an infinite tree for Kingman's
coalescent as a limit of Kingman's $N$-coalescent trees viewed
from a leaf as $N\to\infty$.
We also describe two complementary constructions of an infinite tree
for an infinite extended discrete white noise viewed from a leaf.
Theorem~\ref{main2} is used then to establish the
distributional equivalence of the infinite combinatorial trees for the
Kingman coalescent and an infinite extended white noise.
This also allows us to naturally interpret the values ${\mathcal{N}}_k$
as Horton ratios for Kingman's coalescent tree (or, equivalently,
tree of a white noise) and leads to the following result.
\begin{Mthm3}
Kingman coalescent and an infinite discrete white noise
are root-Horton self-similar with $2\le R \le 4$.
\end{Mthm3}
Finally, numerical experiments in Sect.~\ref{numerics}
(i) suggest that Kingman's coalescent and white noise trees obey a stronger,
geometric, version of Horton self-similarity,
(ii) provide a close estimation of Horton exponent $R=3.043827\hdots$, and
(iii) support a conjecture that these trees enjoy a stronger {\it Tokunaga self-similarity}
defined in Sect.~\ref{Tokunaga}.
\section{Smoluchowski-Horton ODEs for Kingman's coalescent}
\label{SHSODE}
Consider Kingman's $N$-coalescent process with a constant kernel.
In Section~\ref{informal} we informally write Smoluchowski-type
ordinary differential equations (ODEs) for the number of Horton-Strahler branches in the coalescent
tree $T^{(N)}_{\rm K}$ and consider the asymptotic version of these
equations as $N\to\infty$.
Section~\ref{hydro} formally establishes the validity of the hydrodynamic limit.
\subsection{Main equation}
\label{informal}
Recall that we let $K(i,j) \equiv 1/N$ in Kingman's $N$-coalescent process. Let $|\Pi^{(N)}_t|$ denote the total number of clusters (of any mass) at time $t \geq 0$, and let $\eta_{(N)}(t):=|\Pi^{(N)}_t|/N$ be the total number of clusters relative to the system size $N$.
Then $\eta_{(N)}(0)=N/N=1$ and $\eta_{(N)}(t)$ decreases by $1/N$ with each coalescence of clusters at the rate of
$${1 \over N} \, \binom{N\, \eta_{(N)}(t)}{2}={\eta_{(N)}^2(t) \over 2}\cdot N+o(N),\quad{\rm as~} N\to\infty$$
since $1/N$ is the coalescence rate for any pair of clusters regardless of their masses.
Informally, this implies that the limit relative number of clusters
$\displaystyle\eta(t)=\lim_{N\to\infty}\eta_{(N)}(t)$
satisfies the following ODE:
\begin{eqnarray} \label{Aeta_t}
{d \over dt} \eta(t)=-\frac{\eta^2(t)}{2}.
\end{eqnarray}
Next, for any $j \in \mathbb{N}$ we define $\eta_{j,N}(t)$ to be the number of
clusters of Horton-Strahler order $j$ at time $t$ relative to the system size $N$.
Initially, each particle represents a leaf of Horton-Strahler order $1$.
Thus, the initial conditions are set to be, using Kronecker's delta notation,
$$\eta_{j,N}(0)=\delta_1(j).$$
We describe now the evolution of $\eta_{j,N}(t)$ using the definition of Horton-Strahler orders.
Observe that at any time $t$, $~\eta_{j,N}(t)$ increases by $1/N$ with each coalescence of clusters of Horton-Strahler
order $j-1$ with rate
$${1 \over N} \,
\binom{N\, \eta_{(j-1),N}(t)}{2}={\eta_{(j-1),N}^2(t) \over 2} \cdot N+o(N).$$
Thus ${\eta_{(j-1),N}^2(t) \over 2}+o(1)$ is the instantaneous rate of increase of $\eta_{j,N}(t)$.
Similarly, $~\eta_{j,N}(t)$ decreases by $1/N$ when a cluster of order $j$ coalesces
with a cluster of order strictly higher than $j$ with rate
$$\eta_{j,N}(t) \, \left(\eta_{(N)}(t)-\sum\limits_{k=1}^{j} \eta_{k,N}(t) \right)\cdot N,$$
and it decreases by $2/N$ when a cluster
of order $j$ coalesces with another cluster of order $j$ with rate
$${1 \over N} \, \binom{N\, \eta_{j,N}(t)}{2} ={\eta_{j,N}^2(t) \over 2}\cdot N+o(N).$$
Thus the instantaneous rate of change of $\eta_{j,N}(t)$ is
$$\eta_{j,N}(t) \, \left(\eta_{(N)}(t)-\sum\limits_{k=1}^{j} \eta_{k,N}(t) \right)
+\eta^2_{j,N}(t)+o(1).$$
Now we can informally write the limit rates-in and the rates-out for the clusters
of Horton-Strahler order via the following {\it Smoluchowski-Horton system} of ODEs:
\begin{eqnarray} \label{Aeta}
{d \over dt} \eta_j(t)=\frac{\eta_{j-1}(t)}{2}-\eta_j(t)
\, \left(\eta(t)-\sum\limits_{k=1}^{j-1} \eta_k(t) \right)
\end{eqnarray}
with the initial conditions $\eta_j(0)=\delta_1(j)$.
Here we define $\displaystyle\eta_k(t)=\lim_{N\to\infty}\eta_{k,N}(t)$, provided it exists, and
let $\eta_0 \equiv 0$.
Since $\eta_j(t)$ has the instantaneous rate of increase ${\eta_{j-1}^2(t) \over 2}$,
the relative total number of clusters of Horton-Strahler order $j$ is given by
\begin{equation}
\label{cNj}
{\mathcal{N}}_j=\delta_1(j)+\int\limits_0^{\infty} {\eta^2_{j-1}(t) \over 2} dt.
\end{equation}
It is not hard to compute the first terms of sequence
${\mathcal{N}}_k$ by solving equations (\ref{Aeta_t}) and (\ref{Aeta})
in the first three iterations:
$${\mathcal{N}}_1=1, \quad {\mathcal{N}}_2={1 \over 3}, \quad \text{ and }
\quad {\mathcal{N}}_3={e^4 \over 128}-{e^2 \over 8}+{233 \over 384}= 0.109686868100941\hdots$$
Hence, we have
${{\mathcal{N}}_1/ {\mathcal{N}}_2}={3}$ and
${{\mathcal{N}}_2/ {\mathcal{N}}_3}= 3.038953879388\dots$
Our numerical results in Sect.~\ref{numerics} yield, moreover,
$$\lim\limits_{k \rightarrow \infty}\left({\mathcal{N}}_k \right)^{-{1 \over k}}
=\lim\limits_{k \rightarrow \infty}{{\mathcal{N}}_{k} \over {\mathcal{N}}_{k+1}}=3.0438279\dots$$
\subsection{Hydrodynamic limit}
\label{hydro}
This section establishes the existence of the asymptotic ratios
${\mathcal{N}}_k$ as well as the validity of the equations~\eqref{Aeta_t},
\eqref{Aeta} and \eqref{cNj} in a hydrodynamic limit.
We refer to Darling and Norris \cite{RDJN08} for a survey of formal techniques
for proving that a Markov chain converges to the solution of a differential equation.
Notice that {\it quasilinearity} of the
system of ODEs in (\ref{Aeta}) implies the existence and uniqueness.
Specifically, if the first $j-1$ functions $\eta_1(t),\hdots,\eta_{j-1}(t)$
are given, then (\ref{Aeta}) is a linear equation in $\eta_j(t)$.
This makes the following argument less technically involved than
the one presented by Norris \cite{Norris99} for the Smoluchowski equations.
\subsubsection{Hydrodynamic limit for equation (\ref{Aeta_t})}
\begin{lem}
\label{lem2}
The relative total number $\eta_{(N)}(t)$ of clusters converges in probability,
as $N\to\infty$,
to $\eta(t)$ that satisfies equation (\ref{Aeta_t}) with the initial
condition $\eta(0)=1$.
\end{lem}
\begin{proof}
Take $\delta>0$. Consider $\eta_{(N)}(t)-\eta_{(N)}(t+\delta)$ given $\eta_{(N)}(t)$.
The Chernoff inequality bounds the probability that there is more than
${\delta \over N} \binom{N \eta_{(N)}(t)}{2}+N^{2/3}$ coalescing pairs
during $[t,t+\delta]$.
Specifically, we consider the probability that a sum of
${\delta \over N} \binom{N \eta_{(N)}(t)}{2}+N^{2/3}$ exponential inter-arrival
times with the rate not exceeding ${1 \over N} \binom{N \eta_{(N)}(t)}{2}$ adds up to less than $\delta$.
Then, applying Chernoff inequality with $s>0$, we obtain
\begin{eqnarray*}
\lefteqn{P\left(\eta_{(N)}(t)-\eta_{(N)}(t+\delta)>{\delta \over N} \binom{N \eta_{(N)}(t)}{2}+N^{2/3}~\right)}\\
&\leq & {\exp\{s \delta/N\} \over \left(1+{s \over \binom{N \eta_{(N)}(t)}{2}}\right)^{{\delta \over N}
\binom{N \eta_{(N)}(t)}{2}+N^{2/3}}}\\
& \leq & \exp\left\{-{1 \over 2}s N^{2/3}+\delta {s^2 \over N} \over \binom{N \eta_{(N)}(t)}{2}\right\}\\
& \leq & \exp\left\{-{1 \over 2}N^{13/6}+\delta N^2 \over \binom{N \eta_{(N)}(t)}{2}\right\}\\
& & (\text{ taking }s=N^{3/2}~)\\
\\
& \leq & \exp\left\{-N^{1/6}+2\delta\right\}
\end{eqnarray*}
\noindent
for $N$large enough, as $~\ln(1+x) > x-x^2~$ for $x>0$, and $~\ln(1+x) > {1 \over 2}x~$ for $x \in (0,2)$.
In the above inequality, we assume that $\eta_{(N)}(t)$ is bounded from below by a
given $\epsilon_0 \in (0,1)$, and that integer rounding is applied as necessary.
Now that we know with probability exceeding $1-e^{-N^{1/6}+2\delta}$ that there are
no more than ${\delta \over N} \binom{N \eta_{(N)}(t)}{2}+N^{2/3}$ coalescing pairs
during $[t,t+\delta]$, we also know that the exponential rates of inter-arrival times
are greater than ${1 \over N}\binom{N [\eta_{(N)}(t)-\delta \eta_{(N)}^2(t)/2]}{2}$.
Therefore we can use Chernoff inequality to bound the conditional probability that there are
fewer than ${\delta \over N} \binom{N [\eta_{(N)}(t)-\delta \eta_{(N)}^2(t)/2]}{2}-N^{2/3}$ coalescing
pairs in $[t,t+\delta]$.
Specifically, we bound the probability that a sum of
${\delta \over N} \binom{N [\eta_{(N)}(t)-\delta \eta_{(N)}^2(t)/2]}{2}-N^{2/3}$ independent exponential random variables of rate ${1 \over N}\binom{N [\eta_{(N)}(t)-\delta \eta_{(N)}^2(t)/2]}{2}$ is greater than $\delta$.
Chernoff inequality with $s>0$ implies
\begin{eqnarray*}
\lefteqn{P\left(\eta_{(N)}(t)-\eta_{(N)}(t+\delta)<{\delta \over N} \binom{N [\eta_{(N)}(t)-\delta \eta_{(N)}^2(t)/2]}{2}-N^{2/3}~\right)}\\
& \leq & {\exp\{-s \delta/N\} \over \left(1-{s \over \binom{N [\eta_{(N)}(t)-\delta \eta_{(N)}^2(t)/2]}{2}}\right)^{{\delta \over N} \binom{N [\eta_{(N)}(t)-\delta \eta_{(N)}^2(t)/2]}{2}-N^{2/3}}}\\
& \leq & \exp\left\{-s N^{2/3}+\delta {s^2 \over N} \over \binom{N [\eta_{(N)}(t)-\delta \eta_{(N)}^2(t)/2]}{2}\right\}\\
& \leq & \exp\left\{-N^{13/6}+\delta N^2 \over \binom{N [\eta_{(N)}(t)-\delta \eta_{(N)}^2(t)/2]}{2}\right\} \quad (\text{ taking }s=N^{3/2}~)\\
& \leq & \exp\left\{-2N^{1/6}+2\delta\right\}
\end{eqnarray*}
\noindent
as $~-x-x^2 < \ln(1-x) <-x~$ for $x \in \left(0,{1 \over 2}\right)$.
Hence, if we partition $[0,K]$ into $K/\delta$ subintervals, then with probability greater than
$$\left(1-{\delta \over K}e^{-N^{1/6}+2\delta}\right)\left(1-{\delta \over K}e^{-2N^{1/6}+2\delta}\right),$$
for each left partition point $t$,
\begin{equation}\label{psiE}
\Delta_\delta \eta_{(N)}(t)=-{\eta_{(N)}^2(t) \over 2}+\mathcal{E}'(t),
\end{equation}
where $~\Delta_\delta f(x):={f(x+\delta)-f(x) \over \delta}~$ denotes the forward difference,
and the error $~|\mathcal{E}'(t)|<C_1\left(\delta +\delta^{-1}N^{-1/3}\right)$ for some positive $C_1$
as $0 \leq \psi_N(t) \leq 1$.
Next, we let $N \rightarrow \infty$ and $\delta \rightarrow 0_+$ so that $\delta N^{1/3} \rightarrow \infty$. We use the error propagation in (\ref{psiE}) to show $\|\eta_{(N)}-\varphi_\delta \|_{L^2[0,K]} \rightarrow 0$, where $\varphi_\delta$ solves the corresponding difference equation
\begin{equation}\label{phiE}
\Delta_\delta \varphi_\delta (t)=-{\varphi^2_\delta (t) \over 2}
\end{equation}
with the initial condition $\varphi_\delta (0)=1$.
Specifically, we partition $[0,K]$ into subintervals of length $\delta$ each.
The partition points satisfy $~t_{j+1}=t_j+\delta$.
We consider the error quantities $~\varepsilon_j:=\eta_{(N)}(t_j)-\varphi_\delta (t_j)$.
Then
\begin{eqnarray*}
\varepsilon_{j+1} & = & \eta_{(N)}(t_{j+1})-\varphi_\delta (t_{j+1})\\
& = & \left[\varphi_\delta (t_j)+\varepsilon_j-{\Big(\varphi_\delta (t_j)+\varepsilon_j \Big) \over 2}+\delta\mathcal{E}'(t_j)\right]-\left[\varphi_\delta (t_j)-{\varphi^2_\delta (t_j) \over 2}\right]\\
& = & (1-\varphi_\delta (t_j)) \varepsilon_j -{\varepsilon_j^2 \over 2}+\delta\mathcal{E}'(t_j),
\end{eqnarray*}
where $~0 \leq \varphi_\delta (t_j) \leq 1$.
Tracing the above propagation of error results in bounding the error throughout the interval $~[0,K]$,
$$|\varepsilon_j| \leq C_2K\left(\delta +\delta^{-1}N^{-1/3}\right)$$
for all $j$ and some $C_2>0$.
Finally, the same error propagation arguments works to show that $\|\eta-\varphi_\delta \|_{L^2[0,K]} \rightarrow 0$ for $\varphi_\delta$ in (\ref{phiE}) and $\eta$ in (\ref{Aeta_t}).
The expected time for the first $m$ cluster mergers is
$${1 \over \binom{N}{2}}+{1 \over \binom{N-1}{2}}+\dots+{1 \over \binom{N-m+1}{2}}={2m \over (N-m)N},$$
and by Markov inequality, the probability
$$P\Big(\eta_{(N)}(K)>\varepsilon \Big) \leq {2(1-\varepsilon) \over K}(1+2/(\varepsilon N))$$
for any $\varepsilon \in (0,1)$.
Similarly, $~\int\limits_K^{\infty} \eta^2(t) /2 ~dt=\int\limits_K^{\infty} {2 \over (t+2)^2} ~dt={2 \over K+2}$, showing that considering $K$ large enough will suffice for the argument.
Therefore we have shown that $~\|\eta_{(N)}-\eta \|_{L^2[0,\infty)} \rightarrow 0~$ in probability.
\end{proof}
\subsubsection{Hydrodynamic limit for equations (\ref{Aeta})}
Let $G_j(t)=\sum\limits_{i:~i \geq j} N_j(t)$ denote the number of clusters of Horton-Strahler
order $j$ or higher.
Also let $\eta_{j,N}(t):=N^{-1}N_j(t)~$ and $g_{j,N}:=N^{-1}G_j(t)=\eta_{(N)}(t)-\sum\limits_{k:k<j}\eta_{k,N}(t)$.
\begin{lem}
\label{lem3}
For each $j$, the relative number $\eta_{j,N}(t)$ converges in probability,
as $N \rightarrow \infty$, to $\eta_j(t)$ that satisfies system \eqref{Aeta}.
\end{lem}
\begin{proof}
Consider interval $[t,t+\delta] \subset [0,K]$.
As it was shown above, there are
$$m=\delta N \left({\eta^2_{(N)}(t) \over 2}+O(\delta+\delta^{-1}N^{-1/3}) \right)$$
transitions (coalescences) within $[t,t+\delta]$ with the probability
exceeding $$(1-e^{-N^{1/6}+2\delta})(1-e^{-2N^{1/6}+2\delta}) >1-2e^{-N^{1/6}+2\delta}.$$
For a fixed integer $j>1$, this gives us the number $m$ of transitions for the vector
$$\Big(\eta_{1,N}(t), ~\eta_{2,N}(t), ~\hdots, ~\eta_{j-1,N}(t),~g_{j,N}(t) \Big)$$
within the time interval, and the upper and lower bounds on the probabilities.
We apply Chernoff inequality for $m$ independent Bernoulli random variables of $2j-1$ outcomes.
In particular, the transition that results in decreasing $\eta_{k,N}(t)$ by $2/N$ and
increasing $\eta_{k+1,N}(t)$ by $1/N$ happens with the probability bounded above by
${\binom{N \eta_{k,N}(t)}{2}/ \binom{N\eta_{(N)}(t)-m}{2}}$, and bounded below by
${\binom{N \eta_{k,N}(t)-m}{2}/\binom{N\eta_{(N)}(t)}{2}}$.
Similarly, the transition that results in decreasing $\eta_{k,N}(t)$ by $1/N$
happens with the probability bounded from above by
${N^2 \eta_{k,N}(t) g_{k+1,N}(t)/ \binom{N\eta_{(N)}(t)-m}{2}}$,
and bounded from below by ${(N \eta_{k,N}(t)-m) N g_{k+1,N}(t)/ \binom{N\eta_{(N)}(t)}{2}}$.
We obtain the system of difference equations
\begin{eqnarray}\label{eqn:hydr}
\Delta_\delta \eta_{1,N}(t) & = & -\eta_{1,N}(t) g_{1,N}(t) +\mathcal{E}_1 \nonumber \\
\Delta_\delta \eta_{2,N}(t) & = &{\eta^2_{1,N}(t) \over 2} -\eta_{2,N}(t) g_{2,N}(t) +\mathcal{E}_2 \nonumber \\
\vdots & \vdots & \vdots \\
\Delta_\delta \eta_{j-1,N}(t) & = & {\eta^2_{j-2,N}(t) \over 2} -\eta_{j-1,N}(t) g_{j-1,N}(t) +\mathcal{E}_{j-1} \nonumber \\
\Delta_\delta g_{j,N}(t) & = &{\eta^2_{j-1,N}(t) \over 2} -{g^2_{j,N}(t) \over 2} +\mathcal{E}_j \nonumber
\end{eqnarray}
with the initial conditions
$$\Big(\eta_{1,N}(0), ~\eta_{2,N}(0), ~\hdots, ~\eta_{j-1,N}(t),~g_{j,N}(0) \Big)=(1,0,\hdots,0),$$
where each $\mathcal{E}_k=O(\delta+\delta^{-1}N^{-1/3})$.
The errors here are obtained using Chernoff inequality.
Namely, if $S_m$ is a Binomial random variable that represents $m$ Bernoulli trials with
probability $0<p<1$ of success, then for $s>0$,
\begin{eqnarray*}
\lefteqn{
P(S_m \geq mp+m^{2/3}) \leq (pe^s+1-p)^m ~\exp\Big\{-s(mp+m^{2/3})\Big\}} \\
& = & ~\exp\Big\{m\ln(pe^s+1-p)-s(mp+m^{2/3})\Big\} \\
& = & ~\exp\Big\{m\ln(pe^{m^{-1/3}}+1-p)-m^{2/3}p+m^{1/3}\Big\} \quad (\text{ taking }s=m^{-1/3}~)\\
& \leq & ~\exp\Big\{[p^2+p-1]m^{1/3}\Big\} ~ = ~\exp\Big\{-[1-p-p^2]m^{1/3}\Big\}
\end{eqnarray*}
The lower bound on $P(S_m \leq mp-m^{2/3})$ follows symmetrically.
Finally, the same error propagation analysis applies to compare the above
difference equations (\ref{eqn:hydr}) to the difference equations
that correspond to the following system of ODEs
\begin{eqnarray*}
{d \over dt} \eta_1(t) & = & -\eta_1(t) g_1(t) \\
{d \over dt} \eta_2(t) & = &{\eta^2_1(t) \over 2} -\eta_2(t) g_2(t) \\
\vdots & \vdots & \vdots \\
{d \over dt} \eta_{j-1}(t) & = & {\eta^2_{j-2}(t) \over 2} -\eta_{j-1}(t) g_{j-1}(t) \\
{d \over dt} g_j(t) & = & {\eta^2_{j-1}(t) \over 2} -{g^2_j(t) \over 2}, \\
\end{eqnarray*}
where $~g_i(t):=\eta(t)-\sum\limits_{k:~k<i} \eta_k(t)$. Thus we showed that functions $~\eta_{j,N}(t)~$ converge in probability to functions $~\eta_j(t)~$ that solve (\ref{Aeta}).
\end{proof}
\subsubsection{Hydrodynamic limit for asymptotic Horton ratios}
\begin{lem}
\label{lem1}
The Horton ratios $~N_k/N~$ converge in probability to a finite constant ${\mathcal{N}}_k$
given by \eqref{cNj}, as $~N \rightarrow \infty$.
\end{lem}
\begin{proof}
Observe that in the difference equations (\ref{eqn:hydr}), the number of emerging clusters of Horton-Strahler order $j$ within $[t,t+\delta]$ time interval divided by $N$ is
$${\eta^2_{j-1,N}(t) \over 2}\cdot \delta +O(\delta^2+N^{-1/3})$$
with probability greater than $1-2e^{-N^{1/6}+2\delta}$.
Hence, for $j \geq 2$, the total number of emerging clusters of Horton-Strahler order $j$ within $[0,K]$ time interval divided by $N$ is
$$\int\limits_0^K {\eta^2_{j-1,N}(t) \over 2} dt+O(\delta+\delta^{-1}N^{-1/3})$$
with probability exceeding $1-2e^{-N^{1/6}+2\delta}$.
Therefore, the total number of emerging clusters of Horton-Strahler order $j$ within $[0,\infty)$ time interval divided by $N$ is
$$N_j/N=\int\limits_0^K {\eta^2_{j-1,N}(t) \over 2} dt+\mathcal{R}_{\delta,\varepsilon, N},$$
where $\mathcal{R}_{\delta,\varepsilon, N}=O\left(\delta+\delta^{-1}N^{-1/3}+\varepsilon \right)$, with probability exceeding
$$\mathcal{P}_{\delta,\varepsilon,K,N}=\left(1-2e^{-N^{1/6}+2\delta}\right)\left(1-{2(1-\varepsilon) \over K}(1+2/(\varepsilon N)) \right)$$
as $~P\Big(\eta_{(N)}(K)>\varepsilon \Big) \leq {2(1-\varepsilon) \over K}(1+2/(\varepsilon N))$
for any $\varepsilon \in (0,1)$.
So, with probability exceeding $\mathcal{P}_{\delta,\varepsilon,K,N}$,
$$\left|N_j/N-\int\limits_0^\infty {\eta^2_{j-1}(t) \over 2} dt \right| \leq \int\limits_0^K {(\eta_{j-1,N}(t)-\eta_j(t))^2 \over 2} dt+\int\limits_K^\infty {\eta^2_{j-1}(t) \over 2} dt+\mathcal{R}_{\delta,\varepsilon, N},$$
where $~\int\limits_K^\infty {\eta^2_{j-1}(t) \over 2} dt \leq \int\limits_K^{\infty} \eta^2(t) /2 ~dt=\int\limits_K^{\infty} {2 \over (t+2)^2} ~dt={2 \over K+2}.$
Thus, for any $\epsilon>0$ and any $\alpha \in (0,1)$, there exist sufficiently small $\delta>0$ and $\varepsilon>0$ , and sufficiently large $K$ such that the following three inequalities are satisfied for $N$ large enough:
$$ P\left(\int\limits_0^K {(\eta_{j-1,N}(t)-\eta_j(t))^2 \over 2} dt <\epsilon/2 \right) \geq \sqrt{1-\alpha},$$
$${2 \over K+2}+\mathcal{R}_{\delta,\varepsilon, N} <\epsilon/2,$$
and
$$\mathcal{P}_{\delta,\varepsilon,K,N} \geq \sqrt{1-\alpha}.$$
Hence
$$P\left( \left|N_j/N-\int\limits_0^\infty {\eta^2_{j-1}(t) \over 2} dt \right| <\epsilon \right) \geq 1-\alpha$$
for any $\epsilon>0$ and any $\alpha \in (0,1)$, and $N$ large enough.
Thus
$$N_j/N ~\longrightarrow ~ \delta_1(j)+\int\limits_0^{\infty} {\eta^2_{j-1}(t) \over 2} dt \quad \text{ in probability as }~N \rightarrow \infty.$$
\end{proof}
\section{The root-Horton self-similarity and related results}
\label{existence}
We begin this section with preliminary lemmas and propositions,
and then proceed to proving Theorem \ref{main}.
Let $g_1(t)=\eta(t)$ and $g_j(t)=\eta(t)-\sum\limits_{k:~k<j} \eta_k(t)$
be the asymptotic number of clusters of Horton order $j$ or higher at time $t$.
We can rewrite (\ref{Aeta}) via $g_j$ using $\eta_j(t)=g_j(t)-g_{j+1}(t)$ as follows
$${d \over dt}g_j(t)-{d \over dt}g_{j+1}(t)
={g_{j-1}(t)-g_j(t) \over 2}-(g_j(t)-g_{j+1}(t))g_j(t)$$
Observe that
$g_1(t) \geq g_2(t) \geq g_3(t) \geq \dots$
We now rearrange the terms, obtaining for all $j \geq 2$,
\begin{eqnarray} \label{odeF1}
~~~~~~{d \over dt}g_{j+1}(t)-{g^2_j(t) \over 2}+g_j(t) g_{j+1}(t)
={d \over dt} g_j(t)-{g^2_{j-1}(t) \over 2}+g_{j-1}(t) g_j(t).
\end{eqnarray}
One can readily check that ${d \over dt} g_2(t)-{g^2_1(t) \over 2}+g_1(t)\,g_2(t)=0$;
the above equations hence simplify as follows
\begin{eqnarray} \label{odeG}
g'_{j+1}(x)-{g^2_j(x) \over 2}+g_j(x) g_{j+1}(x)=0 \qquad \\ \nonumber \text{ with }~g_1(x)={2 \over x+2}, \text{ and } g_j(0)=0 \text{ for } j \geq 2.
\end{eqnarray}
Notice that the above system of ODEs (\ref{odeG}) is the quasilinearized
Riccati equation \mbox{$g'(x)=-{g^2(x) \over 2}$} with the initial value
$g(0)=0$ that has only a trivial solution.
Next, returning to the asymptotic ratios of the number of order-$j$ branches to $N$,
we observe that (\ref{odeF1}) implies that
$${\mathcal{N}}_j=\int\limits_0^{\infty} {\eta^2_{j-1}(t) \over 2} dt
=\int\limits_0^{\infty} {(g_{j-1}(t)-g_j(t))^2 \over 2} dt
=\int\limits_0^{\infty} {g^2_j(t) \over 2} dt$$
since
$$ {(g_{j-1}(t)-g_j(t))^2 \over 2} ={d \over dt}g_j(t)+{g^2_j(t) \over 2},$$
where $\int\limits_0^{\infty}{d \over dt}g_j(t)dt=g_j(\infty)-g(0)=0$ for $j \geq 2$.
Let $n_k$ represent the number of order-$k$ branches relative to the number
of order-$(k+1)$ branches:
$$n_k:={{\mathcal{N}}_k \over {\mathcal{N}}_{k+1}}={\int\limits_0^{\infty} {g^2_k(x) \over 2} dx \over \int\limits_0^{\infty} {g^2_{k+1}(x) \over 2} dx}={ \|g_k\|^2_{L^2[0, \infty)} \over \|g_{k+1}\|^2_{L^2[0, \infty)}}.$$
Consider the following limits that represent respectively the
root and the ratio asymptotic Horton laws:
$$\lim\limits_{k \rightarrow \infty}\left({\mathcal{N}}_k \right)^{-{1 \over k}}=\lim\limits_{k \rightarrow \infty}\left(\prod\limits_{j=1}^{k} n_j \right)^{-{1 \over k}} \qquad \text{ and } \qquad \lim\limits_{k \rightarrow \infty} n_k=\lim\limits_{k \rightarrow \infty} { \|g_k\|^2_{L^2[0, \infty)} \over \|g_{k+1}\|^2_{L^2[0, \infty)}}.$$
Theorem~\ref{main} establishes the existence of the first limit.
We expect the second, stronger, limit also to exist and both of them to be equal to $3.043827\dots$
according to our numerical results in Sect.~\ref{numerics}.
We now establish some basic facts about $g_j$ and $n_j$.
\begin{prop} \label{prop1}
Let $g_j(x)$ be the solutions to the systems of ODEs (\ref{odeG}). Then\\
\begin{description}
\item[\qquad (a)]
$~\int\limits_0^{\infty} {g^2_j(x) \over 2} dx=\int\limits_0^{\infty} g_j(x)g_{j+1}(x) dx,$\\
\item[\qquad (b)]
$~\int\limits_0^{\infty} g^2_{j+1}(x) dx=\int\limits_0^{\infty} (g_j(x)-g_{j+1}(x))^2 dx,$\\
\item[\qquad (c)]
$~\lim\limits_{x \rightarrow \infty} xg_j(x) =2,$\\
\item[\qquad (d)]
$~n_j={ \|g_j\|^2_{L^2[0, \infty)} \over \|g_{j+1}\|^2_{L^2[0, \infty)}} \geq {2},$\\
\item[\qquad (e)]
$~n_j={ \|g_j\|^2_{L^2[0, \infty)} \over \|g_{j+1}\|^2_{L^2[0, \infty)}} \leq {4}.$\\
\end{description}
\end{prop}
\begin{proof}
Part (a) follows from integrating (\ref{odeG}), and part (b) follows from part (a).
Part (c) is done by induction, using the L'H\^{o}pital's rule as follows. It is obvious that $~\lim\limits_{x \rightarrow \infty} xg_1(x) =2$. Next, suppose $~\lim\limits_{x \rightarrow \infty} xg_j(x) =2$. Then
$$\lim\limits_{x \rightarrow \infty} xg_{j+1}(x)=\lim\limits_{x \rightarrow \infty} {g_{j+1}(x) \over x^{-1}}=\lim\limits_{x \rightarrow \infty} {{g^2_j(x) \over 2}-g_j(x) g_{j+1}(x) \over -x^{-2}}=2\lim\limits_{x \rightarrow \infty} xg_{j+1}(x) -2$$
implying $~\lim\limits_{x \rightarrow \infty} xg_{j+1}(x)=2$.
The statement (d) follows from the tree construction process.
An alternative proof of (d) using differential equations is given in the following subsection.
Part (e) follows from part (a) together with H\"older inequality
$${1 \over 2}\|g_j\|^2_{L^2[0, \infty)}
=\int\limits_0^{\infty} g_j(x)g_{j+1}(x) dx \leq \|g_j\|_{L^2[0, \infty)} \cdot \|g_{j+1}\|_{L^2[0, \infty)},$$
which implies ${\|g_{j}\|^2_{L^2[0, \infty)} \over \|g_{j+1}\|^2_{L^2[0, \infty)}} \leq {4}$.
\end{proof}
\subsection{Rescaling to $[0,1]$ interval}
Let
$$h_k(x)=(1-x)^{-1}-(1-x)^{-2} g_{k+1}\left({2x \over 1-x}\right)$$
for $x \in [0,1]$.
Then $h_0 \equiv 0$, $h_1 \equiv 1$, and the system of ODEs (\ref{odeG}) rewrites as
\begin{equation} \label{ODEh}
h'_{k+1}(x)=2h_k(x)h_{k+1}(x)-h_k^2(x)
\end{equation}
with the initial conditions $h_k(0)=1$.
The above system of ODEs (\ref{ODEh}) is the quasilinearized
Riccati equation $h'(x)=h^2(x)$ over $[0,1)$, with the initial value $h(0)=1$.
Its solution is $h(x)={1 \over 1-x}$.
Here
$$h_k(x) \rightarrow h(x)={1 \over 1-x} \quad \text{ and } \quad n_k={\big\|1-h_{k+1}/h\big\|^2_{L^2[0,1]} \over \big\|1-h_k/h\big\|^2_{L^2[0,1]} }.$$
Observe that $h_2(x)=(1+e^{2x})/2$, but for $k \geq 3$ finding a closed form expression becomes increasingly hard.
Given $h_k(x)$, Eq.~\eqref{ODEh} is a linear first-order ODE in $h_{k+1}(x)$;
its solution is given by $h_{k+1}(x)=\mathcal{H}h_k(x)$ with
\begin{equation}
\label{iter}
\mathcal{H}f(x)=\left[1-\int_0^x f^2(y) e^{-2\int\limits_0^y f(s) ds } dy \right]
\cdot e^{2\int\limits_0^x f(s) ds }.
\end{equation}
Hence, the problem we are dealing with concerns the asymptotic behavior of an iterated
non-linear functional.
Using the setting of (\ref{ODEh}), we give an ODE proof to Proposition \ref{prop1}(d).
To do so, we first need to prove the following lemma.
\begin{lem}\label{half}
$$\big\|1-h_{k+1}/h\big\|_{L^2[0,1]}=\big\|h_{k+1}/h-h_k/h\big\|_{L^2[0,1]}$$
\end{lem}
\begin{proof}
Observe that $$h'_{k+1}(x)+(h_{k+1}(x)-h_k(x))^2=h_{k+1}^2(x).$$
We now use integration by parts to obtain
$$\int\limits_0^1 {(h_{k+1}(x)-h_k(x))^2 \over h^2(x)}dx=\int\limits_0^1 {h_{k+1}^2(x) \over h^2(x)}dx-\int\limits_0^1{h'_{k+1}(x) \over h^2(x)}dx$$
$$=\int\limits_0^1 {h_{k+1}^2(x) \over h^2(x)}dx-\left[{h'_{k+1}(1) \over h^2(1)}-{h'_{k+1}(0) \over h^2(0)}+2\int\limits_0^1 {h_{k+1}(x) \over h(x)}dx\right]$$
$$=\int\limits_0^1 {h_{k+1}^2(x) \over h^2(x)}dx+1-2\int\limits_0^1 {h_{k+1}(x) \over h(x)}dx$$
$$=\int\limits_0^1 {(1-h_{k+1}(x))^2 \over h^2(x)}dx$$
since $1/h(x)=1-x$.
\end{proof}
\vskip 0.2 in
\noindent
\begin{proof} [Alternative proof of Proposition \ref{prop1}(d)]
Notice that
$h \geq \dots\geq h_{k+1} \geq h_k\geq \dots \geq h_0 \equiv 0$,
which follows from
$g_1(t) \geq g_2(t) \geq g_3(t) \geq \dots$
The Lemma \ref{half} implies
$$\big\|1-h_{k+1}/h\big\|_{L^2[0,1]}^2 = \big\|h_{k+1}/h-h_k/h\big\|_{L^2[0,1]}^2= \int\limits_0^1 \left[(1-h_k/h)-(1-h_{k+1}/h) \right]^2dx$$
$$= \big\|1-h_{k+1}/h\big\|_{L^2[0,1]}^2
+\big\|1-h_k/h\big\|_{L^2[0,1]}^2 -2\int\limits_0^1(1-h_k/h)(1-h_{k+1}/h) dx $$
and therefore
$$\big\|1-h_k/h\big\|_{L^2[0,1]}^2 =2\int\limits_0^1(1-h_k/h)(1-h_{k+1}/h) dx$$
$$=2\big\|1-h_{k+1}/h\big\|_{L^2[0,1]}^2
+2\int\limits_0^1(h_{k+1}/h-h_k/h)(1-h_{k+1}/h) dx \geq 2\big\|1-h_{k+1}/h\big\|_{L^2[0,1]}^2$$
yielding $2 \leq {\big\|1-h_{k}/h\big\|_{L^2[0,1]}^2 \over \big\|1-h_{k+1}/h\big\|_{L^2[0,1]}^2}=n_{k}~$ as in Proposition \ref{prop1}(d).
\end{proof}
It is also true that one can improve Proposition \ref{prop1}(d) to make it a
strict inequality since one can check that
$$h(x) > \dots > h_{k+1}(x) > h_k(x)> \dots > h_0(x) \equiv 0 \quad \text{ for } x \in (0,1).$$
\subsection{Proof of the existence of the root-Horton limit}
Here we present the proof of our main Theorem \ref{main}.
It is based on the following two lemmas, Lemma \ref{h1} and Lemma \ref{h1exist},
that will be proven in the following two subsections.
\begin{lem} \label{h1}
If the limit $\lim\limits_{k \rightarrow \infty}{h_{k+1}(1) \over h_k(1)}$ exists, then
$\lim\limits_{k \rightarrow \infty}\left({\mathcal{N}}_k \right)^{-{1 \over k}}=\lim\limits_{k \rightarrow \infty}\left(\prod\limits_{j=1}^{k} n_j \right)^{-{1 \over k}}$ also exists, and
$$\lim\limits_{k \rightarrow \infty}\left({\mathcal{N}}_k \right)^{-{1 \over k}}
=\lim\limits_{k \rightarrow +\infty}\left({1 \over h_k(1)}\right)^{-{1 \over k}}
=\lim\limits_{k \rightarrow \infty}{h_{k+1}(1) \over h_{k}(1)}.$$
\end{lem}
\begin{lem} \label{h1exist}
The limit $\lim\limits_{k \rightarrow \infty}{h_{k+1}(1) \over h_k(1)} \geq 1$ exists, and is finite.
\end{lem}
\begin{thm} \label{main}
The limit
$\lim\limits_{k \rightarrow \infty}\left({\mathcal{N}}_k \right)^{-{1 \over k}}
=\lim\limits_{k \rightarrow \infty}\left(\prod\limits_{j=1}^{k} n_j \right)^{-{1 \over k}}=R$ exists.
Moreover,
$R=\lim\limits_{k \rightarrow \infty} {h_{k+1}(1) \over h_{k}(1)}$, and
$2 \leq R \leq 4$.
\end{thm}
\noindent
\begin{proof}
The existence and finiteness of $\lim\limits_{k \rightarrow \infty}{h_{k+1}(1) \over h_k(1)}$
established in Lemma \ref{h1exist} is the precondition for Lemma \ref{h1} that in turn implies
the existence and finiteness of the limit
$\lim\limits_{k \rightarrow \infty}\left({\mathcal{N}}_k \right)^{-{1 \over k}}$
as needed for the root-Horton law.
Finally, $2 \leq R \leq 4$ follows from Proposition \ref{prop1}.
\end{proof}
\subsection{Proof of Lemma \ref{h1} and related results}
\begin{prop}\label{one}
$$\big\|1-h_{k+1}(x)/h(x)\big\|_{L^2[0,1]}^2
\leq {1 \over h_{k+1}(1)} \leq \big\|1-h_k(x)/h(x)\big\|_{L^2[0,1]}^2.$$
\end{prop}
\begin{proof}
Integrating from 0 to 1 both sides of the equation
$${{d \over dx}h_{k+1}(x) \over h_{k+1}^2(x)}=1-{(h_{k+1}(x)-h_k(x))^2 \over h_{k+1}^2(x)}$$
we obtain $~{1 \over h_{k+1}(1)}=\int\limits_0^1 {(h_{k+1}(x)-h_k(x))^2 \over h_{k+1}^2(x)} dx~$ as $h_{k+1}(0)=1$.
\vskip 0.2 in
\noindent
Hence,
$${1 \over h_{k+1}(1)}=\int\limits_0^1 {(h_{k+1}(x)-h_k(x))^2 \over h_{k+1}^2(x)} dx \geq \int\limits_0^1 {(h_{k+1}(x)-h_k(x))^2 \over h^2(x)} dx =\int\limits_0^1 \left( 1-{h_{k+1}(x) \over h(x)}\right)^2 dx$$
by Proposition \ref{half}, proving the first inequality.
\vskip 0.2 in
\noindent
Now,
$${1 \over h_{k+1}(1)}=\big\|1-h_k(x)/h_{k+1}(x)\big\|_{L^2[0,1]}^2 \leq \big\|1-h_k(x)/h(x)\big\|_{L^2[0,1]}^2$$
thus completing the proof.
\end{proof}
\begin{proof}[Proof of Lemma \ref{h1}]
If the limit $\lim\limits_{k \rightarrow \infty}{h_{k+1}(1) \over h_k(1)}$ exists and is finite, then $\lim\limits_{k \rightarrow \infty}\left({1 \over h_k(1)}\right)^{-{1 \over k}}$ must also exist and be finite. Hence the existence and finiteness of
$$\lim\limits_{k \rightarrow \infty}\left({\mathcal{N}}_k \right)^{-{1 \over k}}=\lim\limits_{k \rightarrow \infty}\left(\int_0^1 \left(1-{h_k(x)\over h(x)}\right)^2 dx \right)^{-{1 \over k}}$$
follows from Proposition \ref{one}.
\end{proof}
\subsection{Proof of Lemma \ref{h1exist} and related results}
In this subsection we use the approach developed by Drmota \cite{MD2009}
to prove the existence and finiteness of
$\lim\limits_{k \rightarrow \infty}{h_{k+1}(1) \over h_k(1)} \geq 1$.
As we observed earlier this result is needed to prove the existence, finiteness, and positivity of
$\lim\limits_{k \rightarrow \infty}\left({\mathcal{N}}_k \right)^{-{1 \over k}}=\lim\limits_{k \rightarrow \infty}\left(\prod\limits_{j=1}^{k} n_j \right)^{-{1 \over k}} $, the root-Horton law.
\begin{Def}
Given $\gamma \in (0,1]$. Let
$$V_{k,\gamma}(x)=\begin{cases}
{1 \over 1-x} & \text{ for } 0 \leq x \leq 1-\gamma, \\
\gamma^{-1} h_k\left({x-(1-\gamma) \over \gamma}\right) & \text{ for } 1-\gamma \leq x \leq 1.
\end{cases} $$
\end{Def}
\noindent
Note that sequences of functions $h_k(x)$ and $V_{k,\gamma}(x)$ can be extended beyond $x=1$.
\noindent
Here are some observations we make about the above defined functions.
\begin{obs}
$V_{k,\gamma}(x)$ are positive continuous functions satisfying
$$V'_{k+1,\gamma}(x)=2V_{k+1,\gamma}(x)V_{k,\gamma}(x)-V^2_{k,\gamma}(x)$$
for all $x \in [0,1] \setminus (1-\gamma)$, with initial conditions $V_{k,\gamma}(0)=1$.
\end{obs}
\begin{obs}
Let $\gamma_k={h_k(1) \over h_{k+1}(1)}$. Then
\begin{equation} \label{gammak1}
V_{k,\gamma_k}(1)=h_{k+1}(1)
\end{equation}
and
\begin{equation} \label{gammak2}
V_{k,\gamma}(1)=\gamma^{-1} h_k(1) \geq h_{k+1}(1) \quad \text{ whenever } \gamma \leq \gamma_k.
\end{equation}
\end{obs}
\begin{obs}
$$V_{k,\gamma}(x) \leq V_{k+1,\gamma}(x)$$
for all $x \in [0,1]$ since $h_k(x) \leq h_{k+1}(x)$.
\end{obs}
\begin{obs}
Since $h_1(x) \equiv 1$ and $\gamma_1={h_1(1) \over h_2(1)}$,
$$h_2(x) \leq V_{1,\gamma_1}(x)=\begin{cases}
{1 \over 1-x} & \text{ for } 0 \leq x \leq 1-\gamma_1, \\
\gamma_1^{-1}=h_2(1) & \text{ for } 1-\gamma_1 \leq x \leq 1.
\end{cases} $$
\end{obs}
The above observation generalizes as follows.
\begin{prop}\label{drmota}
$$h_{k+1}(x) \leq V_{k,\gamma_k}(x)=\begin{cases}
{1 \over 1-x} & {\rm~for~} 0 \leq x \leq 1-\gamma_k, \\
\gamma_k^{-1} h_k\left({x-(1-\gamma_k) \over \gamma_k}\right)
& {\rm~ for ~} 1-\gamma_k \leq x \leq 1.
\end{cases}$$
\end{prop}
In order to prove Proposition \ref{drmota} we will need the following lemma.
\begin{lem}
\label{positivezero}
For any $\gamma \in (0,1)$ and $k \geq 1$, function $V_{k,\gamma}(x)-h_{k+1}(x)$ changes its sign at most once as $x$ increases from $1- \gamma$ to $1$. Moreover, since $V_{k,\gamma}(1-\gamma)=h(1-\gamma) > h_{k+1}(1-\gamma)$, function $V_{k,\gamma}(x)-h_{k+1}(x)$ can only change sign from nonnegative to negative.
\end{lem}
\begin{proof}
This is a proof by induction with base at $k=1$. Here $V_{1,\gamma}(x) ={1 \over \gamma}$ is constant on $[1-\gamma,1]$, while $h_2(x)=(1+e^{2x})/2$ is an increasing function, and
$$V_{1,\gamma}(1-\gamma)=h(1-\gamma)>h_2(1-\gamma)$$
For the induction step, we need to show that if $V_{k,\gamma}(x)-h_{k+1}(x)$ changes its sign at most once,
then so does $V_{k+1,\gamma}(x)-h_{k+2}(x)$.
Since both sequences of functions satisfy the same ODE relation (see Observation 1), we have
\noindent
${d \over dx}\left[(V_{k+1,\gamma}(x)-h_{k+2}(x))\cdot e^{-2\int\limits_{1-\gamma}^x h_{k+1}(y)dy} \right]$
$$\qquad \qquad =(2V_{k+1,\gamma}(x)-V_{k,\gamma}(x)-h_{k+1}(x))\cdot (V_{k,\gamma}(x)-h_{k+1}(x))\cdot e^{-2\int\limits_{1-\gamma}^x h_{k+1}(y)dy},$$
where $h_{k+1}(x) \leq V_{k+1,\gamma}(x)$ by definition of $V_{k+1,\gamma}(x)$, and $V_{k,\gamma}(x) \leq V_{k+1,\gamma}(x)$ as in Observation 3.
\vskip 0.2 in
\noindent
Now, let
$$I(x):=\int\limits_{1-\gamma}^x (2V_{k+1,\gamma}(s)-V_{k,\gamma}(s)-h_{k+1}(s))\cdot (V_{k,\gamma}(s)-h_{k+1}(s))\cdot e^{-2\int\limits_{1-\gamma}^s h_{k+1}(y)dy} ds.$$
Then
$$~(V_{k+1,\gamma}(x)-h_{k+2}(x))\cdot e^{-2\int\limits_{1-\gamma}^x h_{k+1}(y)dy}=V_{k+1,\gamma}(1-\gamma)-h_{k+2}(1-\gamma)+I(x).$$
\vskip 0.2 in
\noindent
The function $2V_{k+1,\gamma}(x)-V_{k,\gamma}(x)-h_{k+1}(x) \geq 0$, and since
$V_{k,\gamma}(x)-h_{k+1}(x)$ changes its sign at most once, then $I(x)$ should change its sign from nonnegative to negative at most once as $x$ increases from $1-\gamma$ to $1$. Hence
$$V_{k+1,\gamma}(x)-h_{k+2}(x)=(V_{k+1,\gamma}(1-\gamma)-h_{k+2}(1-\gamma)+I(x)) \cdot e^{2\int\limits_{1-\gamma}^x h_{k+1}(y)dy}$$
should change its sign from nonnegative to negative at most once as
$$V_{k+1,\gamma}(1-\gamma)=h(1-\gamma)>h_{k+2}(1-\gamma).$$
\end{proof}
\begin{proof}[Proof of Proposition \ref{drmota}]
Take $\gamma=\gamma_k$ in Lemma \ref{positivezero}. Then function $h_{k+1}(x)-V_{k,\gamma_k}(x)$
should change its sign from nonnegative to negative at most once within the interval $[1-\gamma_k,1]$.
Hence, $V_{k,\gamma_k}(1-\gamma_k) > h_{k+1}(1-\gamma_k)$ and $h_{k+1}(1) = V_{k,\gamma_k}(1)$ imply
$h_{k+1}(x) \leq V_{k,\gamma_k}(x)$ as in the statement of the proposition.
\end{proof}
Now we are ready to prove the monotonicity result.
\begin{lem} \label{gamma}
$$\gamma_k \leq \gamma_{k+1} \qquad \text{ for all } k \in \mathbb{N}.$$
\end{lem}
\begin{proof} We prove it by contradiction.
Suppose $\gamma_k \geq \gamma_{k+1}$ for some $k \in \mathbb{N}$.
Then
$$V_{k,\gamma_k}(x) \leq V_{k,\gamma_{k+1}}(x)=\begin{cases}
{1 \over 1-x} & \text{ for } 0 \leq x \leq 1-\gamma_{k+1}, \\
\gamma_{k+1}^{-1} h_k\left({x-(1-\gamma_{k+1}) \over \gamma_{k+1}}\right)
& \text{ for } 1-\gamma_{k+1} \leq x \leq 1
\end{cases}$$
and therefore
$$h_{k+1}(x) \leq V_{k,\gamma_k}(x) \leq V_{k,\gamma_{k+1}}(x) \leq V_{k+1,\gamma_{k+1}}(x)$$
as $h_{k+1}(x) \leq V_{k,\gamma_k}(x)$ by Proposition \ref{drmota}.
Recall that for $x \in [1-\gamma_{k+1},1]$,
$$V'_{k+1,\gamma_{k+1}}(x)=2V_{k,\gamma_{k+1}}(x)V_{k+1,\gamma_{k+1}}(x)-V_{k,\gamma_{k+1}}^2,$$
where at $1-\gamma_{k+1}$ we consider only the right-hand derivative.
Thus for $x \in [1-\gamma_{k+1},1]$,
$${d \over dx}\Big(V_{k+1,\gamma_{k+1}}(x)-h_{k+2}(x)\Big)=A(x)+B(x)\Big(V_{k+1,\gamma_{k+1}}(x)-h_{k+2}(x)\Big),$$
where $A(x)=2V_{k+1,\gamma_{k+1}}(x)-V_{k,\gamma_{k+1}}(x)-h_{k+1}(x) \geq 0$, $B(x)=2h_{k+1}(x) >0$, and $V_{k+1,\gamma_{k+1}}(1-\gamma_{k+1})-h_{k+2}(1-\gamma_{k+1})=h(1-\gamma_{k+1})-h_{k+2}(1-\gamma_{k+1})>0$.
Hence $$V_{k+1,\gamma_{k+1}}(1) - h_{k+2}(1) \geq V_{k+1,\gamma_{k+1}}(1-\gamma_{k+1})-h_{k+2}(1-\gamma_{k+1})>0$$
arriving to a contradiction since $V_{k+1,\gamma_{k+1}}(1) = h_{k+2}(1)$.
\end{proof}
\vskip 0.2 in
\noindent
\begin{cor}
Limit $\lim\limits_{k \rightarrow \infty} \gamma_k$ exists.
\end{cor}
\begin{proof}
Lemma \ref{gamma} implies $\gamma_k$ is a monotone increasing sequence, bounded by $1$.
\end{proof}
\vskip 0.1 in
\noindent
\begin{proof}[Proof of Lemma \ref{h1exist}]
Lemma \ref{h1exist} follows immediately from an observation
that ${h_{k+1}(1) \over h_k(1)}={1 \over \gamma_k}$.
\end{proof}
\section{Relation to the tree representation of white noise}
\label{white}
This section establishes a close connection between the combinatorial tree
of Kingman's $N$-coalescent
and the combinatorial level-set tree of a discrete white noise.
\subsection{Level set tree of a time series}
\label{level}
We start with recalling basic facts about tree representation of time series;
for details and further results see \cite{ZK12}.
Consider a finite time series $X_k$ with discrete time index $k=0,1,\dots,N$
and values distributed without singularities over $\mathbb{R}$.
Let $X_t$ (also denoted $X(t)$) be the time series with continuous time $t\in[0,N]$ obtained from
$X_k$ by linear interpolation of its values.
The level set $\mathcal{L}_{\alpha}\left(X_t\right)$ is defined
as the pre-image of the function values above $\alpha$:
\[\mathcal{L}_{\alpha}\left(X_t\right) = \{t\,:\,X_t\ge\alpha\}.\]
The level set $\mathcal{L}_{\alpha}$ for each $\alpha$ is
a union of non-overlapping intervals; we write
$|\mathcal{L}_{\alpha}|$ for their number.
Notice that
$|\mathcal{L}_{\alpha}| = |\mathcal{L}_{\beta}|$
as soon as the interval $[\alpha,\,\beta]$ does not contain a value of
local maxima or minima of $X_t$ and
$0\le |\mathcal{L}_{\alpha}| \le n$, where $n$ is the number
of the local maxima of $X_t$.
The {\it level set tree} $\textsc{level}(X_t)$ is a planar time-oriented
tree that describes the topology of the level sets $\mathcal{L}_{\alpha}$
as a function of threshold $\alpha$, as illustrated in Fig.~\ref{fig3}.
Namely, there are bijections between
(i) the leaves of $\textsc{level}(X_t)$ and the local
maxima of $X_t$,
(ii) the internal (parental) vertices of $\textsc{level}(X_t)$
and the local minima of $X_t$ (excluding possible local minima
at the boundary points), and
(iii) the pair of subtrees of $\textsc{level}(X_t)$ rooted at a local
minima $X(t_i)$ and the first positive excursions (or meanders bounded
by $t=0$ or $t=N$) of $X(t)-X(t_i)$ to right and left of $t_i$.
Each vertex in the tree is assigned a mark equal to the value
of the local extrema according to the bijections (i) and (ii) above.
This makes the tree time-oriented according to the threshold $\alpha$.
It is readily seen that any function $X_t$ with distinct
values of consecutive local minima corresponds to a binary tree
$\textsc{level}(X_t)$.
Please see \cite{ZK12} for a discussion of some subtleties
related to this construction as well as for further references.
\subsection{Finite case}
\label{finite}
Let $W^{(N)}_k$, $k=1,\dots,N-1$, be a {\it discrete white noise} that is
a discrete time series comprised of $N-1$ i.i.d. random variables
with a common continuous distribution.
Consider now an auxiliary time series $\tilde W^{(N)}_{i}$, $i=1,\dots,2N-1$ such that
it has exactly $N$ local maxima and $N-1$ internal local minima
$\tilde W^{(N)}_{2k}=W^{(N)}_k$, $k=1,\dots,N-1$.
We call $\tilde W^{(N)}_{i}$ an {\it extended white noise};
it can be constructed, for example, as follows:
\begin{equation}
\label{wnt}
\tilde W^{(N)}_{i}=
\left\{
\begin{array}{cc}
W^{(N)}_{i/2},& {\rm for~even~}i,\\
\max\left(W^{(N)}_{\max\left(1,\frac{i-1}{2}\right)},W^{(N)}_{\min\left(N-1,\frac{i+1}{2}\right)}\right)+1,&{\rm for~odd~}i.
\end{array}
\right.
\end{equation}
Let $L^{(N)}_W=\textsc{level}\left(\tilde W^{(N)}_k\right)$ be the level-set tree
of $\tilde W^{(N)}_k$ and
$\textsc{shape}\left(L^{(N)}_W\right)$ be a (random) combinatorial tree
that retains the graph-theoretic structure of
$L^{(N)}_W$ and drops its planar embedding as well as the
vertex marks.
By construction, $L^{(N)}_W$ has exactly $N$ leaves.
\begin{lem}
\label{any_wh}
The distribution of $\textsc{shape}\left(L^{(N)}_W\right)$ on ${\mathcal{T}}_N$
is the same for any continuous distribution $F$ of the values of the
associated white noise $W^{(N)}_k$.
\end{lem}
\begin{proof}
The continuity of $F$ is necessary to ensure that the
level set tree is binary with probability 1.
By construction, the combinatorial level set tree is completely determined by the
ordering of the local minima of the respective time series, independently of
the particular values of its local maxima and minima.
We complete the proof by noticing that the ordering of $W^{(N)}_k$ is the same for any choice of continuous distribution $F$.
\end{proof}
Let $T^{(N)}_{\rm K}$ be the tree that corresponds to a
Kingman's $N$-coalescent with a constant kernel, and let
$\textsc{shape}\left(T^{(N)}_{\rm K}\right)$ be its combinatorial version that
drops the time marks of the vertices.
Both the trees $\textsc{shape}\left(L^{(N)}_W\right)$ and
$\textsc{shape}\left(T^{(N)}_{\rm K}\right)$, belong to the space ${\mathcal{T}}_N$ of
binary rooted trees with $N$ leaves.
\begin{thm}\label{main2}
The trees $\textsc{shape}\left(L^{(N)}_W\right)$ and
$\textsc{shape}\left(T^{(N)}_{\rm K}\right)$ have the same distribution on ${\mathcal{T}}_N$.
\end{thm}
The proof below uses the duality between coalescence and fragmentation
processes \cite{Aldous}.
Recall that a {\it fragmentation process} starts with a single cluster of
mass $N$ at time $t=0$.
Each existing cluster of mass $m$ splits into two clusters
of masses $m-x$ and $x$ at the splitting rate $S_t(m,x)$, $1<m\le N$, $1\le x < N$.
A coalescence process on $N$ particles with time-dependent
collision kernel $K_t(x,y)$, $1\le x,y < N$ is equivalent, upon time
reversal, to a discrete-mass fragmentation process of initial mass
$N$ with some splitting kernel $S_t(m,x)$.
See Aldous \cite{Aldous} for further details and the relationship
between the dual collision and splitting kernels in general case.
\begin{proof}[Proof of Theorem~\ref{main2}]
We show that both the examined trees have the same distribution as
the combinatorial tree of a fragmentation process with mass $N$ and a
splitting kernel that is uniform in mass:
$S_t(m,x)=S(t).$
Kingman's $N$-coalescence with kernel $K(x,y)= 1$
is dual to the fragmentation process with splitting kernel \cite[Table 3]{Aldous}
\[S_t(m,x) = \frac{2}{t\,(t+2)}.\]
This kernel is independent of the cluster mass, which means
that the splitting of mass $m$ is uniform among the $m-1$ possible pairs $\{1,m-1\}$, \mbox{$~\{2,m-2\}$}, $\hdots,\{m-1,1\}$.
The time dependence of the kernel does not affect the combinatorial structure
of the fragmentation tree (and can be removed by a deterministic time change.)
The level-set tree $L^{(N)}_W$ can be viewed as a tree that describes
a fragmentation process with the initial mass $N$ equal
to the number of local maxima of the time series $\tilde W^{(N)}_k$.
By construction, each subtree of $L^{(N)}_W$ with $n$
leaves corresponds to an excursion (or meander, if we treat one of the
boundaries) with $n$ local maxima.
This subtree (as well as the corresponding excursion or meander) splits
into two by the internal global minimum of $\tilde W^{(N)}_k$ at
the corresponding time interval.
The global minimum splits the series $\tilde W^{(N)}_k$ into two,
to the left and right of the minimum, with $M_L$ and $(N-M_L)$ local maxima,
respectively.
Since the local minima of $\tilde W^{(N)}_k$ form a white noise, the
distribution of $M_L$ is uniform on $[1,N-1]$.
Next, the internal vertices of the level set tree of the left (or right) time
series correspond to its $M_L-1$ (or $N-M_L-1$) internal local minima that form
a white noise (with the distribution different from that of the initial
white noise $W^{(N)}_k$).
Hence, the subsequent splits of masses (number of local maxima) continues
according to a discrete uniform distribution.
And so on down the tree.
Hence, the combinatorial level set tree of $\tilde W^{(N)}_k$
has the same distribution as a combinatorial tree of a fragmentation
process with uniform mass splitting.
This completes the proof.
\end{proof}
\begin{Rem}
We notice that the dual splitting kernels for multiplicative and additive
coalescences \cite[Table 3]{Aldous} only differ by their time dependence,
and are equivalent as functions of mass.
Hence, the combinatorial structure of the respective trees is the same.
\end{Rem}
\subsection{Rooted trees with a selected leaf}
\label{sel_leaf}
To construct an infinite tree that represents Kingman's coalescent viewed
from a leaf, we need to introduce some notations.
Consider a space $\Gamma$ of (non-embedded, unlabeled)
{\it rooted binary trees with a selected leaf} $\gamma$.
For any tree $T\in\Gamma$, let $\gamma=\gamma_T$ be the selected leaf and $\rho_T=(\gamma_T,\phi_T)$ denote the ancestral path from the selected leaf $\gamma_T$ to its parent, grandparent, great-grandparent and on towards the tree root $\phi_T$, even if $\phi_T$ is a point at infinity.
The path $\rho_T$ consists of $n_T\le\infty$ edges and $n_T+1$ vertices that
we index by $i\ge 0$ along the path from the leaf $\gamma_T=\rho_T(0)$ to the root
$\phi_T=\rho(n_T)$.
Each tree $T\in\Gamma$ can be represented as a {\it forest attached to the line} $\rho$:
\begin{equation}
\label{forest}
T=\left\{T_i\right\}_{i\ge 1},
\end{equation}
where $T_i\in{\mathcal{T}}$ for $1\le i<n_T$ denotes a subtree of $T$ with the root at $\rho(i)$ and we put $T_i=\emptyset$ for all $i\ge n_T$.
A metric on $\Gamma$ is defined as
\begin{equation}
\label{mu}
\mu(A,B)=\frac{1}{1+\sup\left\{n: ~A_k|n=B_k|n ~~~\forall k\le n\right\}}
\end{equation}
for any $A=\left\{A_i\right\}\in\Gamma$ and $B=\left\{B_i\right\}\in\Gamma$ represented as in (\ref{forest}).
Here $T|n\in \overset{~2^{n-1}}{\underset{i=1}{\bigcup}} {\mathcal{T}}_i$ denotes
the restriction of $T\in{\mathcal{T}}$ to the vertices at the depth less than $n$
from the root.
To show that $\mu(A,B)$ is indeed a metric on $\Gamma$, take trees $A$, $B$, and $C$ in $\Gamma$
such that
$$\kappa=\sup\left\{n: ~A_k|n=B_k|n ~~~\forall k\le n\right\} ~\leq~ \sup\left\{n: ~B_k|n=C_k|n ~~~\forall k\le n\right\}.$$
Then
$$A_k|\kappa=B_k|\kappa=C_k|\kappa \qquad \forall k\le \kappa,$$
\\
and therefore $~\sup\left\{n: ~A_k|n=C_k|n ~~~\forall k\le n\right\} \geq \kappa$. Hence
$$\mu(A,C) ~\leq ~{1 \over 1+\kappa} ~= ~\mu(A,B) ~\leq ~\mu(A,B)+\mu(B,C)$$
and the corresponding equality holds only if $~B=C$. Also, $~\mu(A,B)=0~$ if and only if
$~\sup\left\{n: ~A_k|n=B_k|n ~~~\forall k\le n\right\}=0$, i.e. trees $A$ and $B$ are identical.
So $\mu$ is a metric on $\Gamma$ that compares trees in the neighborhood of the selected leaf
$\gamma$. We denote by $\Gamma_N$ a subspace of $\Gamma$ that contains all trees
with $N\ge 1$ leaves.
\begin{lem}
\label{Gamma}
$\left(\Gamma,\mu\right)$ is a compact Polish space.
\end{lem}
\begin{proof}
The countable dense subset $\Gamma_{<\infty}:=\underset{N \ge 1}{\bigcup}\Gamma_N$ of finite rooted trees with
a selected leaf makes $\Gamma$ a Polish space.
The compactness is readily shown by selecting a subsequence $T^{n_i}$ of an
infinite sequence $T^n=\left\{T^n_i\right\}_{i\ge 1}$ of trees such that $\mu(T^{n_i},A)<i^{-1}$
for some infinite $A\in T^n$, which is always possible since for any $i \in \mathbb{N}$ the number of binary trees of depth $\leq i$ is finite. Thus there is a collection of $i$ binary trees $S_1,\hdots,S_i$, each of depth $\leq i$, such that
$$T^n_k|i=S_k \qquad \forall k\in \{1,2,\hdots,i\}$$
holds for infinitely many $T^n$ of which we select one $T^{n_i}$, and use the rest for selecting $T^{n_{i+1}},T^{n_{i+2}},\hdots$. The resulting limiting tree $A$ of $T^{n_i}$ will satisfy
$$A_k|i=S_k \qquad \forall k\in \{1,2,\hdots,i\}.$$
\end{proof}
\subsection{Kingman's coalescent with a selected particle}
Recall that Kingman's coalescent is constructed in \cite{Kingman82b} as a continuous time Markov process over the set $\mathcal{P}_{\mathbb{N}}$ of all partitions of $\mathbb{N}=\{1,2,\hdots\}$. Here we will use the following special property of Kingman's coalescent: it can be restricted consistently to the set $\mathcal{P}_{[N]}$ of all partitions of $\{1,2,\hdots,n\}$. These restrictions correspond to Kingman's $N$-coalescent processes $\Pi^{(N)}_t$ (e.g., \cite{Kingman82b,Pitman}).
In the above construction let $1$ be the {\it selected particle}, and the block containing $1$ at time $t$ be the {\it selected block}.
When constructing a combinatorial tree for Kingman's $N$-coalescent process, each merger history of $\Pi^{(N)}_t$ corresponds to a tree $T$ from $\Gamma_N$ with the selected leaf $\gamma=\gamma_T$ corresponding to the selected particle $1$ and $\rho_T$ corresponding to the merger history of the selected block. Hence the process induces a distribution on $\Gamma_N$, which we denote ${\mathcal{Q}}^*_N$. The sequence of measures $\left({\mathcal{Q}}^*_N\right)_{N\in\mathbb{N}}$ is {\it tight} on
$\Gamma$ due to the space compactness (Lemma~\ref{Gamma}).
The Prokhorov's theorem implies that this sequence is then {\it relatively compact},
which means that it includes converging subsequences.
By construction, each limit measure ${\mathcal{Q}}^*$ is concentrated on infinite
trees $\Gamma_{\infty}=\Gamma\setminus\Gamma_{<\infty}$.
The uniqueness of the limit follows from the fact that all finite-dimensional
projections of ${\mathcal{Q}}^*$ are uniquely defined by a consistent set of ${\mathcal{Q}}^*_N$. This corresponds to being able to restrict Kingamn's coalescent consistently to $\mathcal{P}_{[N]}$.
We denote the unique limit measure by ${\mathcal{Q}}^{*}$.
It is natural to interpret the space $\left(\Gamma_{\infty},\mu,{\mathcal{Q}}^{*}\right)$ as
a (random) combinatorial Kingman's infinite tree viewed from a leaf.
We emphasize the two features that distinguish our Kingman's tree construction
from the others that exist in the literature (e.g., that of Evans \cite{Evans00}):
(i) only considering the combinatorial part of a tree and
(ii) focusing attention on the vicinity of a selected leaf.
For the extended white noise
an infinite tree can be constructed explicitly as a strong
limit of finite trees $L^{(N)}_W$, as shown in the next section.
\subsection{Infinite case}
\label{infinite}
We describe below two equivalent constructions of an infinite tree
for a discrete time series with infinite number of local maxima.
The first construction involves limit of finite trees in the space
$\left(\Gamma,\mu\right)$; the second introduces an infinite tree as
a random metric space.
\subsubsection{Construction 1: Limit of finite trees}
An infinite level set tree $L^{\infty}_W$ that represents a time series
$W^{\infty}_k$ with an infinite number of local maxima can be constructed
as follows.
Fix a trajectory $W(\omega)$ of $W^{\infty}_k$.
Choose a local maximum of the $W(\omega)$ closest to $k=0$
and assign it index $i=0$.
Now index all the local maxima $W_{(i)}$ and the respective time instants
$t_{(i)}$ in the order of their appearance to the
right or left of this chosen maximum by an integer index $i$, so that
the first local maximum to the right of $i=0$ is assigned index $i=1$,
the first local maximum to the left $i=-1$, etc.
The trajectory on the interval $\left[t_{(i_1)},t_{(i_2)}\right]$ corresponds
to a level-set tree $L^{[i_1,i_2]}_W(\omega)$ with $(i_2-i_1+1)$ leaves.
Notably, if $[j_1,j_2]\subset[i_1,i_2]$ then $[t_{(j_1)},t_{(j_2)}]\subset[t_{(i_1)},t_{(i_2)}]$ and
the level set tree $L^{[j_1,j_2]}_W(\omega)$ is a subtree of
$L^{[i_1,i_2]}_W(\omega)$.
Consider the trajectory $W(\omega)$ on the
interval $[t_{(-i)},t_{(i)}]$ and take the leaf that corresponds to $W_{(0)}$
as the {\it selected leaf} of the tree $L^{[-i,i]}_W(\omega)$, which hence
becomes an element of $\left(\Gamma_{2i+1},\mu\right)$ as defined in Sect.~\ref{sel_leaf}.
We define $L^{\infty}_W(\omega)$ as the limit of
$L^{[-i,i]}_W(\omega)$ as $N\to\infty$ and, in particular,
\[\textsc{shape}\left(L^\infty_W(\omega)\right):=\lim_{i\to\infty}
\textsc{shape}\left(L^{[-i,i]}_W(\omega)\right).\]
The sequence of increasing trees $\textsc{shape}\left(L^{[-i,i]}_W(\omega)\right)$,
$i>0$, converges according to
measure $\mu$ of $\eqref{mu}$ that is concentrated in the vicinity of the
selected leaf, so the limit above is well defined.
\subsubsection{Construction 2: Random metric space}
The limit tree $L^\infty_W(\omega)$ is in fact the {\it tree in continuous path}
of the function $W(t)$ obtained by linear interpolation of the values $W_k$
on $\mathbb{R}$.
Specifically, recall the following definition \cite[Section 7]{Pitman}.
\begin{Def}
\label{tree}
A metric space $(M,d)$ is called a {\it tree} if for each choice of $u,v\in M$
there is a unique continuous path $\sigma_{u,v}:[0,d(u,v)]\to M$ that travels
from $u$ to $v$ at unit speed, and for any simple continuous path $F:[0,T]\to M$
with $f(0)=u$ and $f(T)=v$ the ranges of $f$ and $\sigma_{u,v}$ coincide.
\end{Def}
Let $X(t)\in C\left(I\right)$, the space of continuous
functions from $I\subset\mathbb{R}$ to $\mathbb{R}$,
and
$\underline{X}[a,b]:=\inf_{t\in[a,b]} X(t)$, for any
$a,b\in\,I$.
We define a {\it pseudo-metric} on $I$ as
\begin{equation}
\label{tree_dist}
d_X(a,b):=\left(X(a)-\underline{X}[a,b]\right)
+ \left(X(b)-\underline{X}[a,b]\right),\quad a,b\in\,I.
\end{equation}
We write $a\sim_X b$ if $d_X(a,b)=0$.
The points on the interval $I$ with metric $d_X$
form a metric space $\left(I/\sim_X,d_X\right)$
\cite[Section 7]{Pitman}.
\begin{Def}
\label{ftree}
We call $\left(I/\sim_W,d_W\right)$ the {\it tree embedded in the
continuous path} $W(t)$ on the interval $I$ and denote it by
$\textsc{tree}_W(I)$.
If $W_{(0)}\in I$ and is attained at $t_{(0)}$ then this point is taken to be
the selected leaf of $\textsc{tree}_W(I)$.
\end{Def}
It is readily seen that we have the following equivalence for any $i>0$:
\[L^{[-i,i]}_W(\omega)=\textsc{tree}_W\left(\left[t_{(-i)}(\omega),t_{(i)}(\omega)\right]\right)\]
and
$L^\infty_W(\omega)=\textsc{tree}_W(\mathbb{R},\omega)$, where the second argument
emphasizes that this tree corresponds to a particular trajectory $W(\omega)$.
A probability measure is induced on the space of the trees in continuos path with
a selected leaf by the probability measure on the space of the time series trajectories.
In particular, there exists a probability measure ${\mathcal{Q}}^{*}_{\rm WN}$ on $\left(\Gamma,\mu\right)$
that corresponds to a (random) combinatorial tree of an infinite extended white noise
viewed from a leaf.
\begin{cor}
\label{KWN}
${\mathcal{Q}}^{*}_{\rm WN}(A)={\mathcal{Q}}^*(A)$ for any $A\in\left(\Gamma,\mu\right)$.
\end{cor}
\begin{proof}
The statement follows immediately from the equivalence of the finite-dimensional
distributions generated by an extended white noise and Kingman's $N$-coalescent processes on
$\left(\Gamma_N,\mu\right)$, as in Theorem~\ref{main2}.
\end{proof}
Now we can extend the definition of branch statistics relevant to Horton analyzes
to the infinite trees of Kingman's coalescent and discrete white noise.
Let $B_{k,i}$, $k,i\ge1$ denotes the number of
branches of Horton-Strahler order $k$ in
the tree that corresponds to the interval of an infinite extended white noise
between $W_{(-i)}$ and $W_{(i)}$.
By Theorem~\ref{main2}, it has the same distribution as the number
of branches of order $k$ in a combinatorial tree of Kingman's $N$-coalescent
process with a constant kernel and $2i+1$ particles.
Lemma~\ref{lem1} implies that the following limit exists (in probability)
and is finite
\[\lim_{i\to\infty}\frac{B_{k,i}}{(2i+1)}={\mathcal{N}}_k.\]
This suggests an intuitive interpretation of the
asymptotic ratios ${\mathcal{N}}_k$ as the {\it Horton indices} for an infinite
tree of Kingman's coalescent or extended white noise and proves
the following result.
\begin{thm}
\label{main3}
The random combinatorial infinite trees $\left(\Gamma_{\infty},\mu,{\mathcal{Q}}^{*}\right)$
and $\left(\Gamma_{\infty},\mu,{\mathcal{Q}}^{*}_{\rm WN}\right)$ are root-Horton self-similar.
\end{thm}
\begin{Rem}
The extended white noise was introduced to show the equivalence of the finite
dimensional distributions in Theorem~\ref{main2}.
At the same time, the statement of Theorem~\ref{main3} (Horton self-similarity)
applies as well to the infinite combinatorial tree of a discrete white noise
with a continuous distribution of the values.
This is easily seen if one recalls the operation of tree {\it pruning}
${\mathcal{R}}(T):{\mathcal{T}}\to{\mathcal{T}}$ that cuts the tree leaves and removes possible resulting
nodes of degree 2 \cite{BWW00,ZK12}.
By definition, pruning corresponds to index shift in Horton statistics:
$N_k\to N_{k-1}$, $k>1$.
It has been shown in \cite{ZK12} that
\[{\mathcal{R}}\left[\textsc{level}\left(\tilde W^{(N)}_k\right) \right]=
\textsc{level}\left(W^{(N)}_k\right).\]
Hence, Horton self-similarity for one of these time series implies that
for the other.
\end{Rem}
\section{Numerical results}
\label{numerics}
This section illustrates the theoretical results of the paper and
provides further insight into self-similarity of Kingman's coalescent
in a series of numerical experiments.
\subsection{Horton self-similarity}
Figure~\ref{fig_Horton} shows (by shaded circles) the asymptotic
proportion ${\mathcal{N}}_k$ according to the equation
\begin{equation}
\label{Horton_h}
{\mathcal{N}}_k = \int_0^1 \left(1-\frac{h_{k-1}(x)}{h(x)}\right)^2 dx.
\end{equation}
The integral is evaluated using the numeric solution
$\hat h_k(x)$ to the system \eqref{ODEh}.
The computations are done in Matlab, using the nominal absolute tolerance
of $\epsilon=10^{-16}$ on an irregular time grid with steps decreasing
towards 1 and being as small as $\Delta=10^{-10}$.
The values of the estimated ${\mathcal{N}}_k$ as well as the consecutive
ratios ${\mathcal{N}}_{k}/{\mathcal{N}}_{k+1}$ are reported in Table 1 with 7 digit precision.
We notice that the values in the second column differ from the ratios
of the values in the first column, because of the round-off effects.
The values of ${\mathcal{N}}_k$ are exponentially decreasing in such a way
that the ratio ${\mathcal{N}}_{k}/{\mathcal{N}}_{k+1}$ quickly converges to $R=3.043827\dots$.
This ratio corresponds to the geometric decay
\[{\mathcal{N}}_k = (3.043827\dots)^{-(k-1)}
= 10^{(-0.4834\dots)\,(k-1)}.\]
A black line in Fig.~\ref{fig_Horton} illustrates a geometric decay with this rate
and an arbitrary offset.
The observed convergence of ratios is stronger than the root-convergence
proven in our Theorem~\ref{main}.
We conjecture that in fact the strongest possible geometric-Horton law
also holds for the sequence ${\mathcal{N}}_k$, that is
$ \lim\limits_{k \rightarrow \infty}
\left({\mathcal{N}}_k \, R^k\right) = N_0>0$ for the same Horton exponent $R$ as above.
Table 1 also reports (column 4) the average number $N_k$ of branches
of order $k$ relative to $N_1$ observed in a level-set tree of an
extended white noise with $N=2^{18}=262,144$ local maxima, taken
over 1000 independent realizations.
The typical order of such a tree is $\Omega=11$.
The values are reported with 4 significant digits.
The agreement between the asymptotic statistics for
Kingman's coalescent and those for a finite extended white noise
is very good, in accordance with our equivalence Theorem~\ref{main2}.
Column 5 of Table 1 shows the empirical coefficient of variation
$\rho(N_k)=\sqrt{{\sf Var}(N_k)}/{\sf E}(N_k)$
for the random variables $N_k$.
The values of $\rho(N_k)$
rapidly decrease with the order difference $\Omega-k$; this implies that
the branch statistics for an individual level set tree are very close to
the asymptotic Kingman's values, at least for small orders $k$.
This is illustrated in Fig.~\ref{fig_Horton} that shows the values
of $N_k/N_1$ computed in a single realization of an extended
white noise.
It should be emphasized that the coefficient of variation $\rho(N_k)$
depends on $N$ and decreases for each $k$ as $N$ increases (not illustrated).
\begin{center}
Table 1: Statistics of order-$k$ branches for Kingman's coalescence\\
(columns 2, 3) and a finite extended white noise (columns 4, 5).
\vspace{.2cm}
\begin{tabular}{c|c|c|r|c}
\hline\hline
$k$ & ${\mathcal{N}}_k$ & ${\mathcal{N}}_{k}/{\mathcal{N}}_{k+1}$ & $N_k/N_1$ & $\rho(N_k)$\\
\hline
1 & 1.0000000 & 3.0000000 & 1.0000 &0.001\\
2 & 0.3333333 & 3.0389538 & 0.3333 &0.002\\
3 & 0.1096869 & 3.0432806 & 0.1097 &0.005\\
4 & 0.0360423 & 3.0437674 & 0.0360 &0.008\\
5 & 0.0118413 & 3.0438212 & 0.01183 &0.014\\
6 & 0.0038903 & 3.0438271 & 0.003885 &0.026\\
7 & 0.0012781 & 3.0438277 & 0.001273 &0.044\\
8 & 0.0004200 & 3.0438278 & 0.0004156 &0.074\\
9 & 0.0001380 & 3.0438279 & 0.0001342 &0.142\\
10 & 0.0000453 & 3.0438279 & 0.00004148 &0.311\\
11 & 0.0000149 & - & 0.00001105 &1.591\\
\hline
\hline
\end{tabular}
\end{center}
\subsection{Tokunaga self-similarity}
\label{Tokunaga}
The {\it Tokunaga self-similarity} for trees is based on Tokunaga
indexing \cite{NTG97,Tok78,Pec95}, which extends upon Horton-Strahler
orders (see Fig.~\ref{fig_HST}b).
This indexing focuses on {\it side-branching}, which is the
merging between branches of different order.
Let $N_{ij}$, $1\le i<j\le\Omega$, be the total number
of branches of order $i$ that join branch of order $j$ in a finite tree $T$
of order $\Omega$.
Tokunaga index $\tau_{ij}= N_{ij}/N_{j}$ is the average number
of branches of order $i<j$ per branch of order $j$.
By $\tau^{({\mathcal{Q}}_N)}_{ij}$ we denote the (random) index $\tau_{ij}$
computed for a tree generated according to a measure ${\mathcal{Q}}_N$ on ${\mathcal{T}}_N$.
\begin{Def}
We say that a sequence of probability laws $\{ {\mathcal{Q}}_N\}_{N \in \mathbb{N}}$
has {\it well-defined asymptotic Tokunaga indices} if for each $k \in \mathbb{N}$
random variables $\tau_{ij}^{( {\mathcal{Q}}_N)}$ converge in probability,
as $N \rightarrow \infty$, to a constant value $T_{ij}$,
called the {\it asymptotic Tokunaga index}.
\end{Def}
\begin{Def}
A sequence of measures $\{ {\mathcal{Q}}_N\}_{N \in \mathbb{N}}$
with well-defined Tokunaga indices is said to be
{\it Tokunaga self-similar} with parameters $(a,c)$ if
\[T_{ij}=:T_k=a\,c^{k-1}\quad a,c > 0,~1\le k\le\Omega-1.\]
\end{Def}
To write out the equations for Tokunaga indexes $T_{ij}$ in
Kingman's coalescent, we
observe that the asymptotic ratio of $N_{ij}$ to $N$
is given by the original ODE (\ref{Aeta}) as
$${\mathcal{N}}_{ij} =\int\limits_0^{\infty} \eta_i(t) \eta_j(t) dt
= \int\limits_0^{\infty}\left(g_i(x)-g_{i+1}(x)\right)\left(g_j(x)-g_{j+1}(x)\right)\,dx.$$
This can be rewritten using the rescaled equations in (\ref{ODEh}) as
\[{\mathcal{N}}_{i-1,j-1} = 2\int_0^1 \frac{(h_i(x)-h_{i+1}(x))(h_j(x)-h_{j+1}(x))}{h^2(x)}\,dx.\]
We now use \eqref{Horton_h} to obtain
\begin{eqnarray}
\label{Tok_h}
\lefteqn{\lim_{N\to\infty} T_{i-1,j-1}
= \lim_{N\to\infty}\frac{N_{i-1,j-1}}{N_{j-1}}
= \frac{{\mathcal{N}}_{i-1,j-1}}{{\mathcal{N}}_{j-1}}}\nonumber\\
&=& 2 \, \frac{\int\limits_0^1\frac{(h_i(x)-h_{i+1}(x))(h_j(x)-h_{j+1}(x))}{h^2(x)}\,dx}
{\int\limits_0^1 \left(1-\frac{h_j(x)}{h(x)}\right)^2\,dx}.
\end{eqnarray}
Table 2 reports Tokunaga indices evaluated numerically using \eqref{Tok_h}.
The integrals and functions $h_k(x)$ are evaluated in Matlab with nominal absolute
tolerance $\epsilon = 10^{-20}$ and using the same grid as in computing the
Horton statistics.
The values are reported here with 4 digits precision.
We notice that evaluation of Tokunaga indices for larger ${ij}$ pairs faces
numerical problems because of divergence of $h(x)$, and the associated
``bursts'' of $h_k(x)$, at unity.
The reported values $T_{i,i+k}$ converge to the limit $T_k$
as $i$ increases.
The convergence rate is very fast; the limit value,
within the reported precision, is achieved for $i\ge 4$ or faster.
The reported simulations suggest that the convergence rate increases
with $k$.
\begin{center}
Table 2: Asymptotic Tokunaga indices $T_{ij}$ for Kingman's coalescence
\vspace{.2cm}
\begin{tabular}{ccccccccc}
\hline\hline
& $j=2$ & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
$i=1$ & 0.8196 & 0.5687 & 0.2641 & 0.0993 & 0.0342 & 0.0114 & 0.0038 & 0.0012 \\
2 & & 0.8234 & 0.5720 & 0.2655 & 0.0999 & 0.0344 & 0.0115 & 0.0038 \\
3 & & & 0.8232 & 0.5724 & 0.2657 & 0.0999 & 0.0344 & 0.0115 \\
4 & & & & 0.8231 & 0.5724 & 0.2657 & 0.0999 & 0.0344 \\
5 & & & & & 0.8231 & 0.5724 & 0.2657 & 0.0999 \\
6 & & & & & & 0.8231 & 0.5724 & 0.2657 \\
7 & & & & & & & 0.8231 & 0.5724 \\
8 & & & & & & & & 0.8231 \\
\hline
\hline
\end{tabular}
\end{center}
Figure~\ref{fig_Tokunaga} shows by shaded circles the ``limit'' Tokunaga indices $T_{i\,9}$,
$i=1,\dots,8$, as a function of $k=9-i$.
The figure may suggest that Tokunaga indices form a geometric series,
asymptotic in $k$.
Summing up, we conjecture that
\[
\lim_{i\to\infty}T_{i,i+k} = :T_k\quad{\rm and}\quad
\lim_{k\to\infty}\frac{T_{k}}{c^{k-1}} = a\]
with $a\approx 3\pm 0.02$, $c\approx -0.5\pm 0.02$ obtained by
fitting a geometric series to the last couple of ``limit'' Tokunaga
indices.
We notice that the existing literature only considers the conventional
Tokunaga self-similarity that assumes the geometric form of $T_k$
starting from $k=1$.
We also show in Fig.~\ref{fig_Tokunaga}, by black squares, Tokunaga
indices $\hat T_k$ estimated from the level-set trees of a finite extended white noises.
The values $\hat T_k$ are obtained by averaging the empirical
Tokunaga indices in 100 trees of size $N=2^{17}=131,072$; the typical
order of such trees is $\Omega=11$.
The indices are averaged over different trees and over all
pairs of subindices $\{i,i+k\}$ with $i\ne 1$.
The empirical values match very closely the asymptotic values from
Kingman's process, in accordance with our equivalence Theorem~\ref{main2}.
\section{General coalescent processes}
\label{general}
The ODE approach introduced in this paper can be extended to the
coalescent kernels other than $K(i,j) \equiv 1$.
For that we need to classify the relative number $\eta_j(t)$ of clusters of
order $j$ at time $t$ according to the cluster masses.
Namely, let $\eta_{j,k}(t)$ be the average number of clusters of order $j$
and mass $k \geq 2^j$ at time $t$.
Then
$$\eta_j(t)=\sum\limits_{k=2^j}^{\infty} \eta_{j,k}(t).$$
In the case of a symmetric coalescent
kernel $K(i,j)=K(j,i)$ the Smoluchowski-Horton ODEs can be written asymptotically as
\begin{eqnarray}\label{ODEker}
{d \over dt}\eta_{j,k}(t) & = &\sum\limits_{i=1}^{j-1} \sum\limits_{\kappa=2^j}^{k-2^i} \eta_{j,\kappa}(t)\eta_{i,k-\kappa} K(\kappa, k-\kappa)\\ \nonumber
& + & {1 \over 2} \sum_{\substack{k_1+k_2=k \\ k_1,k_2 \geq 2^{j-1}}} \eta_{j-1,k_1}(t) \eta_{j-1,k_2}(t) K(k_1,k_2)\\ \nonumber
& - & \eta_{j,k}(t)\sum\limits_{\widetilde{k}=2^i}^{\infty} K(k,\widetilde{k}) \left(\sum\limits_{i=1}^{\infty} \eta_{i, \widetilde{k}}(t)\right)
\end{eqnarray}
with the initial conditions $\eta_{1,1}(0)=1~$ and $~\eta_{j,k}(0)=0$ for all $(j,k) \not=(1,1)$.
Observe that when $K(i,j) \equiv 1$, summing the above equations (\ref{ODEker}) over index
$k$ produces the Smoluchowski-Horton ODE (\ref{Aeta}) for the average relative number of
order-$j$ branches $\eta_j(t)$ in Kingman's coalescent process.
\section{Discussion}
\label{discussion}
This paper establishes the root-Horton self-similarity (Theorem~\ref{main}) and
states a numerical conjecture about the asymptotic Tokunaga self-similarity
(Sect.~\ref{numerics}) for
Kingman's coalescent process.
We also demonstrate (Theorem~\ref{main2}) the distributional equivalence of the
combinatorial trees of Kingman's $N$-coalescent
process with a constant collision kernel to that of a discrete
extended white noise with $N$ local maxima, hence extending the
self-similarity results to a tree representation of an infinite white noise
(Sect.~\ref{infinite}, Remark 2).
Our Theorem~\ref{main} establishes a weak root-law convergence of the
asymptotic ratios ${\mathcal{N}}_k$,
while we believe that the stronger (ratio and geometric) forms of convergence
are also valid.
These stronger Horton laws are usually considered in
the literature (e.g., \cite{Pec95,Horton45,DR00,ZK12}).
For instance, a well-known result is that a tree corresponding
to a critical binary Galton-Watson process obey the
geometric Horton law with $R=4$; see
\cite{Shreve66,YM94,Pec95,DK94,BWW00}.
It seems important to show rigorously at least the ratio-Horton law
($\lim\limits_{k \rightarrow \infty}
{\mathcal{N}}_k/{\mathcal{N}}_{k+1}=R>0$)
because the ability to work with asymptotic ratios is necessary to tackle
Tokunaga self-similarity, which provides much stricter constraints on
a branching structure.
The Smoluchowski-Horton equations \eqref{Aeta} that form a core of the
presented method and their equivalents \eqref{odeG} and \eqref{ODEh}
seem to be promising for further more detailed exploration.
Indeed, one may hope that the approach that refers explicitly to the
Horton-Strahler orders might effectively complement conventional analysis
of cluster masses.
The analysis of the Smoluchowski-Horton systems can be done within
the ODE framework, similarly to the present study, or within the
nonlinear iterative system framework (see \eqref{iter}).
The latter approach is still to be explored.
Finally, it is noteworthy that the analysis of multiplicative and additive
coalescents according to the general Smoluchowski-Horton
system~\eqref{ODEker} appears, after a certain series of transformations,
to follow many of the steps implemented in this paper for
Kingman's coalescent, with the ODE system being replaced by a
suitable PDE one.
These results will be published elsewhere.
\bibliographystyle{amsplain}
|
1,116,691,500,813 | arxiv |
\subsection{Semi-supervised Learning}
Semi-supervised learning (SSL) aims to improve generalization on supervised tasks using unlabled data. In the semi-supervised learning framework, a given set of $N = L+U$ ($U >> L$ usually) data can be divided into two parts: historical data $X_{l} = (x_{1},..,x_{l})$ for which corresponding labels $Y_{l} = (y_{1},..,y_{l})$ exist and $Xu = (x_{l+1},..,x_{l+u})$, for which corresponding labels do not exist. The goal of SSL is to improve prediction accuracy of a function learned from $(X_{l}, Y_{l})$, i.e., by taking into account the unlabeled data $X_{u}$. There are several assumptions that a number of SSL theories are based on. These are are smoothness, cluster, and manifold assumptions, each of which is related to the structure of the underlying distribution. While these assumptions are of great theoretical interest and may provide intuitions into SSL, we will not go into further details as our intention is not to uncover theoretical maximum performance gain that SSL approach can bring per se.
There exist numerous algorithms in semi-supervised learning [REF]. Probably the simplest of all is self-learing, which starts by building a base learner from small labeled samples. It then trains again with the labels obtained from the (best) prediction of the base learner, i.e., bootstrapping. This process is repeated until some condition is met. While It is known it often works well in practice [REF/NLP], it is prone to error because prediction mistake can be reinforced through repetitions. Performance gain is thus highly dependent on base learner's prediction confidence (accuracy) [REF].
Another popular set of approaches is based on well-studied probabilistic framework, i..e, generative models. Generative models try to model the conditional density $p(x|y)$ and make a prediction by inferring $p(y|x)$ from Bayes theorem.
It is different from discriminative models in that generative models estimate the density of $x$ as an intermediate step. While there is different techniques such as Gaussian mixture (GMM), naive Bayes, and hidden markov models (HMM), our focus in on one of the generative models with deep architecture (DGM), known as variational autoencoder (VAE). We will give detailed explanation of VAE in the following section.
\subsection{Generative Models}
Considering input variable $x$ and target variable $y$, a discriminative model focuses on $p(y|x)$, whereas a generative mode focuses on $P(x,y)$, which is often parameterized as $p(x|y)p(y)$ [REF-erhan]. Generative models have been recognized less prone to overfitting than discriminative models [REF - erhan, Ng andJordan]. Among different classes of generative models, those models with deep architectures composed of several layers of parameterized non-linear modules are called deep generative models (DGM). Approaches based on neural networks involve unsupervised embedding of the data either as a regularizer or as pre-trained layers for supervised learning [REF: KINGMA].
\subsection{Unsupervised Embedding}
Embedding is a procedure to map $x$ in the original space into a representation, where mapping is obtained from unsupervised (or semi-supervised) learning. Unsupervised embedding is used in SSL by exploiting information about $p(x)$ with a goal improve generalization of a supervised model (learner) [REF - ERHAN page 630]. Projecting into a representation found by unsupervised learning of Principal Component Analysis (PCA) is probably the most commonly used example of an embedding. However, PCA is limited in its capacity as it is linear model whilst many applications (dynamics) require non-linear modeling. The recent applications of DGM in unsupervised embedding have emerged as an exciting direction of SSL as they show state-of-the-art performance (shown promise) in various (unsupervised and) semi-supervised learning tasks.
\subsection{Overview}\label{ourapproach}
In this section, we describe our overall approach for semi-supervised learning for RUL prediction
using neural network framework and a deep generative model described in Section~\ref{dnn_overview} and~\ref{vae}. We provide a description of the implemented architectures for RUL prediction and several important architectural decisions on design parameters in Section~\ref{exp}.
\subsubsection{Non-linear Embedding}
Embedding in semi-supervised learning is a procedure to map $\vec{x}$ in the original space into a lower dimensional space, $\vec{z}$, with the mapping obtained from unsupervised learning. The goal of embedding is to improve generalization of a supervised model. In our approach, we use the VAE to obtain the non-linear embedding. It is done by training the VAE as described in Section~\ref{vae} using $\vec{x}$ from both labeled $\mathcal{D}_{L}$ and unlabeled data $\mathcal{D}_{U}$ to learn a mapping function $f_{embed}$ for latent variable $\vec{z}$. The latent space is defined by the output layer chosen in the encoder of VAE as illustrated in Fig.~\ref{fig:embedding}(a). We choose $k^{th}$ layer of the encoder with $k$ less than $L_{encoder}$ (a total number of layers used in the encoder) to avoid using approximate samples from posterior distribution.
While for the parameterization of the approximate posterior in VAE, feed-forward neural network is conventionally chosen as originally suggested in~\cite{vae2013}. In our case, we choose the RNN-architecture for the parameterization to model the temporal dependency of data. Incorporating the RNN architecture into VAE to include the latent random variables into the hidden state of a RNN requires the estimation of timestep-wise variational lower bound, which is not trivial. We instead simply introduce a weight factor $\alpha$ such that the two terms in Eq.~\ref{eqn:lb2} is differently weighted while modeling the temporal dependence in data with the RNN architecture.
\begin{equation}\label{eqn:lb3}
-D_{KL}(q_{\phi}(\vec{z}|\vec{x}^{(i)})||p_{\theta}(\vec{z})) + \alpha E_{q_{\phi}(\vec{z}|\textbf{x}^{(i)})}\big[\log p_{\theta}(\vec{x}^{(i)}|\vec{z}) \big]
\end{equation}
The weight factor is tuned considering the overall reconstruction efficiency of VAE.
\subsubsection{Reliability Model}
After obtaining the non-linear embedding from the VAE, we train a model (\emph{reliability model}) with the RNN architecture in Eq.~\ref{eqn:rnn_model} using $(\vec{z}^{(i)}, y^{i})$ where $\vec{z}^{(i)}=f_{embed}(\vec{x}^{i})$ and $\vec{x}^{i} \in \mathcal{D}_{L}$. The resulting reliability model is used to predict RUL for a given set of historical CM data. It is noted the resulting model is similar to the \emph{latent feature discriminative model} (M1), which shows the state-of-the-art performance in various classification tasks~\cite{dgm2014}. The biggest difference lies in the way the non-linear embedding is obtained. The M1 model uses approximate samples from the posterior distribution as an embedding. In our case we rather use the $k^{th}$ layer output as an embedding as we want to avoid using the approximate sampling in order to achieve high precision RUL prediction\footnote{We tried with the approximate samples from the posterior distribution as features to train the reliability model and found the prediction precision is lowered because of the stochastic nature of sampling.}. The corresponding procedure is illustrated in the architectures of the embedding network and the reliability model in Fig.~\ref{fig:embedding}(a) and \ref{fig:embedding}(b).
The entire procedure of our approach from generative modeling to supervised learning of reliability model is given in Algorithm~\ref{sslalgo}, where the mini-batch $\vec{x}^{\textit{M}} = \{\vec{x}^{(i)}\}_{i=1}^{\textit{M}}$ is a randomly drawn sample of size $M$. The $\mathcal{L}$ indicates a corresponding loss function with $\widetilde{\mathcal{L}}$ implying that it is calculated for the data points in mini-batch as an approximation to the quantity for the entire data points. For the VAE, for example,
\begin{equation}\label{eqn:mbloss}
\mathcal{L}(\theta,\phi; \vec{x}) \simeq \widetilde{\mathcal{L}}(\theta,\phi; \vec{x}^{M}) = \frac{n}{M}\sum_{i=1}^{n} \mathcal{L}(\theta,\phi; \vec{x}^{(i)})
\end{equation}
\subsubsection{Self-learning Approach}
Self-learning also known as self-labeling is an iterative algorithm that uses supervised learning in repetition~\cite{ssl2010}. As the name suggest, the algorithm tries to ``bootstrap'' using its own prediction. We use the simplest version of self-learning algorithm in order to see if it can help the generalization of supervised learning for the reliability prediction. It is done by first building a reliability model by training a RNN model using the labeled data, $(\vec{x}^{(i)}, y^{i}) \in \mathcal{D}_{L}$. The resulting model, so called \emph{base learner}, is then used to make RUL estimations for the un-labeled data $\vec{x}^{(j)} \in \mathcal{D}_{U}$. The estimated RUL, $\widetilde{y}^{(j)}$, is then used in the next iteration of training such that the base learner is trained again with $(\vec{x}^{(i)}, y^{i})$ and $(\vec{x}^{(j)}, \widetilde{y}^{(j)})$, treating $\widetilde{y}^{(j)}$ as real labels. The iteration is repeated until a certain convergence condition is met. In our study, we use a single iteration since we do not impose any convergence condition for the algorithm.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.99\linewidth]{FIG/rul_dist.pdf} \\ \vspace{-6mm}
\caption{The actual and the estimated RULs in different labeled fraction scenarios. $f$ is the labeled fraction. The green-dotted line indicates perfect estimation where the difference between the actual and the estimated RULs is zero. The blue and red dotted lines indicate the difference of 30 and 50 cycles, respectively. $R^{2}$ score is also shown.}
\label{fig:rul_dist}
\end{center}
\vspace{-3mm}
\end{figure*}
\let\oldnl\n
\newcommand{\renewcommand{\nl}{\let\nl\oldnl}}{\renewcommand{\nl}{\let\nl\oldnl}}
\begin{algorithm}
\caption{Semi-supervised learning based on non-linear embedding with VAE. We use settings $M$, $N$ = 100 and $k$ = 3. The first phase follows the minibatch version of the AEVB algorithm~\cite{vae2013} }\label{sslalgo}
\SetKwProg{Func}{Phase 1}{}{end}
\renewcommand{\nl}{\let\nl\oldnl} \Func{Generative Modeling}{
\KwIn{$\vec{x}^{\textit{M}} = \{\vec{x}^{(i)}\}_{i=1}^{\textit{M}} \in \mathbb{\mathcal{D}}$}
$\theta$, $\phi$ $\leftarrow$ Initialize parameters\;
\Repeat{convergence of parameters ($\theta$,$\phi$)}{
$\vec{x^{\textit{M}}}\leftarrow$ Random mini-batch of size, $M$ \;
$\vec{z^{\textit{M}}} \sim q_{\phi}(\vec{z^{\textit{M}}}|\vec{x^{\textit{M}}})$ \;
$\vec{g} \leftarrow \nabla_{\theta,\phi} \widetilde{\mathcal{L}}^{M}_{VAE}$\;
$\theta$, $\phi$ $\leftarrow$ Update parameters using gradient $\vec{g}$ (e.g., Stochastic Gradient Descent) \;
}
\Return $\theta$, $\phi$\;
}
\vspace*{.2cm}
\vspace*{.2cm}
\SetKwProg{Func}{Phase 2}{}{end}
\renewcommand{\nl}{\let\nl\oldnl} \Func{Supervised Learning with Embedding}{
\KwIn{$(\mathbf{x^{\textit{N}}}, y^{\textit{N}}) \in \mathbb{\mathcal{D}}_{L}$ and $\theta$}
$\gamma$ $\leftarrow$ Initialize parameters\;
Embedding $\vec{x}^{\textit{N}}$ into the latent space defined by the $k^{th}$ output layer of the encoder: $\vec{z^{\textit{N}}_{k}} \leftarrow f_{embed}(\vec{x^{\textit{N}}}; \theta)$ \;
\Repeat{convergence of parameters ($\gamma$)}{
$\vec{z^{\textit{N}}}\leftarrow$ Random mini-batch of size, $N$ \;
$y^{\textit{N}} \sim f_{RNN}(\vec{x^{\textit{N}}}; \gamma)$ \;
$\vec{g} \leftarrow \nabla_{\gamma}\widetilde{\mathcal{L}}^{N}_{RNN}$\;
$\gamma$ $\leftarrow$ Update parameters using gradient $\vec{g}$ (e.g., Stochastic Gradient Descent)
}
\Return $\gamma$\;
}
\label{alg:coltsac}
\end{algorithm}
\begin{comment}
{\bf djw: this section moved to section 2: problem statement}
\subsection{Baseline models}
We use fully connected multi-layer perceptron (MLP) and variants of recurrent neural network (RNN) as our baseline models for RUL estimation. We adopt the RNN architectures because we believe current state is dependent upon previous state(s) and therefore sequential modeling is important for precise modeling of RUL (precise RUL estimation). Considering a deep learning model with $M$ layers of hidden units, a $C$-dimensional output vector is given by
\begin{align}
\begin{aligned}\label{eq:deep_eq1}
f_{i}(x) = \sum_{j=1}^{d}w_{j}^{O,i} h_{j}^{M}(x) + b^{O,i},\ i = 1,...,C
\end{aligned}
\end{align}
and the $k^{th}$ layer is defined as
\begin{align}
\begin{aligned}\label{eq:deep_eq2}
h_{i}^{k}(x) = S(\sum_{j=1}^{d}w_{j}^{O,i} h_{j}^{M}(x) + b^{O,i}),\ k > 1
\end{aligned}
\end{align}
where $w^{O}$ are the weights for the output layer, and $S$ is a non-linear activation function (e.g., $ReLu$). In the RUL estimation, the output dimension is 1-dimensional, or $n$-dimensions that are distributed in time in case of RNN architectures.
\end{comment}
\begin{comment}
\section{Implementation}
In this section we describe how we designed neural net architectures for RUL prediction
with emphasis on several important architectural decisions on design parameters.
\ \\
{\bf jdw: do we need this section ?}
\subsection{Variational Autoencoder (VAE)}
An autoencoder is a neural net with an encoder-decoder architecture [REF:Bengio2007]. Given an input $x$, the autoencoder encodes, $f_{enc}(x)$, and then decodes the encoded, $f_{dec}(z)$, to reconstruct the input as well as possible, i.e., it minimizes the reconstruction error of the form
\begin{align}
\begin{aligned}\label{eq:ae_eq}
|| x - f_{dec}(f_{enc}(x)) ||
\end{aligned}
\end{align}
With some constraints that prevent the encoder and the decoder from being the identity mapping, the autoencoder learns/captures statistical structure while minimizing reconstruction error. It is primarily used for dimensionality reduction and feature learning [Supplement?].
A variational autoencoder (VAE) is a directed probabilistic graphical model that uses learned approximate inference. The VAE basically shares autoencoder's encoder-decoder architecture, but it is different in that the latent representation $z$ of given data $x$ are replaced with stochastic variables.
The encoder and the decoder of VAE are probabilistic, which can be expressed as $q_{\phi}(z|x)$, an approximate posterior, and $p_{\theta}(x|z)$, likelihood of the data $x$ given the latent variable $z$, respectively (note $p_{\theta}(x, z) = p_{\theta}(x|z)p_{\theta}(z)$). Unlike an autoencoder, VAE models the parameters of the distribution rather than the actual value itself. The VAE is trained by optimizing a variational lower bound to the likelihood $\log p_{\theta}(x) = \int p_{\theta}(x,z)$, i.e.,
\begin{align}
\begin{aligned}\label{eq:vae_eq1}
\log p_{\theta}(\textbf{x}^{(i)}) \geq \mathcal{L}(\theta,\phi;\textbf{x}^{i})
\end{aligned}
\end{align}
The variational lower-bound is given by
\begin{align}
\begin{aligned}\label{eq:vae_eq2}
\mathcal{L}(\theta,\phi;\textbf{x}^{i}) \\
= E_{q_{\phi}(\textbf{z}|\textbf{x})}\big[-\log q_{\phi}(\textbf{z}|\textbf{x}) + \log p_{\theta}(\textbf{x}|\textbf{z}) \big] \\
= -D_{KL}(q_{\phi}(\textbf{z}|\textbf{x}^{(i)})) + E_{q_{\phi}(\textbf{z}|\textbf{x}^{(i)})}\big[\log p_{\theta}(\textbf{x}^{(i)}|\textbf{z}) \big]
\end{aligned}
\end{align}
, where $D_{KL}$ is Kullback-Leibeler divergence that measures how different two distributions are. The second equality comes with, as shown in [REF:KINGMA]. The last term can be viewed as the sum of reconstruction error plus a term that acts as a regularization. While the gradients of the encoder is relatively straightforward, that of the decoder is not. This is solved by introducing a reparameterization of $z$ with a deterministic variable such that $z = \mu + \sigma\epsilon$, with $\epsilon \sim \mathcal{N}(0,1)$, known as `reparameterization trick'.
A drawback of VAE is known in image generation. Yet it is known VAE theoretical ground is firm and it is relatively easy to implement. Above all, it results in excellent results in many applications and therefore widely used.
\end{comment}
\begin{comment}
There are three different modes of embedding in deep neural net architectures, as proposed in [REF:DLvisSSE]; output, internal and auxiliary. The output embedding uses the output of the entire network, i.e., output of final layer. The internal embedding uses the output of a hidden (internal) layer. Lastly, the auxiliary embedding shares the first $k$ layers of the original network with new layer(s) added to the shared.
We adopt the internal and auxiliary embedding as the output embedding does not fit in our case since the ouput of VAE reside in the original space. The propose method is first to train the VAE using $X_{L}$ and $X_{U}$, and then embeds $X_{L}$ into a latent space defined by the output of the first $k^{th}$ layers in VAE. (or then uses the first $k^{th}$ layers in VAE in building a network by adding output layers to do supervised learning with the labeled samples.)
\begin{algorithm}
\KwIn{labeled data $(x_{i}, y_{i})$ and unlabeled data}
$\theta$, $\phi$ $\leftarrow$ Initialize parameters\;
\Repeat{convergence of parameters ($\theta$,$\phi$)}{
do these things\;
}
\Return $\theta$, $\phi$\;
\vspace*{.4cm}
\KwIn{labeled data $(x_{i}, y_{i})$ and $\theta$, $\phi$}
$\theta$, $\phi$ $\leftarrow$ Initialize parameters\;
\Repeat{convergence of parameters ($\theta$,$\phi$)}{
do these things\;
}
\Return $\theta$, $\phi$\;
\label{alg:coltsac}
\end{algorithm}
\end{comment}
\subsection{C-MAPSS Turbofan Engine Dataset}
The dataset, \textit{FD001}, consists of multiple multivariate (24 sensors including 3 operational settings) time-series from a fleet of engines with different degree of initial wear and manufacturing variation that are unknown. The training and the testing sets contain 100 such engines for each. For the engines in the training set, run-to-failure time-series trajectories are provided. Whereas for the engines in the testing set, time-series trajectories are truncated prior to failures with true RUL values at the point of truncation given in a separate file for validation. A total number of cycles in the training set and in the testing set reach about 20K and 13K, respectively.
\subsection{Evaluations}
\subsubsection{Overview}
We directly model the engine failure by parameterizing the RNN architecture in Eq.~\ref{eqn:rnn_model} such that the input sensor readings are directly mapped to historical RUL. RUL is assigned to each sequence for a given engine by simply counting the number of cycles left before failure (by definition). In this way, we avoid to create any domain-specific assumptions for the modeling. We fist build a reliability model in a purely supervised setting, where all the training samples are used with their corresponding RULs. We then omit the RULs for a randomly chosen set of engines in the training set and repeat the procedure.
The faction of engines, for which the RUL information is dropped, is varied from 100\% down to 1\% ($f$ = 100, 80, 50, 30, 20, 10, 5, and 1\%). To remove any selection bias, we repeat the random selection five times for each fraction. The performance measures are then averaged over the five different sets of samples except for the 100\% labeled scenario, where no selection is involved. Also in order to take the model variability into account, we always use the homogeneous ensemble of the models obtained from five independent training, which in general produce a better result.
\subsubsection{Preprocessing}
Based on the 3 operational settings, we identify 14 different operational modes that are uniquely defined. We normalize the 21 sensor readings (excluding the 3 operational settings) before feeding into our models to train. The normalization is done for each sensor based on min-max scaling and it is done operational-mode-wise such that each sensor is scaled to have a range between 0 and 1 in each operational mode. The sensors that are constant (i.e., $\sigma_{x}=$ 0) after the normalization are dropped. After checking carefully with its impact on the prediction accuracy, we further drop \emph{discrete} sensors with the number of unique values falling below 20. For the sake of simplicity, the discrete variables were excluded in the training of VAE. For the RUL prediction, we limit the value below a maximum RUL of 140 cycles as suggested in~\cite{lstmed, heimes2008recurrent}. Consequently, we limit the assigned RULs in the training set by setting those that exceed the maximum RUL to be the maximum.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.99\linewidth]{FIG/perf_as_frac_all.pdf} \\ \vspace{-5.5mm}
\subfloat[\label{mae} $MAE$]{\hspace{.25\linewidth}}
\subfloat[\label{mse} $MSE$]{\hspace{.25\linewidth}}
\subfloat[\label{score} $Score$]{\hspace{.25\linewidth}}
\subfloat[\label{r2} $R^{2}$]{\hspace{.25\linewidth}}
\caption{Measured performance of different approaches as a function of the labeled fraction: (a) $MAE$, (b) $MSE$, (c) $Score$, and (d) $R^{2}$. The error bar indicates the standard error of mean at each point.}
\label{fig:finalresult}
\end{center}
\vspace{-3mm}
\end{figure*}
\subsubsection{Metrics}
Different performance metrics are employed to assess the performance of the models. This includes the mean squared error ($MSE$), mean absolute error ($MAE$), and the score metric proposed in~\cite{saxena2008damage}. The proposed score metric is defined such that late predictions are more heavily penalized than early prediction:
\begin{equation}\label{eqn:score}
S = \sum_{i=1}^{n}(\exp{(\alpha|\Delta_{i}|)}-1)
\end{equation}
where $\Delta_{i}$ is the difference between the estimated and the true RUL values, i.e., estimated RUL - true RUL, for the $i^{th}$ engine in the testing set and $\alpha$ is equal to $1/13$ if $\Delta_{i} < 0$ or $1/10$ otherwise. We also use the coefficient of determination also known as $R^{2}$ score as a different measure of regression quality\footnote{$R^{2}$ can yield negative values as models can be arbitrary worse.}.
\subsection{Results}
\subsubsection{Reconstruction with Variational Autoencoder}
We use a three-layer architecture in the encoder (17-8-4) and in the decoder (4-8-17) with batch normalization applied to each layer except for the last layer. The architecture is chosen considering the reconstruction error and the performance of the non-linear embedding. The use of batch normalization is considered essential for training VAE with when the architecture becomes deep (i.e., $L > 2$)~\cite{vaetrain2016}. As mentioned earlier, we incorporate the RNN architecture into the VAE such that latent variables are modeled with the dynamic hidden state of a RNN architecture. This requires a modification in the objective function in Eq~\ref{eqn:lb2} so that sum of loss is averaged in temporal dimension. The model is trained end-to-end using the Adam optimizer~\cite{adam2014} with a mini-batch size of 100.
Fig.~\ref{fig:cycles} shows four sample sensor readings for several engines in sequence overlaid with the reconstructed sensor readings from VAE. In Fig.~\ref{fig:freq}, we show the frequency of the values in histograms to see distributions. The results clearly demonstrate even with the one-fourth of the original input dimension in the bottleneck layer of VAE, the reconstruction is done reasonably well, capturing time dependent features of the signals. The sum of reconstruction loss and the KL-divergence stays below 0.3 when the trained VAE is used to reconstruct the validation sets\footnote{A fraction of the training data is kept from training and used for validation}. This may support the manifold hypothesis that high dimensional data tend to lie in the vicinity of a low dimensional manifold~\cite{dlbook2016}.
\subsubsection{Baseline Results}
Fig.~\ref{fig:rul_dist} shows the baseline results of RNN-based models with a four-layer architecture\footnote{We adopt both GRU and LSTM architecture in the models. However, as we find no quantitative difference in performance, we choose the simpler architecture, namely, GRU for the final results.} trained in a purely supervised setting. It shows the true RUL in the $x$-axis and the estimated RUL in the $y$-axis in different labeled fraction scenarios. From the top left to the bottom right, the labeled fraction, $f$, is varied from 100\% to 1\%. When the full training set is used, the trained model shows good performance with the measured $MSE$, $MAE$, $Score$ and $R^{2}$ of 228, 11.3, 345, and 0.87, respectively. This result is comparable to that of other approaches~\cite{lstmed, babu2016deep, khelif2014rul, peng2012modified, ramasso2014investigating}.
Now as the fraction changes, the prediction accuracy decreases. It can clearly be seen from Fig.~\ref{fig:rul_dist} that the correlation between the actual and the estimated RULs get worse and worse with decreasing fraction, also indicated by the $R^{2}$ score shown in the figure. The performance degradation is expected because the complexity of model requires a large amount of training data. In the current implementation of the RNN-based model, the number of tunable parameters reaches few hundreds. Nevertheless, the performance degradation is not prominent until the fraction is decreased down to 30\%, indicating that only 30\% of the training samples are already sufficient for the model to give a reasonably good result. As such, we rather focus in the region below 30\%, where the performance degradation is significantly showing.
\begin{center}
\begin{table*}[t]
\caption{Comparison of different approaches. The best result is highlighted in bold.}\label{tab:results}
\begin{tabular}{ c c c c c c c c} \toprule
{Metric} \hspace{0.5cm} & {Method} & 1\% & 5\% & 10\% & 20\% & 30\% & 100\%\\ \midrule
& SL & 46.9 $\pm$ 5.3 & 28.5 $\pm$ 3.3 & 19.7 $\pm$ 2.2 & 15.4 $\pm$ 1.7 & 13.7 $\pm$ 1.5 & \textbf{11.3} $\pm$ 1.3 \\
\textit{MAE} \hspace{0.5cm} & Self-SSL & 48.2 $\pm$ 5.5 & 25.8 $\pm$ 2.8 & 19.2 $\pm$ 2.1 & 14.0 $\pm$ 1.6 & 14.2 $\pm$ 1.6 & \textbf{11.3} $\pm$ 1.3\\
& VAE-SSL & \textbf{28.1} $\pm$ 3.4 & \textbf{15.8} $\pm$ 1.8 & \textbf{13.3} $\pm$ 1.5 & \textbf{12.9} $\pm$ 1.5 & \textbf{11.9} $\pm$ 1.4 & \textbf{11.3} $\pm$ 1.3\\ \midrule
& SL & 3309 $\pm$ 441 & 1299 $\pm$ 199 & 642 $\pm$ 80 & 407 $\pm$ 45 & 344 $\pm$ 35 & 228 $\pm$ 27\\
\textit{MSE} \hspace{0.5cm} & Self-SSL & 3483 $\pm$ 493 & 1055 $\pm$ 120 & 618 $\pm$ 74 & 362 $\pm$ 40 & 344 $\pm$ 38 & 228 $\pm$ 27 \\
& VAE-SSL & \textbf{1370} $\pm$ 240 & \textbf{405} $\pm$ 59 & \textbf{294} $\pm$ 36 & \textbf{274} $\pm$ 31 & \textbf{243} $\pm$ 27 & \textbf{221} $\pm$ 26 \\ \midrule
& SL & 482 $\pm$ 121 & 299 $\pm$ 111 & 53.6 $\pm$ 13.9 & 13.6 $\pm$ 2.43 & 6.42 $\pm$ 0.81 &\textbf{3.45} $\pm$ 0.41\\
\textit{Score} [$\times$\num{e-2}] \hspace{0.5cm} & Self-SSL & 610 $\pm$ 173 & 109 $\pm$ 22.4 & 84.9 $\pm$ 41.3 & 10.0 $\pm$ 1.46 & 7.42 $\pm$ 0.93 & \textbf{3.45} $\pm$ 0.41 \\
& VAE-SSL & \textbf{133} $\pm$ 34 & \textbf{8.26} $\pm$ 1.38 & \textbf{7.20} $\pm$ 1.58 & \textbf{6.09} $\pm$ 0.72 & \textbf{5.22} $\pm$ 0.59 & 4.75 $\pm$ 0.56 \\ \midrule
& SL & $-$0.93 $\pm$ 0.28 & 0.24 $\pm$ 0.11 & 0.63 $\pm$ 0.07 & 0.76 $\pm$ 0.09 & 0.81 $\pm$ 0.09 & \textbf{0.87} $\pm$ 0.10 \\
$R^{2} $\hspace{0.5cm} & Self-SSL & $-$1.03 $\pm$ 0.33 & 0.39 $\pm$ 0.05 & 0.64 $\pm$ 0.07 & 0.79 $\pm$ 0.09 & 0.80 $\pm$ 0.09 & \textbf{0.87} $\pm$ 0.10\\
& VAE-SSL & \textbf{0.20} $\pm$ 0.14 & \textbf{0.76} $\pm$ 0.08 & \textbf{0.83} $\pm$ 0.09 & \textbf{0.84} $\pm$ 0.10 & \textbf{0.86} $\pm$ 0.10 & \textbf{0.87} $\pm$ 0.10 \\ \bottomrule
\end{tabular}
\end{table*}
\end{center}
\subsubsection{Semi-supervised Approach}
As discussed in Section~\ref{ourapproach}, we apply semi-supervised learning techniques to see if the performance can be improved when only a small fraction of the training data is labeled. We try two different semi-supervised learning techniques, i.e., self-learning and VAE-based non-linear embedding. Fig.~\ref{fig:finalresult} shows the results of the RUL estimation quantified by the three different metrics plotted as a function of $f$. From left to right, $MAE$, $MSE$, $Score$, and $R^{2}$, are plotted with corresponding statistical uncertainties. The score is plotted in logarithmic scale.
It quantitatively shows the performance degradation becomes increasingly larger as the fraction decreases. The result from the self-learning plotted in a triangular shape reveals that the performance is essentially the same within statistical uncertainty except for the score at $f$ = 5\%, where a distinctive improvement is shown.
Now, the result from the non-linear embedding plotted in a circle shows improvements for $f$ below 30. The level of improvement becomes increasingly larger reaching x1.7, x2.4, and x3.6 improvement, respectively for $MAE$, $MSE$, and $Score$. The $R^{2}$ improvement is also notable. We note the prediction accuracy of the model is kept rather high even down to 5\% fraction. The numerical compilations of the measured performance metrics are given in Table~\ref{tab:results} for all three approaches: $SL$ indicates supervised learning only. $Self$-$SSL$ is the self-learning. $VAE$-$SSL$ is the proposed method.
\section{Approach Overview}
\section{Background}
\label{sec:overview}
\subsection{Problem Statement}
\label{sec:pre}
In our problem, we have multivariate data for sensor measurement and their corresponding RUL.
Let $x^{(i,j)}$ denote $i_{th}$ measurement value of sensor $j$.
Furthermore, let us $\vec{x^{(i)}}$ denote a vector of multivariate sensor measurement
such that $\vec{x^{(i)}}=[x^{(i,1)},\cdots, x^{(i,m)}]$ where $m$ is the number of sensors.
Formally, we describe a sensor measurement matrix by denoting
$\mat{x}=\{\mathbf{x^{(1)}},\cdots, \mathbf{x^{(n)}} \}$ where $\mathbf{x^{(i)}} \in \mathbb{R}^m$
and $n$ is the number of observed measurements.
We also denote the corresponding set of RUL by
$\mathbf{y}= \{y^{(1)},\cdots, y^{(n)} \} $ where $y^{(i)} \in \mathbb{R}^+$.
More compactly, we define the data set $\mathcal{D}$ by $\mathcal{D} =\{(\mathbf{x}^{(i)},y^{(i)})\}_{i=1,\cdots,n}$.
In general, the data-driven prognostics approach is to learn the best predictor of RUL (or equivalent health indicator used to predict RUL) from the previously observed data set $\mathcal{D}^{T}$, i.e.training data set. The challenging problem is to find an appropriate transformation of the raw measurement $\mathbf{X}$ to efficiently learn a reliable RUL predictor. Such problem can be described by finding a mapping function $F$ such that $F :\mathbf{x} \mapsto \mathbf{z}$ where a latent variable $\mathbf{z} \in \mathbb{R}^k$ and $m<k$.
Then the optimal predictor can be defined by a function of $\mathbf{z}$ with parameter $\mathbf{\gamma}$ denoted by
$f_{\mathbf{\gamma}}( \mathbf{z})$ such that $f_{\mathbf{\gamma}}( \mathbf{z})= \underset{y}{argmax}~ p( y~|~\mathbf{z},\mathbf{\gamma}) $.
In traditional machine learning approaches, it first finds appropriate latent variable $\mathbf{z}$ by well known linear transformation methods (e.g. Principal Component Analysis ) often with heuristics.
Then it estimates the optimal parameter $\mathbf{\gamma}$ by training $f_{\mathbf{\gamma}}( \mathbf{z})$ with $\mathcal{D}^{T}$ given a certain simplified assumption for the distribution
$p( y~|~\mathbf{z},\mathbf{\gamma})$(e.g. multivariate normal distribution).
Such conventional approaches inherently has its own limitations in providing a general solution for RUL prediction. First, it is often that the mapping function $F$ for latent variable $\mathbf{z}$ is non-linear and its specific form is largely unknown. Second, any strong assumption on distribution
$p( y~|~\mathbf{z},\mathbf{\gamma})$ often poorly captures complex cross dependency over time and sensors. Meanwhile, deep neural network can provide a generic solution to jointly learn the non-linear mapping function $F$ and the predictor $f_{\mathbf{\gamma}}$ from training data set $\mathcal{D}^{T}$ using directed graph network with various levels of abstraction. We drop the superscript $T$ for the rest of the paper.
In our problem setting, we assume that only a small fraction of the samples has label information. Let $\mathcal{D}^{T}_{L}$ and $\mathcal{D}^{T}_{U}$ denote the part of the data set with labels and the part without the labels, respectively. The primary goal in this paper is to develop deep learning-based approach to predict RUL from $\mathcal{D}^{T}_{L}$ where $|\mathcal{D}^{T}_{L}| \ll |\mathcal{D}^{T}_{U}|$, which is a class of semi-supervised learning. In the semi-supervised learning setting, we assume that $\mathcal{D}^{T}_{L}$ comes from exactly the same distribution of $\mathcal{D}^{T}_{U}$.
Our semi-supervised approach fully exploits $\mathcal{D}^{T}_{U}$ to learn the non-linear mapping function $f$ to get a robust feature representation of $\mathbf{x}$ for stable RUL predictor $f_{\mathbf{\gamma}}$ given $\mathcal{D}^{T}_{L}$.
\subsection{Neural Network Overview}\label{dnn_overview}
\begin{comment}
Semi-supervised learning (SSL) aims to improve generalization on supervised tasks using unlabled data. In the semi-supervised learning framework, a given set of $N = L+U$ ($U >> L$ usually) data can be divided into two parts: historical data $X_{l} = (x_{1},..,x_{l})$ for which corresponding labels $Y_{l} = (y_{1},..,y_{l})$ exist and $Xu = (x_{l+1},..,x_{l+u})$, for which corresponding labels do not exist. The goal of SSL is to improve prediction accuracy of a function learned from $(X_{l}, Y_{l})$, i.e., by taking into account the unlabeled data $X_{u}$. There are several assumptions that a number of SSL theories are based on. These are are smoothness, cluster, and manifold assumptions, each of which is related to the structure of the underlying distribution. While these assumptions are of great theoretical interest and may provide intuitions into SSL, we will not go into further details as our intention is not to uncover theoretical maximum performance gain that SSL approach can bring per se.
\end{comment}
In this section we briefly discuss the definition and the architectures of basic deep neural network and its variants.
\subsubsection{feed-forward neural network} \label{ffnw}
We can formally describe a feed-forward neural network as follows.
Let $f_{l,i}$ and $g_{l,i}$ denote a linear pre-activation and non-linear activation function of $i$th neuron at layer $l$, respectively where $f_{l,i},g_{l,i} \in \mathbb{R}^k$ for $i=1,\cdots, n_l$, $n_l$ is the number of neurons at layer $l$, and $k$ is the dimension of a neuron's outputs.
We can conveniently assume that the output dimension $k$ is the dimension of a single sensor measurement.
In our problem setting, $k=1$ since each sensor measurement is a scalar value.
Then layer $l$ has two function vectors ; $f_l=[f_{l,1}\cdots f_{l,n_l}]^T $ and $g_l=[g_{l,1}\cdots g_{l,n_l}]^T$.
Let $\mathbf{x}_l=[\mathbf{x}_{l,i}]_{i=1,\cdots,n_l}$ denote activation output at layer $l$ where $\mathbf{x}_{l,i}=g_{l,i}\circ f_{l,i}(\mathbf{x}_{l-1})$.
The pre-activation of layer $l$ is computed by linear transformation
\begin{equation}\label{eqn:actfunc}
\mathbf{f}_l(\mathbf{x}_{l-1})=\mathbf{W}_l \mathbf{x}_{l-1}+\mathbf{b}_l
\end{equation}
where $\mathbf{W}_l \in \mathbb{R}^{ n_l \times n_{l-1}}$ and $\mathbf{b}_l \in \mathbb{R}^{n_{l-1}}$.
For the non-linear activation, we use a popular rectified linear unit (ReLU)~\cite{NairH10} where
$g_i(f_i) =\max \{0, f_i\}$ for $f_i \in \mathbb{R}$.
The layer $l$'s parameter, $\theta_l=(\mathbf{W}_l,\mathbf{b}_l)$ is estimated by the\emph{ backpropagation algorithm} \cite{LeCun98} given $\mathcal{D}^{T}$.
Finally, feed-forward neural network of $L$ layers is defined as following :
\begin{equation}\label{eqn:ffnn}
F(\mathbf{x};\theta)=f_{out}\circ h_L(\mathbf{x})
\end{equation}
where $h_l= (g_l \circ f_l)\circ h_{l-1}$ and $\theta=(\theta_{0},\cdots,\theta_L) $.
\begin{figure*}[t!]
\begin{center}
\includegraphics[width=0.95\linewidth]{FIG/sensor_as_cycles_laid.pdf} \\ \vspace{-3mm}
\caption{Normalized sensor values as a function of cycles for four selected sensors $(a)$-$(d)$ from four independent engine: original (blue) and reconstructed (red). The cycles are cumulative. The gray lines drawn to differentiate the units.}
\label{fig:cycles}
\end{center}
\end{figure*}
\subsubsection{Recurrent Neural Networks}
The feed-forward neural network we described in previous section does not allow
temporal dependency of samples to be embedded in its model.
In our problem, the current sample are mostly likely to be dependent on its previous samples.
Hence, we need a more generalized model capable of elegantly incorporating
temporal dependency in it. RNN is such kind of a model that can capture into model the influence of an arbitrary number of previous samples on the current~\cite{dlbook2016}.
The basic RNN structure consists of a cell with a cyclic loop whose internal state evolves over time by the current sample input and its previous state output. The output of RNN is simply a series of cell states.
When we represents the cell state at each time as a neuron, its resultant structure is a non-cyclic horizontally connected neural network where the current neuron's output feeds into the next neuron's input.
Using the horizontal representation of cell states we can formally describe RNN with a simple modification of (\ref{eqn:actfunc}) and (\ref{eqn:ffnn}).
Let us assume that we use the previous $T$ samples for RNN input.
Then the RNN input at time $t$ is the array of multivariate sensor measurements from $t$ to $t_T$ denoted by
$\mat{x}^{t}_{t-T}=[\vec{x}^{t},\cdots,\vec{x}^{t-T}]$.
Let $\vec{s}_t$ denote a cell state at time $t$ such that $\vec{s}_{t} \in \mathbb{R}^{k}$ where $k$ is a dimension of the state. Then pre-activation function of a neuron is defined by a linear mapping of the cell's previous state $\vec{s}_{t-1}$ and the current input $\vec{x}_{t} \in \mathbb{R}^{m}$ at time $t$ as shown in (\ref{eqn:rnn_actfunc}).
\begin{equation}\label{eqn:rnn_actfunc}
f(\vec{s}_{t-1},\vec{x}_{t})=\mat{U}\vec{s}_{t-1}+\mat{W}\vec{x}_{t}+\vec{b}
\end{equation}
where $\vec{b} \in \mathbb{R}^{m}$, $\mat{U} \in \mathbb{R}^{ k \times k}$, and $\mat{W} \in \mathbb{R}^{ k \times m}$.
The cell state at time $t$ is updated by a non-linear activation function $g$ on the $f$ output
such that $\vec{s}_{t}=g\circ f(\vec{s}_{t-1},\vec{x}_{t})=h(\vec{s}_{t-1},\vec{x}_{t})$.
Finally, RNN for $T_{th}$ order time dependency is compactly defined as following :
\begin{equation}\label{eqn:rnn_model}
F(\mat{x}^{t}_{t-T};\mat{\theta})=[ h(\vec{s}_{t'-1},\vec{x}_{t'}) ]_{t'=t,\cdots,t-T}
\end{equation}
where $h = (g \circ f)$ and
$\mat{\theta} =(\mat{U},\mat{W},\mat{b})$.
Note that all neural cells share parameter $\mat{\theta}$ over time.
The RNN parameter $\mat{\theta}$ can be trained using backpropagation through time (BPTT) algorithm~\cite{Werbos90}.
However, RNNs with the BPTT suffer from the \emph{vanishing} and \emph{exploding} gradient problem~\cite{Bengio94} because a recurrent network grows as deep as sequence length. One of popular solutions is to exploit Long Short-Term Memory (LSTM). An LSTM~\cite{Hochreiter97} is a second-order recurrent network that has three additional gate layers called \textit{input}, \textit{forget}, and \textit{update} gates. The gate parameters helps a recurrent network have good information flow over time by alleviating the vanishing gradient. Thus, a recurrent network \textit{with} gates can work better than one \textit{without} gates as more long-term dependencies exist.
GRU~\cite{cho14} is a simpler variant of LSTM which combines \emph{forget input} gate and \emph{read input} gate into \emph{overwrite} gate by setting an input gate to the inverse of forget gate, ,i.e $i_t=1-f_t$. It is shown that the performance of GRU is on par with LSTM but more computationally efficient~\cite{Rafal15}. In our study, we find there is no meaningful difference in prediction accuracy between the two variants.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.9\linewidth]{FIG/method_illustration1.pdf} \\ \vspace{-1mm}
\caption{Overall architecture of VAE with an input dimension of $m$. $\epsilon \sim \mathcal{N}(0,1)$ indicates the reparameterization trick is used in the training of VAE.}
\label{fig:vae}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\subfloat[\label{mae} Embedding Network]{\includegraphics[width=0.48\linewidth]{FIG/method_illustration2.pdf}}
\subfloat[\label{mse} Reliability Model]{\includegraphics[width=0.48\linewidth]{FIG/method_illustration3.pdf}} \\ \vspace{-1mm}
\caption{Embedding followed by supervised learning in latent space. (a) embedding network is taken from the encoder of the trained VAE in Fig.~\ref{fig:vae}. (b) reliability model is built upon the latent space defined by the embedding network. While RNN architecture is used, the recurrence relation is not shown for simplicity.}
\label{fig:embedding}
\end{center}
\end{figure}
\subsection{Variational Autoencoder (VAE)}\label{vae}
An autoencoder is a neural net with an encoder-decoder architecture that is trained to reconstruct its own input. Given an input $\vec{x}$, the autoencoder, $f_{enc}(\vec{x})$, encodes and then the decoder, $f_{dec}(\vec{z})$, decodes the encoded input (a \emph{representation}) back to reconstruct the input as closely as possible by minimizing a loss function of the form~\cite{dlbook2016}:
\begin{equation}\label{eqn:aeloss}
\mathcal{L}(\vec{x}, \vec{x}') = || \vec{x} - f_{dec}(f_{enc}(\vec{x})) ||
\end{equation}
where $f_{enc}$ and $f_{dec}$ are symmetric in topology.
The autoencoder is trained using backpropagation algorithm in the same manner as in the feed-forward neural network in Section~\ref{ffnw}. However, when the architecture becomes deep with many hidden layers, pretraining is necessary to set the initial weights of the autoencoder close to the final ones~\cite{hinton2006}. The VAE inherits autoencoder's encoder-decoder architecture, but it is different in that the latent representation $\vec{z}$ of given data $\vec{x}$ are replaced with stochastic variables. The encoder and the decoder of VAE are probabilistic and given by $q_{\phi}(\vec{z}|\vec{x})$, an approximate posterior, and $p_{\vec{\theta}}(\vec{x}|\vec{z})$, likelihood of the data $\vec{x}$ given the latent variable $\vec{z}$, respectively. The approximate posterior is parameterized by a neural network and the likelihood is given by a multivariate Gaussian whose probabilities are obtained from $\vec{z}$.
The objective function of VAE is the variational lower bound to the marginal likelihood of the data, i.e., $\log p_{\theta}(\vec{x}) = \sum_{i=1}^{n}\log p_{\theta}(\vec{x^{(i)}})$. It can be rewritten for individual data points, $\vec{x}^{(i)}$ as~\cite{vae2013}:
\begin{equation}\label{eqn:likelihood}
\log p_{\theta}(\vec{x}^{(i)}) = D_{KL}(q_{\phi}(\vec{z}|\vec{x}^{(i)})||p_{\theta}(\vec{z}|\vec{x}^{(i)})) + \mathcal{L}(\theta,\phi;\vec{x}^{(i)})
\end{equation}
where $D_{KL}$ is Kullback-Leibeler (KL) divergence. Since $D_{KL} \geq 0$, (\ref{eqn:likelihood}) can be rewritten as:
\begin{equation}\label{eqn:lowerbound}
\log p_{\theta}(\vec{x}^{(i)}) \geq \mathcal{L}(\theta,\phi;\vec{x}^{(i)})
\end{equation}
The left-hand side of (\ref{eqn:lowerbound}), which is called \emph{variational lowerbound}, can further be rewritten using Bayes' rule and the definition of KL divergence as~\cite{vae2013}:
\begin{equation}\label{eqn:lb2}
-D_{KL}(q_{\phi}(\vec{z}|\vec{x}^{(i)})||p_{\theta}(\vec{z})) + E_{q_{\phi}(\vec{z}|\textbf{x}^{(i)})}\big[\log p_{\theta}(\vec{x}^{(i)}|\vec{z}) \big]
\end{equation}
which can be viewed as the sum of reconstruction error plus a term that acts as a regularization.
While obtaining the gradients of the encoder is relatively straightforward, that of the decoder is not. This is because, the reconstruction error term in Eq.~\ref{eqn:lb2} requires the Monte Carlo estimate of the expectation, which is not easily differentiable~\cite{vae2013}. This problem is solved by introducing a reparameterization of $z$ with a \emph{deterministic variable} such that $z = \mu + \sigma\epsilon$, with $\epsilon \sim \mathcal{N}(0,1)$, which is known as ``reparameterization trick''~\cite{vae2013}. With this trick, a differentiable estimator of the variational lower bound can be obtained and the backpropagation algorithm can be applied to train VAE. The overall architecture of VAE is shown in Fig.~\ref{fig:vae}.
\begin{figure}[h]
\begin{center}
\includegraphics[height=0.9\linewidth]{FIG/sensor_hist_all.pdf} \\ \vspace{-1mm}
\caption{Normalized frequency of the values of selected sensors in Fig~\ref{fig:cycles} $(a)$-$(b)$ from all the engines in the dataset: original (blue) and reconstructed (red).}
\label{fig:freq}
\end{center}
\end{figure}
\begin{comment}
We use fully connected multi-layer perceptron (MLP) and variants of recurrent neural network (RNN) as our baseline models for RUL estimation. We adopt the RNN architectures because we believe current state is dependent upon previous state(s) and therefore sequential modeling is important for precise modeling of RUL (precise RUL estimation). Considering a deep learning model with $M$ layers of hidden units, a $C$-dimensional output vector is given by
\begin{align}
\begin{aligned}\label{eq:deep_eq1}
f_{i}(x) = \sum_{j=1}^{d}w_{j}^{O,i} h_{j}^{M}(x) + b^{O,i},\ i = 1,...,C
\end{aligned}
\end{align}
and the $k^{th}$ layer is defined as
\label{subsec:background}
This section briefly reviews semi-supervised learning and deep neural network (DNN) theory [cite xxx]
\begin{align}
\begin{aligned}\label{eq:deep_eq2}
h_{i}^{k}(x) = S(\sum_{j=1}^{d}w_{j}^{O,i} h_{j}^{M}(x) + b^{O,i}),\ k > 1
\end{aligned}
\end{align}
where $w^{O}$ are the weights for the output layer, and $S$ is a non-linear activation function (e.g., $ReLu$). In the RUL estimation, the output dimension is 1-dimensional, or $n$-dimensions that are distributed in time in case of RNN architectures.
\end{comment}
\section{Introduction}
\input{010introduction}\label{intro}
\input{overview}
\section{Our Approach}
\input{030ssl}
\section{Experimental Results}\label{exp}
\input{040experiments}
\section{Discussion}
\input{090discussion}
\vspace{-2mm}
{
\fontsize{7.3}{8.76}\selectfont
\bibliographystyle{abbrv}
|
1,116,691,500,814 | arxiv |
\section{Conclusion}
Verifying that neural networks behave as intended may soon become a limiting factor in their applicability to real-world, safety-critical systems such as those used to control autonomous vehicles and aircraft. Recent work revealing neural networks' vulnerability to adversarial inputs~\cite{szegedy2014}, including in physical-world attacks~\cite{adversarial-physical}, makes meeting this challenge more urgent.
Verification is a promising avenue for mitigating this difficulty, but additional work is required to scale up verification techniques to be practically applicable to modern DNNs. Initial work by us and others points to two complementary avenues that could achieve the sought-after scalability: first, the design of verification algorithms tailored for neural networks (e.g., by enriching the theories used by SMT solvers); and second, the creation and use of design principles for neural networks that produce DNNs that are more amenable to verification (e.g., model topology and activation function selection). We believe that through additional work in these directions, verification could be successfully applied to many real-world deep learning systems.
\section{Designing verification-friendly neural networks}
In addition to improving the scalability of verification tools, a complementary direction for scaling DNN verification is to design the networks themselves in a way that makes them more amenable to verification. When designing a neural network, some of the obvious design decisions are related to the topology of the network,
such as the number of hidden layers and their dimensions. It is not
surprising that, from a verification point of view, smaller networks
are generally easier to handle. Developers of neural networks may
opt to use a smaller network, perhaps achieving lower accuracy,
in order to enjoy the benefits of verification. On the other hand, recent work~\cite{deep-compression} suggests that it is possible to significantly reduce the storage requirements of neural networks without compromising accuracy. Although the motivation for this work was ease of deployment of neural networks in resource-constrained settings rather than ease of verification, these pruned, quantized networks may also be easier for verification tools to handle than uncompressed networks.
Our initial experiments suggest that the size of the network is not necessarily the only factor to consider; the network topology is also important. We have observed that networks with many layers with a few neurons each are generally easier for the solver to handle than networks with few layers, but many neurons in each layer.
An extreme way of applying this principle is by discretizing parts of the neural network in question, effectively turning it into a family of smaller networks. The ACAS Xu network~\cite{policycompression} used this approach due to considerations that did not include verification --- rather, due to hardware constraints, the developers found that many smaller networks were preferable to one large one. However, the discretization step also made it easier to verify properties of each of the smaller networks. A similar approach could facilitate the verification of other systems as well.
Another decision with consequences for the scalability of verification is activation function selection. These choices can have far-reaching effects. For example, some of the more successful verification efforts thus far~\cite{reluplex, ehlers2017} have focused on piecewise-linear activation functions, such as ReLUs or max-pooling layers, while attempts to verify networks with sigmoid activation functions have proved far less scalable.
In addition to network topology and activation functions, there are several other potential avenues to explore. One example is low-precision DNN arithmetic, which is an increasingly popular way to accelerate DNN training and inferencing~\cite{lowprecision}. The simplicity and smaller size of low-precision networks will make them more amenable to verification than full-precision networks~\cite{Narodytska2017,Cheng2017}, as well as more suitable for use on low-power edge devices~\cite{lane:2016:lowpower}. It may even be the case that hardware accelerator techniques that optimize inference on low-precision networks could be used to speed up verification of those same networks.
\section{Introduction}
Machine learning systems, and, in particular, deep neural networks (DNNs), are becoming a widely used and effective means for tackling complex, real-world problems~\cite{goodfellow:2016:dl}. However, a major obstacle to the use of DNNs in safety-critical systems, such as autonomous driving or flight control systems, is the great difficulty in providing formal guarantees about their behavior.
A powerful technique for formal verification of properties of a software artifact is to encode the artifact and the property one wishes to prove about it as a \emph{satisfiability modulo theories} (SMT) formula, and then use an SMT solver to prove that the property holds or find a counterexample showing that it does not. While it is possible to verify properties of neural networks using SMT solvers, until recently the technique only scaled to toy-sized networks of fewer than ten neurons~\cite{pulina2010}.
Yet, for the practical adoption of SMT-based DNN verification, we must be able to verify properties of DNNs of up to thousands (or more) of neurons. To do this, we advocate a two-pronged approach. First, we propose the development of specialized, efficient SMT solvers that are well-suited for DNN verification problems. Second, we propose designing DNNs in ways that make them more amenable to SMT-based verification. These two approaches complement each other, and we observe that design choices that make a DNN more amenable to verification are also desirable for other reasons, such as improved speed of inferencing, smaller memory requirements, and reduced power footprint.
\section{Scaling up SMT-based verification of neural networks}
\begin{figure}[t]
\includegraphics[width=\linewidth]{reluplex-expanded-overview}
\caption{\small Overview of the Reluplex architecture. Reluplex takes as input a network description and a property we wish to prove about the network's behavior, both expressed as an SMT formula. The SMT solver incorporates a domain-specific linear programming (LP) + ReLU theory solver that interacts with an underlying SAT solver and determines whether the formula is satisfiable.}
\end{figure}
A primary focus of this work is in extending the capabilities of automated verification tools such as SMT solvers to formally verify properties of DNNs used for safety-critical systems. A major challenge of verifying properties of DNNs with SMT solvers is in handling the networks' activation functions.
Each neuron of a neural network computes a weighted sum of its inputs according to learned weights. It then passes that sum through an activation function to produce the neuron's final output. Typically, the activation functions (e.g., sigmoid) introduce nonlinearity to the network, making DNNs capable of learning arbitrarily complex functions, but also making the job of automated verification tools much harder, in some cases moving the problem from P to NP.
Using SMT solvers to verify properties of neural networks involves encoding the network and the property in question as formulas in some \emph{theory}, such as the theory of linear real arithmetic. Our work leverages the observation that, apart from their activation functions, neural networks can be expressed using conjunctions of linear real arithmetic formulas, which are straightforward to handle using standard linear programming (LP) solving algorithms. It is also possible to express \emph{piecewise-linear} activation functions, such as rectified linear units (ReLUs), as part of linear arithmetic formulas, but every ReLU in a network then introduces a disjunction to the formula. These disjunctions quickly cause an exponential increase in the state space that the SMT solver must explore to prove properties about the network, thus limiting the applicability and scalability of the approach.
In a recent paper~\cite{reluplex}, we proposed an improved SMT-based algorithm, called Reluplex, capable of verifying properties of networks that are an order of magnitude larger than previously possible. Reluplex mitigates the difficulty posed by activation functions through a \emph{lazy} approach, which often makes it possible to eliminate many activation functions from the problem without changing the result. It extends the theory of linear real arithmetic by introducing a new ``ReLU'' predicate that can be split into disjuncts lazily, making it possible to avoid exploring large parts of the state space. As a result, the Reluplex solver can verify networks that are notably larger than what was previously possible. For example, we used Reluplex to verify safety properties of a DNN used as the controller for a prototype of the ACAS Xu aircraft collision avoidance system~\cite{policycompression}.
The lazy-ReLU-splitting technique that Reluplex uses is an example of the general problem-solving strategy of \emph{exploiting high-level domain-specific abstractions for efficiency} that has proven fruitful in a variety of areas. For example, in the setting of high-performance domain-specific languages, a high-level representation of programmer intent enables compiler optimizations and smart scheduling choices that would be difficult or impossible otherwise~\cite{delite}. The use of high-level abstractions not only does not compromise high performance, but actually enables it. Reluplex's lazy ReLU splitting is another such optimization, made possible by the addition of the high-level ReLU predicate to the theory used by the solver. The higher-level representation makes it possible to determine the satisfiability of a formula more efficiently than if the problem were expressed at a lower level.
An important lesson here for scalable verification is that we have much to gain by not treating SMT solvers as black boxes, but instead developing \emph{domain-specific theory solvers} like Reluplex that are uniquely suited to the verification task at hand. We are currently working on extending Reluplex to handle piecewise-linear approximations of other commonly used activation functions to be able to handle a wider variety of networks.
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